freund-dynamic fracture mechanics

http://www.cambridge.org/9780521303309

This page intentionally left blank

CAMBRIDGE MONOGRAPHS ON

MECHANICS AND APPLIED MATHEMATICS

General Editors

G. K. BATCHELOR, F. R. S.Department of Applied Mathematics, University of Cambridge

C. WUNSCHDepartment of Earth, Atmospheric, and Planetary Sciences,Massachusetts Institute of Technology

J. RICEDivision of Applied Sciences, Harvard University

DYNAMIC FRACTURE MECHANICS

Dynamic Fracture Mechanics

L. B. FREUND

Brown University

CAMBRIDGEUNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-30330-9

ISBN-13 978-0-511-41364-3

© Cambridge University Press 1990

1990

Information on this title: www.cambridge.org/9780521303309

This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

eBook (EBL)

hardback

http://www.cambridge.org

http://www.cambridge.org/9780521303309

CONTENTS

Preface xi

List of symbols xiv

1 Background and overview 11.1 Introduction 1

1.1.1 Inertial effects in fracture mechanics 21.1.2 Historical origins 3

1.2 Continuum mechanics 131.2.1 Notation 131.2.2 Balance equations 161.2.3 Linear elastodynamics 221.2.4 Inelastic materials 29

1.3 Analytic functions and Laplace transforms 301.3.1 Analytic functions of a complex variable 301.3.2 Laplace transforms 33

1.4 Overview of dynamic fracture mechanics 371.4.1 Basic elastodynamic solutions for a

stationary crack 381.4.2 Further results for a stationary crack 401.4.3 Asymptotic fields near a moving crack tip 421.4.4 Energy concepts in dynamic fracture 441.4.5 Elastic crack growth at constant speed 471.4.6 Elastic crack growth at nonuniform speed 491.4.7 Plasticity and rate effects during crack growth 52

2 Basic elastodynamic solutions for a stationary crack 552.1 Introduction 552.2 Suddenly applied antiplane shear loading 602.3 Green's method of solution 65

vi Contents

2.4 Suddenly applied crack face pressure 722.5 The Wiener-Hopf technique 77

2.5.1 Application of integral transforms 782.5.2 The Wiener-Hopf factorization 842.5.3 Inversion of the transforms 912.5.4 Higher order terms 96

2.6 Suddenly applied in-plane shear traction 972.7 Loading with arbitrary time dependence 100

3 Further results for a stationary crack 1043.1 Introduction 1043.2 Nonuniform crack face traction 106

3.2.1 Suddenly applied concentrated loads 1073.2.2 Fundamental solution for a moving dislocation 1103.2.3 The stress intensity factor history 112

3.3 Sudden loading of a crack of finite length 1173.4 Three-dimensional scattering of a pulse by a crack 1233.5 Three-dimensional stress intensity factors 1313.6 Fracture initiation due to dynamic loading 140

3.6.1 The Irwin criterion 1403.6.2 Qualitative observations 1413.6.3 Experimental results 144

4 Asymptotic fields near a moving crack tip 1524.1 Introduction 1524.2 Elastic material; antiplane shear 1554.3 Elastic material; in-plane modes of deformation 160

4.3.1 Singular field for mode I 1614.3.2 Higher order terms for mode I 1694.3.3 Singular field for mode II 1704.3.4 Supersonic crack tip speed 171

4.4 Elastic-ideally plastic material; antiplane shear 1754.4.1 Asymptotic fields for steady dynamic growth 1784.4.2 Comparison with equilibrium results 182

4.5 Elastic-ideally plastic material; plane strain 1844.5.1 Asymptotic field in plastically deforming regions 1874.5.2 A complete solution 1904.5.3 Other possible solutions 1944.5.4 Discontinuities 1974.5.5 Elastic sectors 202

4.6 Elastic-viscous material 206

Contents vii

4.6.1 Antiplane shear crack tip field 2074.6.2 Plane strain crack tip field 214

4.7 Elastic-viscoplastic material; antiplane shear 215

5 Energy concepts in dynamic fracture 2215.1 Introduction 2215.2 The crack tip energy flux integral 224

5.2.1 The energy flux integral for plane deformation 2245.2.2 Some properties of F(T) 227

5.3 Elastodynamic crack growth 2315.3.1 Dynamic energy release rate 2315.3.2 Cohesive zone models of crack tip behavior 2355.3.3 Special forms for numerical computation 240

5.4 Steady crack growth in a strip 2435.4.1 Strip with uniform normal edge displacement 2435.4.2 Shear crack with a cohesive zone in a strip 247

5.5 Elementary applications in structural mechanics 2505.5.1 A one-dimensional string model 2505.5.2 Double cantilever beam configuration 2545.5.3 Splitting of a beam with a wedge 2575.5.4 Steady crack growth in a plate under bending 2615.5.5 Crack growth in a pressurized cylindrical shell 262

5.6 A path-independent integral for transient loading 2645.6.1 The path-independent integral 2645.6.2 Relationship to stress intensity factor 2695.6.3 An application 271

5.7 The transient weight function method 2745.7.1 The weight function based on a particular

solution 2745.7.2 A boundary value problem for the weight

function 2805.8 Energy radiation from an expanding crack 289

6 Elastic crack growth at constant speed 2966.1 Introduction 2966.2 Steady dynamic crack growth 298

6.2.1 General solution procedure 2996.2.2 The Yoffe problem 3006.2.3 Concentrated shear traction on the crack faces 3056.2.4 Superposition and cohesive zone models 3066.2.5 Approach to the steady state 310

viii Contents

6.3 Self-similar dynamic crack growth 3136.3.1 General solution procedure 3146.3.2 The Broberg problem 3186.3.3 Symmetric expansion of a shear crack 3306.3.4 Nonsymmetric crack expansion 3346.3.5 Expansion of circular and elliptical cracks 336

6.4 Crack growth due to general time-independent loading 3406.4.1 The fundamental solution 3426.4.2 Arbitrary initial equilibrium field 3506.4.3 Some illustrative cases 3536.4.4 The in-plane shear mode of crack growth 3556.4.5 The antiplane shear mode of crack growth 356

6.5 Crack growth due to time-dependent loading 3566.5.1 The fundamental solution 3586.5.2 Arbitrary delay time with crack face pressure 3626.5.3 Incident plane stress pulse 365

7 Elastic crack growth at nonuniform speed 3677.1 Introduction 3677.2 Antiplane shear crack growth 3697.3 Plane strain crack growth 378

7.3.1 Suddenly stopping crack 3797.3.2 Arbitrary crack tip motion 3877.3.3 In-plane shear crack growth 392

7.4 Crack tip equation of motion 3937.4.1 Tensile crack growth 3957.4.2 Fine-scale periodic fracture resistance 4017.4.3 Propagation and arrest of a mode II crack 4077.4.4 A one-dimensional string model 4107.4.5 Double cantilever beam: approximate equation

of motion 4217.5 Tensile crack growth under transient loading 426

7.5.1 Incident plane stress pulse 4267.5.2 An influence function for general loading 431

7.6 Rapid expansion of a strip yield zone 4327.7 Uniqueness of elastodynamic crack growth solutions 437

8 Plasticity and rate effects during crack growth 4428.1 Introduction 4428.2 Viscoelastic crack growth 4428.3 Steady crack growth in an elastic-plastic material 448

Contents ix

8.3.1 Plastic strain on the crack line 4518.3.2 A growth criterion 4598.3.3 A formulation for the complete field 4618.3.4 The toughness-speed relationship 4658.3.5 The steady state assumption 467

8.4 High strain rate crack growth in a plastic solid 4698.4.1 High strain rate plasticity 4708.4.2 Steady crack growth with small-scale yielding 4748.4.3 An approximate analysis 4778.4.4 Rate effects and crack arrest 481

8.5 Fracture mode transition due to rate effects 4858.5.1 Formulation 4868.5.2 A rate-dependent cohesive zone 4888.5.3 The crack growth criteria 494

8.6 Ductile void growth 4988.6.1 Spherical expansion of a void 5008.6.2 A more general model 506

8.7 Microcracking and fragmentation 5088.7.1 Overall energy considerations 5098.7.2 Time-dependent strength under pulse loading 512

Bibliography 521

Index 559

PREFACE

This book is an outgrowth of my involvement in the field of dynamicfracture mechanics over a period of nearly twenty years. This sub-branch of fracture mechanics has been wonderfully rich in scope anddiversity, attracting the attention of both researchers and practition-ers with backgrounds in the mechanics of solids, applied mathematics,structural engineering, materials science, and earth science. A widerange of analytical, experimental, and computational methods havebeen brought to bear on the area. Overall, the field of dynamicfracture is highly interdisciplinary, it provides a wealth of challengingfundamental issues for study, and new results have the potential forimmediate practical application. In my view, this combination ofcharacteristics accounts for its continued vitality.

I have written this book in an effort to summarize the currentstate of the mechanics of dynamic fracture. The emphasis is onfundamental concepts, the development of mathematical models ofphenomena which are dominated by mechanical features, and theanalysis of these models. Mathematical problems which are repre-sentative of the problem classes that comprise the area are statedformally, and they are also described in common language in aneffort to make their features clear. These problems are solved usingmathematical methods that are developed to the degree required tomake the presentation more or less self-contained. Experimental andcomputational approaches have been of central importance in thisfield, and relevant results are cited in the course of discussion. Theextraordinary contributions of the few individuals who pioneered thearea of dynamic fracture mechanics occupy prominent positions in thisdiscussion. One hope in preparing this book is that people with newperspectives will be attracted to the field, which continues to provide

XI

xii Preface

fascinating and technically important challenges. Perhaps the bookcan serve as a guide to further development of the area.

The reader is assumed to be familiar with concepts of continuummechanics and methods of applied mathematics to the level normallyprovided through the first year of study in a graduate program insolid mechanics in the United States. A brief summary of relevantresults is included in the first chapter in order to establish notationand to provide a common source for reference in later chapters. Somebackground in equilibrium fracture mechanics would be helpful, ofcourse, but none is presumed. In terms of graduate instruction infracture mechanics, the book could serve as a text for a course devotedto dynamic fracture mechanics or, for a more general course, as asupplement to other books which provide broader coverage of thewhole of fracture mechanics. The overall organization of the bookis evident from the chapter titles. The brief overview included inChapter 1 can serve as a guide to those readers interested in usingthe book as a source of reference for specific results.

The bibliography is an important part of the book. In view ofcontemporary publication practices, the compilation of an all-inclusivebibliography in any technical area is an impossible task. Nonetheless,the bibliography is intended to be comprehensive in the sense that itincludes entries which describe research results on essentially all as-pects of dynamic fracture. With only a few exceptions, the entries areeither articles published in the open literature or relevant textbooksand monographs. Thus, most of the references should be available ina reasonably complete technical library. All references cited in thetext are included in the bibliography, and many additional referencesare included as well. Some judgment was required in the selectionof references for citation in the text, and I have done my best toaccurately identify sources for key steps in the evolution of ideas.

I am indebted to a number of colleagues who read drafts ofvarious chapters of the book. Those who offered suggestions andencouragement in this way are John Hutchinson, Fred Nilsson, AresRosakis, and John Willis. A special thanks goes to Jim Rice, Editorof this series, who generously read a draft of the entire manuscript.It has been among my greatest fortunes to be a member of the SolidMechanics Group at Brown University, which has provided an intellec-tually stimulating and most congenial environment over the years. Iam especially grateful to my colleagues Rod Clifton, Alan Needleman,Michael Ortiz, Fong Shih, and Jerry Weiner for their willingness to

Preface xiii

read and discuss some of the material that is included here. Myown views on the mechanics of dynamic fracture and its fundamen-tal precepts have been formed over a long period of time throughinteractions with people far too numerous to mention individually.This group includes the many colleagues and students with whomI have collaborated and written joint papers; they are identified inthe bibliography. It also includes those who, through questions anddiscussions, showed me where my own understanding of certain pointshad been incomplete.

I also thank Peter-John Leone, Earth Sciences Editor, RhonaJohnson, Manuscript Development Editor, and Louise Calabro Gruen-del, Production Editor, of Cambridge University Press in New Yorkfor the efficient way in which they have managed the preparation ofthe book and for their sensitivity in dealing with my concerns on thematter. It has been a pleasure to be involved in long-term programs ofresearch on fracture mechanics funded by the Office of Naval Researchand by the National Science Foundation. These programs, and thecollaborations that they have fostered, have been invaluable.

Finally, I thank my wife Colleen and our sons, Jon, Jeff, andSteve, who enthusiastically adopted the mission of writing this bookas their own. Their interest and unwavering devotion have lightenedthe task immensely.

L. B. Freund

LIST OF SYMBOLS

Mathematical symbols and functions are defined the first time thatthey are used in the book. Brief definitions of the most frequentlyused symbols and functions are listed below. Some symbols neces-sarily have different definitions in different sections. In such cases,definitions are stated locally and are used consistently within sections.

a Inverse dilatational wave speed c^"1; half length of aGriffith crack

Ai(v) Universal function of crack tip speed for mode I defor-mation (similar for modes II and III)

b Inverse shear wave speed cj1] a material parameterc Inverse Rayleigh wave speed c^1; an elastic wave speedCd Elastic dilatational wave speedco Speed of longitudinal waves in an elastic barCR Elastic Rayleigh surface wave speedcs Elastic shear wave speedCi Dimensionless factor for the mode I asymptotic stress

field (similar for modes II and III)Cijki Components of the elastic stiffness tensord Inverse crack tip speed v~x

dij Components of the symmetric part of the velocity gra-dient tensor

D(v) The quantity 4adas - (1 + a2)2 ; a function of crack tipspeed

E Young's elastic modulusER Far-field radiated energyF(T) Energy flux through contour FG Energy release rate; dynamic energy release rate

xiv

Symbols xv

Ga Crack arrest fracture energyGc Critical value of energy release rateh Thickness of a beam; width of a striphi(x,t) Components of the dynamic weight functionH(t) Unit step functionJjv(F, s) Path independent integral of Laplace transformed fieldsk(v) Universal function of crack tip speed for elastic crack

growth in mode Iku(v) Universal function of crack tip speed for elastic crack

growth in mode II (similar for mode III)Kj Elastic stress intensity factor for mode I (similar for

modes II and III)Kia Value of stress intensity factor at crack arrest; crack

arrest toughness (similar for modes II and III)Kiappi Remotely applied stress intensity factorKic Value of stress intensity factor at fracture initiation;

fracture toughness (similar for modes II and III)Kid Value of stress intensity factor during crack growth; dy-

namic fracture toughness (similar for modes II andin)

/, l(t) Crack length; amount of crack growth/ Instantaneous crack tip speed/± Limit as position x = I is approached through values of

x greater than I (+) or less than I (-)lc Critical crack lengthlo Initial crack lengthm Normalized crack tip speed v/cs or V/CRMijM Components of the elastic compliance tensorn Crack tip bluntness parameter; a material parameterrti Components of unit vector; normal to surface or curve°(f(x)) Asymptotically dominated by / as x —> a limit pointO(f(x)) Asymptotically proportional to / as x —> a limit pointp(x) Pressure distributionp* Magnitude of a concentrated normal forceP(C) Amplitude of $Pc Dimensionless combination of material parameters for a

strain-rate-dependent elastic-plastic solidq* Magnitude of a concentrated shear forceQ(C) Amplitude of *

xvi Symbols

r Polar coordinaterd exp(i9d) Polar form of the complex variable x + ia.dVrp Plastic zone sizer8 exp(i0s) Polar form of the complex variable x + ia8yR Region in space; region in a planeR(() Rayleigh wave functions Laplace transform parameter; a real variableSij Components of the deviatoric stress tensor5±(C) Factors of the Rayleigh wave function that are nonzero

and analytic in overlapping half planest Time coordinateT Kinetic energy densityTtot Total kinetic energy of a bodyT, % Traction vector; components of traction vectorUi Components of particle displacement vector (i = 1,2,3

or i = x, y, z)iii Components of particle velocity vectorHi Components of particle acceleration vectorUy(x,t) Normal surface displacement for Lamb's problemu_(x,t) Displacement distribution for x < 0U Stress work density; elastic strain energy densityUtot Total stress work; total strain energyv Crack tip speedw Displacement for antiplane shear deformationXi Rectangular coordinates (i = 1,2,3)x, y, z Rectangular coordinatesa(C) The function (a2 - C2)1/2

LS The quantities y/l — v2/cj and y/l — v2/c*P(Q The function (ft2 - C2)1/2

7 A material parameter7O Viscosity pa ramete r of a ra te dependent plastic mater ia lT Crack t ip contour; specific fracture energyF(-) G a m m a (factorial) functionF c , F m , F o Cons tan t values of specific fracture energy6t Crack t i p opening displacement6(t) Dirac delta functionA Amplitude of an elastic dislocatione A small real parameterdj Components of the small strain tensor

Symbols xvii

efj Components of the elastic strain tensor€? Components of the plastic strain tensor£ Complex variable; Laplace transform parameter77 Rectangular coordinate9 Polar coordinateK Parameter determined by the behavior of the Rayleigh

wave function as |£| —• 00A Lame elastic constantA Length of cohesive zonefj, Lame elastic constant; elastic shear modulusv Poisson's ratio1/, Vi Unit vector normal to a surface or curve£ Rectangular coordinatep Material mass densitya Mean stress; effective stressc(x) Normal traction within a cohesive zone(Tij Components of the stress tensorcr^ Amplitude of remotely applied tensionao Tensile flow stress of an ideally plastic materiala+(x,t) Normal traction distribution for x > 0a* Magnitude of applied normal tractionEf Angular variation of asymptotic crack tip stress field for

mode I (similar for modes II and III)TQO Amplitude of remotely applied shear tractionro Shear flow stress of an ideally plastic materialr+(x,t) Shear traction distribution on x > 0r* Magnitude of applied shear traction(f> Lame scalar displacement potential function$ Double Laplace transform of (j>t/>, tyi Lame vector displacement potential function-0 Magnitude of ip for plane deformation; local shear angle

in plastically deforming region\I> Double Laplace transform of ?/>fi Potential energyfis Potential energy increase due to creation of free surface{cc.ss.nn) Equation number nn in Section ss of Chapter cc(c.s.r^m The rath equation in a group of equations identified by

the single number (c.s.n)

BACKGROUND AND

OVERVIEW

1.1 Introduction

The field of fracture mechanics is concerned with the quantitativedescription of the mechanical state of a deformable body containinga crack or cracks, with a view toward characterizing and measuringthe resistance of materials to crack growth. The process of describingthe mechanical state of a particular system is tantamount to devisinga mathematical model of it, and then drawing inferences from themodel by applying methods of mathematical or numerical analysis.The mathematical model typically consists of an idealized descriptionof the geometrical configuration of the deforriiable body, an empiricalrelationship between internal stress and deformation, and the perti-nent balance laws of physics dealing with mechanical quantities. Fora given physical system, modeling can usually be done at differentlevels of sophistication and detail. For example, a particular materialmay be idealized as being elastic for some purposes but elastic-plasticfor other purposes, or a particular body may be idealized as a one-dimensional structure in one case but as a three-dimensional structurein another case. It should be noted that the results of most significancefor the field have not always been derived from the most sophisticatedand detailed models.

A question of central importance in the development of a fracturemechanics theory is the following. Is there any particular feature ofthe mechanical state of a cracked solid that can be interpreted as a

2 1. Background and overview

"driving force" acting on the crack, that is, an effect that is correlatedwith a tendency for the crack to extend? A viewpoint that underliesthis important concept is that of the crack as an entity which itselfbehaves according to a "law" of mechanics expressed in terms of arelationship between driving force and motion. The modeling phaseprovides the language for considering the strength of real materials,and it concludes with hypotheses on the behavior of materials. Itis only through observation of the fracturing of materials that thehypotheses can be verified or refuted. This synergistic process has ledto standard methods of practice whereby fracture mechanics is usedroutinely in materials selection and engineering design.

1.1.1 Inertial effects in fracture mechanics

Dynamic fracture mechanics is the subfield of fracture mechanicsconcerned with fracture phenomena for which the role of materialinertia becomes significant. Phenomena for which strain rate depen-dent material properties have a significant effect are also typicallyincluded. Inertial effects can arise either from rapidly applied loadingon a cracked solid or from rapid crack propagation. In the case of rapidloading, the influence of the loads is transferred to the crack by meansof stress waves through the material. To determine whether or not acrack will advance due to the stress wave loading, it is necessary todetermine the transient driving force acting on the crack. In the caseof rapid crack propagation, material particles on opposite crack facesdisplace with respect to each other once the crack edge has passed.The inertial resistance to this motion can also influence the drivingforce, and it must be taken into account in a complete description ofthe process. There is also a connection between the details of rapidcrack motion and the stress wave field radiated from a moving crackthat is important in seismology, as well as in some material testingtechniques.

Progress toward understanding dynamic fracture phenomena hasbeen impeded by several complicating features. The inherent timedependence of a dynamic fracture process results in mathematicalmodels that are more complex than the equivalent equilibrium modelsfor the same configuration and material class. Prom the experimen-tal point of view, the time dependence requires that many accuratesequential measurements of quantities of interest must be made in anextremely short time period in a way that does not interfere with theprocess being observed. In the case of crack growth, the regions of the

LI Introduction 3

boundary of the body over which certain conditions must be enforcedin a mathematical model change with time. Prom the experimentalpoint of view, this feature implies that the place where quantities ofinterest must be measured varies, usually in a nonuniform way, duringthe process.

The question of whether or not inertial effects are significantin any given fracture situation depends on the loading conditions,the material characteristics, and the geometrical configuration of thebody. Circumstances under which they are indeed significant cannotbe specified unambiguously, but some guidelines are evident. Forexample, suppose that there is a characteristic time associated withthe applied loading, say a load maximum divided by the rate of loadincrease. Unless this time is large compared to the time required for astress wave to travel at a characteristic wave speed of the materialover a representative length of the body, say the crack length orthe distance from the crack edge to the loaded boundary, it can beexpected that inertial effects will be significant. In the case of crackgrowth, inertial effects will probably be important if the speed of thecrack tip or edge is a significant fraction of the lowest characteristicwave speed of the material. If the boundaries of the body must betaken into account, then the potential for the body to function as awaveguide must be considered. The group velocity of guided waves isfrequently much less than the characteristic bulk wave speeds of thematerial. For such cases, inertial effects could be important for crackspeeds that are much less than the bulk wave speeds but comparableto relevant group velocities.

1.1.2 Historical origins

Empirical studies concerned with the bursting of military cannon orwith the impact loading of industrial machines were carried out inthe 19th century. For example, an interesting early series of observa-tions on "new" industrial iron-based materials is reported by Kirkaldy(1863). However, no single scientific discovery or other event can beidentified as the stimulus for launching dynamic fracture mechanics asan area of research. Instead, the area as it is summarized in this bookis the conjunction of numerous paths of investigation that were moti-vated by pressing practical needs in engineering design and materialselection, by scientific curiosity about earthquakes and other naturalphenomena, by challenges of classes of mathematical problems, andby advances in experimental and observational techniques.


The area has been under continuous development as a subfieldof the engineering science of fracture mechanics since the 1940s andas a topical area in geophysics since the late 1960s. The view in theearly days of fracture mechanics was that once a crack in a structurebegan to grow the structure had failed (Tipper 1962). The rapidcrack growth phase of failure was viewed as interesting but not par-ticularly important. Beginning about 1970, however, the importanceof understanding crack propagation and arrest was recognized in bothengineering and earth sciences, and significant progress has been madesince then.

A few of the early contributions to the area that provided ideasof lasting significance are cited here. In some cases, these ideas weredeveloped in connection with fracture under equilibrium conditions,but they have been important in the area of dynamic fracture aswell. Among these is the work of Griffith (1920), which is commonlyacknowledged as representing the start of equilibrium fracture me-chanics as a quantitative science of material behavior. In consideringan ideally brittle elastic body containing a crack, he recognized thatthe macroscopic potential energy of the system, consisting of theinternal stored elastic energy and the external potential energy of theapplied loads, varied with the size of the crack. He also recognizedthat extension of the crack resulted in the creation of new cracksurface, and he postulated that a certain amount of work per unitarea of crack surface must be expended at a microscopic level tocreate that area. In this context, the term "microscopic" impliesthat this work is not included in a continuum description of theprocess. It is common to characterize this work per unit area byassuming that a certain force-displacement relationship governs thereversible interaction of atoms or molecules across the fracture plane.The area under the force-displacement relationship from equilibriumto full separation, averaged over the fracture surface, is assumed torepresent this work of separation. This is one way to define thesurface energy of the material. Griffith simply included this workas an additional potential energy of the system, and then invoked theequilibrium principle of minimum potential energy for conservativesystems. That is, he considered the system to be in equilibriumwith a particular fixed loading and a particular crack length. Hepostulated that the crack was at a critical state of incipient growth ifthe reduction in macroscopic potential energy associated with a smallvirtual crack advance from that state was equal to the microscopic

LI Introduction

Figure 1.1. A crack of length I in a rectangular strip. Forces P act to separatethe crack faces and the displacement of each load point is 6.

work of separation for the crack surface area created by the virtualcrack advance. A particular attraction of Griffith's energy fracturecondition is that it obviates the need to examine the actual fractureprocess at the crack tip in detail.

Griffith's original work was based on the plane elasticity solutionfor a crack of finite length in a body subjected to a uniform remotetension in a direction normal to the crack plane. The concept isillustrated here by means of a simple example based on elementaryelastic beam theory. Consider the split rectangular strip shown inFigure 1.1 which is of unit thickness in the direction perpendicular tothe plane of the figure. Suppose that the slit or crack of length I isopened symmetrically by imposing either forces or displacements atthe ends of the arms as shown. In either case, the end force per unitthickness is denoted by P and the displacement of the load point ofeach arm from the undeflected equilibrium position is denoted by 6.Suppose that the isotropic material is elastic with Young's modulusE and that the deformation can be approximated by assuming thateach arm deforms as a Bernoulli-Euler beam of length I cantileveredat the crack tip end. The relationship between the end force per unitthickness and the end displacement is then

(1.1.1)

and the total stored elastic energy per unit thickness in both arms is

The work of separation per unit area of surface created is denoted by

7-Consider first the case of imposed end displacement 6*. What is

the critical crack size for this fixed end displacement according to theGriffith theory? In this case, there is no energy exchange between thebody and its surroundings during the virtual crack extension, and themacroscopic potential energy per unit thickness fi is the stored elasticenergy Q = Eh36*2/Al3. The total work-of-fracture per unit thicknessis 7 times the crack length for both faces, or fts — 27/. The totalwork-of-fracture per unit thickness could equally well be written as 27times the amount of crack extension from some arbitrary initial cracklength, a work measure that would differ from JI5 by a constant.This constant is arbitrary, and it has no significance in consideringcrack advance. The total potential energy is £2 + fi^. According tothe Griffith postulate, the critical crack length for incipient growth isdetermined by the condition

r 3 £ f t 3 < - 2 l l / \ (1.1.3)

The individual potential energies vary with crack length as shownschematically in Figure 1.2. Note that

(1.1.4)

at the critical crack length. This means that the system is stable in thesense of mechanical equilibrium. If the crack length is at the criticalvalue and 6* is slowly increased to a larger value, then the crack willslowly advance under equilibrium conditions to a new length that isrelated to the increased 6* through (1.1.3)2- A useful interpretationof (1.1.3)i is that —dQ/dl is the feature of the mechanical state thatcan be identified as a crack driving force. The driving force is work-conjugate to crack position £, and the fracture mechanics balance lawgoverning crack behavior is —dfl/dl = 27.

Consider next the case of imposed end force P*. What is thecritical crack size for this end condition? The stored elastic energy isagain given by (1.1.2). The external potential energy of the loadingon both arms is —26P* = -813P*2/Eh3, so that the macroscopic

LI Introduction

\ (a) imposed displacement

£ - crack length

(b) imposed force

- crack length

Figure 1.2. The variation of energy with crack length for the split rectangular stripin Figure 1.1 for (a) imposed displacement or (b) imposed force. The variation ofthe continuum potential energy plus the surface energy is shown by the dashedline.

potential energy is fl = -413P*2/Eh3. Imposition of the Griffithcondition in this case yields

- _c ~

,1/2

(1.1.5)

Again, the way in which the individual potential energies vary withcrack size is illustrated schematically in Figure 1.2. If 5*, P*, and

lc are related as in (1.1.1) then the critical lengths in (1.1.3b) and(1.1.5b) are identical. However, there is a fundamental differencebetween the two cases. In the latter case,

d2

— (n + ns)<o (1.1.6)

at the critical crack length. This means that the system is unstablein an equilibrium sense. When / = lc for a given P*, the state isan equilibrium state of incipient crack growth. However, if P* isincreased by any amount from this state, then it is no longer possibleto find an equilibrium configuration. Instead, the crack begins to growrapidly and inertial resistance to motion is called into play to ensurebalance of momentum. This idea is fully developed in Chapter 7 indiscussing dynamic crack growth.

A theoretical framework for including inertial effects during therapid crack growth phase was first proposed by Mott (1948), whoadopted the features of Griffith's analysis as a point of departure. Herecognized that inertial resistance of the material to crack openingcould become significant at high crack speeds. To estimate the crackspeed for a particular loading system, he assumed that the crackgrowth process was steady state, that is, time independent as seenby an observer moving at speed v with the crack tip. Under theseconditions, he obtained an estimate for the total kinetic energy of thesystem TfOt in the form of v2 times a function of crack length /. Hethen argued that the total energy of the system Cl+fls+T is constant,so that

- ( n + rtot

This condition provides a relationship involving the loading parame-ters, the crack length /, and the crack speed v. In implementing thiscondition for the Griffith plane elasticity crack model, Mott assumedthat tt and T could be calculated on the basis of the equilibrium field.

The idea underlying (1.1.7) is correct if the terms are strictlyinterpreted. In fact, it is a special case for steady state crack growth ofthe general energy-rate balance: Rate of work of applied loads = rateof increase of stored elastic energy + rate of increase of kinetic energy— rate of energy loss at the crack tip. However, later work has shownthat the assumptions made by Mott in his original analysis of thedynamic Griffith crack are not valid. Consequently, the conclusions

LI Introduction 9

inferred from this model are generally not valid. For example, thisanalysis has often been cited as establishing that the maximum crackspeed in an ideal brittle material is some fixed fraction, typicallygiven as about one-half, of the shear wave speed of the material.If the model is analyzed more thoroughly, it is found that this isnot the case. Indeed, the result that the theoretical limiting speedof a tensile crack must be the Rayleigh wave speed, as establishedin Chapter 7, was anticipated by Stroh (1957) on the basis of anintuitive argument. Nonetheless, Mott's basic idea of energy balanceduring rapid crack growth was very important to the developmentof the subject. A number of energy concepts pertinent to dynamicfracture are discussed in Chapter 5, and the crack tip equation ofmotion of the type originally sought by Mott is presented in Chapter7.

Another important theoretical idea for applying work and energymethods to fracture dynamics was provided by Irwin (1948). Hewas concerned with cleavage fracture of structural steels, a cleavageprocess that is invariably accompanied by some amount of plastic flowof the material adjacent to the fracture path. Irwin adopted Mott'spostulate of stationary total energy, including a work of fracture,denoted above by Q>s- However, Irwin proposed that this term ftsmay be approximately represented as the sum of two terms, oneproportional to the area of fracture surface and the other proportionalto the volume of material affected by plastic deformation. Withinthe context of the Griffith theory outlined above, this assumption isinvoked by writing f2s = 2(7S + 7P)Z when the thickness of the plas-tically deformed layer adjacent to the crack faces is small. Thus, j8

represents the surface energy of the material associated with cleavagefracture and jp represents the plastic work dissipated in the surround-ing material per unit crack surface area created. Irwin's idea thereforeextended the range of applicability of the energy balance fracturetheory to include materials that undergo some plastic deformationduring crack growth. This particular extension of Griffith's theorywas also proposed for fracture initiation and crack propagation insteels by Orowan (1955). Early experiments aimed at measuring jp

for cleavage of metal single crystals and polycrystals were carried outby Hall (1953). He showed that j p ^ j8 for metals, particularlyfor polycrystals. Furthermore, both Hall and Irwin predicted strongconnections between the fracture behavior of polycrystalline metalsand their microstructures. An extensive survey of early work on the


connection between metallurgical properties and brittle fracture ofwelded steel plates was given by Wells (1961) and Tipper (1962).

The continuum field approach to fracture of solids was launchedwith the introduction of the elastic stress intensity factor, usuallydenoted by K, as a crack tip field characterizing parameter by Irwin(1957). This idea provided an alternate framework for discussingthe strength of cracked solids of nominally elastic material. Irwinproposed that a crack will begin to grow in a cracked body withlimited plastic deformation when K is increased to a value called thefracture toughness of the material. The equivalence of the Irwin stressintensity factor criterion and the Griffith energy criterion for onsetof growth of a tensile crack in a two-dimensional body of nominallyelastic material under plane stress conditions was demonstrated byIrwin (1957, 1960), who showed that

where E is Young's modulus of the material. The quantity on the leftside of (1.1.8) is usually denoted by G and it is called the energy releaserate. Much of the analysis of this book is devoted to determining thestress intensity factor or the energy release rate for various conditions.These concepts are developed more thoroughly in a way relevant todynamic fracture mechanics in the chapters to follow.

Another idea that has been important in the evolution of fracturemechanics concerns the size scale over which different phenomenadominate a fracture process. The idea is implicit in Irwin's stressintensity factor concept and it is a central feature of the crack tip cohe-sive zone model introduced by Barenblatt (1959a). Consider a planarcrack in a body subjected to tensile loading normal to the plane of thecrack. Suppose that the material response is linearly elastic, exceptfor a region adjacent to the crack edge where the response departsfrom linearity. The source of nonlinearity can be plastic deformation,diffuse microcracking, nonlinear interatomic forces, or some otherphysical mechanism. The crack tip region is said to be autonomousat fracture initiation or during crack growth if the following twoconditions are met: (i) the extent of the region of nonlinearity from thecrack edge is very small compared to all other length dimensions of thebody and loading system, and (ii) the mechanical state within this endregion at incipient growth or during growth is independent of loading

1.1 Introduction 11

and geometrical configuration. For the particular case of an elastic-plastic material, the property of autonomy implies that the crack tipplastic zone is completely surrounded by an elastic stress intensityfactor field and that the state within the plastic zone is determinedby the level of stress intensity of the surrounding field. This situation,which was termed small-scale yielding by Rice (1968), is the situationin which the stress intensity factor is a useful fracture characterizingparameter. Furthermore, this close connection between effects atadjacent size scales provided the basis for development of an asymp-totic formulation of nonlinear crack tip field problems in general. Inits simplest form, the asymptotic formulation can be summarized inthe following way. Suppose that, at one scale of observation, themechanical state near a crack edge is dominated by an asymptoticnear crack tip solution with some characterizing parameter. Then,to study the phenomenon at a physical size scale that is an orderof magnitude smaller, a boundary value problem is formulated with acondition that its solution must match the aforementioned asymptoticsolution in the far-field. It is essentially this asymptotic formulationthat underlies the focus on semi-infinite cracks and near tip fields inthis monograph, as well as in fundamental fracture mechanics studiesin general. The central idea of autonomy has been extended to moregeneral material classes than those mentioned in these introductoryremarks.

A good deal of very important experimental research on dynamicfracture of materials was carried out in parallel with the developmentof dynamic fracture mechanics concepts for analyzing cracked solids.These experiments were often done on specimens of glass or other verybrittle materials. The results of these experiments, in the form of bothqualitative observations and quantitative data, have been extremelyimportant in the development of dynamic fracture mechanics. Indeed,a number of specific observations on the behavior of growing cracks inglass were made in the 1940s and 1950s for which generally acceptedexplanations based on physical principles are still incomplete. Forexample, quantitative modeling as a basis for understanding terminalcrack speed in glass or dynamic crack bifurcation under symmetrictensile loading is still in a preliminary stage. In addition to theimportance of the experimental results obtained in the early stages ofdynamic fracture research, the significance of the experimental tech-niques developed in the course of this work must also be recognized.The full-field optical methods of observation and special techniques


of high-speed photography have been particularly notable in dynamicfracture research.

The research group led by Schardin played a major role in thedevelopment of optical methods and high-speed photography. Muchof this work is reviewed by Schardin (1959). For example, in 1937it was found that running cracks in glass seem to have a maximumspeed that depends on the material, but not on loading or geometricalconfiguration. This phenomenon was also observed for fracture ofglass by Edgerton and Barstow (1941). The multiple spark camera,now known as the Cranz-Schardin camera in its most common con-figuration, was developed to obtain a sequence of photographs of afracturing specimen in a very short time interval. It was also shownthat stress waves obliquely incident on the crack path could causethe path to gradually curve toward the local direction of maximumtensile stress. It was observed that there was no delay in the change incrack growth direction due to incident stress waves, suggesting thatthe crack tip itself does not have an effective inertia (cf. Section7.4). This phenomenon was exploited in the 1950s by Kerkhof, whosuperimposed ultrasonic waves on a crack growing rapidly in a brittlebody. Because the frequency of the ultrasonic waves is known, thesmall undulations produced on the fracture surface due to the inter-action of the crack with the ultrasonic waves provided a continuousrecord of crack tip speed (Kerkhof 1970, 1973).

The optical shadow spot method was also introduced through thework of this group (Manogg 1966). This technique made it possibleto infer directly a value of the instantaneous crack tip stress intensityfactor and the crack tip position. When used in conjunction with themultiple spark camera, a record of the stress intensity factor and crackmotion could be obtained for a crack growth process. A multiple sparkarrangement was also used by Wells and Post (1958) to determine theisochromatic fringe pattern representing the near crack tip stress fieldduring rapid crack growth in a photoelastic brittle plastic. A reviewof early work on the formation of fractures in brittle materials bymeans of stress waves is given by Kolsky (1953). Modern surveys ofexperimental results in dynamic fracture of brittle materials are givenby Dally (1987) for photoelastic methods and by Kalthoff (1987) forshadow spot methods.

1.2 Continuum mechanics 13

1.2 Continuum mechanics

In this section, some notational conventions are established and basicequations of continuum mechanics are presented in order to provide aframework for discussing dynamic fracture mechanics. Field equationsand integral relations are stated without derivation; the underlyingdevelopments can be found in the references cited. In presenting theseresults, it is tacitly assumed that the mathematical operations impliedby the expressions can indeed be carried out. Whether or not this isthe case in the study of any particular model must be determined inthe course of analysis.

1.2.1 Notation

Consider a three-dimensional Euclidean point space in which the be-havior of a deformable body is to be described. A rectangular coor-dinate frame is introduced, consisting of a set of mutually orthogonalunit base vectors ei, e2, e3. The position vector p of a point in thespace is

P =

where the summation over the repeated index implied by the last entryin the continued equality is adopted as a convention. The quantities#i, x2, #3 a re the rectangular coordinates of the point. Vector and ten-sor valued quantities are represented by boldface type v throughoutthe book, and components are represented by means of subscripts.

Suppose that v and T are a vector field and a second-order tensorfield, respectively, over the space. At each point, the vector is anelement of a vector space associated with the point space, and thesecond-order tensor is a linear transformation of elements of the vectorspace into possibly another vector space or, as in this case, the sameone. The rectangular components of these fields at any point are

Vi = v • e and l y = e, • T • e^, (1.2.2)

where the dot denotes the binary inner product operator. This nota-tion is commonly called the Gibbs dyadic notation (Malvern 1969).The inner product of two vectors v and u is

u v = v u = mvi. (1.2.3)


Likewise, the inner product from the left (right) of vector v with thetensor T is v -T = v&jej (T-v = eiTijVj). Inner products involvingtensors of second- or higher order are not commutative unless thetensors possess certain symmetry properties. The cross product oftwo vectors u and v is a vector with components

(u x v)i = - ( v x u)i = eijkUjVk , (1-2.4)

where e^* is the alternating symbol with values

{ -1-1 if ijk is an even permutation of 123,—1 if ijk is an odd permutation of 123, (1.2.5)

0 otherwise.The array of components of the second-order unit tensor I in the

rectangular frame is the Kronecker delta

for i = ?',. . , (1-2-6)tor i ^ j .

Thus, for any vector v, v = I • v or Vi = SijVj. The trace of asecond-order tensor T is

tr(T) = e* - T - e{ = Tu . (1.2.7)

Let p represent the set of coordinates Xi,x2,xs. The gradient ofa scalar field /(p), denoted by V/, is the vector with components

(v/) i = %!- = U , (1-2.8)

where the last entry of the continued equality introduces the conven-tion whereby a subscript comma is used to denote partial differenti-ation. The divergence of a vector field v(p), denoted by V • v, is thescalar

V-v=^ = vUi. (1.2.9)

The curl of a vector field v(p), denoted by V x v, is a vector withcomponents

(V x v)i = eijkVk.j . (1.2.10)


The Laplacian operator acting on a scalar field / , defined by V • V/and denoted by V2 / , is

V2 / = /,« (1.2.11)

in indicial notation. If the gradient operator

is viewed as a vector, then useful generalizations of the foregoingresults become obvious. For example, the gradient of a vector fromthe left or the right is the dyad Vv or vV which is a second-ordertensor with rectangular components

(Vv)y = Vj,i or (vV)y = vUj. (1.2.13)

Likewise, the divergence of a second-order tensor from the left V • Tor from the right T • V is a vector with rectangular components

(V • T)j = TijH or (T • V); = Tjiti , (1.2.14)

and so on.Gauss's divergence theorem on integrals of vector fields plays a

key role in continuum mechanics. To state the theorem in its simplestform, consider a vector field v = ^e^ over a region R of the pointspace. Let S be the closed boundary surface of R and let rii be theoutward unit normal to S. Then the divergence theorem states that

/ V v d i ? = / n - v d S or / vUidR= I rijVjdS. (1.2.15)JR JS JR JS

Again, the result is readily generalized to tensor fields of second-and higher order if the divergence and inner product operators areproperly interpreted. For the integral of the divergence from the leftof the second-rank tensor field T, for example, the divergence theoremstates that

I VTdR= f n-TdS or f Tij»dR= ( T^nalS. (1.2.16)JR JS JR JS


Most of the fields to be introduced to represent physical quantitiesdepend on time t as well as on position in a three-dimensional space.In a few situations, it is advantageous to treat the time coordinate inthe same way as a spatial coordinate in a four-dimensional point spacewith coordinates (p,£), as in Section 5.2. For the most part, however,the time coordinate is treated separately from spatial coordinatesin applying the above conventions. Partial differentiation of a timedependent scalar field f(p,t) is denoted by

% - / * . (12.17)

1.2.2 Balance equations

Consider a deformable body occupying the region R of the three-dimensional point space at time t. The closed boundary surface ofR is S with outward unit normal vector n*. The components of thesecond-order Cauchy stress tensor field with respect to the rectangularreference frame are denoted by aij(p,t). Consider any surface in ii,possibly the boundary surface, and let i/j be a unit normal vectorat a particular point on the surface. According to the Cauchy stressprinciple, the vector representing local traction, or force per unit area,as a function of position on the surface is

Ti = ajivj . (1.2.18)

This is the traction exerted on the material on the side of the surfacein the direction of —1/$. If S is the boundary surface, this tractionmay arise from external sources. For a surface interior to ii, therelationship describes the action of the material on one side of thesurface on the material on the other side. The validity of this rela-tionship between the stress tensor and the traction vector does notrequire that the fields be equilibrium fields. No couple stress effectsare included in the discussion of fracture mechanics in this book.

Suppose that a particular fixed configuration of the body is iden-tified as a reference configuration. Material particles are labeled byassociating each with its position in the reference configuration, saythe point with coordinates x\,x*2,x%, represented collectively by p°.The deformation is specified by a nonsingular mapping of points x°ei


in the reference configuration to points xî at time t in the "current"configuration

xi = Xi(x°1,xi,xlt) (1.2.19)

with inversea£=X?(si,X2,*3,t). (1-2.20)

The particle displacement u = u ^ is given by

or ix2,X3,t) (1.2.21)

in a material description or a spatial description , respectively, of thedeformation. The particle velocity field v = vê^ is given by

Vi =dt dt

(1.2.22)po

which is naturally a function of (p°,t) but which can be viewed as afunction of (p,£) through (1.2.20). The particle acceleration a = aê;is given by either

di =

or

~dt

dt

dt2 (1.2.23)

(1.2.24)

The two expressions are both forms of the material time derivative ofparticle velocity, the first in the material description of motion andthe second in the spatial description. The material time derivativewill be denoted by a superposed dot, that is, a — v, which impliesthat di = V{ in the rectangular frame.

Suppose that the mass density of the material occupying R isp(p,t). Conservation of mass during a deformation with particlevelocity field Vi requires that

p + pvi,i = 0. (1.2.25)

If the material is subjected to a body force with rectangular compo-nents fi(p,t) per unit mass, the balance of linear momentum for anarbitrary material volume requires that the fields must satisfy

(1.2.26)

The balance of angular momentum requires that the stress tensor issymmetric, or

<Tij = op . (1.2.27)

If traction and particle velocity are discontinuous across a surfacemoving through the material, then the mQmentum balance equation(1.2.26) is replaced by

[(Tijvi + pVvj] = 0, (1.2.28)

where V is the local speed of the surface in the direction of the unitnormal Vi with respect to material particles instantaneously on thesurface. The double square bracket notation is adopted for the jumpin the value of a function / that is discontinuous across a surfacewhereby

i/] = r - r , (1-2.29)where f± is the limiting value of / on the surface as it is approachedin the direction of =FI/$ .

The instantaneous rate of work of a traction distribution on asurface is defined in the following way. Suppose that 5 is a materialsurface in the body, that is, material particles on the surface at any onetime are on the surface at all times. If 7$ is the traction distributionon the surface and Vi is the velocity of material particles on the surfacethen the instantaneous rate of work being done on the material on the—i/j side of S by the traction is

I TiVidS= IJs Js

(1.2.30)

The surface S may be an interior surface or any part of the boundarysurface of the body.

The kinetic energy per unit volume is defined as

T = \pv • v = \pviVi, (1.2.31)

where p is the material mass density. Total kinetic energy of a materialvolume, say Ttot, is the integral of T over the volume. The stress powerper unit volume is defined as

?-.<£ d.2.32)


Suppose that a certain volume of material instantaneously occupiesa region of space R, and that the boundary surface of the region isS with outward normal n . Then the rate of work being done on thebody by the traction on S and by the body force fi throughout R isequal to the net stress power plus the rate of increase of kinetic energythroughout i2, or

dS+ f pfjVj dR= f {P + f)dR. (1.2.33)JR JR

This relation can be established by applying the divergence theorem(1.2.16) to the first term on the left side, and by incorporating themomentum balance equations (1.2.26), conservation of mass (1.2.25),and the kinematic definitions.

The field equations and integral relations introduced above withrespect to the current configuration of the deformable body can betransformed into equivalent expressions with respect to the referenceconfiguration. Suppose that the body occupies the region Ro of spacewith boundary surface So and outward normal n° in its referenceconfiguration. The mass density in the reference configuration is po-The nonsymmetric nominal stress tensor field (or the transpose of thefirst Piola-Kirchhoff stress tensor) is related to the Cauchy stress by

h ' - (1'2'34>If /? is the body force per unit mass in the reference configuration,then the balance of linear momentum takes the form

Similarly, the work rate balance equation takes the form


where To = ^PoXiitXiit is the kinetic energy per unit volume in thereference configuration. Because the reference configuration is fixed,differentiation with respect to time commutes with integration overRo. Thus, the first term on the right side of (1.2.36) can be rewrittenas

where Uo is the accumulated stress work per unit volume, or the stresswork density

f- dr (1.2.38)

at each material point in the reference configuration.Suppose that u;(p, t) is a time-dependent field which represents

the density per unit mass of some physical quantity throughout adeformable body. Then the net measure of the quantity over a volumeof material occupying a time-dependent region of space R(t) is

n(t) = f p(p,t)uj(p,t)dR. (1.2.39)R(t)

Suppose that S(t) is the time dependent boundary surface of R(t)with outward normal n , and that the velocity of any point on S isvf. The time rate of change of the net measure of $7 for the materialinstantaneously in R(t) is

" = / % ^ d i ? + / P"VndS, (1.2.40)JR(t) Ot JS(t)

where Vn = vfrii is the local normal velocity of the surface S (Trues-dell and Toupin 1960). The result states that the value of f2 for thematerial in the region R(t) changes for two reasons. The first term onthe right side of (1.2.40) is a contribution due to the pointwise rate ofchange of density UJ throughout R. The second term on the right sideof (1.2.40) accounts for the fact that, as the surface S moves, there isa local flux of u through the boundary. The net value of this flux alsocontributes to the rate of change of fi. In the special case when Vn

is the component of velocity of the material particles on S(t) in thedirection of n , or Vn = ^n^, the surface S is a material surface and the


result (1.2.40) is an expression for the rate of change of fi for a fixedmaterial volume. In the latter case, (1.2.40) is called the Reynold'stransport theorem. For the general case when the normal speed Vn

has no connection to the motion of the material particles currently onthe surface, the result will be called the generalized transport theorem.This theorem plays a central role in computing energy flux througha tubular surface surrounding a crack edge and moving with it, forexample.

If the same volume of material is considered in the reference con-figuration of the body, then the statement of the generalized transporttheorem has the form

/ (1.2.41)

where V£ = (vf — Vi)n^x%ii ls the normal speed of the surface Smapped back to the reference configuration. The normal speed wasintroduced as an absolute speed of the surface, so in the referenceconfiguration it assumes the form of a speed with respect to the speedof the particles currently on So.

Finally, some specific deformation measures are adopted. Therate of deformation tensor d is the symmetric part of the particlevelocity gradient in the current configuration, and its rectangularcomponents are

The strain tensor E in terms of displacement (1.2.21) as a function ofcoordinates in the reference configuration has components

Likewise, the strain tensor E* in terms of displacement components asa function of coordinates in the current configuration has components

When all components of displacement gradients in either descriptionare small, the terms involving products of gradient components in


(1.2.43) and (1.2.44) will be small compared to the terms linear ingradient components. The two strain expressions then have the sameform, except that the gradients are taken with respect to materialcoordinates in the former case and spatial coordinates in the lattercase. When the displacements and the displacement gradients aresufficiently small, the distinction between the two descriptions is ig-nored and a single set of coordinates £i,#2>#3 1S used. In this case,the strain tensor components are

A description of deformation based on (1.2.45) is referred to as a smallstrain formulation or an infinitesimal strain formulation. Althoughthe small strain formulation can be viewed as an approximation to atheory of deformable bodies undergoing arbitrary deformations, theprocess of extracting the small strain formulation from the generaltheory of deformation as an asymptotic result is somewhat ambigu-ous (Gurtin 1972; Malvern 1969). An interesting perspective on theconditions necessary for the small strain conditions to be met forelastic-plastic materials is given by Rice et al. (1979). Formulationsbased on small strain concepts are themselves mathematically com-plete, however, and they provide a consistent framework for solvingboundary value problems in mechanics. Most of the results in thisbook are based on small strain concepts. The few exceptions areclearly identified.

1.2.3 Linear elastodynamics

The fundamental set of field equations governing the motion of ahomogeneous and isotropic elastic body consists of the strain dis-placement relation for small strain (1.2.45)

€ = \ (Vu + uV) or ey = \ (Uj,i +Ui,j ) , (1.2.46)

the momentum balance equations (1.2.26) and (1.2.27)

V • a + pi = pn or cry ,i +pfj = pUj , (1.2.47)

and the linear stress-strain relation

a = Altr(e) + 2/xc or cry = X6ijekk + 2/xey , (1.2.48)


where A and // are the positive Lame elastic constants. The fieldequations are to be satisfied throughout a region of space occupied bythe elastic solid in its initial, undeformed configuration. No distinctionis made between the deformed and undeformed configurations in thelinear theory. Thus, fields representing the same physical quantitiesin the current "deformed" configuration (without superscript or sub-script "o") and in the reference "undeformed" configuration in Section1.2.2 are viewed as being indistinguishable, except that a displacementvector is associated with each material particle. The superscript orsubscript "o" designation is dropped in the linear formulation. Thus,for example, the rate of work balance (1.2.33) or (1.2.36) takes theform

(1.2.49)

where the body occupies region R with boundary 5, T is the kineticenergy density, and U is the stress work density for general materialresponse. In the case of linear elasticity, U is the elastic strain energydensity j^yey .

If the momentum equation is written in terms of displacementu by substituting first from (1.2.48) and then from (1.2.46), Navier'sequation of motion

(A + ii)V(V • u) + //V2u + pi = pii (1.2.50)

is obtained. In light of the vector identity V x (V x u) = V(V • u) —V2u, (1.2.50) can be rewritten as

c2dV(V • u) - c2V x (V x u) + f = ii, (1.2.51)

where

and cs = A^ . (1.2.52)V P

Let the body force f be zero for the time being. The result ofoperating on each term of (1.2.51) with the divergence operator V- is

V . u ) = (V-u) ,« . (1.2.53)

Therefore, the dilatation V • u satisfies the elementary wave equationwith characteristic wave speed c<j, called the dilatational wave speed.Similarly, the result of operating on each term of (1.2.51) with thecurl operator Vx is

c*V2(V x u) = (V x u),« (1.2.54)

so that the rotation vector \ V x u satisfies the elementary wave equa-tion with characteristic wave speed cs, called the shear wave speed orthe rotational wave speed. In an unbounded isotropic elastic solid,waves of pure dilatation (rotation) propagate at normal speed Cd (c8).Furthermore, the two types of wave are independent in the interior ofa solid. At boundaries, such as the faces of a crack, for example, thetwo types of wave interact. This is the feature of elastodynamic fieldsthat accounts for most of the complication in solving initial-boundaryvalue problems. A complete discussion of plane wave propagation inunbounded bodies and plane wave reflection from boundaries is givenby Achenbach (1973). He also presents the basic singular solutionsof elastodynamics, principally, Stokes's solution for a time-dependentconcentrated force in an unbounded solid and Love's solution for acenter of dilatation of time-dependent strength in an unbounded solid.

The dilatational and shear wave speeds for an isotropic elasticmaterial are defined in (1.2.52) in terms of mass density and the Lameelastic constants A and //. It is sometimes more convenient to expressthe wave speeds in terms of Young's modulus E and Poisson's ratiov. The relationships among these elastic constants are

(1.2.55)

which can be inverted to yield

A ( L 2 - 5 6 )

In terms of E and */, the wave speeds are

Cd = V r f i f i a ^ ) ' c°= V MTTV) • (L2'57)

Another useful relationship follows from recognizing that the ratiocs/cd has a form depending only on Poisson's ratio,

g-2o^- < L 2 - 5 8 )

To formulate an initial-boundary value problem in elastodynam-ics, an appropriate set of initial and boundary conditions is required.In view of the Navier equation, values of u and u must be specifiedthroughout the body initially, or some conditions equivalent to thesemust be specified. Boundary conditions must be specified on theboundary of the body for all time beyond the initial time. Conditionson the boundary surface may be:

i. the traction is specified,ii. the displacement is specified,

iii. the component of traction in a certain direction is spec-ified and the displacement in the plane perpendicular tothis direction is also specified, or

iv. the component of displacement in a certain direction isspecified and the traction in the plane perpendicular tothis direction is also specified.

The boundary may be divided into any number of parts, but one andonly one of the conditions (i) through (iv) must be specified over eachpart. For an appropriate set of initial and boundary conditions, Neu-mann's uniqueness theorem ensures that the initial-boundary valueproblem has at most one solution, provided that energy is added to thebody or taken from it only through the loaded part of the boundary(Gurtin 1972; Achenbach 1973). This property of uniqueness of solu-tions has enormous practical value, for it means that an elastodynamicfield obtained by any means whatsoever which satisfies the governingequations, initial conditions, and boundary conditions is the only solu-tion of the problem. In obtaining solutions to elastic crack problems,conditions are often assumed in constructing a candidate solutionthat cannot be verified a priori. Nonetheless, if such a candidatesolution can be shown to satisfy the governing equations and otherconditions, then it is certain that it represents the unique solution. Fordynamically growing cracks, the statement of the uniqueness theoremrequires some modification due to the possibility of energy flow outof the body through the crack tip, as discussed in Section 7.7.

The elastodynamic reciprocal theorem provides a useful rela-tionship between solutions of two different boundary value problems

for the same elastic body. Suppose that solutions of two differentboundary value problems for a given body exist, say u and u*. Fur-thermore, suppose that the displacement and particle velocity are zerofor both solutions at the initial time, say t = 0. In addition, supposethat no body forces are present in either boundary value problem.Then, if T and T* are the tractions on the boundary surface, say S,corresponding to the two solutions, then the solutions are related by

/ /Js Jo

= 11Js Jo

T*(p5,t~r).u(p5,r)drd5. (1.2.59)o

The generalization of Betti's reciprocal theorem to elastodynamics,including body forces and general initial conditions, is discussed indetail by Gurtin (1972) and Achenbach (1973). The theorem wasextended to the case of unbounded domains by Wheeler and Sternberg(1968). There is no difficulty in extending the theorem to apply forgeneral initial conditions and body forces.

The practical matter of integrating (1.2.51) to determine partic-ular solutions often hinges on a representation of u in terms of otherfields which satisfy the elementary wave equations with characteristicspeed Cd or c8. The advantage gained through such a representationis that various means are available for solving the wave equation. Adifficulty with this approach concerns the completeness of the repre-sentation. Does every solution of the Navier equation have a repre-sentation of the assumed form? This matter is discussed by Sternberg(1960) for the representations to be introduced here.

A complete representation of a solution of the wave equation isobtained by means of the Helmholtz additive decomposition of thedisplacement vector into the gradient of a scalar field <\> and the curlof a divergence free vector field T/>,

V x ^ or Ui = (f>n +eijkil>k,j , (1.2.60)

where the dilatational displacement potential <\> and the shear dis-placement potential t/> are wave functions satisfying

c2dV

2<f> - 0 = 0, c ? V V - # = 0, V • V = 0. (1.2.61)


In terms of the displacement potential functions (f) and t/>, the dilata-tion of the deformation is V • u = V20 and the rotation vector of thedeformation is ^V x u = — ^V2T/>. The relationship (1.2.60) is calledthe Lame representation of the displacement vector.

Sternberg (1960) also discussed other representations. For exam-ple, the Poisson representation is

u = ud + u s , (1.2.62)

where

V x u d = 0, V-u* = 0 (1.2.63)

and

c2dV

2ud - iid = 0, c2 V2us - ii5 = 0 . (1.2.64)

This representation has been applied effectively in two-dimensionalcases when the elastodynamic fields are steady with respect to amoving observer or are self-similar, so that the wave equations reduceto Laplace's equation for which a general solution is available. For thespecial case of antiplane shear deformation, m = U2 = 0 and u$ =w(xi,X2,t). The out-of-plane displacement satisfies the elementarywave equation

c2sV

2w-w = 0. (1.2.65)

Yet another representation that is shown by Sternberg (1960) tobe equivalent to the Lame representation is

u = 2(1 - v) \c2dV

2G - G ] - V(V • G) , (1.2.66)

where the vector fields G and H satisfy

c2 V2G - G = H , c2dV

2H - H = 0. (1.2.67)

The Lame representation is used extensively in this book, andexpressions for rectangular components of displacement and stress interms of 0 and components of t/> are recorded here for later refer-ence. From (1.2.60), the components of displacement in terms of the


displacement potentials are

«1 =az2 a z 3 '

U2 = S + w9 - Wi' (L2-68)

dip:28x2

If these expressions are substituted into (1.2.48), then the componentsof stress in terms of the displacement potentials are

n - , \ *

r gy av3 a ^ aVal ? dx3dx1\ '

[o a2^ , a V i a2y>3 a 2 ^ ^023 = 2 + +dx\ dx\dx2\

=H \2-^—^ h -£-%• - -K—Q^ dx^ + dx dx '

(1.2.69)

Although no specific solutions are presented for anisotropic elasticmaterial response, some of the general developments are carried outfor an elastic material of general symmetry. The most general stress-strain relation for a material possessing a strain energy function is

*ij = Cijklekl. (1.2.70)


The array of elastic constants satisfies the symmetry conditions

(1.2.71)

so that there are 21 elastic constants at most. In terms of the stiffnessarray Cijki, the strain energy density of the material is

U = \CijkiCijeki = %CijkiUi,j uk,i . (1.2.72)

It is assumed that the relationship is invertible, so that there existsan array M^ki such that

Uj = Mijkiaki. (1.2.73)

The array M^ki of elastic compliances possesses the same symmetriesdoes.

1.2.4 Inelastic materials

The limited results available on dynamic fracture of inelastic mate-rials are also discussed in the chapters to follow. General analyticalprocedures do not exist for solving boundary value problems involvinginelastic material response, except for the case of linear viscoelasticity,and results are obtained on the basis of ad hoc procedures. Therelevant field equations for formulating such problems, aside from theconstitutive equations, are included in Section 1.2.2. Field equationsdescribing material response are presented along with the specificproblems studied. Among these are the determination of asymptoticcrack tip fields for crack growth in a rate independent elastic-plasticmaterial in Sections 4.4 and 4.5. Crack tip fields for rapid crack growthin an elastic-viscous material are considered in Section 4.6, and in anelastic-viscoplastic material in Section 4.7.

Chapter 8 is devoted to the study of plasticity and strain-rateeffects in dynamic fracture. Here, a few results of fairly broad applica-bility are obtained for rapid crack growth in elastic-plastic and elastic-viscoplastic materials. This chapter also includes brief discussions ofductile void growth in plastic materials and the apparent rate sensi-tivity exhibited by brittle materials undergoing rapid microcracking.In all cases, constitutive models are developed as part of the problemformulation.


1.3 Analytic functions and Laplace transforms

Some basic mathematical results to be used subsequently are summa-rized in this section. These results concern the general properties of ananalytic function of a complex variable and the properties of Laplacetransforms, both the one-sided and the two-sided types. Theorems arestated without proof, and only to a degree of generality important tothe applications to follow.

The theory of analytic functions of a complex variable is fullydeveloped by Hille (1959) and, within the context of applications,by Carrier, Krook, and Pearson (1966). A useful summary of mainresults is given by Seebass (1983). Integral transforms are developedby Carrier et al. (1966), and aspects relevant to the Wiener-Hopfmethod are summarized by Noble (1958).

1.3.1 Analytic functions of a complex variable

Consider the function /(£) of the complex variable ( = x + iy = ret0

defined in a neighborhood of a particular point. If /(£) is differentiablewith respect to £ at that point, then it is an analytic function at thatpoint. If /(C) is analytic at every point in a region, it is said tobe analytic in the region. Analyticity of /(C)> in turn, implies thatderivatives of all orders exist.

If /(£) is analytic then, with /(£) = u(x,y) + iv(x,y), it followsthat

du __ dv du _ dv

These relations are called the Cauchy-Riemann equations. If f(() isanalytic in a region, then the Cauchy-Riemann relations imply that

V2u = 0, V2v = 0 (1.3.2)

in that region. Real functions that satisfy Laplace's equation are saidto be harmonic functions. A general solution of the partial differentialequation V2u = 0 over some region is

u = Re{F(C)} or u = Im{G(C)}, (1.3.3)

where F(Q or G(Q is an analytic function in the region that must bedetermined from the boundary conditions.

1.3 Analytic functions and Laplace transforms 31

Among the most useful results on analytic functions for pur-poses of applied analysis are Cauchy's integral theorem and its con-sequences. If /(C) is analytic on and inside a simple closed curve C inthe complex C-plane, then

/ O. (1.3.4)

Similarly, if f(Q is analytic on and outside of a simple closed curve,including at infinity, and if | /(re^) | = o(l/r) as r —> oc then (1.3.4)also applies. Either form of the theorem can be extended to applyfor multiple simple closed curves by interconnecting the simple curveswith paths through regions of analyticity. If these paths are traversedin both directions, then the multiple curves and interconnections canbe viewed as a simple closed curve.

A consequence of Cauchy's theorem is Cauchy's integral formula.Suppose that /(C) is analytic on and inside a simple curve C enclosinga region of the £-plane. If Co is a point inside C then

(1.3.5)

where the path is traversed in the counterclockwise direction. Simi-larly, suppose that /(C) is analytic on and outside of a simple closedcurve C in the C-plane, including at infinity. Then, if |/(re l 0)| = o(l)as r —• oo and if Co is outside of C then

where the curve C is counterclockwise oriented.Various other extensions of the integral formula (1.3.5) are useful

in different circumstances. For example, if /(C) is analytic on andinside the simple closed curve C and Co is inside C then

where the left side denotes the n-th derivative of f((). Cauchy'sintegral formula can also be used to establish other useful properties of


Figure 1.3. Overlapping regions in the complex £-plane used to illustrate the roleof the identity theorem in analytic continuation.

analytic functions. For example, according to the maximum modulustheorem, if /(£) is analytic on and inside a simple closed curve C,then either the maximum value of |/(C)| is attained only on C or/(C) is constant throughout the region. Another very powerful resultis Liouville's theorem, which states that a bounded entire functionof a complex variable is necessarily a constant. A generalization ofLiouville's theorem is stated in Section 2.5.2 in connection with theWiener-Hopf technique. Yet another result, called the zero countingformula, is used to great advantage in Section 2.5.

A theorem which provides the basis for techniques of analyticcontinuation is the identity theorem or uniqueness theorem of analyticfunction theory. Suppose that /(£) is analytic in a region of thecomplex £-plane and that /(£) = 0 on any dense subset of that region,for example, along a segment of a curve or over a small area. Underthese conditions, /(£) is identically zero throughout the region. Thisvery important theorem plays a central role in most techniques ofapplied mathematics based on analytic function theory and on integraltransforms. It follows immediately from the identity theorem that if/i(C) and /2(C) a r e both analytic throughout a region of the £-planeand if /i(C) = /2(C) o n a dense subset of the region, then the twofunctions are equal to each other and, indeed, each represents preciselythe same function over the whole region.

To see how the identity theorem provides the basis for analyticcontinuation, consider functions /i(C) and /2(C) that are analytic inregions i?i and i?2, respectively, of the complex plane. Furthermore,suppose that the two regions have some overlap, as shown schemat-ically in Figure 1.3. Then, if /i(C) = .MC) within the commonregion of overlap, /i(C) or /2(C) represents the analytic continuation of

/2(C) or /i(C) into Ri or i?2, respectively, and the functions togetherrepresent a single analytic function throughout the combined regionRi +#2- This analytic continuation idea is the basis for an importantstep in the Wiener-Hopf technique.

1.3.2 Laplace transforms

Laplace transforms are used in a rather formal way to obtainsolutions to boundary value problems in the chapters to follow. Thetransforms are defined here, and certain properties which are exploitedlater are identified. The equations to be solved are typically partialdifferential equations involving functions of time and one or morespatial variables. For this general discussion, it is sufficient to considera function of time t and a single spatial variable, say x.

Consider a function w(x,t) defined for t > 0 and —oo < x < oo.The Laplace transform on time of w is denoted by a superposed haton the function and is defined by

w(x,s)= w{x,t)e~8tdt, (1.3.7)

where s is the transform parameter. Suppose that w(x, t) is integrableover any finite interval of t and \w(x, t)\ = O (ea+t) as t —• oo,where a+ is a real constant. Then the integral in (1.3.7) convergesfor Re(s) > a+ and it defines some function of s in that half plane.Furthermore, the integral can be differentiated with respect to s, andthe derivative converges for Re(s) > a+. Thus, w(x,s) is an analyticfunction of the complex variable s in Re(s) > a+ for fixed x. In viewof the identity theorem for analytic functions, it is sufficient to vieww(x, s) as a function of a real variable s over some segment of thereal axis in the half plane of analyticity. Once w(x, s) is determinedas an explicit function of s in the course of solving the transformeddifferential equations, the definition of w(x, s) can be extended to theentire complex s-plane, except for isolated singular points, by analyticcontinuation.

The function w(x,t) is obtained by means of the transform in-version integral

-• p<r+ioo

w{x,t) = — \ w(x,s)estds, (1.3.8)2 m Ja-ioo

where a is any real parameter greater than a+. The integration pathis a line parallel to the imaginary axis in the s-plane and within theregion of analyticity of w(x, s).

In the particular case when a transform is factorable into a prod-uct of transforms, say

w(x, s) = {?i(x, 5)^2(^5 s), (1.3.9)

the function w(x11) itself is a convolution of wi(x, i) and W2(x, £), thatis,

fttill f V" 1 —• / Olitft* /• - ^ T Hllftf T ^ I fiT~ 1 1 Q 1 C\ 1

Transform inversion integrals of the form (1.3.8) are often evalu-ated by changing the path of integration from the infinite line parallelto the imaginary axis to a path in a left half plane that encloses allof the singular points of the integrand in the complex s-plane. Thechange in path is accomplished by forming a closed path made upof the inversion path and the path enclosing the singularities pluscircular arcs of indefinitely large radius, and then invoking Cauchy'stheorem (1.3.4). For this process to be successful, it is essential thatthe integrals taken along the circular arcs of large radius vanish as theradius becomes indefinitely large. This will be the case provided thatw{x,s) —• 0 uniformly as |s| —> 00. This result follows from Jordan'slemma as applied to integrals of the form (1.3.8) (Seebass 1983).

The transform of the time derivative dw(x,t)/dt is

/ ^e~8t dt = 8w(x, s) - w(x, 0+). (1.3.11)Jo+ ™

If w is identically zero for t < 0 and if it increases discontinuously att = 0 to w(x,0+) then, in the sense of distributions (Stakgold 1968),

/ *ire~8t dt = wfa s) - (1.3.12)Jo- dt

The utility of the Laplace transform methods in solving linear differen-tial equations is represented by (1.3.11). Multiplication of each termin the differential equation by e~st and integration over t from 0+ to00, in effect, replaces time differentiation with multiplication by the

parameter s. The parameter s can be manipulated algebraically. Oncea solution of the transformed equations is found, the solution in thetime domain is obtained by superposition over the continuous rangeof s represented by (1.3.8). The completeness of the representationof functions in terms of integral transforms is established by meansof Fourier's integral theorem (Carrier et al. 1966). In the presentcase, completeness implies that every function (within an appropriateclass of integrable functions) has the representation (1.3.8), where thedensity function w(x, s) is given by (1.3.7).

Consider now the function w(x, s) for any fixed real s in the regionof convergence of w(x, s) and for —oo < x < oo. The two-sided orbilateral Laplace transform of w(x,s) is denoted by a correspondinguppercase letter and it is defined by the integral

/.oo

s)= w(x, s) e'Z* dx

J—oo

/.0 /.oo

= / w(x, s) e~^x dx + w(x, s) e~^x dx,J—oo Jo

(1.3.13)

where £ is the complex transform parameter. Suppose that w(x, s) isintegrable over any finite interval of x and that

IO(eT+x) as # > + o o ,

) ' (1.3.14)O (e T-x) as x —• - o o ,

where r+ and r_ are real constants. The integral over positive valuesof x in (1.3.13) then converges and defines an analytic function of £ inthe half plane Re(£) > r+, whereas the integral over negative values ofx converges and defines an analytic function in the half plane Re(£) <r_. Thus, W(£,s) exists and represents an analytic function in thestrip r+ < Re(£) < r_. Once sufficient information is available aboutthe dependence of W(£, s) on £ from the boundary value problem,the definition can be extended to the entire complex plane, except forisolated singular points, by means of analytic continuation. If r+ = r_the strip of analyticity degenerates to a line, and if r+ > r_ then thetwo-sided transform does not exist.

The inversion integral for the two-sided Laplace transform is

1 pr+ioow(x, s) = — W(£, s) e«* d£, (1.3.15)

2™ Jr-ioo

where r is a real parameter in the interval r+ < r < r_. Theintegration path is a line parallel to the imaginary axis in the £-planewithin the strip of analyticity of W(£, s).

Consider the special case of a function of x and t that is zero forall x and t < 0 and that vanishes for all x < 0 and t > 0, say w+(x, t).The Laplace transform on time w+(x, s) then also vanishes for x < 0and s in a suitable interval of the real axis. The "two-sided" Laplacetransform of w+ (a;, t) is then no different from a one-sided transform,that is,

W+(f, 5) = / w+(x, s) e-*x dx. (1.3.16)

Evidently, the value of W+(£, s) at any particular value of £depends on w+ over the entire range of x. Likewise, from the in-version integral, the value of w+(x, s) at any x depends on W+ overthe full range of £. However, transform pairs of the kind definedby (1.3.16) have remarkable asymptotic properties relating limitingvalues of a function and its transform. According to van der Poland Bremmer (1955), theorems that yield an asymptotic property ofa transform from a known asymptotic property of the function arecalled Abel theorems. On the other hand, theorems that provide anasymptotic property of a function from a known asymptotic propertyof its transform are called Tauber theorems.

The basic Abel theorem has been useful in elastic fracture me-chanics studies. In this case, direct asymptotic analysis of the dif-ferential equations often leads to the conclusion that a particularfield depends on a spatial coordinate near the crack tip in a certainway. Through an Abel theorem, it is then possible to make definitestatements about the behavior of the transform at remote points inthe transform parameter plane.

For example, suppose that w+(x, s) in (1.3.16) has the behavior

w+{x,s)~xa (1.3.17)

near x = 0, where a is a real constant with a > — 1. The Abeltheorem then states that

X= lim ^aW+(^s), (1.3.18)

where F(-) is the Gamma or factorial function. If the region ofconvergence of the integral defining W+ includes the imaginary axis

1.4 Overview of dynamic fracture mechanics 37

then it can also be established that

f(a; ,a)= lim ^W+(^s) (1.3.19)£ ( ) +

im tRf(a;,a)= limx—>-+oo x" £—•()+

for an appropriate value of /3 > —1.

1.4 Overview of dynamic fracture mechanics

Analytical results of a permanent nature in the area of dynamic frac-ture mechanics are gathered together and summarized in Chapters 2through 8. For the most part, the results are obtained within the smallstrain formulation of continuum mechanics as described in Section1.2.2. Cracks are assumed to be planar and the opposite faces are as-sumed to occupy the same plane in the undeformed configuration. Thegeometrical configurations considered are also very simple, in general.For example, numerous developments are based on the assumption ofa half plane crack in an otherwise unbounded body under conditionsthat result in a two-dimensional state of deformation. Though thisassumption provides the basis for the formulation of a well-posedboundary value problem, it does have practical limitations. Theresults can only be applied in cases when the material points of interestare much closer to the crack edge than to other boundaries of thebody and when the local deformation field is indeed essentially two-dimensional. Other configurations considered include crack growth ina long uniform strip and idealizations of bodies as structural elements,typically beams or strings.

The purpose here is to provide a very brief overview of the mate-rial in the chapters to follow, including a summary of the main resultsand comments on directions for further study. A number of books andmonographs are available to provide background on fracture mechan-ics from various perspectives. For example, the books by Kanninenand Popelar (1985), Cherepanov (1979), and Hellan (1984) developthe topic from the mechanics perspective. The books by Knott (1979)and Lawn and Wilshaw (1975) introduce the subject from a materialsscience perspective, whereas Barsom and Rolfe (1987) discuss fracturewithin the context of materials selection and structural applications.

A number of interesting but specialized issues in dynamic fractureare discussed only briefly in this monograph. Among these topicsare: the existence of an apparent terminal crack speed well below


Figure 1.4. The classification of crack opening modes I, II, and III based on thecomponents of the relative displacement of the crack faces near the crack edge.

the Rayleigh wave speed in glass and some other very brittle mate-rials; the contact interaction of crack faces during dynamic fractureevents in materials; the symmetric bifurcation of a straight, rapidlyrunning crack in a tensile field into two branches; the influence ofheat generated during rapid fracture on the separation process oron the stress distribution; atomistic modeling of dynamic cleavage;and the erosion of materials by impact of solid particles or liquiddroplets. Some experimental research and analytical modeling havebeen done on each of these problems using the approaches describedin the following chapters, but the phenomena are not yet completelyunderstood. Available references are included in the Bibliography.Other topical areas that are not yet fully developed are identified inthe overview to follow or in the later chapters.

1.4-1 Basic elastodynamic solutions for a stationary crack

In Chapter 2, the fundamental stationary crack problems of linearelastodynamics are analyzed. The two-dimensional configuration ofa semi-infinite crack in an otherwise unbounded body is considered.The body is initially stress free and at rest. At time t = 0, equaland opposite spatially uniform tractions begin to act on the crackfaces. Formulated in this way, the problem has neither a characteristiclength nor a characteristic time. The in-plane opening mode of crackdeformation (mode I), the in-plane shearing mode (mode II), andthe antiplane shearing mode (mode III) are treated separately. Thedirection of the relative crack opening displacement for each mode isindicated in Figure 1.4, and the classification is further discussed inSection 2.1.

Two mathematical methods for solving initial-boundary valueproblems are introduced, namely, Green's method based on superposi-tion over a fundamental singular solution of the governing differential


equation and the Wiener-Hopf method based on Laplace transformtechniques. These methods are presented in a formal way, and theyare used in later chapters as well. The main results of the analysisare very simple. Indeed, aside from a scalar multiplier of order unity,the stress distribution near the crack edge can be obtained for eachmode through dimensional analysis. With reference to the coordinatesystem shown in Figure 1.4, if the crack face traction has magnituder* per unit area and if it acts in a direction parallel to the crack edge,then the state of deformation is antiplane shear and the crack opensin mode III. The shear stress on the plane y = 0 directly ahead of thecrack tip, and close to it in some sense, is

azy(x,O,t) ~ ^ ^ , Km{t) = 2r% /2Cs*

where cs is the elastic shear wave speed. This result is derived inSection 2.3 where it appears as (2.3.18). Likewise, if a normal pressureof magnitude a* begins to act on the crack faces at time t = 0 thenthe state of deformation is plane strain and the crack opens in modeI. The normal stress on the plane y = 0 near the crack tip in this case

where Cd is the dilatational wave speed of the material and v isPoisson's ratio. This result is derived in Section 2.5.3 where it appearsas (2.5.44). Lastly, if a shear traction of magnitude r* begins to acton the crack faces at time t = 0 in a direction perpendicular to thecrack edge, then the state of deformation is plane strain and the crackopens in mode II. The shear stress on the plane directly ahead of thecrack tip in this case is

axy(x,0,t)

This result is given in Section 2.6 as equation (2.6.9). The solutionsobtained in Chapter 2 are readily generalized to the case of appliedcrack face loading with arbitrary time variation. Furthermore, thesolution for mode I, II, or III is identical to that for a plane tensile,transverse shear, or horizontal shear wave carrying a simple jump


in stress that is normally incident on the crack plane, except for anelementary one-dimensional plane pulse.

1.4-2 Further results for a stationary crack

The problems analyzed in detail in Chapter 2 have been extremelyimportant in establishing the connection between transient loadingapplied to a cracked elastic body and the time dependence of theresulting stress intensity factor. The fact that the fields are strictlytwo-dimensional and that the loading/configuration systems do notinvolve a characteristic length limits the applicability of the resultsin analyzing certain phenomena. Thus, Chapter 3 is devoted toextensions that overcome some of these limitations.

In Section 3.2, the mode I problem of a pair of concentratednormal forces suddenly applied to the faces of a semi-infinite crack ata fixed distance from the crack tip is considered. The fixed distancefrom the load point to the crack tip is a characteristic length in thiscase. The loading on the crack faces generates a displacement of thesurface, and a moving dislocation problem is formulated and solvedto make it possible to cancel this displacement for points beyond thecrack tip. The exact stress intensity factor history is obtained byconstructing a solution as a superposition over a fundamental movingdislocation field for the configuration. For loads that tend to openthe crack, the stress intensity factor is initially negative, as could beanticipated on the basis of known elastodynamic wave fields. Thesurprising result is that the transient stress intensity factor takeson its long-time equilibrium limiting value instantaneously at a timecorresponding to the arrival time at the crack tip of a Rayleigh wavegenerated at the load point at t = 0, rather than asymptotically ast —• oo. The result is given as equation (3.2.14). With this resultin hand, the stress intensity factor for general time variation of theforces can be obtained as in (3.2.15) or for general spatial distributionof traction as in (3.2.16).

The mode I situation of suddenly applied pressure on the faces ofa crack of finite length is considered in Section 3.3. The characteristiclength in this case is clearly the length of the crack. It is recognizedin this case that the complete solution of the problem can be viewedas a series of solutions of half plane crack problems of ever increasingcomplexity. The first problem in the series is that solved in Section2.5 and the second is formulated and solved in Section 3.3. These twosolutions together give a complete picture of the influence of crack


length through first-order wave scattering. This picture reveals thatthe stress intensity factor at either crack tip increases to a level about30 percent greater than the long-time equilibrium stress intensityfactor before unloading waves from the far end of the crack cause it toagain decrease. This phenomenon, which is illustrated in Figure 3.5,is called dynamic overshoot of the stress intensity factor. Subsequentwave interactions lead to an oscillation of the stress intensity factorthat decays over time to the long-time limit, which is the equilibriumresult for the specified loading.

The problems considered in Sections 3.4 and 3.5 are again con-cerned with the half plane crack configuration. However, in the casesconsidered, the loading is nonuniform in the direction parallel to thecrack edge. Consequently, all field variables, including the stressintensity factor, vary with position along the crack edge. In Section3.4, a plane tensile pulse is obliquely incident on the edge of the crack.If it is noted that the resulting transient deformation field is timeindependent as seen by an observer traveling along the crack edge ata certain speed, then the problem can be reduced to two-dimensionalproblems of the kind solved exactly by the methods introduced inChapter 2. For the particular problem analyzed, the crack openingoccurs as a combination of mode I and mode III, and the transientstress intensity factors are given in (3.4.10) and (3.4.14), respectively.For an incident pulse carrying a step increase in tensile stress and forany given point on the crack edge, the stress intensity factor increasesin proportion to the square root of the time elapsed after the wavefrontreaches that point. The coefficient depends on the angle of incidence,of course.

In Section 3.5, an exact analysis of a truly three-dimensionalelastic crack problem is presented, although for a case withoijt acharacteristic length. A traction is suddenly applied to the crack facesin the form of opposed line loads oriented in a direction perpendicularto the crack edge, with the loads acting in a direction normal to thecrack faces. The transient stress intensity factor at any point alongthe crack edge is obtained by a modification of the transform methodsintroduced in Chapter 2. An expression for the stress intensity factoris given in (3.5.19) and a graph is shown in Figure 3.6. The stressintensity factor is zero until the arrival of the first dilatational wave.It is then negative until just after the arrival of the Rayleigh surfacewave traveling on the crack faces. Thereafter, the stress intensityfactor is positive, and it approaches the equilibrium stress intensity


factor as a long-time limit. Another problem in this same class is thatfor an opposed pair of point forces suddenly applied to the crack facesat a fixed distance behind the crack edge. The availability of this so-lution could have major impact on further study of three-dimensionalelastodynamic crack fields. However, this problem, which is the three-dimensional equivalent of the two-dimensional case studied in Section3.2, has a characteristic length and a solution is not yet available.

Chapter 3 concludes with a discussion of the Irwin stress intensityfactor criterion for fracture initiation from a preexisting crack understress wave loading. According to this criterion, a crack will beginto extend when the crack tip stress intensity factor is increased toa level characteristic of the material, called the fracture toughness.The discussion is based on the elastodynamic solutions in Chapters 2and 3. The important role of stress wave effects in fracture initiationis demonstrated. The duration of an incident pulse, as well as itsamplitude, is important in considering fracture initiation under stresswave loading. Some examples of experimental data showing the strongdependence of fracture toughness on the rate of loading are includedin the discussion.

1.4-3 Asymptotic fields near a moving crack tip

Chapter 4 is concerned with the description of stress and defor-mation fields for points very close to the crack tip compared to anyother length scale in the configuration. Strictly speaking, the resultsare valid only asymptotically as the observation point approaches thecrack tip. However, it is expected that in some cases the asymptoticfield provides a reasonably accurate description of the mechanicalstate near the crack tip over a region large enough to be of somepractical significance for real materials. For example, the size of theregion of validity of an asymptotic field should be large enough tocompletely include within it any finite strain zone or region dominatedby effects beyond the realm of the continuum mechanical models used.

The asymptotic crack tip fields for linear elastodynamic crackgrowth are extracted in Section 4.2 for the antiplane shear mode andin Section 4.3 for the plane strain modes, all for general motion ofthe crack tip in a plane. The analysis is based on standard methodsof asymptotic analysis, applied in a systematic way. As in the caseof equilibrium crack tip fields, the asymptotic solution is found tohave universal spatial dependence. All information about loading andconfiguration are embedded in a scalar multiplier called the dynamic


stress intensity factor. The leading term in the asymptotic expansionin each case, which is the most singular term in the local stress orstrain distribution corresponding to bounded total mechanical energy,is found to vary with radial distance from the crack tip r as r"1/2

and to have a characteristic angular variation. Thus, for any stresscomponent,

as r —• 0, where r, 0 are polar coordinates in the plane of deforma-tion for each mode. The functions prescribing angular variation aregiven explicitly in (4.2.17) for the case of mode III crack growth, in(4.3.11) for mode I crack growth, and in (4.3.24) for mode II. Theangular variation of stress components is found to differ little fromthe corresponding equilibrium results for crack speed less than about40 percent of the shear wave speed c8 of the material, but to showsignificant variation with crack speed for higher speeds. Some stepsare taken toward determining the terms in an asymptotic expansionbeyond the first for nonplanar crack growth and for nonsteady growth,but this is a line of investigation that is not yet fully developed.

The case of steady crack growth in the antiplane shear modethrough an elastic-ideally plastic material is considered in Section 4.4.The material is assumed to be governed by an associated flow rule. Inthis case, the stress components are bounded at the crack tip but thetotal strain components are logarithmically singular. In particular,the total shear strain on the crack line directly ahead of the movingcrack tip is found to vary as lnr, where r is the radial distance fromthe crack tip. This is in contrast to the corresponding result for steadycrack growth under equilibrium conditions for which this shear straincomponent varies as In2 r. This fundamental inconsistency cannotbe resolved through asymptotic analysis alone because the range ofvalidity of asymptotic fields cannot be established without the benefitof a more complete solution. For this particular case, the inconsistencyis eventually resolved in Section 8.3.

The case of steady crack growth under plane strain conditionsthrough an elastic-ideally plastic material is considered in Section 4.5.Again, the material is assumed to respond according to an associatedflow rule. Furthermore, it is assumed that the material is elasticallyincompressible, as well as plastically incompressible. A complete


asymptotic solution is constructed which satisfies all of the imposedconditions. It has the feature that all angular sectors around the cracktip are either regions of active plastic flow or regions of incipient flowin which the stress state satisfies the yield condition. In addition, thisasymptotic solution has the feature that the plastic strain componentsare bounded at the crack tip. One way to admit more general asymp-totic solutions in this case is to permit discontinuities in the angularvariation of stress and particle velocity. However, it is shown thatthis possibility is ruled out if the material is required to satisfy theprinciple of maximum plastic work through the discontinuities. Thepossibility of elastic unloading sectors is also ruled out on the basis ofsimilar arguments. Thus, the fully plastic asymptotic solution withbounded plastic strain components appears to be the only asymptoticsolution, at least for elastic incompressibility. No information on therange of validity of this solution is available, and asymptotic fields ofthis kind require further study.

Chapter 4 concludes with brief sections on the asymptotic fieldsfor dynamic crack growth in elastic-viscous and elastic-viscoplasticmaterials of certain special kinds. It is shown that if the rate sensi-tivity of the material is very strong then the behavior of the materialvery near to the crack tip is dominated by the instantaneous elasticproperties, so the stress is found to be square root singular at the cracktip. For weaker rate sensitivity, on the other hand, the elastic andplastic strain rates are of the same order in magnitude and the stressis found to have a weaker singularity. The material models in thesetwo sections differ from each other only in the fact that viscous effectsare present at all levels of shear stress in the case of elastic-viscousresponse whereas they are present only if the stress state satisfies someyield condition in the case of elastic-viscoplastic response. The cracktip response of such materials is not yet fully understood, and onlyad hoc estimates of the range of validity of the available asymptoticresults exist.

I.4.4 Energy concepts in dynamic fracture

Analytical results concerned with mechanical energy or energy-likequantities are considered in Chapter 5. Following a brief review ofIrwin's calculation leading to the relationship between energy releaserate and stress intensity factor for elastic cracks, suggested by (1.1.8)above and stated for mode I cracks in (5.3.9), the matter of a generalcrack tip energy flux integral is addressed. An expression for energy

flux F through a contour V surrounding the tip of a crack movingnonuniformly through a material with arbitrary mechanical responseis

F(T) = jT [<ry»v - + (U + T) «mj ds,

where the crack grows at speed v in the xi-direction. This resultis derived as (5.2.7). The energy flux includes both recoverable andirrecoverable work done on the material. The development is basedon mechanical quantities alone. However, the restrictions required fora thermodynamic interpretation of the result are outlined. Further-more, it is noted that the energy flux integral applies for deformationswith finite strain with a proper interpretation of the mechanical fieldquantities involved in the integrand. The crack tip integral is pathindependent only in the case of steady state crack growth. It is alsonoted that its limiting value as T is shrunk onto the crack tip isindependent of the shape of T for all known asymptotic fields.

The energy flux integral is first applied to the case of elastody-namic crack growth. The limiting value of the crack tip energy flux asthe integration contour is shrunk onto the crack tip is identified as thedynamic energy release rate G. By means of a direct calculation basedon the asymptotic fields established in Chapter 4, a generalizationof the Irwin relationship between stress intensity factor and energyrelease rate is established for the case of dynamic crack growth. Theresult has the form

for mode I crack growth at speed v, where E and v are the elasticmoduli and Aj{v) is a universal function of instantaneous crack tipspeed. The general result for all three modes of crack advance is givenin (5.3.10).

The property of path independence of the crack tip energy in-tegral for steady crack growth is then exploited to establish severalresults. First, for a situation with small-scale yielding, the connectionbetween some properties of a crack tip cohesive zone and the dynamicstress intensity factor of the elastic field surrounding this nonlinearzone is determined. For example, if the cohesive stress in the zonehas the uniform value <JO, the crack tip opening displacement is 6t,and KiapPi is the remote applied stress intensity factor, then

AT(v)K?appl = Eao6t


under plane stress mode I conditions. The case of a slip weakeningcohesive zone in mode II is also considered. Next, the case of steadycrack growth along an infinite strip with uniform edge conditions isrecognized as an ideal case for application of the path independentenergy integral, and several specific examples are described in Section5.4.

In Section 5.5, some crack growth models based on simple struc-tural elements, such as strings or beams, are considered. The centralidea is to determine the relationship between the crack tip energyrelease rate and crack tip field parameters. Due to the kinematicconstraints on the deformation imposed by the structural model, astress intensity factor field will not develop, in general. Nonetheless,the dynamic energy release rate can be expressed in terms of otherquantities. For example, for a string model G is expressed in termsof the slope of the string at the crack tip in (5.5.5), or for a beammodel it is expressed in terms of the bending moment at the cracktip in (5.5.14). The value of such results is that, even though a stressintensity factor field does not develop in the model at the level of detailemployed, a value of stress intensity factor can be inferred nonethelessby calculating G and then invoking the general relationship betweenthe energy release rate and the stress intensity factor.

For transient elastodynamics, there are no path independent in-tegrals that involve actual work and energy quantities of the system.However, there are path integrals of convolutions of stress and strain,or force and particle velocity, that are path independent. Becausethese integrals do not represent quantities of physical interest, theyare not useful in establishing field characterizing parameters. Instead,their value must be sought in their usefulness in extracting infor-mation from boundary value problems. One such path independentintegral is introduced in Section 5.6 in the Laplace transform domain.The connection between this integral and the Laplace transform of thestress intensity factor of a stationary crack is established in (5.6.22).The integral is used to determine the stress intensity for the problemconsidered in Section 2.5, but without actually solving the boundaryvalue problem in this case.

For elastic crack problems in general, it is reasonable to expectthat the stress intensity factor is a linear functional of the load distri-bution. This idea is developed in Section 5.7 for a body containing astationary crack and subjected to transient loading. The kernel of thelinear functional is called the weight function for the configuration,


and two means of determining this function for mode I cracks areoutlined. In the first method, the weight function is defined in termsof the solution of a particular boundary value problem, whereas in thesecond method a boundary value problem is formulated for the weightfunction itself. Use of the second method is illustrated by determiningthe stress intensity factor for the mode I point load problem consideredin Section 3.2.

The last section of Chapter 5 is concerned with the far fieldenergy radiation from an expanding crack. The radiated energy isdefined as the net work done on a remote material surface, and variousexpressions for this energy flux in terms of crack plane features areobtained. This issue is of central importance in interpreting seismiccrustal faulting data in the earth, and the connection between theradiated energy and certain average fault parameters is discussed.

1.4-5 Elastic crack growth at constant speed

Chapter 6 is concerned with the solution of particular crack growthproblems under the restriction that the crack speed is constant oncegrowth begins. The problems selected are representative of the classesof crack growth problems that are amenable to analysis. The firstclass considered is characterized by the feature that all fields aretime independent as seen by an observer moving with the crack tip.Historically, the most important problem in this class is the Yoffeproblem, consisting of a mode I crack of fixed length traveling atconstant speed under the action of uniform remote tensile loading.The direction of loading is perpendicular to the plane of the crack.For steady crack growth at speeds less than the characteristic wavespeeds, the governing equations are elliptic and the problem can bereduced to a boundary value problem for two analytic functions.Analysis leads to the conclusion that the stress intensity factor, whichis given in (6.2.20), is identical to that for the same configurationwith a stationary crack under equilibrium conditions. However, theangular variation of the near tip stress field has the general propertiesestablished in Section 4.3 for asymptotic mode I cracks. A case ofsteady crack growth in mode II is also analyzed, and the solution isused to construct solutions for steady crack growth accompanied by acrack tip cohesive zone. The length of the cohesive zone for uniformcohesive traction is given in (6.2.33).

The next problem class addressed is that of self-similar crackgrowth. Problems in this class are characterized by the property that


fields at any one time are obtained from those at any other timesimply by rescaling, or by the property that the functions representingmechanical fields are homogeneous functions of spatial coordinatesand time. By assuming that the fields are self-similar, the governingpartial differential equations can be recast into a form that is ellipticbehind the wavefronts but hyperbolic ahead of the wavefronts. Thus,initial data on characteristics are carried to the wavefronts, jumpconditions provide boundary conditions on the elliptic region, andmethods of analytic function theory yield a solution in the ellipticregion. The method is presented in detail for the Broberg problem.The essential features of this problem are that a mode I crack growssymmetrically at constant speed from zero initial length under theaction of remote tensile loading in a direction perpendicular to theplane of the crack. If crack growth begins at time t = 0, it is foundthat the stress intensity factor increases in proportion to \fi". Thefull expression is given in (6.3.48). The higher order terms in theasymptotic field and the amount of work done on the body that goesinto fracture are also considered in Section 6.3. The correspondingsolutions for a mode II shear crack and for nonsymmetric crack growthare outlined as well.

The matter of extension of a preexisting half plane crack in anunbounded body subjected to time independent loading is addressedin Section 6.4. The development proceeds by first finding the full modeI deformation field for constant speed crack growth when an opposedpair of concentrated forces on the crack faces is left behind as thecrack tip begins to move at constant speed v. The analysis is basedon Laplace transform methods and the Wiener-Hopf technique, essen-tially as developed in Chapter 2. With the fundamental solution inhand, the stress intensity factor field for crack advance which negatesany time independent load can be constructed by linear superposition.If p(x) is a tensile traction distribution on the crack plane ahead ofthe crack tip that is to be negated by crack advance, then it is shownin (6.4.31) that the dynamic stress intensity factor at any time afterthe onset of crack growth is

,v) = k(v)J- /V 7T JO

where k(v) is a universal function of crack speed given in (6.4.26). Thisfunction decreases monotonically, and roughly linearly, from k = 1


at speed zero to k = 0 at the Rayleigh wave speed CR. Thus, thedynamic stress intensity factor has the form of a universal functionof crack tip speed multiplied by the corresponding equilibrium stressintensity factor for the same loading conditions and the instantaneouscrack length. The corresponding results for the other modes of crackadvance, which are summarized in the same section, have this samegeneral feature.

The chapter concludes with an analysis of the similar situationof growth of a preexisting crack at constant speed under the actionof transient loading. The approach is illustrated by considering againthe mode I problem analyzed in Section 2.5. Thus, at time t = 0a pressure begins to act on the crack faces and a transient field isradiated from the loaded crack faces. In the present case, however,the crack begins to grow at constant speed at some arbitrary timeinterval, called the delay time, after the loading is applied. For thecase of a normally incident plane stress pulse that carries a jump instress of magnitude a*, the stress intensity factor is shown in (6.5.16)to be

JK}(t,v) = 2(7 k(v)-*—Q v >

where k(v) is the same universal function that appeared for the caseof time independent loading. Note that the resulting dynamic stressintensity factor is independent of the arbitrary delay time, and that ithas the form of a universal function of crack tip speed times the stressintensity factor that would exist if the crack tip has been stationaryand at its instantaneous position for all time. The result is readilygeneralized to arbitrary time variation of the incident wave loading.

1.4-6 Elastic crack growth at nonuniform speed

An objective of dynamic fracture mechanics is to predict the motionof a crack tip under given conditions of loading and geometrical con-figuration. Certainly, if the motion is specified a priori then there is nohope of predicting motion in any literal sense. In order to establishan equation of motion for the position of a crack tip under certainconditions, it is necessary to know the mechanical conditions thatprevail for all possible motions. Then, a crack growth criterion can beimposed to select the "actual" motion according to that criterion fromamong all possible motions. Prom the analytical point of view, thismeans that a solution of the relevant boundary value problem valid


for an arbitrary, nonuniform motion of the crack tip must be known.The few results of this kind that are available are summarized andapplied in Chapter 7.

The case of nonsteady growth of a semi-infinite antiplane shearcrack in an otherwise unbounded body is considered first. This prob-lem can be solved directly for quite general loading conditions bymeans of Green's method as presented in Section 2.3. Some of thedetails are presented in Section 7.2 for the case of time independentloading, and the results for general loading are also summarized. Thesolution is found to have several features that are very unusual amongtwo-dimensional wave propagation problems. For example, it is foundthat the stress intensity factor during nonuniform growth is a universalfunction of crack speed times the equilibrium stress intensity factorfor the specified loading and the instantaneous crack tip position.Furthermore, it is found that if the crack speed is abruptly reducedto zero, then the stress intensity factor changes discontinuously to theappropriate equilibrium value for the given loading and crack position.In fact, it is found that the fully established equilibrium field radiatesout behind the shear wavefront emitted when the crack tip suddenlystops.

It is demonstrated in Section 7.3, by means of an inverse argu-ment, that these same remarkable features are present for the case ofmode I crack growth under general time independent loading, exceptthat the equilibrium field is not fully established in the region behindthe shear wavefront. Instead, it is shown that the equilibrium trac-tion distribution is radiated ahead of the crack tip behind the shearwavefront and the equilibrium displacement distribution is radiatedon the crack faces behind a point moving with the Rayleigh wavespeed. These features are exploited to show that the dynamic stressintensity factor for arbitrary motion of the crack tip, described by thetime dependent amount of crack advance Z(£), say, has the form

, 0),

as given in (7.3.19). That is, the dynamic stress intensity factor hasthe form of a universal function of crack tip speed times the corre-sponding equilibrium stress intensity factor for the given loading andinstantaneous crack tip position. Thus, the dynamic stress intensityfactor depends on instantaneous crack tip position and crack speed,but otherwise not on the history of crack motion. The corresponding

1-4 Overview of dynamic fracture mechanics 51

results for the other modes of crack advance have the same generalfeatures.

With the dynamic stress intensity factor available for nonuniformcrack growth, the dynamic energy release rate is also available throughthe generalized Irwin relationship. Then, if the Griffith energy balancecrack growth criterion is applied in the form G = 27 as discussed inSection 1.1, then an equation of motion for the crack tip is obtainedin the form

2El ^ = AI(i)k(i)2*i-i/cR.

A slightly more general form of this equation of motion appears asequation (7.4.4). If the circumstances of Mott's original analysis areassumed, as outlined in Section 1.1, then this equation of motionimplies that the maximum speed of crack growth is the Rayleigh wavespeed. In addition, it is seen that if the work of separation 7 isa constant then the equation of motion is a first order differentialequation. Thus, if an analogy with the motion of a mass particleis made, the equation of motion implies that the crack tip has noeffective inertia. It is also noted, on the basis of the equation ofmotion, that if an interface with no cohesive strength is suddenlyopened then the opening boundary will travel with the Rayleigh wavespeed CR.

Some illustrations of the equation of motion based on an energybalance crack growth criterion are discussed in Section 7.4. Included isthe situation of crack growth through a material with periodic fractureresistance for which the interpretation of fracture energy is different atdifferent size scales of observation. Also included is the considerationof mode II crack growth in a situation involving spatially nonuniformapplied stress or fracture resistance. This analysis illustrates the im-portant role of nonuniform fields in controlling crack propagation andarrest. Finally, the crack tip equation of motion for some structuralconfigurations is analyzed. Because these configurations are relativelysimple, the implications of the equation of motion can be presentedin greater detail than is possible in general continuum analysis. Thisfeature makes it possible to examine the role of reflected waves oncrack growth and arrest in bounded bodies, for example. The ideaof a crack tip equation of motion is extended to the case of transientloading in Section 7.5.


Chapter 7 concludes with a brief discussion concerning unique-ness of elastodynamic solutions involving dynamic crack growth. Theproof of the standard uniqueness theorem of elastodynamics is basedon a demonstration of the fact that the energy of a difference solutionfor a particular boundary value problem is always zero. Because amoving crack tip acts as a sink or source of energy, the conditionsunder which a solution can be expected to be unique must be recon-sidered. It is shown that the uniqueness theorem applies as long asthe crack speed is less than the Rayleigh wave speed of the material.

1.4-7 Plasticity and rate effects during crack growth

The concluding chapter is a collection of essays on various topicsconcerned with the dynamic fracture of inelastic materials that arerepresentative of the most recent developments in the area. Thetreatment of crack growth in linear viscoelastic materials is very brief.It is noted that for the case of the growth of a sharp crack through aviscoelastic material, the near tip field is determined by the instanta-neous elastic moduli and it is independent of the relaxation propertiesof the material. This paradox is resolved by introducing a failure zoneor cohesive zone of finite length. The crack growth process is then"fast" or "slow" depending on the time required to advance one zonelength compared to a typical relaxation time of the material.

The matter of crack growth in an elastic-plastic material is con-sidered next. The particular problem analyzed in some detail is thesteady growth of an antiplane shear crack through a rate independentelastic-ideally plastic material under small-scale yielding conditions.This is the only case of dynamic elastic-plastic crack growth for whichan exact analytical result is available, and this result is limited tothe distribution of strain on the crack line within the active plasticzone. Nonetheless, this result has been very important in resolvingan inconsistency between equilibrium and dynamic asymptotic fieldsfor the problem, and in guiding the development of computationalapproaches to the various problems in the same general class. Thedistribution of shear strain on the crack line within the active plasticzone is given parametrically in (8.3.20), and a plot of this straindistribution is shown in Figure 8.4. An important conclusion is thatinertial resistance has a strong effect in suppressing the development ofplastic strain ahead of the crack tip. The growth criterion adopted inorder to extract a theoretical dynamic fracture toughness versus crackspeed relationship for the material from the model is a requirement


that the crack must grow in such a way that the strain at a fixeddistance ahead of the crack tip has a certain fixed value. The influenceof inertial resistance on a small scale within the active plastic zonehas a strong effect on the applied driving force required to sustaincrack growth according to the assumed criterion.

The next topic addressed is high strain-rate crack growth in aplastic solid. In this case, it is assumed that the material is vis-coplastic, but that crack growth occurs rapidly enough so that thereis little time for inelastic strain relaxation near the crack tip. In otherwords, the crack is running fast enough through an elastic-viscoplasticmaterial to outrun any substantial accumulation of plastic strain.This is viewed as a model of cleavage crack propagation in a materialthat can undergo a transition in fracture mode from brittle to ductile,with the transition being influenced by the rate of deformation inthe crack tip region. The main point of the analysis is to establishminimum conditions for the sustained growth of a sharp crack in thismaterial in terms of the crack driving force, the crack speed, and thematerial properties. Temperature has an important effect through thetemperature dependence of material properties. The minimum crackdriving force required to sustain crack growth at a given temperature(or for certain values of material parameters) is interpreted as thecrack arrest toughness of the material at that temperature.

The study of the influence of material rate effects in the dynamicfracture of materials is carried over into Section 8.5. Here, steadygrowth of a mode I crack accompanied by a crack tip cohesive zoneis considered. The material response within the cohesive zone istime dependent, in the sense that the cohesive stress depends onthe local rate of opening of the cohesive zone. The point of viewis adopted that, for a given remote applied stress intensity factor,the crack will advance according to either a brittle fracture criterionor a ductile fracture criterion, depending on whichever is satisfied atthe lower level of applied load. Interpretation of the results obtainedby analysis of this model leads to the conclusion that, for a givenlevel of strain rate sensitivity of the material in the cohesive zone,the crack will accelerate from zero speed under rising applied load asa ductile fracture. However, once the speed reaches a certain levelthe stress in the cohesive zone is elevated through rate effects to asufficiently high level to activate a brittle fracture mechanism. Thecrack then accelerates very rapidly to a significantly higher speed as abrittle fracture. Subsequent behavior depends on the details of loading


and other features of the system. In any case, the model permits ademonstration of a rate induced mode transition in dynamic fractureresponse.

The last two sections of the chapter are concerned with mech-anisms of material separation that operate on a microscopic scalein certain materials. The first mechanism considered is the ductilegrowth of voids in a plastic material. It is demonstrated, on the basisof very simple analytical models, that the influence of the inertialresistance of the material is to elevate the apparent resistance todeformation at high rates of straining. For very small voids, however,is seems that material viscosity has a stronger influence on resistancethan does inertia for many materials. In view of the number of com-peting effects and the geometrical complexities of large deformationsand of interacting and coalescing voids, this is an area that requiressignificant further analytical modeling and experimental work.

The last topic considered is the apparent rate sensitivity ex-hibited by a brittle material experiencing distributed microcrackingwhile undergoing a high average rate of straining. The origin of therate dependence appears to be associated with the nucleation andcoalescence of the many microcracks required to form a completeseparation of the solid. Sample calculations based on crude models ofthe process are described to illustrate the qualitative features of thistime dependent strength characteristic. This, too, is a problem areathat requires further work.

2

BASIC ELASTODYNAMIC SOLUTIONS

FOR A STATIONARY CRACK

2.1 Introduction

Consider a body of nominally elastic material that contains a crack.For the time being, the idealized crack is assumed to have no thickness,that is, in the absence of applied loads the two faces of the crackcoincide with the same surface in space. The edge of the crack is asmooth simple space curve, either a closed curve for an internal crackor an open curve intersecting the boundary of the body at two pointsfor an edge or surface crack.

Under the action of applied loads on the boundary of the body oron the crack faces, the crack edge is a potential site for stress concen-tration. If the rate at which loads are applied is sufficiently small, insome sense, then the internal stress field is essentially an equilibriumfield. The body of knowledge that has been developed for describingthe relationships between crack tip fields and the loads applied to asolid of specified configuration is Linear Elastic Fracture Mechanics(LEFM). This a well-developed branch of engineering science whichforms the basis for results to be discussed in this chapter and the next.

If loads are rapidly applied to a cracked solid, on the other hand,the internal stress field is not, in general, an equilibrium field andinertial effects must be taken into account. There is no unambiguouscriterion for deciding whether or not loads are "rapidly" applied in aparticular situation. Instead, the decision must normally be based onqualitative reasoning and experience with known solutions to specific

55

56 2. Basic elastodynamic solutions

problems. For example, if the applied loading has some characteristictime associated with it, say the time for a load to increase from zeroto its final value or the period of a cyclic load, then this time may becompared to the transit time for an elastic wave over a representativedimension of the cracked solid. If the ratio of the former to the latteris much greater than unity, then inertial effects are expected to beminimal. If the ratio is much less than unity, on the other hand, thenwave effects are expected to be significant.

It is worthwhile to think about the phenomenon in general termsat this point in order to appreciate the way in which inertial or stresswave effects can influence the relationship between crack edge stressconcentrations and applied loads. In the case of slowly applied loads,the influence of each load point is felt "simultaneously" throughoutthe body. The body resists the applied loading by means of itsstiffness; that is, the body develops a compatible deformation fieldwhich, through the elastic stress-strain response, corresponds to anequilibrium stress field to balance the loads. In the case of rapidlyapplied loads, on the other hand, resistance of the body to the appliedloads derives not only from material stiffness, but also from inertialresistance of the material to acceleration. Consequently, the influenceof each load point is radiated into the body as a mechanical wave.Different load points first affect the crack tip field at different times,resulting in a transient crack tip field.

But this relationship has another, more subtle, feature whichmay be appreciated by considering some particular situations. Forexample, consider the case of a suddenly applied pressure on thesurface of a spherical cavity in an otherwise unbounded elastic solid(Achenbach 1973). The resulting wave field is spherically symmetric.The equilibrium stress field for this case varies with radial distancer from the center of the sphere as r~3. If attention is focused onthe wavefronts of the wave field resulting from the suddenly appliedpressure, then it is found that the strength of the stress field carriedon the wavefront varies as r~l. This is an illustration of a general,fundamental difference between equilibrium and wave fields. Thedifference arises from the fact that the governing equations are el-liptic in the former case, whereas they are hyperbolic in the lattercase. In a sense, hyperbolic systems are more effective than ellipticsystems in transmitting detailed information about the load pointsthroughout the body. After the initial transients have passed a fieldpoint, the difference diminishes there. In the case of dynamically

2.1 Introduction 57

loaded cracked solids, the effect is manifested in the phenomenon ofdynamic overshoot In some cases, the intensity of the crack tip fieldresulting from a suddenly applied load reaches levels greater than theequilibrium intensity level corresponding to the slow application ofthe same loads. That is, in the early stages of the process, the levelovershoots its long-time limiting value before eventually decaying tothe limiting value. An example of this phenomenon is discussed inSection 3.3 where the situation of suddenly applied pressure on thefaces of a crack of finite length is analyzed.

For points close to the crack edge compared to the principalradius of curvature of the edge and to the distance to the nearestwavefront or boundary, the local fields are essentially two-dimensional.This can be seen by considering the gradient of the stress field atpoints very close to the crack line. In qualitative terms, the gradient ina direction parallel to the crack edge tangent is not sensitive to a crackedge singularity, whereas the gradient in a normal direction is moresingular than the field itself. Consequently, if the governing partialdifferential equations are expressed with reference to coordinate di-rections locally normal and tangential to the crack edge tangent, thenthe local singular field is determined by the terms in the governingequations with the highest order derivatives in the direction normalto the crack edge. Furthermore, the structure of the asymptoticfield does not vary along the crack edge, although the intensity mayvary. This argument is based on the tacit assumption of a crack edgewithout corners or cusps.

It is convenient to follow Irwin (1960) in resolving these localfields into three distinct two-dimensional fields. The basis for classifi-cation is the way in which the components of relative displacement ofinitially contiguous particles on opposite faces of the crack contributeto crack face separation very close to the crack edge: Mode I is thein-plane opening mode due to normal separation of the crack faces,mode II is the in-plane shearing mode due to relative sliding of thecrack faces in a direction locally perpendicular to the crack edge, andmode III is the antiplane (out-of-plane) shearing mode due to relativesliding of the crack faces in a direction tangent to the crack edge(see Figure 1.4). The components of the near tip stress field withreference to a rectangular coordinate system are expressed for each ofthese mode contributions as

i\ . .

(2.1.1)


as r —•> 0, where r, 0 are polar coordinates in a plane perpendicular tothe crack edge. The point r = 0 coincides with the crack edge, andthe line 0 = 0 is tangent to the crack surface at the crack edge and inthe direction ahead of the crack. The dimensionless function £y(0)represents the angular variation of each crack tip stress component.It is a universal function, independent of the configuration of thebody, the details of the applied loads, and the elastic constants. Thisfunction, which is given explicitly below for each stress componentfor each mode, is normalized to satisfy £ij(0) = 1. The multiplierK(t), which has physical dimensions of force/length3/2, is the time-dependent elastic stress intensity factor. The spatial variation of thedominant term in the asymptotic crack tip field is universal, and itis only the factor K(t) which reflects the influence of the geometricalconfiguration of the body and the details of the loading in a specificproblem. The determination of this influence is a central problem infracture mechanics.

The term a\j in (2.1.1) represents a contribution to the cracktip field of order unity. More precisely, in each mode,

Information on this contribution of order unity will be included in thediscussion of problems to be considered in this chapter.

Consider a right-handed rectangular local coordinate system ori-ented so that the X2-axis is normal to the crack surface at a pointon the crack edge (that is, in the direction 9 = TT/2), the #i-axis istangent to the crack surface and in the direction of prospective crackadvance (9 = 0), and the X3-axis is tangent to the crack edge. Thefunctions £^(0) of the angular coordinate 9 for the three modes ofcrack opening with respect to this local coordinate system are givenbelow. For mode I, K(t) = Kj(t) and

Eu(0) = cos ±0{l - sin ±0sin §0} ,

Ei2(0) = cos ±0sin ±0cos §0, (2.1.3)

E22(0) = cos \9{l + sin \9sin §0} .

2.1 Introduction 59

For mode II, K(t) = Kn(t) and

£n(0) = -sin§0{2 + cos§0cos§0},

£12(0) = cos ±0{l - sin ±0sin §0} , (2.1.4)

S22(0) = s in |0cos |0cos |0 .

For mode III, K(t) = Kin(t) and

E31(0) = - sin \0 S32(0) = cos \6. (2.1.5)

Graphs of £^(0) for modes I and II of crack opening are includedwith graphs of the corresponding functions for dynamic crack growthin Figures 4.3-4.7. Concerning the terms of order unity in the cracktip stress field (2.1.2) for the case of traction-free crack faces, theboundary conditions require that cr\2 = 0 for i = 1,2,3. The termsof order unity that are not determined by the crack face boundaryconditions can only be found from a solution for the complete field.

Clearly, a stress distribution which is singular at the crack tip is amathematical idealization. No real material can actually support sucha stress distribution. The rationalization for admitting the singularstress distribution in the study of the fracture of real materials isbased on the universal spatial dependence of the crack tip stress fieldand on the concept of small-scale yielding. The small-scale yieldingsituation arises when the potentially large stresses in the vicinity ofthe crack edge are relieved through plastic flow, or some other inelas-tic process, throughout a region which has lateral dimensions thatare small compared to the crack length and other body dimensions.Under these conditions, the stress distribution in the elastic materialsurrounding the inelastic crack tip zone is adequately described bythe dominant singular term in the elasticity solution based on a sharpelastic crack. This surrounding field is completely determined by thestress intensity factor and, consequently, the stress intensity factorprovides a one-parameter characterization of the load level appliedto the material in the inelastic fracture zone. The stress intensityfactor itself does not provide information on the way in which thematerial in this zone responds to the applied loading. The mainimplication of this viewpoint is the following. Consider two bodies of

the same material, but having different shapes and/or having cracksof different size. Suppose that the two bodies are loaded to resultin the same mode of crack tip deformation. If the loading results inthe same stress intensity factor in the two cases, then the materialin the crack tip region is assumed to respond in the same way in thetwo cases. This idea is exploited in engineering practice by measuringthe stress intensity factor at which a crack will begin to advance ina well-characterized laboratory specimen, and then assuming that acracked structure will experience crack growth at the same level ofstress intensity. This condition for fracture is commonly known asthe Irwin criterion for fracture initiation, after G. R. Irwin, and thecritical value of the stress intensity factor is denoted by a subscript c,for example, Kjc in mode I under plane strain conditions.

Several dynamic fracture problems are examined in the followingsections within the general framework introduced. Attention is limitedto the fundamental dynamic fracture problem for each mode of cracktip deformation separately. In each case, the system studied is ahalf plane crack in an otherwise unbounded body of homogeneousand isotropic elastic material. Loading is spatially uniform and sud-denly applied over the crack faces, resulting in two-dimensional fields.Transient stress intensity factor histories are derived in each case.Extensions of the basic results to other situations of dynamicallyloaded stationary cracks are described in Chapter 3.

2.2 Suddenly applied antiplane shear loading

Consider a body of elastic material that contains a half plane crack,but that is otherwise unbounded. Introduce a right-handed rectangu-lar coordinate system in the body so that the z-axis (or X3-axis) liesalong the crack edge, and the $/-axis (or X2-axis) is normal to the planeof the crack. The crack occupies the half plane i/ = 0, — oo < x < 0;see Figure 2.1.

The opposite faces of the crack are subjected to opposite, sud-denly applied uniform traction in the ^-direction, say Tz(x,z,t) =±T*H{i) on y = ±0, where r* is a constant traction magnitude andH(-) is the unit step function. The remaining traction componentsare Tx(x, z,t) = Ty(x,z,t) = 0 on y = ±0. Thus, the crack faces aretraction-free for time t < 0, and are subjected to uniformly distributedshear traction for t > 0. The material is stress free and at resteverywhere for t < 0.

2.2 Suddenly applied antiplane shear loading 61

Figure 2.1. Configuration for analysis of the antiplane shear and plane straindeformations considered in Chapters 2 and 3. The coordinate axes are alternatelylabeled x,y,z or xi,x%,#3, and the crack is in the plane y = 0 or X2 = 0.

The components of displacement in the #, y, ^-coordinate direc-tions are ux,uyiuz, respectively. Because the complete system isinvariant under translation in the z direction, the displacement isindependent of z. Furthermore, the symmetries of the system can beused to show that ux and uy are zero for all time if they are zeroinitially. To illustrate this point, consider the outcome of reversingthe sense of the applied loading in two different ways. Suppose thatthe displacement in the y-direction for any point (x,y,z) is uy. Ifthe direction of the traction vector on the crack faces is reversed,then, because the response of the system in linear, the displacementat the same point will also be reversed to become — uy. The senseof the traction can also be reversed by reflecting the system in theplane z = 0. Because uy is independent of z, its sense is not changedin this case, so that the displacement at the point of interest willremain uy. Linearity and reflective symmetry can be satisfied atan arbitrary point only if uy = 0. A similar argument leads toux = 0. Consequently, the only nonzero component of displacementis uz(x,y,t) = w(x,y,t), the displacement in the ^-direction. Thisfield is two-dimensional, and the state of deformation at any point isantiplane shear deformation.

If uz = w is the displacement at any point (#,y), then reversalof the sense of the applied loads followed by reflection of the systemin the plane y = 0 leads to the symmetry property that w(x, —y, t) =—w(x,y,t); in particular, w(x, 0,t) = 0 for x > 0. The field over theentire x, y-plane is determined by solving the field equations for y > 0only, subject to this symmetry condition.

Before actually developing a solution to the field equations, it isworthwhile to consider the physical features of the phenomenon to beanalyzed. Prior to application of the loads, the material is stress free


Figure 2.2. Diagram of the wavefronts generated by sudden application of crackface traction acting in the ^-direction. The state of deformation is antiplane shear.

and at rest everywhere. The loading is suddenly applied to the crackfaces at time t = 0 and, according to Huygens's principle, the influenceof each loaded boundary point spreads out from that point behind anelementary wavefront traveling with the characteristic wave speed ofthe material c = cs. The envelopes of the elementary wavefronts arethe actual wavefronts of the transient field, and these wavefronts areshown in a plane z = constant in Figure 2.2. For points near a crackface compared to the distance to the crack edge, the transient fieldconsists only of a plane wave parallel to the crack face and travelingaway from it at speed c. As this wavefront passes a material point, thecomponent of stress azy = ry (or cr32) changes discontinuously fromzero to —r* and the particle velocity dw/dt changes discontinuouslyfrom zero to ±r* /pc for ±y > 0.

Near the crack edge, on the other hand, the field is more com-plex and some nonuniform field exists behind a cylindrical wavefront(circular in two dimensions) of radius ct that is centered on the crackedge (tip in two dimensions). This is the region in which the stressconcentration develops.

It is noteworthy that the process being described involves neithera characteristic length nor a characteristic time with respect to whichthe independent variables may be scaled. That is, the sketch inFigure 2.2 represents the wave propagation process not only at aparticular time, but at any time, provided only that the length scalein the coordinate directions is adjusted to make the radius of thecylindrical wavefront equal to ct. Suppose that rp(x,y,t) is a stresscomponent a$z at point (#, y) in the plane at time t. If the samestress component is considered at another point (ax, ay), where a isany positive real number, then the original value will be recovered if

2.2 Suddenly applied antiplane shear loading 63

it is considered at time at. That is,

Tp{ax, ay, at) = rp(x, y, t) (2.2.1)

for any positive real number a. The observation (2.2.1) shows thatstress is a homogeneous function of its arguments of degree zero. Bymaking a particular choice for a in (2.2.1), say a = t'1, it becomesclear that stress can be expressed as a function of two variables, ratherthan the apparent three. It follows that particle velocity is also ahomogeneous function of degree zero, displacement is a homogeneousfunction of degree one, and so on. It is clearly important in deducing(2.2.1) that the nonzero boundary data are all expressible in termsof a fixed level of stress. If the nonzero boundary data were givenin terms of a fixed level of stress rate, for example, then stress ratewould necessarily be homogeneous of degree zero, stress would behomogeneous of degree one, and so on. The property of homogeneitycan be exploited in developing solution techniques for this problemclass, and an illustration is given in Section 6.3.

For points close to the crack tip, the stress field will be dominatedby the square root singular stress intensity factor field given in (2.1.1).Indeed, the observations that the system is linear, that the fields arehomogeneous, and that the stress varies inversely with the square rootof distance from the crack tip lead directly to the statement that

(2.2.2)

as x/ct —> 0, where Cm is an undetermined dimensionless real con-stant. From the definition of stress intensity factor in (2.1.1),

Km(t) = lim y/2nxTy(x,Q,t) = Cinr*V27rct. (2.2.3)x—•()+

Thus, the stress intensity factor is known up to a dimensionless mul-tiplier Cm, and a main purpose in solving the field equations from thefracture mechanics point of view is to determine the value of Cm- Thefact that the stress intensity factor increases indefinitely with time iscoupled to the fact that the static elasticity problem equivalent to thetransient problem being discussed here does not have a solution.

Finally, it is noted that (2.2.2) is only an asymptotic result.It is reasonable to expect, however, that (2.2.2) provides a good

approximation to the stress component for values of x small comparedto the distance from the crack tip to the cylindrical wavefront c£, theonly length available for comparison with distance from the crack tip.This idea cannot be made more precise until a complete solution tothe problem is available.

The mathematical problem that leads to the solution is statedin the following way. A function w(x,y,t) is sought that satisfies thewave equation in two space dimensions and time,

d2w d2w 1 d2w

in the half plane — o o < x < o o , 0 < y < o o f o r time in the range0 < t < oo. The solution of the wave equation (2.2.4) is subject tothe boundary conditions that

Ty(x,O+>t) =/z—(:r,0+,£) = -r*H(t), -oo < x < 0,dy (2.2.5)

w(z,0V) =0, 0 < x < oo,

for all time, where fx is the shear modulus, and the initial conditionsthat

w(x, y, 0) = 0, - ^ ( z , y, 0) = 0 (2.2.6)

for all points in the half plane. Only solutions that result in stresscomponents that vary inversely with the square root of distance fromthe crack tip are admitted; stresses that have singularities strongerthan an inverse square root are rejected on physical grounds becausethey correspond to deformation states with unbounded elastic energy.The mathematical statement of this requirement, which is alreadyimplicit in (2.2.2), is

lim <oo. (2.2.7)

A solution of this boundary value problem may be obtained throughapplication of any of several techniques of analysis. Solution byGreen's method and by the Wiener-Hopf method are illustrated in

2.3 Green's method of solution 65

Sections 2.3 and 2.5, respectively. It is observed in Section 2.3 thatthe solution is equivalent to the solution for diffraction of a planestress pulse by a crack with traction-free faces. It may be observedthat, once a solution is available for boundary loading with stepfunction time dependence, extension to boundary loading with othertime dependence is quite simple.

2.3 Green's method of solution

The initial-boundary value problem outlined in Section 2.2 and statedin (2.2.4)-(2.2.7) is now solved by means of Green's method. Thisapproach was originally developed for solving problems in potentialtheory, but it has been applied with great success to virtually allcategories of linear boundary value problems in mathematical physics.Suppose that in addition to displacement w(x1y^t)J a second wavefield g(x, 2/, t) is defined over the half plane — oo < x < oo,Q <y < oofor —oo < t < oo. Then Green's second formula for the wave operatorin two space dimensions is

1 f f dg dwY2 ,„ 1 ft2 f°° f dw dg] J J

= - 5 - / \w-7r ~ 9-z-\ dA+- / \g- w-^-\ dxdtc*JA[ dt ydt\ti fijtl J-nVdv dy\y=Q+(2.3.1)

for arbitrary times t t\. The area A is the entire area of thehalf plane, and V2 denotes the Laplacian operator in two space di-mensions. The functions w and g are assumed to have propertiessuch that the integrals in (2.3.1) exist. Integrals taken along pathsin remote parts of the plane have been assumed to vanish in writing(2.3.1) due to the wave propagation character of the fields.

The great utility of (2.3.1) is demonstrated when g is selectedto be a fundamental singular solution, or Green's function, for theproblem. The fundamental singular solution may be defined in anumber of ways, depending on the details of the problem. In thepresent case, g is required to satisfy

- yo)6(t - to), (2.3.2)

where 6(-) is the Dirac delta function, (xo,yo) is an arbitrary pointin the unbounded x, y-plane, to is an arbitrary time, and P is anamplitude that has physical dimensions of force x time/length if gis to have a physical dimension of length. The solution of (2.3.2),which is written as go{x,y,t) to make explicit the special significanceof the point (xo,yo) and the time io> is subject to no boundaryconditions, that is, it is the free space Green's function. To render(2.3.1) particularly useful, however, it is required that

% = 0 for t>to (2.3.3)ot

for all points (x, y).A function that satisfies (2.3.2) and (2.3.3) is

if t < to - R/c,(2.3.4)

Xt>to-R/c,

where R = y/(xo — x)2 + (yo — y)2 > 0. In the three-dimensionalx, y, -space, the function go represents a wave field that is nonzeroinside the cone to — t > R/c with apex at (xo,yo,zo), and thatis identically zero elsewhere. This is the displacement that wouldresult if a line of body forces acting in the ^-direction and uniformlydistributed along x = xo, y = yOi —00 < z < 00 were appliedimpulsively at time t = to, provided that the sense of time wasreversed. For example, this is the interpretation if (to — t) is themeasure of elapsed time.

If g in (2.3.1) is now understood to be the free space Green'sfunction go, and the choices ti = 0+, £2 = *J are made, then

Pw(xo,yoito) = J Lo(x,0,t)-^-(x,0+,t)

^ \ (2.3.5)

where Bo is the portion of the boundary surface y = 0 that fallsinside the cone (£o — t) = \/{xo — x)2 + y2 for time in the range

0 < t < to. The first term on the right side of (2.3.1) vanishesbecause both w and go have initial values of zero. The result (2.3.5) isGreen's representation formula for the solution of the wave equation(2.2.4). It states that the solution at any interior point in the halfspace is determined by boundary values of the wave function and itsderivative (and by the initial conditions if nonzero initial conditionsare specified). Furthermore, it is only those boundary data at pointsthat fall inside the cone (to — t) = R/c with apex at the observationpoint (xoiyo,zo) that influence the field at the observation point.This backward running cone defines the domain of dependence forthe observation point. Both the boundary displacement on the crackfaces w(x,0+,t) for x < 0 and the boundary traction ry(x,0,t) =/j,dw(x,Q+,t)/dy for x > 0 are unknown at this point.

If the observation point in (2.3.5) approaches a boundary pointahead of the crack tip, that is, yo —• 0+ for xo > 0, to > 0, then (2.3.5)becomes

to rxo+c{to-t) <r (r ft/ Ty(X0I /

Jo Jxo-c(to-Q^ ^ > 0 ( 2 3 6 )

(to-t) y/c2(to - t)2 - (xo - x)2

The condition that the displacement vanishes on the crack plane aheadof the tip [equation (2.2.5)2] has been used in writing (2.3.6), andthe observation that dgo/dy = 0 for all x when y = yo = 0 hasbeen incorporated as well. The unknown stress distribution on theboundary ahead of the crack tip may be determined from (2.3.6).With this result in hand, the observation point in the representationformula (2.3.5) can be taken to be on the crack face, which yields anexpression for the displacement distribution on the crack face. Then,having determined the boundary values appearing in the representa-tion formula (2.3.5), this formula provides a complete solution. Forpresent purposes, it is information about the stress ahead of the cracktip that is of primary interest, and such information is now extractedfrom (2.3.6).

The region of the x,t-plane that is of interest in considering(2.3.6) is shown in Figure 2.3, where the time coordinate has beenscaled by the wave speed c for convenience. With reference to Fig-ure 2.3, the area of integration in (2.3.6) is the triangular region OBD.First, it is observed that if the observation point was anywhere in thesector DCE, then the integration area in (2.3.6) would not includeany part of the loaded crack face x < 0. Consequently, the equation


DFigure 2.3. The x,t-plane showing the range of integration for the integral equa-tion (2.3.6). The time axis has been rescaled by the wave speed c for convenience.The observation point O has coordinates xo,cto>

for ry is homogeneous under these conditions and a partial solutionof (2.3.6) is that

ry(a;,0,t) = 0, x>ct. (2.3.7)

This result is obvious from the discussion in the preceding sectionconcerning the wave propagation character of the transient field. Forpoints that are more distant than ct from the nearest loaded boundarypoint, the initial rest state (2.2.6) persists and the stress is zero. Withreference to Figure 2.2, the partial solution (2.3.7) applies for pointson y = 0 ahead of the advancing cylindrical wavefront.

For an observation point in sector EC A in Figure 2.3, on theother hand, the region of integration includes the part of the loadedcrack face in the triangular region ABC in Figure 2.3 and, conse-quently, (2.3.6) is not a homogeneous equation. For this situation,solution of (2.3.6) is facilitated by a change of integration variables to

77 = ct — x . (2.3.8)

The inverse transformation is

1 , . (2.3.9)

The transformation (2.3.8) carries the coordinates of the observationpoint (xo, to) into (£o? Vo), where fo = cto+xo, rjo = cto-xo. In view ofthe partial solution (2.3.7), introduction of the transformation (2.3.8)

into (2.3.6) yields

4 O^M 0, £o>»o>0. (2.3.10)

The name of the variable representing the -component of stress hasbeen preserved as ry in writing (2.3.10) even though it is obviously adifferent function of its arguments than it was in the original equation(2.3.6).

The transformed integral equation (2.3.10) will be satisfied if theinner integral vanishes for all TJ in the range 0 < rj < TJO. This conditioncan be written as a Volterra integral equation of the first kind

« . > » > „ , (2.3.11)

where the traction boundary condition on the crack face [equation(2.2.5)i] for x < 0 has been made explicit. If the range of integra-tion on the left side of (2.3.11) is shifted to make the lower limit ofintegration equal to zero, then (2.3.11) takes the standard form of anAbel integral equation with a square root singular kernel

i: T ( C ) d( = a(Co), Co>0, (2.3.12)

where the function a is given. A solution r(() of (2.3.12) is readilyobtained by application of Laplace transform methods. Denoting theLaplace transform of a function by a superposed hat [equation (1.3.7)],the application of a transform to the terms in (2.3.12) yields

(2.3.13)

where s is the transform parameter. This equation is then solved forT(S) in the form

( 2-3 > 1 4 )

Application of the inverse transform provides the solution of (2.3.12)as

f (2.3.15)

where the prime denotes differentiation.For the problem at hand, the general solution (2.3.15) implies

that the solution to (2.3.11) is

(2.3.16)

for 0 < x < ct, where the rotated coordinates £, ry have been replacedby the physical coordinates x, t.

Some general features of the shear stress distribution on the planey = 0 are evident from (2.3.16). First, ry(x,0,t) is continuous acrossthe wavefront, as could have been anticipated on the basis of thecontinuity of displacement and the fact that the discontinuity of atangential derivative is the tangential derivative of a discontinuity,as deduced from Hadamard's lemma (Truesdell and Toupin 1960).The magnitude of ry increases monotonically with distance from thewavefront toward the crack tip at x = 0. For points very close to thetip,

J (2.3.17)7T V X

The form of this result was anticipated in (2.2.2). With the solutionnow available, it is clear that the undetermined constant in (2.2.2) hasthe value Cm = 2/TT. Thus, from (2.2.3), the transient stress intensityfactor for suddenly applied antiplane shear loading on the faces of ahalf plane crack is

Km(t) = 2r*y/2ct/7r. (2.3.18)

This is the main result of this section.Although the stress intensity factor (2.3.18) was obtained for the

case of crack face loading, it can be given another interpretation. Onceagain, consider the unbounded body containing a half plane crackdepicted in Figure 2.1. Suppose that the crack faces are traction-free,but that a plane horizontal shear pulse described by

w(x,y,t) = — (y - ct)H(ct - y) (2.3.19)A4

is incident on the crack plane. The front of the pulse is parallel tothe crack plane, and it propagates toward the crack plane at speedc, carrying a jump in stress ry from zero to r*. At time t = 0, theincident pulse strikes the crack. For points far to the left of the cracktip (x <C — ct), the pulse reflects back onto itself as a plane wave witha change in sign of the stress. For points far to the right (x > c£), thepulse continues to propagate as a plane wave without being influencedby the crack. The incident pulse is diffracted at the edge of the crack,and the diffracted wave radiates out from the tip behind a cylindricalwavefront of radius ct.

The complete deformation field of this stress pulse diffractionproblem differs from the deformation field of the problem describedin Section 2.2 only by the plane pulse propagating in an uncrackedsolid. Because the governing equations are linear, and because theplane pulse, by itself, results in no stress singularity at the crack tip,it follows immediately that (2.3.18) is also the stress intensity factorhistory for the phenomenon of stress pulse diffraction. Furthermore,the stress distribution on the crack plane ahead of the tip for thediffraction problem is given by (2.3.16) plus the constant r*, that is,

1 - t n - i J V ^ l . (2.3.20)

This distribution, because it is exact, provides the opportunity togain some understanding of the size of the region near the crack overwhich the stress intensity factor field (2.3.17) dominates the completestress distribution in a situation that involves no characteristic length.If the stress distribution (2.3.20) is expanded in powers of the smallquantity x/ct, then

Consequently, the stress intensity factor field describes the full stressfield to within 5 percent of the exact result for points closer to thetip than 10 percent of the distance from the tip to the cylindricalwavefront along y = 0.

This completes the discussion of the basic problem of determin-ing transient stress intensity factors associated with antiplane shear

deformation fields in cracked solids. The discussion is resumed inSection 7.2 where nonuniform crack growth is analyzed by means ofthe same general approach. This basic problem is a useful vehiclefor introducing many ideas of dynamic fracture analysis, both phys-ical and mathematical, that have much broader applicability. Theproblem is relatively simple, and it can be analyzed completely bydirect application of classical methods of analysis. The plane straincounterpart of this basic problem, which is more involved but moresignificant from the practical point of view, is considered next.

2.4 Suddenly applied crack face pressure

Consider once again the body of elastic material that contains a halfplane crack but that is otherwise unbounded (see Figure 2.1). In thepresent instance, the crack faces are subjected to a suddenly applied,spatially uniform pressure of magnitude cr*. In terms of boundarytraction, Tx(x,z,t) = Tz(x, z,t) = 0 and Ty(x, z,t) = ±cr*H(t) ony = ±0, where a* > 0 and i?(-) is the unit step function. Thus, thecrack faces are traction-free for time t < 0, and they are subjected touniform normal pressure for t > 0. The material is stress free and atrest everywhere for t < 0.

The components of displacement in the #, y, z-coordinate direc-tions are uxiuy,uz, respectively. Arguments similar to those usedin Section 2.2 can be applied here to conclude that uz = 0 for alltime if it is zero initially, and that ux and uy are independent of the^-coordinate, so the state of deformation is plane strain. Further-more, the nonzero components of displacement satisfy the symmetryconditions ux(x,-y,t) = ux{x,y,i) and uy(x,-y,t) = -uy(x,y,t).In particular, these conditions imply that both uy(x,0,t) = 0 and&xy(%, 0, t) = 0 for x > 0, due to symmetry.

As in the case of the antiplane shear loading considered in Section2.2, it is helpful at this point to survey the features of the phenomenonto be analyzed. Before application of the loads, the material is stressfree and at rest everywhere. The pressure is suddenly applied to thecrack faces at time t = 0, and the influence of each loaded bound-ary point spreads out from that point behind elementary wavefrontstraveling with the characteristic speeds of the material, the speedof dilatational waves c<j and the speed of shear waves c8, in thiscase. The envelopes of the elementary wavefronts are the anticipatedwavefronts of the transient field, and these wavefronts are shown in a

2.4 Suddenly applied crack face pressure 73

Figure 2.4. The wavefronts generated by sudden application of a spatially uniformpressure on the faces of the crack. The state of deformation is plane strain.

plane z = constant in Figure 2.4. For points near to the crack facecompared to the distance to the crack edge, the transient field consistsonly of a plane wave parallel to the crack face and traveling awayfrom it at speed c^. The uniform pressure loading does not induce acorresponding plane shear wave. As the plane dilatational wavefrontpasses a material point in its path, the component of stress ayy (or 0*22)changes discontinuously from zero to — a* and the particle velocityduy/dt (or 8112/dt) changes discontinuously from zero to ±cr*/pCd for±»>0.

Near the crack edge, on the other hand, the deformation fieldis more complex. A nonuniform scattered field radiates out from thecrack edge behind a cylindrical wavefront of radius c^t that is centeredon the crack edge. Due to the coupling of dilatational and shear wavesat a boundary, this scattered field also includes a cylindrical shearwavefront of radius cst that is centered on the crack edge plus theassociated plane-fronted headwaves traveling at speed c8.

As was the case for antiplane shear loading discussed in Section2.2, the process being described here involves neither a characteristiclength nor a characteristic time. The arguments presented in con-nection with (2.2.1) can be applied here as well to conclude that thecomponents of stress and particle velocity are homogeneous functionsx, y, t of degree zero. An immediate consequence is that, for pointsvery close to the crack edge,

(2.4.1)

where (7/ is an undetermined dimensionless constant. In view of

(2.4.1), the mode I stress intensity factor is also known up to theconstant C/, that is,

#/(<) = lim V27rx(Tyy(x,0,t) = Qa*y/27rcdt. (2.4.2)

The mathematical problem that leads to a solution with theproperties outlined is now stated. The Helmholtz representation ofthe displacement vector in terms of the scalar dilatational potential <\>and the vector shear potential tf> is adopted, as introduced in (1.2.60).For plane strain deformation that is independent of z, the vectorpotential \j) has only one nonzero component; this component is inthe z direction and it will be denoted by xjj. Thus, two functions(j)(x,y,t) and i/j(x,y,t) are sought that satisfy the wave equations intwo space dimensions and time

c2 dt2 " 'dx2 dy2 c2 at2 " ' dx2" dy2 c2dt2 v J

in the half plane — o o < a ; < o o , 0 < y < o o f o r time in the range0 < t < oo. The wave functions satisfying (2.4.3) are subject to theboundary conditions that

(Tyy(x, 0, *) = -<T*H(t) , - 0 0 < X < 0 ,

(Txy(x,O,t) = 0, - o o < : r < o o , (2.4.4)

for all time, where stress and displacement components are interpretedas their representations in terms of displacement potentials <\> and xj).The initial conditions that

= ^ ( * , y , 0 ) = # r , y , O ) = ^ ( * , V , 0 ) = 0 (2.4.5)

for all points in the half plane ensure that the body is stress freeand at rest until the crack face pressure is applied. Finally, onlysolutions that result in stress components that vary inversely with thesquare root of distance from the crack tip are admitted. As before,

2.4 Suddenly applied crack face pressure 75

stresses that have singularities stronger than inverse square root arerejected on the physical grounds that they correspond to states withunbounded total energy. The total energy density is proportional to<Jijeij + piiiUi. The integral of this density over any bounded portionof the body, even a portion including the crack tip, must have a finitevalue.

The solution method to be developed in the next few sections isbased on the application of integral transforms. Solution of partialdifferential equations by means of integral transforms is only a formalprocedure under the best of circumstances. That is, convergenceof the improper integrals of unknown functions must be assumeda priori, and it is only after a solution has been obtained that thevalidity of the assumptions can be checked. Nonetheless, knowledgeof certain qualitative features of the transforms can be used to greatadvantage in guiding progress toward a solution. This is the case inthe Wiener-Hopf technique, which is to be outlined in the followingsection. Indeed, some knowledge of the domains of convergence ofcertain transforms is essential in applying the Wiener-Hopf technique.In the present context, such qualitative features are suggested by thewave propagation character of the fields being analyzed.

The one-sided Laplace transform will be used to suppress depen-dence on time. According to the convention established in (1.3.7), theLaplace transform of 0(x, y, i) is

/»OO

0(x, y, s) = / 0(x, y, t) e~stdt, (2.4.6)Jo

where, for the time being, the transform parameter s is a real, positivevariable. Some understanding of the behavior of (j)(x, y, s), as well asother transformed fields, as the magnitude of x or y becomes verylarge is important in the Wiener-Hopf technique.

Consider any fixed point (x,y) in the half plane. Let t*(x,y)denote the time at which the first wavefront arrives at (x, y) to disturbthe initial rest state. Under these circumstances, the lower limit ofintegration in (2.4.6) can be replaced by t*. If the change of variablet = r + t* is then made, (2.4.6) takes the form

/»O

/

Jo(2.4.7)

Consequently, if the integral has the form of an algebraic function of xand y, then the behavior of the transform for large x or y is determined

by the arrival time t* of the first motion. For example, for any fixedvalue of x in the problem under consideration, the dilatational wavefunction 0 appears to be dominated in y > 0 by the plane wavetraveling at speed Cd away from the crack face. In this case, t*(x, y) =

so that

£(ar, y,s) = o (e'8^1"^^ (2.4.8)

for any fixed x as y —• oo, where e is a small positive real number. Theplane wave itself is described by (j> = — (cr*/2pc%) (cdt — y)2H(cdt —y) for x < 0. A straightforward calculation based on (2.4.7) thenyields </> = — (a* /pc^s3) e~syl°d. As a second illustration involvinga cylindrical wave, consider the wave function 0(#,y,t) describingthe wave response to an impulsive source of strength 2TTC^ per unitmass concentrated at the origin of coordinates in an unbounded plane.Though the relevance of this problem to mode III crack growth couldbe established, only the mathematical features are of interest here.The wave function satisfies

V V - c~d 2<j>,tt = -6(x)6(y)6(t) (2.4.9)

where 6(-) is the Dirac delta function. If <j> and (f>,t are assumed to bezero at time t = 0", then the solution of (2.4.9) is

where if (•) is the unit step function and r = y/x2 + y2. Note thatt*(x, y) = r/cd in this case. The Laplace transform on time of (2.4.10)is TO

/ i lcd), (2.4.11)

where Ko(-) is the modified Bessel function of the second kind. Fromthe asymptotic property that

as r -> oo (2.4.12)

it is evident that the asymptotic behavior of 0(rc, y, s) at remote pointsin the plane is determined by t*(x,j/).

2.5 The Wiener-Hopf technique 77

It is noted again that solution of differential equations by meansof integral transforms is a formal procedure and (2.4.7) has no fun-damental significance. However, it does provide guidance at a keypoint in applying the Wiener-Hopf technique, as will be demonstratedin the next section. Unlike the analysis of harmonic or steady wavephenomena, these transient wave propagation problems do not involveso-called radiation conditions. In a sense, the qualitative conditionssuch as (2.4.11)-(2.4.12), which follow from the wave propagationnature of the mechanical fields, play a role much like the role played bythe radiation conditions in steady wave motion (Courant and Hilbert1962; Kupradze 1963).

2.5 The Wiener-Hopf technique

The mathematical method known as the Wiener-Hopf technique wasdeveloped originally to solve a particular linear integral equation of thefirst kind which arose in the analysis of electromagnetic wave phenom-ena. Two distinguishing features of the integral equation are (i) it isenforced on a semi-infinite interval and (ii) the kernel depends only onthe difference between the independent variable and the integrationvariable. The evolution of the method and a number of extensionsare described in the book by Noble (1958). The application of thetechnique to transient problems in elastodynamics was pioneered byde Hoop (1958) in a study of several half plane diffraction problems,including the one being considered here. He observed that, if time de-pendence was suppressed through application of a Laplace transform,then an integral equation amenable to solution by the Wiener-Hopftechnique could be obtained. Once a solution in the transformeddomain is in hand, the task of inverting the transforms to obtain asolution in the physical domain remains. To execute this step, deHoop (1961) extended an idea introduced earlier by Cagniard (1962).The result, which is now known as the Cagniard-de Hoop technique,is a powerful and elegantly simple method for inverting transforms ina wide range of elastodynamic wave propagation problems.

Although each of these techniques has taken on an existence sep-arate from its foundations, each is perhaps best viewed in the contextof its origins. Each technique is based on the judicious applicationof certain powerful theorems from the theory of analytic functions ofa complex variable, Cauchy's integral formulas and the property ofuniqueness of an analytic function in its domain of analyticity. The

plane strain problem of a crack with suddenly applied pressure loadingon the faces is now used as a vehicle for introducing these techniques.Additional illustrations will be included in subsequent chapters. Theapproach to be used, in fact, will not rely on reformulation of the prob-lem in terms of a standard Wiener-Hopf integral equation. Instead,the governing differential equations and boundary conditions will besolved directly. This direct approach was apparently introduced byJones (1952) for application to scalar wave problems.

2.5.1 Application of integral transforms

Solution of the problem proceeds by application of a one-sided Laplacetransform on time, and then a two-sided Laplace transform on x, tothe governing differential equations and boundary conditions. Theuse of the exponential Fourier transform with a complex transformparameter is completely equivalent to the use of the two-sided Laplacetransform; the choice of one over the other is a matter of taste. It isnoted that the boundary conditions (2.4.4) are defined only on halfof the range of x. Consequently, the two-sided Laplace transformcannot be applied to these boundary conditions as they stand. Toremedy the situation, the boundary conditions must be extended toapply on the full range of x. To this end, two unknown functionsu_(x, t) and a+(x1t) are introduced. The function u_ is defined to bethe (at present unknown) displacement of the crack face y = 0+ inthe y direction for —oo < x < 0 , 0 < t < oo, and to be identicallyzero f o r O < # < o o , 0 < £ < o o . Likewise, a+ is defined to be thetensile stress in the y-direction on the plane y = 0 f o r 0 < : r < o o ,0 < t < oo. With these definitions, the boundary conditions can berewritten as

cryy(x10+,t) = (T+(x,«) - a*H{t)H{-x),

*xy(x,0+,t) = 0, (2.5.1)

for the full range — o o < x < o o , 0 < £ < o o . Obviously, thesubscripts + and — are used at this point to indicate on which half ofthe x-axis a subscripted function is nonzero. The notation is carriedover into the transformed domain, where it is found that the same

subscript symbols are useful for designating a particular half plane ofanalyticity.

Next, the one-sided Laplace transform (2.4.6) is applied to thewave equations (2.4.3), in light of the initial conditions (2.4.5), andto the boundary conditions (2.5.1). Then, the two-sided Laplacetransform defined by

f°° - _ t

>V,s) = / <t>{x,y,s)e 8Qxdx

(2.5.2)

= / (f)(x,y,s)e-s(:xdx+ / <f)(x,y,s)e~s(:xdxJ-oo JO

is applied. The notation in (2.5.2), whereby an upper case symbol isused to denote the double transform of the function represented by thecorresponding lower case symbol, is adopted as a convention for theremainder of this chapter. The transform variable £ is understoodto be a complex variable, and the transform plane is scaled by thereal parameter s simply for convenience. The product s( could beviewed as some other transform parameter, say £', at this point,but eventually the change of variable £' —• s( would be found tobe indispensable. The transform $ in (2.5.2) is written as the sumof two semi-infinite integrals in order to consider the convergenceproperty discussed in Section 1.3.2. In general, the second semi-infinite integral will converge for values of £ in some right half plane.That is, there exists a real number £+ such that the integral convergesfor Re(C) > £+, and, consequently, this integral defines an analyticfunction in that half plane. Likewise, the first semi-infinite integralwill converge for values of C in some left half plane, say Re(C) < f_,and it defines an analytic function there. It is evident that the two-sided transform exists only if £+ < £_. The case £+ = £_ requiresspecial consideration, and it can be avoided in most cases through theintroduction of vanishingly small dissipation in mechanical problems.If £+ < £_, then the doubly infinite integral in (2.5.2) converges anddefines an analytic function of £ in the strip £+ < Re(£) < £_ of thecomplex £-plane.

Returning to the idea concerning asymptotic behavior of trans-forms embodied in (2.4.7), a strip of analyticity of $(C, y, s) for fixedy > 0 and s may now be anticipated. For any x > 0, (/> will be zeroat least until time ** = r/ca > x/c<i. For any x < 0, on the other

hand, <f> takes on nonzero values upon passage of the plane wave attime t* = y/cd- Then, from (2.4.7), it is anticipated that

(,,3)

for arbitrarily small real positive values of e. If the two semi-infiniteintegrals in (2.5.2) are now considered in light of (2.5.3), it is expectedthat the first semi-infinite integral will converge for Re(£) < 0 for anyvalues of y and s, and that the second will converge for Re(£) > —l/c^.These domains of analyticity will be illustrated in Figure 2.5 once allrelevant transforms have been defined. In any case, it is expected that<fr(C,y,s) is an analytic function of C in the strip —1/cd < Re(C) < 0.Similar reasoning applied in the case of other functions leads to theconclusion that this is a domain of analyticity common to all doublytransformed functions in the problem.

To develop further insight into the properties of the transforms,consider once again the wave motion due to an impulsive sourceas introduced in Section 2.4. If the two-sided Laplace transform isapplied to (j)(x,y,s) in (2.4.11), then (Watson 1966)

,»,«) = I" K0(sr/cd)e-^xdxJ-~ (2.5.4)

* -«v(cJ3-C2)1 / ae-«v(cJ

for y > 0. It was shown in Section 2.4 that $ is expected to decayexponentially as y —• oo if (f> is a plane wave propagating in they direction. The result (2.5.4) suggests that the same qualitativebehavior should be expected for a cylindrical wave if C is in the stripof convergence of the transform integral defining $.

If the one-sided Laplace transform is applied to the wave equa-tions (2.4.3), which are subject to the initial conditions (2.4.5), then

V20 - a2s20 = 0, V 2 ^ - b2s2$ = 0, (2.5.5)

where s is the real positive transform parameter. In (2.5.5), and forthe remainder of the chapter, the inverse wave speeds are denoted by

a = l/c<i and b = l/c8. The transformed boundary conditions are

vyy{x, 0+, s) = a+(x, s) - a*H(-x)/s,

^ ( : r , 0 V ) = 0, (2.5.6)

which are valid for — oo < x < oo. Based on the foregoing observationson the wave propagation nature of the fields, it is anticipated thata+(x, s) is bounded by an algebraic function of x times e~a8X as x —•+oo for any positive real value of s. Likewise, u_(x, s) is anticipatedto have algebraic behavior as x —* — oo for any real positive valueof s. In fact, the displacement of the crack faces is dominated bythe plane waves propagating away from the faces for large negativevalues, so that the dominant part of u_(x,s) will be independent ofx as x —• —oo.

Next, the two-sided Laplace transform (2.5.2) is applied to (2.5.5)and (2.5.6). The boundary conditions are considered first, and therepresentation of the left side of each boundary condition in (2.5.6)in terms of the transformed displacement potentials is introduced atthis point. Thus,

(2.5.7)

The special form of the terms on the right side of (2.5.7) has beenselected for convenience, and the selection is based on hindsight, to alarge extent. At this point, it is still possible that

£+(C)=s2 [°°B+(x18)e-<'dx,Jo

(2.5.8)

U-(Q=8S u_(x,s)e-s<xdxJ-OO

82 2, Basic elastodynamic solutions

will depend on s as well as on C- As suggested by the special form of(2.5.7), however, these functions will be found to depend only on £.Prom the foregoing observations on the rate of decay of transformedfields, £+ is expected to be analytic in Re(C) > —a and U_ is expectedto be analytic in the overlapping half plane Re(C) < 0. In particular,both functions are expected to be analytic in the strip —a < Re(£) <0.

Application of the two-sided Laplace transform (2.5.2) to thedifferential equations in (2.5.5) yields linear second order differentialequations in y for $(C>2/> s) and *(C,y,s), involving £ and s as pa-rameters. For values of C in the common strip of analyticity, eachequation has two independent solutions, one growing exponentiallyand one decaying exponentially as y —• oo. The wave propagationnature of the fields precludes the possibility of exponential growthas y becomes large with —a < Re(£) < 0. Consequently, only thesolutions decaying as y —> oo are admitted, so that

, y, s) = lp(C) e-8a\ * (C, V, s) = iQ(C) e~^\ (2.5.9)

where

a = a(C) = (a 2 -C 2 ) 1 / 2 , /? = /5(C) = (b2 - ff2 • (2-5.10)

Again, the special form selected for the undetermined coefficients in(2.5.9) is based on the expectation that P and Q will not depend ons. The functions a and (3 in (2.5.10) are multiple valued functions of£ in the complex £-plane with branch points at C = ±& a n d C = ^&5respectively. The branch of the square root in a is chosen as theone yielding values of a with a positive real part at each interiorpoint of the common strip. For future reference, the definition of ais extended to the entire £-pl&ne. Branch cuts are introduced alonga < |Re(£)| < oo, Im(C) = 0 and the branch of a which is chosen isthe one which has a positive real part everywhere in the cut planeexcept on the branch cuts. If the point £ approaches the cut in theright (left) half plane with Im(C) —> 0+ then the limiting value ofa is a negative (positive) imaginary value. The limiting value onthe opposite side of the branch cut has the same magnitude but theopposite sign. Likewise, /?(£) is defined for the entire C-plane cut alongb < |Re(C)| < oo, Im(C) = 0 and the branch with nonnegative realpart is understood in (2.5.10).

Next, the transformed potentials (2.5.9) are substituted into thetransformed boundary conditions (2.5.7). The result is a system ofthree linear algebraic equations for the four unknown functions P((),Q(C), £+(C)> and LL (£). The system of equations is valid for all valuesof £ in the common strip. As was anticipated, the system of equationsdoes not involve the parameter s. If two of the equations are used toeliminate P and Q in favor of £+ and C/_, then the remaining equation

which is valid in the common strip of analyticity —a < Re(C) < 0,where

R(O = 4<2a(C)/?(C) + (b2 - 2C2)2 • (2.5.12)

The function R is analytic everywhere in the complex plane exceptat the branch points C = ±a and £ = ±b. For the branches of aand {3 that were selected above, R is single valued in the £-plane cutalong a < |Re(C)| < b, Im(C) = 0. The function R is usually calledthe Rayleigh wave function because the two real roots of R = 0, say£ = ±c, are the inverse wave speeds of free surface Rayleigh wavestraveling in opposite directions, each with absolute speed CR = 1/c.In terms of Poisson's ratio v, the value of c is given approximately by

b = 0.862 + 1.141/ • ( 2 > 5-1 3 )

There may be other roots of R = 0 when other branches of themultiple valued function R are considered (that is, roots may exist onother sheets of the Riemann surface of the multiple valued function),but this possibility is of no consequence here.

The principal features of the complex £-plane pertinent to thesolution of (2.5.11) are identified in Figure 2.5. The branch pointsat C = ± a , ± 6 and the corresponding branch cuts are shown, thezeros of the Rayleigh function at C = ±c are shown, and the pole atC = 0 associated with the applied loading in the problem is indicated.The common strip of analyticity is between the dashed line and theimaginary axis, and the overlapping half planes in which the functionslabeled with subscripts (+) or (—) are analytic are also indicated witharrows.

-c -b - a a 6 c

Figure 2.5. The complex £-plane showing the location of the singularities of theanalytic functions involved in the Wiener-Hopf equation (2.5.11), which applies inthe strip -a < Re(Q < 0.

2.5.2 The Wiener-Hopf factorization

The equation (2.5.11) is typical of problems analyzed by means of theWiener-Hopf method. It is a single functional equation that holds ina strip of the complex plane and that involves two unknown functions,each analytic in a half plane with the strip being the region of overlapof the two half planes. The essence of the Wiener-Hopf method is theprocess of determining both of the sectionally analytic functions fromthis single equation by applying certain theorems from the theory ofanalytic functions. So as not to obscure the elegance and simplicityof the steps involved with the details of the problem at hand, theprocedure is first outlined in general form.

Equation (2.5.11) has the form

A(Q + E+(C) = B(QU_(Q (2.5.14)

for all £ in a strip of the complex plane parallel to the imaginaryaxis, say £+ < Re(£) < £_. The specified functions A(£) and B(Qare analytic in the strip and are defined in the entire £-plane, withunique branches identified for multiple valued functions. The un-known functions £+(C) and C/_(C) are analytic in the overlapping halfplanes Re(C) > £+ and Re(C) < £_, respectively. The Wiener-Hopfprocedure requires that B(() must be factored into the product ofsectionally analytic functions

(2.5.15)

where i?+(C) is analytic and nonzero in Re(£) > £+, and £?_(C) isanalytic in Re(C) < £_. Thus,

(2.5.16)

The only "mixed" function is A(C)/B+(Q, and this must be decom-posed into a sum of sectionally analytic functions

(2.5.17)B+(C)

where L+ and L_ are analytic in Re(£) > £+ and Re(£) < £-, re-spectively. These factors are unique up to an additive entire function,that is, any entire function may be added to one factor and subtractedfrom the other factor. With this additive decomposition done, (2.5.16)takes the form

M O + E+(C)/fl+(C) = BM)U-(O ~ M O (2.5.18)

which is valid in the strip £+ < Re(C) < £-• Each side of (2.5.18)is analytic in one of the overlapping half planes, and the sides obvi-ously coincide in the strip of overlap. Consequently, according to theidentity theorem for analytic functions introduced in Section 1.3.1,each side of (2.5.18) is the analytic continuation of the other into itscomplementary half plane, so that the two sides together representone and the same entire function, say E((). Normally, the entirefunction is determined by its behavior as \(\ —• oo, which is relatedto constraints on the variables representing physical quantities nearx = 0. The most common situation is that the sectionally analyticfunctions have algebraic behavior at infinity, so that

= O(Kr) , Re(C) > £+ ,(2.5.19)

\B_(0U-(0 - M O | = O(|Cr) , Re(C) < £- ,

as |C| —• oo. In order to deal with such circumstances, Liouville'stheorem on bounded entire functions has been generalized to the caseof entire functions with no essential singularities, even at infinity.

The result, known commonly as the generalized Liouville's theorem,requires that the entire function E(Q is a polynomial of degree lessthan or equal to the largest integer that is less than or equal to thesmaller of q+ and q_ (Whittaker and Watson 1927). Consequently,S + and [/_ are determined from (2.5.18) up to a finite number ofconstants, the polynomial coefficients.

In some cases, the additive factorization into sectionally analyticfunctions indicated in (2.5.17) can be accomplished by observation.In other cases, the factorization must be done formally. The essentialidea of the formal procedure follows from the statement of Cauchy'sintegral formula,

= L+(Q +L_(C)

where £ is a point in the strip and F is a small simple closed contouraround ( in the strip, executed in the counterclockwise sense. Thecontour F is then "expanded to infinity" by means of Cauchy's the-orem (1.3.4). So as to continue to satisfy the conditions of Cauchy'stheorem, the contour leaves in its wake clockwise contours embracingall the poles and branch cuts of L(C) i n the whole plane as the radius ofthe contour is increased to indefinitely large values. If the collectionof such contours in Re(£) > £_ is called F_ and the collection inRe(C) < £+ is called F+, then

^LMi- (2-5-21)The Cauchy integral defining L± is analytic everywhere except for £approaching certain points on F ± . In view of the fact that these pathsare restricted to the appropriate half planes, the indicated regions ofanalyticity are obvious, and the factorization is complete. It has beentacitly assumed that \L(Q\ —> 0 as |£| —* oo so that the contributionto the integral from the contour at infinity vanishes, but this does notpose a serious restriction on the application of the idea.

The factorization into a product of sectionally analytic functionsindicated in (2.5.15) can be accomplished in the same way by observ-ing that

logB(C) = logB+(C)+logS_(C) = ^ T / l o g B ( 7 ) r f 7 (2.5.22)2m Jr++r- 7 - C


so that

> . (2.5.23)7-C

In writing (2.5.22), it is tacitly assumed that B(Q —> 1 as |£| —• oo,and that B±(Q are not only analytic but also nonzero in the respectivehalf planes. The latter condition ensures that B±(Q and logi?±(C)have a common domain of analyticity. Again, as will be seen, theseconditions do not pose serious restrictions on the application of theidea.

The functional equation (2.5.11) is now considered as a particularcase of (2.5.14). The first step in determining the unknown functionsE+ and U_ is the factorization of R(C) and a(C) into products ofsectionally analytic functions. The requisite factorization of a issimple, and it is given by

a±(C) = (a ± C)V2 , a = a+a_ . (2.5.24)

These factors satisfy all requirements.The Rayleigh wave function R(C) is more complicated than a, and

the formal procedure must be followed for it. To this end, it is essentialto know the behavior of R(C) for large |£|. Through application of thebinomial expansion theorem to the square root terms in (2.5.12), itcan be shown that

R(Q = -2{b2 - a2)C2 + 0(1) as |C| -> oc . (2.5.25)

Furthermore, it is essential to know the locations of all zeros of therelevant branch of i?(C)> not just the locations of the real zeros cor-responding to free surface Rayleigh waves. This information maybe extracted by application of the zero counting formula of analyticfunction theory.

Consider a simple closed curve Z in the ("-plane such that R(C)is analytic everywhere inside and on Z. Then, according to the zerocounting formula, the number N of roots of R(C) — 0 inside Z is

2m

where the prime denotes a derivative. The formula (2.5.26) takesproper account of the multiplicity of roots. Furthermore, Z may be a

set of closed curves that together bound a multiply connected regionin which R(() is analytic. In either case, the sense of traversal of Zis such that the region is on the left. An operational procedure forcalculating N is evident if the mapping from the complex £-plane tothe complex emplane defined by u = R(() is considered. If the integralin (2.5.26) is considered in the o;-plane, then

N = ^- I — , (2.5.27)

where fi is the image of Z according to the mapping. Thus, N issimply the net number of times that the path $7 encircles the originin the u;-plane in a counterclockwise sense as £ moves in a counter-clockwise sense once around Z. A convenient choice of Z, consistingof the three closed paths ABC, DEF, and GHI in the £-plane, andits image in the u;-plane, are shown in Figure 2.6. Evidently, the pathVL encircles the origin in the cj-plane two times in a counterclockwisesense for the choice of Z in the ("-plane. Consequently, the two realzeros of R(C) in the cut ("-plane are the only zeros of the function inthe entire £-plane.

Recall that the factorization (2.5.15) is accomplished most di-rectly for functions that approach unity as |£| —> oo and that haveneither zeros nor poles in the finite plane. A function having theseproperties is

so the general result (2.5.15) is interpreted for this function. Theonly singularities of S(Q are the branch points at £ = ±a, ±6 whichare shared with i?(C)> and it is single valued in the £-plane cut alonga < |Re(C)| < 6, Im(C) = 0- The clockwise contours F+ and F_are closed paths embracing the branch cuts in the left and right halfplanes, respectively. The property that S(Q = S(Q can be exploitedto show that

log Si ( 0 = / tan6 4r?V(??2 - a2)(62 - rj2) dr\

(b2 - 27/2)2

(2.5.29)


Figure 2.6. Mapping of the contour Z in the f-plane into the contour fi in theCJ-plane according to u> = R(C)-

so that £± (0 = exp{log«Sf-b(C)}. It is evident that S+ accounts forthe singularities of 5 in the left half plane, and similarly for S_. Thiscompletes the product factorization, and (2.5.11) now has the form

( ^ ) - S ^ (2.5.30)

where

The final additive factorization (2.5.17) can be carried out byobservation because the only singularity of the mixed function in(2.5.30) in the right half plane is a simple pole at £ = 0. Thissingularity can be removed by requiring the residue to be zero, sothe factorization is

+ VW J + \S>/ + VU / I I I + V / I / O K QO\

s I s J + I s J _

The first (second) term on the right side of (2.5.32) is analytic in theappropriate right (left) half plane, so the particular form of (2.5.18)for the problem at hand is

" £ £ { g (2-5.33)

in the strip —a < Re(C) < 0. As already noted, each side is the uniqueanalytic continuation of the other into a complementary half plane,and together the two sides represent a single entire function E(£).

As |C| -> oo in the respective half planes, |F±(C)| = O (|CI~1/2)-Furthermore, a(x, s) is expected to be square root singular as x —> 0+,and w_(x, s) is expected to vanish as x —• 0~ to ensure continuity ofdisplacement. Consequently, from the Abel theorem (1.3.18) concern-ing asymptotic properties of transforms,

lim x^2a(x,s)^

(2.5.34)lim \x\~qu_(x,s)~ lim \C\1+qU_{()

x 0 - <+oo

for some q > 0. Therefore, each side of (2.5.33) vanishes as |£| —• ooin the corresponding half plane. According to Liouville's theorem, abounded entire function is a constant. In this case, J5(C) is boundedin the finite plane and E(Q —• 0 as |£| —* oo so that the constantmust have the value zero; thus, E(£) = 0.

This completes the solution of the Wiener-Hopf equation (2.5.11).The two unknown functions in that equation have been determinedto be

(2.5.35)

2.5.3 Inversion of the transforms

At this point in the analysis, it would be possible to solve for P(() andQ(C)> in addition to the functions in (2.5.35), and to invert the doubletransforms for (f){x,y,i) and ip(x,y,t) by means of the Cagniard-deHoop method. Attention is focused on the time history of stress aheadof the crack tip in this chapter, however, so that only the inversionof the double transform of cr+(a;,t) will be outlined. Inversion of thetwo-sided Laplace transform is considered first, and the formal inverseis given by the integral

(2.5.36)

where £o is a real number in the interval —a < £o < 0. The central ideaof the Cagniard-de Hoop scheme and, likewise, the central idea here isto convert the integral in (2.5.36) to a form which will allow inversionof the one-sided Laplace transform by observation. To this end, thepath of integration is augmented by two circular arcs of indefinitelylarge radius in the left half plane plus a path running from the remotepoint on the negative real axis inward along the upper side of thebranch cut to ( = —a, around the branch point, and then outwardalong the lower side of the cut to the remote point; see Figure 2.7.These paths, together with the inversion path itself, form a closedcontour, and the integrand of (2.5.36) is analytic inside and on thiscontour.


Figure 2.7. Integration path for evaluation of the integral in (2.5.37), consistingof Too, two remote quarter-circular arcs, and F^ , a path embracing the branchcut along the negative real axis.

According to Cauchy's theorem, the integral of the function alongthe closed contour is zero. An immediate inference is that the integralin (2.5.36) is equal to the integral of the same function taken alongthe alternate path consisting of the two arcs of large radius F ^ andthe branch cut integral

(2.5.37)

The function £+(C) —• 0 uniformly as |C| —> oo in the left half plane.Consequently, according to Jordan's lemma introduced in Section1.3.2, the value of the first integral in (2.5.37) is zero. Furthermore,£+(C) = S+(C) so that the second integral in (2.5.37) can be writtenas the real integral

a+(x,s) = — I °°Im(E + (C))e**dC, (2-5.38)KS J-a I J

where the limiting value of the integrand as Im(£) —• 0 + is understood.If the change of variable 77 = -#C> V ^ 0, is introduced, the integral


takes the form

a+(x,s) = — /°°Im{s+ ( - 5 U e-^dr? (2.5.39)7TSX Jax I V X J J

and, indeed, the inversion of the one-sided transform becomes obvious.That is, (2.5.39) is a product of two transforms, so that cr+(x,t) is aconvolution of the inverses of the two transforms,

a (xt) = -— ( Im(s+ (--)} drj Hit -ax). (2.5.40)KX Jax I V x/J

This completes the process of inversion of the double transform forthe stress distribution on the crack plane ahead of the tip. The resultis in the form of a real integral, and some of its features are nowexamined.

The integrand of (2.5.40) has the property that

ICI - - o o (2.5.41)

along the upper side of the cut in the left half plane. Thus, the integralis divergent as (x/cdt) —> 0+, reflecting the anticipated singularityin stress at the crack tip. The strength of this singularity may bedetermined in a number of ways. For example, application of theAbel theorem (1.3.18) on asymptotic properties of transforms in thepresent situation implies

lim (7vx)1/2a+(x,s)= lim ( « C ) 1 / 2 4 E + ( C ) . (2.5.42)x—•()+

If the indicated limits are taken for the functions at hand, then

K^s) = V2a*F+(0)/s3/2 . (2.5.43)

Thus, by inverting the Laplace transform, the stress intensity factorhistory defined in (2.4.2) is found to be

(2-5.44)

where the coefficient in (2.5.43) has been evaluated in terms of mate-rial parameters. This evaluation is carried out by noting that £+(0) =S_(0) = y/S(0) = b2/cy/K and a+(0) = y/a. If the ratio of wavespeeds b/a is then written in terms of Poisson's ratio v as in (1.2.58),it is found that F+(0) = (1 - ^)~1 \ / cd(1 - 2i/)/2. The dimension-less constant Q that was introduced in (2.4.2) has the value Q =\/2(l — 2I/)/TT(1 — i/). Equation (2.5.44) is the main result of thissection. Note that the stress intensity factor is zero for elastic incom-pressibility, or v —• 1/2. In this case, the stress state is apparently auniform hydrostatic compression for t > 0.

The stress intensity factor (2.5.44) has been obtained for the caseof crack face loading. Just as in the case of antiplane shear loading,however, it may be given another interpretation. Consider once againthe same configuration, but the crack faces are now free of traction.Suppose that a plane tensile pulse propagates toward the crack planeat speed c<j. The front of the pulse is parallel to the crack plane,and it carries a jump in stress avy from its initial value of zero tocr*. The pulse arrives at the crack plane at time t = 0, and it ispartially reflected and partially scattered, or diffracted, upon reachingthe crack.

The complete deformation field of this stress pulse diffractionproblem differs from the field due to crack face loading only by thepulse propagating in an uncracked solid. Because the governing equa-tions are linear, and because the plane pulse, by itself, results inno stress singularity at the crack tip, it follows that (2.5.44) is alsothe stress intensity factor history for the case of stress pulse diffrac-tion. Furthermore, the complete stress distribution on the crack planeahead of the tip is given by (2.5.40) plus the constant a*, that is,

ayy(x, 0, t) = a+(x, t) + a*. (2.5.45)

Unlike the case of antiplane shear in Section 2.3, the expressionfor the complete stress distribution on y = 0, 0 < x < Cdt cannotbe written in terms of elementary functions. The definite integral(2.5.40) can be evaluated numerically in a straightforward manner,however, to obtain a graphical representation of <7+(:r, t). At firstsight, it may appear that the Cauchy integral in the definition ofS+ (C) must be evaluated for points on its path of integration. Thiscan be done by invoking the Plemelj formula for the limiting value of aCauchy integral as the field point approaches the path of integration

2.5-,

2.0-

\ 1.5 H

*bH- 1.0b+

0.5-

0.00.0 0.2 0.4 0.6 0.8

I1.0

x/cdt

Figure 2.8. Normal stress on the plane y = 0 ahead of the crack tip versusnormalized distance x/cat for v = 0.3. The solid curve is the complete stressdistribution (2.5.45) and the dashed curve is the singular field (2.4.1) only.

from either side (see Carrier et al. 1966, for example). If S+(() isreplaced by S(C)/S_(£), however, then this special consideration isunnecessary because the Cauchy integral in the definition of iS_(C)is analytic for all points on the path of integration in (2.5.40). Theresult of numerical evaluation of the ratio (a+ + a*)/a* versus x/cdtis shown in Figure 2.8 for v = 0.3. Also shown in the same figure asa dashed line is a graph of the singular term (2.4.1).

The size of the region, as a fraction of x/cdt or x/c8t, overwhich the stress intensity factor field approximates the complete stressdistribution is slightly smaller than in the case of antiplane shear.The reason is connected with the complexity of wave phenomena inplane strain as compared to antiplane shear deformation. In the caseof plane strain, the initial wave motion radiated out from the crackfaces as the pressure is applied is a negative dilation or compression.The initial tendency of material on the crack plane ahead of the cracktip is to compress slightly, as shown in Figure 2.8. The tensile stresson this plane becomes slightly negative with the first wave arrival,as reported in the experimental study of this configuration by Kim(1985b). It is only after the shear wavefront passes a material pointon this plane that the tensile stress tends to become positive. It doesnot become sufficiently positive for the singular term to dominate thatdistribution until about x/cdt < 0.09, at least for the case of suddenlyapplied loads. That is, the stress intensity factor field differs from the

complete stress distribution (a+ + a*)/a* by less than 10 percent ofthe latter for x/cdt < 0.09. This restriction is mitigated somewhatfor more gradually applied loading.

2.5.4 Higher order terms

The next most significant term in the expansion of <T+(:E, t) for smallvalues of x/cdt, after the square root singular term, is of order unityfor the case of crack face loading. In the course of deriving the resultby an alternate route, Freund (1973a) showed that this term is simplyo-yy' = —cr*. The basic idea is to rewrite (2.5.40) in the alternate form

*+0M) = ~ I Z+(-()d(H(t-ax), (2.5.46)*™ JT(t/x)

where the path T(t/x) embraces the cut in the complex plane runningfrom C = t/x — zO, and around the branch point at £ = a, to C =t/x + iO. By means of Cauchy's theorem, the integral in (2.5.46) canbe shown to be equal to an integral of the same function around a circleof radius t/x plus the residue of the pole at £ = 0. It is the residuewhich leads to the term of order unity in the near tip expansion.Thus, the graph of the first two terms of the near tip expansion maybe obtained from the dashed curve in Figure 2.8 by a unit reductionin the ordinate of each point. Likewise, the graph of a+(x, t)/cr* maybe obtained by a unit reduction in the ordinate of each point of thesolid curve in Figure 2.8. An immediate inference is that the stressintensity factor field, by itself, differs from the complete distributioncr+/(T* by less than 10 percent of the latter only if x/c<it < 0.0025.This observations shows the significance of the terms of order unityin the near tip expansion in some cases.

The stress component axx is also square root singular at thecrack tip. The main contribution to this stress component for pointsnear the tip is given by the universal spatial dependence (2.1.1) andthe stress intensity factor (2.5.44). Like the stress component ayy,the component axx also has a term of order unity if the completedistribution is expanded in powers of r/cdt, where r is the radialdistance from the crack tip. By following the method leading to(2.5.46), Ma and Burgers (1987) showed that the term of order unity

w = o*v < T ( 2 - 6 / C )

I * S(0)S()/l + / '

2.6 Suddenly applied in-plane shear traction 97

The magnitude of aix depends on cr*, of course, and on Poisson'sratio. The algebraic sign is negative for all admissible values of v anda* > 0. For Poisson's ratio of v = 0.3, for example, axj has the value—0.85a*, with the contributions from the two terms in (2.5.47) beingabout equal. The term of order unity in the expansion of the in-planeshear stress axy is zero for this problem.

Consider once again the plane tensile stress pulse propagating inthe y direction through the solid. If the pulse carries a jump in thestress component ayy from zero to a*, then the stress component axx

jumps from zero to a*u/(l — v). But this is precisely the magnitudeof the first term in (2.5.47). Consequently, for the case of diffractionof the stress pulse by a traction-free crack in the plane y = 0, theterm of order unity in the expansion of axx would be just the secondterm in (2.5.47).

2.6 Suddenly applied in-plane shear traction

Consider once again the body of elastic material that contains a halfplane crack but that is otherwise unbounded; see Figure 2.1. In thepresent instance, the crack faces are subjected to suddenly applied,spatially uniform shear traction of magnitude r*. For the boundarytraction, Ty(x, z,t) = Tz(x,z,t) = 0 and Tx(x,z,t) = ±r*H(t) ony = ±0, where r* > 0 and H(t) is the unit step function. Thus, thecrack faces are traction-free for time t < 0, and they are subjected touniform shear traction for t > 0. The material is stress free and atrest everywhere for t < 0.

The components of displacement in the #, 2/, z-coordinate direc-tions are ux,uy,uz, respectively. Arguments similar to those usedin Section 2.2 can be applied here to conclude that uz — 0 for alltime if it is zero initially, and that ux and uy are independent of the^-coordinate, hence the state of deformation is plane strain. Further-more, the nonzero components of displacement satisfy the symmetryconditions ux(x,—y,t) = —ux{x,y,t) and uy(x,—y,t) = Uy(x,y,t).In particular, these conditions imply that both ux(x,0,t) = 0 and0yy(x, 0, t) = 0 for x > 0 due to symmetry.

Before application of the loads, the material is stress free andat rest everywhere. The traction is suddenly applied to the crackfaces at time t = 0. The anticipated wavefronts are shown in a planez = constant in Figure 2.9. For points near to the crack face comparedto the distance to the crack edge, the transient field consists only of

Figure 2.9. Wavefronts generated by sudden application of a spatially uniformin-plane shear traction on the faces of a crack. The state of deformation is planestrain.

a plane wave parallel to the crack face and traveling away from it atspeed c8. The shear loading does not generate a plane dilatationalwave. As the plane shear wavefront passes a material point in itspath, the component of stress axy (or cr12) changes discontinuouslyfrom zero to —r* and the particle velocity dux/dt (or dui/dt) changesdiscontinuously from zero to ±r* /pc8 for ±y > 0. Near the crack edge,a nonuniform scattered field radiates out from the crack edge behindcylindrical wavefronts of radius c&t and c8t that are centered on thecrack edge. The wave pattern includes the associated plane frontedheadwaves traveling at speed cs.

The arguments presented in connection with (2.2.1) can be ap-plied here as well to conclude that the components of stress andparticle velocity are homogeneous functions of #, y, t of degree zero.An immediate consequence is that, for points very close to the crackedge,

m(2.6.1)

where CJJ is an undetermined dimensionless constant. In view of(2.6.1), the shear stress intensity factor is also known up to the con-stant Cn, that is,

Kn(t)= lim y/2nxorxy(x,0,t) = Cnr*yj2-KCdt. (2.6.2)x>0+

The mathematical problem that leads to a solution with theproperties outlined is now stated. The problem differs only in de-tail from the formulation for pressure loading in Section 2.4. TheHelmholtz representation of the displacement vector in terms of thescalar dilatational potential <j) and the single nonzero component ^of the vector shear potential is introduced. Thus, two functions

2.6 Suddenly applied in-plane shear traction 99

<j>(x,y,t) and ip(x,y,t) are sought that satisfy the wave equations intwo space dimensions and time (2.4.3) in the half plane —oo < x < oo,0 < y < oo for time in the range 0 < t < oo. These wave functionsare subject to the boundary conditions that

cryy(x, 0, t) = 0, —oo < x < oo ,

€rXy(x, 0, t) = -r*H(t), -oo < x < 0, (2.6.3)

ux(x, 0, £) = 0, 0 < X < oo,

for all time, where stress and displacement components are under-stood to be their representations in terms of displacement potentials<j> and ij). The initial conditions are given by (2.4.5), as before. Theseconditions ensure that the body is stress free and at rest until the crackface traction is applied. Finally, only solutions that result in stresscomponents that vary inversely with the square root of distance fromthe crack tip are admitted.

The solution of this problem can be obtained by means of theWiener-Hopf method, following the steps outlined in Section 2.5.First, the statement of each of the boundary conditions is extended toapply for the full range of x by introducing two unknown functions.The function u_(x,t) is defined to be the displacement of the crackface y = 0+ in the x-direction for —oo < x < 0, 0 < £ < oo, and tobe identically zero for 0 < x < oo. Likewise, the function r+(x,t) isdefined to be the component of shear stress axy along the plane y = 0for 0 < x < oo, 0 < t < oo, and to be identically zero for —oo < x < 0.With the introduction of these two functions

<Txy(x, 0+, t) = r+(z, t) - r*H(t)H(-x), (2.6.4)

for the full range —oo < x < o o , 0 < t < o o .Application of the one-sided Laplace transform on time and then

the two-sided Laplace transform leads to a standard Wiener-Hopfequation, similar to (2.5.11). The steps are the same as those followedin Section 2.5, and only the final equation is included here,

- (c ) . (2-6-5)

The relationship of T+ and U_ to r+ and u_ is exactly the same as therelationship of £+ and [/_ to cr+ and u__ established in Section 2.5.The solution of (2.6.5) for the doubly transformed shear stress aheadof the crack tip is

T^) (2A6)

where

f£ "" <2-6-7>The shear stress on the crack line ahead of the crack tip is given

by the real integral

(x,t) = -— I ImJT+f-W dr)H{t-ax).™ Jax L \ XJ )

(2.6.8)

It follows from this integral that the stress intensity factor history forthe case of suddenly applied shear traction oh the crack faces is

<2-6-9)

Furthermore, the dimensionless constant that appears in (2.6.1) hasthe value Cn = 2ir-1

y/2cs/(l - v)cd. The integral (2.6.8) givingthe complete stress distribution on the crack line may be evaluatednumerically, and the result is shown in Figure 2.10 for v = 0.3. Thestress intensity factor field along y = 0 is shown as a dashed line in thesame figure. The terms of order unity in the near tip expansions ofstress components in the present case are a^J = CFyy = 0, (Jxy = —T*.

2.7 Loading with arbitrary time dependence

Transient crack tip fields were derived in the foregoing sections underthe assumption that the crack face loading was suddenly applied, andthereafter held at a constant magnitude. This is reflected in the step

2.7 Loading with arbitrary time dependence 101

*

2.5-.

2.0-

1.5-

0.5 -\

0.00.0 0.2 0.4 0.6

x/cdt0.8 1.0

Figure 2.10. Shear stress on the plane y = 0 ahead of the crack tip versusnormalized distance x/cat for v = 0.3. The solid curve represents the completestress distribution and the dashed curve shows only the singular term (2.6.9).

function time dependence of the crack face tractions in the boundaryconditions (2.2.5), (2.4.4), and (2.6.4). Once the time dependence ofthe stress intensity factor is established for suddenly applied loading,the corresponding result for any time history of crack face traction,say p(t) for t > 0, is obtainable by superposition. The boundarycondition (2.2.5)i, (2.4.4)i, or (2.6.4)2 is replaced by

x, 0+,t) = -p{t), -oo < x < 0, (2.7.1)

where (3 is z, y, or x, respectively. The stress intensity factor resultingfrom the boundary loading (2.7.1) may be considered as follows. Ifstep loading is applied to the crack faces at time t = rj instead of timet = 0, and if the infinitesimal magnitude of the step depends on rjaccording to dp(rj) instead of being r* or a*, then the stress intensityfactor is

K(t) = dp(r]) Cy/27rc(t - rj), t > 7], (2.7.2)

where the values of the dimensionless parameter C and the wave speedc appropriate for the particular system being considered are presumed.If (2.7.2) is then integrated with respect to 77 in the range 0 < 77 < t,


understanding that dp{rj) = p'{rj) drj, then

K(t) = C^PhTc f pf(rj)(t - r,)1'2 dr,. (2.7.3)

The result for the step function is recovered if p(t) = r*H{t)^ forexample. Illustrations of other functions p{i) and the stress intensityfactor histories that they produce are given below. In each case, theparameter p* has the physical dimensions of force/length2 and theparameter t* has the dimension of time.

i. Suddenly applied load:

p(t)=p'H(t),t (2-7.4)

K{t) = Cp*VMH{t).

ii. Load increasing linearly in time:

P(t) = P* ( £ ) H(t),(2.7.5)

iii. Impulsively applied load:

p(t)=p*t*S(t),

t\1/2

j ) H(t)

iv. Step load of finite duration t* as a superposition of simple steps:

o (2-7-7)K(t) = - C p * V 2 ^ \ t 1 / 2 H ( t ) - { t - t*)1/2 H(t - « }

2.7 Loading with arbitrary time dependence 103

v. Step load with finite rise time t* as a superposition of linearlyincreasing loads:

P(t) =P*{ (^) £ )l V J V J } (2.7.8)

- (t - n3/2 }

vi. Steady state response to load varying sinusoidally in time withperiod £*, after transients have decayed (t >> t*):

p(t)=p* sin (?£) H(t),(2.7.9)

K(t) « icp* V^F sin - ^ .

The complete solutions of the problems described above can beobtained directly for arbitrary time dependence of the applied loadingby the methods introduced in Sections 2.3 and 2.5. In the case ofGreen's method, the term r* in the integrand of the right side ofthe integral equation (2.3.11) would be replaced by p(.t). Otherwise,the solution procedure is unchanged. In the case of the Wiener-Hopf method, the factor a*/s in (2.5.11) would be replaced by p(s).Likewise, the presence of this factor does not alter the basic solutionprocedure.

3

FURTHER RESULTS FOR A

STATIONARY CRACK

3.1 Introduction

In the preceding chapter, the transient stress intensity factor historyresulting from the application of a spatially uniform crack face trac-tion was examined. The particular situations analyzed represent thesimplest cases of transient loading of a stationary crack for each modeof crack opening. There are many similar situations that can beanalyzed by the methods outlined in Chapter 2. For example, considerthe plane strain situation of a plane tensile stress pulse propagatingthrough the material toward the edge of the crack, which lies in thehalf plane —oo < x < 0, y = 0. Suppose that the x and y componentsof the unit vector normal to the wavefront are — cos 9 and — sin 0,respectively, and that the pulse front reaches the crack edge at timet = 0. Suppose further that the incident pulse carries a jump in thenormal stress component from the initial value of zero to (Tinc, Ifthe solid is uncracked, then the incident pulse will induce a tensiletraction of magnitude a* = crinc{l — (1 — 2v) cos2 0/(1 - v)} and ashear traction of magnitude r* = (Jinc[l — 2z/) sin 0 cos 0/(1 — v) onthe plane y = 0 over the interval —Cdt/ cos 9 < x < 0. This particularstress wave diffraction problem was studied by de Hoop (1958) in hispioneering work on the subject.

The stress intensity factor history can then be determined byanalyzing the situation in which equal but opposite tractions areapplied to the faces of the crack over an expanding interval. The

104

3.1 Introduction 105

analysis differs only in minor details from that presented in Section2.5, and the results for the mixed mode stress intensity factor historiesare

Kj(t) = 2ainc (sin26 + —— cos2o] F+(cos9/cd)J- ,\ \ — V / V 7T

(3.1.1)

smOcos0G+(cos9/cd)y— ,

where F+(-) and <?+(•) are given in (2.5.31) and (2.6.7), respectively.The formulas in (3.1.1) can be evaluated in a straightforward manner,but the matter is not pursued here. However, a comment on thespecial case 6 = 0 is in order. It is well known that if a uniform tensionis applied in the direction of the crack line (in the redirection in thepresent instance) and if no constraint is imposed on the material in theorthogonal direction (the y-direction here), then the stress intensityfactor is zero under conditions of equilibrium. The same is not truein the case of dynamic stress wave loading, as is seen from (3.1.1).The inertial resistance of the material to Poisson contraction in the^/-direction upon arrival of the stress pulse is sufficient to generate atensile stress on the plane y = 0 ahead of the crack tip.

A similar plane strain situation involving an opposed pair ofconcentrated forces moving along the crack faces was considered byAng (1960). The forces, each of magnitude p*, begin to move fromthe crack tip at time t = 0 and at a constant speed v < cs. Theresulting mode I stress intensity factor, which can be obtained bydifferentiation with respect to time of the moving step load problembriefly discussed above, is

(3.1.2)

It was noted in the course of analysis that the problems posedin Chapter 2 have neither a characteristic length nor a characteristictime. The problems mentioned in the foregoing paragraphs are alsoof this type. If the crack face traction distribution is spatially nonuni-form, then a characteristic length is introduced. For example, thecharacteristic length may be the distance of a concentrated load or a

106 3. Further results for a stationary crack

step in traction from the crack tip, or it may be a reference tractionlevel divided by a spatial gradient of traction. The introduction ofa characteristic length necessitates a reconsideration of the approach,as well as the results, for stress intensity factor histories. Two cases ofplane strain deformation involving a characteristic length are studiedin some detail in this chapter. One concerns the sudden loading of thefaces of a crack by a pair of concentrated forces at some fixed distancefrom the crack tip, and the other concerns the sudden loading of acrack of finite length.

There are also several three-dimensional situations of dynamicloading of a stationary crack in an elastic solid that can be analyzedby means of extensions of methods already introduced. Two represen-tative cases are introduced in this chapter. Finally, some implicationsof transient stress intensity factor results for fracture initiation froman existing crack in a body of material subjected to dynamic loadingare summarized.

3.2 Nonuniform crack face traction

For the case of antiplane shear loading, Green's method of solutionas introduced in Section 2.3 goes through without modification fornonuniform crack face traction, although some of the steps are morecomplicated. For example, consider the boundary value problemposed in Section 2.2, except that the crack faces are loaded by anopposed pair of out-of-plane point loads of magnitude q* at a distanceI from the crack tip. More precisely, q* is the force per unit lengthin the ^-direction of the opposed line loads acting on the crack faces.The statement of the mathematical problem is then identical to thatin Section 2.2, except that the boundary condition (2.2.5)i is replacedby

ry{x, 0+, t) = -q*6(x + I) H(t), -oo < x < 0, (3.2.1)

where I > 0 and <$(•) is the Dirac delta function, representing adistribution with respect to x in this case. The argument of thedelta function has the physical dimension of length, so S(x + I) hasthe dimension of length""1. Thus, the multiplier q* does indeed havedimensions of force/length.

The analysis proceeds as in Section 2.3, except that the boundarycondition (3.2.1), rather than (2.2.5)i, is substituted into (2.3.6). The

3.2 Nonuniform crack face traction 107

solution of the resulting Abel integral equation leads to the transientstress intensity factor

Km(t) = q*JlH(cat - I) (3.2.2)

for the case of concentrated antiplane shear crack face loading. Thesteps followed in obtaining (3.2.2) are not included here. The in-terpretation of (3.2.2) is straightforward, but the result is atypicalfor a two-dimensional wave field. The stress intensity factor is zerountil the wavefront generated at the instant of load application attime t = 0 reaches the crack tip; this time is t = l/cs. The stressintensity factor then takes on a value consistent with the equilibriumstress field arising from the applied concentrated forces, and this valueis maintained for all later times. There is no gradual transitionto the long-time value Km (oo) which would be more typical for atwo-dimensional wave field. It will now be demonstrated that thesolution of the equivalent plane strain crack problem possesses thissame remarkable feature.

The means of establishing the properties for the equivalent planestrain situation are not so transparent. In particular, a straightfor-ward application of the Wiener-Hopf method, as outlined in Section2.5, is not successful. Indeed, the exact solution to this problem wasoriginally obtained by an indirect approach based on the superpositionof moving dislocations (Freund 1974b). The transient stress intensityfactor history solution has also been obtained by the weight functionmethod which is introduced in Section 5.7. The original approach isthe more intuitive, and the general viewpoint represented by it hasbeen found useful in attacking a number of other problems in dynamicfracture mechanics. Consequently, this approach is outlined here. Themathematical problem is first stated, and the general approach ismotivated by means of certain observations on the physical process.Next, a fundamental moving dislocation problem, which forms a keyelement in the approach, is solved exactly by the Wiener-Hopf tech-nique. Finally, the exact stress intensity factor history is extracted bymeans of a superposition argument.

3.2.1 Suddenly applied concentrated loads

Once again, consider the elastic solid depicted in Figure 2.1 which iscut by a half plane crack but which is otherwise unbounded. In the

tp'H(t)

x- •

p*H(t)I

Figure 3.1. Sudden application of concentrated loads on the faces of a crack underplane strain conditions. The distance / from the load points to the crack tip isconstant.

present instance, each crack face is subjected to a suddenly appliedline load of intensity p* per unit length in the z-direction. The linesof loading are parallel to the crack edge, and at a distance / from theedge. The loads on the two faces are opposed, tending to open thecrack. For the boundary tractions, Tx(x,z,t) = Tz(x, z,t) = 0 andTy(x,z,t) = ±p*6(x + l)H(t), where p* > 0, H(-) is the unit stepfunction, and <$(•) is the Dirac singular distribution function. Thematerial is stress free and at rest everywhere for time t < 0, and theloading is applied at time t = 0. The two-dimensional configurationis depicted in Figure 3.1. The resulting wave field produces a state ofplane strain deformation at each point.

The statement of the mathematical problem to be solved is sim-ilar to that introduced in Section 2.4. In view of the symmetries ofthe fields, attention can again be limited to the region y > 0. Asolution of the Navier equation (1.2.51) is sought for — oo < x < oo,0 < y < o o , 0 < £ < o o , subject to the conditions that the materialis stress free and at rest at time t = 0, and subject to the boundaryconditions

<Tyy(x, 0, t) = -p*6(x + /) H(t) , -OO < X < 0 ,

axy(x,0,t) = 0, - o o < x < o o , (3.2.3)

uy(x,0,t) = 0, 0 < x < oo,

for all time. The form of the boundary conditions (3.2.3) is similarto (2.4.4), and this observation suggests the Wiener-Hopf method asa means of solving the problem. If integral transforms are applied in

the manner outlined in Section 2.5, however, the steps to be followedto find a solution are not evident. A perusal of the physical processbeing simulated suggests an alternate approach.

If the boundary conditions (3.2.3)i and (3.2.3)2 are imposed overthe entire boundary y = 0+, —oc<x<oo, then the problem reducesto the classical problepi treated by Lamb (1904). The displacement inthe y-direction of the surface y = 0+ for x + / > 0 in Lamb's problemis

v*b2 /%t/(x+/) f a(C) 1<IIL — — 1 _ _ / TTYI } ySJ \ rlf H(i — a(r -I- 7^ C\ 9 4"\

Kfl> Ja

where a(-) is defined in (2.5.10), the Rayleigh wave function R(-) isdefined in (2.5.12), and the other notation is introduced in Section2.5. The limiting value of the integrand as Im(£) —> 0~ is understoodin (3.2.4). The integrand has a pole singularity at £ = c = 1/CR alongthe integration path, so the integral is interpreted in the sense of theCauchy principal value. In the derivation of (3.2.4), the integrationpath includes a small indentation around the pole at £ = c on bothsides of the branch cut of a(£). Each indentation gives rise to acontribution to the contour integral equal in magnitude to one-halfthe residue of the integrand, but the contributions cancel each otherbecause the integrand is equal in magnitude but opposite in sign onopposite sides of the branch cut.

In view of the boundary condition (3.2.3)3, the problem at handis equivalent to Lamb's problem with a concentrated normal load atx = — /, but with the normal displacement Uy of the surface y = 0+

negated for x > 0. The necessary cancellation of normal displacementcan be accomplished by means of the fundamental moving dislocationsolution obtained in Section 3.2.2. Note that Uy is a homogeneousfunction of degree zero of t and x + I. This implies that any givendisplacement level radiates out along the ar-axis at a constant speed.In particular, the displacement level Uy(x,t) propagates in the re-direction at the speed (x + l)/t for t > 0. This feature is exploited inconstructing the requisite moving dislocation solution. The completesolution satisfying the boundary conditions (3.2.3) and the appropri-ate symmetry conditions is then obtained by superposition.

Before constructing a solution, some of the features of the tran-sient stress intensity factor history due to the concentrated loads canbe anticipated on the basis of an examination of Uy in (3.2.4). For the

surface point at x = 0, the displacement Uy is zero until time t = alwhen the cylindrical dilatational wavefront generated at the load pointfirst arrives. Consequently, the transient stress intensity factor will beidentically zero for 0 < t < al. For a < C < c, the integrand of (3.2.4)is positive. Therefore, the displacement Uy is negative for time in therange a < t/l < c, which means that the surface bulges outward. Interms of the crack problem under study, the crack tends to close or thestress intensity factor tends to become negative. This would requirethe unrealistic interpenetration of the crack faces, and this point isreconsidered once quantitative results are in hand. For c < £ < oo, theintegrand of (3.2.4) is negative. Thus, for time t > c/, the crack facestend to move away from each other according to (3.2.4) and a positivestress intensity factor is anticipated. Finally, it is expected that thetransient stress intensity factor will approach the equivalent long-timestress intensity factor, that is, Ki(t) —• p* ^/2/nl as Cdt/l —> oo (Tada,Paris, and Irwin 1985). These qualitative features are indeed exhibitedby the solution to be derived next.

3.2.2 Fundamental solution for a moving dislocation

Consider plane strain deformation of the unbounded elastic solid con-taining a half plane crack that was introduced in Section 3.2.1. Inthe present instance, the faces of the crack are free of traction. Thematerial is initially stress free and at rest. At time t = 0, an elasticdislocation with Burgers vector of magnitude 2A in the y-directionbegins to move at constant speed v in the positive ^-direction fromthe crack tip. In the terminology of elastic dislocation theory, this isan edge dislocation climbing in the ^-direction. The speed is assumedto be in the range 0 < v < Cd for present purposes, although thesolution can be obtained by the same procedure if the speed exceedsthe dilatational wave speed of the material. As was the case in theoriginal problem, the situation is symmetric with respect to the planey = 0. Thus, a solution is sought for —oo < x < oo, 0 < y < oo,0 < t < oo subject to initial conditions appropriate for the materialbeing stress free and at rest. The boundary conditions are

**y(*,0,t) = 0, (3.2.5)

uy(x, 0, t) = w_ (x, t) + AH(vt - x)

for all x and t > 0, where A > 0 and H(-) is the unit step function.The interpretation of the functions a+ and u_ is precisely the sameas in (2.5.1), that is, a+ is the unknown normal stress distribution onthe plane y = 0 for x > 0 and <r+ = 0 for x < 0. Likewise, u_ is theunknown normal displacement distribution on y = 0 for x < 0 andu_ = 0 for x > 0.

The particular moving dislocation problem posed can be solvedby means of the Wiener-Hopf technique, following the steps outlinedin Section 2.5. The details are not included here; only the main resultsare given, using the notation introduced in Section 2.5 unless a specificstatement to the contrary is made. The Wiener-Hopf equation in thepresent case, corresponding to (2.5.11), is

< 3-2 '6 )

where d = 1/v is in the range a < d < oo. This relationship betweensectionally analytic functions is valid in the strip —a < Re(£) < aof the complex £-plane. Application of the factorization proceduresoutlined in Section 2.5 leads to the solution for the double transformof (7+ in the form

where F+(() is defined in (2.5.31). The transform inversion procedureoutlined in Section 2.5.3 leads to the expression

a+(x,t) = —^-Im{S+ {-t/x + iO)} if (t - ax). (3.2.8)TTX

As could have been anticipated from the formulation, the displace-ment components in this problem are homogeneous functions of x, y, tof degree zero, and the stress components are homogeneous of degree—1. The stress intensity factor history due to a moving dislocationof unit magnitude A = 1, say ki(t, v), can be deduced from theasymptotic properties of £+(0 with the result

The stress intensity factor is identically zero for t < 0. In anticipationof the superposition scheme to be introduced in the next subsection,the dependence of the stress intensity factor on dislocation speed v hasbeen indicated explicitly and fcj has been normalized to be the stressintensity factor for A = 1. The fact that ki is negative is consistentwith the viewpoint of the dislocation as an extra layer of material ofthickness equal to two length units inserted in the interval 0 < x < vtalong y = 0.

3.2.3 The stress intensity factor history

Once again, attention is directed to the problem involving the suddenapplication of opposed point loads on the crack faces that was intro-duced in Section 3.2.1. Upon application of the loads at x = — I attime t = 0, a transient field radiates outward from the load point, asdescribed by Lamb (1904). When the leading dilatational wavefrontreaches the crack tip at x = 0 and time t = //c<j, the constraints on thesolid surface change and the complete field can no longer be describedby Lamb's solution. The viewpoint adopted here is that Lamb'ssolution persists for t > l/cd, but that the normal displacementpredicted by Lamb's solution for x > 0 is exactly canceled by emittingdislocations from the crack tip along x > 0, y = 0 in just theappropriate sequence.

It is observed that Uy is a homogeneous function of degree zero oft and x' = rr + Z, that is, it depends on these two independent variablesonly through the ratio t/xf. As noted above, this implies that anygiven displacement level radiates out along the x-axis at a constantspeed. In particular, the displacement level Uy(x'/t) propagates inthe ^-direction at the speed xf /t for t > 0. The speed is in the rangebetween zero and the dilatational wave speed c . Consider a particularvalue of speed v in this range. The time at which the correspondingdisplacement level Uy(v) arrives at x = 0 is r(v) = l/v in this case. Asobserved above, the displacement level Uy (cd) arrives at time r = l/cd-The speed of the displacement level arriving at x = 0 at any instantof time t is l/t. Based on these observations, a superposition integralthat has a value exactly equal but opposite to the normal displacement(3.2.4) of Lamb's problem can be constructed.

Let Ag(x,y,tm,v) represent any scalar field of the fundamentaldislocation solution in Section 3.2.2, such as a stress component or adisplacement component, where the linear dependence on A is madeexplicit. The fundamental solution is based on the condition that an

elastic dislocation with Burgers vector in the y-direction of magnitude2A begins to extend from the crack tip at time t = 0 at speed v. Ifthe dislocation begins moving at time t = T, instead of at t = 0,then the solution is A</(x, y, t — r; v). Furthermore, suppose that thestrength of the dislocation is 2dUy, instead of 2A. The solution ofthis modified problem is g(x, y, t — r; v) dUy. Finally, suppose that Uyand r are prescribed functions of v. Then, the result can be summedover a range of values of v, and the appropriate continuous rangefor the problem at hand has already been identified in the foregoingdiscussion. If G and gL denote the corresponding fields for the crackface loading problem and Lamb's problem, respectively, then

g(x,y,t-T(v);v)—^-!-dv. (3.2.10)

As a check, note that if the displacement component uy is consideredfor x > 0, then g(x,0,t;v) = H{vt — x) and gL = Uy, and the obviousresult that uy(x, 0,£) = 0 is obtained. Thus, the boundary condition(3.2.3)3 is satisfied by solutions constructed according to the proce-dure represented by (3.2.10). With this essential fact established,attention is focused on the stress intensity history for the case ofconcentrated forces acting on the crack faces.

The procedure embodied in (3.2.10) is now applied to determinethe stress intensity factor history according to the boundary condi-tions (3.2.3). The normal stress <Jyy{x, 0, t) is not singular at x = 0, sothat the first term on the right side of (3.2.10) need not be considered.Thus,

/_ r ( w ) ; t , ) — J L L _ i d v , (3.2.11)

cd dv

where the stress intensity factor history for the fundamental disloca-tion solution ki(t;v) is given in (3.2.9) and the surface displacementfrom Lamb's problem Uy is given in (3.2.4). If a change of variableof integration from v to r/, defined by rj = 1/v, is made, the integraldefining the stress intensity factor history (3.2.11) takes the form

Kl(t) = -P-M f Im{? « - y . w l dV (3.2.12)

for t > l/cd, where 77* = t/l. The limiting value of the integrandas Im(r/) —> 0" is understood in (3.2.12). For t < l/cd, no wavesgenerated by the applied loads have arrived at the crack tip, hencethe stress intensity factor is identically zero. The integrand of (3.2.12)has a pole singularity at rj = c, so the integral must be interpreted inthe Cauchy principal value sense for t > cl.

It is helpful to view the integral in (3.2.12) as a line integral inthe complex 77-plane. The singularities of the integrand are a simplepole at 77 = c and branch points at 77 = a,b,rf. For a < 77* < 6, theintegrand is analytic in the entire 77-plane cut along a < Re(77) < 6,Im(77) = 0, except for the pole at 77 = c. On the other hand, for 77* > bthe integrand is analytic in the entire 77-plane cut along a < Re(77) <77*, Im(77) = 0, except for the pole. In the latter case, the path ofintegration can be closed around the upper side of the branch cut,assuming a suitable branch of (77* — 77)x/2, and the integral (3.2.12)can be evaluated by observation. Thus, for 77* > 6, (3.2.12) takes theform

where IX77*) denotes a closed counterclockwise contour embracing thebranch cut of the integrand. Cauchy's integral formula can now beapplied to show that the value of the integral in (3.2.12) is equal to thevalue of the integral taken along a closed circular path of indefinitelylarge radius in the 77-plane plus, if b < rf < c, the negative of theresidue of the integrand at the pole 77 = c, or

Kj{t)

0 if c < 77* < 00 .

The first term on the right side of (3.2.14) is the contribution fromthe closed contour of large radius and the second term is the residuecontribution.

For the time range for which a < rf < 6, the direct evaluationprocedure cannot be applied to this integral. Consequently, the in-tegral (3.2.12) has been evaluated numerically in this range, and thecomposite result is shown in Figure 3.2 for Poisson ratio v = 0.3 in


0.8 1.0 2.2 2.4cdt/l

Figure 3.2. Normalized stress intensity factor (3.2.12) versus normalized timeCdt/l. The stress intensity factor assumes the large-time equilibrium value Kj(oo)= p* yjilitl after the Rayleigh wave generated at the load point reaches the cracktip.

the form of a graph of the stress intensity factor history normalized bythe long time limit Kj (oo) = p*y/2/7rl versus normalized time c&tjl.The history shows the qualitative features anticipated in Section 3.1.1.Upon arrival of the dilatational wavefront generated by the point load,the stress intensity factor takes on a small negative value, reflectingthe tendency for the crack faces to move toward each other for thissort of loading. The small negative values persist until the shearwavefront arrives at time t = l/cs. Thereafter, the stress intensityfactor decreases rapidly to a negative square root singularity at timet = 1/CR, which is the instant of arrival of the Rayleigh wave travelingalong the crack faces from the load points to the crack tip. For timet > 1/CR, the remarkable result is found that the transient stressintensity factor takes on the constant value p*^2/irl, which is theequilibrium stress intensity factor for the specified applied loading.This last observation, which is distinctly uncharacteristic of a two-dimensional wave field, is perhaps the most significant result from thepractical point of view.

The analysis of this section has been carried out under the as-sumption that the crack faces remain traction-free between the load

points and the crack tip. If the body is initially stress free, however,then the faces of an ideal crack are immediately adjacent to eachother, and it has been demonstrated that the faces first tend to movetoward each other. Consequently, they will interfere with each other,thus violating the traction-free condition, or they will interpenetrate,which is impossible. One way out of the dilemma is to insist thatthe transient crack face loading is superimposed on some backgroundstress state which holds the crack faces far enough apart to avoidtheir interference. This is an acceptable viewpoint under certaincircumstances, but more can be stated about crack face interaction.Suppose that the problem formulated in Section 3.2.1 is consideredonce again, except that now the crack is initially closed in the interval—/ < x < 0 between the load points and the crack tip. In this interval,either the normal traction on the crack faces is compressive and they-component of displacement is zero, or the traction is zero and thecrack faces displace away from each other. As before, the crack tip isat x = 0, the opposed forces of magnitude p* are applied at x = — /,and the crack faces are traction-free for x < — I. The exact solutionof this modified problem is presented in Section 7.3, but the relevantresults can be stated here. Upon application of the loads at timet = 0, the crack faces initially press against each other. However,the crack faces are separated behind a point traveling at the Rayleighwave speed of the material CR along the crack faces from the loadpoint to the crack tip. The stress intensity factor remains at its initialvalue of zero until this point arrives at the crack tip at time t = 1/CR,whereupon it takes on its appropriate equilibrium value if/(oo). Thus,the main result that the stress intensity factor takes on its equilibriumvalue a short time after the load is applied is independent of whetherthe crack is originally open or closed. The intermediate situationof interference over a part of the interval — / < x < 0 is unresolved.Prom the practical point of view, a crack artificially cut into a materialwill normally be slightly open even in the absence of applied loading,whereas a crack formed by natural growth in a material withoutextensive plastic deformation will be closed, or very nearly so.

Finally, some observations on more general crack face loadingsare made on the basis of the foregoing results. For example, supposethat the crack faces are loaded by a pair of normal forces appliedat x = — / with time dependent magnitude p*h(t), where h(t) is adimensionless differentiate function of time and h(t) = 0 for t < 0.

3.3 Sudden loading of a crack of finite length 117

The stress intensity factor history for this modified problem is then

K?(t) = / Kx{t - s)h(s) ds, (3.2.15)No-

where Kj{t) is understood to be the stress intensity factor for thecase of step loading of constant magnitude p*. Likewise, if a nor-mal traction distribution p*f(l) is suddenly applied over the intervalli < I < l<i on each crack face, where / has the physical dimensionof length"1, then the stress intensity factor history of the modifiedproblem is

K/(t)= I Kj(t;l)f(l)dl, (3.2.16)

where the dependence of Kj(t) on / has been indicated explicitly. Awide range of transient two-dimensional stress intensity factor solu-tions may be constructed on the basis of the results derived in thissection, either directly or by superposition.

3.3 Sudden loading of a crack of finite length

In the previous section, a transient crack loading configuration involv-ing a characteristic length was considered, where the characteristiclength was associated with the traction distribution on the crack faces.A situation in which the configuration of the solid body itself possessesa characteristic length is considered in this section.

Consider an elastic solid containing a planar slit crack of constantwidth I in one direction and of indefinite length in the perpendiculardirection; the body is otherwise unbounded. A right-handed rectan-gular #, y, z coordinate system is introduced so that y = 0 is the planeof the crack, the 2-axis coincides with one of the edges of the crack,and the x-axis is in the direction of prospective advance of that edge.The configuration is invariant under translation in the ^-direction,and a typical section z = constant is shown in Figure 3.3.

The loading condition to be discussed in some detail is the suddenapplication of a spatially uniform pressure of magnitude a* to thecrack faces. For the boundary traction, Tx(x,z,t) = Tz(x,z,t) = 0and Ty(x,z,t) = ±a*H(t), where a* > 0 and H(t) is the unit stepfunction. Thus, the crack faces are traction-free for time t < 0, andthey are subjected to uniform normal pressure for t > 0. The material

a*H(t)

t t t t t t t X

I \ ITTTlFigure 3.3. A crack of finite length / subjected to a suddenly applied crack facepressure of magnitude a* under plane strain conditions.

is stress free and at rest everywhere for t < 0. The wave field resultingfrom this loading produces a state of plane strain deformation at eachpoint in the body.

The statement of the mathematical problem to be solved is sim-ilar to that introduced in Section 2.4. In view of the symmetries ofthe fields, attention can again be limited to the region y > 0. Asolution of the Navier equation (1.2.51) is sought for —oo < x < oo,0 < y < o o , 0 < £ < o o , subject to the condition that the materialis stress free and at rest at time t = 0, and subject to the boundaryconditions

(Tyyfa 0, t) = ~(T*H(t) , - / < X < 0 ,

(7a.y(x,0,t) = 0, - o o < z < o o , (3.3.1)

Uy(x, 0, t) = 0, —oo < x < — / or 0 < x < o o ,

for all time.The form of the boundary conditions (3.3.1) is similar to (2.4.4),

and this observation suggests the Wiener-Hopf method as a means ofsolving the problem. If integral transforms are applied in the man-ner outlined in Section 2.5, a relationship among sectionally analyticfunctions is obtained which is somewhat more complicated in formthan the standard Wiener-Hopf equations introduced in Chapter 2.In principle, the generalized Wiener-Hopf equations can be solvediteratively to obtain the complete solution at any time in the transientloading process. Essentially, each step in the iteration involves the so-lution of a particular semi-infinite crack problem for which the loadingdepends on the results of the previous iterative steps. In practice,however, only the first step in the iteration process has been carried

out for the problem at hand by Thau and Lu (1971), following thework of Kostrov (1964b) and Flitman (1963), and the result is limitedto the time interval 0 < t < 21/c<i, that is, up until a dilatationalwave has traveled the length of the crack twice. An examinationof the physical wave propagation process being simulated suggests analternate approach based on the moving dislocation solution in Section3.2.2. The alternate approach provides a solution that is valid for thesame time range.

Upon sudden application of the crack face pressure, identicalstress wave fields develop around each edge of the crack as though thatedge was the edge of a semi-infinite half plane crack. In particular,the stress intensity factor is precisely the same as that derived for asemi-infinite crack subjected to the same pressure loading up until thetime at which each crack edge becomes aware of the presence of theother edge. This information is communicated by stress waves. Thattime is t = l/cd, the time of arrival of the cylindrical dilatational wavegenerated at the opposite edge at t = 0. Thus, from (2.5.44),

where the superscript (0) indicates that (3.3.2) is the lowest ordercontribution to the total stress intensity factor history, with higherorder contributions corresponding to subsequent reflections of wavesbetween the edges of the crack.

When the dilatational wavefront generated at time t = 0 atthe left edge of the crack x = — I arrives at the right edge of thecrack x = 0, it carries with it a displacement of the surface in they-direction which violates the boundary condition (3.3.1)3. The view-point adopted here is that this surface displacement persists for x > 0and t > l/cd- To satisfy the boundary condition (3.3.1)3, however, anappropriate sequence of dislocations is emitted from the crack edgeat x = 0 in the x-direction to negate the surface displacement.

It turns out that the surface displacement to be negated is alreadyin hand. It is precisely the inverse of the double transform s~3[/_(C),where £/_(£) is given explicitly in (2.5.35)2, with the coordinate xreplaced by x' = - (x + I). In this case, the inversion steps outlinedin Section 2.5.3 yield

(3.3.3)


0.3 -i

o*b

\

0.2-

0.4 0.6 0.8-x' /cdt

Figure 3.4. Normal displacement of the crack face (3.3.3) versus normalizeddistance —x/cdt. The arrival time of the individual wavefronts at the observationdistance is indicated on the abscissa.

where F±(-) is given in (2.5.31) and the other notation is establishedin Section 2.5. The limiting value of the integrand as Im(£) —> 0+ isunderstood in (3.3.3).

The integral in (3.3.3) has been evaluated numerically for a Pois-son ratio of v — 0.3 and the result is shown in Figure 3.4. Forpoints along the crack face beyond the cylindrical dilatational wavegenerated at the crack edge, that is, for \x'\ > Cdt, the surface ismoving under the action of only the applied pressure. The speed ofthe surface particles is a*c2

s/iiCd, and the displacement is cr*c^//xcd.Thus, u_fi/a*cdt = (c8/cd)

2 = 0.286 for v = 0.3. With the arrival ofthe scattered dilatational wave at t = \xf\/c^ the crack faces beginto move apart at an even greater rate than the uniform motion underpressure loading. This seems to be a by-product of the large hydro-static tension developed ahead of the crack tip. Retardation of themotion of the surface begins with the arrival of the Rayleigh wave atan observation point. When the wave motion represented in Figure 3.4reaches the opposite crack edge, it will first incrementally increasethe stress intensity factor there until the arrival of the Rayleigh wave.Thereafter, it will tend to decrease the stress intensity factor.

The displacement u_ in (3.3.3) is a homogeneous function of x1

and t of degree one. However, the dislocation superposition scheme ismost easily applied for homogeneous functions of degree zero. Con-sequently, the particle velocity distribution w_ = du_/dt, which ishomogeneous of degree zero, is canceled. The cancellation is achievedby superposition over a certain fundamental moving dislocation so-lution. In the present instance, the formulation of the fundamentalmoving dislocation problem is the same as that in Section 3.2.2, exceptthat the boundary condition (3.2.5)3 is replaced by

uy(x, 0, t) = u_(x, t) + A (t - x/v)H(vt - x) (3.3.4)

for all x and t > 0. The constant 2A is the magnitude of a particlevelocity discontinuity, or a particle velocity dislocation, growing fromthe crack tip in the re-direction at speed v. This fundamental solutionwas introduced by Preund (1976a) in the analysis of a particular crackarrest problem.

Analysis of the moving velocity dislocation problem could pro-ceed by means of the Wiener-Hopf technique as in Section 3.2.2.There is no need to repeat the analysis, however. The inhomogeneousterm in the present case, which is the second term on the right sideof (3.3.4), is obtained by integrating over time from t = 0 of theinhomogeneous term in the corresponding boundary condition (3.2.3)3in Section 3.2.2. Because both problems are linear, it follows thatthe complete solution in the present case is the time integral of thesolution obtained in Section 3.2.2. In particular, the specific stressintensity factor here for a dislocation of unit magnitude A = 1 is

(3.3.5)

analogous to (3.2.9). Again, the stress intensity factor is identicallyzero for t < 0.

Prom this point onward, the superposition procedure follows thatdeveloped in Section 3.2.3, except that the superposition is over aparticle velocity distribution rather that over a displacement distri-bution. Thus, the stress intensity factor history over the time intervalof interest is given by

#/(1)(*) = - ( M * - T(V); V) ^ ^ dv , (3.3.6)Jed ^

1.5-i

1.0-

0.5-

0.0

equilibrium crack

^ - - ^ ^semi-infinite crack

0.0I

0.5 1.0 1.5

cdt/l2.0 2.5

Figure 3.5. Transient stress intensity factor (3.3.7) versus normalized time .The dashed curve is the corresponding result for a semi-infinite crack. The long-time limit is KI(OQ) = a* yJirl/2, which is the result for the correspondingequilibrium situation.

where ki(t;v) is given in (3.3.5), r(v) = Z/v, and u_ is the timederivative of (3.3.3). If the change of variable rj = 1/v is incorporated,then (3.3.6) becomes

2(7*F+(0)

7T, (3.3.7)

where rf = t/l, which is valid for l/cd < t < 21/cd- The limitingvalue of the integrand as Im(rf) —• 0+ is understood in (3.3.7). Theintegrand has a pole singularity at t] = c, so the integrand must beinterpreted in the Cauchy principal value sense for rf > c.

The integral (3.3.7) has been evaluated numerically for a Poissonratio of v = 0.3, and the net stress intensity factor is shown inFigure 3.5. The transient stress intensity factor has been normalizedwith respect to Kj(oo) = cr*^7rZ/2, which is the equilibrium stressintensity factor for the specified loading. The variation of Kj ' (t) isshown as a dashed curve, and the variation of Kj(t) + Kj \t) isshown as a solid curve. The stress intensity factor history Kj (t)prevails for time in the range 0 < t < l/cd, that is, before any

3.4 Three-dimensional scattering of a pulse by a crack 123

interaction between the edges of the crack occurs and, consequently,the dashed and solid curves are identical in this time range. Uponarrival of the dilatational wavefront from the opposite edge of thecrack, Kj '(t) becomes nonzero and the net stress intensity factorhistory rises above the value for a semi-infinite crack. This outcomewas anticipated in the examination of Figure 3.4, which showed thatthe effect of the dilatational wave is to cause the faces to move awayfrom each other at an even greater rate than is produced by thepressure loading on the free surface. This trend continues until theRayleigh wave arrives at time t = //CR, whereupon the trend isreversed and the net stress intensity factor reaches a peak shortlythereafter. For a Poisson ratio of v = 0.3, the stress intensity factorpeak is 1.30 Kj(oo), or 30 percent above the long-time limit. This is anillustration of the phenomenon of dynamic overshoot. Unfortunately,the stress intensity factor solution is exact only up to time t = 21 fed*That is the time at which the effect of the scattering, at the oppositeedge, of the leading wavefront generated at one edge of the crackreturns to influence fields at its edge of origin. It is evident fromFigure 3.5 that the scattered dilatational wave has little effect onthe local stress intensity factor, however, and it is reasonable toexpect that the present analysis provides the correct maximum stressintensity factor for the process. Numerical solutions due to Chen andWilkins (1976) and Chen and Sih (1977) suggest that, after achievingthe first peak shown in Figure 3.5, the net stress intensity factorhistory experiences oscillations of diminishing amplitude about theeventual limit of Ki(oo).

3.4 Three-dimensional scattering of a pulse by a crack

Up to this point in the study of the transient stress intensity factorsthat arise from the scattering of a plane stress pulse by a crack, it hasbeen assumed that the edge of the half plane crack is parallel to theplane wavefront of the incident pulse. In this section, mathematicalprocedures are generalized to permit consideration of the scatteringof a stress pulse which is obliquely incident on a crack edge in anelastic solid. The general approach to analysis of three-dimensionalproblems of this type was introduced by Freund (1971) in a study ofthe oblique reflection of Rayleigh surface waves from the edge of a halfplane crack. The scattering of plane harmonic waves from the edge

reflected dilatational wave ^ - incident dilatational wavereflected shear wave / /

/ crackplane

Figure 3.6. Schematic representation of a plane dilatational wave obliquely inci-dent on a half plane crack with traction-free faces. The reflected plane dilatationalwave and the reflected plane shear wave are also shown, but the scattered wavesare not included in the diagram. Point A is the point of intersection of the crackedge with the incident plane wave.

of a half plane crack was also considered by Achenbach and Gautesen(1977).

Consider once again the elastic body depicted in Figure 2.1 thatis cut by a half plane crack but that is otherwise unbounded. Inthe present instance, the body is stress free and at rest except forthe influence of the remotely generated stress pulse incident on thecrack. The pulse is scattered and, at each point along the crack edge,transient stress intensity factors are generated. The purpose here isto determine exact stress intensity factor histories for a particularproblem of this type.

A right-handed rectangular coordinate system x,y, z is intro-duced in the body, oriented so that the z-axis coincides with the crackedge, and the half plane crack occupies y = 0, x < 0. Suppose thatthe incident stress pulse carries a jump in tensile stress of magnitudeGinc in the direction of the normal to the pulse front. Consequently,the pulse is dilatational and it travels with the characteristic speed Cd-In addition, suppose that the unit vector normal to the pulse frontin the direction of propagation has components (0, — sin#, — cos0).Thus, the x-axis is parallel to the pulse front, and 0 is the acute

angle between the negative z-axis and the normal to the front; seeFigure 3.6. Finally, suppose that the pulse front reaches the origin(0,0,0) at time t = 0. Under these conditions, the pulse front is in theplane cjt + ysinO + zcosO = 0. The fact that the subsequent analysisis limited to the case when the pulse front is parallel to the x-axisdoes not reflect a limitation of the analytical procedure. The mainconsequence of the limitation is that the discussion of the analyticalapproach is not obscured by the algebraic complexity of the case ofgeneral oblique incidence.

The diffraction process, which is assumed to have become steadyin the ^-direction, leads to a fairly complicated pattern of wavefronts.For x < 0, the incident dilatational pulse is geometrically reflectedfrom the traction-free crack surface as a combination of a plane di-latational pulse and a plane shear pulse, both of which are shownin Figure 3.6. For x > 0, on the other hand, the incident pulsepropagates past the plane y = 0 without modification. The movementof the intersection point A of the incident pulse front with the crackedge is determined as the motion of the intersection point of thethree planes c^t + ysin9 -I- zcos# = 0, x = 0, and y = 0, that is,cat + z cos 9 = 0. Thus, this point moves with the apparent speedv = Cdl cos 9 along the crack edge and its instantaneous coordinatesare (0,0, —e^t/costf). A scattered field trailing this point (not shownin Figure 3.6) is established around the crack edge. The wavefrontsin the scattered field include a conical dilatational wavefront and aconical shear wavefront, both with apex at (0,0, — Cdt/cosO). Alsoincluded is a pair of plane head wavefronts. These head wavefrontsintersect the dilatational conical wavefronts on the surfaces y = ±0,and they extend into the material from each surface, terminating ontheir lines of tangency with the conical shear wavefronts.

The same sort of superposition arguments discussed in Sections2.5 and 3.1 can be applied here to convert the problem from oneinvolving a wave incident on a traction-free crack to one involvingcrack face tractions but no other loading. In the present case, if thecrack were absent, then the incident wave would induce a normalcomponent of stress a* = ainc{sm2 9 + v cos2 9/(I — v)} and a z-component of shear stress r* = (Tinc(l — 2v) sin9 cos 9/(1 — u) overthe part of the crack plane y = 0 for which c^t + z cos 9 > 0. Thestress intensity factor histories can then be determined by analyzingthe situation in which equal but opposite tractions are applied to thefaces of the crack. The application of the normal stress produces a

transient mode I stress intensity factor for each z, and the shear stressproduces a mode III stress intensity factor for each z. The two modesare analyzed in the same way, and the steps leading to the mode Istress intensity factor history are outlined here. The correspondingresult for the mode III stress intensity factor history is stated withoutderivation.

If the same normal pressure distribution is applied to both facesof the crack, then the wave fields are expected to have reflectivesymmetry with respect to the plane y = 0, and attention can belimited to the region y > 0. The boundary conditions imposed onthe half space y > 0 for x > 0 are conditions that result from thesymmetry of the fields under reflection in the plane y = 0, namely,axy(x,0,z,t) = ayz(x,0,z,t) = 0 and Uy(x,Q,z,t) = 0.

The Helmholtz representation of the displacement vector in termsof the scalar dilatational potential (j>{x, y, z, t) and the vector shearpotential t/?(rc,y, z,£) is introduced as in (1.2.60). In general, (f> andeach rectangular component of tp must satisfy a scalar wave equationin three space dimensions and time. In addition, it is required that thevector potential t/> must be divergence free. In any particular planez = constant, the material is stress free and at rest until time t =—z/v. That is, because the applied loads are moving supersonicallyon the crack faces in the negative ^-direction, no wave motion reachespoints in the plane z = constant before the moving load step reachesthat plane.

It is now observed that the applied loading (which is the onlyinhomogeneous term in the mathematical formulation) depends onz and t only through the combination r = t + z/v. If the scat-tered fields have become steady, then <f> and rj) depend on z andt only through r, as well. Consequently, 0(x,i/,z,t) = (/>(#, ?/,T)and xl>(x,y,z,t) = -0(x, y, r). No confusion should result from thefact that obviously different functions have been identified by thesame symbol. For steady fields, the wave equations governing thedisplacement potentials, or their components, take the form

' ( 2 - 9 J Q_9 ~~ 0 5

(3.4.1)di> (1 1 \ d2^

8x2 dy2 \c2 v2) 8T2

in the half plane y > 0. The condition that %j> must be divergence freebecomes

«*• + « & + !«*« = 0 (3.4.2)ox ay v or

in y > 0, where tyx^v* ifrz are the rectangular components of \j>. Theconditions that the material is stress free and at rest in any planez = constant until the moving load step penetrates that plane takethe form

*foy,0) = |£(s,y,0) = 0, ^(*ly,0) = ^(x,»,0) = 0 (3.4.3)

in y > 0.The boundary value problem represented by the differential equa-

tions (3.4.1) and (3.4.2), the initial conditions (3.4.3), and the bound-ary conditions is similar to the plane strain problem formulated inSection 2.4. There are four unknown dependent variables here, as op-posed to two in the case of plane strain, and the timelike independentvariable r has replaced time t. Furthermore, the inverse characteristicwave speeds c^2 and c~2 in Section 2.4 are replaced here by (c~[2—v~2)and (cj2 — v~2), respectively. There are no essential differences in thetwo formulations, however, and the approach developed in Section2.5 based on the Wiener-Hopf method can be applied here withoutmodification.

The first and last boundary conditions equivalent to (2.5.1)i and(2.5.1)3 are rewritten in a form that is valid over the full range of a:,

<?yy(*, 0, r) = a+(x, r) - a*H(r)H(-x),(3.4.4)

) ()

for —00 < x < 00 and 0 < r. As before, a+ is the unknown normaltraction distribution on x > 0 and a+ = 0 for x < 0. Likewise, w_ isthe unknown displacement in the ^/-direction on x < 0 and u_ = 0 forx > 0. The one-sided Laplace transform with transform parameters is first applied to suppress dependence on r. Next, the two-sidedLaplace transform with parameter s( is applied over x. The boundedsolutions of the resulting ordinary differential equations in y for thedoubly transformed potentials are

,«) = ~P{0e-sav, *K,V,«) = ^Q(C)e-8/3y, (3.4.5)s s

where a and /3 are defined by

a(C, d) = (a2 - d2 - C2)1/2, /?(C, «0 = (62 - d2 - ?)1/2, (3.4.6)

and a = l/cd, b = l/c5, and d = 1/v. Note that d < a for anyangle of incidence 0 > 0. Branch cuts are introduced and branchesare selected so that Re(a) > 0 and Re(/3) > 0 in the entire £-plane.The condition (3.4.2) that the vector potential must be divergencefree takes the form

CQx-PQy + dQ2 = 0, (3.4.7)

where the functions Qx, Qy, Qz are the rectangular components of thevector function Q. At this point, the functions P and Q are unknownand, as in Section 2.5, the coefficients of these functions have beenselected in (3.4.5) so that the functions themselves will not depend ons.

If the Laplace transformed boundary conditions (3.4.4) are im-posed on (3.4.5), the result is a system of five linear equations for thesix unknown functions P, Qx, Qy, Qz, [/_, and £+, where [/_ and £+have precisely the same interpretation here as in (2.5.8). If P, Qx,Qy, and Qz are eliminated, the remaining equation is a Wiener-Hopfequation of the standard type,

where

R({, d) = 4(C2 + d2)a(C, d)p({, d) + (b2 - 2d2 - 2C2)2 . (3.4.9)

The function defined in (3.4.9) is recognized as a modified form ofthe Rayleigh wave function. In particular, it possesses only two zerosin the complex £-plane at £ = ±y/c2 — d2, where c is defined byR(±c, 0) = 0. Note that (3.4.9) is identical to (2.5.12) when d -> 0 (orv -> oo). The strip of analyticity of (3.4.8) is -Va 2 - d2 < Re(C) < 0.

The remaining steps in solving the Wiener-Hopf equation andextracting the transient stress intensity factor history are identicalto those followed in Section 2.5. Therefore, these steps need not be

repeated here, and the stress intensity factor analogous to (2.5.44) isstated without further calculation. In terms of the apparent speed vof the step load moving along the crack faces,

Kl(t) = 4<x*

for any fixed value of z, where a* = crinc{sin20 + */cos2 0/(1 — v)}and v = c<i/cos 9. It can be verified that when the stress pulse isnormally incident on the crack plane, or 9 = TT/2, the expressionin (3.4.10) is exactly the same as that in (2.5.44). The situationof grazing incidence, when the apparent speed of the incident wavealong the crack face approaches the dilatational wave speed v =Cd or when 9 = 0, is a degenerate case. For this situation, thestrip of analyticity of the Wiener-Hopf equation vanishes and thesolution procedure breaks down. That this case could lead to somedifficulty could have been anticipated from the theory of geometricalreflection of plane waves from a traction-free surface. For the caseof grazing incidence of a dilatational pulse, the reflected dilatationalpulse cancels the incident pulse, and the reflected shear pulse has zeroamplitude. This dilemma arises from the combination of assumptionsthat the reflected field has become steady and that the excitationmoves exactly at the dilatational wave speed. In fact, if the loadstep on the traction-free surface moves at the dilatational wave speedQ, it is impossible for dilatational radiation from the applied loadto advance into the material to establish a steady reflected planedilatational wave. The resolution of this dilemma must be soughtin the study of the phenomenon of grazing incidence as a transientprocess (Wright 1968; Freund and Phillips 1968).

The mode III stress intensity factor history resulting from thenegation of the shear stress of magnitude r* induced on the crackplane by the incident wave is briefly considered. In this case, thefields are antisymmetric under reflection in the plane y = 0, andagain attention can be limited to the region y > 0. The boundaryconditions imposed on the half space are

<ryz(x, 0, z, t) = -r*H(t + z/v), -oo < x < 0,(3.4.11)

uz(x, 0, z, t) = 0, 0 < x < oo ,

130 3. Farther results for a stationary crack

for —oo < z < oo and 0 < £, where v = cj/ cos9. The condition onx > 0 results from antisymmetry of the fields under reflection in theplane y = 0.

Although the steps in the derivation of the transient stress inten-sity factor history are not included here, the Wiener-Hopf equationitself is presented to illustrate an interesting feature that is not ob-served in two-dimensional problems of the same general type. Withthe functions C/_ and T+ interpreted as in Section 2.6, the Wiener-Hopf equation is

where (3 is defined in (3.4.6), R is defined in (3.4.9), and

W((, d) = 4C2a/?+2d2(62 - d2 - C2)(3.4.13)

+ ( 6 2 - 2 d 2 - 2 C 2 ) ( & 2 - d 2 - 2 C 2 ) .

The function W has the interesting properties that W((, d)/R((1 d) —*1 as |C| -> oo for any admissible value of d, W(C, 0)/i?(C, 0) = 1for any value of £? and W(C,d) = 0 has two and only two rootsin the complex C-plane, with both roots real. The interpretation ofthese properties in terms of the wave propagation process follows. Forvalues of d = 1/v greater than zero, the applied shear loading givesrise to a surface wave traveling obliquely away from the crack edgeon each crack face. In this case, the zeros of R provide the polesof the sectionally analytic functions corresponding to surface waves.In the limit as v —• oo or d -* 0, the fields reduce to the simplerantiplane shear fields considered in Sections 2.2 and 2.3. In this case,no surface waves are present. The role of the function W is to providean automatic switch for eliminating the Rayleigh wave function fromthe formulation when d -» 0.

If the Wiener-Hopf equation (3.4.12) is solved and the stressintensity factor is extracted following the procedures developed inSection 2.5, the result is

3.5 Three-dimensional stress intensity factors 131

for any fixed value of z, where

v = - ^ r and r* = ainc (Lz^H ) sin0cos0. (3.4.15)cos 0 \ 1 - v )

It can be verified that the expression in (3.4.14) reduces to the corre-sponding two-dimensional mode III result in (2.3.18) in the limit asv —> oo for a fixed value of r*.

3.5 Three-dimensional stress intensity factors

The transient mechanical fields described in Section 3.4 are three-dimensional fields in the sense that the components of stress and de-formation are functions of all three spatial coordinates at any instant.Because of the assumption that the fields have become steady withrespect to an observation frame moving in the negative ^-direction atspeed w, however, the mathematical formulation of the stress analysisproblem could be cast in a form indistinguishable from the formulationof an equivalent two-dimensional problem. Three-dimensional phe-nomena have received relatively little attention in dynamic fracturemechanics, although certain results obtained in the study of self-similar expansion of flat circular or elliptic cracks will be noted inSection 6.3. In this section, a procedure is described for determiningstress intensity factor histories for a class of three-dimensional elas-todynamic crack problems. The geometrical configuration studiedis again a half plane crack in an otherwise unbounded solid, withthe crack faces subjected to tractions that result in variation of thetransient stress intensity factors along the crack edge.

Consider again the elastic body containing a half plane crackdepicted in Figure 2.1. The body is initially stress free and at rest.At a certain instant, tractions begin to act on the crack faces, resultingin a three-dimensional stress wave field in the material. The purposehere is to determine exact expressions for stress intensity factors asfunctions of time and position along the crack edge. To illustrate theapproach, the same transient normal pressure distribution is appliedto each crack face, and the tangential or shear traction componentsare zero. For this type of loading, the mode of deformation is mode Ifor each point along the crack edge. Next, a mathematical formulationof a problem corresponding to the foregoing qualitative description isgiven.

A right-handed rectangular coordinate system is introduced inthe body, oriented so that the z-axis coincides with the crack edge,and the half plane crack occupies y = 0 for x < 0. The normal tractionon the crack faces begins to act at time t = 0. For the time being,consider some general traction variation on y = 0*, say Ty(x,z,t) =^a_(xy z,t), where a_(a;, z,t) is a given function of position on thefaces and of time and <r_ > 0 corresponds to tensile traction. It isassumed that there exists a distance, say zo, such that cr_ is zero for\z\ > zQ, The subscript (-) carries the same interpretation as it did inthe introduction of the Wiener-Hopf method in Section 2.5. The othercomponents of traction are zero, that is, Tx{x,z,t) = Tz(x,z,t) = 0.

In view of the symmetry of the configuration and of the appliedtraction, the wave fields are expected to have reflective symmetry withrespect to the plane y = 0, and attention can be limited to the regiony > 0. The displacement fields satisfy ux(x,—y,z,t) = ux(x,?/,2,£),uy(x,-y,z,t) = -Uy(x,y,z,t), and uz(x,-y,z,t) = uz(x,y,z,t).These, in turn, imply the conditions (Jxy(x,0, z, t) = cryz(x, 0, z, t) = 0and uy (#, 0, z,t) = 0 for x > 0 and for all z, t. Thus, the complete setof boundary conditions to be satisfied by the stress wave fields is

CTyy(x, 0, Z, t) = G_ (x, 2, t) + (7+ (x, Z, t) ,

<Txy(x,0,Z,t) = 0,(3.5.1)

ayz(x,0, z, t) = 0,

Uy(Xj 0, Zj t) = U_ (X, Z, t)

for —oo < x,z < oo and t > 0. The definition of the function cr_,which describes the imposed traction, is extended so that a_ = 0 forall x > 0. As in the study of two-dimensional problems, a+ is theunknown normal component of stress ayy on x > 0 and a+ — 0 forx < 0. Likewise, u_ is the unknown y-component of displacementof material particles on the crack faces for x < 0 and ix_ = 0 for allx > 0 .

The Helmholtz representation of the displacement vector u =V0 + V x tp is adopted, as introduced in Section 1.2.3, where V isthe three-dimensional gradient operator, <j> is the scalar dilatationalwave potential, and rj> is the vector shear wave potential. The wavepotential functions are governed by the partial differential equations

c2dV

2cf> - 0,« = 0, cs2VV - +,u = 0, V • ^ = 0 (3.5.2)

in y > 0. The conditions that the material is stress free and at resteverywhere for t < 0 are expressed in terms of the potential functions

* = ^ = 0 , </> = ^ = 0 (3.5.3)

at t = 0 for all points in y > 0. Finally, it is noted that theboundary conditions (3.5.1) are interpreted as though the stress anddisplacement components were replaced by their expressions in termsof the potentials 0 and i/>.

The mathematical boundary value problem described by the dif-ferential equations (3.5.2), the boundary conditions (3.5.1), and theinitial conditions (3.5.3) is linear, and integral transforms, along withcertain powerful theorems from the theory of analytic functions ofa complex variable, are again used to extract stress intensity factorhistories. In most respects, the procedure differs very little fromthat introduced in Section 2.5 for two-dimensional analysis. Certaininterpretations of the transformed fields at key points in the analysis,however, make it possible to extract exact results for this class ofthree-dimensional problems. The first step is to apply the one-sidedLaplace transform on time (1.3.7) to the differential equations (3.5.2)and the boundary conditions (3.5.1), taking into account the initialconditions (3.5.3). As before, the transform parameter is s and thetransform of any function, say </>(x, y, z, t), is denoted by a superposedhat (/>(x,y,z,s). The parameter s is considered to be a positive realparameter for the time being. Next, a two sided Laplace transformis applied to suppress the dependence on z. The complex transformparameter is s£, and the transform of any function, say 0(x, y, z, s), isdenoted by the corresponding upper case symbol with a superposedhat

/ 0(x, », z, s) e-8** dz . (3.5.4)

Prom the condition that a_ vanishes for \z\ > zo, it can beexpected that

£(*, ?/, z,s) =

as2;-^ ±00 for any e > 0. This condition, in turn, implies that thetransform integral in (3.5.4) converges for —a < Re(£) < a, wherea = 1/crf. Consequently, the integral defines an analytic function in

this strip of the complex £-plane. Prom the property of uniquenessof an analytic function in its domain of analyticity noted in Section1.3.1, the analytic function in the strip is completely specified bythe function defined only on the portion of the real axis in the strip—a < Re(£) < a. Furthermore, it turns out to be of great advantageto restrict the range of £ to this real interval in proceeding with theanalysis. At an appropriate later stage, the definition of the functionsof £ can be extended to the entire complex £-plane by invoking therelevant theorems concerning uniqueness of analytic functions andanalytic continuation from Section 1.3.

Finally, the two-sided Laplace transform that suppresses depen-dence on x is applied. The complex transform parameter is s(, a n dthe transform of any function, say $(x, y, £, s), is denoted by the sameupper case symbol but without the superposed hat, $(£, y, £, s). To beable to continue with the solution procedure, some statement on thedomain of convergence of the transform integrals over x is required atthis point. If the applied loading is nonzero for indefinitely large dis-tances from the crack edge along the crack faces, then the argumentsmade in Section 2.5 can be invoked here, as well, to conclude thatthe integral over negative values of x in the definition of $(£,y,£,s)will converge for Re(£) < 0. For x > 0, the wave fields are zerobeyond a cylindrical wavefront expanding from the y-axis at speed c<*with instantaneous radius zo+y/x2 + z2. To be more specific, considerthe elementary wave field <f>(x, y, z, t) = H(cdt — zo — y/x2 + z2), whereH(-) denotes the unit step function. If the Laplace transform integralsover £, then z, and then x are formed, it is easily demonstrated thatthe triple integral converges if {Re(0} + £2 < a2. This condition,when coupled with the convergence condition Re(C) < 0, provides abasis for expecting that all transforms will be analytic in the strip— \Ja2 — £2 < Re(£) < 0 in the complex ("-plane, with £ real and inthe range —a < £ < a.

It is not possible to consider completely arbitrary distributionsof traction a_(#,z,£) applied to the crack faces. From this pointonward, attention is limited to those distributions for which the tripletransform has the form

J-4E-(C>0, (3.5.6)

-00 J-00 s

where m is a real number and S_(C,£) does not depend on 5. The

condition (3.5.6) essentially restricts a_(x,z,t) to be within a class ofcertain separable or homogeneous functions of x, z, t.

The steps in the solution procedure now follow those outlined inSection 2.5, although the algebraic details are somewhat more com-plicated here. The transforms are applied to the partial differentialequations (3.5.2) to reduce them to ordinary differential equationsin y. The transformed boundary conditions (3.5.1) are imposed onthe solutions of the ordinary differential equations to determine theunknown parameters of integration. As usual in a problem of thisgeneral type, there are more unknown functions than there are alge-braic equations to determine them. However, certain of the unknownfunctions are sectionally analytic functions (of C in this case), and theWiener-Hopf factorization makes it possible to complete the solutionby determining two unknown functions from a single equation. Onlya few of the intermediate steps are included here.

The solutions of the ordinary differential equations obtained byapplication of transforms to the partial differential equations that arebounded as y —• oo are

0 e-Say> * = ^T2 Q(C 0 « - * , (3.5.7)

where the parameters of integration P and Q are unknown functionsof their arguments, the number m follows from (3.5.6), and

a(C,0 = (a2 ~e- C2)1/2 , /?(C,0 = (b2 -e~ C2)1/2- (3.5.8)

The complex ("-plane is cut along y/a2 — £2 < |Re(C)| < oo, Im(£) =0, so that Re(a) > 0 in the entire cut C-plane for each value of £, andlikewise for /?. The transformation of the condition V • -0 = 0 yields

(3.5.9)

where Qx, Qy, Qz are the rectangular components of Q. The rela-tion (3.5.9) is a linear algebraic equation for the unknown functions.The Laplace transformation of the boundary conditions provides fouradditional linear equations for the six unknown functions P, Qx, Qyy

Qz , £/_, and £+, where

U.(C,O = sm+1 (^ f°° u_(x,z,s)eJ — OO J — OO

(3.5.10)/*OO /*OO

J—oo J—oo

These four equations are

(b2 - 2C2 - 2^2)P - 2^/3^,, + 2C/3Q, = S " + E +

y + (b2 - e ~ 2C2)QZ = 0,

2£aP + (b2 - 2i2 - C2)QX + CPQy + tCQz = 0,

If P , Qx, Qy, and Qz are eliminated from the five equations (3.5.9) and(3.5.11), the result is a single equation involving the two remainingunknown functions U_ and S+, namely,

(3-5.12)

where £_(£, ^) is given, a is defined in (3.5.8), and R is defined in(3.4.9). Furthermore, £+ is analytic in Re(£) > — y/a2 — £2 and E/_is analytic in Re(C) < 0. The equation (3.5.12) holds in the strip-y/a2 - £ 2 < Re(C) < 0. Thus, for any fixed value of £, (3.5.12) canbe solved by factorization in much the same way that an equation ofthe standard Wiener-Hopf type is solved.

To make further progress, it is necessary to choose a particu-lar applied traction distribution. A convenient case for purposes ofdemonstrating the general approach is that of a uniform line loadwhich suddenly begins to act along a line perpendicular to the crackedge. Thus, it is assumed that

<r_ (x, z, t) = -p*6(z)H(-x)H(t), (3.5.13)

where H(-) is the unit step function and <$(•) is the Dirac delta func-tion. The amplitude p* has physical dimensions of force/length, andp* > 0 corresponds to traction that tends to separate the crackfaces. The ensemble of wavefronts that results from application ofthe traction specified by (3.5.13) includes a variety of space surfaces.Among the wavefronts are cylindrical dilatational and shear wave-fronts centered on the load lines in x < 0, plus planar head wavefrontsthat intersect the dilatational wavefronts on the surfaces y = ±0 andterminate at the line of tangency with the cylindrical shear wavefronts.There are also spherical dilatational and shear wavefronts centered atthe origin of the coordinates, plus conical headwaves, each with itsapex at a point where the spherical dilatational wave meets the z-axis and its terminus along a circle of tangency with the sphericalshear wavefront. There are also conical headwaves that intersect thespherical wavefront on the surfaces y = ±0 for x < 0 and extend tocircles of tangency with the spherical shear wave.

If the relevant integral transforms indicated in (3.5.10) are ap-plied to (3.5.13), it is found that m = 2 and that

S_(C,O = ^ . (3.5.14)

Consequently, the relationship between the sectionally analytic func-tions stated in (3.5.12) is virtually identical to the relationship givenin (2.5.11), except for the parametric dependence on £ in the presentcase. Though the dependence on £ must be taken into account inthe solution procedure, it requires no special consideration, and thesolution for £+ (C>£) c a n be written immediately as

E (C£)-P* (F+<-°>®

where

(3.5.16)


and

F±(C,0 = , ) ^ • (3.5.17)

If the development in Section 2.5 is followed one step further, it isconcluded that the double transform of Kj(t, z), the variation of stressintensity factor history along the edge of the crack, is

where K = 2(62 — a2), as before. Note that the notational conventionon naming Laplace transforms is not followed for the stress intensityfactor itself. In any case, the procedure for inversion of a doubletransform like that in (3.5.18) has been described in detail in Section2.5.3. The result of applying this procedure to (3.5.18) is (Preund1987a)

(3.5.19)

for z > 0, and Ki(t,—z) = Ki{t,z). The limiting value of theintegrand as Im(£) —• 0+ is understood in (3.5.19).

The integral in (3.5.19) cannot be evaluated in terms of elemen-tary functions. It can be evaluated numerically, however, and theresult of numerical evaluation for a Poisson ratio of v = 0.3 is shownin Figure 3.7. Knowing the asymptotic behavior of the generalizedRayleigh wave function, namely, that i?(0,£) —> — K£2 as £ —• +oo,the integral (3.5.19) implies that

lim #/(*, z) = Kj(oo, z) = - 7 = (3.5.20)*—•oo y/'KZ

for any fixed z > 0. The result of the limiting process is consistentwith the independently calculated equilibrium stress intensity factordistribution along the crack edge for this configuration (Tada et aL

1.0 -i

8 0.5-

N 0.0-

-0.5

-cRt/z =

1.0 2.0 3.0 4.0 5.0 6.0 7.0

Cdl/Z

Figure 3.7. The stress intensity factor (3.5.19) versus the dimensionless time Cdt/z.The singularity corresponds to the arrival of the Rayleigh wave. The long-timelimiting value in this case is Kj (oo, z) = p*//¥~

1985). The stress intensity factor history in Figure 3.7 for any fixedvalue of z is normalized by its long-time limiting value Kj (oo, z).

The general features of Kj(t,z) derive from the nature of thewave fields. For any value of z > 0, the stress intensity factor is zeroup until the arrival of the first dilatational wave at time t = z/cd-Upon sudden application of the compressive line loads on the facesof the crack, the initial response is dilatational and the surfaces atpoints adjacent to the line loads tend to bulge outward. The firstwave arriving at the observation point along the crack edge carriesthis outward bulge. The crack faces tend to move toward each other,and this feature is reflected in the tendency for the stress intensityfactor to become negative following arrival of the leading dilatationalwave. This effect persists until the arrival of the Rayleigh wave attime t = Z/CR (C<I/CR = 2.02 for v = 0.3). The stress intensityfactor history is logarithmically singular at this instant, and it tendsto increase thereafter. These features are qualitatively similar to thoseobserved for the two-dimensional configuration analyzed in Section3.2. The transient stress intensity factor history Kj(t,z) approachesits long-time limit Kj (oo, z) quite slowly, with the ratio of the formerto the latter being about 0.81 at Cdt/z = 20 and about 0.92 at cjt/z =


100. The rate of approach to the long-time limit reflects the timerequired for contributions from more distant segments of the line loadto propagate to the observation point on the crack edge.

This completes the analysis of the three-dimensional stress in-tensity factor history for the particular applied traction distribution(3.5.13). Results could be derived for a number of other tractiondistributions in a similar way. This analysis opens the way for theexamination of other three-dimensional phenomena, but few suchproblems have been considered. A three-dimensional configuration ofparticular interest is that of a pair of opposed collinear concentratedloads acting on the crack faces at a fixed distance from the crack edge.This problem involves a characteristic length, and a solution is not yetavailable. An application of the method to a growing crack problemis discussed by Champion (1988).

3.6 Fracture initiation due to dynamic loading

In the foregoing sections, elastodynamic analysis of cracked solidsis described which leads to the dependence of the time history ofthe crack tip stress intensity factor on the applied loading and cracklength. The analysis was motivated in Section 2.1 by the observationthat the stress intensity factor provides a one-parameter representa-tion of the mechanical load level on material near the edge of thecrack for each mode of crack opening, provided that certain lengthrequirements are met. Thus, the stress intensity factor concept pro-vides a basis for quantifying the resistance of materials to the onset ofgrowth of a preexisting crack, as well as for predicting the initiationof fracture in a cracked, nominally elastic structure.

3.6.1 The Irwin criterion

The engineering science of linear elastic fracture mechanics (LEFM),which has evolved from the notion of the stress intensity factor as afield characterizing parameter, has been profitably extended to situ-ations in which material inertia plays a significant role. Given thesignificance of the stress intensity factor as a single characterizingparameter for each mode of crack opening, a simple criterion for theonset of crack growth is the following: A crack will begin to extendwhen the stress intensity factor has been increased to a materialspecific value, usually called the fracture toughness of the material and

3.6 Fracture initiation due to dynamic loading 141

commonly denoted by Kic for mode I plane strain deformation. Forvalues of the stress intensity factor smaller than the critical value thereis no growth, and values larger than the critical value are inaccessible.This is the Irwin criterion of LEFM in its simplest form. It shouldbe noted that such a criterion is a physical postulate on materialresponse, on the same level as the stress-strain relation or otherphysical postulates on which the mathematical formulation is based.In the statement of this criterion, it should be emphasized that Kjc

is a material parameter and that Kj(t) is a feature of the stressfield. The American Society for Testing and Materials has adopted astandard procedure for establishing the fracture toughness of a metal(Annual Book of Standards, Part 10, Section E399); according tothe specification in this standard, the notation Kic is reserved foran inferred measure of fracture resistance only when certain testingconditions are met.

The foregoing statement of the Irwin criterion may be appliedwithout modification to the study of fracture initiation in nominallyelastic bodies subjected to stress wave loading. That is, if a stressintensity factor field exists under rapidly applied loading then it canbe postulated that crack growth will begin if the value of the stressintensity factor is increased to a certain value. This does not implythat the value of dynamic fracture toughness will be independentof the rate of loading or that dynamic effects do not influence theinferred value of fracture resistance in other ways. Several casesof dynamic loading are considered in this section. The question ofwhether or not the postulated criterion is valid for a range of materialsand/or dynamic testing conditions can only be established throughexperiment.

3.6.2 Qualitative observations

Attention is limited to mode I deformation for the time being. Thesize of the crack is viewed as the physical feature of the configurationof prime importance, and this feature is characterized by the lengthI. The transient applied loading is characterized by some appliedtraction or incident stress wave magnitude a* and some temporalduration t* of the applied loading. The bulk response of the isotropicelastic material is characterized by the Poisson ratio v and by a wavespeed, say the Rayleigh wave speed CR. The fracture resistance isspecified by the fracture toughness Kjc. A complete system is shownschematically in Figure 3.8. The way in which the system responds

t*

t*Figure 3.8. Schematic diagram of a crack of finite length loaded by means of arectangular stress pulse under plane strain conditions. On the right, the stressintensity factor response is indicated qualitatively for the case when the pulseduration is very long or very short compared to a wave transit time along thecrack length.

depends on the relative magnitudes of the system parameters, andthis point is pursued on a qualitative basis in an attempt to achievesome degree of unification in this subsection.

For definiteness, consider a rectangular loading pulse of arbitrarybut fixed duration £*. Suppose that, for a given crack length /, thepulse magnitude a* = cr* is the smallest magnitude that will resultin the Irwin criterion Kj(t) = Kjc being satisfied at some instant.Attention is focused on the relationship between a* and I in thedimensionless form of a*y/l/Kjc versus t*cji/L First, consider thecase t*cjill >> 1, that is, the pulse is long in duration compared tothe transit time of Rayleigh waves over the distance I. The maximumstress intensity factor achieved during the dynamic loading processis about 1.3<T*y 7rZ/2 according to the analysis in Section 3.3. Con-sequently, if the pulse amplitude a* is such that a*y/l/Kjc < 0.6,the initiation criterion cannot be satisfied and fracture will not beinitiated. Thus, for t*CR/l > 1, it appears that a*c\fljKic = 0.6 isthe critical value of the stress pulse amplitude. The critical stresslevel for initiation varies as 1/V7, and the dimensionless amplitude isindependent of £*CR/Z. The criterion yields the same result, except forthe numerical factor, as if it were applied for an equilibrium situation.The result is shown as the horizontal line in Figure 3.9.

Consider next the other extreme situation when the durationof the loading stress pulse is short compared to the transit time forRayleigh waves over the distance /, that is, t*CR,/l <C 1. The maximum

longcrack "*"""response

short[—** crack

response

t*CR /I

Figure 3.9. Schematic diagram of the normalized stress pulse amplitude a*required to satisfy a stress intensity factor fracture criterion versus stress pulseduration t*.

stress intensity factor which can be achieved under these conditionsis approximately 1.6 <r*y/t*CR according to the analysis in Sections2.5 and 2.7. Consequently, if the pulse amplitude cr* is such that(T*y/l/Kic < 0.6 y/l/t*CR, the initiation criterion cannot be satisfiedand fracture will not initiate. (The common factor \fi is not canceledin this expression so that key dimensionless parameter groups canbe retained for establishing important qualitative results.) Thus, fort*CR/l < 1, it appears that a*y/l/Kic = 0.6 y/l/t*CR is the criticalvalue of the stress pulse amplitude for a given value of t*CR/l. Thisresult is also shown in Figure 3.9 as the curved line. A solid curvecould easily be added to the sketch, providing a smooth transitionbetween the two extreme cases considered. No particular significanceshould be attached to the precise values of numerical coefficients in theexpressions for the horizontal and vertical lines in Figure 3.9, althoughit is noteworthy that they are both of order unity. If the extension of athree-dimensional crack due to stress wave loading were to be studiedin the same way, it is expected that the results would be qualitativelysimilar. The coefficients would surely have different values, but it isexpected that they would also be of order unity.

Recall that pulse duration t* is fixed at some arbitrary valuein the diagram in Figure 3.8. Suppose that the curve separatingthe growth conditions from the no-growth conditions in the planeof a*y/l/Kjc versus FCR/1 is considered for two distinct values of£*, with the parameters / and Kjc held fixed. Then the transition

point from one type of response to the other type (the vertical dashedline in Figure 3.9) would be further to the left for the larger of thetwo values of £*, provided that the abscissa scales were identical. Togive some impression of the magnitudes of the physical quantitiesinvolved in this discussion, consider a crack lcm long in a brittleplastic material for which CR = 1000 m/s and Kjc = 1 MPa^/m. Thetransition pulse duration t\, defined as the value of pulse durationthat satisfies t\cRJl = 1, is then t\ — 1/CR = 10/is, and the plateaucritical stress level is a* = 0.6Kic/\/l = 6MPa.

3.6.3 Experimental results

An experimental study of fracture initiation under stress wave loadingconditions was reported by Kalthoff and Shockey (1977) and Shockeyet al. (1983). The material selected was a brittle epoxy with a fracturetoughness under slow loading conditions of Kic = l.lMPa-v/in. Thedilatational wave speed for the material was cj = 2.6 x 103m/s.Shallow cylindrical disk specimens, 50 mm in diameter and 9 mmthick, were cast with a distribution of small circular sheets of mylarembedded within each specimen. Five sizes of the mylar sheets withdiameters I ranging from 0.4 mm to 12.7 mm were used. The tensilestrength of the epoxy-mylar interface was measured to be less than5 percent of the tensile strength of the epoxy. The specimens wereloaded in a plate impact gas gun apparatus. The loading produced acompressive wave which propagated as a uniform plane pulse throughthe specimen, and then reflected from the traction-free back face as atensile pulse of duration 2.04/is. The amplitude of the tensile pulsewas varied by varying the impact loading velocity, resulting in therange 14.9 MPa < a* < 33.7 MPa. The mylar disks were viewed aspenny-shaped cracks, and the experimental situation is as suggested inthe two-dimensional schematic diagram in Figure 3.8. The spacing ofthe disks was such that there was no interaction between them duringapplication of the loading pulse, and each acted like a penny-shapedcrack in an unbounded solid during the time interval of interest.

The epoxy was transparent, allowing visual examination of thespecimens after loading to determine whether or not each embeddeddisk/crack experienced growth. The data based on such examinationare summarized in Figure 3.10. These data are taken directly fromthe work by Shockey et al. (1983) and are replotted in terms ofthe dimensionless variables introduced in Figure 3.9. The dashedcurve is an estimate of the locus of points for which a* = <r*, based


•>*

b

4.0 -i

3.0-

2.0-

1.0-

0.00.0 1.0 2.0 3.0 4.0

cRt*/lFigure 3.10. Data on fracture initiation in epoxy due to short stress pulse loadingreported by Shockey et al. (1983). The filled symbols indicate that crack growthwas observed, whereas the open symbols represent cases of no growth. The curvesrepresent possible interpretations of the result of Figure 3.9 for this experiment.

on the slow loading fracture toughness value of LlMPa-^/in. Thehorizontal portion is determined from Sneddon's (1946) result thata*y/l/Kic = >/TT/2 for an equilibrium field, amplified by a dynamicovershoot of 25 percent. The remaining portion is estimated from thenumerical calculation described by Chen and Sih (1977) for a penny-shaped crack loaded axisymmetrically by a tensile pulse. The stressovershoot for the three-dimensional penny-shaped crack is about thesame as for the plane strain crack. However, the numerical resultsindicate that the stress maximum occurs sooner after loading forthe three-dimensional case than for the two-dimensional case. Thedashed curves in Figure 3.10 capture the trend of the data, but theyoverestimate the values of a*. It was noted by Shockey et al. (1983)that the slow loading fracture toughness Kjc = 1.1 MPa^/m appearedto substantially over-estimate the stress intensity factor level at whichfracture was initiated in this material under rapid loading. Theyestimated that the fracture toughness under the dynamic loadingconditions imposed was about Kid = 0.72MPa>/m, where the sub-script d is used to indicate the special interpretation of this value.If the theoretical estimate of the locus of points for which a* = a*


is recalculated simply by reducing the ordinate of each point on thedashed curves in Figure 3.9 by the ratio Kid/Kici then the solid curvein this figure results. Clearly, the solid curve is more consistent withthe data. Even though the curves are based on plane strain models,the theory captures the essence of the dynamic response of the penny-shaped cracks.

The question of the influence of loading rate on the fracturetoughness of materials will be considered later in this section. Kalthoffand Shockey (1977) and Shockey et al. (1983) also suggested that theelevation of the stress intensity factor to a critical level was a necessarycondition for onset of crack growth, but not a sufficient condition.They proposed that the critical level of stress intensity factor must bemaintained (or exceeded) for a certain minimum time interval. Sucha minimum time has no fundamental significance, and it must dependon material properties and on the operative mechanism of materialseparation in the nonlinear zone at the crack edge. Experimentssimilar to those reported by Shockey et al. (1983) were carried outon specimens of an aluminum alloy, a high strength steel, and astructural steel by Homma, Shockey and Murayama (1983). In theseexperiments, edge cracked strip specimens were used, and the generalconclusions were similar to those reported in the earlier studies.

Ravi-Chandar and Knauss (1984a, 1984b) carried out an exten-sive experimental study of fracture initiation under dynamic loadingconditions in thin sheets of Homalite-100, a brittle plastic material.Their experimental setup is closely modeled as a semi-infinite crack inan otherwise unbounded solid, with the crack faces subjected to uni-form normal pressure. This was accomplished in a large rectangularsheet of lateral dimensions 500 mm by 300 mm and thickness 4.86 mmby cutting a crack through the sheet along the longer symmetry axisto roughly the center of the sheet. The crack tip was artificiallysharpened. A copper ribbon was then inserted in the crack, runningfrom the edge of the sheet to the crack edge along one face. At thecrack edge, the ribbon was folded back onto itself, and it extendedback to the edge of the specimen along the other crack face. The spacewithin the fold of the ribbon, between the crack faces, was filled witha dielectric. A charged capacitor bank was then discharged throughthe ribbon, producing a large current. The pressure on the crack faceswas generated by the interaction of the electric current in each leg ofthe ribbon with the magnetic field induced by the current in the otherleg. The crack face pressure increased within about 25 /xs to a plateau


1.2 -i

|

0.4-

0.00.0 50.0 100.0 150.0

Figure 3.11. Data on the value of the stress intensity factor at initiation ofcrack growth versus time to fracture for a loading pulse of fixed amplitude (Ravi-Chandar and Knauss 1984a).

value, which was maintained at least until fracture initiation. Thetime history of the crack tip stress intensity factor was monitored bymeans of the optical shadow spot method. Among the data presentedby Ravi-Chandar and Knauss (1984a) is the variation of the value ofthe stress intensity factor at the instant of fracture versus the timeelapsed between the application of the crack face loading and theonset of crack growth. This elapsed time varies inversely with themagnitude of the loading pressure. These data are reproduced inFigure 3.11, along with data on the slow loading fracture toughnessof Homalite-100. The data indicate that if the time to fracture isgreater than about 50 JMS, then the critical stress intensity factor isindependent of the time to fracture, and, therefore, independent ofthe magnitude of the applied pressure. Furthermore, this plateaulevel of stress intensity factor is approximately equal to Kjc. On theother hand, for high amplitude loading pulses which produce fracturein times less than 50/xs, the critical level of stress intensity factordepends strongly on the time to fracture. Ravi-Chandar and Knauss(1984a) discuss this point qualitatively on the basis of a mechanisticmodel involving viscous growth and coalescence of microcracks bymeans of a viscous mechanism.

Ravi-Chandar and Knauss (1984a) also reported measurementsof the complete history of the stress intensity factor for applicationof loading to (and beyond) the onset of crack growth. This matterwas also pursued by Kim (1985a, 1985b) who used a different opticalmethod known as the stress intensity factor tracer method. He alsoused large sheet specimens of Homalite-100, as well as the same elec-tromagnetic loading apparatus that was introduced by Ravi-Chandarand Knauss (1984a) to apply pressure over part of the crack faces.With reference to the analysis in Section 3.2, the crack occupied thehalf line y = 0, x < 0 and pressure was uniformly applied over thecrack faces for x < — /. By means of the superposition integrals(3.2.15) and (3.2.16), and the fundamental result in Figure 3.2, hedetermined the expected stress intensity factor history for this appliedloading. The predicted history showed nearly all of the features shownin Figure 3.2, including the fall-off in stress intensity factor betweenarrival of the shear wavefront and arrival of the Rayleigh surface waveat the crack tip. The measured Ki(t) followed the prediction veryclosely. There was even some evidence of crack face interference inthe early stages of loading, as discussed in Section 3.2.

An experimental approach to the study of dynamic fracture ini-tiation in metals was developed by Costin, Duffy, and Freund (1977).In the experiment, a tensile wave was generated in a precracked roundbar by means of an explosion behind a loading head bolted to one endof the bar. The wave produced a fracture at the cracked section withinabout 25 /xs. The load on the ligament at the cracked section and thecrack opening displacement were measured during the course of theexperiment. Because the ligament dimension was small compared tothe length of the incident loading pulse, the mechanical fields at thecracked section were deduced from the measured quantities on thebasis of the assumption that the distribution of field quantities wasthe same as if the loading had been statically applied. The assumptionhas been verified by means of a detailed calculation of the transientfields by Nakamura, Shih, and Freund (1986).

It was noted by Shockey et al. (1983) that the fracture tough-ness under dynamic loading conditions (termed Kid in the foregoingdiscussion) was substantially lower than Kjc for the epoxy used intheir experiments. Indeed, it is frequently found that the rate ofloading on a fracture specimen has an influence on the level of stressintensity factor at which fracture initiates. This was indeed the casefor structural steels in the work reported by Costin et al. (1977).


In some cases, the toughness appears to increase with the rate ofloading whereas in other cases the opposite dependence is found. Theexplanation for a shift in toughness, either upward or downward, mustbe sought in the mechanisms of inelastic deformation and materialseparation in the highly stressed region at the edge of the crack in theloaded body.

To illustrate the fundamental difference in fracture response ofmaterials which gives rise to the two general types of behavior, con-sider two ideal fracture mechanisms. Suppose that materials separatewithin the crack tip zone when either a critical stress is reachedat a material element near the crack edge or when a critical strainis reached there. The critical stress criterion could apply for thecase of cleavage fracture initiation in a polycrystalline metal withintrinsically cleavable grains, such as polycrystalline iron with brittlegrain boundary carbides at or below room temperature. Due to thecombined effects of plastic flow which is inhomogeneous on a grain-to-grain scale and the high degree of stress triaxiality at points ahead ofthe crack tip in mode I, fracture can initiate in such a material whenthe local stress becomes large enough to drive a crack from the brittlephase into a favorably oriented adjacent grain as a cleavage crack. Thecleavage microcrack is presumed to precipitate macroscopic fracturein the ideal case. Thus, only a critical level of stress in the crack tipregion is required for any rate of loading. The critical strain criterion,on the other hand, could apply for the case of locally ductile fractureinitiation in a two-phase material consisting of hard brittle particlesembedded in a ductile matrix, for example, carbon steel at temper-ature above room temperature. Under load, the large strains nearthe crack edge cause separations at the particle-matrix interface orcracking of individual particles, and cause subsequent ductile growthof these cavities to coalescence. Given a distribution of particles toserve as cavity nucleation sites, a critical level of crack tip strain isrequired for fracture initiation (that is, ductile cavity coalescence) forany rate of loading. Inelastic deformation over some small region atthe crack edge is essential in either case.

An effect of increased rate of deformation on the stress-strainresponse of the material is to increase the stress necessary for con-tinued plastic flow at a given level of strain. The magnitude of thiseffect can have enormous variation, depending on the material andits microstructural condition. Consider now two loading situations,one "rapid" and the other "slow." A critical level of stress at a point


150-i

125-

100-

7 5 -

5 0 -

2 5 -

fibrous

cleavage

O KIc = / MPa\fm/s

m KId= 2x106MPay/m/s

-200 -150 -100 -50 0 50

Temperature (C)100

Figure 3.12. Data due to Wilson et al. (1980) showing the influence of loadingrate on the fracture toughness of a 1018 cold-rolled structural steel.

in the crack tip region will be reached at a lower level of strain forrapid loading than for slow loading. Consequently, the stress workdensity (1.2.38) will be less in the case of rapid loading. If less workhas to be done on the material element ahead of the crack tip to bringit to the point of incipient fracture, then the material will appear tobe more brittle. Therefore, for materials in which fracture initiatesby a stress activated mechanism and the flow stress increases withincreasing deformation rate, the effect of increase of loading rate isto reduce the apparent fracture toughness. If the same argument isapplied in the case of the critical strain criterion, it is clear that thestress work density at a point in the crack tip region will be greater forrapid loading than for slow loading when the critical level of strain isreached. The material appears to be tougher in the former case thanin the latter. Therefore, for materials in which fracture initiates by acritical deformation mechanism, the effect of increase of loading rateis to increase the apparent fracture toughness, if it is changed at all.

This brief discussion of ideal fracture initiation mechanisms isintended only to suggest the wide range of responses which are possiblein real materials. The use of energy concepts is somewhat prematurehere; this point will be pursued at greater depth in later discussion.Physical mechanisms of separation are commonly far more complex


than the ideal mechanisms indicated in this discussion. Furthermore,it is possible that the main influence of high loading rate is not tochange the toughness level, but instead to completely change themechanism of fracture. This is evident in the data reported by Wilsonet al. (1980) on fracture initiation in structural steels, which wasobtained by means of the method introduced by Cost in et al. (1977);see Figure 3.12. Their data showed that, at low testing temperaturewhere the fracture initiation mechanism is essentially stress inducedcleavage, the influence of increased loading rate is to reduce the tough-ness level by a small amount. Likewise, at high testing temperaturewhere the fracture mechanism is essentially deformation-controlledductile cavity growth, the influence of increased loading rate is toincrease by a small amount the toughness level. The main effectof increased rate of loading, however, was to shift the temperatureof transition from cleavage to ductile behavior to a position about100 C higher on the temperature scale. This shift reflects a change infracture mechanism induced by an increase in rate of loading.

4

ASYMPTOTIC FIELDS

NEAR A MOVING CRACK TIP

4.1 Introduction

In Section 2.1, it was argued that the fields at an interior point on theedge of a stationary crack in an elastic solid are asymptotically two-dimensional. The convention of resolving the local deformation fieldinto the in-plane opening mode (mode I), the in-plane shearing mode(mode II), and the antiplane shearing mode (mode III) was adopted.It was pointed out that the components of stress have an inversesquare root dependence on normal distance from the crack edge and acharacteristic variation with angular position around the edge. Thisvariation is specified through the functions Ey(-) for each mode inSection 2.1. These general features are common to all configurationsand all loading conditions. The influence of configuration and loadingare included in the asymptotic description of stress only through scalarmultipliers, one for each mode, which are the elastic stress intensityfactors. The role of the stress intensity factor as a crack tip fieldcharacterizing parameter has been discussed in Sections 2.1 and 3.6.

The existence of similar universal fields for growing cracks is con-sidered next. The same convention for categorizing local deformationmodes will be followed. Except where noted explicitly to the contrary,the asymptotic crack tip fields will be determined for variable cracktip speed. Mode III will be considered first, because it is the simplestcase to analyze, and the in-plane modes will be analyzed subsequently.

A rough estimate of the upper limit of the range of crack speedsfor which inertial effects can be ignored in the description of an asymp-

152


totic crack tip field for a given material can be obtained on the basisof an equilibrium field for quasi-static crack growth. To illustratethis idea, two estimates are included here. Consider first the caseof mode I crack growth in an elastic material at speed v for whichthe quasi-static crack tip field is given in Section 2.1. As the cracktip approaches a material particle near its path, the magnitude ofthe particle velocity increases sharply and the kinetic energy densityincreases accordingly. The magnitude of the stress and strain fieldsalso increase sharply as the crack tip approaches, resulting in a rapidincrease in strain energy density. These two energy changes take placeduring comparable intervals of time, so a relative comparison of thetwo energy density values when the material particle is very close tothe crack tip can shed some light on the degree of influence of materialinertia on the local fields.

At a radial distance r from the crack tip, the particle velocity isproportional to vKj/Ey/F, so the kinetic energy density is

to within a multiplier of order unity, where p is the material massdensity and E is Young's modulus. Likewise, the stress componentsare proportional to Ki/y/F, so the stress work density (strain energydensity in this case) at the same point is

to within a multiplier of order unity. The ratio T/U is then

v~m- (4-L3)

The ratio is independent of r, and the result (4.1.3) implies thatinertial effects may not be important in determining the structureof the crack tip field as long as the crack speed v is less than aboutone-third of the elastic wave speed co = y/E/p.

The asymptotic structure of the equilibrium fields for quasi-staticmode I crack advance in an elastic-ideally plastic material under planestrain conditions is also known (Rice et al. 1980), so a similar estimate

154 4- Asymptotic fields near a moving crack tip

can be obtained for this case. Within the angular sectors around thecrack tip in which the plastic strain rate is most singular, the particlevelocity and plastic shear strain in crack tip polar coordinates varywith radial distance as

ur, us ~ veo In (^j , epr0 ~ eo In (^j (4.1.4)

where eo is the tensile yield strain and fp is a constant with dimensionof length that is usually identified with the size of the crack tip plasticzone. If expressions for the kinetic energy density T and the stresswork density U are derived from the fields given by Rice et al. (1980)for 0 = TT/2, say, then the ratio of the two energy measures is

(4.1.5)

to within a multiplier of order unity. The ratio in this case dependson r, and it can be made indefinitely large simply by taking r to besmall enough. This observation may have important mathematicalconsequences in asymptotic analysis. However, it is inconsequentialfrom a practical point of view unless the value of the ratio T/Ubecomes significant compared to unity for values of r/rp that representdistances of relevance on the scale of continuum fracture analysis.For example, if v/co = 0.1 then the ratio T/U is greater than one-tenth if r/rp < 0.3. While the estimate is based on a number ofapproximations, it raises the possibility that the influence of inertialeffects on the local fields may become significant at lower crack speedsin elastic-plastic materials than in elastic materials.

There are several reasons for the study of the asymptotic cracktip field for dynamic growth of a crack in a material. The influence ofmaterial inertia on the distribution of stress and deformation near thecrack edge is of interest in assessing mechanisms of crack advance be-cause these fields represent the environment in which the mechanismsare operative. Furthermore, numerical methods are often the onlymeans for obtaining full field solutions within this problem class. Forpoints very close to the crack edge where gradients are most severethe accuracy of numerical solutions is difficult to assess. The abilityto match computed fields to asymptotic fields valid in this regionestablishes confidence in the numerical results.

4-2 Elastic material; antiplane shear 155

X2

\ >

[l(t),m(t))\ <lM-

Figure 4.1. Crack growing along a smooth curved path under two-dimensionalconditions. The instantaneous crack tip position is xi = l(t), X2 — Tn(t), and theinstantaneous crack speed is v(t) in the local £i-direction.

4.2 Elastic material; antiplane shear

Suppose that the elastic plane in Figure 4.1 is subjected to loadingthat results in antiplane shear deformation. A rectangular coordinatesystem labeled £i,#2,#3 is introduced so that the only nonzero com-ponent of displacement us(xi,X2it) is in the ^-direction. A crackgrows along a curved path x\ = /(£), X2 = m(t) where I and m arecontinuous functions of time. The instantaneous speed of the cracktip in the plane is v(t) = v I2 + m2, where the superposed dot is usedto denote differentiation with respect to time. The crack tip speed isrestricted to the range 0 < v < c8 and it is assumed to be continuousfor the time being. A local rectangular coordinate system £1, 2 isintroduced in such a way that the crack tip velocity is always in the£i-direction. A local polar coordinate system r, 9 is also defined asshown in Figure 4.1, with r = \/£i + £2 an<^ tan0 = £2/^1- Supposethat the crack faces are subjected to equal but opposite shear tractionand that the finite limiting value of the component of the applied stresso"23 (in the local coordinates) at the crack tip is cr^ .

The near tip stress distribution is derived as an interior asymp-totic expansion for the boundary value problem by means of standardasymptotic methods (Freund and Clifton 1974). From Section 1.2.3,out-of-plane displacement U3(xi,x2,t) is governed by the partial dif-ferential equation

dx\ + dx\ c

in regions where it is differentiate. Next, a transformation of coor-dinates from the fixed rectangular system xi,X2 to the rectangular


system £1,^2 fixed with respect to the crack tip is introduced. Thistransformation is defined by

£1 = [xi - /(*)] cos/3(t) + [x2 - m(t)] sin (3(t),(4.2.2)

£2 = - [xi - l(t)] sin (3(t) + [x2 - m(t)]

where (3(t) is the angle from the Xi-axis to the £i-axis, measuredpositively in the counterclockwise sense. Thus, tan/3 = rh/L Thedisplacement 113 is then viewed as a function of position £1,62 inthe moving coordinate system and of time, that is, U3(xi,#2,£) =w(€ii€2it). Under the transformation of coordinates (4.2.2), the par-tial differential equation (4.2.1) becomes

d w d w 1 f d w - - dw ••2" ~* 2" 2 iiij H & + :

(4.2.3)

where £i = /3£2 — v and £2 = — /3£i. Summation over the range 1,2is implied for repeated indices. Obviously, du^/dt is distinct fromdw/dt because xi,#2 is held fixed in the former operation whereas£i,£2 is held fixed in the latter.

To derive the first term in the asymptotic expansion of stresscomponents as r —> 0, a standard device is employed whereby theregion around the crack tip is expanded so that it fills the entirefield of observation. To this end, rescaled coordinates rji = £^/eare introduced, where e is a small parameter. If e is taken to beindefinitely small then all points in the £1, £2-plane except those nearthe crack tip are pushed out of the field of observation in the 771,772-plane. Furthermore, as e becomes indefinitely small, the crack lineoccupies the negative 771-axis in the field of observation.

It is assumed that w has an expansion in powers of e of the form

(4.2.4)

where Wo represents the dominant asymptotic solution, W\ representsthe first order correction to the asymptotic solution, and so on. This

implies that the exponents are ordered so that po < p\ < P2 <The exponent po is expected to be in the range 0 < po < 1 if thedisplacement is to be bounded at the crack tip but the stress is tobe singular. Each term in the expansion (4.2.4) is expected to bebounded as e —• 0+ for any fixed point && 7 0. The expansion (4.2.4)is essentially an assumption that the near tip displacement field can berepresented as a series of homogeneous functions of increasing degree,a feature shared with the more familiar approach based on separationof variables in local polar coordinates. The assumed form of thesolution is substituted into the transformed wave equation (4.2.3) andthe coefficient of each power of e is set equal to zero. The coefficientof the lowest power of e vanishes if WQ satisfies

K) ( 4 .2 .5 )

where v = v(t) is the instantaneous crack tip speed, as before. Thisequation is recognized as Laplace's equation with the coordinate 772rescaled by the factor as = y/l — v2/c*. Thus, a general solution is

W0 = Im{F(C)}, (4.2.6)

where £ = rji+ia8r]2 and F(() is an analytic function in the complex £-plane. The function can depend parametrically on t. A displacementfield that is symmetric with respect to the crack line £2 = 0 resultsin zero stress on the crack line, so there is no loss in generality byassuming that F is further restricted by the symmetry conditionF(C) = F(Q. Note that Wo could equally well be expressed as the realpart of an analytic function. In that case, the analytic function wouldbe subject to a different symmetry condition and the final result forWo would be the same as that obtained below.

In light of the expansion (4.2.4), if the limiting value of the crackface traction at the crack tip, say a^3 , is bounded and if 0 < po < 1then

a^ ± o ( ) M t F , f a ) } = 0 (427)

for 771 < 0, where the prime denotes differentiation of F with respectto its argument. In view of the assumed symmetry condition on F(-),

F'+M + F'M) = KM + FLM = 0 (4.2.8)

on 771 < 0, where the subscript plus or minus indicates a limiting valueof the function as the negative real axis is approached through positiveor negative values of r/2, respectively. The equation (4.2.8) represents astandard Hilbert problem in analytic function theory (Muskhelishvili1953).

The most general analytic function that satisfies (4.2.8) and thatleads to a deformation field with integrable mechanical energy densityis

where A(() is an entire function and a branch cut is provided alongthe negative real axis to render F'(Q single valued in the cut £-plane.The only feature of the function A(() that influences the asymptoticfield as C —* 0 is ^4(0), hereafter denoted simply by A (which canstill depend on time t). The condition F(£) = F(£) further requiresthat A must be a real parameter. Note that if F(() = —F(£), thenthe condition equivalent to (4.2.8) leads to the conclusion that eitherFf(Q is entire or Ff(£) has a pole at the origin. In the former case,the boundedness condition leads only to the trivial solution, whereasthe latter case is ruled out because the singularity is too strong tobe admissible. Neither possibility is of interest in this development.Consequently, the dominant contribution to the near tip field has thesymmetry F(C) = F(C).

In view of (4.2.4) and (4.2.6), the result (4.2.9) implies that thedominant contribution to the near tip displacement field is

n±0a, (4.2.10)

where rs = VCi+oflf , tan0s = ots&lii, and A is an undeterminedparameter that is independent of £1,62- The parameter p0 in (4.2.4)evidently has the value one-half. Prom the stress-strain relation forthe material it follows that

-M^-^T «»§*.. (4.2.11)

If the dynamic stress intensity factor for mode III is defined by

Km(t) = lim y/2^<723(£i, 0, t) (4.2.12)*0


then A = iiji/(t)//xas\/27r. Thus, the feature that the near cracktip field has universal spatial dependence carries over to the case ofnonsteady mode III crack growth as well.

The next term in the expansion (4.2.4) will satisfy the boundaryconditions if p\ = 1. Then, equating coefficients of like powers of e,the boundary condition equivalent to (4.2.7) is

5 p = £23. (4 2.13)

and the differential equation governing the function W\ is

= 0. (4.2.14)

Like Wo, the function W\ can be expressed in terms of an analyticfunction. The relationship between limiting values of this analyticfunction corresponding to (4.2.8) has the nonzero right side 2a^3 '/fnots-It follows immediately that fjWx = rjia^ + r/2^23 » where a)^ is thevalue of the stress component ^13 at the crack tip. Consequently, theasymptotic stress distribution can be summarized as

as r ^ O (4.2.15)

in the presence of crack face tractions. The physical coordinates r, 9are related to r5,68 by

r8 = ry/l - (vsin0/c8)2 = r^8 , tan05 = a s tan0. (4.2.16)

The functions S - in (4.2.15) are

sm\98 cos§08

E ' ( 4 - 2 ' 1 7 )

Upon comparison of (4.2.17) with the corresponding results for astationary crack tip, it is evident that the former reduce to the latteras v —> 0. The distribution of particle velocity in the crack tip region

is also recorded here for future reference. The asymptotic form ofparticle velocity as r —• 0 is

(4.2.18)

The asymptotic forms (4.2.15) and (4.2.18) represent the main resultsof this section. Several observations can be made concerning theseresults. For example, the dominant singular term in the expansion ofstress or particle velocity does not depend on the fact that the crackpath is (or can be) curved. Certainly, the influence of crack pathcurvature will enter through the higher order terms in the expansion.Indeed, examination of the partial differential equation (4.2.3) makesit clear that the terms of order r1/2 in expansions of stress and particlevelocity components will include a contribution with its coefficientproportional to the angular velocity of the crack propagation direction(3. The instantaneous radius of curvature of the crack path is v/$.The asymptotic expansion can be continued to higher order terms byequating the coefficient of the next highest power of e in the differentialequation obtained by substituting (4.2.4) into (4.2.3). This step ispursued for a straight tensile crack (that is, for (3 = 0) in the nextsubsection. For a crack following a curved path, the construction ofthe expansions is conceptually straightforward but the details are notyet available.

4.3 Elastic material; in-plane modes of deformation

Suppose that a planar crack is extending in an elastic body in sucha way that the state of deformation is two-dimensional plane strain.The case of generalized plane stress is identical except for the inter-pretation of the elastic constants. A rectangular £i,£2>#3 coordinatesystem is introduced so that the crack edge is parallel to the #3-axisand the displacement vector of each particle is parallel to the x\,X2-plane. For definiteness, suppose the crack grows in the #i-direction sothat the coordinates of the crack tip in the ^ i^ -p lane are x\ = /(£),x2 = 0, where I is a continuous function of time. The instantaneousspeed of the crack tip is v(t) = /(£). The crack tip speed is restricted tothe range 0 < v < CR and to be continuous for the time being. A localrectangular coordinate system £1,62 with its origin at the crack tip is

4-3 Elastic material; in-plane modes of deformation 161

introduced, where £1 = x\ — /(£), £2 = #2- A local polar coordinatesystem r, 9 is also introduced, where r = \/£i + £ i > t a n ^ = fe/fi- Thenear tip stress distribution is derived for modes I and II, following theprocedure outlined for mode III in the preceding section.

4.3.1 Singular field for mode I

Suppose that the crack faces are subjected to tractions consistent withthe mode I symmetry conditions, and that the finite limiting value ofthe component of applied crack face stress (722 as £1 —+ 0~ at thecrack tip is 032 . The representation of crack tip fields in terms of thedisplacement potentials </> and t/> = ips is adopted as in Section 1.2.3.If (/> is viewed as a function of position (£1, £2) in the local coordinatesystem and of time t, then it is governed by the differential equation

In light of the derivation of (4.2.3) in the previous section, the inter-pretation of the terms in (4.3.1) is obvious. The shear wave potential^(£ijf2»*) must satisfy the same equation with Cd replaced by cs.

The development follows that presented in Section 4.2 for modeIII. First, the rescaled coordinates r\i = &/e are introduced, wheree is an arbitrary small parameter. It is then assumed that <\> has anexpansion in powers of e of the form

(4.3.2)

where $0 represents the main contribution to the asymptotic solution,$1 represents the first order correction, and so on. The powers of eare ordered so that po < Pi < P2 < • • •, and po is expected to bein the range 1 < p0 < 2 if displacement is to be bounded but stressis to be singular at the crack tip. The assumed expansion (4.3.2) issubstituted into the governing equation (4.3.1) and the coefficient ofeach power of e is set equal to zero. The coefficient of the lowest powerof e vanishes if $0 satisfies

arii • 84 = 0 ' ( 4 3-3 )

where <*<* = y/l — v2/cj. Thus, $o is viewed as a function of 771,772with the possibility that it also depends on time t as a parameter. Ageneral solution of this equation is

$o = Re{F(C*)}, (4.3.4)

where Q = Vi + i<*dV2 and F(() is an analytic function everywhere inthe complex £-plane except along the nonpositive real axis. For modeI deformation, F must satisfy F(£) = F(£). Similarly, the leadingterm in the expansion for xp has the form

*o = Im{G(G)}, (4.3.5)

where £« = 771 + iasr)2 and G(Q has the same domain of analyticityand symmetry as does -F(C). The powers of e in the expansion for %/>must be the same as for </> because neither type of wave function alonecan satisfy the boundary conditions on the crack faces.

If the expansions of <f> and ip are substituted into the stress bound-ary conditions on the crack faces, and the coefficients of independentpowers of e are set equal to zero, then

2s) [*?fai) + Fl'fox)] + 2a8 [G'l(Vi) + G'L(m)] = 0,

(4.3.6)Fffax) - F!!{m)] + (1 + a2.) [G'Km) ~ G'i(m)] = 0 •

The solution of this pair of equations that leads to singular stress butintegrable mechanical energy density is

(4.3.7)

where A is independent of £ (although it can depend on time) and

D = Aadas - (1 + a28)

2 . (4.3.8)

Note that D —> 0 as v —• c^, where c# is the Rayleigh wave speedfrom Section 2.5. In light of (4.3.7), the exponent po in (4.3.2) hasthe value 3/2.


If the dynamic stress intensity factor for mode I crack growth isdefined by

Ki(t)= lim y/2icticr22(£i,0,t) (4.3.9)

then A = Ki(t)/'/iV^r. The dominant contribution to the near tipfield is therefore completely determined up to the scalar multiplierKi(t) for nonuniform crack tip motion.

The next term in the expansion (4.3.2) can be determined byfollowing steps similar to those for mode III. The details are notincluded here, but it is noted that the exponent p\ takes on the value 2and the stress components derived are locally uniform. Consequently,the leading terms in the asymptotic expansion of the crack tip stressfield can be summarized by

as r ^ O . (4.3.10)

The functions Sy(0,v) that represent the angular variation of stresscomponents for any value of instantaneous crack tip speed v are

j _ 2ad{l + aa) \sm\9d sin\0s

12

(4.3.11)where

7rf = y l — (v sin 0/cd)2 , tan 0d = cxd tan 0,(4.3.12)

7s = \A "~ (vsm9/cs)2 , tan68 = a8 tan0 .

The terms of order unity denoted by a^' in (4.3.10) have relatively

simple interpretations. The term cr^ = 0 to conform with the


symmetry of a mode I deformation field. The term cr^ is determinedfrom the crack face loading conditions as the limiting value of normalcrack face traction at the crack tip. If the crack faces are traction-free during crack growth, then this term has value zero. Finally, theterm a[i cannot be determined from the asymptotic solution. Itmust be determined as a feature of the complete stress distributionin any particular stress analysis problem. An elastic field consistingonly of a uniform tensile stress in the direction of the crack line (the^-direction, in this case) can always be added to any other stressfield without violating the boundary conditions on the crack faces,provided that the deformation in the transverse direction is completelyunconstrained.

It can be verified by direct calculation that the terms in theexpansion (4.3.10) do indeed reduce to the equivalent results for equi-librium crack tip fields given in Section 2.1 as the crack speed becomesvanishingly small. Some care is required in the calculation becauseboth the term D in the denominator of each expression and each of thequantities enclosed in brackets in (4.3.11) vanish as v —• 0. However,in each case both numerator and denominator vanish as v2 so theratio has an unambiguous limit as v —• 0 and the Irwin-Williamsasymptotic field (2.1.3) is recovered.

It is noteworthy that the mode I crack tip field for nonuniformmotion of the crack tip involves the crack tip motion only through theinstantaneous crack tip speed v(t). An immediate consequence is thatthe near tip field for nonuniform motion is identical to that for steadystate crack growth in the same material, at least for the singular andlocally uniform contributions investigated up to this point. This wasdemonstrated by Preund and Clifton (1974), Nilsson (1974b), andAchenbach and Bazant (1975), each of whom compared asymptoticsolutions for nonuniform crack growth with earlier results based onthe assumption of steady growth obtained by Cotterell (1964), Rice(1968), and Sih (1970). It is shown in the next subsection that theexpansions do indeed differ if terms beyond those included in (4.3.10)are examined. Such higher order terms are used to interpret exper-imental data obtained by optical sensing methods applied over someregion around the tip of a rapidly growing crack, so the interpretationof such higher order terms is of practical significance.

The variation of stress components near the crack tip can berepresented in any number of ways. Here, the circumferential tensilestress (the "hoop" stress or the tensile stress in the 0-direction), the

4.3 Elastic material; in-plane modes of deformation 165

-0.4120 150 180

6 (degrees)Figure 4.2. Variation of the circumferential tensile stress (4.3.13)i with angle 9around the crack edge for several values of normalized crack speed. Crack growthis in mode I.

largest principal stress, and the maximum shear stress are plottedagainst the angular position 0 around the crack tip for a fixed valueof Ki(t)j\j2/Kr. These three stress measures are denoted by

iV) (circumferential tensile stress),

^g, v) (largest principal stress), (4.3.13)

rmax = —•j=YiIrnax(g, v) (maximum shear stress),V2

respectively. The angular variations of the circumferential tensilestress, the largest principal stress, and the maximum shear stress areshown in Figures 4.2 through 4.4. Note that the definition of stressintensity factor (4.3.9) implies that S22 = 1.

For crack speeds less than about 60 percent of the shear wavespeed of the material, the hoop stress or transverse tensile stressshown in Figure 4.2 has a maximum value along g = 0, that is,in the direction of crack growth. For crack speeds greater than about

166 > Asymptotic fields near a moving crack tip

3.0 n

2.0-

30 90

9 (degrees)120 150 180

Figure 4.3. Variation of the maximum principal stress (4.3.13)2 with angle 0around the crack edge for several values of normalized crack speed. Crack growthis in mode I.

0.6 c5, however, this stress measure develops a maximum in a directioninclined at about 60° to the direction of crack growth. The feature wasfirst observed by Yoffe (1951) who suggested that this inertia inducedmodification of the local singular stress field could account for thetendency of rapidly growing cracks in very brittle materials to developroughened fracture surfaces and to bifurcate into several branchedcracks. Though this feature does provide a mechanism by whicha growing crack can sample propagation directions inclined to thestraight ahead direction, it is not able to fully explain crack branchingand bifurcation behavior in brittle materials. In general, bifurcationhalf angles (one-half the total angle between the two branches ofa bifurcated crack) tend to be substantially smaller than 60° andthe attainment of a critical speed appears to be neither a necessarynor a sufficient condition for a fast running crack to bifurcate. Thebifurcation behavior of fast running fractures appears to depend onproperties of the stress field not included in the singular term in(4.3.10) alone (Cotterell 1964; Ramulu and Kobayashi 1983). Thispoint will be discussed further in connection with crack propagationbehaviors.

The same crack tip singular field that was found by Yoffe (1951)


in a study of a steady crack growth process was also found in a studyof a transient crack growth problem by Baker (1962). It was only laterin the development of the subject that this field was established as auniversal crack tip field for any instantaneous stress intensity factorand instantaneous crack tip speed. Baker suggested that it was notthe hoop stress or transverse tensile stress that was of fundamentalsignificance in analyzing crack advance in brittle materials. Instead,he suggested that if the local condition for fracture is the attainmentof a critical tensile stress on some plane, then the maximum principalstress is the stress measure of significance. The angular variation ofthis stress measure is shown in Figure 4.3. For low crack speeds, thisangular variation has a shallow maximum between about 60° and90°. For very high speeds, on the other hand, the variation showslocal maxima at both 0 = 0 and for some value of 9 beyond 90°.

The variation of maximum shear stress for the singular field in(4.3.10) with angle 9 around the crack edge is shown in Figure 4.4.It is evident from this figure that there is a monotonic increase inmaximum shear stress with increasing crack speed for a fixed level ofstress intensity factor. Thus, the driving force for inelastic modes ofdeformation that are shear driven, such as dislocation glide, twinning,or viscous relaxation, is increased with increasing speed. Along 9 = 0,the in-plane principal stresses within the singular field are equal forzero crack speed and nearly equal for low crack speeds. However, theratio C^/CTII on 9 = 0 is a rapidly decreasing function of speed forvalues of v greater than about 50 percent of the shear wave speed. Thechange from the corresponding equilibrium result in the nature of theasymptotic field accounts for the dramatic increase in maximum shearstress along 9 = 0 with crack speed. Thus, at high crack speeds, thelarger in-plane principal stress ahead of the crack edge acts on planesperpendicular to the direction of growth, and the largest in-planeprincipal stress in the regions near the crack edge acts at points off ofthe crack plane. These effects may underlie the observed tendency forrapidly growing cracks in brittle materials to develop rough fracturesurfaces.

Finally, it is noted that the elastodynamic particle velocity fieldcorresponding to the square root singular stress field in (4.3.10) isalso square root singular at the crack edge. The dominant spatialdependence of the particle velocity components near the crack edge is

2.0 -i

1.5-

1.0-

0.5-

0.030 60 90 120 150 180

6 (degrees)Figure 4.4. Variation of the maximum shear stress (4.3.13)3 with angle 0 aroundthe crack edge for several values of normalized crack speed. Crack growth is inmode I.

given by

vKi(t)

/j,DV2wr I{(1 +y/ld }•

(4.3.14)

The limiting velocity field as the crack tip speed becomes vanishinglysmall is

(4.3.15)

where the right side is intended to represent a feature of the Irwin-Williams equilibrium asymptotic field.

It is evident from (4.3.10) and (4.3.14) that for crack tip speeds inthe range 0 < v < c8 the local stress and particle velocity vary as theinverse square root of radial distance from the crack tip. Due to theappearance of the factor D in the expressions for these fields, however,the algebraic sign of the coefficient of the square root singularity

4.3 Elastic material; in-plane modes of deformation 169

depends on whether the crack tip speed is less than or greater than theRayleigh wave speed of the material CR. Suppose that the crack facesseparate behind the advancing crack tip. Inspection of (4.3.10) and(4.3.14) leads to the conclusion that the traction on the prospectivefracture plane ahead of the tip is tensile if 0 < v < CR but that it iscompressive if CR < v < cs. This feature implies that there is a netflux of energy out of the advancing crack tip for CR < v < cs, whereasthere is a net flux of energy into the crack tip for crack tip speeds lessthan the Rayleigh wave speed. This result is established in Section5.3.

4.3.2 Higher order terms for mode I

The next term in the expansion (4.3.2) for 0 which can yield a non-trivial result has exponent P2 = 5/2. If the expansion is substitutedinto the governing equation (4.3.1) and the coefficient of each powerof e is set equal to zero, then

J ' <4-3-16>where the function 3>o(7?i,7?25£) was determined in the previous sub-section. Again, $2 is viewed as a function of 771,7/2 that possiblydepends on t as a parameter. The equation is more convenient toconsider in the distorted polar coordinates pd = \Jr\{ +arctan(ad772/77i), so that for constant crack speed

^ V y cos 1 , (4.3-17)p\ duj d 2

where

Note that A* is the rate of change of a quantity that depends on theinstantaneous stress intensity factor Kj(t) and on the crack tip speedv. A general solution of (4.3.17) for mode I deformation is

$ 2 = p5J2 (IA* cos \ud + A2 cos \ud) , (4.3.19)

where A2 is a constant of integration to be determined from theboundary conditions.

Likewise, the next term in the asymptotic expansion of ip is givenby

tf2 = - p V2 ( i B * s i n i ^ + B2sin 1^) (4.3.20)

for mode I deformation, where B2 is a constant to be determined fromthe boundary conditions, p8 = y/rfi + a^rfe, u8 = arctan(asr/2/^i),and

*• = *f 3 l-l=*f ^*H')I • (4-3-21)The condition of traction-free crack faces will be satisfied to

corresponding order if A2 and B2 are related by 5 (30^2 + A*) =8 (15i?2 + B*). The coefficients A2 and B2 are otherwise unrestrictedby the near tip conditions, and a second relationship between themcan only follow from remote conditions. The mean in-plane stressderived from (4.3.19) and (4.3.20) is

+ A^^r]/2 cos \eA . (4.3.22)^ Cd

Only the second term in each of (4.3.19) and (4.3.20) is commonlyviewed as the next most significant term, after the term correspondingto locally uniform stress, in the expansion of </> or \j). While this isindeed the case for steady state crack growth, it is not correct wheneither the crack speed or the stress intensity factor is changing withtime. Prom their definitions in (4.3.18) and (4.3.21), it is evident thatA* and B* are nonzero, in general, when either v ^ 0 or Ki(t) ^ 0.When this is the case, the first term in each of (4.3.19) and (4.3.20)must be included in the expansion. The existence of these contri-butions associated with nonsteady situations has not been generallyrecognized in dynamic fracture studies, and their importance relativeto the other terms of the same order in the expansion of crack tipfields has not been studied systematically.

4-3.3 Singular field for mode II

The asymptotic field for mode II deformation can be analyzed byfollowing the steps outlined for the case of mode I deformation. The

results for the leading term in the crack tip field are given here;details of the analysis are omitted. For mode II deformation, thestress distribution near the crack edge has the form

ai5 = ^ 2 s g ( 0 , v) + o(l) as r -> 0+ , (4.3.23)v27rr

where Kjj(t) is the instantaneous mode II stress intensity factor. Inthis case, the functions Efj" representing angular variation of stresscomponents for any value of instantaneous crack tip speed v are

EE = --

2_;-i 9 — " "~

H 2as(l-\-al) (sin^0d sin|6

The angular variations of the circumferential tensile stress, the largestprincipal stress, and the maximum shear stress as determined from(4.3.24) are shown in Figures 4.5 through 4.7. The definition of stressintensity factor adopted in (4.3.23) implies that Ej^O,!?) = 1.

The elastodynamic particle velocity field corresponding to thesingular stress field (4.3.24) is given by

va8Kn(t) f n dZ — ( 1 -j-

(4.3.25)vKnit) ( cos \Q<i 2 cos|6

I d S yjld S y/Ts

4^3.4 Supersonic crack tip speed

In the foregoing analyses of crack tip singular fields, it has beenassumed that the crack speed is subsonic, that is, v < cs in all cases.The observation of crack speeds greater than cs or even greater than


0.5 -iv/c

-2.090 120

6 (degrees)150 180

Figure 4.5. Variation of the circumferential tensile stress with angle 0 around thecrack edge for five values of normalized crack speed. Crack growth is in mode II.

1.0-

0.5-

0.0-

-0 .5-

-1.0-

-1.5 —j—

30 60 906 (degrees)

—I—120 150

—I180

Figure 4.6. Variation of the maximum principal stress with angle 0 around thecrack edge for five values of normalized crack speed. Crack growth is in mode II.

Cd in structural materials has been limited to cases where the loadingis applied directly at the crack tip by means of a fluid under high

2.0 -i

0.0120 150 180

6 (degrees)Figure 4.7. Variation of the maximum shear stress with angle 6 around the crackedge for several values of normalized crack speed. Crack growth is in mode II.

pressure flowing into the opening crack or by some other extremecondition. Winkler, Curran, and Shockey (1970a, 1970b) focused anintense laser beam at a point in the interior of a sodium chloridecrystal. The vaporized material expanding in the resulting cavityinduced supersonic crack growth. Analysis of seismic data takenduring crustal earthquakes has led to the conclusion that a shearfracture on a preexisting fault can propagate at a speed greater thanthe shear wave speed of the crustal material (Archuleta 1982). Fewanalytical results are available for crack growth at a speed greaterthan cs or c . In the truly supersonic case where v > Cd the crack isgrowing into material that has no advance warning that the crack isapproaching. Thus, it seems that such a process in a brittle-elasticmaterial must be driven by crack face loading that acts up to thecrack tip. The crack tip field in this case consists of plane wavefrontstrailing the crack edge, with the deformation distribution determinedby the distribution of traction on the crack faces. Indeed, the solutionis exactly that for a supersonically moving pressure step load on thesurface of an elastic half space (Cole and Huth 1958). The case of asupersonic step pressure load on the surface of an elastic-plastic halfspace was analyzed by Bleich and Matthews (1967). The situation forthe intermediate case with cs < v < Cd (sometimes called transonic)

is less clear; this case is treated very briefly and representative resultsare derived in this subsection.

Suppose that a planar crack grows in a body subjected to trac-tions consistent with the mode I symmetry conditions. The represen-tation of the crack tip fields in terms of the displacement potentials<\> and t/> as functions of local coordinates £1,^2 and time t is againadopted. Thus, </>(£i,£2>*) satisfies (4.3.1) and V>(fi>£2>*) satisfies thesame equation with Cd replaced by c8. Following the development formode III, the rescaled coordinates rji = &/e are introduced, wheree is an arbitrary small positive parameter. Then, expansions of theform (4.3.2) are assumed. The main contributions to the potentialsare again denoted by 3>o(f7i>f72>£) and *o(?7i> V2^)^ and they satisfythe partial differential equations (4.3.3) and

respectively, where as = y/v2/c* — 1. Equation (4.3.26) is the el-ementary wave equation in two independent variables. The generalsolution of (4.3.3) is given in (4.3.4) and the general solution of (4.3.26)which satisfies the principle of causality and which has the symmetriesconsistent with mode I deformation is

a.*> «*2o,

where G is an arbitrary function with the property that G(s) = 0if s > 0. If the expansions of <f> and ip are substituted into thestress boundary conditions on the crack faces, and the coefficientsof independent powers of e are set equal to zero, then

(2 - v2/c2s) [FZ(m) + F-(Vi)] ~ 4StG"(m) = 0,

(4.3.28)ad[F?(Vl) - Jt'(ifr)] + (2 - v2/c2

s) G"(Vl) = 0

for — oo < r/i < 0. The solution of this pair of equations that cor-responds to singular stress but integrable mechanical energy densityis

4-4 Elastic-ideally plastic material; antiplane shear 175

where A is independent of position in the plane and

The exponent q depends on crack speed v in the following simple way.It has the value q = 0 when v — cs, it increases monotonically toq = 1/2 when ^ = cs^/2, and it decreases monotonically to # = 0when v = Cd-

In terms of crack tip coordinates £i,£2> a representative pair ofstress component and particle velocity component is

- 2) A

(4.3.31)

where H(•) is the unit step function.Thus, A is also a stress intensity factor but the singularity in

(4.3.31) is weaker, in general, than the inverse square root singularityin the case of subsonic crack growth. The shear wave contribution tothe above expressions is fundamentally different from that in the caseof v < cs. Because the crack speed is greater than the shear wavespeed, no shear wave radiated from the crack edge can propagateahead of the running crack tip. Instead, shear wave motion can existonly behind plane wavefronts trailing from the advancing crack edge.In the local coordinate system, these plane wavefronts coincide withthe lines £1 + 2S|£2| = 0 and the step function in (4.3.31) ensuresthat the shear wave fields are confined to the material points behindthe wavefronts. Some particular aspects of transonic crack growth inelastic materials were considered by Burridge (1973), Burridge, Conn,and Preund (1979), Preund (1979a), and Simonov (1983).

4.4 Elastic-ideally plastic material; antiplane shear

Suppose that the plane in Figure 4.8 is subjected to loading that re-sults in antiplane shear deformation. A rectangular coordinate system

Figure 4.8. Schematic diagram of the near tip region for steady crack growththrough an elastic-plastic material at speed v in the x\-direction. The length rp isthe extent of the active plastic zone in the direction of crack growth. The dashedline is a shear line with local slope tp.

labeled x\, x<i, x% is introduced so that the only nonzero component ofdisplacement u$(x\,xi,i) is in the ^-direction. A crack in the plane#2 = 0 grows in the #i-direction with steady crack tip speed v in therange 0 < v < c8. The crack faces are free of traction, and the drivingforce is assumed to be applied remotely.

The equation ensuring momentum balance is

9(723 92Us= P ( 4 4 1 )dxi 8x2 dt2

where p is the material mass density. The nonzero strain compo-nents are ei3 and 623, and these strain components are subject to thecompatibility condition

d*u3 2de2sdx2

0x2The total strain is assumed to be the sum of elastic strain and

plastic strain,«y = «« + c y - (4-4-3)

The material time rate of change of elastic strain is assumed to beproportional to the material stress rate

where /x is the elastic shear modulus. In plastically deforming regionsof the plane, the stress state at each point is assumed to satisfy theMises yield criterion

^ 3 + 4 * = ro2, (4.4.5)

where ro is the flow stress in homogeneous plastic shearing. For thecase of antiplane shear, the Tresca yield criterion takes on the samesimple form. The crack tip field for the case of a yield conditionrepresented by a diamond-shaped yield locus in stress space has beenconsidered by Nikolic and Rice (1988).

The plastic strain rate is assumed to depend on stress at eachpoint through the associated flow rule

k% = Xsij , (4.4.6)

where Sij is the deviatoric stress S{j — G^ — (Jkk^ij and A is anonnegative factor proportional to the plastic work rate. For thecase of antiplane shear, the stress state is deviatoric. Furthermore,if (4.4.6) is written for the two nonzero strain rate components andthe unknown parameter A is eliminated from the resulting equations,then

2/i<h3 ~ ^"13 _ 2/J623 ~ <^23 / , . -x

0*13 ^23

must hold in a plastically deforming region.Finally, for steady growth in the x\-direction at speed v, any

field quantity, say f(xi,X2,t), depends on X\ and t only through thecombination x\ — vt. Consequently, the material time derivative of/ can be replaced by the spatial gradient —v(df/dx\). The spatialgradient form for the time rates will be assumed in the remainder ofthis section. Without loss of generality, it is assumed that the cracktip passes the point x\ = 0 at time t = 0.

The yield condition (4.4.5) is satisfied identically at each pointwithin a plastically deforming region if the nonzero deviatoric stresscomponents are represented in terms of a single function ip(xi,X2, i),commonly called the local shear angle, according to

cri3 = — ro sin ip , cr23 = TO cos ip . (4.4.8)

If <Ti3 and (723 are viewed as the components of the shear traction vec-tor in the xi, X2-plane, (4.4.8) implies that this vector has magnitudeTO and direction at a counterclockwise angle xj; from the X2-axis. Theshear angle field is assumed to have the symmetry ^(#i,— #2,£) ——ip(xi,X2,t). Thus, if ifr is continuous across the crack line X2 = 0,then ip = 0 on the crack line. If (4.4.8) is introduced into both the

momentum equation (4.4.1) and the incremental constitutive equation(4.4.7), and 623 is eliminated by means of the compatibility equation(4.4.2), then the first order system of partial differential equationsgoverning the remaining dependent variables if) and 7 = 2/xei3/ro is

+ W 0,

(4.4.9)dip . , dij) o ^7X\ OX2 OX 1

where m = v/c8 is the ratio of the crack tip speed to the elasticshear wave speed. The system of partial differential equations (4.4.9)is quasi-linear and hyperbolic with two families of real characteristiccurves in a plastically deforming region. Because the equations arequasi-linear (the nonlinear equations are linear in the derivatives),the characteristic network cannot be established without knowing thesolution beforehand. Characteristic curves will be discussed in Section8.3; they are not required to extract asymptotic fields in this section.

4-4-1 Asymptotic fields for steady dynamic growth

To aid in the construction of an asymptotic solution, a polar coordi-nate system r, 0 centered at the crack tip in the plane is introduced,with r = \/{x\ — vt)2 + x\ and 0 = t a n " 1 ^ / ^ — vt)]. In themoving crack tip coordinate system, fields depend only on the relativeposition r,0. Because the stress components a^ and <r23 must bebounded and single valued at the crack tip, the shear angle ip(r,0)must have the same properties. Thus, there exists a function of 0,denoted by p(0), such that tjj(r,8) —• p(0) as r —> 0+. The derivativesof ip in (4.4.9)2 are no more singular in r than r""1 as r —• 0, so thatthe derivatives of 7 must have the same property. It follows that

-y(r,9)~A\nr + g{e) (4.4.10)

for small r, where A is a constant and g{8) = —g(—9) is a function tobe determined. For antiplane deformation with reflective symmetrywith respect to the crack plane, 7(1% 0) = 0 so that A = 0.

The pair of partial differential equations (4.4.9) is now reducedto a pair of ordinary differential equations for p(0) and g(0). This is

accomplished by writing the partial derivatives as their polar coor-dinate equivalents, multiplying the terms in each equation by r, andtaking the limit as r —> 0. The result is

p'(9) sin 9 - g'(0) sin(p - 9) = 0,(4.4.11)

p'{9) sin(p - 9) - m2g'(9) sin 9 = 0,

where the prime denotes differentiation with respect to 9. There aretwo solutions of the pair of equations. Either

p'(0) = O, g'(0) = 0 (4.4.12)

separately, or the determinant of coefficients of p' and g' vanishes sothat

p(9) = 9±sin"1 (rasin9). (4.4.13)

The former solution leads to an asymptotically uniform field; thelatter solution represents a nonuniform field similar to a centered fan.The choice of sign in the latter case is dictated by the requirement ofpositive plastic dissipation.

An asymptotic expansion for A in a region where (4.4.13) holdscan be obtained as follows. Form the product of Sjk with each termin (4.4.6) and contract over the indices i and fc, with the result that

\TQ = -v~-—<7i3 - v——<723 > (4.4.14)OX\ OX\

where (4.4.3) has been used to express the difference between totalstrain and plastic strain. Next, the partial derivatives in (4.4.14)are transformed to their polar coordinate equivalents, both sides of(4.4.14) are multiplied by r, and then the equation is examined as rbecomes indefinitely small. The result is

Xr - -^V(0)cos(p - 9). (4.4.15)

The factor g'{9) can be obtained from (4.4.11) in light of (4.4.13), andcos(p — 9) = T V 1 — m2 sin2 9, where the upper (lower) algebraic signcorresponds to the upper (lower) sign in (4.4.13). Thus,

Ar— ~ T — Vl - m2 sin2 9 - cos9 (4.4.16)v m

as r —• 0. The magnitude of the first term is greater than or equal tothe magnitude of the second term for any ra > 0 and for any value of0 in the range 0 < 0 < ?r, so that the condition of nonnegative plasticdissipation is ensured if the lower (positive) sign is selected in (4.4.16)and, consequently, the lower (negative) sign must be selected in thedefinition of p(0) in (4.4.13).

The asymptotic stress distribution in a plastically deforming re-gion corresponding to the asymptotic form of the stress function ip in(4.4.13) is

0"i3 ~ —To ( v 1 — m2 sin2 0 — ra cos 0 j sin 9,(4.4.17)

0"23 ~ T~O (cos9v 1 — m2 sin2 9 + rasin2 9) .

This stress distribution satisfies the symmetry condition with respectto the plane 9 = 0, but it does not satisfy the boundary condition ofzero traction on the crack faces at 9 = TT. The stress distribution isuniform for the other type of region (4.4.12). The particular choice

0"13 ~ —To 5 0"23 ~ 0 (4.4.18)

does satisfy the boundary condition of vanishing traction on the crackface. Furthermore, if (4.4.17) holds for 0 < 9 < 0* and (4.4.18) holdsfor 9* < 9 < 7T, then the two stress distributions satisfy continuity oftraction across the radial line 9 = 9* provided that p(9*) = TT/2 or

0* = tan" 1 ( -1 /m) . (4.4.19)

(The question of continuity of traction in the asymptotic field will beexamined in some detail in Section 4.5.) Thus, the stress distributiondescribed satisfies all conditions imposed, and it therefore representsa possible asymptotic stress distribution. This is the distributiondetermined originally by Slepyan (1976).

The corresponding asymptotic deformation field can be estab-lished in a similar manner. Prom (4.4.12) and (4.4.13), the distribu-tion of the strain component ei3 is given by

t0-s in~ 1 (msin0) i f O < 0 < 0 *

44 Elastic-ideally plastic material; antiplane shear 181

This distribution satisfies the symmetry condition on 9 = 0, and itimplies continuity of particle velocity across 9 = 0*. Furthermore,because 2ei3 = du^/Oxi, it implies that the crack tip opening dis-placement is zero, that is,

X\lim ti3(xi,0,t) = 0 (4.4.21)

t

and the crack tip opening angle is

lim p-{xiAt) = - ^ - . (4.4.22)

A particularly significant feature of the deformation field is thedistribution of shear strain £23 ahead of the advancing crack tip. Forpresent purposes, it suffices to extract the variation of 623 with r =x\—vt on 9 = 0. From (4.4.20) and the compatibility equation (4.4.2),it is evident that

dc13 9623 TO COS 9 \ 1 + COS 9 , ^

dx2 dxi 2// r [ m >/l - rn2 sin2 9

for 0 < 9 < 9*. Therefore, on 9 = 0,

9e23 ro(l - m )axi 2\xm\x\ — vt)

so that

(4.4.24)

r o l - ra 1 r o l -m 123 ~ -z In T -r = In - (4.4.25)

2/x m (J;I — vt) 2/x m r

as r — 0 on 0 = 0 for any speed m > 0. Thus, the shear strain on theprospective fracture plane is logarithmically singular. The equation(4.4.23) can be integrated to determine the asymptotic form of e23 asa function of 9; the result is given by Slepyan (1976) who showed thatthe strain is logarithmically singular throughout the sector of activeplastic flow. This point will be dealt with in greater detail whenmore complete solutions of elastic-plastic crack growth problems areconsidered.

4.4.2 Comparison with equilibrium results

For purposes of comparison, some features of the crack tip solutionsfor a stationary crack or a steadily growing crack in the same elastic-plastic material but under equilibrium conditions are included here.The governing equations once again are (4.4.1) through (4.4.6). Themain point for comparison is the strength of the singularity in plasticstrain e^ ahead of the crack tip because of its significance for crackadvance studies. Other features of the fields are also noted.

Consider first a stationary crack in a body subjected to mono-tonically increasing loading resulting in antiplane shear deformationwith the symmetry us (r, —0) = — ( r , 0) in terms of crack tip polarcoordinates. This situation was analyzed by Hult and McClintock(1956) and Rice (1968). In the absence of inertial effects, the equilib-rium equation and the yield condition lead to the result that the shearlines are straight. Furthermore, in anticipation of a strain singularityahead of the crack tip within the plastic zone, the shear lines are takento be radial. Actually, choices are severely restricted by the conditionsthat no two shear lines may intersect (some stress component wouldexceed ro in magnitude at the intersection point) and shear lines mayintersect the traction-free crack faces only at right angles. If the shearlines are radial, then the stress distribution in the active plastic zoneis <7i3 = — ro sin 0 and cr23 = rocos0. From (4.4.8) it is evident that

tl> = 0, 0 < 0 < TT/2 , (4.4.26)

over the plastic sector ahead of the tip, and the material remainselastic for TT/2 < 0 < TT. Along any radial shear line, erz — 0 becausethe stress vector in the xi,#2-plane is always transverse to the radialline. Thus, particle displacement is uniform, that is,

us = us(9), 0 < 0 < TT/2 . (4.4.27)

The shear strain within the plastic zone is

TOR(0) sing TOR(0) cos 02fi r 2fi r

where R(0) is the distance from the crack tip to the elastic-plasticboundary in the direction from the tip defined by the angle 9. Thus,


the singularity in plastic strain 633 ahead of the crack tip is pro-portional to r"1 , which is much stronger than that for the steadilygrowing dynamic crack in the same material.

To complete the summary of results for the stationary crackproblem, it is noted that the crack tip opening displacement is notzero, as it is in the case of a growing crack. Instead, it has the value

r r/2

lim u3(xu0+) = — / R{6)d6. (4.4.29)xi— 0- fJL Jo

The crack tip opening angle or slope in this case has the value

lim p-(Xl,0+) = - ^ . (4.4.30)

a?i-+0- OX\ 2/i

For a crack growing steadily under equilibrium conditions in thesame material, (4.4.9) can be applied with m = 0. Then (4.4.9)2implies that the gradient of %j) in the direction defined by the angle \j)is zero, or that the shear lines are straight. For the reasons outlinedin the foregoing paragraphs, therefore, the shear lines are taken tobe radial lines focused at the crack tip in the region ahead of theadvancing crack, so that ip = 9, Then (4.4.9) 1 implies that thegradient of 7 in the radial direction is equal to —dip/dxi or

The result of integrating with respect to r and imposing the conditionthat e i3 = — (To/2/i)sin0 on the elastic-plastic boundary at r = R{9)is

7(r, 9) = - sin 0 11 + In ^ ^ 1 . (4.4.32)L r J

Then, upon differentiation with respect to #2 and incorporation of thecompatibility equation (4.4.2),

de23

2/xr{1 - cos2 0 f2 + In 1 - sin 9 cos 9^-) . (4.4.33)

Consider the strain distribution along 9 = 0 directly ahead of thecrack tip. Along this line, (4.4.33) can be integrated subject to the


condition that €23 = TO/2/X when r = R{0) = i?o> with the result that

(4.4.34)

This result, which was presented by Rice (1968), provides not onlythe asymptotic strain distribution, but also the full distribution onthe crack line within the active plastic zone.

An immediate observation is that the crack tip strain singularityfor the case of a steadily growing crack is much weaker than thatfor a stationary crack subjected to monotonic loading in the samematerial. In the latter case, the stress state at a material particle nearthe crack tip deep within the plastic zone is fixed and plastic strainaccumulates without a change in the direction of plastic straining.In the former case, on the other hand, the crack is advancing intoplastically deformed material. The stress state, and consequentlythe direction of plastic straining, rotates continuously at a materialpoint as the crack tip moves by, and a lesser strain concentration isestablished.

Comparison of the crack tip strain distributions along 9 = 0 forsteady growth under equilibrium and dynamic conditions reveals aparadox at this stage of development of elastic-plastic crack growth.For steady quasi-static growth, the strain 623 near the crack tip canbe extracted from (4.4.34), whereas for steady dynamic growth theasymptotic result

e23^^iz^inl (4.4.35)2/i m r

was obtained in (4.4.25). It would be natural to expect that the formof the former result for small r with m = 0 would be the same asthe form of the latter result for m —• 0 with small r, but this isnot the case. The asymptotic solutions do not provide a basis forpursuing the matter, and the paradox can be resolved only through amore complete dynamic solution. The analysis provided in Section 8.3reveals that the range of validity of this asymptotic solution vanishesas m —» 0.

4.5 Elastic-ideally plastic material; plane strain

Consider a mode I tensile crack growing steadily at speed v in anelastic-ideally plastic material under plane strain conditions. The

4-5 Elastic-ideally plastic material; plane strain 185

material is assumed to be fully incompressible. Both a rectangularcoordinate system a?i, #2 and a polar coordinate system r, 0 are intro-duced with a common origin at the tip of the crack. The coordinatesystems translate with the crack tip which moves steadily in the xi-direction at speed v. The equations of momentum balance are

where cr - are the rectangular components of the Cauchy stress tensor,iii are the rectangular components of the particle velocity, and p isthe mass density of the material. As in the previous section, it isconvenient to introduce the dimensionless crack tip speed m = v/c8,where cs = y/Jifp and ji is the elastic shear modulus of the material.Furthermore, steady state conditions imply that the material timederivative of any field quantity / can be replaced by the scaled spatialderivative / = —vdf/dxi.

In terms of the particle velocity, the rectangular components ofthe small strain rate tensor are

duj\£ ) ( 4 - 5 - 2 )

and additive decomposition of strain into elastic and plastic partsis assumed as in (4.4.3). The rectangular components of the elasticstrain rate, plastic strain rate, and deviatoric stress rate tensors areefj, ej7-, and Sij, respectively. The elastic strain components arerelated to the deviatoric stress components Sij = &ij — \(Jkkfiij throughHooke's law,

(4.5.3)

The plastic strain rate is assumed to be proportional to thedeviatoric stress,

$ - = A a y , (4.5.4)

where the proportionately factor A is a nonnegative function of po-sition to be determined as part of the solution. Combining (4.5.2)through (4.5.4) yields

which describes the material response in plastically deforming regions.The yield condition in plane strain is assumed to be

!(<r i i -a 2 2 ) 2 + ^ 2 = ro2, (4.5.6)

where ro is the constant flow stress in shear. This condition statesthat in plastically deforming regions the maximum shear stress in theplane of deformation has the value ro. Such a yield condition resultsfor an isotropic material exhibiting plastic incompressibility when theelastic deformation is also incompressible. The Tresca and Mises yieldconditions are particular cases under these assumptions.

The deformation field possesses the symmetry

6i(r,0) = 6i(r, -0) , *i2(r,0) = -ii2(r, -0) (4.5.7)

corresponding to mode I crack opening. The crack faces are traction-free, which is ensured if

<722(r, ±TT) = 0, <7i2(r, ±TT) = 0 (4.5.8)

for all r > 0.It is assumed that that the out-of-plane stress 033 is the interme-

diate principal stress, an assumption that can be verified subsequentlyfrom the solution. Then there is no plastic strain in the out-of-planedirection, that is, €33 = 0. (It is unlikely that this feature can becarried over to the case of elastic compressibility.) In light of (4.5.4),this implies that

511 = ~522 , (4.5.9)

which allows the yield condition (4.5.6) to be expressed as

4> + s\2 = rl. (4.5.10)

This condition, with (4.5.9), implies that the deviatoric stress compo-nents are bounded for all values of r > 0. If, in addition, it is assumedthat the hydrostatic stress a = \(Tkk is bounded, then the momentumequations can be used to discern the form of iii. A discussion ofthe conditions under which a can be shown to be bounded has beenpresented by Leighton, Champion and Preund (1987).

4-5.1 Asymptotic field in plastically deforming regions

A momentum equation for steady crack growth is

d<Tn day _ _ 9^1 , vdx

If the dependent variables are treated as functions of r and 0, deriva-tives with respect to the rectangular coordinates can be replaced byderivatives with respect to the polar coordinates by means of a simplechange of variables, for example,

If the stress components are bounded as r —• 0, then in view of(4.5.12), the left side of (4.5.11) is O( l / r ) . The right side of (4.5.11)must therefore be O( l / r ) . In this way, (4.5.11) can be used to de-termine the form of iii that gives rise to the strongest allowablesingularity. If the particle velocity is logarithmically singular as r —* 0,then the first term in (4.5.12) is (^(r"1). In order to avoid a strongersingularity arising from the second term in (4.5.12), the coefficient ofthe lnr term must be independent of 0. Alternately, if the particlevelocity is a function of 0 as r —> 0, then the second term of (4.5.11) is(^(r"1) whereas the first term is o(r~x). In summary, it appears thatthe two possibilities leading to the strongest allowable singularity inparticle acceleration are

"-(!)•

dAin ~ vAi In ( ^ ) , —± = 0 or in ~ vB^O), (4.5.13)

where R is a parameter with the dimensions of length which is as-sumed to be the same order of magnitude as the maximum extentof the active plastic zone. The plane strain condition implies thatB3(0) = 0 and A3 = 0.

It is assumed that the behavior of iii can vary from one angularsector to another around the crack tip. Symmetry requires thatA2 = 0 in the upstreammost sector, and therefore the condition thatthe velocity components must be continuous (to be obtained subse-quently) implies that 112 cannot be logarithmically singular in any

angular sector about the crack tip. Thus (4.5.13)2 is the asymptoticrepresentation of ii2 for 0 < 0 < n. From here on, A\ is replaced byA.

The incompressibility condition implies that the particle velocityis divergence free, which in plane strain is expressed by

This implies that the velocity components are derivable from a streamfunction <fr(r, 0) according to

d<t> . dc/>U t W U = V (4515)

If the velocity components are taken to be a combination of the formsindicated in (4.5.13),

in = vAln (- J + vBx{0), u2 = vB2(0), (4.5.16)

then the divergence-free condition (4.5.14) requires that A, I?i(0), andB2(0) satisfy the relation

Bi(0)tan0 = B2(0) -A, (4.5.17)

where the prime denotes differentiation with respect to 0. This rela-tion allows integration of (4.5.15) to obtain

0(r,0) = Ax2 \ In I - ) + 1 > + x2B1(0) - xxB2{0) (4.5.18)

to within an arbitrary constant.It should be pointed out that the asymptotic form of iii is actually

a s r - ^ O , (4.5.19)

where F(r) = o(lnr) does not contribute to the leading order terms inany of the governing equations, and therefore cannot be determined

4*5 Elastic-ideally plastic material; plane strain 189

without a higher order analysis. The function F(r) has been neglectedhere. If such a function were part of the complete asymptotic formof iii it would not affect the results presented here for stress, strainrate, or u2. The results for the strain and iii would require impor-tant modification, however. That there is no similar function in theasymptotic form of u2 follows from the foregoing argument that ledto the exclusion of a logarithmic term from u2.

The material strain rates are obtained from (4.5.2) and (4.5.16)as

en = - e 2 2 = -^-B'2(0)sm26,

(4.5.20)e12 = ^-[B'2(0)cos20-A\.

2X2

The strain rate components are therefore O(r~l) as r —> 0.Throughout any region of the body in which the material is at

a state of yielding, the yield condition (4.5.10) can be satisfied iden-tically if the two deviatoric stress components appearing in (4.5.10)are derived from a stress function ip(0) according to

$22 = ro cos [20 - I/J{0)] , 512 = - T O sin [20 - ip(0)] . (4.5.21)

Because the hydrostatic stress a is taken to be bounded, it is expressedas a function of 0 where

<T(0) = § ( < T I I + < 7 2 2 ) . (4.5.22)

In terms of the functions B2(0), </>(#)> a n d 0"(0)» the symmetry condi-tions (4.5.7) and the boundary conditions (4.5.8) are

()(4.5.23)

<0(7r) = 7r, a ( 7 r ) = r o ,

respectively, for the range 0 < 0 < n. The system of ordinarydifferential equations that these functions must satisfy is obtainednext.

In addition to the incompressibility condition (4.5.14), the gov-erning equations that remain to be satisfied are the two nontriv-ial momentum balance equations (4.5.1) and the two rate equations

(4.5.5) describing the material model. If the representations (4.5.16),(4.5.21), and (4.5.22) are substituted into these four equations then astraightforward manipulation of the results leads to

2- 2] [cos2 rft(O) - m2 sin2 0] = —Acos(tp(9) - 26),

(4.5.24)

B[(9) tan0 = B'2{0) - A, </(0) = ro fy'(9) - 2] sin

These differential equations now are considered for the case (4.5.13)2,that is, for A = 0. The case of A ^ 0, which was considered by Gaoand Nemat-Nasser (1983a), will be examined subsequently.

4-5.2 A complete solution

A solution to the system of equations (4.5.24) is sought for the casewhen J4 = 0. This condition is equivalent to the assumption thatthe particle velocity components have the asymptotic form (4.5.13)2-Attention is first focused on the function ^{9) which, according to(4.5.24), must satisfy

[1>'{0) - 2] [cos2 ip(9) - m2 sin2 9] = 0 (4.5.25)

in the interval 0 < 9 < TT, varying from t/>(0) = 0 to ^(TT) = fl". Clearly(4.5.25) is satisfied if either t/>'(0) = 2 or cosip(9) = ±rasin0. Thecurves defined by cosip(9) = ±rasin0 are shown in Figure 4.9. Anystraight line in this plane with a slope of two is an integral curve of0(0) = 2.

A complete solution is constructed in the following way. Thecurve il)(9) = 29 satisfies the boundary condition at 9 = 0, andthe curve i/)(9) = 29 — TT satisfies the boundary condition at 9 = TT.The two curves can be joined by cos^(0) = ±rasin0 with eithersign, and the choice is governed by the requirement that the plasticflow factor A(0) given by (4.5.24)2 must be nonnegative. The factorip'(9) — 2 is negative along both possible joining curves. Furthermore,

TX/2

- cos = -msinflcos = +msin0

/ ' "

/

m-0,9 /

m = 0.5 \ /

m = 0.1

rn^0.5 y

m = 0.9

0 TT/2

eTT

Figure 4.9. Solution curves ip(0) that satisfy the condition (4.5.25), includingcurves cost/)(0) = ±msin# and lines with slope of two.

tan ip(9) sinO = ±m~1 sin/0(0), where ra"1 sin-0 is nonnegative alongeither curve. Therefore, the curve cos ip(9) = — rasinfl must be chosento render A nonnegative. For the case m = 0.5, the complete solutionfor ip(0) is shown as the solid line in Figure 4.9, and it is the onlycontinuous solution satisfying the boundary conditions. The interpre-tation of this solution is straightforward. The stress and deformationfields are spatially uniform within the angular sector 0 < 9 < 0J, thefields are nonuniform within the sector 9\ < 9 < 9%, and again thefields are uniform within the sector 9% < 9 < TT. That the upstreamand downstream regions are uniform follows directly from ip'(9)—2 = 0and (4.5.21). The transition angles bounding the region of nonuniformdeformation are determined to be

.* . _i \m + V8 + m2

; = s i n

(4.5.26)

9% =TT — sin. _! - r a + y/8 + m2

and the deviatoric stress components are determined by means of

(4.5.21) from

{ 20 if 0 < 0 < 0J ,

cos"1 ( - m sin 0) if 0J < 0 < 9\ , (4.5.27)

2(9 - 7T if 0$ < 0 < 7T .With ^(0) determined the remaining differential equations in (4.5.24)can be integrated. The first integrals are constant in 0 < 0 < 9\ andin 05 < 0 < 7T. In the intermediate region, after application of theboundary conditions, it is found that

B±(9) = A [(i _ m2)F(0; m) - E(9; m) - (1 - m2)F(0*; rn)

sin0J + 5?,

B2(0) = A [xW - X(»i) - 2m cos 0 + 2mcos0j| ,

a{0) = ro [1 + m sin 0 - 2£(0; m) - m sin 0 + 2£(0£; m)] ,

(4.5.28)

where fc = ro//x, x(^) = \A — m2sin0, and F(6;m) and E(9;m) areelliptic integrals of the first and second kind, respectively, both withparameter m. SJ is a constant whose value cannot be determinedfrom the asymptotic analysis.

With the functions f?j(0) determined by (4.5.28), the expressionsfor strain rate components in terms of Bi(9) in (4.5.20) can be in-tegrated along a line #2 = constant from x\ = #2 cot 9\ throughdecreasing values of x\ to determine the strain distribution in thenonuniform region 9\ < 9 < 02. The results of this integration, afterapplication of the boundary conditions, are

en(0) =-€22(0) =-B^O),

k [m2 /[1-X(fl)]sinfln= ^ ITln {[i-x(ei)}sine) + x{e)

-2m cos 0 - x(0£) + 2m cos 0*~m ln ( a n ^ ^ ) . (4.5.29)\ tan 2 0 i / J

£ 2H

- 230 60 90 120 150 180

eFigure 4.10. Variation of stress components with angle 0 around the crack edgefor m = 0.1 and m = 0.9.

The strain components are uniform in the angular sectors 0 < 9 < 6\and 01 < 6 < IT.

The angular variations of the stress components for dimensionlesscrack speeds m = 0.1 and 0.9 are shown in Figure 4.10. The influ-ence of inertia, aside from increasing the transition angles 0J and 0\somewhat, is merely to cause a moderate reduction of the hydrostaticstress a in the upstream region. Of course, this implies that both anand 0*22 are similarly reduced. Figure 4.11 shows the variation of thestrain components for m = 0.1, 0.5, and 0.9. Inertial effects are moresignificant here than for stress; the downstream value of ei2 is reducedalmost to the upstream level, and the variations of both €12 and €22are reduced in the transition region. This results in very little strainvariation for large m. Note that the unknown constant B° has beenchosen to be unity for displaying €22- -B? can only be determined bymatching the asymptotic solution to the outer solution. However, itis possible that B° —• 0 as m —• 1.

The solution presented here includes no elastic unloading in anyangular sector about the crack tip, and therefore it cannot reduce tothe corresponding quasi-static solution of Rice, Drugan, and Sham(1980). A similar situation arises in mode III, where the dynamicsteady state solution fails to reduce to the corresponding quasi-static


12

180

Figure 4.11. Variation of strain components with angle 9 around the crack edgefor m = 0.1, 0.5, and 0.9.

solution. However, this issue can be resolved only by a completesolution of the problem. For the case of antiplane shear crack growth,a complete crack line solution is discussed in Chapter 8. In this case,the domain of validity of the dynamic asymptotic solution vanisheswith vanishing crack tip speed. Carrying this observation over to themode I problem, it appears that here, too, the domain of validityof the dynamic asymptotic field probably vanishes with vanishingcrack tip speed. The stress components reduce to those observedin the Prandtl punch field as m —• 0, as noted by Achenbach andDunayevsky (1981). The plastic strain components are nonsingularas r —• 0 for a fixed, positive value of m, whereas the correspondingplastic strain components in the quasi-static analysis by Rice et al.(1980) are logarithmically singular at the crack tip. The componentsof plastic strain are unbounded a s m - > 0 , although ei2 remains zeroin the upstream sector. The region where €22 is singular as m —• 0depends on the behavior of B° as m —* 0.

4*5.3 Other possible solutions

Next, the possibility of constructing an asymptotic solution with log-arithmic dependence of ii\ on radial coordinate r is considered, that

is, the constant A in (4.5.16) is taken to be positive. With reference to(4.5.24), the stress function ip(0) must satisfy the nonlinear ordinarydifferential equation

m2

[t/j'(O) - 2] [cos21/>(0) - m2 sin2 0] = —ACOS(I/J(0) - 20) , (4.5.30)K

with boundary conditions

V>(0) = 0 , ip(n) = 7r. (4.5.31)

In this case, the plastic flow factor is

9 J ' ( }

Recall that the conditions (4.5.30)-(4.5.32) correspond to a stressstate satisfying the yield condition over the full angular range aroundthe crack tip. A continuous solution for I/J(0) is inadmissible on thegrounds that it would necessarily pass through a range of 0 for whichthe plastic work rate is negative. This fact is demonstrated next.

Prom (4.5.32) it is noted that the sign of the flow parameter A(0)depends on the relative signs of the functions

N(0) = 2 cos 0 cos ip + m2 sin 9 sin(t/> - 26) ,(4.5.33)

D(0) = cos2 %l) - m2 sin2 9.

The curves N(0) = 0, denoted by (I,F), and the curves D{0) = 0,denoted by (1,1')> are shown in Figure 4.12; the algebraic sign of theplastic flow parameter is indicated in each region.

Consider (4.5.30) and the first boundary condition (4.5.31). Thesolution curve I/J(0) is well defined up to its point of intersection cwith the curve cos^ = rasin0 (curve I in Figure 4.13), at whichpoint ip' —+ oo. Equation (4.5.30) can be used to show that

tl>'(0) > 2 , ij)"(0) > 0 (4.5.34)

for 0 < 0 < 0C, where 0C is the value of 0 at the point c. Thisindicates that the solution of (4.5.30) up to the point c is of the form


-n/2

Figure 4.12. Diagram of the regions in which the plastic flow factor A is positive(+) or negative (—) for m = 0.5 and A ^ 0.

7T/2

Figure 4.13. Curves that separate regions where the plastic flow factor A is eitherpositive or negative (see Figure 4.12), and a possible solution curve M. The curveID was obtained by numerical integration of the governing equation for t/j(B) form = 0.5 and A = 1. Curve IV is the line V = 20.

4.5 Elastic-ideally plastic material; plane strain 197

shown by curve IE in Figure 4.13, where the line \j) = 20 (curve IV inFigure 4.13) is shown for comparison. In particular, (4.5.34) showsthat the solution curve must lie above the line i\) = 20, at least imtilpoint c, where it encounters the curve cos^ = msin0 (curve E'ii>Figure 4.13) with ip'(0) —• oc. The point c will always have theproperty that ip(0c) < n/2 and, therefore, the segment of the solutioncurve El must always lie inside the triangular region Odb defined bythe lines 0 = 0, ip = TT/2, and I/J = 20. Point b in Figure 4.13is the intersection of line IV with i\) = ?r/2. The only way for thesolution to be extended continuously as a plastic state beyond thepoint c is through a region of negative plastic work rate, which isinadmissible. A possible remedy for this difficulty exists in the formof a discontinuity in xp(0) initiating at the point c. However, thisapproach is shown later in this section to be inconsistent with theprinciple of maximum plastic work, so that discontinuities in fieldvariables are inadmissible within the context of the present mechanicalproblem.

Seemingly, the only way to construct a continuous asymptoticsolution for which the velocity components depend logarithmicallyon r and which satisfies the maximum plastic work principle wouldinvolve the insertion of an elastically deforming sector starting at somepoint on the curve M. This possibility is also considered below, whereit is shown that an elastic sector cannot initiate at any point insidethe region Odb, and therefore on curve EL The consequence is that asolution with logarithmic behavior in the velocity components is notadmissible, and the velocity components must therefore be bounded.

4.5.4 Discontinuities

Possible discontinuities in field variables across radial lines from thecrack tip are considered next. For quasi-statically moving surfaces ofstrong discontinuity in elastic-plastic solids, Drugan and Rice (1984)showed that all stress components are continuous, although certain ve-locity components in the plane of the discontinuity may suffer jumps.The following analysis leads to the conclusion that for the dynamicproblem considered in this section all field quantities must be contin-uous.

Consider a hypothetical line discontinuity S propagating withthe crack and inclined at an angle 0^ to the crack line, as shown inFigure 4.14. Define local rectangular coordinates x,y such that they-direction is along the line of discontinuity. The jump in a quantity

y

Figure 4.14. A radial line of discontinuity E with local rectangular coordinates x, y.The upstream and downstream regions are identified by (+) and (—), respectively.

/ is denoted by [/] = / + — / " , where the superscripts plus and minusdenote limiting values of / on the leading and trailing sides of thediscontinuity, respectively. Application of momentum balance acrossa moving surface of discontinuity gives rise to the following jumpconditions (Truesdell and Toupin 1960):

[<Tij]nj + pc*lui\ = 0 , (4.5.35)

where rij is the normal to the surface of discontinuity, and cE is thespeed of the surface of discontinuity in the direction of the normal.From Figure 4.14, cE = vsin0E and (4.5.35) can be rewritten as

\°xx\ + pv sindslua.] = 0 , \<*xy\ + Pv sin0E[tiy] = 0. (4.5.36)

In terms of the particle displacement components ux and uy, thevelocity components are

ux =v sin0E—•£ - cos0E -^- ,\ dy dy J

( • a duV • a duv\Uy = -v I sin0E—— + c o s 0 s - ^ ,

\ dx dy J

(4.5.37)

where the incompressibility condition dux /dx + duy jdy = 0 has beenincorporated.

Displacement is continuous across E so the gradient of displace-ment along E is also continuous, or

Substitution of (4.5.38) into (4.5.37) reveals that

[ux] = 0 , [uy] = -vsin0E 1 ^ 1 = -2vsm6xlcxy} . (4.5.39)

In view of (4.5.39), the jump relations (4.5.36) reduce to

[crXx] = 0 , [asy\ = 2pv2 sin2 0E [exy] . (4.5.40)

The constitutive equations combined with (4.5.40) imply that

,] , (4.5.41)

where q = (m2 sin2 0E)/(1 - m2 sin2 0E).Suppose that material particles follow a continuous path in strain

space through the discontinuity. Furthermore, suppose that.if thejump in any quantity is zero, then that quantity remains constantalong the strain path through the discontinuity. In the followingdiscussion the phrase "inside the jump" is understood to mean "alongthe strain path from strain state et. to c . within the discontinuity."Then, (4.5.40) and (4.5.41) can be written in the differential form

daxx = 0 , daxy = 2qiidepxy , dayy = ~^dep

yy (4.5.42)

for states inside the jump.The maximum plastic work inequality is

(4.5.43)

where a^ is any stress state on or inside the yield surface in stressspace. Following the analysis of Drugan and Rice (1984) for the quasi-static case, the maximum plastic work inequality is integrated acrossthe discontinuity to obtain

W= I + {<Tij - a^d^ > 0. (4.5.44)

W is now written in the form W = Wp — W*, where

W* = IJ a^de?-. (4.5.45)

The incremental relations (4.5.42) imply that

P — —I xx 1 I xy 1 I VV

Wp = vxx depxx + - I axydaxy - - I (Jyydayy

ht «J«ty ihu ( 4 5 4 6 )1 , , 1

°xy) ) [gjltrwl " 2 ^

where the superscript S denotes the value of a continuous field quan-tity at the discontinuity. Use of (4.5.41) in (4.5.46) results in

U j ^ M (4-5-47)

The integral W* can also be evaluated with the result that

W = - * & [ $ ] . (4.5.48)

Finally, combining (4.5.47) and (4.5.48) yields

W = |(2a?. - oti ~ *y )[<*] > 0 • (4-5-49)

Because the material is nonhardening, any stress state in thebody can be chosen for cr* . Choosing a*j = crj and a*j = cr^ in(4.5.49) gives [<Jyl[e?-J > 0 and [a^][e^] < 0, respectively. Theserelations are satisfied for a plastically incompressible material if andonly if

[*«][«&] = 0. (4.5.50)

Using (4.5.40) and (4.5.41) with (4.5.50) implies that

[sxy\ = ±y/q[8yy] . (4.5.51)

As noted above, any state inside the jump is a possible candidate fora\j in equation (4.5.43) and the limits of integration in (4.5.49) may


be any pair of states between et. and e^ inclusive. Thus, (4.5.51)must hold for any pair of states inside the jump. This implies that

sxy = ±y/qsyy + C (4.5.52)

inside the jump, where C is an unknown constant. In addition,(4.5.40) and (4.5.41) show that no part of the strain path throughthe discontinuity may be purely elastic, and hence the yield conditionmust be satisfied at every state inside the jump.

Use of (4.5.52) in the yield condition (4.5.6) shows that

K y ] = [**y]=0. (4.5.53)

These results, together with (4.5.39) through (4.5.41), show that allfield quantities must be continuous across the discontinuity, that is,

K - ] = 0 , [tii] = 0, [€£1 = 0. (4.5.54)

In particular, (4.5.54) requires that [sij] = 0, and therefore that ip(9)is continuous. In a related study of the motion of a smooth punchalong the surface of a rigid-perfectly plastic material, Spencer (1960)concluded that the velocity field in the immediate vicinity of the punchedge must be continuous. His dynamic solution was obtained in theform of a perturbation from the equilibrium slip line field.

To summarize, if it is required that the strain histories mustabide by the principle of maximum plastic work through the jump,then discontinuities are ruled out. Consequently, only asymptoticsolutions that exhibit continuous angular variations in stress andparticle velocity are in full compliance with the maximum plastic workprinciple. A general result on the existence of discontinuities of fieldquantities in the material model considered here has been presentedby Drugan and Shen (1987).

The question of whether or not the strain paths through thejumps should be required to satisfy the principle of maximum plasticwork could be raised, of course. Within the context of gas dynamics,Courant and Priedrichs (1948) show that, for a mechanical analysis ofthe propagation of discontinuities, the sequence of mechanical statesexperienced by a material particle as a discontinuity propagates acrossit must be the same as though the transition from the initial state tothe final state has occurred in a smooth simple wave. The reason for


this outcome, in a somewhat simplistic form, is that in a mechanicaldescription of the phenomenon (as opposed to a more complete ther-modynamic description) the behavior of the material does not havethe flexibility aflForded by thermodynamic properties to depart fromthe mechanical constitutive law imposed. These observations can becarried over to the present case of elastic-plastic crack growth. Themodel is strictly mechanical so it may be expected that the behaviorof the material in a jump discontinuity must also be describable asthe limit of a smooth wave as the slope of the smooth wave becomesindefinitely steep. If this is so, then it follows immediately that thesequence of states in the jump must be consistent with the principleof maximum plastic work.

It should be pointed out that Courant and Friedrichs (1948) onlystate that for a mechanical description of the transition through a dis-continuity, the initial and final states are the same as if the transitionhad occurred in a simple wave. However, their proof of this statementdemonstrates that the sequence of states within the discontinuity mustalso be the same as if the transition had occurred in a simple wave.In order to describe the state of a material particle passing throughthe discontinuity, they used essentially the assumption that was usedby Drugan and Rice (1984) and that was used in this section: If thejump in some quantity is zero, then that quantity remains constantalong the path through the discontinuity.

For the case of antiplane shear crack growth in a material witha diamond-shaped yield locus, Nikolic and Rice (1988) showed thatthe near tip solution consists of sectors which carry constant stress atyield levels corresponding to adjacent vertices on the diamond-shapedyield locus, and which are joined along an elastic-plastic discontinuity.All plastic flow occurs within the discontinuity. Plastic strains andparticle velocity are found to be finite at the crack tip. The specialfeatures of this asymptotic solution arise from the existence of verticeson the yield locus in stress space. The corresponding results fordynamic plane strain crack growth are not yet available.

4-5.5 Elastic sectors

The possibility of including an elastic sector in the asymptotic field,initiating at some angle 0* on curve II in Figure 4.13, is considered inthis section. In an elastic sector, e - = 0 and, consequently, the stress

rate and strain rate are related by

e y = ^-Sij . (4.5.55)

Substitution of the incompressibility condition and (4.5.55) into themomentum equations reveals that the stream function <t>{r,0) mustsatisfy

<^i \s*id <t> 9 (b"\ , . _ __x

V \p — j " 2 = ^ (4.5.56)

where (32 = 1 — m2. As can be seen from (4.5.18), the stream functionhas the general asymptotic form

0(r, 0) = rg{0) - Ar sin 0 In r. (4.5.57)

The representation (4.5.57) implies that

= —— , (4.5.58)

where

G{0) = (1 - m2 sin2 0) [g{0) + g"(0) - 2A cos(0)] . (4.5.59)

Using (4.5.58) in (4.5.56) shows that G(0) must satisfy the ordinarydifferential equation G"{0) + G{0) = 0 which, with (4.5.59), showsthat g(0) satisfies the inhomogeneous equation

g"{e) + gifi) = ^ i n ^ 6 0 c o ^ + 2Acos^1 — m 2 sm 0

where CLQ and bo are constants to be determined. The general solutionof this equation has the form

(4.5.61)


where the ai are constants, and a$ = —A The functions g%{6) aregiven by

01 (0) = cos0, g2(0) = sin0, gz(0) = 0cos 0,

ff4(0) = x(0)cos0 + sin0 [0-/?tan~1(/?tan0)] , (4.5.62)

05(0) = -x(0)sin0 + cos0 [0-/r1 tan"1(/?tan0)] ,

where x(0) = In \/l — m2 sin2 0. The strains can be obtained from thestream function (4.5.57) by integration along X2 = constant. However,ei2 can be determined only to within an arbitrary function of x^. Thisunknown function is determined by the boundary conditions at 0*.Expressions for the deviatoric stress components are obtained fromthe constitutive equations

*«/M = 2[6y-€£(*2)] , (4.5.63)

and the hydrostatic stress a is obtained from the momentum equations(4.5.1). The resulting expressions are

= -2a2 + 2a4 [/3tan"1(/?tan0) - 0]

+ a4 [™2 lnr + (2 - m2)x(0)] (4.5.64)

5 [20-(/3 + /3"1)tan"1(/?tan0)j -2A9 ,

- ra2a40 + m2(A - a5) lnr - 2c

where so and ao are constants.By means of (4.5.59), boundedness of su in the elastic sector

requires that^11( 2) A ln(r sin0) + eo , (4.5.65)

where eo is a constant. Also, because su must be bounded, (4.5.64)2shows that a4 = 0. Continuity of the hydrostatic stress at an elastic-plastic boundary implies that a is bounded in elastic regions, and,

therefore, as = —A(2 — m2)/m2. The resulting expressions for thestress components are

= ci - A 2 ln(sin0) + ^ - ln(l - m2 sin2 0) ,m J

a//x = C3 — 2A ln(sin 0) ,

where ci, c2, and C3 are constants.With reference to Figure 4.13, the possibility of an elastic sector

initiating on curve M at some angle 0 = 9*, with 0 < 9* < 9C, isconsidered. The point is to determine if it is possible to avoid theregion of negative plastic work at 9 = 9C through introduction of anelastic sector. In any such elastic sector the unloading condition

^(«22 + « i 2 )<0 (4.5.67)

must hold in some neighborhood of 9 = 9*. Hence, the inequality

f2(O < 0 (4.5.68)

must hold, where the superscript e indicates that quantities are eval-uated on the elastic side of the elastic-plastic boundary.

From continuity of stress at an elastic-plastic boundary, (4.5.21)implies

se22(0*) = Tcos(29*-r),

where V* denotes ip(6*). The use of (4.5.66) and (4.5.69) in theinequality (4.5.68) shows that ip* must satisfy

(4.5.70)

It can be shown that this inequality is not satisfied anywhereinside the triangle Odb of Figure 4.13. But, as was mentioned earlier


in this section, the point (V>*,0*) must lie on curve H, which in turnmust lie inside triangle Obd. Hence an elastic sector cannot be insertedinto the plastic field to circumvent the region of negative plastic work.This implies that there is no solution curve if the region of negativeplastic work, indicated in Figure 4.13, is present. Therefore, waysto eliminate it must be sought. Apparently, this can be accomplishedonly by choosing A = 0, and therefore the velocity component u\ maynot have a logarithmic singularity in r.

For A = 0, it can be seen from (4.5.66) that the only possibleelastic sector is a region of constant stress which must satisfy the yieldcondition in light of full continuity at an elastic-plastic boundary.But such constant state regions are included in the solution presentedabove. It is concluded that the solution above is unique within thesolution class considered. Finally, it is noted once again that the rangeof dominance of an asymptotic solution cannot be established withouthaving a more complete solution in hand. It must be recognized thatthe size of the region near the crack tip in which these asymptoticresults provide an accurate description of the local fields may be toosmall to be of practical significance.

4.6 Elastic-viscous material

In this section, some results on the nature of near tip fields for dynamicgrowth of a crack in an elastic-viscous material are considered. Thestudy is based on a material model that is a special type of viscoelas-tic material commonly called a generalized Maxwell material. Theessential properties of the model are that the total strain rate is asum of the elastic strain rate and the viscous strain rate, the elasticstrain rate depends on the stress rate through the rate form of Hooke'slaw, and the viscous strain rate depends on the current stress but isindependent of stress history. A stress-strain relation having theseproperties is

where E and v are Young's modulus and Poisson's ratio of the elas-tically isotropic material, Sij is the deviatoric stress, a = y/SsijSij/2is the effective stress, and g(a) is the uniaxial viscous response func-tion for isotropic, incompressible viscous behavior. This function is

4.6 Elastic-viscous material 207

assumed to have the properties that

g'(a)>0 for < T > 0 ,

g(a) - • 0 as a -* 0 + , (4.6.2)

g(a) = O{an) as a —• oo ,

where n is a nonnegative real number. Some special cases of interestinclude the situation with g proportional to a, which is a linearMaxwell material, and g proportional to <jn, which is the nonlinearpower law viscous material. Condition (4.6.2)2 ensures that the vis-cous strain rate is zero when the effective stress is zero, and (4.6.2)3provides information on the strength of the singularity in viscousstrain rate as the stress becomes unbounded, such as in a crack tipsingular field.

Some features of the stress and deformation distributions on thecrack line ahead of the crack tip are considered in the subsectionsto follow. Several assumptions are made in order to obtain specificresults. For one, crack tip fields are assumed to be steady as seenby an observer moving with the crack tip. Though this assumptionresults in simpler mathematical expressions than those for nonsteadymotion, the results are asymptotically correct for nonsteady motionas well. The reason for this situation is that, in a local crack tipcoordinate system, the material time derivative is a spatial gradientin local coordinates plus a time rate of change at a fixed point in localcoordinates. For any singular crack tip field, the former contributionis sensitive to local gradients whereas the latter is not, so that theformer dominates the latter. The assumption that fields are steadyleads to the result that the latter contribution is not merely negligiblebut that it is zero. A second assumption is that the asymptotic stressdistribution is a separable function in crack tip polar coordinates.Though it cannot be established that the crack tip fields must havea separable form, it is nonetheless possible to satisfy all conditionsimposed on the basis of this assumption.

4-6.1 Antiplane shear crack tip field

Suppose that a two-dimensional elastic-viscous solid is subjected toloading that results in antiplane shear deformation. A rectangularxi,X2,X3 coordinate system is introduced so that the only nonzero

component of displacement 1x3(^1, #2, *) is in the ^-direction. A crackgrows in the plane X2 = 0 in the Xi-direction with steady crack tipspeed v in the range 0 < v < c8. The crack faces are free of traction,and the driving force is assumed to be applied remotely and to resultin a field with the symmetry u3(x\, — X2,t) = — u$(xi,X2,t).

The equation ensuring momentum balance, in a form appropriatefor steady antiplane shear deformation fields, is

where p is the material mass density. For steady fields, the stress-strain relation (4.6.1) simplifies to

dei3 v dai3 g(a)

-"a^-^a**—*« (4-6-4)

for i = 1,2, where /x = E/2(l + v) is the elastic shear modulus anda = ^3(a2

13 + <rlz).Polar coordinates r, 0 centered at the crack tip and translating

with it are also introduced. The crack tip is at r = 0 and the directionof crack growth is 9 = 0. For the asymptotic analysis, suppose thatthe stress components have the separable form

<7 i 3 ~^LEi (M) , * = 1,2, (4.6.5)

as r —• 0, where p is a positive real number and 5^(0, v) representsthe angular variation of the singular stress field. Order argumentsare next outlined which lead to conclusions on the value of p. Thereasoning follows that introduced by Hui and Riedel (1981) for crackgrowth under equilibrium conditions and by Lo (1983) and Brickstad(1983b) for crack growth under dynamic conditions in an elastic-viscous material. According to the assumption (4.6.5), the stresscomponents are singular at the crack tip. Consequently, only theform of g(a) for large cr, say

9(<7)~b0O<rn, (4.6.6)

where &oo is a constant, is required.

4-6 Elastic-viscous material 209

First, suppose that the elastic strain rate dominates the viscousstrain rate in the near tip region, that is, suppose that

(4.6.7)

The implication of (4.6.7) is that the asymptotic fields are governedby (4.6.3) and (4.6.4) with the second term on the right side of(4.6.4) being of negligible magnitude. The remaining equations areessentially the elasticity equations, which implies that

>,«) (4.6.8)rl/2'

or that p = 1/2. But with this result in hand, consistency with theoriginal hypothesis (4.6.7) must be checked. Withp = 1/2, the elasticstrain rate is singular as r~3/2 whereas from (4.6.6) the viscous strainrate is singular as r~n/2. Thus, (4.6.7) holds only if n < 3. Otherwise,the elastic strain rate cannot dominate the viscous strain rate.

Suppose now that the viscous strain rate is much larger than theelastic strain rate in the near tip region. Then (4.6.4) implies that

dai:

for % = 1 or 2. But if this is true, then (4.6.3) implies that

de13

—— ~ 0

(4.6.9)

(4.6.10)

which is inconsistent with (4.6.9). The hypothesis leads to a contra-diction and it is therefore false. Thus, the viscous strain rate cannotdominate the elastic strain rate in the near tip region for any value ofn.

Having shown that the viscous strain rate cannot dominate theelastic strain rate for any value of n, and that the elastic strain ratecannot dominate the viscous strain rate unless n < 3, it must beconcluded that the two strain rates are singular of the same order ifn > 3. In view of the fact that the elastic strain rate is singular as


r-(p+i) an (j that the viscous strain rate is singular as r~~np, it followsthat

p = l / ( n - l ) (4.6.11)

for n > 3.With the strength of the singularity in stress determined, it re-

mains to determine the angular variation Y,i(0,v) of the stress com-ponents. The momentum equation and the compatibility conditionfor antiplane shear deformation are given by (4.4.1) and (4.4.2), re-spectively. The particular forms of the momentum equation and thestress-strain relation for steady crack growth in the xi-direction aregiven in (4.6.3) and (4.6.4), respectively. For the present calculation,it is assumed that

g(a) = ban (4.6.12)

for n > 1 and all a > 0 in the general constitutive equation (4.6.4).The parameter 6 has the physical dimensions of length271 x time" 1 xforce"71. The qualitative features of material response included insuch a power law dependence of inelastic strain rate on stress areevident. The apparent stiffness of the material increases with the rateof deformation. The coefficient b is a measure of the viscosity of thematerial.

It is convenient to express the governing equations in terms ofdimensionless quantities. Thus, the normalized coordinates and fieldquantities

(4.6.13)7 = 2e i3,

are introduced, where cs is the elastic shear wave speed. Greek indicesrange over 1,2 throughout, and repeated indices are summed. Inlight of the notation conventions established, the constitutive equationtakes the form

where €32 has been eliminated by means of the compatibility equation.The balance of linear momentum takes the form

The auxiliary conditions in this problem include boundary con-ditions to ensure traction-free crack faces, continuity conditions oncertain fields, and remote asymptotic conditions to render the cracktip fields consistent with the outer fields. The conditions that thecrack faces are traction-free and that the displacement field is anti-symmetric with respect to the crack plane are

for & < 0 ,(4.6.16)

0) = 0 for £x>0,

where it has been assumed that the fields are continuous across £2 = 0for £1 > 0. It can be argued that the regularity requirement of Huiand Riedel (1981) is equivalent to the second of these conditions. Allfields are assumed to be continuous.

This completes a formulation of the problem of determining thecrack tip asymptotic field. In general, the nature of the solution willdepend on three parameters, namely, the crack tip speed t>, the rateexponent n, and the remote loads. Both the rate coefficient 6, whichis the material property representing the ratio of inelastic strain rateto rn , and the shear modulus // have already been absorbed into thedimensionless physical quantities. The remote asymptotic conditionswill not be considered for the time being, but any indeterminateparameters in the description of the local crack tip fields will be in-terpreted as parameters whose value can be determined only throughcoupling with the remote mechanical fields.

Attention is now focused on the near tip fields for steady crackpropagation. A local polar coordinate system r, 9 is introduced, withthe origin of coordinates moving with the crack tip and r = \/£i + £f>9 = tan"1 (£2/^1 )• For n > 3, it was shown above that, for small valuesof r,

ri(r,0) ~r-*Ti(0), T2(r,9) ~ r-pT2(0) (4.6.17)

plus terms with less singular dependence on r, where p = l/(n —1). The strength of the singularity was obtained by balancing thedominant singularities of elastic and plastic strain rate. The stressfield is anticipated to be uniform with respect to r for any crack speedm or rate exponent n. In contrast, the dependence of the deformationfield on r is expected to be nonuniform in n and ra, but this can be


demonstrated only if a full solution is known. Likewise, the straincomponent appearing in the reduced equations (4.6.14) and (4.6.15)is assumed to have the asymptotic form

7(r,0) - r~pG(0) (4.6.18)

plus terms less singular in r as r —> 0.If the assumed asymptotic forms are substituted into the field

equations and dominant terms in r are collected, then a quasi-linearsystem of ordinary differential equations is obtained in the matrixform

pcosO psinO= | pcose + T""1 0 -poos* T2 I , (4.6.19)

0 pcose + T"-1 -psine J { G) (

where T(9) = y/Ti{9)2 +T2(9)2 and a prime denotes differentiationwith respect to 9. This system is subject to the boundary conditions

Ti(0) = 0, G(0) = 0, T2(7r) = 0. (4.6.20)

The first two conditions in (4.6.19) result from symmetry of the fieldsplus continuity across 9 = 0, and the third equation ensures a traction-free crack face at 9 = ?r.

Note that the three differential equations (4.6.19) are subject tothree boundary conditions (4.6.20). Consequently, it appears that asolution of the equations will be completely determined without freeparameters for any specific values of m and n. In other words, thesomewhat surprising result is obtained (for n > 3) that the asymptoticfield is completely determined by the imposed local boundary condi-tions for a given crack speed v. The asymptotic field involves no freeparameter to be determined from remote conditions and to representthe influence of the applied loads. In this sense, the asymptoticsolution is independent of the applied loads. Suppose that some crackgrowth criterion is applied to this asymptotic solution. In view of thecharacter of the asymptotic field, it appears that a crack in an elastic-viscous material will grow at just the right speed v so that a crack

growth criterion is satisfied within the near tip region, no matter whatthe applied loading might be. However, it must be kept in mind thatthe asymptotic field has limited implication for crack growth unlessthe asymptotic solution is an accurate representation of the actualfield over a region of size greater than some characteristic dimensionof the physical separation process. This matter cannot be addressedon the basis of an asymptotic analysis alone.

At first sight, the boundary value problem represented by (4.6.19)and its boundary conditions seems to be of a standard type for whicha solution can be obtained by numerical methods. Upon closer ex-amination, however, it is seen that the coefficient matrix on the leftside of (4.6.19) is singular, or its determinant vanishes, at 9 = 0.Consequently, evaluation of the differential equation at 9 = 0 doesnot provide values of T{(0) and G'(0) which are essential in startinga numerical procedure, say a finite difference method. Thus, thefeatures of the solution near 9 = 0 are examined in order to finda way around this difficulty.

Consider the local fields in the asymptotic power series form

72ra(Q)

1 '*' (4.6.21)

for £i > 0 and £2 < £1? where K is a constant. These functions havethe symmetry and continuity properties anticipated in any solutionand they satisfy the boundary conditions (4.6.20) 1 and (4.6.20)2 on9 = 0. Note that T^{0) is expected to be zero from symmetry of thefields. If these functions are substituted into the differential equation(4.6.19), and if the lowest order terms in powers of £2 are required tobalance, then it is found that

An expression for K in terms of Ti (0) can also be found, but it is ofno consequence.


With the above values of T[(0) and G'(0) in terms of the initialvalue of T2(0) in hand, an approximate solution can be obtained byapplication of a standard numerical integration algorithm. Initialvalues are assigned as (0, T2(0),0) and the solution can then proceedby means of (4.6.19) from 9 = 0 up to 9 = ?r. For an arbitrarychoice of T2(0), it is found that any procedure leads to T2(TT) ^ 0, ingeneral. Thus, to satisfy the boundary condition at 9 = TT, a shootingmethod for solving a two-point boundary value problem is adoptedto systematically adjust the value of T2(0) for the next numericalintegration from 9 = 0 to 6 = TT on the basis of the value of T2(TT)

obtained from the previous integration, until a value is found for whichT^TT) = 0. An asymptotic solution has been found when this is thecase. This numerical analysis has been carried out by Lo (1983).

The distribution of viscous strain on the crack line 9 = 0 aheadof the crack tip was found by Lo (1983) to have the form

(4.6.23)

where T2(0) is the dimensionless constant introduced above. By nu-merical calculation, he found that the amplitude factor T2(0) de-creased from 0.662 for m = 0 to 0.470 for m = 0.7 if n = 4, andthat it decreased from 1.08 to 0.890 over the same speed range forn = 10. The result (4.6.23) illustrates features of the asymptotic fieldthat have already been discussed.

4-6.2 Plane strain crack tip field

The arguments presented in Section 4.6.1 for antiplane shear deforma-tion apply equally well in considering the case of crack growth in anelastic-viscous material under plane strain opening mode conditions.For the plane strain form of the stress-strain relation (4.6.1), the neartip stress distribution in crack tip polar coordinates is found to be

i/M)

where i or j = 1,2. Numerical results showing the dependence of theangular variation of stress components with angle 9 are presented byLo (1983).

4-7 Elastic-viscoplastic material; antiplane shear 215

4.7 Elastic-viscoplastic material; antiplane shear

The general discussion on crack growth in rate-dependent materialsbegun in the previous section is continued here by considering thesituation of steady crack growth in an elastic-viscoplastic material.In particular, an elastic-viscoplastic material is considered that showsthe same power law dependence of inelastic strain rate on stress forlarge stress magnitudes as suggested in (4.6.2), but that retains theproperties of existence of an elastic region in stress space and a yieldcondition on stress state for moderate stress levels. With reference tothe stress-strain relation (4.6.1), the distinction is made more preciseby noting that the dependence of effective viscous strain rate ev =

V.ey/3 on effective stress a indicated in Section 4.6 has the form

e« = lg(a) = jban (4.7.1)

for an elastic-viscous material, where the exponent n is a materialparameter indicating stress sensitivity of viscous strain rate. On theother hand, an elastic-viscoplastic material model having essentiallythe same dependence of inelastic strain rate on stress for large stressmagnitude, but retaining the characteristics of plastic yield at mod-erate stress levels, is ev = f#(cr), where

{ 0 for a < ao ,

* . - ( )

The material parameter n has the same interpretation as in (4.7.1)but the reference stress cro can now be interpreted as a flow stressfor slow deformations. This constitutive assumption was introducedby Malvern (1951) in a study of one-dimensional stress waves in anelastic-viscoplastic material, and it was later generalized to three-dimensional stress states by Perzyna (1963). The general structure ofconstitutive laws of this form was considered by Rice (1970).

The notation established in Section 4.6 is followed here unlessnoted otherwise. The momentum equation and the compatibilitycondition for antiplane shear are given by (4.4.1) and (4.4.2), re-spectively. For steady crack growth, the governing equations aremost conveniently expressed in terms of the normalized convected

coordinates and functions of position as

j v , m = v/c8 ,(4.7.3)

7 = 2/iei3/ro , r a = aa3/ro ,

where cs is the elastic wave speed and ro = ao/y/Z is the flow stress inshear for very slow deformation. In addition, the normalized effectivestress r = y/rara is introduced. The constitutive equation takes theform

^T~ T<*-H7-=0' « = 1,2, (4.7.4)

where €32 has been eliminated by means of the compatibility relation.The momentum balance is identical to (4.6.15) but it must be re-membered that the coordinates and fields have been normalized in aslightly different way here. Finally, the conditions that the crack facesare traction-free and that the displacement field is antisymmetric withrespect to the crack plane are

for £ i < 0 ,(4.7.5)

0) = 0 for & > ( ) .

It is instructive to consider the elastic-viscoplastic constitutiverelationship in a very simple uniaxial form so that the qualitativefeatures of the response become clearer. To this end, let T2 = 0 in(4.7.4), so that |TI| = r. Furthermore, —£1 is identified as a timelikeparameter and 7 is the magnitude of the normalized :Ei-componentof strain. Thus, with r and 7 as the measures of stress and strain,respectively, the relationship is

7 = r + < T - l ) n , (4.7.6)

where the superimposed dot denotes differentiation with respect to thetimelike parameter and the angle brackets (•) are used to indicate thatthe value of the term is zero if the enclosed quantity is nonpositive.

4-7Elastic-viscoplastic material; antiplane shear 217

7P

a,p

high strain rate —7

low strain rate

high strainrate

low strain rate

Figure 4.15. Schematic diagram of material response according to (4.7.6) underconditions of imposed uniform strain rate from zero initial strain.

Consider material response according to (4.7.6) under conditionsof imposed strain rate from zero initial strain. If 7 has a very smallconstant value greater than zero, then f is also very small. The stressrate f is proportional to 7 until r = 1, and r is close to one thereafter.The response is essentially an elastic-perfectly plastic response, asindicated in Figure 4.15. On the other hand, if 7 has a very largeconstant value, then f is equal to 7 up to r = 1, and thereafter rasymptotically approaches the stress level 1 + 71/71 as 7 becomes verylarge. This behavior is also indicated in Figure 4.15.

Consider now the response to a nonuniform strain rate havingthe general continuous variation with strain shown in Figure 4.16.The dependence has a maximum at some strain level, and it has azero at some higher strain level. The strain necessarily begins todecrease once the strain rate becomes negative. The response to suchan imposed strain rate according to (4.7.6) is shown in the lower partof Figure 4.16 where the algebraic signs of the elastic rate Y = f andthe viscoplastic strain rate 7P = (r — l) n vary along the stress strainpath. For example, j e = 7 > 0 and j p = 0 in oa; 7 > 0, j e > 0,and 7^ > 0 in ab; 7 > 0, Y < 0, and j p > 0 in 6c; 7 < 0, Y < 0,

Figure 4.16. Schematic diagram of material response according to (4.7.6) underconditions of imposed nonuniform strain rate from zero initial strain.

and 7P > 0 in cd; and 7 = Y < 0 and 7** = 0 in de. A materialparticle close to the crack path in the crack growth problem understudy is expected to experience a stress-strain history having thesegeneral features, but under multiaxial conditions. The stress may besingular in a region of elastic-viscoplastic response, but the materialcan unload elastically only from a stress level of r = 1.

The understanding of the asymptotic crack tip field for the situa-tion under discussion is far from complete, so this section is concludedwith some general observations in light of the results described previ-ously for the case of elastic-viscous response. Suppose that the term(r — l)n in the constitutive equation is expanded by means of thebinomial theorem, so that

In any region near the crack tip in which r > 1 the first termin this expansion is much larger than any of the succeeding terms.Indeed, if (r — l)n is approximated by rn in such a region, thenthe constitutive equation (4.7.4) is identical to the equation (4.6.14)for the elastic-viscous case. The elastic-viscous material admits an

4-7 Elastic-viscoplastic material; antiplane shear 219

asymptotic solution that is singular as r~p, where p = l/(n — 1), forn > 3, so the elastic-viscoplastic material admits a singular solutionof exactly the same strength over a sector of some angular extentaround the crack tip. For the elastic-viscous material, the asymptoticsolution was found to be singular over the entire range of 0 in theinterval 0 < 0 < n. Is the same true for the elastic-viscoplastic case?

For the case of elastic-viscoplastic response, suppose that

r(r,e)~r-PT(6) (4.7.8)

near 0 = 0, where the notation of the previous section has beenadopted, and suppose that T(0) > 0. This seems to be reasonablebecause plastic strain is concentrated ahead of the tip for mode IIIdeformation. As 0 increases from zero, it appears that (4.7.8) providesthe correct asymptotic form for the solution as long as T(0) > 0. Theform (4.7.8) could give way to an unloading sector or some other typeof region, say at 0 = 0*, only if T(0*) = 0. This matter has not beenthoroughly studied, and a complete description of the asymptotic fieldwhich accounts for all possibilities is not yet available.

It was observed by Hui and Riedel (1981) for the case of crackgrowth through an elastic-viscous material under quasi-static condi-tions and by Lo (1983) for dynamic crack growth through the samematerial that the computed asymptotic fields do not seem to con-verge to the commonly accepted asymptotic solution for steady crackgrowth through a rate independent elastic-plastic material reportedby Chitaley and McClintock (1971) as n —• oo. In a sense, this isnot too surprising because the limiting response of the elastic-viscousmaterial model as n becomes indefinitely large is not the same asthe rate independent elastic-perfectly plastic response. For example,it does not exhibit an elastic region in stress space. This elasticregion plays a significant role in the rate independent case. It isalso interesting to consider this general question from the point ofview of asymptotic analysis. For either the elastic-viscous or elastic-viscoplastic models considered here, the dominant term in the cracktip asymptotic solution has the radial dependence r~p. The nextterm in the expansion in either case appears to be of order unity asr —• 0. However, the observation that the equations determining thiscontribution are slightly different for the two cases, due to the secondterm in the expansion (4.7.7), is potentially significant. In either case,the term that is O(r~p) as r —• 0 will dominate the second term of


order unity only over a region with radial coordinates r for whichr-p ^> i As n —> oo or as p —• 0, the size of this region of dominancevanishes. In other words, as n becomes large, the second term in theasymptotic expansion appears to be as significant as the first term.Perhaps a study of the higher order terms in the crack tip asymptoticsolution will help to resolve the apparent discrepancy among limitingcases as n —> oo. Results on this question are not yet available.

5

IN DYNAMIC FRACTURE

5.1 Introduction

Analytical methods based on the work done by applied loads and thechanges in the energy of a system that accompany a real or virtualcrack advance have been of central importance in the developmentof fracture mechanics. These methods have provided a degree ofunification of seemingly diverse ideas in fracture mechanics, and theyhave led to procedures of enormous practical significance for the char-acterization of the fracture behavior of materials. In addition, someof the most elegant theoretical analyses in the field have been thoseassociated with energy methods. In this chapter, energy concepts thatare particularly relevant to the study of dynamic fracture processesare considered.

The importance of the variation of energy measures during crackgrowth was recognized by Griffith (1920) in his pioneering discussionof brittle fracture, as outlined in Section 1.1.2. The extension of acrack requires the formation of new surface, he reasoned, with itsassociated surface energy. Consequently, a crack in a brittle solidshould advance when the reduction of the total potential energy of thebody during a small amount of crack advance equals the surface energyof the new surface thereby created. For an elastic body containing acrack, the negative of the rate of change of total potential energywith respect to crack dimension is called the energy release rate. Thisquantity, which is usually denoted by the symbol G, is a function

221

222 5. Energy concepts in dynamic fracture

of crack size, in general. Prom its definition, G is the amount ofenergy, per unit length along the crack edge, that is supplied by theelastic energy in the body and by the loading system in creating thenew fracture surface. In this sense, G is a generalized force that iswork-conjugate to the amount of crack advance. Consequently, Ghas also become known as the crack driving force. These ideas havebeen exploited and extended in a host of ways in fracture mechanics,including dynamic fracture mechanics.

Irwin (1957) recognized the role of the elastic stress intensityfactor as a characterizing parameter for the elastic crack tip field and,on this basis, he proposed the critical stress intensity factor fracturecriterion as discussed in Section 3.6. The energy criterion and thestress intensity factor criterion were then shown to be equivalent foran elastic-brittle material under equilibrium conditions by means ofan energy analysis presented by Irwin (1957, 1960). The essentialfeatures of the analysis are summarized here. Consider a planar crackin a homogeneous and isotropic elastic solid under plane strain modeI conditions. If x is a spatial coordinate in the direction of prospectivecrack advance with its origin at the crack tip position, then the tensilestress distribution on the prospective fracture plane for points veryclose to the crack tip compared to any physical dimension of theconfiguration is

£ (5.1.1)

where Kj is the mode I stress intensity factor; see (2.1.1). Imaginethat the crack then advances a small distance Ax under equilibriumconditions. After advance, the outward normal displacement of eithercrack face near the crack tip is

for x < Ax, where E and v are the elastic constants and AKj is thechange in stress intensity factor during the crack advance. The energythat was released from the body and loading system in going from theconfiguration before crack advance to the configuration after advanceis the work that must be supplied to return the latter configurationto the former. In view of the linearity of the system, that work is

AW = / Xa(x)u(x)dx = ^—-^IK} (l + ^±) Ax. (5.1.3)Jo ^ \ Ki J


The energy release rate is then obtained in terms of the stress intensityfactor by means of its definition above as

E(5.1.4)V }

where it has been assumed that AKJ/KI —• 0 as Ax —• 0. The rela-tionship (5.1.4), which is commonly known as the Irwin relationship,is of fundamental importance in fracture mechanics. It provides anunambiguous link between stress intensity factor and energy releaserate for elastic crack problems. Furthermore, if crack growth criteriaare postulated in terms of critical values of these parameters, then(5.1.4) implies that values of these material specific parameters cannotbe independent. The form of (5.1.4) with (1 — v2) replaced by unity,which is appropriate for two-dimensional plane stress, was given as(1.1.8).

The relationship (5.1.4) is also at the root of the so-called compli-ance methods of fracture testing introduced by Irwin and Kies (1954).As noted above, the energy release rate is the change in potentialenergy of a system with respect to crack length. This change can beexpressed in terms of the overall compliance of the body and the ap-plied loading. Likewise, the energy release rate is related to the cracktip stress intensity factor through (5.1.4). Thus, if the complianceversus crack length can be estimated or measured separately, thenthe energy approach provides a direct relationship among load level,crack length, and stress intensity factor. For example, for the splitbeam configuration used to motivate the energy approach in Section1.1.2, the relation G = -dtt/dl implies that

* - #

in view of the Irwin relationship.The path-independent J-integral introduced by Eshelby (1956)

and Rice (1968) provides a more general means of relating crack tipfields to remote loading conditions in cracked solids. The path inte-gral method was generalized to the case of nonlinear elastic materialresponse for which Hutchinson (1968) and Rice and Rosengren (1968)showed that the value of J provides a crack tip field characterizingparameter, similar to the stress intensity factor in linear fracture


mechanics. These key ideas involving energy concepts in equilibriumfracture mechanics provide the basis for the development of similarideas in dynamic fracture, as described in the sections to follow.

5.2 The crack tip energy flux integral

A crack tip contour integral expression for elastodynamic energy re-lease rate was proposed by Atkinson and Eshelby (1968) who arguedthat the form for dynamic crack growth should be the same as forquasi-static growth with the elastic energy density replaced by thetotal mechanical energy density, that is, the elastic energy plus thekinetic energy. The integral expression for dynamic energy release ratein terms of crack tip stress and deformation fields was subsequentlyderived directly from the field equations for general material behaviorby Kostrov and Nikitin (1970) and for elastodynamics by Preund(1972c). They enforced an instantaneous energy rate balance for thetime-dependent volume of material bounded by the outer boundaryof the solid, the crack faces, and small loops surrounding each movingcrack tip and translat ng with it. By application of the generalizedtransport theorem (1.2.41) and the divergence theorem, an expressionfor crack tip energy flux in the form of a path integral of field quantitiesalong the crack tip loops was obtained. A result that is applicableto a broad range of material response and that contains the earlierelastodynamic result as a special case is derived here. Although avariety of seemingly different path-independent energy integrals havebeen proposed, each can be extracted from this general result byinvoking the appropriate restrictions on material response and cracktip motion.

5.2.1 The energy flux integral for plane deformation

In the absence of body forces, the equation of motion in terms ofrectangular components of stress &ij and displacement ui is

where p is mass density. If the inner product of (5.2.1) with theparticle velocity dui/dt is formed, then the resulting equation can be

5.2 The crack tip energy flux integral 225

Figure 5.1. A crack tip moving with the instantaneous speed v. The contour Ttranslates with the crack tip.

written in the form

where U is the stress work density and T is the kinetic energy density,that is,

at any material point, as defined in Section 1.2.2. There are no con-stitutive assumptions implicit in the definition of stress work density.

If the time coordinate is now treated in the same way as aspatial coordinate, then (5.2.2) is the requirement that a certain vectorexpression is divergence-free in a particular "volume" of space-time,that is, it represents a mechanical conservation law. Thus, if theexpression is integrated over this volume and the divergence theoremis applied, the integral of the inner product of the divergence-freevector with the surface unit normal vector, taken over the boundingsurface of this volume, vanishes. This idea is developed below.

Consider a two-dimensional body that contains an extendingcrack. Suppose that the outer boundary of the body is the planecurve C. Rectangular coordinates are introduced so that the planeof the body is the xi,X2-plane, the crack plane is x^ = 0, and crackgrowth is along the xi-axis. The instantaneous crack tip speed isv. A small contour T begins on one traction-free face of the crack,surrounds the tip, and ends on the opposite traction-free face. Thiscrack tip contour is fixed in both size and orientation in the cracktip reference frame as the crack grows; see Figure 5.1. Consider the

volume in the three-dimensional #1, #2, £-space that is bounded by theplanes t — t\ and t — t^^ where t\ and £2 are arbitrary times, by theright cylinder swept out by C between the two times, by the planesswept out by the crack faces between the two times, and by the tubularsurface swept out by the small contour F as the crack advances in theXi-direction with increasing time. The "bottom" of the volume is theconfiguration of the body at time t\, the "top" of the volume is theconfiguration of the body at time £2, and the lateral surface is the timehistory of C. The outward unit normal vector to the bounding surfaceof the volume of space-time is denoted by v which has componentsi> 2 5 vt in the coordinate directions. A second unit normal vector is

defined for the plane body at any fixed time. This vector is denotedby n and it is directed out of the material for points on C or on thecrack faces, and it is directed away from the crack tip for points onF. Note that n\ ^ v\ and n<i ^. v^, in general, but these componentsare equal at points where vt — 0. The general approach is similar tothat followed in deriving energy integrals in the theory of hyperbolicpartial differential equations (Courant and Hilbert 1962).

If the quantity (5.2.2) is integrated over the volume in space-timeand the divergence theorem is applied to the result, then

f ajirij^dCdt- f (U + T)dA + f (U + T)dAJC Ut JA(t2) JA(ti)

= - I [<W ^ T - (U + T)ut] dS, (5.2.4)JTube L dt J

where A(t) is the cross-sectional area of the volume on any plane t =constant and the integral on the right-hand side is the surface integralover the portion of the tube between times t\ and £2- Because the tubeis formed by translating the fixed contour F in the #1-direction atinstantaneous speed v, the components of v are related by vt = —vv\.Thus, the right side of (5.2.4) takes the form

/ Wlir l i dS'Tube L Ot J

where <5 is the Kronecker delta operator (1.2.6). Suppose that thesurface of the tube is parameterized in terms of surface coordinatest and 5, where the arclength s is measured along F at constant t

from some arbitrary point. Then, from the theory of surfaces, thearea element Vj dS is the element — rij ds dt in terms of the surfacecoordinates. The relationship (5.2.4) becomes

T)dA

ds dt'

The term on the left side of the equal sign in (5.2.6) is the totalwork done on the body by tractions applied to either C or the crackfaces in the time interval of interest, and the first two terms on theright side represent the net stress work done plus kinetic energy addedduring the same time. Consequently, the last term in (5.2.6) is thetotal energy lost from the body due to flux through F during theinterval t\ < t < t^- Because this time interval is completely arbitrary,it follows immediately that the crack tip line integral

f2 f (7jin~dCdt= ( (U + T)dA^ f

I \a3in3 1& + (U + T)VT

F(T) = J lajiTij £ + (U + T)vm] ds (5.2.7)

is the instantaneous rate of energy flow through F toward the cracktip. From its definition, F is the flux of total energy. It is the fluxof kinetic energy plus free energy only under the special conditions ofadiabatic and isentropic deformation. The energy flux has the physicaldimensions of energy per unit time per unit width, or force/time. Thisresult has also been obtained by Willis (1975).

5.2.2 Some properties of F(T)

The interpretation of the terms in (5.2.7) is straightforward. Thefirst term is the rate of work done on the material inside of F bythe tractions acting on F. If the curve F were a material line, thenthis would be the only contribution to the energy flux. The curve ismoving through the material, however, and the second term in (5.2.7)represents the contribution to the energy flux due to transport ofmaterial through F, with its associated energy density U' + T.

The energy flux integral expression (5.2.7) was derived within theframework of infinitesimal deformations, and no distinction was made


between the deformed and undeformed configurations of the body.In fact, the integral is valid for finite deformation if the terms areproperly interpreted. Suppose that a deformable body is undergoinga deformation of arbitrary magnitude with respect to some referenceconfiguration. A common rectangular coordinate system is assumed,and components of vectors and tensors are defined in reference to thecoordinate directions. Suppose that Ui denotes the time-dependentdisplacement of the particle at location Xi in the reference configura-tion, so that dui/dt is the velocity of the particle at Xi in the referenceconfiguration. Suppose that F is a surface with unit normal vector U{in the reference configuration that moves with respect to the materialparticles instantaneously on it with normal speed v. If p is the massdensity of the material in the reference configuration and cr is thetranspose of the nominal stress (or the first Piola-Kirchhoff stress)then (5.2.7) represents the instantaneous energy flux through F. Thework rate balance for a continuum undergoing finite deformation isdiscussed in a form useful for considering the energy flux by Ogden(1984), as outlined in Section 1.2.2. The energy flux integral for theparticular case of finite deformation of a hyperelastic material wasdiscussed by Gurtin and Yatomi (1980).

The integral defining F itself is not path-independent, in general,as can be seen by means of a direct check. Consider the closed pathformed by two crack tip contours Fi and F2, and by the segments ofthe crack faces that connect the ends of these contours. Applicationof the divergence theorem to the energy flux integral for this entireclosed path leads to the result that

where A\2 is the area within the closed path. Without loss of gen-erality, the algebraic signs in (5.2.8) are chosen to be consistent withthe normal vector to Fi directed into A\% and the normal vector toF2 directed away from A\2- Except in special circumstances, theintegrand of (5.2.8) is not zero, so that F(Fi) / F(r2) in general.The path dependence can be understood quite readily in terms of thewave character of the fields represented by the integrand. For any


choice of I \ and F2, transient stress fields can be introduced whichinfluence the energy flux through one contour but not through theother.

One special case for which the integral is indeed path-independentis that of steady crack growth. If the mechanical fields are invariantin a reference frame traveling with the crack tip in the x\ -direction ata uniform speed v, then any field quantity depends on X\ and t onlythrough the combination £ = x\ — vt. In this case the area integral in(5.2.8) vanishes at each point in the plane, implying that the energyflux integral (5.2.7) is independent of path F for steady crack growthsituations. The energy flux integral takes the special form

F ( r ) = v I \{U + T)m - (TjiUj^-11 dT , (5.2.9)

where, in this case,

This property of path-independence is exploited in analyzing crackgrowth problems in Sections 5.3, 5.4, and 8.3.

The energy flux integral can be extended to the case of three-dimensional deformation fields. If F is reinterpreted as a surfacein space with local unit normal vector with components rii and ifthe surface is translating in the direction defined by the unit vectorwith components rrii with instantaneous speed v, then the energy fluxthrough this surface is

F{Y) = jT \ a ^ + (U + T)vm^ nj dV, (5.2.11)

where the integration is taken over the surface F. In this case, F hasthe physical dimensions of forcexlength/time. The result (5.2.7) or(5.2.11) is a purely mechanical result that does not depend on theexistence of a thermodynamic equation of state. All applications tofollow are based on this mechanical result.

An expression similar to (5.2.11), including internal energy andheat flux quantities, was presented by Kostrov and Nikitin (1970)


on the basis of the first law of thermodynamics, and the result wasfurther discussed by Nilsson and Stahle (1988). The existence of aninternal energy presumes an underlying thermodynamic equation ofstate (Malvern 1969). If an equation of state exists as a constitutiveassumption, then an equivalence between (5.2.11) and the expressionproposed by Kostrov and Nikitin (1970) can be established. In theabsence of heat conduction or any other nonmechanical energy trans-fer modes, the two forms are the same if the quantity U in (5.2.11) isidentified as the internal energy of the material per unit volume. Inthe presence of heat conduction (but no other nonmechanical energytransfer modes), the results are equivalent if U is related to theinternal energy per unit mass e by

where p is the material mass density and qj are the componentsof the heat flux vector. This expression states that the internalenergy density and the stress work density at a point differ by anamount equal to the net heat flow away from that point. When(5.2.12) applies, terms in the energy-momentum equation (5.2.2) canbe regrouped as

Application of the divergence theorem then yields the energy fluxexpression given by Kostrov and Nikitin (1970).

Finally, it is noted that the energy flux integral does not dependon the fact that the energy flux is associated with crack propagation.Rather, the expression (5.2.7) is a general expression for energy fluxthrough a surface translating through a deforming solid. If the totaldeformation rate, that is, the symmetric part of d2v,i/dxjdt, is viewedas the sum of an elastic deformation rate and an inelastic deformationrate, then the stress work density U can also be viewed as the sum ofan elastic (recoverable) part and an inelastic (dissipated) part. Thenthe flux of recoverable energy through the contour T is given by (5.2.7)with U replaced by the elastic part. An example of this separation ofU into elastic and inelastic parts appears in Section 8.3.

5.3 Elastodynamic crack growth 231

5.3 Elastodynamic crack growth

Applications of the general energy flux integral F{T) for the case ofelastodynamic crack growth are considered next. For linear elasticmaterial response, the stress work density is the strain energy den-sity U = \(Jijeij = \Cijkieijtki = \MijkiGi:j(jki, where Mijki is thearray of elastic compliance parameters for the material. For isotropicresponse

v 1M^ki — — , —zdijSki + x-(^fc^j/ + SiiSjk) (5.3.1)

in terms of the shear modulus fi and Poisson's ratio v. The elasticwave speeds are defined in terms of elastic moduli and the massdensity in Section 1.2.

5.3.1 Dynamic energy release rate

The dynamic energy release rate is defined as the rate of mechanicalenergy flow out of the body and into the crack tip per unit crackadvance. The energy flux, or energy flow per unit time, must bedivided by v, the crack movement per unit time, in order to obtain theenergy released from the body per unit crack advance. The dynamicenergy release rate is denoted by G and it is defined by

I v J(5.3.2)

drv Jv

where ni is the unit normal vector to F directed away from the cracktip and the limit implies that the contour F is shrunk onto the cracktip. For two-dimensional fields, the quantity G is the mechanicalenergy released per unit crack advance per unit thickness of the bodyin the direction perpendicular to the plane of deformation, so that thephysical dimensions of G are force/length.

For the energy release rate concept to have a fundamental signif-icance, the value calculated according to (5.3.2) must be independentof the shape of F. For linear elastic material response, the integral

(5.2.8) expressing the difference in values of the energy flux for twoarbitrary crack tip paths Fi and F2 becomes

where the notation is as established in connection with (5.2.8). It isclear from the asymptotic analysis in Section 4.3 that, in the near tipregion, ui depends on xi and t only through x\ — vt. If both contoursFi and F2 are within the near tip region, then the terms in parenthesesin (5.3.3) vanish and the value of F(T) is indeed path-independent forall paths within the near tip region. Consequently, the value of G willnot depend on the choice of path F used to compute it.

In view of its definition (5.3.2), G is determined by the near cracktip mechanical field. In addition, the near tip spatial distributionof mechanical fields was established in Chapter 4 for each mode ofdeformation. Only the scalar multiplier, the elastic stress intensityfactor, was undetermined. Consequently, the terms in (5.3.2) canbe expressed in terms of their near tip universal spatial descriptionsand the integral can be evaluated to obtain a relationship betweenthe energy release rate and the stress intensity factor. For modeI deformation, the near tip stress and particle velocity componentshave the form

(5.3.4)

in terms of crack tip polar coordinates r, 0, where v is the instanta-neous crack tip speed and the functions E^ and V* are given explicitlyin (4.3.11) and (4.3.14), respectively. Thus, the integrand of (5.3.2)is vK$B(0)/2nrp, where

B(0) = XijUjVi + iniMM<i*iEyEiM + ^nMVi. (5.3.5)C8

The integrand has several notable features. For instance, the crackspeed v appears as a common factor, canceling the l/v multiplier in


Figure 5.2. A particular choice of the contour F in (5.3.2) that is useful inestablishing the relationship between energy release rate and stress intensity factorin (5.3.9).

the definition of G. Furthermore, the integrand depends on r onlythrough the common factor 1/r. If F is chosen to be a circular pathof radius r centered at the crack tip then

/J—

B{9) d6. (5.3.6)

Thus, G is proportional to K\j\x and the dimensionless proportion-ality factor depends on the instantaneous crack speed v and on Pois-son's ratio for the material. The integral (5.3.6) was evaluated for aparticular case by Atkinson and Eshelby (1968).

The value of the proportionality factor is obtained here by choos-ing the contour F to be the rectangle shown in Figure 5.2. This is aconvenient choice because rii = 0 along the segments parallel to thexi-axis. Furthermore, if the rectangle F is shrunk onto the crack tipby first letting 62 —• 0 and then 61 —• 0, there is no contribution toG from the segments parallel to the X2-axis. Consequently, G canbe computed by evaluating only the first term of (5.3.2) along thesegments parallel to the #i-axis, that is,

G = 2 lim I lim /6l-0 |^2->Oi/j_01

0 2 ,dt

, (5.3.7)

where I is the instantaneous re 1-coordinate of the crack tip and thefactor 2 is introduced to account for the side of the rectangle at X2 =—62 by symmetry.


For the case of plane strain mode I deformation, the integrand of(5.3.7) is

^ 47T/,

where D = 4ofdas - (1 + a2)2 , as in (4.3.8). If the integral in (5.3.7)is then evaluated, the result is

If the computation is repeated for mode II (plane strain) and modeIII deformations, then it is found that

G = —-^- [Aj{y)Kj + An(v)Kjj] + —Ani{v)KjU , (5.3.10)

where

= —• (5.3.11)a(1 — v)c2

sD (1 — v)c%D a

The functions Aj, AJJ, and Am are also universal functions, in thesense that they do not depend on the details of the applied loading oron the configuration of the body being analyzed. They do depend onthe crack speed and on the properties of the material. Each functionin (5.3.11) has the properties that A —• 1 as v —> 0+, dA/dv —• 0as v —> 0+, and A = O[(CR — v)'1] as v —• CR for plane strain, orA = O[(cs — v)~x] as v —•> cs for antiplane strain. The variation ofeach function A(v) with v is monotonic. For plane stress conditions,the factor 1 — v2 in (5.3.10) is absent and the wave speeds must beadjusted to account for the lesser in-plane stiffness of the material.

As noted in Section 5.1, the relationship between stress intensityfactor and energy release rate under equilibrium conditions (v —» 0)was first obtained by Irwin (1957, 1960). The result (5.3.10) thusgeneralizes the Irwin relationship to the case of elastodynamics. Thequantity G introduced by Irwin, as well as the corresponding elastody-namic quantity introduced here, refers to the total mechanical energy


change that is not taken into account in a continuum description ofthe crack advance process. Although G is sometimes called the elasticenergy release rate, this terminology is strictly correct only if thecrack advances under equilibrium conditions and under the fixed gripboundary condition. The fixed grip condition precludes the exchangeof mechanical energy between the elastic body and its surroundings.If the deformation during crack advance is always at equilibrium,then the only global energy measure that can change its value is theelastic energy and, consequently, the energy release rate is exactlythe elastic energy change due to crack growth. For crack growth withmore general boundary loading conditions and with inertial resistancetaken into account, energy variations may include exchanges with thesurroundings and/or changes in kinetic energy. Under these generalconditions, G cannot be related to elastic energy variations alone.Instead, G is the total mechanical energy release rate. For example,for dynamic crack growth and fixed grip conditions, there can be anexchange between elastic and kinetic energy of the body through wavereflection at a remote boundary. This variation in elastic energy hasno immediate influence on the energy release rate G. The significanceof the energy release rate concept, and on its role in crack growthcriteria, will be discussed further in Chapters 7 and 8.

5.3.2 Cohesive zone models of crack tip behavior

The study of the advance of a sharp crack edge on the basis of anelastodynamic model leads naturally to the result that the stress issingular at the crack edge. The result cannot be accepted literally,of course. Instead, the result is accepted on the basis that the sizeof the region over which the material behavior deviates from linearelasticity is too small to be detected on the scale of the model. Thecohesive zone idea provides a simple yet useful device for examiningcrack tip phenomena at a level of observation for which deviationsfrom linearity in material response are admitted.

The essential idea of the cohesive zone model for plane strainmode I crack growth under the simplest circumstances is as follows.A tensile crack advances in a body under the action of some appliedloading, and the configuration is such that a stress intensity factor fieldof the form (4.3.10) is fully established around the crack edge. Thestress intensity factor due to this field alone is called the applied stressintensity factor Kjappi. To simulate material response beyond thelinear elastic range, the opening of the crack behind the mathematical

crack tip at x\ = l(t) is resisted by a traction distribution calledthe cohesive traction or the cohesive stress, say <J(£) for —A < £ <0, where £ = xi — Z(£), acting on both crack faces. The cohesivetraction leads to a negative stress intensity factor Kicoh at £ = 0.For a given cohesive stress, the total stress intensity factor is the sumKiappi + Kicoh = Ki- It is now observed that the total stress can berendered nonsingular by choosing the length of the cohesive zone Ato satisfy the condition Ki = 0. It is noted here that for the case ofmode III, the use of this device to eliminate the crack tip singularitydoes not result in all stress components being bounded everywhere.

To pursue the cohesive zone concept somewhat further, con-sider the specific case of crack growth by simple cleavage of a ma-terial. That is, the advance of the crack is viewed as the breaking ofatomic bonds between pairs of atoms that end up on opposite facesof the advancing crack. A unique relationship between bonding forceand separation distance is assumed. This situation was studied byBarenblatt (1959a, 1959b) on the basis of a cohesive zone model forequilibrium conditions, and the same idea was subsequently appliedto the situation of dynamic cleavage crack growth by Barenblatt,Salganik, and Cherepanov (1962). In this case, the cohesive stress(j depends on position £ with respect to the mathematical crack tiponly through the separation of the crack faces at that position, say£(£). The crack opening displacement is zero at the mathematicalcrack tip, so that 6(0) = 0. The opening increases with distancebehind the mathematical tip, and it reaches some final value at theend of the cohesive zone, that is, at £ = —A, where the cohesive stressis effectively reduced to zero. The end of the cohesive zone is thephysical crack tip in this model, and the opening displacement therehas special significance. It is denoted here by <$(—A) = 6f.

Suppose that, on a laboratory scale, the cleavage crack appearsto be a perfectly sharp crack and to be growing at speed v underthe action of an applied stress intensity factor Kiappi as defined inSection 4.3. On a finer scale of observation, the same crack appearsto open gradually from the leading edge of the cohesive zone (themathematical crack tip at £ = 0 where the limit of linearity of materialbehavior is presumably reached) to its full opening 6t at the physicalcrack tip at £ = —A. The final opening 6t is determined by theinteratomic force law assumed for the material. There must be aconnection between the applied stress intensity factor Kiappi and theparameters of the cohesive zone model, and it is the purpose here

Figure 5.3. Two crack tip contours for the evaluation of the energy flux integral(5.2.7) in order to relate the crack tip parameters to the remote loading parame-ters.

to illustrate this connection on the basis of energy arguments. Thisequivalence was first examined and the conditions under which it holdswere established by Willis (1967a) by means of direct stress analysis.

Consider two crack tip contours as shown in Figure 5.3. Thecontour Tcoh embraces the cohesive zone, whereas the other contourFappi is circular in shape, with center on the mathematical crack tipand radius large compared to A. Suppose that the larger contourpasses through the region where the fields are accurately described bythe stress intensity factor field. In view of (5.3.10), the energy fluxthrough Tappi is

F(Tappl) = l-=^LvAl{v)K?c . (5.3.12)

On the other hand, the fundamental energy flux integral (5.2.7) leadsimmediately to the result that the energy flux through the cohesivezone contour is

F(Tcoh) = 2 / <T2jUjdZ, (5.3.13)J-A

where the symmetry property of mode I deformation has been incorpo-rated. But in the cohesive zone, 6(£,t) = 2w2(£, 0, t), <x12(£, 0,t) = 0,and <722(£?0, t) = cr{8). Therefore,

F(Tcoh)= f <r(6)6(M)dt, (5.3.14)J-A

where the superposed dot denotes a material time derivative. Thus,

) = ^ - v ^ , (5.3.15)

so that the energy flux takes the form

F(Tcoh) = / a(6)— d£ + v / <r(0)d<5. (5.3.16)J-A ut Jo

The second term in the energy flux is the rate of work that must besupplied per unit surface area in order to separate the crack faces toa separation distance 6t, thereby overcoming the cohesive stress. Thefirst term reflects the fact that the rate of work of the cohesive stressdepends on internal adjustments associated with nonuniform growthwithin the cohesive zone.

If the crack growth process is steady as seen by an observermoving at uniform speed v with the crack tip, then the first termon the right side in (5.3.16) is identically zero. Furthermore, underthese conditions, the energy flux integral is path-independent. Thus,in the case of steady crack growth, F(Tcoh) = F(Tappi) or

1 - v2 o f6t

A!(v)Kfappl = I a(6)d6. (5.3.17)E Jo

This result establishes a relationship between the prevailing appliedstress intensity factor and the cohesive property of the material, rep-resented by the interaction law embodied in cr(6). The plane stressresult corresponding to (5.3.17) is obtained by a simple redefinitionof the material constants.

If the notion is adopted that the cohesion arises from atomicinteraction during growth of a cleavage crack, then the term on theright side of (5.3.17) is twice the surface energy of the material. Thispoint, which was mentioned briefly in Section 1.1.2, will be pursued inSection 7.4, where crack growth criteria are considered. The matter ofdetermining the size of the cohesive zone A is a stress analysis issue,and it is considered in Section 6.2. However, it is emphasized herethat (5.3.17) is valid only for a cohesive zone that is small enoughto be completely surrounded by a fully established stress intensityfactor field characterized by Kiappi. Strictly speaking, it is validasymptotically as the length of the cohesive zone becomes vanishinglysmall.

A cohesive zone model of crack tip plasticity was introducedby Dugdale (1960). The physical phenomenon of interest was thehighly localized plastic deformation ahead of the tip of a crack in a

thin sheet of a ductile material subjected to tension in the directionperpendicular to the crack line. The mathematical model is similar tothat already introduced above and, because the model circumvents thepath dependent nature of plastic flow, it can be applied for growingas well as stationary cracks. The cohesive stress is taken as the tensileflow stress ao of an ideally plastic solid. In the case of steady dynamiccrack growth under plane stress conditions the energy flow per unitcrack advance into the cohesive zone is simply ao8t. If the cohesivezone is completely surrounded by a stress intensity factor field withparameter Kiappi, then

ii (5.3.18)

where the plane stress form of Ai(v) is assumed.Yet another cohesive zone model with particular relevance to

shear cracks was introduced by Ida (1972, 1973) to analyze dynamicearth faulting phenomena and by Palmer and Rice (1973) to analyzestability of a slope under gravitational loading. This model alsohas the feature that the cohesive stress representing the interactionbetween the crack faces depends on the relative slip between thefaces. For a plane strain mode II crack occupying the half planeX2 = 0, x\ < vt advancing in the xi-direction steadily at speed v,the relative coordinate £ = x\ — vt is again introduced. The relativeslip between the crack faces is <5(£) = wi(£, 0+) — Mi(£,0~) and thecohesive shear traction is denoted by r(6). The essential features ofthe model are as follows. The shear stress ai2 in the unslipped portionof the plane £ > 0 is assumed to increase to a characteristic value r8fas £ decreases toward the mathematical crack tip at £ = 0. Themagnitude of r8f may be loosely identified with the static frictionalstrength of the interface along which the shear crack advances. At themathematical crack tip the relative slip is zero, that is, 6(0) = 0. As£ decreases further, the crack faces begin to slide so that 6 increases.As 6 increases, the cohesive shear stress r(6) decreases monotonically.When the relative slip has increased to the level 6t, the cohesive stresshas been reduced to a characteristic value r^/. The latter value hasbeen identified with the dynamic friction strength of the interface.For values of 6 greater than 6t the relative motion of the crack faces isresisted by a uniform shear traction of magnitude r<#\ The cohesivezone properties are depicted in Figure 5.4. This model is commonlycalled the slip-weakening model because, once the strength limit is


vT,'df

6t 6

Figure 5.4. The relationship between cohesive zone traction r and the amount ofrelative slip 6 for the slip-weakening model. The shaded area is the energy releaserate in (5.3.19).

achieved on the fracture plane, the cohesive stress magnitude falls offas slip progresses.

The excess of the shear stress over and above the residual levelTdf is the cohesive stress that must be overcome to advance the crack.For steady growth of a plane strain mode II shear crack under theseconditions, the energy that must be supplied per unit crack advanceis

G= / [T{6) - rdf] d6 , (5.3.19)Jo

which is shown as the shaded area in Figure 5.4.

5.3.3 Special forms for numerical computation

In elastodynamic crack growth analysis based on numerical finiteelement or finite difference methods, it is difficult to calculate thedynamic energy release rate by evaluating the expression for G fromthe fundamental definition in (5.3.2). The difficulty arises from thefact that G is given in (5.3.2) in terms of fields evaluated at pointsarbitrarily close to the crack tip. In numerical approaches, valuesof fields are available only at discrete points and these values areleast accurate within the high gradient region around the crack tip.Two computational algorithms that have been successfully applied forcalculating G are briefly described here.

Consider first a method based on (5.2.8) which is known as thearea domain integral method (Kishimoto, Aoki, and Sakata 1980c;Nishioka and Atluri 1983; Nakamura, Shih, and Freund 1985a). Therelation (5.2.8) is valid for any contours Fi and F2 encircling a simplyconnected region free of body forces under conditions for which the


various quantities can be evaluated. With a view toward computa-tional problems, consider the particular case for which Fi is shrunkonto the crack tip and F2 = F encircles the crack tip at a sufficientdistance from it so that accurate numerical results can be expectedfor points along F. Then, in light of (5.3.2),

where F is a contour encircling the crack tip and ^4(F) is the areaenclosed by F and the crack faces up to the crack tip. The integrandof the area integral appears to be singular as r~2 for r —• 0, in whichcase the integral would be divergent. As noted by Nakamura et al.(1985a), however, this singularity is only apparent and the integraldoes indeed have a finite value for boundary conditions of interest.

Many different forms of the type (5.3.20) can be obtained byapplying the divergence theorem and by recognizing that df/dt «—vdf/dxi for any field / very near to the crack tip, but all of theserelationships have the form of a line integral along a crack tip contourF plus a surface integral over the area bounded by F and the crackfaces up to the crack tip. Even though the value of G computed in thisway does not depend on the particular choice of F, the descriptive term"path-independent integral" is not applicable due to the presence ofthe area term. Advantages and disadvantages of some particular formsof (5.3.20) are discussed by Nakamura et al. (1985a), who presentnumerical finite element applications of the area domain method ofcomputing G. An application based on a finite difference numericalmethod is described by Ravichandran and Clifton (1989).

A second algorithm that has been useful in extracting valuesof G for elastodynamic crack growth in finite element simulationsis the node release method. The main idea underlying this methodcan be seen in the expression (5.3.13) for the rate of energy fluxout of the body when a crack advances by gradual reduction of thetractions restraining the crack faces as the crack opens (Keegstra1976; Rydholm, Predriksson, and Nilsson 1978). Consider planarcrack growth in a two-dimensional body covered by a finite elementmesh so that the crack path extends from node to node along straightelement boundaries. A finite element node ahead of the moving cracktip is viewed as being two coincident nodes that are kinematicallyconstrained against relative displacement until the crack tip arrives.


When the crack tip passes the material point occupied by the nodepair, the nodes are allowed to separate dynamically under the actionof equal but opposite nodal forces acting to resist the opening. Theinitial magnitude of these nodal forces is typically assumed to be themagnitude of the nodal constraint force at the instant that the cracktip passes the node pair. The kinematic constraint on the node pairis released at the instant the crack tip passes, and the nodal forcemagnitude is assumed to gradually diminish from its initial valueto zero as the crack tip propagates to the next node pair. Severalschemes for reducing the nodal force magnitude have been introducedbut, typically, the magnitude is always proportional to the initialvalue, with the proportionality factor depending instantaneously onthe fraction of the distance to the next node pair that the crack tip hastraversed along an element side. For example, consider crack growthin a single mode and suppose that the nodal constraint force at aparticular node pair has magnitude Qo when the crack tip arrives.Let H denote the distance to the next node pair along the fracturepath ahead of the crack tip. If the crack tip is instantaneously at adistance h beyond the separating node pair, then the instantaneousvalue of the restraining force there, say Q, is assumed to have theform

Q = Qoq(h/H), (5.3.21)

where q(-) is a monotonically decreasing function of its argumentwith q(0) = 1 and q(l) = 0. Some particular choices that havebeen considered are q{x) = (1 — x)1/2 by Rydholm et al. (1978),q{x) = (1 - xf'2 by Malluck and King (1980), and q{x) = (1 - x)by Kobayashi, Emery, and Mall (1976). A comparative study wasreported by Malluck and King (1980) in which some sensitivity of theresults to the form of q(-) was found.

The two computational approaches for finding G noted above donot rely on special interpolation functions in the crack tip region forinferring field values between nodes in the finite element or finite dif-ference discretization. There are other numerical approaches in whichthe knowledge of the crack tip asymptotic field for elastodynamiccrack growth is exploited. Typically, the displacement interpolationfunctions very near to the crack tip are chosen to be precisely thedisplacement distributions corresponding to the asymptotic field; thecommon amplitude of these distributions is the stress intensity factorunder single mode deformation conditions. This general approach, ofwhich there are a number of variations, is usually called the singular

5.4 Steady crack growth in a strip 243

element method in finite element analysis (Aoki et al. 1978; Kishimoto,Aoki, and Sakata 1980a; Nishioka and Atluri 1980a, 1980b). Althoughthe idea is to compute the stress intensity factor directly, withoutthe use of its relationship to energy release rate (5.3.10), it must berecognized that the variational work/energy principle underlying thefinite element formulation must account for the flux of energy out ofthe body during crack growth.

5.4 Steady crack growth in a strip

Determination of the dependence of the dynamic stress intensity fac-tor on body configuration and applied loading during rapid crackgrowth is a principal objective of stress analysis in dynamic fracture.Some direct methods of stress analysis for elastodynamic crack growthare introduced and illustrated in Chapter 6. One point that willbecome clear, if it is not already so, is that exact analytical resultscan be extracted for only a few situations. Furthermore, analyti-cal results for configurations with boundaries, either constrained orfree, other than the crack faces are rare. A particular situation forwhich exact stress intensity factor results can be obtained is that ofsteady crack growth along the length of a strip subjected to simpleboundary loading on its edges. This configuration has been used byNilsson (1974a) in experiments on rapid fracture in high strength steelplates. The full boundary value problem can be approached with aview toward integrating the field equations, thereby establishing thedependence of the stress intensity factor on loading, configuration,and crack speed. This approach was described by Nilsson (1972) andit is outlined in Section 6.2. Certain results can be obtained moredirectly by application of energy arguments, however, and the energyapproach is summarized in this section.

5.4-1 Strip with uniform normal edge displacement

Consider a homogeneous and isotropic elastic strip in the region ofthe xi,^2-plane with — oc < x\ < +oo, —h < X2 < +h. Suppose thebody is a thin sheet or plate, and that the elastodynamic fields are ad-equately described by the two-dimensional theory of generalized planestress. The analysis for plane strain differs only in the interpretationof elastic constants. Suppose that the edges of the strip at x2 = ±/iare displaced away from each other a distance 2wo, that is,

(5-4.1)

Figure 5.5. Steady crack growth at speed v in a strip of width 2h. The edgesof the strip are given a uniform normal displacement wo, and the relationshipbetween the boundary loading and the crack tip stress intensity factor is obtainedby means of the path-independent integral evaluated on the near tip and remotecontours shown.

Specification of the complementary component of traction or displace-ment is postponed, but the second boundary condition will be either

= 0 or = 0. (5.4.2)

Suppose that a crack with traction-free faces grows steadily at speedv < CR in the ffi-direction along the symmetry line #2 = 0 of thestrip; see Figure 5.5. Without loss of generality, the crack is assumedto pass x\ = 0 at time t = 0. If the crack growth process is steady,then the elastic fields depend on x\ and t only through £ = x\ — vt.

Both far ahead of the crack tip and far behind the crack tip, fieldsare uniform in the xi-direction. The kinematic boundary condition(5.4.1) requires that the extensional strain in the #2-direction farahead of the tip is 622 —• eo as ^ —• +00, where CQ = uo/h. Farbehind the crack tip, the only stress distribution that is uniform inthe xi-direction and that is consistent with the condition that thecrack face is traction-free has the property that

as (5.4.3)

To demonstrate this for one of the stress components, consider balanceof momentum in the xi-direction which requires that

'11 da 12

dx\ dx2 = P-dt2 (5.4.4)

For fields that are steady in the xi-direction, this balance equationreduces to dau/dx2 = 0, which implies that a^i is a constant. The

5.^ Steady crack growth in a strip 245

boundary condition that o\2 = 0 on the crack faces then requires thatthe constant is zero.

Suppose that the limiting value of any field as £ —• ±oo isindicated by a superposed (+) or (—) sign. For example, c^&a^) =eo or 0*22 (f>#2) — 0- Then, the stress-strain relations imply that

E°22 = l - I / 2

l - I / 2 '

Consider now the particular case (5.4.2) 1 so that the edges of thestrip are constrained against displacement in the xi-direction. Then,en(f, #2) ~ 0 a nd ^2(4^2) = 0. The value of the energy release ratefor a remote contour F ^ shown in Figure 5.5 is

u)ni - w g ] dr = ^ . (5.4.6)

The only contribution to the integral arises from the segment of FQOparallel to the #2-axis at f —> +00. The path independence of theenergy release rate integral for steady crack growth implies that

G(r0) = G(roo), (5.4.7)

where Fo is a small contour surrounding the crack tip within the regionwhere the asymptotic stress intensity factor field prevails. In view of(5.3.10), the stress intensity factor is

Kl = U°E (5.4.8)/h(l*)A()

for any crack speed v. The stress intensity factor depends linearly onu0, as it must in view of linearity of the system. The nature of thedependence on E and h could also have been anticipated on the basisof dimensional considerations.

Although the result (5.4.8) was derived through formal applica-tion of the energy release rate integral, the same result could havebeen obtained even more directly by means of an intuitive argument.The mechanical energy density of the system per unit length in theXi-direction far ahead of the crack tip is e^hE^l — v2). Far behind thecrack tip, the energy density per unit length in the xi-direction is zero.During advance of the crack tip a unit distance in the rri-direction,a length of material with the former energy density is essentiallyreplaced by an equal length with zero energy density. There is noenergy exchange of the body with its surroundings and overall energymust be balanced for a steady state situation. Consequently, themechanical energy lost from the body must flow out through the cracktip. It follows immediately that

The argument is quite general, and it can be applied to advantagein a variety of situations. Some equilibrium situations are consideredfrom this point of view by Rice (1968).

Suppose for the moment that the boundary condition (5.4.2)2 ap-plies, that is, suppose that cr\2(£, ±/i) = 0 is considered. Tn this case,the body is unconstrained in the £i-direction so that an additionalcondition must be imposed to ensure overall momentum balance inthe xi-direction. For the symmetries associated with mode I defor-mation and for fields uniform in the xi-direction at remote points,this condition takes the form

/ K i " *n " /™2(4i " en)] dx2 = 0. (5.4.10)-h

For purposes of illustration, suppose that all limiting fields areuniform across the strip so that the integrand of (5.4.10) vanishespointwise in x2- Consider the particular case with <J^ = 0, whichimplies from (5.4.5) that e\x = —ue^. Thus, global momentum balance(5.4.10) requires that a^ = pv2(e^1 + ve0). In light of (5.4.5),

With this result in hand, either the energy release rate integral or theenergy balance argument leads to the result that the crack tip energyrelease rate is

where co = y/E/p. Note that v is always less than CR. Otherparticular cases, such as e\x = 0, can be considered as well.

5.4-2 Shear crack with a cohesive zone in a strip

As a second example of steady dynamic crack growth in a strip,consider the propagation of a mode II shear crack in a slab of materialof thickness 2h under plane strain conditions; see Figure 5.6. Underthe action of applied displacements

i*i(a;i,±M) = ±^0, u2(xli±h,t) = 0 (5.4.13)

the mode II crack grows in the #i-direction at speed v < CR. Thecoordinate x\ is eliminated in favor of £ through £ = x\ — vt. Therectangular coordinate system £,#2 has its origin at the crack tipand it translates with the mathematical tip. Although a singularityin the stress component vyi at the leading edge of the crack canbe considered, a linear slip-weakening cohesive zone is introducedinstead. The features of the slip-weakening cohesive zone model arediscussed in Section 5.3.2. The slip across the plane x 6t. According to this model, the physical processof slipping begins at £ = 0, it continues throughout the interval—A < £ < 0, and it terminates with total relative displacement of8T at £ = — A. Relative slipping is resisted by the cohesive/frictionaltraction <Ji2(£, 0) = rsf — (rsf—Tdf)6(^)/St in the interval —A < £ < 0,and by the uniform traction rdf in the interval —A < £ < —A. Thefrictional stress might be due to Coulomb friction arising from asuperimposed uniform compressive stress in the ^-direction. If sucha stress were to be introduced by specifying the alternate boundarycondition ^2(^1, ±/i, t) = T (b however, it would have no influence onthe main results to be obtained in this subsection.

A notable feature of this particular problem is that it representscrack propagation under displacement control or very stiff loading

Figure 5.6. Steady growth of a shear crack at speed v in a slab of thickness 2/i.The faces of the slab are given a uniform relative tangential displacement 2UQ.The relationship between the boundary loading and the features of a crack tipslip-weakening zone is obtained by means of the energy integral (5.2.9) evaluatedon the contours shown.

conditions, in contrast to crack propagation under stress control orvery compliant boundary loading as modeled by growth of a crackin an otherwise unbounded solid. Displacement controlled conditionsallow for the consideration of stress relief by crack extension, that is,the shear stress on the slip plane at some point far behind the slippingregion, say <7{"2> will be less than the shear stress on the fault planefar ahead of the slipping region, say a\2. In terms of the boundarydisplacement and the total residual offset,

a12 = - ST/2)/h. (5.4.14)

The relationship among the various stress magnitudes and slip magni-tudes is obtained by direct application of the path independent energyintegral (5.2.9).

The two choices of contour F for evaluation of the energy integralare shown in Figure 5.6. The value of i^Foo) will be consideredfirst. For points far ahead of and far behind the crack tip the onlynonzero stress component is 0-12(^x2) = cr^2

o r a\2-> respectively, andthe only nonzero displacement gradient is dui(£,X2)/dx2 = ^12//^ o r

(j^/zx, respectively. These components are essentially uniform overremote regions. For the portion of the remote contour along X2 = ±/i,the displacement components are uniform and n\ = 0 so that theintegrand of (5.2.9) vanishes identically and there is no contributionto the value of F. The only contributions to F arise from the segmentsof FQO parallel to the X2-axis in the remote uniform regions. Thus,

= v(*l2 + <Tn)6T/2 . (5.4.15)

As is indicated in Figure 5.6, the other choice of contour To embracesthe crack and n\ = 0 at virtually all points of this path. Thus,

(5.4.16)

= v(rsf - rdf)6t/2 + vrdf6T .

In view of the property of path independence, i ^ F ^ ) = F(T0) or

6T = rsJ-rd} ( g

0t (T12 + (T12 — 2rdf

This analysis introduces two stress differences that can be associatedwith stress drops in the seismological characterization of earthquakes.These stress differences are the dynamic stress drop (a^2 — rdf) andthe total stress drop (a^2 — <7{"2). The second of these is particularlyinteresting because it satisfies the relationship a^2 — a\2 — H8T/2}I. Ifh can be identified in some way with the total distance of travel of theedge of the slip region, then this expression coincides with one of thestandard definitions of stress drop in seismology. It is noteworthy that0- 2 and a{2 do not represent frictional properties of the slip surface.These magnitudes are arbitrary except that a^2 + a^2 > 2rdf anda\2 > di2 for the process to occur at all. Unfortunately, the energyintegral approach provides no information on the magnitudes of thephysical lengths A and A in the model.

The strip problems considered in this chapter are representativeof steady crack propagation problems that can be analyzed by meansof energy arguments. In none of these cases have complete stress anddeformation fields been determined by solution of the correspondingboundary value problems. As long as a crack edge is moving moreor less steadily, a steady state analysis yields relationships among thevarious parameters used to characterize the process without referenceto a growth criterion. For example, this approach can provide re-lationships among the parameters used to describe a crack growthprocess at different scales of observation, as noted in Section 5.3.2.The study of the way in which a crack tip or slip zone edge movesaccording to a particular crack propagation condition must be basedon a transient analysis of the propagation process, and such analysisis considered in Chapters 7 and 8.

5.5 Elementary applications in structural mechanics

The energy flux integral (5.2.7) was derived under the assumption ofa general field description of the mechanical behavior of a continuum.The concepts are more general than the derivation suggests, however.In this section, several applications of the same energy concepts areconsidered within the framework of elastic structural analysis. Theapplications illustrate the relationship between stress resultants at thelevel of structural analysis and crack tip parameters. Furthermore, insome particular cases, energy arguments can be applied to determinecrack tip parameters in terms of applied loading for dynamic crackgrowth in structural elements.

Analytical models of dynamic crack growth involving a single spa-tial dimension have been developed in connection with dynamic frac-ture toughness testing by Kanninen (1974), Bilek and Burns (1974),and Preund (1977), in connection with earthquake source modelingby Knopoff, Mouton, and Burridge (1973) and Landoni and Knopoff(1981), and in connection with peeling of a bonded layer by Hellan(1978a, 1978b). These models have been remarkably successful inenhancing insight into the particular physical process of interest, andthe few models for which complete mathematical solutions exist pro-vide rare opportunities to see all aspects of a dynamic crack growthevent in a common and relatively transparent framework.

5.5.1 A one-dimensional string model

The discussion here will focus on the near tip fields for a modeldeveloped to illustrate the influence of reflected waves on crack growthin a double cantilever beam fracture specimen (Preund 1977). Thedescription can be recast in terms of the dynamics of an elastic string,an even simpler conceptual model, and it has been analyzed withnumerous variations from this point of view by Burridge and Keller(1978).

Consider a stretched elastic string lying along the positive x-axis, that is, along 0 < x < oo. The string has mass per unit lengthp and characteristic wave speed c. The small transverse deflectionis w(x,t), the elastic strain is 7(#,£) = dw/dx, and the transverseparticle velocity is f](x,t) = dw/dt. Initially, the string is free oftransverse loading in the interval 0 < x < Zo and it is bonded to a rigid,flat surface for x > Zo- A boundary condition in the form of a conditionon the deflection w(0, i), the slope 7(0, £), or a linear combination of

5.5 Elementary applications in structural mechanics 251

the two is required; specific cases will be considered in a later sectionwhen the full motion of the string during a crack growth process isconsidered. Finally, suppose that the string is initially deflected but atrest, so that w(x, 0) = wo(x) and 7?(x, 0) = 0, where WQ{X) is specified.At time t = 0, the string begins to peel away from the surface, so thatat some later time t > 0 the free length is 0 < x < l(t). Thus, thefield equations

dj_drj 2dj_dr}

at "a*' cte~m (5-5*1}

are to be satisfied in the time-dependent interval 0 < x < l(t) subjectto the stated initial and boundary conditions. If l(t) is specified, thenthe solution of the governing differential equation is subject to the"crack tip" condition that w(l(t),t) = 0 or, in rate form, that

/ 7 + 77 = 0 at x = l{t). (5.5.2)

The field equations and boundary conditions can be satisfied for a hostof crack motions /(£). If a crack growth criterion is also specified atx = /(£), then a part of the solution procedure is to find that particularcrack motion l(t) for which the growth criterion is satisfied pointwisein time. Some cases of crack motion will be discussed in Chapter 7after the issues of crack tip singular field and energy release rate havebeen considered within the framework of the dynamic string model.

Ahead of the crack tip the strain and particle velocity are zero,whereas behind the tip they have nonzero values, in general. Thus,the crack tip carries a propagating discontinuity in strain and particlevelocity. Because the equation governing motion of the string is theelementary wave equation, discontinuities propagate freely only at thecharacteristic wave speed of the string c. But the speed of the cracktip l(t) is not restricted to be equal to c, hence the propagating cracktip must carry a momentum source or sink. In view of the fact thatthe only degree of freedom is the transverse deflection, the momentumsource must be a generalized transverse force that is work-conjugateto it;, namely, a concentrated force acting at x = l(t) and travelingwith that point. This is the crack tip singular field for this simplestructure. The elastic energy density and kinetic energy density forthe string are

U(x, t) = \pcW , T(x, t) = \pt? . (5.5.3)


lr Th $ x

Figure 5.7. A crack tip contour for evaluation of the energy release rate in a modelof crack growth based on the elastic string.

Suppose that the string is now viewed as a two-dimensional struc-ture with the internal deformation fields constrained to be consistentwith the string idealization. Thus, for example, all points on a crosssection have a velocity in the direction transverse to the x-axis equalto 77 and no velocity in the direction of the :r-axis. The componentof traction on a cross section in the direction transverse to the x-axisis ±pc27, where pc2 is the tension in the undeflected string and thealgebraic sign is determined by the direction of the outward normalon the cross section in the indirection.

Consider now a contour F as shown in the sketch in Figure 5.7.The contour begins on the crack face indefinitely close to the cracktip at x = Z~, it traverses the cross section there, then extends alonga portion of the upper traction-free surface of the material to x = l+,and then traverses the cross section to the rigid surface. Ahead ofthe crack tip, both 7 and 77 are zero. Thus, the portion of F ahead ofthe tip yields no contribution to the energy flux. Along the segmentfrom x — l~ to x — l+ the traction is zero and the outward normalvector component n\ is zero, so no contribution is obtained from thissegment of F. Therefore, the only contribution to energy flux arisesfrom the portion of F along the cross section at x = l~. Applicationof the energy flux integral (5.2.7) yields

F = Gl = -pc2jrj - \pc272l - \frq2l (5.5.4)

and, in light of the kinematic boundary condition (5.5.2) at the cracktip,

2 i 2 2 t ) 2 . (5.5.5)

This expression provides the energy release rate G for an arbitrarycrack growth rate l(t) in terms of the value of a crack tip parameter7(Z~,£). If this model is developed to simulate the double cantileverbeam situation mentioned above then the factor one-half in (5.5.5)would be absent to account for energy contributions from both halves


of the symmetric specimen. Note that G has the physical dimensionsof energy/length for this simple one-dimensional model.

The observation made in the foregoing discussion that the trac-tion exerted on the string by the rigid surface is a concentrated re-straining force Q acting at the moving point x = l(t) provides thebasis for an alternate derivation of the result (5.5.5). In general, therate of work of a concentrated force on a body is the inner productof the force with particle velocity of the point on the body at whichthe force instantaneously acts. (In order to account for the possibilitythat the force moves with respect to the body on which it acts, thisdefinition of the rate of work of a force is slightly more elaborate thanis often necessary in continuum mechanics. In the present situation,the point of application of the force moves in a direction perpendicularto the direction of the force, so a naive definition of the rate of workof a force, based on the motion of its point of application as opposedto the particle velocity at its point of application, would lead to theincorrect conclusion that the rate of work is zero.) In the presentcase, the velocity rj is discontinuous at x = l(t) so that the rate ofwork is \Qr]{l~,t). The present approach requires the evaluation ofthe integral / ^ 6(x)H(x) dx, where 6 and H are the Dirac deltafunction and the unit step function, respectively. The integral can beevaluated formally by noting that 6(x) = dH(x)/dx. A discussion ofintegrals of this type is presented by Fisher (1971) within the contextof the theory of distributions. Application of the principle of impulseand momentum to the portion of the string enclosed in the contour Tof Figure 5.7 leads to the conclusion that

Q = -pc2j(l-, t) - plr,(l-, t) = -pc2{I - i2/c2)j(l~, t). (5.5.6)

The energy flux is, by definition, the negative rate of work of theconcentrated force, and the result (5.5.4) follows immediately.

Finally, it is noted that the result can also be derived by directapplication of the overall energy rate balance. That is, if P is the rateof work of the boundary load at x = 0 then

F = P-(Utot + fiot), (5.5.7)

where Utot and Ttot are the integrals over the length of the string ofthe energy densities defined in (5.5.3). In the present notation, the


work rate is P = -pc2rj(0,t)^(0,t). Then

F = -pc2^, t)7(0, t) - £ [ p c 27 ^ + prj^ dx

[ $ ? 2 ±t)2]i. (5.5.8)

If the terms d^/dt and drj/dt in the integrand are replaced by drj/dxand c2dj/dx from (5.5.1) then the integrand is an exact differential.The result of evaluating the integral is once again the expression(5.5.4) for energy flux.

5.5.2 Double cantilever beam configuration

A specimen configuration that has been employed for some crackpropagation experiments is the split rectangular plate shown in thediagram in Figure 5.8. Because of its configuration, the specimenhas been analyzed by a strength of materials approach, whereby thearms of the specimen are viewed as beams cantilevered at the cracktip end. For this reason, the configuration has become known as thedouble cantilever beam. The origin of the descriptive term for thisspecimen configuration is not clear, but Benbow and Roessler (1957),Gilman, Knudsen, and Walsh (1958), and Burns and Webb (1966)were among the first to take advantage of its simplicity. The deforma-tion and associated stress distribution obtained by applying the usualassumptions of beam theory represent internally constrained planestrain or plane stress fields, and the general energy flux expressioncan be applied directly to compute the energy release rate. Supposethat the plane of deformation is the rri,rc2-plane, that the crack isin the plane x2 = 0, and that the tip is at x\ = l(t). The crack isassumed to open symmetrically in mode I. The contour F in the planeof deformation begins on one traction-free crack face at x\ = l(t) — 0.It runs along a cross section of one beam arm to the traction-freeboundary at x2 = /i, along this boundary until X\ = l(t) + 0, andthen along the cross section to x2 = — h. It returns to the other crackface in such a way that F has reflective symmetry with respect to thecrack plane.

Suppose that the energy flux integral (5.2.7) is now interpretedwithin the framework of elastic Bernoulli-Euler beam theory. Thetransverse deflection of the neutral axis of the upper beam arm at


Figure 5.8. The split rectangular plate configuration, commonly known as thedouble cantilever beam configuration, showing a contour for evaluation of theenergy release rate during crack growth.

x2 = h/2 is denoted by w(xut). Ahead of the crack tip xi > l(t),curvature and velocity are both zero and there is no contribution tothe energy flux from this region. Along the portion of F with x2 = h,the traction is zero and n\ = 0, so there is no contribution from thisportion of F either. Therefore, the energy flux integral is reduced to

F(F) = 2 / Lijrij ^- + (U + T)ini]Jo L dt Jxi=|_

dx2 (5.5.9)

For a Bernoulli-Euler beam cantilevered at x\ = l(t),

dt(5.5.10)

Furthermore, the slope of the neutral axis is zero there, that is,dw(l~ ,t)/dxi — 0. The rate form of this condition is

d2w- d2w+dx\

_ (5.5.11)

asThe normal stress cru(l~,X2,t) appearing in (5.5.9) is related to

the bending moment per unit width in the beam M(l~,t) through


The bending moment per unit width is related to the local curvatureof the neutral axis d2w/dx2 through

M(x,t) = ^ ( x , t ) (5.5.13)

for plane stress deformation. Each term in (5.5.9) can now be writtenin the form of (#2 - h/2)2 times M{l~,t)2. Because the dependenceon #2 is explicit, the integral can be evaluated with the result thatthe energy release rate is

where M(l , t) is the bending moment per unit thickness in each beamarm at the crack tip end and c = E/p. The second term in parenthe-ses in (5.5.14) is the kinetic energy contribution due to the velocitydistribution in (5.5.10). This "rotary inertia" contribution is notincluded in a derivation of the equation of motion of a Bernoulli-Eulerbeam, and it should be omitted if the energy release rate expressionis to be interpreted strictly within the framework of Bernoulli-Eulerbeam theory. It is interesting to note that if the corresponding energyrelease rate expression is derived within the framework of Timoshenkobeam theory, with both rotary inertia and shear deformation effectsincluded, and the deformation is then constrained so that each crosssection remains normal to the neutral axis as the beam deforms, theresult is (5.5.14). If the result is to be applied to a state of plane strainrather than plane stress then a factor (1 — u2) should be inserted inthe numerator of (5.5.14).

It can be argued, as in the case of compliance methods of equi-librium fracture mechanics, that the energy being supplied to thecrack tip cross section is actually being absorbed at the crack tip.Then (5.3.10) and (5.5.14) yield an expression for the crack tip stressintensity factor for the specimen in terms of the internal bendingmoment at x = l(t), namely,

(5.5.15)

Figure 5.9. Schematic diagram of a long rectangular plate being split in half by awedge. The process is analyzed as a steady crack growth process.

At the level of the beam approximation, the singularity at the crack tipon the fracture plane is a concentrated force. On the level of the planeelastic field, however, it is the square root singular stress distribution.A number of other connections between strength of materials modelsand elastic field models can be established in a similar manner.

5.5.3 Splitting of a beam with a wedge

The results of the previous subsection can be applied to study crackpropagation phenomena in structures for which the beam idealizationis appropriate. Consider the situation in which a long uniform elasticstrip is split by driving a wedge at speed v along the symmetry lineof the strip; see Figure 5.9. The stress state is assumed to be two-dimensional plane stress, the fields are steady as seen by an observermoving with the wedge or with the crack edge, and the system hasthe symmetries typical of mode I deformations. The total wedge angleis 2a, where a <C 1. The crack tip is located a distance / ahead ofthe point on the advancing wedge at which the arms first come intocontact with the wedge. The purpose of this brief analysis is to obtainrelationships among the wedge angle a, the crack length Z, the speed v,and a crack tip characterizing parameter. The length / is supposed tobe much greater than the beam height h so that the use of Bernoulli-Euler beam theory is suitable. Generalization of the procedure tomore sophisticated beam or plate theories is possible.

The ^-coordinate is eliminated in favor of £ = x\ — vt and,without loss of generality, the position of the crack edge in the planeof deformation is taken to be £ = 0, X2 = 0. Only the half of thestrip with X2 > 0 is considered, and the influence of the other beamarm is included by symmetry. The beam arm is modeled as a beamcantilevered at the crack tip end, and the transverse deflection of

the neutral axis of the beam is w(£). For £ > 0, each half of thespecimen is constrained against transverse deflection. Thus, withinthe context of elementary beam theory, the internal bending momentvanishes for £ > 0. As the crack tip passes a section of the beam, thetransverse constraint is relieved and the two arms of the split stripbegin to separate. The crack must open from the crack tip with boththe deflection w(0) and the slope dw(0)/d£ of the neutral axis beingzero. In the interval — I < £ < 0 the beam arms separate freely, andno transverse load acts in this interval. At £ = —/, the surfaces ofthe beam arms come into contact with the wedge, and the arms areassumed to remain in contact with the wedge for £ < —/. Thus, thebeam arms have no curvature and, consequently, no internal bendingmoment for £ < — I. However, the arms arrive at the contact pointwith nonzero curvature. Therefore, the contact point is a point ofdiscontinuity in the internal bending moment per width in the beamM(£) which, in turn, implies that the reaction force of the wedge oneach beam arm is a concentrated force per unit width at £ = —/.Suppose that the magnitude of the concentrated force per unit widthis Q, positive in a direction tending to open the crack. The wedgeis assumed to be smooth for the time being, but the effect of slidingfriction is considered subsequently.

Consider the energy integral (5.2.9) along a path enclosing thedeformed portion of the beam in Figure 5.9. Because the processis steady state and the path is closed, the value of the integral iszero. If the terms in the integrand are interpreted as representingtwo-dimensional fields constrained to be consistent with beam theory,as in the previous subsection, then the value of the integral is

-\phv2a2 + Qa- 6M(0~)2/Eh3 = 0, (5.5.16)

where p is the mass density of the material, E is the elastic mod-ulus, and the form of the energy release rate strictly applicable fora Bernoulli-Euler beam is assumed. The relationship is a statementof energy rate balance and the interpretation of the individual termsis straightforward. The first term accounts for the residual kineticenergy the material has as it slides up the wedge face beyond thecontact point. The second term is the rate of work of the wedgereaction force on the material, and the last term on the left side ofthe equation is the energy flow into the crack tip region as given in(5.5.14).


The equation of motion for the beam governing the transversedeflection under steady state conditions is

w"" + / ?V ' = 0 (5.5.17)

subject to the boundary conditions

w(0) = 0, ti/(0) = 0, w'(-l) = -a, w"(-l) = O, (5.5.18)

as already noted, where (32 = I2pv2/Eh2. The solution of this bound-ary value problem is

= a^cos^-fsin^-sin^ + Q](3(1 — cos (51)

which implies that

The result for Q can be substituted into (5.5.16) to obtain

where G is the crack tip energy flux from (5.5.14).Suppose that the crack grows in such a way that the energy

release rate is a constant Gc, as in pure cleavage, for example. Thispoint of view was adopted in a study of the cleavage strength of micaunder equilibrium conditions by Obreimoff (1930), independently ofthe earlier work by Griffith on the connection between the energyrelease rate and the surface energy. With a constant energy releaserate, (5.5.21) provides an expression for the length / in terms of othersystem parameters in the form

h ~ 2^3 v COS [Gc/Eh + a>(v/co)*

where co = \jEjp is the speed of one-dimensional stress waves inan elastic bar. Some special cases can be extracted from this general


result. For example, for vanishing v/co —• 0 and any Gc/Eh ^ 0,(5.5.22) implies that

i

If a = 10° and Eh/Gc = 104 then l/h « 17. On the other hand, ifthe fracture resistance of the material is relatively low, then it can beshown from (5.5.22) that

' * * « L as | i - > 0 . (5.5.24)v Eh v J

as |h 2y/S v Eh

If a = 10° and v/co = 0.1 then l/h « 2.4.The influence of frictional interaction between the wedge and the

beam arms can also be included in several ways. In the simplest ap-proach, the transverse force Q gives rise to a force fQ in the directionof motion, where / is the coefficient of sliding friction between thewedge and beam. If buckling of the beam arms in the column modeis not considered to be important then the equations of motion canbe written in the undeformed configuration for small deflections. Theenergy balance equation (5.5.16) is modified by including the term—fQ on the left side to account for the work dissipated in frictionalsliding. The solution of the relevant beam boundary value problemcorresponding to (5.5.17) can also be obtained in a straightforwardmanner. The beam arms are assumed to be axially inextensible andthe last boundary condition in (5.5.18) is replaced by

±^L (5.5.25)

to account for the bending moment at the contact point due to theeccentricity of the axial frictional force. The magnitude of the shearforce Q corresponding to (5.5.20) is found to be

Q = (l ( 5 5 2 6 )

The significance of the frictional effect can be seen by examining thelimiting case of crack growth with constant energy release rate Gc and

with vanishing speed v. Under these conditions,

as f -0 . (5.5.27)

As / —> 0, this result approaches the corresponding result in (5.5.23).

5.5.4 Steady crack growth in a plate under bending

Although the energy release rate idea has been applied for cases ofplane strain, plane stress, or antiplane shear deformations, the conceptitself can be applied equally well to analyze crack propagation in platesand shells. A few simple illustrations of the application of energyconcepts are included here.

Consider a thin isotropic elastic Kirchhoff plate under bending.The undeformed mid-plane of the plate lies in the a:i,X2-plane andit occupies the region —00 < x\ < 00, — b < #2 < b. Suppose thata through-crack grows steadily at speed v along the centerline of theplate in the positive xi-direction. That is, the crack surface extendsacross the full thickness of the plate, and it occupies the portion ofthe xi-axis with —00 < x\ < vt. The edges of the plate X2 = ±6are constrained to remain in the plane, but the mid-plane normals onopposite edges are rotated an angle a away from each other. There isno transverse loading on the plate. The transverse deflection dependson x\ and t only through £ = x\—vt and it is denoted by w(xi,X2,t) =

)- The boundary conditions imposed on the deflection are

, ±b) = 0, f^(£, ±b) = Ta . (5.5.28)0x

Far ahead of the crack tip the plate is at rest and in a state of purebending across the width of the strip. The deflection there is

(5-5.29)

Far behind the crack tip, the plate is also at rest but completely stress-free. Thus, by adopting the intuitive argument on energy balanceput forward in Section 5.4 the crack tip energy release rate can bedetermined directly. Indeed, the crack tip energy release rate is equal


to the elastic energy per unit length in the xi-direction stored in theplate far ahead of the advancing crack tip. Thus, the energy releaserate is

The stress resultants are singular at the crack tip, but the matterof crack tip stress singularities is not pursued further here. A fullstress analysis of a plate problem of this type has been presented byFossum (1978) who found the energy release rate to be identical to(5.5.30) even though he included rotary inertia effects in the model ofthe plate. The reason is that the energy release rate depends only onthe energy stored in the plate far ahead of the crack tip, and not onwave propagation in the plate. Indeed, an analysis of this situationbased on any of the standard bending theories of plates will lead to thesame energy release rate. However, the crack tip stress distributioncorresponding to this energy release will depend on the exact natureof the bending model.

5.5.5 Crack growth in a pressurized cylindrical shell

For propagation of a sharp crack in a cylindrical membrane shell atinstantaneous crack tip speed v the rate of energy flow into the cracktip is

F(T) = lim / [vaNapup + \v(Napeap + phupup + phw2)] dT,r~>° Jv(5.5.31)

where the subscripts have the range 1,2, F is a contour in the middlesurface passing from one crack face to the other around the cracktip, vp is the unit normal to F in the tangent plane of the middlesurface, Nap are the in-plane stress resultants referred to orthogonalcoordinates, cap are the work-conjugate strains of the middle surface,up are the in-plane displacement components, w is the displacementin the direction normal to the undeformed middle surface, h is the wallthickness, and the superposed dot denotes a material time derivative.This energy flux integral can be deduced from (5.2.7) or from directapplication of the overall energy rate balance to the shell configura-tion. For a membrane shell the crack tip stress distribution is identicalto the plane stress distribution, so that (5.5.31) can be evaluated in


terms of a stress intensity factor. On the other hand, if the crack tipregion is modeled by a cohesive zone then the results of Section 5.3can be applied.

The particular case of an axial crack propagating steadily along agenerator of the cylindrical shell under the action of internal pressurep provides an instructive situation. Again, for steady propagation,all field quantities including the pressure depend on the axial coordi-nate x and time t only through the combination £ = x — vt. Forsteady growth, and in the absence of internal pressure, it can beshown that the energy integral (5.5.31) is path-independent for allcrack tip contours, analogous to the plane strain result established inSection 5.2. In the presence of internal pressure, the integral is pathdependent, and the nature of this path dependence can be illustratedby considering the following special case.

Far ahead of the crack tip the material is at rest and the internalpressure is at some reference level, say p^. Far behind the cracktip, the material is also at rest and the internal pressure has beenreduced to zero, perhaps due to the escape of an initially pressurizedgas through the opening crack. Then, by applying the divergencetheorem to the integral (5.5.31) for the closed path consisting of Varound the crack tip, the traction-free crack faces, and circumferentialcircles far ahead of and far behind the crack tip, it follows that theenergy release rate is

where a is the shell mean radius and A is the entire middle surfaceof the shell enclosed within the contour I\ The first term in (5.5.32)is the contribution to the energy release rate from the elastic energystored in the shell per unit length in the axial direction far aheadof the advancing crack tip (free axial expansion is tacitly assumed),whereas the second term is the contribution to the energy release ratefrom the work done on the shell by the escaping gas as the crackopens. From this perspective, the shell is treated as a closed systemand the gas as an external agent. The energy stored initially in thepressurized gas that is carried away through the opening crack, aswell as the way in which the pressure decays behind the advancingcrack tip, must be determined by the dynamics of the gas flow andescape along the cylinder. Some elaboration on this point is given by


Kanninen, Sampath, and Popelar (1976) and by Parks and Preund(1978). Because the gradient dw/dx vanishes far ahead of the tip andthe pressure p vanishes far behind the tip, the second term in (5.5.32)will exist, in general, even though the range of integration is infinitein the axial direction.

In any case, once G is computed from the pressure distribution,it can be converted to a stress intensity factor or a crack tip openingdisplacement. It is observed that the form of G in (5.5.32) is indepen-dent of the specific shell model assumed, but the actual value of Gwill depend on the details of the solution through dw/dx and possiblythrough the influence of deformation on the pressure distribution.

5.6 A path-independent integral for transient loading

The energy flux integral and its variations which were discussed in thepreceding sections of this chapter have been useful in several ways indynamic fracture studies. The crack tip energy flux provides a usefulone parameter characterization of the mechanical fields in the cracktip region in some cases. If the integral definition of such a parameteris also independent of the path used to evaluate the integral, thenthis property can be exploited to relate the near tip field to remoteloading conditions without the need to solve the field equations forthe problem.

The crack tip energy integral (5.2.7) was noted to be path depen-dent, in general, even for the case of linear elastodynamics. However,Nilsson (1973) introduced a path-independent integral for plane elas-todynamics that can be used to advantage in certain transient crackproblems. This integral is defined and applied in this section. Theintegral has been found useful in the study of two-dimensional planestrain, plane stress, or antiplane shear deformation problems involvingthe transient loading of a body containing a crack that does notextend. The integral is first introduced, and its use is then illustratedby application to a transient mode I problem that has already beenanalyzed in Chapter 2.

5.6.1 The path-independent integral

The path-independent integrals of equilibrium fracture mechanics,especially the J-integral due to Eshelby (1956) and Rice (1968), haveplayed a central role in the development of fracture mechanics. The

5.6 A path-independent integral for transient loading 265

path-independent integrals can be interpreted as consequences of theconservation laws of elastostatics. These conservation laws, in turn,are consequences of the principle of stationary potential energy andthe invariance of the potential energy functional over the elastic fieldunder certain classes of transformations. These ideas are generalizedand made precise by Noether's theorem on invariant variational prin-ciples, as described by Giinther (1962) and Knowles and Sternberg(1972). Here, the connection between a path-independent integraland the underlying variational principle is illustrated by means of asimple approach introduced by Eshelby (1970).

Suppose that the configuration of a mechanical system is de-scribed by a vector function of position over a volume of space, sayR. The rectangular components of the vector are denoted by Ui andthe spatial gradient of the vector has components dui/dxj = Ui,j.Suppose that the field is governed by a variational principle of theform 6$[ui,R] = 0, where

]= / L{ui, Ui,j )dR + boundary terms (5.6.1)JR

and R is held fixed as Ui is varied. Thus, $[ui, R] is a functional of thefield Ui over the region of space R, and the integrand L is assumed tobe a differentiate function of its arguments. The functional $ is sta-tionary under variations in Ui that are consistent with all constraintson the admissible functions within the range of $. Under suitableassumptions, a necessary condition on Ui for $ to be stationary isthat Ui must satisfy the Euler equations

) 0 (5>6-2)

in R. In view of its arguments, the spatial gradient of L is

dL (dL d dL \ d ( dL \ ,r x

= ( o 7T-3 ^fc+Tr- o m* - (5.6.3\oui dxjduiijj dxj \dui,j )

dxk

In light of (5.6.3), the terms of (5.6.2) can be rearranged to yield

= 0, k = 1,2,3. (5.6.4)


A particularly noteworthy feature of (5.6.4) for present purposes isthat, for each value of &, the divergence of a certain vector is equalto zero. Therefore, if (5.6.4) is integrated over R or any subregion ofR, an application of the divergence theorem leads immediately to themechanical conservation law

/.[•or

Lnk - Ui,knAdS = 0, (5.6.5)s L du J\

where S is the closed bounding surface of the region or subregion, andUi are the components of the outward unit normal vector to S.

The result (5.6.5) applies for any three-dimensional region S inwhich (5.6.2) holds. The equivalent result for two-dimensional fieldstakes virtually the same form, where the counterpart of S is the closedcontour surrounding the region of the plane in which (5.6.2) holds.To see this, introduce a rectangular coordinate system £i, £2,0:3 andsuppose that the field ui is independent of £3. Take the surface in(5.6.5) to be a right cylinder with its generator in the ^-direction.The end faces are planes of constant £3 at an arbitrary distance apart,and the right section of the cylinder is bounded by the curve C. Fork = 1 or 2 there is no contribution to the surface integral from the endfaces, and for k = 3 the contributions from the two end faces cancel.Consequently, in two dimensions,

/ \Lnk - -^-uhknA dC = 0, k = 1,2, (5.6.6)Jc I 9uuj

J]

where C is any closed curve in the plane bounding a region in which(5.6.2) holds.

As a simple illustration of the conservation law (5.6.6), considerthe case of a two-dimensional elastodynamic field in the £i,£2-planethat is steady with respect to a reference frame translating in the£i-direction at speed v. The momentum equation can be written interms of the displacement Ui{x\ — ^ ,£2) as

( 5-6-7 )

where Cijpq is the array of elastic constants. This is a specific exampleof the field equation (5.6.2). The variational statement underlying this


equation is

6 if (^CijpqUiijUpiq-^Pv^iaUia) dfl + . . . | =0, (5.6.8)

where R is now understood to be any region of the plane in which(5.6.7) holds. Consider (5.6.6) for k = 1; the integrand is

Lni - - uia rij = (T + U)ni - (7yiii,i rij (5.6.9)OUi j

and the conservation law

/ [(T + U)m - (JijUia nj] dC = 0 (5.6.10)Jc

follows immediately.Consider steady growth of a crack with traction-free faces in the

a;i-direction. Form a closed path consisting of two contours, say Fiand F2, each of which begins on one crack face, encircles the crack tip,and terminates on the other face, plus the two crack face segmentsnecessary to complete the closed path. The first term in the integrandof (5.6.10) vanishes on the crack faces because n\ = 0 there, and thesecond term vanishes because (JijUj = 0 for i = 1 or 2 to satisfy thetraction condition. It follows that

G(T) = f [|(T + U)m - crania nj] dT (5.6.11)JT

is independent of path F for all paths beginning on one crack face,surrounding the tip, and ending on the other face. By custom, n* isthe unit normal vector to F directed away from the tip. This integralis a special case of (5.2.9) and both its property as an energy releaserate and its property of path independence have been exploited in thepreceding sections of this chapter.

To obtain the main result of this subsection as a second exampleof a path-independent integral, consider the case of a plane strainelastodynamic field represented by the displacement Ui(x\, X2, t). Thecomponents of the displacement vector satisfy


where Cijpq is the array of elastic constants. Assume that the dis-placement and particle velocity components are zero initially, thatis,

Ui(x!, a:2,0) = 0, 5 ? ( * i , *2,0) = 0. (5.6.13)

If the Laplace transform (1.3.7)

/»OO

Ui(xu z2 , s)= Ui(xux2, t)e~st dt (5.6.14)Jo

is applied to (5.6.12), then

Btrps2%- (5-6-15)To a certain extent, the introduction of the Laplace transform canbe motivated through the following argument. The hyperbolic natureof the partial differential equations governing the transient elastody-namic response precludes the existence of path-independent integrals.Furthermore, the mechanical systems for which path-independent in-tegrals have been found are governed by elliptic partial differentialequations. It is well known that the hyperbolic equations of elasto-dynamics can be converted to elliptic equations by application of theLaplace transform over time, and this is the conversion represented by(5.6.15). The elliptic equation (5.6.15) is the Euler equation impliedby the underlying variational statement of the problem,

6Q[ui,R] = 0, (5.6.16)

where

*[ui,R] = / [\Cijpqup,qui,j+\ps2uiui] dR (5.6.17)JR

plus any boundary terms. The two-dimensional conservation law

/ [\@ifiiij +P52Si2i)ni - dijTijUia ] dC = 0 (5.6.18)Jc

follows immediately, where 5^ = Cijpqupiq are the Laplace trans-formed stress components. The integration path C in (5.6.18) is anyclosed path in the plane of deformation bounding a region in which(5.6.15) is satisfied.

Consider now a body containing a planar crack with traction-freefaces that is deformed under plane strain conditions. Rectangularcoordinates labeled £i,£2>#3 are introduced so that the displacementis independent of £3, and the plane of the crack is #2 = 0. A cracktip contour F is introduced starting on one crack face, encircling thecrack tip in the plane, and ending at some point on the other crackface. Then a line integral along F is defined as

JN(T;s)= / [±(aijuiij+ps2UiUi)n1--aijnjUi,1] dT, (5.6.19)Jr

where n; is a unit vector normal to F and directed away from thecrack tip. Because the integrand of (5.6.19) vanishes pointwise on thecrack faces, this conservation law implies that JN(T] S) is independentof the particular choice of F used to compute its value. Again, F mustbe a material line, and the region between F and the crack tip mustbe simply connected and free of body force. This path-independentcrack tip integral was introduced by Nilsson (1973), who noted thatit applies for transient viscoelastic fields as well as elastic fields. Asnoted by Gurtin (1976), the path-independent integral (5.6.19) hasan expression in the time domain equivalent to its expression in theLaplace transformed domain.

5.6.2 Relationship to stress intensity factor

The path-independent integral JN(T; S) does not represent the Laplacetransform of an energy measure or other physically significant featureof the elastic field. For the integral to be useful, it must be related tosome quantity of physical significance. That quantity is the dynamicstress intensity factor, and the connection is established in this sub-section. The case of mode I deformation under plane strain conditionsis considered here.

Again, a rectangular coordinate system xi,#2,#3 is introducedin the body. The deformation is independent of £3, and the crack liesin the plane x<i = 0. Consider a crack tip contour F that is arbitrarilyclose to the crack tip. For a mode I plane strain deformation and a

stationary crack tip, the stress and displacement fields very near tothe crack tip are

n Kl{t)y m

(5.6.20)

as r —• 0, in terms of polar coordinates r, 0 centered at the cracktip. The direction 0 = 0 coincides with the direction of prospectivecrack advance, and // is the elastic shear modulus. The functions£ij(0) are given in (2.1.3) and the Ui(9) are readily determined fromthe limiting values of the angular variations in (4.3.14) as v —> 0+.Taking the Laplace transform of (5.6.20), the near tip distributions ofaij and S< are

Ki(s) ^ KT(s)

But these near tip fields are exactly the same as the correspondingresults for equilibrium fields with Kj(s) in place of the equilibriumstress intensity factor. The term in JN(F;S) that is proportional toUiUi makes no contribution to the integral along F as the path shrinksonto the crack tip because it is proportional to r as r —• 0. Each ofthe remaining terms is proportional to r"1 and, consequently, eachcontributes to the value of Jjv(F; &) as T is shrunk onto the crack tip.However, the remaining terms together are identical to those makingup the integrand of Rice's J-integral as applied in equilibrium elasticfracture mechanics (Rice 1968). Consequently,

lim JN(T; s) = l^Kjis)2 (5.6.22)r — • o JLJ

for an isotropic elastic material with Young's modulus E and Poisson'sratio v under plane strain conditions.

Evidently, the value of JN(T]S) for F very near the crack tip isrelated to the value of the Laplace transform of the stress intensityfactor Kj(s). But J/v(F;s) is path-independent. Therefore, if thevalue of JN can be determined in terms of the Laplace transformedboundary loading by evaluating the integral for a contour F far from

the crack tip, then the property of path independence can be invokedto obtain a relationship between the applied loading and the stressintensity factor. Such an application is illustrated in the next subsec-tion.

5.6.3 An application

To see how the conservation law (5.6.19) can be applied to extract astress intensity factor history for a transient crack problem withoutsolving the field equations, consider the problem formulated in Sec-tion 2.4. This is the situation of a half plane crack in an otherwiseunbounded elastic solid. The faces of the crack are subjected to auniform pressure of magnitude cr*, suddenly applied at time t = 0. Arectangular coordinate system £1,0:2, £3 ls introduced with the crackoccupying the half plane x2 = 0, x\ < 0. The deformation is planestrain with all field quantities being independent of the ^-coordinateand the crack opens with mode I deformation.

As noted in Section 2.4, the transient stress distribution resultingfrom this loading consists of cylindrical waves diverging from the crackedge plus plane dilatational waves propagating away from the crackfaces. As the plane wave passes a point with coordinate x2i the normalstress component changes discontinuously according to

*22(*i,x2,t) = -(r*H{cdt-\x2\), (5.6.23)

where H(-) is the unit step function. Likewise, the normal displace-ment changes according to

*^2(^1,^2,^) = ±—-2(cdt T x2)H(cdt T x2) (5.6.24)

Pcd

for ±x2 > 0.The conservation law (5.6.18) is now applied to determine the

stress intensity factor for this loading without solving the boundaryvalue problem. For this purpose, the Laplace transform of the fieldis considered, that is, Ui(xi,x2,s) for s in some small interval ofthe positive real axis. The path C is chosen to be the closed curveshown in Figure 5.10. The path AAf is a circular crack tip contourof indefinitely small radius. The paths AB and A'B' coincide withthe crack faces, and B and B' are far from the crack tip compared to

Figure 5.10. Diagram of the path C for evaluation of the integral (5.6.18) in orderto relate the applied loading to the transient crack tip stress intensity factor forthe case of pressure suddenly applied to the faces of the crack. A diagram of thewavefronts appears in Figure 2.4.

the distance Cd/s for any fixed s in the real interval. The remainderof C is the rectangle BCDD'C'B' with all four edges far from thecrack tip compared to Cd/s. The Laplace transforms of the divergingcylindrical waves decay rapidly with distance from the crack edge, asdiscussed in Section 2.4. In fact, the decay is rapid enough so thatthese transformed quantities make no contribution to the integral.

The remaining features of the wave field that could contributeare the plane waves traveling in the ±#2-directions, the crack faceloading, and the crack tip field. The contributions can be workedout as follows. In light of (5.6.22), the nontrivial contributions aresummarized as

1 - t / 2

EKI(s)2

JA

LC

B= 0. (5.6.25)

Along AB, <722 is independent of x\ so the first term is proportionalto the value of U2 at point £?, which is known. All points on BC arefar enough from the crack edge so that the diverging cylindrical wavesmake no contribution, and the integral is given in terms of the planewave traveling away from the crack faces. From (5.6.23) and (5.6.24)


the relevant Laplace transforms are

^ / (5.6.26)

for ±^2 > 0. If the integrals in (5.6.25) are evaluated for the case whenthe xi-coordinate of B and the a^-coordinate of C are indefinitelylarge, it is found that

(5.6.27)E

With the identity pc2d = (l-u)E/[(l-2v)(l+v)] this result simplifies

to

where only the positive root is retained on physical grounds. Theinverse Laplace transform is

(5.6.29,

which is identical to the result (2.5.44) obtained from a complete so-lution of the field equations. Another application of the path integralto an elastodynamic strip problem is given by Nilsson (1973).

Finally, it is noted that the path integral idea applies in the caseof steady vibratory deformation, as well as for Laplace transformedtransient fields. For example, suppose that an elastodynamic field hasthe form

Ui(x1,x2, t) = Ui(xi, x2)eiut. (5.6.30)

Such a field might represent the steady harmonic response of a crackedelastic solid to a periodic applied traction with frequency CJ. Thequantity Ui represents the amplitude field of the resulting harmonicresponse. Under these conditions, the conservation law (5.6.18) ap-plies with s replaced by to;. The stress intensity factor for a mode Icrack under these conditions has the form

y"* (5.6.31)


and the value of the integral taken along a crack tip contour analogousto (5.6.22) is

1 ,,2-KT{UJ)2 . (5.6.32)

l - i / 2

E

5.7 The transient weight function method

Up to this point, dynamic stress intensity factor results have beenobtained on the basis of a solution of a particular boundary valueproblem or the application of a path-independent integral to a par-ticular problem. Here, the notion of an influence function for stressintensity factor is introduced. For a body of any given shape, thestress intensity history for a linear elastodynamic crack problem musthave the form

Kj(t)= [ fJo JB

(5.7.1)

where B is the spatial boundary of the body, including the crack faces,p is the time-dependent loading applied to the boundary which beginsto act at time t = 0, and xB symbolically represents points on theboundary B. Thus, the stress intensity factor for any applied loadingp is determined by the function h{xB,t — T). This function is calledthe influence function, or, alternatively, the weight function, for thedynamic stress intensity factor for a particular configuration.

5.7.1 The weight function based on a particular solution

In Section 5.6 it was shown that the stationary character of thefunctional 3>[ui;i2] given in (5.6.17), where the argument Ui is theLaplace transform on time of a displacement field, leads to the path-independent crack tip integral JN(T; s) given in (5.6.19). The notationestablished in that section is followed without modification. In partic-ular, the plane of deformation is the xi, X2— plane and the crack opensin the plane strain opening mode. There are two properties of thispath-independent integral that are exploited in this development. Thefirst is that the integral JN has a direct relationship with the Laplacetransform of the stress intensity factor, as stated in (5.6.22), and thesecond property concerns the change in $ due to a change in cracklength. The latter property is discussed next.

5.7 The transient weight function method 275

Suppose that the solution to the crack problem Ui can be foundfor any fixed position of the crack tip, say at X\ = /, X2 = 0. Thenui depends in some way on Z, as well as on position in the plane, and$ can be viewed as a function of I. Suppose that the total derivatived$/dl is evaluated. The region R in (5.6.17) is now considered to bethe entire two-dimensional body exterior to a small but arbitrary looparound the crack tip of interest. For present purposes, it is sufficientto consider the outer boundary of R to be constrained against workingdisplacement during the increment dl, that is, dui/dl = 0 at all pointson the outer boundary on which nonzero traction acts. (If this condi-tion is not assumed then it is necessary to add appropriate boundaryterms to the functional in (5.6.17); the final result is unaffected.) Toevaluate d$/dl, the parameter / can be viewed as a timelike parameter,and the generalized transport theorem (1.2.41) can be applied to yield

IT = r « ~ 5 i +ps Ui~m~\ dR~ I ^VrijUiij+ps'uiU^riidT,dl JR I Ol 01 J JT

(5.7.2)

where rti is the unit normal vector to F directed away from the cracktip. An application of the divergence theorem yields

The derivatives with respect to / in this expression are computed withXi held fixed. For points very close to the crack tip,

ai i, - Axi r (5-7'4)

C/6 Ixi UX\ x\—I

Comparison of (5.6.19) with the result (5.7.3) then leads to the secondmain property of the path-independent integral, namely,

-^ = - JN • (5.7.5)

In a study of linear elastostatic crack tip fields, Rice (1972)employed Irwin's relation between energy release rate and the stress

intensity factor to show that if the displacement field and stress in-tensity factor are known as functions of crack length for any onesymmetrical load system acting on a linear elastic body in planestrain, then the stress intensity factor for any other symmetrical loadsystem acting on the same body can be determined directly. As shownby Preund and Rice (1974), an analogous result can be obtained fordynamic stress fields in an elastic solid containing a crack. Thisapproach is developed and illustrated in this section.

Consider an elastic solid containing a planar crack of fixed lengthunder conditions of plane strain. The body is assumed to be symmet-rical with respect to the plane of the crack, and only loading resultingin mode I deformation is considered. Consider any particular loadsystem on the body and suppose that the crack length is determinedby the parameter /. For time t < 0, the material is stress-free andat rest. At time t = 0, a traction distribution begins to act on theboundary B of the body. (A body force distribution can also betaken into account, but only the final result is stated with body forcesincluded.) The boundary traction is given at any later time t > 0 byT = Q\Tw{xB,t) for xB on B. As noted in Section 1.2.1, the bold-face characters in this development represent vector quantities. Theparameter Qi, which appears as a scale factor, will subsequently beviewed as a generalized force. Suppose that the resulting displacementfield u = Qiii^(xB^t) and stress intensity factor Kj = Q\K^\t)are known as functions of I. These quantities are necessarily linearin the load scale factor <3iult will now be shown that if the Laplacetransforms on time u and Kj are known as functions of I for a rangeof positive real values of the Laplace transform parameter s, thenthe Laplace transform on time of the stress intensity factor for anyother load system acting on the same body can be determined. Thisdevelopment hinges on the two properties of the path-independentintegral JN identified above. These properties imply that

—— = K (5 7 6)dl I fixed displ. E I ' \ ' ' )

where fixed displacement means that the increment in $ due to anincrement in I is evaluated with the loaded portion of the boundaryheld fixed in position.

Consider now a second load system characterized by the gener-alized force Q2? in addition to the loading system introduced above.

The complete solution for the case of the loading system representedby the generalized force Q\ is assumed to be known. Generalizeddisplacements q\ and q^ are associated with any displacement field uby

Qi= f fW-uda;*- (5.7.7)JB

If both load systems act simultaneously, linear superposition yields

j = / f W • u«> <teB , (5.7.8)Qi =

where summation over the repeated index is implied and Q\U^ andQ2U^ are the separate transformed displacement fields. It is clearfrom (5.7.8) that the generalized displacements qi are work-conjugateto Qi with respect to the energy like quantity $.

When both loading systems are applied to the body simultane-ously, then $ can be viewed as a function of #i, (?25 and I with

where Ki = QiK^%\ It is then possible to write the variation ofdue to a small variation in each of its arguments as

6* = Q16q1 + Q26q2 - ( — — 1 KjSl. (5.7.10)

An interchange of the dependent and independent variables, com-monly known as a Legendre transformation, yields the equivalentcomplete differential

6{Qiqi

Because (5.7.11) is an exact differential, the relationships

must be valid. Furthermore, these relationships must hold for arbi-trary values of Qj so that

^ = 2 (ydl \ E

= 2 (1-y^) K^kW . (5.7.13)\ E )

But C12 = C21 and K^ are known functions of I. Thus, making useof (5.7.8), relation (5.7.13) with i = 2 and j = 1 can be solved for

to yield

T<a> • dxB , (5.7.14)dl "' v y

where the derivative with respect to / is taken with XB and 5 heldfixed.

As in the case of equilibrium fields, the stress intensity factor forthe load system represented by Q2 should not depend in any way onthe particular choice of loading system represented by Qi- Hence,there must exist a function, say h, which is defined in terms of anyparticular solution by

< 5 " 5 )

but which is a universal function for a cracked body of given geo-metrical configuration, regardless of the way in which the body isloaded. The uniqueness of h in the equilibrium case is established byRice (1972). It is then evident that if T and F are the transformsof any particular boundary traction distribution and body force dis-tribution, respectively, that result in a mode I deformation field thenthe corresponding transformed stress intensity factor is

/ T • hdxB + [ F- hdxR (5.7.16)B JR

in terms of the weight function h. In all cases, of course, the taskof finding the stress intensity factor history is not complete until theLaplace transform has been inverted into the time domain. A specificcase is examined next.

The weight function method as developed above is now appliedto a particular plane strain case of transient loading considered inSection 3.2. A semi-infinite crack in an otherwise unbounded bodyis subjected to an opposed pair of concentrated loads of magnitudep* acting at some fixed distance / from the crack tip and tending toseparate the crack faces. If the crack is in the plane #2 = 0 and thecrack occupies the interval — 00 < x\ < Z, the boundary condition onnormal stress on the crack faces is

^22(^1,0*,*) = -p*«(a?i)ff(t), -00 < Xl < I. (5.7.17)

The other boundary conditions on the region x 0 are given in(3.2.3)2 and (3.2.3)3.

The stress intensity factor for this case will be determined directlyfrom the results for the same body subjected to a suddenly appliedcrack face pressure of magnitude a* at time t = 0. Except for a minorcoordinate transformation, this is precisely the problem analyzed inSection 2.5. The Laplace transform of the stress intensity factor forthat problem is given in (2.5.43) as

M (5.7.18)

in terms of quantities defined in Section 2.5. The Laplace transformof the displacement of the crack face is u_(xi,s) and this functionis determined by inverting the transform expression in (2.5.35)2- Ifthe minor change in coordinates between the two cases is taken intoaccount, then it is found that

^ ( 0 ± ) ± ( )

= ±<T*PF+{0)

(5.7.19)

where, again, the various quantities appearing in (5.7.19) are definedin Section 2.5. The weight function is required on the boundary wheretraction is applied, and it is given by

All of the entries in the general expression (5.7.14) have nowbeen identified. If they are substituted into (5.7.14) and the resultingexpression is simplified, then it is found that

^ F _ ( C ) e - ^ d C , (5.7.21)

where —a < £o < a. The constant coefficient of this expressionsimplifies considerably if it is noted that b2 /K = (1 — v). Uponinversion of the Laplace transform on time, the integral expressionfor Ki(t) is exactly the same as the expression obtained as (3.2.12).The features of the stress intensity factor history are described there,and they are interpreted in terms of the process of load transfer bymeans of stress waves. The purpose here has been to illustrate theapplication of the weight function method for a case that was alreadyfamiliar.

5.7.2 A boundary value problem for the weight function

The discussion of the preceding subsection emphasized the construc-tion of the weight function for a particular configuration from any oneknown solution. However, it is also possible to pose a boundary valueproblem for the weight function itself, as discussed within the contextof elastostatic crack fields by Bueckner (1970) and Rice (1972). It isevident from (5.7.15) that the weight function vector hi must satisfythe same field equations as does the displacement vector itself. Theinitial and boundary conditions on the solution of the displacementequation of motion, as well as the nature of the crack tip singularityfor the weight function, are considered here. The analysis proceeds byinvoking the elastodynamic reciprocal theorem (1.2.59) as it appliesfor the field for which the dynamic stress intensity factor is soughtand the weight function field for the same geometrical configuration.The conditions under which the statement of the reciprocal theoremreduces to (5.7.16) are identified, and the weight function meetingthese conditions is determined. The development presumes a modeI deformation field, but it can be carried out equally well for othermodes of deformation.

The particular situation to be studied here is the transient loadingof a cracked body under plane strain opening mode conditions. Theplane of deformation is the xi,#2-plane, and the notation of the

previous subsection is adopted. A crack occupies the half plane X2 =0, —oo < x\ < 0. The material is stress-free and at rest for timet < 0. At time t = 0 a pressure distribution p{x\,t) begins to acton the crack faces tending to open the crack, and the objective isto determine the resulting stress intensity factor history Kj(t). Inview of the mode I symmetry with respect to the crack plane, onlythe half plane #2 > 0 is considered. The elastodynamic field withinthis region, represented by the displacement n , say, is subject to theboundary conditions

), -oo < xi < 0,

,0+,£) = 0, - o o < z i < o o , (5.7.22)

u2(xi,0+,t) = 0, 0 < xi < oo.

Consider a second elastodynamic field u* for this body that alsosatisfies the mode I symmetry conditions, so that this field is subjectto the boundary conditions (5.7.22)2 and (5.7.22)3 as well. For thetime being, it is also assumed that u* = 0 for t < 0, and that acrack face pressure distribution p*(xi,t) begins to act at time t = 0.This distribution p* has no relationship to p, except that its range ofdefinition is the same, and no restrictions are placed on the asymptoticcrack tip behavior of u*.

The elastodynamic reciprocal theorem is stated in (1.2.59). Inthe present case, the region of the body is the half plane x2 > 0except for a cylindrical cut-out of arbitrarily small radius r centeredon the crack edge, and the boundary of the region is made up of theintervals —oo < x\ < — r and r < x\ < oo of the line #2 = 0> plus thecylindrical surface x\ + x\ = r2, #2 ^ 0. In the present instance, thereciprocal theorem implies that

/ /Jo J-

2(xiJ0+,t — r)dx\ dr

/ lim / ay(r,0,r)nj(flX(r,0,t - r)rd6 drJo r~*°Jo

rt rO

= P*(xi->T) ^2(^150+ , t — r)dx\ drJo J-00


+ I lim / (TUr.Orfnjifiuiir&t - r)rdOdr. (5.7.23)Jo r^°Jo

This relationship incorporates the boundary conditions on both elas-todynamic fields.

Guidance on a direction for further development is provided by(5.7.23). The first term on the left side has the form of (5.7.16),which suggests that u\ itself may be the influence function. Thefirst term on the right side, on the other hand, is the integral of anarbitrary function p* times a displacement distribution that is notknown a priori. This term can be eliminated by taking advantageof the arbitrariness of p* to require that p* = 0. At first sight,this step might seem to be self-defeating because it suggests thatu\ is an elastodynamic field satisfying homogeneous boundary condi-tions. The uniqueness theorem of elastodynamics states that the onlyelastodynamic field satisfying homogeneous boundary conditions is adeformation-free motion. In the present case, this would imply thatu\ = 0 everywhere. However, the uniqueness theorem applies for fieldsfor which total mechanical energy is bounded. It is advantageous toconsider u\ to be outside this class of fields, for reasons to becomeevident, and thus p* = 0 is assumed.

Evaluation of the remaining two terms in (5.7.23) involves inte-gration of stress and deformation fields along a semicircular arc atan arbitrarily small radial distance from the crack tip. The cracktip stress field for the problem of interest aij varies with r as r"1/2.The integral over 0 appearing as the second term on the left sideof (5.7.23) will have a finite nonzero value only if u* also varies asr"1/2 for arbitrarily small r. This suggests that the elastodynamic"displacement" field should be square root singular at the crack tip.It is not necessary for u\ to represent a physically realizable field, sothere can be no fundamental objection to this singularity. The cracktip displacement field for the problem of interest U{ varies with r asr1/2. Thus, the remaining integral on the right side of (5.7.23) willhave a finite nonzero value only if a^ varies as r~3/2 for arbitrarilysmall r. This suggests that the "stress" field a^ which is derived fromthe "displacement" field u\ is singular in r to the power —3/2, whichis consistent with the singularity in u^. Thus, this singular behavioris adopted as a requirement on the field u£.

Next, the time dependence of the terms in the integrands of thesesame two integrals in (5.7.23) is considered. The time dependenceof the stress &ij is reflected in the stress intensity factor history

Ki(t) which is to be determined. The integral on the left side isa convolution of KJ(T) with u*. Thus, if the time dependence of u\ isrequired to be an impulse, or a delta function in time, then the siftingproperty of the delta function will render the second term on the leftside of (5.7.23) proportional to Kj(t). Likewise, for the second termon the right side, the time dependence of ui is Kj(t — r), hence if<j*j has time dependence 6(r) then this term will also be proportionalto Kj(t). Thus, it is required that the asymptotic crack tip fieldof u* must have delta function time dependence. If a function u\can indeed be found which satisfies the requirements identified, then(5.7.23) will have precisely the form (5.7.1) with u\ being proportionalto the influence function.

The crack tip singular field of u\ is presented next. The spatialstructure of the singular field is dictated by the highest order spatialderivatives in the governing differential equations, so that the struc-ture for a stationary crack tip is identical to that for an equilibriumfield. If the tensile "stress" on the crack plane directly ahead of thecrack tip has the form

22 ~ Wi (5-7-24)for arbitrarily small r, then the angular variations of the "stress" and"displacement" components near the crack tip are

w 3sin|0-3sin§0

. 7cosf0-3cos|<9> (5.7.25)

\ — 8u) cos \9 + cos | #

iu — 9) sin \0 + sin \9

If these expressions are substituted into (5.7.23) and the pertinentintegrals are evaluated, it is found that, as r —• 0,

Jo Jo

f [" a* (rArfaJo Jo

A4

(5.7.26)

In a similar analysis, Burridge (1976) circumvented this integra-tion step by adopting a result from the theory of generalized functionsdue to Fisher (1971). If the reciprocal theorem is applied in a directmanner on the boundary x*i = 0 without taking special precautionto deflect the integration path around the crack tip then the secondterm on the right side of (5.7.23) is

Jo^22(^1? 0+} T}U<){X\, 0 + , t — T) dx\ dT

0 c ~"" "

IKl(t)lim f x-1'2xZ1/2dx, (5.7.27)

where, for any real A,

x f o r z > 0 , Y (0 f o r x > 0 ,

f o r z < 0 , X- = \ M * f o r x < 0 .

Fisher (1971) showed that

" + 2sin(7rA) v J

within the framework of generalized functions. This approach leadsto precisely the same results as in (5.7.26). By either approach, theconditions imposed on u^ reduce (5.7.23) to

= T ^ - = / / p(x1,r)uk2(xuO

+,t-T)dx1dr.(1 - i/)V2?r Jo J-oc

(5.7.30)

The remaining step is to actually determine ^ ( ^ I , 0+, t), and thisprocess is greatly facilitated by the analysis in Section 2.5. There,the case of a suddenly applied crack face pressure of magnitude a*for the same geometrical configuration is considered in some detail,and the solution of the boundary value problem is reduced to thesolution of the functional equation (2.5.11) by means of the Wiener-Hopf technique. The corresponding functional equation in the present

case must be identical, except that a* — 0. Therefore, in the presentcase,

C) = - £ £ § , (5.7.31)where all quantities are defined as in Section 2.5. The functions U_(Qand £+(C) a r e proportional to the double transforms of the unknowncrack face normal displacement for x\ < 0 and the unknown crackline normal stress for x\ > 0, respectively. In the present context,

(5.7.32)

This expression differs from the corresponding expression in (2.5.8)where the exponent on s is fixed by the nature of the applied loading.Here, a factor as a power of s remains in anticipation of a homogeneoussolution, but the exponent 7 is left arbitrary until it can be fixed byone of the conditions to be imposed on the field u*. Likewise, in thepresent context,

(^-1E+(C) = - s 7 / / aZ2(x',0,s)dx'e-*iXl dxx, (5.7.33)

where the special form of this expression is dictated by the presenceof a nonintegrable singularity in d\2 at x\ — 0.

Attention is now focused on (5.7.31) with the interpretations(5.7.32) and (5.7.33) in mind. Each side is the analytic continuationof the other into its complementary half plane, so each side equalsone and the same entire function. The Abel relationships betweenasymptotic properties of functions and their transforms (2.5.34) implythat both (7_(C) and C"1S+(C) in (5.7.31) are O(C~*) as |C| -> 00and, consequently, the entire function is at most a constant, say Eo,according to Liouville's theorem. If the steps involved in invertingthe transformed solution that are outlined in Section 2.5 are followedhere, then it is found that

, 0- t) -

for t > 0 and x\ < 0, where the limiting value of the integrand asIm(C) —» 0+ is assumed. IX7) is the gamma function, and it has been

assumed that 7 is in the range 0 < 7 < 1. The inversion is similarfor values of 7 outside this range but, based on experimenting withdifferent values of 7, it is concluded that this is the pertinent rangefor present purposes.

Indeed, suppose that the choice 7 = \ is made and that (5.7.34)is rewritten as

(5.7.35)

where C\ is an integration path that begins at £ = —t/xi—iO, and thenruns along the lower side of the real axis to the branch point at £ = a,around the branch point, and along the upper side of the real axis to£ = —t/x\ + iO. If —t/x\ > c then the integration path is augmentedby small semicircular loops around £ = c on both sides of the branchcut along the real axis. These loops make no net contribution to thevalue of the integral. The function (—£ — tjx\)xl2 is made single-valued by cutting the £-plane along the real axis from £ = —tjx\ to£ —• oo, and the branch with a positive real value at £ = 0 is assumed.

Observe that if — t/xi > c then the path of integration can beclosed around the branch point at £ = —tjx\ and that all singularitiesof the integrand are inside this path. Consequently, according toCauchy's integral theorem, the value of the integral along C\ is equalto the value along a clockwise circular path of indefinitely large radiuscentered at the origin. The latter integral can be determined from theasymptotic properties of the integrand to be

(5.7.36)

for 0 < t < —CRX\. If this result is compared with (5.7.25) then itis evident that Eo = —2y/¥. Thus, in view of (5.7.1),

pt pO

Ki(t)= / p(x1,T)h(xut-T)dx1dT, (5.7.37)JO J-oo

where

h(xut) = J^- I- { o^ / F~^ i dQ • (5.7.38)

This is the problem-independent weight function for transient modeI crack problems.

Two simple illustrations of the application of (5.7.37) to situa-tions considered previously are noted. First, if p(xi,t) = p*6(x\ +l)H(t) for some positive constant distance /, then the steps leadingto the stress intensity factor are very simple. A second exampleconsidered in Section 2.5 is that when p(x\,t) = a*H(t). In thiscase, p(x\,t) is independent of x\ and it is advantageous to write(5.7.38) in the form

where Coo is the limiting case of C\ as x\ —»• 0~. Then, from (5.7.37),

= 2<T*F_ (0) -v/2*/7r. (5.7.40)

This result is identical to (2.5.44).Finally, a special property of /i(#i, t) is noted. From the foregoing

development, it is evident that, for any time t > 0, the functionh(xi,t) is nonzero on the crack faces only for — Cdt < xi < —CRt.In other words, the impulse singularity acts at time t = 0 and thedisplacement of any material point on the crack face, in the directionof the normal to the face, is zero for all time after the Rayleigh waveemitted from the tip at time t = 0 has passed that point.

The corresponding normal stress G\2 ahead of the crack tip is,following inversion of (5.7.33) for 7 = ^,

Figure 5.11. The parts of the xi,£-plane over which u\ and cr£2 are nonzero,shown as the shaded regions. The lines labeled d, s, and R represent trajectoriesof the dilatational, shear, and Rayleigh waves, respectively, emitted from xi = 0at t = 0.

for t > 0 and xi > 0, where the path Ci now begins at ( = —t/xi—iO,and then extends along the lower side of the real axis to the branchpoint at £ = — a, around this point, and along the upper side ofthe real axis to C = — t/xi + iO. It is readily verified by means ofcalculations parallel to those performed above in studying u\ that

022 (si,6(t)

x3/2

(5.7.42)

for t > 0 and arbitrarily small values of x\ > 0. In view of the factthat <722(#i,0, t) = 0 on x\ < 0, it can be shown from (5.7.41) thatthis same condition holds for 0 < x\ < cst for all t > 0. The regionsof the Xi,t—plane over which u\ and a\2

a r e nonzero are shown inFigure 5.11. The remarkable feature is that both of these fields areidentically zero for — CR£ < x\ < cst for all time t > 0. Thus, eventhough the present situation presumes that the crack tip is stationaryat xi = 0, it could equally well meander according to xi = /(£), with1(0) = 0 for t > 0, and the fields denoted by a superimposed starwould still apply. This feature was exploited to great advantage byBurridge (1976) in his study of dynamic crack growth.

5.8 Energy radiation from an expanding crack 289

5.8 Energy radiation from an expanding crack

In this section, expressions for the energy radiated from an expandingplanar crack in an elastic solid are considered. The discussion is lim-ited to bodies that are unbounded except for the crack surface, and thematerial is assumed to be an isotropic and homogeneous linear elasticmaterial. The body may be subjected to some initial equilibriumstress distribution, but any transient stress field is assumed to arisefrom expansion of the crack in the body.

The energy radiated from an expanding crack is of particularinterest in the field of seismology, where the energy radiated froma dynamically expanding fault in the earth, inferred from transientdisplacement measurements at remote points, is used to infer a valuefor the energy released in the faulting process. Alternatively, thisenergy is estimated from changes in the equilibrium deformation ofthe earth's surface at points relatively near the epicenter of the earth-quake. These energies are equal only under certain conditions, and thepresent model provides a framework for considering this equivalenceand other issues associated with radiated seismic energy (Rudnickiand Freund 1981).

Consider an expanding crack that has a characteristic dimension/. The total energy radiated from this crack is defined as

ER = aijTijUidSdt, (5.8.1)J-oo Js

where Ui is the change in displacement from the initial equilibriumdistribution and Gij is the corresponding change in stress. The surfaceS is a spherical surface of radius r centered at the expanding crack,with r > /, and n« is the unit normal to S pointing toward the crack.The integrand of (5.8.1) is the rate of work done on the materialoutside S by the tractions on S. If the stress change cr and theparticle velocity iii are each proportional to r*"1 as r becomes verylarge, as is the case in some point source representations for far-fieldmotion, then ER is expected to have a finite nonzero value.

Kostrov (1973) has shown that the energy radiated from thecrack can be expressed in terms of the crack surface traction andparticle velocity. Let Sf be the surface coinciding with the crackfaces, excluding the edge, let St be a tube enclosing the crack edge,and let n* be a unit vector normal to 5, S/, and St and pointing into


the region bounded by these surfaces. A work rate balance for thisregion implies that

- / VijTijiii dS - t -V" = T(t) + I WijUjiii dS, (5.8.2)Js Jsf

where T and U are the rates of increase of kinetic energy and strainenergy, respectively, and F(t) is the rate of energy flow to the extend-ing crack edge through St. The total stress is

*y = * « + * y (5.8.3)

and the total particle displacement is

u{ = Ui + v% , (5.8.4)

where v% is the initial equilibrium displacement and o^j is the cor-responding stress. The left side of (5.8.2) is the excess of the totalpower input over the increase in internal energy, and the right sideis the energy dissipation at the crack edge and over the crack sur-faces. Adapting the general energy flux expression (5.2.11), the rateof energy flow into the crack edge is

= lim / {(TijTijiii + \ (WijUiij +puiiii) vriknik} dS, (5.8.5)st-*ojStF(t)

where v is the local normal speed of the crack edge in the plane of thecrack and mk is the local normal direction of crack front motion.

In view of (5.8.3) and the definition of radiated energy, (5.8.2)becomes

-ER- (a?jnjuidS-f-Tj = F(t)+ f aijtijiiidS. (5.8.6)Js Jsf

If this expression is integrated over all time then

ER+ f aijnju{inal dS + (Ufinal - Uinitial)Js

r°° _ f°° r= - F(t)dt- / WijUjUidSdt, (5.8.7)

J—oo J—oo J Sfoo J Sf


where Tfinai = Tiniuai = 0 has been assumed. For crack expansionunder equilibrium conditions, ER = 0 and (5.8.7) reduces to a formulafor the change of strain energy in the body, that is,

Ufinai - Uinitial + / ^ûf* n a l dSJs

/•oo /»oo p

= - / F0(t)dt- / [WijrijUi^ dSdt, (5.8.8)J-oo ./-oo «/£/

where the subscript zero denotes values during an equilibrium expan-sion. If (5.8.8) is subtracted from (5.8.7), then

ER= /J-oo JSf

{F0(t)-F(t)}dt, (5.8.9)

where the superposed bar may be omitted in this equation. Theintegrand of the second term is the difference between values of energyflux to the crack edge for quasi-static and dynamic propagation of thecrack edge, and the integrand of the first term is the difference betweenvalues of the work rate of the crack surface traction away from theedge for quasi-static and dynamic expansion. For bounded crack facetraction, the first term in (5.8.9) can be integrated by parts with theresult that

f°° fER -= I I \ \rij(JijUi\ — îîWjL r dSdt

J-oo J Sf

r°° _ _/ {F0(t)-F(t)}"dt, (5.8.10)

J—oo

/oo J Sf

+

where the integrated terms are zero because both the initial and finalstates are time-independent. This form of ER makes clear the roleof the time-dependent surface tractions in radiating energy. It showsthat it is impossible to obtain an unambiguous expression for radiatedenergy in terms of only changes from the initial equilibrium state ofthe medium to the final equilibrium. Furthermore, the first term of

(5.8.10) makes clear the important role for the radiated energy oftransients in fault plane traction due to variable interaction of thesurfaces undergoing relative sliding (Kostrov 1973).

An alternate form for the radiated energy can be obtained byusing (5.8.3) in the first term on the right side of (5.8.8), with theresult that

/ / [WijTijUi^ dSdt =J-oo JSf

( I [<Tijnjui]odSdt+ [ o?jnju{inal dS. (5.8.11)./-oo JSf Jsf

If the stress cr depends on time only through its dependence on u ,then

f°° f /* r "i/ / [crijnj/^Li\o dS dt = I \(oijnj)avguia dS, (5.8.12)

J-oo JSf JSf •" "*

where this equation defines the average crack face traction (cijnj)avg.In the special case where (Tijtij is proportional to Ui

(5.8.13)

Note that Wijnj]initiai is n°t the traction existing ahead of the crackon the prospective fracture plane, but rather that which exists on thecrack surface just after passage of the edge. If (5.8.12) is substitutedinto (5.8.11) and if the relationship

(aijnj)avg = (&ijnj)avg ~ ^jnj (5.8.14)

is incorporated, then

/ / [Wijnjiii]odSdt= f ( [(vijn^aygUi^ dS. (5.8.15)J—oo J Sf J — oo J Sf

Consequently, (5.8.8) can be written as

Ufinal ~ Uinitial + / ^njU^1 dSJs

= - / F0(t)dt+ / [((Tijn^avgiii^dSdt. (5.8.16)J—oo J—oo J Sf


If Fo and the integral over S are assumed to be negligible in the partic-ular application of dynamic earth faulting, then this equation reducesto the formula for strain energy change associated with faulting asdiscussed by Aki and Richards (1980). Yet another expression for theradiated energy is obtained by subtracting (5.8.16) from (5.8.7) withthe result that

ER= / {[(^ijnj)avgUi]0 - [VijfljUi]} dS dtJ-oo JSf

{F0(t)-F(t)}dt, (5.8.17)

where, as before, the superposed bars can be omitted.If the crack surface traction is time-independent, except possibly

in the small region very near the crack edge, then the first term in(5.8.9), (5.8.10), or (5.8.17) vanishes and the radiated energy is

r°° _ER= \F0(t)-F(t)] dt. (5.8.18)

«/ —oo

A special case of the expression (5.8.18) was used by Husseini andRandall (1976) in a study of the efficiency of a seismic source inconverting strain energy released to radiated wave energy.

The concept of the seismic moment of an earthquake sourcewas introduced by Aki (1966) to provide a quantitative connectionbetween source properties and the far-field, long wavelength energyradiated by the source. Under the simplest circumstances of relativeslip on a planar surface in a homogeneous and isotropic elastic body,the seismic moment is defined to be

M = n[u]AMp, (5.8.19)

where JJL is the shear modulus of the material, Asup is the total area ofslip, and [uj is the spatial average of the relative slip or displacementdiscontinuity, say [ity.J, across the fault plane

M = -J— IAslip JA8H

dS. (5.8.20)

The radiated energy ER is also a parameter of fundamental signifi-cance in the study of earthquake sources. It can be measured remotely,essentially according to (5.8.1), and its value is representative of phys-ical characteristics of the fault region. The foregoing analysis providesthe basis for observations on the possible connection between M andER.

The qualitative notion of seismic efficiency has been used inthe past to establish such a relationship. In terms of quantitiesalready introduced above, seismic efficiency is defined as the ratio7/ = ER/ \Ufinai — UiniUai\- The strain energy change of the materialin the volume bounded by the surfaces S + St + Sf which appearsin (5.8.7) can be expressed in terms of the elastic compliance of thematerial Mijki by

Ufinal - Uinitial = / hJv/ Mijkl

v

= IJv

= f - I \a{;nal + o*J njU{inal dS. (5.8.21)Js+st+sf

l Jt+sf

The last step in the calculation follows from the relation 2Mijki&ij =v>k<,i+ui,k and the divergence theorem. Suppose that the contri-butions from the remote surface S and the crack edge surface Stare assumed to be negligibly small compared to the fault surfacecontribution. If f^inal and r° denote the average values of the tractionvectors <r{jnalnj and c^jfij over Sf projected onto Sf, and if [uj isthe spatial average of the discontinuity in displacement across the slipsurface of area Asup, then

\Ufinal ~ Uinitial| « \ ( T / W + f°) [u]AMp . (5.8.22)

From the definition of seismic efficiency,

ER =T] \Ufinal(5.8.23)

^final _ ^ | u | = lv ^finol + ^o) M / / x

The quantity r}{r^lnal + r°)/2 was termed the "apparent stress" byWyss and Brune (1968). This idea is discussed more completely byKostrov and Das (1989).

Another approximate relationship between M and ER that doesnot rely on the idea of radiation efficiency can be derived directly from(5.8.7). Again, suppose that the contributions to this relationshipfrom the remote surface S and the crack edge surface St are negligiblysmall in comparison to the fault plane contributions, so that

ERtt- WijUjili dSdt - (Vfinai - Uinitial) . (5.8.24)J — oo

If the fault plane traction is time-independent then the time integra-tion in the first term on the right side of (5.8.24) is readily evaluated.This equation becomes

ER = - I off" V f " " ' dS + I § (*{jnai + <7°) njU{inal dS,Jsf JSf v 7

(5.8.25)

where the second term follows from (5.8.21). If the various fields arereplaced by their spatial averages, then

ER = \ (rfinal - r°) [u] = \ {rfinal - T°) M/fi. (5.8.26)

Thus, the ratio of radiated energy to the seismic moment is equal tothe ratio of one-half the average fault plane stress drop (r^inal — r°)to the shear modulus. Comparison of (5.8.23) with (5.8.26) suggeststhat the seismic efficiency is the ratio of the stress drop to twice thefault plane effective stress (jfinal + ^°)/2, that is,

(5-8-27)

as noted by Kostrov (1973). Though these results on seismic mo-ment and its connection to radiated energy are commonly used incharacterizations of earthquake sources, it must be recognized thatthey do not account for time dependence in the fault plane stressdistribution, for any transient aspects of the dynamic faulting process,or for inhomogeneities in material properties on the fault plane.

6

ELASTIC CRACK GROWTH

AT CONSTANT SPEED

6.1 Introduction

The focus in this chapter, as well as in the following chapter, is on an-alytical models of crack growth phenomena based on nominally elasticmaterial response. The analysis of Chapter 4 provides information onthe nature of crack tip fields during rapid crack growth for severalcategories of material response, and parameters that characterize thestrength or intensity of these fields are also identified. A main purposein formulating and solving boundary value problems concerned withcrack propagation is to determine the dependence of the crack tipfield characterizing parameters on the applied loading and on theconfiguration of the body.

For any particular crack growth process, it is often the case thatits analytical modeling can be pursued at several scales of observation.For example, consider the case of crack growth in an engineeringstructure. The full structure can be analyzed to determine the stressresultants that are imposed at the fracturing section. Under suitablecircumstances, these stress resultants can, in turn, be viewed as theapplied loads in a two-dimensional plane stress or plane strain problemfrom which the dynamic stress intensity factor can be determined. Ata finer scale of observation, the elastic stress intensity factor repre-sents the applied loading in an analysis of nonlinear processes veryclose to the crack edge. By focusing on dominant effects at adjacentlevels of observation in this way, various aspects of dynamic fracturephenomena can be analyzed.

296


In this chapter, attention is limited to crack growth at constantspeed in a nominally elastic material. The imposed condition of con-stant speed results in significant simplification of the boundary valueproblems to be solved for the governing partial differential equations.Under certain circumstances, the restriction to constant speed crackgrowth permits a reduction of the number of independent variablesby one. Thus, problems involving two space dimensions and time canbe reduced to boundary value problems in two independent variables.It is shown in this chapter that solutions of the equations of elastody-namics for certain crack growth problems can be expressed in termsof one or more general solutions of elementary partial differentialequations in two independent variables. The specific solutions aredetermined by the boundary conditions.

The first problem class considered is steady state crack growth.Here, the crack tip has been moving at constant speed for all time andthe mechanical fields are invariant with respect to an observer movingwith the crack tip. The prototype problem in this category is thetwo-dimensional Yoffe problem of a crack of fixed length propagatingin a body subjected to uniform remote tensile loading. The secondclass of problems considered is self-similar crack growth. In this case,the crack tip has been moving at constant speed since some initialinstant, and certain mechanical fields (to be identified in the contextof a particular problem) are invariant with respect to an observermoving steadily away from the process being observed. The prototypeproblem in this category is the two-dimensional Broberg problem ofa crack that suddenly expands symmetrically from zero initial lengthat constant speed in a body subjected to a uniform remote tension.

Next, the situation of the sudden constant speed extension of apreexisting crack in a body initially at rest and subjected to time-independent loading is considered. The particular cases analyzed arealso self-similar, but they are solved by means of integral transformmethods rather than by direct appeal to similarity arguments. Thisproblem is significant because the indirect analytical approach de-veloped to analyze it, based on superposition over a fundamentalelastodynamic solution and inverse arguments, opens the way foranalysis of certain problems of crack growth at nonuniform speed,which are described in Chapter 7.

As already noted, the objective of the analysis of crack growthis to understand the connection between crack tip motion in a bodyof given material and the loading applied to that body. Obviously,

298 6. Elastic crack growth at constant speed

the potential for improved understanding is severely diminished if themotion of the crack is assumed to be uniform from the outset. How-ever, the analysis of constant speed crack growth has been importantfor two main reasons. In some cases, the prevailing conditions aresuch that crack growth in a real material is indeed steady for a longenough time so that these constant speed results are useful in theinterpretation of the phenomenon. In addition, the progress in thestudy of crack growth at constant speed has resulted in the analyticaltools that have been successful in the analysis of crack growth atnonuniform rates.

6.2 Steady dynamic crack growth

Consider a planar crack extending in an elastic solid under conditionsof two-dimensional plane strain deformation. Suppose that the cracktip of interest moves in the plane of deformation at constant speedv. Furthermore, suppose that both the configuration of the bodyand the traction distribution are time invariant in a reference frametranslating with the crack tip, and that this situation has prevailedfor a "long time." In the typical configuration assumed in steadycrack growth analysis, the body is either unbounded or it has straightboundaries parallel to the crack plane. (Other configurations can beadopted for special purposes, say to develop numerical procedures fordynamic crack growth analysis, but the boundary conditions mustthen be chosen to simulate the property of translational invarianceof the configuration in the direction of crack growth.) Under theseconditions a steady state solution of the boundary value problem issought. The assumption that a steady state has been achieved permitsthe reduction of the number of independent variables in the problemfrom three to two, and the analysis is simplified considerably. It mustalways be kept in mind, of course, that some features of the crackgrowth process that are physically significant are overlooked in thisapproach, and the results must be assessed accordingly.

A rectangular coordinate system labeled x, y, z is introduced inthe body in such a way that the plane of deformation is the x, y-plane,and the crack is in the plane y = 0. The crack tip of interest advancesin the x-direction at constant speed and, without loss of generality,it is assumed that the crack tip passes x = 0 at time t = 0. If othercrack edges are present, each must be at a fixed distance from theedge of primary interest. Let /(x, y, t) denote any field over the body.

6.2 Steady dynamic crack growth 299

If this field is an element of a steady state solution then

/(*,y,t) = /(&»), (6.2.1)

where £ = x — vt. The functions on opposite sides of the equationin (6.2.1) are different, of course, but the values of the functionsrepresent one and the same physical quantity, so there is little dangerof confusing the two functions. The £, y-coordinate system translatesat speed v in the x- or ^-direction with the crack tip of primaryinterest, and this crack tip is at the origin of coordinates in thissystem.

6.2.1 General solution procedure

The main properties of the mechanical fields which result from thesteady state condition (6.2.1) are that, for any field / ,

^(x,y,t) = ^(t,y), %(x,y,t) = -v^foy), (6.2.2)

where the interpretation of the function is made clear by includingthe arguments explicitly. Following Sneddon (1952), the basic fieldquantities are chosen to be the dilatational and shear displacementpotentials (f>{£,y) and ip(£,y), respectively. An alternative approachbased on stress potential functions was introduced independently byRadok (1956), who considered applications to moving punch anddislocation problems, as well as crack problems. The equivalence ofthe two methods for the problem at hand was noted by Sneddon(1958). A representation in terms of stress functions that has beenuseful in the solution of some steady state problems for anisotropicelastic materials was introduced by Stroh (1962).

The displacement potentials satisfy their respective scalar waveequations (1.2.61) which, in the case of steady state motion, take theform

2

over the region of the £, y-plane occupied by the body, where a<j =y/l — v2 j(?d and as = y/l — v2/c2 . As noted in Section 1.3, theequations have the general solutions

4>(t,y) = Re{F(O)} , 1>(t,v) = Im{G(C)} , (6.2.4)

where Q = £ + iotdV and Cs = £ + ia sy, and the functions F andG are analytic functions of their complex arguments in the interiorregion of the body. These functions must be determined from bound-ary conditions. For the particular case of mode I deformation, thepotentials have the symmetries <t>(^—y) = </>(£,y) and ip(£,—y) =

)i whereas for mode II deformation 0(£, -y ) = —(f>{€,y) and> —y) — V>(£>2/)- These symmetries imply that

and G(C) = ±G(C) (6.2.5)

for mode I (upper sign) and mode II (lower sign), where the bardenotes the complex conjugate.

Displacement and stress components are readily expressed interms of the unknown functions F and G by means of (1.2.68) and(1.2.69). For example,

uy&y) = -Im{adF'({d) + G'({s)} ,

*yy& V) = -MRe{(l + a2)F"(&) + 2a.G"(C.)} , (6.2.6)

**„(& V) = -/iIm{2adF"(Cd) + (1 + a2s)G"(Cs)} ,

where the prime denotes differentiation with respect to the argument.The foregoing mathematical results are now applied to obtain

the solution of representative steady crack growth problems. The firstboundary value problem considered is that formulated by Yoffe (1951)in a pioneering continuum analysis of fracture dynamics. The secondcase studied is the mode II propagation of a half plane crack under theaction of crack face traction. The mode I counterpart of the secondproblem was first considered by Craggs (1960). The section concludeswith a brief discussion of the concept of crack tip cohesive zone, withinthe context of steady crack growth, as it has been applied to simulatecrack tip plasticity in ductile materials, interatomic interaction in purecleavage, and various frictional interactions between the sliding facesof a shear crack. The essential ideas of the cohesive zone models wereintroduced in Section 5.3.

6.2.2 The Yoffe problem

The plane strain problem analyzed by Yoffe (1951) is that of a crackof fixed length 2a gliding through an otherwise unbounded solid at

t t *- t t t

y

i i i i iFigure 6.1. Diagram of the plane strain problem considered by Yoffe (1951),showing a crack of fixed length moving steadily at speed v through a bodysubjected to uniform remote tension of magnitude (Too •

speed v, with the crack opening at the leading crack tip at £ = 0,y = 0 and closing at the trailing crack tip at £ = —2a, y = 0; seeFigure 6.1. The body is loaded by means of a uniform remote tensionof magnitude a^ in the ^/-direction. The crack faces are tractionfree. It is convenient (but not essential) to superimpose a uniformcompressive stress of magnitude a^ in the ^/-direction. Thus, thecrack faces are pushed apart by a uniform normal traction, Tx (£) = 0and Ty(£) = ±(Joo on y — ±0, —2a < £ < 0. All stress componentsvanish as £2 + y2 —• oo. The boundary conditions are

^ 2 / ( ^ 0 ± ) = -aoo, ^ ( £ , 0 * ) = 0 (6.2.7)

for —2a < £ < 0. The problem possesses the symmetry characteristicof mode I deformation, and all fields are continuous across y = 0 for£ > 0 and £ < -2a .

In terms of the unknown functions F and G, the boundary con-ditions take the form

+2as [K U)] /(6.2.8)

2ad [F?(O - F'_!(O] + (1 + a2.) [G?(O - &!_{£)] =0

for —2a < £ < 0, where the prime denotes differentiation with respect

to the argument,F"(C*), (6.2.9)

and the symmetries noted in (6.2.5) have been incorporated.The second boundary condition implies that the function

E(C) = 2adF"(() + (1 + c%)G"(Q , (6.2.10)

which is analytic everywhere in the plane of the body except possiblyalong the crack line, is continuous across the crack line. The onlysingularities that E(Q could have are poles on the crack line, butthis possibility is precluded by the boundary conditions and the tacitassumption that concentrated forces do not act at the crack tips.Consequently, E(() is an entire function. Furthermore, the vanishingof stress at remote points implies that E(Q —• 0 as |£| —> 0. Liouville'stheorem on bounded entire functions then implies that E(Q = 0, orthat

G"(O = - f ^ 2 F " ( O (6.2.11)

throughout the region of the body.In light of (6.2.11) the first boundary condition (6.2.8) takes the

form

F " ( O + F!!(t) = - 2 ( 1 t a s ) — > ~2a < £ < o> (6-2-12)

where D is given in (4.3.8) as 4a<jas — (1 + OL2S)

2. This equation hasthe homogeneous solution

where P and Q are polynomial functions of (. The C-plane is cutalong the crack line, and the branch of ^/£(C + 2a) that is positiveand real on the positive real £-axis is assumed. The polynomial Q(C)may have zeros only at the ends of the crack line, but this possibilityis precluded by the condition that there can be no concentrated forcesor dislocations there. Thus, Q(() = 1 without loss of generality. Theequation (6.2.12) has the particular solution

F'm = JJ^1^f • (6-2.14)


Then, to ensure that the stress vanishes at remote points, P(£) mustbe a polynomial of degree one, that is, P(() = Po + PiC with theconstant Pi having the value

1 + OL28 (JQQ f f iOl^

P ( 6 2 1 5 )

To ensure continuity of displacement across the crack plane behindthe advancing crack,

Po = aPi. (6.2.16)

This condition renders F"(C) symmetric with respect to the center ofthe crack. The function F"(C) is completely determined as the sumof the homogeneous and particular solutions,

v ' D /i

which implies from (6.2.11) that

(6.2.17)

With the functions F"(() and G;/(C) determined, features of thesolution of physical interest can be extracted. For example, from(6.2.6)2, the normal stress distribution (Tyy(£,0) on the crack lineahead of the advancing crack tip is

f - l . (6.2.19)

For the original problem of uniform remote tension in the directionperpendicular to the crack plane and traction-free crack faces, the sec-ond term in brackets is absent. It is noteworthy that the distributiondoes not depend on the crack speed v. From (4.3.9), the dynamicstress intensity factor is

/i[(t,0) = a^y/Tm', (6.2.20)

which is identical to the corresponding equilibrium result for any v.By integrating (6.2.17) and (6.2.18), the crack opening displacementis found from (6.2.6)i to be

—la < £ 0, the coefficient v2otdlc2

sD —• (1 — v),so the expression (6.2.21) reduces to the corresponding result for theclassical Griffith crack, and the magnitude of displacement increasesmonotonically with increasing v for fixed <7oo-

Finally, it must be recognized that the mathematical problemconsidered in this section is not a realistic model of a physical situationbecause of the feature that the crack closes at one end at the same rateat which it opens at the other end. In her study of dynamic fracturephenomena, on the basis of this analysis Yoffe (1951) drew conclusionsonly from those features of the solution that are independent of thefictitious crack length 2a. In particular, she was interested in theangular variation of the near tip stress field. As was mentioned inSection 4.3, this variation was established some years later to beuniversal for a given material. Features of the solution of this problemor of similar problems that do depend on the fictitious crack lengthmust be viewed as having limited significance. Another unrealisticfeature of the system emerges if it is considered from the energy flowpoint of view. For 0 < v < CR, the leading edge of the advancingcrack of fixed length absorbs a certain amount of energy from thebody according to the result in Section 5.3. The system is in a steadystate, however, so the total mechanical energy cannot change and,indeed, it does not. If the energy flux at the trailing crack edge isexamined, it is found that mechanical energy is added to the bodythere at exactly the same rate that it is lost at the leading crackedge. Because the crack opening is symmetric with respect to thecenter of the crack, the tractions acting on the crack faces do no network. For CR < v < e5, the roles of the two crack tips as sourceand sink of energy are reversed. For this case, if the crack is indeedopen then the normal stress on the crack plane ahead of the leadingtip is compressive and the local energy flow is out of the crack tip.Similarly, the trailing crack tip absorbs energy at the rate at which itis radiated at the leading tip.

9*

Figure 6.2. Diagram of steady crack growth in mode II due to the action of anopposed pair of concentrated forces acting on the crack faces at a fixed distance Ibehind the crack tip.

6.2.3 Concentrated shear traction on the crack faces

As a second illustration of the analysis of a steady crack growthproblem, consider the plane strain situation of a semi-infinite crackalong y = 0, £ < 0 with the crack edge at £ = 0 advancing inthe ^-direction at speed v in an otherwise unbounded body. Forthe time being suppose that there are no remote tractions on thebody but that the crack faces are subjected to concentrated loadstangential to the crack faces at a distance I from the crack edge, thatis, Tx(£) = ±q*6(£ + /), Ty(€) = 0 on y = ±0, f < 0. The magnitudeof each of the concentrated forces is q* and the point of application ofeach moves along the faces with the same speed v as the crack edge.The problem possesses the symmetries of a mode II deformation field.The problem is depicted in Figure 6.2 and a steady state solution issought.

In terms of stress components, the boundary conditions are

<rxy(Z, ±0) = -q*6(£ + I), *yy(Z, ±0) = 0 (6.2.22)

for — oo < £ < 0. Stress vanishes at points far from the crack tipcompared to /. A general solution exists in the form of two analyticfunctions of a complex variable through (6.2.4). These two functionsare again denoted by F and G. The conditions on these functionsimposed by the boundary conditions (6.2.22) are

(1 + eft [F!l(O - F'!{0] + 2as [G'KO - G"_{Ci\ = 0,

2ad [F?(O + F'_!(0] + (1 + a2.) [G'KO + G'L(O] = 2iq'6(t + /)//*,(6.2.23)

where the symmetry properties (6.2.5) for mode II deformation havebeen incorporated.

The first of the two equations (6.2.23) implies that

G"(O = - ^ 7 ^ " ( O (6-2.24)

throughout the region of the body. The arguments leading to thisconclusion are similar to those presented in the preceding subsection.The function G can be eliminated from (6.2.23)2 by means of (6.2.24),with the result that

Kit) + FH(O = % ^ « ( * + I) (6.2.25)

on —oo < £ < 0. A solution of this equation that is consistent withthe constraints of the physical problem is

With F"(C) determined throughout the region of the body, andconsequently G"(() determined, any feature of the solution can beexamined. For example, the shear stress distribution aXy(€i 0) o n

crack line ahead of the tip £ > 0 is found to be

h) <6-2-27)and the mode II stress intensity factor is

j ^ (6.2.28)

Again, the feature that both the stress distribution on the crack lineand the stress intensity factor are independent of the crack speed v isobserved.

6.2.4 Superposition and cohesive zone models

As noted in Section 4.3, the study of the advance of a sharp crackon the basis of an elastodynamic model leads naturally to the result

that the stress is unbounded at the crack edge. At a certain levelof observation, the result is accepted on the basis that the size ofthe region over which the material behavior deviates from linear elas-ticity is small enough to admit the use of the stress intensity factorconcept. The cohesive zone idea provides a simple yet useful devicefor examining crack tip phenomena at a level of observation for whichdeviations from linearity in material response are included. On thebasis of energy concepts, it was determined in Section 5.3.2 that stressintensity factor in a mode I deformation field can be related to theconstitutive characteristics of the cohesive zone through (5.3.18), forexample. However, it was not possible to determine the size of thecohesive zone in terms of the remote loading and the constitutiveproperties of the zone by means of the global energy arguments.Instead, the determination of the size A must follow from a studyof the stress distribution.

The main ideas of cohesive zone models are simpler to conveyfor mode I deformation than for mode II deformation. Although theconcentrated force problem analyzed in the preceding subsection wasa case of mode II, the mode I results corresponding to (6.2.27) and(6.2.28) are obvious. The first is

) < 6 - 2 - 2 9 )

for normal stress on the crack line £ > 0, where p* is the magnitudeof each concentrated force acting at £ = — I to open the crack. Thecorresponding stress intensity factor is

Ki=p*ylrr (6-2-30)The essential idea of the cohesive zoile model for plane strain

mode I crack growth under the simplest circumstances is outlinedin Section 5.3.2, and the notation of that section is followed here.The stress intensity factor due to the cohesive stress is determined bysuperposition. The concentrated force p* in (6.2.29) is first replacedby a traction — a(l) distributed over the infinitesimal interval between£ = — I and £ = — (I + dl), and the result is then integrated over therange 0 < / < A to yield

The total stress is nonsingular or Kj = 0 if A is chosen to satisfy

for given a(l). The weighted integral of the cohesive stress on theright side of (6.2.32) is sometimes called the modulus of cohesion, aterminology apparently due to Barenblatt (1959a, 1959b).

To make further progress with the cohesive zone idea, it is nec-essary to be more specific about the nature of the cohesive stresscr(l). The simplest case of a uniform cohesive stress was introducedby Dugdale (1960) to represent a plastic zone near a crack tip in athin sheet of an elastic-plastic nonhardening material under tension.He assumed the cohesive stress to be the tensile flow stress ao of anideally plastic material. Then, from (6.2.32), the traction on the planey = 0 will be less than or equal to ao everywhere provided that A is

( 6 < 2 . 3 3 )

The size of the crack tip inelastic zone scales with the length pa-rameter (Ki/ao)

2 which could have been anticipated on the basisof dimensional considerations. With A determined, the crack lineloading is completely specified and any feature of the solution can beextracted by following the steps outlined in the foregoing subsections.It should be noted that the physical crack tip is at £ = —A and notat £ = 0. The fact that the crack tip is initially taken to be at £ = 0is simply a mathematical artifice of the approach.

Although the result (6.2.33) is based on an assumption that thecrack growth process is steady, it is reasonable to expect that it will beaccurate for nonsteady growth provided that cohesive zone length issmall compared to overall crack size and that the crack speed does notchange appreciably over growth distances equal to several times thezone length. As an application for nonsteady crack growth, considerthe result (6.2.32) for the case when the cohesive stress a dependson position I only through a dependence on the local crack openingdisplacement 6(1). Another relationship between the applied dynamicstress intensity factor and the cohesive stress is given in (5.3.17) onthe basis of energy considerations. If Kiappi is eliminated from the

two equations, it follows that

where Aj(v) is given in (5.3.11). If the crack is assumed to growwith a fixed value of crack opening displacement 6t then the left sideof (6.2.34) is essentially a material constant. Suppose now that thecrack grows with an ever increasing remote driving force so that vcontinues to increase. The function Ai(v) increases monotonicallywith speed i/, becoming unbounded as v —• CR, SO that the integral insquare brackets in (6.2.34) must decrease as the crack speed increases.This suggests that A must decrease as v increases such that A —• 0as v —• CR. Because the magnitude of particle displacement mustincrease from zero to 6t over a distance A, large displacement gradientsand thus potentially enormous stress components must be generatedat points near the cohesive zone. This observation was developed byRice (1980) within the context of steady crack growth analysis appliedto earthquake rupture mechanics.

To pursue the matter a bit further, suppose that the interactionstress has the uniform value cro everywhere within the cohesive zone(or slip zone for the case of shear modes). The relation (6.2.34) canbe rewritten as

A/Ao = A(v)- 1 , (6.2.35)

where Ao is the length of the cohesive zone or slip zone when the crackgrowth condition of fixed crack tip opening displacement is satisfiedunder equilibrium conditions. The function A(v) in (6.2.35) is writtenwithout a subscript because the relationship is valid for each of thethree modes of crack tip opening. The form of A(v) for each modeis given in (5.3.11). For the present case, it is evident that A doesindeed become smaller as v increases. For mode II deformation andthe coordinate system established in Figure 6.2, Rice (1980) notedthat the average value of stress axx, say {(7XX), along the stretchedside of the slip zone is

<O = i3^~> = T^Tfc = V/(")TO, (6.2.36)

where ro is the uniform shear traction in the zone, A = nKjappl/8r*,and (1 — v2)An(v)K?appi = Ero6t for mode II. If (crxx} becomes


large enough, local tensile fracturing off the main crack plane mayoccur. This effect can contribute to making shear fracture of thekind thought to model seismic faulting unstable at high crack speed.Though stresses at points off the main crack plane may become largeduring rapid crack growth for a number of reasons, the influence ofoff-plane stresses has received only limited consideration in dynamicfracture studies.

Finally, it is observed that some specific results obtained areidentical to the corresponding equilibrium results for the same con-figurations, but that they are nonetheless exact for steady dynamiccrack growth. Once again, it is noted that the cohesive zone ideawas introduced above with the feature that this zone was completelyembedded within a stress intensity factor field. This is the small-scale yielding situation introduced in Section 1.1.2. It is not essentialthat this be the case for the idea to be useful, and an example forwhich the situation is relaxed is given in Section 7.6. For a crackof finite length with cohesive zones in a body subjected to uniformremote tension of magnitude a^ in a direction transverse to the crackplane, Rice (1968) showed that the cohesive zones may be viewed asbeing embedded within a stress intensity factor field provided thatthe ratio of applied stress to uniform cohesive zone stress CTOO/CTO isless than about 0.2. Because the stress distribution on the crack planeis independent of crack speed for steady elastodynamic crack growth,presumably the same restriction applies for this case as well.

6.2.5 Approach to the steady state

A steady state situation for a physical system should always be viewedas a state that is approached in some limiting sense. In consideringsteady state crack propagation solutions, it is important to have someunderstanding of the rate of approach of a transient field to its steadystate limit, assuming that such a limit does indeed exist. It is adifficult issue to resolve unambiguously. If transient solutions couldbe found in cases of interest to permit examination of their long-timebehaviors, then there would be little motivation for considering thecorresponding steady state problems. Fortunately, a limited numberof transient solutions that have steady state limits can be found, thusproviding some guidance in the matter.

It appears that configurations that do not have a characteristiclength will not approach a long time limit of interest. This is suggestedby the dimensional argument leading up to (2.4.1). In the absence of

a characteristic length, the limit as t —* oo is indistinguishable fromthe limit as the observation point moves toward the crack tip. Thus,as time becomes very large, a feature of the solution is that stress andparticle velocity increase without bound everywhere in the finite partof the plane. For this reason, attention in this section is limited toconfigurations with a characteristic length in considering the questionof approach to the steady state.

For a configuration with a single characteristic length, say lo, andno characteristic time, an important time period is the time for a waveto travel the distance Zo at speed c, where c = c , cs, or CR. Supposethat a transient process having these features begins at time t = 0.A rule of thumb in wave propagation analysis is that the solution isapproximated by its long-time limit for time t greater than two orthree times such a wave transit time, that is, for t > nlo/c, wheren = 2 or 3. For example, suppose that the Yoffe crack in Section6.2 is initially at rest in an unbounded body subjected to uniformremote tension. The characteristic length in this case is the lengthof the crack 2a. At time t = 0, the crack tips both begin to moveat speed v < CR in the x-direction. On the basis of the qualitativeargument above, it is expected that the steady state stress intensityfactor (6.2.20) is a good approximation to the actual transient stressintensity factor for time t > ncR/2a for n = 2 or 3. The speed CR ischosen for this case because the main interaction between the crackedges occurs through Rayleigh waves propagating on the crack faces,as suggested by the analysis of Section 3.3.

A simple but instructive mode III crack propagation problemwhich sheds further light on the matter of approach to the steadystate and which can be solved completely is the following. A halfplane crack exists in a body that is initially at rest and stress free. Interms of rectangular coordinates, the plane of the crack is y = 0, andthe crack occupies x < 0. At time t = 0 the crack begins to move atspeed v < c = c8. Simultaneously, a traction distribution given by

- r* if -lo < x - vt < 0 ,(6.2.37)

0 if x < lo

f1

begins to act on the crack faces. The traction on the crack faces isuniform over an interval of length lo behind the moving crack tip,and it is zero elsewhere. The steady state stress intensity factor forthis configuration is the equilibrium stress intensity factor Km (oo) =

This transient crack propagation problem can be solved exactlyby means of Green's method introduced in Chapter 2. The completetraction distribution on the plane y = 0 between the leading wavefrontat x = ct and the crack tip at x = vt is found to be

, (6.2.38)

where

{ ct — xif ct — lo < x < ct,

X~vt (6.2.39)— — if vt < x < ct - lo .x — vt

The corresponding dynamic stress intensity factor is

~ ^ ' (6.2.40)

In this case, the transient stress intensity factor assumes its long-time, steady state limit at t = lo/{c — v), and it maintains this valuethereafter. The time t = lo/(c — v) is the time at which the wavefrontemitted at x = — lo and t = 0 overtakes the moving crack tip. This isa case where the rule of thumb argument put forward above actuallyunder estimates the time required for the transient stress intensityfactor to approach its steady state limit.

Experience with various two-dimensional transient elastodynamiccrack solutions leads to the following conjecture for this class of prob-lems. Consider the wavefronts generated through sudden applicationof boundary loading or through scattering of waves at crack edges andother geometrical features. For a particular situation, if all wavefrontsare generated within a finite time interval, then the transient stressintensity factor assumes its long-time limit immediately upon gener-ation of the last wavefront. The mode III crack propagation problemdiscussed in this section provides an illustration of this behavior. Onthe other hand, if new wavefronts continue to appear as time goes on,then the transient stress intensity factor gradually approaches its longtime limit in a oscillatory manner. The case of transient loading of acrack of finite length exhibits this kind of behavior. This conjecture isbased only on the few solutions that have been found, and no generalmathematical argument is available to support it.

6.3 Self-similar dynamic crack growth 313

6.3 Self-similar dynamic crack growth

Once again, consider a planar crack growing dynamically in an elasticsolid under two-dimensional plane strain conditions. To introducethe notion of self-similarity in concrete terms, attention is focused onthe symmetric expansion of a crack at constant rate from zero initiallength. The crack grows in an otherwise unbounded body which issubjected to uniform tensile stress at remote points. The magnitudeof the remote stress is a^ and the tensile direction is normal to thecrack plane. Due to the linearity of the problem, it can be assumedthat the crack grows in a body that is stress free and at rest prior tothe onset of crack growth, and to grow under the action of compressivenormal traction of magnitude a^ on the crack faces.

A rectangular coordinate system is introduced so that the planeof deformation is the x, /-plane, and the crack begins to grow at timet = 0. The system of differential equations, boundary conditions,and initial conditions is a linear system. Consequently, any stresscomponent has the representation

aij(x,y,t) =croofij(x,y,t), (6.3.1)

where fij is a dimensionless function of its arguments.Next, it is observed that the physical system involves neither a

characteristic length nor a characteristic time. The crack length scaleswith time £, so it is not a characteristic length. It follows that if fij isdimensionless and if there is no characteristic length or time againstwhich the independent variables can be scaled then / can only dependon dimensionless combinations of x, y, and t. For example, it can beassumed that

fij(x,y,t) = fij (ct/x,ct/y) , (6.3.2)

where c is a wave speed. Thus, the function fij is a homogeneousfunction of its argument of degree zero. Solutions of boundary valueproblems having this property are commonly called homogeneous so-lutions and the fields that they describe are called self-similar fields.The latter term derives from the fact that when such fields are com-pared at two diflFerent times they are found to be identical except fortheir spatial scale. Evidently, the property of self-similarity reducesthe number of independent variables from three to two in this case andthe mathematical problem becomes more manageable. Some early

examples of the use of homogeneous solutions of the scalar wave equa-tion are reviewed by Bateman (1955). Applications to elastodynamicproblems involving stationary cracks were presented by Miles (1960)and Papadopoulos (1963). A thorough discussion of the analysis ofself-similar mixed boundary value problems in elastodynamics waspresented by Willis (1973). He considered problems in both two andthree space dimensions, and his general results embody virtually allknown self-similar elastodynamic solutions. The approach exploitedhere has evolved from the study of external flows in steady supersonicaerodynamics, and a general discussion from this point of view ispresented by Ward (1955) under the heading of Busemann's methodof conical fields in supersonic flow. In that case, the axial coordinateof the conical region is a spatial coordinate, whereas in the presentapproach the axial coordinate is time.

The idea of self-similar fields was motivated by the introductionof a particular crack growth problem that will be solved subsequently.It is an idea of broad applicability that can be used to great ad-vantage. As a second elementary illustration within plane strainelastodynamics, suppose that two parallel edge dislocations that areinitially coincident begin to move away from each other at a uniformrate. If the magnitude of the Burgers vector of each dislocation is A,then each component of displacement has the form

Ui{x,y,t) = A9i (ct/x,ct/y) , (6.3.3)

where <fo is a homogeneous function of x, y, and t of degree zero.Furthermore, the derivative with respect to any of the original inde-pendent variables of a function that is homogeneous of degree m isitself homogeneous of degree m — 1. Consequently, if the displacementfield is homogeneous of degree zero, then the velocity field and thestress field are homogeneous of degree — 1, the acceleration field andthe stress rate field are homogeneous of degree —2, the displacementpotential fields are homogeneous of degree 4-1, and so on.

6.3.1 General solution procedure

Suppose that f(x, y, t) is a homogeneous function of its arguments ofdegree zero of the type introduced above. Further, suppose that / isa wave function with characteristic wave speed c, that is,

°- (6'3-4)

As already noted, the property of homogeneity permits reduction ofthe number of independent variables from three to two. The choice ofthe two reduced independent variables is not unique, and for presentpurposes the particular choice is

), (6.3.5)

where

= y/x2 + y2 . (6.3.6)

Again, the functions on opposite sides of the equal sign in (6.3.5) aredifferent but they are represented by the same variable name, withlittle likelihood of confusion.

If the derivatives appearing in (6.3.4) are formed by differentiat-ing both sides of (6.3.5), taking into account the implicit dependenceof £ and 9 on x, y, and t, then the partial differential equation takesthe form

2 *fOf.*f «0t d&=

This second-order equation in two independent variables is ellipticif £ > 1 and hyperbolic if 0 < £ < 1. Note that the discriminatingcondition £ = 1 is the equation of a cone in x, y, t-space. An additionaltransformation of independent variables is sought that will transformthe differential equation to Laplace's equation in the former case andto the elementary wave equation in the latter case. To this end,introduce a new variable s that is related to £ through a function s =u;(£) that is to be determined. Then, /(£,0) = /(u;(£),0) = f(s,0)and the partial differential equation takes the form

% + % = 0• (6.3.8)

If (6.3.8) is to reduce to Laplace's equation for £ > 1 then the coeffi-cient of d2 f Ids2 must be unity, or

u/(£) = ± ( £ 2 - l ) - 1 / 2 - (6.3.9)

Fortuitously, for this choice of u/(£), the coefficient of df/ds in (6.3.8)vanishes identically and the differential equation is indeed reduced to

The function u;(£) itself is determined by integration of (6.3.9) withthe result that

w(0 = ± cosh"1 £, £ > 1, (6.3.11)

where an inconsequential constant of integration has been taken aszero. The transformation (6.3.11) is commonly known as the Chaply-gin transformation. Likewise, the differential equation is reduced tothe wave equation

w--^=0 (6-3-12)if u;(£) = sin"1 £ or cos"1 £ for 0 < £ < 1.

A major advantage is gained by means of the transformation, inthe sense that general solutions are known for both Laplace's equationand the wave equation in two independent variables. For example, asolution of (6.3.10) is

f = Re{F1(9 + is)}, (6.3.13)

where Fi is an analytic function of its argument in a semi-infinitestrip of the complex 9 + is-plane. If the plus sign is chosen in (6.3.11)then the strip extends over 0 < s < oo and an appropriate range of0. If the minus sign is chosen in (6.3.11) then the semi-infinite stripextends over — oo < s < 0 and the same range of 9. The latter case ischosen here for a reason to be clear in the subsequent development.Furthermore, 9 is assumed to be in the interval —?r < 9 < TT. Othercases, such as 0 < 9 < TT, can be handled with minor modifications.Next, it is observed that the strip — oo < s < 0, -n < 9 < n of thecomplex 9 + is-plane is simply a rectangle with one edge at infinity.Thus, a degenerate Schwarz-Christoffel transformation can be used toconformally map the part of the semi-infinite strip with 9 < 0 intothe lower half of the complex C~plane and the part with 9 > 0 intothe upper half of the C-plane. The transformation is

C = - cos(0 + is) = - cos(0 - i cosh"1 £), (6.3.14)c c

where the factor 1/c appears only for convenience. If the cosineexpression is expanded, and 9 and £ are written in terms of physicalcoordinates, then

(=^t + i^y/t*-ryc> , - > 1 . (6.3.15)rz r£ r

6,3 Self-similar dynamic crack growth 317

Thus, the portion of the physical plane y > 0 (y < 0) is mappedinto the upper (lower) half of the complex £-plane, a correspondenceassured by the choice of the negative sign in (6.3.11). It is noted thatthe wavefront r = ct — 0 for y > 0 (y < 0) maps into the portion of thereal axis with £ = x/cr+iO (£ = x/cr—iO), where —1 < x/r < 1. Also,the portion of the plane y = 0* bounded by the wavefront maps intothe portion of the real axis with £ = t/x±iO1 where —1/c < t/x < 1/c.The corners of the rectangle in the 0 + is-plane are singular pointsof the Schwarz-Christoffel transformation, and these points map intoRe(C) = ±c~ \ Im(£) = 0, which are branch points in the complexC-plane. A branch cut between these two points is introduced, a stepthat could have been anticipated when it was noted that two pointson the wavefront in the physical plane map into the same point onthe real axis of the £-plane. The points on the wavefront in the upper(lower) half of the physical plane are mapped into the points on theupper (lower) side of this branch cut. For future reference, it is notedthat the expression (6.3.15) for £ is a root of the equation

(6.3.16)

where the branch of \Jc~2 — £2 with positive (negative) real valueson the upper (lower) side of the cut is assumed.

Because the mapping (6.3.14) is conformal, the wave function /has the representation

/ = Re{F2(C)}, (6.3.17)

where F^ is an analytic function of £ for y > 0 and y < 0. The functionF2 may have singularities on the real axis other than the branchpoints at £ = ±c~x. Thus, the determination of the wave functionsatisfying (6.3.4) has been reduced by the property of self-similarity todetermination of an analytic function of a complex variable in either ahalf plane or a whole plane. The analytic function must be determinedfrom its boundary values, a process greatly facilitated by the factthat the complex variable is related to the physical coordinates in thesimple way shown in (6.3.15). Next, the solution for a particular crackgrowth problem is constructed by taking advantage of self-similarity.The general approach is as follows. First, the field quantities that arehomogeneous of degree zero are identified by anticipating certain qual-itative features of the solution. Then these fields are represented as

sums of wave functions, each of which can be expressed in terms of ananalytic function. The geometrical symmetries, boundary conditions,and other constraints on the solution are then invoked to determinethe analytic functions, and thereby the complete solution.

6.3.2 The Broberg problem

The scope of mathematical solutions available for interpretation ofdynamic fracture phenomena was extended significantly with the so-lution of the problem of the self-similar expansion of a crack from zeroinitial length in a uniform tension field by Broberg (1960). This is theproblem described in the opening remarks of this section which pro-vided the motivation for pursuing the implications of self-similarity inanalysis of certain boundary value problems. Broberg used solutionsof a sequence of particular problems to produce a complete solution tothe crack expansion problem of interest. Though the property of self-similarity played a role in his procedure, the approach to be followedhere is more direct and, in the course of describing it, a generalapproach for solving a wide range of problems having the propertyof self-similarity emerges. A number of illustrations are presented byCherepanov and Afanasev (1973), who developed a general frameworkfor analysis of self-similar fields.

A rectangular #, -coordinate system is introduced in the planeof deformation, oriented so that crack growth occurs in the planey = 0. The crack begins to expand symmetrically from zero initiallength at time t = 0 with each tip moving at a constant speed v thatis less than the Rayleigh wave speed of the material. At any latertime, the crack occupies the interval —vt < x < vt. Both crack facesare subjected to compressive normal traction of magnitude a^. Theshear traction is zero on the crack faces, and no other loads act onthe body. The material is stress free and at rest for t < 0. Thus,displacement potentials 0 and ij) that satisfy the wave equations

dy* c\ di* " U ' dx* + dy* <*W " U K }

are sought subject to the boundary conditions

cryy(x, 0±,t) = -(Too , axy(x, 0±,t) = 0 for \x\ < vt. (6.3.19)


The solution is expected to have a symmetry with respect to reflectionin the x-axis typical of mode I fields, namely,

ux(x, -y , t) = ux(x, y, t), uy(x, -y, t) = -uy(x, y, t). (6.3.20)

As anticipated in (6.3.2), the stress and particle velocity componentsare homogeneous functions of #, y, and t of degree zero, and a solutionis sought on this basis.

The in-plane stress components are given in terms of the dis-placement potentials by

c\ dx2 \c2s ) By2 dxdy '

oxoy( 6 . 3 . 2 1 )

^2 |

/x dxdy dy2 dx2 '

where // is the elastic shear modulus. Likewise, the particle velocitycomponents are

i ^ 3 ^ ^ ! ^ ( 6 3 2 2 )Ux ~ dxdt + dydt' Uy ~ dydt dxdt • {b'6-zl)

The left side of each equation in (6.3.21) is a homogeneous functionof degree zero which has certain symmetry properties with respectto reflection in the plane y — 0. These properties will be assured ifsolutions of the wave equations (6.3.18) are sought such that each termon the right side of each equation in (6.3.21) has the same propertiesas the function on the left side. Thus, in light of the discussion inSection 6.3.1, the function d2(t>/dx2 can be represented as the realpart of a function, say FXX1 of the complex variable

-r2/c2, °-f>l, (6.3.23)

that is analytic in the complex Cd-plane except at certain singular

points on the real axis. In a similar way,

^ = Re{Fxx(Cd)} , jf-%- = Im{Fxy(<;d)} ,dx2 dxdy y

d2<f>2

dy

d2d d26

and, likewise,

dy2

(6.3.24)

= lm{Gyy(Cs)}, (6.3.25)

where

C ^ ^ + V^^W, f>l- (6-3.26)

The required symmetries of the stress field with respect to the crackplane are assured if

(6.3.27)

for a, (3 as any two of x, y, or t, where the overbar denotes complexconjugate.

The functions F x x , Fxy, Fyyj and Fyt are all derivable from thesame function <f> according to (6.3.24), so they are not independent.Indeed, it is evident from (6.3.24) that the functions Fxx and Fxy arecompatible only if

(6.3.28)

which is assured if

y/"2-C2Kx(Q = -KKy(O (6.3.29)

everywhere in the complex plane, where a = c^1 and the primedenotes differentiation with respect to the argument. In establishing(6.3.29), it is necessary to evaluate the partial derivatives of (d(x, V-, t)with respect to x and y. This is done most conveniently by differen-tiating the relevant special case of (6.3.16) with respect to x and y tofind that

a2-CJdx x^/a* -Q- yd ' dy Xy/a? -Q- yd

Likewise, it is found that

-iVtf^PKyiC) = CFyy(C) = iCV^eF^O (6.3.31)

identically in £, and

= C2G'yy(O = -C(b2 - C2)G'xt(O • (6.3.32)

As a first step in the determination of the various analytic func-tions in terms of which the physical fields are represented, considerthe dilatational contribution to iiy, namely, d2cj)/dydt in (6.3.22).Due to symmetry of the fields, this function vanishes on y = 0 forvt < \x\ < Cdt. This interval maps into the portion of the real axisa < |Re(Cd)| < h in the complex ^-plane, where h = v~x.

Consider next the behavior of d2(/)/dydt at the wavefront r =Cdt. It is evident that this quantity is zero everywhere ahead of thewavefront, that is, for r > Cdt. It is well known from the theory oflinear plane strain elastodynamics that sudden application of a forceof constant magnitude in the elastic plane produces a dilatationalwavefront that carries a square root singularity in particle velocityand stress. Sudden application of a force with a magnitude thatincreases linearly in time from zero initial magnitude produces adilatational wavefront across which the particle velocity and stress are

continuous. The applied loading in the present problem is even lesssingular than the second of these two cases because it is essentiallya self-equilibrating pair of linearly increasing forces. Consequently,it is anticipated that stress and particle velocity will be continuousacross the wavefront r = Cdt in the present problem. An immediateimplication of the initial conditions is that d2</>/dydt = 0 on r = Cdt—0for — 7T < 0 < 7T. This wavefront is mapped by (6.3.23) into theupper and lower sides of the real axis in the C<j-plane in the interval-a < Re(Cd) < a.

The implication of the foregoing arguments based on symmetryand the initial conditions is that

= 0 for -h<£<h (6.3.33)

on the real axis of the complex C -plane. An analytic function Fyt(Q)that satisfies this condition and that results in a particle velocity thatis square root singular at the crack tips is

Fvt(C) = PFP , (6.3.34)

where PF(C) a nd QF(O a r e polynomial functions with real coeffi-cients. The square root function in the denominator is single valuedin the plane cut along h < |Re(£)| < oo, Im(£) = 0 and the branchwith positive real value at the origin is selected. Similar argumentsapplied to d2^/dxdt lead to a representation for Gxt(C) in the complexC-plane with the same general features,

Gxt(0 = PG^} . (6.3.35)

Because neither of these functions represents a physical quantityby itself, anticipated features of the solution cannot be used to deter-mine the nature of the polynomial functions. However, the differenceof these functions represents the particle velocity component uy. Thepolynomial functions QF(C) an (i QG(Q c a n have zeros only on the realaxis. But if such zeros exist, they can correspond only to concentratedforces or dislocations in the physical plane. Thus, the polynomials are

at most constants and, without loss of generality, QF(O = QG(() = 1-The symmetry property that iiy{—x, —y, t) = —iiy(x, y, t) implies thatthe polynomials Pp(C) a n d PG(O must be odd functions of £. Finally,if the particle velocity is to have a finite limiting value at any pointin the physical plane as t —• oo, then the polynomials can be at mostlinear in £. Consequently, the difference of the two analytic functionsthat comprise the velocity component uy has the form

F y t ( 0 - G x t ( Q = h / ^ e , (6-3-36)

where a is a real dimensionless constant to be determined from theboundary conditions. Although a is constant with respect to £, itmay depend on crack speed or other system parameters. As |£| —• oo,the right side of (6.3.36) has the form of a constant plus terms thatare O(C~2). The presence of terms that are O(C -1) would suggest alogarithmic singularity in the long time behavior of stress in the finiteplane.

The condition that the shear stress vanishes on the plane y = 0along with continuity of stress across the wavefronts implies that

lm{2Fxy(t) + Gyy(t)-Gxx(O} = 0 for - oo < £ < oo (6.3.37)

for Im(£) —> 0*. Thus, in view of the property of analyticity of thefunctions involved, this condition is assured if

2Fxy(O + Gyy(O - GXX(O = 0 (6.3.38)

identically in £• Actually, the right side of (6.3.38) could be somesuitable rational function with poles on the real axis, but other ar-guments can be used to reduce it to the result shown in (6.3.38).If this equation is differentiated with respect to C> and the resultingderivatives are eliminated in favor of either. Fyt(Q or G'xt(() by meansof (6.3.31) or (6.3.32), respectively, then

2C2^t(C) + (b2 - 2(2)G'xt(O = 0. (6.3.39)

The two relations (6.3.39) and (6.3.36), following differentiation ofthe latter, provide two linear equations for i^t(C) and G'xt(C). Thesolution is

_ a(b2-2C2)h 2aC2h_ £2)3/2 ' G *tK) - & 2(^2_ £2)3/2 •

Except for evaluating the constant a, the solution of the boundaryvalue problem is complete.

The normal stress ayy{x, 0, t) on the plane y = 0 is given by

= Re ( 4 ^ y ( O + (4 - 2 ) F - ( O - 2G*y(t)) (6-3-41)

for a < \t/x\ < oo. If this expression is differentiated with respect to£, each term can be expressed in terms of one or the other functionin (6.3.40). The result is

( o - 4 2 )

where R(O = (b2 - 2£2)2 + 4£2yja2 - £2 y/b2 - £2 is the Rayleighwave function introduced in Section 2.5. Continuity of stress at thewave front r = erf implies that cryy(x, 0, t) = 0 for x = t/a = c<it.Consequently, the normal stress cryy{x, 0, i) at any point in the intervala < t/x < h is given by the definite integral

\Ja

t/x R(e\ 1V U (6.3.43)

Finally, the constant a is determined by applying the boundary con-dition (6.3.19)i to this integral expression. Because of the term (h2 —£2)3/2 in the denominator of the integrand, the integral cannot beevaluated as a line integral along the real axis of the C-plane fort/x > h. The path of integration is readily distorted into the complexplane by means of Cauchy's integral theorem in order to provide analternate representation for &yy(x, 0, £), avoiding the strong singularityat C = h. Thus, the value of a can be determined from

: dr). (6.3.44)(h2 -\-f]2)6f2y/a2 + \

This expression is rewritten as

a = ^I(b/h), (6.3.45)


0.40.0

Figure 6.3. Graph of the function I(v/cs) introduced in (6.3.45) versus thenormalized crack speed v/cs for v = 0.3.

where the value of I(b/h) = I(v/c3) is determined by numericalevaluation of the integral in (6.3.44). The result is shown graphicallyin Figure 6.3. From the property that R(irj) —> 2r)2(b2 — a2) as rj —• oo,it can be shown that I(b/h) -> b2/2(b2 - a2) = (1 - v) as h/b -» ooor as v/cs —• 0.

For points close to the crack tip at x = vt or £ = fo, the factorR(0/(h + 03/2Va2 ~ £ 2 i n (6.3.42) is analytic and it can be approxi-mated by the first term of its Taylor series expansion. Thus, for smallbut positive x — vt or h — £,

aR(h)2b2{h - - a2a2)

(6.3.46)

For points close to the crack tip, the stress component ayy varies as

aR(h)(6.3.47)

plus contributions that are bounded as £ —• br. From the definition

of dynamic stress intensity factor, it follows that

,t;)= lim <Tyy(x,0,t)y/2ir(x-vt)x—+vt+

< 6 - 3 - 4 8 )

The mode I stress intensity factor for the equivalent quasi-static crackproblem with total crack length I is obtained by letting v —> 0 andt —• oc in such a way that vt —• 1/2. If the resulting stress intensityfactor is denoted by Kjo, then the result that Kjo = aoQy/irl/2 isobtained. Consequently, the dimensionless ratio

Kj{t,v) _ I(b/h)R(h)KIo

is of interest because it indicates the influence of inertial effects inthe process. It is evident from the asymptotic behavior of a or 7 ash/b —* oo or v/cs —• 0 that Ki(t,v)/Kio —> 1 as v/cs —• 0, which isconsistent with the corresponding quasi-static result if vt is identifiedwith the half crack length 1/2. This ratio is plotted as a function ofv/cs in Figure 6.4.

The crack tip singular field is completely determined by (6.3.49)once the stress intensity factor is determined. The next most signifi-cant contributions to the crack tip stress field are those terms in thelocal expansion that are bounded and nonzero in the limit as the cracktip is approached. These terms are denoted by a\-' in the developmentin Section 4.3. For the Broberg problem, ayj = —(Too for a crack facepressure of magnitude cr^ or ayj = 0 for remote tension of magnitude(JQO with traction-free crack faces, and aiy = 0 due to symmetry ofthe fields. The remaining term of order unity a^J must be worked outfrom the complete solution. The field axx(x,0,t) can be determinedfrom (6.3.40) in precisely the same way that the expression for ayy in(6.3.43) was obtained. The result is

h [ ft/x

( M J S 0 )

1.0

0.0

- 0.5

•fej

0.01.0

Figure 6.4. The stress intensity factor for the Broberg problem, normalized by thecorresponding equilibrium stress intensity factor for a crack of length 2vt, versusnormalized crack speed v/c3 from (6.3.49). Also shown is the ratio of the rate ofenergy flux into the moving crack tips to the rate of work done on the body bythe applied tractions versus v/c3 for v = 0.3.

Prom the observation that the integrand is zero for t/x > h, it is clearthat axx is a constant on the crack faces. Thus, this constant mustbe the value of axx . By following the arguments leading to (6.3.44),it can be shown that the value of axx is given by

2(b2 - a2)(b2 + 2T?2) -drj. (6.3.51)

For the equilibrium limit of a finite length crack subjected to uniformcrack face pressure of magnitude (Too, the stress component parallelto the crack line that is of order unity is — a^. A plot of the ratio—<Txx/<?oo versus crack speed is shown in Figure 6.5. It is emphasizedthat this result applies for the case of pressure loading on the crackfaces.

In light of (6.3.22) and (6.3.40), the distribution of the particlevelocity of points on the crack faces at any time t is

ily(X,0±,t) = M (6.3.52)C=t/x±iO

0.40.0

Figure 6.5. Graph of the nonsingular tensile stress component <Txx normalized by(Too versus dimensionless crack speed v/c8 for v — 0.3.

for x2 < (vt)2. The normal displacement of the crack faces is obtainedby integration with respect to time from zero initial displacement,with the result that

uy{x,0±,t) = ±avty/l - (x/vt)2 , x2 < (vt)7 (6.3.53)

Thus, the crack opening profile is elliptical with the axis in the planey = 0 having length 2vt and the transverse axis having length 2avt.For the corresponding quasi-static crack problem with vt —> 1/2 thatwas discussed above, the total opening displacement at the midpointof the crack is uyo = ^^/( l — i/)/2/z. The ratio of %(0,0+, t) = avt tothis corresponding quasi-static result is I(b/h)/(l—u)^ which indicatesthe influence of inertia on the magnitude of crack opening at a givencrack length. This ratio is the quantity plotted versus crack speed inFigure 6.3 for v = 0.3.

Finally, some global energy rates for this crack growth processare considered. Because the crack face traction is constant and thecrack face velocity in the direction of the applied traction is knownexplicitly, the rate of work being done on the solid (per unit thicknessin the ^-direction) is

W(t) = 2 iJ-vt

(6.3.54)


where the fact that the rates of work on the upper and lower crackfaces are identical has been incorporated. Substitution of (6.3.52) into(6.3.54) and evaluation of the integral yields

W(t) = 27ra2ooI(v/c8)v

2t/fjL, (6.3.55)

where I(v/c8) is defined in (6.3.45).The rate at which energy is absorbed into the moving crack

edges (per unit thickness in the ^-direction), each moving at speedv, was found by Broberg (1967). Here, the result is obtained bysubstituting the explicit expression for stress intensity factor (6.3.48)for this problem into the general energy rate expression (5.3.9). Theresult is

( 6 . 3 > 5 6 )

2v2y/l-v2/c%

where, as before, D(v) = Ay/1 - v2/c2d y/l - v2/c2

s - (2 - v2/c2s)

2.The expression has been written so that W(t) is an obvious factor,and the ratio of the energy flux rate into the crack edges to the rateof work on the body is

E(t) = I(v/c8)D(v)c2 = I(b/h)R(h)

W(t) 2v2y/l-v2/c2 2b2hVh2 - a2

for any crack speed v = 1/h. For the limiting case of v/cs —• 0, thisratio approaches one-half, a feature of the corresponding quasi-staticsolution that can be verified independently. As the crack speed v/cs

increases from zero, this ratio decreases monotonically, approachingzero as the crack speed approaches the Rayleigh wave speed. It isnoted that this ratio is exactly one-half of the ratio Kj(t, v)/Kjo givenin (6.3.49), so it is also shown in Figure 6.4 with an adjusted scale.

The physical interpretation of the ratio (6.3.57) is that it is thefraction of the energy being added to the body that is dissipated inthe fracture process at a fixed crack speed. For example, if v/cs = 0.5then about one-third of the work being done on the body is beingconsumed in the fracture process. The remaining two-thirds of thework done is deposited as a distribution of strain energy and kinetic

energy radiating out behind the wavefronts. The question of whetheror not this constant speed crack growth process is physically realizabledepends on whether or not a crack growth criterion is satisfied, amatter that will be addressed in Chapter 7.

6.3.3 Symmetric expansion of a shear crack

The self-similar expansion of a finite length crack under in-plane shearloading may be analyzed in the same way as the corresponding modeI crack treated in the preceding subsection. Only a few of the detailsare included here. As before, the material is assumed to be stress freeand at rest for time t < 0. A rectangular #, y-coordinate system isintroduced in the plane of deformation, oriented so that crack growthoccurs in the plane y = 0. The crack begins to expand symmetricallyfrom zero initial length at time t — 0 with each tip moving at aconstant speed v that is less than the Rayleigh wave speed of thematerial. At any later time, the crack occupies the interval —vt <x < vt. The crack faces are subjected to opposed shear traction ofmagnitude r^ . The normal traction is zero on the crack faces, andno other loads act on the body. Thus, displacement potentials <j> andtp that satisfy the wave equations (6.3.18) are sought subject to theboundary conditions

ayy(x,0±,t) = 0, axy(x10±,t) = -Too for \x\ < vt. (6.3.58)

The solution is expected to have a symmetry with respect to reflectionin the :r-axis typical of mode II fields, namely,

u>x{x, "V, t) = -ux{x, y, t), uy(x, -y, t) = uy{x, y, t). (6.3.59)

Following the reasoning of the preceding section, the stress and parti-cle velocity components are anticipated to be homogeneous functionsof x, y and t of degree zero, and a solution is sought on this basis.The solution of the corresponding problem of symmetric growth of acrack with traction-free faces under the action of a remotely appliedshear traction is obtained by adding a uniform shear stress field tothe present solution.

In light of the symmetries in the mode II case, the second deriva-tives of displacement potentials, in terms of which the stress compo-nents are represented in (6.3.21), are expressed in terms of analytic


functions as

J ^ = Re{Fxy(Ca)},

(6-3.60)

and, likewise,

(6.3.61)

where the complex variables Q and (s are defined in terms of physicalcoordinates in (6.3.23) and (6.3.26), respectively. Only two of thevarious functions labeled Fa@ and Gap above are independent. Byfollowing the development of the preceding section, it can be shownthat

(aa-Ca)*S*(O = «= C2Fyy(O = -C(«2 - C2)Kt(O • (6.3.62)

A similar compatibility condition can be written for the functionsdesignated Gap.

The complete solution, except for evaluating a real constant /3from the boundary condition (6.3.58)2 is embodied in the two func-tions

_ ^2)3/2 '

The shear stress axy{x, 0, t) at any point in the interval a < t/x <h is given by the definite integral

t/x !}(£} I

, d$ \ . (6.3.64)

The value of the constant f3 is determined from the condition that

==dV. (6.3.65)

The similarities between these results and the corresponding expres-sions (6.3.43) and (6.3.44) for mode I are obvious. The expression(6.3.65) is rewritten as

P=—In(b/h), (6.3.66)

where the value of the quantity /// is determined by numerical evalu-ation of the integral in (6.3.65).

Prom the definition of dynamic stress intensity factor, it followsthat

KII(t1v)= lim <jxy(x,0,t)y/2<K{x-vt)x—>vt+

The mode II stress intensity factor for the equivalent quasi-static crackproblem with total crack length I is obtained by letting v —> 0 andt —• oo in such a way that vt —• 1/2. If the resulting stress intensityfactor is denoted by Kno, then the result that Kno = r ^ \Ar//2 isobtained. Consequently, the dimensionless ratio

Kn(t,v) _ In(b/h)R(h) {63M)

is of interest because it indicates the influence of inertial effects inthe process. It is evident from the asymptotic behavior of (3 or /// ash/b —• oo or v/c8 —> 0 that Kn(t,v)/Kno —• 1 as v/cs —• 0, which is

6.3 Self'Similar dynamic crack growth 333

1.0

0.00.0

Figure 6.6. Graph of the mode II stress intensity factor for symmetric expansion ofa shear crack, normalized by the corresponding equilibrium stress intensity factorfor a crack of length 2vt, versus dimensionless crack speed v/c3 for v = 0.3 from(6.3.68).

consistent with the corresponding quasi-static result if vt is identifiedwith the half crack length 1/2. This ratio is plotted as a function ofv/cs in Figure 6.6.

Burridge (1973) made an interesting observation on this solutionfor the special case when the crack speed is the Rayleigh wave speed,that is, when v = CR or h = c. In this case the shear stress onthe crack plane is nonsingular at the crack tip, as can be seen fromthe result that the stress intensity factor vanishes as v —» CR. Theshear stress axy has some finite value at the crack tip which must be—TOQ. By direct calculation, he showed that the distribution of axy

has a positive peak in amplitude at the shear wavefront x = c8t. Thisfeature is indeed suggested by the form of (6.3.64).

Suppose that (6.3.64) is differentiated with respect to £ = t/x.Then it can be shown by substitution that daxy/d£ is square rootsingular with a negative coefficient at £ = 6 + 0, and that it isalso square root singular with a negative coefficient at £ = c — 0.Demonstration of the latter result depends on the observation thatdR(c)/d£ < 0. Furthermore, daxy/d£ has no apparent zeros in theinterval between these two values. It is therefore concluded that axy

increases very rapidly from x = CRt to x = cst. Burridge (1973)

reports the magnitude of axy at the shear wavefront to be about1.63 Too. The implication is that if a slip plane has very little resistanceto fracture and if the crack rapidly accelerates up to the Rayleigh wavespeed, then this observation identifies a possible mechanism by whichthe crack can induce secondary fracture ahead of the main crack tipand thereby precipitate crack growth at a speed beyond the Rayleighwave speed. This mechanism was suggested by the numerical resultsreported by Andrews (1976b) on the expansion of a shear crack alonga weak interface. The interpretation of the 1979 Imperial Valleyearthquake by Archuleta (1982) suggests that this event may haveexhibited such propagation of the boundary of the slip zone at a speedin excess of the shear wave speed of the crustal material.

6.3.4 Nonsymmetric crack expansion

In the preceding subsections, the basic problems of self-similar crackexpansion in modes I and II were considered. In both cases, it wasassumed that the expansion is symmetrical, that is, that the two crackedges move with the same crack tip speed. This is not an essentialfeature for the fields to be self-similar. To result in self-similar fields,it is only necessary that the crack edges each move at constant speedas the crack expands from zero initial length. The constant speedsneed not be the same, nor is it essential that the crack tips movein opposite directions. Such a situation of nonsymmetric growth isconsidered in this subsection, but only briefly.

Consider once again the growth of a crack in a uniform remotetensile field of intensity (Too, as in Section 6.3.2. The equivalentsituation of crack growth with crack face pressure is discussed here.A rectangular x, /-coordinate system is introduced in the plane ofdeformation, oriented so that crack growth occurs in the plane y = 0.The crack begins to expand from zero initial length at time t = 0 witheach tip moving at a constant speed which is less than the Rayleighwave speed of the material. The tip advancing in the positive re-direction has speed v+ > 0 and the tip advancing in the negativex-direction has speed v_ > 0; see the inset in Figure 6.7. The caseof v_ < 0 can be handled in a similar manner, but this case is notconsidered here. At any time t > 0, the crack occupies the interval—v_t < x < v+t. Both crack faces are subjected to compressivenormal traction of magnitude a^. The shear traction is zero on thecrack faces, and no other loads act on the body. The material is stressfree and at rest for t < 0.

The governing equations are developed in Section 6.3.2, and thesesteps are not repeated here. However, arguments similar to thosepresented there lead to the conclusion that the difference of the twoanalytic functions that comprise the velocity component iiy has theform

~ Gxi{Q = t , (6.3.69)2/hh{h Q{h + C)

where h± = l/v± and a is a real dimensionless function of v+/cs andv+/v_ which must be determined from the boundary conditions.

If the steps leading up to (6.3.43) are followed, then it is foundthat the normal stress or

yy(x, 0,i) on the plane y = 0 ahead of thecrack tip advancing in the direction of increasing x is

IT ^ 7=4\Ja {h+-W2{h-+i)*l*y/#=? Iif / i >.\ o /n / i . >-\ o / o / o Zo~ ^ I V * * /

for a<t/x < h+, where A = an(h+ + h_)2/4b2y/h+h_ . The value ofa is determined for any values of v+/v_ and v+/cs by evaluating theintegral in the equation

, (6.3.71)

where p > h+ is a point on the real axis and Tap is any simple path inthe complex £-plaiie from £ = a to £ = p along which the integrandis analytic.

The stress intensity factor is

(b2 - 2ft*)2 -

where Kjo = cr^ y/7r£(?;+ + -)/2 is the equilibrium stress intensityfactor for any crack length t(v+ +v_). Graphs of stress intensity factorversus v+/cs are shown in Figure 6.7 for v_/v+ < 1, which includesall possible cases through interchange of v+ and v_.

1.2 -i

0.00.0

Figure 6.7. Graph of the dynamic stress intensity factor for nonsymmetrical self-similar expansion of a crack normalized by the corresponding equilibrium stressintensity factor for a crack of the same length versus dimensionless speed of therightmost crack tip v+/cs for v — 0.3 from (6.3.72). The speed of the left cracktip is V-.

6.3.5 Expansion of circular and elliptical cracks

Suppose that an unbounded, homogeneous, and isotropic elastic solidis subjected to a uniform remote tensile stress of magnitude (Too in aparticular direction, and that the material is initially at rest. A singlecomponent of stress is nonzero and the stress field is initially uniform.At a certain instant of time, say t = 0, a circular crack begins togrow from nominally zero initial radius in a plane transverse to thedirection of the initial tension. The faces of the crack are free oftraction, and the crack speed v is assumed to be constant and lessthan the Rayleigh wave speed of the material.

Due to the linearity of the boundary value problem, the situationcan be analyzed by considering the same configuration with the uni-form normal pressure a^ acting on the faces of the expanding crackand with the material initially stress free and at rest. As in the caseof the Broberg problem, the complete solution is then obtained bysuperimposing a uniform tensile stress field on the result. Considercylindrical coordinates r, 0,z with the crack occupying the circularregion r < vt of the plane z = 0 for t > 0. The configuration and load

distribution are invariant with respect to rotation about the z-axis,so the component of displacement in the ^-direction is zero at everypoint and all fields are independent of 6. The nonzero components ofstress with reference to cylindrical coordinates are <rrr, CTQQ, azz, andarz. The boundary conditions on the crack faces are

crZz(r,0±,t) = - a o o , arz(r,0±,t) = 0 for r<vt. (6.3.73)

In view of the symmetry with respect to reflection in the planez = 0,

ur(r, -z, t) = ur(r, z, t), uz(r, -z, t) = -uz(r, z, t) (6.3.74)

everywhere, and

uz(r,0,t) = 0, (?rz{r,0,t) = 0 for r > vt (6.3.75)

on the crack plane.The anticipated solution of this problem is a function of the three

coordinates r,z,t. The geometrical configuration of the body at anytime is obtained by self-similar expansion of the configuration at anyearlier time and, furthermore, the boundary values of the specifiedstress components are time-independent. Under these conditions, asolution for which the stress and particle velocity components arehomogeneous functions of r, z, t of degree zero may be anticipated.A solution of the problem was obtained on this basis by Kostrov(1964a) and Craggs (1966). The steps in the solution procedure arenot included here, but the principal results are summarized.

The normal velocity of a point on the crack surface is found tobe

2^ ( r , 0 V ) = ^ - 7 = f H = , r<vt,. (6.3.76)

where 7 is a parameter depending on v/cs and Cd/cs which is tobe determined from the boundary conditions. The function on theright side of (6.3.76) is indeed a homogeneous function of degree zeroand it exhibits the characteristic square root singularity as r —• vt~.The crack opening displacement can be obtained from (6.3.76) byintegration with the result that

uz(r,0+,t) = <JOO7AA2*2 ~r2 III. (6.3.77)


According to Kostrov (1964a), the value of 7 is determined from therelation

r ( i+2,)WfrT^rrry „ . ±. (6.3.78)

The mode of crack opening near the crack edge is mode I, andthe stress intensity factor is

l£(t,i;)= lim azz(r,0,t)27ry/r-vt , (6.3.79)

which is uniform around the crack edge. In terms of parametersalready defined, the stress intensity factor for the expanding circular

Kl(t,v) =

where D(v) = 4y/l - v2/c2d y/l - v2/c2

s - (2 - v2/c2s)

2 is defined in(4.3.8). The equilibrium stress intensity factor for a uniform pressureaoo acting on the faces of a circular crack of radius vt in an unboundedelastic solid is (Tada, Paris, and Irwin 1985)

KJo = 2^S^L . (6.3.81)

7T

Thus, the ratio of the dynamic stress intensity factor to the equilib-rium stress intensity factor for the same loading and crack size is

• ( 6 - 3 - 8 2 )

This ratio is shown for Poisson's ratio v = 0.3 in Figure 6.8. Bystandard asymptotic methods, it can be established that 7 —» 2(1 —u)/ir as v/c3 —• 0, so the ratio (6.3.82) also approaches unity asv/c8 -> 0.

Much of the discussion following the analysis of the self-similarexpansion of a crack under plane strain conditions in Section 6.3.2applies in the present case as well. The dependence of stress intensityfactor on speed v is similar, and the stress intensity factor vanishes


1.2 -i

0.0

Figure 6.8. Ratio of the dynamic stress intensity factor for self-similar expansion ofa circular crack with uniform crack face pressure to the equilibrium stress intensityfor the same loading and crack size versus crack speed, according to (6.3.82).

as the speed approaches the Rayleigh wave speed. Energy variationswith time may also be considered for the circular crack. One dif-ference between the two-dimensional and three-dimensional cases isthat the length of the crack edge increases in the latter case (it is thecircumference of the circle) but it is constant in the former case. Inboth cases, the rate at which energy is absorbed from the material perunit length of crack edge through advance of the crack is proportionalto time t\ see (6.3.56), for example. In the case of a circular crack,however, the length of the crack also increases in proportion to timet, so that the total rate of energy dissipation is proportional to t2.

Although few exact analytical solutions are available for three-dimensional crack growth, several other studies are noted. The self-similar expansion of a circular crack in an unbounded body subjectedto remote homogeneous shear stress was also studied by Kostrov(1964b). Burridge and Willis (1969) obtained the solution for self-similar expansion of an elliptical crack in an unbounded anisotropicbody subjected to general remote homogeneous stress. They essen-tially anticipated the shape of the crack opening and then verifiedthat all conditions of the boundary value problem could be satisfiedexactly. The particular case of an expanding elliptical shear crackwas studied by Richards (1973), who presented a thorough discussion

340 6, Elastic crack growth at constant speed

of the radiation field and the crack edge singularities with a viewtoward applications in seismology. A general, systematic study of self-similar problems in elastodynamics, including both crack and punchproblems, was carried out by Willis (1973). This work provides aunified mathematical framework for all of the self-similar problemsdiscussed previously, and it offers the prospect of finding new solu-tions, particularly within the realm of three-dimensional crack growth.

The particular situation of planar crack growth in an unboundedhomogeneous elastic solid is ideal for reduction of the linear fieldequations and boundary conditions to an integral equation on thecrack plane, analogous to the procedure in Section 2.3 for a relativelysimple two-dimensional problem. An efficient numerical solution tech-nique of the type commonly known as boundary integral methods wasdeveloped by Das (1980) and Das and Kostrov (1987) for this class ofproblems. This technique and related techniques are reviewed by Das(1985), and the area of study is fully developed in the monograph byKostrov and Das (1989).

6.4 Crack growth due to general time-independent loading

Suppose that a crack exists in a body of elastic material, and that thematerial is subjected to time-independent loads in the form of surfacetractions or body forces. The mechanical fields prior to crack growthare equilibrium fields. If the loading is increased to a sufficiently largemagnitude, then the crack will begin to extend. It is a purpose offracture mechanics to predict the way in which the crack will growon the basis of the laws of continuum mechanics and a crack growthcriterion. The special case in which the crack suddenly begins togrow dynamically at a fixed speed is considered in this section; themore general case of nonuniform crack tip motion is studied in thenext chapter. The general approach to the problem of the suddenextension of a preexisting crack under the action of arbitrary time-independent loading is illustrated for the case of the extension of aplanar mode I crack under plane strain conditions. The analysis iscarried out for a semi-infinite crack in an otherwise unbounded body,so it is strictly applicable to crack growth in a bounded body only untilwaves reflected from the boundaries perturb the fields in the cracktip region. The method of analysis is based on integral transformtechniques and the Wiener-Hopf technique. Important steps in thesolution of dynamic crack problems by means of this approach were

6.4 Crack growth due to general time-independent loading 341

introduced by Baker (1962), who studied a particular situation ofcrack growth under stress wave loading. The problem studied byBaker will be discussed in Section 6.5. The application of integraltransform methods and the Wiener-Hopf technique in the analysis ofstationary crack problems was discussed in Chapter 2, which providesa framework for the present approach.

Consider crack growth under two-dimensional plane strain con-ditions. A rectangular x, y, -coordinate system is introduced in thebody in such a way that the crack is in the plane y = 0, the edge ofthe crack is parallel to the z-axis, and the crack edge moves in thepositive ^-direction. Thus, the plane of deformation is the rr, y-plane.The crack tip is initially at x — 0 and it begins to move at constantspeed v at time t = 0. The position of the crack tip at any time t > 0is then x = vt.

Suppose that the configuration of the body and the applied load-ing system have reflective symmetry in the plane y = 0, so that thedeformation field prior to crack growth has the symmetries repre-sentative of mode I crack fields. The applied loads induce a tractiondistribution on the crack plane ahead of the crack tip, and the processof crack growth is essentially the negation of this traction distribution.This idea is exploited to obtain a complete solution for general loadingby means of superposition. Denote the tensile traction distributionon y — 0, x > 0 prior to crack growth by ayy(x,0) = p(x). Theshear traction on the crack plane is zero due to symmetry. If asolution can be found for elastodynamic crack growth which satisfiesthe conditions that the initial stress and deformation are zero, thecrack edge moves with position x = vt for t > 0, and the crack facesare subject to a compressive traction of magnitude p(x) over 0 < x <vt, then the superposition of this result and the initial equilibrium fieldprovides the complete solution to the problem of sudden growth of thecrack under general time-independent loading. The initial equilibriumsolution is assumed to be known, and the requisite elastodynamiccrack growth fields are determined next. This is accomplished in twosteps (Preund 1972a). First, the situation of crack growth with a pairof opposed concentrated forces acting on fixed material points on thecrack faces is analyzed, giving rise to a very useful result called thefundamental solution for the problem. Then, the corresponding fieldquantities for any distribution of tractions on the crack faces can bedetermined directly by superposition over this fundamental solution.The fundamental solution is derived in the next subsection.

6.4-1 The fundamental solution

For time t < 0, the crack tip is at x = 0 and the material is stressfree and at rest everywhere. At time t = 0, the crack tip begins tomove in the positive ^-direction at speed v. As the tip moves awayfrom the origin of the coordinate system, it leaves behind a pair ofconcentrated forces, each of magnitude p* (force per unit length in the^-direction) and tending to separate the crack faces. The crack facesare traction free at points other than x = 0. As the crack advances,the crack faces separate, the forces do work, and an elastodynamicfield is generated. This field is the fundamental solution. In view ofthe symmetry of the problem, the solution is required only in the halfplane — o o < r r < o o , 0 < 2 / < o o . Thus, displacement potentials <\>and tp that satisfy the wave equations

,dx2 dy2 c2dt2 ' Ox2 dy2 c2dt2 { }

are sought subject to the boundary conditions

cryy(x, 0, t) = —p*6(x)H(t) for — oc < x < vt,

axy(x,0,t) = 0 for - oo < x < oo , (6.4.2)

uy(x,0,t) = 0 for vt<x<oo,

where S(-) and H(•) represent the Dirac delta function and the unitstep function, respectively. The last condition in (6.4.2) follows fromthe symmetry properties of displacement fields for mode I deforma-tion, namely,

ux(x, -y, t) = ux(x, y, t), uy(x, -y, t) = -uy(x, y, t). (6.4.3)

It is advantageous to change variables from material coordinatesx, y in the plane of deformation to crack tip coordinates £, y, where

£ = x-vt, (6.4.4)

as shown in Figure 6.9. The displacement potentials are then soughtas functions of position in the crack tip coordinate system. It should

y

Figure 6.9. Diagram of the boundary value problem leading to the fundamentalsolution described in Section 6.4.1.

be noted that, at any material point, 0(£, y,t) depends on time ex-plicitly through its last argument and implicitly through £. The waveequation governing <f> in crack tip coordinates is

h2) dy2 h d£dt dt2 = 0, (6.4.5)

where a = l/cd and h = 1/v. The shear wave potential ip(^y,t) isgoverned by the same partial differential equation except that a isreplaced by b = l/cs.

In the moving coordinate system, the boundary conditions (6.4.2)take the form

, 0, t) = *+ (£, t) ~ P* Vt)H(i)H(-£) ,

(6.4.6)

for —oo < £ < oo, where cr+ and u_ have the same roles as thecorresponding functions in (2.5.1). In particular, they are unknownfunctions which may take on nonzero values only for positive or neg-ative values of £, as indicated by the subscript + or —. In thisform the boundary conditions have certain important characteristicsin common with the corresponding boundary conditions (2.5.1), whichsuggests the Wiener-Hopf technique as a method of analysis. Indeed,from this point onward, the analysis parallels that in Section 2.5 al-though the details differ somewhat. Only the main steps are includedhere.

The dependence of field quantities on time t is suppressed byapplication of the Laplace transform on time defined by

-i: (6.4.7)

where the transform parameter s is viewed as a positive real parameterfor the time being. Then the two-sided Laplace transform

= /J —

(6.4.8)

is applied to the governing equations and boundary conditions, wherethe transform parameter is s£, the factor s being simply a scalingparameter. The notation for transforms follows that established inSections 2.4 and 2.5. Furthermore, based on the anticipated far-fieldasymptotic properties of </>(£, ?/, s), it is anticipated that the transform(6.4.8) will converge in the strip —a_ < Re(£) < a+, where a± =a/(l±a/h).

Application of the transforms to the differential equations (6.4.1)yields two ordinary differential equations whose solutions, bounded asy —> 00, a re

*(C, V, s) = s~3P(O e~s<*y , *(C, y, s) = s-3Q(0 e-to , (6.4.9)

where

a(C) = (a2 - C2 + a2C2/^2 ~ 2a2C//i)1/2 (6.4.10)

and /3(£) is the same function of £ except that a is replaced by b.Furthermore, for (6.4.9) to be admissible for any C, the branch of aor (3 with Re(a) > 0 or Re(/?) > 0 must be selected. This is assured ifbranch cuts are introduced in the complex ^-plane, running from thebranch points —a_, a+, — fe_, and 6+ outward along the real axis (awayfrom the origin) to ±oo, and if the branches with positive real valuesat the origin are selected. The functions P(C) &nd Q(C) a r e unknownat this point, and the coefficients of these functions in (6.4.9) havebeen selected in anticipation of a solution with stress and particlevelocity fields that are homogeneous functions of degree negative one.

The transformed boundary conditions (6.4.6) are

= 0 ,

_ MO

(6.4.11)

The special form of the terms on the right side of (6.4.11) with powersof s as factors has been selected for convenience. At this point, it isstill possible that the transforms

(6.4.12)2 / ~

will depend on s as well as on £. As suggested by the special form of(6.4.11), however, these functions will be found to depend only on £.Based on a consideration of the far-field behavior of the transformedfields, £+ is expected to be analytic in Re(£) > — a_ and U_ isexpected to be analytic in the overlapping half plane Re(C) < a+.In particular, both functions are expected to be analytic in the strip- a_<Re(C)<a + .

Next, the transformed potentials (6.4.9) are substituted into thetransformed boundary conditions (6.4.11). The result is a system ofthree linear algebraic equations for the four unknown functions P(C)>Q(£), £+(C)> and C/_ (C)- The system of equations is valid for all valuesof £ in the common strip. As was anticipated, the system of equationsdoes not involve the parameter s. If two of the equations are used toeliminate P and Q in favor of £+ and U_, then the remaining equationis

(C-h)

which is valid in the common strip of analyticity —a_ < Re(£) < a+,where

R(O = 4C2a(C)/3(C) + (2C2 - b2 - b2C2/h2 + 2b2(/h)2 (6.4.14)

is a modified form of the Rayleigh wave function. The function Ris analytic everywhere in the complex plane except at the branchpoints £ = ±a± and £ = ±b±. For the branches of a and /3 thatwere selected above, R is single valued in the £-plane cut along a± <|Re(C)| < b±, Im(C) = 0. The similarity between the Wiener-Hopfequations (6.4.13) and (2.5.11) is obvious, but it must be emphasizedthat the functions a, /?, and R here are different from their counter-parts introduced in connection with (2.5.11). The solution proceedsaccording to the outline in Section 2.5.

The factorization of the function a(Q into a product of section-ally analytic functions a+a_ can be carried out by inspection withthe result that

a± = {a ± C(l T a/h)}1/2 . (6.4.15)

The function R must also be factored and, to do so, the behavior ofthis function at infinity, as well as the locations of its zeros, must beknown. Repeated application of the binomial expansion theorem forlarge values of |C| yields the result that R(() ~ — 4 as |C| —• oo,where

K = 4y/(l-a2/h2)(l-b2/h2) - (2 - b2/h2)2 (6.4.16)

is real and positive for all crack speeds between zero and the shearwave speed. Furthermore, it can be shown by direct calculation thatR(C) vanishes for £ = /i, c+, —c_, where c± = c/(l±c/h) and c = c^1

is the inverse Rayleigh wave speed. The zero at £ = h has multiplicitytwo, whereas the other zeros are simple zeros. The zero countingformula of complex integration theory (2.5.26) can be applied to provethat the function R has no other zeros in the entire complex £-plane.With the step in (2.5.28) as a guide, it is convenient here to introducean auxiliary function S(() defined by

This function has the important properties that S(Q —• 1 as |£| —• ooand that it has neither zeros nor poles in the complex £-plane. In lightof these properties, the factorization theorem (2.5.23) can be applieddirectly with the result that

(6.4.18)

The factors S±(C) have a simple relationship to the correspondingfactors for v = 0 as obtained in (2.5.29). Suppose that the latter aredenoted here and subsequently by S!j!(C) *° ren*ec*

VimS±(O (6-4.19)

identically in £. If the change of variable r/ —> —r)/(l + rj/h) isintroduced into (6.4.18) then it can be shown that

S° )^ (6-4-20)

With the factors (6.4.15) and (6.4.20) in hand, the Wiener-Hopfequation (6.4.13) is conveniently rewritten in terms of the compositefunction

F±(C)= i

Then (6.4.13) becomes

hp*F+(h)

which is valid in the strip —a_ < Re(£) < a+ of the complex plane.The left side of (6.4.22) is analytic in the half plane Re(C) > - a_

and the right side is analytic in the overlapping half plane Re(C) <a+. Therefore, by the identity theorem of analytic functions notedin Section 1.3.1, each side provides the analytic continuation of theother side into the complementary half plane. Together the two sidesrepresent one and the same entire function. Just as in Section 2.5,the conditions that the displacement must be continuous at x = 0 andthat the stress must be only square root singular there imply that thevalue of this entire function is zero. It follows that

(6.4.23)

A number of features of the solution will be examined in theremainder of this chapter and in the next chapter, but the crack tipstress intensity factor will be extracted first. It is evident from (6.4.21)that F±(C) = O(C"1/2) as |C| -> oo, hence E+(C) = O (C"1/2) as|C| —• oo. This implies that the stress will indeed be square rootsingular at the crack edge. The strength of the singularity can bedetermined from the transformed solution by direct application of theAbel theorem (1.3.18) governing asymptotic properties of functionsand their transforms,

lim («C)1/2-£+(C) = l i m O r O 1 / 2 ^ , * ) = - i * j ( « , t ; ) , (6.4.24)C—oo S £—0+ V 2

where Ki(s, v) is the Laplace transform on time of the stress intensityfactor as viewed by the crack tip observer moving at speed t/. Directapplication of this general result to the solution of the Wiener-Hopfequation (6.4.23) yields

(6.4.25)

where the function of crack speed k(h) is given by

k{h) = 0- - c/fc) . (6.4.26)


1.0

a.2 0.8afe °*6

13

E 0.4

M 0.20.0

k(v)

0.0 0.2 0.4 0.6 0.8 1.0vie R

Figure 6.10. Graph of the universal function k(v) versus crack tip speed V/CR forv = 0.3 from (6.4.26). Also shown is the universal function g(v) from (6.4.35).

This function of crack speed recurs throughout the study of elasto-dynamic crack tip fields. The factor S+(h) = 1/S°_(h) is not toodifferent from unity over the full range of its argument, and a usefulapproximation to k(h) for most practical purposes is k(h) « (1 —c/h)/y/l — a/h or k(v) « (1 — V/CR)/ y/\ — v/cd- The functiondepends on the material properties through the elastic wave speeds,but it is independent of the loading on the body. In this sense, it is auniversal function of crack tip speed. It will be written as a functionof inverse crack speed h = 1/v, or, alternatively, as a function ofcrack speed, that is, as k(v), depending on the context. However,the function defined in (6.4.26) is understood in all cases. A graphof the dimensionless function k versus crack speed normalized by theRayleigh wave speed V/CR is shown in Figure 6.10. Its general featuresare evident from its definition or from the graph. In particular, it hasthe value k = 1 when V/CR = 0, has the value k = 0 when V/CR = 1and is monotonic between these values.

The Laplace transform of the mode I stress intensity factor in(6.4.25) is readily inverted. The result is

KI(vt,v)=p*xl—k(v), (6.4.27)

where the arguments of the function have been shown as the amount of

crack growth and the crack tip speed in anticipation of more generalresults to follow. The form of the stress intensity factor result isconsistent with the expectation that the stress and particle velocityfields are homogeneous functions of degree negative one. Furthermore,the result has the correct form in the limit of vanishing inertial effects.If vt is identified with the distance from the concentrated loads ofmagnitude p* to the crack tip, say /, under equilibrium conditions,then (6.4.27) implies that Kj = p* y/2/nl for V/CR —• 0, which is thecorrect equilibrium result.

Although the focus here has been on the determination of thedynamic elastic stress intensity factor, the full elastodynamic fieldcan be determined from the transformed solution. Once S+(C) andC/_(C) have been found by means of the Wiener-Hopf procedure, thefunctions F(C) and Q(C) introduced in (6.4.9) are also known. Forthe problem at hand, these functions are found to be

hp*(b2 - 2C2 + b\2/h2 - 2b\lh)F+{h)

C)(6.4.28)

2hp*a(QCF+(h)fi(C-h)R(OF+(C)'

The function </>(£, y, i) itself is given by the double Laplace inversionintegral

ao+*oo

(6.4.29)

where so is a positive real number and Co is a positive real numberbetween —a_ and a+. Such a transform inversion integral is mosteffectively evaluated by means of the Cagniard-de Hoop technique(de Hoop 1961).

6.4-2 Arbitrary initial equilibrium field

Attention now is returned to the case of an arbitrary initial equi-librium stress field. Under the action of an applied loading on thecracked solid which results in mode I deformation, a normal traction

p(x) is induced on the prospective fracture plane ahead of the cracktip, where the rr-direction is the crack growth direction. The process ofcrack growth is essentially the negation of this traction distribution,as described in the introduction of this section. An elastodynamicsolution is sought for time t > 0 satisfying the condition of zero initialstress and particle velocity in the body and the condition that thenormal compressive traction on the newly created crack faces, sayover 0 < x < vt, is p(x). Such a solution, when superimposed onthe initial equilibrium solution, is the desired complete solution ofthe problem. Superposition of solutions is not valid, in general, formoving boundary problems even though they are governed by linearequations. However, if two solutions can be found for the same motionof the boundary, then a linear sum of the two solutions will also be asolution. The solution for arbitrary p(x) is now constructed from thefundamental solution obtained in the preceding subsection.

Consider the same problem as in Section 6.4.1, except that thepair of concentrated loads appears on the crack faces at x = x' > 0instead of at x = 0. The crack tip still begins to move at constantspeed v from the position x = 0 at time t = 0. The tip thereforepasses the point x1 at time t = x1 /v. Let p*f(x,y, t) denote a scalarcomponent of any field quantity in the fundamental solution. Then,the solution of the modified problem for the same physical quantityis p*f(x — x1\y,t — x'/v). Furthermore, suppose that the load thatappears on the crack faces behind the crack tip as it passes the pointx = x1 is not a concentrated load of magnitude p* but that it is aload of intensity p(x') distributed over the infinitesimal interval fromx1 to x' + dx1. The solution for the same physical quantity for thisproblem is f(x — x', y,t — x'/v)p(x') dxf to first order in dx'. Finally,the solution for the case of a distributed traction appearing throughthe crack tip and giving rise to the normal traction p(x) on 0 < x < vtis given by the superposition integral

/ f(x - x1, y, t - x'/v) p(x') dx'. (6.4.30)Jo

It is obvious by direct substitution that the integral in (6.4.30) is awave function if f(x,y,t) is a wave function, and that the integralsatisfies the correct initial conditions. Furthermore, by choosing thefunction / to be any of the particular fields that are subject to the cor-rect boundary conditions, the superposition integral itself satisfies the

correct boundary conditions. This superposition provides a solutionto the problem of interest for the entire physical plane. Of specialinterest for the time being is the stress intensity factor for generalcrack face loading. This result is extracted next.

If the stress intensity factor for general loading is sought, thenfrom (6.4.27) it is evident that the function f(x,y,t) in the superpo-sition integral (6.4.30) should be selected to be k(v)y/2/7rvt. Thus,the stress intensity factor for general loading is

K!(vt,v) = k(v)\ - (V , P ( X ) dx (6.4.31)V 7T Jo y / v t — X

for t > 0. Note that the result has the form of the universal func-tion k(v) times a function that depends on v and t only throughthe combination vt, that is, through the amount of crack growthsince t = 0. Indeed, the factor of k(v) in (6.4.31) is exactly theequilibrium stress intensity factor for a crack tip at x = / = vt andcrack face pressure p(x) over 0 < x < I Thus, the dynamic stressintensity factor (6.4.31) is simply the universal function k(v) timesthe equilibrium stress intensity factor for the given applied loadingand the instantaneous amount of crack growth,

Kj(vtj v) = k{v)K!{vt, 0). (6.4.32)

With the stress intensity factor in hand for general loading p(x),the energy release rate can be calculated by means of the generalizedIrwin relationship (5.3.9), with the result that

G(vt, v) = ^—^- Mv) Kj(vt, v)2 . (6.4.33)

In view of the special form of the stress intensity factor result, theenergy release rate expression can be rewritten as

G(vt, v) = ^—^- Ki{vt, 0)2 A!{y)k{yf . (6.4.34)hi

This form is immediately recognized as the corresponding equilibriumenergy release rate times a universal function of crack tip speed g(v),so that

, v) = g(v) G{vt, 0), g(v) = Aj{v)k{v)2 . (6.4.35)

Thus, the dynamic energy release rate also has the form of a universalfunction of crack tip speed times the equilibrium energy release ratefor the specified crack face loading and a crack tip at the positioncorresponding to the instantaneous amount of dynamic growth. Thedimensionless function g(v) is also shown in Figure 6.10 as a functionof crack speed normalized by the Rayleigh wave speed of the materialV/CR. It has the features that g = 1 for V/CR = 0 and g = 0 forV/CR = 1. It varies monotonically between these limiting values, andit can be approximated by the linear function g(v) « 1 — V/CR forvirtually all applications.

The stress wave fields radiated by a crack once it begins to growdynamically have not been considered in any detail here. However,these fields are of central importance in methods used in seismologyand acoustic emission techniques of materials testing. It was noted byPreund (1972e) that a mode I crack radiates a finite discontinuity instress and particle velocity fields if it suddenly begins to grow at a con-stant speed. The issue of stress wave radiation from a growing crackwas pursued in much greater depth by Rose (1981), who consideredradiation from a semi-infinite mode I crack due to suddenly starting orstopping crack growth, and from a three-dimensional crack. He alsoconsidered the influence of multiple scattering of the kind studied inSection 3.3 on the radiation from a growing crack of finite length.

6.4-3 Some illustrative cases

As a simple illustration of the general stress intensity factor result,consider the special case in which the advancing crack relieves anequilibrium stress intensity factor field as it begins to grow from restat speed v. Thus, it is assumed that

p(x) = -7=2= for 0 < x < vt, (6.4.36)

where Ko is the initial equilibrium stress intensity factor. The dy-namic stress intensity factor for this case is

Ki(vt, v) = *(*>)— T , dX x = k(v) Ko. (6.4.37)

^ J0 y/x{vt — X)

Thus, at the onset of crack growth at constant speed v, the stressintensity factor changes discontinuously from its initial value of KQ

to its dynamic value h(v)Ko. Because k{v) < 1 for V/CR > 0,the stress intensity factor decreases discontinuously. Likewise, theenergy release rate experiences a discontinuous drop at the instantcrack growth begins. For the case of time-independent loading, thissuggests that crack growth under equilibrium or slowly rising appliedloading probably does not begin abruptly with finite crack speed but,instead, that the crack tip will accelerate from zero initial speed underdynamic conditions. For example, suppose that a crack is assumed togrow with a fixed, material specific value of the stress intensity factor.Then the applied loading must be increased to a sufficient level inorder to bring the stress intensity factor to the critical level. If thecrack then begins to grow with a constant nonzero speed, however,the dynamic crack tip stress intensity factor will jump to a valuebelow the critical value. On the other hand, if the process of fractureinitiation requires that a sharp crack must be induced to grow from aninitial blunt crack, a sudden drop in stress intensity factor cannot beruled out. Indeed, this situation has been exploited in certain crackpropagation and arrest experiments. A more thorough discussion ofthis situation must be postponed until the solution for nonuniformmotion of the crack tip is developed in the next chapter.

As a second illustration, consider the case when

p(x) = p*6(x) - ^^JZH{X - I), (6.4.38)

where I is a positive parameter with the dimension of length and H(-)is the unit step function. The loading function (6.4.38) represents aconcentrated force p* acting at x = 0 to open the crack plus a squareroot singular normal traction distribution acting over I < x < vtand tending to close the crack. Again, the crack begins to grow fromx = 0 at time t = 0 with constant speed v, and the normal compressivetraction (6.4.38) appears on the crack faces behind the crack tip. Forvt < Z, the solution for this special case is simply the fundamentalsolution. For vt > Z, on the other hand, the dynamic stress intensityfactor Ki(vt,v) is

l jg f dx 1 Q . (6.4.39)l Xy/{X-I){vt-X)\

Thus, the dynamic stress intensity factor corresponding to the loading(6.4.38) is identically zero for all time t > l/v. This remarkable result


arises in a natural way in the discussion of nonuniform crack growthin Section 7.2.

Finally, the special case when the traction p(x) is a spatiallyuniform pressure of magnitude a* is considered. In this case, thedynamic stress intensity factor is

Ao~ fvt J

KI(vtJv) = k(v)(T*\ - / =9/r*fe(ti)%/9fi*/*r (6.4.40)V 7T Jo y/vt — X

The stress intensity factor therefore increases indefinitely with timeas the crack tip advances at constant speed v.

6.4-4 The in-plane shear mode of crack growth

The situation of dynamic growth of a semi-infinite crack at constantspeed under the action of an applied loading that produces a state ofmode II deformation can be handled in the same way as the foregoingmode I situation. The analysis has been summarized by Fossum andFreund (1975). Of particular note is the way in which the dynamicstress intensity factor depends on the initial loading. Suppose thatthe distribution of shear traction on the prospective fracture plane is—q{x) prior to crack growth. Then the significant step in the analysisis the solution for the case of the shear stress q(x) appearing oneach crack face as the crack advances at speed v. It is found thatthe dynamic stress intensity factor for this situation has the samegeneral properties that were established above for the case of mode Ideformation. In particular, it is found that

Kn(vt, v) = kn{v) Kn{vt, 0), (6.4.41)

where

M1 - c/h 1 - v/cRv) — . = ; . (6.4.42)

S+(h)y/l-b/h S+iv'^/l/

As before, h = l/v and S+(h) is given in (6.4.18). That is, thedynamic stress intensity factor has the form of a universal function ofcrack tip speed for mode II deformation times the equilibrium stressintensity factor for the given initial load distribution represented bythe function q(x) and the instantaneous amount of growth vt, namely,

Kn(vt, 0) = Jl ( S^L- dx. (6.4.43)


For purposes of calculation, the function in (6.4.42) can be approxi-mated by kn(v) « (1 — V/CR)/y/l — v/cs for most practical purposes.

6.4-5 The antiplane shear mode of crack growth

The corresponding situation of dynamic growth of a semi-infinite crackat constant speed under the action of time-independent loading thatresults in a state of mode III deformation can also be analyzed bymeans of the integral transform methods outlined above. Supposethat the distribution of shear traction on the prospective fractureplane ahead of the crack tip prior to crack advance is given as afunction of position x by —q(x). At time t = 0, the crack begins togrow at the constant speed v. It can be shown that

Km(vt, v) = km(v)Km(vt, 0), (6.4.44)

where Km is also given by the right side of (6.4.43) and

km(v) = y/l - v/c8 = yjl - b/h . (6.4.45)

This result is derived for arbitrary crack motion in Section 7.2, fol-lowing an approach introduced by Kostrov (1966). The result for thespecial case of constant speed crack growth is included here for thesake of completeness.

6.5 Crack growth due to time-dependent loading

In the preceding section, the situation of a semi-infinite crack in anotherwise unbounded body subjected to general time-independentloading was considered. Here, the same physical system is againconsidered but the material in the region of the crack is assumed to bestress free and at rest initially. At a certain instant of time, say t = 0,the crack tip region is stressed due to the passage of a plane stresswave or to the sudden application of uniform crack face traction. Thecrack will begin to extend at some later time, and the purpose is toexamine the mechanical fields associated with such a crack extensionprocess. The development of this section is less general than thatof the preceding section, in the sense that only a uniform crack facepressure or a normally incident plane stress pulse is considered. The

6.5 Crack growth due to time-dependent loading 357

time-independent loading of the preceding section, by comparison,is quite general in its spatial variation. The restriction to relativelysimple spatial distributions of loading is not essential, but its adoptionpermits an examination of the structure of the elastodynamic crackfields that is not possible when a completely general time-dependentloading is considered. The more general case is discussed briefly inSection 7.5.2.

The material is stress free and at rest for time t < 0. At timet = 0, a uniform compressive normal traction of magnitude a* beginsto act on each face of the crack [problem (b) in Figure 6.11]. Theconnection between this situation and loading by an incident stresspulse is discussed in a subsequent subsection. A rectangular #, y, z-coordinate system is introduced in the body in such a way that thecrack is in the plane y = 0 for x < 0, the edge of the crack is parallel tothe z-axis, and the crack edge moves in the positive x-direction onceit begins to advance. Thus, the plane of deformation is the #, y-planeand the mode of deformation is mode I. The crack tip is initially atx = 0. Prior to crack growth, this is exactly the situation studied inSection 2.5, where it was observed that the crack edge stress intensityfactor is proportional to a* due to linearity and that it must increasein proportion to yjc^t on the basis of dimensional considerations. Ifthe material is of limited strength, then the crack will begin to extendat some later time, say t — r, where r is the delay time for theprocess. It is the purpose of this section to study the situation ofcrack growth at constant speed v beginning at time t = r due to thesudden application of crack face pressure at time t = 0. The positionof the crack tip at any time t > r is then x = v(t — r) . The specialcase of constant speed crack growth with r = 0 was first analyzed byBaker (1962). The possibility of crack growth at a nonuniform rateafter an arbitrary delay r will be considered in Chapter 7.

The suddenly applied crack face pressure induces a transienttraction distribution on the crack plane ahead of the crack tip, andthe process of crack growth is essentially the negation of this tractiondistribution. As in the preceding section, this idea is exploited toobtain a complete solution by means of superposition. Denote thetensile traction distribution on y = 0, x > 0 prior to crack growth by

ayy(x,0) =p(x/t) = a+(x,t). (6.5.1)

This stress distribution was determined in Section 2.5, where it wasdenoted by a+(x, t). The particular formp(x/t) in (6.5.1) is selected to

make clear the important property that this function is a homogeneousfunction of x and t of degree zero. The shear traction on the crackplane is zero due to symmetry. If a solution for elastodynamic crackgrowth can be found which satisfies the conditions that the initialstress and deformation are zero everywhere, the crack edge moveswith position x = v(t — r) for t > r, and the crack faces are subject tocompressive traction of magnitude p(x/t) over 0 < x < v(t — r), thenthe superposition of this result [problem (c) in Figure 6.11] and theinitial transient field of problem (b) provides the complete solution tothe problem of the growth of the crack under suddenly applied crackface pressure loading over — oo < x < 0. The requisite elastodynamiccrack growth fields are determined next. This is accomplished by firstderiving a very useful result for a special distribution of traction on thecrack faces called the fundamental solution for the problem (Preund1973a). Then, the corresponding field quantities for the distributionof traction on the crack faces given in (6.5.1) can be determinedby superposition over this fundamental solution. The fundamentalsolution is derived in the next subsection.

6.5.1 The fundamental solution

As a result of the sudden application of the crack face pressure,the normal stress distribution (6.5.1) is induced on the prospectivefracture plane x > 0, y = 0. When the crack begins to grow at timet = r, it must in effect negate this traction distribution. Stated inanother way, if a solution can be found for zero initial stress andparticle velocity everywhere in the body and for crack growth atspeed v beginning at time t = r with compressive normal crack facetraction of magnitude p(x/t) over 0 < x < v(t — r), then the sum ofthis solution and the solution of Section 2.5 yields the desired result.These two problems whose solutions are to be superimposed will becalled simply the stationary crack problem (b) and the moving crackproblem (c) in this section.

The mathematical property of homogeneity of the distribution(6.5.1) has an important implication for this analysis. It implies thatany fixed stress level in the scattered field radiates out along the #-axisat constant speed, that is, the stress level p{u) travels from the cracktip at x = 0 at speed u for time t > 0. The speed u lies in the rangefrom zero to the longitudinal wave speed c<j. The ^-coordinate attime t of the stress levels moving with speeds u and u + du are ut and(u + du)t, respectively, where du is understood to be an infinitesimal


(a)

\y

\ \ \ \ \ \ \

TTTTTT1t P(

I t t

(d) y\\ \ \ f 1 1

X

YTT\ I IFigure 6.11. Schematic representation of the various boundary value problemsconsidered in constructing the solution for constant-speed crack growth understress pulse loading.

PO+P,

Po+P,

y

Figure 6.12. Diagram of the boundary value problem leading to the fundamentalsolution described in Section 6.5.1.

increment in speed. Thus, to within first order in the infinitesimalquantity du, the total force (per unit length in the ^-direction) dueto all stress levels with speeds between u and u + du is tp(u) du, andthis force acts at x = ut.

Suppose now that the crack tip begins to move at time t — r > 0with speed v, and that the compressive normal force element tp(u) dubegins to act on each crack face as the crack tip passes its point ofapplication. If the time and place of this encounter are denoted byt' and x' then, from the geometry of the situation as depicted inFigure 6.12, the coordinates of this point are t1 = vr/(v — u) andxf = UVT/(V — u). The force element continues to act on each crackface and it continues to move in the redirection at speed u. The forceelement is rewritten as

tp(u) du = (t- t')p(u) du + t'p{u) du. (6.5.2)

This particular choice is made because the second term in the ex-pression (6.5.2) is the magnitude of the force element as it appearsat the crack tip at time t = £', and the first term is the increasein the magnitude after it appears. These observations provide themotivation for the formulation of the following problem.

For time t < 0, the crack tip is at x = 0 and the material isstress free and at rest everywhere. At time t = 0, the crack tip beginsto move in the positive x-direction at speed v. As the tip movesaway from the origin of the coordinate system, it leaves behind apair of concentrated forces tending to separate the crack faces, eachof magnitude po (force per unit length in the ^-direction). Each ofthese forces increases in magnitude at rate pi, and the forces movein the positive ^-direction at speed u for t > 0. The crack faces aretraction free at points other than x = ut. As the crack advances, thecrack faces separate, the forces do work, and an elastodynamic fieldis generated. This field is the fundamental solution. In view of thesymmetry of the problem, the solution is required only in the halfplane —oo < # < oo, 0 < y < oo. Thus, displacement potentials (j>and %j) which satisfy the wave equations (6.4.1) are sought subject tothe boundary conditions

(jyy(x, 0, t) = -(po + Pit)S(x - ut)H{t), -oo < x < vt,

(Txy{x,0,t) = 0, - o o < z < o o , (6.5.3)

uy(x, 0, t) = 0, vt < x < oo ,

where <$(•) and H(•) represent the Dirac delta function and the unitstep function, respectively. The last condition follows from the sym-metry properties of displacement fields for mode I deformation (6.4.3).

It is again advantageous to change variables from material coor-dinates x, y in the plane of deformation to crack tip coordinates £, y,where £ = x — vt. The solution procedure for this particular boundaryvalue problem follows that in the preceding section quite closely, andthe details are not included here. The solution procedure is simplifiedif two separate problems are treated, one with po ^ 0 and pi = 0 andthe other with po = 0 and pi ^ 0. A solution for the full physicalplane can be found, but the only elements of the solution includedhere are the normal stress on the prospective fracture plane ahead ofthe crack tip and the normal component of displacement of the crackface at y = 0+. The double transforms of this stress component anddisplacement component, £^(£,0,5) and Uy((,0+,s), are

*.... = n.. WF^W>> - n, W<1 l i l t ! ! ! ] '

(6.5.4)

where w = l/(v — u) is the inverse relative speed and the otherparameters are defined in the preceding section. In particular, thefunction F±{Q is given in (6.4.21).

Of special interest is the stress intensity factor for the funda-mental solution. The behavior of the Laplace transform of the stressintensity factor Kj(s) can be determined by examining the asymptoticbehavior of Y,yy(C, 0, s) as £ —* oo. The result can be inverted byinspection with the result that

(6.5.5)- a/h

If the time dependence of the concentrated load in (6.5.3)i is gener-alized to po + Pit + P2t2 + • • • then the stress intensity factor can bedetermined to correspondingly higher fractional powers in t, with thecoefficients depending on higher order derivatives of F+(w). The fun-damental solution as derived, however, is adequate to determine thedynamic stress intensity factor for the situation of suddenly appliedcrack face pressure described above and, if desired, to determine thefull elastodynamic field. This is demonstrated next.

6.5.2 Arbitrary delay time with crack face pressure

Let Po/o(£> y-> t; u) + Pitfi(£, y, t\ u) represent any element of the fun-damental solution, such as a stress component or a displacementcomponent. It is evident from the linearity of the problem that onepart of the solution will always be proportional to po and anotherpart to pi. The dependence of the solution on the load point speedu is made explicit for this development. Consider now the sameproblem as that studied in Section 6.5.1 except that the crack beginsto move at time t = r instead of time t = 0. The concentratedloads still appear on the crack faces at x = 0 at the instant thatthe crack begins to extend. The solution of this modified problemis then pofo(€, V, t - r; u) + Pi(t - r)/i(f, y, t - r; u), where £ is nowunderstood to be £ = x — v(t — r). Furthermore, suppose that theconcentrated loads appear on the crack faces as the crack tip passesthe point x = x' instead of at x = 0. The crack tip still beginsto move at time t = r at the point x = 0. Then, the solution isPofoit, y,t-r- x'/v; u) + pi(t - r - x1/v)fi(£, y,t-r- x'/v; u ) .

With a view toward constructing the solution for the case ofa compressive pressure p(x/t) on the crack faces, x1 ju is chosen ast1 = vr/(v — u) = wr/h. Furthermore, with (6.5.2) in mind, po is setequal to t' p{u) du and p\ is set equal to p(u) du. The solution for themoving crack face load (6.5.2) is then

/o(£, y,t-t';u)p(u) du + (t- t')h(£, yit-t';u)p{u) du. (6.5.6)

The complete field representation for the element of the solution underconsideration can then be found by superposition over the appropriaterange of u. In view of the interpretation of u as the speed at whichthe stress level p{u) moves from the crack tip in the stationary crackproblem, the range of u is the range of x/t between x = 0, t = r andx = v(t — r), t = £, that is, the range of u is 0 < u < v(t — r)/t. Thus,the element of the solution is given by

r

/Jo

v(t-r)/t

[/o(£,y,* - t';u)p(u) + (t -

(6.5.7)

The superposition integral (6.5.7) can be applied to determine thestress intensity factor for the moving crack problem with the crack tip

at x = v(t — T) and the compressive normal traction p(x/t) acting onthe crack faces. In view of the fact that the various functions in (6.5.7)depend on u mainly through w = l/(v — u), it is more convenient tointegrate with respect to w. Thus, the change of variable of integrationfrom u to w is effected with du = dw/w2. The range of integrationover w is then h < w < h*, where h* = th/r. Following somemanipulation of the resulting expression, it is found that

p(w)dw, (6.5.8)

where it should be noted that p(u) has been replaced by p(w). Al-though p is a different function of u than it is of w, the notation shouldnot cause any confusion because the function always represents thesame traction distribution. The integral in (6.5.8) can be evaluated,but it requires special care because the term p(w) is singular at w = h[u = 0) and its factor in the integrand is singular at w = h*.

The explicit form of p(w) is required to carry out the evaluation.Prom (2.5.40), this function is given by

Wh/(w-h)

where F® is the function defined in (2.5.31). The superscript 0 hasbeen added to make clear the distinction between functions defined inSection 2.5 for the stationary crack problem and functions representedby the same letter in the present section. The connection betweenthese functions is simply

+(C) (C), +(C) = o»(C), (6.5.10)

and so on. These various functions satisfy some identity relationshipsthat are important for extracting the principal result. These identitiesare not obvious, but they can be established by direct manipulationof the quantities involved. For example, it can be shown that

This follows directly from the identity (6.4.20). The relationship(6.5.11) is important, in turn, in establishing that

where k(h) is the universal function of crack speed defined in (6.4.26).The integral in (6.5.7) is evidently convergent and, at first sight,

it appears to be ideally suited for evaluation by integration by parts.This is not possible, however, as was already noted previously. Tocircumvent the difficulty, the lower limit of integration h is replacedby h+e, where e is a positive number that is arbitrarily small comparedto h. The resulting integral can be integrated by parts to yield

Kj(t) =

rh* uZ/2,h* _wy/2-I w(w -dw\ - Vvt - vr } (6.5.13)

plus terms that are o(l) as e —• 0. The surprisingly simple form ofthe expression in the integrand of (6.5.13) is a result of applying theidentity (6.5.11). The remaining integral is easily evaluated in termsof algebraic functions with the result that

fh h?'2{h* - w)1'2 J rt h(h*-h)/ T M372- dw = 2V

Jh+e w(w-h)3/2 V e

6*

plus terms that are bounded as e —• 0. Thus, the divergent part of theintegral (6.5.13) exactly cancels the divergent term resulting from theintegration by parts. The remainder of the expression is unaffectedby letting e —> 0. The stress intensity factor for the moving crackproblem is therefore

Kl{t,v) = 2<r*k(v)yfl [ ^ J ^ 7 2 - A / ^ T T ] (6.5.15)

for t > T. The universal function k(v) is shown in the graph inFigure 6.10. For 0 < t < r, the stress intensity factor is given by(2.5.44).

6.5.3 Incident plane stress pulse

Attention is turned once again to the case of a plane pulse incidenton the crack edge. Suppose that a plane pulse parallel to the crackplane propagates through the body so that the wavefront reachesthe crack plane at time t = 0. Suppose, further, that this pulsecarries a jump in tensile stress ayy from its zero initial value to a*.If the material was not cracked on the plane y = 0 then the stresspulse would induce a tensile traction of magnitude a* on this plane,as already discussed in Section 2.5; see problem (a) in Figure 6.11.The complete elastodynamic field for an incident stress pulse can beobtained by superimposing the elementary solution for the uncrackedsolid and the solution for a suddenly applied crack face pressure ofmagnitude cr*.

This superposition scheme carries over without modification tothe case of crack growth. The stress intensity factor (6.5.15) alreadyincludes the influence of the crack face pressure a* over the originalsurface x < 0 [problem (6)], and it includes the influence of thediffracted wave field [problem (c)]. To extract the stress intensityfactor for the case of an incident step stress pulse, the only additionaleffect that must be taken into account is the uniform crack facepressure a* on the newly created crack surfaces 0 < x < v(t — r)[problem (d) in Figure 6.11]. From the expression (6.4.40), the stressintensity due to this contribution is 2a*k(v)y/2v(t — T)/TT . If thiscontribution is added to (6.5.15), then it is found that the stressintensity factor for an incident step stress pulse of magnitude a* anddelay time r is

(6.5.16)

This result has a couple of noteworthy features. First, it was derivedfor arbitrary delay time r but it is found to be independent of r,except for the condition that it applies only for t > r. Second, thestress intensity factor has the form of the stress intensity factor for astationary crack subjected to the same stress wave loading multipliedby the universal function of crack speed k(v). That is, it has the form

#}(*, v) = fc(v)/&(t, 0), (6.5.17)

which is a remarkable property shared with the corresponding resultfor time-independent loading obtained in Section 6.4.

The foregoing analysis has been carried out for step stress loadingof the crack plane at time t = 0. The stress intensity factor result iseasily generalized to the case of an incident stress pulse of generalprofile, following the procedure in Section 2.7. Suppose that thewavefront of a plane stress pulse parallel to the plane of the crackreaches y = 0 at time t = 0 and that the profile of the pulse is givenby

if t < 0,if t> 0.

Then the transient stress intensity factor for the onset of crack growthat speed v at any time t = r is

f2yi f(1 - v) Jo-

(6.5.19)

Numerous examples of Ki(t, 0) are given in Section 2.7 for variousprofiles p(£), and the corresponding result for growth is obtained from(6.5.19) by multiplication by k(v). As in the case of time-independentloading, the sudden onset of crack growth at some speed v alwaysimplies a discontinuous drop in the level of the stress intensity factor.As in that case, if the level of stress intensity is thought to control theprocess of crack growth and a critical level is achieved just prior to theonset of growth, then it is unlikely that growth will commence at alarge crack speed. Instead, the more likely behavior is for crack growthto begin gradually and the speed to increase with the amount of crackgrowth. The actual response depends on the details of the growthcriterion and the applied loading, and several cases are examined inChapter 7.

7

ELASTIC CRACK GROWTH

AT NONUNIFORM SPEED

7.1 Introduction

The restriction to constant crack speed in the early analytical work ondynamic crack propagation, as reflected in the contents of Chapter 6,was not motivated by physical considerations. Instead, the restrictionwas imposed in order to render the mathematical models tractable.In this chapter, progress toward relaxing this restriction for rapidcrack growth in nominally elastic materials is described. Results arelimited, but sufficient to provide a relatively complete conceptual basisfor analysis of crack propagation under two-dimensional conditions orin simple structural elements.

The study of a problem involving crack growth at nonuniformrate proceeds in two steps. First, the underlying boundary valueproblem is considered for arbitrary motion of the crack tip, witha view toward obtaining a full description of the mechanical fieldsnear the crack edge during growth. Then, an additional physicalpostulate in the form of a crack propagation criterion is imposed onthe mechanical fields in order to determine an equation of motion forthe crack tip. This approach is illustrated for antiplane shear crackgrowth, for plane strain crack growth under quite general loadingconditions, and for one-dimensional string and beam models. Thetechnologically important issue of crack arrest (Bluhm 1969) is anessential feature in this development; arrest is identified through thecrack tip equation of motion as the point beyond which crack growthcannot be sustained.

367

368 7. Crack growth at nonuniform speed

The difficulty in extending the analytical approaches that wereso successful for stationary cracks and cracks growing at a constantrate to the case of nonuniform crack growth are evident from theanalysis in Chapters 2, 3, and 6. The boundary value problemsconsidered are strictly linear and, for the most part, the mechanicalfields obtained are steady state or self-similar. In such cases, solutionprocedures based on the properties of analytic functions of a complexvariable and on integral transform techniques are very effective. Forthe case of crack growth at a nonuniform rate, however, the solutiondepends on the crack tip motion in a way that precludes the useof these techniques. The mechanical fields are neither steady norself-similar. Consequently, these same analytical methods cannot beapplied directly. A solution procedure that is an exception to thisobservation is Green's method, as described in Section 2.3, whichrelies on the linearity of the governing differential equations but noton a particular form of the boundary conditions. Indeed, the firstexact stress intensity factor solution for two-dimensional transientcrack growth under conditions of antiplane shear deformation, ormode III crack propagation, was obtained by Kostrov (1966) throughapplication of this method. The approach and principal conclusionsare presented in Section 7.2. Direct approaches to the correspondingplane strain crack growth problem have been elusive. However, certainfeatures of the solution obtained by Kostrov (1966), as well as a similarsolution obtained by Eshelby (1969a), suggested an indirect approachbased on the notion of fundamental solutions (Preund 1972a, 1972b).This approach is described in Section 7.3 for the case of mode Ideformation and general time dependent loading. The correspondingmode II analysis was reported by Fossum and Freund (1975). Theapproach was generalized by Freund (1973a), Kostrov (1975), andBurridge (1976) to other cases of loading.

The chapter concludes with two sections devoted to ancillaryresults. In Section 7.6, it is shown that the mathematical solutionfor growth of a mode I crack at nonuniform speed can be appliedto determine the speed at which a strip yield zone grows from acrack tip in a body subjected to dynamic loading. Finally, in thelast section, the question of uniqueness of elastodynamic crack growthsolutions is addressed. The proof of the classical uniqueness theoremfor elastodynamics due to Neumann is based on consideration of globalenergy variations associated with a difference solution of the governingequations. Because a moving crack tip is a sink of mechanical energy,

7.2 Antiplane shear crack growth 369

it is essential that possible implications for the conditions of theuniqueness theorem are considered.

7.2 Antiplane shear crack growth

Consider an elastic body which contains a half plane crack but whichis otherwise unbounded. Introduce a rectangular x,y, z coordinatesystem so that the z-axis lies along the crack edge. The crack is in theplane y = 0 and occupies the half plane x < 0. Suppose that the ma-terial is initially at rest, and that time-independent loading is appliedwhich produces a state of antiplane shear deformation in the body.Thus, the only nonzero component of displacement is the z-componentuz(x, y, t) = w(x, y, t). The displacement field is independent of z andit has the symmetry property that w(x,—y,t) = —w(x1y,t). Priorto crack advance, the stress distribution on the prospective fractureplane ahead of the crack tip is ayz(x,0) = ry(x, 0) = q(x).

In Section 6.4, the process of crack advance in such a situation wasviewed as the process of negating the traction q{x), which suggesteda superposition scheme for construction of solutions to crack growthproblems. The same general point of view is adopted here. Supposethat at time t = 0 the crack begins to advance in the x-directionso that the position of the crack edge is x = l(t) for t > 0. Thefunction l(t) specifies the amount of crack advance at any instant oftime, and the instantaneous crack tip speed is l(t). The function l(t)is continuous and its rate of change is restricted by

0 < l(t) <c for 0 < t < oo , (7.2.1)

where c denotes the elastic shear wave speed cs iji this development;otherwise, the function is arbitrary. Following the reasoning in Section6.4, it is assumed that the material is stress free and at rest for t < 0.As the crack advances for t > 0, it does so under the action of atraction distribution on the crack faces that is equal and opposite tothe initial traction, that is,

yz(x,0±,t) = ry(x,0±,t) = -q(x), 0 < x < l(t). (7.2.2)

No other loads act on the body. The solution of this problem, whensuperimposed on the original equilibrium state, provides a complete

solution for the situation of crack growth in a body subjected to someinitial time-independent loading. Also, as demonstrated in Section6.4, the principal features of the solution are exhibited by a relativelysimple canonical problem, or fundamental solution. Thus, the pro-cedure is developed for the particular case of opposed concentratedforces on the crack faces of magnitude #*, that is,

q(x) = q*6(x). (7.2.3)

More general loadings are then handled by superposition.The general framework for a solution procedure based on Green's

method is outlined in Section 2.3. The component of displacementw(x, ?/, t) satisfies the wave equation in two space dimensions and time,

If attention is restricted to the half plane — o o < : r < o o , 0 < ? / < o othen Green's formula (2.3.1) leads to a representation for the valueof w at any point in the body and any time in terms of the Green'sfunction and the values of w and dw/dy on the boundary y = 0+.The representation formula appears as (2.3.5) and the correspondingGreen's function is given in (2.3.4). This representation formula wasapplied to solve a stationary crack problem in Section 2.3. Thatproblem is identical to the one being considered here except that/(£) = 0 in the former case. It was observed in Section 2.3 that ifthe field point in the representation theorem approaches the boundaryy = 0 ahead of the crack tip, then the condition that the displacementvanishes on y = 0, x > 0 due to symmetry yields an integral equationfor the unknown traction on the crack plane ahead of the crack tip.For a stationary crack tip, this integral equation is (2.3.6). In thepresent situation involving crack growth according to x = /(£), therepresentation theorem (2.3.5) is likewise applicable. Furthermore,the symmetry condition that

w(x, 0, t) = 0 for x > l(t) (7.2.5)

implies that

rto rxo+c(to-t) r (r ft/ Ty[X^/ /

Jo Jxo-c(to-(to-t) y/c2(to - t)2 - (xo - x)2 (7.2.6)

Figure 7.1. A portion of the x,t-plane showing the region of integration in (7.2.6).The time axis has been rescaled by wave speed c for convenience. The crack tippath in this plane is x = l(t).

for xo > /(to). The integral equation (7.2.6) is identical to (2.3.6),of course, except that the location of the field point is restricted byxo > I(to) rather than xo > 0. A solution of this integral equationcan be found by following the procedure outlined in Section 2.3, asadapted from the work of Kostrov (1966) on antiplane shear crackgrowth at a nonuniform rate. With reference to (7.2.6), it is notedthat ry(x, 0±, t) = —q*6(x) for x < l(t) but that ry(x, 0, t) is unknownfor x > l(t). Thus, the first step is to determine this stress distributionahead of the crack tip.

Consider the portion of the x, £-plane shown in Figure 7.1, wherethe time coordinate has been scaled by c for convenience. This isthe plane y = 0+ in the three-dimensional x, y, t-space of the solutionw(x,y,t). A generic crack path x = l(t) is shown, and a field point(xo, to) ahead of the crack tip is chosen where condition (7.2.6) applies.The lines x + ct = xo + cto and x — ct = xo — cto running backward intime from (xo, to) are shown as dashed in Figure 7.1. The solution at(xo, to) depends on boundary values and initial values only within thedomain of dependence enclosed by the lines for t < to. The solutionprocedure is facilitated by a change of integration variables in (7.2.6)to

= ct + x , = ct — x. (7.2.7)

The £, 77-coordinate axes are also shown in Figure 7.1. This transfor-mation carries the point (xo, to) into (£o> Vo) and the integral equation

(7.2.6) becomes

f ° -=L= /?° ^ M « *» = 0, Co > 9(Vo), (7.2.8)

where the function g(-) satisfies

$ ti V}) (7.2.9)

identically. Thus, £ = g(rj) is the trajectory of the crack tip in termsof the £, 77-coordinates. The variable representing the y-componentof stress has been preserved as ry in writing (7.2.8) even though itis obviously a diflFerent function of its arguments than it was in theoriginal equation (7.2.6). The partial solution of this integral (2.3.7)implied by the initial conditions has been incorporated into (7.2.8).

The integral equation (7.2.8) will be satisfied if the inner integralvanishes for all 77 in the interval 0 < 77 < rjo. This condition can bewritten as the Volterra integral equation of the first kind

f / (7.2.10)9(T)O) V s o s J-r)o

for £o > 9(Vo)' In writing (7.2.10), the boundary condition on traction(7.2.2) has been incorporated. Finally, for the special case in whichthe boundary traction is the pair of opposed concentrated forces atx = 0, namely, q(x) = q*6(x), the integral equation (7.2.10) becomes

io T ( g 0 * 7 ) 2g*(7.2.11)/c c

V£o ^ 4

where the scaling property of the delta function 6(x/2) = 26(x) hasbeen used. With a change of integration variable to £ = €—g(r)o) this isagain an integral equation with the standard form of an Abel integralequation with a square root singular kernel as given in (2.3.12). Thegeneral solution is given in (2.3.15). In the present case, the solutionfor any point with £0 >

To transform this result from the coordinates £, 77 to the coordi-nates x, £, let the physical coordinates of the point £ = g(r}o), V = Vobe (#*,£*). The special significance of this point is that if the cracktip emits an elementary wave signal at time t* when it is at the placex* = l(t*) and if the signal radiates with speed c, then the signal willarrive at position x at time t. Prom the geometry of Figure 7.1, it isevident that

= 2 [ x 0 - J ( O ] > (7-2.13)

£0 — Vo = 2xo •

Therefore, in terms of the physical coordinates, the stress distributionahead of the crack tip (7.2.12) is

ry(x,0,t) = - ^ - V \> , x>l(t), (7.2.14)

where the subscript "o" has been dropped because its use is no longeressential. For any fixed x, t and any crack motion, the parameter t*is a solution of

which is simply a statement of the fact that a signal traveling at speedc that is emitted at the position l(t*) at time <* will reach position xat time t.

The solution (7.2.14) has a number of noteworthy features. First,it is remarkably simple. Indeed, the mathematical expression is ex-actly the same as the corresponding expression for the equilibriumstress distribution ahead of a crack tip at x = /o due to a pairof opposed concentrated loads q* applied at x = 0, except thatthe quantity l(t*) replaces the constant lo in the equilibrium case.Another surprising feature emerges for the situation where l(t) ^ 0for 0 < t < ta but l(t) = 0 for t > ta, that is, if crack growth stopsat the arrest time t = ta. The result (7.2.14) implies that the correctequilibrium field for the given loading and crack length l(ta) radiatesout from the crack tip following its arrest, and that the equilibriumfield is fully established behind a point moving with the wave speedc, that is, for l(ta) < x < c(t — £a). This is an extraordinary feature

for a two-dimensional elastodynamic field. Such fields are normallycharacterized by long transient "tails" after any abrupt change inloading. This feature was observed and discussed in some detail byEshelby (1969b), who noted that the corresponding stress distributionfor a nonuniformly moving dislocation does not share this remarkablefeature. Some broad implications of this feature will play a centralrole in the discussions in subsequent sections in this chapter.

The dynamic elastic stress intensity factor is readily extractedfrom the complete stress distribution (7.2.14). If x — l(t) is vanishinglysmall, then (< — £*) is also small. Thus, for points close to the crackedge in this sense,

x - l(t*) = x- l(t) + l(t)(t - t*) + o(t - t*) (7.2.16)

as t* —> t. In light of (7.2.15), this can be rewritten as

( 7-2-1 7 )

for points x near l(t). Therefore, the dynamic stress intensity factoris

= lim

(7.2.18)2 1 1 -

\

The equilibrium stress intensity factor for this same loading appliedto the faces of a stationary crack at a fixed distance /o behind thecrack tip is

(7.2.19)

Therefore, the dynamic stress intensity factor for arbitrary nonuni-form motion of the crack edge has the form of a function of instanta-neous crack speed times the corresponding equilibrium stress intensityfactor for the given applied loading and an amount of crack growthequal to the instantaneous value of l(t),

(7.2.20)


Although the specific results obtained so far in this section applyonly for a very special loading situation, generalization to a morecomplex loading situation can be accomplished by means of the su-perposition arguments presented in Section 6.4. For the case of generalcrack face traction —q(x) behind the advancing crack edge, the resultfor the dynamic stress intensity factor is

(7.2.21)

Again, the expression has the form of the universal function of crackspeed times the corresponding equilibrium stress intensity factor forthe specified loading and a crack length equal to the instantaneouslength l(t). In effect, the result (7.2.21) provides the dynamic stressintensity factor for the most general time-independent loading con-ditions. For example, the function q(x) could be the initial stressdistribution on the prospective crack plane ahead of a finite lengthcrack. The result (7.2.21) then provides the stress intensity factor fordynamic advance of a finite length crack up until the time at whichwaves generated at the advancing crack edge are reflected back ontothe moving crack tip from the other, still stationary, edge of the crack.The matter of interaction between the ends of a crack of finite lengthwhen both are moving was considered by Rose (1976c).

The discussion of antiplane shear crack growth at nonuniformrate has been developed under the assumption of time-independentloading, the situation for which the features of the analysis and theinterpretation of the results are most transparent. This is not anessential restriction, and cases of time-dependent loading of the crackfaces can be handled in much the same way. For example, the situationin which the crack faces are subjected to suddenly applied, spatiallyuniform shear traction ry{x, 0±, t) = —r*H(t) was studied in Sections2.2 and 2.3 for the case of a stationary crack tip. The dynamic stressintensity factor in that case was found to be if///(t, 0) = 2r* ^2ct/n.If the crack tip actually moves according to x = /(£), then an analysissimilar to that just presented yields the result that

KIH(t, i) = Jl-i/cKmtt, 0). (7.2.22)

In other words, the dynamic stress intensity factor for arbitrary mo-tion of the crack tip is again the same universal function of crack

tip speed that appeared in (7.2.20) times the time-dependent stressintensity factor that the crack tip would have if it were not mov-ing. Solutions for more general cases of time-dependent loading andoblique stress wave loading have been given by Achenbach (1970a,1970b).

The displacement representation theorem takes the form

c rto ,xo+c(tot) ( 0 ^ dx dt

w{xo, yo, to) = —- / / *v ' ' ' == (7.2.23)27T/i J0 JXo_c{to_t) y/C2{to - t)2 ~ R2

for the problem at hand, where R2 = (xo — x)2 + (yo — y)2. The stresscomponent ry(x, 0, t) is known on the entire boundary y = 0+, eitherthrough the boundary condition (7.2.2) or the result (7.2.14) for thestress distribution ahead of the crack tip, so the displacement can beevaluated according to (7.2.23) anywhere in the body. The particularcase of w(xo, 0, to) for the pair of opposed concentrated forces is nowdeveloped in some detail for purposes of illustration.

For the situation that has been under consideration in this sec-tion, namely, crack growth due to crack face traction that is operativefor t > 0, the change of integration variables (7.2.7) yields

, 0, Vo) = - J - f ° -?J== f'° T~§^4 K dr,. (7.2.24)

The point (xo,to) is assumed here to be on the crack face behind thetip that is advancing according to x = l(t). In view of (7.2.8), theinner integral vanishes for all rj < rj* = cb* — #*, where the point(#*, t*) is defined as above. The integral is now evaluated for the casewhen ry(x, 0,t) = —q*6(x) for x < l(t) and for an observation pointfor which 0 < xo < l(to). In this case, the integral becomes

(7.2.25)

This integral is readily evaluated in terms of elementary functions.The result can be expressed in terms of physical coordinates by notingthat 77* = 770 — 2(x* — xo) and x* = /(£*), so that

for 0 < xo < l(to). The displacement field on the crack face behindthe moving crack tip has the same property exhibited by the stressdistribution ahead of the tip, that is, the expression for the displace-ment is the same as the corresponding equilibrium expression exceptthat the quantity Z(£*) appears in place of the crack length parameter.An immediate consequence is that, if the crack tip stops at some time£a, then the equilibrium crack face displacement appropriate for thegiven crack face loading and the crack length parameter l(ta) = la

radiates out along the crack face from the crack tip behind a pointtraveling with the speed c. Although the result is established for asimple case of crack face loading, the solution for any other case ofcrack face loading can be constructed by linear superposition over thissolution, so that all solutions for time-independent loading share thisremarkable property.

The features of the elastodynamic fields for growth of a semi-infinite crack in the antiplane shear mode at a nonuniform rate canbe summarized as follows, whether the crack face loading is time-independent or time-dependent. The stress intensity factor has theform of a universal function of crack tip speed times the stress intensityfactor that would be found for the same loading but with a stationarycrack tip. Furthermore, if the crack tip stops at a certain instant oftime, then the complete field for the given loading and the amount ofcrack growth at that instant radiates out from the crack tip behind awavefront traveling at speed c, and it is fully established everywherebehind this wavefront. Finally, the results apply equally well for thecase of extension of a crack of finite length or extension of a crackin a body of finite extent, but only for times up until waves reflectedfrom the remote boundary, reflected from the other stationary cracktip or emitted from the other crack tip reach the crack tip of interest.Thus, in any practical situation, the solutions are strictly valid onlyfor a short time, roughly twice the travel time for a wave to propagatefrom the crack tip to the nearest boundary or to the other crack tip,if the other tip is stationary.

An interesting application of Kostrov's solution was developed byNilsson (1977a). He considered a mode III crack that was propagatingunder steady state conditions up until a certain instant of time. Theclass of steady state elastodynamic crack problems was considered inSection 6.2. At a certain instant of time, the motion of the crack tipwas assumed to become nonuniform. Nilsson (1977a) determined themode III stress intensity factor for the nonuniformly moving crack tip


for both acceleration and deceleration from the steady state speed.The resulting stress intensity factor was shown to depend on thefeatures of the steady state problem and on the difference in distancetraveled by the crack tip between the steady state and nonuniformcrack tip motions. The solution provides a framework for discussingissues connected with the arrest of a rapidly growing crack.

The results of this section were obtained by following the anal-ysis of Kostrov (1966) for mode III crack growth based on Green'smethod. For the particular case of general time-independent loading,these same results were obtained by Eshelby (1969a) who followed acompletely different approach. In a study of electromagnetic radiationdue to moving sources in two space dimensions and time, Bateman(1955) established that if a function of position w(x,y) satisfies theLaplace equation V2w = 0 and is homogeneous of degree one-half inx, y, then w(x — l(r),y) satisfies the wave equation (7.2.4) with

c(t -T) = y/[x-l(r)]* + y* . (7.2.27)

The "time" r(x, y, t) satisfying (7.2.27) has the following interpreta-tion. A point moves in the rr-direction so that its position at any timet is x = l(t). As it moves, the point continuously radiates cylindricalwavelets which expand at speed c. The particular wavelet sensed atlocation x, y and time t was emitted by the moving source at time rwhen the source was located a tx = /(r),j/ = 0. For any given motionl(t) and any particular #,y,£, this time r is specified by (7.2.27).By means of this result, Eshelby (1969a) showed that the essentialstructure of the near tip fields for the stationary mode III crack arepreserved for an arbitrarily moving crack. The solution for completecrack fields for general time-independent loading was obtained bybuilding up general crack motions from a sequence of infinitesimalconstant speed segments. This constructive approach to the problemprovided some motivation for pursuing the analysis of mode I crackgrowth at arbitrary rate presented in the following sections.

7.3 Plane strain crack growth

In Section 6.4, the extension of a tensile crack in a body subjectedto general time-independent loading was considered. It was assumedthat the crack tip is stationary up until the instant of onset of crack

1.3 Plane strain crack growth 379

growth, and that the speed of propagation of the crack tip is constantfor all later time. The process of crack extension was viewed asthe negation of traction on the prospective fracture plane, and theproblem was reduced to examination of a loading situation involvingcrack face traction but no other loading. The case of crack extensionat constant rate was solved in two steps. First, the relatively simplecase in which a pair of opposed concentrated normal forces of constantmagnitude acting on the crack faces are left behind the crack tip as itadvances at speed v through an otherwise unloaded body was studied.The forces continue to act on the same material particles as the cracktip moves away. The elastodynamic field generated in this way iscalled the fundamental solution of the problem. Then, the solution forgeneral time-independent loading was obtained from the fundamentalsolution by means of linear superposition.

The discussion of this same physical problem is resumed here,since the features of this problem indicate a path to determining thestress intensity factor history for the same physical system when thecrack tip propagates at a nonuniform rate. For nonuniform propa-gation under the assumed plane strain conditions, the crack edge isat x = l(t) for t > 0, where l(t) is a general continuous function oftime. Motivated by the results of Section 7.2 on the correspondingantiplane shear problem of crack growth at a nonuniform rate, aninverse method is developed that leads to the desired stress intensityfactor result. The development proceeds in several steps. First, thesituation when the constant speed crack of Section 6.4 suddenly stopsis examined. The solution for this case exhibits the remarkable featurethat the stress intensity takes on the correct equilibrium value for thegiven applied loading and the new crack length immediately upon thestopping of the crack. Furthermore, the corresponding equilibriumstress distribution radiates out ahead of the crack tip behind thewavefront traveling with the shear wave speed. These features ofthe response provide a basis for construction of the stress intensityfactor for general motion, again by linear superposition. The notationestablished in Section 6.4 is followed here, except that variables andfunctions in the fundamental solution are denoted by a superposedstar * to distinguish them from corresponding variables and functionsrepresenting the same physical quantities in the present case.

1.3.1 Suddenly stopping crack

Recall the general features of the constant speed analysis in which

crack growth under two-dimensional plane strain conditions is con-sidered. A rectangular coordinate system x, y, z is introduced in thebody in such a way that the crack is in the plane y = 0, the edge ofthe crack is parallel to the 2-axis, and the crack edge moves in thepositive x-direction. Thus, the plane of deformation is the x, y-plane.The crack tip is initially at x = 0 and it begins to move at constantspeed v at time t = 0. The position of the crack tip at any time t > 0is then x = vt.

For time t < 0, the crack tip is at x = 0 and the material is stressfree and at rest everywhere. At time t = 0, the crack tip begins tomove in the positive x-direction at speed v. As the tip moves awayfrom the origin of the coordinate system, it leaves behind a pair ofconcentrated forces, each of magnitude p* (force per unit length inthe z-direction) and tending to separate the crack faces. The crackfaces are traction free at points other than x = 0. The field generatedis the fundamental solution.

Now suppose that the crack tip advances a distance la at theconstant speed v and then suddenly stops, that is, the speed of thecrack tip changes abruptly from v to zero. The fundamental solutionstill solves the problem for vt < la but some modification is requiredfor vt > la. For vt > la the field equations (6.4.1) remain unchanged,of course, but the boundary conditions (6.4.2) are replaced by

cryy(x, 0, t) = —p*6(x)H(t) for — oo < x < la ,

(Txy(x, 0, t) = 0 for - oo < x < oo , (7.3.1)

uy(x, 0, t) = 0 for la < x < oo .

As before, only the region y > 0 must be considered due to symmetryof the fields.

Suppose for the moment that the crack tip does not actuallystop at x = Ia but that it continues to advance at speed v. Further-more, suppose that a time-independent distribution of normal tractionayy(x,0, t) = p(x) appears on the crack faces behind the advancingcrack tip for x > Za. If the solution for a stopping crack is sought,then the stress intensity factor at the moving crack tip must be zerofor x > la, and the crack face normal displacement must be zero forla < x < vt. What must this distribution p(x) be so that the stressintensity factor for x > la is identically zero? This is one feature that

the solution must have if it is to actually represent the field for thesuddenly stopping crack. The choice of a time-independent tractiondistribution is motivated by the observation that the correspondingtraction distribution for the mode III problem analyzed in Section 7.2is time-independent.

The answer to the question on p(x) is provided by the stressintensity factor expression (6.4.31) for general crack face loading. Inthe present case, the crack face traction is — p*8(x) +p(x) for — oo <x < vt, where p(x) = 0 for vt <la. If this expression is substituted forp(x) in (6.4.31) and the condition that the resulting stress intensityfactor is identically zero is imposed, then the integral equation

vt>la, (7.3.2)

is the result. With a change of integration variable to x' = x — Za,this equation takes the standard form of an Abel integral equation asgiven in (2.3.12). The unique solution based on (2.3.15) is

x>la. (7.3.3)

This distribution is recognized as the equilibrium normal tractiondistribution on y = 0, x > la in a body containing a half plane crackoccupying y = 0, x < la with the crack faces subjected to concentratednormal forces of magnitude p* at x = 0. The stress intensity factorfor this equilibrium distribution is p* y/2/7rla at x = la> It is thereforeestablished that if the traction distribution (7.3.3) acts on the crackfaces behind the crack tip as it advances at constant speed then thestress intensity factor is zero for x > la- It cannot be concluded thatthe crack tip has actually stopped at x = Za, however, because it isnot yet known if the boundary condition (7.3.1)3 is satisfied. Thepossibility that uy(x,Q,t) = 0 for x > la is examined next.

To address the matter, it is more convenient to consider thegradient of normal crack face displacement duy/dx rather than thedisplacement itself. If this displacement gradient for the fundamentalsolution is the wave function f(x,y,t) in the general superpositionintegral (6.4.30) and if the particular traction distribution (6.4.38)is assumed, then the complete crack face displacement gradient on

y = 0+ is

(7.3.4)

From (6.4.12) the crack face displacement gradient is

du* If 1-*-(x,0+,t) = — Im{ CC -(C) \ (7-3.5)OX 7TC I J s -

for 0 < — a+£ < t or — c&t < x < vt, where the function £/_(C) is givenin (6.4.23).

It is important to recall some of the properties of the function£/_(C). The factor S_(Q has branch points at ( = a+ and C = &+,and it is single valued in the £-plane cut along the real axis betweenthese points. It is analytic everywhere except at these two branchpoints. Furthermore, 5_(C) -> 1 as |C| -> oo, so C/_(C) = O(C3/2) as|£| —• oo. The function £/_(C) also has simple poles at £ = c+ andC = h. The relative magnitudes of the singular points are such thata+ < b+ < c+ < h.

If the expression (7.3.5) is substituted into the integral term in(7.3.4) and the change of variable of integration x' = v£(( + t/£) ismade, the resulting integral is

where a* = —(t + lati)l£. For la < x < vt, the upper limit ofintegration is in the range h < a* < —t/£. Because C/_ (C) has poleson the path of integration, the integral in (7.3.6) is interpreted in theprincipal value sense.

The integral (7.3.6) is a real integral. By viewing it as a lineintegral in the complex £-plane, however, its evaluation becomes asimple matter. With this objective in mind, the definition of (a* —C)1/2 is extended to the complex plane and the branch with nonneg-ative real part everywhere in the £-plane is chosen. The functionC?7-(C)/(a* — C)1^2 is single valued in the plane cut along Im(£) = 0,

Figure 7.2. The path of integration F in the complex £-plane used to evaluate(7.3.7) encloses the branch cut running from a+ to a*.

a+ < Re(C) < a*. The integral (7.3.6) is now viewed as a complexline integral along the upper side of this cut, as shown in Figure 7.2.When evaluated at £, the integrand is the complex conjugate of itsvalue at £, so the path of integration can be converted to the closedcontour F embracing the branch cut, as shown in Figure 7.2. Notethat each of the small semicircular arcs added to the contour near apole of the integrand makes a contribution to the value of the integralequal in magnitude to one-half the residue of the pole. Because ofthe branch cut passing through the pole in each case, however, thealgebraic signs of the contributions differ on opposite sides of the cutso that the net contribution is zero. The integral (7.3.6) becomes

(7.3.7)

The integral can be evaluated by direct application of Cauchy's the-orem (1.3.4). The only singularity of the integrand exterior to thecontour F is a simple pole at ( = —t/%. The value of the integrandtaken along F is equal to the value of the residue of this pole plus thevalue of the integral along a circle of indefinitely large radius in the £-plane. The integrand is O(C~2) as |£| —> oo, so the latter contribution

is zero. Therefore, the value of the integral is

for 0 < — a+£ < £ or —c t < x < vt.If this result is compared with (7.3.5) then it is seen immediately

that the displacement gradient due to the action of the traction distri-bution p(x) on the crack faces behind the advancing crack tip exactlycancels the displacement gradient due to the concentrated loads inthe fundamental solution for x in the range la < x < vt. The totaldisplacement is thus a constant in this interval, and the value of thisconstant must be zero because the normal crack face displacement iszero at the crack tip x = vt. Therefore, the fundamental solution hasbeen extended to the case where the crack tip grows at constant speedv up to the point x = la and then suddenly stops.

The foregoing analysis leads to the result that if the constantspeed crack suddenly stops, then the stress intensity factor instan-taneously takes on the value appropriate for the given crack lengthand applied loading under equilibrium conditions. Furthermore, theequilibrium stress distribution is fully established on the extension ofthe crack line ahead of the crack tip behind the shear wavefront, afeature that has been verified experimentally by Vu and Kinra (1981).This is a remarkable result for a two-dimensional elastodynamic field,and it indicates that the stress intensity factor result for generalmotion may have the same simple features that were observed forthe case of antiplane shear crack growth in the preceding section. Ifthis is to be the case, however, other special properties will have tobe demonstrated. In particular, it must be shown that the appropri-ate equilibrium stress distribution radiates out along the crack linebehind a front traveling with an elastic wave speed when the cracktip suddenly stops, that is, at a speed beyond any possible crack tipspeed in the present analysis. This point is considered next, alongwith the nature of the crack face displacement that is radiated outfrom the crack edge upon sudden stopping of the crack tip at x = la.

Consider the displacement of the crack faces first. Suppose thatthe crack tip suddenly stops at x = la and that a Rayleigh pulse isradiated out along the crack faces with speed CR = 1/c. Attention isnow focused on points on the crack faces between the crack tip andthis Rayleigh pulse. Thus, at any time £, the value of x is restricted

1.3 Plant strain crack growth 385

to be in the range la — (t — la/v)cR < x < Za. Thus, the parametera* introduced as the limit of integration in (7.3.6) is in the rangec+ < a* < h. The steps outlined in going from (7.3.6) to (7.3.8)can again be followed in extracting the surface displacement gradientin this case. The only difference here is that the integrand has twosingularities exterior to the contour F, namely, simple poles at £ =—t/£ and at £ = h. The value of duy/dx in the range of interest istherefore the residue of the pole at £ = /i, which is

(7.3.9)

where v is Poisson's ratio and fj, is the shear modulus of the material.This displacement is the equilibrium displacement gradient of the

crack face y = 0+ of a semi-infinite crack on y = 0, x < la dueto an opposed pair of concentrated loads of intensity p* acting atx = 0 and tending to open the crack. Because uy(la,Q,t) = 0, thedisplacement itself is the appropriate equilibrium displacement. Ifthe same procedure is followed for the case when x is farther fromthe crack tip than the Rayleigh pulse that is emitted when the tipsuddenly stops, then it is found that the surface displacement hasa complicated transient variation. Thus, it is concluded that if theconstant-speed crack propagation described by the fundamental so-lution suddenly stops, then the equilibrium displacement appropriatefor the specified load and new crack length radiates out along thecrack faces behind a front traveling with the Rayleigh wave speedof the material. Though this has been established here only for thecase of the fundamental solution, the generalization to general time-independent loading conditions is straightforward and is discussed inwhat follows.

Next, the distribution of normal stress ayy on the crack line aheadof the crack tip is considered for times after the crack tip suddenlystops. Based on the discussion leading to (7.3.4), the expression forthis stress distribution is

-L V a;y(x-x',O,t-x'/v)p(x')dx',

(7.3.10)

where ayy is the stress component indicated from the fundamentalsolution. By means of an analysis paralleling that presented above in

the discussion of crack face displacement, it can be shown that

(7.3.11)nx " ~ ' v '

for vt < x < la + (t — lah)cd- The expression (7.3.11) is recognizedas the equilibrium stress distribution on the plane y = 0, x > vtfor a semi-infinite crack occupying y = 0, x < la with the crack facessubjected to an opposed pair of concentrated forces of magnitude p* atx = 0. It was shown in deriving (7.3.3) that precisely the same stressdistribution prevails over the region la < x < vt. Thus, the importantconclusion is reached that if the constant speed crack represented bythe fundamental solution suddenly stops, then the equilibrium normalstress distribution appropriate for the given loads and the new cracklength radiates out from the tip along the prospective fracture planeand it is fully established behind a point traveling with the shear wavespeed of the material c8. Again, the result can be extended to anycase with time-independent mode I loading. This strong result makesit possible to construct the stress intensity factor for a crack extendingat a nonuniform rate.

The foregoing analysis applies only in the case of concentratedloads of magnitude p* left behind the crack tip as it grow from x = 0.Suppose now that a normal traction distribution — p(x; 0) acts on thecrack faces behind the crack tip as it advances at constant speed fromx = 0 to x = vt. This traction distribution might result from theapplication of general time-independent loading on the cracked body,resulting in the normal stress distribution p(x; 0) ahead of the cracktip for t < 0. After the moving crack tip passes the point x = /a, anadditional traction distribution p(x;la) is left behind the advancingcrack tip on la < x < vt. If the net stress intensity factor is to be zerofor x > la or t > la/v then a linear superposition over (7.3.3) yieldsthe relationship

p(x; la) = p(x; 0) - - J _ t PfoOMT^ ^ (? 3TTy/X — la Jo X' — X

for x > la. The important observations made above concerning theradiation of equilibrium fields from the crack tip upon sudden stoppingof the advancing crack represented by the fundamental solution canbe extended to the case of general loading by means of (7.3.12). Of


special interest is the stress intensity factor for the case of a cracktip that advances at constant speed under general time-independentloading and then suddenly stops. If the traction left behind thecrack tip as it advances from x = 0 is p(x;0) and if the normalstress distribution that develops ahead of the tip following stoppingis p(x;la) then the stress intensity factor following stopping is

= lim_ v2?r(a; - /o) p(x; la)

(7.3.13)

This is precisely the equilibrium stress intensity factor for the givenapplied loads when the crack tip is at x = la.

The stress intensity factor history for the constant speed growthsegment from x = 0 to x = la can be summarized as follows. Ini-tially, say at time t = 0, the stress intensity factor has some time-independent value Ki (0,0). Then, when the crack begins to ad-vance at speed v, negating the traction distribution p{x\ 0), the stressintensity factor changes discontinuously to k(v)Ki(0,0) and it hasthe value k(v)Kj(vt,0) thereafter, as demonstrated in Section 6.4.Then, upon stopping at x = la, the crack tip stress intensity factorchanges discontinuously from k(v)Ki(la, 0) to Ki(la, 0), and this valueis maintained indefinitely. Thus, for the entire growth segment, thestress intensity factor has the form of the universal function of cracktip speed k(v) times the equilibrium stress intensity factor for thegiven time-independent loading and the instantaneous crack length.The stress intensity factor history for growth of a crack at generalnonuniform rate is next constructed as a sequence of constant speedsegments.

7.3.2 Arbitrary crack tip motion

Consider once again the elastic body containing a half plane crack butthat is otherwise unbounded, as described in the foregoing section.The body is under the action of time-independent loads that producea state of mode I plane strain deformation. The coordinate systemintroduced earlier in this section is used once again. Initially the cracktip is at x = 0 on the plane y = 0. The normal stress distribution on

Figure 7.3. The piecewise linear curve L(t) with vertices at t^ = kAt, k =0 ,1 ,2 , . . . , approximates the actual crack tip trajectory l(t) ever more closely, interms of both position and speed, as At —• 0.

the prospective fracture plane x > 0, y = 0 is given by ayy = p(x; 0)for time t < 0. If the same time independent loading acts on the bodybut the crack tip is at x = Z, then the normal stress distribution onthe plane ahead of the tip is p(x; Z). At time t = 0 the crack beginsto extend, and for £ > 0 the crack tip position is given by x = l(t).The function Z(£) is continuous, nondecreasing, and restricted by thecondition that the crack speed is always less than the Rayleigh wavespeed of the material, 0 < l(t) < CR. Otherwise, the crack tip motionis arbitrary.

The function l(t) is now approximated by a polygonal curveL(t) which has its vertices lying on the original curve, as shown inFigure 7.3. Successive times tk = fcAt, k = 0,1,2,... are markedoff, and the vertices of L(t) are located at (0,0), (*i,/i), and so on,where Ik = l(tk) = L(tk). According to this approximation, the cracktip moves with constant speed vo = (h — 0)/(ti — 0) during the timeinterval 0 < t < ti, with speed v\ = (Z2 — fi)/(<2 " *i)> during thetime interval ti < t < £2, and so on.

If the crack tip motion is actually described by the polygonalcurve, then the analysis of the preceding subsection provides the exactstress intensity factor as a functional of L(t). Initially, the crackextends by negating the normal stress p(x\ 0), or, in other words, thecrack advances under the action of crack face normal traction — p(x; 0).

From (6.4.31), the stress intensity factor is then given by

(7.3.14)

for 0 < t < ti.Suppose that the crack tip suddenly stops at time t = ti —

0. According to the analysis in Section 7.3.1, the normal stressdistribution p(x;li) is radiated out along y = 0, x > l\ behind apoint traveling with the shear wave speed of the material c8. In orderto extend further the crack must negate the normal stress p(x; li) onthe prospective fracture plane. In view of the remarkable fact thatthis distribution is also time-independent, the original solution can beapplied once again. Thus, at time t = ti + 0 the crack tip begins tomove with speed v\ and the stress intensity factor is

y/L{t) ~ Xdx (7.3.15)

for ti < t < t2.If the relationship (7.3.12) is multiplied by (L(t) - x)'1'2 and

integrated with respect to x then it follows that

dx. {73le)lh y/L(t) ~ X

In light of this result, (7.3.15) can also be written in the form

Ki(L{t), wi) = k(vx)Ki{L(t), 0). (7.3.17)

This procedure can be continued indefinitely with the result that thestress intensity factor has the exact form

,vk) = k(vk)Kj(L(t),0) (7.3.18)

in any time interval tk < t < tk+i- It is clear that the polygonalcurve L(t) approaches the actual crack tip trajectory l(t) as At —• 0.Furthermore, the approach is not only in magnitude but also in slope,


that is, L(t) also approaches l(t) pointwise in time as At —> 0. Inview of the fact that the change in the stress intensity factor at adiscontinuity in crack speed is determined completely by the cracktip speed before the discontinuity, the discontinuity in speed itself,and the crack length at the instant of the jump, it is anticipated thatthe stress intensity factor for crack tip motion L(t) approaches thestress intensity factor for crack tip motion l(t) as At —> 0.

Thus, the conclusion is reached that the stress intensity factorfor arbitrary motion of the crack tip depends on the crack tip motiononly through the instantaneous amount of crack advance l(t) and theinstantaneous crack tip speed l(t). Furthermore, the dynamic stressintensity factor is given as a functional of crack tip motion by

KI(lJ) = k(l)KI{l,0). (7.3.19)

In other words, the dynamic stress intensity factor for arbitrary mo-tion of the crack tip has the form of the universal function of instan-taneous crack tip speed defined in (6.4.26) and shown in Figure 6.10times the equilibrium stress intensity factor for the specified time-independent loading and an amount of crack growth equal to theinstantaneous crack growth. The result (7.3.19) implies that if thecrack tip speed changes abruptly from l~ to l+ for some particularcrack length /, then the ratio of the stress intensity factor immedi-ately after the jump to the value immediately preceding the jump isindependent of the applied loading and is given by

(7.3.20)K,(l,l-) *(!+)

This relationship carries over to all modes of crack growth for bothtime-independent and time-dependent loading conditions.

In deriving the general result (7.3.19), the process of crack ad-vance was viewed as the negation of a stress distribution on theprospective fracture path. There is another class of elastodynamiccrack problems which can be analyzed by considering a displacementdistribution, rather than a stress distribution, to be negated, butwhich leads to exactly the same result for a suddenly stopping crack.The approach is based on the use of continuously distributed movingdislocations, and illustrations were presented in Sections 3.2 and 3.3where stress intensity factor solutions were obtained by implementing


this idea. The same method can be used to determine the transientstress intensity factor history for particular cases of sudden stopping ofa constant speed crack tip. This was demonstrated by Freund (1976a)for the sudden stopping of a symmetrically expanding crack, and byNilsson (1977b) for the sudden stopping of a crack growing steadilyin a uniform strip. In each case, the results provided not only thevalue of the stress intensity factor immediately after the stopping ofthe crack according to (7.3.20), but also the transient stress intensityfactor for some time after stopping.

The result (7.3.19) shows that the main property noted for an-tiplane shear crack growth at a nonuniform rate in Section 7.2 isshared by the corresponding mode I crack growth process. The prop-erty is unusual for two reasons. The dynamically advancing crackcreates a stress wave field due to a sudden release of traction thatradiates out from the crack edge. The crack tip propagates throughthis stress wave field, presumably sensing waves that were radiatedwhen the crack tip was at relatively long past positions. However, thecrack tip field shows no sensitivity to this wave field in its dominantsingular term, in the sense that the instantaneous stress intensity fac-tor depends on the instantaneous crack tip position and speed but noton the history of crack tip motion. This is unlike the correspondingsituation in dislocation dynamics when a dislocation moves rapidlyand passes through the stress wave field created by its motion. Inthis case, there is a stronger sensitivity of the singular field to thepast positions of the dislocation. A second reason for surprise atthis property becomes evident when the two-dimensional process isviewed as a three-dimensional situation. If the dynamically advancingcrack edge suddenly stops, a material particle near a particular pointon the crack edge receives information concerning the stopping ofthe crack edge from ever more distant points along the edge. Agradual decay toward the long-time limiting field near the crack edgecould be anticipated, and this is indeed the common situation in two-dimensional elastodynamics. In this case, however, the field near thecrack edge takes on its long time value instantaneously as the crackedge stops.

With this particular solution in hand, the way is opened forobtaining a number of other results on crack tip fields for nonuni-form crack tip motion. Some of these are discussed subsequently.However, the great value of such a solution is that it makes possiblethe development of a crack tip equation of motion and this idea is

pursued after a brief summary of corresponding results for mode IIcrack growth.

1.3.3 In-plane shear crack growth

The analysis of growth of a mode II in-plane shear crack parallels thatof mode I crack growth, and only the main results are summarizedhere. This analysis is based on the growth of a half plane crack in anotherwise unbounded elastic solid. The mode II crack has particularsignificance for the area of earthquake source modeling. In seismology,more attention has been devoted to the study of dynamic faults oflimited extent than to half plane crack models. The obvious reason forthis emphasis is that actual faults have finite dimensions. If primaryinterest is on the details of seismic radiation due to fault motion, forexample, then the fault dimensions are of fundamental importance.On the other hand, if the primary interest is on the fracture processthen the actual fault dimensions are of lesser importance and thesemi-infinite crack models appear to be suitable. In general, for aspecific characterization of fracture resistance the main influence oncrack tip motion is the increased area over which the stress drop actsas the crack increases in size, and this effect can be included in thesemi-infinite plane crack models. The influence on crack tip motionat some point on the fault edge due to stress waves generated at someother points on the fault edge seems to be minor by comparison, andthis effect is precluded in the half plane crack models. With regard tothree-dimensional shear cracks, the edge deformation can be resolvedinto a combination of mode II and mode III deformations. With thispoint of view, the results obtained from analysis of plane models havebearing on three-dimensional plane crack propagation as well.

The specific mode II crack problem considered in this section issimilar to the mode I problem analyzed in detail above. At the initialtime t = 0 the crack begins to grow in the plane y = 0 from some initialposition 1(0) = Zo > 0, and at time t the edge has advanced to theposition x = /(£). Equal but opposite shear tractions correspondingto crxy(x^Q,t) = — q(x) act on the crack faces over 0 < x < l(t)as shown. The traction may be viewed as the difference between aremotely applied shear stress and a frictional stress on the crack facesdue to relative sliding of the faces. In this sense, q(x) may be viewedas a stress drop associated with crack advance, and it acts over anever increasing area as the crack advances.

The stress intensity factor for crack growth at the constant speed

7.4 Crack tip equation of motion 393

I = v for t > 0 is given in Section 6.4.4. The stress intensity factorsolution for growth at nonuniform speed in the range 0 < l(t) < CRwas determined by Fossum and Preund (1975). The result is

Kn(lJ) = kn(i)Kn(l,0), (7.3.21)

where ku is the universal function of crack speed defined in (6.4.42)and JK}J(/,0) is the equilibrium stress intensity factor for the givenapplied loading and crack length. The equilibrium stress intensityfactor for this case is given in (6.4.43). This property is identical inform to the corresponding mode I result in (7.3.19), and the discussionfollowing that result applies in this case as well. The generalizationto time-dependent loading also parallels that for mode I deformationin Section 6.5.

7.4 Crack tip equation of motion

In the study of the propagation of a crack through a solid within thecontext of continuum mechanics, the field equations can be solved forany motion of the crack edge, in principle. That is, if the motion ofthe crack edge is specified, along with the configuration of the bodyand the details of the loading, then the resulting mechanical fieldscan be determined. In the development of a theoretical model ofa dynamic crack growth process, however, the motion of the crackedge should not be specified a priori but, instead, it should followfrom the analysis. Unless the constitutive equation for bulk responsealready includes the possibility of material separation in some sense,the ingredient that must be added to the governing equations is amathematical statement of a crack growth criterion. Such a criterionmust be stated as a physical postulate on material behavior, sepa-rate from the kinematical theorems governing deformation and theother physical postulates governing momentum balance and materialresponse at a point in the continuous medium. The most commonform for such a criterion is the requirement that the crack must growin such a way that some parameter defined as part of the crack edgemechanical field maintains a value that is specific to the material.This value represents the resistance of the material to advance of thecrack, and it must be a material property.

In the study of crack growth processes in materials which failin a purely brittle manner, or in which any inelastic crack tip zone


is contained within a surrounding elastic crack tip zone, the mostcommon crack growth criteria are the generalizations of Irwin's criticalstress intensity factor criterion and Griffith's critical energy releaserate criterion. According to the generalized Griffith criterion, forexample, the crack must grow in such a way that the crack tip en-ergy release rate is always equal to the dynamic fracture energy ofthe material. The energy release rate represents the effect of theapplied loading, the geometrical configuration of the body, and thebulk material parameters on the material in the crack tip region forany motion of the crack tip; it is a property of the local mechanicalfields. The dynamic crack propagation energy or fracture energy, onthe other hand, represents the resistance of the material to crackadvance; it is assumed to be a property of the material and its valuescan be determined only through laboratory measurement. In thesimplest case, the amount of energy that must be supplied per unitcrack advance and per unit length along the crack edge to sustaingrowth is a constant, say F. The parameter F is termed the specificfracture energy. It is a material property with physical dimensions ofenergy/area or force/length, and its values can be determined only byexperiment.

Under the simplest circumstances of two-dimensional deforma-tion, the local energy release rate depends on the instantaneous cracklength Z(t), the instantaneous crack tip speed Z(£), the applied loading,the geometrical configuration, and the bulk elastic moduli. Then, thestatement of the growth criterion is

G(Z, Z, £, loading, configuration, moduli,...) = F, (7.4.1)

which is in a deceptively simple form. This relation is called anequation of motion for the crack tip because it has the form of anordinary differential equation for the crack tip position l(t) as a func-tion of time, analogous to the equation of motion for a particle inelementary dynamics. Similar statements can be made concerning thegeneralization of Irwin's stress intensity factor criterion to the case ofdynamic crack propagation. In this criterion, the material specificresistance to crack advance is called the dynamic fracture toughness.At any instantaneous crack speed v, the dynamic energy release rateand the dynamic stress intensity factor are related through (5.3.10).Consequently, if either the energy criterion or the stress intensity cri-terion is considered to provide an acceptable postulate for describing


crack growth then either the specific fracture energy or the dynamicfracture toughness characterizes the resistance of the material to crackgrowth, while the other material function is derived from (5.3.10).Specific examples of the equation of motion are available for severalcases of rapid crack advance, and some of these are discussed in theremainder of this section. The nature of the crack growth resistancewill be assumed to be very simple in these illustrations. Qualitativeobservations on the dynamic fracture response of real materials willbe discussed in parallel with the development of equations of motion,and again in Chapter 8.

The way in which a crack tip equation of motion is used dependson the particular approach to a crack growth problem. For example,if a problem is approached analytically as a boundary value problemmodeling some process, then the objective is to determine the cracktip characterizing parameter as a functional of crack tip motion forall possible motions. The role of the growth criterion is then to selectthe actual crack tip motion, according to this criterion, from theclass of all possible motions. On the other hand, if a crack growthproblem is approached by numerical methods, then the objective isto incrementally integrate the field equations while holding the valueof the crack tip characterizing parameter at its critical value. Theequation of motion is not solved directly but, instead, it is appliedincrementally. And if a crack growth phenomenon is approachedexperimentally, then the applied loads and crack motion are measuredor inferred from measurements. In this case the equation of motionitself provides a means for determining the crack growth resistancefrom the measurements.

7.4-1 Tensile crack growth

For crack growth in the plane strain tensile opening mode, or modeI, the dynamic energy release rate G is given in terms of the dynamicstress intensity factor Kj for arbitrary nonuniform motion of the cracktip in (5.3.9). If the energy balance crack growth criterion is adoptedand isothermal conditions are assumed, then G = F, where, as before,F is the specific fracture energy. The specific fracture energy maydepend on crack length or crack speed, and it is viewed as a propertythat characterizes the resistance of the material to crack growth.Suppose that the stress intensity factor is known for arbitrary motionl(t) of the crack tip. For example, it might be expressible as a functionof time £, instantaneous crack length /(£), and instantaneous crack tip


speed l(t). The equation of motion of the crack tip according to theenergy balance growth criterion is then

(7.4.2)

where v and E are the elastic constants and Ai(l) is a universalfunction of crack speed defined in (5.3.11). If F is a function ofI and /, for example, then (7.4.2) provides an ordinary differentialequation for l(t) that can be identified as an equation of motion forthe crack tip under the stated conditions. The relationship (7.4.2) isquite general in the sense that it applies for any case for which theconditions evident in the statement are satisfied. To summarize, therelationship is based on the assumptions of nominally elastic responseof a homogeneous and isotropic material, plane strain opening modeof deformation, a sharp crack tip, a region around the crack tip inwhich the mechanical field is dominated by a stress intensity factorfield, and applicability of the energy balance idea. Validity of (7.4.2)does not hinge on the crack plane being semi-infinite or the body beingotherwise unbounded; it applies without modification for a boundedbody containing an edge crack or an internal crack.

To draw conclusions from (7.4.2) on the motion of a crack tip,some knowledge of the way in which Kj depends on its arguments isrequired. For the particular case of a semi-infinite crack growing at anarbitrary rate in an unbounded body subjected to time-independentloading, it was found in Section 7.3 that Ki depends on time t onlythrough / and /, and that Kj has the separable form of a universalfunction of crack tip speed k(l) times an equivalent stationary crackstress intensity factor. This property is embodied in (7.3.19). It willbe shown subsequently in this chapter that this special property existsfor virtually any transient mode I crack growth situation. That is, anydynamic stress intensity factor for general crack face loading whichresults in mode I deformation has the form

KI(t,l,i) = k{i)KI(t,l,O), (7.4.3)

where Kj(t, Z,0) is the time-independent stress intensity factor thatwould have resulted from the applied loading if the crack tip hadalways been at its instantaneous position represented by I. Virtuallyany crack growth problem, including one with transient boundary

tractions, body forces, and a complex geometrical configuration, canbe reduced through superposition to a situation involving generalcrack face tractions on a semi-infinite crack in an otherwise unboundedbody. Consequently, the equation is relatively unrestricted within theclass of mode I crack growth problems.

In light of (7.4.3), the crack tip equation of motion (7.4.2) can berewritten as

2 . y , , y n . a = MlWf • (7.4.4)

The effects of the bulk properties of the material are included in theelastic constants, the effects of loading and geometrical configurationare represented by Kj(t, Z,0), and the resistance of the material tocrack growth is represented by F. The equation of motion requiresthat the crack must move in such a way that the combination of theseeffects on the left side of (7.4.4) always equals the universal functionof crack speed

« 1 - l/cR (7.4.5)

on the right side, where CR is the Rayleigh wave speed. A graph of g(l)versus 1/CR is shown in Figure 6.9 for v = 0.3. It is evident that thelinear approximation to g in (7.4.5) is very accurate over the wholerange of crack speed 0 < / < CR. This approximation is used wheneverthe function g(l) is required in subsequent analysis. Of course, if theleft side of (7.4.4) is greater than unity then crack growth will notoccur.

Some general conclusions can be drawn from the equation ofmotion in the form (7.4.4). For example, consider the case in whichF is a constant and crack growth is unstable in the equilibrium sense,that is, Kj(t, /, 0) is an increasing function of /. These are features ofthe original crack tip equation of motion modd developed by Mott(1948). If the loading is increased to a level sufficiently great to initiatecrack growth, then the left side of (7.4.4) decreases as / increases and,according to the equation of motion, g(l) must also decrease. Asthe crack length becomes large, the crack tip speed approaches theRayleigh wave speed asymptotically. Because the analysis leadingto the result that CR is the upper limit on crack speed is exact, itmust be concluded that inertial effects are not responsible for thefact that the greatest attainable crack speed for any known materialappears to be considerably less than CR. Thus, although Mott's (1948)qualitative description of crack tip motion on the basis of an energy


criterion has been important to the development of the subject, theapproximations invoked are valid only for very small crack speedand the attempts to quantify this description (for example, Robertsand Wells 1954) are inconsistent with the above exact results. Theagreement between predicted terminal velocities much less than CRand the experimental results is fortuitous. The fact that crack speedsare usually substantially less than CR must be attributed to influencesthat are not taken into account in the model at this level. This topicwill be discussed further in Chapter 8.

As a second general conclusion, consider the form of the equationof motion (7.4.4) when F depends on crack speed /, and possiblyon crack length I. It is noteworthy that the ordinary differentialequation for l(t) is a first-order equation, that is, it does not involvesecond and higher derivatives of l(t). If the analogy between themoving crack tip and an elementary mass particle is recalled, thisobservation implies that the crack tip moves like a massless particle.In other words, the crack tip speed changes in phase with the applieddriving force, whereas for a mass particle the acceleration changes inphase with the driving force. This does not imply that inertia of thematerial is not important in dynamic crack propagation; materialinertia can have a strong influence on momentum transfer in thematerial. Instead, it is an observation on the relationship betweenthe crack driving force and the crack tip motion. This observationis perhaps relevant in considering phenomena such as crack arrest,crack propagation through periodic media or other materials withnonuniform fracture resistance such as weldments, or in situationswhere the crack direction changes abruptly such as in bifurcation. Thefeature of no apparent mass of a crack tip moving rapidly through abrittle material was anticipated in the observations made by Kuppers(1967) on crack growth in glass using the ultrasonic wave fractographictechnique developed by Kerkhof (1970, 1973). By using modulatingultrasonic waves of two different frequencies during crack growth ina glass plate subjected to a large quasi-static load, he was able tomodify the local stress amplitude with the lower frequency wave andto discern the response to the stress amplitude change by means ofthe higher frequency wave. He found that the crack speed changed inphase with the local driving force.

A third general observation on (7.4.4) concerns the separation ofa cohesionless interface. Suppose that flat surfaces of identical elasticbodies are pressed against each other and then, by means of some

local action, the interface is forced to separate very rapidly underplane strain conditions. If the leading edge of the separation regionis regarded as a crack edge with zero specific fracture energy, thensubstitution of F = 0 into (7.4.4) implies that I = CR for the edge. Inother words, interface separation with no energy absorption occurs atthe Rayleigh wave speed of the material. This idea was discussed byPreund (1974b) in an analysis of the problem described in Section 3.2,and it was incorporated into a very sophisticated analysis of diffractionof a stress pulse by a crack by Papadopoulos (1963), who took intoaccount the possibility that the crack faces could be in contact overportions of the boundary.

As a specific illustration of the equation of motion (7.4.4), con-sider the following plane strain situation. A half plane crack existsin an otherwise unbounded isotropic elastic solid on the half planeV = 0, x < IQ. The crack faces are subjected to an opposed pair ofconcentrated forces of magnitude p* at x = 0 that tend to separate thecrack faces, and no other loads are applied to the body. The specificfracture energy is assumed to be a constant Fc, but the initial cracktip is slightly blunted so that an energy release rate of nTc is requiredto initiate crack growth, where n > 1. The parameter n used in thisway may be called a bluntness parameter. The force magnitude p* isgradually increased until the crack tip energy release rate is nFc, andthe crack then grows with G = Fc and with no further increase in p*.The total distance from the crack tip to the load point is denoted by

The equilibrium stress intensity factor for this loading case is2/nl. At the onset of crack growth

henceI <nV F!<rrJn

(7.4.7)

during growth. The equation of motion (7.4.4) during growth is

l/nl0 « 1 - l/cR , (7.4.8)

subject to the initial condition that 1(0) = Zo- The initial speed of thecrack tip is therefore CR(TI — l)/n. As the crack grows, it continuously

decelerates from this speed, and it comes to rest when / = nlo. Thisis also the equilibrium length for a sharp crack subjected to the givenloading. The complete crack tip trajectory, that is, the integral of(7.4.8), is

l(t) = Jo [n - (n - 1) exp(-cRt/nl0)]. (7.4.9)

Note that TIIQ/CR is the characteristic time for the transient response.As a second illustration of the equation of motion (7.4.4), consider

again a crack occupying the half plane y = 0, x < lo in an otherwiseunbounded elastic solid. The crack faces are subjected to pressure ofmagnitude a* over 0 < x < 1$. At time t = 0, the crack begins togrow and the crack tip position is x = l(t) for t > 0. The pressure a*acts over 0 < x < l(i) for t > 0 so that the equilibrium stress intensityfactor due to the applied loading increases with crack length. Thisequilibrium stress intensity factor for any amount of crack growth isgiven in (6.4.40) as iij(Z,O) = 2a*y/2l/n. Suppose that the fractureresistance varies with crack speed according to

r = r c /( /) , /(/) = i + pi/cR, (7.4.io)

where /? is a constant. The fracture resistance of the material increases(decreases) with speed if /? > 0 (/?• < 0), and /3 = 0 is the caseconsidered in the preceding illustration. In the present case, theequation of motion of the crack tip is

- , (7.4.11)CR

where lc = 7rETc/8a*2(l — v2) and the approximation (7.4.5) has beenused. It is evident from (7.4.11) that crack growth will begin only iflo > lc and the initial speed will be I = CR(1$ — lc)/(lo + Plc)- Theintegration of the differential equation subject to the initial condition1(0) = l0 yields

(7.4.l2)

The crack speed approaches CR asymptotically as t —• oo for any /?,but the rate of approach increases as (3 decreases. The acceleration

1.0-

0.8-

0.6-

0.4-

0.2-

0.00.0 0.2 0.4 0.6 0.8 1.0

Figure 7.4. Schematic diagram of the functions g(l) and f(l) for /9 <C —1, plus asmooth function /*(/) that is assumed to approach /(/). For any particular /*(/)the equation of motion can be solved only for speeds I > lp.

to speed CR is instantaneous for (3 = — 1, and it appears that nophysically acceptable solution to the differential equation exists for

This peculiar result on the nature of the solution for (3 < —1can be resolved in the following way. Note that /(/) in (7.4.10) isnegative for —CR/(3 < i < CR. Suppose that the definition of /(/)is extended for (3 < — 1 so that it is given by (7.4.10) for 0 < / <—CR/'(3 and that /(/) = 0 for —CR/(3 < I < CR. NOW, considerthe case when T = rc/*(Z), where /* is the smooth function shownschematically as the solid curve in Figure 7.4. For this case, theequation of motion is satisfied by having the crack begin to grow withinitial speed lp defined by the condition g{lp)/f*(lp) = 1. The cracktip then accelerates from this speed to CR. AS the function /* becomescloser to / , as suggested by the dashed curve in Figure 7.4, the speedlp approaches CR. Thus, it appears that the case with (3 < — 1 isindistinguishable from the case of a cohesionless interface discussedabove.

7.4-2 Fine-scale periodic fracture resistance

In the analysis of dynamic fracture phenomena or in the interpretationof experiments, it is commonly assumed that the crack tip speed


varies monotonically with time over most of its growth. This is thecase if the fracture resistance of the material appears to be spatiallyuniform. Such motion is termed the apparent motion in the presentdiscussion. Suppose that the fracture resistance of the material isnonuniform on a finer scale of observation, so that the crack speedwill actually vary nonmonotonically. This appears to be the case inbrittle fracture of polycrystalline materials or composite materials. Ifa crack is perceived from a macroscopic point of view to be advancingat a uniform rate in a material, but it is known that the materialhas some structure on a smaller scale that leads to local variation ofthe fracture resistance, then what features of the physical process offracture can be discerned at the macroscopic level? This question isconsidered on the basis of a simple model of crack advance througha material with fine-scale periodic resistance, and some implicationsfor the apparent features of the process at the macroscopic level areexamined (Preund 1987b).

Consider rapid growth of a planar crack through a nominallyelastic isotropic material under plane strain conditions. The tensileopening mode of deformation, or mode I, is assumed. The elasticmodulus and Poisson's ratio of the material are E and i/, respectively.Attention is focused on points close to the crack tip compared to thein-plane dimensions of the body in which the crack grows, that is,the case of a semi-infinite crack in an otherwise unbounded body isconsidered. At any instant, the mechanical field surrounding the cracktip is characterized by the stress intensity factor and the speed of thecrack tip. The instantaneous crack length measured with respect tosome arbitrary reference point is denoted by l(t) and the instantaneouscrack tip speed is l(t). It is assumed that the crack grows accordingto an overall energy balance criterion. Thus, the crack tip equationof motion (7.4.4) based on this criterion applies.

On a macroscopic scale of observation, the material appears tohave a constant spatially homogeneous fracture resistance Fo. Whenobserved on a fine scale, however, the material actually has a periodic(not necessarily sinusoidal) resistance to crack growth, say

r = rm 7( /) , (7.4.13)

where Tm is a constant and 7(Z) is a periodic function of crack tipposition with spatial period A, defined for all /. Suppose 7(Z) has the

properties that

max 7(0 = 1, min 7(1) > 0 . (7.4.14)o<z<A w o<i<\ w "

Thus, the constant F m is the maximum value of the nonnegativeperiodic specific fracture energy F.

The actual crack speed as observed at the finer level of observationis determined by examining the crack tip equation of motion (7.4.4)for the periodic specific fracture energy (7.4.13),

The applied stress intensity factor A}(Z,0) = KO is assumed to havelittle variation over distances equal to many times A or during timesequal to many times t\, SO it is taken to be a constant for purposes ofthis discussion. The quantity t\ is the time required for the crack tipto travel a distance equal to one wave length A, and its dependenceon loading level is obtained by integration as

where B is the dimensionless combination of parameters ETm/(l —v2)K% which has the range 0 < B < 1. The extreme values of B —• 1"and B —• 0+ correspond, respectively, to the applied stress intensityfactor at a level just large enough to push the crack tip past the peaksin the fracture resistance and at a level many times greater than thisminimum level.

Consider now the same process at a macroscopic level at whichthe crack tip appears to move along steadily at constant speed, sayvo, under the action of the uniform applied stress intensity factor Ko.This crack speed must be the average speed in the periodic fine-scalefracture resistance, that is,

t\. (7.4.17)

If the crack tip equation of motion is applied at the macroscopic level,then

E r 2 w- V2 Kl CR

where Fo is the macroscopically uniform specific fracture energy. Theparameters on the left side of (7.4.18) are conveniently expressed interms of B, so the ratio of the apparent macroscopic fracture energyto the maximum in the fine scale periodic variation is

r o _ 1 - vo(B)/cR _ 1 - VO/CR ,-A^(Q

m B o\V0)

This is the main result for the apparent fracture energy in terms ofthe details of the periodic variation of the fine-scale resistance and theapplied load level. It is written in two ways to make clear that theratio can be expressed in terms of the load parameter B or the averagecrack speed vo, these two quantities being related through (7.4.17). Arelationship similar to (7.4.53) was derived by Dahlberg and Nilsson(1977) in considering fluctuations in velocity measurements for rapidcrack propagation in metals.

The limiting values of the ratio (7.4.19) for very slow and veryfast crack growth are easily deduced for arbitrary 7(Z). Recall that forvery slow crack growth, B —• 1~. In view of the property (7.4.14)i it isclear that t\ —• oo as B —• 1~. This, in turn, implies that vo(B) —• 0,hence

To /rm -+ 1 as vo/cR -+ 0 or B-> 1 . (7.4.20)

On the other hand, for severely overdriven crack growth, B « 1.Thus, to first order in small values of 1?,

cRtx « / {1 + #7(0} dl = A(l + Bj), (7.4.21)./o

where 7 < 1 is the average value of j(l) over one wave length A. Theuniform macroscopic speed is then approximately (1 — B7y)cR1 so

r o / r m ^ 7 as VO/CR-^1 or B^O. (7.4.22)

1.25 -i

1.00

0.75-

0.50-

0.25-

0.00

l/XFigure 7.5. Periodic variation of the specific fracture energy of a material withposition along the fracture path. The horizontal dashed lines represent the ap-parent fracture energy for the two extreme cases of very slow and very fast crackgrowth represented by (7.4.20) and (7.4.22), respectively.

These extreme cases (7.4.20) and (7.4.22) can be interpreted asfollows. Consider first the case when B is only slightly less than unityor VO/CR <C 1. Suppose that the variation in fracture energy hasthe qualitative features shown in Figure 7.5. The crack tip drivingforce is barely large enough to push the crack tip past the maximain the fracture energy and, according to the equation of motion, thecrack tip moves slowly as it passes a maximum in fracture energy.Between the maxima, the crack tip rapidly accelerates to a relativelyhigh speed and then decelerates as it approaches the next maximumin 7(Z). Of the total time required for the crack tip to traverse awavelength A in fracture energy variation, the crack tip spends mostof its time in regions where the fracture energy is Tm. If this resultis now interpreted in terms of a uniform crack tip speed vo, then theapparent time rate of energy flux into the crack tip is Tmvo for mostof the time. This is the conclusion reached in (7.4.20). Of course, theactual energy absorbed per wavelength A in the fracture process differsfrom the apparent energy absorbed by the amount represented as ashaded area in Figure 7.5. This amount of energy, which is includedin the macroscopic energy flow into the crack tip region, is radiatedoutward from the crack tip as it accelerates and decelerates. The

spectrum of this radiation is dominated by wavelengths on the orderof A and it is not accounted for in the macroscopic crack tip energyflux.

The other extreme case, which is perhaps less interesting froma physical point of view, is when B has a value only slightly greaterthan zero. This is the situation when the driving force is far greaterthan the minimum required to push the crack tip past the maxima inthe fracture resistance. In this case, the crack speed is always nearthe ideal upper limit CR and there is little acceleration or decelerationdue to the variation in the fracture energy. Thus, the crack motionis essentially uniform with the crack tip spending about equal timeat each resistance level within a wavelength. Thus, the macroscopicinterpretation of the process occurring with uniform crack speed vo

leads to the conclusion that the apparent fracture energy is the wave-length average of fine-scale periodic fracture energy, as in (7.4.22).

In general, the difference between the apparent rate of energyflow into the crack tip and the actual energy consumed in the fractureprocess is radiated as high frequency wave motion. The amount ofenergy radiated per unit crack advance is

ER = \ ( {To-Yml{l)}dl, (7.4.23)* Jo

which is the difference in area under the graphs of the apparentfracture energy and the actual fracture energy versus distance.

Although the main results are evident in (7.4.20) and (7.4.22)for arbitrary periodic variation of the specific fracture energy, it isinstructive to consider some special cases of 7(Z). For example, sup-pose that 7(Z) = COS2(TT//A). This variation obviously satisfies theconditions in (7.4.14). The integral in (7.4.17) can be evaluated withthe result that

B 1 + vo/cR '(7.4.24)

It is evident that this special case has the general properties in (7.4.20)and (7.4.22). A second instructive example is the piecewise constantfunction 7(Z) = 1 for 0 < / < nX and 7(Z) = q for n\ < I < A, where0 < n, q < 1. Evaluation of the integral in (7.4.17) for the particular


case with n = q = \ yields

r o = 6(vo/cR) + 2 ^ 4 ^ 4(tio/cfl) + 9(vo/cR)*

Tm 9(vo/cR) - 2 + 3^ /4- 4(vo/cfl) + 9(vo/cH)2 '

Again, the general features already discussed are evident in the re-sult. A piecewise constant fracture resistance was used by Das andAki (1977b) in their study of strength inhomogeneities on a crustalfault in the earth during seismic slip. In such an application, it isanticipated that three-dimensional aspects of the fracture resistancewould significantly influence the results.

It is noted that the general features of the analysis here wouldbe the same if a stress intensity factor criterion for crack growth hadbeen adopted, instead of the energy criterion. One issue raised bythis simple calculation is whether or not the macroscopic fractureenergy of a material perceived through brittle crack propagation ex-periments has a direct relationship with the true surface energy of thematerial. If the fracture energy of the material is inhomogeneous ona fine scale, then the answer would appear to be negative. Instead,the macroscopically perceived fracture energy represents a maximum,rather than an average, of the fine-scale fracture resistance. That is,if the fracture resistance T has a periodic variation in the material,then the maximum of this variation governs crack growth behavior forspeeds near v = 0 but the average of the variation governs behavior forcrack speeds v approaching the Rayleigh wave speed of the material

7.4-3 Propagation and arrest of a mode II crack

In this section, the results of Section 7.3.3 on mode II crack growthat nonuniform speed are coupled with a growth criterion to considerpropagation and arrest of a shear crack due to a spatially varyingstress drop, defined by q(x) in Section 7.3.3, and varying fracture re-sistance in the growth condition (Husseini et al. 1975; Preund 1979a).The discussion of the growth of a tensile mode I crack in Sections 7.4.1and 7.4.2 was based on the assumption that crack growth is governedby an energy growth condition, which was stated symbolically in(7.4.1). It was noted in the introduction to Section 7.4 that similarresults are obtained if a critical stress intensity factor growth conditionis assumed, and the discussion of this section is based on the latter


postulate. Furthermore, in discussing mode I crack growth, it wasassumed that the specific fracture energy was either constant or that itvaried with crack tip speed. In this case, the parameter characterizingfracture resistance of the material is not speed dependent. Instead,it is assumed to depend on position along the fracture path. Witha view toward modeling the earthquake faulting process, this is oneway to account for variability of slip resistance along the fault surfaceas the edge of a shear crack advances. Thus, it is assumed that thecrack tip equation of motion follows from the condition that

Ku(h 0 = hi(i)Kn(l, 0) = C(l), (7.4.26)

where C(l) is a function which represents the resistance of the materialto mode II fracture at the position x = / on the fracture plane (referto Figure 7.6).

Under the conditions described, a running fracture can be ar-rested because either the driving force is reduced to a subcriticallevel or the fracture resistance is increased to a supercritical levelas the crack advances. These two cases are considered qualitativelyfor the equation of motion based on the critical stress intensity factorcriterion. For mode II crack growth, this equation of motion is

kn(l)J- / --PL dx = C(l), (7.4.27)

subject to the initial condition 1(0) = IQ. It is assumed that themagnitudes of the various parameters involved are appropriate forfracture initiation at crack position x = Zo-

Consider first the case in which the critical stress intensity factoris a constant, independent of position along the fracture path, but inwhich the applied stress distribution q(x) (the stress drop) decreasesto negative values on the crack faces as the crack advances. This situ-ation is represented schematically in Figure 7.6a. The linear decreasein q(x) implies that the integral in (7.4.27) will first increase with I toa local maximum, and it will then decrease. The equation of motionrequires that the product of &//(/) and if//(Z,O) is a constant. Thus,so long as Kn(l,0) increases, kn(l) must decrease and the crack tipaccelerates to ever larger crack speeds, as shown in Figure 7.6a. Like-wise, as X//(Z,0) decreases beyond x = /C, the crack tip decelerates.

7-4 Crack tip equation of motion 409

Kn(l,0)

Figure 7.6. A representation of the variation along the fracture path of the physicalquantities appearing in the crack tip equation of motion (7.4.27) for a mode IIcrack, (a) The critical stress intensity factor is constant C, but the crack facestress distribution q(x) decreases with distance along the fracture path, (b) Thecritical stress intensity factor increases along the fracture path, but the crack facestress distribution is uniform.

Growth is arrested at x = Za, where the equilibrium stress intensityfactor has been reduced to the level satisfying Kn(la,0) = C.

A similar illustration of the crack tip equation of motion is repre-sented schematically in Figure 7.6b for the case when the stress dropq(x) = qo is uniform over the faces of the expanding crack but thecrack tip encounters ever increasing fracture resistance C{1) as it ad-vances. Under these conditions, the equilibrium stress intensity factorJK//(Z,O) increases parabolically with increasing crack length. Withreference to the equation of motion (7.4.27), the factor &//(/) will de-crease and the crack will accelerate so long as the ratio C{l)/Kn{h 0)decreases. Thus, the crack tip will accelerate to the position x = lc onthe fracture plane at which C'(lc)/C(lc) = Kij(lc,0)/Kn(lc10), wherethe prime denotes the derivative with respect to the crack tip position

coordinate. For points on the crack path beyond x = Zc, the cracktip decelerates. According to the equation of motion, crack growthis arrested at the point x = la where C(la) = lfjj(Za,O), as shown inFigure 7.6b.

7.4-4 A one-dimensional string model

The simple string model for dynamic crack growth introduced inSection 5.5.1 is one of the few cases in which a complete mathematicalsolution of the field equations can be found, and it provides the rareopportunity to see all aspects of a dynamic crack growth event ina simple framework. It was noted there that the string model isequivalent to a shear beam model of crack propagation in a doublecantilever beam specimen, and the results obtained through analysisof the string model shed light on observed crack propagation phenom-ena in the double cantilever beam specimen (Freund 1977, 1979b).

The notation of Section 5.5.1 is adopted here without modifi-cation. Recall that a stretched elastic string lies along the positivex-axis, that is, along 0 < x < oo. The string has mass per unitlength p and characteristic wave speed c. The transverse deflectionis w(x,t), the elastic strain from the undeflected configuration is7(x,£) = dw/dx, and the transverse particle velocity is rj(x,t) =dw/dt. Initially, the string is free to move in the interval 0 < x < loand it is bonded to a rigid, flat surface for x > IQ. A particularboundary condition on w(0,t) is introduced here, namely, that

w{0,t) = wo, (7.4.28)

where WQ is a constant. With this boundary condition, there is noenergy exchange between the deformable body and its surroundings.Consistent with this boundary condition of fixed end deflection, theinitial deflected shape is w(x,0) = wo(l — x/lo). If the string isinitially at rest, then the initial conditions in terms of strain andparticle velocity are r/(x,0) = 0 and 7(#,0) =70 = —Wo/Zo- At timet = 0, the string begins to peel away from the surface, so at somelater time t > 0 the free length is 0 < x < l{t). The field equationsand the condition to be satisfied at the moving crack tip are given as(5.5.1) and (5.5.2), and the crack tip energy release rate is found in(5.5.5) to be

2 i 2 2 t ) 2 . (7.4.29)

x = l(t)

x

Figure 7.7. The x,£-plane for the string problem showing the path of the cracktip x = l(t). At any instant of time, the string is free over 0 < x < l(t) and it isbonded to a rigid foundation for x > l(t).

If a symmetrical structure, such as a double cantilever beam, is con-sidered then both halves would contribute the same energy supply andthe factor of one-half in the energy release rate expression would beabsent. With reference to the foregoing discussion concerning the in-gredients essential to deriving an equation of motion for the crack tip,this energy release rate expression provides a crack tip characterizingparameter as a functional of crack motion for all possible motionsprovided that 7(/~, i) can be found for arbitrary motion of the cracktip from the field equations. Thus, 7(Z~,£) is determined next.

The governing partial differential equations (5.5.1) form a first-order hyperbolic system of equations. The characteristic curves arestraight lines in the x,t-plane with slope c"1 or - c " 1 , and the char-acteristic relations are

drj = 0 along dx/dt = ±c. (7.4.30)

Local momentum balance requires that discontinuities in strain andparticle velocity, denoted by [7] and [77], satisfy

[77] ± c[7] = 0 along dx/dt = ±c. (7.4.31)

A sketch of the x,£-plane is shown in Figure 7.7 and values of strainand particle velocity can be determined at any time and any pointon the string by a straightforward application of the characteristicrelations and jump conditions for arbitrary crack tip motion withspeed in the range 0 < I < c. Only certain pertinent results areincluded here.


If the crack begins to grow with a nonzero speed I A , then dis-continuities in 7? and 7 are generated and these propagate along thecharacteristic line AB in Figure 7.7. These quantities jump from theirinitial values of 7/ = 0 and 7 = 70 to the values 77 = — Z*A7O/[1 + W c ]and 7 = 70/[I + W c ] across AB. The discontinuities reflect from thefixed end at B and then propagate back toward the crack tip alongBC. At an arbitrary point P in the region bounded by ABC, thestrain and particle velocity have the values

J Q 7 O _ P _ 7oP _

where the subscripts indicate the points in the #,£-plane, accordingto Figure 7.7, at which the various quantities are evaluated and PQis a segment of a characteristic line.

The discontinuities propagating along BC reflect from the mov-ing crack tip at C and propagate back toward the fixed end alongCD. If R is any point in the region BCD then

R= 70[JT-is\ R=

where UR, RS and TV are segments of characteristic lines. Thesolution procedure can be continued to indefinitely large times, butthe results obtained so far are sufficient for present purposes. Next,an equation of motion for l(t) is derived.

The crack propagation criterion that is adopted here is the energycriterion according to which the crack must propagate in such a waythat the energy release rate G is always equal to a material specificfracture energy F. At first, this fracture energy is assumed to beconstant, but the possibility that it depends on crack tip speed isconsidered subsequently. Furthermore, suppose that the crack tipis initially blunt, that is, that the quasi-static energy release raterequired to cause the onset of growth, which is Go = c2p7o/2, exceedsthe level of G required to sustain growth of the sharp crack, which isF. The relationship between the initiation level and the propagationlevel of the fracture energy is conveniently expressed through the valueof the bluntness parameter n = Go/T > 1. Given a value of n, therelation n = GQ/T actually specifies the end displacement WQ which

must be imposed to initiate the process at t = 0. For subsequent timet > 0 the crack must propagate with G = F, or else Z = 0 with G < F.

By letting the observation point P in (7.4.32) approach the crackedge, it is evident that the crack tip strain appearing in the energyrelease rate expression (7.4.29) is given by

at any point in the segment AC in Figure 7.7. The energy growthcriterion then implies that

2(1 + l/cf *

for all time until the reflected wave overtakes the crack tip. If theparameter 70 is eliminated in favor of the equivalent parameter n,then the simple result that

*- = 4 (7.4.36)c n + 1 v J

follows immediately. Thus, the energy criterion implies that the crackspeed is constant in the interval AC and its value depends only onthe initial bluntness parameter. It is noted in passing that if a growthcriterion based on a critical value of the force Q in (5.5.6) were usedinstead of the energy criterion, then the crack tip speed would alsobe found to be constant. The force criterion is the counterpart ofthe stress intensity factor criterion within the context of this simpleone-dimensional model.

What happens after the reflected wave overtakes the crack tip atpoint C in Figure 7.7? The two crack speeds £5 and IT that appearin (7.4.33) are both equal to the constant value (7.4.36), hence in theregion BCD,

rj = 0, 7 = v^F/npc2 (7.4.37)

for the actual motion. At time t = tc, the body is therefore entirelyat rest and the crack tip strain is below the level required to meet thecrack growth criterion (7.4.35) for a positive crack speed. Because thefields (7.4.37) represent an equilibrium solution, it is concluded that

the crack growth is arrested at point C. From the geometry of thex, t-plane the arrest length of the crack is seen to be lc = nlo, whichis significantly larger than the equilibrium value of the length y/n forend displacement wo. From the fact that the instant of crack arrestcorresponds to the instant at which the reflected wave BC overtakesthe crack tip, it is evident that He = lc ~ lo and that etc = lc + 'o-If tc is eliminated then a relationship between the crack speed up toarrest and the arrest length is obtained in the form

l < 7 4 3 8 >

It is instructive to compare the actual dynamic energy releaserate as a function of crack length with the corresponding equilibriumenergy release rate for fixed end displacement w$. The equilibriumrate is simply pc2/y2/2 = nT(lo/l)2. The dynamic energy release rate,on the other hand, has the value nT for / = 1$. It then drops abruptlyto F as the crack begins to grow, and this value is maintained untilarrest of the growth process. Upon arrest, the energy release rateagain drops abruptly to T/n when I = l$n, the crack length at arrest.The two energy release rate variations have three points in common.Because the initial and final fields are equilibrium fields, the twoenergy release rates must coincide for the corresponding crack lengths,as indicated in Figure 7.8. The intermediate intersection point alsohas some significance. It occurs at I = loy/n and the energy releaserate there is the equilibrium value for the end displacement w$. If thecrack tip were initially sharp and the end displacement were increasedvery slowly to its final value of WQ, then the final energy release ratewould be F at crack length / = lo\/n. These features seem to capturethe essence of data on rapid crack propagation and arrest in doublecantilever beam specimens of a brittle plastic obtained by Kalthoff,Beinert, and Winkler (1976, 1977), although the quantitative detailsare beyond the capabilities of the model.

The reason why the crack tip propagates beyond this equilibriumlength in the dynamic growth situation can be found in the wavepropagation character of the mechanical fields. Initially, the crackgrows as though the material that is strained to the level 70 extendsindefinitely far behind the tip. If this were indeed the case, then therewould be a large reservoir of energy available to drive the crack growthprocess, and the crack advances accordingly. The wave generated at

n

1

1/n

G/T

\ ^ dynamic growth

IS\ / equilibrium growth

1 Vn nFigure 7.8. A graph of the normalized energy release rate G/T versus amountof crack growth l/lo for the string problem. The darker line is the actual energyrelease rate variation for crack growth governed by a constant critical energyrelease rate criterion, whereas the lighter curve is the equilibrium energy releaserate for the imposed end displacement WQ.

the onset of growth (AB in Figure 7.7) which is ultimately reflectedback toward the crack tip (BC in Figure 7.7) carries the informationthat the body is bounded and that the energy supply is limited. Bythe time that this information reaches the crack tip, however, the tiphas already advanced beyond the equilibrium position for the availableenergy supply. The crack tip mechanical field is subcritical thereafter,and the crack growth process arrests abruptly with a correspondingdrop in energy release rate, as shown in Figure 7.8.

Because of the simplicity of the solution in this case, both the to-tal strain energy Utot and the total kinetic energy Ttot of the specimencan be calculated as explicit functions of crack length or, equivalently,as functions of time. Expressions for the energy densities are given in(5.5.3) and the result of this calculation is

UtsL

Uo= 1 2n

-tot " r (7-4.39)

for 0 < t < lo/c, and

Utot n2 + 2n - 1 n +

Uo

2n2

(n-1)2

2n2n —

(7.4.40)

1.0-

0.8-

0.6-

0.4-

0.2-

0.0- 1 1 1 I '

/u0

1 '

^IT/UO

1 ' 1 ' 1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure 7.9. The variation of potential energy, kinetic energy and fracture energywith amount of crack growth for the string problem, as given by (7.4.39) and(4.4.40), for the case of n = 4.

for lo/c < t < (n + l)/o/c, where UQ = \pc2^l$ is the strain energystored in the body at crack growth initiation. The time lo/c corre-sponds to point B in Figure 7.7 and the time (n + 1)IQ/C is the timeof crack arrest.

The variation of energy with crack length is shown in Figure 7.9for the case of n = 4. For this case, the crack speed up to arrest is3c/5, the arrest length is 4/Q, and the crack length at time lo/c is 8/o/5.The interpretation of the energy variations in Figure 7.9 in terms ofthe wave propagation process is straightforward. At the instant offracture initiation, the strain 7 is reduced at the crack tip from 70 tothe value appropriate to satisfy the crack growth criterion, and thisreduction in strain implies a reduction in strain energy. Because ofthe rate of this reduction in 7, however, the inertia of the materialcomes into play. The value of 7 is reduced behind the wavefrontAB traveling toward the fixed end of the string. The strain energydecreases and the kinetic energy increases as the wave engulfs moreand more of the free length of the body. Because energy is drawnfrom the body at the crack tip, the decrease in Utot is not balancedby the increase in Ttot. When the stress wave reflects from the fixedend x = 0, the fixed grip condition requires that it must do so in justthe right way so as to cancel the particle velocity. Thus, as the wave


reflects back onto itself, the particle velocity rj is reduced to zero andthe strain 7 is further reduced behind the wavefront BC travelingback toward the crack tip. The kinetic energy decreases after wavereflection, and the strain energy also continues to decrease but at arate slower than before. Until the reflected wave overtakes the cracktip, the tip moves as though it was advancing in an unbounded body.The strain level carried by the reflected wave brings the crack tipstrain to a value below that required to sustain crack growth, andthe motion is abruptly arrested. It should be noted that the factthat the kinetic energy is exactly zero at the instant of arrest is aspecial feature of this model. For more realistic models of the doublecantilever beam specimen, the kinetic energy typically increases andthen decreases with time, but not necessarily to zero.

Up to this point in the analysis of the one-dimensional model, ithas been assumed that the specific fracture energy is independent ofcrack speed. The extension of the main ideas to the case of a fractureenergy that is speed dependent is straightforward, and some resultsare included here. A particular speed dependence that models thebehavior of some materials is

( \ m—1

where Fc is a constant and m > 1. The fracture energy is constant ifm = 1.

Suppose that the same crack growth situation that was examinedabove is considered again for the speed-dependent fracture energy(7.4.41). If the energy growth criterion is applied then it is foundthat the crack will grow with the constant speed

(7.4.42)

where a^ = (1 — 2/m)k~1/m, k = 1,2,3,..., from time t = 0 until thetime at which the wave generated at the initiation site reflects fromthe end x = 0 and overtakes the moving crack tip. The parametern is a measure of bluntness of the initial crack tip, as before. Atthe instant when the reflected wave overtakes the crack tip, the bodyis completely at rest and the energy release rate takes on the value

n i-2/mpc Thus, if 1 < ra < 2 then this value of energy release rateis less than or equal to the minimum level necessary to sustain crackgrowth. The process is arrested with a crack length of n1/mZo. Asbefore, the energy release rate decreases abruptly at arrest except forthe special case ra = 2.

If ra > 2, then the crack propagation criterion implies that thecrack grows with the constant speed

• n«2 _ i£QJ, fiaiin < / < n^ai^"a2^ln

1 lZ*\ (7.4.43)— for n^ai+a2Ho /0 ,

— = <c

and so on. The crack speed is piecewise constant in time, and itdecreases gradually with each wave reflection to zero as time becomesvery large. The crack length itself approaches a finite limit, however,that is given by

lim l(t) = Z0n ( a i+ a 2 +- ) = loVn, (7.4.44)

which does not depend on ra. It is interesting to note that the cracklength (7.4.44) is exactly the length that would result from slow crackgrowth under equilibrium conditions to the final state.

As a final point on this problem, the condition that the endx = 0 of the specimen is held rigidly fixed is relaxed to introducethe possibility of flexibility in the loading apparatus. A commonmanner of loading the double cantilever beam specimen, which wasused to motivate this simple model, is to gradually force a pair ofwedges between loading pins that pass through the specimen armsnear the free ends. If these wedges are extremely stiff compared to theflexibility of the arms of the specimen then the assumption of fixedgrip loading is quite realistic (unless the pins lose contact with thewedges at some time in the process). This might be the case for hardsteel wedges and pins used to load a brittle plastic specimen. If thespecimen is also very stiff, then the flexibility of the loading apparatuscan influence the crack growth process. Thus, the boundary conditionof constant displacement at the end x = 0 is replaced by the conditionthat the transverse shear force at the end — pc27(0, t) varies linearlywith the transverse deflection at the end of the specimen, for example,

-/9C27(0, t) = k[w0 - w(0, t)) - pc27o , (7.4.45)

where k is the stiffness of an elastic spring. This boundary conditioncorresponds to the case of a linear elastic spring of stiffness k thatis compressed to a length WQ by a force of magnitude pc27o thatmaintains the initial equilibrium deflection of the string and that actsaccording to (7.4.45) in a direction transverse to the string at x = 0.

The condition (7.4.45) is a nonstandard boundary condition forthe application of the method of characteristics to determine the solu-tion for arbitrary motion of the crack tip, but it can be converted intoa standard form in the following way. The result of time differentiationof (7.4.45) is

| ^ ) . (7.4.46)

With reference to Figure 7.10, for any point R on the boundary suchthat ts <tR < tn the method of characteristics yields

where 70 and n are related as before. Substitution of rjR into (7.4.46)yields a differential equation for j R that has the solution

7* = \(n - 1) exp {-(* - Z0/c)/r} + ll 70/n (7.4.48)

which satisfies the initial condition 7(0, lo/c) = 70, where r is thecharacteristic time pc/k. The condition (7.4.48) is a standard bound-ary condition, and the method of characteristics can be applied todetermine the values of 7 and 77 at point P in terms of the unknowncrack length and speed there. Imposition of the constant energyrelease crack growth criterion then yields the differential equation for

/(*) _ 1 - m(t)c ~ l + m(t) ' ( 7 4 4 9 )

m(t) = n [l + 2(n - 1) exp {[/(*) + /„ - ct]/cr}] ~2

for tc <t <tE, subject to the initial condition that / = nlo when t =(n + l)lo/c. This equation has been integrated numerically for n = 4,


X

Figure 7.10. The x,£-plane for the string problem, showing the path of thecrack tip moving according to the critical energy release criterion when loadingcompliance is taken into account.

and the qualitative features of the result are shown in Figure 7.10.Of particular interest is the additional amount of crack growth due toloading apparatus flexibility. It was found that this amount of crackgrowth is 0.31Z0, 0.06/0, and 0.01/0 for cr/l0 = 0.5, 0.25, and 0.10,respectively.

With reference to Figure 7.10, the observation that the crackspeed increases at C before decaying to zero can be explained in thefollowing way. When the unloading wave AB reaches the boundaryx = 0, the particle velocity of the load point can change instan-taneously without affecting the applied force. This is a feature ofdead weight loading with an indefinitely large energy supply, andthe crack tip responds to the waves carrying this information byaccelerating. After some time has elapsed, however, the load pointactually displaces and the load level decays with characteristic timer. This information is also communicated to the crack tip by meansof reflected waves, and the crack tip responds by decelerating. If theloading is very stiff compared to the specimen, then r is very smalland the reduction of the crack speed to zero is very rapid.

The differences between the crack tip motion predicted here andthe motion found in Section 7.4.1 point out the importance of theshape of the elastic body in which a crack is growing on the nature

of the crack tip motion. For example, in the case of the string model,the crack grows at constant speed and then is arrested with a jumpin speed. In the illustration presented near the end of Section 7.4.1,on the other hand, the crack speed decreases gradually to zero fromits initial value. Furthermore, the final length of the crack in thestring model is found to be much greater than the equilibrium lengthfor the imposed end displacement, whereas in the plane strain casethe final length is found to be identical to the equilibrium length.These differences are apparently due to the way in which waves areradiated from, and subsequently reflected back onto, the moving cracktip. In one-dimensional structures, loads can be transferred over longdistances without dilution of their eflFect and waves are guided alongthe length. In the plane strain case, on the other hand, the influence ofan applied load becomes increasingly diffuse with increasing distancefrom its point of application and waves are continuously scatteredrather than guided.

7.4-5 Double cantilever beam: approximate equation of motion

The crack tip energy release rate for rapid crack growth along thesymmetry line of a body in the double cantilever beam configurationwas determined in Section 5.5.2 and an application to steady statecrack growth was considered in Section 5.5.3. Here, a transient crackgrowth problem for the double cantilever beam configuration is ana-lyzed. The analysis is approximate; no exact solution is available forcrack growth at an arbitrary rate in this configuration. With referenceto the notation established in Section 5.5, a solution is required forthe equation governing the transverse deflection w(x,t) of a uniformbeam

over the interval 0 < x < /(£), where E is the elastic modulus, / isthe area moment of inertia of the beam cross section, A is the areaof the beam cross section, and p is the mass density of the material.The solution of this differential equation is subject to the boundaryconditions

«;(/,<) = 0, -^(Z,<) = 0 (7.4.51)ox

at the crack tip. Appropriate end conditions must be specified at

x — 0, say the conditions of fixed displacement with zero moment,

w(0, t) = w0 , * - T ( 0 , *) = ° (7.4.52)

for 0 < £ < oo. Initial conditions on deflection and particle velocityare also required; for example,

« ( . , 0 ) - f (n -s f + j j f ) , f=(.,0>-0 (7.4.53,

for 0 < # < l(t). The function /(£) is a continuous, monotonicallyincreasing function that specifies the position of the crack tip as afunction of time.

The crack tip dynamic energy release rate for this crack growthprocess is, from (5.5.14),

( 7 . 4 . 5 4 )

where M(l~,t) is the total internal bending moment in the beam atthe crack tip end. This form takes into account energy contributionsfrom both halves of the specimen, and it is the form of the energyrelease rate that is strictly consistent with Bernoulli-Euler beam the-ory. As given by (7.4.54), G is the mechanical energy released from thebody per unit crack advance, taking full account of the width of thespecimen, and it has the physical dimension of force. The difficulty inwriting an exact equation of motion for this case is that the quantityM(l~ ,t) which appears in (7.4.54) cannot be determined for arbitraryl(t). Thus, an approximate approach based on Hamilton's principleis considered. The problem represented by the equation (7.4.50), theboundary conditions (7.4.51) and (7.4.52), and the initial conditions(7.4.53) is examined to make the development specific, but the sameapproach can be applied to a range of problems.

The kinetic energy and strain energy of the symmetrically de-formed double cantilever beam specimen at any time t are

= / pA(w,t)2dx, Utot= / EI(w,xxfdx (7.4.55)

Jo Joltot

in terms of quantities already defined. The familiar factor one-halfis absent to account for both halves of the symmetric body. For anygiven crack tip motion /(£), Hamilton's principle of stationary ac-tion (Gelfand and Fomin 1963) requires that the differential equation(7.4.50) and the boundary conditions (7.4.51) and (7.4.52) must besatisfied if the space of admissible deflections includes all continuousand sufficiently differentiate functions w(x, t) which satisfy the kine-matic or essential boundary conditions. If the crack length l(t) is alsoviewed as a generalized coordinate, then the additional condition that

d_ fd(Ttot-Utot)\ _ d(Ttot-Utot) = _Q ( ? 4 5 g )

is obtained, where the dynamic energy release rate is the "dissipative"generalized force that is work-conjugate to l(t). \iw(x,t) is a solutionof the complete boundary value problem represented by the partialdifferential equation (7.4.50) plus its initial and boundary conditionsthen (7.4.56) reduces to (7.4.50). As already noted, however, suchan exact solution is not available for arbitrary /(£), so the followingapproximate approach based on variational methods is included here.

The approach is developed for the particular crack growth prob-lem introduced above, and extensions to other cases are evident. TheBernoulli-Euler idealization of a beam does not admit propagationof sharp-fronted stress waves, so a modal approach appears to bereasonable for some purposes. If the tracking of discrete wave pulsesis important in the analysis, then the Bernoulli-Euler idealization itselfis deficient and a higher order beam idealization is required. Supposethat at any time t the deflection of the beam is given by

/ 3 \ °°

w(x,t) = Y ( 2 " 3 ? i ) + W F ) + ^ a n { t ) +»(xm')' (7-4-57)

where l(t) is the instantaneous crack length, an(t) is an unknownamplitude, and </>n(x/l) is the nth normal mode shape for free vibra-tion of a beam subjected to homogeneous boundary conditions. Inthe present case, these homogeneous boundary conditions are that0n(O) = 0n(°) = ^n(l) = 0U1) = ° for anY n> w h e r e t h e Pr ime

denotes differentiation with respect to the argument. This form issuggested because it satisfies all boundary conditions, it satisfies the

initial conditions if an(0) = 0 for all n, and the set of functions <j>n('),n = 1,2,..., is a complete orthogonal set. Thus, it appears that anysolution can be represented by (7.4.57), where the functions an(-) aredetermined from

d fd(Ttot-Utot)\ d(Ttot - Utot)

according to Hamilton's principle.Although this approach offers some promise for analytical treat-

ment of problems in this class, it has not been extensively developed.Some preliminary steps were taken by Benbow and Roessler (1957)and Gilman, Knudsen, and Walsh (1958) for crack growth underequilibrium conditions, and by Burns and Webb (1970a, 1970b) fora case of dynamic crack growth somewhat different from that beingconsidered here. In all three cases, the analysis was carried out inorder to interpret fracture propagation experiments in which a wedgewas driven into the end of a relatively long single crystal to inducecleavage crack growth. Other perspectives on this class of problemsare provided by Berry (1960a, 1960b) and Sih (1970). The presentanalysis is restricted to the case where an = 0 for all n in (7.4.57) and,in this sense, it is similar to the analysis of Burns and Webb (1970a,1970b).

If the deflection is assumed to be given by the first term in(7.4.57) then the kinetic energy and the potential energy expressionsin (7.4.55) can be evaluated to yield

— WTj > Utot - p { )

If the energy balance crack growth criterion G = T is assumed, whereF is the constant specific fracture energy, then substitution of Ttot —Utot into (7.4.56) yields the differential equation for l(t)

l~ 2 I + 4 pAP 12 wlpA' [(A-™}

Just prior to the onset of crack growth, Ttot = 0, Utot = Uo, and theenergy release rate is 9WQEI/IQ, and it is assumed that this is equal tonT to account for some initial bluntness of the crack tip. In this sense,


2.5 -r

2.0-

r 1.0

oJ/KFigure 7.11. The crack tip motion predicted by the approximate equation ofmotion (7.4.61) for the double cantilever beam configuration when n = 4 andIQ = 1.45/i. The kinetic energy and strain energy of the body versus time areshown as the dashed curves.

n is essentially the same bluntness parameter that was introduced inSection 7.4.1. The differential equation then becomes

I / 2 105 El fl I

This is the equation of motion for the crack tip, subject to the initialconditions that 1(0) = IQ and 1(0) = 0. The latter condition followsfrom the assumption that the total kinetic energy of the system isinitially zero. The equilibrium length of the crack is evident from theequation of motion when / = / = 0, namely, / = Ion1/4. Based on theresults of the previous section, crack growth beyond this length dueto inertial effects might be anticipated.

The differential equation (7.4.61) has been integrated numericallyfor n = 4 and Zo = 1.45/i. The result is shown in Figure 7.11 in thedimensionless form of l(t)/lo versus cQt/lo, where co = y/E/p. Theequilibrium length of the crack in this case is 1.414 Zo, but it is evidentthat it has grown to a final length of about 2.2 Zo- The reason for thisdynamic overshoot can be seen from the equation of motion (7.4.61).Initially, the term in parentheses on the right side is positive for any


n > 1 and 1(0) = 0. The crack therefore accelerates from its initialposition. As it does so, the term in parentheses on the right sidedecreases while the first term increases. This situation persists untilthe term in parentheses becomes sufficiently negative to override thepositive contribution from the first term and cause the crack tip todecelerate. Because 12/Io is nonnegative, this can happen only forI > Ion1/4. Also shown in Figure 7.11 are the variations of kineticenergy and strain energy of the body with time. The similarities withthe results in Figure 7.9 are obvious.

This same double cantilever beam problem has been analyzed byKanninen (1974) and by Hellan (1981), who both based their analyseson the Timoshenko beam idealization. Furthermore, both used a fullnumerical approach to solve the governing differential equations andboundary conditions. The values of final crack length and averagecrack tip speed obtained here are similar to those obtained by meansof the more complete numerical analyses for comparable values of thesystem parameters n and h/lo. Finally, it is noted that generalizationof the analysis of this section to cases for which the specific fractureenergy F depends on crack position or crack speed is straightforward.

7.5 Tensile crack growth under transient loading

In this section, the matter of calculating the transient stress intensityfactor history as a feature of the stress distribution is discussed for thecase of a crack subjected to time-dependent loading. For background,the situation that is equivalent to the present case except for the crackedge being stationary was analyzed in Sections 2.4 and 2.5, and thecase of constant speed crack growth with an arbitrary delay time wasanalyzed in Section 6.5. The notation established in these sectionsis adopted here without modification. The plane wave loading casediscussed next is the simplest example of nonuniform plane straincrack growth under transient loading conditions (Freund 1973a). Ananalysis which generalized this result to include most of the results ofthis chapter as particular cases was outlined by Kostrov (1975) whoconsidered all three modes of crack deformation fields and crack faceloading varying in both position and time.

7.5.1 Incident plane stress pulse

Consider once again a body of elastic material which contains a halfplane crack but that is otherwise unbounded. In the present instance,

7.5 Tensile crack growth under transient loading 427

the material is stress free and at rest everywhere for t < 0, and thecrack faces y = ±0, x < 0 are subjected to uniform normal pressure ofmagnitude a* for t > 0. For points near to the crack face compared todistance to the crack edge, the transient field consists only of a planewave parallel to the crack face and traveling away from it at speed c«j.Near the crack edge, on the other hand, the deformation field is morecomplex. It was shown in Section 2.5 that the stress intensity factorfor this dynamic loading situation is

= lim y/27cxcryy(x,0,t) = CjcrV27rcd* , (7.5.1)x—•()+

where Q = ^/2(1 — 2v) /TT(1 — v) in terms of Poisson's ratio v.Although this stress intensity factor was obtained for the case of crackface loading, it was given an interpretation in terms of an incidentplane stress wave in Section 2.5.

Next, suppose that the crack edge remains stationary at x = 0for some time after the load begins to act, but at some later timet = r > 0 the crack edge begins to advance in the x-direction atconstant speed v < CR. The time r is called the delay time. An exactstress intensity factor solution for this situation has been provided inSection 6.5 with the result that

= lim ^2n(x - vt) ayy(x,0,t) = k{v)Ci<r* y/2ircdt . (7.5.2)x>vt+x

This result, which appears as (6.5.16), differs from the correspondingresult for a stationary crack only through the dimensionless factor&(w), which is the universal function of instantaneous crack tip speedintroduced in (6.4.26).

The stress intensity factor expression equivalent to (7.5.2) for thecase of arbitrary crack tip motion x = l(t) was obtained by solvinga sequence of special boundary value problems (Freund 1973a). Thekey step was showing that if the constant velocity crack describedin Section 6.5 suddenly stops, say at position x = Za, then a normalstress distribution is radiated out along the crack plane ahead of thecrack tip, behind a point traveling with the shear wave speed of thematerial, that is precisely the normal stress distribution that wouldhave resulted if the crack tip had always been at its instantaneousposition x = la and if the crack had been struck by the incident stresspulse at time t = 0. This stress distribution is denoted by p[(x — la)/t]

in Section 6.5, indicating the fact that it is a homogeneous functionof distance ahead of the crack tip x — la and elapsed time t after thestress pulse reaches the crack plane. In other words, it is the stressdistribution appropriate for the given incident wave and the new cracklength, and it is independent of the fact that the crack tip had beenin motion in the past. It follows that, upon sudden stopping of thecrack tip, the stress intensity factor jumps immediately from the valuegiven by (7.5.2) to the stationary crack value given by (7.5.1).

With this strong result in hand, it becomes a simple matter toconstruct the stress intensity factor history for any crack motion byapproximating the actual crack speed history by a piecewise constantcrack speed history, as was done for the case of time-independentloading in Section 7.3. A piecewise constant crack speed history canbe found that is arbitrarily close to the actual history, in terms ofboth position and speed. It is concluded that the instantaneous valueof the stress intensity factor for any motion of the crack tip dependson crack motion only through time and the instantaneous value ofcrack speed l(t). This dependence has the separable form

= lim y/2n(x-l) cryy{x,0,t) = k(l)Ci<r*\f2ncdt , (7.5.3)x*l+

where k(l) is again the universal function of crack tip speed given in(6.4.26) and shown graphically in Figure 6.10.

This result thus broadens the class of situations for which thetransient stress intensity factor has this separable form, as has alreadybeen suggested in (7.4.3). In Section 6.5, the stress intensity factorresult for an incident stress pulse carrying a jump in tensile stress isgeneralized to the case of an incident tensile pulse of arbitrary stressprofile. This generalization extends without modification to the caseof arbitrary crack tip motion. Consequently, for arbitrary crack tipmotion described by l(t) and for an incident plane stress pulse withgeneral tensile stress variation, the stress intensity factor history is

KI(tJ) = k(l)KI(t10). (7.5.4)

That is, at any instant of time, the stress intensity factor has the valuegiven by the universal function of crack tip speed k(l) times the valuethe stress intensity factor would have if the crack tip had been at theinstantaneous position on the fracture plane x = l(t) for all time. A

0.0

Figure 7.12. The solid line shows the stress intensity factor versus time for thecase of a rectangular tensile stress pulse of magnitude a* and duration t* strikinga long crack in a body initially at rest, as shown in the inset, assuming that crackgrowth does not occur. If the crack grows according to a critical stress intensityfactor criterion Ki = Kic, then U and ta indicate the times of initiation and arrestof crack advance.

number of specific examples of stress intensity history Kj(t, 0) for astationary crack tip have been introduced in Section 2.7.

To conclude this section, a specific example of stress wave loadingis considered. In particular, the equation of motion for the cracktip for the case of loading by a rectangular stress pulse is obtainedand its implications for dynamic crack growth according to a stressintensity factor growth criterion under these conditions are studied.This configuration has been realized experimentally by Ravichandranand Clifton (1989). Consider a tensile rectangular stress pulse ofmagnitude a* and duration t* normally incident on a traction-freecrack. If the crack does not extend, then it is clear from (7.5.1) thatthe crack tip stress intensity factor will increase in proportion to y/tfor 0 < t < t*, and it will decrease in proportion to y/t — y/t — t* fort* < t < oo. The time dependence of Kj without extension is shownin Figure 7.12 as the solid line. The largest value of stress intensityfactor without extension is Qa*y/2ircdt* , so the crack will grow onlyif Ci<T*y/27rcdt* > Kjc, the critical value of stress intensity factor forthe material being studied. It is assumed that the crack does indeed

0.4 n

\ 0.2-

Figure 7.13. The crack tip motion corresponding to the case of stress wave loadingof a crack as shown in Figure 7.12.

grow, and the simplest possible crack growth criterion, namely,

l(t) = 0 with JK}(<,0) < KIc

or

i(t) >0 with KI(tJ) = KIc

(7.5.5)

is adopted. It is observed that the growth criterion is first satisfied attime t = ti, where

This time is the initiation time for the process, and the crack willgrow for t > U. The growth criterion imposes the condition that

KIc =if U < t < t*

* if t* < t < tQ

(7.5.7)

where the function k was introduced in (7.5.4) and

(7.5.8)

is the arrest time, that is, the time at which the decreasing Kj(t, 0)passes the value Kjc\ see Figure 7.12. The relationship (7.5.7) is anordinary differential equation for the position of the crack tip l(t) asa function of time and, as such, it is an equation of motion for thecrack tip. It is another example of the kind discussed in Section 7.4.This equation is subject to the initial condition 1(0) = 0 and it appliesduring ti <t <ta>

The features of the solution of (7.5.7) are evident. The cracktip begins to move at time t = U, it accelerates for U < t < £*, itdecelerates for t* < t < ta, and it arrests at time ta- This response isdepicted in Figure 7.13. Although these results have been obtained bystudy of the case of a semi-infinite crack in an otherwise unboundedbody, they apply for other situations as well. For example, for a crackof finite length or for a crack near a boundary of the body, if the arresttime ta is less than the time required for waves from the far end of thecrack or from the remote boundary to significantly perturb the cracktip field, then these results apply without modification. For a finitelength crack the important time is the transit time for a Rayleighwave along the length of the crack.

7.5.2 An influence function for general loading

The results for the dynamic stress intensity factor for arbitrary motionof the crack tip in (7.3.19) and (7.5.4) are special cases due to therestrictions on the loading that were imposed. In the first case, theloading was allowed to have a general spatial variation but it wasrestricted to be time-independent. In the second case, the loadingwas allowed to have general time variation but it was restricted tohave the spatial uniformity of a plane wave.

In spite of the fact that the loading was fundamentally differentin the two cases, the final mathematical forms of the stress expres-sions were quite similar. Burridge (1976) reasoned that these specialforms (7.3.19) and (7.5.4) had some common origin, and he deriveda function, which he termed the influence function for the problemclass, that gave the dynamic stress intensity factor directly as a linearfunctional over general applied crack face traction. Burridge's unify-ing analysis was based on the central idea developed in Section 5.7 onthe weight function method, and only his final result is included here.

Suppose that the crack grows in the plane strain opening mode inthe x-direction. The body is stress free and at rest for time t < 0, andthe crack tip moves according to x = l(t) for t > 0. The crack faces aresubjected to normal traction ayy(x, 0, t) = -p(x, t) for x < l(t), t > 0.Burridge (1976) showed that the dynamic stress intensity factor forthis situation is given by

,0), (7.5.9)

where the stress intensity factor Ki(t, Z,0) for the same crack faceloading p(x, t) applied to a stationary crack for which the tip is located

at x = l(t) is given in (5.7.37). In the present notation, this stressintensity factor expression is

, /, 0) = f p(x, t1) h[x - l(t), t - *'] dx dt', (7.5.10)

where A is the area of the £,£'-plane for which l(t) — Cd(t — t1) <x < l(t) — cji(t — t1) for any fixed t > 0. Furthermore, the functionof crack speed k(l) in (7.5.9) is the universal function defined in(6.4.26). While this result is complicated and difficult to apply inmany situations, the main point is that the properties of the dynamicstress intensity factor results in (7.3.19) and (7.5.4) are common toall problems. That is, for any situation of mode I crack advance atan arbitrary rate under general loading the dynamic stress intensityfactor is the universal function of instantaneous crack tip k(l) timesthe stress intensity factor appropriate for a crack with a stationarytip at x = l(t).

7.6 Rapid expansion of a strip yield zone

As a relatively simple application of the results on elastodynamiccrack growth at nonuniform rate discussed previously in this chapter,the growth of a strip yield zone or cohesive crack tip zone is considered.The concept of a crack tip cohesive zone was introduced in Sections5.3.2 and 6.2.4 as a device for relaxing the singular stress distributionon the crack plane ahead of the crack tip. The main idea is to thinkof the elastic crack as extending some distance ahead of the physicalcrack tip. A self-equilibrating cohesive stress acts over this distance,tending to close the crack. The size of the region over which thecohesive stress acts is determined by the condition that the net stressintensity factor at the tip of the extended crack due to both the appliedloading and the cohesive stress is zero. The same idea is applied hereto estimate the rate of growth of a crack tip plastic zone under stresswave loading. The cohesive stress is identified with the tensile yieldstress of the material, and the length of the cohesive zone is the plasticzone size.

Consider a crack tip region under plane strain mode I conditions.The material is elastic-ideally plastic with tensile flow stress ao. Aplane tensile pulse with wavefront parallel to the crack plane carries a

7.6 Rapid expansion of a strip yield zone 433

jump in stress from zero to a* < cro, and the pulse strikes the crack attime t = 0. It was shown in Section 2.5 that, if the material is elastic,then the tensile stress on the crack plane is square root singular fort > 0 with a stress intensity factor that increases in proportion tot1/2. If the stress carrying capacity of the material ahead of the tip isconstrained to be cro, then a zone in which this stress level has beenreached must grow out from the stationary crack tip for t > 0. Thelength of this zone will be denoted by rp(t). The way in which rp(t)varies with time will be determined from the elastodynamic crackgrowth solutions already in hand.

Consider a crack in the plane y = 0 that initially occupies thehalf plane x < 0 under plane strain conditions. The material in thecrack tip region is initially stress free and at rest. At time t = 0 aplane stress pulse traveling at speed Cd in the y-direction strikes thecrack. The stress pulse carries a jump in the stress component ayy

from zero to cr* > 0. Instantaneously, the extended crack tip beginsto move in such a way that its position along the x-axis for t > 0 isx = rp(t). The physical crack tip remains stationary. The crack facesremain free of traction for — oo < x < 0 and t > 0, but

(Tyy(x,0±,t) = ao, axy(x,0±,t) = 0 for 0 < x < rp(t). (7.6.1)

Thus, whereas the incident loading pulse tends to separate the crackfaces, the cohesive traction acting over 0 < x < rp(t) tends to holdthe crack closed.

In general, the tensile stress ayy on the crack plane ahead ofthe crack tip is singular as x —• r +. The value of the stress intensityfactor for arbitrary rp(t) is obtained by superimposing the results from(7.5.3) for loading by a stress pulse of magnitude cr* and from (7.3.19)for loading by a time-independent crack face traction- of magnitudeao. If rp(t) here is identified with the crack growth rate / in the citedexpressions, then the net stress intensity factor is

Ki = (T*CIk{rp)y/2TTCdi - 2aok(rp)^2rpt/ir (7.6.2)

for arbitrary rp(t), where k is the universal function of crack tip speedgiven in (6.4.26) and Q = y/2(l-2v)/ic(l - i/).

If the strength of the material on the plane y = 0 ahead of thecrack tip is indeed limited, then the stress distribution cannot be

singular there. Thus, Kj = 0 for t > 0, which provides a condition fordetermining rp(t). In the present case, setting Kj = 0 in (7.6.2) leadsto the condition that

cd -\ao) 2 ( 1 - ^ <7-6-3)

Thus, the rate of growth of the plastic zone size rp is constant forthe special case of an incident step stress pulse. The stress analysesof Section 6.5 leading to the stress intensity factor results used herewere based on the assumption that / < c#, SO the present result issubject to the restriction that rp < CR. This limitation is of littleconsequence, however, because the right side of (7.6.3) is less thanTT"1 for all values of Poisson's ratio in the range 0 < v < 0.5 and alla* < ao. For v = 0.3, for example, (7.6.3) gives rp/cd = 0.004, 0.102for cr*/(Jo = 0.1, 0.5, respectively.

For an incident pulse with stress profile more complicated thana simple step, the corresponding rate of growth of the plastic zoneis also more complicated. In general, this rate is not constant. Forexample, if the profile of the loading pulse is such that the stress at apoint increases linearly in time as the wave passes, then the first termon the right side of (7.6.2) is replaced by a term proportional to t3/2

rather than to t1/2. Consequently, it is found that the rate of increaseof the plastic zone is rp ~ t2 rather than a constant.

Note that the validity of the result (7.6.3) does not depend inany way on the plastic zone size being small compared to c&t. If itis the case that rp <C c<j£, however, then the result can be written ina somewhat different form. If rp <C c^t, which is the case if a* <0.2cro, then the plastic zone is completely enclosed within the cracktip singular field associated with the stationary physical crack tip atx = 0 and the incident pulse of magnitude cr*. This singular field ischaracterized by the stress intensity factor

K; = a^QV^c^t (7.6.4)

and the plastic zone size can be written as

1.6 Rapid expansion of a strip yield zone 435

This result is an elastodynamic equivalent of the equilibrium resultwhich is commonly known as the small-scale yielding result for plasticzone size based on the strip yield model (Rice 1968). Even thoughthere is no physical length in the present case with which the plasticzone size can be compared, it is still required that the plastic zonesize must be small compared to the distance from the crack tip to thenearest wavefront. The latter distance here is on the order of c3t. Fora material with a shear wave speed of about cs = 2000 m/s = 2 mm//xsthe distance from the crack tip to the closest wavefront when t = 10 /zsis 20 mm. Thus, the small-scale yielding approximation is valid onlyif rp is less than about 2 mm at this time.

Another feature of the process of development of a crack tipplastic zone that is significant is the crack tip opening displacement.Within the framework of the strip yield model being discussed here,the crack opening displacement is the relative displacement of oppositefaces of the crack at the physical crack tip. If this total displacementis denoted by £*(£), then

St(t) = uy(0,0+, t) - Uy(0,0", t) (7.6.6)

for mode I deformation. For the strip yield zone model discussedabove, an expression for 6t(t) can be worked out for the case ofstep stress wave loading. The expression involves rather complicatedintegrals that cannot be expressed in terms of elementary functions,however, and the implications for fracture initiation under dynamicloading are difficult to extract. The corresponding result for antiplaneshear deformation has been obtained by Achenbach (1970c), and thisresult is included here. The antiplane shear problem with cohesivezone has also been studied by Glennie and Willis (1971).

Consider the mode III crack problem with a strip yield zone thatcorresponds to the mode I situation studied above. The geometry ofthe physical system is unchanged. The incident wave with wavefrontparallel to the crack plane is now a horizontally polarized shear wave.It propagates at speed cs in the ^/-direction, and it carries a jump inshear stress ayz from zero to r*. The wave arrives at the crack planeat time t = 0, whereupon a strip yield zone begins to grow from thecrack tip in the rr-direction. The cohesive shear traction in the yieldzone has magnitude ro.

This problem can be solved for an arbitrary rate of growth of thestrip plastic zone rp(t) by means of the integral equation approach

developed in Sections 2.4 and 6.2. Indeed, a number of the steps inthe analysis can be constructed directly from the discussion in thesesections. Only certain pertinent results are included here. First of all,it is found that the rate of growth of the plastic zone is constant foran incident step wave, with

This result shows that rp/c8 depends on T*/TO for mode III deforma-tion in the same way that it depends on a*/cro for mode I deformation.Furthermore, the numerical values of the coefficients in the two casesare similar. This comparison provides some hope that the crackopening displacement rates in the two cases will be similar as well,but there is no basis for direct comparison.

The crack opening displacement for the case of mode III is

St(t) = uz(0,0+, t) - uz(0,0-, t). (7.6.8)

For the problem at hand, the elastodynamic solution yields the resultthat

St(t) = i£^tan-1(->) (7.6.9)7T H \TOJ

for t > 0. This result is valid for any value of r* in the range 0 <r* < ro or for plastic zone size in the range 0 < rp < c8t. Theopening displacement scales with the accumulated displacement dueto elastic strain cstr* //i and it depends on T*/TQ only through thefactor tan~1(r*/ro).

A simple ductile fracture initiation condition is that the crackwill begin to grow when the crack opening displacement achieves acritical material specific value, say 6^. In light of (7.6.9), this criterionprovides a relationship between time to fracture initiation tf and theintensity of applied loading in the dimensionless form

(7.6.10)7T » dcr \_TO \TOJ]

For example, the dimensionless term on the right side of (7.6.10) isabout 11.5 for T*/TO = 0.3. Thus, for a material for which /x/r0 = 200,

7.7 Uniqueness of elastodynamic crack growth solutions 437

cs = 2mm//iS and 6cr = 0.1 mm the predicted fracture initiation timetf is about 90 /is. For such a short initiation time, the magnitude of ro

should probably be adjusted to account for the influence of materialstrain rate on the flow stress.

Finally, the particular form of the mode III results for small-scaleyielding are examined. If rp <C cst, which is the case when r* is lessthan about 0.25 ro, then the strip yield zone is completely embeddedwithin the elastodynamic crack tip singular field. The stress intensityfactor for this field is

K;n = 2r* y/2cat/n , (7.6.11)

from (2.3.18). Then, if r* is eliminated from the expression for plasticzone size in favor of Kfn,

IT (K* Y= 8 \~)

(7.6.12)

which gives the plastic zone size in terms of the transient stressintensity factor. Likewise, the crack opening displacement has thesimple expression

«,-££. (7.6.13)

Both of these expressions are consistent with the corresponding equi-librium results.

7.7 Uniqueness of elastodynamic crack growth solutions

A solution of the equations of linear elastodynamics with suitableboundary conditions is unique according to a classical theorem due toNeumann (Achenbach 1973). An extension of the theorem to includeunbounded domains is given by Wheeler and Sternberg (1968). Ineach case, the statement of the theorem specifies the function classesof which the rectangular components of stress (jy, of displacement Ui,and of body force fa are members. In particular, these functions arerequired to be sufficiently smooth so that the governing differentialequations are valid everywhere in the body and that the divergencetheorem can be applied as needed. In many cases, however, the most


common smoothness conditions are violated due to concentrated loadson the boundary of the body, the existence of discrete wavefrontscarrying discontinuities or singularities in certain field quantities, orgeometrical features that result in stress concentrations. Some ofthese cases have received careful attention. For example, a version ofthe uniqueness theorem that takes into account concentrated loads isstated and proved by Wheeler and Sternberg (1968). At wavefrontsthat carry discontinuities in stress and particle velocity the govern-ing differential equations must be replaced by kinematic and dy-namic jump conditions. If the wavefronts carry algebraic singularitiesin stress and particle velocity, which is common in two-dimensionalfields, the requisite conservation laws are satisfied if the strengthsof the algebraic singularities satisfy the same conditions as jumpdiscontinuities. The uniqueness theorem can be extended to includepropagating discontinuities or algebraic singularities on wavefrontsin the manner discussed for acoustic fields by Friedlander (1958).The case of discontinuities in elastodynamic fields was considered byBrockway (1972).

The purpose here is to consider modifications of the statementand proof of the elastodynamic uniqueness theorem which are requiredto include crack growth solutions within its range of applicability.Attention is limited to plane elastodynamic solutions for a body ofhomogeneous and isotropic elastic material of limited extent. Exten-sion to unbounded regions or to three-dimensional fields can be carriedout as in the discussion by Wheeler and Sternberg (1968). The cracksurface may be curved, but the crack tip trajectory is supposed tohave a continuously turning tangent.

Rectangular coordinates are introduced in such a way that theparticle displacement is in the xi, #2-plane, as shown in Figure 7.14.The region of the plane occupied by the body is R and the outerboundary of R is S. The inner boundary of R is made up of thecrack faces So except near the ends of the crack where the boundaryis augmented by vanishingly small loops S\ and 52 surrounding thecrack tips. The shape of each loop is arbitrary, but the shape is fixedwith respect to an observer moving with the crack tip. The outerboundary S is subjected to traction and/or displacement boundaryconditions which result in a traction distribution % on S. The facesof the crack are free of traction for simplicity.

The stress and displacement fields satisfy the differential equa-


Figure 7.14. Schematic diagram of a two-dimensional body containing a rapidlygrowing crack. The small loops surrounding each crack tip are fixed in size andshape, and they translate with the crack tips.

tions

+Pfi =

ij = Cijkiuk,i in i?,

Tjifij = Ti on S,

in R,

(7.7.1)

where p is the material mass density, Cyw is the array of elasticconstants, and n* is the unit vector normal to the boundary of iZ,directed away from R. The dot denotes material time differentiationand the notation is consistent with Section 1.2.1 for the rectangularcomponents of tensors.

It was shown in Section 4.3 that the near tip fields for crackgrowth have universal spatial dependence in a local crack tip coordi-nate system for each of the three modes of deformation. These modesmay coexist, in which case the local fields are the sums of the fieldsfor the individual modes. The only feature of the crack tip field thatvaries from case to case is the elastic stress intensity factor, or thevarious stress intensity factors for mixed mode conditions. The totalenergy flow into the crack tip as it advances through the material isgiven in terms of the stress intensity factors by the generalization ofthe Irwin relationship in (5.3.10). This rate of energy flux into thecrack tip plays a central role in the proof of uniqueness of crack growthsolutions. The analysis is carried out under the assumption that thetotal mechanical energy of the body, kinetic energy plus potentialenergy, is finite, which implies that the mechanical energy densitymust be integrable. The motion of the crack tip is arbitrary, except


that the crack tip speed is assumed to be less than the Rayleigh wavespeed of the material.

Under the stated assumptions it can be established that, for anygiven motion of the crack tips, the equations of elastodynamics haveat most one solution for which the total mechanical energy is finite(Freund and Clifton 1974). The elastic moduli are assumed to be suchthat the mechanical energy density is a positive definite functional ofthe elastic field. The procedure by which the assertion is establishedis based on the Neumann proof, but it differs in the respect that thechange of energy due to crack growth must be taken into account.

Suppose that two solutions exist for a particular crack growthproblem, each satisfying the same initial and boundary conditions.The difference between these two solutions is also a solution of thegoverning differential equations (7.7.1) without body force and subjectto homogeneous boundary and initial conditions. If the inner productof the momentum equation (7.7.1)i with the velocity u* is formed andthe result is integrated over the region i?, then

/ idR = 0. (7.7.2)R

If the first term in the integrand is rewritten as (aijiii),j —(TijUi,jand the divergence theorem is applied in light of the homogeneousboundary conditions, then

/ (TijTijiiidS — / (JijUi,jdR— I piliUidR = 0. (7.7.3)Js1-\-s2 JR JR

The order of time differentiation and integration over R can be inter-changed, provided that the flux of the integrand through the time-dependent boundary of R is taken into account as in (1.2.41). Thus,

pUiiiidR=— / \pUiiiidR- IJR dt JR Js!

/ GijUi,j dR= — / \(JijUi,j dR- IJR

dt JR JS

(7.7.4)

where vn is the velocity of any point on either Si or S2 in the localdirection of n». In view of these relationships, (7.7.3) becomes

- / [(TijUjUi + ^((TijUi.j +piiiUi)vn] dSJS!+S2

+ li I \mu%dR+l: I \<TijUiij dR = O. (7.7.5)at JR at JR

In the limit as Si and S2 are shrunk onto the crack tips, the first termin (7.7.5) becomes the crack tip energy flux (5.2.7) for the particularcase of elastic response. The second and third terms are the timederivatives of the total kinetic and potential energy, respectively, andthese energy rates are denoted here by ttot(t) and Utot(t)- Thus, inthe limit as Si and S2 are shrunk onto the crack tips, the equation(7.7.5) is

ftot(t) + Utot(t) = 0. (7.7.6)

For all time up until the instant that the crack begins to extend,F = 0 and the standard uniqueness theorem applies. In this timerange, Ttot + Utot = 0. The instant that the crack tip begins to extend,F{t) > 0 and therefore ttot + Utot < 0 from (7.7.6). This implies thatthe value of the total mechanical energy is decreasing from zero or,in other words, it is taking on negative values. But this violates thebasic hypothesis on the positive definiteness of the mechanical energy.Thus, (7.7.6) is valid only if F(t) = 0, which implies that either thecrack is not extending or that the stress intensity factor differences arezero. In either case, the standard uniqueness theorem leads directlyto the desired result.

It is noted that for crack speed in the range between the Rayleighwave speed and the shear wave speed of the material, the energy fluxF(t) is negative for singular mode I and mode II fields. Consequently,with reference to (7.7.6), the difference solution for crack speeds inthis range may correspond to increasing mechanical energy Ttot + Utot,which does not violate any of the assumptions made. Thus, unique-ness of solutions cannot be established by means of the Neumannargument for speeds in this range. It is unclear whether or not thisobservation has any implications of practical consequence.

8

PLASTICITY AND RATE EFFECTS

DURING CRACK GROWTH

8.1 Introduction

In this chapter, the role of material inelasticity is considered, in theform of irreversible plastic flow, dependence of the material responseon the rate of deformation, or microcracking. The study of issues indynamic fracture mechanics concerned with these effects is at an earlystage. Consequently, the sections are not integrated to any significantdegree. Instead, each section is intended to give an impression ofthe present stage of development of analytical modeling in the areascovered.

8.2 Viscoelastic crack growth

The study of crack growth in a linear viscoelastic material has beenmotivated primarily by interest in modeling the fracture process inrelatively brittle polymeric materials, although other materials maybe idealized as linear viscoelastic under some circumstances. Thereare numerous specific linear viscoelastic models available for stressanalysis, but only some general properties are considered here forcrack growth analysis. Time-dependent material response, of the kindon which the theory of viscoelasticity is based, may be of interestin the analysis of fracture phenomena at two different levels. Onthe one hand, the bulk properties of the body in which the crackis propagating are important in determining the way in which the

442

8.2 Viscoelastic crack growth 443

effect of applied loads is transferred to the crack tip region or theway in which stress is redistributed due to crack growth. If this isthe case, then the entire body is assumed to exhibit time-dependentresponse, the crack is assumed to be sharp, and the nature of thecrack tip mechanical fields is represented by a fracture characterizingparameter. Experimental data on crack propagation in Homalite-100which suggests that this effect is significant is described by Dally andShukla (1980). In any case, this is the domain of linear viscoelasticstress analysis in dynamic fracture. On the other hand, if the primaryfocus is on the influence of material rate effects in the crack tip regionitself, then this region is better modeled as a cohesive zone with time-dependent cohesive stress, as in Section 7.6, or, better still, as a regionof viscoplastic flow, as in Section 8.4. Linear viscoelastic boundaryvalue problems motivated by the former objective have been solved,and some of these are described here. The solutions of these problemsare very difficult to obtain, in general, and they have had a limitedimpact on the interpretation of experiments in dynamic fracture or onthe advance of the underlying theory. Consequently, the discussion isbrief.

The most common aspects of linear viscoelastic response are em-bodied in the convolution integral relationship between stress c^ (p, t)and strain e^ (p, t) at a spatial point represented symbolically by p andat time t in an isotropic material, namely,

Gij(p, t) = 6ij I X(t - r)ekk(p, r)dr+ / 2/i(t - r)ei<7-(p, r) dr,J— oo J — oo

(8.2.1)

where X(t) and n(t) are the two relaxation functions that characterizethe response of the material. Each function is defined for all posi-tive values of its argument t, and each is assumed to be a positive,monotonically decreasing function of its argument. If the material ishomogeneous, then these functions are independent of p, as suggestedby the form of (8.2.1).

Consider the special case of homogeneous deformation, that is,when the strain e^ (p, t) is independent of p. Furthermore, supposethat the strain is zero up to time t = 0, and it is a constant e°-thereafter. Thene^p,*) = <*•«(«) and ( p , t) = \{t)e°kk6ij+2ti(t)e°j.The functions X(t) and fi(t) characterize the transient stress relaxation

444 8. Plasticity and rate effects

process under these conditions, thus accounting for the use of thedescriptive term relaxation functions.

The instantaneous response to the suddenly imposed strain de-pends only on Ao = A(0+) and /lo = /i(0+), the limiting values ofthese functions as t —• 0+. These values are called the instantaneouselastic moduli, the short-time moduli, or the glassy moduli of thematerial. Likewise, the stress response at a long time after the straine^ is imposed on the material depends only on Aoo = A(oo) andMoo = /x(oo). These limiting values are called the long-time modulior the rubbery moduli. It is customary to refer to the material asa fluid if //QO = 0 and as a solid otherwise. A characteristic timerepresentative of the rate of decay of a relaxation function from itsinstantaneous value to its long time value is commonly introducedas the relaxation time for the response. These basic properties arethoroughly developed by Christensen (1982).

Consider the situation of advance of a sharp crack through aviscoelastic material under conditions of essentially two-dimensionaldeformation. The plane of deformation is the x,?/-plane. It wasshown in Section 4.6 that, for a class of elastic-viscous materials, theasymptotic field near the crack tip is dominated by the instantaneouselastic response. The same result carries over to the case of a linearviscoelastic material under quite general conditions. To see that thisis indeed the case, consider the integral relation in (8.2.1) under thecondition of steady motion in the ^-direction at speed v. The stressand strain depend on x and t only through the combination £ = x — vt.If the variable of integration r is replaced by ry, where r\ = x — VT,then it is found that

(8.2.2)where q = (77 — £)/v. An integration by parts for any fixed y yields

-2fjL00eij(ooJy)-v 16ij A' (q)ekk(r],y)dr]

ft-v-1 2n'(q)eij(n,y)dri, (8.2.3)

Joo

where the prime denotes the derivative of a function with respect toits argument. It is evident from this relationship that if the stress andstrain are singular at £ = y = 0, then the nature of this singular fieldis established by the relationship

*«(& V) ~ xotkk(Z,y)6ij + 2jxoe«(f,y) (8.2.4)

alone, provided that A'(0+) < oo and //(0+) < oo. If the strain hasan integrable singularity as £ —• 0 then the integral term in (8.2.3)will be nonsingular if A'(0+) and //(0+) are finite. This condition onthe limiting values of the relaxation moduli can be weakened withoutaffecting the main result. This matter is not pursued here, but it isconsidered in some detail by Kostrov and Nikitin (1970).

In any case, if the nature of the crack tip field is established by(8.2.4) then the main features of this singular field are evident fromthe elasticity analysis of Chapter 4. The essential part of the stress-strain relationship is identical to that in linear elasticity with theelastic moduli replaced by the instantaneous or short-time viscoelasticmoduli. Consequently, the near tip mode I field for crack growth atspeed v in a linear isotropic viscoelastic material is given by (4.3.10)with c8 and cj replaced by

/Ao + 2/io> (8.2.5)

respectively, where Ao and /x0 are the short-time viscoelastic modulicorresponding to the elastic moduli A and //.

The result that the asymptotic crack tip field for a dynamicallygrowing crack in a linear viscoelastic solid has the same form as theelastic crack tip field, provided that the limiting values of relaxationmoduli as t —• 0 replace the elastic moduli, leads to a somewhatparadoxical conclusion (Kostrov and Nikitin 1970). Suppose that theform of the Griffith energy release rate condition or the Irwin stressintensity factor condition appropriate for dynamic crack growth isadopted. Then the only material properties appearing in the growthcondition are the short-time moduli, so the growth condition doesnot reflect in any direct way the time-dependent nature of the ma-terial response. (For specified applied loads on the boundary of theviscoelastic body, the time-dependent bulk material properties could


influence the way in which load is transferred from the boundary tothe crack tip region. The relaxation properties could enter indirectlyin this way.) This seems to be a consequence of the assumptionthat the crack tip is sharp, or that fracture occurs at a point. Forany finite nonzero crack speed, the physical separation process at asharp crack tip is infinitely fast and only the instantaneous moduliof the material are relevant. This paradox is resolved by introducinga separation zone of finite extent in the form of a cohesive zone ofsome length, say A, ahead of the moving crack tip. The idea of acrack tip cohesive zone has been developed in Sections 5.3, 5.4, 6.2,and 7.6. With a cohesive zone accompanying the crack tip, the crackopens gradually against the resistance of some cohesive stress withinthis zone. The time required for a crack tip to advance a distanceequal to A at crack speed v introduces a process time A/v that mustbe compared to a characteristic relaxation time of the material todetermine if the separation is "slow" or "fast." A growth criterionbased on a given level of cohesive stress and a fixed level of cracktip opening displacement leads to a relationship between crack speedand crack driving force that does include the relaxation propertiesof the material. This has been demonstrated in detail for mode Icrack growth under equilibrium conditions by Kostrov and Nikitin(1970) and Christensen (1982), and for dynamic antiplane shear crackgrowth in a general linear viscoelastic material by Walton (1987) andin a Maxwell material by Goleniewski (1988). Some of the importantfeatures of these models arise in the study of the growth of a crackwith a viscous cohesive zone in an otherwise elastic body in Section8.5.

A main objective of stress analysis in viscoelastic crack growthproblems is to establish the dependence of the crack tip stress in-tensity factor on the configuration of the body in which the crack isgrowing and on the boundary conditions. Quasi-static crack growthin a viscoelastic material is discussed by Christensen (1982) and byKanninen and Popelar (1985). Few exact solutions are availablefor the case of dynamic crack growth in a viscoelastic solid. Willis(1967b) analyzed the situation of steady propagation at speed v of asemi-infinite antiplane shear crack in an infinite body idealized as astandard linear solid. The only loading on the body was an arbitraryintegrable distribution of shear traction acting on the crack faces andfixed with respect to an observer moving with the crack tip. By meansof the Wiener-Hopf technique, Willis (1967b) showed that the stress

intensity factor is independent of crack speed and that it depends onthe applied loading in the same way as in the corresponding quasi-static elastic problem if v < c5OO, whereas the stress intensity factordepends on crack speed and material parameters for csoo < v < cso.As before, c8o = \Jnolp and csoo = y^oo/p? so that cso and csoo arethe elastic wave speeds corresponding to the short-time and long-timemoduli, respectively. The general results obtained by Willis have beenextended by Walton (1982) to the case of general linear viscoelasticresponse.

Several plane strain steady state dynamic viscoelastic problemswere considered by Atkinson and Coleman (1977). They assumedthe material to be characterized by a single relaxation time r, forexample, /j,(t) = (/x0 — ^oo)^"^7" + /ioo? and the configuration of thebody to be characterized by a single length parameter h. Then, underthe assumption that the parameter vr/h, where v is the steady statecrack propagation speed, is small compared to unity, Atkinson andColeman (1977) used the method of matched asymptotic expansionsto obtain mode I stress intensity factors. They considered the steadygrowth of a semi-infinite crack along the centerline of a very long stripof width /i, which is the viscoelastic equivalent of the elastic problemdiscussed in Section 5.4. The strip is loaded by imposing a uniformnormal displacement on each edge. They found that the viscoelasticstress intensity factor is identical to the elastodynamic result (5.4.8)with the elastic moduli replaced by the long-time equivalent viscoelas-tic relaxation moduli. These results were subsequently refined byPopelar and Atkinson (1987), who formally solved the strip problemby means of the Wiener-Hopf technique, following the approach ofNilsson (1972) for the elastic problem. They obtained expressions forthe stress intensity factor in terms of sectionally analytic functionsthat arise in the Wiener-Hopf factorization process by application ofthe appropriate Abel theorem (1.3.18). The factors were available inthe form of complex integrals which had to be evaluated numerically.

Finally, two studies that represent departures from the situationof steady growth of a sharp crack in a viscoelastic body are mentioned.Both studies are based on the antiplane shear mode of crack advance.The first is that reported by Atkinson and List (1972) concerningthe transient problem of the sudden onset of constant-speed crackgrowth at a certain instant of time in a material described by thestandard linear solid material model. A uniform traction acted on theentire newly created crack surface. They obtained the transient stress


intensity factor history for several cases, and they showed that thestress intensity factor increases indefinitely or approaches a limitingvalue, depending on the ratio of crack speed to the long-time elasticwave speed. In the second study, Walton (1987) introduced a cracktip cohesive zone with rate-independent cohesive traction within theviscoelastic body. For growth of a crack with a cohesive zone througha quite general viscoelastic material, he determined the net energyflow into the cohesive zone in terms of crack speed and materialparameters. Details were presented for both a power law materialand a standard linear solid.

8.3 Steady crack growth in an elastic-plastic material

The discussion in Chapters 6 and 7 of rapid crack advance througha material was based on the assumption of nominally elastic materialresponse. This assumption led to the notion of stress intensity factoror energy release rate as a crack tip field characterizing parameter, asdiscussed in Chapters 4 and 5. These ideas, in turn, when consideredin light of experience with rapid crack advance in real materials, ledto the proposal that the actual value that the stress intensity factoror the energy release rate assumes during dynamic crack growth, sayas a function of crack speed, can be interpreted as representing theresistance of the material to crack advance. These material-specificresistance measures were termed dynamic fracture toughness and dy-namic fracture energy in Section 7.4. With a representation of crackdriving force in hand, as either the applied level of stress intensityfactor or energy release rate, along with a description of crack growthresistance as a constitutive postulate, a balance equation emerges thatwas termed the crack tip equation of motion in Section 7.4. Theseideas collectively provide a framework for prediction of crack advancein a body of known fracture resistance subjected to specified loads,or for extraction of fracture resistance information from a laboratoryexperiment in which both driving force and crack motion are measuredor inferred. The theoretical framework is relatively complete, in thissense.

The purpose here is to focus on a phenomenon at a finer scaleof observation. In particular, it was noted above that the level ofresistance to crack advance that is exhibited by a material may dependon the instantaneous crack tip speed. For materials that do notundergo a transition in fracture mode with increasing crack tip speed

8.3 Steady crack growth in an elastic-plastic material 449

250-

200-

150-

100-

50-*

double cantilever beam specimen

three point bend specimen

: ' ' *

0 200 400 600 800 1000 1200

v (m/s)Figure 8.1. Dependence of dynamic fracture toughness on crack speed for AISI4340 steel (Zehnder and Rosakis 1989).

and that have a relatively low rate of strain hardening in the plasticrange, the dynamic fracture toughness Kd is often reported to dependon crack tip speed v in a characteristic way. An illustration of thisdependence for mode I is shown in Figure 8.1 from the work of Zehnderand Rosakis (1989) for rapid crack growth in a 4340 steel. Otherexamples of this kind of dependence are described by Hahn et al.(1975) and Hahn, Hoagland, and Rosenfield (1976), by Kobayashiand Dally (1979), by Bilek (1978, 1980), by Rosakis, Duffy, andPreund (1984), and by Angelino (1978), all for 4340 steel in a hardcondition. The detailed description of material condition for eachseries of experiments is given in the original articles. Still otherexamples are reported by Dahlberg, Nilsson, and Brickstad (1980)and Brickstad (1983a) for a high strength carbon steel, by Kanazawaet al. (1981) for a ship hull steel, and by Shukla, Agarwal, andNigam (1988) for 7075-T6 aluminum and 4340 steel. Most of thesedata are summarized and compared by Rosakis and Zehnder (1985).The loading conditions and methods of measurement leading to thetoughness-speed data varied widely among the experimental projects.Similar dependence of dynamic toughness on crack tip speed has alsobeen reported for rapid crack propagation in hard plastics, but thetrends in this case show features that are not yet fully understood(Ravi-Chandar and Knauss 1984c). In any case, such materials are


not suitably modeled as elastic-ideally plastic materials, which is thematerial model upon which the discussion of this section is based.

The most significant feature of the dependence in Figure 8.1 isthe increasing sensitivity of dynamic toughness to crack speed withincreasing speed. Although this sensitivity might be attributed, atleast in part, to strain rate dependence of the material response,it is noteworthy that the feature persists even for materials whichappear to exhibit little strain rate dependence in their bulk response.Furthermore, the feature cannot be attributed to crack speed depen-dence of the elastic field surrounding the crack tip plastic zone. Thesurrounding elastic field shows little dependence on crack speed forspeeds less than about 50-60 percent of the shear wave speed, whereasthe sharp upturn in the variation of toughness with speed has beenobserved for speeds in the range of 25-30 percent of the shear wavespeed. It is shown in this section that the behavior represented inFigure 8.1 can be attributed to inertial effects within the crack tipplastic zone. This conclusion is drawn from analysis of a particularantiplane shear crack problem.

Suppose that a crack grows under two-dimensional conditionsthrough an elastic-plastic material. The potentially large stress veryclose to the crack edge that is predicted on the basis of an elasticmodel of the process must be relieved by some inelastic process.It is assumed here that the process is plastic flow, and that thematerial behavior can be idealized as elastic-perfectly plastic. Thecrack advances steadily at a constant speed v and the mechanicalfields, as seen by an observer translating with the crack tip, aretime-independent. The crack grows under the action of a remotelyapplied stress intensity factor field which is applied at a distancefar from the crack tip compared to the size of the region of activeplastic deformation near the crack tip. A permanently deformed butunloaded layer is left in the wake of the active plastic zone along eachcrack face.

The procedure to be followed within the setting described isstraightforward. A solution of the problem posed is sought in theform of stress and deformation fields that satisfy the field equations insome sense, with material inertia effects included explicitly. With thissolution in hand, a crack growth criterion motivated by the physicsof the separation process is imposed at the level of the crack tipplastic zone. The result is a condition on the applied stress intensityfactor and the crack speed that must be satisfied if the crack is to

Figure 8.2. Active plastic zone region for steady dynamic growth of a mode IIIcrack in an elastic-plastic material.

advance according to this growth criterion. This condition generates atheoretical dynamic fracture toughness versus crack speed relationshipfor the model problem.

The procedure is now followed for the case of dynamic elastic-plastic crack growth in the antiplane shear mode, which is the onlycase for which relatively complete results are available. Summaryremarks on the less developed case of plane strain crack growth areincluded when corresponding results are available.

8.3.1 Plastic strain on the crack line

The active plastic zone is depicted in Figure 8.2. The crack propagatessteadily at speed v through the material in the #i-direction. The crackhas traction-free faces and it occupies the half plane x\ < 0, x<i — 0.The active plastic zone extends a distance rp ahead of the crack tipin the xi-direction. The flow stress of the material in simple shear isro and the shear angle ^(^1^2) at any point is shown in Figure 8.2.

The equations governing the stress and deformation fields in theactive plastic zone are given as (4.4.1)-(4.4.6), and the same notationis followed here. This same problem, but with the inertial effectsneglected, was studied by Rice (1968) and Chitaley and McClintock(1971). Each employed incremental plastic stress-strain relationsbased on the associated flow rule and a yield condition suitable forthe antiplane deformation mode in a nonhardening material. Theyshowed that the principal shear lines in the active plastic zone arestraight, and Rice (1968) determined the distribution of plastic strainon the line directly ahead of the tip in terms of the unknown distancebetween the tip and the elastic-plastic boundary. Chitaley and Mc-Clintock (1971) went on to integrate the field equations numerically.

In the latter study, it was implicitly assumed that the principal shearlines in the active plastic zone are all members of a centered fan.This assumption was questioned by Dean and Hutchinson (1980) onthe basis of a numerical finite element analysis of the steady quasi-static growth problem. If material inertia is taken into account in thissteady state problem, then the system of governing partial differentialequations is hyperbolic in the active plastic zone, as in the caseof quasi-static growth. However, the local directions of principalshear lines are no longer characteristic directions. Instead, there existtwo families of characteristic curves which coalesce to a single familycorresponding to the shear lines in the limit as crack speed v —» 0.No clear picture of the structure of the characteristic net within theplastic zone for v > 0 is yet available.

An important step in this analysis is finding the shear strain 623on the crack line X2 = 0, 0 < x\ < rp within the active plastic zone(Preund and Douglas 1982; Dunayevsky and Achenbach 1982). Forthis purpose, the dimensionless coordinates

x = xi/rp, y = x2/rp (8.3.1)

are introduced. Symmetry requires that <TI3 = €13 = 0 on the liney = 0. From the definition of the shear angle i/>, a convention isadopted whereby ij) = 0 on the crack line. It should be noted that theplastic zone size rp is not known beforehand, and the crack tip plasticzone analysis alone provides no means for determining it.

For steady crack growth, the governing system of partial differen-tial equations can be reduced to (4.4.9). In terms of the dimensionlesscoordinates (8.3.1), these two first-order equations for the shear angle%j) and the x-component of shear strain 7 are

(8.3.2)

where m = v/c8. Equations (8.3.2) form a quasi-linear first-orderhyperbolic system with two families of real characteristic curves inthe x, y-plane. The system is equivalent to the ordinary differentialequations

d(ip ± mi) = 0 (8.3.3)

dtp——— 1 r*c\dx

COSI/J——h iox

• s V ^ + sin^

• ,dip 2SIIIWT:—1- Til

dy


along the characteristic curves

dy sin ip

dx cos i\) ± m(8.3.4)

in the x,i/-plane. The differential equations (8.3.4) for the char-acteristic curves depend on the shear angle ^(x,y), which is notknown a priori, and, consequently, the characteristic network cannotbe determined without determining the complete solution.

The equations in (8.3.2) are linear in the first derivatives, which isthe distinguishing feature of a quasi-linear system, and the coefficientsare functions of the dependent variables. A special feature is thatthe coefficients do not involve the independent variables. Equationssuch as these can be transformed into a linear first-order systemby interchanging the roles of dependent and independent variables.Transformations of this type are often grouped under the heading ofhodograph transformations, and this transformation is especially effec-tive in cases in which a boundary with unknown location is involved(Courant and Priedrichs 1948). If the transformations

) (8.3.5)

are introduced into (8.3.2), then the linear equations

dg dg df- cos-0—- + sin ij)— = 0,c>7 di\) di\)

(8.3.6)dg df 9 dg

c o s ^ - s i n ^ mz— = 0

are obtained. In the ?/>, 7-plane, the system (8.3.6) is equivalent to theordinary differential equations (8.3.4) along the straight characteris-tics curves (8.3.3). For this interpretation, dy/dx is understood to bedg/df. The ^ , 7-plane is shown in Figure 8.3.

At any point on the portion of the boundary of the active plasticzone that is advancing into the elastic region, the material is in astate of incipient flow. In particular, both the yield condition and theelastic constitutive relation are satisfied, hence

7 + sinV> = 0. (8.3.7)

Figure 8.3. The ip,j-p\a.ne showing the straight characteristics which satisfy(8.3.3) and the curve (8.3.7) which represents the leading boundary of the plasticzone.

This relation is also graphed in Figure 8.3. If the position of apoint (/,#) on the elastic-plastic boundary were known in terms ofip (or 7) on the boundary (8.3.7), then a straightforward numericalintegration in the characteristic directions in the ^,7-plane wouldyield a complete description of the stress and deformation fields withinthe active plastic zone. Unfortunately, this information is not knowna priori.

Note that the entire crack line 0 < x < l , j / = 0 in the physicalplane maps into the origin of the ^7-plane. If the polar coordinates

V ~ + 72 a; = tan 1(— (8.3.8)

are introduced, then the equations governing / and g take the form

-sm(pcosu;) . \ \ £ 1 \^ ' \ sin a; cos a; ) { fiuf /p )

, x fcosa; — sinu; 1 ( g,v+ cos(pcosu;) . /

I, sina; COSUJ ) Vg^l

+ f T" T " ] ( 9», } =0, (8.3.9)^ mz cos UJ —mz sin u ) ( giu) /p ) v /

where / , p = df/dp and so on. Since the entire crack line withinthe active plastic zone maps into the origin in the p, o;-plane, it is

assumed that points near the crack line map into points near p =0. For p « 1, the asymptotic approximations cos(pcosu;) « 1 andsin(pcosu;) « pcosw are incorporated, and the equations (8.3.9) thenadmit a separable solution of the form

f=pnF(uj), g=pn+1G(uj) (8.3.10)

near p = 0, where n is a real number. The only value of n for whichthe inverse mapping will carry points near p = 0 into points near theline 0 < x < l , 2 / = 0 i n the physical plane is n = 0. With n fixed,it remains to determine the functions F and G. They are governedby a pair of first-order ordinary differential equations obtained bysubstituting (8.3.10) into (8.3.9) and then letting p -+ 0. Theseequations take the form

dG 1 + (1 + m2) sin a; cos a;du (cos2 u — m2 sin2 u)

(8.3.11)dF 1 - r a 2 ^ „

du cos a;(cos2 u> — m2 sin2 UJ)

A solution of this coupled pair of equations is sought subject to asuitable set of boundary conditions.

Points in the active plastic zone map into points above the curve7 + s i n ^ = 0 in Figure 8.3, so the differential equations (8.3.11) applyover a range of UJ near p = 0 that increases from UJ = TT/4. AS thepoint x = 1, y = 0 is approached along the boundary of the plasticzone in the physical plane, both 7 and ip approach zero. In view of(8.3.7), however, they do so in such a way that — 7 / ^ —• 1. Theboundary condition

F(TT/4) = 1 (8.3.12)

follows immediately. Values of u larger than TT/4 near p = 0 cor-respond to points along y = 0+ with x < 1. Therefore, the upperlimit of the range of the equations (8.3.11) is some value of a;, saya;*, for which x has decreased to zero, corresponding to the crack tip.Therefore, a second condition on the solution (8.3.11) is

F(CJ*) = 0 . (8.3.13)


The value of u/* is not known a priori so an additional conditionis required to produce a unique solution of (8.3.11). According toSlepyan's asymptotic result (4.4.25) for this problem, \j) and 7 aremultiple valued at the crack tip for any value of m. For the presentsolution to have this feature, it is necessary that

G(w*) = 0. (8.3.14)

Upon integrating the differential equations (8.3.11) and enforcing thethree conditions (8.3.12)-(8.3.14), it is found that w* = tan"1(l/m)and

^ ~

(8.3.15)m (cosa;-msina;)(1+m)/2m

(1 - m2)7(m) (cosu>

where / is the definite integral

/•(I—z)/(l+z) (l-m)/2m

I(z)= du. (8.3.16)Jo 1 + u

The integral can be evaluated in terms of elementary functions forany value of m for which (1 — ra)/2ra is a rational number, and it canbe evaluated by numerical means for any value of m of interest.

Of primary interest for present purposes is the variation of thestrain component €23 along the crack line in the active plastic zone.The compatibility relation (4.4.2) implies that

dxi dx2 \2/irp

If (8.3.17) is differentiated with respect to y, viewing i\) and 7 asfunctions of x and y, then it is seen that

%- = - ^ . (8.3.18)dy G(u) K '


Prom (8.3.17) it follows that

If the functions F and G are substituted from (8.3.15) into (8.3.19)and the integral is evaluated, then

1 — m2 tan2 u \ 1

(8.3.20)J(ratancj)

This result is an exact, parametric representation for 623(^1,0) withinthe active plastic zone for the problem at hand. Although this analysisof the fields within the active plastic zone was motivated by discussionof the response due to the surrounding elastic field characterized by astress intensity factor, the result is more general. That is, the resultdoes not depend on the surrounding elastic field, and it provides adescription of the crack line plastic strain no matter what the natureof the fields surrounding the plastic zone might be. In particular, itdoes not depend in any way on whether or not the surrounding fieldmeets the conditions of the small-scale yielding hypothesis.

Graphs of the strain distribution (8.3.20) are shown in Figure 8.4in the dimensionless form of 2/ie23(#i,0)/ro versus X\jrv for severalvalues of crack tip speed m. The main features of the variation areevident from the graphs. The total dimensionless strain at a materialparticle on the crack line is unity at the instant that the particle isengulfed by the active plastic zone, and the strain increases monoton-ically as the crack tip approaches the particle. The strain is singularat the crack tip, as already shown through the asymptotic analysesin Section 4.4. The most significant feature of the results plotted inFigure 8.4 is that the level of plastic strain at a given distance ahead ofthe crack tip as a fraction of plastic zone size is significantly reducedas a result of the inertial resistance of the material. For materialswhich fail by means of a locally ductile mechanism of separation,this suggests that an increasing driving force is required to sustaingrowth at increasing crack tip speed. This intuitive idea is put into aquantitative form in the next subsection.

v/c3= 0.0

v/c8 = 0.3

v/c8 = 0.5

Figure 8.4. The shear strain e23 on the crack line within the active plastic zoneversus distance ahead of the crack tip for three values of crack growth speed.

Finally, recall the apparently contradictory asymptotic results forthis problem obtained by Rice (1968) for m = 0 and any x\ in therange 0 < X\ < rp and by Slepyan (1976) for any ra > 0 and "small'values of x\jrv. These asymptotic results are given in (4.4.34) and(4.4.25), respectively. With the complete solution in hand in the form(8.3.20), this paradox can be pursued (Freund and Douglas 1982).The integral I(z) in (8.3.16) has the form

= m Il/2m

o(m) (8.3.21)

as m —• 0 for any z in the range 0 < z < 1. It follows from (8.3.20)that

xi/rp ~ exp(l - tana;) (8.3.22)

as m —» 0. If tana; is eliminated from (8.3.20) in favor of xi accordingto (8.3.22) and the limit as m —• 0 is taken, then Rice's result (4.4.34)is obtained. On the other hand, for points very near the crack tip,mtano; is very near unity and the integral I(z) has the form

( 1 + m) / 2 m

( 8 A 2 3 )

as z -> 1". Then, from (8.3.15),

i \ (l+m)/2m

— ratanu; \ v " (8.3.24)

where L is an indeterminate scale factor. An expression for tana; interms of X\jrv can be determined from (8.3.24) and substituted into(8.3.20). If, for any fixed value of m in the range 0 < m < 1, thevalue of x\ is sufficiently small that

/ \2m/(l+m)

f — J < 1, (8.3.25)

then Slepyan's result (4.4.25) is recovered. The condition (8.3.25)indicates the range of validity of the asymptotic result (4.4.25). Evi-dently, the range of validity vanishes as m vanishes, so the equilibriumcrack growth result (4.4.34) is not recovered as m —• 0. Generally, form < 0.3, the range of x\ for which (8.3.25) can be satisfied appearsto be smaller than rp/100. The assumptions of a sharp crack, smallstrains, and homogeneous material are subject to question on thisscale. The result (8.3.25) points out a difficulty in using asymptoticsolutions alone to describe mechanical fields in fracture mechanics,namely, the range of validity of an asymptotic solution can be toosmall for that solution to be a useful approximation of crack tip fields.

8.3.2 A growth criterion

The generation of a theoretical fracture toughness versus crack speedrelationship is now pursued along the lines outlined in the introductionto this section. That is, the crack grows in a body in such a way thatthe active plastic zone is surrounded by a region of elastic deforma-tion and that the elastic field is characterized by the elastic stressintensity factor Km. The stress intensity factor field for antiplaneshear deformation is given in (4.2.15). This situation is the small-scale yielding situation of nonlinear fracture mechanics. Then, aductile crack growth criterion is applied at the level of the cracktip plastic zone to determine the relationship that must be satisfiedbetween the applied stress intensity factor and the crack speed forcontinued growth to be sustained. The particular level of appliedremote stress intensity that satisfies this condition is denoted by Knid,


and it is called the dynamic fracture toughness of the material. Thegrowth criterion adopted for this calculation is the critical plasticstrain criterion proposed by McClintock and Irwin (1965). Accordingto this criterion, a crack will grow with a critical, material specificlevel of plastic strain at a point on the crack line at a characteristicdistance ahead of the crack tip. Let the critical level of plastic strain€33 be denoted by 7^ro/2/x, so that 7*? is the actual critical shearstrain normalized by the yield strain. Likewise, let the characteristicdistance be denoted by rc = xcrp, so that xc is the characteristicdistance normalized by the plastic zone size rp. To complete thespecification of the growth criterion, the crack cannot grow for levelsof plastic strain below 7*? at x = xci and levels of plastic strain abovethe critical level at x = xc are inaccessible.

Though this criterion is selected for the present calculation pri-marily for its simplicity, it should also be noted that the criterioncaptures the essence of the separation process for materials that fail bya local ductile mechanism. For example, suppose that the microstruc-tural mechanism of crack advance in a material is the nucleationof microvoids and their subsequent ductile growth to coalescence.Assume that there is a certain distribution of void nucleation sites, saysecond phase particles within a certain size range, and that the flowbehavior of the material is independent of the rate of deformation.Then, for a material volume in the crack tip region, a certain level ofinelastic strain will be required to grow the voids from nucleation tocoalescence no matter how rapidly the crack advances or how intensethe applied loading might be.

For present purposes, the characteristic length xc is eliminatedin favor of the level of applied stress intensity factor that is requiredto initiate growth of a stationary crack in the same material underslowly applied monotonic loading. This level of applied stress inten-sity is denoted by Kmc. Rice (1968) showed that the plastic straindistribution on the crack line in the plastic zone in this case is

(8-3.26)§3 (f

If the critical strain criterion is applied, then

7^(1 + 7?) = ^ ! • (8.3.27)


Let q be the value of ratanu; in the parametric relationship between623 and x\ defined in (8.3.20) when the growth criterion is satisfied.Then

Ltt lv \ -L lib J J. \l I v I

The parameter xc can be eliminated from (8.3.28) by means of (8.3.27)but the result will still depend on the unknown plastic zone size rp.Because there is no characteristic length in the problem consideredin Section 8.3.1, the plastic zone size must scale with K2

n/r2. Fur-

thermore, it must depend on crack speed in some way. Thus, assumethat

rp = Xp{m)^f. (8.3.29)

Then, for steady growth with the critical plastic strain criterion sat-isfied, (8.3.27) can be rewritten as

(KIIId\2_

UW ~ ( ? + l)Xp(m) ' (8-

where, from (8.3.28),

q2 = 1 - (1 - m2) exp[-27£m2 / ( l - m2)] . (8.3.31)

Equation (8.3.30) provides the relationship between fracture tough-ness and crack speed that was sought, obtained on the basis of thecritical plastic strain criterion. The dependence of plastic zone sizeon crack speed represented by Xp(m) is as yet unknown, and thisdependence can only be determined with certainty from a completesolution of the problem.

8.3.3 A formulation for the complete field

The governing equations are now cast into a form appropriate fordetermination of the complete elastic-plastic field under small-scaleyielding conditions. The dimensionless stress and strain componentsdefined by

T~x = Cm/To , Ty = C T 2 3 / T O ,

(8.3.32)x = 2fie13/ro , jy /


are introduced for this purpose. Dimensionless coordinates (differentfrom those used in Section 8.3.2) and a dimensionless displacement ware defined by

x = xxrl/Kfn , y = x2ar^/KI2

II, w = fjLTou3/K?n . (8.3.33)

The parameter a is \ / l — m2 .In terms of the dimensionless variables, the equation of motion

has the form

*£+*£_„,.£-. (8.3.34)

The total strain is decomposed additively into elastic and plastic parts

7/8 = 7! + ^ , (8-3.35)

where (3 = x or y, and the elastic strain is related to stress accordingto

(8.3.36)

By making use of the strain-displacement relations

and (8.3.36), the equation of motion can be rewritten in the form

d2w dw _ 1 0r£ , 1 H (R o o8^oxz oyz az ox a oy

Thus, the differential equation governing the total displacement wis reduced to the Poisson equation with the right side depending onthe unknown plastic strain distribution. According to the small-scaleyielding hypothesis, the stress components are required to have far-field behavior that is the same as the near tip elastic solution (4.2.15).In terms of the dimensionless variables,

sin \9 cos \6 . n .& # ( a 3 3 9 )

iv

p

X

crack lineFigure 8.5. Coarse representation of the finite element mesh used to solve (8.3.38).The stress history of particle P is stored in its forward path-line to the boundary.

as r —• oo for 0 < 9 < TT, where 9 = arctan(2//x). Any plastic wakeregion of finite thickness is formally excluded from the asymptoticcondition by enforcing (8.3.39) up to but not including 9 = TT.

To provide a basis for application of a numerical finite elementprocedure, the governing equation (8.3.38) is recast into a variationalform. A representation of the finite element mesh covering the regionR over which the equation was solved is shown in Figure 8.5 (Freundand Douglas 1982). In order to solve the equations (8.3.38) for valuesof displacement at the mesh node points, the plastic strain must beknown throughout R. This strain distribution is not known before-hand, however, so an iterative scheme is developed. The iterativescheme is based on the fact that the deformation fields are steady asseen by an observer traveling with the crack tip. This implies thatthe stress history of any material particle in the region is recordedon the forward path-line of that particle. For the particle P inFigure 8.5, for example, the stress history is the spatial variationof stress along the forward path-line from point P to the boundaryof the region. Therefore, if the stress variation along this path-lineis known, then the plastic strain at point P can be computed byintegration of the incremental relation between plastic strain andstress from the boundary along the path-line to P. The cycles in theiterative procedure are then (i) finding a plastic strain distribution foran estimate of the stress distribution throughout R, and (ii) finding atotal displacement field for an estimate of plastic strain throughout R.


v/cs= 0.0

v/cs= 0.3

v/cs = 0.5

0.0

Figure 8.6. The discrete symbols show the strain distribution on the crack linewithin the active plastic zone obtained by the numerical procedure described inSection 8.3.3. The exact results from Figure 8.5 are included as the solid curves.

Results of calculations based on an array of 1800 rectangular elements,each comprised of four constant strain triangles, were reported byFreund and Douglas (1982). Nodal points were concentrated near thecrack tip in anticipation of large field gradients there. The size ofthe smallest element in these calculations was about 0.3 percent ofthe maximum extent of the crack tip plastic zone, and the nearestboundary point (other than on the crack plane) was at a distancefrom the crack tip greater than ten times the maximum extent of theplastic zone. Thus, Figure 8.5 presents a very coarse representationof an actual element array.

The strain distributions on the crack line within the active plasticzone determined by means of the computational scheme are shownas discrete points in Figure 8.6 for three values of crack tip speed.The strain and distance ahead of the crack tip are normalized in thesame way as in Figure 8.4, and the analytical results from the latterfigure are included for comparison. The computed strain agrees quitewell with the analytical result for strain up to more than ten timesthe plastic yield strain. The quality of the comparison is significantbecause the analytical result for strain on the crack line in the activeplastic zone is the only exact result available to serve as a check onthe numerical procedure outlined above. Once the algorithm has been

tested in this way, it can be applied with greater confidence in casesfor which there is no certain basis for comparison.

In order to complete the discussion of a dynamic fracture tough-ness versus crack speed relationship begun in Section 8.3.2, an es-timate of the extent of the plastic zone in the direction of crackgrowth is required. More specifically, the dependence of the size of theplastic zone on the crack speed m represented by Xp(m) in (8.3.29)is required. From the numerical results, this dependence is estimatedto be

Xp(m) « (0.3 - 0.5 m2) (8.3.40)

for 0 < m < 0.6. As is evident from Figure 8.6, the variation ofplastic strain with distance ahead of the crack tip near the elastic-plastic boundary is small for the higher crack speeds, which impliesthat the estimate (8.3.40) is less reliable for the higher crack speeds.

8.3.4 The toughness-speed relationship

With the mechanical fields in the active plastic zone related to theremote loading through (8.3.29) and (8.3.40), the theoretical dynamicfracture toughness versus crack tip speed relationship is completelyspecified by (8.3.30) for any values of Kmc and 7*?. These are thetwo parameters that specify the growth criterion. The normalizeddynamic fracture toughness Knid/Knic versus dimensionless crack tipspeed m is shown in Figure 8.7 for three values of the critical plasticstrain 7*?. The surface shown represents all states of the appliedremote stress intensity factor and the crack speed for which steadypropagation can be sustained according to the critical plastic straincriterion. The variable intercept at m = 0 is due to the overallincrease in plastic deformation with increasing critical plastic strain.The plastically deformed material in the wake region has a stabilizinginfluence on the growth process, and the magnitude of this effectincreases with increasing critical plastic strain. The intercept valueof Kind at m = 0 corresponds to the so-called steady state toughnessvalue of the theory of stable crack growth, that is, with the level ofthe asymptote of the resistance versus crack growth curve for a largeamount of growth. The spread of intercepts on the m = 0 axis resultsfrom the use of the common factor Kmc to normalize all data. Theintercept represents the lowest level of applied stress intensity factorthat can cause crack growth; in this sense, it is the arrest toughnessof the material.


10.0 -I

8.0-

6.0-

\

§ 4.0

2.0-

0.00.0 0.2 0.4 0.6

m = v/c0.8

Figure 8.7. The dynamic fracture toughness versus crack speed relationship ob-tained by imposing the critical strain criterion described in Section 8.3.4.

The ratio of Kmd/Knic is a monotonically increasing function ofcrack speed for fixed critical strain, and this function takes on largevalues compared to unity for moderate values of m. Although thereis no unambiguous way to associate a maximum attainable velocitywith these results, they suggest such a terminal velocity well belowthe elastic wave speed of the material.

Perhaps the most significant observation on the result in Fig-ure 8.7 is that the variation of toughness with crack speed is due toinertial effects alone. The material response is independent of therate of deformation, and the crack growth criterion that is enforcedinvolves no characteristic time. If inertial effects were neglected inthis analysis, then the calculated toughness would be completely in-dependent of speed for any given level of plastic strain. Because thistheoretical result exhibits the main features observed in rapid crackgrowth in materials that are relatively strain rate insensitive in theirbulk response and that fail in a locally ductile manner, it is concludedthat material inertia on a small physical scale plays a significant role inestablishing the toughness-speed relation observed in such materials.An explanation for this effect can be found in the fact that particledisplacements, and therefore particle velocities, are much greater inthe crack tip region for the case of elastic-plastic response than for

elastic response. The greater particle velocities lead to greater inertialresistance to motion, and the model problem leads to quantitativeestimates of this effect.

The equivalent plane strain problem of dynamic crack growth inan elastic-ideally plastic material has not been so fully developed. Astudy of a smooth punch moving along the surface of a rigid-ideallyplastic solid was carried out by Spencer (1960). The field equa-tions were referenced to a system of moving curvilinear coordinates,which were viewed as a generalization of the slip line coordinates inplane strain plasticity under equilibrium conditions. A solution ofthe equations of dynamic plasticity was obtained as a perturbationfrom known equilibrium slip line fields. The connection betweenequilibrium fields near the edges of a smooth punch and mode I cracktip fields was established by Rice (1968), but the matter has notbeen pursued for dynamic fields beyond the work of Spencer (1960).A numerical calculation leading to a fracture toughness versus crackspeed relationship, analogous to Figure 8.7, has been described byLam and Preund (1985). They adopted a critical crack tip openingangle growth criterion and derived results for dynamic mode I crackgrowth on the basis of the Mises yield condition and the J2 flow theoryof plasticity. Whereas in the case of antiplane shear deformation theplastic deformation is concentrated ahead of the crack tip, in thiscase the plastic deformation is concentrated in lobes off to the sidesof the advancing crack, and the accumulated effect of this plasticstrain is a net opening rate for the crack behind the tip. The re-gion directly ahead of the crack tip is one of high hydrostatic stressbut low deviatoric stress in mode I deformation. Rice and Sorensen(1978) proposed the condition that the crack must grow with theseparation between the crack faces at some material specific distance,say rm, behind the crack tip equal to a critical value, say 6C. Resultsof calculations for several values of the dimensionless combinationof parameters 2?<$c/<7orm are shown in Figure 8.8, where E is theYoung's modulus of the material and ao is the uniaxial tensile flowstress. Finally, it is noted that the deformation fields obtained in thisnumerical simulation appeared to be qualitatively consistent with theasymptotic solution presented in Section 4.5, but the matter has notbeen resolved with certainty.

8.3.5 The steady state assumption

The analysis in this section has been carried out for steady state


22.5

Figure 8.8. The fracture toughness versus crack speed relationship obtained byapplication of the numerical procedure described in Section 8.3.4 to the case ofmode I crack growth in an elastic-ideally plastic material. The growth criterionimposed in this case was that the crack face opening displacement at a fixeddistance behind the crack tip rm assumed the constant value 6C.

crack growth with constant remote stress intensity factor, but thehope is that the result can be used for nonsteady growth providedthat the remote stress intensity factor does not vary too rapidly withtime or position. Thus, a rough estimate of conditions under whichthe results of the steady state analysis can be applied for nonsteadyconditions is sought. Suppose attention is focused on a particularmaterial particle in the path of the crack tip plastic zone. For thecase of a crack approaching steadily with a constant stress intensityfactor, an expression for the plastic strain rate is given in (8.3.20).For purposes of comparison, consider a stationary crack at the sameposition in the same material and at the same level of applied stressintensity factor. If the stress intensity factor has some rate of change,then the rate of plastic strain can be calculated here as well. It mightbe anticipated that the steady state approximation will be useful inthe nonsteady cases for which the former strain rate dominates thelatter. If these rates are actually determined for the mode III problemconsidered in this section, using the formula (8.3.26) for the formerand (8.3.20) for the latter, then it is found that the steady state

8.4 High strain rate crack growth in a plastic solid 469

approximation will be useful according to this criterion as long as

±ir(l - m)Xp(m)^- » ^ . (8.3.41)rp Km

The factor depending on m on the left side is of order one for the speedrange of practical interest, so the condition simply suggests that thesteady state approximation will be valid as long as the change in thestress intensity factor during the time required for an elastic wave totraverse the plastic zone is small compared to the magnitude of thestress intensity factor. This is not a severe constraint, so the steadystate approximation is expected to be broadly useful.

8.4 High strain rate crack growth in a plastic solid

A particularly interesting class of dynamic fracture problems are thoseconcerned with crack growth in materials that may or may not ex-perience rapid growth of a sharp cleavage crack, depending on theconditions of temperature, stress state, and rate of loading. Thesematerials can fracture by either a brittle or ductile mechanism onthe microscale, and a focus of work in this area is on establishingconditions for one or the other mode to dominate. The phenomenonis most commonly observed in ferritic steels, but it has also beenobserved in tungsten (Liu and Shen 1984a, 1984b) and other materi-als. Such materials show a dependence of flow stress on strain rate,and the strain rates experienced by a material particle in the pathof an advancing crack are potentially enormous. Consequently, themechanics of rapid growth of a sharp macroscopic crack in an elastic-viscoplastic material that exhibits a fairly strong variation of flowstress with strain rate is of interest.

The general features of the process as experienced by a materialparticle on or near the fracture path are straightforward. As theedge of a growing crack approaches, the stress magnitude tends toincrease due to the stress concentrating influence of the crack edge.The material responds by flowing at a rate related to the stress level inorder to mitigate the influence of the crack edge. It appears that theessence of cleavage crack growth is the ability to elevate the stress to acritical level before plastic flow can accumulate to defeat the influenceof the crack tip. In terms of the mechanical fields near the edge of


an advancing crack, the rate of stress increase is determined by theelastic strain rate, whereas the rate of crack tip blunting is determinedby the plastic strain rate. Thus, an equivalent observation is thatthe elastic strain rate near the crack edge must dominate the plasticstrain rate in the crack tip region for sustained cleavage crack growth.This concept of elastic rate dominance is central to the model thatfollows. It is implicit in this approach that the material is intrinsicallycleavable, and the question investigated is concerned with the way inwhich energy can be supplied to the immediate vicinity of the cracktip.

8.4-1 High strain rate plasticity

A material particle in the path of a rapidly advancing crack may besubjected to very high rates of plastic strain for a short period of time.For example, as a rough estimate, suppose that a crack grows in anelastic-plastic material at a speed v of about 103 m/s. Furthermore,suppose that a material particle in the crack path experiences a plasticstrain j p of about 10~3 during the time that the active plastic zoneof extent rp « 10~4 mm sweeps past the particle. An estimate of theplastic strain rate is then the accumulated plastic strain divided bythe time that the particle is included within the active plastic zone,

which has the value of 104s~1 in this case. Thus, constitutive lawsappropriate for the high strain rate response of materials must beadopted for a description of crack tip processes in rate-dependentmaterials.

At a temperature well below the melting temperature, manymetals exhibit a relationship between flow stress in simple shear rand the corresponding rate of plastic strain j p having the featuresillustrated schematically in Figure 8.9. The scale on the plastic strainrate axis is logarithmic. This high strain rate response apparentlycan be divided into two regimes. For strain rates above a certainlevel, hereafter called the transition strain rate 7t, the response isdominated by effects that can be represented as viscous effects, atleast phenomenologically. For purposes of analysis, the response inthis regime is described by

7P = 7t + 7 o ( r - r t r / / i for r>rt, (8.4.2)


Low/Moderate Rate Region

High Rate Region

I I I I l l l l | 1 I I I l l l l | I I I I l l ! l | I I I I i l l l | I I I I I ITT] I I M

10° 101 102 103 104 105

Plast ic S t ra in Rate - yPFigure 8.9. Schematic diagram of the dependence of flow stress on plastic shearstrain rate for rapid deformation in simple shear.

where // is the elastic shear modulus, rt is the transition flow stressdefined as the flow stress when j p = 7$, n is a material constant, and7o is a material constant representing the viscosity of the material.Hereafter, it will be identified as the material viscosity. The relation(8.4.2) is a particular case of the inelastic strain rate dependenceintroduced in Section 4.7. All further analysis in this section is basedon the case n = 1 in (8.4.2).

The origin of a constitutive equation of the form (8.4.2) can befound in the "physical" theories of plasticity based on dislocationdynamics. Orowan (1934) showed that the macroscopic plastic strainrate due to N mobile dislocations per unit area with Burgers displace-ment b and average speed Vd is

Y = bNvd. (8.4.3)

It is assumed that steady motion of a dislocation at speed Vd throughthe crystalline lattice requires an applied shear stress r given by

rb = Bvd, (8.4.4)

where B is a drag coefficient. The origin of the drag on the dislocationis the background thermal motion of the lattice (Hirth and Lothe

1982). Elimination of Vd from (8.4.4) and (8.4.3) yields a linearrelationship between r and 7P, as suggested in (8.4.2). It should benoted that this simple dislocation dynamics description breaks down ifthe plastic strain rate is significantly affected by the changing numberof mobile dislocations with ongoing plastic strain. With reference to(8.4.3), if the value of N increases rapidly due to high applied stress,then the plastic strain rate will increase more rapidly with appliedstress than is suggested by (8.4.2). Indeed, experimental results onhigh rate response at relatively large plastic strain show this to bethe case (Clifton 1983). In the study of crack growth, however, themain interest is on situations where the magnitude of plastic strainis about the same as the magnitude of elastic strain, and where theplastic strain accumulates in a very short period of time. Thus, it isassumed that the response is governed by dislocation motion, ratherthan generation, and the linear form (8.4.2) is adopted. Campbell andFerguson (1970) carried out a series of experiments on mild steel; theygive j t = 5 x 103s~1 and % = 3 x 107s~1 as values of the constantsin (8.4.2).

For plastic strain rates below the transition strain rate 7 , thestress required to produce plastic flow is relatively insensitive to vari-ations in strain rate. The plastic strain rate may drop many or-ders of magnitude with little reduction in flow stress. The physicalmechanism of flow in this regime is believed to be the thermallyactivated motion of dislocations past discrete obstacles in the materiallattice. For the case of iron, Frost and Ashby (1982) have proposed arelationship between flow stress and plastic strain rate of the form

for T <Tt, where f is the flow stress at a temperature of absolute zero,T is the absolute temperature in degrees Kelvin, k is Boltzmann'sconstant, and AFP is an activation energy. Representative values ofthe constants at T = 300 K are given by Frost and Ashby (1982) as

f//io = 10"2 , cp = 1011 s"1, AFp/fiob3 = 0.1, (8.4.6)

where /io is the value of the shear modulus at T = 300 K and b =2.48 x 10~10 m is a lattice-spacing parameter.


Kr

rWake h

1, T • i i High R a t e

Crack Line I \ I % n e

KActive Plastic

Zone

Figure 8.10. A schematic representation of steady crack growth at speed v underplane strain conditions in a rate-dependent plastic solid due to a remotely appliedelastic stress intensity factor field.

To generalize the constitutive relation to a multiaxial stress staterepresented by ay, let

T = (±SijSij)1/2 , (8.4.7)

where Sij is the stress deviator. Represent the simple shear viscoplas-tic relation identified jointly by (8.4.2) and (8.4.5) by

jP = F(T) . (8.4.8)

Then the plastic strain rate under multiaxial states of stress is

^i=F{T)8^, (8.4.9)

which reduces to (8.4.8) for simple shear. When r drops below theslow loading yield stress ry, the plastic strain rate is zero. This cutoffis somewhat arbitrary, but the cutoff value has little influence on theresults that are obtained.

The material is assumed to be elastically isotropic so the elasticstrain rate is given by

cy = (8.4.10)

where E is Young's modulus and v is Poisson's ratio. The total strainrate is the sum of the elastic and plastic strain rates,

= ««


8.4-2 Steady crack growth with small-scale yielding

Suppose that a planar crack advances through the elastic-viscoplasticmaterial described in Section 8.4.1 with speed v in the xi-directionunder conditions of plane strain in the tensile opening mode, or modeI. The speed is constant, and it is assumed that the crack tip hastraveled a distance that is large compared to the size of the activeplastic zone. Furthermore, the active plastic zone is assumed to besmall compared to the region of dominance of a surrounding elasticsingular field. This elastic field is described by (4.3.10) and it ischaracterized by the dynamic stress intensity factor Kj, which isassumed to be maintained at a constant level. Under these conditionsthe steady state, small-scale yielding situation depicted in Figure 8.10is considered. The active plastic zone travels with the crack tip leavingbehind a wake of plastic strain and, in general, an accompanying resid-ual stress distribution. Outside of the wake region, the remote stressfield in this asymptotic small-scale yielding problem is the dynamicsingular field specified by Kj or, equivalently, by the dynamic energyrelease rate G by means of (5.3.10). This situation is analyzed withinthe framework of small strain theory under the assumption that themechanical fields are constant as seen by an observer traveling withthe crack tip. Thus,

'---!£• (8-4-i2)where the superimposed dot denotes material time differentiation and/ is any scalar field or any rectangular component of an array.

An observation that is central to the analysis of this situationconcerns the behavior of the stress and deformation fields near thecrack tip deep within the active plastic zone. For the viscoplastic law(8.4.2), the considerations of Section 4.7 lead to the conclusion that

Eo-(0,m) (8.4.13)

as r —• 0, where r, 6 are crack tip polar coordinates and m = V/CR.The functions £^(0, m) representing the angular variation of thestress distribution deep within the plastic zone are identical to the cor-responding functions representing the angular variation of the stressdistribution in the remote elastic field. The crack tip stress intensityfactor Kiup, however, differs from the remote stress intensity factor


Kj. The main purpose of the analysis is to establish the connectionbetween the two stress intensities.

Because the elastic strain rate dominates the plastic strain ratenear the crack tip, the local mechanical field is square root singularand there is a net energy flux out of the body at the tip which,according to (5.3.10), is

Gtip = Aj{v)^^-Klip. (8.4.14)

The rate of energy supply per unit crack advance to the crack tipregion which is provided by the remote loading is G. Some of thismechanical energy is dissipated through active plastic flow, and someis locked in the wake region as stored elastic energy due to the devel-opment of incompatible plastic strain in the wake. The remainder isGup. It will be assumed subsequently that a certain level of Gtip mustbe maintained if the crack is to advance as a sharp crack. Thus, therelationship between G and Gup is of central interest. The importantquestion concerns how much of the applied energy release rate G isavailable to drive the crack in the form of Gup- In other words, thequestion concerns the extent to which the intervening plastic flow iseffective in screening the crack tip from the applied loading. It isnoted in passing that Gtip = 0 for rate-independent plastic response,as well as for viscoplastic response (8.4.2) with n > 3.

The overall work rate balance, which is tacit in the foregoingarguments used to motivate the study of the relationship between Gand Gup, can be written as

Gup = G-- I an^ dR- I u; dx2 , (8.4.15)vJR J-h

where R is the area of the active plastic zone in the rri,a:2-plane, his the thickness of the wake on either side of the crack plane in the#2-direction, and U*(x2) is the elastic energy density locked in thewake as X\ —• —oo. This relationship states that the energy releaserate at the crack tip deep within the active plastic zone equals therate of energy supply to the crack tip region reduced by the rate ofmechanical energy dissipation due to active plastic flow and furtherreduced by the rate at which energy is locked into the wake.


In light of the restriction to steady fields, the relationship (8.4.15)is evident from an overall work rate balance. However, as was demon-strated by Freund and Hutchinson (1985), the relationship can also bederived by direct application of the path-independent integral (5.2.9).For a path of integration which is very close to the crack tip comparedto the size of the active plastic zone and which begins on one traction-free crack face, surrounds the tip, and ends on the opposite crack face,the value of the integral is Gup. On the other hand, for a path ofintegration that is far from the crack tip compared to the size of theplastic zone, the value of the integral is

fh

G- U*dx2, (8.4.16)

J-h

where U*{x2)= lim U(xux2) (8.4.17)X\ —• — OO

is the stress work density (5.2.3) in the wake region far behind thecrack tip. In view of the path independence of (5.2.9),

fh

Gtip = G- / U*dx2. (8.4.18)J-h

To see that this is identical to (8.4.15), split the total stress workdensity U into elastic and plastic parts according to

Ue = [ aijetj dt', Up = f n ^ dt'. (8.4.19)J — oo J—oo

The area integral in (8.4.15) can then be rewritten as

/ Giji?- dR= f UpdR=-v f ^ d xJR JR JR OXI

=v lim Up(xi, x2)dx2 = v / U* dx2 .J —h ^ J —h

\dX2

(8.4.20)

Substitution of (8.4.20) into (8.4.15) then yields (8.4.18) once again.In any case, the expression (8.4.15) is exact.

8.4-3 An approximate analysis

A complete solution of the field equations for the asymptotic small-scale yielding problem is not available, although an approximate nu-merical solution of the field equations has been described by Preund,Hutchinson, and Lam (1986) and Mataga, Preund, and Hutchinson(1987). In this section, an approximate analysis that results in anexplicit relationship between G and GtiP is outlined. The approxi-mations that are incorporated are based on the idea of elastic ratedominance of the crack tip fields, and they are exact in the limit ofvanishing plastic dissipation. The overall work rate balance (8.4.15)is the starting point for the approximate analysis.

The term in (8.4.15) representing the elastic energy stored inthe wake is considered first. For steady growth of a mode I crackthrough a rate-independent elastic-plastic material under small-scaleyielding conditions, the elastic energy locked into the wake is roughly10 percent of the total rate of energy supply, represented here byG (Dean and Hutchinson 1980). Both inertial effects and materialrate effects tend to suppress the accumulation of plastic strain inthe active plastic zone which, in turn, implies smaller incompatibleplastic strain in the wake. In particular, for the present case the elasticenergy stored in the wake is expected to be a very small percentage ofG, and this contribution to the balance in (8.4.15) will be neglected.The validity of this assumption has been supported by the numericalanalyses cited above.

The plastic dissipation term in (8.4.15) is considered next. If thedistribution of stress cr throughout the plastic zone were known, thene?j could be determined from the constitutive equation (8.4.9) and theintegral over the region R in (8.4.15) could be evaluated. Of course,the stress can be determined only by solving the field equations. Thestress distribution deep within the active plastic zone has the asymp-totic property (8.4.13). The stress takes on its largest magnitude closeto the crack tip, and the rate of plastic strain is also largest where thestress is largest. Consequently, the plastic dissipation in (8.4.15) canbe estimated on the basis of the assumption that the stress everywherein the active plastic zone is given by (8.4.13). This is the approachfollowed originally by Freund and Hutchinson (1985). In fact, thestress field must exhibit a transition from a remote field given by(4.3.10) to the near tip field (8.4.13) through the active plastic zone,where Kitip < Kj. Consequently, the stress level in the active plastic


zone given by (8.4.13) is expected to result in an estimate of the plasticdissipation that is too small.

Alternatively, it could be assumed that the stress in the plasticzone is given by (8.4.13) with Kiuv replaced by Kj. This approxi-mation overestimates the stress magnitude everywhere in the plasticzone. A compromise approximation which overestimates the stressmagnitude at the crack tip but which might be expected to underes-timate it at points further from the tip is

(8-4.21)

everywhere within the active plastic zone. Though the assumptionis motivated only by an interest in exhibiting the qualitative featuresof crack growth under the conditions described, it has been verifiedby numerical calculation that the assumption provides an accurateestimate of plastic dissipation (Mataga et al. 1987).

The integrand of the plastic dissipation term in (8.4.15) is

a y c ? . = r F ( r ) , (8.4.22)

where r is the effective shear stress defined in (8.4.7). In light of theassumption (8.4.21), r is given by

T = V ^ F " W m ) ) (8-4-23)

where B(O,m) is defined by

B2 = |E?i + £?2 + sS22 + i (2i/2 + 2i/ - 1) ( E n + E22)2 (8.4.24)

everywhere in the active plastic zone. If rp(9) is the radial distancefrom the crack tip to the boundary of the active plastic zone, wherer = ry, then

(8.4.25)

The plastic dissipation term in (8.4.15) is

- / (TiigidR=- dO rF(T)rdr. (8.4.26)v JR %3 v J_w Jo

Next, using (8.4.23) to change the variable of integration from r to r ,it is found that

- I «& dR = -L-K?UpK?C(m) I" T-4F(T) dr, (8.4.27)v JR ^n V Jry

where

C(m)= f B(9,m)4d0. (8.4.28)J — 7T

The integral with respect to r in (8.4.27) is independent of crackspeed ra. It is convenient to separate this integral into contributionsfrom the high strain rate regime with rt < r < oo and from the low ormoderate regime with ry < r < Tf. The former contribution is easilyevaluated because F(r) is a linear function in this regime, whereas thelatter contribution is small by comparison for most realistic values ofmaterial parameters and it is therefore neglected. Thus,

+ - ^ . (8.4.29)

With this estimate of plastic dissipation, the work rate balance takesthe form

^ ( ^ ) D(m)GGtip , (8.4.30)

where the stress intensity factors Kj and Kiup have been replaced byequivalent expressions in terms of energy rates G and Gup by meansof (5.3.10), and

7r2m[(l-iy)AI(m)Y

Values of the function D(m) have been determined numerically forv — 0.3 and the result is shown in Figure 8.11. The function hasa minimum at m = 0.55, which is an observation that has somesignificance for the details to follow.

Suppose that the criterion for the crack to propagate steadily isthat the crack tip energy release rate Gup must take on a materialspecific value,

GuP = Gctip . (8.4.32)


0.6-

0.4

0.2-

0.00.0 0.2 0.4 0.6

m = v/cR

0.8 1.0

Figure 8.11. The function D(m) defined in (8.4.31) versus dimensionless crackspeed m = V/CR for v = 0.3.

For present purposes G\iv is assumed to be a material constant, inde-pendent of temperature and crack speed. The quantity Gtip representsthe energy per unit crack advance that flows out of the body atthe crack tip and that is not otherwise taken into account in thecontinuum analysis. The growth criterion (8.4.32) implies that this isthe energy per unit crack advance that is absorbed by the materialseparation process. The quantity Gc

iiv should perhaps be assumed tobe a function of v and T, but not enough is known about this processto provide a basis for assigning dependence. With (8.4.32) imposed,relation (8.4.30) yields the value of G required to drive the crack atspeed v as

IU (8-4-33)where Pc is the dimensionless collection of material parameters

(8.4.34)Pc =tip

J

The term in the expression for Pc that is perhaps the most stronglytemperature dependent is Tt, which decreases with increasing temper-ature for iron, for example. In this sense, Pc is a temperature-likeparameter in this model.


The ratio GjG\ip is plotted against crack speed m for three valuesof Pc in Figure 8.12. For the corresponding value of Pc, each curveindicates all states represented by G and m for which steady crackgrowth can be sustained under the conditions assumed. The generalshape of the curve for any fixed value of Pc could have been anticipatedfrom the basic features of the model. At low crack speed, there issufficient time for plastic strain to accumulate, so a larger remoteenergy release rate is required to maintain a critical crack tip energyrelease rate. This effect diminishes with increasing crack speed, whichaccounts for the falling branch of the curve with increasing m. At highcrack speed, on the other hand, inertial resistance of the material iscalled into play. Again, a high driving force is required to sustainsteady growth, and this effect accounts for the rising branch of thecurve with increasing m. The combination of these effects leads tothe variation of driving force versus speed with a minimum as shown.The fact that the curves become asymptotically infinite as v —> 0 is ashortcoming of the model. As the crack speed becomes small, there istime for plastic strain to accumulate. However, there is no provisionin the model for this case (it is based on the assumption of elastic ratedominance), and the limiting response as the rate sensitivity vanishesis not rate-independent plastic response, in the sense considered inthe preceding section. Modeling of the transition between the crackpropagation situations in which elastic strain dominates, as in thissection, and in which plastic strain dominates, as in the precedingsection, is incomplete, although some ideas are summarized in Section8.5.

8.4-4 Rate effects and crack arrest

Although the analysis is based on the assumption that the processis steady state, some implications for nonsteady crack propagationcan be inferred. For example, for any state above the curve for theappropriate value of Pc, the crack is overdriven and it will tend toaccelerate toward the rising branch of the curve. In this sense, therising branch of each curve represents states that are stable underperturbations in driving force, whereas the falling branch representsunstable states. As a second example, consider the situation in whicha crack is initiated in a material at values of G and m that correspondto a point above the steady state curve for the prevailing value ofPc, and the crack then propagates under loading that produces adynamic energy release rate that decreases as the crack becomes


6 - 1

4 -

2-

0.0 0.2 0.4 0.6

m = v/cR

0.8I

1.0

Figure 8.12. The ratio of remote crack tip driving force G to the crack tip drivingforce G%- versus dimensionless crack speed m as given in (8.4.33) for three valuesof the dimensionless parameter Pc in (8.3.34). The dashed line is a possiblerepresentation of the G-v history for fracture initiation at state L if the crackgrows into a region of decreasing driving force.

longer. Point L in Figure 8.12 indicates such a starting point. Thecrack tip apparently accelerates toward the stable branch of the steadystate curve with G decreasing, as suggested by the dashed curve inthe figure. Once the stable branch is reached, the crack can continueto grow but with diminishing speed by following this branch in thedirection of decreasing G toward the minimum in the steady statecurve. The existence of this minimum is perhaps the most significantfeature of the result shown in Figure 8.12. It implies that, for a givenvalue of Pc, there is some minimum level of applied energy releaserate below which rapid growth of a sharp crack cannot be sustained.

Although the model being developed here does not include eventsafter the driving force falls below this minimum, it is possible tospeculate on the matter. For G below this minimum, the growthcriterion (8.4.32) can no longer be satisfied, which implies that growthmust be arrested. The graphs in Figure 8.12 suggest that this occursfrom a relatively high crack propagation speed, hence arrest is abruptrather than gradual. The abruptness of crack arrest in rate sensitivematerials was observed by Clark and Irwin (1966) in hard plastics,and by Ishikawa and Tsuya (1976) and deWit and Fields (1987) in


Figure 8.13. Surface of G/G%ip over the plane of dimensionless crack speed mand dimensionless material parameter Pc according to (8.4.33). The surface istruncated arbitrarily at = 6 to permit better visualization of its features.

structural steels. Once the crack tip motion is stopped, it is no longerpossible for elastic deformation rates to dominate plastic rates, soplastic deformation can develop freely. The crack tip is expectedto remain stationary until some ductile growth criterion is satisfiedor until the stress is elevated within the plastic zone, presumablyby a combination of strain hardening and rate effects, to a levelsufficiently great to reinitiate cleavage fracture. It is emphasized thatthis sequence of events is speculative; the model itself provides nodirect information on events that might occur when the propagationcondition can no longer be satisfied.

The graph in Figure 8.13 gives the locus of state combinationsof G, i>, and Pc for which steady state propagation of a sharp crackcan be sustained. As was discussed in connection with Figure 8.12,the implication is that if a cleavage crack can be initiated for a statecombination G,v,Pc that is above the surface, then the crack willaccelerate to a state on the stable branch of the surface (that is, theside with increasing G at fixed Pc). If the driving force diminishesas the crack advances, or if the local value of Pc increases as thecrack advances due to a temperature increase, say, then the statecombination will move toward the minimum point on the surface at


lO-i

Figure 8.14. The crack arrest fracture energy Ga versus material parameter Pcimplied by (8.4.33). The arrest toughness Kja can be determined by means of(5.3.10).

the local value of Pc. If the driving force is further decreased, or if Pc

is further increased, then growth of a sharp cleavage crack cannot besustained according to the model. The implication is that the crackwill arrest abruptly from a fairly large speed, and a plastic zone willthen grow from the arrested crack.

Of special significance is the observation that, at any given tem-perature, the variation of required driving force with crack speed hasan absolute minimum, say Ga/G^ip. This implies that it is impossibleto sustain cleavage crack growth at that value of Pc with a drivingforce below this minimum. Thus, this minimum as a function of Pc canbe interpreted as the variation of the crack arrest toughness for thematerial with the local value of Pc or, for most practical purposes,with material temperature. This minimum is plotted against Pc

in Figure 8.14. The dependence of Pc on temperature depends onthe material properties, as seen from (8.4.34). Perhaps the mostsignificant variation arises from the temperature dependence of thetransition flow stress rt. Based on high strain rate data for mildsteel reported by Campbell and Ferguson (1970), the parameter Pc

increases monotonically from a value of about one to a value of aboutnine as the temperature increases from 0 K to 300 K for this material.Crack arrest data for A533 grade B pressure vessel steel is reported

8.5 Fracture mode transition due to rate effects 485

and discussed by Pugh et al. (1988). For this material, arrest of rapidfracture coincides with the end of cleavage crack growth.

Finally, it should be noted that the foregoing argument on crackarrest due to the combined roles of rate and temperature effects isbased on global energy considerations, and is not concerned withthe details of the microstructural mechanism by which rapid cleavagecrack growth gives way to slower ductile growth. The accumulationof plastic strain ahead of the advancing crack to mitigate the stressconcentrating effect of the crack edge, with the ductile growth of voidsleading to coalescence as a consequence, is one possibility. Anotherpossibility for poly crystalline materials is that, as plastic strain ac-cumulation ahead of the cleavage crack tip increases for whateverreason, not all grains in the crack path can be cleaved. Some grainsare left behind the main crack front as ligaments connecting the crackfaces (Hoagland et al. 1972). With the relief of stress triaxiality, theseligaments then must be extended plastically to open the crack. Asthe distance of a ligament behind the main crack increases, so doesits restraining effect on further crack advance, and a distribution ofthese ligaments may be responsible for arrest of a running cleavagecrack.

8.5 Fracture mode transition due to rate effects

In Section 8.3 a model of the process of dynamic crack growth throughan elastic-plastic material is developed for the case when some cracktip plastic deformation measure is required to take on a materialspecific value by the "ductile" growth criterion. Similarly, in Section8.4, a model is developed that provides a description of rapid crackgrowth in a rate-dependent elastic-plastic material under conditionsthat permit crack advance in a cleavage mode. The notion of elasticrate dominance within the crack tip field is identified as a neces-sary condition for advance of a sharp crack in a "brittle" cleavagemode through an elastic-viscoplastic material. The descriptive terms"brittle" and "ductile" are used here to suggest material separationmechanisms that are stress-controlled and deformation controlled,respectively. These connotations, which embody gross idealizationsof actual physical processes of material separation, are not universalbut they are adopted for the discussion in this section.

Both of these lines of study of nonlinear fracture phenomenahave led to conceptual pictures of rapid crack advance by means


of stress-controlled and deformation-controlled mechanisms that arequite complete. However, the two lines of study have few intersections,in the sense that models have not been developed for which either typeof criterion can be applied, depending on the prevailing conditions.The availability of such models would make it possible to analyze theprocess of fracture mode conversion as it is observed to occur, mostcommonly in fracture initiation and crack arrest. The purpose of thissection is to outline a modest step in the direction of developing sucha model. The analysis is based on the simple one-dimensional stripyield model of crack tip plasticity. In spite of the limitations, theform of the results establishes a framework for pursuing the matterthrough computational or other less direct methods in the future. Thedeficiencies of the strip yield model of the crack tip plastic zone fordescribing fracture processes that occur under essentially plane strainconditions have been identified elsewhere. As was noted in Section8.3, the influence of the wake of the active plastic zone is very smallin the high strain rate crack growth regime, so the absence of a wakeeffect with the strip yield model is not viewed as a serious drawbackin considering processes in this regime.

The presentation is organized in the following way. A bound-ary value problem is formulated that models the process of steadydynamic growth of a crack under plane strain, small-scale yieldingconditions. The influence of material strain rate sensitivity is in-cluded by assuming that the resistance to opening within the stripyield zone depends on the opening rate. The analysis follows thework of Glennie (1971b), who considered the case of general yielding.Then, it is assumed that the crack grows steadily according to acritical crack tip opening displacement criterion or a critical stress ata characteristic distance criterion to infer the relationship that mustbe satisfied between the remotely applied stress intensity factor (ordynamic fracture toughness) and the crack speed in order to sustaingrowth. Finally, the criteria are imposed simultaneously and, underthe assumption that the criterion that is easiest to satisfy will be theone that is operative, the complete toughness-speed relationship isgenerated.

8.5.1 Formulation

Consider a planar crack extending in an otherwise unbounded elasticsolid under two-dimensional plane strain conditions. The crack tipmoves at constant speed v and the traction distribution on the crack

8,5 Fracture mode transition due to rate effects 487

vFigure 8.15. A representation of steady growth of a crack under plane strainconditions with a cohesive zone of length A.

faces is time invariant in a reference frame translating with the cracktip. Under these conditions, a steady state solution of the governingequations is sought. A rectangular x, y-coordinate system is estab-lished in the plane of deformation with its origin fixed at the movingcrack tip. The crack grows in the plane y = 0 and the crack occupiesthe half plane x < 0; see Figure 8.15. The general solution procedureis outlined in Section 6.2.1 and the notation of that section is adoptedhere.

The boundary conditions imposed on the crack faces that providethe essential features of the strip yield model are

(x) for -A < x < 0,f o r - o o < z < - A , ( g 5 1 )

<7xy(x, O"*1) = 0 for - oo < x < 0.

Thus, a(x) represents the cohesive stress in the strip yield zone oflength A. In terms of the unknown functions F and G introducedin Section 6.2 for analysis of steady elastodynamic crack growth, theboundary conditions take the form

\-2as[G'l(x) + G"_{xj\

= -2a(x)H(A + x)/fi, (8.5.2)

for —oo < x < 0, where H(-) is the unit step function and the sym-metries consistent with mode I crack growth have been incorporated.The functions are further restricted by the requirement that their far-field behavior must match the square root singular stress intensityfactor field for steady dynamic growth (4.3.10).

The solution of the problem is thus reduced to the solution ofa Hilbert arc problem for F and G. A general solution is presentedby Muskhelishvili (1953) and only the result is presented here. Thegradient of the opening displacement uy(x, 0+) in the ^-direction alongthe crack face y = 0+ is found to be

duyvad 1 [l r y / ^ v W ) * , Kj][ J d J ( }y_v2ad 1 [l r A y/^vW) * ,

~ c2D ^ 7 ^ [n Jo x-x> dXdx

where Ki is the remotely applied dynamic stress intensity factor andD = \oLdOL8 — (1 + a2

s)2 as in (4.3.8). For the strip yield zone model,

it is required that the displacement gradient must be nonsingular atx = 0, the leading edge of the zone. Therefore, the cohesive stressa(x) is related to the applied stress intensity factor by

[T f° <r{x)(8.5.4)

It is convenient to normalize the spatial coordinate in the directionof crack growth by the length of the yield zone. In terms of thedimensionless coordinates

i = -x'/A and 77 = -x/A (8.5.5)

the displacement gradient (8.5.3) becomes

^(*,0+) = W(,) = 4 ^ f -jf^-di. (8.5.6)dx cj D pic Jo T(£ V)

This result provides a single relationship between the opening in thecohesive zone and the cohesive traction resisting the opening; thisresult follows from the elastodynamic field equations outside of theyield zone. This relationship must be augmented by a constitutiveassumption on the response within the yield zone.

8.5.2 A rate-dependent cohesive zone

A fundamental objective in selecting a constitutive relation for theresponse in the yield zone is to incorporate the salient features of


viscoplastic material behavior while still retaining a mathematicalstructure that is amenable to analysis. This objective is met bychoosing the cohesive traction at any point a(r]) such that it dependson the local rate of opening in the cohesive zone uy according to

a(r)) = *o(l + uy/u°y) (8.5.7)

for iiy > 0, where ao and u° are material constants. For very slowdeformations, the response is ideally plastic with flow stress ao. Forrapid deformation, the local magnitude of the cohesive stress is el-evated by an amount determined by the reference opening rate u°.The arguments for selecting a value for it® that will have relevance toplane strain crack growth are oblique at best. In any case, this modelpermits a qualitative illustration of the phenomenon of interest.

An argument is proposed that is based on the idea that the"average cohesive stress" in the yield zone should be increased aboveao by a factor of two or three under the most extreme conditionsof crack growth. The actual numerical value of such a factor is oflittle significance, but existence of such a factor is essential to theargument. As an estimate of "average opening rate" within the yieldzone, consider the crack tip opening displacement 6 = 2uy{—A,0+)divided by the time required for the crack to advance one plastic zonelength A at speed v, that is,

Under quasi-static conditions of small-scale yielding

A=?f—) > 6=Q-^-£-, (8.5.9)

where v is Poisson's ratio, so that in this case the opening rateestimate is

This combination of terms with a numerical coefficient of order unityis used to set the scale for opening rate, and the reference openingrate ii°y is chosen to be

( 8 ' 5 ' n )

where /? is a dimensionless viscosity parameter. In terms of /?, therelation (8.5.7) becomes

} (8.5.12)

or, for steady crack growth at speed v,

<j(r?) = (To [l - p—-u(v)] • (8.5.13)

Thus, for a crack speed that is a modest fraction of cs and an "averagecrack opening" over the length of the yield zone that is on the orderof cro//x (in the range from 0.01 to 0.001, say) the material rate effecton the flow stress will be significant if (3 has a value greater thanabout two or three. The limiting value of /3 = 0 corresponds to theinviscid ideally plastic case. The relationship (8.5.13) is used in theanalysis to follow, and numerical results are presented for /3 in therange 0 < J3 < 10.

If uj(r)) is eliminated from (8.5.6) and (8.5.13), the result is thesingular integral equation

1 , 0 < , < l , (8.5.14)

where

f(fl) = ^ L , tan(7r7) = /S^IT •

The parameter 7 is in the range 0 < 7 < | . For v/cs <C 1, the pa-rameter has the asymptotic form tan(vr7) = (1 — u)/3v/cs + O(^3/c^),and 7 —• \ as V/CR —• 1. A solution of (8.5.14) is sought which meetsthe conditions that <r(0) = ao and a(rj)y/l — r] /ao —» 0 as rj —>> 1~.

A general solution of the integral equation (8.5.14) subject to theend conditions of interest here is given by Muskhelishvili (1953). Oneform of this solution is

Jo» (1-rip Jo si(l-mi) d, ( M J 6 )

\

4 -

3 -

1-

-1.0 -0.8 -0.6 -0.4

x/ A-0.2 0.0

Figure 8.16. Distribution of the traction (T(X) within the cohesive zone from(8.5.17) for five values of parameter 7.

for 0 < 77 < 1. The integral in (8.5.16) is identified as a hypergeometricfunction 2F\ which, in turn, has a representation in terms of theGamma function F(-) . Consequently, alternate representations ofthis result are

_ sin(7T7) r/7+2 557 + §;

(8.5.17)

V-

The infinite series converges for 0 < 77 < 1. For 7—^0, the right sideof (8.5.16) is one. The dependence of a(rj)/ao on 77 for five nonzerovalues of 7 is shown in Figure 8.16. It is evident from Figure 8.16that, as the influence of rate sensitivity becomes greater with otherfactors held fixed, the strength of the crack edge singularity increasesand the fraction of the yield zone over which the singular solutiondominates becomes greater. These same features were suggested bythe finite element numerical results reported by Freund and Douglas(1983) for dynamic mode III crack growth in a viscoplastic material.

With the cohesive stress determined, the relationship betweenthe yield zone size A and the applied stress intensity factor Kj can be

492

1.0-

0.8-

0.6-

0.4-

0.2-

0.0

8. Plasticity and rate effects

0.0 0.1 0.2 0.3 0.4 0.5

7Figure 8.17. Normalized length of the cohesive zone £(7) versus 7 according to(8.5.18).

found by means of (8.5.4). The power series representation of a{r})can be integrated term-by-term with the result that

A 8 | v , O

A - ( -r^7T 7T

1/2)12

^ 1 ) J(8.5.18)

The quantity £(7) is the ratio of the yield zone size for any Kj, (3 andv/cs to the zone size for any Ki but with /? —• 0 and v/c3 —> 0. Theratio £(7) is plotted for the full range of 7 in Figure 8.17 where it isevident that the function decreases monotonically from L(0) = 1 andthe limiting value is L(l/2) = 4/?r2. This is the distance from thecrack tip at which the stress distribution <jyy for the case of elasticmaterial response, that is, for (3 —• 00, equals the yield stress cro.

According to (5.3.9), the rate of energy flux into the crack tipregion per unit crack advance is

= {l-v)AI{v)K?/2n. (8.5.19)

For 7 < | , all of this energy is dissipated within the yield zone, but itis interesting to note the part of the yield zone in which most energy

o.o-1.0

Figure 8.18. The fraction of total energy flux into the crack tip region G that isdissipated in the leading —x/A fractional part of the cohesive zone according to(8.5.20).

is dissipated. To illustrate this distribution, values of the ratio

wiri) = - -^ \ a(x')uyix (x'i0+) dx'

= / 1 dX2 tan(7T7) JQ cro [ <ro J

(8.5.20)

have been computed for several values of 7. In short, w(rj) is thefraction of the energy G that is dissipated between the leading edge ofthe yield zone at x = 0 and the intermediate position x = —r]A. Thus,w(rj) must increase monotonically from w(0) = 0 to w(l) = 1. Thefunction has been evaluated numerically for several values of 7, andthe results are shown in Figure 8.18. From the figure, it is evident that,as material rate effects become more significant, the energy flowinginto the crack tip region is dissipated in a smaller fractional intervalnear the physical crack tip.

The crack tip opening displacement 6 = 2uy(—A,0+) is obtainedby integrating (JO(TJ) from r] = 0 to 77 = 1. The resulting expression is


an integral representation of a combination of Gamma functions, so

where Ai(v) is defined by (5.3.11). An interpretation of the relation-ship (8.5.21) is given in the next section.

8.5.3 The crack growth criteria

In this section, the two growth criteria suggested in the introductionto Section 8.5 are applied at the level of the crack tip plastic zone inorder to infer the relationship between crack speed and applied stressintensity factor that is required to sustain crack growth accordingto one or the other criterion. The inferred value of applied stressintensity factor is once again the theoretical dynamic fracture tough-ness. The main focus is on the theoretical fracture toughness versuscrack speed relationship because the results of experiments are mostcommonly reported in terms of the toughness-speed relationship, andit is the input required in order to do fracture mechanics calculationsto predict crack growth.

The two criteria adopted are the critical crack tip opening dis-placement criterion and the critical stress at a characteristic distancecriterion. The two criteria are applied separately at first, and thenlater they are evaluated with respect to each other. For there to bea basis for comparison, the criteria must be expressed in terms of thesame set of material parameters. The important parameters here areao, the slow loading flow stress, and 6C, the critical crack tip openingdisplacement for very slow crack growth. A critical stress intensityfactor Kjc is related to these parameters through

ao6c=^-^Kfc. (8.5.22)

This connection follows from overall energy balance under the con-dition when all of the energy flowing into the crack tip region isdissipated in the cohesive zone with uniform cohesive stress ao andcrack tip opening displacement 6C.

The crack tip opening displacement 6 is given in terms of theapplied stress intensity factor in (8.5.21). A crack growth criterion

usually associated with a "ductile" mode of crack advance is therequirement that 6 = 6C at all times. If the relation (8.5.21) isincorporated into this condition and if the result is viewed as anequation for Ki, then

( 8-5-2 3 )

This is the principal result for the critical crack tip opening displace-ment criterion. The right side depends only on the crack speed v/cs

and the viscosity parameter /?. For /3 = 0, the well-known resultthat Ki/Kic = [^/(i;)]"1/2 is recovered. This result follows directlyfrom the conservation of energy statement that G = ao6c. For somefixed value of (3 > 0, the dynamic toughness is an increasing functionof crack speed. As the crack moves more rapidly, the material isdeformed more rapidly and a larger cohesive stress is required in orderto achieve the requisite crack tip opening displacement. This increaseof stress in the crack tip region translates into an increased likelihoodof cleavage initiation in those materials that are intrinsically cleavable.This point of view leads to the second growth criterion.

As a simple cleavage crack growth criterion, it is required that thestress at a point in the yield zone at some fixed distance ahead of thecrack tip must always have a material specific value. In order to havea basis for comparison of the two criteria, the distance is expressedas a multiple, say q, of 6C and the critical stress is expressed as amultiple, say p, of <ro. The distance ahead of the tip q6c is expectedto be a small fraction of A and the critical stress level is expected tobe significantly larger than ao, so only the singular part of the cracktip stress distribution is used, that is,

The mathematical statement of the growth criterion is then

cr(rf) = pao when 77 = 1 — q8c/A (8.5.25)

orKj _ Uq(l - v)

V sin(7r7)

Numerical results are presented below for q = 2 and p = 3. Asis evident from (8.5.26), a value of cro//x is also required for thecalculation, and 0.005 was chosen as a representative value.

The interpretation of the result (8.5.26) is quite straightforward.If either the crack speed v/c8 or the viscosity (3 is small, then there islittle elevation of the stress in the yield zone due to rate effects. Con-sequently, a large applied stress intensity factor is required in order tosatisfy the stress-controlled growth criterion. As the crack tip speedincreases for some fixed level of viscosity, the local stress is elevateddue to rate sensitivity and a smaller applied stress intensity factor isrequired. This effect becomes stronger as the viscosity increases.

It should be noted that the definition of 11(7) in (8.5.24) is notcompletely unambiguous. The form shown was obtained by evaluatingthe coefficient of (1 — /?)~7 at rj = 1 for any value of 7, and then byusing the identity

Win n-h-rt r( f tl)r(&l 0 1 0 2 ) / o . 9 7 N2 Fi(a i , a2; &i; 1) = =77 ,r(h —7 (8.5.27)

r(& a)r(& - a2)which applies for b\ > ai + a2, which is the case here. This asymptoticexpansion about 77 = 1 is not uniform in 7. The fact that S(7)(l—r/)~7

takes on the correct value of a(ri)/cro as 7 —• 0 for any 77 < 1 appearsto be fortuitous.

It is clear that the relationships (8.5.23) and (8.5.26) have op-posite trends. For the deformation-controlled criterion, KijKic takeson minimum values when either the crack speed or the viscosity (orboth) is small, and this ratio increases as either v/cs or /3 increases.On the other hand, for the stress-controlled criterion, KI/KIC takeson its largest values when either the crack speed or the viscosity issmall, and this ratio decreases as either v/cs or p increases. Thus, forsmall viscosity and/or for low crack speed, the level of Kj requiredto satisfy the deformation-controlled criterion is less than the levelrequired to satisfy the stress-controlled criterion. For larger crackspeed or viscosity, however, the relative magnitudes of the requisiteKj levels for the two criteria are reversed.

If conditions of the body are such that crack growth can occur ineither mode, and that the actual mode is established as the one whichbecomes operative at the lower level of applied Kj, then a compositesurface of theoretical toughness versus crack speed and viscosity canbe constructed. This surface is obtained by selecting the lower KI/KIC


Figure 8.19. Surface of dynamic fracture toughness Kj — Kjd versus crack speedv/cs and viscosity parameter (3. The ridge corresponds to a change in growthcriterion.

value of the two relationships for any given values of v/cs and /?.The result is shown in Figure 8.19. With reference to Figure 8.19,the following crack growth behavior is represented. Suppose that acracked body characterized by a particular value of /3 is loaded so thatthe crack begins to advance from speed v = 0, and that the appliedstress intensity factor increases as the crack advances. The resultimplies that the crack will accelerate with the separation on a localscale occurring according to a ductile mechanism. In terms of thesurface in Figure 8.19, the state of Kj and v follows the surface alonga path for which /3 = constant, starting from Kj = Kic and v = 0.The crack accelerates until a speed corresponding to the position ofthe ridge in the surface is attained. At this point, separation convertsto a stress controlled mechanism due to the rate-induced elevationof flow stress. The only way for the applied stress intensity factorto further increase is for the crack speed to suddenly become verylarge, so the state ends up on the rapidly rising portion of the surfaceassociated with inertial effects (not shown in Figure 8.19). Thereafter,if the applied stress intensity factor decreases, then the state falls tothe minimum in the path for fixed /?. Further decrease of the appliedstress intensity factor implies crack arrest, or no further growth can


be sustained according to either of the possible criteria. These generalfeatures are consistent with the observations of deWit and Fields(1987) in their experiments on crack arrest in pressure vessel steels.

The results of applying the critical crack tip opening displacementcriterion and the critical stress criterion separately show the samequalitative features that were found in the corresponding continuumplasticity results reported in Sections 8.3 and 8.4, respectively. There-fore, it is expected that if the continuum plasticity result analogousto Figure 8.19 could be generated, then it too would have these samefeatures. Of course, this result has been obtained by assuming thesimplest possible crack growth criterion, and by selecting particularvalues of the parameters p and q in (8.5.25). A choice of otherparameters would yield results with differing magnitudes, but thequalitative features appear to arise nonetheless within the frameworkof the model.

8.6 Ductile void growth

The examination of fracture surfaces of a wide range of metals andother materials following tensile crack growth under plane strain con-ditions leads to the conclusion that the process of crack advanceis essentially the nucleation of voids or cavities at material inho-mogeneities, and the subsequent ductile growth of these voids tocoalescence. The voids nucleate at second phase particles in theductile matrix, due to either decohesion at particle-matrix interfacesor cracking of the particles. The process of material failure by voidgrowth to coalescence has been reviewed by Tvergaard (1989). Cer-tain features of the elastic-plastic crack tip field, which provides theenvironment in which the mechanism operates, are important for thisprocess. Among these are the high triaxial stress condition within thesmall strain region ahead of the crack tip that serves to nucleate voidsand the zone of large plastic straining directly ahead of the tip thataccommodates the ductile expansion of voids necessary for coalescencewith the main crack. The mechanical process of the ductile growth ofcylindrical and spherical voids in plastic materials has been describedby McClintock (1968) and Rice and Tracey (1969), respectively, whoshowed the strong influence of the mean normal stress component onthe rate of growth of an isolated void.

McClintock (1968) and Rice and Johnson (1970) developed mod-els of the void growth process within the crack tip field with a view

8.6 Ductile void growth 499

toward relating microstructural features to fracture mechanics pa-rameters for fracture initiation and stable crack growth. Thoughthese models can be viewed only as rough approximations, they doappear to capture the essence of the hole growth mechanism whenit is not terminated prior to coalescence by strain localization. Itshould be emphasized that the assumption of a fully ductile separationmechanism on the microscale does not necessarily imply extensiveplastic flow in the body containing the crack. Indeed, it is quitepossible to observe fully ductile separation at a crack tip in a materialfor which the plastic zone size is small and conditions of small-scaleyielding are satisfied. Likewise, extensive plastic flow in the body doesnot imply that the separation mechanism is a ductile mechanism. Itcould just as well be cleavage induced by material rate effects or strainhardening as a result of the plastic flow.

Consider the situation deep within the active plastic zone nearthe edge of a crack advancing rapidly in the plane strain tensile modethrough an elastic-plastic material. Based on the asymptotic analysisdescribed in Section 4.5 and on the calculations of Lam and Freund(1985), it appears that there is a region of high stress triaxiality aheadof the advancing crack which serves as a driving force for nucleationof voids in the material. However, the concentration of plastic strainat the crack tip is not nearly as great for crack growth, even quasi-static growth, as it is for a stationary crack tip subject to monotonicloading. For example, the crack tip opening displacement obtainedfrom a small strain continuum plasticity analysis is zero for steadygrowth, whereas it has a finite value proportional to (Ki/ao)

2 for astationary crack under small-scale yielding conditions, where Kj is theremote elastic stress intensity factor and <ro is the tensile yield stress.

This difference is significant, for it implies that the macroscopicplastic field cannot provide the large strain zone ahead of the crack tipthat is necessary to accommodate void growth. Thus, if a large strainregion is to exist, and it seems that it must, then an alternative is thatthe void growth process itself must lead to such a region. This requiresthe sympathetic growth of voids in the crack tip region to form a smallregion of cavitated material adjacent to the tip. Suppose that voidsgrow by ductile expansion to coalescence without the intervention ofsome other mode of joining. Then, for a given distribution of voidnucleation sites, the weaker plastic strain concentration for growthimplies that a greater remote driving force is required to sustaingrowth than to initiate growth. If the crack is growing rapidly, then


the velocity of material particles on the surfaces of the expanding voidsare potentially enormous, although occurring over a small volumeof material. If the inertial resistance to this motion is manifestedmacroscopically as an effect on the applied driving force required tosustain fracture, then the apparent dynamic fracture toughness shouldbe an increasing function of crack speed, even without the influenceof material rate sensitivity. This was indeed suggested by the analysispresented in Section 8.3 for the case of antiplane shear crack growth,and the purpose of this section is to examine the influence of inertiaof individual voids undergoing rapid ductile expansion. Some obser-vations on the role of material strain rate effects are also included.

8.6.1 Spherical expansion of a void

In order to examine the influence of material inertia on a small-scaleon the ductile void growth process, a simple spherically symmetricmodel introduced by Carroll and Holt (1972) is adopted. This modelwas further developed within the framework of dynamic spall fractureby Johnson (1981). Consider a thick spherical shell of incompressiblematerial with inner and outer radii r = a(t) and r = &(£), respectively,where r is the eulerian coordinate representing distance from the fixedcenter of the shell. The material is initially at rest with void radiusa0 = a(0). At time t = 0, a uniform normal traction of magnitude atbegins to act on the outer surface. The inner surface is traction free.It is assumed that elastic effects are negligible and that the magnitudeof at, is sufficient to produce ductile expansion of the shell.

With reference to the discussion of ductile void growth and coales-cence as a mechanism of crack advance, the parameter ao is identifiedwith a representative physical dimension of the inhomogeneity thatcauses nucleation of the void, and 26O = 26(0) is identified with thespacing of void nucleation sites. If a body with a periodic array of suchvoids is subjected to a mean normal stress a^ resulting in sphericalgrowth of the voids, then the body "fractures" at some later time, sayt = £*, for which a(t*) — b$. The objective of this model calculationis to determine the relationship between ab and t*. With referenceto a crack growth process, 2bo/t* and at, provide crude estimates ofcrack speed and crack tip field intensity, respectively.

The velocity in the radial direction of a particle with instan-taneous radial coordinate r is vr(r, t). Material incompressibility


requires that

- ^ + 2— = 0 , (8.6.1)or r

or thatvr(r,t) = aa2/r2 . (8.6.2)

Furthermore, the inner and outer radii are related by

b3 - bl = a3 - al, (8.6.3)

identically in time.The momentum balance equation in terms of true stress compo-

nents in the eulerian coordinate r is

dcrr 2(ar - at) _dr + r

where p is the material mass density, aT is the radial component ofnormal stress, and at is the component of normal stress in any direc-tion transverse to the radial direction. Substitution of the expression(8.6.2) for vr into (8.6.4) and integration of the result with respect tor from a to b yields

- ^r ) - 2

(8.6.5)

This result is independent of the constitutive description of the ma-terial, except for the assumption of incompressibility. It is tacitlyassumed in writing (8.6.5), however, that the normal stress differencear — at is completely determined by the velocity field by means ofa constitutive description of flow for either rate-independent or rate-dependent material response.

To examine the influence of inertia on void growth, suppose thatthe material is perfectly plastic with tensile flow stress of magnitudeao. Note that material elements in the radial direction shorten as thespherical shell expands, so ar — at = —cro in this case and

dr = -ao In - . (8.6.6)

Figure 8.20. Relationship between the tensile traction applied to the surface r =b(t) and the time t* defined by a(t*) = bo, obtained by numerical integration of(8.6.5).

Thus, the minimum applied stress a^ required to produce flow in thesphere is {pb)min = 2<j0ln(&o/ao)- For any given value of 07,, therelation (8.6.5) provides a second-order ordinary differential equationfor a(t) subject to the initial conditions that

o(0) = a0 , d(0) = 0. (8.6.7)

This equation can be solved numerically for any values of bo/a>o > 1and (Jbl{pb)min > 1- A particular result is shown in Figure 8.20 forthe case of bo/ao = 20 in the form of a graph of <Tb/(&b)min versus60y/p/&o /t*. Recall that t* is defined by the condition that a(t*) =b0. It is evident from the result that any applied stress only slightlylarger than (ab)min will result in growth of the void to critical sizein a time t* > \b$\Jp/cro. However, if the void must be grown tocritical size in a shorter time, then a larger stress o\> is required toovercome the inertial resistance.

Based on this simple model calculation, the time ^boy/p/cr0

appears to have particular significance for assessing the influence ofmaterial inertia on the physical scale of the ductile hole growth mecha-nism. This time can be estimated for high strength steel or aluminumalloys for which the crack growth mechanism is predominantly ductile


void growth. Of greater interest, perhaps, is the "speed" 2bo/t* which,as noted above, is a crude estimate of crack propagation speed if theinitial void nucleation site spacing is 2&o and the time required togrow the voids to the critical size is t*. Thus, microscale inertia issignificant for bo/a>o = 20 if

26o/** > W°o/p • (8.6.8)

If the parameter \Joolp has a value of 300 m/s, which is typical forhigh strength alloys, then (8.6.8) implies that the crack speed at whichlocal inertial effects result in an observable influence on macroscopicdynamic fracture toughness is about 1200 m/s. This speed is some-what greater than the observed crack speed at which the measured orinferred dynamic fracture toughness for such materials begins to showa dramatic increase with speed, which is usually in the range of 500-1000 m/s. The model can provide only a very rough approximation,and a systematic study of the influence of microstructural features onits implications is not yet available.

Up to this point, the spherical expansion model has not includedan influence due to material rate effects. Some information on themagnitude of material viscosity effects can be obtained in the followingway. Suppose that the rectangular components of the plastic stretch-ing rate tensor d^ depend on the instantaneous deviatoric stress Sijcomponents according to

V ^ (8.6.9)

where j o is a material parameter with dimensions of strain rate, nis a material constant, and r is given in terms of s^ by (8.4.7). Theparameter j o is essentially the strain rate at which rate effects becomesignificant for homogeneous straining.

For the case of spherical expansion, d% = dvr/dr = —2df and

( \ l/n

— — ^ r ) • (8.6.10)7o r3 J

In this case, the last term of (8.6.5) becomes, • \ l/n

f crt-ar b aon / 2V3 a \ r 3 / n i/ dr = aoln- + —- — 1 - (a/6) ' .

Ja r a 3 y 70 aj I J(8.6.11)

It is evident that (8.6.11) reduces to the rate-independent limitwhen the viscosity parameter j o —• oc. The question of the relativemagnitudes of the two terms on the right side of (8.6.11) has notbeen considered in detail. A rough estimate of the magnitude of theviscosity term relative to the strength term can be obtained in thefollowing way. Suppose that b/b is identified as a representative overallreference plastic strain rate, say 7 ^ , for the voiding material. Promthe incompressibility condition (8.6.3) it follows that a « j^^/a2.For the early stages of voiding when a <C 6, the ratio of the viscosityterm to the strength term is roughly equal to

(8.6.12)

For n = 5, ^vre± = 7o/10, and b/a = 25 this ratio has a value of about

2.9, implying that viscosity effects are relatively large compared tostrength effects in the early stages of void expansion at relativelymodest values of overall strain rate. The factor (b/a)3/n has a stronginfluence on the ratio (8.6.12). For example, if the viscosity is linear,that is, if n = 1, but the values of other parameters are unchangedthen the ratio has a value of about 560. For ductile materials, thestrains around expanding cavities are quite large and an exponent nin the range from 3 to 10 might be reasonable (Clifton 1983).

A similar conclusion on the magnitude of viscosity effects com-pared to the inertial effects was reached by Curran et al. (1987). Theyconsidered the spherical expansion model under the assumption oflinear viscous material resistance to flow and no residual strength,that is, they assumed n = 1 and ao = 0 in (8.6.11). In this case,the process is insensitive to the value of b as long as b ^ a, and thedifferential equation governing void expansion (8.6.5) becomes

3 d 2 16 // a ab /Q* i -na + h —— « = — . (8.6.13)

2 a 3 fryo a2 pa

They included an effect of material strength indirectly by assumingthat 0-5 is actually the excess of the applied stress over a thresholdstress given by | a o [1 + ln(2///ao)], where // is the elastic shear mod-ulus. This threshold stress, which was obtained by Hill (1950), is theremote isotropic tension required to produce yielding at the traction-free surface of a spherical cavity in an incompressible elastic-ideally


plastic material. Through numerical integration of this equation formaterial parameters corresponding to a particular aluminum alloy,they concluded that in dynamic ductile fracture the threshold stressand the material viscosity are dominant; inertial effects become sig-nificant only for quite large voids, greater than about 10 /im or so forthe particular material considered.

Although this area requires extensive quantitative modeling tosort out the complicated features of the interconnected phenomenainvolved, some qualitative observations on the process of rapid growthof a macroscopic crack through a material that separates by meansof a void growth mechanism can be made. As the speed of the crackincreases, voids must be grown at an ever increasing rate from verysmall size to coalescence in order to accommodate crack advance. Thespherical expansion model suggests that an ever increasing appliedstress ((Jb in the model) is required to drive this process, and this stressis identified with the crack driving force. The increase in crack drivingforce with crack speed for materials that separate by the mechanismis surely evident in observations.

There are other ramifications of this stress increase that areconnected with the prevailing crack tip stress fields. For one thing,the presence of higher stress levels in the crack tip region implies thatvoid nucleation sites that might have remained dormant during slowcrack growth can be activated during rapid crack growth. A secondobservation is based on the spatial structure of the crack tip fields.To illustrate this point, consider the plane strain elastic crack tipfield for which the stress intensity factor is the driving force. In thiscase, if the crack driving force is doubled, then the linear dimensionof the region around the crack tip over which the stress magnitudeexceeds some given value is increased by a factor of four and the areaof this region is increased by a factor of sixteen. The implication ofthe observation is that the increased resistance of the material nearthe crack tip to high rate voiding causes the region of high stress, andconsequently the region over which damage accumulates, to spreadover a significantly larger volume of the material around the crack tip.This suggests that the size of the region near a crack tip over whichthe physical process of voiding progresses is larger and more diffuseduring rapid crack growth than during slow growth. This increasedsize results in a more textured fracture surface. It also implies aconnection between crack tip stress fields and the size of the regionover which the physical separation mechanism is operative. A tacit

assumption in the fracture mechanics approach is that this scale isset by the material microstructure and that it is independent of thesurrounding stress field. This point has been discussed qualitativelyby Broberg (1977).

8.6.2 A more general model

The process of dynamic growth of a void in a plastic material sub-jected to a far-field strain rate and mean normal stress was studiedby Glennie (1972), who applied the results to a model of dynamicfracture by the void growth mechanism. The features of the modeland the principal results are summarized here. Consider the flow ofan isotropic rigid-ideally plastic material that deforms according toan associated flow rule. Suppose that an unbounded body of thismaterial experiences a spatially uniform stretching rate, or rate ofstrain with respect to the instantaneous configuration, say dij. Thedeviatoric stress is determined from the strain rate as

y/3dpqdpq/2(8.6.14)

where ao is the flow stress in simple tension. If Vi is the velocity fieldwith respect to spatial or eulerian coordinates #*, then the momentumequation

f dvi\ .^ ) ( 8-6-1 5 )

yields the mean normal stress field, where cr = s^ + \<Jkkt>ij is thetrue stress and p is the material mass density. The particle velocityfield is

Vi = dijXj , (8.6.16)

where one material point has been fixed arbitrarily at the origin.The material spin rate is zero everywhere. The momentum balanceequation is satisfied when the mean normal stress is

jj iXk , (8.6.17)

where a is an arbitrary function of time.With the uniform deformation rate field as a reference, suppose

that a spherical void with traction-free surface is introduced into


the body, centered at the origin of the coordinate system, withoutchanging the remote conditions on deformation rate or mean normalstress. Two assumptions are then made concerning the deformationfield in the presence of the expanding void. In a somewhat simplifiedform, it is assumed that the void is small enough that the backgroundstrain it experiences is spatially uniform and that the velocity fieldin the presence of the void rapidly approaches the velocity field for auniform rate of deformation as distance from the origin becomes largecompared to the void radius.

On the basis of these assumptions, a variational principle wasapplied in order to determine the rate of growth of the void in termsof the applied mean normal stress a and the background uniformdeformation rate, say d^. The procedure is similar to that developedby Rice and Tracey (1969) for quasi-static void expansion.

Under these conditions, which do not correspond to sphericalsymmetry, the void does not grow in a self-similar fashion. However,at this level of approximation, the size of the void is described bya single length parameter, say a(t), which can be interpreted as themean void radius, as the radius of some equivalent spherical void,or as the cube root of the void volume. With this understanding,Glennie (1972) found that the expansion of the void is governed byan ordinary differential equation of the form

d = 0.372Asinh | - ^ - {a - p (ad + | d 2 - ±A2a2)} 1 ,L2^o J

(8.6.18)

where A = JM^df?/2 represents the influence of the backgrounddeformation rate and a is the remotely applied mean normal stress.

Implications of the result (8.6.18) for fracture dynamics are exam-ined by Glennie (1972) through numerical solution of the differentialequation and the development of a model of crack tip processes.Conditions under which material inertia is important are evident from(8.6.18). It is known from the results of McClintock (1968) and Riceand Tracey (1969) that the presence of a mean normal stress has astrong influence on void growth rate, as can be seen from the expo-nential dependence of a on a/ao in (8.6.18) when p = 0. It is evidentfrom this equation that the effect of inertia is to counteract the meannormal stress. Consequently, inertial effects become significant whenthe variation of a(t) is such that the inertial term in the argument


of the hyperbolic sine function in (8.6.18) becomes comparable to themean normal stress.

Consider the particular case when the void grows at essentiallyconstant rate from a relatively small initial size to a final size b0 in atime £*, that is, a(t) « bot/t*. The notation of the previous subsectionis adopted here. Furthermore, suppose that the mean normal stressa is about 3cro, as implied by the small strain crack tip solution foran ideally plastic material. This, in turn, implies that inertial effectsare important when 3ao « ^p(bo/t*)2 or

2bo/t* » 4.83 V W P , (8.6.19)

which is quite similar to the result obtained in (8.6.8).While these ductile void growth models provide only rough ap-

proximations to the physically important mechanism of crack advance,they do suggest that inertial resistance to material motion on the smallscale may be important in the interpretation of the apparent dynamicfracture resistance of real materials under realizable conditions. Otherinfluences, such as void interaction, strain hardening, and materialrate sensitivity, will require thorough study before these effects canbe understood. Another complicating factor in examining the processis that voids can join by mechanisms other than simple expansion upto coalescence. For example, fracture surfaces reveal some evidenceof plastic strain localization in ligaments between voids, resulting incoalescence by shearing of the ligament. The mechanism appears torequire movement of less material than continued void expansion, soit may be promoted by inertial effects.

8.7 Microcracking and fragmentation

If a macroscopically homogeneous body of brittle material fracturesdue to quasi-static loading or stress wave loading of low intensity,the process normally consists of the formation of a single dominantcrack that grows across the load carrying section of the body, therebyunloading adjacent material. Apparently, this crack growth ariseswhen a particular flaw in the body is stressed to a critical condition,and the growth process becomes catastrophic before growth of anyother potential fracture site is activated. On the other hand, if abody of brittle material is subjected to intense impulsive loading or to

8.7 Microcracking and fragmentation 509

radiant deposition of a nonmechanical form of energy that is rapidlyconverted to mechanical energy, then a higher density of fracturescan be nucleated and they can grow to a significant size withoutarresting each other. That is, the loading is sufficiently intense tocause more than one flaw, and possibly very many flaws, to precipitatecrack growth before these potential fracture sites are unloaded bystress waves from a single growing crack. This seems to require aninitial energy density, in the form of kinetic energy or elastic energywith predominantly compressive stresses, that is much greater thanthe body could sustain as elastic energy with predominantly tensilestresses. The result of many cracks growing is the development ofdistributed damage in the material. If many cracks grow to inter-sections with other cracks or if they grow to the body surface, thenthe body is split into many fragments. The fracture resistance ofrocks, ceramics, and other brittle materials under impulsive fractureconditions is commonly observed to be significantly greater than theresistance under slow loading conditions. Consequently, the process isoften viewed as rate dependent on the macroscopic scale. The originof the rate dependence appears to be inertial and frictional effects onthe microscale.

The fragmentation process involves a number of physical mech-anisms. It most cases, the process is too complex to be viewed asbeing completely deterministic, and the evolution of the process mustbe considered in terms of statistical characteristics. Pioneering workincorporating the statistical aspects for ductile fragmentation wasreported by Mott (1947). The statistical and geometrical aspects ofbrittle fragmentation have been considered by Grady and Kipp (1985).The process is extremely complex and no theoretical models of broadapplicability are available. The phenomenon is important in mining,materials processing, and other applications, and some steps towarda rational basis for observed phenomena have been provided. Oneconceptual approach based on overall energy balance is discussed hereon the basis of a particular model and followed by a brief discussionof a more mechanistic approach.

8.7.1 Overall energy considerations

A specific example is considered in order to obtain an estimate ofthe size of a fragment due to impulsive fracture of a brittle mate-rial. Consider the process of fragmentation of a thin circular ring orcylinder due to sudden radial expansion. Suppose that the ring has


mass density p and a uniform cross-sectional area A. At a certaininstant, the ring is impulsively loaded so that it begins to expandsymmetrically from rest with radial speed vo. If the mean radiusof the ring is r, then the tensile strain rate in the circumferentialdirection in the ring is eo = vo/r. Suppose that the ring expandsand then splits into n equal-sized fragments, each of circumferentiallength d such that nd = 2nr. If the initial loading is very intense, thenthe fragment size can be estimated in terms of material parametersand initial conditions by means of an approach introduced by Grady(1982). The loading is intense, for example, if the initial kinetic energyin the ring is very large compared to the elastic energy in the ring whencracks begin to form.

Consider one fragment of length d, and view the velocity of anyparticle in the fragment as the sum of the velocity of the centerof mass of the fragment and the circumferential velocity relative tothe translation of the center of mass in the radial direction. Thekinetic energy associated with the translation of the center of mass isinitially very large. The work of the forces producing the fractures issmall by comparison, so the contribution to the kinetic energy due totranslation of the center of mass does not change significantly. Thus,the final radial speed of the center of mass of each fragment is onlyslightly less than the initial radial speed.

The principal source of energy available to drive the fracturesis the kinetic energy of the material arising from motion relative tothe center of mass. If a fragment is viewed as a straight rod forpurposes of estimating this kinetic energy, then the relative velocityis approximately vrei(x) = eox, where x is the coordinate along therod measured from the center of mass. The associated kinetic energyis

fd/2Trel = \Ap \ e2

ox2 dx = ±Ape2

od3 . (8.7.1)

J-d/2For purposes of comparison, the kinetic energy associated with radialmotion is about \pAde^r2 = 12Treir

2/d2. Suppose that Tc is theaverage specific fracture energy for the material and that the area ofeach fracture surface is the cross-sectional area A. Then a fractureenergy of ^TCA is consumed at each end of the fragment from theenergy supply within the fragment. If this is set equal to the kineticenergy supply in (8.7.1) then

(8.7.2)

Thus, an estimate of fragment size d in terms of the initial staterepresented by eo and the material parameters F c and p is

<8-7-3)

If F c is eliminated in favor of a fracture toughness through therelationship F c « K^/pc2, where IQr is a representative fracturetoughness for high speed crack growth which is several times largerthan Kjc for the material, then

- ^ \ , (8.7.4)

where co = y/E/p is the bar wave speed for the ring material. Toconsider a specific case, consider a material for which the relevantcrack growth toughness KcT is about 50MPav/nT, the mass densityp is about 7000 kg/m , and the pertinent elastic wave speed c isabout 5000 m/s. If the ring is subjected to an axially symmetric,radial impulse that induces a circumferential strain rate of 5 x 104 s"1

then the estimate of fragment size on the basis of (8.7.4) is d «3 mm. If the initial radius of the cylinder r is about 3 cm then thekinetic energy associated with radial motion is about 100 times largerthan that due to motion of the material in each fragment relativeto the fragment center of mass. These parameters are within therange of parameters reported by Weimer and Rogers (1979) for aseries of experiments on fragmentation of a high strength tool steel.The specimen configuration was a long thin-walled cylinder, and thespecimens were loaded by internal explosive charges. Grady (1982)examined the data in terms of the energy balance model, with theconclusions that the formula (8.7.4) overestimates the fragment sizeby about 30 percent and that the observed variation of fragment sizewith toughness measure seemed to follow the 2/3 power dependencesuggested by (8.7.4).

A feature of the process that was exploited above, namely, thatthe energy consumed in fracturing is extremely small compared tothe overall energy of the fragmenting solid, often emerges as a prin-cipal difficulty in analyzing fragmentation by means of overall energymethods. For example, Tilly and Sage (1970) studied the process of


impact of glass spheres against various surfaces. For values of theparameters in a representative impact, suppose that a glass particlewith diameter of 3 mm is projected against a steel surface at a speed of275 m/s. The kinetic energy of the particle is then about 1.3 J. Uponimpact, the glass sphere shatters into about 8000 fragments of meandiameter 150/xm. The total surface area of the fragments is about6 x 10~4m2. If the specific fracture energy is taken to be 10 J/m2,then the net energy dissipated in fragmentation is only about 0.006 J.Even if the area of fracture surface created is underestimated by anorder of magnitude, the energy dissipated in fracturing is still smallcompared to other energies involved in the process. Relatively largeamounts of energy must be dissipated through frictional interactionof the fragments or it must be radiated into the impacted body bymeans of stress waves.

8.7.2 Time-dependent strength under pulse loading

Brittle materials such as rocks, glasses, and ceramics exhibit virtuallyno macroscopic plastic deformation when subjected to tensile loading.Furthermore, for loading that is applied relatively slowly, the responseup to and including fracture is insensitive to the rate of loading.The deformation appears to be elastic for the early stages of loading.Because these materials are incapable of plastic deformation except atelevated temperature, their response is very sensitive to the existenceof structural flaws in their microstructure. Examples of such flawsin rocks and ceramics are poorly bonded grain boundaries, inclusionparticles, and discontinuities in elastic stiffness at grain boundaryjunctions. In glass and brittle plastics, the flaw structure, aside fromsurface cracks, has not yet been fully characterized.

Once the level of applied loading is increased to the point whereone flaw can be extended in a cracklike manner, it grows catastroph-ically into a macroscopic crack completely splitting the sample. Thesize and severity of these "critical flaws" in nominally identical spec-imens can vary, accounting for the scatter of experimental data onthe strength of brittle materials. Similarly, the likelihood of a mate-rial sample including a flaw that is critical at any given stress levelincreases with the absolute physical size of the sample, accountingfor the "size effect" of experimental data on the strength of brittlematerials. Simply stated, if the tensile strengths of two samplesof nominally the same material are measured, the smaller sampletypically will have greater strength because the larger sample typically


will contain a more severe strength limiting flaw than the smallersample. Such observations are meaningless when actually appliedto only two samples, and they must be understood as statisticalinferences based on measurements on many samples.

The statistical approach to brittle fracture was pioneered byWeibull (1939). He imagined that all of the flawed elements in amaterial sample were arranged in series, and that the sample wouldfracture when the common load was increased to the strength levelof the critically flawed "weakest link". Furthermore, he proposed astatistical distribution of flaw sizes throughout the material to accountfor the size effect of strength mentioned above. The distribution, asample of which will be used in the development to follow, is expressedin terms of parameters that can be determined for a given materialonly through an appropriate series of experiments. The Weibull statis-tical approach is discussed within a general framework by Freudenthal(1968).

The situation is quite different for intense pulse loading of a brit-tle material. In this case, the weakest link concept is not applicable.Suppose that a sample of material is deformed very rapidly so thata stress level significantly greater than that required to precipitategrowth of the most severe flaw is achieved in a short time. Once thestress level necessary to produce fracture at the most severe flaw isreached, a finite additional time may elapse before fracture can actu-ally begin. This delay time was called the incubation time in Section7.5, and an illustration of the idea is given in (7.5.6). During thistime, the macroscopic stress on the sample can continue to increaseas the macroscopic deformation proceeds. As it does so, the criticallevel of stress to produce crack growth can be reached at many otherdefects in the material. Thus, crack growth can be initiated at verymany flaws if the loading is applied rapidly.

As the flaws begin to grow as microcracks and as the overalldeformation proceeds, the stress at first continues to rise. As themicrocracks become longer, however, the effective stiffness of thesample diminishes and, correspondingly, the rate of increase of stresscorresponding to a fixed rate of deformation diminishes. At someinstant of time, the rate of change of stress on the sample becomeszero even though the deformation rate is maintained at a fixed level.Thereafter, the stress level falls as the deformation proceeds. Eventu-ally, the growing microcracks coalesce into one or more macroscopiccracks. In an average sense, this occurs when each growing microcrack


has extended to a length equal to the initial spacing of activatedflaws. The maximum stress acting on the sample during this processis called the impact strength as at constant strain rate, and the timeat which the maximum occurs after stressing begins is identified asthe minimum duration of the high rate deformation that is required toproduce a macroscopic failure. These qualitative ideas are next givena simple mathematical structure for a particular plane strain model.

Consider a sample of brittle material initially at rest and stressfree. At a certain time instant, the sample begins to undergo homoge-neous tensile straining at a constant strain rate eo. The macroscopictensile stress required to sustain this rate of deformation is cr(t), andthe elastic modulus of the material in the absence of microcracks is E.For purposes of this discussion, the instantaneous elastic modulus ofthe material with many microcracks is denoted by E. This modulusis defined as the elastic modulus that would be perceived for thesample for a given microcrack distribution if the sample were deformedhomogeneously under equilibrium conditions with no extension of themicrocracks. This definition probably results in an estimate of theapparent stiffness that is too low because it neglects the finite timerequired for dynamically growing microcracks to interact by means ofstress waves.

If the sample is large enough that its macroscopic response isrepresentative of the material, then the size of the sample cannotenter the relationship between macroscopic stress and strain. But ifthe effective stiffness is assumed to depend on an average microcracklength, say a, then an additional length scale must be identifiedon dimensional grounds. This length is assumed to be the averageinitial spacing between flaws that are activated or, equivalently, theinitial density of activated flaws per unit volume (per unit area in twodimensions).

In his original statistical approach, Weibull assumed that therewas some stress level, say awi below which no microcracks would growfor a given material. For stress levels a > aw, he assumed that thenumber of flaws per unit volume (or per unit area in two dimensions)having a strength equal to or less than a is Poisson distributed. Onthe basis of probability arguments he concluded that the density offlaws with strength less than or equal to a has the form

where bw is a material length scale, m is a positive material con-stant, and a =1, 2, or 3 for flaw distributions in one, two, or threedimensions, respectively. The parameters aw, bw and m can onlybe determined through experimental measurements. Although theweakest link idea was used heuristically by Weibull (1939) in arrivingat (8.7.5), the idea has been shown to lead more directly to the sameresult on the basis of the asymptotic theory of extreme values in largesamples of a statistical population. In any case, the characteristiclength parameter for purposes of comparison to the average micro-crack size is [n{a)]~1/°t.

On the basis of features identified up to this point, the time-dependent stress acting on the sample for constant strain rate is

a(t) = E (a(t)y/n(a)^ eot + aw (8.7.6)

for a = 2, where the dependence of E on the microcrack distributionhas been made explicit. It is tacitly assumed in writing (8.7.6) thatcr(O) = aw. This equation can be solved for a as a function of time ifa(t) is known. A general framework for writing equations of motionfor crack tips is presented in Section 7.4. For example, with referenceto (7.4.2), if the equilibrium stress intensity factor X/(£, a, 0) for amicrocrack of length 2a(t) acted upon by stress a(t) is assumed to be

, then an evolution equation for a(t) is

where F is the specific fracture energy. The initial conditions forthis differential equation are established by the initial flaw size. Thisequation of motion implies that if a crack grows to twice the initialflaw size under constant stress then the speed has increased to 0.75 CRif F is a constant. Because microcracks are expected to grow to manytimes their initial sizes, the initial length is relatively unimportant(especially at this crude level of modeling) and the growth speed islarge for most of the growth time interval. Consequently, a largeconstant speed is assumed here, that is, a(t) « vt is assumed for thetime dependence of the average crack length, where v is a constant inthe range 0 < v < CR.

A specific form of the dependence of E on its argument is re-quired for a calculation. Budiansky and O'Connell (1976) presented


results for the elastic moduli of materials with a dilute distributionof randomly oriented elliptic microcracks. The analysis, which wasbased on the self-consistent theory of material inhomogeneities, wasextended to the case of dilute distributions of microcracks with non-random orientations by Hoenig (1979). For present purposes involvingone-dimensional deformation based on plane strain crack analysis,the effective modulus result of Delameter, Herrmann, and Barnett(1975) is adopted. They determined the effective modulus of a two-dimensional elastic solid containing a rectangular array of identicalcracks, each perpendicular to the tensile stress axis. For the case ofcracks of length 2a with centers spaced at a constant interval b fromeach other on planes separated by a distance h, the effective modulusgiven by Delameter et al. (1975) is

This expression does not presume that the microcrack concentrationis dilute, but it is based on a microcrack distribution with an unreal-istically high degree of geometric regularity.

For this estimate of the elastic modulus of the microcrackedmaterial, the expression (8.7.6) for stress becomes

*(t) = e o t E / a w + 1=

aw 1 - i In (cos [(irvt/bw) (a(t)/aw - l)m]) *

To illustrate the time dependence of stress, values of the variousparameters must be selected. For several rock types (granodiorite,basalt, and limestone) Vardar and Finnie (1977) concluded that main (8.7.5) has a value in the range from 3 to 4.5. Here, a = 2, som = 2 is selected. They also concluded that aw has a value in therange 2 to 20MPa, and bw has a value that ranges from 10 m to2 cm, with the lower portion of the range being more representative.Values of the strain rate parameter

are chosen to be 1, 2, 5, 25, and 125. Thus, if v = 103m/s, bw =100 /im and E/crw = 2 x 104, the value r = 1 corresponds to a strain

5-i

4 -

b 3H

2 -

1-

r = 125

0.0 0.5 1.0

cRt/\1.5 2.0

Figure 8.21. Stress a(i) acting on a sample of microcracking material deformingat constant strain rate eo versus time t according to (8.7.9). The rate parameterr is defined in (8.7.10).

rate of 500 s *. Evaluation of (8.7.9) leads to the stress historiesshown in Figure 8.21, where CR is the Rayleigh wave speed. Thestress variation has the general features anticipated in the preliminarydiscussion.

The maximum stress for a given value of eo, which was termedthe impact strength as at constant strain rate, is readily found from(8.7.9) by the condition that

tot -^-(t8) = 0. (8.7.11)

Differentiation of (8.7.9) with respect to time with da/dt = 0 yieldsthe equation

1 - ^ In(cos7r£) - 4£tan7r£ = 0, (8.7.12)7T

where £ = (cr/aw — l)mvt/bw. The only root of this equation in therange 0 < £ < 0.5 is £ = 0.302. Therefore, the relationship

^ = i + (0.302Maw V vts)

l/m

(8.7.13)


4 -

2 -

r 0.3

- 0.2

- 0.1

0.0100 200

r300 400

Figure 8.22. Impact strength as versus loading rate parameter r implied by theconstant strain rate model. Also shown by the dashed curves are ts, the timerequired for the stress to reach the maximum level, and b(ts), a measure of theamount of crack growth at that time.

defines the locus of stress maxima in the graph of a(t) versus t; seeFigure 8.21 for example. The dependence of as and ts on strain rate eo

can be established by solving (8.7.9) and (8.7.11) for these quantities.No simple expressions are evident, but numerically solving for thevalues of the parameters used to generate Figure 8.21 yields the resultshown in Figure 8.22. Also shown in the figure is b(ts) versus strainrate. This quantity can be interpreted as an estimate of the spacingof activated flaw sites at the time that.the stress reaches its maximumvalue as.

It is interesting to note that Tuler and Butcher (1968) proposeda criterion for dynamic fracture due to damage accumulation that,in effect, anticipated the form deduced from mechanistic modeling.They suggested that if a{t) is a tensile stress pulse amplitude and t8

is the pulse duration, then a necessary condition for fracture due todamage is

dt = constant, (8.7.14)

where aw is the stress threshold for damage accumulation and {3 is apositive constant. If it is assumed that stress increases linearly from


aw at t = 0 to as at t = ts in the present model, then the integral

does indeed have a constant value if f3 = m. The observation followsimmediately from integration of (8.7.15) and substitution of (8.7.13).

The analysis of this simple model has been presented to illustratethe qualitative features of transient microcrack damage accumulationin a brittle material under intense stress pulse loading. The physicalprocesses involved are very complex, and a number of importantobservations have been made on the basis of detailed modeling andexperimental studies. Shockey et al. (1974) carried out impact ex-periments on a polycrystalline quartzite rock. They characterizedthe microcrack structure of the damaged material, and developed amodel of fragmentation on the basis of these observations. The modelwas embedded in a computational wave propagation code so that theimpact experiment could be simulated numerically. Vardar and Finnie(1977) considered experiments on several rocks in which the materialnear the surface of the samples was heated rapidly by means of shortbursts of energetic electrons. The resulting expansion of the materialnear the traction-free surface produced severe microcrack damage.They analyzed their results in the same spirit as the model describedabove to infer values of dynamic strength. Impact methods were alsoused by Grady and Kipp (1979) to create controlled dynamic fracturein polycrystalline quartzite rock samples. They showed that theirobservations were consistent with a micromechanical fracture modelbased on dynamic elastic fracture mechanics. Detailed computationson microcrack-induced damage evolution in brittle rock under pulseloading were reported by Taylor, Chen, and Kuszmaul (1986). Theyhave proposed detailed constitutive equations for large-scale numeri-cal simulation of rock fracture behavior.

Many aspects of the process of microcrack growth and coales-cence have been neglected up to this point in the study of the phe-nomenon. For example, the dynamic interaction of growing micro-cracks by means of stress waves is not well understood, nor is theprocess of crack coalescence. On the latter point, the experimentsdescribed by Melin (1983) on the coalescence of initially coplanartensile cracks suggest that the process may consume more energy thanis commonly assumed. Another issue that is not well understood


is the influence of the inertial resistance to opening of microcrackson the apparent effective stiffness of a crack elastic body. Crackbranching and bifurcation are other phenomena that are of potentialimportance in the fragmentation process but for which a mechanisticunderstanding is not yet available. The prospect for more completetheories of dynamic microcracking and fragmentation of rocks, ce-ramics, concrete, and glasses hinges on progress on these and relatedissues.

Finally, it is noted that some one-dimensional wave propagationprocesses may be analyzed on the basis of the simple model outlinedabove. As an example, consider the experimental procedure describedby Gran, Gupta, and Florence (1987). Suppose that a long cylindricalrod of a brittle material is loaded in static triaxial compression. Theaxial pressure is then released suddenly and simultaneously at theends of the rod. A stress relaxation wave propagates away from eachend and, barring any dissipation in the material, an axial tensile stressequal in magnitude to the initial compression is induced at the centerof the rod according to one-dimensional wave theory. If the level oftensile stress is sufficiently high, then microcracking is induced nearthe center of the rod as the zone of tensile stress expands. If the knownrise time of the stress pulse is assumed to determine the strain rateat the central cross section of the rod, then the result in Figure 8.21provides an indication of the stress level at which fracture will occurat that section and the time at which it will occur. One-dimensionalcomputations of the kind described by Taylor et al. (1986) provide ameans of studying the evolution of damage as a function of time andposition along the rod.

BIBLIOGRAPHY

ABDEL-LATIF, A. I. A., BRADT, R. C, AND TRESSLER, R. E. (1977), Dynamics offracture mirror boundary formation in glass, International Journal of Fracture13, pp. 349-59.

ABOUDI, J., AND ACHENBACH, J. D. (1981), Rapid mode III crack propagationin a strip of viscoplastic work-hardening material, International Journal ofSolids and Structures 17, pp. 879-90.

ABOUDI, J., AND ACHENBACH, J. D. (1983a), Arrest of fast mode I fracturein an elastic-viscoplastic transition zone, Engineering Fracture Mechanics 18,pp. 109-19.

ABOUDI, J., AND ACHENBACH, J. D. (1983b), Numerical analysis of fast modeI fracture of a strip of viscoplastic work-hardening material, InternationalJournal of Fracture 21, pp. 133-47.

ACHENBACH, J. D. (1970a), Crack propagation generated by a horizontallypolarized shear wave, Journal of the Mechanics and Physics of Solids 18,pp. 245-59.

ACHENBACH, J. D. (1970b), Extension of a crack by a shear wave, Zeitschrigt furangewandte Mathematik und Physik 21, pp. 887-900.

ACHENBACH, J. D. (1970c), Brittle and ductile extension of a finite crack bya horizontally polarized shear wave, International Journal of EngineeringScience 8, pp. 947-66.

ACHENBACH, J. D., AND NUISMER, R. (1971), Fracture generated by a dilatationalwave, International Journal of Fracture 7, pp. 77-88.

ACHENBACH, J. D. (1973), Wave Propagation in Elastic Solids. Amsterdam:North-Holland.

ACHENBACH, J. D. (1974), Dynamic effects in brittle fracture, in MechanicsToday, Vol. 1, ed. S. Nemat-Nasser. Elmsford, NY: Pergamon, pp. 1-57.

ACHENBACH, J. D., AND VARATHARAJULU, V. K. (1974), Skew crack propagationat the diffraction of a transient stress wave, Quarterly of Applied Mathematics32, pp. 123-35.

ACHENBACH, J. D. (1975), Bifurcation of a running crack in antiplane shear,International Journal of Solids and Structures 11, pp. 130-1314.

ACHENBACH, J. D., AND BAZANT, Z. P. (1975), Elastodynamic near-tip stressand displacement fields for rapidly propagating cracks in orthotropic media,Journal of Applied Mechanics 42, pp. 183-9.

ACHENBACH, J. D., AND GAUTESEN, A. K. (1977), Elastodynamic stress inten-sity factors for a semi-infinite crack under 3-D loading, Journal of AppliedMechanics 44, pp. 243-9.

521

522 Bibliography

ACHENBACH, J. D. (1979), Dynamic crack tip fields according to deformationtheory, Journal of Applied Mechanics 46, pp. 707-8.

ACHENBACH, J. D., AND KHETAN, R. P. (1979), Kinking of a crack under dynamicloading conditions, Journal of Elasticity 9, pp. 113-29.

ACHENBACH, J. D., KEER, L. M., AND MENDELSOHN, D. A. (1980), Elastody-namic analysis of an edge crack, Journal of Applied Mechanics 47, pp. 551-6.

ACHENBACH, J. D., AND DUNAYEVSKY, V. (1981), Fields near a rapidly propagat-ing crack tip in an elastic-perfectly plastic material, Journal of the Mechanicsand Physics of Solids 29, pp. 283-303.

ACHENBACH, J. D., KANNINEN, M. F., AND POPELAR, C. H. (1981), Crack tipfields for fast fracture of an elastic-plastic material, Journal of the Mechanicsand Physics of Solids 29, pp. 211-25.

ACHENBACH, J. D., AND NEIMITZ, A. (1981), Fast fracture and crack arrestaccording to the Dugdale model, Engineering Fracture Mechanics 14, pp. 385-95.

ACHENBACH, J. D., Kuo, M. K., AND DEMPSEY, J. P. (1984), Mode III and mixedmode I—II crack kinking under stress wave loading, International Journal ofSolids and Structures 20, pp. 395-410.

ACHENBACH, J. D., AND KUO, M. K. (1985), Conditions for crack kinking understress wave loading, Engineering Fracture Mechanics 22, pp. 165-80.

ACHENBACH, J. D., Li, Z. L., AND NISHIMURA, N. (1985), Dynamic fields gener-ated by rapid crack growth, International Journal of Fracture 27, pp. 215-27.

ACHENBACH, J. D., AND NISHIMURA, N. (1985), Effect of inertia on finite neartip deformation for fast mode III crack growth, Journal of Applied Mechanics52, pp. 281-6.

ACHENBACH, J. D., AND NISHIMURA, N. (1986), Large deformations near apropagating crack tip, Engineering Fracture Mechanics 23, pp. 183-99.

ADLER, W. F., AND HOOKER, S. V. (1978a), Water drop impact damage in zincsulfide, Wear 48, pp. 103-19.

ADLER, W. F., AND HOOKER, S. V. (1978b), Rain erosion mechanisms in brittlematerials, Wear 50, pp. 11-38.

ADLER, W. F., AND HOOKER, S. V. (1978c), Rain erosion behavior of poly methyl-met hacry late, Journal of Materials Science 13, pp. 1015-25.

AHMAD, J., JUNG, J., BARNES, C. R., AND KANNINEN, M. F. (1983), Elastic-plastic finite element analysis of dynamic fracture, Engineering FractureMechanics 17, pp. 235-46.

AKI, K. (1966), Generation and propagation of G-waves from the Niigata earth-quake of June 16, 1964. Estimation of earthquake movement, released energyand stress-strain drop from G-wave spectrum, Bulletin of the EarthquakeResearch Institute 44, pp. 23-88.

AKI, K., AND RICHARDS, P. G. (1980), Quantitative Seismology. New York:Freeman.

AKITA, Y., AND IKEDA, K. (1961), Theory of brittle crack initiation and prop-agation - A theoretical analysis of ESSO test, Welding Journal ResearchSupplement 40, pp. 138-44s.

ANDREWS, D. J. (1976a), Rupture propagation with finite stress in antiplanestrain, Journal of Geophysical Research 81, pp. 3575-82.

ANDREWS, D. J. (1976b), Rupture velocity of plane strain shear cracks, Journalof Geophysical Research 81, pp. 5679-87.

ANDREWS, D. J. (1985), Dynamic plane strain shear rupture with a slip weak-ening friction law calculated by a boundary integral method, Bulletin of theSeismological Society of America 75, pp. 1-21.

Bibliography 523

ANG, D. D. (1960), Elastic waves generated by a force moving along a crack,Journal of Mathematics and Physics 38, pp. 246-56.

ANG, D. D., AND KNOPOFF, L. (1964), Diffraction of scalar elastic waves by afinite crack, Proceedings of the National Academy of Science 51, pp. 593-8.

ANGELINO, G. C. (1978), Influence of the geometry on unstable crack extensionand determination of dynamic fracture mechanics parameters, in Fast Fractureand Crack Arrest, Special Technical Publication 627, eds. G. T. Hahn andM. F. Kanninen. Philadelphia: American Society for Testing and Materials,pp. 392-407.

ANTHONY, S. R., CHUBB, J. P., AND CONGLETON, J. (1970), The crack-branchingvelocity, Philosophical Magazine 22, pp. 1201-16.

AOKI, S., KISHIMOTO, K., KONDO, H., AND SAKATA, M. (1978), Elastodynamicanalysis of crack by finite element method using singular element, InternationalJournal of Fracture 14, pp. 59-68.

AOKI, S., KISHIMOTO, K., IZUMIHARA, Y., AND SAKATA, M. (1980), Dynamicanalysis of cracked linear viscoelastic solids by finite element method usingsingular element, International Journal of Fracture 16, pp. 97-109.

AOKI, S., KISHMOTO, K., AND SAKATA, M. (1987), Finite element computationof dynamic stress intensity factor for a rapidly propagating crack using J-integral, Computational Mechanics 2, pp. 54-62.

AOKI, S. (1988), On the mechanics of dynamic fracture, JSME InternationalJournal, Series 7 31, pp. 487-99.

ARCHULETA, R. J., AND BRUNE, J. N. (1975), Surface strong motion associatedwith a stick-slip even in a foam rubber model of earthquakes, Bulletin of theSeismological Society of America 65, pp. 1059-71.

ARCHULETA, R. J., AND FRAZIER, G. A. (1978), Three-dimensional numericalsimulations of dynamic faulting in a half space, Bulletin of the SeismologicalSociety of America 68, pp. 541-72.

ARCHULETA, R. J., AND DAY, S. M. (1980), Dynamic rupture in layered medium:The 1966 Parkfield earthquake, Bulletin of the Seismological Society of Amer-ica 70, pp. 671-89.

ARCHULETA, R. J. (1982), Analysis of near source static and dynamic measure-ments from the 1979 Imperial Valley earthquake, Bulletin of the SeismologicalSociety of America 72, pp. 1927-56.

ASHURST, W. T., AND HOOVER, W. G. (1976), Microscopic fracture studies inthe two-dimensional triangular lattice, Physical Review B14, pp. 1465-73.

ATKINSON, C. (1965), The propagation of fracture in aeolotropic materials,International Journal of Fracture Mechancs 1, pp. 47-55.

ATKINSON, C, AND HEAD, A. K. (1966), The influence of elastic anisotropy onthe propagation of fracture, International Journal of Fracture 2, pp. 489-505.

ATKINSON, C. (1967), A simple model of a relaxed expanding crack, Arkiv furFysik 35, pp. 469-76.

ATKINSON, C. (1968), On axially symmetric expanding boundary value problemsin classical elasticity, International Journal of Engineering Science 6, pp. 27-35.

ATKINSON, C., AND ESHELBY, J. D. (1968), The flow of energy into the tip of amoving crack, International Journal of Fracture 4, pp. 3-8.

ATKINSON, C , AND LIST, R. D. (1972), A moving crack problem in a viscoelasticsolid, International Journal of Engineering Science 10, pp. 309-22.

ATKINSON, C. (1974), On the dynamic stress and displacement field associatedwith a crack propagating across the interface between two media, InternationalJournal of Engineering Science 12, pp. 491-506.

524 Bibliography

ATKINSON, C, AND COLEMAN, C. J. (1977), On some steady-state movingboundary problems in the linear theory of viscoelasticity, Journal of theInstitute for Mathematics and Applications 2, pp. 85-106.

ATKINSON, C. (1979), A note on some dynamic crack problems in linear viscoelas-ticity, Archiwum Mechanik Stosowanej 31, pp. 829-49.

ATKINSON, C, AND POPELAR, C. H. (1979), Antiplane dynamic crack propaga-tion in a viscoelastic strip, Journal of the Mechanics and Physics of Solids 27,pp. 431-9.

ATKINSON, C, AND SMELSER, R. E. (1982), Invariant integrals for thermo-viscoelasticity and applications, International Journal of Solids and Structures18, pp. 533-49.

ATKINSON, C, BASTERO, J. M., AND MIRANDA, I. (1986), Path-independentintegrals in fracture dynamics using auxiliary fields, Engineering FractureMechanics 25, pp. 53-62.

ATLURI, S. N., AND NlSHlOKA, T. (1983), Path independent integrals, energyrelease rates and general solutions of near-tip fields in mixed mode dynamicfracture mechanics, Engineering Fracture Mechanics 18, pp. 1-22.

ATLURI, S. N., AND NISHIOKA, T. (1985), Numerical studies in dynamic fracturemechanics, International Journal of Fracture 27, pp. 245-61.

AYERS, D. J. (1976), Dynamic plastic analysis of ductile fracture - the Charpyspecimen, International Journal of Fracture 12, pp. 567-78.

BAKER, B. R. (1962), Dynamic stresses created by a moving crack, Journal ofApplied Mechanics 29, pp. 449-58.

BARBEE, T. W., JR., SEAMAN, L., CREWDSON, R., AND CURRAN, D. (1972),Dynamic fracture criteria for ductile and brittle metals, Journal of Materials7, pp. 2812-25.

BARENBLATT, G. I. (1959a), The formation of equilibrium cracks during brittlefracture: General ideas and hypotheses, axially symmetric cracks, AppliedMathematics and Mechanics (PMM) 23, pp. 622-36.

BARENBLATT, G. I. (1959b), Concerning equilibrium cracks forming during brittlefracture: The stability of isolated cracks, relationship with energetic theories,Applied Mathematics and Mechanics (PMM) 23, pp. 1273-82.

BARENBLATT, G. I., AND CHEREPANOV, G. P. (1960), On the wedging of brittlebodies, Applied Mathematics and Mechanics (PMM) 24, pp. 993-1014.

BARENBLATT, G. I., SALGANIK, R. L., AND CHEREPANOV, G. P. (1962), On thenonsteady motion of cracks, Applied Mathematics and Mechanics (PMM) 26,pp. 469-77.

BARENBLATT, G. I., AND GOLDSTEIN, G. V. (1972), Wedging of an elastic bodyby a slender wedge moving with a constant super-Rayleigh subsonic velocity,International Journal of Fracture Mechancs 8, pp. 427-34.

BARSOM, J. M., AND ROLFE, S. T. (1987), Fracture and Fatigue Control ofStructures, 2nd edition. Englewood Cliffs, NJ: Prentice-Hall.

BARSTOW, F. E., AND EDGERTON, H. E. (1939), Glass fracture velocity, Journalof the American Ceramic Society 22, pp. 302-7.

BASS, B. R., PUGH, C. E., AND WALKER, J. K. (1987), Elastodynamic fractureanalysis of large crack arrest experiments, Nuclear Engineering and Design 98,pp. 157-69.

BATEMAN, H. (1955), Electrical and Optical Wave Motion. Mineola, NY: Dover.BAZANT, Z. P., GLAZIK, J. L., AND ACHENBACH, J. D. (1976), Finite element

analysis of wave diffraction by a crack, Journal of the Engineering MechanicsDivision, ASCE 102, pp. 479-96.

BENBOW, J. J., AND ROESSLER, F. C. (1957), Experiments on controlled fractures,Proceedings of the Physical Society (London) 70, p. 201.

Bibliography 525

BERGKVIST, H. (1973), The motion of a brittle crack, Journal of the Mechanicsand Physics of Solids 21, pp. 229-39.

BERGKVIST, H. (1974), Crack arrest in elastic sheets, Journal of the Mechanicsand Physics of Solids 22, pp. 491-502.

BERRY, J. P. (1960a), Some kinetic considerations of the Griffith criterion forfracture - I. Equations of motion at constant force, Journal of the Mechanicsand Physics of Solids 8, pp. 194-206.

BERRY, J. P. (1960b), Some kinetic considerations of the Griffith criterion forfracture - II. Equations of motion at constant displacement, Journal of theMechanics and Physics of Solids 8, pp. 207-16.

BERTOLOTTI, R. L. (1973), Fracture toughness of polycrystalline AI2O3, Journalof the American Ceramic Society 56, p. 107.

BIENIAWSKI, Z. (1967), Stability concept of brittle fracture propagation in rock,Engineering Geology 2, pp. 149-62.

BILBY, B. A. (1980), Tewksbury Lecture: Putting fracture to work, Journal ofMaterials Science 15, pp. 535-56.

BILEK, Z. J., AND BURNS, S. J. (1974), Crack propagation in wedged doublecantilever beam specimens, Journal of the Mechanics and Physics of Solids22, pp. 85-95.

BILEK, Z. J. (1978), The dependence of the fracture toughness of high strengthsteel on crack velocity, Scripta Metallurgica 12, pp. 1101-6.

BILEK, Z. J. (1980), Some comments on dynamic crack propagation in a highstrength steel, in Crack Arrest Methodology and Applications, Special Tech-nical Publication 711, eds. G. T. Hahn and M. F. Kanninen. Philadelphia:American Society for Testing and Materials, pp. 240-7.

BIRCH, M. W., AND WILLIAMS, J. G. (1978), The effect of rate on the impactfracture toughness of polymers, International Journal of Fracture 14, pp. 69-84.

BLEICH, H. H., AND MATTHEWS, A. T. (1967), Step load moving with super-seismic velocity on the surface of an elastic-plastic half space, InternationalJournal of Solids and Structures 3, pp. 819-52.

BLUHM, J. I., AND MARDIROSIAN, M. M. (1963), Fracture arrest capabilities ofannularly reinforced cylindrical pressure vessels, Experimental Mechanics 3,pp. 57-66.

BLUHM, J. I. (1969), Fracture arrest, in Fracture, Vol. 5, ed. H. Liebowitz. NewYork: Academic, pp. 1-63.

BOATWRIGHT, J., AND FLETCHER, J. B. (1984), The partition of radiated energybetween P and S waves, Bulletin of the Seismological Society of America 74,pp. 361-76.

BODNER, S. R. (1973), Stress waves due to fracture of glass in bending, Journalof the Mechanics and Physics of Solids 21, pp. 1-8.

BOHME, W., AND KALTHOFF, J. F. (1982), The behavior of notched bendspecimens in impact testing, International Journal of Fracture 20, pp. 139-43.

BORZYKH, A. A., AND CHEREPANOV, G. P. (1980), On the theory of fracture ofsolids submitted to powerful pulsed electron beams, Applied Mathematics andMechanics (PMM) 44, pp. 801-7.

BOWDEN, F. P., AND BRUNTON, J. H. (1961), The deformation of solids by liquidimpact at supersonic speeds, Proceedings of the Royal Society (London) A26,pp. 433-50.

BOWDEN, F. P., AND FIELD, J. E. (1964), The brittle fracture of solids byliquid impact, by solid impact, and by shock, Proceedings of the Royal Society(London) A28, pp. 331-52.

526 Bibliography

BOWDEN, F. P., BRUNTON, J. H., FIELD, J. E., AND HAYES, A. D. (1967),Controlled fracture of brittle solids and interruption of electric current, Nature216, pp. 38-42.

BRADLEY, W. B., AND KOBAYASHI, A. S. (1970), An investigation of propagatingcracks by dynamic photoelasticity, Experimental Mechanics 10, pp. 106-13.

BRICKSTAD, B., AND NILSSON, F. (1980), Numerical evaluation by FEM of crackpropagation experiments, International Journal of Fracture 16, pp. 71-84.

BRICKSTAD, B. (1983a), A FEM analysis of crack arrest experiments, Interna-tional Journal of Fracture 21, pp. 177-94.

BRICKSTAD, B. (1983b), A viscoplastic analysis of rapid crack propagationexperiments in steel, Journal of the Mechanics and Physics of Solids 31,pp. 307-27.

BROBERG, K. B. (1960), The propagation of a brittle crack, Archiv fur Fysik 18,pp. 159-92.

BROBERG, K. B. (1964), On the speed of a brittle crack, Journal of AppliedMechanics 31, pp. 546-7.

BROBERG, K. B. (1967), Discussion of fracture from the energy point of view, inRecent Progress in Applied Mechanics, eds. K. B. Broberg, J. Hult, and F.Niordson. New York: Wiley, pp. 125-51.

BROBERG, K. B. (1977), On the effects of plastic flow at fast crack growth, inFast Fracture and Crack Arrest, Special Technical Publication 627, eds. G. T.Hahn and M. F. Kanninen. Philadelphia: American Society for Testing andMaterials, pp. 243-56.

BROBERG, K. B. (1978), On transient sliding motion, Geophysical Journal of theRoyal Astronomical Society 52, pp. 397-432.

BROCK, L. M. (1983), The dynamic stress intensity factor due to arbitrary screwdislocation motion, Journal of Applied Mechanics 50, pp. 383-9.

BROCK, L. M., JOLLES, M., AND SCHROEDL, M. (1985), Dynamic impact over asubsurface crack: Applications to the dynamic tear test, Journal of AppliedMechanics 52, pp. 287-90.

BROCK, L. M., AND ROSSMANITH, H. P. (1985), Analysis of the reflection ofpoint force induced crack surface waves by a crack edge, Journal of AppliedMechanics 52, pp. 57-61.

BROCK, L. M. (1986), Transient dynamic Green's functions for a cracked plane,Quarterly of Applied Mathematics 44, pp. 265-75.

BROCK, L. M., AND JOLLES, M. (1987), Dislocation-crack edge interaction indynamic brittle fracture and crack propagation, International Journal of Solidsand Structures 23, pp. 607-19.

BROCKENBROUGH, J. R., SURESH, S., AND DUFFY, J. (1988), An analysis ofdynamic fracture in microcracking brittle solids, Philosophical Magazine A58,pp. 619-34.

BROCKWAY, G. S. (1972), On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics, Archive for Rational Mechanicsand Analysis 48, pp. 213-44.

BRUNE, J. N. (1973), Earthquake modelling by stable slip along pre-cut surfacesin stressed foam rubber, Bulletin of the Seismological Society of America 63,pp. 2105-29.

BUDIANSKY, B., AND O'CONNELL, R. J. (1976), Elastic moduli of a cracked solid,International Journal of Solids and Structures 12, pp. 81-97.

BUECKNER, H. F. (1970), A novel principle for the computation of stress intensityfactors, Zeitschrigt fur angewandte Mathematik und Mechanik 50, pp. 529-46.

Bui, H. D., EHRLACHER, A., AND NGUYEN, Q. S. (1980), Propagation de fissureen thermoelasticite dynamique, Journal de Mecanique 19, pp. 698-723.

Bibliography 527

BULL, T. H. (1956), The tensile strengths of liquids under dynamic loading,Philosophical Magazine 1, pp. 153-65.

BURGERS, P. (1980), An analysis of dynamic linear elastic crack propagationin antiplane shear by finite differences, International Journal of Fracture 16,pp. 261-74.

BURGERS, P., AND FREUND, L. B. (1980), Dynamic growth of an edge crack in ahalf-space, International Journal of Solids and Structures 16, pp. 265-74.

BURGERS, P. (1982), Dynamic propagation of a kinked or bifurcated crack inantiplane strain, Journal of Applied Mechanics 49, pp. 371-6.

BURGERS, P., AND DEMPSEY, J. P. (1982), Two analytical solutions for dy-namic crack bifurcation in antiplane strain, Journal of Applied Mechanics 49,pp. 366-70.

BURGERS, P. (1983), Dynamic kinking of a crack in plane strain, InternationalJournal of Solids and Structures 19, pp. 735-52.

BURGERS, P., AND DEMPSEY, J. P. (1984), Plane strain dynamic crack bifurca-tion, International Journal of Solids and Structures 20, pp. 609-18.

BURNS, S. J., AND WEBB, W. W. (1966), Plastic deformation during cleavage ofLiF, Transactions of the Metallurgical Society of AIME 236, pp. 1165-74.

BURNS, S. J. (1968), Transverse fracture markings generated by unsteady cleavagevelocities, Philosophical Magazine 18, pp. 625-35.

BURNS, S. J., AND WEBB, W. W. (1970a)^ Fraeture surface energies and disloca-tion processes during dynamical cleavage of LiF. I. Theory, Journal of AppliedPhysics 41, pp. 2078-85.

BURNS, S. J., AND WEBB, W. W. (1970b), Fracture surface energies and disloca-tion processes during dynamical cleavage of LiF. II. Experiments, Journal ofApplied Physics 41, pp. 2086-95.

BURNS, S. J. (1972), Fracture surface energies from dynamical cleavage analysis,Philosophical Magazine 25, pp. 131-8.

BURRIDGE, R. (1965), The effect of source rupture velocity on the ratio of Sto P corner frequencies, Bulletin of the Seismological Society of America 65,pp. 667-75.

BURRIDGE, R., AND WILLIS, J. R. (1969), The self-similar problem of theexpanding elliptical crack in an anisotropic solid, Proceedings of the CambridgePhilosophical Society 66, pp. 443-68.

BURRIDGE, R., AND HALIDAY, G. S. (1971), Dynamic shear cracks with frictionas models for shallow focus earthquakes, Geophysical Journal of the RoyalAstronomical Society 25, pp. 261-83.

BURRIDGE, R. (1973), Admissible speeds for plane-strain self-similar shear crackswith friction but lacking cohesion, Geophysical Journal of the Royal Astro-nomical Society 35, pp. 439-55.

BURRIDGE, R., AND LEVY, C. (1974), Self similar circular shear cracks lackingcohesion, Bulletin of the Seismological Society of America 64, pp. 1789-808.

BURRIDGE, R. (1976), An influence function for the intensity factor in tensilefracture, International Journal of Engineering Science 14, pp. 725-34.

BURRIDGE, R., AND KELLER, J. B. (1978), Peeling, slipping and cracking -some one-dimensional free boundary problems in mechanics, SIAM Review20, pp. 31-61.

BURRIDGE, R., CONN, G., AND FREUND, L. B. (1979), The stability of a planestrain shear crack with finite cohesive force running at intersonic speeds,Journal of Geophysical Research 84, pp. 2210-22.

BURRIDGE, R., AND MOON, R. (1981), Slipping on a frictional fault plane in threedimensions: A numerical simulation of a scalar analogue, Geophysical Journalof the Royal Astronomical Society 67, pp. 325-42.

528 Bibliography

BYKOVTSEV, A. S., AND CHEREPANOV, G. P. (1980a), On one model of sourceof tectonic earthquake, Doklady Akademii Nauk SSSR 251, pp. 1353-6.

BYKOVTSEV, A. S., AND CHEREPANOV, G. P. (1980b), On modelling the earth-quake source, Applied Mathematics and Mechanics (PMM) 44, pp. 557-64.

BYKOVTSEV, A. S. (1983), On conditions for starting two collinear dislocationruptures, Applied Mathematics and Mechanics (PMM) 47, pp. 700-3.

BYKOVTSEV, A. S., AND TAVBAEV, J. S. (1984), On star-like system of propa-gating dislocation ruptures, Applied Mathematics and Mechanics (PMM) 48,pp. 148-51.

BYKOVTSEV, A. S., AND KRAMAROVSKII, D. B. (1987), The propagation of acomplex fracture area - the exact three-dimensional solution, Applied Mathe-matics and Mechanics (PMM) 51, pp. 89-98.

CAGNIARD, L. (1962), Reflection and Refraction of Progressive Seismic Waves,translated by E. A. Flinn and C. H. Dix. New York: McGraw-Hill.

CAMPBELL, J. D., AND FERGUSON, W. G. (1970), The temperature and strainrate dependence of the shear strength of mild steel, Philosophical Magazine21, pp. 63-82.

CARDENAS-GARCIA, J. F., AND HOLLOWAY, D. C. (1988), On the Lamb solutionand Rayleigh wave induced cracking, Experimental Mechanics 28, pp. 105-9.

CARLSSON, A. J. (1962), Experimental studies of brittle fracture propagation,Transactions of the Royal Institute of Technology, Stockholm, Number 189.

CARLSSON, A. J. (1963), Method for continuous measurement of instantaneouscrack propagation velocities, Transactions of the Royal Institute of Technology,Stockholm, Number 207.

CARRIER, G. F., KROOK, M., AND PEARSON, C. E. (1966), Functions of aComplex Variable. New York: McGraw-Hill.

CARROLL, M. M., AND HOLT, A. C. (1972), Static and dynamic pore-collapserelations for ductile porous materials, Journal of Applied Physics 43, pp. 1626-36.

CHAMPION, C. R. (1988), The stress intensity factor history for an advancingcrack under three-dimensional loading, International Journal of Solids andStructures 24, pp. 285-300.

CHANG, S. J. (1971), Diffraction of plane dilatational waves by a finite crack,Quarterly Journal of Mechanics and Applied Mathematics 24, pp. 423-43.

CHATTERJEE, A. K., AND KNOPOFF, L. (1983), Bilateral propagation of aspontaneous two-dimensional antiplane shear crack under the influence ofcohesion, Geophysical Journal of the Royal Astronomical Society 73, pp. 449-73.

CHATTERJEE, A. K., AND KNOPOFF, L. (1984), Spontaneous growth of an in-plane shear crack, International Journal of Solids and Structures 20, pp. 963-78.

CHAUDHRI, M. M., WELLS, J. K., AND STEPHENS, A. (1981), Dynamic hardness,deformation and fracture of simple ionic crystals at very high rates of strain,Philosophical Magazine 43, pp. 643-64.

CHEN, E. P., AND SIH, G. C. (1973), Running crack in an incident wave field,International Journal of Solids and Structures 9, pp. 897-919.

CHEN, E. P., AND SIH, G. C. (1975), Scattering of plane waves by a propagatingcrack, Journal of Applied Mechanics 42, pp. 705—11.

CHEN, E. P., AND SIH, G. C. (1977), Transient response of cracks to impact, inElastodynamic Crack Problems, ed. G. C. Sih. Leyden: NoordhofF Publishing,pp. 1-58.

Bibliography 529

CHEN, Y. M., AND WILKINS, M. L. (1976), Stress analysis of crack problems witha three-dimensional, time dependent computer program, International Journalof Fracture 12, pp. 607-17.

CHEREPANOV, G. P., AND SOKOLINSKY, V. B. (1972), On fracturing of brittlebodies by impact, Engineering Fracture Mechanics 4, pp. 205-14.

CHEREPANOV, G. P., AND AFANASEV, E. F. (1973), Some dynamic problemsof the theory of elasticity - a review, Applied Mathematics and Mechanics(PMM) 37, pp. 584-606.

CHEREPANOV, G. P., AND AFANASEV, E. F. (1974), Some dynamic problemsof the theory of elasticity - a review, International Journal of EngineeringScience 12, pp. 665-90.

CHEREPANOV, G. P. (1979), Mechanics of Brittle Fracture, translated by R. deWit and W. C. Cooley. New York: McGraw-Hill.

CHEREPANOV, G. P. (1980), On the theory of fracture of brittle bodies byexplosion, Mechanics of Solids (Mechanika Tverdogo Tela) 15, pp. 155-68.

CHI, Y. C, LEE, S., CHO, K., AND DUFFY, J. (1989), The effects of temperingand test temperature on the dynamic fracture initiation behavior of an AISI4340 VAR steel, Materials Science and Engineering A114, pp. 105-26.

CHITALEY, A. D., AND MCCLINTOCK, F. A. (1971), Elastic-plastic mechanics ofsteady crack growth under antiplane conditions, Journal of the Mechanics andPhysics of Solids 19, pp. 147-63.

CHRISTENSEN, R. M. (1982), Theory of Viscoelasticity: An Introduction. NewYork: Academic Press.

CHRISTIE, D. G. (1952), An investigation of cracks and stress waves in glassand plastics by high-speed photography, Transactions of the Society of GlassTechnology 36, pp. 74-89.

CHRISTIE, D. G., AND KOLSKY, H. (1952), The fractures produced in glass andplastics by the passage of stress waves, Transactions of the Society of GlassTechnology 36, pp. 65-73.

CHRISTIE, D. G. (1955), A multiple spark camera for dynamic stress analysis, TheJournal of Photographic Science 3, pp. 153-9.

CLARK, A. B. J., AND IRWIN, G. R. (1966), Crack propagation behaviors,Experimental Mechanics, pp. 321-30.

CLAYTON, J. Q. (1987), Modes of cleavage initiation and propagation in spheri-odized steels, Scripta Metallurgica 21, pp. 993-6.

CLIFTON, R. J. (1983), Dynamic plasticity, Journal of Applied Mechanics 50,pp. 941-52.

COHEN, L. J., AND BERKOWITZ, H. M. (1971), Time dependent fracture criteriafor 6061-T6 aluminum under stress wave loading in uniaxial strain, Interna-tional Journal of Fracture Mechancs 7, pp. 183-96.

COLE, J. D., AND HUTH, J. H. (1958), Stresses produced in a half plane by movingloads, Journal of Applied Mechanics 25, pp. 433-6.

CONGLETON, J., AND PETCH, N. J. (1965), The surface energy of a running crackin AI2O3, MgO and glass from crack branching measurements, InternationalJournal of Fracture 1, pp. 14-9.

CONGLETON, J., AND PETCH, N. J. (1967), Crack branching, PhilosophicalMagazine 16, pp. 749-60.

COOK, R. F., Crack propagation thresholds: A measure of surface energy, Journalof Materials Research 1, pp. 852-60.

COSTIN, L. S., DUFFY, J., AND FREUND, L. B. (1977), Fracture initiation inmetals under stress wave loading conditions, in Fast Fracture and Crack Arrest,Special Technical Publication 627, eds. G. T. Hahn and M. F. Kanninen.Philadelphia: American Society for Testing and Materials, pp. 301-18.

530 Bibliography

COSTIN, L. S., AND DUFFY, J. (1979), The effect of loading rate and temper-ature on the initiation of fracture in a mild, rate sensitive steel, Journal ofEngineering Materials and Technology 101, pp. 258-64.

COSTIN, L. S., SERVER, W. L., AND DUFFY, J. (1979), Dynamic fractureinitiation: A comparison of two experimental methods, Journal of EngineeringMaterials and Technology 101, pp. 168-72.

COTTERELL, B. (1964), On the nature of moving cracks, Journal of AppliedMechanics 31, pp. 12-6.

COTTERELL, B. (1965), On brittle fracture paths, International Journal ofFracture Mechancs 1, pp. 96-103.

COUQUE, H., ASARO, R. J., DUFFY, J., AND LEE, S. H. (1988), Correlations ofmicrostructure with dynamic and quasi-static fracture in plain carbon steel,Metallurgical Transactions 19A, pp. 2119-206.

COURANT, R., AND FRIEDRICHS, K. O. (1948), Supersonic Flow and ShockWaves. New York: Springer-Verlag.

COURANT, R., AND HILBERT, D. (1962), Methods of Mathematical Physics, Vol.II. New York: Interscience.

CRAGGS, J. W. (1960), On the propagation of a crack in an elastic-brittle solid,Journal of the Mechanics and Physics of Solids 8, pp. 66-75.

CRAGGS, J. W. (1966), The growth of a disk shaped crack, International Journalof Engineering Science 4, pp. 113-24.

CROSLEY, P. B., AND RIPLING, E. J. (1969), Dynamic fracture toughness of A5333steel, Journal of Basic Engineering 91, pp. 525-34.

CROSLEY, P. B., AND RIPLING, E. J. (1977), Characteristics of a run-arrestsegment of crack extension, in Fast Fracture and Crack Arrest, SpecialTechnical Publication 627, eds. G. T. Hahn and M. F. Kanninen. Philadelphia:American Society for Testing and Materials, pp. 203-27.

CROSLEY, P. B., AND RIPLING, E. J. (1980), Significance of crack arrest toughness{Kia) testing, in Crack Arrest Methodology and Applications, Special Tech-nical Publication 711, eds. G. T. Hahn and M. F. Kanninen. Philadelphia:American Society for Testing and Materials, pp. 321-37.

CROSLEY, P. B., AND RIPLING, E. J. (1980), Comparison of crack arrest method-ologies, in Crack Arrest Methodology and Applications, Special TechnicalPublication 711, eds. by G. T. Hahn and M. F. Kanninen. Philadelphia:American Society for Testing and Materials, pp. 211-27.

CROUCH, B. A., AND WILLIAMS, J. G. (1987a), Application of a dynamic numer-ical solution to high speed fracture experiments - I. Analysis of experimentalgeometries, Engineering Fracture Mechanics 26, pp. 541-51.

CROUCH, B. A., AND WILLIAMS, J. G. (1987b), Application of a dynamicnumerical solution to high speed fracture experiments - II. Results and athermal blunting model, Engineering Fracture Mechanics 26, pp. 553-66.

CURRAN, D. R., SHOCKEY, D. A., AND SEAMAN, L. (1973), Dynamic fracturecriteria for a polycarbonate, Journal of Applied Physics 44, pp. 4025-38.

CURRAN, D. R., SEAMAN, L., AND SHOCKEY, D. A. (1987), Dynamic failure ofsolids, Physics Reports 147, pp. 253-388.

DAHLBERG, L., AND NILSSON, F. (1977), Some aspects of testing crack propaga-tion toughness, in International Conference on Dynamic Fracture Toughness.Cambridge: The Welding Institute, pp. 281-91.

DAHLBERG, L., NILSSON, F., AND BRICKSTAD, B. (1980), Influence of specimengeometry on crack propagation and arrest toughness, in Crack Arrest Method-ology and Applications, Special Technical Publication 711, eds. G. T. Hahn andM. F. Kanninen. Philadelphia: American Society for Testing and Materials,pp. 89-108.

Bibliography 531

DALLY, J. W., AND KOBAYASHI, T. (1978), Crack arrest in duplex specimens,International Journal of Solids and Structures 14, pp. 121-9.

DALLY, J. W. (1979), Dynamic photoelastic studies of fracture, ExperimentalMechanics 19, pp. 349-61.

DALLY, J. W., AND SHUKLA, A. (1980), Energy loss in Homalite-100 during crackpropagation and arrest, Engineering Fracture Mechanics 13, pp. 807-17.

DALLY, J. W., FOURNEY, W. L., AND IRWIN, G. R. (1985), On the uniquenessof the stress intensity factor-crack velocity relationship, International Journalof Fracture 27, pp. 159-68.

DALLY, J. W. (1987), Dynamic photoelasticity and its application to stress wavepropagation, fracture mechanics and fracture control, in Static and DynamicPhotoelasticity and Caustics, ed. A. Lagarde. New York: Springer-Verlag,pp. 247-406.

DALLY, J. W., AND BARKER, D. B. (1988), Dynamic measurements of initiationtoughness at high loading rates, Experimental Mechanics, pp. 298-303.

DAS, S., AND AKI, K. (1977a), A numerical study of two-dimensional spontaneousrupture propagation, Geophysical Journal of the Royal Astronomical Society50, pp. 643-68.

DAS, S., AND AKI, K. (1977b), Fault plane with barriers: a versatile earthquakemodel, Journal of Geophysical Research 82, pp. 565-70.

DAS, S. (1980), A numerical method for determination of source time functionsfor general three-dimensional rupture propagation, Geophysical Journal of theRoyal Astronomical Society 62, pp. 591-604.

DAS, S. (1981), Three-dimensional spontaneous rupture propagation and implica-tions for the earthquake source mechanisms, Geophysical Journal of the RoyalAstronomical Society 67, pp. 375-93.

DAS, S. (1985), Application of dynamic shear crack models to the study of theearthquake faulting process, International Journal of Fracture 26, pp. 263-76.

DAS, S., AND KOSTROV, B. V. (1987), On the numerical boundary integralequation method for three-dimensional dynamic shear crack problems, Journalof Applied Mechanics 54, pp. 99-104.

DATTA, S. K. (1979), Diffraction of SH-waves by edge cracks, Journal of AppliedMechanics 46, pp. 101-6.

DATTA, S. K., SHAH, A. H., AND FORTUNKO, C. M. (1982), Diffraction of mediumand long wavelength horizontally polarized shear waves by edge cracks, Journalof Applied Physics 53, pp. 2895-903.

DAVISON, L., AND STEVENS, A. L. (1973), Thermomechanical constitution ofspalling elastic bodies, Journal of Applied Physics 44, pp. 668-74.

DAVISON, L., STEVENS, A. L., AND KIPP, M. E. (1977), Theory of spall damageaccumulation in ductile materials, Journal of the Mechanics and Physics ofSolids 25, pp. 11-28.

DAVISON, L., AND GRAHAM, R. A. (1979), Shock compression of solids, PhysicsReports 55, pp. 255-379.

DAY, S. M. (1982a), Three-dimensional finite difference simulation of fault dynam-ics: Rectangular faults with fixed rupture velocity, Bulletin of the SeismologicalSociety of America 72, pp. 705-27.

DAY, S. M. (1982b), Three-dimensional simulation of spontaneous rupture: Theeffect of nonuniform pre-stress, Bulletin of the Seismological Society of Amer-ica 72, pp. 1881-902.

DE HOOP, A. T. (1958), Representation Theorems for the Displacement in anElastic Solid and Their Application to Elastodynamic Diffraction Theory,Doctoral Dissertation, Technical University of Delft.

532 Bibliography

DE HOOP, A. T. (1961), A modification of Cagniard's method for solving seismicpulse problems, Applied Scientific Research B8, pp. 349-56.

DE WIT, R., AND FIELDS, R. J. (1987), Wide plate crack arrest testing, NuclearEngineering and Design 98, pp. 149-55.

DE WIT, R., LOW, S. R., AND FIELDS, R. J. (1988), Wide-plate crack arresttesting: Evolution of experimental procedures, in Fracture Mechanics: Nine-teenth Symposium, Special Technical Publication 969. Philadelphia: AmericanSociety for Testing and Materials, pp. 679-90.

DEAN, R. H., AND HUTCHINSON, J. W. (1980), Quasi-static steady crack growthin small scale yielding, in Fracture Mechanics: Twelfth Symposium, SpecialTechnical Publication 700. Philadelphia: American Society for Testing andMaterials, pp. 383-405.

DELAMETER, W. R., HERRMANN, G., AND BARNETT, D. M. (1975), Weakening ofan elastic solid by a rectangular array of cracks, Journal of Applied Mechanics42, pp. 74-80.

DEMPSEY, J. P., KUO, M. K., AND ACHENBACH, J. D. (1982), Mode III crackkinking under stress wave loading, Wave Motion 4, pp. 181-90.

DEMPSEY, J. P., Kuo, M. K., AND BENTLEY, D. L. (1986), Dynamic effects inmode III crack bifurcation, International Journal of Solids and Structures 22,pp. 333-53.

DIAZ, M., AND LUND, F. (1989), The inertia of a crack near a dislocation,Philosophical Magazine A60, pp. 139-45.

DMOWSKA, R., AND RICE, J. R. (1983), Fracture theory and its seismological ap-plications, in Continuum Theories in Solid Earth Geophysics, ed. R. Teisseyre.Warsaw: PWN-Polish Scientific Publishers, pp. 187-255.

DOLL, W. (1973), An experimental study of the heat generated in the plasticregion of a running crack in different polymeric materials, Engineering FractureMechanics 5, pp. 259-68.

DOLL, W. (1975a), Investigations of the crack branching energy, InternationalJournal of Fracture 11, pp. 184-6.

DOLL, W. (1975b), A molecular weight dependent fracture transition in poly-met hylmethacry late, Journal of Materials Science 10, pp. 935-42.

DOLL, W. (1976), Application of an energy balance and an energy method todynamic crack propagation, International Journal of Fracture 12, pp. 595-605.

DOLL, W., WEIDMANN, G. W., AND SCHINKER, M. (1977), Isothermal andadiabatic effects associated with crack propagation in poly met hylmethacry lateand polystyrene, Journal of Non-Crystalline Solids, pp. 612-7.

DOORNBOS, D. J. (1984), On the determination of radiated seismic energy andrelated source parameters, Bulletin of the Seismological Society of America74, pp. 395-416.

DRUGAN, W. J., AND RICE, J. R. (1984), Restrictions on quasi-statically mov-ing surfaces of strong discontinuity in elastic-plastic solids, in Mechanics ofMaterial Behavior, eds. G. J. Dvorak and R. T. Shield. New York: Elsevier,pp. 59-73.

DRUGAN, W. J., AND SHEN, Y. (1987), Restrictions on dynamically propagatingsurfaces of strong discontinuity in elastic-plastic solids, Journal of the Mechan-ics and Physics of Solids 35, pp. 771-87.

DUFFY, A. R., MCCLURE, G. M., EIBNER, R. J., AND MAXEY, W. A. (1969),Fracture design practices for pressure piping, in Fracture, Vol. 5, ed. H.Liebowitz. New York: Academic, pp. 159-232.

DUGDALE, D. C. (1960), Yielding of steel sheets containing slits, Journal of theMechanics and Physics of Solids 8, pp. 100-4.

Bibliography 533

DULANEY, E. N., AND BRACE, W. F. (1960), Velocity behavior of a growing crack,Journal of Applied Physics 31, pp. 2233-6.

DUNAYEVSKY, V., AND ACHENBACH, J. D. (1982), Boundary layer phenomenonin the plastic zone near a rapidly propagating crack tip, International Journalof Solids and Structures 18, pp. 1-12.

DVORAK, G. J. (1971a), Formation of plastic enclaves at running brittle cracks,International Journal of Fracture 7, pp. 251-67.

DVORAK, G. J. (1971b), A model of brittle fracture propagation: Part L-Continuum aspects, Engineering Fracture Mechanics 3, pp. 351-79.

DVORAK, G. J., AND TANG, H. C. (1973), Influence of material propertieson dynamic fracture toughness of steels, Engineering Fracture Mechanics 5,pp. 91-106.

EDGERTON, H. E., AND BARSTOW, F. E. (1941), Further studies of glass fracturewith high-speed photography, Journal of the American Ceramic Society 24,pp. 131-7.

EFTIS, J., AND KRAFFT, J. M. (1965), A comparison of the initiation with therapid propagation of a crack in a mild steel plate, Journal of Basic Engineering87, pp. 257-63.

EMBLEY, G. T., AND SIH, G. C. (1973), Sudden appearance of a crack in a bentplate, International Journal of Solids and Structures 9, pp. 1349-59.

ENGEL, P. A. (1978), Impact Wear of Materials. New York: Elsevier.ERDOGAN, F. (1968), Crack propagation theories, in Fracture, Vol. 2, ed. H.

Liebowitz. New York: Academic, pp. 498-586.ERMAK, A. A. (1978), The temperature field in the vicinity of a moving crack,

Soviet Mining Science, pp. 28-32.ESHELBY, J. D. (1953), The equation of motion of a dislocation, Physical Review

90, pp. 248-55.ESHELBY, J. D. (1956), The continuum theory of lattice defects, in Progress

in Solid State Physics, Vol. 3, eds. F. Seitz and D. Turnbull. New York:Academic, pp. 79-144.

ESHELBY, J. D. (1969a), The elastic field of a crack extending nonuniformly undergeneral anti-plane loading, Journal of the Mechanics and Physics of Solids 17,pp. 177-99.

ESHELBY, J. D. (1969b), The starting of a crack, in Physics of Strength andPlasticity, ed. A. S. Argon. Cambridge, MA: MIT Press, pp. 263-75.

ESHELBY, J. D. (1970), Energy relations and the energy-momentum tensor incontinuum mechanics, in Inelastic Behavior of Solids, eds. M. F. Kanninen etal. New York: McGraw-Hill, pp. 77-115.

ETHERIDGE, J. M., DALLY, J. W., AND KOBAYASHI, T. (1978), A new methodof determining the stress intensity factor K from isochromatic fringe loops,Engineering Fracture Mechanics 10, pp. 81-93.

EVANS, A. G. (1974), Slow crack growth in brittle materials under dynamic loadingconditions, International Journal of Fracture 10, pp. 251-9.

EVANS, A. G., AND WILSHAW, T. R. (1977), Dynamic solid particle damage inbrittle materials: an appraisal, Journal of Materials Science 12, pp. 97-116.

FIELD, F. A., AND BAKER, B. R. (1962), Crack propagation under shear displace-ments, Journal of Applied Mechanics 29, pp. 436-7.

FiNKEL', V. M., MURAVIN, G. B., LEZVINSKAYA, L. M. (1981), Energy fluxdensity when a longitudinal shear crack propagates, Soviet Journal of Nonde-structive Testing, pp. 485-9.

FINKEL', V. M., KOROLEV, A. P., AND SAVEL'EV, A. M. (1979), Self-arrestpotential of rapid cracks in ferrosilicon at low temperatures, Strength ofMaterials (Problemy Prochnosti), pp. 1138-43.

534 Bibliography

FISHER, B. (1971), The product of distributions, Quarterly Journal of Mathemat-ics 22, pp. 291-8.

FISHER, J. C , AND HOLLOMON, J. H. (1947), A statistical theory of fracture,Transactions of AIME 171, pp. 546-61.

FLITMAN, L. M. (1963), Waves generated by sudden crack in a continuous elasticmedium, Applied Mathematics and Mechanics (PMM) 27, pp. 938-53.

FONSECA, J. G., ESHELBY, J. D., AND ATKINSON, C (1971), The fracturemechanics of flint-knapping and allied processes, International Journal ofFracture Mechancs 7, pp. 421-33.

FORWOOD, C. T., AND FORTY, A. J. (1965), The interaction of cleavage crackswith inhomogeneities in sodium chloride crystals, Philosophical Magazine 11,pp. 1067-82.

FORWOOD, C. T. (1968), The work of fracture in crystals of sodium chloridecontaining cavities, Philosophical Magazine 17, pp. 657-67.

FOSSUM, A. F., AND FREUND, L. B. (1975), Nonuniformly moving shear crackmodel of a shallow focus earthquake mechanism, Journal of GeophysicalResearch 80, pp. 3343-7.

FOSSUM, A. F. (1978), Influence of rotary inertia on steady-state crack propa-gation in a finite plate subjected to out-of-plane bending, Journal of AppliedMechanics 45, pp. 130-4.

FOURNEY, W. L., HOLLOWAY, D. C, AND DALLY, J. W. (1975), Fractureinitiation and propagation from a center of dilatation, International Journalof Fracture 11, pp. 1011-29.

FREDRICKS, R. W., AND KNOPOFF, L. (1960), The reflection of Rayleigh wavesby a high impedance obstacle on a half space, Geophysics 25, pp. 1195-202.

FREDRICKS, R. W. (1961), Diffraction of an elastic pulse in a loaded half space,Journal of the Acoustical Society of America 33, pp. 17-22.

FREUDENTHAL, A. M. (1968), Statistical approach to brittle fracture, in Fracture,Vol. 2, ed. H. Liebowitz, New York: Academic, pp. 592-619.

FREUND, L. B., AND PHILLIPS, J. W. (1970), Near front stress singularity forimpact on an elastic quarter space, Journal of the Acoustical Society ofAmerica 47, pp. 942-3.

FREUND, L. B. (1971), The oblique reflection of a Rayleigh wave from a crack tip,International Journal of Solids and Structures 7, pp. 1199-210.

FREUND, L. B. (1972a), Crack propagation in an elastic solid subjected to generalloading. I. Constant rate of extension, Journal of the Mechanics and Physicsof Solids 20, pp. 129-40.

FREUND, L. B. (1972b), Crack propagation in an elastic solid subjected to generalloading. II. Nonuniform rate of extension, Journal of the Mechanics andPhysics of Solids 20, pp. 141-52.

FREUND, L. B. (1972C), Energy flux into the tip of an extending crack in an elasticsolid, Journal of Elasticity 2, pp. 341-9.

FREUND, L. B. (1972d), Surface waves guided by a slit in an elastic solid, Journalof Applied Mechanics 39, pp. 1027-32.

FREUND, L. B. (1972e), The initial wave emitted by a suddenly extending crackin an elastic solid, Journal of Applied Mechanics 39, pp. 601-2.

FREUND, L. B. (1973a), Crack propagation in an elastic solid subjected to generalloading. III. Stress wave loading, Journal of the Mechanics and Physics ofSolids 21, pp. 47-61.

FREUND, L. B. (1973b), The response of an elastic solid to nonuniformly movingsurface loads, Journal of Applied Mechanics 40, pp. 699-704.

FREUND, L. B., AND CLIFTON, R. J. (1974), On the uniqueness of elastodynamicsolutions for running cracks, Journal of Elasticity 4, pp. 293-9.

Bibliography 535

FREUND, L. B., AND RICE, J. R. (1974), On the determination of elastodynamiccrack tip stress fields, International Journal of Solids and Structures 10,pp. 411-7.

FREUND, L. B. (1974a), Crack propagation in an elastic solid subjected to generalloading. IV. Obliquely incident stress pulse, Journal of the Mechanics andPhysics of Solids 22, pp. 137-46.

FREUND, L. B. (1974b), The stress intensity factor due to normal impact loading ofthe faces of a crack, International Journal of Engineering Science 12, pp. 179-89.

FREUND, L. B., PARKS, D. M., AND RICE, J. R. (1975), Running ductile fracturein a pressurized line pipe, in Mechanics of Crack Growth, Special TechnicalPublication 590. Philadelphia: American Society for Testing and Materials,pp. 243-62.

FREUND, L. B., AND HERRMANN, G. (1976), Dynamic fracture of a beam or platein plane bending, Journal of Applied Mechanics 43, pp. 112-6.

FREUND, L. B. (1976a), The analysis of elastodynamic crack tip fields, inMechanics Today, Vol. Ill, ed. S. Nemat-Nasser, Elmsford, NY: Pergamon,pp. 55-91.

FREUND, L. B. (1976b), Dynamic crack propagation, in The Mechanics ofFracture, ed. F. Erdogan. New York: American Society of MechanicalEngineers, pp. 105-34.

FREUND, L. B. (1977), A simple model of the double cantilever beam crackpropagation specimen, Journal of the Mechanics and Physics of Solids 25,pp. 69-79.

FREUND, L. B., LI, V. C. F., AND PARKS, D. M. (1979), An analysis of awire-wrapped mechanical crack arrester for pressurized pipelines, Journal ofPressure Vessel Technology 101, pp. 51-8.

FREUND, L. B. (1979a), The mechanics of dynamic shear crack propagation,Journal of Geophysical Research 84, pp. 2199-209.

FREUND, L. B. (1979b), A one-dimensional dynamic crack propagation model,in Mathematical Problems in Fracture, ed. R. Burridge. Providence, RI:American Mathematical Society, pp. 21-37.

FREUND, L. B., AND PARKS, D. M. (1980), Analytical interpretation of runningductile fracture experiments in gas pressurized linepipe, in Crack ArrestMethodology and Applications, Special Technical Publication 711, eds. G. T.Hahn and M. F. Kanninen. Philadelphia: American Society for Testing andMaterials, pp. 359-78.

FREUND, L. B. (1981), Influence of reflected Rayleigh waves on a propagatingedge crack, International Journal of Fracture 7, pp. R83-6.

FREUND, L. B., AND DOUGLAS, A. S. (1982), The influence of inertia on elastic-plastic antiplane shear crack growth, Journal of the Mechanics and Physics ofSolids 30, pp. 59-74.

FREUND, L. B., AND DOUGLAS, A. S. (1983), Dynamic growth of an antiplaneshear crack in a rate-sensitive elastic-plastic material, in Elastic-Plastic Frac-ture Mechanics, Special Technical Publication 803, eds. C. F. Shih and J.Gudas. Philadelphia: American Society for Testing and Materials, pp. 5-20.

FREUND, L. B., AND HUTCHINSON, J. W. (1985), High strain-rate crack growth inrate-dependent plastic solids, Journal of the Mechanics and Physics of Solids33, pp. 169-91.

FREUND, L. B., HUTCHINSON, J. W., AND LAM, P. S. (1986), Analysis of highstrain rate elastic-plastic crack growth, Engineering Fracture Mechanics 23,pp. 119-29.

536 Bibliography

FREUND, L. B. (1987a), The stress intensity factor history due to three-dimen-sional loading of the faces of a crack, Journal of the Mechanics and Physicsof Solids 35, pp. 61-72.

FREUND, L. B. (1987b), The apparent fracture energy for dynamic crack growthwith fine scale periodic fracture resistance, Journal of Applied Mechanics 54,pp. 970-3.

FRIEDLANDER, F. G. (1958), Sound Pulses. Cambridge: Cambridge UniversityPress.

FROST, H. J., AND ASHBY, M. J. (1982), Deformation Mechanism Maps. Elms-ford, NY: Pergamon.

FULLER, K. N. G., FOX, P. G., AND FIELD, J. E. (1975), The temperature riseat the tip of fast-moving cracks in glassy polymers, Proceedings of the RoyalSociety (London) A341, pp. 537-57.

FYFE, I. M., AND RAJENDRAN, A. M. (1980), Dynamic pre-strain and inertiaeffects on the fracture of metals, Journal of the Mechanics and Physics ofSolids 28, pp. 17-26.

GAO, Y. C , AND NEMAT-NASSER, S. (1983a), Dynamic fields near a crack tipgrowing in an elastic-perfectly plastic solid, Mechanics of Materials 2, pp. 47-60.

GAO, Y. C, AND NEMAT-NASSER, S. (1983b), Near-tip dynamic fields for acrack advancing in a power-law elastic-plastic material: modes I, II, and III,Mechanics of Materials 2, pp. 305-17.

GAO, Y. C, AND NEMAT-NASSER, S. (1984), Mode II dynamic fields near a cracktip growing in an elastic-perfectly plastic solid, Journal of the Mechanics andPhysics of Solids 32, pp. 1-19.

GAO, Y. C. (1985), Asymptotic dynamic solution to the mode I propagating cracktip field, International Journal of Fracture 29, pp. 171-80.

GELFAND, I. M., AND FOMIN, S. V. (1963), Calculus of Variations. EnglewoodCliffs, NJ: Prentice-Hall.

GILATH, I., ELIEZER, S., DARIEL, M. P., AND KORNBLIT, L. (1988), Brittle-to-ductile transition in laser induced spall at ultrahigh strain rate in 6061-T6aluminum, Applied Physics Letters 52, pp. 1207-9.

GILATH, I., SALZMANN, D., GIVON, M., DARIEL, M., KORNBLIT, L., AND BAR-NOY, T. (1988), Spallation as an effect of laser-induced shock waves, Journalof Materials Science 23, pp. 1825-8.

GILLIS, P. P., AND GILMAN, J. J. (1964), Double-cantilever cleavage mode of crackpropagation, Journal of Applied Physics 35, pp. 647-58.

GILMAN, J. J., KNUDSEN, C, AND WALSH, W. P. (1958), Cleavage cracks anddislocations in LiF crystals, Journal of Applied Physics 29, pp. 601-7.

GILMAN, J. J., AND TULER, F. R. (1970), Dynamic fracture by spallation inmetals, International Journal of Fracture Mechancs 6, pp. 169-82.

GLENN, L. A. (1976), The fracture of a glass half space by projectile impact,Journal of the Mechanics and Physics of Solids 24, pp. 93-106.

GLENN, L. A., AND JANACH, W. (1977), Failure of granite cylinders under impactloading, International Journal of Fracture 13, pp. 301-17.

GLENN, L. A., GOMMERSTADT, B. Y., AND CHUDNOVSKY, A. (1986), A fracturemechanics model of fragmentation, Journal of Applied Physics 60, pp. 1224-6.

GLENNIE, E. B., AND WILLIS, J. R. (1971), An examination of the effects of someidealized models of fracture on accelerating cracks, Journal of the Mechanicsand Physics of Solids 19, pp. 11-30.

GLENNIE, E. B. (1971a), A rate-independent crack model, Journal of the Mechan-ics and Physics of Solids 19, pp. 255-72.

Bibliography 537

GLENNIE, E. B. (1971b), The unsteady motion of a rate-dependent crack model,Journal of the Mechanics and Physics of Solids 19, pp. 329-38.

GLENNIE, E. B. (1972), The dynamic growth of a void in a plastic material andan application to fracture, Journal of the Mechanics and Physics of Solids 20,pp. 415-29.

GODSE, R., RAVICHANDRAN, G., AND CLIFTON, R. J. (1989), Micromechanismsof dynamic crack propagation in an AISI 4340 steel, Materials Science andEngineering A112, pp. 79-88.

GOLDSMITH, W., AND KATSAMANIS, R. (1979), Fracture of notched polymericbeams due to central impact, Experimental Mechanics 19, pp. 235-43.

GOL'DSHTEIN, R. V., AND KAPTSOV, A. V. (1983), IntegrodifFerential equationsof the three-dimensional nonsteady dynamic problem of a crack in an elasticmedium, Mechanics of Solids (Mechanika Tverdogo Tela) 18, pp. 74-9.

GOLENIEWSKI, G. (1988), Dynamic crack growth in a viscoelastic material,International Journal of Fracture 37, pp. R39-44.

GORITSKII, V. M. (1981), Ductile-brittle transition and nonmetallic inclusionsin materials with a bcc lattice, Strength of Materials (Problemy Prochnosti),pp. 96-105.

GRADY, D. E., AND HOLLENBACH, R. E. (1979), Dynamic fracture strength ofrock, Geophysics Research Letters 6, pp. 73-6.

GRADY, D. E., AND KIPP, M. E. (1979), The micromechanics of impact fractureof rock, International Journal of Rock Mechanics and Mining Science 16,pp. 293-302.

GRADY, D. E., AND LIPKIN, J. (1980), Criteria for impulsive rock fracture,Geophysics Research Letters 7, pp. 255-8.

GRADY, D. E. (1981), Fragmentation of solids under impulsive stress loading,Journal of Geophysical Research 86, pp. 1047-54.

GRADY, D. E. (1982), Local inertial effects in dynamic fracture, Journal of AppliedPhysics 53, pp. 322-5.

GRADY, D. E., AND BENSON, D. A. (1983), Fragmentation of metal rings byelectromagnetic loading, Experimental Mechanics 23, pp. 393-400.

GRADY, D. E., AND KIPP, M. E. (1985), Geometric statistics and dynamicfragmentation, Journal of Applied Physics 58, pp. 1210-22.

GRADY, D. E. (1988), The spall strength of condensed matter, Journal of theMechanics and Physics of Solids 36, pp. 353-84.

GRAN, J. K., GUPTA, Y. M., AND FLORENCE, A. L. (1987), An experimentalmethod to study the dynamic tensile failure of brittle geologic materials,Mechanics of Materials 6, pp. 113-25.

GREENWOOD, J. H. (1972), The measurement of fracture speeds in silicon andgermanium, International Journal of Fracture 8, pp. 183-93.

GREGORY, R. D. (1966), The attenuation of a Rayleigh wave in a half space bya surface impedance, Proceedings of the Cambridge Philosophical Society 62,pp. 811-27.

GRIFFITH, A. A. (1920), The phenomenon of rupture and flow in solids, Philo-sophical Transactions of the Royal Society (London) A221, pp. 163-98.

GUNTHER, W. (1962), Uber einige randintegrale der elastomechanik, Abhandlun-gen der Braunschweigischen Wissenschaftlichen Gesellschaft 14, pp. 53-72.

Guo, Q., Li, Z., AND LI, K. (1988), Dynamic effects on the near crack line fieldsfor crack growth in an elastic perfectly-plastic solid, International Journal ofFracture 36, pp. 71-81.

GUR, Y., JAEGER, Z., AND ENGLMAN, R. (1984), Fragmentation of rock bygeometrical simulation of crack motion - I. Independent planar cracks, En-gineering Fracture Mechanics 20, pp. 783-800.

538 Bibliography

GURTIN, M. E. (1972), The linear theory of elasticity, in Handbuch der Physik,Vol. VIa/2, ed. S. Fliigge. Berlin: Springer-Verlag, pp. 1-295.

GURTIN, M. E. (1976), On a path-independent integral for elastodynamics,International Journal of Fracture 12, pp. 643-4.

GURTIN, M. E., AND YATOMI, C. (1980), On the energy release rate in elasto-dynamic crack propagation, Archive for Rational Mechanics and Analysis 74,pp. 231-47.

Guz', I. S., AND KATANCHIK, V. N. (1981), Interaction of a transverse wavewith a stationary macrocrack, Strength of Materials (Problemy Prochnosti),pp. 1046-53.

HAHN, G. T., SARRATE, M., KANNINEN, M. F., AND ROSENFIELD, A. R. (1973),A model for unstable shear crack propagation in pipes containing gas pressure,International Journal of Fracture 9, pp. 209-22.

HAHN, G. T., HOAGLAND, R. G., ROSENFIELD, A. R., AND SEJNOHA, R. (1974),Rapid crack propagation in a high strength steel, Metallurgical Transactions5, pp. 475-82.

HAHN, G. T., HOAGLAND, R. G., KANNINEN, M. F., AND ROSENFIELD, A. R.(1975), Crack arrest in steels, Engineering Fracture Mechanics 7, pp. 583-91.

HAHN, G. T., HOAGLAND, R. G., AND ROSENFIELD, A. R. (1976), Influence ofmetallurgical factors on the fast fracture energy absorption rates, MetallurgicalTransactions 7, pp. 49-54.

HAHN, G. T., AND KANNINEN, M. F. (1981), Dynamic fracture toughness param-eters for HY80 and HY130 steels and their weldments, Engineering FractureMechanics 14, pp. 725-40.

HAHN, G. T. (1984), The influence of microstructure on brittle fracture toughness,Metallurgical Transactions 15A, pp. 947-59.

HALICIOGLU, T., AND COOPER, D. M. (1986), A computer simulation of the crackpropagation process, Materials Science and Engineering 79, pp. 157-63.

HALL, E. O. (1953), The brittle fracture of metals, Journal of the Mechanics andPhysics of Solids 1, pp. 227-33.

HALL, W. J., KIHARA, H., SOETE, W., AND WELLS, A. A. (1976), Brittle Fractureof Welded Plate. Englewood Cliffs, NJ: Prentice-Hall.

HANSON, M. E., AND SANFORD, A. R. (1971), Some characteristics of a prop-agating brittle tensile crack, Geophysical Journal of the Royal AstronomicalSociety 24, pp. 231-44.

HANSON, M. E. (1975), Some effects of macroscopic flaws on dynamic fracture pat-terns near a pressure-driven fracture, International Journal of Rock Mechanicsand Mining Science 12, pp. 311-23.

HASSON, D. F., AND JOYCE, J. A. (1981), The effect of a higher loading rateon the Jic fracture toughness transition temperature of HY steels, Journal ofEngineering Materials and Technology 103, pp. 133-41.

HAWONG, J. S., KOBAYASHI, A. S., DADKHAH, M. S., KANG, B. S.-J., ANDRAMULU, M. (1987), Dynamic crack curving and branching under biaxialloading, Experimental Mechanics 27, pp. 146-53.

HELLAN, K. (1978a), Debond dynamics of an elastic strip - I. Timoshenko beamproperties and steady motion, International Journal of Fracture 14, pp. 91-100.

HELLAN, K. (1978b), Debond dynamics of an elastic strip - II. Simple transientmotion, International Journal of Fracture 14, pp. 173-84.

HELLAN, K. (1981), An alternative one-dimensional study of dynamic crackgrowth in deb test specimens, International Journal of Fracture 17, pp. 311-9.

HELLAN, K. (1984), Introduction to Fracture Mechanics. New York: McGraw-Hill.

Bibliography 539

HILL, R. (1950), The Mathematical Theory of Plasticity. Oxford: OxfordUniversity Press.

HlLLE, E. (1959), Analytic Function Theory. Vol. I, New York: Blaisdell.HlRTH, J. P., AND LOTHE, J. (1982), Theory of Dislocations, 2nd edition. New

York: McGraw-Hill.HOAGLAND, R. G., ROSENFIELD, A. R., AND HAHN, G. T. (1972), Mechanisms

of fast fracture and arrest in steels, Metallurgical Transactions 3, pp. 123-36.HOENIG, A. (1979), Elastic moduli of a non-randomly cracked body, International

Journal of Solids and Structures 15, pp. 137-54.HOFF, R., RUBIN, C. A., AND HAHN, G. T. (1985), Strain-rate dependence of the

deformation at the tip of a stationary crack, in Fracture Mechanics: SixteenthSymposium, Special Technical Publication 868, eds. M. F. Kanninen and A. T.Hooper. Philadelphia: American Society for Testing and Materials, pp. 409-30.

HOFF, R., RUBIN, C. A., AND HAHN, G. T. (1987), Viscoplastic finite elementanalysis of rapid fracture, Engineering Fracture Mechanics 26, pp. 445-61.

HOMMA, H., USHIRO, T., AND NAKAZAWA, H, (1979), Dynamic crack growthunder stress wave loading, Journal of the Mechanics and Physics of Solids 27,pp. 151-62.

HOMMA, H., SHOCKEY, D. A., AND MURAYAMA, Y. (1983), Response of cracksin structural materials to short pulse loads, Journal of the Mechanics andPhysics of Solids 31, pp. 261-79.

HORI, M., AND NEMAT-NASSER, S. (1988), Dynamic response of crystalline solidswith microcavities, Journal of Applied Physics 64, pp. 856-63.

HUANG, N. C. (1983), The mechanics of splitting a strip through penetration witha sharp wedge, International Journal of Fracture 23, pp. 23-5.

HUDAK, S. J., DEXTER, R. J., FITZGERALD, J. H., AND KANNINEN, M. F. (1986),The influence of specimen boundary conditions on the fracture toughness ofrunning cracks, Engineering Fracture Mechanics 23, pp. 201-13.

HUDSON, G., AND GREENFIELD, M. (1947), The speed of propagation of brittlecracks in steel, Journal of Applied Physics 18, pp. 405-7.

Hui, C. Y., AND RIEDEL, H. (1981), The asymptotic stress and strain field nearthe tip of a growing crack under creep conditions, International Journal ofFracture 17, pp. 409-25.

HULL, D., AND BEARDMORE, P. (1966), Velocity of propagation of cleavage cracksin tungsten, International Journal of Fracture Mechancs 2, pp. 468-87.

HULT, J. A., AND MCCLINTOCK, F. A. (1956), Elastic-plastic stress and straindistributions around sharp notches under repeated shear, Ninth InternationalCongress for Applied Mechanics, Brussels 8, pp. 51-8.

HUSSEINI, M. I., JOVANOVICH, D. B., RANDALL, M. J., AND FREUND, L. B.(1975), The fracture energy of earthquakes, Geophysical Journal of the RoyalAstronomical Society 43, pp. 367-85.

HUSSEINI, M. I., AND RANDALL, M. J. (1976), Rupture velocity and radiationefficiency, Bulletin of the Seismological Society of America 66, pp. 1173-87.

HUSSEINI, M. I. (1977), Energy balance for motion along a fault, GeophysicalJournal of the Royal Astronomical Society 49, pp. 699-714.

HUTCHINSON, J. W. (1968), Plastic stress and strain fields at a crack tip, Journalof the Mechanics and Physics of Solids 16, pp. 337-47.

IDA, Y. (1972), Cohesive force across the tip of a longitudinal shear crack andGriffith's specific surface energy, Journal of Geophysical Research 77, pp. 3796-805.

IDA, Y., AND AKI, K. (1972), Seismic source time function of propagatinglongitudinal shear cracks, Journal of Geophysical Research 77, pp. 2034-44.

540 Bibliography

IDA, Y. (1973), Stress concentration and unsteady propagation of longitudinalshear cracks, Journal of Geophysical Research 78, pp. 3418-29.

IRWIN, G. R. (1948), Fracture dynamics, in Fracturing of Metals. Cleveland:American Society of Metals, pp. 147-66.

IRWIN, G. R., AND KIES, J. A. (1952), Fracturing and fracture dynamics 38,Welding Journal Research Supplement, pp. 95-6s.

IRWIN, G. R., AND KIES, J. A. (1954), Critical energy rate analysis of fracturestrength, Welding Journal Research Supplement 33, pp. 193-8s.

IRWIN, G. R. (1957), Analysis of stresses and strains near the end of a cracktraversing a plate, Journal of Applied Mechanics 24, pp. 361-4.

IRWIN, G. R. (1958), Fracture, in Encyclopedia of Physics, Vol. VI, ed. S. Fliigge.New York: Springer-Verlag, pp. 551-90.

IRWIN, G. R. (1960), Fracture mechanics, in Structural Mechanics, eds. J. N.Goodier and N. J. Hoff. Elmsford, NY: Pergamon, pp. 557-91.

IRWIN, G. R. (1964), Crack-toughness testing of strain-rate sensitive materials,Journal of Engineering for Power 86, pp. 444-50.

IRWIN, G. R. (1969), Basic concepts for dynamic fracture testing, Journal of BasicEngineering 91, pp. 519-24.

IRWIN, G. R., AND PARIS, P. C. (1971), Fundamental aspects of crack growth andfracture, in Fracture, Vol. 3, ed. H. Liebowitz, New York: Academic, pp. 1-46.

IRWIN, G. R., DALLY, J. W., KOBAYASHI, T., FOURNEY, W. L., ETHERIDGE,M. J., AND ROSSMANITH, H. P. (1979), On the determination of the d-Krelationship for birefringent polymers, Experimental Mechanics 9, pp. 121-8.

ISHIKAWA, K., AND TSUYA, K. (1976), A note on brittle crack deceleration instructural steel, International Journal of Fracture 12, pp. 95-100.

IVES, K. D., SHOEMAKER, A. K., AND MCCARTNEY, R. F. (1974), Pipe defor-mation during a running shear fracture in a line pipe, Journal of EngineeringMaterials and Technology 96, pp. 309-17.

JAHANSHAHI, A. (1967), A diffraction problem and crack propagation, Journal ofApplied Mechanics 34, pp. 100-3.

JAIN, D. L., AND CONWAL, R. P. (1972), Diffraction of a plane shear elasticwave by a circular rigid disk and a penny shaped crack, Quarterly of AppliedMathematics 30, pp. 283-97.

JOHNSON, J. N. (1981), Dynamic fracture and spallation in ductile solids, Journalof Applied Physics 52, pp. 2812-25.

JOHNSON, J. N. (1983), Ductile fracture of rapidly expanding rings, Journal ofApplied Mechanics 50, pp. 593-600.

JOHNSON, J. N., AND ADDESSIO, F. L. (1988), Tensile plasticity and ductilefracture, Journal of Applied Physics 64, pp. 6699-712.

JOHNSON, J. W., AND HOLLOWAY, D. G. (1966), On the shape and size of thefracture zones on glass fracture surfaces, Philosophical Magazine 14, pp. 731-43.

JOHNSON, J. W., AND HOLLOWAY, D. G. (1968), Microstructure of the mist zoneon glass fracture surfaces, Philosophical Magazine 18, pp. 899-910.

JONES, D. S. (1952), A simplifying technique in the solution of a class of diffractionproblems, Quarterly Journal of Mathematics 3, pp. 189-96.

KALTHOFF, J. F. (1971), On the characteristic angle for crack branching in brittlematerials, International Journal of Fracture Mechancs 7, pp. 478-80.

KALTHOFF, J. F., WINKLER, S., AND BEINERT, J. (1976), Dynamic stressintensity factors for arresting cracks in deb specimens, International Journalof Fracture 12, pp. R317-9.

KALTHOFF, J. F., BEINERT, J., AND WINKLER, S. (1977), Measurements ofdynamic stress intensity factors for fast running and arresting cracks in deb

Bibliography 541

specimens, in Fast Fracture and Crack Arrest, Special Technical Publication627, eds. G. T. Hahn and M. F. Kanninen. Philadelphia: American Societyfor Testing and Materials, pp. 161-76.

KALTHOFF, J. F., AND SHOCKEY, D. A. (1977), Instability of cracks under impulseloading, Journal of Applied Physics 48, pp. 986-93.

KALTHOFF, J. F., WINKLER, S., AND BEINERT, J. (1977), The influence ofdynamic effects in impact testing, International Journal of Fracture 13,pp. 528-31.

KALTHOFF, J. F., BEINERT, J., WINKLER, S., AND KLEMM, W. (1980), Exper-imental analysis of dynamic effects in different crack arrest test specimens,in Crack Arrest Methodology and Applications, Special Technical Publication711, eds. G. T. Hahn and M. F. Kanninen. Philadelphia: American Societyfor Testing and Materials, pp. 109-27.

KALTHOFF, J. F. (1985), On the measurement of dynamic fracture toughness - Areview of recent work, International Journal of Fracture 27, pp. 277-98.

KALTHOFF, J. F. (1986), Fracture behavior under high rates of loading, Engineer-ing Fracture Mechanics 23, pp. 289-98.

KALTHOFF, J. F. (1987), The shadow optical method of caustics, in Staticand Dynamic Photoelasticity and Caustics, ed. by A. Lagarde. New York:Springer-Verlag, pp. 407-522.

KAMEDA, J. (1986), A kinetic model for ductile-brittle fracture mode transitionbehavior, Ada Metallurgica 12, pp. 2391-8.

KANAZAWA, T., MACHIDA, S., TERAMOTO, T., AND YOSHINARI, H. (1981),Study on fast fracture and crack arrest, Experimental Mechanics 21, pp. 78-88.

KANEMITSU, Y., AND TANAKA, Y. (1987), Mechanism of crack formation in glassafter high-power laser pulse radiation, Journal of Applied Physics 62, pp. 1208-11.

KANEMITSU, Y., TANAKA, Y., AND HARADA, Y. (1987), Surface transformationsin glass initiated by laser-driven shock, Japanese Journal of Applied Physics26, pp. 212-4.

KANNINEN, M. F. (1968), An estimate of the limiting speed of a propagatingductile crack, Journal of the Mechanics and Physics of Solids 16, pp. 215-28.

KANNINEN, M. F. (1973), An augmented double cantilever beam model forstudying crack propagation and arrest, International Journal of Fracture 9,pp. 83-91.

KANNINEN, M. F. (1974), A dynamic analysis of unstable crack propagation andarrest in the deb test specimen, International Journal of Fracture 10, pp. 415-30.

KANNINEN, M. F., SAMPATH, S. G., AND POPELAR, C. H. (1976), Steady-statecrack propagation in pressurized pipelines without backfill, Journal of PressureVessel Technology 98, pp. 56-64.

KANNINEN, M. F. (1985), Application of dynamic fracture mechanics for theprediction of crack arrest in engineering structures, International Journal ofFracture 27, pp. 299-312.

KANNINEN, M. F., AND POPELAR, C. H. (1985), Advanced Fracture Mechanics.Oxford: Oxford University Press.

KATSAMANIS, F. G., AND DELIDES, C. (1988), Fracture surface energy measure-ments of PMMA: A new experimental approach, Journal of Physics D: AppliedPhysics 21, pp. 79-86.

KEEGSTRA, P. N. R. (1976), A transient finite element crack propagation model fornuclear pressure vessel steels, Journal of the Institution of Nuclear Engineers17, pp. 89-96.

542 Bibliography

KELLEY, J. M., AND SUN, C. T. (1979), A singular finite element for computingtime dependent stress intensity factors, Engineering Fracture Mechanics 12,pp. 13-22.

KERKHOF, F. (1970), Bruchvorgange in Gldsern. Frankfurt: Verlag der DeutschenGlastechnischen Gesellschaft.

KERKHOF, F. (1973), Wave fractographic investigations of brittle fracture dy-namics, in Dynamic Crack Propagation, ed. G. C. Sih. Ley den: Noordhoff,pp. 3-35.

KIES, J. A., SULLIVAN, A. M., AND IRWIN, G. R. (1950), Interpretation of fracturemarkings, Journal of Applied Physics 21, pp. 716-20.

KlKUCHl, M. (1975), Inelastic effects on crack propagation, Journal of Physics ofthe Earth 23, pp. 161-72.

KIM, K. S. (1985a), A stress intensity factor tracer, Journal of Applied Mechanics52, pp. 291-7.

KIM, K. S. (1985b), Dynamic fracture under normal impact loading of the crackfaces, Journal of Applied Mechanics 52, pp. 585-92.

KING, W. W., MALLUCK, J. F., ABERSON, J. A., AND ANDERSON, J. M. (1976),Application of running-crack eigenfunctions to finite-element simulation ofcrack propagation, Mechanics Research Communications 3, pp. 197-202.

KING, W. W. (1978), Toward a singular element for propagating cracks, Interna-tional Journal of Fracture 14, pp. R7-R10.

KlNRA, V. K. (1976), Stress pulses emitted during fracture in tension, Interna-tional Journal of Solids and Structures 12, pp. 803-8.

KlNRA, V. K., AND KOLSKY, H. (1977), The interaction between bending fracturesand the emitted stress waves, Engineering Fracture Mechanics 10, pp. 423-32.

KlNRA, V. K., AND Vu, B. Q. (1980), Brittle fracture of plates in tension: Virginwaves and boundary reflections, Journal of Applied Mechanics 47, pp. 45-50.

KIPP, M. E., GRADY, D. E., AND CHEN, E. P. (1980), Strain rate dependentfracture initiation, International Journal of Fracture 16, pp. 471-8.

KlPP, M. E., AND GRADY, D. E. (1985), Dynamic fracture growth and interactionin one dimension, Journal of the Mechanics and Physics of Solids 33, pp. 399-415.

KlRKALDY, D. (1863), Experiments on Wrought Iron and Steel, 2nd edition. NewYork: Scribner.

KISHIMOTO, K., AOKI, S., AND SAKATA, M. (1980a), Computer simulation offast crack propagation in brittle material, International Journal of Fracture16, pp. 3-13.

KISHIMOTO, K., AOKI, S., AND SAKATA, M. (1980b), Dynamic stress intensityfactors using J-integral and finite element method, Engineering FractureMechanics 13, pp. 387-94.

KISHIMOTO, K., AOKI, S., AND SAKATA, M. (1980C), On the path independentJ-integral, Engineering Fracture Mechanics 13, pp. 841-50.

KLEPACZKO, J. R. (1984), Loading rate spectra for fracture initiation in metals,Theoretical and Applied Fracture Mechanics 1, pp. 181-91.

KLEPACZKO, J. R. (1985), Fracture initiation under impact, International Journalof Impact Engineering 3, pp. 191-210.

KNAUSS, W. G., AND RAVI-CHANDAR, K. (1985), Some basic problems in stresswave dominated fracture, International Journal of Fracture 27, pp. 127-43.

KNIGHT, C. G., SWAIN, M. V., AND CHAUDHRI, M. M. (1977), Impact of smallsteel spheres on glass surfaces, Journal of Materials Science 12, pp. 1573-86.

KNOPOFF, L., AND GILBERT, F. (1959), First motion methods in theoreticalseismology, Journal of the Acoustical Society of America 31, pp. 1161-8.

Bibliography 543

KNOPOFF, L., MOUTON, J. O., AND BURRIDGE, R. (1973), The dynamics of aone-dimensional fault in the presence of friction, Geophysical Journal of theRoyal Astronomical Society 35, pp. 169-84.

KNOPOFF, L., AND CHATTERJEE, A. K. (1982), Unilateral extension of a two-dimensional shear crack under the influence of cohesive forces, GeophysicalJournal of the Royal Astronomical Society 68, pp. 7-25.

KNOTT, J. F. (1979), Fundamentals of Fracture Mechanics. London: Butter-worth.

KNOWLES, J. K., AND STERNBERG, E. (1972), On a class of conservation laws inlinearized and finite elastostatics, Archive for Rational Mechanics and Analysis44, pp. 187-211.

KNOWLES, J. K. (1977), The finite anti-plane shear field near the tip of a crackfor a class of incompressible elastic solids, International Journal of Fracture13, pp. 611-39.

KOBAYASHI, A. S., WADE, B. G., BRADLEY, W. B., AND CHIU, S. T. (1974),Crack branching in Homalite-100 plates, Engineering Fracture Mechanics 6,pp. 81-92.

KOBAYASHI, A. S., AND CHAN, C. F. (1976), A dynamic photoelastic analysis ofdynamic tear test specimen, Experimental Mechanics 16, pp. 176-81.

KOBAYASHI, A. S., EMERY, A. F., AND MALL, S. (1976), Dynamic finite elementand dynamic photoelastic analyses of two fracturing Homalite-100 plates,Experimental Mechanics 16, pp. 321-8.

KOBAYASHI, A. S., AND MALL, S. (1978), Dynamic fracture toughness of Homa-lite-100, Experimental Mechanics 18, pp. 11-8.

KOBAYASHI, A. S., EMERY, A. F., LOVE, W. J., AND CHAO, Y.-H. (1988), Subsizeexperiments and numerical modeling of axial rupture of gas transmission lines,Journal of Pressure Vessel Technology 110, pp. 155-60.

KOBAYASHI, T., AND DALLY, J. W. (1979), Dynamic photoelastic determinationof the d-K relation for 4340 steel, in Crack Arrest Methodology and Applica-tions, Special Technical Publication 711, eds. G. T. Hahn and M. F. Kanninen.Philadelphia: American Society for Testing and Materials, pp. 189-210.

KOBAYASHI, T., YAMAMOTO, H., AND MATSUO, K. (1988), Evaluation of dy-namic fracture toughness of heavy wall ductile cast iron for container, Engi-neering Fracture Mechanics 30, pp. 397-407.

KOLSKY, H., AND CHRISTIE, D. G. (1952), The fractures produced in glass andplastics by the passage of stress waves, Transactions of the Society of GlassTechnology 36, pp. 65-73.

KOLSKY, H. (1953), Stress Waves in Solids. Mineola, NY: Dover.KOLSKY, H., AND RADER, D. (1969), Stress waves and fracture, Fracture, Vol. 1,

ed. H. Liebowitz. New York: Academic, pp. 533-69.KOSTROV, B. V. (1964a), The axisymmetric problem of propagation of a tensile

crack, Applied Mathematics and Mechanics (PMM) 28, pp. 793-803.KOSTROV, B. V. (1964b), Self-similar problems of propagation of shear cracks,

Applied Mathematics and Mechanics (PMM) 28, pp. 1077-87.KOSTROV, B. V. (1966), Unsteady propagation of longitudinal shear cracks,

Applied Mathematics and Mechanics (PMM) 30, pp. 1241-8.KOSTROV, B. V., AND NIKITIN, L. V. (1970), Some general problems of mechanics

of brittle fracture, Archiwum Mechaniki Stosowanej 22, pp. 749-75.KOSTROV, B. V. (1973), Seismic moment, earthquake energy and seismic flow of

rocks, Publications of the Institute of Geophysics, Polish Academy of Science62, pp. 25-47.

KOSTROV, B. V. (1974), Seismic moment and energy of earthquakes and seismicflow of rocks, Izvestia: Earth Physics 1, pp. 23-40.

544 Bibliography

KOSTROV, B. V. (1975), On the crack propagation with variable velocity, Inter-national Journal of Fracture 11, pp. 47-56.

KOSTROV, B. V., AND OSAULENKO, V. I. (1976), Crack propagation at arbitraryvariable rate under the action of static loads, Mechanics of Solids (MechanikaTverdogo Tela) 11, pp. 76-92.

KOSTROV, B. V., AND DAS, S. (1989), Principles of Earthquake Source Mechan-ics. Cambridge: Cambridge University Press.

KOTOUL, M., AND BILEK, Z. (1987), Waves of fracture in brittle bodies, Engi-neering Fracture Mechanics 27, pp. 517-38.

KOVALEVA, I. N., AND KOSMODEM'YANSKI, A. A. (1980), Plane fracture waves inbrittle solids, Mechanics of Solids (Mechanika Tverdogo Tela) 15, pp. 118-21.

KRAFFT, J. M., SULLIVAN, A. M., AND IRWIN, G. R. (1957), Relationship betweenthe fracture ductility transition and strain hardening characteristics of a lowcarbon steel, Journal of Applied Physics 28, pp. 379-80.

KRAFFT, J. M., AND SULLIVAN, A. M. (1963), Effects of speed and temperature oncrack toughness and yield strength in mild steel, Transactions of the AmericanSociety of Metals 56, pp. 160-75.

KRAFFT, J. M., AND IRWIN, G. R. (1965), Crack velocity considerations, in Frac-ture Toughness Testing and Its Applications, Special Technical Publication381. Philadelphia: American Society for Testing and Materials, pp. 114-32.

KRASOWKSY, A. J., KASHTALYAN, YU. A., AND KRASIKO, V. N. (1983), Brittle-to-ductile transition in steels and the critical transition temperature, Interna-tional Journal of Fracture 23, pp. 297-315.

KUHN, G., AND MATCZYNSKI, M. (1974), A dynamic problem of a crack in a platestrip, Engineering Transactions, Polska Akademia Nauk 22, pp. 469-85.

KULAKHMETOVA, S. A., SARAIKIN, V. A., AND SLEPYAN, L. I. (1984), Planeproblem of a crack in a lattice, Mechanics of Solids (Mechanika TverdogoTela) 19, pp. 102-8.

KULAKHMETOVA, S. A. (1985), Dynamics of a crack in an anisotropic lattice,Soviet Physics 30, pp. 254-5.

KUMAR, A. M., HAHN, G. T., RUBIN, C. A., AND X U , N. (1988), A crackarrest toughness test for tough materials, Engineering Fracture Mechanics 31,pp. 793-805.

KUNDU, T., AND MAL, A. K. (1981), Diffraction of elastic waves by a surfacecrack on a plate, Journal of Applied Mechanics 48, pp. 570-6.

KUPPERS, H. (1967), The initial course of crack velocity in glass plates, Interna-tional Journal of Fracture Mechancs 3, pp. 13-7.

KUPRADZE, V. D. (1963), Dynamical problems in elasticity, in Progress in SolidMechanics, Vol. Ill, eds. I. N. Sneddon and R. Hill. New York: North-Holland,pp. 1-258.

LAM, P. S., AND FREUND, L. B. (1985), Analysis of dynamic growth of a tensilecrack in an elastic-plastic material, Journal of the Mechanics and Physics ofSolids 33, pp. 153-67.

LAMB, H. (1904), On the propagation of tremors over the surface of an elastic solid,Philosophical Transactions of the Royal Society (London) A203, pp. 1-12.

LANDONI, J. A., AND KNOPOFF, L. (1981), Dynamics of one-dimensional crackwith variable friction, Geophysical Journal of the Royal Astronomical Society64, pp. 151-61.

LARDNER, R. W. (1974), Mathematical Theory of Dislocations and Fracture.Toronto: University of Toronto Press.

LAWN, B. R., AND WILSHAW, T. R. (1975), Fracture of Brittle Solids. Cambridge:Cambridge University Press.

Bibliography 545

LEE, C. S., LIVINE, T., AND GERBERICH, W. W. (1986), The acoustic emissionmeasurement of cleavage initiation near the ductile brittle transition temper-ature in steel, Scripta Metallurgies, 20, pp. 1137-40.

LEIGHTON, J. T., CHAMPION, C. R., AND FREUND, L. B. (1987), Asymptoticanalysis of steady dynamic crack growth in an elastic-plastic material, Journalof the Mechanics and Physics of Solids 35, pp. 541-63.

Li, Z. L., YANG, J. L., AND LEE, H. (1988), Temperature fields near a runningcrack tip, Engineering Fracture Mechanics 30, pp. 791-9.

LIN, B.-S. (1985), Elastic perfectly-plastic fields at a rapidly propagating crack tip,Applied Mathematics and Mechanics (trans, from Chinese) 6, pp. 1017-25.

LIN, I. H., AND THOMSON, R. M. (1986), Dynamic cleavage in ductile materials,Journal of Materials Research 1, pp. 73-80.

LINGER, K. R., AND HOLLOWAY, D. G. (1968), The fracture energy of glass,Philosophical Magazine 18, pp. 1269-80.

Liu, J. M., AND SHEN, B. W. (1984a), Dependence of dynamic fracture resistanceon crack velocity in tungsten: Part I. Single crystals, Metallurgical Transac-tions 15A, pp. 1247-51.

Liu, J. M., AND SHEN, B. W. (1984b), Dependence of dynamic fracture resistanceon crack velocity in tungsten: Part II. Bicrystals and polycrystals, Metallur-gical Transactions 15A, pp. 1253-8.

Lo, K. K. (1983), Dynamic crack-tip fields in rate sensitive solids, Journal of theMechanics and Physics of Solids 31, pp. 287-305.

LOEBER, J. F., AND SIH, G. C. (1975), Diffraction of antiplane shear waves by afinite crack, Journal of the Acoustical Society of America 44, pp. 90-8.

MA, C. C , AND BURGERS, P. (1986), Mode III crack kinking with delay time: Ananalytical approximation, International Journal of Solids and Structures 22,pp. 883-99.

MA, C. C , AND FREUND, L. B. (1986), The extent of the stress intensity factorfield during crack growth under dynamic loading conditions, Journal of AppliedMechanics 53, pp. 303-10.

MA, C. C , AND BURGERS, P. (1987), Dynamic mode I and mode II crack kinkingincluding delay time effects, International Journal of Solids and Structures 23,pp. 897-918.

MA, C. C. (1988), Initiation, propagation, and kinking of an in-plane crack,Journal of Applied Mechanics 55, pp. 587-95.

MA, C. C , AND BURGERS, P. (1988), Initiation, propagation and kinking of anantiplane crack, Journal of Applied Mechanics 55, pp. 111-9.

MACHIDA, S., YOSHINARI, H., AND KANAZAWA, T. (1986), Some recent experi-mental work in Japan on fast fracture and crack arrest, Engineering FractureMechanics 23, pp. 251-64.

MADARIAGA, R. (1976), Dynamics of an expanding circular fault, Bulletin of theSeismological Society of America 66, pp. 639-66.

MAKOVEI, V. A. (1984), Determination of the dynamic characteristics of fracturetoughness on ring specimens in loading using a vertical impact tester, Strengthof Materials (Problemy Prochnosti), pp. 887-90.

MAL, A. K. (1968a), Diffraction of elastic waves by a penny shaped crack,Quarterly of Applied Mathematics 26, pp. 231-8.

MAL, A. K. (1968b), Dynamic stress intensity factors for a non-symmetric loadingof a penny shaped crack, International Journal of Engineering Science 6,pp. 725-33.

MAL, A. K. (1970a), Interaction of elastic waves with a Griffith crack, Interna-tional Journal of Engineering Science 8, pp. 763-76.

546 Bibliography

MAL, A. K. (1970b), Interaction of elastic waves with a penny shaped crack,International Journal of Engineering Science 8, pp. 381-8.

MALL, S., KOBAYASHI, A. S., AND URABE, Y. (1978), Dynamic photoelastic anddynamic finite element analyses of dynamic tear test specimens, ExperimentalMechanics 18, pp. 449-56.

MALLUCK, J. F., AND KING, W. W. (1977), Simulations of fast fracture in thedeb specimen using Kanninen's model, International Journal of Fracture 13,pp. 655-65.

MALLUCK, J. F., AND KING, W. W. (1980), Fast fracture simulated by conven-tional finite elements: a comparison of two energy release algorithms, in CrackArrest Methodology and Applications, Special Technical Publication 711, ed.by G. T. Hahn and M. F. Kanninen. Philadelphia: American Society forTesting and Materials, pp. 38-53.

MALVERN, L. E. (1951), Plastic wave propagation in a bar of material exhibitinga strain rate effect, Quarterly of Applied Mathematics 8, pp. 405-11.

MALVERN, L. E. (1969), Introduction to the Mechanics of a Continuous Medium.Englewood Cliffs, NJ: Prentice-Hall.

MANNION, L. F., AND PIPKIN, A. C. (1983), Dynamic fracture of idealized fiberreinforced materials, Journal of Elasticity 13, pp. 395-409.

MANOGG, P. (1966), Investigation of the rupture of a plexiglass plate by meansof an optical method involving high speed filming of the shadows originatingaround holes drilled in the plate, International Journal of Fracture 2, pp. 604-13.

MANSINHA, L. (1964), The propagating fracture of constant shape, Journal of theMechanics and Physics of Solids 12, pp. 353-60.

MASLOV, L. A. (1980), Motion of a crack in a discrete medium, Mechanics ofSolids (Mechanika Tverdogo Tela) 15, pp. 106-9.

MATAGA, P. A., FREUND, L. B., AND HUTCHINSON, J. W. (1987), Crack tipplasticity in dynamic fracture, Journal of the Physics and Chemistry of Solids48, pp. 985-1005.

MAUE, A. W. (1959), Die beugung elastischer wellen an der halbebene, Zeitschrigtfur angewandte Mathematik und Mechanik 12, pp. 1-10.

MCCLINTOCK, F. A., AND SUKHATME, S. P. (1960), Travelling cracks in elasticmaterials under longitudinal shear, Journal of the Mechanics and Physics ofSolids 8, pp. 187-93.

MCCLINTOCK, F. A., AND IRWIN, G. R. (1965), Plasticity aspects of fracture me-chanics, in Fracture Toughness Testing and Its Applications, Special TechnicalPublication 381. Philadelphia: American Society for Testing and Materials,pp. 84-113.

MCCLINTOCK, F. A. (1968), A criterion for ductile fracture by the growth ofholes, Journal of Applied Mechanics 35, pp. 363-71.

MCLACHLAN, N. W. (1963), Complex Variable Theory and Transform Calculus.Cambridge: Cambridge University Press.

MCMAHON, C. J., JR., AND COHEN, M. (1965), Initiation of cleavage in polycrys-talline iron, Ada Metallurgica 13, pp. 591-604.

MECHOLSKY, J. J., RICE, R. W., AND FREIMAN, S. W. (1974), Prediction offracture energy and flaw size in glasses from measurements of mirror size,Journal of the American Ceramic Society 57, pp. 440-3.

MECHOLSKY, J. J., FREIMAN, S. W., AND RICE, R. W. (1976), Fracture surfaceanalysis of ceramics, Journal of Materials Science 11, pp. 1310-9.

MELIN, S. (1983), Why do cracks avoid each other?, International Journal ofFracture 23, pp. 37-45.

Bibliography 547

MEYERS, M. A., AND AIMONE, C. T. (1983), Dynamic fracture (spalling) ofmetals, Progress in Materials Science 28, pp. 1-96.

MIKUMO, T., AND MIYATAKE, T. (1979), Earthquake sequences on a frictionalfault model with a non-uniform strength and relaxation time, GeophysicalJournal of the Royal Astronomical Society 59, pp. 497-522.

MILES, J. W. (1960), Homogeneous solutions in elastic wave propagation, Quar-terly of Applied Mathematics 18, pp. 37-59.

MIYATAKE, T. (1980), Numerical simulations of earthquake source process by athree-dimensional crack model. Part I. Rupture process, Journal of Physicsof the Earth 28, pp. 565-98.

MOCK, W., AND HOLT, W. H. (1983), Fragmentation behavior of Armco iron andHF-1 steel explosive-filled cylinders, Journal of Applied Physics 54, pp. 2344-51.

MOTT, N. F. (1947), Fragmentation of shell cases, Proceedings of the Royal Society(London) A300, pp. 300-8.

MOTT, N. F. (1948), Brittle fracture in mild steel plates, Engineering 165, pp. 16-8.

MUSKHELISHVILI, N. I. (1953), Singular Integral Equations. Leyden: Noordhoff.NAIT ABDELAZIZ, M., NEVIERE, R., AND PLUVINAGE, G. (1987), Experimental

method for Jjc computation on fracture of solid propellants under dynamicloading conditions, Engineering Fracture Mechanics 28, pp. 425-34.

NAKAMURA, T., SHIH, C. F., AND FREUND, L. B. (1985a), Computationalmethods based on an energy integral in dynamic fracture, InternationalJournal of Fracture 27, pp. 229-43.

NAKAMURA, T., SHIH, C. F., AND FREUND, L. B. (1985b), Elastic-plastic analysisof a dynamically loaded circumferentially notched round bar, EngineeringFracture Mechanics 22, pp. 437-52.

NAKAMURA, T., SHIH, C. F., AND FREUND, L. B. (1986), Analysis of a dynami-cally loaded three-point-bend ductile fracture specimen, Engineering FractureMechanics 25, pp. 323-39.

NAKAMURA, T., SHIH, C. F., AND FREUND, L. B. (1988), Three-dimensionaltransient analysis of a dynamically loaded three-point-bend ductile fracturespecimen, in Nonlinear Fracture Mechanics, Special Technical Publication 995.Philadelphia: American Society for Testing and Materials, pp. 217-41.

NiGAM, H., AND SHUKLA, A. (1988), Comparison of the techniques of transmittedcaustics and photoelasticity as applied to fracture, Experimental Mechanics 28,pp. 123-33.

NlKOLAEVSKll, V. N. (1980), Dynamics of fracture fronts in brittle solids, Me-chanics of Solids (Mechanika Tverdogo Tela) 15, pp. 93-102.

NlKOLAEVSKll, V. N. (1981), Limit velocity of fracture front and dynamic strengthof brittle solids, International Journal of Engineering Science 19, pp. 41-56.

NIKOLIC, R. R., AND RICE, J. R. (1988), Dynamic growth of antiplane shearcracks in ideally plastic crystals, Mechanics of Materials 7, pp. 163-73.

NlLSSON, F. (1972), Dynamic stress-intensity factors for finite strip problems,International Journal of Fracture 8, pp. 403-11.

NILSSON, F. (1973), A path-independent integral for transient crack problems,International Journal of Solids and Structures 9, pp. 1107-15.

NILSSON, F. (1974a), Crack propagation experiments on strip specimens, Engi-neering FYacture Mechanics 6, pp. 397-403.

NILSSON, F. (1974b), A note on the stress singularity at a nonuniformly movingcrack tip, Journal of Elasticity 4, pp. 73-5.

NILSSON, F. (1977a), Steady mode III crack propagation followed by non-steadygrowth, International Journal of Solids and Structures 13, pp. 543-8.

548 Bibliography

NILSSON, F. (1977b), Sudden arrest of steadily moving cracks, in DynamicFracture Toughness. Cambridge: The Welding Institute, pp. 249-57.

NILSSON, F., AND BRICKSTAD, B. (1985), Dynamic fracture mechanics - rapidcrack growth in linear and nonlinear materials, in Elastic-Plastic FractureMechanics, Special Technical Publication 803, eds. C. F. Shih and J. P. Gudas.Philadelphia: American Society for Testing and Materials, pp. 427-78.

NILSSON, F., AND STAHLE, P. (1988), Crack growth criteria and crack tip models,Solid Mechanics Archives 13, pp. 193-238.

NISHIOKA, T., AND ATLURI, S. N. (1980a), Numerical modeling of dynamic crackpropagation in finite bodies, by moving singular elements - Part I. Formulation,Journal of Applied Mechanics 47, pp. 570-6.

NISHIOKA, T., AND ATLURI, S. N. (1980b), Numerical modeling of dynamic crackpropagation in finite bodies, by moving singular elements - Part II. Results,Journal of Applied Mechanics 47, pp. 577-82.

NISHIOKA, T., AND ATLURI, S. N. (1982), Finite element simulation of fastfracture in steel deb specimen, Engineering Fracture Mechanics 16, pp. 157-75.

NISHIOKA, T., AND ATLURI, S. N. (1983), Path-independent integrals, energyrelease rates, and general solutions of near tip fields in mixed mode dynamicfracture mechanics, Engineering Fracture Mechanics 18, pp. 1-22.

NOBLE, B. (1958), Methods Based on the Wiener-Hopf Technique. Elmsford, NY:Pergamon.

NuiSMER, R. J., AND ACHENBACH, J. D. (1972), Dynamically induced fracture,Journal of the Mechanics and Physics of Solids 20, pp. 203-22.

O'DONOGHUE, P. E., PREDEBON, W. W., AND ANDERSON, C. E. (1988), Dy-namic launch process for preformed fragments, Journal of Applied Physics 63,pp. 337-48.

OBREIMOFF, J. W. (1930), The splitting strength of mica, Proceedings of theRoyal Society (London) A127, pp. 290-7.

OGDEN, R. W. (1984), Non-Linear Elastic Deformations. Chichester: Horwood.OH, K. P. L., AND FINNIE, I. (1970), On the location of fracture in brittle solids -

II. Due to wave propagation in a slender rod, International Journal of FractureMechancs 6, pp. 333-9.

OROWAN, E. (1934), Crystal plasticity - III. On the mechanism of the glideprocess, Zeitschrift fur Physik 89, pp. 634-44.

OROWAN, E. (1949), Fracture and strength of solids, Reports of Progress inPhysics 12, pp. 185-232.

OROWAN, E. (1955), Energy criteria of fracture, Welding Journal ResearchSupplement 34, pp. 157-60s.

OSTLUND, S., AND GUDMUNDSON, P. (1988), Asymptotic crack tip fields fordynamic fracture of linear strain-hardening solids, International Journal ofSolids and Structures 24, pp. 1141-58.

OWEN, D. R. J., AND SHANTARAM, S. (1977), Numerical study of dynamic crackgrowth by the finite element method, International Journal of Fracture 13,pp. 821-37.

PALMER, A. C , AND RICE, J. R. (1973), The growth of slip surfaces in theprogressive failure of overconsolidated clay, Proceedings of the Royal Society(London) A332, pp. 527-48.

PAPADOPOULOS, M. (1963), Diffraction of plane elastic waves by a crack, withapplication to a problem of brittle fracture, Journal of the Australian Mathe-matical Society 3, pp. 325-39.

PARKS, D. M., AND FREUND, L. B. (1978), On the gas dynamics of running ductilefracture in a pressurized linepipe, Journal of Pressure Vessel Technology 100,pp. 13-7.

Bibliography 549

PASKIN, A., SOM, D. K., AND DIENES, G. J. (1983), The dynamic properties ofmoving cracks, Acta Metallurgica 31, pp. 1841-8.

PAXSON, T. L., AND LUCAS, R. A. (1973), An experimental investigation of thevelocity characteristics of a fixed boundary fracture model, in Dynamic CrackPropagation, ed. G. C. Sih. Leyden: Noordhoff, pp. 415-26.

PERZYNA, P. (1963), The constitutive equations for rate sensitive plastic materi-als, Quarterly of Applied Mathematics 20, pp. 321-32.

PIPKIN, A. C. (1984), Crack speeds in an ideal fiber reinforced material, Quarterlyof Applied Mathematics 42, pp. 267-74.

PONCELET, E. F. (1946), Fracture and comminution of brittle solids, AmericanInstitute of Mining, Metallurgical and Petroleum Engineers 169, p. 37.

POPELAR, C. H., AND ATKINSON, C. (1981), Dynamic crack propagation in aviscoelastic strip, Journal of the Mechanics and Physics of Solids 28, pp. 79-93.

POYNTON, W. A., SHANNON, R. W. E., AND FEARNEHOUGH, G. D. (1974),The design and application of shear fracture propagation studies, Journal ofEngineering Materials and Technology 9J6, pp. 323-9.

PRATT, P. L., AND STOCK, T. (1965), The distribution of strain about a runningcrack, Proceedings of the Royal Society (London) A28, pp. 73-82.

PUGH, C. E., AND WHITMAN, G. D. (1987), HSST crack arrest studies overview,Nuclear Engineering and Design 98, pp. 141-7.

PUGH, C. E., NAUS, D. J., BASS, B. R., NANSTAD, R. K., DE W I T , R., FIELDS, R.J., AND Low, S.R. (1988), Wide-plate crack arrest tests utilizing a prototypicalpressure vessel steel, International Journal of Pressure Vessels and Piping 31,pp. 165-85.

RADER, D. (1967), On the dynamics of crack growth in glass, Proceedings of theSESA 24, pp. 160-7.

RADOK, J. R. M. (1956), On the solution of problems of dynamic plane elasticity,Quarterly of Applied Mathematics 14, pp. 289-98.

RAJENDRAN, A. M., AND FYFE, I. M. (1982), Inertia effects on the ductile failureof thin rings, Journal of Applied Mechanics 49, pp. 31-6.

RAMULU, M., KOBAYASHI, A. S., AND KANG, B. S.-J. (1982), Dynamic crackcurving and branching in line-pipe, Journal of Pressure Vessel Technology104, pp. 317-22.

RAMULU, M., AND KOBAYASHI, A. S. (1983), Dynamic crack curving - a photoe-lastic evaluation, Experimental Mechanics 23, pp. 1-9.

RAMULU, M., KOBAYASHI, A. S., KANG, B. S.-J., AND BARKER, D. (1983),Further studies on dynamic crack branching, Experimental Mechanics 23,pp. 431-7.

RAVENHALL, F. W. (1977), The application of stress wave emission to crackpropagation in metals - a crack propagation model, Acustica 37, pp. 307-16.

RAVI-CHANDAR, K., AND KNAUSS, W. G. (1982), Dynamic crack-tip stresses un-der stress wave loading - a comparison of theory and experiment, InternationalJournal of Fracture 20, pp. 209-22.

RAVI-CHANDAR, K., AND KNAUSS, W. G. (1984a), An experimental investigationinto dynamic fracture: I. Crack initiation and arrest, International Journal ofFracture 25, pp. 247-62.

RAVI-CHANDAR, K., AND KNAUSS, W. G. (1984b), An experimental investigationinto dynamic fracture: II. Microstructural aspects, International Journal ofFracture 26, pp. 65-80.

RAVI-CHANDAR, K., AND KNAUSS, W. G. (1984C), An experimental investigationinto dynamic fracture: III. On steady state crack propagation and crackbranching, International Journal of Fracture 26, pp. 141-54.

550 Bibliography

RAVI-CHANDAR, K., AND KNAUSS, W. G. (1984d), An experimental investigationinto dynamic fracture: IV. On the interaction of stress waves with propagatingcracks, International Journal of Fracture 26, pp. 189-200.

RAVICHANDRAN, G., AND CLIFTON, R. J. (1989), Dynamic fracture under planewave loading, International Journal of Fracture 40, pp. 157-201.

REGAZZONI, G., JOHNSON, J. N., AND FOLLANSBEE, P. S. (1986), Theoreticalstudy of the dynamic tensile test, Journal of Applied Mechanics 53, pp. 519-28.

RICE, J. R. (1968), Mathematical analysis in the mechanics of fracture, inFracture, Vol. 2, ed. H. Liebowitz. New York: Academic, pp. 191-311.

RICE, J. R., AND ROSENGREN, G. F. (1968), Plane strain deformation near acrack tip in a power law hardening material, Journal of the Mechanics andPhysics of Solids 16, pp. 1-12.

RICE, J. R., AND TRACEY, D. M. (1969), On the ductile enlargement of voidsin triaxial stress fields, Journal of the Mechanics and Physics of Solids 17,pp. 201-17.

RICE, J. R. (1970), On the structure of stress-strain relations for time dependentplastic deformation in metals, Journal of Applied Mechanics 37, pp. 728-37.

RICE, J. R., AND JOHNSON, M. A. (1970), The role of large crack tip geometrychanges in plane strain fracture, in Inelastic Behavior of Solids, eds. M. F.Kanninen et al. New York: McGraw-Hill, pp. 641-72.

RICE, J. R. (1972), Some remarks on elastic crack tip stress fields, InternationalJournal of Solids and Structures 8, pp. 751-8.

RICE, J. R., AND SORENSEN, E. P. (1978), Continuing crack tip deformation andfracture for plane strain crack growth in elastic-plastic materials, Journal ofthe Mechanics and Physics of Solids 26, pp. 163-86.

RICE, J. R., MCMEEKING, R. M., PARKS, D. M., AND SORENSEN, E. P. (1979),Recent finite element studies in plasticity and fracture mechanics, ComputerMethods in Applied Mechanics and Engineering 17, pp. 411-42.

RlCE, J. R. (1980), The mechanics of earthquake rupture, in Physics of the Earth'sInterior, ed. E. Boschi. Amsterdam: North Holland, pp. 555-649.

RICE, J. R., DRUGAN, W. J., AND SHAM, T. L. (1980), Elastic-plastic analysis ofgrowing cracks, in Fracture Mechanics: Twelfth Symposium, Special TechnicalPublication 700. Philadelphia: American Society for Testing and Materials,pp. 189-219.

RICHARDS, P. G. (1973), The dynamic field of a growing plane elliptical shearcrack, International Journal of Solids and Structures 9, pp. 843-61.

RICHARDS, P. G. (1976), Dynamic motions near an earthquake fault: A three-dimensional solution, Bulletin of the Seismological Society of America 66,pp. 1-32.

RIPLING, E. J., CROSLEY, P. B., AND WIERSMA, S. J. (1986), A review of staticcrack arrest concepts, Engineering Fracture Mechanics 23, pp. 21-33.

ROBERTS, D. K., AND WELLS, A. A. (1954), The velocity of brittle fracture,Engineering 178, pp. 820-1.

ROBERTSON, I. A. (1967), Diffraction of a plane longitudinal wave by a pennyshaped crack, Proceedings of the Cambridge Philosophical Society 63, pp. 229-38.

ROBERTSON, T. S. (1953), Propagation of brittle fracture in steel, Journal of theIron and Steel Institute 175, pp. 361-74.

ROSAKIS, A. J., AND FREUND, L. B. (1981), The effect of crack tip plasticity onthe determination of dynamic stress intensity factors by the optical method ofcaustics, Journal of Applied Mechanics 48, pp. 302-8.

Bibliography 551

ROSAKIS, A. J., DUFFY, J., AND FREUND, L. B. (1984), The determination ofdynamic fracture toughness of AISI 4340 steel by the shadow spot method,Journal of the Mechanics and Physics of Solids 32, pp. 443-60.

ROSAKIS, A. J., AND ZEHNDER, A. T. (1985), On the dynamic fracture ofstructural metals, International Journal of Fracture 27, pp. 169-86.

ROSE, L. R. F. (1976a), An approximate (Wiener-Hopf) kernel for dynamiccrack problems in linear elasticity and viscoelasticity, Proceedings of the RoyalSociety (London) 349, pp. 497-521.

ROSE, L. R. F. (1976b), Recent theoretical and experimental results on fast brittlefracture, International Journal of Fracture 12, pp. 799-813.

ROSE, L. R. F. (1976C), On the initial motion of a Griffith crack, InternationalJournal of Fracture 12, pp. 829-41.

ROSE, L. R. F. (1981), The stress wave radiation from growing cracks, Interna-tional Journal of Fracture 17, pp. 45-60.

ROSENFIELD, A. R., AND MAJUMDAR, B. S. (1987), A micromechanical modelfor cleavage-crack reinitiation, Metallurgical Transactions 18A, pp. 1053-9.

ROSSMANITH, H. P., AND SHUKLA, A. (1981), Dynamic photoelastic investigationof interaction of stress waves with running cracks, Experimental Mechanics,pp. 415-22.

ROSSMANITH, H. P. (1983), How mixed is dynamic mixed-mode crack propa-gation? - A dynamic photoelasticity study, Journal of the Mechanics andPhysics of Solids 31, pp. 251-60.

RUDNICKI, J. W., AND FREUND, L. B. (1981), On energy radiation from seismicsources, Bulletin of the Seismological Society of America 71, pp. 583-95.

RUDNICKI, J. W. (1980), Fracture mechanics applied to the Earth's crust, AnnualReviews in Earth and Planetary Science 8, pp. 489-525.

Russo, R. (1986a), Theorems in linear elastodynamics for plane cracked bodies,Theoretical and Applied Fracture Mechanics 6, pp. 187-95.

RUSSO, R. (1986b), On the dynamics of an elastic body containing a moving crack,Journal of Elasticity 16, pp. 367-74.

RYDHOLM, G., FREDRIKSSON, B., AND NILSSON, F. (1978), Numerical investiga-tions of rapid crack propagation, in Numerical Methods in Fracture Mechanics,eds. A. R. Luxmore and D. J. R. Owen. Swansea: University of Swansea,pp. 660-72.

SANDERS, W. T. (1972), On the possibility of a supersonic crack in a crystallattice, Engineering Fracture Mechanics 4, pp. 145-53.

SCHARDIN, H. (1959), Velocity effects in fracture, in Fracture, eds. B. L. Averbachet al. Cambridge, MA: MIT Press, pp. 297-330.

SCHINDLER, H. J. (1981), Path-stability of a crack crossing a beam in bending,Journal of Applied Mathematics and Physics (ZAMP) 32, pp. 570-81.

SCHINDLER, H. J., AND KOLSKY, H. (1983), Multiple fractures produced by thebending of brittle beams, Journal of the Mechanics and Physics of Solids 31,pp. 427-37.

SCHINDLER, H. J., AND SAYIR, M. (1984), Path of a crack in a beam due todynamic flexural fracture, International Journal of Fracture 25, pp. 95-107.

SCHIRRER, R. (1978), The effects of a strain rate dependent Young's modulusupon the stress and strain fields around a running crack tip, InternationalJournal of Fracture 14, pp. 265-79.

SEAMAN, L., CURRAN, D. R., AND SHOCKEY, D. A. (1976), Computational mod-els for ductile and brittle fracture, Journal of Applied Physics 47, pp. 4814-26.

SEAMAN, L., CURRAN, D. R., AND MURRI, W. J. (1985), A continuum model fordynamic microfracture and fragmentation, Journal of Applied Mechanics 52,pp. 593-600.

552 Bibliography

SECOR, G. (1972), Dynamic fracture of simulated solid propellant, InternationalJournal of Fracture Mechancs 8, pp. 299-309.

SEEBASS, A. R. (1983), Functions of a complex variable, in Handbook of AppliedMathematics, ed. C. E. Pearson. New York: Van Nostrand Reinhold, pp. 226-70.

SHAH, A. H., WONG, K. C , AND DATTA, S. K. (1986), Dynamic stress intensityfactors for buried planar and non-planar cracks, International Journal of Solidsand Structures 22, pp. 845-57.

SHAH, A. H., CHIN, Y. F., AND DATTA, S. K. (1987), Elastic wave scattering bysurface-breaking planar and nonplanar cracks, Journal of Applied Mechanics54, pp. 761-7.

SHAND, E. B. (1954a), Experimental study of fracture of glass: I. The fractureprocess, Journal of the American Ceramic Society 37, pp. 52-60.

SHAND, E. B. (1954b), Experimental study of fracture of glass: II. Experimentaldata, Journal of the American Ceramic Society 37, pp. 559-72.

SHAND, E. B. (1959), Breaking stress of glass determined from dimensions offracture mirrors, Journal of the American Ceramic Society 42, pp. 474-7.

SHANNON, R. W. E., AND WELLS, A. A. (1974), Brittle crack propagation in gasfilled pipelines - A model study using thin walled unplasticised PVC pipe,International Journal of Fracture 10, pp. 471-86.

SHAR, E. N. (1980), Stress state of a straight isolated cut, loaded from withoutby concentrated forces and growing at a constant rate, Journal of AppliedMechanics and Technical Physics (USSR), pp. 144-152.

SHMUELY, M., AND ALTERMAN, Z. S. (1973), Crack propagation analysis by finitedifferences, Journal of Applied Mechanics 40, pp. 902-8.

SHMUELY, M., AND PERETZ, D. (1976), Static and dynamic analysis of the debproblem in fracture mechanics, International Journal of Solids and Structures12, pp. 67-79.

SHMUELY, M. (1977), Analysis of fast fracture and crack arrest by finite differ-ences, International Journal of Fracture 13, pp. 443-54.

SHOCKEY, D. A., CURRAN, D. R., SEAMAN, L., ROSENBERG, J., AND PETERSEN,C. F. (1974), Fragmentation of rock under dynamic loads, InternationalJournal of Rock Mechanics and Mining Science 11, pp. 303-17.

SHOCKEY, D. A., SEAMAN, L., DAO, K. C , AND CURRAN, D. R. (1980), Kineticsof void growth in fracturing A533B tensile bars, Journal of Pressure VesselTechnology 102, pp. 14-21.

SHOCKEY, D. A., KALTHOFF, J. F., AND ERLICH, D. C. (1983), Evaluation ofdynamic crack instability, International Journal of Fracture 22, pp. 217-29.

SHOCKEY, D. A., KALTHOFF, J. F., KLEMM, W., AND WINKLER, S. (1983),Simultaneous measurements of stress intensity and toughness for fast-runningcracks in steel, Experimental Mechanics 23, pp. 140-5.

SHOEMAKER, A. K., AND ROLFE, S. T. (1969), Static and dynamic low temper-ature K behavior of steels, Journal of Basic Engineering 91, pp. 512-8.

SHOEMAKER, A. K., AND ROLFE, S. T. (1971), The static and dynamic low tem-perature crack toughness performance of seven structural steels, EngineeringFracture Mechanics 2, pp. 319-39.

SHOEMAKER, A. K., AND MCCARTNEY, R. F. (1974), Displacement considera-tions for a ductile propagating fracture in line pipe, Journal of EngineeringMaterials and Technology 96, pp. 318-22.

SHOEMAKER, A. K., AND SEELEY, R. S. (1983), Summary report of round robintesting by the ASTM Group E24.01.06 on rapid loading plane strain fracturetoughness testing, Journal of Testing and Evaluation 11, pp. 261-72.

Bibliography 553

SHUKLA, A., AGARWAL, R. K., AND NIGAM, H. (1988), Dynamic fracture studiesof 7075-T6 aluminum and 4340 steel using strain gages and photoelasticcoatings, Engineering Fracture Mechanics 31, pp. 501-15.

SHUKLA, A., AND CHONA, R. (1988), The stress field surrounding a rapidly prop-agating curving crack, in Fracture Mechanics: Eighteenth Symposium, SpecialTechnical Publication 945, eds. D. T. Read and R. P. Reed. Philadelphia:American Society for Testing and Materials, pp. 86-99.

SIH, G. C. (1968), Some elastodynamic problems of cracks, International Journalof Fracture 4, pp. 51-68.

SIH, G. C , AND LOEBER, J. F. (1968), Flexural wave scattering at a throughcrack in an elastic plate, Engineering Fracture Mechanics 1, pp. 369-78.

SIH, G. C , AND LOEBER, J. F. (1969), Normal compression and radial shear wavesscattering at a penny shaped crack in an elastic solid, Journal of the AcousticalSociety of America 46, pp. 711-21.

SIH, G. C. (1970), Dynamic aspects of crack propagation, in Inelastic Behavior ofSolids, eds. M. F. Kanninen et al. New York: McGraw-Hill, pp. 607-39.

SIH, G. C , AND CHEN, E. P. (1972), Crack propagating in a strip of material underplane extension, International Journal of Engineering Science 10, pp. 537-51.

SIH, G. C , EMBLEY, G. T., AND RAVERA, R. S. (1972), Impact response of afinite crack in plane extension, International Journal of Solids and Structures8, pp. 977-93.

SIH, G. C , AND EMBLEY, G. T. (1973), Sudden twisting of a penny shaped crack,International Journal of Solids and Structures 9, pp. 1349-59.

SIMONOV, I. V. (1983), Behavior of solutions of dynamic problems in the neigh-borhood of the edge of a cut moving at transonic speed in an elastic medium,Mechanics of Solids (Mechanika Tverdogo Tela) 18, pp. 100-6.

SLEPYAN, L. I. (1976), Crack dynamics in an elastic-plastic body, Mechanics ofSolids (Mechanika Tverdogo Tela) 11, pp. 126-34.

SLEPYAN, L. I. (1978), Calculation of the size of the crater formed by a high-speedimpact, Soviet Mining Journal 14, pp. 465-71.

SLEPYAN, L. I. (1981a), Dynamics of brittle fracture in lattice, Soviet PhysicsDoklady 26, pp. 538-40.

SLEPYAN, L. I. (1981b), Crack propagation in high-frequency lattice vibrations,Soviet Physics Doklady 26, pp. 900-2.

SLEPYAN, L. I. (1982), Antiplane problem of a crack in a lattice, Mechanics ofSolids (Mechanika Tverdogo Tela) 16, pp. 101-14.

SLEPYAN, L. I. (1984), Dynamics of brittle fracture in media with a structure,Mechanics of Solids (Mechanika Tverdogo Tela) 19, pp. 114-22.

SMITH, E. (1968), Crack bifurcation in brittle solids, Journal of the Mechanicsand Physics of Solids 16, pp. 329-36.

SMITH, E. (1981), The origin of dynamic effects during the arrest of a propagatingcrack, Archive for Mechanics 33, pp. 313-8.

SMITH, E. (1987), The effective fracture energy associated with cleavage crackgrowth in b.c.c. iron, Materials Science and Engineering 85, pp. 53-8.

SMITH, G. C , AND KNAUSS, W. G. (1975), Experiments on critical stress intensityfactors resulting from stress wave loading, Mechanics Research Communica-tions 2, pp. 187-92.

SNEDDON, I. N. (1946), The distribution of stress in the neighborhood of a crack inan elastic solid, Proceedings of the Royal Society (London) A187, pp. 229-60.

SNEDDON, I. N. (1952), The stress produced by a pulse of pressure movingalong the surface of a semi-infinite solid, Rendiconti del Circolo Matematicodi Palermo 1, pp. 57-62.

554 Bibliography

SNEDDON, I. N. (1958), Note on a paper by J. R. M. Radok, Quarterly of AppliedMathematics 16, p. 197.

SOKOLINSKY, V. B. (1981), On the nature of dynamic strength of brittle solids,Engineering Fracture Mechanics 14, pp. 753-8.

SOKOLOWSKI, M. (1977), On a one-dimensional model of the fracture process,Engineering Transactions (Rozprawy Inzynierskie) 25, pp. 369-93.

SPENCER, A. J. M. (1960), The dynamic plane deformation of an ideal plastic-rigidsolid, Journal of the Mechanics and Physics of Solids 8, pp. 262-79.

STAHL, B., AND KEER, L. M. (1972), Vibration and stability of cracked rectangularplates, International Journal of Solids and Structures 8, pp. 69-91.

STAKGOLD, I. (1968), Boundary Value Problems of Mathematical Physics. Vol.11. New York: MacMillan.

STEPANOV, G. V., AND MAKOVEI, V. A. (1982), Determination of fracture tough-ness characteristics under a pulsed loading, Strength of Materials (ProblemyProchnosti), pp. 1252-6.

STEPANOV, G. V. (1984), Unsteady crack propagation under dynamic loading,Strength of Materials (Problemy Prochnosti), pp. 1432-6.

STEPANOV, G. V., AND MAKOVEI, V. A. (1984a), Effect of loading rate onthe fracture toughness of quenched steel, Strength of Materials (ProblemyProchnosti), pp. 802-6.

STEPANOV, G. V., AND MAKOVEI, V. A. (1984b), Propagation of a rapidcrack in annular specimens of brittle steel, Strength of Materials (ProblemyProchnosti), pp. 1255-60.

STERNBERG, E. (1960), On the integration of the equations of motion in theclassical theory of elasticity, Archive for Rational Mechanics and Analysis 6,pp. 34-50.

STEVENS, A. L., DAVISON, L., AND WARREN, W. E. (1972), Spall fracture inaluminum monocrystals: a dislocation dynamics approach, Journal of AppliedPhysics 43, pp. 4922-7.

STEVERDING, B. (1969), Brittleness and impact resistance, Journal of the Amer-ican Ceramic Society 52, pp. 133-6.

STEVERDING, B., AND LEHNIGK, S. H. (1970), Response of cracks to impact,Journal of Applied Physics 41, pp. 2096-9.

STEVERDING, B., AND LEHNIGK, S. H. (1971), Collision of stress pulses withobstacles and dynamics of fracture, Journal of Applied Physics 42, pp. 3231-8.

STOCK, T. A. C. (1967), Stress field intensity factors for propagating brittle cracks,International Journal of Fracture Mechancs 3, pp. 121-9.

STOCKL, H., AND AUER, F. (1976), Dynamic behavior of a tensile crack: finite dif-ference simulation of fracture experiments, International Journal of Fracture12, pp. 345-57.

STROH, A. N. (1954), The formation of cracks as a result of plastic flow,Proceedings of the Royal Society (London) A223, pp. 404-14.

STROH, A. N. (1955), The formation of cracks in plastic flow II, Proceedings ofthe Royal Society (London) A232, pp. 548-60.

STROH, A. N. (1957), A theory of the fracture of metals, Advances in Physics,Philosophical Magazine Supplement 6, pp. 418-65.

STROH, A. N. (1960), A simple model of a propagating crack, Journal of theMechanics and Physics of Solids 8, pp. 119-22.

STROH, A. N. (1962), Steady state problems in anisotropic elasticity, Journal ofMathematics and Physics 41, pp. 77-103.

SUNG, J. C , AND ACHENBACH, J. D. (1987), Temperature at a propagating cracktip in a viscoplastic material, Journal of Thermal Stresses 10, pp. 243-62.

Bibliography 555

SUZUKI, S., HOMMA, H., AND KUSAKA, R. (1988), Pulsed holographic microscopyas a measurement method of dynamic fracture toughness for fast propagatingcracks, Journal of the Mechanics and Physics of Solids 36, pp. 631-53.

TADA, H., PARIS, P. C , AND IRWIN, G. R. (1985), The Stress Analysis of CracksHandbook. St. Louis: Del Research Corporation.

TAKAHASHI, K. (1987), Dynamic fracture instability in glassy polymers as studiedby ultrasonic fractography, Polymer Engineering and Science 27, pp. 25-32.

TAKAHASHI, K., AND ARAKAWA, K. (1987), Dependence of crack acceleration onthe dynamic stress intensity factor of polymers, Experimental Mechanics 27,pp. 195-200.

TAKEUCHI, H., AND KIKUCHI, M. (1973), A dynamical model of crack propaga-tion, Journal of Physics of the Earth 21, pp. 27-37.

TANG, H. C , AND DVORAK, G. J. (1973), The configuration of moving crack tipzones, Engineering Fracture Mechanics 5, pp. 79-90.

TAYLOR, J. W., HARLOW, F. H., AND AMSDEN, A. A. (1978), Dynamic plasticinstabilities in stretching plates and shells, Journal of Applied Mechanics 45,pp. 105-10.

TAYLOR, L. M., CHEN, E. P., AND KUSZMAUL, J. S. (1986), Microcrack induceddamage accumulation in brittle rock under dynamic loading, Computer Meth-ods in Applied Mechanics and Engineering 55, pp. 301-20.

THAU, S. A., AND LU, T. H. (1971), Transient stress intensity factors for a finitecrack in an elastic solid caused by a dilatational wave, International Journalof Solids and Structures 7, pp. 731-50.

THOMSON, R. (1986), Physics of fracture, Solid State Physics 39, pp. 1-129.TILLY, G. P., AND SAGE, W. (1970), The interaction of particles and material

behavior in erosion processes, Wear 16, pp. 447-65.TIPPER, C. F. (1962), The Brittle Fracture Story. Cambridge: Cambridge

University Press.TRUESDELL, C , AND TOUPIN, R. A. (1960), The classical field theories, in

Handbuch der Physik, Vol. III/l, ed. S. Fliigge. Berlin: Springer-Verlag,pp. 226-793.

TSAI, Y. M., AND KOLSKY, H. (1967), A study of the fractures produced in glassblocks by impact, Journal of the Mechanics and Physics of Solids 15, pp. 263-78.

TSAI, Y. M. (1973a), Exact stress distribution, crack shape and energy for arunning penny shaped crack in an infinite elastic solid, International Journalof Fracture 9, pp. 157-69.

TSAI, Y. M., AND CHEN, Y. T. (1978), Propagation of a flexure crack: experimentand analysis, International Journal of Fracture 14, pp. 281-91.

TSAI, Y. M. (1981), Initiation and propagation of a penny shaped crack in a finitelydeformed incompressible elastic medium, Engineering Fracture Mechanics 14,pp. 627-36.

TULER, F. R., AND BUTCHER, B. M. (1968), A criterion for the time dependenceof dynamic fracture, International Journal of Fracture 4, pp. 431-7.

TURNER, R. D. (1956), The diffraction of a cylindrical pulse by a half plane,Quarterly of Applied Mathematics 14, pp. 63-73.

TVERGAARD, V., AND NEEDLEMAN, A. (1988), An analysis of the temperatureand rate dependence of Charpy v-notch energies for a high nitrogen steel,International Journal of Fracture 37, pp. 197-216.

TVERGAARD, V. (1989), Material failure by void growth to coalescence, inAdvances in Applied Mechanics, eds. J. W. Hutchinson and T. Y. Wu. NewYork: Academic, in press.

556 Bibliography

VAN DER POL, B., AND BREMMER, H. (1955), Operational Calculus, 2nd edition.Cambridge: Cambridge University Press.

VAN DER ZWAAG, S., AND FIELD, J. E. (1982a), Liquid jet impact damage on zincsulphide, Journal of Materials Science 17, pp. 2625-36.

VAN DER ZWAAG, S., AND FIELD, J. E. (1983b), Rain erosion damage in brittlematerials, Engineering Fracture Mechanics 17, pp. 367-79.

VAN DER ZWAAG, S., AND FIELD, J. E. (1983c), Indentation and liquid impactstudies on coated germanium, Philosophical Magazine A48, pp. 767-77.

VARDAR, 0. , AND FINNIE, I. (1977), The prediction of fracture in brittle solidssubjected to very short duration tensile stresses, International Journal ofFracture 13, pp. 115-31.

VASUDEVAN, N., AND KNAUSS, W. G. (1988), An approximate analysis of theeffect of micro fractures on the caustic of a dynamically moving crack tip,International Journal of Fracture 36, pp. 121-35.

VELIKOTNYI, A. V., AND SMETANIN, B. I. (1975), On the problem of steadyvibrations of a plane with a slit, Applied Mathematics and Mechanics (PMM)39, pp. 179-82.

VICKERS, G. W., AND JOHNSON, W. (1972), The development of an impressionand the threshold velocity for erosion damage in a-brass and perspex due torepeated jet impact, International Journal of Mechanical Sciences 14, pp. 765-77.

Vu, B. Q., AND KINRA, V. K. (1981), Brittle fracture of plates in tension: Staticfield radiated by a suddenly stopping crack, Engineering Fracture Mechanics15, pp. 107-14.

WALTON, J. R., AND NACHMAN, A. (1979), The propagation of a crack by a rigidwedge in an infinite power law viscoelastic body, Journal of Applied Mechanics46, pp. 605-10.

WALTON, J. R. (1982), On the steady-state propagation of an antiplane shearcrack in an infinite general linearly viscoelastic body, Quarterly of AppliedMathematics 40, pp. 37-52.

WALTON, J. R. (1985), The dynamic steady-state propagation of an antiplaneshear crack in a general linearly viscoelastic layer, Journal of Applied Mechan-ics 52, pp. 853-6.

WALTON, J. R. (1987), The dynamic energy release rate for a steadily propagatingantiplane shear crack in a linearly viscoelastic body, Journal of AppliedMechanics 54, pp. 635-41.

W A R D , G. N. (1955), Linearized Theory of Steady High-Speed Flow. New York:Cambridge University Press.

WATSON, G. N. (1966), A Treatise on the Theory of Bessel Functions. Cambridge:Cambridge University Press.

W E B B , D. (1969), A note on a penny shaped crack expanding under nonuniforminternal pressure, International Journal of Engineering Science 7, pp. 525-30.

WEIBULL, W. (1939), A statistical theory of the strength of materials, IngeniorsVetenskaps Akademien Handlingar, Number 151.

WEICHERT, R., AND SCHONERT, K. (1974), Heat generation at the tip of a movingcrack, Journal of the Mechanics and Physics of Solids 22, pp. 127-33.

WEICHERT, R., AND SCHONERT, K. (1978), On the temperature rise at the tipof a fast running crack, Journal of the Mechanics and Physics of Solids 26,pp. 151-62.

WEIMER, R. J., AND ROGERS, H. C. (1979), Dynamic fracture phenomena inhigh-strength steels, Journal of Applied Physics 50, pp. 8025-30.

WEINER, J. H., AND PEAR, M. (1975), Crack and dislocation propagation in anidealized crystal model, Journal of Applied Physics 46, pp. 2398-405.

Bibliography 557

WELLS, A. A., AND POST, D. (1958), The dynamic stress distribution surroundinga running crack - a photoelastic analysis, Proceedings of the SESA 16, pp. 69-92.

WELLS, A. A. (1961), Influence of residual stresses and metallurgical changes onlow stress brittle fracture in welded steel plates, Welding Journal ResearchSupplement 40, pp. 182-92s.

WESENBERG, D. L., AND SAGARTZ, M. J. (1977), Dynamic fracture of 6061-T6aluminum cylinders, Journal of Applied Mechanics 44, pp. 643-6.

WHEELER, L. T., AND STERNBERG, E. (1968), Some theorems in classicalelastodynamics, Archive for Rational Mechanics and Analysis 31, pp. 51-90.

WHITTAKER, E. T., AND WATSON, G. N. (1927), Modern Analysis. Cambridge:Cambridge University Press.

WIEDERHORN, S. M. (1969), Fracture surface energy of glass, Journal of theAmerican Ceramic Society 52, pp. 99-105.

WIEDERHORN, S. M., AND HOCKEY, B. J. (1983), Effect of material parameterson the erosion resistance of brittle materials, Journal of Materials Science 18,pp. 766-80.

WILLIAMS, D. P. (1973), Relation between hysteresis and the dynamic crackgrowth resistance of rubber, International Journal of Fracture 9, pp. 449-62.

WILLIAMS, J. G. (1987), The analysis of dynamic fracture using lumped mass-spring models, International Journal of Fracture 33, pp. 47-59.

WILLIAMS, J. G., AND ADAMS, G. C. (1987), The analysis of instrumented impacttests using a mass-spring model, International Journal of Fracture 33, pp. 209-22.

WILLIAMS, M. L. (1957), On the stress distribution at the base of a stationarycrack, Journal of Applied Mechanics 24, pp. 109-14.

WILLIS, J. R. (1967a), A comparison of the fracture criteria of Griffith andBarenblatt, Journal of the Mechanics and Physics of Solids 15, pp. 151-62.

WILLIS, J. R. (1967b), Crack propagation in viscoelastic media, Journal of theMechanics and Physics of Solids 15, pp. 229-40.

WILLIS, J. R. (1973), Self-similar problems in elastodynamics, PhilosophicalTransactions of the Royal Society (London) 274, pp. 435-91.

WILLIS, J. R. (1975), Equations of motion for propagating cracks, The Mechanicsand Physics of Fracture, The Metals Society, pp. 57-67.

WILSON, M. L., HAWLEY, R. H., AND DUFFY, J. (1980), The effect of loading rateand temperature on fracture initiation in 1020 hot-rolled steel, EngineeringFracture Mechanics 13, pp. 371-85.

WINKLER, S., CURRAN, D. R., AND SHOCKEY, D. A. (1970a), Crack propagationat supersonic velocities, International Journal of Fracture 6, pp. 151-8.

WINKLER, S., CURRAN, D. R., AND SHOCKEY, D. A. (1970b), Crack propagationat supersonic velocities, International Journal of Fracture 6, pp. 271-8.

WRIGHT, T. W. (1969), Impact on an elastic quarter space, Journal of theAcoustical Society of America 45, pp. 935-43.

Wu, K. C. (1988), Stress intensity factors due to non-elastic strains and bodyforces for steady dynamic crack extension in an anisotropic elastic material,International Journal of Solids and Structures 24, pp. 805-15.

WYSS, M., AND BRUNE, J. N. (1968), Seismic moment, stress, and source dynam-ics for earthquakes in the California-Nevada region, Journal of GeophysicalResearch 73, pp. 4681-94.

YATOMI, C. (1981), The effect of surface energy on temperature rise around a fastrunning crack, Engineering Fracture Mechanics 14, pp. 759-62.

YATOMI, C. (1982), On Irwin's and Achenbach's expressions for the energy releaserate, International Journal of Fracture 82, pp. 233-6.

558 Bibliography

YOFFE, E. H. (1951), The moving Griffith crack, Philosophical Magazine 42,pp. 739-50.

ZEHNDER, A. T., AND ROSAKIS, A. J. (1989), Dynamic fracture initiation andpropagation in 4340 steel under impact loading, International Journal ofFracture 30, in press.

ZENER, C, AND HOLLOMON, J. H. (1944), Effect of strain rate upon plastic flowof steel, Journal of Applied Physics 15, pp. 22-32.

ZIMMERMANN, C, KLEMM, W., AND SCHONERT, K. (1984), Dynamic energyrelease rate and fracture heat in polymethylmethacrylate (pmma) and a highstrength steel, Engineering Fracture Mechanics 20, pp. 777-82.

ZLATIN, N. A., AND IOFFE, B. S. (1973), Time dependence of the resistance tospalling, Soviet Physics: Technical Physics 17, pp. 1390-3.

INDEX

Abel integral equationapplication, 69, 372general solution, 70

Abel asymptotic theorem, 36-7application, 90, 348

acceleration of a particle, 17analytic continuation, 32-3, 85analytic function, 30-2

identity theorem, 32, 85angular momentum, balance of, 18area domain integral, 240arrest, see crack arrestassociated flow rule, 177, 185autonomy of crack tip field, 10-11

Barenblatt cohesive zone, see alsocohesive zone model

Bateman's theorem, 378Bernoulli-Euler beam, 5-8, 254, 421bifurcation, Yoffe criterion, 166blunt ness parameter, 399, 412brittle-ductile transition, 149-51,

485-6Broberg problem, 318-26

energy rate balance, 328-9stress intensity factor, 326

Buseman's method of conical fields,314

Cagniard-de Hoop technique, 77, 91,350

Cauchy stress principle, 16Cauchy's integral formula, 31Cauchy's integral theorem, 31caustics, method of, 12

Chaplygin transformation, 316characteristic length, 105characteristics, method of

ideally plastic material, 176, 186,452-3

string problem, 411-13circular crack

energy rate balance, 339stress intensity factor, 145, 338

circumferential tensile stress, 164cohesionless interface, 398-9cohesive zone model, 235-40, 306-10

cohesive traction, 236crack opening displacement, 236,

435-7rate-dependent, 485-6

slip-weakening zone, 239, 247-9zone length, 308, 432

completeness of representations, 26compliance method, 223, 256compliance of loading system, 418-

21computational methods

area domain integral, 240iteration for steady state problem,

463-4node release method, 241-2singular finite element, 243

conservation law, 264-6contact of crack faces, 38, 116, 139crack arrest, 367, 481-5

arrest fracture toughness, 414, 484arrest length, 399-400, 409-10,

413-14arrest time, 413, 430

559

560 Index

due to reflected waves, 413due to variable resistance, 408-10

crack growth criterion, see fracturecriterion

crack tip equation of motion, seeequation of motion

crack tip opening angle, 181, 183fracture criterion, 467

crack tip opening displacement, 181,183, 236, 309, 435-7

fracture criterion, 436, 494criterion for fracture, see fracture

criterioncritical crack length, 6-8cross product, 14curl of a vector, 14curved crack path, 160

delay time, 357, 426diffraction of waves, 70, 94, 125dilatation, 24, 27dilatational wave speed, 23dislocation, moving, 110-12, 121displacement components, in terms

of potentials, 28displacement of a particle, 17displacement potentials

Lam£ representation, 26Poisson representation, 27

divergenceof a vector, 14of a tensor, 15

divergence theorem, 15double cantilever beam, 254

approximate equation of motion,424-5

energy release rate, 256rotary inertia, 256

driving force, 2,6Dugdale plastic zone, 238; see also

cohesive zone modeldyadic notation, 13-15dynamic energy release rate, 231; see

also energy release ratedynamic stress intensity factor, 58;

see also stress intensity factor

earthquake rupture mechanicsradiated energy, 289, 405-7seismic efficiency, 294-5seismic moment, 293-5shear crack models, 247, 293, 309-

10, 330^, 407

slip-weakening zone, 239-40, 247-9

strength inhomogenieties, 407-10stress drop, 249, 295transonic crack propagation, 173

elastic compliance, 29, 231elastic constants, relations among,

24elastic rate dominance, 470elastic stiffness, 29elastic stress intensity factor, see

stress intensity factorelastic-plastic material, 177, 185elastic-viscoplastic material, 215elastic-viscous material, 206elastodynamic boundary value prob-

lem, 25elliptical crack, 339-40energy, bounded total, 64, 74-5energy, fracture, see fracture energyenergy flux integral, 224-7energy radiation

periodic fracture resistance, 401-5seismic, 289

energy rate balance, 5dynamic, 8, 19, 475

energy release rate, 10, 221, 231equation of motion, crack tip, 367

double cantilever beam, 424-5shear crack growth, 407-10string model, 413tensile crack growth, 396, 430

finite deformation, 16-21, 227-8,500-3

fracture criterioncrack tip opening angle, 467crack tip opening displacement,

436, 494energy criterion, 149, 394

plastic strain, 459-60stress, 495stress intensity factor criterion,

60, 394fracture energy, specific, 394

periodic, 401-7position dependent, 407speed dependent, 400-1, 417-18

fracture initiationcrack tip bluntness, 399, 412dynamic loading, 140-51, 426-30,

436-7fracture toughness, 10, 140

Index 561

fracture toughness, dynamic, 141versus crack speed, 448-50, 465-7

fragmentation, 508-12, 518fundamental solution, 342, 358

and superposition, 341, 350-2,362-3

gradientof a scalar, 14of a vector, 15

grazing incidence, 129Green's function, 65Green's method of solution, 65-7,

370Griffith energy criterion, 4-8

Hadamaard's lemma, 70Hamilton's principle, 422-4heat flux vector, 230Helmholtz decomposition, 26higher order terms, 58, 94-7, 169-70hodograph transformation, 453homogeneous function, 63homogeneous solutions, method of,

313Hooke's law, 22hoop stress, see circumferential ten-

sile stresshyperbolic system of partial differ-

ential equations, 178, 251, 411,452

ideally plastic material, 177, 186identity theorem of analytic func-

tions, 32impact strength of brittle material,

514incompressibility condition, 188inertia of crack tip, 398inertial effects, 2-3, 153-4, 235influence function, 431-2; see also

weight functioninner product, 13Irwin fracture criterion, 10, 60, 140Irwin relationship, 10, 222-3Irwin-Williams asymptotic stress

field, 58-9

J-integral, 223, 264Jones's method, 78Jordan's lemma, 34jump conditions, 18, 198, 411

kinetic energy, 8, 18, 225

Lamb's problem, 109Lame" displacement potentials, 26

displacement in terms of, 28stress in terms of, 28

Lame* elastic constants, 22-3Laplace transform, 33-6

Abel asymptotic theorem, 36-7convolution formula, 33inversion formula, 33, 35one-sided transform, 33two-sided transform, 35

Laplace's equation, 30general solution, 30

linear elastic fracture mechanics, 55linear elastodynamics, 22-9linear momentum, balance of, 17

propagating discontinuity, 18, 199,411

Liouville's theorem, 32, 91

mass of crack tip, apparent, 398maximum plastic work inequality,

200microcracking, 508-9

fragment size, 511time-dependent strength, 512

modes of deformation, 38, 57modulus of cohesion, 308moving loads on crack faces, 105

Navier's equation, 23Neumann's uniqueness theorem, see

uniqueness theoremnode release method, 241-2nonuniform crack tip motion

mode I stress intensity factor, 390,428

mode II stress intensity factor,393 '

mode III stress intensity factor,374, 375

numerical techniques, see computa-tional methods

optical measurement methods, 12overshoot, dynamic, 57, 123

path-independent integral, 223, 229,264-9

penny-shaped crack, see circularcrack

562 Index

photoelasticity, 12plane wave, incident on crack, 70,

94, 123-31, 142-4, 356-8, 365-6, 375-6, 426-30

effect on growth, 365-6, 413-15normal incidence, 70-1, 94oblique incidence, 104, 123

plastic strain fracture criterion, 459-61

plastic work rate parameter, 177,185, 197

plastic zone, 238-9, 432-7, 459-65, 492; see also cohesive zonemodel

plate in bending, 261Poisson representation, 27Poisson's ratio, 24potential energy, 4principal stress, maximum, 165, 172

radiated seismic energy, 289, 406-7radiated stress fields, 353, 373, 381radiation conditions, 77rate of deformation, 21Rayleigh wave function, 83

modified, 128, 130, 346Wiener-Hopf factorization, 87-8zeros of, 83

Rayleigh wave speed, 83reciprocal theorem, elastodynamic,

26, 281rotation, 24, 27

seismic efficiency, 294-5seismic moment, 293-5seismic stress drop, 249, 295self-similar crack growth, 313-18,

336shadow spot method, 12, 147shear beam model, 250shear stress, maximum, 165, 173shear wave speed, 23shell, pressurized, 262-4singular element method, 242size effect for brittle materials, 513slip weakening model, 239, 247-9;

see also cohesive zonesmall-scale yielding, 11, 59, 310,

434-7, 462, 474stability of equilibrium, 6-8steady state crack growth, 298-300

approach to steady state, 310-12,467-9

strain, 21infinitesimal, 22

strain energy, 29, 231strain rate effects, 470-2, 485stream function, 188stress components and Lame* poten-

tials, 28stress drop, 249, 295stress fracture criterion, 495stress intensity factor

concept, 10, 140-1dynamic, 58, 158, 163, 171

stress work density, 20, 225string model, 250-1

arrest length, 413-14crack tip equation of motion, 413energy release rate, 252energy variations, 415-17

strip yield zone, see cohesive zonesummation convention, 13surface energy, 4, 9, 259symmetry of fields, 61, 72, 186

temperature effect, 151, 481-5terminal crack tip speed, 9, 38, 397three-dimensional scattering, 123time to fracture, 141-4, 147, 430,

436-7, 512-15transition strain rate, 470transonic crack growth, 173transport theorem, generalized, 20-1triaxiality, 167, 193, 485, 498

ultrasonic waves, 12, 398uniqueness theorem, 25

for crack propagation, 437-41universal function

A(v), 234g{v), 352, 397k(v), 348-9kn(v), 355kmiv), 356

universal spatial dependence, 59, 167

variational methods, 265velocity of a particle, 17

vibration of cracked solid, 273viscoelastic material, 443-4

sharp crack paradox, 445-6viscosity, 471void growth, ductile, 498-500

inertial effect, 500-3, 507rate-dependent material, 503-5

Index 563

wave equation, 26Green's solution, 66homogeneous solutions, 62-3, 313-

17wave loading, see plane wavewave speeds, elastic, 23wavefronts, 61, 73Weibull statistical approach, 513weight function, 274; see also influ-

ence functionWiener-Hopf technique, 77

analytic continuation, 85

factorization, 84-5functional equation, 83, 345

work, rate of, 18work of fracture, 4-9

yield condition, 176, 186Yoffe problem, 300-4

energy rate balance, 304stress intensity factor, 303

Young's modulus, 24

zero counting formula, 87-8

freund-dynamic fracture mechanics

Documents