elastoplasticity theory - friedrich pfeiffer, peter wriggers.pdf

Lecture Notes in Applied and Computational Mechanics Volume 42

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Elastoplasticity Theory

Koichi Hashiguchi

123

Koichi Hashiguchi 3-10-10-201, Ohtemon Fukuoka, 810-0074 Japan E-mail: [email protected]

ISBN: 978-3-642-00272-4 e-ISBN: 978-3-642-00273-1 DOI 10.1007/978-3-642-00273-1

Lecture Notes in Applied and Computational Mechanics ISSN 1613-7736

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Preface

Contents

Recent advancements in the performance of industrial products and structures are quite intense. Consequently, mechanical design of high accuracy is necessary to enhance their mechanical performance, strength and durability. The basis for their mechanical design can be provided through elastoplastic deformation analyses. For that reason, industrial engineers in the fields of mechanical, civil, architectu-ral, aerospace engineering, etc. must learn pertinent knowledge relevant to elasto-plasticity.

Numerous books about elastoplasticity have been published since “Mathemati-cal Theory of Plasticity”, the notable book of R. Hill (1950), was written in the middle of the last century. That and similar books mainly address conventional plasticity models on the premise that the interior of a yield surface is an elastic domain. However, conventional plasticity models are applicable to the prediction of monotonic loading behavior, but are inapplicable to prediction of deformation behavior of machinery subjected to cyclic loading and civil or architectural struc-tures subjected to earthquakes. Elastoplasticity has developed to predict deforma-tion behavior under cyclic loading and non-proportional loading and to describe nonlocal, finite and rate-dependent deformation behavior.

The author has been lecturing on applied mechanics and has been investigating elastoplasticity for nearly a half century, during which time elastoplasticity has made great progress. Various lecture notes, research papers, review articles in English or Japanese, and books in Japanese on these subjects are piled at hand. At present, the author is continuing composition of a monograph on elastoplasticity that has been published serially in a monthly journal from June 2007, to be com-pleted at the end of 2009, although it will unfortunately have been written entirely in Japanese. Based on those teaching and research materials, this book compre-hensively addresses fundamental concepts and formulations of phenomenological elastoplasticity from the conventional to latest theories. Especially, the subloading surface model falling within the framework of the unconventional plasticity model is delineated in detail, which enables us to predict rigorously the plastic strain rate induced by the rate of stress inside the yield surface, comparing it with the other unconventional models. The viscoplastic model is also presented; it is applicable to prediction of deformation behavior in the wide range of strain rate from the quasi-static to the impact loads. Explicit constitutive equations of metals and soils are given for practical application of the theories. In addition, constitutive models of friction are described because they are indispensable for analysis of boundary

VI Preface value problems. Various theories proposed by the author himself are included among the contents in this book in no small number. Their detailed explanations would be possible but, on the other hand, they would easily fall into subjective explanations. For that reason, particular care was devoted to retaining objectivity in the presentation.

The main purpose of this book is to expedite the application of elastoplasticity theory to analyses of engineering problems in practice. Consequently, the salient feature of this book is the exhaustive explanation of elastoplasticity, which is in-tended to be understood easily and clearly not only by researchers but also by be-ginners in the field of applied mechanics, without reading any other book. There-fore, mathematics including vector-tensor analysis and derivatives and the fundamentals in continuum mechanics are first explained to the degree necessary to understand the elastoplasticity theory described in subsequent chapters. For that reason, circumstantial explanations of physical concepts and formulations in elas-toplasticity are given without a logical jump such that the derivations and trans-formations of all equations are described without abbreviation. Besides, a general formulation unlimited to a particular material is first addressed in detail since de-formations of materials obey fundamental common characteristics, which would provide the universal knowledge for deformation of materials more than describ-ing a formulation for a particular material. Thereafter, explicit constitutive equa-tions of metals, soils, and friction phenomena are presented in detail, specifying material functions involved in the general formulation. Without difficulty, readers will be able to incorporate the equations included in this book into their computer programs. The author expects that a wide audience including students, engineers, and researchers of elastoplasticity will read this book and that this work will thereby contribute to the steady development of the study of elastoplasticity and applied mechanics.

As a foundation, the mathematical and the physical ingredients of the contin-uum mechanics are treated in Chapters 1, 2, 3, and 4. Chapter 1 addresses tensor analysis. The physical quantities used in continuum mechanics are tensors; conse-quently, their relations are described mathematically using tensor equations. Ex-planations for mathematical properties and rules of tensors are presented to the ex-tent that is sufficient to understand the subject of this book: elastoplasticity. In chapters 2–4, physical quantities for the description of deformation of solids, con-taining stress (rate) and strain (rate), are described with their fundamental laws. Chapter 2 is devoted to the description of motion and strain (rate) and their related quantities. Chapter 3 presents conservation laws of mass, momentum, and angular momentum, and equilibrium equations and virtual work principles derived from them. In addition, their rate forms used for constitutive equations of inelastic de-formation are explained concisely. Chapter 4 specifically addresses the objectivity of constitutive equations, which is required for the description of material proper-ties. The objectivities of various stress, strain and rotation measures are described by examining their coordinate transformation rules. Then, it is shown that the ma-terial-time derivative of state variables has no objectivity. The corotational rate tensors fulfilling the objectivity are described in detail.

Preface VII

Chapter 5 specifically examines the description of elastic deformation. Elastic constitutive equations are classified as either hyperelasticity, Cauchy elasticity, or hypoelasticity depending on their levels of reversibility. The mathematical and physical characteristics of these equations are explained prior to the description of elastoplastic constitutive equations in the subsequent chapters.

Elastoplastic constitutive equations are described comprehensively in Chapters 6–8. Chapter 6 presents the basic concepts and formulations of plastic constitutive equation, e.g. the elastic and the plastic strain rates, the consistency condition, the plastic flow rule, and the loading criterion. Then, the elastoplastic constitutive equation is formulated based on them. A description of anisotropy and the tangen-tial inelastic strain rate are also incorporated. However, they fall within the framework of conventional plasticity on the premise that the interior of the yield surface is an elastic domain. Therefore, they are incapable of predicting a smooth transition from the elastic to plastic state and a cyclic loading behavior of real ma-terials. In Chapter 7, the continuity and the smoothness conditions are described first. They are the fundamental requirements for the constitutive equations for ir-reversible deformation. Then, various unconventional elastoplastic constitutive equations are proposed, with the intention of describing the plastic strain rate in-duced by the rate of stress inside the yield surface. Among those equations, only the subloading surface model, assuming the subloading surface passing through the current stress point and similar to the yield surface, fulfils the requirements for elastoplastic constitutive equations: continuity and smoothness conditions. Then, the inelastic strain rate attributable to the stress rate tangential to the subloading surface is pertinently incorporated, which is indispensable for the prediction of non-proportional loading behavior observed often in plastic instability problems. In Chapter 8, the initial subloading surface model explained in Chapter 7 is shown to be incapable of describing cyclic loading behavior appropriately because it pre-dicts only an elastic strain rate in the unloading process. Therefore, excessive strain accumulation with open hysteresis loops is predicted in the cyclic loading process. Various cyclic plasticity models have been proposed to date. Among them, only the extended subloading surface model, making a similarity center of the normal-yield and the subloading surfaces move with a plastic strain rate, can represent cyclic loading behavior appropriately, fulfilling the continuity and smoothness conditions. Notable advantages of the extended subloading surface model are presented in comparison with the other cyclic plasticity models.

Chapter 9 presents a viscoplastic constitutive equation for describing rate-dependent deformation induced for the stress level over the yield surface. A perti-nent viscoplastic constitutive equation is described, in which the concept of the subloading surface is incorporated into the overstress model. It is applicable to the prediction of rate-dependent deformation behavior from quasi-static to impact loads.

In Chapters 10 and 11, based on the elastoplastic constitutive equations de-scribed in the preceding chapters, specific constitutive equations of metals and soils are formulated. They are typical elastoplastic materials related to engineering practice. Specific yield conditions, evolution rules of hardening and softening,

VIII Preface anisotropy, stagnation of isotropic hardening, up/degradation of structure, etc. for these materials are incorporated in their formulations.

Special issues related to elastoplastic deformation behavior are discussed in Chapters 12–14. Chapter 12 specifically examines corotational rate tensors, the necessity of which is suggested in Chapter 4. Mechanical features of corotational tensors with various spins are examined comparing their simple shear deformation characteristics. The pertinence of the plastic spin is particularly explained. Chapter 13 opens with a mechanical interpretation for the localization of deformation in-ducing a shear band. Then, the approaches to the prediction of shear band incep-tion condition, inclination and thickness, e.g. eigenvalue analysis and the gradient theory are explained, and the smeared model, i.e., the shear-band embedded model for the practical finite element analysis, is described. Chapter 14 verifies first the distinguished ability of the subloading surface model for the numerical calculation in the yield state. However, it is limited to the yield state. Then, the basic equa-tions for the return-mapping method are shown for subyield state in the subloading surface model.

Chapter 15 describes the prediction of friction phenomena. All bodies except those floating in a vacuum are contacting with other bodies so that the friction phenomena occur on their surfaces. Pertinent analyses, not only of the deforma-tion behavior of bodies but also of friction behavior on the contact surface, are necessary for the analyses of boundary-value problems. A constitutive equation of friction is formulated similarly to the elastoplastic constitutive equation. It is sub-sequently extended to describe the transition from a static to a kinetic friction at-tributable to plastic softening and the recovery of the static friction attributable to creep hardening.

The author wishes to express his thanks to the colleagues at Kyushu University, who have discussed and collaborated for a long time during work undertaken until retirement: Professor M. Ueno (currently at The University of the Ryukyus), and T. Okayasu, S. Tsutsumi, and S. Ozaki (currently at Yokohama National Univ.). In addition, Professor T. Tanaka of The University of Tokyo, Professor Yatomi, C. of Kanazawa Univ., Professor F. Yoshida of Hiroshima Univ., Professor M. Kuroda of Yamagata Univ., Y. Yamakawa of Tohoku Univ., Dr. T. Ozaki of Kyu-shu Electric Engineering Consultants Inc., and Mr. T. Mase of Tokyo Electric Power Services Co., Ltd. are appreciated for their valuable discussions and col-laborations.

Furthermore, the author would like to express his sincere gratitude to Professor A. Asaoka and his colleagues at Nagoya University: Professor M. Nakano and Professor T. Noda who have appreciated and used the author’s subloading surface model widely in their analyses and who have offered discussion continually on de-formation of geomaterials. In addition, the author thanks Professor T. Nakai and Professor T.F. Zhang of the Nagoya Institute of Technology, for their valuable comments.

The author is deeply indebted to Professor Bogdan Raniecki and H. Petryk of the Inst. Fund. Tech. Research, Poland, who have visited Kyushu University sev-eral times to deliver lectures on applied mechanics. Bogdan gave me valuable comments and suggestions by the critical reading of the manuscript, which highly

Preface IX contributed to the correction. Further, the author thanks Professor I.F. Collins of the University of Auckland, Professor O.T. Bruhns of Ruhr Univ., Bochum, Pro-fessor A.C. Aifantis of Michigan Tech. Univ. and Professor I. Vardoulakis of Natl. Univ. Tech. Athens, who have also stayed at Kyushu Univ., delivering lectures and engaging in valuable discussions related to continuum mechanics.

The author wishes to acknowledge deeply Professor P. Wrrigers of Hanover Univ. who recommend me to publish this book in the series of Lecture Notes in Applied and Computational Mechanics, Springer. Finally, the author would like to state that the enthusiastic support of Mrs. Heather King from the Springer Publish-ing Company and SPS data processing team for their help in editing this book.

Fukuoka December 2008

Koichi Hashiguchi

Contents

Contents

1 Tensor Analysis ……………………………………………………… 1

1.1 Conventions and Symbols………………………………………... 1 1.1.1 Summation Convention…………………………………... 1 1.1.2 Kronecker’s Delta and Permutation Symbol……………... 1 1.1.3 Matrix and Determinant…………………………………... 2 1.2 Vector…………………………………………………………...... 8 1.2.1 Definition of Vector………………………………………. 8 1.2.2 Operations for Vectors……………………………………. 8 1.2.3 Component Description of Vector………………………... 9 1.3 Tensor……………………………………………………………. 14 1.3.1 Definition of Tensor………………………………………. 14 1.3.2 Quotient Law……………………………………………… 15 1.3.3 Notations of Tensors……………………………………… 17 1.3.4 Orthogonal Tensor………………………………………... 18 1.3.5 Tensor Product and Component…………………………... 19 1.4 Operations of Second-Order Tensor……………………………... 20 1.4.1 Trace……………………………………………………… 20 1.4.2 Various Tensors…………………………………………... 21 1.5 Eigenvalues and Eigenvectors………………………………….... 26 1.6 Calculations of Eigenvalues and Eigenvectors…………………... 31 1.6.1 Eigenvalues……………………………………………….. 31 1.6.2 Eigenvectors………………………………………………. 32 1.7 Eigenvalue and Eigenvectors of Skew-Symmetric Tensor………. 33 1.8 Cayley-Hamilton’s Theorem…………………………………….. 35 1.9 Positive Definite Tensor…………………………………………. 35 1.10 Polar Decomposition…………………………………………....... 36 1.11 Isotropic Tensor-Valued Tensor Function……………………….. 37 1.12 Representation of Tensor in Principal Space…………………….. 40 1.13 Two-Dimensional State………………….……………………….. 44 1.14 Partial Differential Calculi……………..……………………….... 47 1.15 Time Derivatives……………………..………………………..…. 50 1.16 Differentiation and Integration in Field……………………..…… 51

2 Motion and Strain (Rate)……………………………………………. 57

2.1 Motion and Deformation…………………………………………. 57 2.1.1 Material, Spatial and Relative Descriptions………………. 57 2.1.2 Deformation Gradient and Deformation Tensors………… 59

XII Contents

2.2 Strain Tensor……………………………………………………... 64 2.3 Strain Rate and Spin Tensors…………………………………….. 70 2.4 Various Simple Deformations……………………………………. 81 2.4.1 Uniaxial Loading………………………………………….. 81 2.4.2 Simple Shear……………………………………………… 84 2.4.3 Combination of Tension and Distortion…………………... 94 2.5 Surface Element, Volume Element and Their Rates……………... 97

3 Conservation Laws and Stress Tensors………………………..…… 101

3.1 Conservation Law of Mass………………………………………. 101 3.2 Conservation Law of Momentum………………………………... 101 3.3 Conservation Law of Angular Momentum………………………. 102 3.4 Stress Tensor……………………………………………………... 102 3.5 Equilibrium Equation…………………………………………….. 105 3.6 Equilibrium Equation of Moment………………………………... 107 3.7 Virtual Work Principle…………………………………………… 108

4 Objectivity and Corotational Rate Tensor…………………………. 111

4.1 Objectivity………………………………………………………... 111 4.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities…………………………………………………………

112

4.3 Rate of State Variable and Corotational Rate Tensor……………. 114 4.4 Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative…………………………………….

119

4.5 Various Objective Stress Rate Tensors…………………………... 122 4.6 Work Conjugacy…………………………………………………. 124

5 Elastic Constitutive Equations…….………………………………… 127

5.1 Hyperelasticity…………………………………………………… 127 5.2 Cauchy Elasticity………………………………………………… 130 5.3 Hypoelasticity……………………………………………………. 131

6 Basic Formulations for Elastoplastic Constitutive Equations…….. 135

6.1 Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate…………………………..

135

6.2 Conventional Elastoplastic Constitutive Equations……………… 142 6.3 Loading Criterion……………….………………………………... 148 6.4 Associated Flow Rule……………………………………………. 151 6.4.1 Positivity of Second-Order Plastic Work Rate: Prager’s Interpretation ……………………………………………...

151

6.4.2 Positivity of Work Done During Stress Cycle: Drucker’s Hypothesis…………………………………………………

151

6.4.3 Positivity of Second-Order Plastic Relaxation Work Rate..………………………………………………………

152

6.4.4 Comparison of Interpretations for Associated Flow Rule... 153

Contents XIII

6.5 Anisotropy………………………………………………………... 156 6.5.1 Definition of Isotropy.…………………………………….. 156 6.5.2 Anisotropic Plastic Constitutive Equation………………... 157 6.6 Incorporation of Tangential-Inelastic Strain Rate………………... 159

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory……………………………………………………….……

165

7 Unconventional Elastoplasticity Model: Subloading Surface

Model………………………………………..........................................

171 7.1 Mechanical Requirements………………………………………... 171 7.1.1 Continuity Condition……………………………………… 171 7.1.2 Smoothness Condition……………………………………. 172 7.2 Subloading Surface Model……………………………………….. 174 7.3 Salient Features of Subloading Surface Model…………………... 181 7.4 On Bounding Surface and Bounding Surface Model…………….. 184 7.5 Incorporation of Anisotropy……………………………………... 186 7.6 Incorporation of Tangential Inelastic Strain Rate………………... 187

8 Cyclic Plasticity Model: Extended Subloading Surface Model…… 191

8.1 Classification of Cyclic Plasticity Models……………………….. 191 8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening………………………..………...………….

191

8.2.1 Multi-surface Model……………………………………… 191 8.2.2 Two-Surface Model………………………………………. 194 8.2.3 Infinite-Surface Model……………………………………. 195 8.2.4 Nonlinear Kinematic Hardening Model…………………... 195 8.3 Extended Subloading Surface Model…………………………….. 196 8.4 Modification of Reloading Curve………………………………... 205 8.5 Incorporation of Tangential-Inelastic Strain Rate………………... 208

9 Viscoplastic Constitutive Equations………………………………… 211

9.1 History of Viscoplastic Constitutive Equations………………….. 211 9.2 Mechanical Response of Ordinary Overstress Model……………. 214 9.3 Modification of Overstress Model: Extension to General Rate of Deformation……………………………………………………

215

9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model…………………………………………………

217

10 Constitutive Equations of Metals………………………………….... 221

10.1 Isotropic and Kinematic Hardening…………………………….. 221 10.2 Cyclic Stagnation of Isotropic Hardening……………………… 225 10.3 On Calculation of the Normal-Yield Ratio……………………... 232 10.4 Comparisons of Test Results…………………………………… 232 10.5 Orthotropic Anisotropy…………………………………………. 238

XIV Contents

10.6 Representation of Isotropic Mises Yield Condition……………. 244 10.6.1 Plane Stress State……………………………………….. 245 10.6.2 Plane Strain State……………………………………….. 248

11 Constitutive Equations of Soils……………………………………… 249

11.1 Isotropic Consolidation Characteristics………………………… 249 11.2 Yield Conditions………………………………………………... 253 11.3 Isotropic Hardening Function…………………………………... 259 11.4 Rotational Hardening…………………………………………… 261 11.5 Extended Subloading Surface Model…………………………... 265 11.6 Partial Derivatives of Subloading Surface Function……………. 267 11.7 Calculation of Normal-Yield Ratio……………………………... 271 11.8 Simulations of Test Results…………………………………….. 275 11.9 Simple Subloading Surface Model……………………………... 281 11.10 Super-Yield Surface for Structured Soils in Natural

Deposits………………………………………………………..

291 11.11 Numerical Analysis of Footing Settlement Problem………….. 301

12 Corotational Rate Tensor…………….…………….…………….….. 309

12.1 Hypoelasticity…………….…………….…………….………… 309 12.1.1 Jaumann Rate…………….…………….………………. 309 12.1.2 Green-Naghdi Rate…………….…………….………… 311 12.2 Kinematic Hardening Material…………….…………………… 313 12.2.1 Jaumann Rate…………….…………….………………. 315 12.2.2 Green-Naghdi Rate…………….…………….………… 316 12.3 Plastic Spin…………….…………….…………….………….... 317

13 Localization of Deformation………………………………………… 327

13.1 Element Test…………….…………….…………….………….. 327 13.2 Gradient Theory…………….…………….…………….………. 328 13.3 Shear-Band Embedded Model: Smeared Crack Model………… 331 13.4 Necessary Condition for Shear Band Inception………………… 333

14 Numerical Calculation………………………………………………. 337

14.1 Numerical Ability of Subloading Surface Model………………. 337 14.2 Return-Mapping Algorithm Formulation for Subloading

Surface Model…………………………………………………..

340 15 Constitutive Equation for Friction………………………………….. 349

15.1 History of Constitutive Equation for Friction…………………... 349 15.2 Decomposition of Sliding Velocity………………….................. 350 15.3 Normal Sliding-Yield and Sliding-Subloading Surfaces……….. 354 15.4 Evolution Rules of Sliding-Hardening Function and Normal

Sliding-Yield Ratio………..………………….............................

355 15.4.1 Evolution Rule of Sliding-Hardening Function……….. 355 15.4.2 Evolution Rule of Normal Sliding-Yield Ratio………... 356

Contents XV

15.5 Relations of Contact Traction Rate and Sliding Velocity……… 357 15.6 Loading Criterion………………….............................................. 359 15.7 Sliding-Yield Surfaces…………………....................................... 360 15.8 Basic Mechanical Behavior of Subloading-Friction Model…….. 365 15.8.1 Relation of Tangential Contact Traction Rate and

Sliding Velocity…………………...………………….....

366 15.8.2 Numerical Experiments and Comparisons with Test

Data………………….......................................................

367 15.9 Extension to Orthotropic Anisotropy…………………................ 375 Appendixes……………………………………………………………. 387 Appendix 1: Projection of Area………………………………………... 387 Appendix 2: Proof of ( / ) / = 0jjA xF J ∂∂ ................................................ 388 Appendix 3: Euler’s Theorem for Homogeneous Function…………… 388 Appendix 4: Normal Vector of Surface………………………………... 389 Appendix 5: Convexity of Two-Dimensional Curve………………….. 390 Appendix 6: Derivation of Eq. (11.19) ……………………………….. 391 Appendix 7: Numerical Experiments for Deformation Behavior Near

Yield State………………………………………………...

392

References……………………………………………………………... 395 Index…………………………………………………………………… 407

K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 1–56. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

Chapter 1 Tensor Analysis 1 Tensor Analysis

Physical quantities appearing in deformation mechanics of solids belong mathematically to tensors. Therefore, their relations are described using tensor equations. Before studying the main theme of this book, elastoplaticity theory, the mathematical properties of tensors and the mathematical rules on tensor operations are explained to the extent necessary to understand elasoplasticity theory. The orthogonal Cartesian coordinate system is adopted throughout this book.

1.1 Conventions and Symbols

Some basic conventions and symbols appearing in the tensor analysis are described in this section.

1.1.1 Summation Convention

We first introduce the Cartesian summation convention. A repeated suffix in any term is summed over numbers that the suffix can take. For instance,

1 1

1

= =

=

= , =

=

n n

r r r r r rir irr r

n

rr rrr

u v u v T v T v

T T

⎫⎪⎪⎬⎪⎪⎭

∑ ∑

∑ (1.1)

where the range of suffixes is 1, 2, , n• •• . Because of = ,r r s su v u v

= , =isir r s rr ssT v T v T T a letter of the repeated suffix is arbitrary. It is therefore

called as the dummy index. The convention described above is also called Einstein’s summation convention.

Hereinafter, we assume that a repeated suffix obeys this convention, except for the particular case in which it is stated that this convention is not applied.

1.1.2 Kronecker’s Delta and Permutation Symbol

The symbol ( , =1,2, , )ιj i j nδ … defined in the following equation is called the

Kronecker’s delta.

2 1 Tensor Analysis

1:

0 : iji = j

=i j

δ ⎧⎨ ≠⎩

　

(1.2)

for which it holds that

, ir rj ij iiδ δ = δ δ = n (1.3)

Furthermore, the symbol 21 np p pε … defined by the following equation is called the alternating (or permutation) symbol or Eddington’s epsilon or Levi-Citiva “e” tensor.

21

1 2 are different from each other( 1) :

0 : othersp p p

ln

n

p p p=ε ⎧ −⎨⎩

…... 　

(1.4)

where l is the number of replacement required to obtain the permutation

1 2 , , , np p p... from the regular permutation 1, 2, , n... . Equation (1.4) is

written for the third degree ( = 3)n as

1 : for is a cyclic permutation of 123

1 : for is an anticyclic permutation of 123

0 : others

ijk

ijk

= ijkε⎧⎪−⎨⎪⎩

(1.5)

The number of permutations that the suffixes 1 2 , , , np p p... in 21p p pnε … take

different values from each other is !n and, needless to say, the square of

21 ( 1 or 1)=p p pnε −… is +1. Therefore, it holds that

2 21 1 !p p p p p pn n n=ε ε… … (1.6)

1.1.3 Matrix and Determinant

When the quantity T possessing n n× components ijT is expressed in the

arrangement

111 12

21 22 2

1 2

= [ ]

n

n

ij

nnn n

T T T

T T TT =

T T T

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

T

　

……

… (1.7)

the expression of T in this form is called a matrix. For the two matrices T and S ,

their product TS is defined by the matrix having the following components.

1.1 Conventions and Symbols 3

( )ij rjir= T STS (1.8)

Further, the quantity defined by the following equation is called the determinant of T and is shown by the symbol det T , i.e.

1 2 1 21 2detnnp p p p p pn= T T TεT … … (1.9)

which for = 3n becomes

1 2 3det p qqp rr= T T TεT (1.10)

Therefore, the determinant is the number obtained using the process by which i) one makes product by selecting elements in different rows from all lines (number of products made in this process is !n ) and ii) sums them up with signs which are plus for even permutation and minus for odd permutation, i.e. multiplying the permutation symbol.

Now, consider the rearrangement of the elements

1 2 1 221 2 1 nnp p p nn rr rT T T T T T→… …

changing the order such that the second suffices becomes the regular permutation, while the first suffices was the regular order before the rearrangement. Here, the

numbers of times in the replacement to obtain the permutations 1 2, , , np p p…

and 1 2, , , nr r r… from the regular permutation 1, 2, , n... are the same. Then,

Eq. (1.9) can be rewritten as follows:

1 2 1 221det n nnr r r r r r= T T TεT … … (1.11)

Eventually, “the value of determinant does not change even if the lines and the rows are mutually substituted”, i.e.

det detT =T T (1.12)

where ( )T designates the transpose, i.e. the mutual replacement of lines and rows.

Here, the number of permutations that the suffixes 1 2 , , , np p p... in 21p p pnε …

can take is !n . Therefore, Eqs. (1.9) and (1.11) can be written collectively as

1 2 1 2 1 1 2 2

1det! n n nnp p p p p pr r r r r r= T T T

nε εT … … … (1.13)

which is written for the third degree matrix as

1det3! abc pq ap cbqr r= T T Tε εT (1.14)

4 1 Tensor Analysis Eq. (1.13) is rewritten as

1 1det , det Trs rs= T =n nΔT T TΔ (1.15)

where

1 1 21 2 1 1 1 2 2 111

( 1)! n np pij j p pp p nni rr r rr rT T TnΔ ε ε

− −− −≡ − …… … (1.16)

which is called the cofactor for the i -line and the j -row. Equations. (1.13) and (1.16) are written for the third degree matrix as

1 1det , 3 2 pqabij pqabrsrs r s= T T TΔ ε εΔ ≡T (1.17)

Because of 1 2 j np p p p piε … … … 1 2 j np p p p pi= ε− … … … meaning the change

of sign and 1 21 2p pT T j npp pni jiT T T… … … 1 21 2p p= T T …jpjT piiT…

npnT… it follows that

1 21 2 1 2 nj n jp p p p p p p pp p ni i jiT T T T Tε … … … … … …

1 21 2 1 2 nj n jp p p p p p p pp p ni ij i= T T T T Tε− … … … … … …

(1.18)

Then, “the sign of the determinant changes if two lines or two rows ( i -line and

- rowj in the above equation) are mutually substituted”. This property engenders the fact “the determinant having two lines or rows that are mutually same is zero”.

Multiplying 1 2 nr r rε … to both sides in Eq. (1.9), one has

1 2 1 2 1 2 1 21 2det =n n n np p p p p pnr r r r rr T T Tε ε εT… … … …

1 2 21 1 2= n nn

p p p p p prr rT T Tε … … (1.19)

The transformation from the second side to the third side in Eq. (1.19) results from

the fact that the determinant is zero if the same number exists in 1 2 nr rr… , as

described in Lemma (1.18), and is detT and det− T if 1 2 nr rr… respectively signify the even permutation and odd permutation. Here, note that the expression of

the determinant in Eq. (1.13) is obtained by multiplying 1 2 nr r rε … to both sides in Eq. (1.19) and noting Eq. (1.6).

The additive decomposition of the elements jpiT in the determinant 1 2 j np p p pε … …

1 21 2 j npp p pniT T T T… … into =j jjp ppi iiT A B+ leads to

1 2 1 21 2j nj jnp p p p p p pp p ni iT T A B Tε ( + )… … … …

1 2 1 21 2j njnp p p p p p pp ni= T T A Tε … … … …


1 2 1 21 2j njnp p p p p p pp niT T B Tε+ … … … … (1.20)

Therefore, the value of determinant in which elements in a line (or row) are decomposed additively is the sum of the two determinants made by exchanging the line (or row) of the original determinants into the decomposed elements.

Since it is obtained from Eqs. (1.8), (1.9), (1.11) and (1.19) that

1 2 1 1 1 2 2 21 2( )( ) ( )n n nnp p p p p pnr r r r r rT S T S T Sε … …

1 2 1 2 1 1 2 21 2= n n n np p p p p pnr rr rr rT T T S S Sε … … …

1 2 1 21 2 detn nnr rr r r r= T T Tε S… … (1.21)

the following product law of determinant holds.

det( ) = det detTS T S (1.22)

The partial derivative of a determinant is given from Eq. (1.13) as

1 2 1 2 1 1 2 2

1 11 2 1 2 2 2

2 21 2 1 2 1 1

1!det =

1 = (!

n n nn

n n nn

n n nn

p p p p p p

pp p p p p

pp p p p p

ijij

ji

ji

r r r r r r

rr r r r r

rr r r r r

T T Tn

TT

T Tn

T T

ε ε

ε δε δ

ε δε δ• • •

∂∂∂∂

++

T… …

… …

… …

…

……

2 21 2 1 2 1 1

2 2 2 2

1 3 31 1 1

1 2 1 2 1 1 2 2 1 1

1 = (!

n nn n

n n nn

n n nn

n nj

p pp p p p

p p p pj

p p p p pj

p p p p p

j ji i

i

i

r rr r r r

r r r r

r r r r r

r r r r ri

T

T Tn

T T T

T T T

ε δ δε δ δ

ε ε

ε ε

ε ε − −

• • •

+

++

+

… …

… …

… …

… …

……

…

…

1 211 2 1 1 1 2 2 1 1 = ( 1)! n np p p pp p n njr rr ri r rn T T Tn

ε ε−− − −− …… …

which leads to

det det= , =jij

iΔT∂ ∂∂ ∂

T T ΔT (1.23)

6 1 Tensor Analysis Equation (1.23) is derived for the third degree matrix as follows:

13!det =

1 = ( )3!

1 = ( )3!

1 = (3!

abc pq ap cbq

abc jpq jp bqia ap icjqibapc cbq

a ci abiq pj pqjbc bqj apapbq cc

qcbiq jbc j bq c

r r

ij ij

rr r r

r r rr

r r

i

i

T T T

T T

T TT T T T

T TT T T T

T T

ε ε

ε ε δ δ δ δ δ δ

ε ε εε ε ε

ε εε ε

∂∂∂ ∂

+ +

+ +

+

T

)

1 = =2!

cbi qj bqq crb c

qbc j cbq

r rr

ijr ri

T TT T

T T

ε ε

ε ε Δ

+

The permutation symbol in the third order, i.e. ijkε appears often hereinafter. It

is related to Kronecker’s delta by the determinants.

1 1 1 1 2 3

22 2 1 2 3

3 3 13 2 3

= =

i j k i ii

ijk i j jk j j

i j k k k k

δ δ δ δ δ δε δ δ δ δ δ δ

δ δ δ δ δ δ

(1.24)

Here, the second side in Eq. (1.24) is expanded as

1 1 1

2 22 2 1 2 2 1 33 1 3

3 33

= =

i j k

ijk i j i j i j j ik k k k

i j k

δ δ δε δ δ δ δ δ δ δ δ δ δ δ δ

δ δ δ+ +

22 3 1 1 21 33j i i j j ik k kδ δ δ δ δ δ δ δ δ− − −

We can confirm this relation by

1 1211 22 13 2 32 12 23 22 31 11 23 32 2133 31 13 33123 = = 1ε δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ+ + − − −

21 13 31 11 23 21 12 23 3133 22 32 13 32 11 22 3312213 = = 1ε δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ+ + − − − − for instance. The third side in Eq. (1.24) could be confirmed as well.


The following relation holds from Eqs. (1.24) and (1.22).

1 1 1 1 1 11 12 3 2 3

2 21 2 2 1 2 22 3 2 3

3 3 3 31 13 32 3 2 3

= =

q qp pr ri ii ii i

pqr p q p qijk r rj jj jj j

p pq qr rk kk k k k

δ δ δ δ δ δδ δ δ δ δ δε ε δ δ δ δ δ δ δ δ δ δ δ δ

δ δ δ δ δ δ δ δ δ δ δ δ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

=

sp s q ssi i s i sr

sp sj q ssj s j sr

sp qs ss ss k k rk

δ δ δ δ δ δδ δ δ δ δ δδ δ δ δ δ δ

=

irip iq

jp jq jr

kqkp kr

δ δ δδ δ δδ δ δ

(1.25)

from which further one has

iip iq k

jp jqpqk jk

kqkp kk

ijk =

δ δ δε ε δ δ δ

δ δ δ

jq jp jq jpj i kq i j kqip iq ip iqkk kp k k kp kkk k = δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ+ + − − −

3 3jq p p jq jq jpj jip iq iq ip ip iq = δ δ δ δ δ δ δ δ δ δ δ δ+ + − − −

jq jpip iq = δ δ δ δ−

ii ij iq

qjp jjjiijq

pjp pqi

ij =

δ δ δε ε δ δ δ

δ δ δ

q qii ij j pj ii j pj ijjj pq p pq jj piq iqi ji ji i = δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ+ + − − −

9 3 3 3 2pq pq pqip pq pqiq pi iq = =δ δ δ δ δ δ δ δ δ+ + − − −

2 6ijk ijk kk= =ε ε δ Consequently, the following relation holds.

2 , 6

ip jq jppqk iqkpq

pqijqijp

ijk kij

ijkijk

= =

= =

ε ε ε ε δ δ δ δ

ε ε δ ε ε

− ⎫⎪⎬⎪⎭

(1.26)

The last equation can also be obtained directly from Eq. (1.6).

8 1 Tensor Analysis 1.2 Vector

1.2.1 Definition of Vector

A quantity having only magnitude is defined as a scalar. On the other hand, a quantity having direction and sense in addition to magnitude and fulfilling the following three properties is defined as a vector. A vector is expressed using lowercase letters in boldface to distinguish it from a scalar.

Equivalence: The vectors having same magnitude, direction and sense are equivalent. Here, equivalence of two vectors u and v is expressed by =u v .

Addition: The addition of vectors is given by the parallelogram law.

Multiplication with scalar: The multiplication of vector and scalar induces a vector whose magnitude is given by the multiplication of the original vector and the scalar, direction is identical to that of the original vector, and sense is same and opposite to that of the original vector if the scalar is positive and negative, respectively.

By virtue of the properties presented above, the commutative, distributive, and the associative laws hold as follows:

, ( ) ( )

( ) ( ) ( ), ( ) ( ) , ( )

= =

a b = ab = b a a b = b a a = a a

+ + + + + + ⎫⎬+ + + + ⎭

u v v u u v w u v w

v v v v v u v u v

　　

(1.27)

where ,a b are arbitrary scalars. The magnitude of vector is denoted by || ||v . In particular, the vector whose

magnitude zero is called the zero vector and is shown as 0 . The vector whose magnitude is unity is called the unit vector.

1.2.2 Operations for Vectors

1) Scalar product Denoting the angle between the two vectors ,u v by θ when they are translated to the common initial point, the scalar (or inner) product is defined as || |||| ||cosθvu and it is denoted by the symbol •u v , i.e.

|| |||| ||cosθ• ≡ vuu v (1.28)

The magnitude of vector is expressed using the scalar product in Eq. (1.28) as follows:

|| ||= •v v v (1.29)

The quantity obtained by the scalar product is a scalar and the following commutative, distributive and associative laws hold.

, ( ) , ( ) ( )= = a = a• • • • • • •+ +u v v u v u w u v u w u v u v　 (1.30)

1.2 Vector 9

2) Vector product The operation obtaining a vector having 1) magnitude identical to the area of the parallelogram formed by the two vectors ,u v , provided that they are translated to the common initial point, and 2) direction of the unit vector n which forms the right-hand bases , ,u v n in this order is defined as the vector (or cross) product and is noted by the symbol ×u v . Therefore, denoting the angle between the two vectors ,u v by θ when they are translated to the common initial point, it holds that

|| |||| || || ||sin ( 1) =× ≡ vuu v n n (1.31)

The vector product is not commutative, i.e.

= −× ×u v v u (1.32)

On the other hand, the distributive and the associative laws hold as follows:

, ( ) ( ) = ( )= a b ab×( + ) × + × × ×u v w u v u w u v u v (1.33)

The operation defined by the following equation for the vector and the scalar products of three vectors is called scalar triple product.

[ ] ( ) •≡ ×uvw u v w (1.34)

The commutative law for the scalar triple product will be shown in the subsequent section.

In addition to the scalar and the vector products, the tensor product is defined as will be described in 1.3.5.

1.2.3 Component Description of Vector

The component description of vector is explained here prior to the description of component description of general tensor.

1) Component description Consider the set of the normalized orthonormal vectors ie , , ( = 1, 2 )i n… .

Here, the “normalized” means the unit vector and “orthonormal” means that they

are mutually orthonormal. Then, adopt the coordinate system O ix−, in which

the directions of axes are chosen to the directions of normalized orthonormal

vectors ie . Hereinafter, the set ie of vectors is called the normalized

orthonormal base. The scalar and the vector products between the base vectors for = 3n are given from Eq. (1.2), (1.28) and (1.31) as follows:

j iji = δ•e e (1.35)

10 1 Tensor Analysis

=j ijr ri ε×e e e (1.36)

Vector v is described in the linear associative form as follows:

21 1 2 ( )=r r n n= v v v v+ + +v e e e e… (1.37)

where 1 2, , , nv v v… are the components of v . Denoting the angle of the

direction of vector v from the direction of the base vector ie by iθ ,

cos = iiθ •n e is called the direction cosine by which the component of v is

given as || || || ||cos= = = iiii θv • •v v n e ve (1.38)

The magnitude of vector v and its unit direction vector n are given from Eqs.

(1.29), (1.35), (1.37) as follows:

|| || , || || || ||r

r r rv

v v =≡ ≡ vv n ev v (1.39)

Because of sr rss sr r= u v = u v δ• •u v e e the scalar product is expressed using the components as

rr= u v•u v (1.40)

The vector product is expressed from Eqs. (1.36), (1.37) as follows:

2 3 3 2 1 3 1 1 3 2 1 2 2 1 3( ( ) ( ) ( ) )j j kj iijki i=u v u v u v u v u v u v u v u v= =ε× × − + − + −u v e e e e e e

(1.41)

For sake of Eq. (1.41) the scalar triple product defined in Eq. (1.34) is expressed in a component form as

[ ] = ( ) j j jkk k rkr kijr i ijr i iijku v w u v w u v w= = =ε ε εδ• •×uvw u v w e e

(1.42)

or in a matrix form as follows:

1 1 1

2 22

33 3

[ ] =

u v w

u v w

u v w

uvw (1.43)

Furthermore, from these equations the following equation holds.

[ ] = [ ] = [ ] = [ ] = [ ] = [ ]− − −uvw vwu wuv vuw wvu uwv (1.44)

1.2 Vector 11 Here, note that [ ]uvw designates the volume of a parallelepiped formed by

choosing , , u v w as the three sides to produce the right-handed coordinates in this

order.

2) Coordinate transformation Adopt the other normalized orthogonal coordinate system O - ix∗ with the bases

* ie in addition to the normalized orthogonal coordinate system O ix− with

the bases ie (Fig. 1.1). Noting ( )= =j j j jv •v ve e e in general, the

following relations hold in these bases.

* *( )=

* *( )=

j j

j j

i i

i i

•

•

⎫⎪⎬⎪⎭

e e e e

e e e e (1.45)

*=

* =

i ri r

i ir r

Q

Q

⎫⎪⎬⎪⎭

e e

e e (1.46)

where the coordinate transformation operator ijQ is defined by

* *cos(angle between and ) =i iij j jQ •≡ e e e e (1.47)

v

θ

v

v1

2

x1

x2

e2

0

2*e

1e

1*e

1*v

2*v

2*x

1*x

v

θ

v

v1

2

x1

x2

e2

0

2*e

1e

1*e

1*v

2*v

2*x

1*x

Fig. 1.1 Coordinate transformation of vector in a two-dimensional state

12 1 Tensor Analysis Moreover, because of

( ( (* * * * * *) ) )

( (* * * *) ) )(rj

j j j

j j j

jr r r r rir i i i

ri r r ri ir i

Q Q = = =

Q Q = = =

• • • • •

•• • • •

⎫⎪⎬⎪⎭

e e e e e e e e e e

ee e e e e e e e e

it holds that

rj ijjr riirQ Q Q Q= = δ (1.48)

It is assumed for a while that the relative (parallel and rotational) motion does not

exist between the above-described coordinate systems, and that their origins

mutually coincide. Then, denoting the component on the base *ie by *( ) , the

coordinate transformation rule, i.e. the transformation rule of the components of v

on these coordinate systems is given by

=i ij jv Q v∗ (1.49)

noting * * * *= ji j ir rv v• •e e e e

and based on

**= =j j r rv vv e e (1.50)

Furthermore, noting = =ri ri sr rs issQ v Q Q v vδ∗ , the inverse relation of Eq. (1.49)

is given as

=i ji jv Q v∗ (1.51)

Equations (1.49) and (1.51) are expressed in matrix form as

11

2 2

33

11 12 13

21 22 23

31 32 33

*

* =

*

v Q Q Q v

v Q Q Q v

vQ Q Qv

⎧ ⎫ ⎡ ⎤⎧ ⎫⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎩ ⎭⎪ ⎪ ⎣ ⎦⎩ ⎭

,

11

2 2

33

11 21 31

12 22 32

13 23 33

*

*

*

vQ Q Qv

v Q Q Q v

v Q Q Q v

⎧ ⎫⎡ ⎤⎧ ⎫ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎪ ⎪⎣ ⎦ ⎩ ⎭

(1.52)

which are often expressed simply as

* * = [ ] , = [ ] Tv Q v v Q v (1.53)

1.2 Vector 13 Needless to say, an equation involving *( ) or jiQ does not describe the relation between different physical quantities, but describes the relations between components when a certain physical quantity is described by two different coordinate systems.

As known from the following equation, the vector of magnitude is not influenced by the coordinate transformation, whilst it is the basic property of the scalar quantity, as described later.

* r s r s r s r ris rsisir ir|| || = Q v Q v = Q Q v v = v v = v v = || ||δv v

The relations expressed above are shown below.

1 2 1 2

1 2 1 2

1 1 1 1

2 2 2 2

_

* * * * ( ) ( ) cos sin[ ] = = =

sin cos* * * * ( ) ( )Q

θ θθ θ

• •

• •

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

e e e e e e

e e e e e e

_

_cos sin cos sin 1 0

= = =sin cos sin cos 0 1

jrir ijQ Q

θ θ θ θδ

θ θ θ θ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

2 2 21 21 1 1= ( ) ( ) =i i i i iQ Q• •∗ ∗ ∗+ +e e e e e e e e e ,

1 21

2 21

= cos sin

= sin cos

θ θθ θ

∗ ⎫+ ⎪⎬∗ − + ⎪⎭

e e e

e e e

1 1 2 2 1 21 2= ( ) ( ) = i ii i i Q Q•• ∗ ∗ ∗ ∗ ∗ ∗+ +e e e e e e e e e

1 1 2 2 1 1 2 2= =v v v v∗ ∗ ∗ ∗+ +v e e e e

11

22

cos sin= _* sin cos

v v

vv

θ θθ θ

∗⎧ ⎫ ⎧ ⎫⎡ ⎤⎪ ⎪⎨ ⎬ ⎨ ⎬⎢ ⎥

⎣ ⎦ ⎩ ⎭⎪ ⎪⎩ ⎭,

1 1

22

_cos sin *=

sin cos

v v

vv

θ θθ θ

⎧ ⎫⎧ ⎫ ⎡ ⎤ ⎪ ⎪⎨ ⎬ ⎨ ⎬⎢ ⎥ ∗⎣ ⎦⎩ ⎭ ⎪ ⎪⎩ ⎭

Choosing the position vector x as the vector v , it holds from Eqs. (1.49) and (1.51)that

=

=

i ir r

i rri

x Q x

x Q x

∗ ⎫⎪⎬∗ ⎪⎭

(1.54)

14 1 Tensor Analysis from which one has

*

* *

= = = =

= = = =

i

r r

ii i

ir r rir ir jr ij

j j j

j rjrj rj ir ij

x Q x xQ Q Qx x x

Q xx xQ Q Qxx x

δ

δ

⎫∂ ∂ ∂ ⎪∂ ∂ ∂ ⎪⎬

∗ ∗∂∂ ⎪∂⎪∗∂∂ ∂ ⎭

(1.55)

Consequently, ijQ can be also described as

= j

ij j

i

i

xxQ =

x x∗ ∂∂

∗∂ ∂ (1.56)

1.3 Tensor

The vector described in the foregoing possesses the direction in first order but there exit quantities possessing the direction in high order. They are collectively called the tensor. The general definition and mathematical properties of tensor are described in this section.

1.3.1 Definition of Tensor

When the set of mn functions 1 2 ( , , , )mp p pT ... , where each of the suffixes

1 2 , , , mp p p... takes the number 1, 2, , n , is described in the coordinate

system O - ix , this set of functions is defined as the th-orderm tensor in the th - dimensionn if the set of functions is observed as

1 1 2 22 21 1*( , , ) = ( , , )

m mm mp q p q p qT p p p Q Q Q T q q q (1.57)

or

21

1 2

2 21 1*( , , ) = ( , , )m

m

q qq

p p pmm

x x xT p p p T q q q

x x x

∗ ∗ ∗∂ ∂ ∂∂ ∂ ∂

(1.58)

in the other coordinate system, provided that only the directions of axes are different but the origin is common and the relative motion does not exist. It is denoted by the symbol 1 2 mp p pT ⋅⋅⋅ . Then, Eq. (1.58) is expressed as

1 2 1 1 12 2 2=m m m m

p p p p q p q p q q q qT Q Q Q T∗ ⋅⋅⋅ ⋅⋅⋅… (1.59)

1.3 Tensor 15 or

211 2 1 2

1 2

=m

m m

mp p p q q qqq q

p p px x xT Txx x

∗ ∗ ∗∂ ∂ ∂∗∂∂ ∂⋅⋅⋅ ⋅⋅⋅ (1.60)

Noting that

1 1 12 2 21 21 1 1 12 2 2 2=

m m m m mmm m p p q p q p q q q qrp p pp p p p pr rr r rTQ Q Q Q Q Q Q Q Q T• • •

∗ ⋅⋅⋅⋅⋅⋅

11 1 1 1 22 2 2 2 ( )( ) (= )

mm m m mp p q p p q p p q q q qr r rQ Q Q Q Q Q T ⋅⋅⋅

1 1 12 2 2 =

m m mq q r q q q qr r Tδ δδ • • • ⋅⋅⋅

the inverse relation of Eq. (1.59) is given by

1 2 1 21 1 2 2=

m mm mrr r p p pp p pr r rT Q Q Q T ∗⋅⋅⋅ ⋅⋅⋅ (1.61)

While the transformation rule of the first-order tensor, i.e. vector is given by Eqs. (1.49) and (1.51), the transformation rule of the second-order tensor is given by

, = =ijir js rs jsri rsijT TQ Q T T Q Q∗ ∗ (1.62)

The transformation between the coordinate systems without relative motion is considered above in the definition of the tensor, whereas the transformation in the form of (1.59) or (1.61) is called as the objective transformation. A tensor that obeys the objective transformation even between the coordinate systems with the relative motion is called an objective tensor.

1.3.2 Quotient Law

One has a convenient law, called the quotient law, which is used to judge whether or not a quantity is a tensor. It will be explained below.

Quotient law: “If a set of functions 21( , , , )mT p p p• • • becomes +1 +2 ml lp p pB ⋅⋅⋅

( - thm l− order tensor lacking the suffices 1 lp p∼ ) by multiplying it by

1 2 lp p pA ⋅⋅⋅ ( - thl order tensor ( )l m≤ ), the set is a - thm order tensor”.

(Proof) The proof is achieved by showing that the quantity 21( , , , )mT p p p• • • is

the - thm order tensor when it holds that

21( , , , )mT p p p• • •1 2 l

p p pA ⋅⋅⋅ =+1 +2 ml lp p pB ⋅⋅⋅ (1.63)


which is described in the coordinate system O - ix∗ as follows:

21( , , , )mT p p p∗ • • •1 2 mp p pA∗ ⋅⋅⋅ =

+1 +2*

ml lp p pB ⋅⋅⋅ (1.64)

Here, 1 2 lp p pA ⋅⋅⋅ is the - thl order tensor and +1 +2 ml lp p pB ⋅⋅⋅ is the - thm l−

order tensor,. Therefore, the following relation holds.

+1 +22 2+1 +1 + ++1 +2*

ml lml l l l mml l p r rp p rr r rp p p Q Q QB B=⋅⋅⋅ ⋅⋅⋅• • •

=1 22 2+1 +1 + + 1 2( , , , )

m ml l l l lr rp rp p mr r rQ Q Q T r r r A ⋅⋅⋅• • • • • •

1 222 2 1 2+1 +1 + + 121= ( , , , )m m l l ll l l l

pp r pp r p p pp p rr r r mQ Q Q Q Q QT r r r A∗ ⋅⋅⋅• • • • • • • • • (1.65)

( 1 ) (1 )l m l+ ∼ ∼

Substituting Eq. (1.64) into Eq. (1.65) yields

21 21 1 21 2 1 2 ( , , , ) ( , , , ) = 0m m l

p p pmm p p prr rQ Q QT p p p T r r r A∗ − ⋅⋅⋅• • • • • • • • •

(1.66) from which it holds that

221 1 22 11( , , , ) ( , , , )= m mm mpp prr rQ Q QT p p p T r r r∗ • • • • • • • • • (1.67)

Therefore, noting the definition of tensor in Eq. (1.67), the quantity

1 2( , , , )mp p pT • • • is the - thm order tensor. (End of proof)

According to the proof presented above, Eq. (1.63) can be written as

1 2 mp p pT ⋅⋅⋅ 1 2 lp p pA ⋅⋅⋅ = +1 +2 ml lp p pB ⋅⋅⋅ (1.68)

For instance, if the quantity ( , )jT i transforms the first-order tensor, i.e. vector iv to the vector iu by the operation ( , ) =j ijT i v u , one can regard ( , )jT i as the

second-order tensor. Eventually, in order to prove that a certain quantity is a tensor, one needs only to

show that it obeys the tensor transformation rule (1.59) or that the multiplication of a tensor to the quantity leads to a tensor by the quotient rule.

Tensors fulfill linearity as follows:

1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

( ) =

( ) =

m m mlll l

m ml l

p p p p p p p p p p p p p p p p p p p p p

p p p p p p p p p p p p

T G H T G T H

T aA aT A

+ + ⎫⎪⎬⎪⎭

⋅⋅⋅⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅

where a is an arbitrary scalar. Therefore, the tensor has the function to transform linearly a tensor to the other tensor and thus it is called the linear transformation. The operation that lowers the order of tensor by multiplying the other tensor is called the contraction.

1.3 Tensor 17

1.3.3 Notations of Tensors

When we express the tensor T as

21 21 mmp p p pp p= T ⊗⊗⋅⋅⋅T e e e• • • (1.69)

in a similar form to the case of vector in Eq. (1.37), Eq. (1.69) is called the

component notation with bases, defining 21 mpp p ⊗⊗e e e• • • as the base of - thm order tensor. The transformation of T between the bases in Eq. (1.46)

leads Eq. (1.69) to

1 2 211 1 2 2 mm mmp p p p p pr r rr r r= T Q Q Q•••∗ ∗ ∗⊗ ⊗⊗T e e e• • •

21221 1 1 2=

m mm m p p ppp pr r r r r rQ Q Q T ∗ ∗ ∗⊗ ⊗⊗⋅⋅⋅ e e e• • • • • •

211 2=

mmr r r r r rT •••∗ ∗ ∗∗ ⊗ ⊗⊗e e e• • • (1.70)

The following various notations are used for tensors.

Component notation: 1 2 mp p pT ⋅⋅⋅

Component notation with base: 1 2 1 2m mp p p p p pT ⊗⊗⋅⋅⋅ e e e⋅⋅⋅

Symbolic (or direct) notation: T

Matrix notation: Eq. (1.7)

The matrix notation holds only for a vector or a second-order tensor or for a fourth-order tensor if it is formally expressed by two suffixes. For instance, the stress-strain relation can be expressed in matrix notation by expressing the stress and the strain of second-order tensors as a form of vector and the stiffness coefficient of fourth-order tensor as a form of second-order tensor.

Various contractions exist in the operation of higher-order tensors and thus the symbolic notation is not useful in general. For instance, which of the following does ST mean: , , , , , , jijk ijk ijk ijk ijk ijk ijkij jijk kj l kl ilS T S T S T S T S T S T S T ? In other words, symbolic notation has its limit. On the other hand, component notation with bases holds always without setting any special rule.

Introducing the notation

1 1 1 1m2 2 2 2

1 21 1 1 2 2m2

( )

( )m m m

mm m

T

p p p p q p p qq q q q

q q qpp p p p q q p q

Q Q Q T

Q Q Q T

• • •

• • •

⎫≡ ⎪⎬≡ ⎪⎭

⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅

Q T

TQ (1.71)

18 1 Tensor Analysis Eqs. (1.59) and (1.61) can be expressed by the symbolic notation as follows:

* =

= T

⎫⎪⎬

∗ ⎪⎭

QT T

TT Q (1.72)

In particular, transformations of the vector and the second-order tensor are expressed by

= , = T∗ ∗v vQv v Q (1.73)

** = , =T TT QTQ T Q T Q (1.74)

1.3.4 Orthogonal Tensor

The coordinate transformation operator jiQ described in 1.2.3 plays an important role in the coordinate transformation and is called the orthogonal tensor. The component notation with bases is obtained from

( ( ) = ( ) (j j j j jij r ri i i i i i iQ = =• • •∗ ∗ ∗ ∗ ∗⊗ ⊗ ⊗ ⊗e e e e e )e e e e e e e e )e

rri i= Q ∗ ∗⊗e e (1.75) as follows:

j ij jij i i= Q = Q ∗ ∗⊗ ⊗Q e e e e (1.76)

Furthermore, considering Eq. (1.46), the direct notation of Q is given by

i i= ∗⊗Q e e (1.77)

Because of

= = = = ) = ( )

= = = = ) = ( )

i r s sir r is rs rs r is rs r i rs r i

i s sir r sr r sr r si sr r i sr r isi

Q Q Q Q Q Q

Q Q Q Q Q Q

δ

δ

•

•

∗ ∗ ⎫⊗ ⎪⎬∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎪⊗ ⎭

e e e e e (e e e e e

e e e e e (e e e e e

it holds that

= , = Ti i i i

∗ ∗e Qe e Q e (1.78) Furthermore, changing Eq. (1.48) to the direct notation or noting the relation

( )= = =

( )= = =

T j j jj j ij

Tj jj j jij

i i i i i

i i i i i

δ

δ

•

•

∗ ∗ ∗∗ ⎫⊗ ⊗ ⊗⊗ ⎪⎬∗ ∗ ∗ ∗ ∗ ∗⊗ ⊗ ⊗⊗ ⎪⎭

e ee e e e e e e eQQ

e e ee e e e e e eQ Q

1.3 Tensor 19 obtained from Eq. (1.77), it holds that

= =T TQQ Q Q I (1.79) from which one has

1=T −Q Q (1.80)

Moreover, from Eqs. (1.12), (1.22) and (1.79), it is obtained that

det = det = 1T ±Q Q (1.81)

Further from Eq. (1.79) one obtains ( ) = ( )T T− − −Q I Q Q I

Making the determinant of this equation and noting Eqs. (1.12), (1.22) and (1.81), it holds that

det( )= det( ) det( ) = 0− − − → −Q I Q I Q I (1.82)

Then, it is known that one of the principal values of the orthogonal tensor is unity as known from the fact described in 1.5.1.

1.3.5 Tensor Product and Component

Based on the three vectors (1) (2) ( ), , , mv v v⋅⋅⋅ , one can make the - thm order

tensor as follows:

21 21

(1) (2) ( ) (1) (2) ( )=m mp pp

m mp p pv v v ⊗⊗⊗ ⊗v v v e e e⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ (1.83)

For the two vectors, one has the second-order tensor

= =j ji j i ji iu v u v⊗ ⊗⊗u v e e e e (1.84)

which is expressed in matrix form

21 1 1 1 3

2 2 31 2 2

33 2 33 1

u v u v u v

u v u v u v

u v u v u v

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(1.85)

As described above, one can make a tensor from two vectors. After the scalar product •u v and the vector product ×u v for the two vectors , u v , one calls

⊗u v as the tensor (cross) product or dyad which means one set by two. Particularly, it holds for three arbitrary vectors that


( ) ( ) =ir r i r ru v w⊗u v w which can be expressed in the symbolic notation as

( ) = ( )•⊗u v w u v w

and thus the following expression holds.

= ( •⊗u v u v (1.86)

The component of vector is expressed by the direct notation in Eq. (1.38). Here, consider the component of second-order tensor in the direct notation. The second-order tensor T is expressed from Eq. (1.69) as

ij ji= T ⊗T e e (1.87)

from which, noting Eq. (1.86) it holds that j ji i rs r s= T• • ⊗e e e eT e e

jrs ir s= T δ δ and thus, the component of T in the direct notation is given as

=ij jiT • Te e (1.88)

As known from Eq. (1.88), the orthogonal projection of the vector jTe to the base

vector ie is the component of the tensor T . Especially, ( )ij i = jT and ( ) ij jiT ≠ are called the normal component and the shear component,

respectively.

1.4 Operations of Second-Order Tensor

As described in 1.3.3, tensor operations must be expressed by component notation

in general. While the operations Tv , TS , TΞ appear often in the

elastoplasticity, let them be meant the following operations.

( ) = , ( ) = , ( ) = ijir r ij ir rj klkli ijT v T v TΞTv TS TΞ (1.89)

Various operations of the second-order tensor are described below.

1.4.1 Trace

An operation taking the sum of the components having the same suffixes, i.e. the

sum of diagonal components in the matrix notation is called the trace and is expressed as

11 22 33tr = = =rrrs rsT T T T Tδ + +T (1.90)

1.4 Operations of Second-Order Tensor 21

11 11 12 21 13 31tr( ) = =ir riT S T S T S T S+ +TS

21 12 22 22 23 32T T T T T T+ + +

33 3332 2331 13T T T T T T+ + + (1.91)

The following relations hold for the trace.

t r ( ) = t r t r , t r ( ) = t r , t r ( ) = t r ( ), t r ( )= a a •+ + ⊗T S T S T T TS ST u v u v

(1.92)

1.4.2 Various Tensors

1) Zero tensor A tensor, the components of which are all zero, is called the zero tensor and is denoted by the symbol 0 , i.e.

( ) = 0ij0 (1.93) The zero tensor transforms any tensor to a zero tensor.

2) Identity tensor The tensor which transforms a tensor to itself is called the identity tensor. The second-order identity tensor has the components expressed by the Kronecker’s delta and is denoted by the symbol I , i.e.

=ij ijI δ (1.94)

The fourth-order identity tensor has the components given by the following equation and is denoted by the symbol I .

1 ( )2 ik jl il jkijklI δ δ δ δ≡ + (1.95)

3) Transposed tensor

The tensor TT fulfilling

( )Tjiij = TT (1.96)

is called the transposed tensor of T.

It holds from ij( ) =TrijrT STS that

( ) = T TTTS S T (1.97)

22 1 Tensor Analysis Further, the following relation holds for the trace.

t r = t rTT T (1.98)

The magnitude of tensor is defined as the square root of the sum of the squares of each components and thus it is expressed using Eq. (1.96) as

= = tr ( )Tij ijT TT TT (1.99)

The tensor whose magnitude is unity is called the unit tensor.

It holds for arbitrary vectors ,u v from Eq. (1.96) that

= , ( )T Tij j ji ji iT u v = u v• •Tu v u T v T (1.100)

Further, it holds from Eq. (1.100) that T= ••Tu Sv u T Sv (1.101)

4) Inverse tensor The tensor 1−T fulfilling the following relation is defined as the inverse tensor of

the tensor T . 1 =−TT I , 1( ) =rj ijirT δ−T (1.102)

It holds from Eq. (1.15) with Eq. (1.3) for = 3n that

1detij ij ij s si j i js s= T = Tnδ δ δ Δ ΔT

from which one has

= , =detdet

T jiji

ssΔ

T δT ITT

ΔΔ (1.103)

Consequently, 1−T is given by

11 = , ( ) =det det

jiTij

Δ− −T TT T

ΔΔ (1.104)

Then, det 0≠T is required in order that 1−T exists, while the tensor fulfilling this condition is called the non-singular tensor. The partial derivative of Eq. (1.23)

is rewritten by Eq. (1.104) as

det det= ( ) T−∂∂

T T TT

(1.105)

The derivation of Eq. (1.105) starting from the definition of the total differential equation has been often described in some literatures (cf. Leigh, 1964) but it needs cumbersome manipulations. Compared with it, the derivation shown above would be concise and straightforward.


The following relation holds for the inverse tensor.

1 1 1 11 ( ) = ( ) ( ), ( ) = T T T− − − −− −≡T T T TS S T (1.106)

because of

1 1 1(( ) =) ( ) = = ( )T T T T T− − −TT T T I T T

1 1 1( ) = ( ) =− − −→TS TS I S TS T

Now, when we regard the transformation of the vector v to the vector u by the tensor T , i.e.

, == ijij uT vTv u (1.107)

as the simultaneous equation in which the components of v are the unknown

numbers, solution exists for ≠u 0 if det 0≠T and is given by = T−v T u as

= , =det det

T jijiv u

ΔΔv uT T

(1.108)

Here, T must be the non-singular tensor fulfilling det 0≠T in order that the

non-trivial solution ≠v 0 exists for ≠u 0 . On the other hand, T must be the

singular tensor fulfilling det = 0T in order that the solution ≠v 0 exists for

=u 0 .

5) Symmetric and skew-symmetric tensors Tensors ST and AT fulfilling the following relations are defined as the symmetric

and the skew (or anti)-symmetric tensor, respectively.

= , S SST Sji ijT = TT T (1.109)

and

= , A AAATji ijT = T− −T T (1.110)

An arbitrary tensor T is uniquely decomposed into the symmetric and the skew-symmetric tensors.

S A= +T T T (1.111)

11 ( ), = ( ) 2 2

S T A T= + −T T T T T T (1.112)


while the components of ST and AT are often denoted by ( )ijT and [ ]ijT ,

respectively. Eq. (1.111) is called the Cartesian decomposition, following the

decomposition of a complex number to a real and an imaginary parts. It holds that

no sum( ) = , ( ) = (( ) = 0, )S T A T A ASii−T T T T T (1.113)

and

t r = t r , t r = 0S AT T T (1.114)

The diagonal components of the skew-symmetric tensor are zero, and thus its determinant is zero, i.e.

det = 0AT (1.115)

6) Mean and deviatoric components

When the tensor T is decomposed as follows:

= m '+T T T (1.116)

1, (tr )

)tr ( 0=

m m m

m

T T n

T' '

⎫≡ ≡ ⎪⎬⎪−≡ ⎭

T I T

T T I T (1.117)

mT and 'T are called the mean (or spherical) part and the deviatoric part of the

tensor T . Noting Eq. (1.113), the skew-symmetric tensor A'T of the deviatoric

tensor 'T is given by

=A A'T T (1.118)

Then, the symmetric part of the deviatoric tensor is given by

= =A ASmT' ' −− −T T T T I T (1.119)

from which one has

= ASmT '+ +T I T T (1.120)

The decomposition of T into the mean component mT I , the deviatoric symmetric

component S'T and the skew-symmetric component AT is called triple

decomposition.


7) Axial vector

The skew-symmetric tensor AT has three independent components in the

three-dimensional state. Therefore, vector At having the following components is

called the axial vector.

1=2

A Arsii rst Tε− (1.121)

Inversely from Eq. (1.121) it is obtained that

3 2

1

0

= , = 0

ant. 0

A A

A A A Aijr ijij r

t t

T t T tε⎡ ⎤−⎢ ⎥

− −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(1.122)

Furthermore, noting Eq. (1.41) and the relation

= =A A Airs rs irs srir r v vT v t tε ε− (1.123)

the following relation holds.

=A A×T v t v (1.124)

The relation of AT and At is shown in Fig. 1.2 in the case that At is the angular velocity vector and v is the position vector of particle. The quantity in Eq. (1.124)

v

At

=A A×T v t v

θ

sinθ|| ||v

n= ( || |||| ||sin )A θ nt v

0

v

At

=A A×T v t v

θ

sinθ|| ||v

n= ( || |||| ||sin )A θ nt v

0 Fig. 1.2 Meaning of axial vector in case of rotation


designates the peripheral velocity vector, while AT is called the spin tensor which

induces the peripheral velocity by undergoing the multiplication of the position

vector.

1.5 Eigenvalues and Eigenvectors

The tensor T is expressed in the component notation having only normal

components by choosing the coordinates with special bases. In what follows,

consider the special direction for the second-order tensor in the thn dimensional space.

The unit vector e fulfilling

= , = ij j iT e Te T Te e (1.125)

i.e. (T ) = 0, ( ) = ij jijΤ e Τδ− eT 0− Ι (1.126)

for the second-order tensor is called the eigenvector (or principal or characteristic or proper vector) and the scalar T is called the eigenvalue (or principal or characteristic or proper) value.

The necessary and sufficient condition that e has a non-zero solution in the simultaneous equation (1.126) is given by

= 0, det( ) = 0ij ijT T Tδ− −T I (1.127)

noting the description in 1.6. Eq. (1.127) is called the characteristic equation of the tensor, which is regarded as the equation of thn degree of T . Unit vectors

, , , Ne e eⅠ Ⅱ… are derived for each of solutions , , , NT T TⅠ Ⅱ

… from Eq. (1.126).

Eigen values are real numbers and eigenvectors are mutually orthogonal for eigenvalues that differ from each other in the second-order real symmetric tensor, the components of which are real and which fulfill the symmetry =ij jiT T . The simple proof for this fact is presented below.

Regarding T and ie as complex numbers in general and taking the conjugate

complex number of both sides in Eq. (1.125), one has the following equation, denoting the conjugate complex number of T and ie by T and ie (note

=ab a b for arbitrary complex numbers , a b ).

=jij ieT eT (1.128)

Multiplying both sides of Eqs. (1.125) and (1.128) by ie and ie , respectively, and taking the difference in both sides, one has

1.5 Eigenvalues and Eigenvectors 27

( )=ij ji ijj i i ie eT e e T e e T T− − (1.129)

The left-hand side in Eq. (1.129) becomes zero because of =ij jiT T . Thereby, one

has =T T which means that T must be a real number because 0iie e ≠ . Furthermore, from the two equations for the two solutions

= , =T TTe e Te eⅠⅠ Ⅱ Ⅱ ⅡⅠ

(1.130)

one has = ( )T Τ• • •− −e Te e Te e eⅡ Ⅰ Ⅱ Ⅱ Ⅰ ⅡⅠ Ⅰ

(1.131)

The left-hand side of Eq. (1.131) is zero because of the symmetry of the tensor T . Therefore, one has

( ) = 0T Τ •− e eⅠ Ⅱ Ⅰ Ⅱ (1.132)

Consequently, it can be concluded that eⅠ and eⅡ are mutually orthogonal for

T T≠Ⅰ Ⅱ . Based on the result described above, denoting the eigenvectors by ( , , , )J J = Ne …ⅠⅡ and the corresponding eigenvalues as JT , one can write

= (no sum)J J JTTe e (1.133)

In addition, noting that the shear component on the coordinate system with the base

vector Je is zero, i.e.

( ) = 0JKT J K≠ (1.134)

the tensor T is expressed by

= J J JT ⊗T e e (1.135)

If tensor T having eigenvector Je has the same eigenvalues as tensor T , it holds that

= (no sum)J J JTTe e (1.136)

where the orthogonal tensor Q between the eigenvectors of these tensors is given by

= IIJ JQ •e e , = , = , = , TJ JJ J J J⊗Q Q Qe ee e e e (1.137)

Applying Q to Eq. (1.133), one has

( =) =TT T TJ J J JTQ T Q TQQ Qe e e (no sum)

28 1 Tensor Analysis from which, considering Eqs. (1.136) and (1.137), it holds that

= = (no sum)TJ J J JTQ TQ Te e e (1.138)

Then, one obtains the relation

= TT Q TQ (1.139)

As presented above, tensors having identical eigenvalues can be related by the orthogonal tensor; they are called the similar tensor mutually. The coordinate transformation rule (1.74) of a certain tensor and the relation (1.139) of two tensors having identical eigenvalues but different eigenvectors, are of mutually opposite forms.

If the function f of tensor , , • • •A B is observed to be identical independent of observers, i.e. if it fulfills the relation

( , , ) = ( , , )f f• • • • • •A BA B Q Q (1.140)

using the symbol in Eq. (1.71), f is called the isotropic scalar-valued tensor function, which is none other than the invariant. In particular, The isotropic scalar-valued tensor function ( )f T of single second-order tensor T fulfills

( ) = ( )Tf fT QTQ (1.141)

( )f T is expressed by three principal values in the three-dimensional case,

involving them in symmetric form so as to be identical even if they are exchanged

to each other. Then, there exist three independent invariants for a single tensor. Their explicit forms are presented below.

The expansion of the characteristic equation (1.127) of T for = 3n leads to

11 12 13

21 22 23

3231 33

T Τ T T

T T Τ T

T T T Τ

−−

−

11 22 12 21 32 1333 23 31 = ( )( )( )T Τ T Τ T Τ T T T T T T− − − + +

11 32 22 31 13 33 12 2123( ) ( ) ( )T Τ T T T Τ T T T Τ T T− − − − − −

3 2

11 22 11 22 22 11 11 22 1233 33 33 33 23 31

12 21 32 13 11 32 22 13 33 12 2123 31 23 31

=

+ ( )

Τ T T T Τ T T T T T T Τ T T T T T T

T T T T T T T T T T T T T T T T

− + ( + + ) − ( + + ) + + 2+ + − − −

3 211 22 11 22 22 11 12 21 32 1333 33 33 23 31

11 22 11 32 22 13 33 12 21 1233 23 31 23 31

= Τ T T T Τ T T T T T T T T T T T T Τ T T T T T T T T T T T T T T T

− + ( + + ) − ( + + − − − )+ − − − + 2

1.5 Eigenvalues and Eigenvectors 29

3 211 22 33 = Τ T T T Τ− + ( + + )

2 2 211 22 22 1133 3322 3311 2( )Τ Τ Τ T T T T T T1 − [ + + + + +

2

2 2 212 21 32 1323 3122 3311 2Τ Τ Τ T T T T T T Τ− + + + ( + + )]

11 22 12 11 32 22 13 33 12 21 1233 23 31 23 31 23 31T T T +T T T T T T T T T T T T T T T+ − − − + 2 3 2

11 22 33 = Τ T T T Τ− + ( + + )

2 2 211 22 12 21 32 1333 23 3122 3311 2T T T Τ Τ Τ T T T T T T Τ21 − [( + + ) − + + + ( + + )]

2

2 2 23323 22 31 12 2311 22 11 12 3133 2 = 0 T T T T T T T T T T T T+ − − − +

from which the characteristic equation is given as

3 2 = 0T T Τ+ −− I Ⅱ Ⅲ (1.142) where

2211 33 tr= =iiT T T T≡ + + TⅠ (1.143)

2 21311 22 23 1112

3321 22 32 33 31

1 1 (= ( )= )= = tr tr2 2 rs srrr ssii ii

T T T TT TT T T TΔD

T T T T T T− −≡ + + T TⅡ

(1.144)

33 2

11 12 13

31 221 22 23

31 32 33

11 1tr tr tr tr = det = = 26 3

rst tr s

T T T

T T TT T T

T T T

ε − +≡ T T T TTⅢ

(1.145)

The direct notation of III is derived by taking the trace of the expression of the Cayley-Hamilton theorem described in the next section and substituting the direct notations of I and II into it.

On the other hand, the characteristic equation (1.142) is expressed using the principal values as follows:

( )( )( ) = 0T T T T T T− − −Ⅰ Ⅱ Ⅲ (1.146)

Comparing Eqs. (1.151) and (1.155), coefficients I, II, and III are described as

=

= +

T T T

TT T T Τ T

T T T

⎫⎪+ ⎬⎪⎭

Ⅰ Ⅱ Ⅲ

Ⅰ Ⅱ Ⅰ Ⅱ Ⅲ Ⅰ

Ⅰ Ⅱ Ⅲ

Ⅰ + +

Ⅱ

Ⅲ=

(1.147)


Eq. (1.147) can also be derived by substituting 11 22= , = , T T T T

1233 23 31= , = = = 0T T T T TⅢ in the determinants in Eqs. (1.143)-(1.145). Since

I, II, and III are the symmetric functions of principal values, they are the invariants

and are called the principal invariants. Next, consider the deviatoric tensor 'T . The characteristic equation of 'T is

given by replacing T to 'T in Eq. (1.142) as follows:

3 = 0T' ' ' '− − (1.148) noting

t r 0=' '≡ TⅠ (1.149)

where

21 1= = tr =2 2 rs srii iiD T TΔ' ' ' ' ''≡ TⅡ

22 22 22

11 22 33 12 23 311 ( )= 2

T T T T T T' ' ' ' ' '+ + + + +

2 2 2 2 2 2

12 312311 22 22 33 33 111( ) ( ) ( ) = 6

T T T T T T T T T'' '− + − + − + + +

2 2 2 22 21 1( ) ( ) ( ) ( ) = =2 6

T T T T T TT T T' ' '+ + − + − + −Ⅰ Ⅱ Ⅱ Ⅲ Ⅲ ⅠⅠ Ⅱ Ⅲ(1.150)

3

1211 13

21 22 23

31 32 33

1 1det= = tr = 3 3

rs sr rt

T T T

T T TT T T

T T T

' ' '

' ' '' ''' ' '

' ' '

≡ T TⅢ

2 2231 12 12 3111 22 33 11 23 22 33 232= T T T T T T T T T T T T' ' ' ' ' ' ' ' ' ' ' '+− − −

32 2 3 31 ( )= = = 3T T T T T T T T T T'' ' ' ' ' ' ' '' +− − +Ⅰ Ⅱ Ⅲ Ⅰ Ⅱ Ⅰ Ⅱ Ⅰ Ⅱ Ⅲ (1.151)

The direct notation in (1.151) is derived by taking the deviator and the trace of the expression of Cayley-Hamilton’s theorem described in the next section and substituting the direct notations of (=0) 'Ⅰ and 'Ⅱ in Eqs. (1.149) and (1.150) into that.

1.6 Calculations of Eigenvalues and Eigenvectors 31 1.6 Calculations of Eigenvalues and Eigenvectors

The symmetric tensor T can be represented in the eigendirections as

1

=N

JJ JJ

T=

⊗∑ eT e (1.152)

which is called the spectral representation. To express the tensor in the eigendirections, one must calculate the eigenvalues and the eigenvectors of the tensor. The solutions for them (cf. Hoger and Carlson, 1984 and Carlson and Hoger, 1986) are shown in this section.

1.6.1 Eigenvalues

In order to obtain eigenvalues, one must solve the characteristic equation which is the cubic equation having the coefficients as the functions of invariants. Now, infer the form

4= cos

3T '' ψⅡ

(1.153)

for the eigenvalues of deviatoric part of tensor T . The substituting Eq. (1.153) into Eq. (1.148), we have

1 / 23/ 2 3 44 cos cos = 033( )( ) '' ' 'ψ ψ− −ⅡⅡ

Ⅱ Ⅲ (1.154)

which reduces to

/ 234 cos3 = 03 3

'' ψ −ⅢⅡ (1.155)

using the trigonometric formula

3 1 cos cos(cos 3 3 )= 4ψ ψψ + (1.156)

It is obtained from (1.155) that

3/ 23 3=cos32

''

ψ Ⅲ

Ⅱ (1.157)

Noting that the cosine is the periodic function with the period 2π , the angle ψ is

expressed by the following equation in general.

13/ 2

3 31= cos 23 2 )( J J

''

ψ π− −Ⅲ

Ⅱ (1.158)

32 1 Tensor Analysis Substituting Eq. (1.158) into Eq. (1.153) and adding the isotropic component / 3Ⅰ , the eigenvalues of T are given as follows:

13/ 2

41 3 31= cos cos 233 3 2[ ] )( ( )JT J

' ''

π−+ −Ⅱ Ⅲ

ⅡⅠ (1.159)

1.6.2 Eigenvectors

Eq. (1.152) can be expressed as follows:

1

=N

J JJ

T=∑T E (1.160)

while the tensor JE is called the eigenprojection of T , which is defined by

(no sum)JJ J≡ ⊗E e e (1.161)

fulfilling

1

(= ) =N

m mJJ =

⊗ + ⊗ + + ⊗∑ E e e e e e e I…Ⅱ ⅡⅠ Ⅰ (1.162)

for ==

for , t ) =r( J

JKJ K IJ

J K

J Kδ

≠E

E E E E0

(1.163)

It holds that

1 1

11

= = = (no sum for )

= = =

( ) ( )

( ) ( )

N N

J JJ J K J JK K K J JKK K

N N

J J K K J KJ J K JK JJKK

T T T

J

T T T

= =

==

⎫⊗⊗ ⊗ ⊗ ⎪⎪⎬⎪⊗⊗ ⊗ ⊗ ⎪⎭

∑ ∑

∑ ∑

e eTE E e e e e e e

e eE T e e E e e e e

and thus one has

(no sum for )= =J J J JT JTE E T E (1.164)

On the other hand, it holds from Eq. (1.162) that

1 1

=J J

N N

K KJJ JT T Τ= =

− −∑ ∑T I E E

1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor 33 and thus it is obtained that

1

= ( )J

N

K KJ JT T Τ=

− −∑T I E (1.165)

from which one has

1

1 1

( ) = ( )N N

JK KK K

N

J KK JT ΤTθ θ =≠ ≠

= =

−− ∑∏ ∏ ET I (1.166)

The right-hand side in Eq. (1.166) is rewritten as

11 1

( ) = ( )N N

JK KK K

N

J K KJT Τ T Τθ θ

θ θ=≠ ≠

= =

− −∑∏ ∏E E (1.167)

noting Eq. (1.163). Then, considering =θE I for the case of = 1N , i.e. a single

root, the following Sylvester’s formula is obtained.

1

for >1=

for =1

N

K KK

KT NT T

N

θ θθ ≠=

⎧ −⎪ −⎨⎪⎩

∏ T I

E

I

(1.168)

For instance, 2 ( =2)θE in the popular case of = 3N is obtained by the above-mentioned method as follows:

311, 3

( ) = ( )( )K

KT T T=∏ − − −T I T I T I

3

1, 3 1

1 1 1 1 1 1 12 3 3 3 3 33 32 2 2

1 13 32 2 22 2 2 221, 3

( )

= ( ) ( ) ( ) ( ) ( ) ( )

=( )( ) ( )( ) = ( ) =

K J

K

J K J

K

T Τ

T Τ T Τ T Τ T Τ T Τ T Τ

T Τ T Τ T Τ T ΤT Τ

==

=

−

− − − − − −+ + + +

− − − −−

∏

∏

∑ E

E E E E E E

E E E

1, 3 312

2 1 32 21, 3

( )( )( )

= =( ) ( )( )

K

KK

KTT T

T T T Τ T Τ=

=

− − −− − −

∏∏

T IT I T IE

1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor

The characteristic equation of skew-symmetric tensor is given by substituting AT into Eq. (1.142) as follows:


223 2 1 ( )tr trtr det = 02A A A AT T Τ−+ −− T TT T (1.169)

Noting t r det= = 0A AT T in Eqs. (1.114) and (1.115), Eq. (1.169) leads to

22 t r(2 ) = 02A T

T − T (1.170)

from which we have

2t r | / 2= = and 0 A A|T i i|| ||± ±T t (1.171)

noting

2

1 32t r =A A A At t t2 2 2−2( + + ) < 0T

obtained from (1.122). It is known from Eq. (1.171) that the real eigenvalue of skew-symmetric tensor is zero.

Here, if select one of the principal direction eⅢ to the one with the zero principal value, it holds that

T* = [ ][ ][ ]A AQ QT T

cos sin 0cos sin 0 0 0

= sin cos 0 0 0 sin cos 0

0 0 1 0 0 1 0 0 1

θ θθ θ ωθ θ ω θ θ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

__

2 2

2 2

sin cos sin cos sin cos 0

= cos sin sin cos sin cos 0

0 0 1

ω θ θ ω θ θ ω θ ω θω θ ω θ ω θ θ ω θ θ

⎡ ⎤− + +⎢ ⎥_ − −⎢ ⎥⎢ ⎥⎣ ⎦

0 0

= 0 0 =

0 0 1

A

ωω

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

T (1.172)

meaning that the components do not change in the coordinate transformation. It is caused from the fact that the independent component of skew-symmetric tensor is only one when the one of base in the coordinate system is chosen to the principal direction of skew-symmetric tensor.

1.8 Cayley-Hamilton’s Theorem 35 1.8 Cayley-Hamilton’s Theorem

Denoting the eigenvector of the tensor T by unit vector e , it holds from Eq.

(1.125) and 1 1 1= / = /T T T− − −T e T e T Te 1= T −e that

=r rTT e e (1.173)

for = 1r ± . Now, assuming that Eq. (1.173) holds for =r k , one has

1 1 1 1 1 1= = = = =k k k k k kT T T T T± ± ± ± ± ±T e T T e T e T e e e

Consequently, Eq. (1.173) holds also for = 1r k ± . Then, it is verified that Eq. (1.173) holds for all integers r by mathematical induction. Eq. (1.173) means that the principal value of rT is rT and the principal directions of the tensors T and

rT are identical mutually, while the tensors having an identical set of principal directions are called to be coaxial or said to fulfill the coaxiality. Then, the linear associative function ( )f T of T is coaxial with T and the principal values are given by ( )f T .

For = 3n , the multiplication of the eigenvector e to the characteristic equation (1.142) leads to

3 2( ) =− −+ eT T T I 0Ⅰ Ⅱ Ⅲ

noting Eq. (1.173). Because of ≠e 0 , the following Cayley-Hamilton theorem holds.

3 2 =− + −T T T I 0Ⅰ Ⅱ Ⅲ (1.174)

It follows from the Cayley-Hamilton theorem that 22 24 2 3 ( )= ( ) = = ++ − ++− − −T T IT T T TT T T I T T Ⅰ Ⅱ ⅢⅢ Ⅱ ⅢⅠ Ⅱ Ⅲ Ⅰ Ⅱ Ⅰ

2 2 = ( ) ( )− − − +T T IⅠ Ⅱ ⅠⅡ Ⅲ ⅠⅢ (1.175)

1 2=− − +T T T IⅢ Ⅰ Ⅱ (1.176)

It is concluded that the power of the tensor T is expressed by the linear associative

of 2, , T T I with coefficients consisting of the principal values.

1.9 Positive Definite Tensor

When the second-order tensor P is symmetric and fulfills

0• >Pv v (1.177)

for an arbitrary vector ( )≠v 0 , P is called the positive definite tensor.


Denoting the principal value and direction of P as JP and Je , respectively, it

holds that 2= = || || 0 (no sum)J J J J J J JP P• • >Pe e e ee (1.178)

noting Eq. (1.177). Then, it is known that the principal value of positive definite tensor is positive. Taking this fact into account for Eq. (1.147)3, it holds that

det > 0=P Ⅲ (1.179)

The positive definite tensor U having the same principal directions and principal

values 1/ 2JP is defined as the square root of P , i.e. 2 =U P or 1/ 2=U P .

1.10 Polar Decomposition

Assuming that the second-order tensor T is not singular ( det 0≠T ), it holds that

≠Tv 0 for an arbitrary vector ( )≠v 0 as described in 4) of 1.4.2 and thus using

Eq. (1.101), one obtains

0T • •= >T Tv v Tv Tv (1.180)

where TT T is the symmetric tensor and thus it is the positive-definite tensor as

described in 1.9. Denoting the square root of TT T by U , one can write

1/2 2= ( ) ( = ), =T T TU T T U T T U U (1.181)

Then, U is the positive definite tensor. Furthermore, defining the tensor R as

1= −R TU (1.182)

noting Eq. (1.106), one has

11 1 1 1 2 1)( )= ( = = ( ) = =T T T T TT −− − − − − −RR TU TU TU U T T U T TT T T I

(1.183) Therefore, R is the orthogonal tensor. Furthermore, similarly to Eqs. (1.181) and

(1.182), consider 1/ 2 2= ( ) ( = )T TV TT V T T (1.184)

from which we have

2 2= = ( ) ( ) = = = ( )T T T T T TV TT RU RU RUUR RUR RUR RUR (1.185)

1.11 Isotropic Tensor-Valued Tensor Function 37 noting Eq. (1.182), and thus it holds that

= , =T TV RUR U R VR (1.186)

Then, one can write

= =T RU VR (1.187)

Consequently, an arbitrary non-singular tensor T can be decomposed into two

forms in terms of the positive definite tensors or U V and the orthogonal tensor

R . Here, based on Eqs. (1.181), (1.182), (1.184), and (1.185), R is expressed by

the original tensor T as follows:

1/ 2 1/ 2= ( ) = ( )T T− −R TT T T T T (1.188)

Based on (1.186), ,U V are the mutually similar tensors, as described in 1.5.1. For

that reason, they have same positive principal values, denoted as , ,U U UⅠ Ⅱ Ⅲ , and

their unit principal vectors , ( = , , )J J Ju v ⅠⅡⅢ are mutually related by

= , =J J J J⊗v Ru R v u (1.189)

according to Eqs. (1.137), (1.139) and (1.186). Equation (1.187) is called the polar decomposition in similarity to the polar form

= (iθ θZ Z e : phase angle) which expresses the complex number by the

decomposition into the magnitude and the direction in the polar coordinate system.

Actually, RU and VR are respectively called the right and the left polar

decompositions.

1.11 Isotropic Tensor-Valued Tensor Function

If the function f of tensors , , • • •T S fulfills the following equation, it is called the isotropic function.

)( , , = ( , , )• • • • • •Q T S f Q Q STf (1.190)

where the symbol in Eq. (1.72) is used. If f is a scalar, it is to be the invariant

defined in Eq. (1.140) and if it is a tensor, it is called the isotropic tensor-valued

tensor function.

Now, consider the isotropic second-order tensor function B of a single

second-order tensor A , i.e.


= ( )B f A (1.191) where f fulfills

( ) = ( )T Tf QAQ Qf A Q (1.192)

First introducing the coordinate system with the bases , , ee eⅡI Ⅲ , which are the

normalized eigenvector of the tensor A and further adopting the another coordinate system rotated 180°around the base eⅢ , the orthogonal tensor between

the bases of these coordinate systems is given by

0

1 0 0

= 0 1 0

0 0 1

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

Q (1.193)

where 0Q fulfills 0 0= TQ Q resulting in the symmetric tensor and it holds that

1 0 0 0 0

0 1 0 0 = 0

0 0 1 1 1

−⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥− ⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎣ ⎦ ⎩ ⎭ ⎩ ⎭

, i.e. 0 =Q e eⅢ Ⅲ (1.194)

Then, it is known that eⅢ is one eigenvector not only of A but also of 0Q . Furthermore, denoting the principal values of A by , , α α αⅢⅠ Ⅱ , it holds that

T0 0

0 0 0 01 0 0 1 0 0

0 1 0 0 0 0 1 0 = 0 0 , i.e. =

0 0 1 0 0 1 0 0 0 0

α αα α

α α

− −⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

Q AQ AⅠ Ⅰ

Ⅱ Ⅱ

Ⅲ Ⅲ

(1.195) and thus it holds that

T0 0( ) = ( ) =f Q AQ f A B (1.196)

On the other hand, from Eq. (1.192) one has

T T0 0 0 0( ) =f Q AQ Q BQ (1.197)

Then, the commutative law

0 0=Q B BQ (1.198)

1.11 Isotropic Tensor-Valued Tensor Function 39 holds from Eqs. (1.196) and (1.197), and further, noting Eq. (1.194), the following relation is obtained.

0 0= =e e eQ B BQ BⅢ Ⅲ Ⅲ (1.199)

which means that eB Ⅲ is the eigenvector of 0Q and thus it has the same direction

as eⅢ . Then, denoting the principal value B for the eigenvector eⅢ by βⅢ , one

can writes

= βeB eⅢ Ⅲ Ⅲ (1.200)

Performing the similar manipulations also for eⅠ and eⅡ , it can be concluded that

the tensor B has the same eigenvectors as the tensor A , leading to the coaxility.

Therefore, the principal values , , ββ β ⅢⅠ Ⅱ of the tensor B can be represented

in unique relation to the principal values , , α α αⅢⅠ Ⅱ of the tensor A . Now, noting the results obtained above, i.e. the coaxiality and the representation

of , , ββ β ⅢⅠ Ⅱ by , , α α αⅢⅠ Ⅱ in view of the coaxiality, consider the

following equation by way of trial.

20 1 2

20 1 2

20 1 2

, , ( ) =

, , ( ) =

, , ( ) =

α α αβ φ φ α φ α



⎫+ +⎪⎪+ + ⎬⎪

+ + ⎪⎭

ⅢⅠ ⅡⅠ Ⅰ

ⅢⅠ ⅡⅡ Ⅱ

ⅢⅠ ⅡⅢ Ⅲ

Ⅰ

Ⅱ

Ⅲ

(1.201)

where 0 1 2φ φ φ, , are symmetric functions of , , α α αⅢⅠ Ⅱ . While Eq. (1.201) is regarded as the representation of the relation of the tensors A and B in their

common principal coordinate system, it is expressed in the direct notation of tensor as

20 1 2= φ φ φ+ +B I A A (1.202)

in which 0 1 2φ φ φ, , are functions of invariants of the principal tensor A . When we regard Eq. (1.201) to be the simultaneous equation for the unknown values

0 1 2φ φ φ, , , we know that the Vandermonde’s determinant is not zero for mutually

different values of , , α α αⅢⅠ Ⅱ as follows:

2

2

2

1

1 ( )( )( ) 0=

1

α αα α αα α α α αα α

− − − ≠Ⅰ Ⅰ

ⅢⅡ Ⅲ ⅠⅡⅠ ⅡⅡ

Ⅲ Ⅲ

(1.203)

40 1 Tensor Analysis Therefore, 0 1 2φ φ φ, , are scalar functions which are uniquely determined by

, , α α αⅢⅠ Ⅱ and then , , ββ β ⅢⅠ Ⅱ are uniquely determined by , , α α αⅢⅠ Ⅱ , while 0 1 2φ φ φ, , are invariants of A since they are scalar functions of

, , α α αⅢⅠ Ⅱ , i.e. A . Thus, we can conclude that Eq. (1.202) is the correct explicit form of Eq. (1.191). Although Eq. (1.202) would not be the unique explicit form of Eq. (1.191), one can say that the isotropic tensor-valued tensor function of a single tensor reduces to this simplest form. Eventually, Eq. (1.202) is the most concise from of the isotropic second-order tensor-valued function of a single second-order tensor. This fact can also be verified using Cayley-Hamilton’s theorem for the special case that f is the linear associative form of the power of A . However, for the case in which f is the general function of A , one must depend on the proof given in this section.

In the particular case in which f is the linear function of the tensor A , Eq. (1.202)reduces to

= (tr )a b+B A I A (1.204)

where a and b are the material constants. Equation (1.204) is rewritten as

=B CA (1.205) where

( ( )ijijkl ik il jkkl ila b C a bδ δ δ δ δ δ1≡ ⊗ + ≡ + +2C I I I (1.206)

While the second-order isotropic tensor-valued tensor function of single tensor is considered above, the representation theorem of the second-order isotropic tensor-valued tensor function f of two tensors A and B is sown below (cf. e.g., Spencer, 1971).

2 2 240 2 3 51, = ( )ϕ ϕ ϕ ϕ ϕ ϕ+ + + + + +f(A B) I A B A B AB BA

2 2 2 2 2 2 2 26 87( ( (ϕ ϕ ϕ + + ) + + ) + + )A B BA AB B A A B B A (1.207)

where 0 81 , , , ϕ ϕ ϕ• •• are scalar functions of invariants

32 23

2 2 2 2

tr , tr , tr , tr , tr , tr

tr( ), tr( ), tr( ), tr( )

⎫⎪⎬⎪⎭

A A A B B B

AB AB A B A B (1.208)

1.12 Representation of Tensor in Principal Space

Let the second-order tensor T be expressed by the vector from in terms of the

eigenvectors , , ee eⅡ ⅢI and the principal values , , T T TⅡI Ⅲ as follows:

= T T T+ +ee eT ⅡⅡI I ⅢⅢ (1.209)

1.12 Representation of Tensor in Principal Space 41 Equation (1.209) is called the representation of tensor in principal space (Fig. 1.3) by which the second-order tensor can be visualized in the three dimensional space. Equation (1.209) is rewritten by decomposing T into the mean and the deviatoric components as follows:

= ( ) ( ) ( ) =m m mmT T T T T T' ' ' '+ + + + + +ee eT T TⅡ ⅢI ⅢⅡI (1.210)

where

= ( ( ) )3 /3m m m mm mT T T T T T≡ ≡ + +IT e ⅢI Ⅱ (1.211)

1 , (|| || = 1) 3

mm mm+ +≡ ≡e e e eI eIⅡI Ⅲ (1.212)

= || ||T T T' ' ' ' ' '≡ + +ee eT T tⅡ ⅢII Ⅱ Ⅲ (1.213)

, , m m mT T TT TT T T T' ' '≡ ≡ ≡− − −I I Ⅱ Ⅱ Ⅲ Ⅲ (1.214)

2 2 2 222 13

|| || (|| || = 1)

= ( ) ( ) ( )|| || =

/

T T T T T T TT T

''' '

''' '

⎫≡ ⎪⎬

+++ + − − − ⎪⎭

Tt T t

T Ⅲ Ⅲ IIⅡ Ⅱ ⅡⅢI

(1.215)

Space diagonal

Deviatoric plane

0TI

TⅡ

TⅢ

'T

't

mT

me

T

eIeⅡ

eⅢ

T'ⅠT'Ⅱ

T'Ⅲ

Space diagonal

Deviatoric plane

0TI

TⅡ

TⅢ

'T

't

mT

me

T

eIeⅡ

eⅢ

T'ⅠT'Ⅱ

T'Ⅲ

Fig. 1.3 Representation of second-order tensor in principal space


'T

θ

( T'−)_Ⅲ

0

TI TⅡ

TⅢ

T '_Ⅱ

T' _Ⅰ

'T

θ

( T'−)_Ⅲ

0

TI TⅡ

TⅢ

T '_Ⅱ

T' _Ⅰ

Fig. 1.4 Coordinate system in deviatoric ( ) plane

Whereas the deviatoric tensor 'T lies on the -π plane (Fig. 1.4), the orthogonal projection of 'T to the three oblique axes, which are the orthogonal projections of the orthogonal axes , , T T TI ⅢⅡ to the -π plane, are given as

|| ||= cos ,

|| ||= cos ,

|| ||= cos

)(

)(

T

T

T

' '

' '

' '

θ

θ π

θ π

⎫⎪⎪⎪2− ⎬3 ⎪⎪2+ ⎪3 ⎭

T

T

T

Ⅱ

Ⅲ

I

(1.216)

On the other hand, the deviatoric components , , T T T'' ' ⅢⅡI are the components

on the orthogonal coordinates ( , , )T T TI ⅢⅡ of the diviatoric tensor 'T (see

Fig. 1.3). Denoting the angle contained between the coordinate axis, e.g. eI and its

projected line onto the -π plane by the symbol α (see Fig. 1.5), it holds that

1.12 Representation of Tensor in Principal Space 43

'T

θ

TⅠ

α

T ' _Ⅰ

IT'

cos = 2/3α

0Ie

|| || cos=T' ' θ _

TⅠ

me

T T T' ' ' '+≡ +T e e eⅠ ⅡⅠ Ⅲ ⅢⅡ

cos=T T'' α _

Ⅰ Ⅰ

A

B

C

0ABC: Rectangular triangular pyramid

π-plane

'T

θ

TⅠ

α

T ' _Ⅰ

IT'

cos = 2/3α

0Ie

|| || cos=T' ' θ _

TⅠ

me

T T T' ' ' '+≡ +T e e eⅠ ⅡⅠ Ⅲ ⅢⅡ

cos=T T'' α _

Ⅰ Ⅰ

A

B

C

0ABC: Rectangular triangular pyramid

π-plane

Fig. 1.5 Relation of 1T' and =T'I

1= , = cos23

( )m mπ α• • −e e e eI I (1.217)

resulting in

cos = 2/3α (1.218)

Substituting Eq. (1.218) into Eq. (1.216) it is obtained that

2 2 || ||= = cos ,3 3

2 2 || ||= = cos ,3 3

2 2 || ||= = cos3 3

)(

)(

T T

T T

T T

' ' '

' ' '

' ' '

θ

θ π

θ π

⎫⎪⎪⎪⎪2− ⎬3 ⎪⎪2+ ⎪3 ⎪⎭

T

T

T

I

ⅡⅡ

Ⅲ Ⅲ

I

(1.219)

The product of the three components is given by

2 2 1= cos cos cos = cos33 3|| || || |||| || 3 6

) )( (T T T' ' ''' '

θ θ π θ π θ2 2− +3 3T TT

Ⅱ ⅢI

(1.220) Considering Eq. (1.151) to Eq. (1.220), it follows that

36cos3 = tr 'θ t (1.221)

44 1 Tensor Analysis It holds from Eq. (1.216) that

2 2= = tan )(T T T T T T

T TT T

' ''' '

μ θ π− − − − 1≡ −3 6−−Ⅲ ⅢⅡ Ⅱ

ⅢⅢ

I I

II

(1.222)

which is called the Lode’s variable. Assuming the principal value

12 || ||= cos3

T '' θT (1.223)

and substituting it into in the characteristic equation of deviatoric tensor (1.148) while, taking account of 0='Ⅰ , one has the following equation

3 2 31 12 2|| || || || || ||cos cos t r = 033 2 3

( ) ( ) '' ' 'θ θ− −T T T T

resulting in

334cos 3cos t r =06 'θ θ− − t

from which one can derive Eq. (1.221) and further reach Eq. (1.219) by tracing the above-described process inversely.

1.13 Two-Dimensional State

Consider the two-dimensional state in which the components related to the 3e

direction in the coordinate system 1 32( , , )x x x with the bases 1 32( , , )e e e are

zero, i.e. 233133 = = = 0T T T . Furthermore, introduce the coordinate system

1 2 3* **( , , )x x x with the bases 1 2 3 3* * *( , , ( ))=e e e e which is rotated by the angle α

in the counterclockwise direction around the axis 3x as shown in Fig. 1.6. The

orthogonal tensor between these bases is given from Eq. (1.47) as follows:

cos sin 0

[ ] = sin cos 0

0 0 1

Q

α αα α

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

　　

　 (1.224)

1.13 Two-Dimensional State 45

1*x2*x

2*x

1*x

11 12( , )T T

1211* *( , )T T

1222* *( , )T T

22 12( , )T T

11 22

2T T+ 1x

2x

nT

tT

P

01x

2x

α

α

2α2 pα

TⅠTⅡ

TR

Material

(a) Physical plane (b) 　　　plane( , )n tT T

1*x2*x

2*x

1*x

11 12( , )T T

1211* *( , )T T

1222* *( , )T T

22 12( , )T T

11 22

2T T+ 1x

2x

nT

tT

P

01x

2x

α

α

2α2 pα

TⅠTⅡ

TR

Material

(a) Physical plane (b) 　　　plane( , )n tT T Fig. 1.6 Mohr’s circle

Substituting Eq. (1.224) into Eq. (1.62), one has

2 211 11 22 12

2 222 11 22 12

2 212 22 11 12

* = cos sin 2 cos sin

* = sin cos 2 sin cos

* = ( )cos sin (cos sin )

T T T T

T T T T

T T T T

α α α α

α α α α

α α α α

⎫+ +⎪⎪+ − ⎬⎪

− + − ⎪⎭

(1.225)

which is rewritten as

11 12

22 12

12 12

* = cos 2 sin 2

* = cos 2 sin 2

* = sin 2 cos 2

m

m

T T T T

T T T T

T T T

α α

α α

α α

⎫+ +⎪⎪− − ⎬⎪

− + ⎪⎭

(1.226)

where

11 22 11 22, 2 2

mT T T T

T T+ −≡ ≡ (1.227)

Furthermore, it holds from Eq. (1.226) that

11 22 11 22* * =T T T T+ + (1.228)

11 2212 12

* ** *= 2 , = 2

T TT Tα α

∂ ∂ −∂ ∂

(1.229)


While the axis 3 3*(= )x x is one of the principal directions, the other principal

directions exist on the plane 1 2( , )x x . Denoting the principal direction from the

1-x axis by α , it is obtained by putting 11 12* */ = = 0T Tα∂ ∂ or 22* / = 0T α∂ ∂

in Eq. (1.229) with Eq. (1.226) that

12tan 2 =pT

Tα (1.230)

from which one obtains

12

2 212

= cos 2 , = sin 2

=

p pT T

T

R T RT

ΤR T

α α ⎫± ± ⎪⎬⎪+ ⎭

(1.231)

Substituting Eq. (1.231) into the upper two of Eq. (1.226) with specifying α as pα , the maximum and the minimum principal values TⅠ and TⅡ are described by

= mT

T RT

⎫±⎬

⎭Ⅰ

Ⅱ

(1.232)

Equation. (1.232) can also be derived directly from the quadratic equation

2211 22 11 22 12( ) = 0T T T T T T T− + + −

which is obtained by inserting 211 22 11 22 12= , = ,T T T T T+ − = 0 Ⅲ

2333 31( = = = 0)T T T in Eq. (1.142).

Furthermore, denoting α for the extremum of 12*T as sα , it holds by taking

12* = 0/T α∂ ∂ in Eq. (1.226) that

12tan 2 =

2sTT

α −

(1.233)

Equations (1.230) and (1.233) yield the relation = 4ps πα α ± /

( tan 2 tan 2 = 1p sα α − ）and thus there exist the two directions for the extremum

of 12*T and they divide the two principal directions into two equal angles, i.e. /4π .

The extremum of 12*T denoted by MT is given from Eq. (1.231)2 as follows:

= TMT R± (1.234)

1.14 Partial Differential Calculi 47 which is also expressed by Eq. (1.232) as

=2M

T TT −± Ⅰ Ⅱ (1.235)

Designating the normal stress ( , = 1,2; = )ijT i j i j∗ by nT and the shear

stress ( )ijT i j∗ ≠ by tT , the following equation is derived from Eqs. (1.226) and

(1.231)3. 2 2( ) =n m TtT T T R− + (1.236)

Consequently, the stress on an arbitrary plane is expressed by the point on the circle

with the radius TR centering at ( , 0)mT in the two-dimensional plane ( , )n tT T

as shown in Fig. 1.5. This circle is called the Mohr’s circle. Substituting Eq. (1.231) into Eq. (1.226), we have the expressions

11

22

12

* = cos(2 ),

* = cos(2 ),

* = cos(2 )

pm T

pm T

pT

T T R

T T R

T R

α α

α α

α α

⎫+ − 2⎪⎪− − 2 ⎬⎪

− − 2 ⎪⎭

(1.237)

Therefore, 11*T and 12*T are shown by the values in the ordinate and abscissa axes,

respectively, of the point rotated 2α counterclockwise from point 11 12,T T on the

Mohr’s circle as shown in Fig. 1.5, provided that the definition for the sign of shear

stress is altered to be positive when it applies to the body surface in the clockwise

direction, in the Mohr’s circle. As shown in Fig. 1.5, the intersecting angle of the two straight lines drawn

parallelly to the physical plane 1x and 1*x stemming from the points 11 12( , )T T

and 1211* *( , )T T , respectively, on the Mohr’s circle is α which is the angle of

circumference of Mohr’s circle and thus the intersecting point lies on the circle. This

point is called the pole. Generally speaking, the normal stress nT and the shear stress

tT applying to a certain physical plane are given by the intersecting point of Mohr’s

circle and the straight line drawn parallel to that physical plane from the pole.

1.14 Partial Differential Calculi

Partial derivatives of symmetric tensors appearing often in elastoplasticity are shown below.


1= ( ) , =2

ijjkjlik il

kl

T

Tδ δ δ δ∂ ∂+

∂∂T IT

(1.238)

1 1 tr13 31 1= = , ==3 3 3

( () )rsrsmijir js rs

ij ij

TT

T T

δδδ δ δ

∂ ∂∂∂ ∂ ∂

TI

T (1.239)

( ) 11 1= = ( ) , = 32 3ij m ijij

jkjlik il klijkl kl

TTT

T T

' 'δδ δ δ δ δ δ

∂ −∂ ∂ − ⊗+ − ∂∂ ∂T I IIT

(1.240)

1/ 2 1/ 2( )1 1= ( ) = ( ) 2 = ,2 2

rs rs rs rsrs rs rs rs rs ijir js

ijij

T T T TT T T T T

T Tt' ' ' '' ' ' ' ' '

''δ δ− −∂ ∂

∂ ∂

|| ||=

|| ||' ' '''

∂ ≡∂

T TT T

t (1.241)

= =

ij kljl ijik pq pq

pq pqpq pqij

pq pqkl kl

T TTT T

T TT T

T TT T

t

' ''' '

' '' ''' '' '

δ δ∂ −∂∂ ∂

1 1= ( ) ,2

jkjlik il ij kl

pq pqT Tt t' '

' 'δ δ δ δ+ −

1= ( )|| ||

' ' '' '∂ ⊗−∂ IT T

t t t (1.242)

11 1= = ( )2 3

( )ij ij rsjs is jr slir rk klrsij rs

kl rs klpq pq

TT T T T T

t tt t

' ' '' ''' ' '

δ δ δ δ δ δ δ δ∂ ∂ ∂ + −−∂ ∂ ∂

1 1 1 = ( )2 3 jlik il jk ij kl ij kl

pq pqT Tt t' '

' 'δ δ δ δ δ δ+ −− ,

1 1= || || 3( )'

' ''∂

− ⊗− ⊗∂ I I IT Tt

t t (1.243)

1.14 Partial Differential Calculi 49

3cos3 6t r 'θ ≡ t (1.244)

6cos3 = = 3 = 3 ,6 6pq qr rp

ip jq qr rp ir rjijij

t t tt t t t

tt' ' '

'' ''' 'θ δ δ∂∂

∂∂

2cos3 = 3 6 ''θ∂

∂tt

(1.245)

cos3 cos3=rsij ij

rsttT T

''

θ θ ∂∂ ∂∂∂ ∂

11 1= 3 ( )6 2 3 js is jr ijir rsrt ts ij rs

pq pqT Tt t t t' ' ' '

' 'δ δ δ δ δ δ+ − −

1 1= 3 ( )6 3 ijir rj rt tr rs st tr ij

pq pqT Tt t t t t t t t' ' ' ' ' ' ' '

' 'δ− − ,

2 2cos3 1 || || )3 6= cos36 3|| ||( '' '

'θ θ∂ − + −∂ tt I tT T

(1.246)

( ) ( )( ) = =tr ( )T

rsrs

f fddf dT

T∂ ∂

∂∂T T TT T

(1.247)

Differentiating 1 =ir rs isT T δ− , one has

1 1 11 1 1 1( ) = = 0 = 0ir rs ir ir

rk sl rk slrs ir rs sj ir sjkl klkl

T T T TT T T T T TT TT

δ δ δ δ− − −

− − − −∂ ∂ ∂+ → +∂ ∂∂

which leads to 1

1 11 11= ( )2

jijl jkilik

kl

TT TT T

T

−− −− −∂

− +∂

Differentiating Eq. (1.145), it holds that

2=∂ − +∂

IT TTⅢ

ⅡⅠ


Then, noting Eq. (1.176), one has

2 1 1det ( )det= = = =− −∂∂ − +∂ ∂

T IT T T TTT T

ⅢⅡⅠ Ⅲ

which was derived also in Eq. (1.105). For asymmetric tensors, the tensor ij klδ δ has to be used instead of the

symmetric identity tensor ( ) /2jkjlik ilδ δ δ δ+ in the above equations.

1.15 Time Derivatives

Let the tensor T , describing a physical quantity, be denoted by ( , )tT x in the

spatial description fixed in a space and by ( , )tT X in the material description

moving with a material, where t is a time. The time derivative in the former, i.e.

( , )tt

∂∂

T x (1.248)

describes the rate of the tensor T in the fixed point in the space and then it is called the spatial-time (or local) derivative. In many cases of fluid mechanics, a mechanical variation of individual particle of fluid is not important since deformation of fluid from the initial state does not influence the movement of fluid and thus the spatial-time derivative is often adopted. On the other hand, the time derivative in the material description, i.e.

( , )tt

∂∂

T X (1.249)

describes the rate of the tensor T in the particular particle of material and thus is called the material-time derivative. It is denoted by the symbol

( , ) ( , ) or

t tDt Dt t

• ∂ ∂≡ ≡∂ ∂

T TX XTT (1.250)

In solid mechanics, the rate of deformation in individual particles of a solid is important and thus the material-time derivative is used usually.

Because of ( ( , ), ) = ( , )t t tT X x T x the material-time derivative in Eq.

(1.250) and the spatial-time derivative in Eq. (1.248) is related as follows:

( , ) ( , )t tt

• ∂ ∂≡ +∂ ∂

xxT T vT x (1.251)

where / t≡ ∂ ∂v x is the velocity vector of material particle. The first term in the right-hand side signifies the non-steady (or local time derivative) term describing

1.16 Differentiation and Integration in Field 51 the change with the elapse of time and the second term signifies the steady (or convective) term describing the change attributable to the movement of material, which results from the existence of a gradient in the mechanical state.

Rate-type constitutive equations describing the irreversible deformation, e.g. the viscoelastic, the elastoplastic and the viscoplastic deformation, must be described by the material-time derivative pursuing a material particle because it describes the relation of physical quantities in a material particle. Further, as described in 4.3, the corotational derivative based on the rate of mechanical state observed by the coordinate system rotating with a material must be used in general.

1.16 Differentiation and Integration in Field

Scalar s , vector v , and tensor T are called the scalar field, the vector field, and the tensor field, respectively when they are functions of the position vector x . Their differentiation and integration in fields are shown below, in which the following operator, called the nabra or Hamilton operator, is often used.

=rrx

∂ ∂≡ ∂ ∂xe∇ (1.252)

1) Gradient

Scalar field: grad = rr

= xss s ∂∇ ∂ e (1.253)

Vector field:

: rear (right) formgrad =

: front left( ) form

iii jj i

j j

jji ij j

i i

vv= =x x

vv= =x x

⎧ ∂ ∂⊗ ⊗ ⊗⎪ ∂ ∂⎪⎨

∂∂⎪ ⊗ ⊗ ⊗∂ ∂⎪⎩

v

v

v

e e e e

e ee e

　

　

∇

∇(1.254)

Second-order tensor field:

: rear (right) formgrad =

: front left ( ) form

ijj jk iij i k

kk

jki ij jjk k k

i i

TT= =x x

T= =Tx x

⎧ ∂∂⊗ ⊗ ⊗⎪ ⊗ ⊗∂ ∂⎪⎨

∂∂⎪ ⊗ ⊗ ⊗ ⊗ ⊗⎪ ∂ ∂⎩

TT

T

e e e e e e

e ee e e e

　

　

　

∇

∇

(1.255) 2) Divergence

Vector field: div (= ) iii j

j i

v= v= =x x• •• ∂ ∂

∂ ∂v vv e e

　

∇ ∇ (1.256)

52 1 Tensor Analysis Second-order tensor field:

: rear right( ) formdiv =


irj k iij i

rk

rii j ijk k ri

TT= =x x

T= =Tx x

• •

• •

⎧ ∂ ∂⊗⎪ ∂ ∂⎪⎨

∂ ∂⎪ ⊗∂ ∂⎪⎩

TT

T

e e e e

e e e e

　

　

　

　

∇

∇

(1.257)

3) Rotation (or curl)

Vector field:

: rear right ( ) formrot =


ii ijk ki j

j j

jijki ijk kj j

i i

vv= =x x

vv= =x x

ε

ε ε

⎧ ∂ ∂× ×⎪ ∂ ∂⎪⎨

∂∂⎪ × ×∂ ∂⎪⎩

v

v

v

e e e

e e e

　

　

　

∇

∇(1.258)

noting Eq. (1.36). Second-order tensor field:

( ) : rear right( ) formrot =

( ) : front left( ) form

j kij ik

ij ijj jkri rik

k k

i jjk ki

jk jkji ijr rk ki i

T= x

T T = =x x

= Tx

T T = = x x

ε

ε

⎧ ∂× ×⊗⎪ ∂⎪⎪ ∂ ∂⎪ ⊗ × ⊗∂ ∂⎪⎨

∂⎪ × × ⊗∂⎪⎪

∂ ∂× ⊗ ⊗∂ ∂⎩

T

TT

e e e

e e e e e

e e e

e e e e e

　

　

　

　

∇

∇

⎪⎪

(1.259)

The symbol ∇ is regarded as a vector, and the scalar product of itself, i.e.

22 = =r s

r r rsx x xx

• •∂ ∂∂Δ ≡ ≡∇ ∂ ∂ ∂∂e e∇ ∇ (1.260)

has the meaning of 2 ( ) div(grad( ) )∇ ≡ . The symbol Δ is called the Laplacian

or Laplace operator, which is often used for scalar or vector fields as

2

=r rx xss ∂Δ ∂ ∂ (1.261)

1.16 Differentiation and Integration in Field 53

2

=s

sr r

vx x∂Δ ∂ ∂

v e (1.262)

The following relations hold between the above-mentioned operators.

div( ) = div grad ,

div( ) = rot rot ,

rot( ) = (grad ) (grad ) (div ) (

grad( ) = grad grad ,

div( ) = grad div ,

div( ) = div tr

div ) ,

gr

( gr

ad( ) = (grad ) (grad ) ro

a

t ot ,

d

rT

s s s

s s

s

s

s s

•

•

•

•

•

+× −× − +

×

+

−+ ×

⊗v v v

u v v u u v

u v u v v u v u u v

u v v u +

v v v

T T + T

Tv v T +

+ v u

T

u v u v

)v

(1.263)

4) Gauss’ divergence theorem Consider the physical quantity T ( )x in the zone surrounded by a smooth surface

inside a material. Then, suppose the slender prism cut by the four planes perpendicular to the 2x -axis and 3x -axis in infinitesimal intervals from a zone

inside the material. The following equation holds for the prism.

1

11 2 3 2 3 2 3

1 1

= = [ ] = ( )xxv v

T Tdv dx dx dx dx dx dx dxT T Tx x

+− + −∂ ∂ −

∂ ∂∫ ∫ (1.264)

where ( )+ and ( )− designate the values of physical quantity at the maximum

and the minimum 1x -coordinates, respectively. The neighborhood of the surface cut by the prism is magnified in Fig. 1.7.

Consider the infinitesimal rectangular surface PQRS of the prism exposed at the

surface in the maximum 1x -coordinate and the infinitesimal rectangular section

PQ R S∗ ∗ ∗ cut by the plane passing through the point P and perpendicular to the

1x -axis by the prism. Then, denoting QQ == , SSQ Sdxdx∗ ∗ , the vectors

PQ, PS→→

are given by

Q 1 12 2 3 3 SPQ = , PS =dx dx dx dx→→

+ +e e e e (1.265)

and thus it holds that

Q 1 12 2 3 3 S= PQ PS = ( ) ( )da dx dx dx dx+ + →→× + × +n e e e e


P

Q

R

S

Q∗

R∗

S∗

3dx

sdx

Qdx

1e

2e

3e

0

n+

P

Q

R

S

Q∗

R∗

S∗

3dx

sdx

Qdx

1e

2e

3e

0

n+

P

Q

R

S

Q∗

R∗

S∗

3dx

sdx

Qdx

1e

2e

3e

0

n+

Fig. 1.7 Infinitesimal square pillar cut from a zone in material

Q 1 12 3 2 3 3 3 S 2 2 = dx dx dx dx dx dx× + × + ×e e e e e e

Q 32 3 3 S 21 2 = dx dx dx dx dx dx− −e e e (1.266)

Comparing the components in the base 1e on the both sides in Eq. (1.266), one has

2 31 =n da dx dx+ + (1.267)

In a similar manner for the surface of the prism exposed on surface in the minimum 1x -coordinate, one has

2 31 =n da dx dx−− − (1.268)

The general expression of projected area is given in Appendix 1. Adopting Eqs. (1.267) and (1.268) in Eq. (1.264), it holds for the prism that

1 1 1 11

= ( ) =v

T dv n da n da n da n daT T T Tx+ − + −− −+ + + +− −∂ − − +

∂∫ (1.269)

Then, the following equation holds for the whole zone.

11=

avT dv n daTx

∂∂∫ ∫ (1.270)

In a similar manner also for the 2x -direction and 3x -direction, the following Gauss’ divergence theorem holds.

= iiv a

T dv n daTx∂∂∫ ∫ (1.271)

1.16 Differentiation and Integration in Field 55

The following equations for the scalar s , the vector v and the tensor T hold from Eq. (1.271).

= , =i

iav v avdv n da d da

xs s ss∂

∂∫ ∫ ∫ ∫ n∇ (1.272)

= , =ii

iiv a v a

v dv v n da dv dax

• •∂∂∫ ∫ ∫ ∫v v n∇ (1.273)

= , =ijij

jiv a v a

T dv n da dv daTx

•∂∂∫ ∫ ∫ ∫ nTT ∇ (1.274)

5) Material-time derivative of volume integration Supposing that the zone of material occupying the volume v at the current moment ( = )t t changes to occupy the volume v vδ+ after the infinitesimal time

( = )t t tδ+ , the material-time derivative of the volume integration

( , )v vT t d∫ x of the physical quantity ( , )T tx involved in the volume is defined

by the following equation.

0

1= lim ( , ) ( , )( ) tv v v v

v T t t dv T t dvT dt δδ

δδ +→

• + −∫ ∫∫ x x

0

1= lim ( , ) ( , ) ( , ) [ ]t v v

T t t T t dv T t t dvt δδ

δ δδ→++ − +∫ ∫x x x

(1.275)

The integration of the first term in the right-hand side in Eq. (1.275) is transformed as

0

( , )1lim ( , ) ( , ) =t v v

T tT t t T t dv dvttδ

δδ→∂+ − ∂∫ ∫ xx x (1.276)

On the other hand, the second term in Eq. (1.275) describes the influence caused by the change of volume during the infinite time. Here, the increment of volume, vδ , is given by subtracting the volume going out the boundary of the zone from the volume going into the boundary, which is the sum of ( )=dv da tδ•v n over the whole boundary surface (Fig. 1.8). Therefore, substituting the Gauss’ divergence theorem (1.271) and ignoring the second-order infinitesimal quantity, the integration of the second term in the right-hand side of Eq. (1.275) is given by

0 0

0

1 1lim ( , ) lim ( , )

( , )1 = lim = =( , )( , )

t t

t

rr r r r

r

v v

a a v

T t t dv T t dvt t

T t vv n da t v n da dvT tT t xt

δ δδ δ

δ

δδ δ

δδ

→ →

→

+ ≅

∂∂

∫ ∫

∫ ∫ ∫

x x

xxx


v v vδ+

tδv

da

ad tδv

n

n

aa

v v vδ+

tδv

da

ad tδv

n

n

aa

Fig. 1.8 Translation of a zone in material

The sum of the first term in the right-hand side in this equation and the Eq. (1.276) is equal to the material-time derivative of ( , )T tx because of Eq. (1.251) and thus

Eq. (1.275) is given by

( , ) ( , )( , ) = ( , ) = ( , )div ( ) r

rv v vvt tT t v T T t T T td dv dvx

• •• ∂+ +∂∫ ∫∫ x xx x x v

(1.277)

which is called the Reynolds’ transportation theorem. Equation (1.277) can be obtained also by the following manner.

( , ) = ( , ) = ( ( , ) ( , ) )( ) ( )Vv v

T t v T t V T t T t Vd Jd J dJ• •• • +∫ ∫ ∫x X X X

( , ) = ( , ) r

rvvtT T t dvx

• ∂+∂∫ x x

where V is the initial volume of zone and it is set that J dv/dV≡ . Here, it holds

that ( )= / rr xvJ J• ∂ ∂ as will be described in 2.5.

For the physical quantity T kept constant in a volume element, Eq. (1.277) leads to

( ( , ) ( , )div ) = 0v

tT T t dv•

+∫ x x v (1.278)

The local (weak) form of Eq. (1.278) is given as

( , ) ( , )div = 0tT T t•

+x x v (1.279)


Chapter 2 Motion and Strain (Rate)

2 Motion and Strain (Rate)

The tensor analysis providing the mathematical foundation for the continuum me-chanics is described in Chapter 1. Basic concepts and quantities for continuum mechanics will be studied in the three chapters up to Chapter 4. The description of motion is required in the continuum mechanics. Various measures for the motion and strain (rate) of solids are adopted for the description of reversible and irre-versible deformation of materials. Main measures among them will be explained in this chapter.

2.1 Motion and Deformation

General, basic quantities and manners for describing the motion and deformation will be shown in this section.

2.1.1 Material, Spatial and Relative Descriptions

A material body is assembly of material particles (or material elements). The posi-tions of all material particles in a laboratory (Euclidean space) is referred to as the configuration of the material body. Here, the configurations in the initial time

= 0t and the current time t are called the initial configuration and the current configuration, respectively, and the position vectors of material particle in the initial and the current configurations are designated by X and ( )tx , respectively.

Here, X is fixed and thus it can be regarded as a label of each material particle. At fixed time the unique relation exists between X and x provided that the material does not overlap or separate. The mapping may be symbolically described as

1= ( , ), = ( , )t t−x X X xχ χ (2.1)

As discussed in 1.15, a physical quantity, say T , representing the state of the body changes with the position of material particle, i.e., with the change of the body-configuration and the time. The special reference configuration is usually selected to identify the material particles of the body and it is called the reference configuration. Here, we adopt the initial configuration to be the reference con-figuration. When X is regarded as an independent variable, the field of physical quantity is described by ( , )tT X . This type of description of material state and

properties is called the Lagrangian (or material) description (cf. Section 1.5). On

58 2 Motion and Strain (Rate)

the other hand, when we adopt current position of a material particle as independent variable, then the field of physical quantity in the domain of current configuration is described by ( , )txT . This way of description is called the Eulerian description or

spatial description, as already mentioned in 1.5. The former follows the variation of the state of a material particle, whereas the latter observes the variation of physical quantity at a fixed point of a laboratory.

A configuration at a particular time =t τ , i.e. ( )τx can be described from

Eq. (2.1) as follows:

1( ) = ( , ), = ( ( ), )τ τ τ τ−x X X xχ χ (2.2)

The state of physical quantity in a body may be described in terms of ( )τx as

1( ( ), ) ( ), ( , ) = tt ττ τ τ− x xT Tχ (2.3)

Here, the set of ( )τx is called the relative configuration and the descrip-

tion presented in Eq. (2.3) is called the relative description. Specifically,

( ), ( = )t t t tτxT leading to ( , )txT is called the updated Lagrangian de-

scription. In this case the current configuration is regarded as the reference con-figuration and, for the sake of clarity the type of description ( , )tT X will be called

the total Lagrangian description. The fact that a material does not overlap or separate by the deformation requires

the existence of the one-to-one correspondence between X and x ( x is uniquely

determined for X and vice versa) so that 1 1 2 3 2 1 2 3( ), ( , , )x X X X x X X X and

3 1 2 3( )x X X X must be mutually independent. Now, assume that 1 2 3, ,x x x are not

mutually independent. Then, there exists a function f such that

1 1 2 3 2 1 2 3 3 1 2 3( ( ), ( , , ), ( )) = 0f x X X X x X X X x X X X (2.4)

from which one has

31 2

1 1 2 1 3 1

31 2

1 2 2 2 3 2

31 2

1 3 2 3 3 3

= 0

= 0

= 0

xf x f x f

x X x X x X

xf x f x f

x X x X x X

xf x f x f

x X x X x X

⎫∂∂ ∂ ∂ ∂ ∂+ + ⎪∂ ∂ ∂ ∂ ∂ ∂ ⎪⎪∂∂ ∂ ∂ ∂ ∂ ⎪+ + ⎬∂ ∂ ∂ ∂ ∂ ∂ ⎪⎪∂∂ ∂ ∂ ∂ ∂+ + ⎪

∂ ∂ ∂ ∂ ∂ ∂ ⎪⎭

(2.5)

2.1 Motion and Deformation 59

Eliminating 1 2 3/ , / , /f x f x f x∂ ∂ ∂ ∂ ∂ ∂ from Eq. (2.5), one gets

1 2 3det = = 0( )iIJK

J I KJ

xx x xJ

X X X Xε∂ ∂∂ ∂≡ ∂ ∂ ∂ ∂

(2.6)

on account of Eq. (1.10), where J is called the functional determinant or Jacobian. This result shows that functions 1 2 3, ,x x x are not independent. In order that they

are independent, it must hold that

det 0( )i

J

xX∂ ≠∂

(2.7)

The transformation between x and X is called the admissible transformation, if

1 2 3, , f f f in 1 1 1 2 3 2 2 1 2 3= ( ), = ( , , )x f X X X x f X X X , 3 3 1 2 3= ( )x f X X X

are single-valued and continuous functions and the Jacobian is not zero at all points. Further, if the Jacobian is positive at all points, a right-hand coordinate system is transformed to other right-hand one, and it is called the positive transformation. In-versely, if the Jacobian is negative at all points, a right-hand coordinate system is transformed to a left-hand one, and it is called the negative transformation. Admissible and positive transformation is assumed throughout this book.

2.1.2 Deformation Gradient and Deformation Tensors

At the initial state of deformation ( = 0t ), consider the material particle, the posi-tion vector of which is X , and let the position vector of the adjacent material point be d+X X . Furthermore, consider the current state ( =t t ) in which these ma-terial particles moved to the space points having position vectors x and d+x x . The line elements before and after the deformation are described as

= , ( ) = ( ) ( )AA i id dX d t dx t tX e x e (2.8)

where the different bases are assumed for the initial and current configurations so that the observer can move with a deformation.

Here, define the deformation gradient

,( )( )

( ) , ( ) ( )iA A i A Ai i iiA

A

tt xt F t x tX

∂∂≡ ⊗ ≡ ⊗ ≡ ⊗∂ ∂xF e e e e e eX

(2.9)

Then, ( )d tx is described by dX as follows:

( ) = ( ) , ( ) = ( ) Ai iAd t t d dx t F t dXx F X ,

( ) ( ) = ( ) = ( )A AB Bi i i iiA iAdx t t F t dX F dX t⊗e e e e e (2.10)


Equation (2.7) is written in terms of the deformation gradient as

det 0=J ≠F (2.11)

Therefore, the inverse tensor 1−F exists, which is derived from ( ))(/ =/∂ ∂∂ ∂x IXX x as

1 1,= , ( ) = ( )A

A A Ai i iA Ai ii

X X tx

− − ∂∂ ⊗ ⊗ ≡ ⊗∂ ∂XF F e e e e e ex

(2.12)

Here, consider the unit cubic cell (a parallelepiped) whose sides at initial con-

figuration are given by the triad Ae . It deforms to the cell whose sides are formed

by the triad Ae while Ae are not unit vectors in general but are given from Eq.

(2.10) as follows:

= ( = = )i iA AA iA AF • •e Fe e Fe e e (2.13)

Applying the polar decomposition in 1.10 to the deformation gradient F , we have

= =F RU VR ( = = )rAiRiA RA irF R U V R (2.14)

where

1/2 2= ( ) ( = ) ( = )T T TU F F U F F U U (2.15)

1/ 2 2 )(= ( ) ( = ) =T T TV FF V FF V V (2.16)

1 1/ 2 1 1/ 2= = ( ) , = = ( )T T− − − −R FU F F F R V F FF F (det = 1)R (2.17)

= , =T TV RUR U R VR (2.18)

Since Eq. (2.18) holds and R is the orthogonal tensor, U and V are the

similar tensors as discussed in 1.5.2. Therefore, they possesses the principal values

( = 1, 2, 3)αλ α . Denoting the bases for the principal directions of U and V

by ( )tαN and ( ) ( )tαn , respectively, it can be written as

3 3

1 1= =

( ) ( ) ( ) ( )= ( ) ( ) = ( ) ( ),t t t tα α α αα α

α αλ λ⊗ ⊗∑ ∑U N N V n n (2.19)

where the relation of ( ) ( )tαN and ( ) ( )tαn is given from Eq. (1.189) as follows:

( ) ( ) ( ) ( )( ) ( ) ( ), ( ) ( ) ( )= = Tt t t t t tα α α αn R N N R n (2.20)


with ( ) ( )( ) = ( ) ( )t t tα α⊗R n N (2.21)

( ) ( )tαN and ( ) ( )tαn are called the Lagrangian triad and the Eulerian triad,

respectively.

Substituting Eqs. (2.19) and (2.21) into Eq. (2.14), F is described by 3

1=

( ) ( )( ) = ( ) ( ) ( )t t t tα αα

αλ ⊗∑F n N (2.22)

Let the mechanical meanings of ,U V and R be examined below. The variation of infinitesimal line-element is given by the polar decomposition

=F RU noting ( ) ( )= (no sum)α ααλUN N as follows: 3 3

1 1= =

( ) ( )= = = =d d d dX dXα αα αα

α αλ∑ ∑x F X RU X RU N R N (2.23)

Equation (2.23) means that the infinitesimal line-elements ( )dX αα N (no sum)in

the principal directions ( )αN are first stretched αλ times to ( )dX αααλ N

(no sum) and then undergoes the rotation R as shown in Fig. 2.1. On the other hand, the change of the infinitesimal line-element by the polar

decomposition VR is described as 3 3 3

1 1 1= = =

( ) ( ) ( )= = = =d d dX dX dXα α α

α α αα α ααλ∑ ∑ ∑x VR X R N n nV V

(2.24)

Equation (2.24) means that the infinitesimal line-elements ( )dX αα N

(no sum)in the principal directions ( )αN first becomes ( )dX αα n (no sum) by

rotation R and then are stretched αλ times to ( )dX αααλ n (no sum), noting ( ) ( )=α α

αλVn n (see Fig. 2.1). As described above, ,U V designates the deformation and R the rotation.

αλ is called the principal stretch and U and V are called the right and left stretch tensor, respectively.

Letting LR and ER designate the rotations of the Lagrangian triad ( ) αN

and the Eulerian triad ( ) αn , respectively, from the fixed base αe

( = 1,2,3)α , they are given by 3 3

1 1= =

( ) ( ), EL α ααα

α α≡ ≡⊗ ⊗∑ ∑R N e R n e (2.25)

where the following relations hold.


dXUdX

dX1

λ 1 dX1

X1

X2

0

0

dX2

λ2 dX2

RdX dx=FdX=RUdX =VRdX

R

N(1)N(2)

n(1)n(2)

n(2)

n(1)

N(1)

N(2)

e1

e2

LR

ERR

N(2)

n(1)

N(1)

N(2)

e1

e2

LR

R

λ2λ1

N(1)

U

N(2)

e1

e2

n(2)ER

n(1)

n(2)

R

U

R

N(1) λ1

λ 2

1

1

1

1

dXUdX

dX1dX1

λ 1 dX1dX1

X1X1

X2X2

0

0

dX2dX2

λ2 dX2λ2 dX2dX2

RdX dx=FdX=RUdX =VRdX

R

N(1)N(1)(1)

N(2)N(2)(2)

n(1)

n(1)n(2)

n(2)

n(2)

n(1)

n(1)

N(1)N(1)(1)

N(2)N(2)(2)

e1e1

e2

LR

ERR

N(2)N(2)(2)

n(1)

n(1)

N(1)N(1)(1)

N(2)N(2)(2)

e1e1

e2e2

LR

R

λ2λ2λ1λ1

N(1)N(1)(1)

U

N(2)N(2)(2)

e1e1

e2e2

n(2)

n(2)ER

n(1)

n(1)

n(2)

n(2)

R

U

R

N(1)N(1)(1) λ1λ1

λ 2λ 2

1

1

1

1

Fig. 2.1 Polar decomposition of the deformation gradient


( ) ( ), = = ELα αα αN R e n R e (2.26)

=E LR R R (2.27)

Considering the particle P and the adjacent particles 'P and ''P , we designate their position vectors before and after the deformation by , + , +d δX X X X X

and , + , +d δx x x x x , respectively. Then, noting (1.100), one has

= = =Td d d dδ δ δ δ• • ••x x F X F X F F X X C X X (2.28)

1 1 1= = Td d dδ δ δ−− − −• • •X X F x F x F F x x

1 1= ( ) =T d dδ δ− −• •FF x x b x x (2.29)

where

2 == ( = ),T TAk kBAB F F≡ CC F F U C C (2.30)

and

2 ( = )= (= ) , =T T Tij iA jAFb F≡b b bFF V RCR (2.31)

are the metric tensors describing how the scalar product of two line-element vectors passing through a material point is influenced by a deformation. They are called the right and left Cauchy-Green deformation tensor. In accordance with Eq. (2.19) they are described by

32

1=

( ) ( )=α

α ααλ ⊗C N N , 2

1=

3( )( )=

α

αααλ ⊗nb n , (2.32)

The principal values aλ are obtained by the solutions of the characteristic equation

3 2 = 0c c cλ λ λ− − +Ⅰ Ⅱ Ⅲ (2.33)

where

33 22 11 11 tr tr tr trtr , (tr tr ), 26 32c c c − +≡ ≡ − ≡ CC C CC C CⅠ Ⅱ Ⅲ (2.34)

The principal values and directions are calculated by the method described in 1.5.2.

Using the relative description (2.3),the relative deformation gradient on the reference configuration ( )τx is defined as

( ),

( ), =( )

tt τ

τττ τ

∂∂χ x

F xx

(2.35)


which is hereinafter shown in the abbreviated notation as follows:

( ) ( ), t tτ τ τ≡F F x (2.36)

( )tτF is related to the deformation gradient 0( )( ( ))t t≡F F as

( ) ( ) ( )( ) = = = ( ) ( )( )

t tt tττ ττ

∂ ∂ ∂∂ ∂∂x x xF F FX Xx

(2.37)

and is further expressed in the polar decomposition as

( ) = ( ) ( ) = ( ) ( )t t t t tτ τ τ τ τF R U V R (2.38)

where ( ), ( )t tτ τC b defined by 2

2

( ) = ( ( )) ( ) = ( )

( ) = ( ) ( ( )) = ( )

T

T

t t t t

t t t t

τ τ τ τ

τ τ τ τ

C F U

Fb

F

F V (2.39)

are called the relative right, left Cauchy-Green tensors.

2.2 Strain Tensor

Taking the subtraction of Eqs. (2.28) and (2.29), one has

= 2 ( = 2 )AAB Bd d d E dX Xδ δ δ δ• • •−x x X X E X X

= 2 ( = 2 )ij i jd e dx xδ δ•e x x (2.40)

where

1 1

1 1

1 1 1( ) = ( ) =2 2 2

1 1( ) =2 2

1 1 1( ) = ( ) =2 2 2

1 1) = ( )2 2

( ) ( )

( )

( )

TT

k kAk kBAB AB AB

A B

TT

K Kjij ij ijKK

jii

xxE F F

X X

X Xexx

δ δ

δ δ

− −

− −

−

∂ ∂≡ − − −∂ ∂∂∂≡ − −

∂ ∂

∂ ∂≡ − − −∂ ∂

∂ ∂≡ − ( ) −∂∂

x xC I F F I IEX X

X Xe I I F F Ix x

F (F

b

(2.41)

Applying the quotient law described in 1.3.2 to Eq. (2.40), it is confirmed that E and e are the second-order tensors.

2.2 Strain Tensor 65

If a deformation is not induced, the triangle PP P' '' has the same shape as in the initial state and thus the scalar quantities in the left-hand side in Eq. (2.40) are zero so that E and e are independent of rigid-body rotation. Conversely, if

,≠ ≠E 0 e 0 , the scalar quantities in the left-hand side in Eq. (2.40) are not zero so that the shape of the triangle is not same as in the initial state. Therefore, E and e are the quantities describing the deformation independent of rigid-body rotation and called the Green (or Lagrangian) strain tensor and the Almansi (or Eulerian) strain tensor, respectively. Using the displacement vector

= = i iu−u x X e (2.42)

they are expressed by

11= , =2 2

11= , =2 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

T T A K KBAB

B BAA

T T ji k kij

jj ii

u u u uE

X X X X

uu u ue

xx x x

⎫∂ ∂ ∂ ∂∂ ∂ ∂ ∂+ + + + ⎪∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎪⎬∂∂ ∂ ∂∂ ∂ ∂ ∂ ⎪+ − + − ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎪⎭

u u u uEX X X X

u u u ue x x x x

(2.43) The following relation exists between them.

= , =TjAB ijiA BE F F eeE F F (2.44)

Now, limiting to the infinitesimal deformation and rotation, d d≅x X holds

and thus the third term becomes the second-order infinitesimal quantity in Eq.

(2.43) so that the difference between E and e can be ignored. Then, denoting E

or e by ε, it can be written that

1 1= ,2 2

( ) ( ) ( )TS A BAB

AB

u uXX

ε⎧ ⎫ ∂ ∂∂ ∂ ∂≡ + ≡ +⎨ ⎬∂ ∂ ∂ ∂∂⎭⎩u u uε X X X

(2.45)

or

1 1= ,2 2

( ) ( ) ( )jTS iij

j i

uux x

ε⎧ ∂⎫ ∂∂ ∂ ∂≡ + ≡ +⎨ ⎬∂ ∂ ∂ ∂∂⎭⎩u u uε x x x

(2.46)

ε is called the infinitesimal strain tensor. ε is not an exact measure to describe a

deformation for finite deformation and rotation since it does not describe the rela-

tion of the infinitesimal line-elements dX and dx directly, and thus it possesses

various impertinence as will be described in 2.3.


Consider the same infinitesimal line-element for PP' and PP'' , i.e.

= , =d dδ δX X x x and denoting its direction vector by (|| ||= 1)N N , it holds

from Eq. (2.40) that 2 2 2|| || || || || ||2=d d d•− N Nx X E X (2.47)

Selecting the 1X -axis for this line-element, 321( , , ) = (1, 0, 0)N N N holds and thus we have

2 2

11 2

|| || || |||| || || || || ||1 1 1= =2 2|| || || || || ||( ))(d dd d dE

d d d− − +x XX x xX X X

(2.48)

from which the ratio of the line-elements before and after the deformation is given by

11|| ||

1 2=|| ||d

Ed

+xX

(2.49)

In the case that the variation of the length of the line-element is infinitesimal || || || || )( / 1d d ≅x X , Eq. (2.48) becomes

11 11|| |||| ||

= || ||ddE

dε −≅ Xx

X (2.50)

Then, 11E becomes to describe the rate of elongation coinciding with the normal

strain in the infinitesimal strain ε. On the other hand, denoting the direction vectors of the two distinct infinitesimal

line-element PP' and PP'' as 'N and ''N , respectively, and the angles con-tained by them as θ , it holds from Eq. (2.40) that

0 || |||| ||cos|| |||| || 2|| || || ||cos = ddd ' '' δθθ δδ •− X XN NEX Xx x (2.51)

i.e.

0|| || || || cos 2 2cos = =|| || || || ij i j

d N NEd

' '' ' ''δ θθδ•−x x N NE

X X (2.52)

where 0θ is the initial value of θ . Here, assuming that the infinitesimal

line-elements PP' and PP'' were mutually perpendicular before a deformation

( 0 = /2θ π ), and taking the 1X - and 2X -axes to their directions, i.e.

1 2 3 1 2 3( , , ) = (1, 0, 0), ( , , ) = (0, 1, 0)N N N N N N' ' ' '' '' '' , it holds that

12|| || || || || || || ||

= cos = sin( /2 )|| || || || || || || ||

)( d dEd d

δ δθ π θδ δ1 1 −2 2

x x x xX X X X

(2.53)


which describes the half of decrease in the sine of angle contained by the

two line-elements which were mutually perpendicular before deformation when

the changes in lengths of these line-elements are infinitesimal

( || || || || || |||| ||/ 1, / 1d d δδ≅ ≅x x XX ). Furthermore, when the change in the angle

formed by these line-elements is infinitesimal ( /2θ π≅ ), one has

12 12 tan( /2 ) ( /2 ) / 2=E ε π θ π θ≅ (1/2) − ≅ − (2.54)

Consequently, 12E describes half of the decrease in the angle contained by the two line-elements which were perpendicular before deformation.

In addition to the Lagrangian and Eulerian strain tensors defined above, we can

define various strain tensors in terms of U or V , fulfilling the condition that they

are zero when = =U V I as follows (Seth, 1964; Hill, 1968):

1 1( ), ( ) for 0

lnln , for = 0

m m mm m

m

⎫− − ≠ ⎪⎬⎪⎭

U I V I

U V (2.55)

where m is the integer (positive or negative). The Green strain tensor is obtained

by choosing = 2m in the equation of U and the Almansi strain tensor is obtained

by choosing = 2m − in the equation of V in Eq. (2.55). The Biot strain tensor

−U I (Biot, 1965) is given by = 1m . The strain tensors in Eq. (2.55) are coaxial

with U or V and their principal values are given by

1 ( 1) for 0( ) =

ln for = 0

m mmfm

αα

α

λλ

λ

⎧ − ≠⎪⎨⎪⎩

(2.56)

The function ( )f αλ fulfills

(1) = 0, (1) = 1f f ' (2.57)

and ( ) 0f s' > (2.58)

where s is an arbitrary positive scalar quantity. The function ( )f αλ is shown in Fig. 2.2.

(Note) it holds for =m 0 Eq. (2.56) that

0 0 0

exp( ln ) exp( ln ) ln11lim ( 1) = lim = lim = ln1m

m m m

m mm m

α α αα α

λ λ λλ λ→ → →

−−


( )f αλ

αλ

= 0m

= 0m

=1m

= 2m

= 1m −

= 1m −

1 2

1

1−

0.5

0.5−

1.5

0 0.5

= 2m

( )f αλ

αλ

= 0m

= 0m

=1m

= 2m

= 1m −

= 1m −

1 2

1

1−

0.5

0.5−

1.5

0 0.5

= 2m

Fig. 2.2 Function of general principal strain measures

Further, adopt the second-order tensor function ( )f U which is coaxial with the

right stretch tensor U and has the principal values ( )f αλ . Therefore, we can

define the general strain tensor in the spectral representation as follows: 3

1=

( ) ( )( ) = ( )f α αα

αλ ⊗∑f U N N (2.59)

In addition, for the left stretch tensor V , we can define the following strain tensor.

3 3

1 1= =

( ) ( ) ( ) ( )( ) = ( ) = ( ) = ( ) Tf fα α α αα α

α αλ λ⊗ ⊗∑ ∑f V n n RN RN Rf U R

(2.60)

The relation ( ) = =ij ir irjs s s jsr rT u T u T u u T⊗Tu Tv = ( ( ) )T ij⊗T u v T leading to ⊗Tu Tv ( )= T⊗T u v T is used in the derivation of the last side in Eq. (2.60).


In the particular case of = 0m , noting 0, 0U Vα α≥ ≥ , ( )f U and ( )f V defined by the following equation are called the Hencky strain tensor.

3( ) ( )

1=

3( ) ( )

1=

1Right - strain tensor : Hencky = ln = ln2

1Left - strain tensor : Hencky = ln = ln2

l

r α αα

α

α αα

α

λ

λ

⊗ ≡

⊗ ≡

N N U C

n bn V

(2.61) where

1 1ln = ln ( = , = )= ln = ln2 2

U C b U V CbV αα α α α α αα α≡ (2.62)

which are mutually related as follows.

= r lTR Rλλ (2.63)

When the principal directions of C and b are fixed, choosing the coordinate axes to these directions = 0 ( )iAF i A≠ ), the following equations hold.

= ln (no sum) xX

αα

αλ ∂

∂ (2.64)

= = (no sum)= = ( ) ( )/x x xDxX X

α α αα α ααα α α

λ λ•• •

•∂ ∂ ∂∂ ∂ ∂ (2.65)

where αλ in Eq. (2.64) is the logarithmic strain and (no sum)Dαα is the normal component in the strain rate tensor defined in the next section. It holds from Eq. (2.64) that

3 3 3

I II III= =1

3 3 3

I II III=1 =1

1 1

1

1tr = = ln ln = ln( )=2

1tr = = ln ln = ln( )=2l

r C U U U U

V V V Vb

α α αα=α α

αα αα=α α

λ

λ (2.66)

which is identical to the logarithmic volumetric strain, i.e.

3

=1

ln = ln = ln =tr tr= = vr l x vJ

X Vααα

ε∂∂∑λ λ (2.67)


2.3 Strain Rate and Spin Tensors

The idealized deformation process in which the deformation is uniquely determined by the state of stress independent of the loading path is called the elastic deforma-tion process. To describe it, it is required to introduce the strain tensor describing the deformation from the initial state and relate it to the stress. Here, since the su-perposition rule does not hold in the strain tensor, the null stress state is chosen as the reference state of strain.

On the other hand, the deformation is not determined uniquely by the state of stress depending on the loading path and thus it cannot be related to the stress in the irreversible deformation process, e.g. the viscoelastic, the plastic and the vis-coplastic loading processes. Therefore, it is obligatory to relate the infinitesimal changes of stress and deformation and to integrate them along the loading path in order to know the current states of stress and deformation.

Here, introduce the velocity gradient tensor defined as

, jij ij

ivL vx

∂∂≡ ≡ ≡ ∂∂ ∂vLx

(2.68)

Noting = / = / ( = )d d•• •∂ ∂ ∂ ∂X v X v XFxF and the chain rule of derivative,

Eq. (2.68) can be rewritten as

1= = ( = ), = Aij

jA

i XvL

X x• ∂∂∂ ∂

∂ ∂ ∂ ∂-v XL LF FF F

X x• (2.69)

Now, differentiating Eq. (2.36) and choosing the current state as the reference

state, i.e. ( ) =t tF I ( = tτ ), L can be expressed in the updated Lagrangian

description as follows:

= ( )t t•

L F (2.70)

Further, taking the time-derivative of Eq. (2.38) and noting

( ) = ( ) = ( ) =t t tt t tR U V I , it follows that

( ) = ( ) ( ) = ( )t t tt tt t t t+ + RUF R V• •• •• (2.71)

Decomposing L into the symmetric and the skew-symmetric parts and noting Eqs. (2.69)-(2.71), it is obtained that

= +L D W (2.72)

2.3 Strain Rate and Spin Tensors 71

where

)(

1 1= ( ) = = ( ) = ( )22

1= 2

( )

( )

TT

jij iij ij

j i

st t

s

t t

vvD Lx x

v

• • ⎫∂ ∂≡ + + ⎪∂ ∂ ⎪⎬∂∂ ⎪≡ ≡ ∂ +∂ ∂ ⎪⎭

v vD L L L U Vx x

(2.73)

[ ]

1 1= ( ) = = ( )22

1= 2

( )

( )

TA T

jA ij iij

j i

t

ij

t

vvW Lx x

v

• ⎫∂ ∂≡ − − ⎪∂ ∂ ⎪⎬∂∂ ⎪≡ ≡ ∂ −∂ ∂ ⎪⎭

v vW L L L Rx x

(2.74)

where D is called the strain rate tensor or the deformation rate tensor or stretching

and W is called the (continuum) rotation rate tensor or (continuum) spin tensor.

Here, note that D is not a time-derivative of any strain tensor but is defined in-

dependently as the rate variable although it is called the strain rate tensor.

Substituting Eqs. (2.14) and (2.69) into Eqs. (2.73) and (2.74), D and W are

described by , U R as follows:

1 1 1

1 1 T

1 1 1

1 1

1 1= ( ) = [( ) ( ) ( ) ( ) ]2 2

1 = ( )2

1 1= ( ) = [( ) ( ) ( ) ( ) ]2 2

1 = 2

TT T

TT T

T T

• • • •

• •

• • • •

• • •

− − − −

− −

− − − −

− −

⎫+ + ⎪⎪⎪+ ⎪⎪⎬⎪− −⎪⎪⎪+ −⎪⎭

D F F F F RU RU RU RU

R U U U U R

W F F F F RU RU RU RU

R R R U U U U R

(2.75) Consequently, we obtain

1= ( )2

1= ( )2

T T

TR T

• •

• •

⎫+ ⎪⎪

⎬⎪−+ ⎪⎭

D R U U R

W Ω R U U R

(2.76)

where 1−

• •≡U U U (2.77)

R T•≡Ω R R (2.78)

72 2 Motion and Strain (Rate) RΩ is called the relative (or polar) spin tensor. Further, D and W are described

by , V R as follows:

1

1 1 1

1

1 1 1 1

1= [( ) ( ) ( ) ( ) ]2

1 1 = ( ) ( )2 2

1= [( ) ( ) ( ) ( ) ]2

1 1 = ( ) ( )2 2

T T

TTT T

T T

T T

−

• •− −

− −−

− −

− − − −

• • • •

• •

• • • •

⎫+ ⎪⎪⎪−+ + ⎪⎪⎬⎪− ⎪⎪⎪+ + −⎪⎭

D VR VR VR VR

V V V V V R R V V R R V

W VR VR VR VR

V R R V V R R V V V V V

(2.79)

and thus

1 1= ( ) ( )2 2

1 1= ( ) ( )2 2

T TR R

T TR R

• •

• •

⎫++ + ⎪⎪

⎬⎪− + − ⎪⎭

D V V Ω Ω

W Ω Ω V V

(2.80)

where

1−• •≡V V V (2.81)

1R R −≡Ω VΩ V (2.82) It holds that

( ) ( )= ( ) ( )L TL L Tα βα β

• •≡ ⊗ ⊗Ω R R N e N e

( )( ) ( ) ( )( ) ( ) = ( )αα α βα β• •⊗ + ⊗ ⊗eN e N e N

( ) ( ) = α α•

⊗N N (2.83)

and thus one has ( ) ( )= Lα α• Ω NN (2.84)

noting ( ) =α•e 0 since ( ) αe is the fixed base.

Therefore, LΩ describes the spin of the Lagrangian triad ( ) αN of the right

stretch tensor U and is called the Lagrangian spin tensor. On the other hand, it holds that

( ) ( )( ) ( )= ( ) ( )TEE E Tα βα β• •≡ ⊗ ⊗e eΩ R R n n

( ) ( ) = α α• ⊗n n (2.85)


and thus one has

( ) ( )= Eα α•n nΩ (2.86)

Therefore, EΩ describes the spin of the Eulerian triad ( ) αn of the right

stretch tensor V and is called the Eulerian spin tensor. Here, it holds that

( ) ( )( ) ( )= ( ) ( )T Tα βα β• •⊗ ⊗R R n N n N

( )( ) ( ) ( )( ) ( ) = ( )αα α βα β•• ⊗ + ⊗ ⊗n N n N N n

( )( ) ( ) ( ) ( ) ( )( ) ( ) = ( ) γγα αα βα β•• •⊗ + ⊗ ⊗nn n N N N N n

( )( ) ( ) ( ) ( ) ( )( ) ( ) = γγα αα βα β•• ⊗ + ⊗ ⊗ ⊗nn n N N N N n

( )( ) ( ) ( )( ) ( ) ( ) ( ) =γ γα βα α βα •• − ⊗ ⊗ ⊗⊗n n N N N N nn

and thus the following relations hold.

( ), , = = =T T T E RL EE L LR R− −+ Ω ΩR R R R R RΩ Ω ΩΩ Ω Ω Ω

(2.87)

In the rigid-body rotation ( = , = =F R U V I ), it holds from Eqs. (2.73), (2.76), (2.82) and (2.87) that

= , = = =E LRL 0ΩW Ω Ω (2.88)

In what follows consider the physical meanings of D and W . The relative velocity of the particle points P and P' , the position vectors of

which are x and d+x x , respectively, is given by

=d dv L x (2.89)

from Eq. (2.68) and it is additively decomposed as

= drd d d+v v v (2.90)

where

dd d≡v D x (2.91)

rd d≡v W x (2.92)


Here, denoting ddv in the infinitesimal line-element (= ;i i id dxx e

no sum) in the direction of the orthogonal bases ie as d

idv and substituting Eq.

(2.91) into = = (no sum)/ ij j ji i iD d dx• •e e x eD D based on Eq. (1.88), one

has

= (no sum)d

jiji

idD dx •v e (2.93)

while didv is not parallel to the line-element idx in general. Therefore, ijD is the

orthogonal projection of (no sum)/diid dxv onto the je -direction. Eventually,

( (no sum)= )dii i iiD dx d • ev describes the relative velocity in the direction of

the line-element idx and (= ) ( no sum),d jij i iD dx d ji• ≠ev describes the

peripheral velocity around ( , )k k i j≠e , as shown in Fig. 2.3. Adopting the bases

( , , )e e eⅠ ⅢⅡ which are the principal directions of D , the direction of Iddv coin-

cides with Ie , i.e.

IIdd ∝v e (2.94)

On the other hand, denoting the axial vector described in Eq. (1.121) for the

skew-symmetric tensor W by w , it holds that

1= , =2 rsii ij ijrrs rw W W wε ε− − (2.95)

and thus Eq. (2.92) is rewritten from Eq. (1.124) as

= , ( = ) =isris s s irs r srri rd d dv dx w dx w dxW ε ε× −v w x (2.96)

11 1D dx1dx

12 1D dx

1x

2x

P

P'

1ddv

1rdv

1dv

12w D+Ⅲ

1e

2e 11 1D dx1dx

12 1D dx

1x

2x

P

P'

1ddv

1rdv

1dv

12w D+Ⅲ

1e

2e

Fig. 2.3 Extension and rotation of the line-element


Ix

xⅡ

xⅢ

I ID dx

D dxⅡ Ⅱ

D dxⅢ Ⅲ

w

Ix

xⅡ

xⅢ

I ID dx

D dxⅡ Ⅱ

D dxⅢ Ⅲ

w

Fig. 2.4 Deformation and rotation for principal directions of strain rate

Therefore, the arbitrary line-element dx rotates in the peripheral velocity rdv

and angular velocity w , called often the spin vector, whereas 2w is called the

vorticity. The total angular velocity of arbitrary line-element due to D and W is given

by the equation = ( )jki iw D i j k iω + ≠ ≠ ≠ (2.97)

w describes the angular velocity of line-element in the principal directions of D , i.e. the mean angular velocity since the second term in Eq. (2.97) is zero in these

directions. The movement of the infinitesimal line-element 1dx around one of the

principal directions, eⅢ , of D is shown in Fig. 2.3. The state of deformation and

rotation is shown in Fig. 2.4 for the the principal directions of D .

The rate of the scalar product of the vectors dx and δ x of the infinitesimal

elements connecting the three particle points P, P , P' '' with the position vectors

, ,d δ+ +x x x x x , respectively, is given noting Eq. (1.100) as follows:

( ) =d d dδ δ δ•• • • vx x v x + x

= =[ ( ) ]Td d dδ δ δ•• •∂ ∂ ∂ ∂+∂ ∂∂ ∂

v v v vx x + x x x xx xx x

d δ•= 2D x x (2.98)


If the vicinity of the particle P undergoes the rigid-body rotation, the quantity in Eq. (2.98) for an arbitrary scalar quantity d δ•x x is zero and thus =D 0 has to hold. Inversely, if =D 0 , the quantity in Eq. (2.98) for the scalar quantity d δ•x x of arbitrary line-element vectors becomes zero and thus it can be stated that the vicinity of the particle P does not undergo a deformation. Then, =D 0 is the necessary and the sufficient conditions for the situation that a deformation is not induced, allowing only a rigid-body rotation.

Denoting the lengths of the line-elements PP' and PP'' as dS and Sδ and the angle contained by them as θ , it holds that

( ) = ( cos )d dS Sδ δ θ• • •x x

( ) ( )

cos sin[ ]dS SdS S

dS S

δ θ θ θ δδ

• • •= + − (2.99)

Further, denoting the unit vectors in the directions of the line-elements PP' and

PP'' as μ and ν , respectively, and noting =d dSx μ , δ x = Sδν , it holds

from Eqs. (2.98) and (2.99) that

( ) ( )cos sin = 2 ( = 2 ) srs r

dS SD μ ν

dS Sδ θ θ θδ

• • ••+ − Dμ ν (2.100)

If the particles P' and P'' mutually coincide ( 0)θ = , it holds from Eq.

(2.100) that ( )

=dSdS

••μ μD (2.101)

The left-hand side of Eq. (2.101) designates the rate of extension of the line-element. Therefore, the rate of extension is given by the normal component of D in the relevant direction, noting Eq. (1.88).

On the other hand, choosing the line-element PP'' to be perpendicular to the line-element PP' ( = /2)θ π , it holds that

= 2 ( = 0)θ•

• •− μ μD ν ν (2.102)

The left-hand side of Eq. (2.102) designates the decreasing rate of the angle con-tained by the two line-elements mutually perpendicular instantaneously and is called the shear strain rate.

Next, the relations of the rate •E of Green strain tensor E and the rate

•e

of the Almansi strain tensor e to the strain rate tensor D are formulated below. The material-time derivative of Eq. (2.40) is given by

( ) = 2d dδ δ••••x x E x X (2.103)


It is obtained from Eqs. (2.98) and (2.103) that

=d dδ δ•

• •D x x E XX (2.104)

Furthermore, substituting Eq. (2.10) into Eq. (2.104) and noting Eq. (1.101), we have the relation of Green strain tensor to the strain rate tensor as follows:

= T•E F DF , 1= T− −•

D F EF (2.105) which is obtained also from

1 1= ( ) = ( )2 2

TT T• • ••− +E F F I F F F F

11 ) ( = ( ) 2

TT TT • •− −+F F F F F F F F

1 1 = ( ) = ( )2 2

T T TT T+ +F L F F LF F L L F

= TF DF (2.106)

Next, the time-differentiation of Eq. (2.41)2 leads to

1 11= ( ) ( ) 2

T T− − − −• • •− +e F F F F (2.107)

where

1 11( ) = = =

( ) ( ) ( )( )t

t t

− −− •

∂ ∂ ∂∂∂ ∂ ∂∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂+ + + −∂ ∂ ∂ ∂∂ ∂ ∂ ∂

X X XX vF F x x x vXv vF

x x x xx x

= ( )t

∂ ∂ ∂ ∂∂+ −∂ ∂∂ ∂ ∂vX XX v

x xx x (2.108)

The inside of the bracket ( ) in the last side of Eq. (2.108) is the material-time derivative of the initial configuration X and thus it is zero. Then, it holds that

1 1( ) =− −• −F F L (2.109)


1 1 11= ( )

2T T• − − − −+e F L F F F L

1 11 1 1 1=

2 2 2 2 ( ) ( )T T T− − − −− − + − −L I I F F I I F F L

1 1=2 2

( ) ( )T − + −L I e I e L (2.110)


from which one has the relation of the rate of the Almansi strain tensor to the strain rate tensor:

T•= − −e D L e eL (2.111)

Equation (2.111) is rewritten as

1= ( ) ( ) ( ) ( )

2T T T T•

− + − − + + −e D L L L L e e L L L L1−2

Consequently, we obtain

=•e We + eW D De eD− − − (2.112)

where

∇ •≡e e We + eW− (2.113)

is called the Jaumann rate of Almansi strain tensor, while the Jaumann rate will be explained in 4.3.

In the initial state ( = , = =F I E e 0) , it holds that = = =• ∇•

E e e D and

thus all the strain rates mutually coincide. In what follows, let D and dtD be designated as

•ε and dε , respectively. If the direction of the material line-element always coincides with the xⅠ-axis,

the principal strain rate in this direction is given by

= = =( ) 1

uuu X

x X u uX

ε

•

•••

∂∂∂ ∂

∂ ∂ + ∂+ ∂

Ⅰ

Ⅰ ⅠⅠ

Ⅰ

Ⅰ Ⅰ Ⅰ Ⅰ

Ⅰ

(2.114)

The time-integration of Eq. (2.114) leads to

= 1n 1+ = 1n = 1n(1+ )( )u xX X

ε ε∂ ∂∂ ∂Ⅰ Ⅰ

Ⅰ Ⅰ

ⅠⅠ

(2.115)

Therefore, the integration of principal strain rate, εⅠ, differs from the infini-

tesimal (nominal) strain εⅠ

. Putting 0 0, x , X l l u l l∂ → ∂ → ∂ → −Ⅰ Ⅰ Ⅰ , where 0l and l are the lengths of the line-element in in the initial and the current states,

respectively, it holds that


0 0

0 0 0= 1n = 1n 1+ , =( )l l l l l

l l lε ε− −Ⅰ Ⅰ

(2.116)

As described above, the material-time integration of the strain rate ε•Ⅰ leads

to the logarithmic (or natural) strain. On the other hand, the infinitesimal

strain coincides with the nominal strain. It results that = 0 : = 1 l ε −Ⅰ 0and =2 : = 1l l εⅠ in the nominal strain, whereas = 0 : = andl ε −∞Ⅰ

0e=2 : = 1n 2 ( 0.693)l l ε ≅Ⅰ in the logarithmic strain. Then, the nominal strain

would be impertinent to use for constitutive equation describing the large deformation.

In the logarithmic strain one obtains

1 2

0 0 1 1=n nl l l l

nl l l l

dl dl dl dl

l l l l−+ + +∫ ∫ ∫ ∫

i.e. 1 2

10 0 11n = 1n 1n 1n( ) ( ) ( ) ( )n n

n

l l l l

l l l l −+ + + (2.117)

Consequently, the superposition rule

0 1 0 2 10 = n nnε ε ε ε −+ + +～～～～

Ⅰ Ⅰ Ⅰ Ⅰ (2.118)

holds, while a bε ～

Ⅰ designates the normal strain in the xⅠ

-direction when the length

of the line-element changed from al to bl , provided that the principal direction of strain rate is fixed.

When the direction of principal strain rates are fixed, it holds that

0 0 0 0 0 01n 1n 1n 1n( ) ( ) ( ) ( )l l l l l l

l l l l l l= + +

Ⅲ Ⅲ

ⅠⅡⅢ Ⅰ Ⅱ Ⅲ

Ⅰ Ⅱ Ⅰ Ⅱ

(2.119)

where , , l l lⅠ Ⅱ Ⅲ are the lengths of line-elements in the directions of three prin-

cipal strain rates and thus the logarithmic volumetric strain νε is given by the sum

of the principal strains:

=

= ln = JJ

vVνε ε∑

Ⅲ

Ⅰ

(2.120)

It will be shown in 2.5 that the time-integration of the volumetric strain rate

trvD ≡ D coincides with the logarithmic volumetric strain νε in general.


On the other hand, in the nominal strain, the above-mentioned convenient properties are not possessed and thus the error increases as the deformation be-comes large.

In what follows, consider the rotation of the material line-element in the plane motion, where the velocity of particles is given by

1 1 2 2 31 2 1 2( , ), ( , ),== = 0v v x x x xv v v (2.121)

Substituting Eq. (2.121) into Eqs. (2.73) and (2.95), one obtains

3

121 32

= 0 ( =1, 2, 3)

= = 0, =

jD j

w w w W

⎫⎪⎬

− ⎪⎭ (2.122)

Denoting 33, we by , weⅢ Ⅲ while 3e is the principal direction of D , the total

angular velocity ω around eⅢ is given from (2.97) as

12= w Dω +Ⅲ (2.123)

By choosing D and ,n sD D for T and ,n sT T described in 1.13, the relation

of the rate of extension and the rate of rotation is shown in Fig. 2.5. It is depicted by

the circle of relative velocity with the radius 2 2

11 22 12[( ) ]/ 2D D D+ + cen-

tering in 11 22(( ) / 2, )D D w+ Ⅲ in the two-dimensional plane ( ,nD ω ).

1211P ( , )D D w' + Ⅲ

12 (= )w Dω +Ⅲ

wⅢ

DⅠDⅡ nD2211

2D D+

P22 12( , )D D w+ Ⅲ

P''

0

1211P ( , )D D w' + Ⅲ

12 (= )w Dω +Ⅲ

wⅢ

DⅠDⅡ nD2211

2D D+ 2211

2D D+

P22 12( , )D D w+ Ⅲ

P''

0

Fig. 2.5 Circle of relative velocity

2.4 Various Simple Deformations 81

The angle contained by the line-elements PP' and PP" changes as the dif-ference of angular velocity depicted by the difference of the ordinates of the inter-secting point of the relative velocity circle and the two straight parallel lines to PP' and PP" stemming from the point P as the pole. In particular, the angle con-tained by the line-elements in the directions of the maximum and the minimum shear strain rates, which are mutually perpendicular momentarily, changes most

quickly by D D−Ⅰ Ⅲ . On the other hand, the angle contained by the line-elements

for the maximum and the minimum strain rate does not change. In Fig. 2.5 it is

confirmed that the mean angular velocity is given by 12 ( )=w W−Ⅲ which is

identical to the angular velocity of the line-elements in the direction of the principal strain rates.

2.4 Various Simple Deformations

Let various strain (rate) and stress (rate) described in the foregoing be shown ex-plicitly and let their relation be described for various simple deformations. These deformations are often observed in experiments for measurement of material properties. Homogeneous and isotropic deformation is assumed therein.

2.4.1 Uniaxial Loading

For a cylindrical specimen with the initial length L and the initial radius R , suppose that the length and the radius changes to l and r (Khan and Huang, 1995). Choosing the XⅠ-axis to the axial direction of cylinder, it holds that

= , = , =x X x X x Xλ λ λⅡ Ⅱ Ⅱ Ⅲ Ⅱ ⅢⅠ Ⅰ Ⅰ (2.124)

where

= , = = rlL R

λ λ λⅡⅠ Ⅲ (2.125)

from which one has

1

1 1 2

1

0 0 0 0

= = = 0 0 , = 0 0 = det =

0 0 0 0

=

, J

λλλ λ λ λ

λ λ

−

− −

−

⎫⎡ ⎤⎡ ⎤⎪⎢ ⎥⎢ ⎥⎪⎢ ⎥⎢ ⎥ ⎬⎢ ⎥⎢ ⎥⎣ ⎦ ⎪⎣ ⎦⎪⎭

F U V F F

R I

ⅠⅠ

Ⅱ Ⅱ Ⅰ Ⅱ

Ⅱ Ⅱ

(2.126)


The aforementioned measures of deformation (rate) are given as

2

2 2

2 2

2

2 1 2 1

0 0 0 0

= , = 0 0 , = = = 0 0

0 0 0 0

T T

λ λλ λ

λ λ

− − − −

−

−

−

bC U F F V F F

(2.127)

2

2

2

1 0 01 1= ( ) = 0 1 02 2

0 0 1

λλ

λ

−

− −

−bE I ,

2

2

2

1

1 0 01 1= ( ) = 0 1 02 2

0 0 1

λλ

λ

−

−

−

−

−

− −

−be I (2.128)

ln 0 0 ln( / ) 0 0

= = 0 ln 0 = 0 ln( / ) 0

0 0 ln( / ) 0 0 ln

r l

l L

r R

r R

λλ

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

λ λⅠ

　 Ⅱ

Ⅱ

　

(2.129)

11

1 1 1

11

0 0 0 0 0 0

= = 0 0 0 0 = 0 0

0 0 0 0 0 0

λλ λλλ λ λλ

λλ λλ

−−

− −

−−

• •

•• •

• •

⎡ ⎤⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

L F F

Ⅰ ⅠⅠⅠ

　Ⅱ ⅡⅡ Ⅱ

ⅡⅡ Ⅱ Ⅱ

　

　

−

1

1

1

0 0

= 0 0 = = =

0 0

lr

ll

r r

r r

−

−

−

•

• ••

•

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

D λ λ (2.130)

=W 0 (2.131)


For the nominal strain E for the description of longitudinal deformation one has

1

2

2

1 0 0

( ) / 0 0 0 1 0ˆ = 0 ( ) / 0 =

0 0 ( ) / 0 0 1

l L L

r R R

r R R

λ

λ

λ

−⎡ ⎤⎢ ⎥⎢ ⎥−⎡ ⎤⎢ ⎥−⎢ ⎥− ⎢⎢ ⎥⎢⎢ ⎥−⎣ ⎦ ⎢ −⎢⎢⎣ ⎦

E = −⎥⎥⎥⎥⎥

V I

(2.132)

/ 0 0

ˆ = 0 / 0

0 0 /

l L

r R

r R

•

• •

•

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

E (2.133)

by which the strain rate tensor is described as

1ˆ ˆ= ( )−•

+D E E Ι (2.134)

Denoting the axial load as F , various stresses (cf. Chapter 3) are shown as follows:

22 220

0 0 0 0 0 0

= 0 0 0 = 0 0 0 = 0 0 0

0 0 0 0 0 0 0 0 0

FFFr ARπ λλ π

⎡ ⎤⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

σⅡⅡ

(2.135)

200

2

0 0 0 0

= = 0 0 0 = 0 0 0

0 0 0 0 0 0

F FAA

J

λλ

λ λ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

τ σ

Ⅰ

Ⅱ

Ⅰ Ⅱ (2.136)

1 200

2 1

1

0 0 /( ) 0 0 / 0 0

0 0 0 0 0 0= = 0 0 =

0 0 0 0 0 0 0 0

F A F A

J

λ λλ λ λ

λ

−

− −

−

1

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎣ ⎦

Π F σⅠ Ⅱ

Ⅰ Ⅱ Ⅱ

Ⅱ

　　

　　

(2.137)


0

0 0

= = 0 0 0

0 0 0

T

FA

Jλ

− −1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

S F Fσ Ⅰ

　　　

　　　

　

(2.138)

2.4.2 Simple Shear

As shown in Fig. 2.6, consider the simple shear in which the shear deformation is induced parallelly to the 1x -axis.

2 3 31 1 2 2= , = , =x X X x X x Xγ+ (2.139)

where 12 (=2 )Dγ is the engineering shear strain. Denoting the shear angle by θ ,

it holds that

2= tan , = secγ γθ θ θ••

(2.140)

It holds in this situation that

1

1 0 1

= = 0 1 0 , = , =det =1

0 0 1

i

j

xJ

X

γ γ−

− 0⎡ ⎤ ⎡ ⎤⎡ ⎤∂ ⎢ ⎥ ⎢ ⎥0 1 0⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎢ ⎥0 0 1⎣ ⎦ ⎣ ⎦

F F F

(2.141)

2 2=X x

x

1X 1x

2Xγθ−

1e

2e

X2 2=X x

x

1X 1x

2Xγθ−

1e

2e

X

Fig. 2.6 Simple shear


where the inverse tensor 1−F is derived using (1.104). The components in the third

line and those in the third row are zero except for unity in the third line and the third

row in all tensors appearing hereinafter for the simple shear deformation. Then, for simplicity, let them be expressed by the matrix with two lines and two rows.

1 1 = , =

0 1

γ γ−1 −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥0 1⎣ ⎦ ⎣ ⎦

F F (2.142)

from which it is obtained that

1 0 0 0 = = =0 10 0 0 0

γγ γ−• •• ⎡ ⎤ ⎡ ⎤−⎡ ⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

L F F (2.143)

0 1 0 1= , =2 21 0 1 0

γ γ• •⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

D W (2.144)

Further, it holds that

2

2

1 ( = = ) =

1 +T

γγ γ⎡ ⎤⎢ ⎥⎣ ⎦

C U F F ，

21 2 1 1 )( = = =T

γ γγ

− − − ⎡ ⎤+ −⎢ ⎥− 1⎣ ⎦

C U F F (2.145)

22 1+

( ) = = = 1

T γ γγ

Fb V F 1 2 1

2

1 ( ) = = =

1+T

γγ γ

− −−

V F Fb− − −

(2.146)

2

1 1 1= ( ) =2 2

γγγ

−E C I , 1

2

01 1= ( ) =2 2

γγ γ

−−

−b−−

,

(2.147)


Next, derive the principal stretches αλ and the eigenvectors ( )αn ( = , α Ⅰ Ⅱ

in the present two dimensional state) of V in Eq. (2.19). The following relation holds from Eq. (1.125).

( ) ( ) ( ) ( ) ( ) ( )= (|| || =|| || = 1, = )βα α α β ααβαλ δ•Vn n n n n n (2.148)

where the principal stretches αλ of V（and U ）must fulfill the following char-

acteristic equation based on Eq. (1.151).

11 1211 22 12 21

21 22

= ( )( )

V VV V V V

V V

αα α

α

λλ λ

λ−

− − −−

11 22 11 22 12 212= ( )V V V V V Vα αλ λ− + + −

2= (tr ) det = 0α αλ λ− + VV (2.149)

where, it holds from Eqs. (2.146) and (2.140) that

2 2 2det = det = 1+ = 1γ γ−V V (2.150)

and

2 2 2t r = 2 = 2 tanγ θ+ +V (2.151)

Here, denoting the principal values of the second-order tensor T in the

two-dimensional state as , T TⅡⅠ with = 0TⅢ , it holds that

2 2 222tr = = ( ) 2 = (tr ) 2detT T T T T T+ + − −T T TⅠ Ⅰ Ⅱ Ⅰ ⅡⅡ (2.152)

in general and thus we have

2 2 2t r = tr 2det = 4 = 4 tanΓ γ θ≡ + + +V V V (2.153)

where it holds that

11= 2 =2

γγ γΓ Γ γΓ−• • • (2.154)

Substituting Eqs. (2.150) and (2.151) into Eq. (2.149), the principal stretches

λ+ and λ− are given by

1= ( ) ( = 1 = 1 0 for = 0 )2

γλ Γ γ λ λ+± −± → +∞, → → ∞

(2.155)


Furthermore, multiplying the identity tensor =

( ) ( )( )= α αα

⊗∑I n nⅢ

Ⅰ to both

sides of the last equation in Eq. (2.149), it holds that

2 (tr ) det ) =− + (V V V V I 0 (2.156)

Substituting Eqs. (2.146)，(2.150)，(2.153) into Eq. (2.156), one has

22 1 01 det ) 1= =tr 0 1 1

)( γ γΓ γ

⎡ ⎤+ ⎡ ⎤+ ( +⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

V V IV

V (2.157)

resulting in 2 22 tan tan2 1 1= , = =

2 tan 2ij ijjiV V

θ θγ γΓ Γγ θ

⎡ ⎤+⎡ ⎤+⎡ ⎤⊗ ⎢ ⎥⎢ ⎥⎣ ⎦

⎣ ⎦ ⎣ ⎦V e e

(2.158)

for which the inverse tensor of V is given noting Eq. (1.104) as

1

2 2

2 2 tan1 1= = 2 tan 2 tan

γ θΓ Γγ γ θ θ

− − ⎡ ⎤−⎡ ⎤⎢ ⎥⎢ ⎥− + − +⎣ ⎦ ⎣ ⎦

V (2.159)

Substituting Eqs. (2.142), (2.159) into (2.17), R is described as follows:

1

2

2 1 2 1 1= = =0 1 2 2

γ γ γΓ Γ γγ γ

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−− + ⎣ ⎦ ⎣ ⎦⎣ ⎦

R V F (2.160)

Here, setting

12 11tantan ( = ) = = ( = 2 tan )/

2 2R RR R

γ θθ γ θ (2.161)

R is also expressed from Eqs. (1.88), (2.21), (2.160) and (2.161) as

( ) ( )cos sin

= )( ) =sin cos

R R

i jR R

α α θ θθ θ

• •⎡ ⎤

⎡ ⎤ ⎢ ⎥⎣ ⎦ −⎣ ⎦R (e n e N

　

(2.162)

It holds from Eq. (2.161) that

22 22

2 22( ) , = =tan= = 1cos 2 ( / 2) 1

R R RRR

γγ θ θ θ θ γΓθ γ

•• •• • •

++

(2.163)


The relative spin in Eq. (2.78) is given from Eqs. (2.162) and (2.163) as follows:

sin cos cos sin=

cos sin sin cos

R R R RRR

R R R R

θ θ θ θθ

θ θ θ θ•⎡ ⎤ ⎡ ⎤− −

⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

Ω

2

0 1 0 12= =1 0 1 0

Rθ γΓ

••⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

(2.164)

Next, denoting the expression of V in the principal direction by pV , it holds from Eq. (1.73) that

=p E TEQ VQV ，

( ) ( )= ( ) ( )p sir jij rsV V• •e en n (2.165)

where

0=

0 p λ

λ+

−

⎡ ⎤⎢ ⎥⎣ ⎦

V (2.166)

1 11 2

2 221

( ) ( )( )

( ) ( )

cos sin= = =

sin cos

E EE i j

E E

θ θ − θ θ

• ••

• •

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎣ ⎦⎣ ⎦

e en neQ n

e en n (2.167)

where Eθ is the rotation angle of the eigenvector 1 2( ) ( ), n n of V from the bases

21, e e measured in a counterclockwise direction.

On the other hand, the components of ER are given from Eq. (2.25) as follows:

3

=

( ) ( )= = = = , =E E E Tj E Ej ji i iij jiR Q

α

α α• • • •⊗∑e e e e eR n e n R Q1

(2.168) The following expressions are obtained by substituting Eq. (2.165) into

Eq. (2.168).

= , =Tp p E EE Eij rs sjriV R V RR RV V (2.169)

= , =p pTE E E E

ij jsrsirV R V RV R RV (2.170)

Since 1 2V V≥ always, choosing the maximum principal value λ+ in the direc-

tion of 1( )n , it holds from Eqs. (2.158), (2.166)，(2.168), (2.170) that

2 0cos sin cos sin2 1 = 0 2 sin cos sin cos

E E E E

E E E E

λθ − θ θ θγ γΓ λγ θ θ − θ θ

+

−

⎡ ⎤ ⎡ ⎤⎡ ⎤+ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦


cos sin cos sin=

sin cos sin cos

E E E E

E E E E

−λ θ λ θ θ θλ θ λ θ − θ θ

+ −

+ −

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

2 2

2 2

cos sin ( )sin cos=

( )sin cos sin cos

E E E E

E E E E

λ θ λ θ λ − λ θ θλ − λ θ θ λ θ λ θ

− −+ +

− −+ +

⎡ ⎤+⎢ ⎥

+⎢ ⎥⎣ ⎦ (2.171)

Substituting Eq. (2.155) into Eq. (2.171), one obtains

22 1 2

γ γΓ γ⎡ ⎤+⎢ ⎥⎣ ⎦

2 2

2 2

1 1( )cos ( )sin sin cos2 2

=1 1 sin cos ( )sin ( )cos2 2

E E E E

E E E E

γ θ γ θ γ θ θΓ Γ

γ θ θ γ θ γ θΓ Γ

⎡ ⎤+ + −⎢ ⎥⎢ ⎥⎢ ⎥+ + −⎢ ⎥⎣ ⎦

cos2 sin2=

sin2 cos2

E E

E E

γ θ γ θΓγ θ γ θΓ

12⎡ ⎤+⎢ ⎥

−⎣ ⎦ (2.172)

from which it holds that

2

11 22

1 2= sin2 sin2 =2

( =) = cos2 =cos2

E E

E EV V

γ γ θ θ ΓΓγ γγ θ θΓ Γ

⎫→ ⎪⎪⎬⎪− →⎪⎭

and thus it can be obtained that

2

( ) ( )/2 /21 1sin = 1 = = = =2

1cos = 1 = =

tan = = 2

( )

( )

E

E

E

ZZ

Z ZZ

ZZ

γ γγ ΓΓ Γθ Γ Γ ΓΓ

θ Γ ΓγΓθ

−+

+−−

−+

⎫− −− ⎪⎪⎪⎪− ⎬⎪⎪−⎪⎪⎭

(2.173)


with

( ) / 2γΖ Γ Γ± ≡ ± (2.174)

where the double signs ± take in the same order. The following relations hold for Ζ ± .

22 2 2 2

2 2 2 2

=

, = =2 2

= , =

1 1 1 11 = =

Ζ Ζ ΓΖ γ γΖΓ Γ

ΖΖγΓ ΓΖ Ζ Ζ ΖγΓΖ Ζ Ζ Ζ

+ −

+ −

+−

+ +− −

+ +− −

⎫⎪−+ ⎪⎪⎪⎬

+ − ⎪⎪

+ , − ⎪⎪⎭

(2.175)

The substitution of Eq. (2.167), (2.168), (2.173) into Eq. (2.168) yields.

1 1

1= = 1 1

E Ζ Ζ Ζ ΖΓ Ζ Ζ

Ζ Ζ

−+ − +

− +

+ −

⎡ ⎤−⎢ ⎥−⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

R (2.176)

From Eqs. (2.154) and (2.173) we have

2 22

( ) /21 1 1= , ==1 12 2cos( () )

EE

E

γ γ γΓ Γθ γγ γθΓ Γθ Γ Γ

•• •• •+

− − −

(2.177)

Substituting Eqs. (2.167), (2.168), (2.177) into Eq. (2.85), it is obtained that

sin cos cos sin=

cos sin sin cos

E E E EE E

E E E E

θ θ θ θθ

θ θ − θ θ•⎡ ⎤ ⎡ ⎤− −

⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

Ω

2

0 1 0 12 1= = = 21 0 1 0

RE γθ Γ• •−⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦Ω (2.178)

The following expression of U is obtained in a similar manner to that used in Eq. (2.157)

2

2

1 1 0det ) 1= =tr 0 1 1 +

)( γΓ γ γ

⎡ ⎤ ⎡ ⎤+ ( +⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

U U IU

U



22

2 cos sin1= = 2+ sin (1 sin ) / cos

γ θ θΓ γ γ θ θ θ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ +⎣ ⎦ ⎣ ⎦

U

(2.179)

In order to obtain the rotation LR of the eigenvector ( )αN of U , denoting the

angle measured in counterclockwise direction from 1 2, e e to 1 2( ) ( ), N N by Lθ ,

one has

= TpL LU R V R (2.180) where, setting

1 21 1

1 222

( ) ( )( )

( ) ( )

cos sin= = =

sin cos

L LL ji

L L

θ − θ θ θ

• ••

• •

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎣ ⎦⎣ ⎦

e N e NeR N

e N e N (2.181)

and substituting Eqs. (2.166), (2.179), (2.181) into Eq. (2.180), we have

2

2 0 cos sin cos sin1 = 0 2 sin cos sin cos

L L L L

L L L L

γ λθ θ θ − θΓ λγ γ − θ θ θ θ

+

−

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2.182)

The substitution of Eq. (2.155) into Eq. (2.182) leads to

2

2 cos2 sin21 = 2 sin2 cos2

L L

L L

γ γ θ γ θΓΓ γ γ γ θ γ θΓ

12⎡ ⎤+ −⎡ ⎤⎢ ⎥⎢ ⎥+ − −⎣ ⎦ ⎣ ⎦

(2.183)

from which one has

2

11 22

2sin2 (= sin2 ) =

( =) = cos2 =cos 2

L E

L LU U

θ θ Γγ γγ θ θΓ Γ

⎫− ⎪⎪⎬⎪− − → −⎪⎭

(2.184)

and

2

2

11sin 1= = =2

1cos 1= = =

2 1tan = = =tan

)(

( )

L

L

LE

ZZ

Z ZZ

ZZ

γθ Γ Γ

θ Γ Γ

θ γΓ θ

+

−

+ −+

+−

⎫+ ⎪

⎪⎪⎪− ⎬⎪⎪⎪− ⎪⎭

(2.185)


The substitution of Eq. (2.185) into Eq. (2.181) reads:

1 1 1 = =

1 1

LZ ZZ Z

Z ZZ Z

Γ+ −− +

−+

− +

⎡ ⎤−⎢ ⎥−⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

R (2.186)

Substituting Eqs. (2.176) and (2.186) into = TLER R R based on Eq. (2.27), it holds that

2

2 1 11 = = 2

Z Z Z Z Z Z

Z Z Z ZZ Z

Ζ ΖΓ Γ ΓΖ Ζ

2 2

2 2

−+ − −+ + + −

− + −+ −++ −

− ⎡ ⎤−⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥− − +⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

R

2 1=2

γΓ γ⎡ ⎤⎢ ⎥−⎣ ⎦

(2.187)

which coincides with Eq. (2.160) obtained by the different approach. Substituting

Eqs. (2.162), (2.167), (2.168), (2.181) into = TLER R R , one obtains

cos sin cos sin cos sin=

sin cos sin cos sin cos

R R L LE E

R R L LE E

θ θ θ − θ θ θθ θ θ θ − θ θ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

　

cos cos sin sin cos sin sin cos =

sin cos cos sin sin sin cos cos

L L L LE E E E

L L L LE E E E

θ θ θ θ θ θ θ θθ θ θ θ θ θ θ θ

⎡ ⎤+ −⎢ ⎥

− +⎣ ⎦

cos( sin( ) =

sin( ) cos(

L LE E

L LE E

θ θ θ θθ θ θ θ

⎡ ⎤− ) − −⎢ ⎥

− − )⎣ ⎦ (2.188)

from which the following relation is obtained.

= L REθ θ θ− (2.189)

The rotations of ( )αn and ( )αN are shown in Fig. 2.7. Furthermore, it is derived from Eqs (2.154) and (2.185) that

2 2 2

( ) / 21 1 1= =1 , 1 =2 2cos( () )

LL

L

γ γ γ−Γ Γθ γ γθ γΓ Γ Γθ Γ

••• • •+ +

(2.190)


( )E tθ( )L tθ

( )R tθ

( ) (0)nⅠ

1e

( ) ( )0nⅡ

2e

( ) (0)NⅠ( ) ( )0NⅡ

( ) ( )∞N Ⅱ

( ) ( )∞nⅡ

( ) ( )∞nⅠ

( ) ( )∞NⅠ ( ) ( )tNⅠ

( ) ( )tN Ⅱ

( ) ( )tn Ⅱ

/4π

( ) ( )tnⅠ

( )E tθ( )L tθ

( )R tθ

( ) (0)nⅠ

1e

( ) ( )0nⅡ

2e

( ) (0)NⅠ( ) ( )0NⅡ

( ) ( )∞N Ⅱ

( ) ( )∞nⅡ

( ) ( )∞nⅠ

( ) ( )∞NⅠ ( ) ( )tNⅠ

( ) ( )tN Ⅱ

( ) ( )tn Ⅱ

/4π

( ) ( )tnⅠ

Fig. 2.7 Rotation of Lagrangian triad ( )αN and Eulerian triad ( )αn

Substituting Eqs. (2.181), (2.190) into Eq. (2.83), we obtains

sin cos cos sin=

cos sin sin cos

L LL LL L

L LL L

θ θ θ θθ

θ θ − θ θ•⎡ ⎤ ⎡ ⎤− −

⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

Ω

2

0 1 0 1 12= = = 21 0 1 0RL

γθ Γ • •− −⎡ ⎤ ⎡ ⎤

−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Ω (2.191)

It can be confirmed easily that the three kinds of spins , , LR EΩ Ω Ω fulfill

the relation (2.87) by Eqs. (2.160), (2.164), (2.178) and (2.191).

Note here that a simple relation between D and =r l• •

λ λ does not hold since

their principal directions does not coincide mutually. Denoting 12=τ σ , various stress tensors are described as follows:

11

22

=

σ ττ σ

⎡ ⎤⎢ ⎥⎣ ⎦

σ (2.192)


11 11 221

22 22

1 = = =

0 1 J

σ τ σ γτ τ γσγτ σ τ σ

− − −− ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦Π F σ

(2.193)

11 221

22

1 0= =

1TJ

σ γτ τ γστ σ γ

− − − −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦

S F Fσ

211 22 22

22 22

=

σ γ σ γτ τ γστ γσ σ

⎡ ⎤− − 2 −⎢ ⎥−⎣ ⎦

(2.194)

2.4.3 Combination of Tension and Distortion

Consider a thin cylindrical specimen subjected to the combination of tension and distortion described by the following equation in the polar coordinate system.

= , = , =zr R Z Zα θ Θ ω λ+ (2.195)

where ( , , Z)R Θ signifies the initial configuration, and α , ω and λ denote

the proportionality factors depending on the deformation, while ω is described by the relative distortion angle φ between both ends as follows:

/ Lω φ≡ (2.196)

L being the length of the specimen. Variables depending on the deformation are given as follows:

= =

R Z

R Z

R Z

r r r

z z z

r rrRR ZF F F

r r rF F FRR Z

F F Fz zz

R ZR

Θ

θ θΘ θ

Θ

Θθ θ θ

Θ

Θ

⎡ ⎤∂ ∂∂⎢ ⎥∂ ∂ ∂⎡ ⎤ ⎢ ⎥

⎢ ⎥ ∂ ∂ ∂⎢ ⎥⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎢ ⎥ ⎢ ⎥⎣ ⎦ ∂ ∂∂⎢ ⎥

∂ ∂⎢ ⎥∂⎣ ⎦

F

0 0

( ) ( ) ( ) = = 0

0 0

R R RRR Z

r r rZ Z Z RRR Z

Z Z ZR ZR

α α α αΘ

Θ ω Θ ω Θ ω α ωαΘλ λ λ

λΘ

⎡ ⎤∂ ∂ ∂ ⎡⎢ ⎥∂ ∂ ∂⎢ ⎥

+ + +∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂⎢ ⎥

∂ ∂ ∂⎢ ⎥∂ ∂⎢ ⎥∂⎣ ⎦

⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2.197)


1

1 0 0

1= 0

10 0

R

α

ωα λ

λ

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

F (2.198)

0 0

= 0 cos sin sins

0 sin cos

R

αα φ αω φ λ φ

λ φ λ φ

⎡ ⎤⎢ ⎥+⎢ ⎥⎢ ⎥⎣ ⎦

V (2.199)

0 0

= 0 cos sin

0 sin sin cosR

αα φ α φα φ αω φ λ φ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥+⎣ ⎦

U (2.200)

where

2 2 2 2cos = , sin =

( ) ( ) ( ) ( )

R

R R

λ α αωφ φα λ αω α λ αω

++ + + +

(2.201)

R being the mean radius ( )R R≅ of the thin cylindrical specimen in the ini-

tial state. Here, R is given by Eq. (2.162).

2

22

2 2 2 2 2

0 0

= = 0

0

T R

R R

αα ωα

ωα λ ω α

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥+⎣ ⎦

C F F (2.202)

2

2

1 12

2

2 2

1 0 0

1= = 0

1 0

T R

R R

α

ωα αλ

ω ωλ λαλ

− −

− +

Fb F− − (2.203)


1

0

= =

0 0

Z

RZ

α ωα

ωααω α λ

λλ

• •

• •• •

•

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L F F− (2.204)

0 0 0 0

= 0 , = 0 2 2

0 0 02 2

Z

R RZ

R R

αα ω

ωα ωαα ωα λ λ

ωα λ ωαλ λ λ

••

•• • •

• ••

⎡ ⎤⎢ ⎥

−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −⎢ ⎥⎣ ⎦

D W

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2.205)

Stresses in various definitions are described by the following equations, desig-

nating the normal stress zzσ and the shear stress rθσ applied to the traverse sec-

tion of the cylinder by σ and τ , respectively.

0 0 0

= 0 0

0

ττ σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

σ (2.206)

1 2

1 0 0 0 0 0

1= = 0 0 0

10 0 0

RJ

α

ωα λ τα λ

τ σλ

−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

Π F σ

2.5 Surface Element, Volume Element and Their Rates 97

22

2 2

0 0 0

= 0

0

R Rα ωτ αλτ α ωσα τ α σ

⎡ ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥⎣ ⎦

(2.207)

2 2

2 2

1 0 00 0 0

1= = 0 0 0

0 10

T R R

R

α

α ωτ αλτ α ωσ αα τ α σ

ωλ λ

−

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦

⎢ ⎥−⎢ ⎥⎣ ⎦

S ΠF

　

2 2 22

22

0 0 0

= 0

0

RRR

R

α ω σ α ωσαωτ ατλ λαα ωσατ λ λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥− 2 + −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

(2.208)

2.5 Surface Element, Volume Element and Their Rates

Presuming that the line-elements , , a cbd d dX X X change to , , a cbd d dx x x by

the deformation, the following relation holds for the volume element before and after the deformation from Eqs. (1.22), (1.42), (1.43), (2.6), (2.9) and (2.11).

= ( ) = = deta cb a cbjijk kidv d d d dx dx dx dε•×x x x x

1 1 1

1 2 2

3 1 3 3

1 1 1

2 2 2

3 3

= =

a ca b c bR R R R R R

a b c a b cR R R R R R

a b c c b cR R R R R R

F dX F dX F dXdx dx dx

dx dx dx F dX F dX F dX

dx dx dx F dX F dX F dX

1 1 111 12 13

21 22 23 2 2 2

31 32 33 33 3

= =

a b c

a b c

a b c

dX dX dXF F F

F F F dX dX dX JdV

F F F dX dX dX


from which one has 0= det = =dvJ

dVρρF (2.209)

where 0ρ and ρ are the initial and the current densities. On the other hand, de-

noting the areas and the unit normal vectors of the surface elements formed by the

line-elements , a bd dX X and , a bd dx x as dA，N and da , n , respectively,

we have

= [ ] = ( ) =

= [ ] = ( ) = = = T

a ac c cb b

b c b c c c ca a

dV d d d d d d d dA

dv d d d d d d d da d da d da

⎫× • • ⎪⎬

× • • • • ⎪⎭

X X X X X X X N

x x x x x x x n F X n X F n

(2.210) noting Eq. (1.100). The following Nanson’s formula is derived from Eqs. (2.209) and (2.210).

1= , =T Tda J dA dA daJ

−n F N N F n (2.211)

or

1= , =T Td J d d dJ

−a F A A F a (2.212)

where , d da d dA≡ ≡a n A N (2.213)

Further, noting

1det = = 2cb

q ppq jjk qbc kp

d dx dxx

ε εΔ∂∂

x

derived from Eqs. (1.17) and (1.23), one has

1det det( )det( ) == = ( ) = 2qq q cb

q qp

pp p jjk qbc kp p

rr

vd dddv dx dv dx dx dxxx xd dε ε• •• ∂∂ ∂

∂∂ ∂x xx

1 1= =2 2

c cb a a biijk ijkabc abcj jk k irr rr

vdx dx dx dx dx dx Lxε ε ε ε∂∂

1 1= =6 6c ca b a b

ijk ijkabc abcj j vvk kir ir irdx dx dx L dx dx dx Lε ε ε εδ

1= =6

ca bijk abc vvj vvki d Dvdx dx dx Dε ε

or

1det (= tr = tr( ) = tr ) = trdet( ) T T T TTJ J J− −• •••∂

∂F F F LFF FFF

2.5 Surface Element, Volume Element and Their Rates 99

from which the following relation holds for the rate of volume element, noting

t r = trTT T and t r = 0W .

( )= t r = =v

dv Jdv J

ε••• D or ( ) = t r =dvdv dVJ

•• D (2.214)

Then, the time-integration of the volumetric strain rate leads to the logarithmic volumetric strain.

t r = lnvdvtdV

ε δ≡ ∫ D (2.215)

Moreover, it is obtained from Eq. (2.69) and the Nanson’s formula (2.211) that

( ) = ( ) = ( )TT Tda J dA J dA J dA−− −• •• • +n F N F F N

= (t r ) T T T J dA− −•+D I F F F N

= (tr ) TT da− •− nD I F F

= (tr ) T da− nD I L (2.216)

On the other hand, noting = 0••n n from 1=•n n for the unit vector n , it holds

that

( ) = ( ) ( ) ( )= =da da da da da•• • • •• • •−n n n n n n n (2.217)

Substituting Eq. (2.216) into Eq. (2.217), one obtains the rate of the current infini-tesimal area as follows:

)tr( ) = ( Tda da• − nn D I L (2.218)

which reduce to

)tr( ) = ( da da• •− n nD I D (2.219)

Further, it holds from Eqs. (2.216) and (2.219) that

( ) ( )=da da da• • •−n n n

) tr( (tr )= T da da•− −− n nn D n DD I L (2.220)

Then, the rate of the unit normal of the current surface element is given by

)= ( T• • −n nn n D I L (2.221)


Chapter 3 Conservation Laws and Stress Tensors

3 Conservation Laws and Stress Tensors

Conservation laws must be fulfilled for mass, momentum, angular momentum, etc. during a deformation. These laws are described first in detail. Then, the Cauchy stress tensor is defined and further, based on it, various stresses are derived. Introducing the stress tensor, the equilibrium equations of force and moment are formulated from the conservation rules. The virtual work principle required for the analyses of boundary value problems are also described in this chapter.

3.1 Conservation Law of Mass

Denoting the field of density in a material as ( , )tρ x , the mass in a region v is

given as ( , )= vt dvm ρ∫ x . Therefore, the following conservation law of mass

must hold.

( , ) = = 0( ( ) )vt dvm ρ •• ∫ x

(3.1)

from which, noting the Reynolds’ transportation theorem of Eq. (1.277), one has the

continuity equation.

div = 0ρ ρ•+ v , = 0r

r

vx

ρ ρ• ∂+ ∂ (3.2)

Further, setting ( , )T t ρφ≡x , where φ is a physical quantity per unit mass, it

holds from Eqs. (1.277) and (3.2) that

= =( ) ( )r

rv v vvdv dv dvx

ρ ρ ρρ φφ φφ ρ φ• • •• ∂+ +∂∫ ∫ ∫ (3.3)

3.2 Conservation Law of Momentum

The momentum possessed by a region v in a current state is given by v

dvρ∫ v . On

the other hand, denoting the traction (or stress vector) applied to the unit surface area of the region as t, the traction applied to the surface of the region is given as

102 3 Conservation Laws and Stress Tensors

a da∫ t and, denoting the body force per unit mass as b, the body force applied to

the region is given by v

dvρ∫ b . The rate of momentum has to be equivalent to the

sum of the traction and the body force applied to the region. Therefore, Euler’s first law of motion (or conservation law of momentum) is given as

=( )v a vadv d dvρ ρ• +∫ ∫ ∫t bv

or

=v a vadv d dvρ ρ• +∫ ∫ ∫t bv (3.4)

by virtue of Eq. (3.3).

3.3 Conservation Law of Angular Momentum

The angular momentum possessed by a region v in a current state is given as

( )v dvρ ×∫ x v . On the other hand, denoting the moment of traction and the

moment of body force applied to the region as ( )a ad×∫ x t and ( )v

dvρ ×∫ x b ,

respectively, Euler’s second law of motion, i.e. conservation law of angular momentum is described as

=( ) a vvdv dvdaρ ρ• ×× +×∫ ∫ ∫ xx v bx t

=( )j j jk k kijk ijk ijkv vaax dv x d x dvv btρε ε ρε• +∫ ∫ ∫

which reduces to

= a vvdvdv daρ ρ• ×× +×∫ ∫ ∫ xx v bx t

=j j jk kijk k ijk ijkv vaax x d x dvv bdv tρε ε ρε• +∫ ∫ ∫ (3.5)

noting ( ) = =• ••× × × ×+x v v v x v x v and Eq. (3.3).

3.4 Stress Tensor

When the infinitesimal force vector df applies to the surface with infinitesimal

area da and the unit normal vector n, the stress vector is given as shown below.

dda

≡ ft (3.6)

3.4 Stress Tensor 103 Now, introduce the following second-order tensor σ fulfilling the relation

= , =Tjii jt nσt nσ (3.7)

by the quotient law described in 1.3.2. The components of the tensor σ are given by Eq. (1.88) as

= = = Tij j j ji i iσ • • •e e e e e eσ σ σ (3.8)

if σ is the symmetric tensor. Here, when we choose ie to the unit normal vector n

of the surface on which t applies, the following equation holds by substituting Eq.

(3.7) with = in e into Eq. (3.8).

= jijσ • et (3.9)

Therefore, i jσ is the component in the direction of je for the stress vector t applying on the surface element having the outward-normal ie . The tensor σ is called the Cauchy stress tensor. Equation (3.7) is called the Cauchy’s fundamental theorem or Cauchy’s stress principle. It holds from the conservation law of angular momentum described in 3.6 that

( = )= T jij i σ σσ σ (3.10)

which means that σ is the symmetric tensor. Various stress tensors are defined in addition to the Cauchy stress tensor

described above. Some of them, which are often used in continuum mechanics, are presented below.

The tensor τ defined by the following equation is called the Kirchhoff stress tensor.

= Jτ σ (3.11)

The vector t defined by the following equation is called the nominal stress vector (see Fig. 3.1).

ddA

≡ ft (3.12)

TensorΠ , which is related to t by the following equation, is called the nominal stress tensor or the first Piola-Kirchhoff stress tensor.

( )AAiit Π N≡ ≡t ΠN (3.13)

Here, substituting Eqs. (2.211) and (3.13) into Eq. (3.12), we have

1 1( = )i iA AΤ

rrid dA= da df N dA= F n daJ JΑ ΑΠ Π≡f ΠN ΠF n (3.14)

104 3 Conservation Laws and Stress Tensors On the other hand, the substitution of Eq. (3.7) into Eq. (3.6) yields

= Td daf nσ (3.15)

Comparing Eqs. (3.14) and (3.15), it holds that

T da = dAn ΠNσ (3.16)

and the relation of σ and Π is obtained noting Eq. (2.211) as shown below.

1

1 1 1 1 ( ) ( = = )

( = ( ) ) )(

ij j jAA

A

T T

T T

i i

ri ir

= = F FJ J J J

= JJ

Α Α

Α

σ Π Π

Π σ− −

⎫⎪⎬⎪≠ ⎭

FΠ ΠF

Π F F Π Π

σ

σ (3.17)

F

dfn

Ndf

dAda

dX=d dx F XdV dv

F

dfn

Ndf

dAda

dX=d dx F XdV dv

Fig. 3.1 Variation of configuration

Further, the tensor S defined by the following equation is called the second

Piola-Kirchhoff stress tensor.

=t SN (3.18)

where

11=d

dA

−−≡ F ft tF (3.19)

Using Eq. (2.211) into these equations, one has the following expression.

1 Td dA= daJ

≡f FSN FSF n (3.20)

3.5 Equilibrium Equation 105 Comparing Eqs. (3.15) and (3.20), it holds that

1 11

1 1( = )

)( ( ) ( ) ( )=

A AB Bj

AB jA jB

T

T T

iij

ii

= F S FJ J

=J= J S

σ

σ−− −−

⎫⎪⎬⎪⎭

FSF

S SS F F FF

σ

σ (3.21)

The following stress is called the convected stress.

)( =T Tc c c≡ F Fσσ σ σ (3.22)

The relations of various stress tensors defined above are summarized in Table 3.1.

Table 3.1 Relations of various stress tensors

Names, Notations

(= )T (= )T ( )T≠ (= )TS S ( )= Tc c

Cauchy 1J

1 TJ

F 1 T

JFSF 1T c− −F F

KirchhoffJ TF TFSF 1T cJ − −F F

Nominal(1st Piola-Kirchhoff) TJ −F T−F FS 1T TcJ − − −F F F

2nd Piola-KirchhoffS

1 TJ − −F F T−1 −F F TT −F 1 1T TcJ − − − −F F F F

Convected c

TF F 1 T

JF F 1 T T

JFF F 1 T T

JF FF FS

(Note) 1

1, , =dd d

da dA dA

−−≡ ≡ ≡ff F ft t t tF

=t nσ , =Jt nτ , =t ΠN , =t SN

3.5 Equilibrium Equation

Substituting Eq. (3.7) into Eq. (3.4) for the conservation law of momentum and noting Eq. (3.10), the following equation is obtained.

= =, ( ) ( )T r ri iiv va v v aa n adv d dv dv d dvv bσρ ρ ρ ρ• •+ +∫ ∫ ∫ ∫ ∫ ∫n bv σ

(3.23)

The right had side is given from Eq. (3.3) as

=( )v vdv dvρ ρ ••∫ ∫ vv

106 3 Conservation Laws and Stress Tensors and the first term in the right-hand side of Eq. (3.23) is given by Eq. (1.267) of Gauss’ divergence theorem as

= irr rir

van a dvd

xσσ ∂

∂∫∫

By this equations and Eq. (3.3) the local form of Eq. (3.23) is given as

,= = ij

i ij

b vxσρ ρρ ρ• •

•∂+ +∂

b vσ ∇ (3.24)

This equation is called the Cauchy’s first law of motion, i.e. the equilibrium equation.

On the other hand, substituting Eqs. (2.209) and (3.16) into Eq. (3.23), one has

0 0=( ) AV VA dVd dVρ ρ• +∫ ∫ ∫v ΠN b (3.25)

which is rewritten by Gauss’ divergence theorem as follows:

0 0= VV VV dVd dVρ ρ• • +∫ ∫ ∫Xv Π b∇ (3.26)

where )( / = /AAX≡ ∂ ∂ ∂ ∂X Xe∇ . The local form of this equation is given as

0 0 0 0= =, iAii

Avb

Xρ ρ ρ ρ• •• ∂+ +

∂vb∇X

(3.27)

The equilibrium equation in a rate form is required in constitutive equations for irreversible deformation including elastoplastic deformation. The time-differentiation of Eq. (3.27) engenders the following rate-type (or incremental-type) equilibrium equation, provided that the acceleration does not change, i.e. =••v 0 .

0 0= =, 0iAi

Ab

Xρ ρ

•• ••• ∂+ +

∂X 0b∇ (3.28)

Denoting 1 1( ) = = TT TJ J

• • •∗ ≡ ΠF FΠσ σ (3.29)

and noting Eq. (3.17), we have

1 ) == (T TT TT T JJJJ JJ

− −− −•• ••• •∗ ++ ++ F FF F FF σσσ σσσ σ

Substituting ( ) = == T T TT TTT T T− −− −• • •• + +F 0F F LF F F F F due to Eq.

(2.69) and Eqs. (2.73) and (2.214) to this equation, the following holds.

3.6 Equilibrium Equation of Moment 107

t r t r= =T•∗ ++ − + −LL D D Wσ σ σσ σσ σσ (3.30)

Therein, ∗σ is designated as the nominal stress rate, whereas

• − +≡ w wσ σ σ σ is the Jaumann stress rate which will be described in detail in the next chapter.

The partial derivative of Eq. (3.29) by jx noting ( / ) / = 0jjA xF J ∂∂ (see

Appendix 2) leads to the following. *

1 1 1= = =ji i iijjA

j j A Aj

xF

J J Jx x x X XΑ Α Ασ Π Π Π• • •∂∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

Substitution of this relation into Eq. (3.28) yields the rate-type equilibrium defined in the current configuration:

*= =, 0

iji

jbx

σρ ρ •••

∗ ∂+ +∂0b∇ (3.31)

This equation is derived also by the following manner. From Eq. (3.23) one has

=( ) ( ) ( )Ta vv

ad dvdv ρρ • • •• +∫ ∫∫ n bv σ

0 0=( ) ( ) ( )TaV V

add dV Vρ ρ• • •• +∫∫ ∫nv bσ

0( )=0 T T

a a Vada d dVρ •• •+ +n n b

(tr )=0TT T

va ada vdda ρ •• +−+ n bn D I L

t r )(=0TT T T

v vdv vdρ •• •−+ +D L b∇

which results in Eq. (3.31), noting Eqs. (1.274), (2.216) and = Tσ σ .

3.6 Equilibrium Equation of Moment

Substituting Eq. (3.7) into Eq. (3.5) of the conservation law of angular momentum and noting (3.10), one has

=k jj rkrijk jijk kijka vvxx n av dv d x dvbερε σ ρε• +∫ ∫ ∫ (3.32)

Because the first term in the right-hand side of this equation is rewritten as

= = ( )j kr krj rkrijk ijk ijk j ijkk jr ra a v

xx n ad xdv dvx xσ σε ε ε εσ σ∂ ∂+

∂ ∂∫ ∫ ∫

108 3 Conservation Laws and Stress Tensors Eq. (3.32) leads to

= 0 ( )krijk j ijkk kj krv

bx dvvxσε εσ ρ ρ •∂ −+ +∂∫

Noting the equilibrium equation (3.24) to this equation, it holds that = 0jijk kε σ from which we have the symmetry of Cauchy stress tensor, i.e.

=ij jiσ σ (3.33)

3.7 Virtual Work Principle

The stress (rate) field fulfilling the equilibrium equation and the boundary condition of stress is called the statically admissible filed. On the other hand, the displacement (velocity) field fulfilling the geometrical requirement

(= =/ ) /A Ai iiA iAF uX Xx δ∂ ∂∂ ∂+ or = ( /ij jiD v x∂ ∂ ) / 2/j iv x+∂ ∂ and

the boundary condition of displacement (velocity) is called the kinematically-admissible field. Denoting arbitrary statically admissible stress field

and kinematically-admissible velocity field by ( )Δ and ( )∇ respectively, one has the following equation from Eq. (3.24).

= =0, 0( ) ( )ijii i

jv vdv b v u dv

xσρ ρ ρ ρ

ΔΔ ∇∇• •• •

∂+ +− −∂

b uv∇

(3.34) Using the Eq. (1.264) of Gauss’ divergence theorem, we have

( )= ij ijiiiij

j j jv v v

uu udv dv dvx x x

σ σσ

Δ Δ∇∇Δ ∇∂∂∂ −

∂ ∂ ∂∫ ∫ ∫

= ij

ij j jii ij it jv va a

n nu uda u da dvxσσ σ

ΔΔ∇ ∇∂

+ −∂∫ ∫ ∫

= ij

j iij ij iit jv va ant da u da u dvu

xσσ

ΔΔ ∇∇

∂+ −

∂∫ ∫ ∫

where ( )− designates the given boundary condition, and ta and va specify the surfaces of the body on which the traction (rate) and the displacement (velocity) are given, respectively. Substituting Eq. (3.34) into this equation, the following virtual work principle described by the quantities in the current state is obtained.

=i ij i ii ij i iiij tj vv v va au u dvnu udv t da u da b dv vx

ρ ρσσ∇ ∇ΔΔ ∇∇ •∂ + + −

∂ ∫∫ ∫ ∫ ∫

(3.35)

3.7 Virtual Work Principle 109

Similarly, the following virtual work principle can be described by the quantities in the initial state from Eq. (3.27).

0 0=iiJ iiA i i ii iiA t vAV V VA A

u uu udV dA N u dA b dV v dVtXρ ρΠΠ

∇ ΔΔ ∇ ∇∇ •∂ ++ −∂ ∫∫ ∫ ∫ ∫

(3.36) Furthermore, one has the rate-type virtual work principle from Eqs. (3.28) and (3.31) as follows:

= j iijiiij iiij t vv va anv vD dv t da da v dvbρσσ ΔΔ ∇ ∇ ∇•• •+ +∫∫ ∫ ∫• (3.37)

0= iiA iA iAi iAi iVt vV A AΠv vΠ F dV t d N dA b v dVA ρ ∇

ΔΔ ∇ ∇• •• •• ++∫ ∫∫ ∫ (3.38)


Chapter 4 Objectivity and Corotational Rate Tensor

4 Objectivity and Corotational Rate Tensor

The mechanical properties of a material are independent of the reference frame of the observer. Therefore, a constitutive equation describing the properties must be invariant under the change of coordinate system mutually rotating with respect to each other. The physical meaning of this type of the objectivity of constitutive equations and the general form of corotational rate of a tensor that has to be used in stead of the usual material-time derivative (under the fixed coordinate system) in order to fulfill the invariance are described in this chapter.

4.1 Objectivity

When a material is subjected to a constant stress, i.e. a stress the components of which are observed to be constant from the material itself, the deformation has not to be induced even if a rigid-body rotation is added. However, the components of stress in a coordinate system fixed in the laboratory change so that the stress rate is observed to be induced in the fixed coordinate system if the material is subjected to a rotation. On the other hand, the strain rate described in 2.3 is given by the symmetric part of the velocity gradient excluded its skew-symmetric, i.e. rotational part and thus its components are not influenced by the rotation. The one-to-one correspondence between the stress and the strain does not exist, and thus the stress rate and the strain rate have to be related mutually in an irreversible constitutive equation leading to the rate-type equation, e.g. the viscoplastic, the elastoplastic and the viscoplastic constitutive equations. Therefore, if the material-time derivative of stress observed in the fixed coordinate system is adopted in these constitutive equations, the incorrect prediction is given such that the strain rate is induced and thus the material deforms even if the stress observed by material itself is constant. Let the general interpretation for this fact and how to remedy this defect be considered below.

As already mentioned, the nechanical properties of materials are independent of the mutual rigid body motion of observers. Therefore, the deformation characteristics must be described uniquely in an identical equation independent of the relative position and motion to the coordinate systems by which it is described. This fact is called the principle of objectivity or principle of material-frame indifference or simply objectivity (Oldroyd, 1950). The explicit problem occurs in the deformation analysis of materials subjected to the rotation, while the coordinate system describing a constitutive equation is fixed usually. Therein, the check is

112 4 Objectivity and Corotational Rate Tensor required as to whether or not the deformation behavior is formulated such that it is not influenced by the rotation. As the preliminary examination, how the physical quantities relevant to the stress and deformation are influenced by the rotation will be first considered in the next section. Consequently, how to remedy this defect for the quantity, which is observed to be constant by the material itself but is observed to change in the fixed coordinate system, will be considered. Further, the explicit mathematical process that the material-time derivative of arbitrary scalar-valued tensor function is transformed directly to its corotational derivative is shown, while this transformation is required in the consistency condition which is derived from the yield function on the formulation of plastic constitutive equation described in the later chapter.

Mechanical elements are often subjected to a rotation independent of the occurrence of deformation, as seen in gears, wheels, etc. in the case of machinery. Soils near the side edges of footings, at the pointed ends of piles, etc. undergo a large rigid-body rotation in soil structures. Therefore, a formulation of constitutive equations independent of rigid-body rotation is of great importance in practical engineering problems.

4.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities

In order to check whether or not a constitutive equation is formulated in accordance with the objectivity principle, it is expedient first to examine the influence of rigid-body rotation on variables used in constitutive equations. To this end, it is sufficient to derive the transformation rules for the components of constitutive variables under the change of coordinate system from one with the fixed base

ie

into the other with the rotating base ( ) ( )Tii t t∗ =e eQ , which coincide with base

ie in initial instant and then rotates with the elapse of time. Designating the infinitesimal line-element vectors observed in the

above-mentioned coordinate systems before and after the deformation as , d dX x and , d d∗ ∗X x , the relations

= ( = for = 0), =d d t d d∗ ∗X X Q I x Q x (4.1)

hold where i i= ∗⊗Q e e as shown in Eq. (1.77) and they are related by

, ( )= = = =d d d d d d∗∗x F X x Q x QF X F X (4.2)

and consequently, we have

* =F QF (4.3)

It is known from Eq. (4.3) that the deformation gradient F transforms as the second-order tensor between two fixed coordinate systems, but it would be observed as if it were a vector when a rigid-body rotation is superposed. In other

4.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities 113 words, it obeys the objective transformation as the first-order tensor for the superposition of rigid-body rotation. It is called the two-point tensor because it is described using the two bases in the initial and the current states.

Substituting Eq. (4.3) into Eqs. (2.15)-(2.16) and (2.41), the following relations hold for various quantities describing a deformation.

* =R QR (4.4)

, ( (= = ) = == )T T T TT∗ ∗ ∗∗ C C F F QF QF F Q QFU FU F (4.5)

( (= ) = ),= = T TT TT T ∗ ∗∗∗ Q Q Q Q F F QF QF QFF Q V V b b (4.6)

=∗E E , = T∗e QeQ (4.7) Noting the relation

1 1 1 1( ) ( ) = ( ) = ( )T T− − − −• •• •• + −Q QQF QF Q F Q Q Q QF FF F

and Eq. (4.3), it holds for the velocity gradient 1( )−•≡L FF that

= ( ) =T T∗ +−L Q L Ω Q QLQ Ω (4.8)

where Ω and Ω are given by

,

,

T

T

jri rj

jjrir

i

i

Q Q

Q Q

••

••

⎫≡≡ ⊗ ⎪⎪⎬⎪≡≡ ⊗ ⎪⎭

e eΩ ΩQ Q

e eΩ ΩQ Q (4.9)

and they are related by

=T T•

−≡ QQ QΩQΩ (4.10)

where •Q is given by

r r=• •∗⊗Q e e (4.11)

Substituting Eq. (4.11) into Eq. (4.9)1, we have

** rr•≡ ⊗eΩ e (4.12)


* *=i i•e eΩ (4.13)

It is seen from Eq. (4.13) that Ω is the spin of the base i∗e .

114 4 Objectivity and Corotational Rate Tensor

The substitution of Eq. (4.8) into Eqs. (2.73) and (2.74) yields the following relations.

= T∗D QDQ (4.14)

= ( ) =T T∗ − +W Q W Ω Q QWQ Ω (4.15)

The conclusions concerning the influence of rigid-body rotation and following from Eqs. (4.5)-(4.15) are:

1) The right Cauchy-Green deformation tensor C and the Green strain tensor E are observed to be unchangeable, obeying the transformation rule of scalar quantities independent of the rigid-body rotation. On the other hand, the left Cauchy-Green deformation tensor b and the Almansi strain tensor e obey the transformation rule of second-order tensors.

2) The strain rate tensor D obeys the transformation rule of the second-order tensor. On the other hand, the velocity gradient tensor L and the continuum spin tensor W are directly subjected to the influence of rate of rigid-body rotation.

The following relations hold for stress tensors described in Chapter 3.

= ( = = = = )T T∗ ∗ ∗ ∗ ∗nQ Q t Qt Q Q Q n nσ σ σ σ σ (4.16)

= T∗ Q Qτ τ (4.17)

=∗Π QΠ

( ( ) )T T T T TT T= = = = =J J J J− − − − −∗ ∗ ∗Π F Q Q QF Q Q Q F Q F QΠσ σ σ σ

(4.18)

( ( ) ( ) )T= = = = = =− − − −1 1 1 1∗ ∗ ∗ ∗S S S F Π QF QΠ F Q QΠ F Π S

(4.19)

Then, the Cauchy stress tensor σ and the Kirchhoff stress tensor τ are observed as the second-order tensor. On the other hand, the nominal stress tensor Π is observed as the first-order tensor which is the two-point tensor and the second Piola-Kirchhoff stress tensor S is observed as the scalar under the superposition of rigid-body rotation.

4.3 Rate of State Variable and Corotational Rate Tensor

It was stated in 1.15 that the material-time derivatives of state variables pursuing the material particle must be used in rate-type constitutive equations. In addition, as described in 4.1 for the stress rate tensor, if the material-time derivative of physical

4.3 Rate of State Variable and Corotational Rate Tensor 115 quantity observed in the fixed coordinate system is adopted in a constitutive equation, the irrational prediction is resulted. Instead of the material-derivative, considering the rate of the quantity which is observed not only moving but also rotating with the material, we would have to adopt its translation to the fixed coordinate system describing the constitutive equation. The explicit formulation for this quantity will be shown below (Hashiguchi, 2007).

Consider the transformation of the material-time derivative of a state variable obeying the objective transformation (1.59) or (1.61). Their material-time derivative reads:

1 2 1 1 12 2 2 21 1 1 2 2=m

m m m m m mp p p p q p q p q p qq q q q q qp q p q

T Q Q Q T Q Q Q T• • • • • •

• • •• • •

∗ +⋅⋅⋅ +⋅⋅⋅ ⋅⋅⋅

1 21 1 1 12 2 2 21 2 mm m mm mq q qp q p q p q p q p qq q qp qQ Q Q T Q Q Q T• • • • • •

• •+ + ⋅⋅⋅⋅⋅⋅ (4.20)

1 m2 m m 12 m 22 2 1 1 11 1 2 2= m mm

p p p q pp q q qq q p q pq p q q pq qT Q Q QQ Q Q T T••• ••••••• • •• • •

• ••• • •∗+ +

1 21 1 2 2 1 1 2 2 1 2mm m m m mq q qq p pq q p p q pq p q q q qQ Q Q Q Q QT T••• •••

• • • • • ••• ∗ ∗+ +

(4.21)

Noting the relation = =i ii i i ip pp q q qt ts ss sQ QQ Q Qδ• • •

= =i i i ip q p qt t tt ssQ Q QQ Ω•− −

and replacing iqt → , i iq r→ , then Eqs. (4.20)

and (4.21) can be rewritten as follows:

1 2 211 1 2 2 21 1 1 12 2 2= (m mm m m m

p p p q q qp q p q p q q q qq q qrr r rT Q Q Q T T TΩ Ω• • •

• •∗ − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 1 2)

m mq q qr rTΩ• • •− − ⋅⋅⋅ (4.22)

1 2 1 12 1 21 2 21 1 12 2 2= (m m mm m m

p p p q q q qq q qq p q p rq q qp r r rT Q T TQ Q T Ω Ω••• ••• ••• •••

• •• • • ∗ ∗∗ − −

1 2

)m m mq q q rr TΩ• • • ∗− − ⋅⋅⋅ (4.23)

It is known from Eqs. (4.20)-(4.23) that the material-time derivative cannot be

adopted in constitutive equations, since the components 1 2 mp p pT •••

• in the fixed

coordinate system changes even when the components 1 2 mp p pT• ∗ ⋅⋅⋅ observed in the

coordinate system rotating with the material does not change. Eqs. (4.20)-(4.23) are expressed for scalar S , vector v and second-order tensor T in symbolic notation as follows:

=S S• •∗ (4.24)

( )= ( ), = T• • • •∗ ∗− − ∗Qv Ω v vQv v v Ω (4.25)


( ( )) == , TT • •• • ∗∗ ∗∗− − −Ω Q QQ T + TΩ Q ΩT TT T T ΩT (4.26)

Now, consider the tensor T having the components obtained using the inverse transformation from the components observed by the coordinate system rotating with the material. Then, using Eq. (4.22), it is represented by

1 2 11 1 22 2=

m mm mp p p qq p p q p q qqQ Q Q TT ••• •••• • • • ∗

1 1 1 12 2 2 2=

m m m mq p p q pq q q qt ttQ Q Q Q Q Q• • •• • •

21 21 1 1 1 1 12 2 2 2( )m mm m mqtt t t rt t tr t r t r r t t rT T T TΩ Ω Ω•

• • •− − − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 2 21= m mp p pt t tδ δ δ• • •

21 21 1 1 12 2 2( m mm tt t t rt tr t r t r tT T TΩ Ω•

• • •− − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 1 1 2)

m mt r t t rTΩ− ⋅⋅⋅

1 12 1 2 21 1 2m2= m m

pp p p p pp p pr r rr T TT Ω Ω••• ••••••

• − − 21m m mp p pr rTΩ •••• • •− −

(4.27)

While various spins describing explicit rotational rates of material would be assumed, denoting them by the symbol ω , let the following corotational rate

tensor T be introduced, following the form of T in Eq. (4.27), i.e.

1 12 2 2 21 1 1 12 2=m m m m

p p p p p p p pp p p pr rr r T T TT ω ω•• • •− − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 2 m m mp p p rr Tω− ⋅⋅⋅ (4.28)

which must obey the objective transformation rule

1 2 1 21 1 2 2

* =m mm mp p p q q qp q p q p qQ Q QT T• • •⋅⋅⋅ ⋅⋅⋅ (4.29)

between an arbitrary and the reference coordinate systems with the bases i∗e and ie . Thus, substituting Eq. (4.28) into Eq. (4.29) and noting Eq. (4.22), it must hold that

1 2 2 21 1 1 1 12 2 2* * * * ** *m m mm m mp p p p p pp p p p p pr r rr r r T TT Tω ω ω•

• • •− − − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 1 2 2=

m mp q p q p qQ Q Q• • •

21 21 1 1 1 1 12 2 2 2( )m mm m m

q q q q q qq qq q q qrr r r r rT T T TΩ Ω Ω•• • •− − − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

2 21 1 1 1 11 2 2 1 2 2 12 2* *

m m m mm mp pp qq p q p q q p q q qq q q qr rr r Q Q Q Q Q QT Tω ω• • • • • •− −⋅⋅⋅ ⋅⋅⋅

4.3 Rate of State Variable and Corotational Rate Tensor 117

1 1 2 2 1 2

* *m m m m mqp p q p q q q qrr Q Q Q Tω • • •• • • −− ⋅⋅⋅

1 1 2 2=


1 12 11 2 2 2 1 2 1 2

( )m m m m m mqp p p r q qq q q q q q rrr r r T TT Tω ω ω•

• • •− − − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

from which it obtained that

1 1 2 2 21 1 1 1 12 2 2 2

( )m m m mm m m

p q p q p q q q qq qq q q qrr r r r rQ Q Q T T TΩ Ω Ω• • • • • •+ + +⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

2 21 1 1 1 11 2 2 1 2 2 12 2

* *m m m mm m

p pp qq p q p q p p q qq q q qr r rr r Q Q Q Q Q QT Tω ω• • • • • •+ +⋅⋅⋅ ⋅⋅⋅

1 1 2 2 1 2* *

m m m m mp qp p q p q q qrr Q Q Q Tω • • •• • •+ + ⋅⋅⋅

1 1 2 2=


1 11 2 2 2 1 2 1 2

( )m m m m mq r q qq q q q q q rrr r r T T Tω ω ω• • •+ + +⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

which may be arranged into the form

1 1 11 2 2 1 2

1 11 1 2 2 1 21 1

1 2 2 1 1 1 2

1 1 2 2 1 2

1 1

*

(= )

(= )

(= )

=

m m m

m m m

m m m

m m m

p q p q p q q q q

qp q p q p q r q qq

p p q p q q q q q q

qp p q p q q q q

p

r

r

r

r

r

Q Q Q T

Q Q Q T

Q Q Q T

Q Q Q T

Q Q Q

αα α

αα β αβ β

βα

ω

ω Ω

ω Ωωδ Ω

• • •

• • •

• • •

• • •

−

−

−

⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅1 1 2 2 1 2

1 1 1 1 2 2 1 2

( )

(= )

m m m

m m m

q p q p q q q q

p q p q p q q q q

r

r r

Q Q T

Q Q Q QQ T

ααβ β

β αα αβ β

ω Ωω Ω

• • •

• • •

−

−⋅⋅⋅⋅⋅⋅

Eventually, ω must obey the following transformation rule.

= ( ) = ( ) , T rs rsij ir jsQ Qω ω Ω∗∗ −−ω ωQ Ω Q (4.30)

The following equations for scalar S , vector v and the second-order tensor T hold from Eqs. (4.28).

= =S S S• ∗ (4.31)

; ;= = = T• ∗∗−v v v v v v vQ Qω (4.32)

= ,= =; T T• ∗ ∗−T T + T QT T Q Q T Q TTω ω (4.33)


Now, introduce the symbols for the coordinate transformation.

1 2 m m1 1 12 2 2

1 2 1 21 1 2 2

( )

( )m m

m mm m

T

p p p p q p p qq q q q

p p p q q qp q pq q p

Q Q Q T

Q Q Q T

• • •

• • •

⎫≡ ⎪⎬≡ ⎪⎭

⋅⋅⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅

Q T

Q T (4.34)

Then, the following expressions for an arbitrary tensor T hold from Eqs. (1.59) and (4.29).

,= = T∗ ∗QT T T Q T (4.35)

and

= , =T∗ ∗TQ QT TT ( )=,=

T •• ∗∗ ∗ Q TT TT

(4.36)

The tensor is calculated by the following time-integration by Eq. (4.28), (4.32) and (4.33).

1 2 1 2 2 21 1 1 12 2= (m m m m

p p p p p p p pp p p pr rr rT T T Tω ω+ +⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅∫

1 2)m m mp p p rr T dtω• • •+ + ⋅⋅⋅ (4.37)

i.e.

(= dt)+∫v v vω (4.38)

(= )dt+ −∫ TT T Tω ω (4.39)

after calculating 1 2 mp p pT ⋅⋅⋅ or v or T by the constitutive relation.

While the corotaional rate T in Eq. (4.28) following the tensor T in Eq. (4.27) is introduced above, note that various objective corotational rate have been

proposed, which do not belong to T , as will be described in 4.5 for stress rate. When the continuum spin W in Eq. (2.74) is adopted for the material spin ω,

the corotational rate tensor T is called the Jaumann rate (Jaumann, 1911) and then

let it be designated by the symbol T , i.e.

1 12 2 1 2 21 1 2 1 2

=m m m mp p p p p p p p p p p prr r r T W T TT W•••

•• • •− − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 2m m mp p p rrW T− ⋅⋅⋅ (4.40)

which is represented for the second-order tensor as

4.4 Transformation of Material-Time Derivative 119

=• −T T + TT WW (4.41)

The spin W obeys the transformation rule (4.30) as shown in Eq. (4.15). Furthermore, the corotational rate tensor adopting the relative spin

( = )TR •Ω R R in Eq. (2.78) is called the Green-Naghdi rate tensor (Green-Naghdi, 1965) or Dienes rate tensor (Dienes, 1979) and is represented by

⊕T , i.e.

1 12 2 1 2 21 1 2 1 2

=m m m m

R Rp p p p p p p p p p p prr r r T T TT Ω Ω•••

⊕ •• • •− − −⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 2m m m

Rp p p rr TΩ− ⋅⋅⋅ (4.42)

which is represented for the second-order tensor as

= R R⊕ • −T T + TT Ω Ω (4.43)

Note that RΩ obeys Eq. (4.30) as known from

( ) ( ) ( )= = =TR T T T• •• •∗ ∗ ∗ +QR QR Q R Q R R QΩ R R

( ) )= = = ( T T T R T R T• • •+ + −Q R Q R R Q QQ Q Q Q QΩ Ω Ω (4.44)

noting Eqs. (4.4) and (4.9). When the deformation can be ignored, all the material spins in the corotational

rate described above mutually coincide as described in Eq. (2.88). Here, they are determined solely by a geometrical change of outward appearance, which is independent of individual properties of given material. On the other hand, the spin which influences on the mechanical response is the spin of substructure (microstructure) in the material. For that reason, we must adopt the corotational rate with the material spin which depends on the material property reflecting a microstructure and the deformation process. Generally speaking, the spin of the microstructure is not so large as that given by the continuum spin. An explicit form of the spin of substructure in the elastoplastic deformation will be described in Chapter 12. 4.4 Transfor mation of Material-Time Derivative

4.4 Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative

4.4 Transfor mation of Material-Time Derivative As in the plastic constitutive equation described in the later chapters, one takes first the material-time derivative of yield condition as a scalar function and must then transform it to the corotational derivative to formulate the plastic strain rate. Here, needless to say, any scalar function and its material-derivative are observed to be identical to each other independent of coordinate systems and thus it would be

120 4 Objectivity and Corotational Rate Tensor expected that the material-time derivative is directly transformed to the corotational derivative as shown in Eq. (4.31). On the other hand, the explicit mathematical process by which the material-time derivative of scalar function is transformed to the form that all the involved material-time derivatives are replaced to the corotational derivatives obeying Eq. (4.36) is shown below (Hashiguchi, 2007).

Here, adopting the symbol

1 12 2Tr( )

m mp p p p p pTS≡ ⋅⋅⋅ ⋅⋅⋅ST

(4.45)

for arbitrary tensors S and T , the scalar-valued tensor function ( , , )f • • •A B

of arbitrary tensors , , • • •A B is described as shown below.

( , , )( , , ) Tr( , , ) = Tr ( )( ) fff

• • • •• • • • • • • •• • •∂∂ ++

∂∂A BA B BA B A BA

(4.46)

Here, because ( , , )f • • •A B is the isotropic function of the arguments , , • • •A B , it holds that

( , , ) = ( , , )

( , , ) = ( , , )

f f

f f

• • • • • •

• •• • • • • •

∗ ∗

∗ ∗

⎫⎪⎬⎪⎭

A B A B

A B A B (4.47)

where the following pertains.

( , , ) ( , , )( , , ) = Tr Tr( ) ( )f f

f• • • • • • •• •

• • • • • •∗ ∗ ∗ ∗ ∗∗∗ ∗

∗∗∂ ∂ ++ ∂∂

A B A BA B A B

A B

( , , ) ( , , )= Tr Tr( ) ( )f f• • • • • •• •

• • •∗∗∂ ∂ ++∂ ∂A B A B

Q QA BBA

( , , ) ( , , )= Tr tr( ) ( )T Tf f• • • • • • •• • • •∗ ∗∂ ∂ ++

∂ ∂A B A B

Q Q BABA

( , , ) ( , , )=Tr T r( ) ( )f f• • • • • •

• • •∂ ∂+ +

∂ ∂A B A B

A BBA

(4.48)

Equations (4.35), (4.36) and the following equation for arbitrary tensors T and S

in the same order are considered in the derivation of Eq. (4.48).

4.4 Transformation of Material-Time Derivative 121

11 2 21 1 2 2Tr( ) = ( ) m m mmp q q q q p p pp q p qQ Q Q T S• • • ⋅⋅⋅ ⋅⋅⋅Q T S

11 2 21 1 2 2( )= m m mm p qq q q p p pp q p qQ Q QT S• • •⋅⋅⋅ ⋅⋅⋅

)= Tr( TQT S (4.49)

Here, if ,T S are vectors ,u v , it holds that = = T• • •∗ ∗ ∗ ∗Qu v u v Qu v ,

and if they are the second-order tensors ,U V , it holds that t r( )∗ ∗U V

t r( )t r( ) == TT ∗∗ QU V QQ QU V . The following expression is obtained from Eqs. (4.47)2 and (4.48).

( , , ) ( , , )( , , ) = Tr Tr( ) ( )f f

f• • • • • • •

• • • • • •∂ ∂+ +

∂ ∂A B A B

A B A BBA

(4.50) Therefore, the material-derivative of scalar function is transformed directly to the corotational derivative.

Here, if function f involves only a single second-order tensor A , the

transformation to the corotational derivative can be verified using the following

equation based on Eq. (1.192), noting that t r( ) = tr( )n nA Aω ω .

2

0 21

( )( )tr = tr( )( ) = 0 f φφ φ∂ − − − −

∂A A A I A A A A

Aω ω ω ω

(4.51) Furthermore, for the case in which two tensor variables are involved, the transformation to the corotational derivative is verified using Eq. (4.51) if they are replaced to a single variable, as in the kinematic hardening of metals and the rotational hardening of soils, which will be described in later chapters. However, in the general case for which function f involves plural tensor variables, it consists of many invariants made by them. The scalar function involving two independent tensor variables as the simplest example consists of a lot of scalar variables in Eq. (1.198) in general. Therefore, one has the inequality

( , , ) ( , , )Tr Tr( ) ( )f f• • • • • ••∂ ∂≠

∂ ∂A B A B

A AA A

for individual derivative terms. Then, the simple proof by individual transformation of each variable is inapplicable in this case but proofs for the transformation to the corotational derivative are necessary for all invariants including tensor operations between tensors. They usually require quite complicated calculations, while these would be the limit of itself. On the other hand, the mathematical process described

122 4 Objectivity and Corotational Rate Tensor above gives rise to the general proof for the transformation of the material-time derivative of scalar function to the corotational derivative without such complicated processes.

4.5 Various Objective Stress Rate Tensors

Various stress rate tensor with objectivity are shown below. The following Jaumann rate and Green-Naghdi (Dienes) rate of Cauchy stress

tensor based on Eqs. (4.40) and (4.42) are often adopted in constitutive equations. • − +≡ W Wσ σ σ σ (4.52)

( ) =T T R R⊕ •• − +≡ R R R R Ω Ωσσ σ σ σ (4.53)

from which various objective stress rates are obtained as follows:

( )

= = =( )JJ J J

J JJ J

•• +−≡ W Wτ +τ τ σ σ σττ

t r tr= = ++ L Dσσ σ σ (4.54)

1( ) = ( ) = ( )TT T

v JΔ − −• •≡ SF F F F F Fσ σ σL

1 11 1 )(= =

TTT T T T−−− −− − − −• • • •• •+ + + +F F F F F F F F F F F Fσ σ σ σ σ σ

1 1 1= = ( )TT T− − −−• • • •• •− −− −+ +F F F F F F F Fσ σ σ σ σ σ

= T• −− LLσ σ σ (4.55)

1 T( )( )= =

T

J

Tv JJ

J JJ

− −Δ

• •≡

F F F FFSF σσσ L

11 1 1( )=

TTT T TJ J JJJ

− −− − − − − −• ••• + + +F F F F F F F F F Fσσ σ σ

1 1) )(= ( TJ

J− −

•• ••+ + +F F FFσ σσ σ

1 1) ) tr( (= tr=T T

− −• •• •+− −− +−−DFF FF DLLσσ σσ σ σ σσ

= tr− +−D D Dσ σσ σ (4.56)

4.5 Various Objective Stress Rate Tensors 123

1( )( ) = =T T T

v•Δ − − •≡ F F F FFSF τττ L

1 1 1T ) )( (= =T T− − −−• • • •• •+ + + +− −FF F F FF FFτ τ τ τσ σ

= T•+− − LLτ τ τ

) )( (= =T• •+ + +− − − + − −D D D DW W W Wτ τ τ τ τ τ τ τ

= − −D Dτ τ τ (4.57)

1 1( )=( ) = T T T

v c− −− −∇ • •≡ F FF FF F σσ σ σL

1( )=

T T T T −− • ••F + + F FF F F FF σ σ σ

= T• +L + Lσσ σ (4.58)

In the above equations, the symbol 1( ) ( )T Tv

− − •≡T F F TF FL (cf. Marsden

and Hughes, 1983 for general definition) designates the Lie derivative, i.e. the

push-forward operation TFTF of the material-time derivative of the pull-back

operation of 1 T− −F TF of the contravariant tensor. In addition, the symbol 1( ) ( )T T

v−− •≡T F F TF FL designates the push-forward operation 1T −−F TF

of the material-time derivative of the pull-back operation of TF TF of the

covariant tensor. The objective stress rate tensors described above are listed in Table 4.1.

Table 4.1. Various stress rate tensors

1

Jaumann rate of Cauchy stress :

Green - Naghdi or Dienes stress rate : ( ) =

Jaumann rate of Kirchhoff stress : tr =

Oldroyd stress rate : ( ) (= = ( ) T

T T R R

J

J v

J

J−

⊕

Δ

•

•

•

•

− +≡

− +≡

≡ +

≡

w w

R R R R Ω Ω

τ D

SF F F F

σ σ σ σ

σσ σ

τ σσ

σσ

σ σ

L

1

) =

( ) ( )Truesdell stress rate : tr== =

= tr

:Truesdell rate of Kirchhoff stress ( ) ==

Convec

T

TT T

T

T T

v

v

JJ J

− − ⊗Δ

Δ

•−

•

•

•

•

−−

+≡

− +−

−≡ −

F F LL

FS F F F F F S D

D D D

D DFS F

σ σ σ σ

σ τ σσ

σ σσ στ ττ ττ

L

L

1 1( ) =ted or Cotter - Rivlin stress rate : ( )=

=

T TT

T

v c−− −∇ • −

•

•≡ FF F F F F

+L + L

σ σσ

σσ σ

σL

(4.59)


The corotational stress rate tensors obtained directly from the basic equation (4.28) of corotational tensor fulfill the following relation, noting tr( ) = tr( )TT T Tω ω .

2t r( ) = 2t r( ) = 2t r( ) = 2t r( )

⊕•σ σσ σσ σσ (4.60)

In addition, the corotational rate of deviatoric tensor becomes deviatoric leading to

* *t r( ) = t r( ) = 0⊕σ σ (4.61)

Other objective stress tensors do not possess these properties. Although the objective rates are shown above for the stress rate, the objective

rate must be adopted also for rates of internal variables.

4.6 Work Conjugacy

Designating the volume in a specific region of material in the initial and the current configurations as V and v , respectively, the work rate W

• done in this region is

given from Eqs. (2.30)，(2.41)，(2.69)，(2.73)，(2.105)，(3.11)，(3.17) and (3.21) as follows:

= tr( ) = tr ( ) = tr( )v v v

W dv JdV dV•

∫ ∫ ∫D D Dσ σ τ (4.62)

1= tr( ) = tr( ) t r( )2v v

TW dv dv•

+∫ ∫ L σLDσ σ

1= tr( ) t r( ) =2

tr( )T TTv v

T dv dv+∫ ∫σLσ L σL

1= tr ) = tr( ) = tr( )( T

V V V

T T TJdV dV dVJ− −• • •∫ ∫ ∫FF F F Fσ σ Π

(4.63)

1 1= tr( ) = tr( ) = tr( )T T

v v vW dv dv dv− − − −

• ••∫ ∫ ∫Dσ σ σF EF F F E

1 1= tr( ) = tr( ) = tr( )V V V

T TJdV dV dVJ− − − −• • •

∫ ∫ ∫ SEσ σF F E F F E

(4.64)

21 1= tr( ) = tr( ) = tr ( )

2 2V V VW dV dV dV

•• • •∫ ∫ ∫SC S USE

1 1 tr( )= tr ( ) = tr( ) =2 2V V V

dV dV dV•• • •

∫ ∫ ∫S U U U U SU US U U+ + Ζ (4.65)

4.6 Work Conjugacy 125 where

1 ( )2

≡Ζ SU + US (4.66)

which is called the Biot stress tensor. The rate of work done per unit volume element in the reference configuration is

given as

t r( )= t r( ) = t r( ) = t r( ) = t r( /2) =Tw•• ••• τD ΠF SC ΖUSE (4.67)

The observation follows directly from the following equation (see Fig. 4.1).

( )= = = = Tr rrr rrr rw d

• • •• •• • • •e e e e e ef e Π F Π F F Π

t r( ) = ( ) = t r( ) =T T Trr

• • •F Π F Π ΠF (4.68)

noting Eqs. (2.10), (3.14) and (4.67) with = 1dA . The unit cubic cell having the

unit orthonormal outward-normal vectors re deforms to the current cell having the

outward-normal vectors re which are changed from re , while the tractions rdf

are applied to the cell in Fig. 4.1. The work rate influencing the constitutive property is not the one done for a

current unit volume but the one done for a reference volume element, i.e. a unit volume element at the initial state of deformation. The pairs of stresses and strain rates (or rates of deformation gradient) shown in Eq. (4.67) are called the work-conjugate pair. Formulation of a constitutive equation relating them pertinently is necessary for the description of finite deformation with finite rotation.

F

1df

1e

2e

1e

2e

2df2e

1e

3e 3e

3df

3d− f

1d− f

2d− f

−−

−

F

1df

1e

2e

1e

2e

2df2e

1e

3e 3e

3df

3d− f

1d− f

2d− f

−−

−

Fig. 4.1 Variation of the initial unit cell due to the deformation


Chapter 5 Elastic Constitutive Equations 5 Ela stic Constitutive Equatio ns

Elastic deformation is induced by the reversible change of distances between ma-terial particles without a mutual slip between them. They therefore exhibit high stiffness. Elastic constitutive equations are classifiable into several types depending on their levels of reversibility. As preparation for the study of elastoplasticity after the next chapters, they are explained in this chapter.

5.1 Hyperelasticity

The elastic material which has the strain energy function so that the one-to-one correspondence between stress and strain exists and the work done by the stress is independent of the loading path is called the hyperelastic material or Green elastic material. The constitutive equation of hyperelastic material is given as the following relation between the first Piola-Kirchhoff Π and the deformation

gradient F .

0

( )=

ψρ ∂∂

FΠ F (5.1)

under the condition [ / ] = , ( ) = 0ψ ψ∂ ∂ F=RF 0 R , where ( )ψ F is the specific

strain energy (per unit mass). Substituting Eq. (5.1) into Eq. (4.63) or (4.68), the mechanical work done by a surface loading on an enclosure of certain region of the initial volume V is described as

00 0

= =( )t r(t r( ) ] [ ) ][ T T

V V

t tW dVdt dt dV

ψρ •• ∂∂∫ ∫∫ ∫

FΠF FF

00

= ( () )=[ ]V V

tdVdt dVψ ρ ψ•

∫ ∫∫ F F (5.2)

Consequently, the work done in a specific region is independent of the loading path and therefore ψ has the status of the potential energy. The energy dissipation is not induced during the stress or strain cycle.

Substituting Eq. Eq. (5.1) into Eqs. (3.11), (3.17), (3.21), we obtain various ex-pressions of the hyperelasticity by the deformation gradient as follows:

128 5 Elastic Constitutive Equations

0

0

0

10

( () )1

( )

( )=

( )

T T

T

= =J

=

=

ψ ψρ ρ

ψρ

ψρ

ψρ −

∂ ∂∂ ∂

∂∂

∂∂∂

∂

F FF FF F

F FF

FΠ F

FFS F

σ

τ (5.3)

Furthermore, noting

= = = ( )QPQ

A QAri riP Q PPQ PQ PQiA iA iA

Pr rr r

F FCF F

F C F C F Cδ δ δ δ∂ ∂∂ ∂ ∂ ∂ +

∂ ∂ ∂ ∂ ∂ ∂

= = 2P Pi iQiAQ PA PA

F FFC C C∂ ∂ ∂+

∂ ∂ ∂ (5.4)

it holds that

1

1

2

1 12 2

2

= =

= =

= =

ψ ψ ψ

ψ ψ ψ

ψ ψ ψ

−

−

⎫∂ ∂ ∂⎪∂ ∂ ∂ ⎪⎪∂ ∂ ∂ ⎬∂ ∂ ∂ ⎪⎪∂ ∂ ∂⎪∂ ∂ ⎭∂

F FF C E

F FC E

F F CE

(5.5)

Then, substituting Eq. (5.5) into Eq. (5.3), the hyperelasticity is expressed in terms of the Green strain E as follows:

0 0

0 0

0 0

( () )= =2

( () )2= =

( () )= =2

( () )= =2

T T

T T

ψ ψρ ρ

ψ ψρ ρ

ψ ψρ ρ

ψ ψρ ρ

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

C EF F F FC EC EF F F F

C EC EF F

C EC E

S C E

σ

τ

Π (5.6)

5.1 Hyperelasticity 129

It holds from Eq. (1.192) for the isotropic material that

220 1

( )= E E E

ψφ φ φ∂ + +

∂E

I E EE

(5.7)

where 20 1, , E E Eφ φ φ are the functions of invariants of E . Eq. (5.7) reduces to the

following equation for the linear elastic material.

( )= ( t r ) EEa b

ψ∂ + 2∂

EE I E

E (5.8)

where , E Ea b are the material constants. The time-differentiation of Eq. (5.6) 4 leads to

0

2 ( )=

ψρ •• ∂∂ ∂

ES E

E E (5.9)

Substituting Eq. (2.106) and using the Truesdell rate of Kirchhoff stress Δτ in

Eq. (4.57), Eq. (5.9) is rewritten as

0

2 ( )= Τ ΤψρΔ ∂∂ ∂

E DF F FFE E

τ 0

2 ( )=( )ij lDC

AB CDkliA jB kF DFF F

E E

ψρτΔ ∂∂ ∂

E

(5.10) Here, Δτ is related to the Jaumann rate of Cauchy stress as

= )(det ( tr )Δ − − +F D D Dτ σ σ σ σ (5.11)

The relation between the Jaumann rate of Cauchy stress and the strain rate is given by

20 ( )

= trdetΤΤρ ψ∂ + + −

∂ ∂EF F FF D D DDF E E

σ σ σ σ (5.12)

which is expressed as

= EDσ (5.13)

where the elastic modulus tensor E is given by

20 ( )

det ijjBiAijkl kC lD ijkl klAB CD

F F F FE E Eσ

ρ ψΣ δ∂≡ + −

∂ ∂E

F (5.14)

130 5 Elastic Constitutive Equations with

1 ( ) ( = = = )2ijkl ik jl il jk jk il jl ik ijkl klij jikl ijlkσ δ σ δ σ δ σ δΣ Σ Σ Σ Σ≡ + + +

(5.15)

5.2 Cauchy Elasticity

The elastic material which does not have a strain energy function but has a one-to-one correspondence between the Cauchy stress and a strain is called the Cauchy elastic material. Here, the stress tensor is given by an equation of strain tensor and thus the equation involves six strain components. The equation of six strain components fulfills the condition of complete integration leading to the strain energy function, i.e. the hyperelasticity only in special cases. Then, the work done by the stress is generally dependent on the deformation path. For that reason, an energy dissipation is induced during the stress or strain cycle.

In the above-mentioned definition, the Cauchy elastic material is described as

= ( )f eσ (5.16)

Equation (5.16) reduces to the following equation by virtue of Eq. (1.202) for the isotropic material.

220 1= e e eφ φ φ+ +I e eσ (5.17)

where 20 1, , e eeφ φ φ are functions of invariants of e . Furthermore, for an isotropic

linear elastic material, Eq. (5.17) reduces to

= ( t r )a b+ 2e I eσ (5.18)

where , a b are the material constants. Limiting the infinitesimal strain leading to

≅e ε , Eq. (5.18) results in t r= ( )a b+ 2Iε εσ (5.19)

Here, substituting Eq. (5.19) into Eq. (3.24), the Navier’s equation is obtained as follows:

2 2, ) =) ( ) (=(

j ii i

jj ji

u u bb v a b ba b xx xxρρρρ •••

∂ ∂ +++ Δ + ++ ∂∂ ∂∂b vu u∇ ∇

(5.20) noting

2 2212

=( ) jik

ijjk j ji i

jj jj ji i

uu ua bx x u ux ua b bxx x xx x x

δ ∂∂∂ + +2∂ ∂∂ ∂ ∂∂ ∂+ +∂∂ ∂ ∂∂ ∂∂

5.3 Hypoelasticity 131 5.3 Hypoelasticity

The following material, for which the Jaumann rate of Cauchy stress is related linearly to the strain rate, is referred to as the hyperelastic material by Truesdell (1955).

= , = ijklij klH DσHDσ (5.21)

where H is the function of stress and internal variables. If Eq. (5.21) exhibits the isotropic rate-linearity, it is described as follows:

= (t r ) 2L G+D I Dσ (5.22)

Therein, , L G are the material parameters corresponding to the Lame coefficients in Hooke’s law for the infinitesimal deformation of isotropic linear elastic material. The following relation is obtained from Eq. (5.22), and thus G is called the shear modulus.

= 2ij ijGD' 'σ

(5.23)

where ( )' designates the deviatoric part described in 1.4.2. Equation (5.22) is expressed as follows:

= =, ij ijkl klE DσEDσ

(5.24)

where

( )2 , jlijijkl kl ik il jkL G E L Gδ δ δδ δ δ≡ ⊗ + ≡ + +E I I I (5.25)

is the tangent elastic modulus tensor, which hereinafter will be simply called the elastic modulus tensor. Equation (5.25) is further rewritten as shown below.

= (t r ) 2K G '+D I Dσ (5.26)

where

K L G2+≡ 3 (5.27)

Using these elastic moduli, the elastic modulus tensor E is given as

1

23

( ),

1 1 1 1( ) ( )= 423 3

)(

( ) ij kl ik jl il jkijkl

ijijkl kl ik jl il jkE = K G G

G GK

δ δ δ δ δ δ

δ δ δ δ δ δ−

⎫⎪⎪⎬

− ⎪⎪⎭

− + +

+ +E (5.28)

in the form of Hooke’s law.

132 5 Elastic Constitutive Equations

The following equation is derived from Eq. (5.26) and thus K is called the bulk modulus.

=m vKDσ•

(5.29)

where ( )m designates the mean part described 1.4.2.

Furthermore, the following is obtained from Eqs. (5.23) and (5.29).

1 1= , =3 2 3 2 ijij ijm m D

K G K G' 'σσ σ δ• •1 1+ +D I σ (5.30)

from which we have

1 1= , = ijij ijm m DE E E Eν ν ν ν σσ σ δ• •+ +−3 + −3 +D I σ (5.31)

where

9 3 2, )2(3 3

KG K GEK G K G

ν −≡ ≡+ + (5.32)

The expressions below hold from Eq. (5.31) in the uniaxial loading process

( for = 1= 0 ij jiσ ≠ ).

2211 11 1122

11

1= =, =DD D

E E Dνσ σ ν− → − (5.33)

E is the ratio of the axial stress rate to the axial strain rate and is called the Young’s modulus, and ν is the ratio of the magnitude of lateral strain rate to that of axial strain rate and is called the Poisson’s ratio.

The isotropic linear hypoelastic material has two independent material parame-ters, as described above. They are listed in Table 5.1.

Table 5.1 Relationships between two independent material constants

,E ν ,G ν ,E G ,E K ,G K ,L G

E E 2(1 )Gν+ E E 93

KGK G+

( )3 2GLL G

μ ++

G 2(1 )E

ν+ G G 39

EKK E− G G

K 3(1 )2E

ν−2(1 )

(1 )3 2Gν

ν+− ( )3 3

EGG E− K K 2

3L G+

ν ν ν 22

E GG

− 36

K EK− 3 2

2(3 )K G

K G−

+ 2( )L

L G+

L (1 )(1 )2Eν

ν ν+ −21

Gνν−

( )23E GGG E

−−

( )3 39

K K EK E

−−

23K G− L

5.3 Hypoelasticity 133

The following equation in which the Jaumann rate of Cauchy stress is related nonlinearly to the strain rate is called the hypoplastic material (Kolymbas and Wu, 1993)．

= ( , ), = ( , )ijij kl klf D σσf Dσ σ

(5.34)

where ijf is the nonlinear function of klD and the stress, and for rate-independent

deformation it is the homogeneous function of klD in degree-one fulfilling

( ) = ( )ij ijkl klf D f Ds s which implies ( / ) =ij kl ijklf D D f∂ ∂ on account of

Euler’s theorem for homogeneous function (see Appendix 3). While the tree popular elastic materials are described above, the special elastic

material, called the Cosserat elastic material, is advocated by Cosserat and Cosserat (1909). The couple stress is related to the rotational strain in this material. It has been applied to the prediction of localized deformation (e.g. cf. Mindlin, 1963; Muhlhaus and Vardoulaskis, 1987).


Chapter 6 Basic Formulations for Elastoplastic Constitutive Equations

6 Basic Formulations for Elastoplastic Constitutive Equations

As described in Chapter 5, the elastic deformation is induced microscopically by the deformations of the material particles themselves, which returns to the initial state if the applied stress is removed. Therefore, it has a one-to-one correspondence to the stress. On the other hand, when the stress reaches the yield stress, slippage between material particles is induced and they do not return to the initial state even if the stress is removed, which leads macroscopically to the plastic deformation. For that reason, one-to-one correspondence between the stress and the strain, i.e. the stress-strain relation, observed in the elastic deformation does not hold in the elas-toplastic deformation process. Therefore, one must formulate the constitutive equation as a relation between the stress rate and the strain rate in that process. This chapter presents a description of the basic concept and formulation for elastoplastic constitutive equations within the framework of conventional plasticity (Drucker, 1988) premised on the assumption that the inside of the yield surface is a purely elastic domain for the introductory to elastoplasticity. The unconventional plasticity describing the plastic strain rate induced by the rate of stress inside the yield surface will be described in subsequent chapters.

6.1 Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate

6.1 Mult iplicative Decomposition of Deformation Gradie nt

Variation in characteristics of the mechanical responses of materials arises from irreversible changes of their internal structures. The changes of internal structures are induced macroscopically by plastic deformation but they must be free from elastic deformation. Therefore, the rigorous decomposition of deformation into the elastic and the plastic deformations is necessary for description of elastoplastic deformation. Strictly speaking, the elastic deformation must be described by the purely elastic, i.e., hyperelastic constitutive equation, in which the elastic defor-mation returns to an initial state during a stress cycle and the work done during that cycle is zero, as described in Chapter 5. The decomposition of deformation into elastic and plastic deformations is the basic concept of elastoplasticity.

On the premise of the definite decomposition of the deformation into the elastic and the plastic deformations as described above, the individual constitutive rela-tions are formulated for these deformations in the elastoplasticity. Then, one must first decompose the current deformation into the elastic and the plastic deformations

136 6 Basic Formulations for Elastoplastic Constitutive Equations

explicitly. The elastic deformation is known from the current stress since it is re-lated uniquely to the stress, then the plastic deformation is known as that remainder. This is realized through an explicit process: incorporating the intermediate con-figuration unloaded obeying the elastic constitutive relation from the current con-figuration, one regards deformation from the initial to the intermediate configura-tion as plastic deformation and the deformation from the intermediate to the current configuration as the elastic deformation. Here, note that a constitutive relation concerns with a homogeneous state of deformation. However, residual stresses exist even if a load is removed from the current configuration of a body undergoing an inhomogeneous deformation because a plastic deformation which is not re-moved by an unloading is induced non-homogeneously in general. Therefore, in order that all material points reach the stress-free state, individual material points must undergo different amount of de-stressing. Then, intermediate configuration can be defined only locally (point-wise). Thus, we suppose conceptually virtual intermediate configurations for each infinitesimal material elements by cutting a material into them. Based on this idea, the multiplicative (or Lee) decomposition of the deformation gradient F into the elastic deformation gradient eF and the plastic deformation gradient pF was proposed by Eckart (1948), advanced by Kroner (1960) and further developed by Lee and Liu (1967), Lee (1969), Mandel (1972), Lubarda and Lee (1981), etc. (see Fig. 6.1) as follows:

= == , ( )p

A A

iie x x XX XX

α

α

∂ ∂∂ ∂ ∂∂∂ ∂∂ ∂∂ ∂

x x XF F F X XX

(6.1)

where

, = = = = ( () )pi pi A

A

ee x XF FX X

αα α

α

∂ ∂∂ ∂∂∂ ∂ ∂

x XFF XX (6.2)

X represents the position vector of material particle in the intermediate configu-ration and the mapping is restricted to the class of homogeneous deformation. F is

termed the Eulerian–total Lagrangian two-point tensor, eF the Eulerian-mobile

Lagrangian two-point tensor, pF the mobile Lagrangian-total Lagrangian tensor. When the stress reached the current state, different plastic deformations are in-

duced depending on the loading path along which the current stress is realized and the elastic and plastic deformations are induced in mixed order in real unloading process. Further, a purely elastic unloading path does not necessarily exist in a real loading process. However, it is noteworthy that the real existence of purely elastic unloading process is not required since the intermediate configuration is known by the elastic constitutive relation.

It holds from Eq. (6.1) with Eq. (1.22) that

det = det( ) = det detp pe eF F F F F (6.3)

On the other hand, presuming that line-elements , , a b cΔ Δ ΔX X X in the initial

state change to , , a b cΔ Δ Δx x x in the current state and to , , a b cΔ Δ ΔX X X in

6.1 Multiplicative Decomposition of Deformation Gradient 137

X

F

eF

pF

1 1 1, , X X x

02 2 2, , X X x

3 3 3, , X X x

x

dXdX

dx

XX

F

eF

pF

1 1 1, , X X x

02 2 2, , X X x

3 3 3, , X X x

x

dXdX

dx

XX

Fig. 6.1 Multiplicative decomposition of deformation gradient

the intermediate state, and denoting the initial, current and intermediate volume

elements as V , v and V , respectively, it holds that

= ( ) = = deta cb a cbjijk kiv x x xε•Δ Δ × Δ Δ Δ Δ Δ Δx x x x

1 1 1

2 2 2

3 3 3 3

1 1 1 1 1 111 12 13

2 2 2 21 22 23 2 2 2

31 32 33 33 3 3 3

= = = =

a ca b c a b cbR R R R R R

a b c a b c a b cR R R R R R

a b c a b cc b cR R R R R R

F X F X F Xx x x X X XF F F

x x x F X F X F X F F F X X X J V

F F Fx x x X X XF X F X F X

Δ Δ ΔΔ Δ Δ Δ Δ Δ

Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ

Δ Δ Δ Δ Δ ΔΔ Δ Δ

1 1 1 11 12 13

21 22 232 2 2

3 31 32 333 3 3

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3

= = =

a b ca b c a b cR R R R R R


a b c aa b cR R R R R R

e e e e e e

e e e e e e

e e ee e e

F X F X F Xx x x X X XF F F

v x x x F X F X F X F F F X X X

F F Fx x x XF X F X F X

Δ Δ ΔΔ Δ Δ Δ Δ Δ

Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ

Δ Δ Δ ΔΔ Δ Δ 33

=b c

eJ V

X X

Δ

Δ Δ

1 1 1 11 12 13

21 22 232 2 2

3 31 32 333 3

11 1 11 1

2 22 2 22

2 3 3 3

= = =

a b ca b c a b cR R R R R R


a b c aa b cR R R R R R

p p p p p p

p p p p p p

p p pp p p

F X F X F XX X X X X XF F F

V X X X F X F X F X F F F X X X

F F FX X X XF X F X F X

Δ Δ ΔΔ Δ Δ Δ Δ ΔΔ Δ Δ Δ Δ Δ Δ Δ Δ Δ

Δ Δ Δ ΔΔ Δ Δ 3 3

=b c

pJ V

X X

Δ

Δ Δ

It obtained from these equations that

= peJ J J (6.4)


where

det = , det = , det =ppeev v VJ J JV VV

Δ Δ Δ≡ ≡ ≡Δ ΔΔF F F (6.5)

Equation (6.4) engenders the additive decomposition of the logarithmic volu-

metric strain vε into the elastic logarithmic volumetric strain evε and the plastic

logarithmic volumetric strain pvε , i.e.

= pev v vε ε ε+ (6.6)

where

ln = ln ln = ln ln = lnppeev v v

v v VJ J JV VV

ε ε ε≡ , ≡ , ≡ (6.7)

for a homogeneous deformation. It holds from Eqs. (6.7) that

= = = = = =ppp

eeev vv

v vJ J V J Vv vJ J JV V

ε ε ε• •• • •• •• • •, − , (6.8)

Substitutions of Eqs. (6.1) into the velocity gradient L in Eq. (2.69) leads to

11 1= =p p pe e ee e− − −• •+ +L F F F F F F L L (6.9)

where

1

1

1

=

= =

=

( )

p p p

p p

e ee

ee

−

−

−

••

• ••

•

⎫∂ ∂ ⎪≡ ∂∂ ⎪⎪⎪∂∂ ∂ ⎬≡ ∂ ⎪∂ ∂⎪

∂ ∂ ⎪∂≡⎪∂∂ ∂ ⎭

x XL F F xX

XX XL F F X X X

x X XL F FLxX X

(6.10)

Furthermore, , , ppeL L L can be decomposed additively into the symmetric and skew-symmetric parts as follows:

=

=

=

p pp

p p p

e ee + ⎫⎪+ ⎬⎪

+ ⎭

D WL

L D W

L D W

(6.11)


where

1

1

= ( ) = ( ) ,

= ( ) = ( )

e e e e

e e e e

s s

aa

−

−

•

•

⎫⎪⎬⎪⎭

D L F F

W L F F

(6.12)

1

1

= ( ) = ( ) = ,

= ( ) = ( ) =

( )

( )

pp p p

pp p p

s s s

aa a

−

−

••

••

⎫∂ ⎪

⎪∂⎬⎪∂ ⎪

∂ ⎭

XD L F FX

XW L F FX

(6.13)

1 11

1 11

) )( (= ( ) = ( ) = ,

) )( (= ( ) = ( ) =

p p p pp

p p p pp

e e ee ee

e ee eee

s ss s

aaaa

− −−

− −−

⎫+ ⎪⎬

+ ⎪⎭

DD L F L F F F F W F

DW L F L F F F F W F

6.14)

where ( )s and ( )a denotes the symmetric and the skew(anti)-symmetric parts,

respectively. In the similar manner as in Eq. (2.98) it holds that

( ) =d d dδ δ δ•• •• • •X X X X + X X

= =[ ( ) ]Td d dδ δ δ

• • • •

•• •∂ ∂ ∂ ∂+∂∂ ∂ ∂

X X X XX X + X X X XX X X X

1( )= = Tp pe ed dδ δ−−• •2 2D X X F D F x x (6.15)

The strain rate D and the spin W are additively decomposed by substituting Eq. (6.9) into Eqs. (2.73) and (2.74) as follows:

11

1 1

)()= (

)()= (

) )( (= =

) )( (= =

p p p

p

p

p p p

e ee e e e

e ee e

e

e e e

s ss s

aaaa

−−

− −

+ + + +

++ + +

DD D D D F F F W F

DW W W W F F F W F

L L

L L (6.16)

Furthermore, the following equations hold for the volumetric strain rate from Eqs. (2.73), (6.10), (6.12)-(6.14).


1 2 3

1 2 3

1 2 3

1 2 3

( )t r = t r = = = = ,

( )t r = t r = = = = ,

t r = t r = t r = = =

t r = t r t r = =

( )

i

i

pp p i

i

pp p p

p eee

v v

vv

vv v

vv

x x xx vD vx x x x

X X XX VDVX X X X

VD DV

v VD v V

ε

ε

ε

ε

• • •• •

• •• • •

•• ••

•• •

⎫∂ ∂ ∂∂∂≡ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂∂∂≡∂∂ ∂ ∂ ∂

∂∂ ∂ ∂≡ ∂∂ ∂ ∂

−≡ −

xDx

XDX

x XX XD xX X X

D D D

⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(6.17)

from which the additive decomposition holds exactly for the volumetric strain rate, i.e.

= pev v vD D D+ (6.18)

Now, let ( )tQ be a time-dependent rotation superimposed on the current con-figuration, by which deformation gradient is given as

=∗F QF (6.19)

Here, let it be postulated that the initial and the intermediate configurations are fixed so that they are not affected by the superposition of rigid-body rotation (Dashner, 1986). Then, it holds that

= ; = , =p pp ee e ∗∗ ∗ ∗∗F F F F QF F F (6.20)

The elastic part of the right Cauchy-Green tensor in Eq. (2.30) defined in the in-

termediate configuration, denoted eC , is given by Eq. (6.20) as

= = ( ) =T TTe e e eee ee∗ ∗ ∗ ≡C F F QF QF F F C (6.21)

Therefore, the hyperelastic constitutive equation described by Eq. (5.6)4 , i.e.

0 (= )2 /e eρ ψ ∂∂ CS C holds independent of the superposition of rigid-body

rotation, noting =∗S S in Eq (4.19), where S is the second Piola-Kirchhoff

stress defined in the intermediate configuration ( F is replaced to eF ), the explicit expression of which will be shown in 6.7. Eq. (6.20) imposes the basis describing elastic deformation, i.e. the intermediate configuration to be unaffected by the rigid-body rotation. Thus, the multiplicative decomposition is required to obey Eq. (6.20) for the fulfillment of objectivity of elastic constitutive equation.


Now, consider the influence of the rigid-body rotation on the main rate variables for the multiplicative decomposition fulfilling Eq. (6.20). The substitution of Eq. (6.20) into Eq. (6.10) leads to

1 1( ) ( ) ( )= =e e e ee − −• •∗ ∗ ∗ Q QF F F FL

1 1( )= ee e − −• •+Q Q QF FF

1= TTe e −• •+Q QQQF F

)(= Te −Q Ω QL (6.22)

where Ω is the spin of the base i∗e rotating with a material as shown in Eq.

(4.12). Then, we have

1 )( == 2

1 ) )(( == 2

T T

T T

ee e e

ee ee

⎫∗ ∗ ∗+ ⎪⎪⎬∗ ∗ −∗ − ⎪⎪⎭

QD QD L L

Q Ω QL L WW

(6.23)

Next, substituting Eq. (6.20) into Eq.(6.10), one has

1 11

=

(( ) )= = = =

p p

p p ppp T Te ee e ee − −−

⎫∗ ⎪⎬∗ ∗∗ ∗⎪⎭

L L

Q Q Q Q QL F L F F L F F L F QL

(6.24)

Then, It holds from Eq. (6.13) and (6.14) that

= , = p p p p∗ ∗D D WW (6.25)

= , = p p p pT T∗ ∗D QD Q Q QW W (6.26)

Consequently, the elastic strain rate eD and the plastic strain rate pD indicate the

objectivity. Apart from the exact formulation conforming to the hyperelasticity, let it be

postulated that the unloading process to the intermediate configuration is a purely elastic deformation (Lee, 1969) and the elastic deformation is infinitesimal (Dafalias, 1985). Then, it holds in the polar decomposition of elastic deformation

gradient =e e eF V R that

= , =e e e≅F V I R I (6.27)


In this case, substituting Eq. (6.27) into Eq. (6.16), the following equation holds.

1

11

= = )( ) , ( )( = ==

( ( ) ( ) )== , = =

pp p pp

pp p p p

e eee

eee e a a

ss−

−−

•• •

• •

⎫+ ⎪⎬⎪+ ⎭

D D F FD V VD D D

W W W W V V W W F F

(6.28) with an error of first order in small elastic strain. Here, note that the rigid-body

rotation is attributed to the variation of pF in direct opposition to the

above-mentioned Dashner’s (1986) notion in Eq. (6.20) and that pW differs sub-stantially from the so-called plastic spin (Dafalias, 1985) as will be described in Chapter 12. The additive decomposition of strain rate D into the elastic strain rate

eD and the plastic strain rate pD in Eq. (6.28) is adopted in the formulation

of elastoplastic constitutive equations with the hypoelasticity for eD in the subsequent sections.

6.2 Conventional Elastoplastic Constitutive Equations

First, let the elastic strain rate be given by the following hypoelastic constitutive equation from Eq. (5.24).

1=e −D E σ (6.29)

Next, the plastic strain rate must be formulated. As described in the beginning, let the conventional elastoplastic constitutive equations (Drucker, 1988) be de-scribed below.

Now, to formulate the plastic strain rate, consider first the following isotropic yield condition exhibiting the isotropic hardening/softening.

( ) ( )=f F Hσ (6.30)

where F is the function of the isotropic hardening variable H and is called the

hardening function. Hereinafter, assume that the yield stress function ( )f σ is the

function of stress invariants and is the homogeneous function of σ in degree-one. Therefore, it holds that

( ) = ( )f fs sσ σ (6.31)

for the arbitrary positive scalar s and

( )tr = ( )( )f

f∂∂σ σ σσ (6.32)

for sake of Euler’s theorem for homogeneous function in degree-one (see Appendix 3). Then, it holds from Eqs. (6.30) and (6.32) that

6.2 Conventional Elastoplastic Constitutive Equations 143

( )t r

( ) ( )= == ) ) )tr( tr( tr(

( )ff f F

∂∂ ∂

∂N N N

N N N

σ σσ σσσ σ σ σ

(6.33)

where N is the normalized outward-normal of the yield surface (see Appendix 4), which is described as

( ) ( ) ( = 1)

f f∂ ∂≡ ∂ ∂ NNσ σσ σ

(6.34)

Here, the yield surface in Eq. (6.30) retains the similar shape and orientation with respect to the origin of stress space by virtue of homogeneity of function ( )f σ .

Taking the material-time derivative of Eq. (6.30) and considering the transfor-mation to the corotational derivative (Hashiguchi, 2007) described in Chapter 4, one has the consistency condition:

( )tr =( )f

F H'•∂

∂σ σσ

(6.35)

where ( 0)/F HdF d' ≡ ≥ (6.36)

H•

is assumed to be the function of stress, internal variables and the plastic strain rate, i.e.

;( ),= pih HH

•Dσ (6.37)

i H ( 1, 2, )=i • •• signifies internal variables collectively. Here, note that h has

to be the homogeneous function of pD in degree-one fulfilling ;( , ) iph H sDσ

( , );= ipHhs Dσ for the rate-independent deformation behavior, while, need-

less to say, h is a nonlinear equation of pijD in general as seen in metals described

later. The substitution of Eq. (6.33) into the consistency condition (6.35) leads to

( )t r ) tr(=F HF

' •NN σσ (6.38)

Further, assume the plastic flow rule

(|| || 1)=p λ ≠M MD (6.39)

where λ is the positive proportionality factor and M is the second-order tensor function of stress and internal variables. Substituting the flow rule (6.39) into the consistency condition (6.38) with Eq. (6.37), one has


;( ) ( )tr ) tr( ,= iF HhF' λN MN σ σσ

from which it is derived that

t r( )p=

Mλ Nσ

,

t r( )pp=

MND Mσ (6.40)

where pM is called the plastic modulus and is given by

( );( ), t r p

iF h HM F

′≡ NM σσ (6.41)

Substituting Eq. (6.29),

(6.40) into Eq. (6.23)1, the strain rate is given by

1 t r( )= pM

− + N MD Eσσ (6.42)

The proportionality factor described in terms of strain rate, denoted by Λ instead of λ , in the flow rule (6.39) is given from Eq. (6.42) as follows:

t r( )=

tr( )pMΛ +

EDNNEM

(6.43)

Actually, Eq. (6.43) can be derived directly by substituting the flow rule (6.39) into the following consistency condition in terms of the strain rate obtained by substituting Eqs. (6.23), (6.29), (6.33) and (6.37) into the consistency condition (6.38) in terms of stress rate.

;) ) ( )tr tr( ,( = p

iF HhF' λ− N MNE D D σ σ (6.44)

The inverse relation of Eq. (6.42) is given by using Eq. (6.43) as follows:

t r( ) ==tr( ) t r( )

( )p pM M⊗

+ +DNE EM ENEM DED E

NEM NEMσ − −

= pqpq mn mnij ijrs rs ijrs rsp

cdab abcd

N E DE D E MN EM M

σ − + (6.45)

Using the plastic relaxation modulus tensor

t r( )p

pr

M⊗≡ +

ME ENK NEM,

p ijrs pq pqrs klijkl p cdab abcd

r E N EMKN E MM

≡ + (6.46)


the plastic relaxation stress rate pσ is described as

= pp p r≡ DKEDσ

Furthermore, using the elastoplastic stiffness modulus tensor

= , tr( )pp

pe r

M⊗≡ − −

+EM ENE K EK

NEM

(6.47) = ijrs pq pqp rsp klijkl ijklijkl ijkl p cdab abcd

e r E N EMK KE EN E MM

≡ − − +

the stress rate can be described as

= pe DKσ As known from Eqs. (6.46) and (6.47), the plastic relaxation modulus tensor and

the elastoplastic stiffness modulus tensor are the asymmetric tensors, i.e.

, p p p pT Te e≠≠K K K K

in general. As described in the next section, however, these stiffness modulus ten-sors become symmetric when the direction of plastic strain rate is chosen to be the outward-normal of yield surface, i.e. =M N . In this case the plastic flow rule is called the associated flow rule or the normality rule because the direction of plastic strain rate is given using the yield function as the plastic potential function. Here-inafter, the explicit equations of the above-mentioned constitutive equations adopting the associated flow rule are shown collectively as a support for the explanation of subsequent formulations.

=p λ ND , t r( )p

p=MN NDσ (6.48)

;( ) ( ), t r ip F h HM F

′≡ NN σσ (6.49)

1 t r( )= pM

− + N ND Eσσ (6.50)

t r( )=

tr( )pMΛ +

EDNNEN

(6.51)

t r( ) ==tr( ) t r( )

( )p pM M⊗

+ +DNE EN ENEN DED ENEN NEN

σ − − (6.52)


t r( )p

pr

M⊗≡ +

EN ENKNEN

，

t r( )p

pe

M⊗≡ −

+EN ENK E

NEN (6.53)

The elastoplastic constitutive equations for isotropic materials are described above. The constitutive equation of metals is shown below, which has made an important contribution to the development of elastoplasticity. The following von Mises yield condition with the associated flow rule is assumed for metals.

3( ) = , || ||2

2( ) = ( ), = || ||3

=

pp p

e e

e e

f

F H F H dt

'σ σ

ε ε

⎫⎪≡ ⎪⎪⎪⎬≡ ⎪⎪⎪⎪⎭

∫ D

M N

σ σ

(6.54)

In the monotonic simple tension ( = 0ijσ except for = = 1ji , 1122 33= = / 2p p pD D D−

(tr =0)pD , = 0pijD ( )i j≠ ) it holds that

1111 11 11

2 21111 11 11

= 3/2 ( ) 2(0 ) =/ 3/ 3

2 2( / 2) = =3pp p pp

e

e dt D dtD D

σσ σ σ σ

ε ε

⎫− + − ⎪⎬−≡ + ⎪⎭

∫ ∫ (6.55)

Then, eσ and peε coincide with the axial stress and the axial plastic strain in that

loading and thus are called the equivalent stress and the equivalent (or accumu-

lated) plastic strain, respectively. Substituting Eq. (6.54) into Eqs. (6.50)-(6.52)

and using the relations

( ) 3 || ||= , = , t r( ) =2 || || || ||

3 3tr= = tr( )2 || || 2

=

2 2 2; ( , ( ;) , ) || ||= = =|| ||, 3 3 3

2 2|| ||= =3 3

( )

i i

p p

p p

p

e

f

H h HH

FM FF

' ' '' '

' '''

' ''

σ•

•

⎫∂ ⎪

⎪∂⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

N N

N

D D

D ND N

σ σ σ σ σσ σ σ

σ σσσ

σ σ

σ

(6.56)


the constitutive equation of the isotropic Mises material is given as follows:

1 1 1

23tr( ) 1= = =|| ||22

33

ee

eFFF

' '' '''

σσσ

− − −

••3+ + + 2

N ND E E Eσ σ σσ σ σσ

(6.57) which is called the Prandtl-Reuss equation. The plastic work rate of this material is described as

t r( t r || || || |||| ||) = = = = =|| ||( )p p p pe ee F' ' ''σλ λ ε ε• •D Dσ σ σσ σ σ

(6.58)

which is the product of the hardening function and the equivalent plastic strain and thus the hardening attributable to the equivalent plastic strain is also called the work hardening.

The traction t acting on the plane having the normalized outward-normal vector

me inclined identically to the principal stress direction, called the octahedral

plane, is given by

1 1 1= = ( )3 3 3

( )m σ σ σ+ + + +⊗ ⊗ ⊗t e e e e e e e e eeσ Ⅲ Ⅲ Ⅲ ⅢⅠ Ⅰ Ⅰ Ⅱ Ⅱ Ⅱ Ⅰ Ⅱ

1 = ( )3

σ σ σ+ +e e eⅢ ⅢⅠ Ⅰ Ⅱ Ⅱ (6.59)

noting Eq. (3.7). Then, the normal component octσ and the tangential component

octτ of the traction t is given as

oct1 1 1 1= ( )3 3 3 3

( )mσ σ σ σ• •≡ + + + +t e e e e e ee Ⅲ Ⅲ ⅢⅠ Ⅰ Ⅱ Ⅱ Ⅰ Ⅱ

1= ( ) =3

mσ σ σ σ+ + ⅢⅠ Ⅱ (6.60)

2 2oct oct|| ||τ σ−≡ t

22 2 21 1 = ( ) ( )93σ σ σ σ σ σ−+ + + + ⅢⅠ ⅡⅠ Ⅱ Ⅲ

2 2 22 = ( )9

σ σ σ σ σ σ σ σ σ−+ + + +Ⅲ ⅢⅠ Ⅱ Ⅱ ⅠⅠ Ⅱ Ⅲ

2 2 22

1 2 = ( ) ( ) ( ) =3 3

Jσ σ σ σ σ σ− + − + −Ⅲ ⅢⅠ Ⅱ Ⅱ Ⅰ (6.61)


where octτ is called the octahedral shear stress and 2J is given by choosing the deviatoric Cauchy tensor 'σ as the second invariant of the deviatoric tensor 'T in Eq. (1.150), i.e.

222

1 1 1|| || t r= =2 2 2 rs srJ ' ' '' σ σ≡ σ σ

22 22 22

11 22 33 12 23 311 ( )= 2 ' ' ' ' ' 'σ σ σ σ σ σ+ + + + +

2 2 2 22 21 1( ) ( ) ( ) ( ) = =2 6' ' 'σ σ σ σ σ σ σ σ σ+ + − + − + −Ⅰ Ⅱ Ⅱ Ⅲ Ⅲ ⅠⅠ Ⅱ Ⅲ

(6.62) It is interpreted from Eqs. (6.54), (6.61) and (6.62) for the Mises yield condition

that the yielding is induced when the octahedral shear stress reaches a certain value.

Eqs. (6.60) and (6.61) are also derived as the components in the directions

= ( )3m mI e and 3 't , i.e. mT and || || / 3'T of mT and 'T , respectively.,

regarding T as σ in Eq. (1.210).

6.3 Loading Criterion

The judgment of whether or not the plastic strain rate is induced for a given incremental loading is required for the elastoplastic deformation analysis. The criterion for this judgment is called the loading criterion. In what follows, this criterion is formulated (Hashiguchi, 2000).

1. It is required that = 0λ Λ > (6.63)

in the loading (plastic deformation) process p ≠D 0 .

2. It holds that

t r( ) 0≤Nσ (6.64)

in the unloading (elastic deformation) process =pD 0. Further, substituting

= eD D leading to = tr( )tr( ) tr( )= e NED EDN N σ into Eq. (6.43), Λ is described in this process as

t r( )=t r( )pM

Λ +N

NEMσ

. (6.65)

in this process. 3. The plastic modulus pM takes both positive and negative signs in general.

On the other hand, noting that the elastic modulus E is the positive definite tensor and thus it holds that t r ( ) pMNEM in general and postulating that the plastic relaxation does not proceed infinitely, let the following ine-quality be assumed.

6.3 Loading Criterion 149

t r( ) 0pM >+ NEM . (6.66)

The ultimate state fulfilling t r( ) 0pM →+ NEM leading to p

ijklrK → ∞ is

illustrated for the uniaxial loading process in Fig. 6.2.

Then, in the unloading process =pD 0 , the following inequalities hold from

Eqs. (6.40) and (6.63)-(6.66), depending on the sign of the plastic modulus pM leading to the hardening, perfectly-plastic and softening state.

0 and 0 when 0

or indeterminate and 0 when = 0

0 and 0 when 0

p

p

p

M

M

M

λ Λλ Λλ Λ

⎫≤ ≤ >⎪

→ −∞ ≤ ⎬⎪<≥ ≤ ⎭

(6.67)

Consequently, the sign of λ at the moment of unloading from the state 0pm ≤ is

not necessarily negative. On the other hand, Λ is necessarily negative in the unloading process. Thus, the distinction between a loading and an unloading proc-ess cannot be judged by the sign of λ but it can be done by that ofΛ . Therefore, the loading criterion is given as either

: and 0( ) ( )=

= : otherwise

p

p

f F H Λ ⎫≠ ⎪⎬⎪⎭

>D 0

D 0

σ (6.68)

ε

σ

0

0pM >

t r( ) 0t r( ) 0>

>NNEDσ

t r( ) 0t r( ) 0

=>

NNEDσ

t r( ) 0t r( ) 0<

<NNEDσ

0=pM

t r( ) 0t r( ) 0<

<NNEDσ

t r( ) 0t r( ) 0<

<NNEDσ

t r( ) 0t r( ) 0>

<NNEDσ 0pM <

t r ( ) 0p

pijkl

rMK

+ →→ ∞

NEM

ε

σ

0

0pM >

t r( ) 0t r( ) 0>

>NNEDσ

t r( ) 0t r( ) 0>

>NNEDσ

t r( ) 0t r( ) 0

=>

NNEDσ

t r( ) 0t r( ) 0

=>

NNEDσ

t r( ) 0t r( ) 0<

<NNEDσ

t r( ) 0t r( ) 0<

<NNEDσ

0=pM

t r( ) 0t r( ) 0<

<NNEDσ

t r( ) 0t r( ) 0<

<NNEDσ

t r( ) 0t r( ) 0<

<NNEDσ

t r( ) 0t r( ) 0<

<NNEDσ

t r( ) 0t r( ) 0>

<NNEDσ

t r( ) 0t r( ) 0>

<NNEDσ 0pM <

t r ( ) 0p

pijkl

rMK

+ →→ ∞

NEM

Fig. 6.2 Signs of t r( )Nσ t r( )Nσ t r( )Nσ and t r( )NED t r( )NED t r( )NED in uniaxial loading


or

: and tr( ) 0( ) ( )=

= : otherwise

p

p

f F H ⎫≠ ⎪⎬⎪⎭

>EDD 0 N

D 0

σ (6.69)

(Hill, 1967, 1983) in lieu of Eq. (6.66). Limiting to the hardening process, Eq. (6.69) leads to

tr( ): and 0( ) ( )=

= : otherwise

>p

p

f F H⎫⎪≠⎬⎪⎭

ND 0

D 0

σσ (6.70)

Denoting the infinitesimal total strain and plastic strain by ε and pε , respec-tively, the elastic strain

=e p−ε ε ε (6.71)

is related uniquely to the stress. Then, substituting Eq. (6.71) in to Eq. (6.30), the yield surface can be described in the strain space as follows:

) )( ( ) or ( ( )= =peg gF H F H−ε ε ε (6.72)

setting ) )( ( ( ) e eg f≡ε εσ . Then, one has

1 1) ) ) ) )( ( ( ( (= = = =)(

p p

p

e e eee e

g g g gf − −− −∂ ∂ ∂ ∂ ∂∂∂ ∂ ∂−∂ ∂ ∂

D E Eσ ε ε ε εε εε

εσ σε ε ε εσ (6.73)

The substitution of Eq. (6.73) into t r( ) 0>EDN in Eq. (6.69) leads to

)(t r( ) 0

pg −∂∂ >Dε εε (6.74)

Therefore, it can be interpreted that loading occurs when the strain rate is di-rected the outward-normal of the yield surface in the strain space.

In elastoplastic deformation analysis, suppose to calculate first σ and D by

either of the elastic or the elastoplastic constitutive equation. Then, check the sign of t r( )NED . If the sign conflicts with the loading criterion, it is required to re-

calculate them using other constitutive equation. Here, it would be efficient to

calculate first by the elastoplastic constitutive equation since the monotonic loading process in which the elastoplastic deformation process continues is seen often in

practical engineering problems.

6.4 Associated Flow Rule 151

6.4 Associated Flow Rule

The associated flow rule holds under some assumptions for a wide range of mate-rials. Some mechanical interpretations for the associated flow rule are delineated in this section.

6.4.1 Positivity of Second-Order Plastic Work Rate: Prager’s Interpretation

Prager (1949) reported that the associated flow rule must hold to fulfill the posi-tivity of the second-order plastic work rate. i.e.

t r( ) / 2 0ppw ≡ ≥Dσ (6.75)

However, Eq. (6.75) holds only for the hardening process in which the stress rate is directed the outwards of the yield surface.

6.4.2 Positivity of Work Done During Stress Cycle: Drucker’s Hypothesis

Drucker (1951) postulated “the work done during the stress cycle by the external agency is positive”. It is described mathematically as follows:

0t r ) 0dtσ

( − ≥∫ Dσ σ (6.76)

where 0σ denotes the initial stress. The following inequality is obtained from

Eq. (6.76) under the assumption that the inside of the yield surface is elastic domain

(see Fig. 6.3).

0 ) 0py( − ≥Dσ σ (6.77)

where yσ designates the stress on the yield surface in which the plastic strain rate pD is induced. The followings should hold in order to fulfill Eq. (6.77).

1) The plastic strain rate is directed outward-normal of the yield surface. Then, the associated flow rule must hold, provided that the direction of plastic strain rate is determined solely by the current stress and internal variable but independent of stress rate.

2) In this occasion the yield surface has to be the convex surface (see Fig. 6.4).

The result 1) is called the associated flow rule or the normality rule and the result 2) is called the convexity of yield surface.


σ

σ

σy

0

ε0

0t r( ) 0ypdt− ≥Dσ σ

pdtD

σ

σ

σy

0

ε0

0t r( ) 0ypdt− ≥Dσ σ

pdtD

Fig. 6.3 Positive work done by external agency in Drucker’s (1951) postulate (Illustration in uniaxial loading process)

0 σij , ε ijp.

Elastic region

Yield surface

Normality ruleConcave yield surface:

Violation of Drucker (1951)’s postulate.

pD

0

y

0

0

0

0

0y

0t r( ) 0yp− <D

pD

0 σij , ε ijp.

Elastic region

Yield surface

Normality ruleConcave yield surface:

Violation of Drucker (1951)’s postulate.

pD

0

y

0

0

0

0

0y

0t r( ) 0yp− <D

pD

Fig. 6.4 Associated flow (normality) rule and convexity of yield surface based on the Drucker’s (1951) postulate

6.4.3 Positivity of Second-Order Plastic Relaxation Work Rate

Ilyushin (1961) postulated that “the work done during the strain cycle is positive”. Limiting to the infinitesimal deformation process, it leads to the postulate “the


second-order work rate w is smaller than the second-order elastic stress work rate sew calculated by presuming that the strain rate is induced elastically” or “the

second-order plastic relaxation work rate prw , i.e. the work rate done during the infinitesimal strain cycle is positive” (Hill, 1968; Hashiguchi, 1993a; Petryk, 1993; see Fig. 6.5). It is described mathematically as follows:

, 0pse rww w≤ ≥ (6.78)

where

t r( ) / 2 = tr( ) / 2

tr( ) / 2 = tr( ) / 2

tr( ) / 2 = tr( ) / 2

e

p pp

e

es

r

w

w

w

⎫≡ ⎪⎪

≡ ⎬⎪⎪≡⎭

D D ED

D DED

D D ED

σ

σ

σ

(6.79)

with

, p pe ≡ ≡ED EDσ σ (6.80)

eσ and pσ are called the elastic stress rate and the plastic relaxation stress rate,

respectively.

It should be noted that the associated flow rule must hold for Eq. (6.78) 2 to meet

the loading condition in Eq. (6.68).

6.4.4 Comparison of Interpretations for Associated Flow Rule

Prager’s (1949) interpretation of the associated flow rule is concerned only with hardening materials as described previously. On the other hand, the interpretation of the positivity of work done by the external agency, i.e. the additional stress during the stress cycle by Drucker (1951) and the positivity of the second-order plastic relaxation work rate are based on postulates of the dissipation energy of materials independent of hardening behavior.

Here, Drucker’s (1951) postulate is related to the stress cycle but the postulate of the second-order plastic relaxation work rate is related with the infinitesimal strain cycle. Now, compare below the pertinence of these postulates.

(1) The strain cycle can be realized always. However, the stress cycle cannot be made in the softening state in which the stress cannot be returned to the initial state if the plastic strain rate is induced. It is based on the fact that any defor-mation can be given, but a stress cannot be given arbitrarily to materials since strength of materials is limited.


edε

dσ

1

E

σ

ε

E

1a

b

c

d

efσ

0

edε

dσ

0

ab

c

d

e f

σ

ε

σ

pdε

1

Epdσ

edσ

E

1

pdσedσ

dε

pdεdε

Hardening process ( )

prw

prw

0pM >

Softening process ( )0pM <

edε

dσ

1

E

σ

ε

E

1a

b

c

d

efσ

0

edε

dσ

0

ab

c

d

e f

σ

ε

σ

pdε

1

Epdσ

edσ

E

1

pdσedσ

dε

pdεdε

Hardening process ( )

prw

prw

0pM >

Softening process ( )0pM <

Fig. 6.5 Positiveness of second order work rate (Illustration in uniaxial loading)


(2) Limiting to the infinitesimal cycles, consider the stress and strain cycles. The

second-order work increment done during the infinitesimal stress cycle is given

by ( )t r / 2pdt dtDσ ( abeΔ in the hardening process in Fig. 6.5). On the

other hand, the additional work increment tr( ) / 2p pdt dtD ED ( aecΔ in the

hardening process in Fig. 6.5) must be done to close the strain cycle, whilst tr( ) / 2 0p pdt dt >D ED holds because of the positive-definiteness of the elastic modulus tensor E. Therefore, the work done during the infinitesimal

stress cycle is far smaller than the work during the infinitesimal strain cycle. In

other words, Drucker’s (1951) postulate holds for the materials fulfilling a more restricted condition, i.e., more particular materials than the materials

fulfilling the positivity of the second-order plastic relaxation work rate. (3) The strain (increment) has one-to-one correspondence to the displacement

(increment) induced in the material. Therefore, the configuration of material returns to the initial state only if the strain in any definition returns to the initial value. In other words, if the strain cycle in any definition of strain (logarithmic strain, nominal strain for instance) closes, the strain cycle in the other definition also closes. On the other hand, the configuration in the end of stress cycle differs from the initial configuration depending on the loading path chosen during the cycle and on the definition of stress (Cauchy stress, nominal stress for instance). Eventually, the strain cycle possesses the objectivity, but the stress cycle does not possess it.

(4) The assumption that the interior of the yield surface is the purely elastic domain is adopted in Drucker’s postulate. On the other hand, it is not required by the postulate of the positivity of second-order plastic relaxation work rate, which holds on the quite natural premise that the purely elastic deformation is induced at the moment of unloading.

Eventually, it can be stated that postulate of the positivity of second-order plastic relaxation work rate is more general than Drucker’s postulate. However, even the former is based on the premise that the direction of the plastic strain rate is de-pendent on the normal component but independent of the tangential component of stress rate to the yield surface. It is observed in the test result that the inelastic de-formation is induced even by the tangential component. The inelastic strain rate induced by the tangential component will be described in 6.6.

Regarding the case in which the plastic strain rate is directed the outward-normal of convex yield surface, i.e. the case in which the associated flow rule holds, the

plastic work rate done by the actual stress yσ on the convex yield surface is greater

than that done by any statically-admissible stress *σ as depicted in Fig. 6.6. It is called the principle of maximum plastic work. It is beneficial for the formulation of

variational principles.


0

yσ

N

ijσ

Yield surface

*σ

0

yσ

N

ijσ

Yield surface

*σ

Fig. 6.6 Principle of maximum plastic work

6.5 Anisotropy

The plastic strain rate described in 6.2 concerns the yield condition with the func-tion f involving only the stress invariants. Therefore, it is limited to the materials exhibiting the isotropy in the plastic deformation behavior. In what follows, first the isotropy in constitutive equation is defined. Then, the plastic strain rate extended to the anisotropy will be explained in this section.

6.5.1 Definition of Isotropy

An isotropic material is defined as one exhibiting identical mechanical response that is independent of the chosen direction of material element or of the coordinate system by which the response is described. Here, the input/output variables are the stress rate and the strain rate in the irreversible deformation.

The rate-type constitutive equation in a rate form is described in general as follows:

)( , =iH , , Df 0σ σ (6.81)

When the following equation holds by giving coordinate transformations only for stress (rate) and strain rate tensors in the function f , it can be stated that Eq. (6.81) describes the constitutive equation of isotropic material.

) )( (, , =T T T Ti iH H , , , ,D Df Q Q Q Q Q Q Qf Qσ σ σ σ (6.82)

In the plastic constitutive equation formulated incorporating the yield and/or plastic potential function, the isotropy holds if the yield and/or plastic potential

6.5 Anisotropy 157

function is given by the function of stress invariants and scalar internal variables. Then, designating these functions by f , it must fulfill the equation.

) )( (=T Ti iH H f f, ,Q Q Q Qσ σ (6.83)

In contrast, the anisotropic plastic constitutive equation is described by incor-porating the yield and/or plastic potential function involving tensor-valued parameters in addition to the stress invariants and scalar internal variables. Then, Eqs. (6.82) and (6.83) do not hold in anisotropic constitutive equations.

6.5.2 Anisotropic Plastic Constitutive Equation

If the monotonic loading proceeds towards a certain direction in the stress space, the hardening develops in that direction but the yield stress lowers in the opposite di-rection. This phenomenon is induced by the statically-indeterminable deformation of internal structure and is called the Bauschinger effect. To reflect this effect in the elastoplastic constitutive equation, the translation or the rotation of the yield surface is adopted widely. The translation of the yield surface, called the kinematic hard-ening, is realized by introducing the back stress developing towards the loading direction and replacing the stress tensor with the tensor given by subtracting the back stress tensor from the stress tensor. On the other hand, soils, which is the as-sembly of particles with weak adhesion between them, can bear a far larger com-pression stress than the tensile stress. Therefore, they exhibit a strong frictional property that the deviatoric yield stress increases with the pressure, while the yield surface only slightly includes the origin of the stress space. Therefore, once the yield surface translates leaving the origin, it can never come back to include the origin again because the yield surface contracts with the plastic volume expansion. Therefore, the kinematic hardening cannot be applied but the rotation of the yield surface, i.e. the rotational hardening, is pertinent to soils.

Now, let the yield condition (6.30) be extended to describe the anisotropy by introducing the internal variables of second-order tensors as follows:

( ) ( )=ˆf F H,βσ (6.84)

where ˆ ≡ −σ σ α (6.85)

α being the back stress (kinematic hardening variable) (Prager, 1956). Here, it is

assumed that f in Eq. (6.84) is also the function of σ in the homogeneous de-

gree-one fulfilling ( ) ( )=ˆ ˆf fs s, ,β βσ σ . Then, the yield surface (6.84) main-

tains the similar shape and orientation with respect to =σ α when = const.β

β is the second-order tensor of non-dimension describing the rotation of the yield


surface by replacing the deviatoric stress ratio tensor p' /σ into p' / −βσ and is

called the rotational hardening variable (Sekiguchi and Ohta, 1977). The isotropic

deformation, i.e. expansion/contraction would not influence the anisotropy.

Therefore, the anisotropic hardening variables α and β are assumed to be the

deviatoric tensors, the evolution rules of which are given by

( (|| || , || ||) )== i ip pH H ', ,βD Dba σ σα (6.86)

where a and b are functions of iH ,σ and are deviatoric, fulfilling

tr = tr = 0ba . The symbol for deviatoric tensor is eliminated in (6.86)1 because

of tr = 0pD in metals. The material-time derivative of Eq. (6.84) leads to the consistency condition

( ) ( ) ( )tr tr tr =

ˆ ˆ ˆˆ ˆ

( ) ( ) ( )f f fF H

•, , ,∂∂ ∂+− ′∂∂ ∂β β β ββ

σ σ σσ ασ σ (6.87)

Substituting Eqs. (6.37) and (6.86) into Eq. (6.87), one has

( ) ( ) ( )|| || || ||t r tr trˆ ˆ ˆˆ ˆ

( ) ( ) ( )p pf f f ', , ,∂∂ ∂+− ∂∂ ∂β β β

D Dβ bσ σ σaσσ σ

;( )= , p

iF h H′ Dσ (6.88)

Further, substituting the associated flow rule ( λ ：positive proportionality factor)

ˆ ˆ=p λ ND ,

( ) ( )ˆ ˆ ˆ|| ||/

f f, ,∂ ∂≡ ∂ ∂β β

Nσ σσ σ (6.89)

into Eq. (6.87) and noting ( ) ( )/ /=ˆ ˆ ˆf f, ,∂ ∂∂ ∂β βσ σσ σ , it follows that

( ) ( ) ( ) ˆˆ ˆ|| ||t r tr trˆ ˆ ˆˆ ˆ

( ) ( ) ( )f f f' λλ

, , ,∂∂ ∂+− ∂∂ ∂β β β

Nβ bσ σ σσ aσ σ

ˆˆ ;( )= iHhF λ ,′ Nσ (6.90)

Taking account of the Euler’s theorem for homogeneous function in order-one, i.e.

( )

t r( ) ( )ˆ ˆ ˆ= == ˆ ˆ ˆ) ) )tr( tr( tr(

ˆˆ ˆˆˆ ˆ ˆ ˆ

( )ff f F

,∂, ,∂∂

∂

ββ β

N N NN N N


(6.91)

6.6 Incorporation of Tangential-Inelastic Strain Rate 159

into Eq. (6.90), it is obtained that

( )ˆ ˆˆ ˆ ˆ|| ||t r( ) tr( ) trˆ )ˆ tr()tr(

ˆˆˆ

( )fFF 'λ λ,∂+− ∂β

N N NβNNb

σσ aσσ

ˆˆ ; ( )= iHhF λ ,′ Nσ (6.92)

from which we obtains

ˆt r( )ˆ =( )1 ˆ ˆ ˆˆˆ; )|| ||( ) ) tr(tr( )tr( +ˆ ˆˆ ( )i

fF HhF F 'λ

′ ,∂, − ∂

Nβ

N N NNN β b

σσσ a σσ

ˆtr( ) =

( )1ˆ ˆˆ ;( ) || ||tr +ˆ ˆ( )( ][ )( i

fF HhF F ',′ ∂, − ∂

NβN NN β b

σσσ aσ

(6.93)

Consequently, we have

ˆtr( )ˆ =ˆ pM

λ Nσ，

ˆt r( ) ˆ=ˆ

ppM

ND Nσ

(6.94)

where

( )1ˆˆ ˆ ;( )tr +|| ||ˆ ˆ( )( ][ )(p i

fF HhMFF '

,∂′ ,≡ −∂

βNN β Nbσσ aσ

(6.95) and

1ˆtr( ) ˆ= ˆ pM

− + N ND Eσσ (6.96)

ˆtr( )ˆ =ˆ ˆ ˆtr( )pM

Λ+

EDNNEN

(6.97)

ˆ ˆ ˆt r( ) ˆ == ˆ ˆ ˆ ˆ ˆ ˆ) )t r( t r(( )

p pM M⊗

+ +DNE NE ENNE DED ENEN NEN

σ − − (6.98)

6.6 Incorporation of Tangential-Inelastic Strain Rate

As presented in Eqs. (6.40), (6.48) and (6.94), the inelastic strain rate, i.e. the plastic strain rate in the traditional constitutive equation has the following limitations.


(i) It depends solely on the stress rate component normal to the yield surface, called the normal stress rate, but is independent of the component tangential to the yield surface, called the tangential stress rate, since it is derived merely based on the consistency condition.

(ii) The direction is determined solely by the current state of stress and internal variables but it is independent of the stress rate.

(iii) The principal directions of plastic strain rate tensor coincide with those of stress tensor, exhibiting the so-called coaxiality, in the case of isotropy in which the direction of plastic strain rate depends only on the direction of stress by the fact described in 1.11.

On the other hand, it has been verified by experiments that an inelastic strain rate that influences deformation considerably is induced by the deviatoric tangential stress rate, and it is called the tangential inelastic strain rate, in the non-proportional loading process deviating from the proportional loading path normal to the yield surface. Here, the mean part of the tangential stress rate does not induce an inelastic strain rate, as Rudnicki and Rice (1975) verified based on the fissure model. The tangential inelastic strain rate is induced considerably in the plastic instability phenomena with the strain localization induced by the generation of the shear band and it influences the macroscopic deformation and strength characteristics even in the macroscopically proportional loading process. To rem-edy these insufficiencies, vairous models have been proposed to date as follows:

1) Intersection of plural yield surfaces: Various models assuming the intersection of plural yield surfaces have been proposed (Batdorf and Budiansky, 1949; Koiter, 1953; Bland, 1957; Mandel, 1965; Hill, 1966; Sewell, 1973, 1974). The Koiter’s (1953) model has been adopted by Sewell (1973, 1974), but it is in-dicated that the applicability of the model is limited to the inception of uniaxial loading. Models in this category cannot describe the latent hardening perti-nently and are not readily applicable to general loading processes (cf. Christof-fersen and Hutchinson (1979)).

2) Corner theory: The singularity of outward-normal of the yield surface is intro-duced by assuming the conical corner or vertex at the stress point on the yield surface. Therefore, the direction of plastic strain rate can take a wide range surrounded by the outward-normal of the yield surface (Christoffersen and Hutchinson, 1979; Ito, 1979; Gotoh, 1985; Goya and Ito, 1991; Petryk and Thermann, 1997). There exist the two kinds of models: One kind is based on the assumption of an imaginary infinitesimal vertex and the other subsumes a finite projecting cone. The evolution rule of the cone cannot be formulated and the reloading from the cone surface after partial unloading cannot be described pertinently by the latter models. It was described by Hecker (1972; Ikegami, 1979) that the yield surface projects towards the loading direction generally but the formation of the so-called vertex is doubtful.

3) Hypoplasticity: This term was first used by Dafalias (1986) in the analogy to the

term hypoelasticity introduced by Truesdell (1955) described in 5.3. Models in


this category are classified into models in which the direction of plastic strain

rate depends on the direction of the stress rate || ||/σ σ (Mroz, 1966; Dafalias

and Popov, 1977; Hughes and Shakib, 1986; Wang et al., 1990; Hashiguchi, 1993a) and the models in which the direction of the plastic strain rate depends

on the direction of stress rate || ||/D D (Hill, 1959; Simo, 1987; Hashiguchi,

1997; Kuroda and Tvergaard, 2001). The singularity in the filed of direction of plastic strain rate is introduced algebraically into these models, although it is

done geometrically in the models described in 1 ) and 2 ). However, the mag-

nitude of the plastic strain rate is derived from the consistency condition. Therefore, the plastic strain rate diminishes when the stress rate is directed

tangentially to the yield surface, as in the traditional constitutive equations

without the vertex.

The constitutive equations described in 1)–3) possess the following problems.

i) A formulation of pertinent model which fulfills the consistency condition and is applicable to the general loading process is difficult.

ii) The stress rate-strain rate relation becomes nonlinear. Therefore, the inverse expression cannot be derived, which renders deformation analysis as difficult.

4) 2J - deformation theory: Budiansky (1959) and later Rudnicki and Rice (1975) incorporated the tangential-inelastic strain rate into the constitutive equation (6.57) with the isotropic Mises yield condition as follows:

1 1 t r=

|| || || ||( ) e

eeF

' '' ''' ' 'σ φ σσ

−•3+ + −( )2D E σ σσσ σ σσ σ (6.99)

which can be rewritten as

1 1= 2 / 32 / 3

eee

e eF''''

σ φ σ σσ σ−

• •3+ + −( )2D E σσ σ σ

1 1= )(e

e eeF ' ''

σφ φσ σσ−

•3+ +( ) ( )−2E σσ σ (6.100)

where the rate-linearity is retained.

On the other hand, Hencky’s deformation theory (Hencky, 1924) is described as

1= e 'φ σ− + ( )E σε σ (6.101)

The material-time derivative of Eq. (6.101) leads to

1= ee e '' 'φ φσ σ σ− •+ +( ) ( )E σ σε σ (6.102)

162 6 Basic Formulations for Elastoplastic Constitutive Equations Comparing Eq. (6.100) with Eq. (6.102), choosing eF σ( ) so as to fulfill

13= 2pe e

e e eF'

'ε σ φ φσ σ σ

( )( ) ( )+ ( ) (6.103)

and regarding ε as D , it is known that the 2J - deformation theory (6.99) coin-

cides with Hencky’s deformation theory (6.101).

In what follows, extend the 2J - deformation theory for the general yield con-

dition unlimited to the Mises yield condition (Hashiguchi, 1998, 2005; Hashiguchi

and Tsutsumi, 2001,2003)．

First, assume that the strain rate is decomposed additively into elastic and plastic

strain rates in Eq. (6.28) and further assume the tangential-inelastic strain tD as

= pe t+ +D D D D (6.104)

where tD is induced by the deviatoric tangential stress rate 'σ which is decom-

posed into the deviatoric normal stress rate n'σ and the deviatoric tangential stress

rate t'σ (Fig. 6.7):

= n t' '' +σ σσ (6.105)

where

1 rt= 3

' −σ σ σ (6.106)

ˆˆ ˆ ˆtr( )=

ˆˆ (tr( ) )0= =

n

n tt

'' ' ''

''' ''

⎫≡ ⊗ ⎪⎪

⎬⎪≡ − ⎪⎭

nn n n

NI

σ σσ

σσ σ σ σ

(6.107)

ˆ( ) ( )ˆ ˆ(|| || 1) = = ˆ|| ||

ˆ ˆf f '' '' ''

, ,∂ ∂≡ ∂ ∂( ) ( )β β Nn nN

σ σσ σ

(6.108)

1 1ˆˆ ( )ˆˆ ˆ ˆ 2 3( )ij klij klik jl il jkijklI n n' ' '' ' ' ' δ δ δ δ δ δ≡ + − −−≡ ⊗nnI I (6.109)

1 1 1( ) 3 2 3( )ij klik jl il jkijklI '' δ δ δ δ δ δ≡ − ⊗ ≡ + −I II I (6.110)

In these expressions, the fourth-order tensor 'I plays the role of transforming

an arbitrary second-order tensor into its deviatoric second-order tensor. Therefore,

it might be called the deviatoric transformation (or projection) tensor. Further-

more, the fourth-order tensor 'I plays the role of transforming an arbitrary


second-order tensor into its deviatoric second-order tensor tangential to the yield

surface. For that reason, it might be called the deviatoric tangential (or projection)

tensor. Hereinafter, the deviatoric tangential tensor is denoted as ( )t' , i.e. ˆ

t' '≡T TI for arbitrary second-order tensor T .

Yield surface

ij'σ

σ

0

ˆ'n

'σn'σ

t'σ

Yield surface

ij'σ

σ

0

ˆ'n

'σn'σ

t'σ

Fig. 6.7 Normal and tangential stress rates in the deviatoric stress plane

Now, assume that the tangential inelastic strain rate tD is related linearly to the

tangential deviatoric stress rate t'σ .

= 2 tt T

G 'D σ (6.111)

where T is a function of stress and internal variables in general. Substituting Eqs. (6.29), (6.94), (6.111) into Eq. (6.104), the strain rate is given

by

1ˆtr( ) ˆ= 2ˆ p t

TGM

'− + +ND E N

σσ σ

1 ˆ ˆ ˆ= 2ˆ( )

p

TGM

'− ⊗+ +N NE I σ (6.112)

For derivation of the inverse expression of Eq. (6.112), let the elastic modulus

tensor (5.28) in the Hooke’s type be adopted. Therefore, note that Λ is given by

Eq. (6.97) itself since it holds that tr( tr() 2 )= = 0t tG' 'EN Nσ σ because of

Eq. (6.107)2.

164 6 Basic Formulations for Elastoplastic Constitutive Equations Noting (5.28), it holds from Eq. (6.112) that

ˆt r( )1 ˆ= 22 ˆ p t

TGG M

''' ' ++ ND N

σσ σ (6.113)

from which one has the following relation, considering

ˆˆ ˆ ˆ ˆˆ ˆtr( ) = =t ' ' '' ' '−≡ nN nN I 0N N .

1 (1 )= 2t tTG

' +D σ' (6.114)

where

ˆ ˆ ˆtr( ) =t' ' ' ' ' '−≡ n nD D DI D (6.115)

Substituting further Eq. (6.114) into Eq. (6.111), the tangential inelastic strain

rate is given by

= 1 tt T

T '+ DD (6.116)

The stress rate is derived from Eqs. (6.29), (6.89), (6.97) and (6.116) as follows:

ˆ( )t r 2ˆ= 1ˆ ˆ ˆ( )t rp tGT

TM'− − ++

NED ENED DNEN

σ (6.117)

ˆ 2ˆ ˆ=1ˆ ˆ ˆ( )t r

( )p

GTTM

'− −⊗ ++ENE DEN I

NEN (6.118)

Here, since there exists the relation

( )( ) ( ) ( )trtr tr = ==( ) ( ) pe tff fdF

∂∂ ∂ − −∂∂ ∂

D DE DEDσσ σσ σσ σ

(6.119) it holds that

( )tr = 0 tf∂

∂ DEσσ (6.120)

Then, it is known that the tangential inelastic strain rate tD does not influence the hardening behavior. Furthermore, the loading criterion is given by Eq. (6.69) as it is, and the mathematical structure is rate-linear identically to the common elas-toplastic constitutive equation, whilst it is called sometimes the linear comparison material since it is linear in the continuation of loading. Consequently, no difficulty is brought in the analysis of boundary value problems.

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory 165 The tangential inelastic strain rate has no loading criterion and is induced only if

the deviatoric tangential stress rate is induced, falling within the framework of hypoelasticity in which the complete integrability condition does not hold and the time-integration depends on the loading path. However, the continuity condition is violated because it falls within the framework of the conventional plasticity as-suming the interior of the yield surface to be an elastic domain. The tangential inelastic strain rate is induced suddenly when the stress reaches the yield surface. Therefore, the range of application is limited to the proportional loading process in which the tangential component of stress rate is far smaller than the normal component.

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain

Theory

Strictly speaking, the elastic strain rate has to be formulated based on the hypere-lastic constitutive equation in which the elastic deformation is determined uniquely for a given stress and the elastic work done during the stress cycle is zero. On the other and, these basic requirements are not fulfilled exactly in the hypoelasticity as described in Chapter 5. In addition, the stress is calculated directly for a given elastic deformation in the hyperelastic constitutive equation. On the other hand, the hypoelastic constitutive equation must be integrated in time in order to calculate the stress. Therein, it is required to adopt an appropriate corotational stress rate in order to predict the stress pertinently. One of hyperelastic-plastic constitutive equations is shown below, which is based on the postulate of Eq. (6.20) (cf. Mandel, 1973, 1974; Simo, 1998; Belytschko et al., 2001).

The hyperelastic constitutive equation is given by replacing F to eF in Eq. (5.9) for the unit initial density as

2 ( )=

ee

e eψ• •∂

∂ ∂ES E

E E (6.121)

where

Te e−1 −≡ F FS τ (6.122)

1 1) ),= 2 2T Te ee ee ee ≡≡ FF F F(C I ( I CE − − (6.123)

S is interpreted to be the pull-back of the Kirchhoff stress τ to the intermediate

configuration. Here, note that = Te e −−1∗ ∗ ∗∗F FS τ ( ) ( )= T Te e−1 −Q Q Q QF Fτ

= Te e−1 −F Fτ = S leading to the frame-indifference of the hyperelastic constitutive

relation.


One can formulate the following relation from Eq. (6.19).

= pe +D D D (6.124) where

, = , p pT TT ee e e e ee e e•

≡ ≡ ≡F FD F D D E F D F D F D (6.125)

Here, note that pD is expressed as

1= ( )2

Tp p pe e+D C L L C (6.126)

because of

1 11 1= ( ) = ( )2 2

Tp p p p pTT T Te e ee e e e ee e− −+ +D F L F F L F F L LF F F F F

The following equation is obtained from Eq. (6.121), adopting Eq. (6.124).

( )= = peleel•

−C D DCS D (6.127) where

2 ( )le

ee e

ψ∂≡∂ ∂

EC

E E (6.128)

Now, we adopt the following yield condition for the sake of simplicity of explanation, while the extension to the anisotropy is not difficult.

( ) ( )=f F HS (6.129)

while f is the homogeneous function of S in degree-one. The material-time dif-

ferentiation of Eq. (6.129) leads to

( )tr =( )f

F H'• •∂

∂S SS

(6.130)

Noting the relation

( )tr( ) ( ) ( )= = =

tr( ) tr( )tr( )

( )S S S

S SS

ff f F H

∂∂ ∂

∂

S SS SS N N N

N S NS N S S (6.131)

( ) ( ) ||||/S

f f∂ ∂≡∂ ∂

S SNS S

(6.132)

the consistency condition is given from Eq. (6.130) as

tr( ) = tr( )S SF HF'• •

N NS S (6.133)

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory 167

Here, let the plastic flow rule be given by

( )=p λL Sr (6.134)

where ( )Sr stands for the direction of pL , which is function of S . The substitution of Eq. (6.134) into Eq. (6.133), one has

tr( ) = ( )tr( )S SF hF' λ

•N NS Sr (6.135)

where ( )h S is related to H•

as = ( )H hλ•S . It holds from Eqs. (6.134) and

(6.135) that

tr( ) tr( )= ( ), =

( ) ( )tr( ) tr( )

S Sp

S SF Fh hF F' '

λ• •

N NS SSL

N NS SS Sr (6.136)

pD is given from Eqs. (6.126) and (6.136) by

tr( ) 1= ( )2( )tr( )

p S T

S

e e

F hF'

•

+N SD C r r CN SS

(6.137)

D is given from Eqs. (6.124), (6.127) and (6.137) as

1 tr( ) 1= ( )2( )tr( )

Sel T

S

e e

F hF'

−•

•+ +N SD C r r CC S

N SS

(6.138)

from which the proportionality factor in the flow rule in Eq. (6.134) is described in terms of D as

tr( )=

tr ( )tr( )( ) / 2

lS

l TSS

e

e eeF hF'

Λ+ +

N DC

N C r r CN S CS (6.139)

It is obtained from Eqs. (6.138) and (6.139) that

1 tr( )( )2=tr ( )tr( )( ) / 2

[ ]l T l

Sl

l TSS

e ee ee

e eeF hF'

• + ⊗−

+ +

NC r r CC CCS D

N C C r r Cr N S (6.140)

On the other hand, Lie derivative of S due to pF defined in 4.5, is given by

=p p pTv

•− −SS S L SLL (6.141)


noting 1= ( )p Tp p p pT

v•− −S F F SF FL

1 1 1= ( )p TpT Tp p p p p pT•• •− −− − − −+ +F F SF F S F F S F F

1 1) )( (=p pp p T

•• •− −+ +− −SF F S S F F

The substitution of Eqs. (6.134) and (6.140) into Eq. (6.141) leads to

=p pev CS DL (6.142)

where

1 tr( )( ) ( )2tr ( )tr( )( ) / 2

l T lSp l

l TSS

e ee ee e

e eeF hF'

+ ++ ⊗−≡

+ +

Nr rC r r CC S S CCC

N C C r r CN SS

(6.143)

It holds from Eqs. (3.21), (4.57) and (6.122) for the Truesdell rate of Kirchhaff stress that

1= = ( ) TTv

Δ •− −F F F Fττ τL

1 1(= )p p p pTTTT Teee e e e•− −− −S F FF F F F F F F F

1( )= ) =p p p p pT T T Tee e e

v•− − (SF F F F F F F S FL (6.144)

Substituting Eqs. (6.125) and (6.142), it is derived that

( )= = =ep ep epTT Tee e ee eΔC C F FF DF F D F C Dτ (6.145)

= ( )

= ( ) =

pij pq pqim jnmn

p ppq pqim jn im jnmn mnkp lq qkpkl l kl

ee e

e ee e e e e e e e

F C D F

F C F D F F F F F F C D

τΔ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

where

=p p

pqijkl im jn q mnpk le ee e e eC F F F F C (6.146)

Noting Eq. (4.57), Eq. (6.145) is rewritten as

( = ) ( )= peΔ+ + +D D DΣCτ τ τ τ (6.147)

by the Jaumann rate of Kirchhoff stress or, noting Eq. (4.54),

1tr == ( ) ( )peJJ

− + − ⊗τ D Σ DC Iσσ σ (6.148)

by the Jaumann rate of Cauchy stress, where Σ is defined in Eq. (5.15).

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory 169

In the above-mentioned constitutive equation, the flow rule (6.134) for pL has

to be formulated rigorously so as to provide the plastic strain rate pertinently noting

the associated flow rule. Further, it has to be done also for the spin due to the plastic

deformation, i.e. = ( ( ))paλW r S . Further discussions for the finite elastoplastic

Experiment

Elastic state

Elastoplastic statePrediction by conventional

plasticity

0 0Hardening

Almost real prediction

Unrealistic prediction:Excessively high peak stress

Softening

Stress Stress

Strain Strain

Experiment

Elastic state


plasticity

0 0Hardening

Almost real prediction

Unrealistic prediction:Excessively high peak stress

Softening

Stress Stress

Strain Strain

Fig. 6.8 Prediction of monotonic loading behavior by conventional plasticity

Cyclic loading with constant stress amplitude

Stress

Strain0

Experiment

Elastic state


plasticity

Cyclic loading with constant stress amplitude

Stress

Strain0

Experiment

Elastic state


plasticity

Fig. 6.9 Cyclic loading behavior: Inability of conventional plasticity


strain theory are referred to the literatures (cf. Kleiber and Raniecki,1985; Raniecki and Mroz, 1990; Raniecki and Nguyen, 2005; Raniecki et al., 2008).

The conventional elastoplasticity described in this chapter is premised on the assumption that the interior of yield surface is a purely elastic domain. Therefore, the relation of stress rate vs. strain rate is predicted to change abruptly at the mo-ment when the stress reaches the yield surface. Therefore, the smooth stress-strain curve observed in real materials is not predicted as shown in Fig. 6.8. This results in the serious defect in the prediction of softening behavior. Further, only an elastic deformation is repeated for the cyclic loading of stress below the yield stress. In fact, however, plastic deformation is accumulated for stress cycles less than the yield stress and the strain is amplified leading to the failure as depicted in Fig. 6.9. Therefore, it has various limitations in the application to the mechanical design of machines and structures in engineering practice.


Chapter 7 Unconventional Elastoplasticity Model: Subloading Surface Model

7 Unconventional Elastoplasticity Model: Subloading Surface Model

Elastoplastic constitutive equations with the yield surface enclosing the elastic domain possess many limitations in the description of elastoplastic deformation, as explained in the last chapter. They are designated as the conventional model in Drucker’s (1988) classification of plasticity models. Various unconventional elas-toplasticity models have been proposed, which are intended to describe the plastic strain rate induced by the rate of stress inside the yield surface. Among them, the subloading surface model is the only pertinent model fulfilling the mechanical re-quirements for elastoplastic constitutive equations. These mechanical requirements are first described and then the subloading surface model is explained in detail.

7.1 Mechanical Requirements

There exist various mechanical requirements, e.g., the thermodynamic restriction and the objectivity for constitutive equations. Among them, the continuity and the smoothness conditions are violated in many elastoplasticity models, although their importance for formulation of constitutive equations has not been sufficiently rec-ognized to date. Before formulation of the plastic strain rate, these requirements will be explained below (Hashiguchi, 1993a, b, 1997, 2000).

7.1.1 Continuity Condition

It is observed in experiments that “the stress rate changes continuously for a con-tinuous change of the strain rate”. This fact is called the continuity condition and is expressed mathematically as follows (Hashiguchi, 1993a, b, 1997; 2000).

lim ; ;) )( ( =i i H Hδ

δ→

, ,D 0

D D D+σ σ σ σ (7.1)

where iH ( =1, 2, 3, )i • •• collectively denotes scalar-valued or tensor-valued

internal state variables. In addition, ( )δ stands for an infinitesimal variation. The

response of the stress rate to the input of strain rate in the current state of stress and internal variables is designated by ; )( , iH Dσ σ . Uniqueness of the solution is not guaranteed in constitutive equations violating the continuity condition, predicting

172 7 Unconventional Elastoplasticity Model: Subloading Surface Model

jump jump

:Input D:putOut σ

jump jump

:Input D:putOut σ

Fig. 7.1 Violation of continuity condition

different stresses or strains. The violation of this condition is schematically shown in Fig. 7.1. Ordinary elastoplastic constitutive equations, in which the plastic strain rate is derived obeying the consistency condition, fulfill the continuity condition. As described later, however, no elastoplastic constitutive equation fulfills it except for the subloading surface model when they are extended to describe the tangential inelastic strain rate.

The concept of the continuity condition was first advocated by Prager (1949). However, a mathematical expression of this condition was not given. The condition was defined as the continuity of strain rate to the input of stress rate by Prager (1949) inversely to the definition given above. However, the identical stress rate directed the inward of yield surface can induce different strain rates in loading and unloading states in a softening material. Here, it is noteworthy that a stress rate cannot be given arbitrarily since there exists a limitation in strength of materials although a strain rate can be given arbitrarily. For that reason, the Prager’s (1949) notion does not hold in the general loading state including softening and the per-fectly plastic states (Fig. 7.2).

7.1.2 Smoothness Condition

It is observed in experiments that “the stress rate induced by the identical strain rate

changes continuously for a continuous change of stress state”. This fact is called the

smoothness condition and is expressed mathematically as follows:

; ) ; )lim ( (+ = i iH Hδ

δ→

, ,0σ

D Dσ σ σ σ σ (7.2)

The rate-linear constitutive equation is described as

( )= peiH,M Dσ σ (7.3)

7.1 Mechanical Requirements 173

σ

εdε dε

dσ

0

(<0) (>0)

(<0)Input:

Output:

loadingunloading

Input : σOutput : D

Tangent plane of yield surface

σ

εdε dε

dσ

0

(<0) (>0)

(<0)Input:

Output:

loadingunloading





Fig. 7.2 Impertinence of Prager’s (1949) continuity condition (illustrated for softening state) where the fourth-order tensor peM is the elastoplastic modulus, which is a func-tion of the stress and internal variables, can be described generally as

=pe ∂∂M

Dσ . (7.4)


Consequently, Eq. (7.2) can be rewritten as

) )lim ( (+ =p pe ei iH H

δδ

→, ,

0σM Mσ σ σ (7.5)

Constitutive equations violating the smoothness condition cannot predict a smooth stress-strain curve. Therefore, they cannot describe softening behavior pertinently, as depicted in Fig. 6.8. Further, it cannot predict the strain accumulation for cyclic loading of stress amplitude less than the yield stress, as depicted in Fig. 6.9. Among the existing constitutive models only the subloading surface model always fulfills the smoothness condition.

7.2 Subloading Surface Model

The basic concept and equations for the subloading surface model (Hashiguchi and Ueno, 1977; Hashiguchi, 1978, 1980, 1989) are described below. This is the only model fulfilling the mathematical requirements described in 7.1.

For the formulation of plastic strain rate induced by the rate of stress inside the yield surface let the following assumptions be incorporated for the loading process

( )p ≠D 0 , which would be quite natural properties in plastic deformation.

a) Only an elastic strain rate is induced up to a certain stress level. Therefore, the stress changes at infinite rate up to that stress level even if the plastic strain rate is infinitesimal.

b ) The stress always approaches the yield surface. c ) The ratio of the magnitude of stress rate to that of plastic strain rate decreases as

the stress approaches the yield surface. d ) The stress changes along the yield surface in the plastic loading process when it

reaches the yield surface. It does not go out from the yield surface. e ) A conventional elastoplastic constitutive equation holds when the stress lies on

the yield surface.

To formulate an unconventional elastoplastic constitutive equation based on these assumptions, it is necessary to adopt an appropriate measure describing the degree of approach to the yield state. Then, let the subloading surface, be intro-duced, which always passes through the current stress point and has similar shape and orientation to the yield surface, while the yield surface is renamed the nor-mal-yield surface. Here, the similar shape and orientation of surfaces imply the following geometrical properties.

i ) All lines connecting an arbitrary point on or inside the subloading surface and it conjugate point on or within the normal-yield surface join at the similarity-center.

ii ) All ratios of length of an arbitrary line-element connecting two points on or in-side the subloading surface and that of the conjugate line-element connecting two conjugate points on or inside the normal-yield surface are identical. The ratio is called the similarity-ratio which coincides with the ratio of the sizes of these surfaces.


Consequently, the subloading surface (see Fig. 7.3) can be described as

( ) = ( )f RF Hσ (7.6)

where, (0 1)R R≤ ≤ is the ratio of the size of the subloading surface to that of normal-yield surface, called as the normal-yield ratio. It plays the role of de-scription for the approaching degree of stress to the normal-yield surface. The state

= 0R ( = 0f ) corresponds to the null stress state in which a purely elastic de-

formation behavior occurs, the state 0 1R< < ( 0 f F< < ) to the sub-yield

state, and the state = 1R ( =f F) to the normal-yield state in which the stress

lies on the normal-yield surface for which the plastic strain rate has been formu-lated in the conventional plasticity. It should be noted that the subloading surface is not assumed independently from the normal-yield surface but is merely the shadow surface projected from the normal-yield surface. Therefore, the subloading surface coincides completely with the normal-yield surface when they contact mutually, leading to = 1R . Eventually, there exists only one independent surface, i.e. the normal-yield surface in the subloading surface model identically to the conventional plasticity. On the other hand, plural independent yield (loading) surfaces are assumed and they contact mutually at one point, resulting in the dis-continuity of plastic modulus, in unconventional models, e.g. the multi surface model (Mroz, 1967; Iwan, 1967) and the two surface models (Dafalias, 1975; Krieg, 1975) and the infinite surface model (Mroz et al., 1981) delineated in the next chapter.

Normal-yield surface

Subloadingsurface

( ) ( )=f R F Hσ

ijσ

( ) ( )=f F Hσ

σ

0

1RRe

Elasticregion


Subloadingsurface

( ) ( )=f R F Hσ

ijσ

( ) ( )=f F Hσ

σ

0

1RRe

Elasticregion

Fig. 7.3 Normal-yield and subloading surfaces


Regarding the plastic strain rate formulation, it is noteworthy that the following equation should be fulfilled to incorporate assumptions a) and e) described above, noting Eqs. (6.40) and (6.41).

for =0

)tr(for =1

;( ) )tr(i

p

R

= RF h HF

'

⎧⎪⎪⎨⎪ ,⎪⎩

0

ND NNN

σ σσ

(7.7)

The simplest equation fulfilling Eq. (7.7) would be given by the following equation if adopting the interpolation method.

)tr(=

( ; ) )tr(i

p nRF h HF' ,

ND N

NN

σ σσ

(7.8)

However, Eq. (7.8) cannot be adopted for materials exhibiting softening be-havior induced by the plastic volumetric strain as seen in soils in which when the

stress reaches the critical state line, it holds that

( ; ) t r= = 0i h H, N Nσ , and

thus the stress cannot go up that line. In stead, we can assume various equations other than Eq. (7.8) fulfilling Eq. (7.7), e.g.

)tr(

;( ) )tr( ( )i

p = F h H RSF' +,

NND

NN

σ σσ

(7.9)

)tr(

)( ; ) ( ) tr(i

p =F h H RSF

' , +N ND

NN

σ σσ

(7.10)

where ( )RS is the monotonically-decreasing function of R fulfilling

for 0( )

= 0 for = 1

RRS

R

→ +∞ →⎧⎨⎩

(7.11)

Here, it remains ambiguous which equation among them is rigorous. In what fol-lows, let a pertinent equation for plastic strain rate be formulated by incorporating the consistency condition for the subloading surface.

The material-time derivative of Eq. (7.6) is given as

( )tr =( )f

RF H R F'• •∂ +

∂σ σσ (7.12)


Equation (7.12) itself cannot play the role of the consistency condition for the

derivation of plastic strain rate because it includes the rate variable R•

, which is not

related to the plastic strain rate at this stage, although H•

is related to the plastic strain rate in Eq. (6.37). To embody Eq. (7.12) as the consistency condition, let the

evolution rule of R , i.e. R•

be formulated to fulfill the following conditions in the

loading process ( )p ≠D 0 , based on the assumptions a )-e ) described in the

foregoing. a) The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate

is infinite up to a certain value of normal-yield ratio: || ||/ pR•

→∞D for

eR R≤ , where ( )<1 eR is the material constant describing the elastic limit

of R (Tsutsumi et al, 2006), whereas 20.eR ≥ for many metals but it can be

put that = 0eR for soils. b) The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate

is positive before the subloading surface coincides with the normal-yield sur-

face: || ||/ 0pR• >D for 10 R << .

c) The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate

decreases monotonically as the normal-yield ratio increases: || ||/ pR• D is the

monotonically-decreasing function of R . d) The subloading surface does not expand over the normal-yield surface. Then,

the ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate becomes zero when the subloading surface coincides with the normal-yield

surface || ||/ =0pR• D for 1=R .

e) A conventional elastoplastic constitutive equation holds as it is in the nor-mal-yield state: Eq. (6.40) with Eq. (6.41) holds for = 1R .

f) The ratio of the rate of normal-yield ratio to the magnitude of plastic strain rate becomes negative if the subloading surface becomes larger than the nor-

mal-yield surface: || ||/ 0pR• <D for >1R .

Based on these assumptions, let the following evolution equation of the nor-mal-yield ratio R be assumed.

|| ||( ) for = ppRR U

• ≠D D 0 (7.13) where U is a monotonically decreasing function of R , fulfilling the following conditions (see Fig. 7.4).

for 0 (quasi - elastic state)

0 for < 1 (sub - yield state),( )

= 0 for = 1 (normal - yield state),

0 for 1 (over normal - yield state)

e

e

R R

R < RU R

R

R

→ +∞ ≤ ≤⎧⎪>⎪⎨⎪⎪⎩< >

(7.14)


The stress-controlling ability attracting stress to normal-yield surface is furnished in this model incorporating Eq. (7.14). Let the function U satisfying Eq. (7.14) be simply given by

=( ) cot 2 1( )e

e

RRuRU Rπ − ⟩⟨

− (7.15)

where u is a material constant and ⟨ ⟩ is the McCauley’s bracket defined by = ( + | |)/2s s s⟨ ⟩ , i.e. == for 0 and 0 for s<0s s ss ⟨ ⟩⟨ ⟩ ≥ for an arbi-

trary scalar variable. Equation (7.13) with Eq. (7.15) can be integrated analytically as

1 00

0

0

( )

1

2 1 cos= cos exp 11 22

cos1 22( )1 ln=

cos2

)( ) (

( )

( )

p p

p p

ee e

ee

e

ee

e

e

R R

R RR

R RR u RRR R

RR

ε ε

ε ε

πππ

π

π π

− +

−−

⎫−− ⎪− − −− ⎪⎪−⎬−−− ⎪⎪⎪⎭

(7.16)

under the initial condition 00 =:=pp R Rε ε , where || || ( : time)pp dt tε ≡ ∫ D .

The following equation for U has been used after Hashiguchi (1989) but the analytical integration cannot be obtained.

( ) = ln1

e

e

RRU R uR

− ⟩⟨− − (7.17)

eR R10

|||| ( )=p RUR

•

D

eR R10

|||| ( )=p RUR

•

D

eR R10

|||| ( )=p RUR

•

D

Fig. 7.4 Function ( )RU in the evolution rule of normal-yield ratio


The value of R is calculated by substituting the current values of stress and internal variables into the equation of the subloading surface. It can also be calcu-lated by integrating the evolution equation of R , i.e. Eq. (7.13) in the loading

process ( )p ≠D 0 . However, it must be calculated directly from the equation of

the subloading surface in the unloading process because the evolution equation, Eq. (7.13), does not hold in that process.

The substitution of Eq. (6.37) and (7.13) into Eq. (7.12) yields the consistency condition extended for the subloading surface:

( )tr || ||( ; ) ( )=( ) p pi

f Rh H FRF U∂ +,′∂ D D

σ σ σσ (7.18)

Here, note that Eq. (7.18) reduces to the consistency condition (6.43) in the con-ventional plasticity for = 1R . Therefore, assumption e) in the foregoing was also fulfilled.

Further, substituting the associated flow rule

=p λ ND (7.19)

into Eq. (7.18) and using the relation

( )t r

( ) ( )= == ) ) )tr( tr( tr(

( )ff f RF

∂∂ ∂

∂N N N

N N N


(7.20)

based on the Euler’s theorem of the homogeneous function in degree-one, it follows that

;( ) ( )(tr ) =)tr(

iRF Rh H FRF Uλ λ+,′ NNN

σσσ

from which one has

t r( )p=

Mλ Nσ

(7.21)

Consequently,

t r( )pp=

MN NDσ

(7.22)

where

;( ) ( )tr( )i

p F U HhM RF′ +,≡ NN σ σ

(7.23)

Substituting Eq. (6.29) and (7.22) into Eq. (6.28), the strain rate is given by

1 t r( )= pM

− + N ND E σσ

(7.24)


from which we have

t r( )=

tr( )pMΛ +

DNENEN

(7.25)

The stress rate is described from Eqs. (7.24) and (7.25) as

t r( )=

tr( )pM +⟨ ⟩EDN NEEDNEN

σ −

(7.26)

The plastic strain rate is induced depending on the direction of strain rate, even in the subyield surface. Therefore, a judgment of whether or not the yield condition is satisfied is not required. Therefore, the loading criterion (6.68) is simplified as

: 0

= : otherwise

p

p

Λ ⎫≠ ⎪⎬⎪⎭

>D 0

D 0 (7.27)

The following relation is obtained from Eq. (7.26) in the loading state

( )p ≠D 0 .

t r( )=tr( )tr( )

p

pM

M +EDN N

NENσ

(7.28)

It is known from Eq. (7.28) that the subloading surface expands or contracts de-pending on the sign of

pM . Then, let it be defined that

0 tr( ) 0 : subloading hardening

= 0 tr( ) = 0 : subloading nonhardening

0 tr( ) 0 : subloading softening

p

p

p

M

M

M

⎫> → > ⎪⎪⎪→ ⎬⎪⎪< → <⎪⎭

N

N

N

σ

σ

σ

(7.29)

On the other hand, the normal-yield surface expands or contracts depending on the

sign of the rate of isotropic hardening function F•

. Then, let it be defined alterna-

tively that

0 : normal - isotropic hardening

= 0 : normal - isotropic nonhardening

0 : normal - isotropic softening

F

F

F

•

•

•

⎫> ⎪⎪⎪⎬⎪⎪<⎪⎭

(7.30)

On the other hand, a similar equation to Eq. (7.28) holds also for the conven-

tional plasticity. Nevertheless, the signs of t r( )Nσ and F•

are the same because

7.3 Salient Features of Subloading Surface Model 181

of Eq. (6.35) of the consistency condition. Further, the positiveness of λ in Eq.

(6.40) engenders the coincidence of the signs of t r( )Nσ and pM . Therefore,

the sings of t r( )Nσ , pM and F

• coincide mutually. Eventually, the distinction

between Eqs. (7.29) and (7.30) does not exist in the conventional plasticity.

7.3 Salient Features of Subloading Surface Model

The subloading surface model reduces to the conventional plasticity model with the yield surface enclosing the purely elastic domain by putting u → ∞ or 1= eR . It is improved substantially from the conventional model and has noticeable advan-tages compared even with the other unconventional model, as described below.

i) It predicts a smooth response in a smooth monotonic loading process, i.e. a smooth relation of axial stress and axial logarithmic strain in uniaxial mono-tonic loading for example, as shown in Fig. 7.5.

0 0

2σ

1σ

3σ

1σ

1ε

σSubloading surface modelE

last

ic

eR

Conventional plasticity( ) model u → ∞


Subloadingsurface

R 1

0 0

2σ

1σ

3σ

1σ

1ε

σSubloading surface modelE

last

ic

eR

Conventional plasticity( ) model u → ∞


Subloadingsurface

R 1

Fig. 7.5 Smooth stress-strain curve predicted by the subloading surface

The influence of the material constant u in the evolution rule of the nor-mal-yield ratio R on the curvature of stress-strain curve is depicted in Fig. 7.6. The stress-strain curve for u → ∞ coincides with that predicted by the con-ventional constitutive equation. For the smaller value of u , a more gentle


transition from the elastic to plastic state, called the elastic-plastic transition, is

predicted. The decrease in the value of eR also engenders the prediction of

smoother elastic-plastic transition. The smoothness condition is fulfilled al-ways even for 0eR ≠ , because the ratio of the rate of normal-yield ratio to the

magnitude of plastic strain rate, i.e. || ||/ pR• D , changes continuously from the

infinite value at = eR R to zero in the normal-yield state ( )=1R . On the other hand, a non-smooth response is predicted by the conventional

constitutive model, violating the smoothness condition at the yield point. The nonsmooth response is predicted also by the unconventional plasticity models other than the subloading surface model, e.g. the multi surface model (Mroz, 1967; Iwan, 1967), the two surface model (Dafalias, 1975; Krieg, 1975) and the nonlinear kinematic hardening model (Armstrong and Fredericson, 1966; Chaboche et al., 1979) because the small yield surface enclosing the purely-elastic domain and/or plural subuield surfaces with different sizes are assumed and thus the smoothness condition is violated when the stress reaches theses surfaces, exhibiting the discontinuous mechanical response.

ii) The stress always lies on the subloading surface which plays the role of the loading surface. Therefore, only the judgment for the sign of the propor-tionality factor Λ is required in the loading criterion for the subloading surface model. On the other hand, for the plasticity model assuming the yield surface enclosing the purely-elastic domain, a judgment whether or not the stress lies on the yield surface is required in addition to a judgment for the sign of Λ . Moreover, a judgment on which loading surface the current stress lies is required in the unconventional models assuming the subyield surface(s).

iii) A stress is automatically drawn back to the normal-yield surface even if it goes out from that surface because it is formulated that 0R

• < for 1R > (over normal-yield state) in Eq. (7.13) with condition (7.14) (see Fig. 7.7). A stable and robust calculation can be executed even by rough loading steps as will be verified quantitatively in Chapter 14. On the other hand, the numerical calcu-lation by the conventional plasticity model requires the algorithm to pull back the stress to the yield surface. Unconventional models other than the subloading surface model also requires the operation to pull back the stress to a current loading surface, as described in the next chapter.

Eventually, the subloading surface model can describe pertinently the monotonic loading process. Besides, it is necessary to describe rigorously the deformation behavior of materials undergoing softening behavior with the plastic volumetric strain, e.g. soils as will be delineated in Chapter 11. In addition, the subloading surface model possesses the high ability in numarical calculation, while this distinguishable advantage will be quantitatively verified in Chapter 14.

7.3 Salient Features of Subloading Surface Model 183

0

u =∞

u decreases

Subloading surface modelConventional plasticity model

σ

ε0

u =∞

u decreases

Subloading surface modelConventional plasticity model

σ

ε

Fig. 7.6 Variation in curvature of stress-strain curve due to material constant u in evolution rule of normal-yield ratio

σ

0

( ) || || for = p pRR U• ≠D D 0

R>1: R<0•

R<1: R>0•

R=1: R=0•

σ

pε

for 0 (almost elastic state)




e

e

R R

R < RU R

R

R

→ +∞ ≤ ≤⎧⎪>⎪⎨⎪⎪⎩< >

σ

0

( ) || || for = p pRR U• ≠D D 0

R>1: R<0•

R<1: R>0•

R=1: R=0•

σ

pε

for 0 (almost elastic state)




e

e

R R

R < RU R

R

R

→ +∞ ≤ ≤⎧⎪>⎪⎨⎪⎪⎩< >

Fig. 7.7 Stress-controlling function in subloading surface model: Stress is automatically attracted to yield surface in the plastic loading process


7.4 On Bounding Surface and Bounding Surface Model

The terms bounding surface and bounding surface model are widely used for models falling within the framework of unconventional plasticity describing the plastic strain rate induced by the rate of stress inside the yield surface. This no-menclature was named by Y. F. Dafalias (1975), who also coined the terms plastic spin (Dafalias, 1985) and hypoplasticity (Dafalias, 1986). The only concrete model proposed by Dafalias as the bounding surface model is the two-surface model (Dafalias and Popov, 1975), in which a small subyield surface is assumed inside the yield surface, which encloses the purely elastic domain and translates maintaining the constant ratio of the size to the size of yield surface, and the single surface model (Dafalias and Popov, 1977), in which the subyield surface shrinks to a point.

However, note the following facts.

1) The so-called bounding surface is no more than the yield surface that has been used historically in the field of plasticity. The term yield surface has a clear physical meaning that the plastic deformation begins when stress reaches it; it also has the geometrical meaning that the stress cannot go out from it in the quasi-static deformation process. In contrast, the phrase bounding surface has only a geometrical meaning but has no physical meaning.

2) The yield surface always exists. However, the stress goes over the yield surface in the deformation process at a high rate as represented by the overstress model in the viscoplastic constitutive models. Therefore, no surface exists which bounds the stress except for the quasi-static deformation process. Consequently, the phrase bounding surface has no generality.

3) The bounding surface is to be the yield surface itself. Therefore, the term bounding surface model induces the confusion that all unconventional plastic-ity models inheriting the yield surface belong to the bounding surface model. Words expressing salient features of each model should be used for names of models to avoid confusion. In fact, researchers aside from Dafalias, e.g. Krieg (1975) uses the term limit surface to describe the novel loading surface in his two surface model, Mroz (1967) uses outmost surface in his multi surface model, and Hashiguchi (1989) uses normal-yield surface in his subloading surface model instead of yield surface. However, they use these words only in a limited sense for naming elements in their models: they never use these words as names of their proposed models such as the limit surface model, the outmost surface model, or the normal-yield surface model.

Furthermore, Dafalias uses the phrase bounding surface model with a radial mapping (Dafalias and Herrmann, 1980). Nevertheless, the model has a physical and mathematical structure that differs from that of a two-surface model but has similar structure to that of the subloading surface model proposed in 1977 three years earlier than 1980 when Dafalias began to write the articles on the bounding surface model with a radial mapping. Furthermore, note that it involves various immature and impertinent formulations as described below.

7.4 On Bounding Surface and Bounding Surface Model 185

In the bounding surface model with a radial mapping, any surface other than the yield surface is adopted, but the following ratio is adopted as the measure to de-scribe the degree of approaching the yield surface.

y || |||| ||b /≡ σ σ (7.31)

which is the ratio of the magnitude of current stress σ to the magnitude of conju-gate stress yσ on the yield surface. Then, the plastic modulus pM in the plastic

strain rate of the conventional plasticity, i.e. Eq. (6.41) is modified as

for 1ˆ = for 1=

p pp

e

e

b bbHM Mb b M b

≥→ ∞⎧⎛ ⎞−+→ ⎨⎜ ⎟− ⎜ ⎟⎩⎝ ⎠⟨ ⟩ (7.32)

where H is the function of stress and internal variables and eb is the value of the

variable b at the elastic limit. Here, note the following facts.

i. The variable (1 )b b≤ ≤ ∞ is merely the inverse of normal-yield ratio

(0 1)R R≤ ≤ in the subloading surface model.

ii. The plastic modulus pM in the bounding surface model with a radial mapping is given by the interpolation rule between the null stress and the conjugate stress on the yield surface, where no consistency condition is introduced as

confirmed from the statement “No consistency condition = 0f•

is required for

stress points inside = 0F , since now = 0f is always defined at any ijσ .”

(P. 978: Dafalias, 1986)), whereas the consistency condition for the subloading surface is introduced into the subloading surface model. Various equations other than Eq. (7.32) can be assumed for the plastic modulus if an easy-going interpolation method is adopted. It might be readily apparent that Eq. (7.32) differs substantially from the plastic modulus of Eq. (7.23) which is derived rigorously from the consistency condition obtained by incorporating the as-sumption that the normal-yield ratio approaches unity in the plastic loading process.

iii. Therefore, it is not guaranteed that the stress approaches the yield surface in the plastic loading process. On the other hand, the subloading surface model pos-sesses a stress controlling function to attract the stress to the yield surface in the plastic loading process even if the stress goes out from the yield surface in the numerical calculation by the finite stress increments.

iv. A formulation for describing cyclic loading behavior has not been given for the bounding surface model with a radial mapping. On the other hand, it has been attained in the subloading surface model as the extended subloading surface model (Hashiguchi, 1989) by making the similarity-center of the normal-yield and the subloading surfaces move with the plastic strain rate as described in detail in the next chapter.


Eventually, it can be concluded for the bounding surface model with radial mapping as follows:

I) The bounding surface is substantially the synonym of the yield surface al-though it does not express any physical meaning. Therefore, the bounding surface model would cause confusion as if all models adopting the yield surface belong to this model, while in fact it is insisted by Dafalias that “the model with “stress reversal surfaces” (infinite surface model of Mroz et al., 1981) pro-posed for soils can be classified as a radial mapping model” (p. 981: Dafalias, 1986) in addition to the impertinent assessment on the subloading surface model. It is desirable to make effort for concrete formulation of pertinent model rather than only coining new terms. For that reason, the words yield surface should not be replaced with the term bounding surface which would not have to be used for the steady and sound development of elastoplasticity.

II) The bounding surface model with radial mapping falls within the framework of the subloading surface model but it is not formulated rationally, whereas the rigorous formulations including the description of cyclic loading behavior have been given by the subloading surface model.

7.5 Incorporation of Anisotropy

A subloading surface based on the yield surface in Eq. (6.84) with the kinematic and rotational hardening is described as

( ) ( )=ˆf RF H,βσ

(7.33)

The material-time derivative of Eq. (7.33) is given as

( ) ( ) ( )tr tr tr =

ˆ ˆ ˆˆ ˆ

( ) ( ) ( )f f fR F RF H• •, , ,∂∂ ∂+− ′+∂∂ ∂

β β β ββσ σ σσ ασ σ

(7.34) Substituting Eq. (6.37), (6.86), (7.13) and the associated flow rule

ˆ=p λ ND

(7.35)

(7.34) is rewritten as

( ) ( ) ( ) ˆ|| ||t r tr trˆ ˆ ˆˆ ˆ

( ) ( ) ( )f f f'λλ, , ,∂∂ ∂+− ∂∂ ∂

β β βNβ bσ σ σσ aσ σ

ˆ ;( )= ih H FURF λ λ+,′ Nσ

(7.36)

Noting

( )t r

( ) ( )ˆ ˆ ˆ= == ˆ ˆ ˆ) ) )tr( tr( tr(

ˆˆ ˆˆˆ ˆ ˆ ˆ

( )ff f RF

,∂, ,∂∂

∂

ββ β

N N NN N N


(7.37)

7.6 Incorporation of Tangential Inelastic Strain Rate 187

Eq. (7.36) is rewritten as

ˆ )tr( ( )ˆ ˆ ˆ|| ||) ) t rtr( tr(ˆ ˆ( )f

RF 'λλ,∂− +

∂N β

N N Nβ bσ σaσ

ˆ ˆ ;( ) )tr(= ˆ)( i

UF h H RFλ λ′ +, N Nσ σ

(7.38)

from which one has

ˆ ˆtr( ) tr( ) ˆ= , =pp pM M

λ N ND Nσ σ

(7.39)

where

ˆ|| || ( )ˆˆ ( );tr

ˆˆ([ ) ][ ]p

ifUF HhM

R RFF' ,∂′≡ , + − +∂

βNNN β bσσ aσ

(7.40)

7.6 Incorporation of Tangential Inelastic Strain Rate

Let the tangential inelastic strain rate be incorporated into the above-mentioned subloading surface model in the following (Hashiguchi, 1998, 2005; Hashiguchi and Tsutsumi, 2003; Hashiguchi and Protasov, 2004; Khojastepour and Hashiguchi, 2004a, b).

Assume that T in Eq. (6.111) in 6.5 is a monotonically-increasing function of R in addition to the stress and the internal variable, i.e.

T Rτξ≡

(7.41)

where τ is the material constant and ξ is the function of the stress σ and the in-

ternal variables iH as

( , )= iHξ ξ σ

(7.42)

Adding Eq. (6.111) with Eq. (7.41) into Eq. (7.24) yields

1 ˆtr( ) ˆ= 2p tRGM

'τξ− + +ND E N

σσ σ

(7.43)

or

1 ˆtr( ) ˆˆ= 2p

RGM

'τξ− + +ND E IN

σσ σ

(7.44)

1 ˆ ˆ ˆ= 2 )( pRGM

'τξ− ⊗+ +N N ID E σ

(7.45)


from which the stress rate is given as

ˆ( )t r ˆ=ˆ ˆ 1( )t rp t

RGRM

'τ

τξξ

2− −++

⟨ ⟩NED ENED DNEN

σ (7.46)

or

ˆ=( )t r 1

( )p

RGRM

'τ

τξξ

2− −⊗+ +

ENEN IE D NEN

σ (7.47)

The deviatoric normal and tangential stress rates for the subloading surface model are shown in Fig. 7.8. This model fulfills the continuity and the smoothness condition as shown in Fig. 7.9. On the other hand, all models other than the subloading surface model violate the smoothness condition. Therefore, they also violate the continuity condition as illustrated for the 2J -deformation model of Rudnicki and Rice (1975) in Fig. 7.9.

The subloading surface model has been applied to metals (Hashiguchi, 1980; 1989; Hashiguchi and Yoshimaru, 1995; Hashiguchi and Tsutsumi, 2001; Hashiguchi and Protasov, 2004; Khojastehpor et al., 2006; Tsutsumi et al., 2005) and soils (Hashiguchi and Ueno, 1977; Hashiguchi, 1978; Topolnicki, 1990; Kohgo et al., 1993; Asaoka et al., 1997; Hashiguchi and Chen, 1998; Hashiguchi et al. 2002; Khojastehpor and Hashiguchi, 2004a,b; Hashiguchi and Tsutsumi, 2006; Hashiguchi and Mase, 2007; Wongsaroj et al., 2007). Consequently, its capability has been verified widely.


Subloadingsurface

ij'σ

σ

0

ˆ'n

'σn'σ

t'σ


Subloadingsurface

ij'σ

σ

0

ˆ'n

'σn'σ

t'σ

Fig. 7.8 Normal and tangential stress rates for subloading surface model in deviatoric stress plane

7.6 Incorporation of Tangential Inelastic Strain Rate 189

1σ

Input : σ

2σ

0

Yield surface

σ

1σ

3σ

2σ

1σ

tD

Normal-yieldsurface

0 0

tD

Subloading surface model

2σ 3σ 3σ

Yield surface

Conventional plasticity model:Rudnicki and Rice model

1σ

Input : σ

2σ

0

Yield surface

σ

1σ

3σ

2σ

1σ

tD

Normal-yieldsurface

0 0

tD


2σ 3σ 3σ

Yield surface

Conventional plasticity model:Rudnicki and Rice model

Fig. 7.9 Incorporation of tangential inelastic strain rate


Chapter 8

Cyclic Plasticity Model: Extended Subloading Surface Model

8 Cyclic Pla sticity Model: Extended Subloading Surface Model

The subloading surface model described in Chapter 7 would be the only pertinent unconventional model for the description of monotonic loading behavior but it cannot describe the cyclic loading behavior pertinently by the formulation itself shown therein, predicting the unrealistically large plastic strain accumulation because only elastic deformation is predicted in the unloading process. Various unconventional plasticity models aimed at describing the cyclic loading behavior, i.e. the cyclic plasticity models have been proposed, while models other than the extension of the subloading surface model involve serious defects individually. The basic features of these models will be reviewed first and thereafter the formulation of the extended subloading surface model is described in detail in this chapter.

8.1 Classification of Cyclic Plasticity Models

Various cyclic plasticity models have been proposed to date. They are classifiable into the two types. One type is based on the idea of kinematic hardening, i.e., the translation of subyield surface assumed inside the yield surface: the size retains a constant ratio to the yield surface. The other type is based on the idea of a subloading surface which expands/contracts keeping similarity to the yield surface as the stress approaches/leaves the yield surface described in the last section, in which a single independent surface is assumed. The classification is shown schematically in Fig. 8.1.

8.2 Translation of Subyield Surface(s): Extension of Kinematic

Hardening

The description of plastic deformation induced in the subyield state has been initiated through the extension of kinematic hardening concept as described below.

8.2.1 Multi-Surface Model

Mroz (1967) and Iwan (1967) proposed the multi surface model which assumes the multiple encircled subyield surfaces which are pushed out by the current stress,

192 8 Cyclic Plasticity Model: Extended Subloading Surface Model

Multi surfacemodel

Mroz (1966, 1967)Iwan (1967)

Two surface modelKrieg (1975)

Dafalias & Popov (1975)

Infinite surface modelMroz et al. (1981)

Single surface model

(nonlinear kinematichardening model)

Armstrong-Fredericson (1966)Chaboche et al. (1979)

Subloading surfacemodel

Hashiguchi-Ueno (1977)Hashiguchi (1980)

Extended subloading surface model

Hashiguchi (1986, 1989)

Translation of subyield surface

Cyclic plasticity model

Simplification

Expansion/contractionof loading surface

Translation ofSimilarity-center

Mathematicalsimplification

Multi surfacemodel

Mroz (1966, 1967)Iwan (1967)

Two surface modelKrieg (1975)

Dafalias & Popov (1975)

Infinite surface modelMroz et al. (1981)

Single surface model

(nonlinear kinematichardening model)

Armstrong-Fredericson (1966)Chaboche et al. (1979)

Subloading surfacemodel

Hashiguchi-Ueno (1977)Hashiguchi (1980)

Extended subloading surface model

Hashiguchi (1986, 1989)

Translation of subyield surface

Cyclic plasticity model

Simplification

Expansion/contractionof loading surface

Translation ofSimilarity-center

Mathematicalsimplification

Fig. 8.1 Classification of cyclic plasticity models

retaining the ratios of their sizes to the size of the yield surface as constant. The plastic modulus depends on the ratio for the subyield surface on which the current stress lies.

In this model, the stress translates by the difference of the half sizes of subyield surfaces and then contacts with a larger subyield surface in order in the initial loading process from the initial isotropic state where the subyield surfaces are concentric with respect to the origin of stress space. On the other hand, it translates by the difference of the sizes of subyield surfaces and then contacts with a larger subyield surface in turn in the unloading-reverse loading process. Therefore, the Masing rule (Masing, 1926) meaning that the curvature of stress-strain curve in the unloading-reverse loading decreases compared to the curvature of initial loading curve is described exactly and simply. On account of this mechanical feature, this model has been used widely. However, it includes the following serious defects.

1) Variations in curvatures of stress-strain curves in test data are not as great as those described by this model.

2) Subyield surfaces with different sizes having different plastic moduli contact mutually at one point. Then, the singular point of plastic modulus is induced at the contact point. Numerical calculation of cyclic loading behavior in the vicinity of contact point becomes unstable.

8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening 193

Normal-yield state

0

σ

0

σ

Constant stress amplitude

No strain accumulation(Trace of fixed loop)

Normal-yield state

Nonh-hardening state

Hardening state

Elastic state

ε p

ε p


No strain accumulation(Gradual contraction of loop)

Elastic state

Normal-yield state

0

σ

0

σ


No strain accumulation(Trace of fixed loop)

Normal-yield state

Nonh-hardening state

Hardening state

Elastic state

ε pε p

ε pε p


No strain accumulation(Gradual contraction of loop)

Elastic state

Fig. 8.2 Prediction of cyclic loading behavior by the multi surface model under constant one-side stress amplitude

3) Plastic modulus decreases suddenly at the moment when the stress reaches a larger subyield surface. Therefore, the smoothness condition (Hashiguchi, 1993a, b, 1997, 2000) is violated at that moment. Needless to say, the smooth stress-strain curve cannot be described.

4) Plastic strain accumulation in cyclic loading cannot be described for the stress amplitude within the smallest subyield surface enclosing a purely elastic domain.

5) In the cyclic loading process with the constant amplitude of the positive or negative one side stress, the plastic shakedown is induced immediately for non-hardening (sub)yield surfaces and after several cycles for hardening (sub)yield surfaces, tracing the fixed loop cyclically (Fig. 7.2). In other words, the accumulation of plastic strain for a cyclic loading under positive or negative one side stress amplitude, called the mechanical ratcheting effect, cannot be described at all by this model. Therefore, the deformations of machinery and structures subjected to cyclic loading are predicted to be unrealistically small.


6) The continuity condition (Hashiguchi, 1993a, b, 1997, 2000) is also violated at the moment when the stress transfers to a larger subyield surface if the tangential inelastic strain rate described in 6.6 is incorporated.

7) Judgment on which subyield surface among multiple subyield surfaces the current stress lies is required in the loading criterion. In addition, deformation analysis by this model is complicated because it is necessary to calculate all movements of multi-subyield surfaces. It increases memory usage and calculations necessary for numerical analysis. It becomes more serious in the analysis of cyclic loading behavior.

8.2.2 Two-Surface Model

Dafalias and Popov (1975) and Krieg (1975) proposed the simplification of the multi surface model by assuming only one subyield surface enclosing a purely elastic domain. Then, it assumes two surfaces, i.e. the yield and the subyield

0

σ

ε p

0

σ

Elastic state

Cyclic loading inside subyield surface.

ε p

Constant one-side stress amplitude


No strain accumulation(Elastic deformation)

Normal-yield state

Elastic state

0

σ

ε p

0

σ

Elastic state

Cyclic loading inside subyield surface.

ε pε p

Constant one-side stress amplitude


No strain accumulation(Elastic deformation)

Normal-yield state

Elastic state

Fig. 8.3 Prediction of cyclic loading behavior by the two surface model under constant one-side stress amplitude

8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening 195

surfaces and thus it is called the two surface model, while the yield surface is called the bounding surface by Dafalias and the limit surface by Krieg as described in 7.4. The plastic modulus is assumed to depend on the distance from the current stress on the subyield surface to the conjugate stress on the yield surface, whereas the outward normal of these surfaces have the same direction. Here, it is required that the subyield surface must translate so as not to intersect with the yield surface. The pertinent translation rules have been given by Hashiguchi (1981, 1988). This model have been adopted widely for the prediction of deformation behavior of metals (cf. e.g. Dafalias and Popov, 1976; McDowell, 1985, 1989; Ohno, 1982; Ohno and Kachi, 1986; Ellyin, 1989; Hassan and Kyriakides, 1992; Yoshida and Uemori, 2002, 2003).

This model is accompanied with the defects 2), 3), 4), 6) and a part of 7 ) described for the multi surfaced model. Further, the plastic modulus is determined only by the distance from the current stress to the yield (bounding) surface in this model. Then, the stress-strain curve with a unique curvature is predicted independent of the loading processes in this model in direct opposition to the multi surface model. In other words, the curvatures of the initial, the reloading and the unloading-inverse loading curves cannot be distinguished. Therefore, the open hysteresis loop is predicted in the unloading-reloading process. On account of this feature, this model cannot predict the cyclic loading behavior pertinently, describing excessively large accumulation of plastic strain as shown in Fig. 8.3.

The single surface model is proposed by Dafalias and Popov (1977), in which the subyield surface shrinks to a point in the two surface model. It would describe a smooth response and the dependence of the direction of plastic strain rate on the direction of stress rate but it would not be applicable to the cyclic loading behavior, predicting the open hysteresis loop.

8.2.3 Infinite-Surface Model

Modification of the multi surface model was proposed by Mroz et al. (1981), in which infinite number of subyield surfaces are incorporated inside the yield surface in contrast to the two surface model. It is called the infinite surface model. The smoothness condition is fulfilled in the monotonic loading process but it is violated at the moment when the stress passes through the starting point of unloading in the reloading process after the partial unloading, whereas the plastic modulus is singular at that point, since subyield surfaces with different sizes contact mutually. All the defects 1)-7) described in the multi surface model except for the fulfillment of smoothness condition in the monotonic loading process are retained in the infinite surface model.

8.2.4 Nonlinear Kinematic Hardening Model

The nonlinear kinematic hardening model proposed by Armstrong and Fredericson (1966) quite overestimates the mechanical ratcheting effect as the two surface model does. In order to improve this defect, the superposition of several nonlinear


Experiment

Elastic state

Elastoplastic statePrediction:

(no strain accumulation)

0

⎫⎫⎬⎬⎭⎭

σ

ε

Experiment

Elastic state

Elastoplastic statePrediction:

(no strain accumulation)

0

⎫⎫⎬⎬⎭⎭

σ

ε

Fig. 8.4 Prediction of cyclic loading behavior after slight unloading by the multi, the two and the nonlinear kinematic hardening models having a purely elastic domain

kinematic hardening rules was proposed by Chaboche et al. (1979). The mathematical structure seems quite simple at first view, while it is one of polynomial approximation so that a fitting to test data might be easy. On the other hand, it needs to use many material constants without clear mechanical meaning. The mechanical behavior of this model is identical to that of the two-surface model as examined by Chaboche (1989). A singularity of the plastic modulus is not induced because only one yield surface is assumed; however, the smoothness condition is violated when the stress reaches the yield surface. Therefore, similar defects as those in the two-surface model are involved in this model. Needless to say, the continuity condition is also violated if the tangential inelastic strain rate is incorporated. Eventually, this model is physically and mathematically quite irrational contrary to the repeated long reviews by Chaboche (1989, 2008) and Lamaitre and Chaboche (1990).

A small (sub)yield surface enclosing the elastic domain is assumed in all the unconventional plasticity models described above. Then, the accumulation of plastic strain observed in experiments shown in Fig. 8.4 cannot be described for cyclic loading of partial unloading-reloading even immediately after the saturation of hardening. Therefore, it results in risky mechanical designs.

8.3 Extended Subloading Surface Model

As described in the beginning, the subloading surface model formulated in Chapter 7, called the initial subloading surface model hereinafter, is incapable of describing cyclic loading behavior appropriately, predicting an open hysteresis loop in an unloading-reloading process and thus overestimating a mechanical ratcheting

8.3 Extended Subloading Surface Model 197

effect. This defect has been remedied by making the similarity-center of the normal-yield and the subloading surfaces move with the plastic strain rate after Hashiguchi (1986, 1989) as will be explained below.

Now, adopt the normal-yield surface with the anisotropy in Eq. (6.84), i.e.

( ) ( )=ˆf F H,βσ (8.1)(re-record)

for the formulation of the extended subloading surface model. Denoting the similarity-center by the symbol s of the normal-yield and the subloading surfaces, the conjugate point in the subloading surface for the back stress α in the

normal-yield surface by α , the conjugate point on the normal-yield surface for the

current stress σ in the normal-yield surface by yσ and the ratio of the size of

extended subloading surface to that of the normal-yield surface by R , the following relations hold (see Fig. 8.5).

= ˆ yRσ σ (8.2)

= = ˆ, ˆ RR −α s ss s (8.3) where

ˆ y y≡ −σ σ α， ≡ −σ σ α (8.4)

ˆ ≡ −s s α ， ≡ −s s α (8.5)

It holds from Eqs. (8.3) and (8.5)2 that

ˆR≡ +σ σ s (8.6)

where ≡ −σ σ s (8.7)

Regarding σ as yσ and substituting Eq. (8.2) into Eq. (8.1) of the normal-yield

surface, the expression of the subloading surface is given as

( , ) = ( )f RF Hβσ (8.8)

By substituting Eq. (8.6) into Eq. (8.8), the subloading surface is rewritten as follows:

( , ) = ( )ˆf R RF H+ βσ s (8.9)

The normal-yield ratio R can be calculated from Eq. (8.9), while there does not exist an analytical solution in general. The explicit equations and calculation methods for metals and soils will be shown in Chapters 10 and 11, respectively.


yσ

σ

s

αα

s

N

N( )=ˆ y /Rσ σ

σ

ijσ0

Normal-yieldsurface

Subloadingsurface

−

−− −

−

−

yσ

σ

s

αα

s

N

N( )=ˆ y /Rσ σ

σ

ijσ0

Normal-yieldsurface

Subloadingsurface

−

−− −

−

−

Fig. 8.5 Normal-yield and subloading surfaces


( , ) ( , ) ( , )tr tr tr =( ) ( ) ( )f f f

R F RF H• •∂ ∂ ∂− + ′+∂ ∂ ∂

β β β ββσ σ σσ ασ σ

(8.10)

where it holds from Eq. (8.3) that

(1 )= ˆ RRR•

+ −−α sα s (8.11)

If the similarity-center s is located beyond the normal-yield surface, the subloading surface intersects with the normal-yield surface, resulting in the impertinence that the outward-normal of the subloading surface differs from that of the normal-yield surface in the intersecting point. Therefore, the similarity-center must lie inside the normal-yield surface. Consequently, the following inequality must hold.

( , ) ( )ˆf F H≤βs (8.12)

The time-differentiation of Eq. (8.12) at the limit state that s lies on normal-yield surface (see Fig. 8.6) leads to

( , ) ( , ) for ( , ) = ( )0t r ( ) t r ˆ ˆ ˆˆ ( )f f f F HF

•∂ ∂ −− +∂∂

≤β β βββs s sαss



( , ) ( , ) ( , )1t r ( ) t r t rˆ ˆ ˆˆˆ ˆ

( )f f fF

∂ ∂ ∂− +∂∂ ∂

β β β ββs s sαs ss s

( , )1 for ( , ) = ( )t r 0 ˆ ˆˆˆ f

f F HFF

•∂−∂

≤β βs sss (8.13)

noting the relation t r( ( , )ˆf∂ βs )/ ˆˆ∂ ss = F due to the Euler’s homogeneous function of degree-one. Further, Eq. (8.13) is rewritten as

( , ) ( , )1t r t r 0ˆ ˆ ˆˆ

( ) ( )[ ]f fF

F•∂ ∂− + −

∂∂≤β β ββ

s sαs ss

for ( , ) = ( )ˆf F Hβs (8.14) Equations (8.12) and (8.14) are called the enclosing condition of

similarity-center. Let the following equation be assumed, which fulfills Eq. (8.14) since

( ) /ˆ ˆf∂ ∂s s and y −σ s produces the obtuse angle as shown in Fig. 8.6.

••

•

•

Subloading surface


( , )ˆˆ

f∂∂

βss

NN

yσ

σσ s

αs

α

s/Rσ

ˆ yσ

ijσ0

σ

− −

−−

−

••

•

•

Subloading surface


( , )ˆˆ

f∂∂

βss

NN

yσ

σσ s

αs

α

s/Rσ

ˆ yσ

ijσ0

σ

− −

−−

−

Fig. 8.6 Limit state that similarity-center lies on normal-yield surface


1 ( , )tr =

ˆ ˆ ( ) py

fF

F|| ||c•∂− + −∂

β β Dβs s σαs (8.15)

where

y y≡ −σ σ s (8.16)

and c is the material constant. The evolution rule of the similarity center is given from Eq. (8.15) as follows:

1 ( , )= tr

ˆ ˆ( )p fF HF FR'c|| ||

• ∂+ + −∂

βD ββsσ αs s (8.17)

Although Eq. (8.17) is formulated so as to avoid the protrusion of stress from the normal-yield surface, it is conceivable that s approaches the current stress in general

as shown for the non-hardening case, i.e. = =β 0α and = 0H•

in Fig. 8.7. The

close approach of s to the normal-yield surface can be avoided by replacement of

/Rσ to ˆ/ / MR ℜ− sσ ( ( 1) :Mℜ < material constant prescribing the maximum

value of ( , ) /ˆs f Fℜ ≡ βs ) in Eq. (8.17) when it is required (cf. Hashiguchi,

1986, 1989).

•

•

s

yσ

s

αα

σ

•

•

Subloadingsurface

Normal-yieldsurface

ijσ0

−

•

•

s

yσ

s

αα

σ

•

•

Subloadingsurface

Normal-yieldsurface

ijσ0

−

Fig. 8.7 Direction of translation of the similarity- center in the general state (illustration for

non-hardening state:

F•

= 0, = =β 0α )


0 0paε

0as

as

F

0 03exp( ) 1 ( )2 [ ]

p pa a aF cs ε ε− − −∓

1

3 ( )2 aFc s−

as

paε p

aε

1−

3 ( )2 aFc s− +

F

0 0paε

0as

as

F

0 03exp( ) 1 ( )2 [ ]

p pa a aF cs ε ε− − −∓

1

3 ( )2 aFc s−

as

paε p

aε

1−

3 ( )2 aFc s− +

F

Fig. 8.8 Similarity-center vs. plastic strain curve in uniaxial loading process for non-harden-ing state

For the non-hardening case the relation of the axial components as and paD of

s and pD is given from Eq. (8.17) as follows:

3 ( )( )= (upper: , lower: ) < 002

p ppa aa aaD DF Dcs s± ± − > (8.18)

in the uniaxial loading process under the plastically-incompressible state. The time-integration of Eq. (8.18) is given as

0 003exp( ) 1 (= )2 [ ]

p pa a a a aF cs s sε ε± − +− −∓ (8.19)

which is shown in Fig. 8.8. Substituting Eq. (8.17) into Eqs. (8.11), one has

1 ( , )(1 )= tr

ˆ ˆˆ( )[ ]p F fRR RF FRc|| ||

• •∂+ − + + −−∂

βD ββsσα αα ss

(8.20)


and further substituting Eq. (8.20) into Eq. (8.10), it follows that

( , )t r

( , )(1 ) t r

( , )1 ( , ) tr =tr ˆ ˆˆ

( )

( )( )

[

]

[ ]

p

f

fRR

R

fF f F R FRRF F

c|| ||

• • ••

∂∂

∂ + −− +∂

∂∂ +++ −− ∂∂

β

βD

ββ ββ ββ

σ σσσ σα ασ

σs ss　

which, using the relation

( , ) ( , ) ( = 1)

f f∂ ∂≡ ∂ ∂β β

N Nσ σσ σ

(8.21)

( , )tr ( , )( , )

= = =) ) )tr( tr( tr(

( )fff RF

∂∂ ∂

∂

βββ

N N NN N N

σ σ σσσσ σσ σ

(8.22)

based on the Euler’s theorem for the homogeneous function in degree-one, is rewritten as follows:

)tr()tr(

(1 ) t r)tr(

1 ( , )( , )tr t r =ˆ ˆ ˆ( ) ( )

[

]

[

]

p

RF

RF RR

F ffFR R FRF F

c|| ||

•• • •

+ −−

∂∂+ − +− +∂ ∂

NN

DNN

ββ β ββ β

σσ

σασ

s σs s

i.e.

( , )1) trtr (1 )tr(

( , )1tr tr = 0

ˆˆ

ˆ

( )

( )

][

[

] ][

pfF

R F FR

f R FRFRRF

c|| ||•

• ••

∂+ −− + − ∂

∂ −− −+∂

β βN N D β

β βN β

sσσ α s

σ σs

i.e.

11( ) 1(1 ) )tr( tr ˆˆ )([ pF R RF RRc || ||

• •− −++ − + +− DN N σ ss σσσ α

( , ) ( , )11 tr = 0trˆ

ˆ( ) ( ) ]f fRFR F

∂ ∂−− − ∂∂β β ββ ββ

sσ σ s


This equation, with the relations

( )(1 ) = =

11 1( )= =

ˆˆ

ˆ ˆˆ

R

RRR R

+ − − + − − − ⎫⎪⎬−− + ⎪⎭

σ α s α s α σsσ

s s σσ σ s

on account of Eqs. (8.4) and (8.6), is transformed as

1 1)tr( tr ˆ )( [ pRF || ||F R R

c••

+ −+ +−N DN σσ σ σα

( , ) ( , )11 tr = 0trˆ

ˆ( ) ( ) ]f fRFR F

∂ ∂−− − ∂∂β β ββ ββ

sσ σ s (8.23)

Substituting Eqs. (6.37)， (6.86) and (7.13), regarding R as R , into this equation, the following consistency condition is obtained.

)tr(Nσ ;( ) ( )tr || ||ˆ[ i ip pF H Hh

F′ +, ,− DDN σ σaσ

( ) 1 ( , )11 ||( ) ||t r)( ( ))ip pU R f

HRR FR

c || || '∂−++ − ,∂

βD Dβ bσσ σ σ

( , )1 ||( ) || = 0trˆ

ˆ)( ) ]ipfR H

F'∂− ,−

∂β

Dβ bs σ s (8.24)

Substituting the associated flow rule

( > 0)=p λ λD N (8.25)

into Eq. (8.24), it holds that

( );( )tr)tr( ˆ[ ii

F Hh HFλ

′ ,+,− NN N σaσ σσ

( , )( ) 1 11 ||( ) ||t r() ) )( ifU R H

R FRRc '

∂− − ,++∂

βNβ bσ σσ σ

( , )1 ||( ) || = 0trˆ

ˆ)( ) ]ifR H

F'∂− ,− ∂

βNβ b

s σ s

from which one has

t r( )= pM

λ Nσ, t r( )= p

p

MND Nσ (8.26)


σ

pε

σ

ε p

σSubloading surface:

expands

Contracts

Expands

Normal-yieldsurface

(a)

(b)

(c)

(d)

(e)

σ

σ

s

pε

0

0 0

pε

σ

0 0

0

σ

0

σ

pε

σ

0

σ

pε

σ

(f)

0

σ

Expands

0

0

σs

σ

α

σs

α

σ

α

αsσ

σs

α

ασ

σαs

σασ sss

s

( = )pD 00

sσ

( )=0s

( )=0ss α

σContracts( = )pD 0

σ

pε

σ

ε p

σSubloading surface:

expands

Contracts

Expands

Normal-yieldsurface

(a)

(b)

(c)

(d)

(e)

σ

σ

s

pε

0

0 0

pε

σ

0 0

0

σ

0

σ

pε

σ

0

σ

pε

σ

(f)

0

σ

Expands

0

0

σs

σ

α

σs

α

σ

α

αsσ

σs

α

ασ

σαs

σασ sss

s

( = )pD 00

sσ

( )=0s

( )=0ss α

σContracts( = )pD 0

Fig. 8.9 Prediction of uniaxial loading behavior by extended subloading surface model: (a) initial state, (b) initial loading process, (c) unloading process until similarity-center, (d) unloading-inverse loading process after passing similarity-center, (e) reloading process until similarity-center and (f) reloading process after passing similarity-center. ( Stress, Similarity-center, Center of subloading surface)

where

; ( )( )tr ˆ[pii

F HM h HF

′ + ,,≡ NN σaσ σ

||( ) || ( , )1 1 ( )t r)( ( ) ( ifU R

HRR FR

'c ∂−++ − ,∂

βNβ bσ σσ σ

( , )( )1( )tr

ˆˆ( ) )]i

fHR R

∂− ,∂

ββ b

s σ+ s (8.27)

The strain rate is given from Eqs. (6.28), (6.29) and (8.26) as

1 t r( )= pM

− + ND E Nσσ (8.28)

from which the proportionality factor described in terms of the strain rate, denoted

by Λ instead of λ , in the flow rule (8.25) is given as follows:

8.4 Modification of Reloading Curve 205

t r ( )=

t r ( )pMΛ

+NED

NEN (8.29)

Using Eq. (8.29), the stress rate is given from Eq. (8.28) as follows:

t r ( )=t r ( )pM +

− NEDED ENNEN

σ

(8.30)

The loading criterion is given by

: , 0

= : otherwise

p

p

Λ≠ > ⎫⎪⎬⎪⎭

D 0

D 0 (8.31)

The relation of axial stress vs. plastic strain is illustrated for the uniaxial loading process of non-hardening material in Fig. 8.9. The closed hysteresis loop is depicted in this figure.

8.4 Modification of Reloading Curve

The relation 0 0( )= p pfR R ε ε− − holds in the loading process from Eq. (7.13)

with the function U of only normal-yield ratio R , where it is set that

pa~bε

aRbR

pa~bε pε

σ

s

= 1R

(similarity - center)

Monotonic

loading Reloading

Same

− −−

pa~bε

aRbR

pa~bε pε

σ

s

= 1R


Monotonic

loading Reloading

Same

− −−

Fig. 8.10 The defect of past subloading surface model: Unrealistically gentle returning to monotonic loading curve


= 1R

pε

σ

s

pε

σ

s

= 1R

Modification

(similarity-center)

(similarity-center)

−

−

= 1R

pε

σ

s

pε

σ

s

= 1R

Modification

(similarity-center)

(similarity-center)

−

−

Fig. 8.11 Description of cyclic loading behavior in the neighborhood of yield surface

|| ||pp dtε ≡ ∫ D , and 0R and 0pε are the initial values of R and pε , and thus the

inverse relation is given by 10 0( )=p p f R Rε ε −− − . Therefore, the

accumulation of the magnitude of plastic strain rate for a certain change of R is identical independent of loading processes, i.e. the initial loading process, the reloading process after a complete or partial unloading and the inverse loading process. This results in the prediction that the returning of the reloading stress-strain curve to the monotonic loading curve is unrealistically too gentle as shown in Fig. 8.10. In particular, the excessively large plastic strain accumulation is predicted for the cyclic loading process in the neighborhood of yield surface as shown in Fig. 8.11.

It can be stated that the curvature of unloading-inverse loading curve is smaller than that of the initial loading curve but the curvature of reloading curve is larger than it as has been described in the Masing rule (Masing, 1926). In order to describe this behavior, let the material constant u in Eq. (7.13) for the evolution rule of the normal-yield ratio be extended as follows:

8.4 Modification of Reloading Curve 207

1 ) u is more different from the value in the initial isotropic state as the similarity-center is nearer to the normal-yield surface. In order to describe the approaching degree to the normal-yield surface, supposing the similarity-center surface which passes through the current similarity-center and is similar to the normal-yield surface, the similarity-center yield ratio sℜ describing the ratio

of the size of the similarity-center surface to that of the normal-yield surface is introduced. The similarity-center surface is described by the equation

( , ) = ( )ˆ sf F Hℜβs by the replacement , , sR ℜ→ → →σ s α α in

the yield surface (6.84). Then, the similarity-center yield ratio is described by

= ( , )/ ( )ˆs f F Hℜ βs (0 1)sℜ≤ ≤ (8.32)

in terms of the known variables , , , Fβs α .

2 ) The similarity-center yield ratio is large and the deviatoric stress lies outside of the similarity-center surface in the reloading process. Conversely, the similarity-center yield ratio is also large but the deviatoric stress lies inside of the similarity-center surface in the inverse loading process. Which the deviatoric stress lies outside or inside of the similarity-center surface can be judged by the sign of the following variable Sσ .

ˆtr|| ||( )r

rsSσ ≡ n σσ ( ) Sσ− 1≤ ≤1 (8.33)

=

⎧⎪⎪⎨⎪⎪⎩

3 1ˆ for for= = =| ||| || 3

ˆtr ˆfor tr 0= || ||

( )

( )

m m

ms

s s

'

''

σ σσ

± ± ±n0 I

n n

σσσσ

where

1ˆfor =1 3ˆtr =3 ˆfor tr 0=

mr

ss

s

''

σ⎧ ±⎪+≡ ⎨⎪⎩

n In I

n

σσ σ

σ (8.34)

( , ) ( , )ˆ ˆ ˆ|| ||/sf f∂ ∂≡

∂ ∂β βn s s

s s (8.35)

Introducing these variables, let the material parameter u in Eq. (7.13) be modified as follows:

0

0 0

0

exp( ) for = 1 and = 1

= exp( ) = for = 0 or = 0

exp( ) for = 1 and = 1

s

s

s

s

ss

s

u u S

u u u S u S

u u S

σ

σ σ

σ

ℜℜℜℜ

⎧⎪⎨⎪ − −⎩

(8.36)


0=u u

pε

σ

s

s

0<u u

0=u u

= 1R0>u u


s

(improved)

(improved)

−−

0=u u

pε

σ

s

s

0<u u

0=u u

= 1R0>u u


s

(improved)

(improved)

−−

Fig. 8.12 Stress-plastic strain curve predicted by the modified subloading surface model: Rapid recovery to preceding monotonic loading curve

where 0u and su are the material constants, while the former denotes the mean value of u . Then, u increases in the loading direction but inversely it decreases in the opposite direction. By this modification, the phenomenon that in the reloading curve after a partial unloading the stress returns rapidly to the monotonic loading curve can be described realistically as shown in Fig. 8.12. Then, the plastic strain accumulation for the cyclic loading process in the neighborhood of yield surface is suppressed as shown in Fig. 8.11.

8.5 Incorporation of Tangential-Inelastic Strain Rate

Incorporating the tangential inelastic strain rate into Eqs. (8.28) and (8.30), the strain rate and the stress rate are given as follows:

1 t r( )= 2p t

RGM

'τξ− + +ND E N

σσ σ (8.37)

t r ( )=t r ( ) 1p t

RGRM

'τ

τξξ

2−++

− NEDED EN DNEN

σ (8.38)


where

tr( )=

(tr( ) )0= =

n

n tt

'' '' '

''' ''

⎫≡ ⊗ ⎪⎪

⎬⎪≡ − ⎪⎭

n n n n

NI

σ σσ

σσ σ σσ

(8.39)

tr( ) =t '' '' '−≡ nD D n DI D ' (8.40)

(|| || 1)= || ||

'' ''

≡ Nn nN

(8.41)

' '' '−≡ ⊗n nI I (8.42)


Chapter 9 Viscoplastic Constitutive Equations

9 Viscoplastic Constit utive Equations

Deformation of solids exhibits the dependence on the rate of loading or deforma-tion, i.e. the time- or rate-dependence. However, that dependence was deliberately ignored in preceding chapters. The history of development of the viscoplastic con-stitutive equation describing the rate-dependence of plastic deformation induced in the state of stress over the yield surface is first reviewed briefly. Then, the vis-coplastic constitutive equations falling within the framework of the overstress model are described exhaustively, which would be applicable to the description of deformation for the wide range of strain rate from the quasi-static to the impact loads.

9.1 History of Viscoplastic Constitutive Equations

Before pertinent formulation of the overstress model, the development of the vis-coplastic constitutive equations is reviewed below. An overview of the history of the development of viscoplastic constitutive equations is portrayed in Fig. 9.1.

The elastic constitutive equation extended so as to describe the rate-dependence is called the viscoelastic constitutive equation and one of the typical models is the Maxwell model, in which the spring and the dashpot are connected in series.

Therefore, the strain rate ε• is additively decomposed into the elastic strain rate 1=e Eε σ−• • and the viscous strain rate 1=v με σ−• , where σ designates the

stress, E is the elastic modulus and μ is the viscous coefficient. This model is concerned with the rate-dependent deformation at the low stress level below the yield stress. On the other hand, the elastoplastic constitutive equation can be schematically expressed by the Prandtl model in which the dashpot is replaced with the slider in the Maxwell model, whereas the slider begins to move in the state that the stress σ goes over the yield stress yσ , by which the plastic strain rate is

induced. Furthermore, the model which describes the rate-dependent plastic strain rate pvε• induced for the state of stress over the yield stress, called the viscoplastic strain

rate, was introduced by Bingham (1922), assembling the above-mentioned Maxwell model and Prandtl model so as to connect the dashpot and the slider in parallel as

shown in Fig. 9.1, where μ is the viscoplastic coefficient and n is the material constant. The Bingham model is the origin of the overstress model based on the concept that the viscoplastic strain rate is induced for the stress over the yield stress.

212 9 Viscoplastic Constitutive Equations

Maxwell’s viscoelastic model

1 1= =e v E με ε ε σ σ− −• • • •+ +

, σ ε , σ ε

Bingham’s viscoplastic model (Bingham, 1922): Overstress model for one-dimension

11= = fp ne v E μ σε ε ε σ σ−−• • • •+ + < − >

, σ ε , σ ε

E μ

E

μ

fσ

Prager’s overstress model (Prager, 1961): Overstress model for Mises metals

11= = 1|| ||( )

pp

e nveeF

''

σμε

−−+ + −⟨ ⟩D D D E σσ σ

General overstress model (Perzyna,1963): Overstress model for general materials

11 ( )= = 1

( )p nve f

HFμ −−+ + −⟨ ⟩D D D E N

σσ

Prandtl’s elastoplastic model

1

11

= for =

for = =

y

p py

e

e

E

E M

σε σ σε

σ σε ε σ σ

−

−−

• ••

• • • •

⎧<⎪

⎨⎪ ++⎩

, σ ε, σ εE

yσ

+

−

−

−

−

Maxwell’s viscoelastic model

1 1= =e v E με ε ε σ σ− −• • • •+ +

, σ ε , σ ε

Bingham’s viscoplastic model (Bingham, 1922): Overstress model for one-dimension

11= = fp ne v E μ σε ε ε σ σ−−• • • •+ + < − >

, σ ε , σ ε

E μ

E

μ

fσ

Prager’s overstress model (Prager, 1961): Overstress model for Mises metals

11= = 1|| ||( )

pp

e nveeF

''

σμε

−−+ + −⟨ ⟩D D D E σσ σ

General overstress model (Perzyna,1963): Overstress model for general materials

11 ( )= = 1

( )p nve f

HFμ −−+ + −⟨ ⟩D D D E N

σσ

Prandtl’s elastoplastic model

1

11

= for =

for = =

y

p py

e

e

E

E M

σε σ σε

σ σε ε σ σ

−

−−

• ••

• • • •

⎧<⎪

⎨⎪ ++⎩

, σ ε, σ εE

yσ

+

−

−

−

−

Fig. 9.1 History of viscoplastic model

The above-mentioned Bingham model for the one-dimensional deformation was extended by Prager (Hohenemser and Prager, 1932; Prager, 1961a) to describe the three-dimensional deformation of metals, adopting the Mises yield condition as

9.1 History of Viscoplastic Constitutive Equations 213

shown in Fig. 9.1. In this model, the strain rate D is additively decomposed into

the elastic strain rate eD and the viscoplastic strain rate pvD , i.e.

= pve +D D D (9.1)

with the viscoplastic strain rate pvD

1= 1|| ||( )

pp

e nvvF

''

σμε

− −⟨ ⟩D σσ

(9.2)

where || ||2 / 3 pp vv dt'ε ≡ ∫ D is the equivalent viscoplastic strain given by

replacing the plastic strain rate pD to the viscoplastic strain rate pvD in the plastic

equivalent strain || ||2 /3 ppe dt'ε ≡ ∫ D . Furthermore, the viscoplastic strain rate in the Prager’s overstress model was

extended by Perzyna (1963, 1966) for materials having the general yield condition unlimited to the Mises yield condition as

1 ( )= 1

( )p nv f

HFμ − −⟨ ⟩D N

σ (9.3)

Then, substituting Eq. (6.29) and (9.3) into Eq. (9.1), we have

11 ( )= 1

( )nf

HFμ −− + −⟨ ⟩D E N

σσ (9.4)

and thus

1 ( )= 1

( )nf

HFμ −− −⟨ ⟩ED EN

σσ (9.5)

where μ depends on stress, internal variables and temperature. The simplest yield

condition ( ) = ( )f HFσ in Eq. (6.30) is used for sake of simplicity in explana-

tion. Internal variables prescribing the variation of yield surface evolve with the inelastic strain rate. Then, the evolution rule of the isotropic hardening variable H is given as

( ), ,= ipvh HH

•Dσ (9.6)

by replacing the plastic strain rate pD to the viscoplastic strain rate pvD in the

evolution rule of the isotropic hardening variable in Eq. (6.37) for the plastic con-

stitutive equation. Here, the hardening function ( )HF signifies the size of yield

surface and ( )f σ does the size of the surface which passes through the current stress point and has similar shape and orientation to the yield surface, whereas the latter surface is called the dynamic-loading surface.


On the other hand, the creep model, which also aims at describing the inelastic deformation in the state of stress over the yield stress as well as the overstress model, has been studied (cf. Norton, 1929; Odqvist and Hult, 1962; Odqvist, 1966) by extending the yield condition such that the yield surface expands with the creep

strain rate cD as follows:

1

0

/|| ||( ) ( )= ( ) m

cc

f F HDDσ

(9.7)

where ( 1)m is the material constant and 0cD is the reference creep strain rate.

It is obtained from Eq. (9.7) that

0

( )|| || = ( )( )m

c c fD F HD

σ (9.8)

In addition, adopting the associated flow rule, the creep strain rate is given by

0

( )|| ||= = ( )( )m

cccf

D F HN ND Dσ (9.9)

and then the strain rate is given by

01 ( )

= = ( )( )mce c

fD F H

−+ +E ND D Dσσ (9.10)

The creep model has different structures from the overstress model because it has no threshold value for the generation of the viscoplastic strain rate. Therefore, this model cannot describe appropriately the deformation behavior at a low rate since it does not reduce to the elastoplastic constitutive equation in a quasi-static deforma-tion. In fact, it predicts the viscoplastic strain rate leading to the overshooting stress-strain curve at the moment of unloading in the quasi-static loading process. Consequently, this model is going to be disused and then it will be replaced over time by the overstress model.

Furthermore, various constitutive models involving the time itself elapsed after a certain physical event begins/stops have been proposed to date. Here, note that the estimation of time when a physical event begins/stops depends on the subjectivity of observers, especially in a fluctuating rate of deformation. Therefore, these models are impertinent, lacking an objectivity.

9.2 Mechanical Response of Ordinary Overstress Model

The development of rate-dependent elastoplastic constitutive equation is reviewed above and it is described that the overstress model would have a pertinent basic structure. Here, let the mechanical responses at the infinitesimal and the infinite rates of deformation be examined in order to clarify the basic property of this model.

9.3 Modification of Overstress Model: Extension to General Rate of Deformation 215

Equation (9.4) is rewritten using the notation d dt≡ Dε as follows:

1 1 ( )= 1

( )nf

d d dtHF

μ− −+ −⟨ ⟩E Nσε σ (9.11)

In the infinitesimal rate of deformation, i.e. the quasi-static deformation state, Eq. (9.11) reduces to

1 ( )1

( )nf

dtHF

μ −≅ + −⟨ ⟩0 0 Nσ (9.12)

Then, the stress varies fulfilling the yield condition ( ) ( )=f HFσ so that Eq.

(9.12) approaches the response of the elastoplastic constitutive relation in the in-finitesimal rate of deformation. On the other hand, in the infinite rate of deforma-tion, Eq. (9.11) reduces to

1d d−≅ E + 0ε σ (9.13)

thereby approaching the elastic response. Therefore, it predicts the unrealistic re-sponse that the material can bear an infinite load.

Eventually, the existing overstress model in Eq. (9.4) describes pertinently the deformation behavior in a low rate. However, it is inapplicable to the prediction of deformation at a high rate. The material constant n included as the power form in Eq. (9.11) is usually selected to be larger than five, but the fitting to the test data for impact load is impossible even if n is selected as one hundred which, needless to say, results in the inappropriate prediction of deformation in a slow loading process. In addition, the inclusion of a high power in the equation induces difficulty in nu-merical calculations.

9.3 Modification of Overstress Model: Extension to General

Rate of Deformation

The ordinary overstress model in Eq. (9.4) is inapplicable to the description of deformation at a high rate, as described above. We modify it to exclude this defect. Then, let the following equation be adopted instead of Eq. (9.3).

=pv CD N (9.14)

where C is the material function fulfilling the following conditions (see Fig. 9.2).

( ) ( )= 0 for

( ) ( ) for m

f F HC

f R F H

≤⎧⎨ →→ ∞⎩

σσ (9.15)

mR is the material constant designating the maximum value of the ratio

( ) ( )/f F Hσ , whereas 1mR would hold usually. The simplest equation fulfilling the condition (9.15) is given by


0 1 ( )mR F H

Modified overstress model

1 ( ) ( )= ( ) ( )m

f F HC f F H f

μ − −⟨ ⟩−

σσ

1 ( )1=

( )nf

CF H

μ − −⟨ ⟩σExisting overstress model

( )f σ

C

−

−

0 1 ( )mR F H

Modified overstress model

1 ( ) ( )= ( ) ( )m

f F HC f F H f

μ − −⟨ ⟩−

σσ

1 ( )1=

( )nf

CF H

μ − −⟨ ⟩σExisting overstress model

( )f σ

C

−

−

Fig. 9.2 Function CC prescribing the magnitude of viscoplastic strain rate in the existing and the modified overstress models

1( ) ( )

=( ) ( )m

f F HC

R F H fμ −

−⟨ ⟩−

σσ (9.16)

Adopting Eq. (9.16) in Eq. (9.14), the viscoplastic strain rate is given as

1( ) ( )

=( ) ( )N

p

m

v f F HR F H f

μ −−⟨ ⟩

−ND

σσ (9.17)

Then, it follows that

1 1( ) ( )

= ( ) ( )m

f F HR F H f

μ− −−⟨ ⟩+

− ND Eσ

σ σ (9.18)

1( ) ( )

= ( ) ( )m

f F HR F H f

μ −−⟨ ⟩−

− ENEDσ

σ σ (9.19)

The stress-strain curves predicted by the existing overstress model (9.4) and the modified model (9.19) are depicted schematically in Fig. 9.3. As seen in this figure, the viscoplastic strain rate is induced infinitely as ( )f σ approaches ( )mR F H in the modified model. Therefore, the defect in Eq. (9.4) that the stress increases elastically for the infinite rate of deformation is resolved in Eq. (9.19).

9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model 217

σ

ε0

|| || 0≅D

|| ||D

Stress-strain curve for impact load predicted by the existing overstress model

( || || )→ ∞D

increases

Quasi-static load

Stress-strain curve for impact load predicted by the modified overstress model

( || || )→ ∞D

( )/ = 1: yield state( )f HFσ

( )/ =( ) mf RHFσ

|| ||→ ∞DImpact load

σ

ε0

|| || 0≅D

|| ||D

Stress-strain curve for impact load predicted by the existing overstress model

( || || )→ ∞D

increases

Quasi-static load

Stress-strain curve for impact load predicted by the modified overstress model

( || || )→ ∞D

( )/ = 1: yield state( )f HFσ

( )/ =( ) mf RHFσ

|| ||→ ∞DImpact load

Fig. 9.3 Stress-strain curve predicted by the existing and the modified overstress models

The necessity of incorporation of the plastic strain rate depending on the mag-nitude of strain rate in addition to the elastic and the viscoplastic strain rates has been inferred in order to describe the deformation at a high rate (cf. Lamaitre and Chaboche, 1990; Hashiguchi et al., 2005a, Chaboche, 2008). However, it induces the physical contradiction that the plastic strain rate is described twice in the two terms of the viscoplastic and the plastic strain rates in the quasi-static deformation process. In addition, inclusion of the magnitude of strain rate leads to the rate non-linearity of the constitutive equation, thereby causing difficulty in numerical calculations.

9.4 Incorporation of Subloading Surface Concept: Subloading

Overstress Model

The overstress model formulated in 9.3 falls within the framework of conventional plasticity on the premise that the interior of yield surface is an elastic domain (Drucker, 1988). Consequently, it exhibits an abrupt transition from the elastic to the viscoplastic state, violating the smoothness condition. Moreover, it violates not only the smoothness but also the continuity conditions (Hashiguchi, 1993a, b, 1997, 2000) when the tangential inelastic strain rate is introduced. This defect in the


existing overstress model would be remedied by introducing the concept of the subloading surface (Hashiguchi, 1980, 1986, 1989). Explicitly, we would have only to modify such that the viscoplastic strain rate is induced by the expansion of the dynamic-loading surface from the subloading surface, whereas it is induced by that from the yield surface in the existing overstress model.

Let the ratio of the size of dynamic-loading surface to that of normal-yield sur-face be called the dynamic-loading ratio. It is denoted by the same symbol R as the normal-yield ratio in the rate-independent subloading surface model, noting that the dynamic-loading surface passes through the current stress. On the other hand, let the ratio evolving by the following equation which is given by replacing the plastic strain rate to the viscoplastic strain rate in Eq. (7.13) for the rate-independent subloading surface model be called the subloading ratio, denoted by the symbol

(0 1) s sR R≤ ≤ in stead of the normal-yield ratio R , and let the surface having this ratio to the normal-yield surface be again called the subloading surface.

|| ||( ) for =

for = ( = )

p p

p

v v

v

ss

s

RUR

R R R

••

⎧ ≠⎪⎨⎪⎩

0D D

D 0 (9.20)

The stress-controlling ability attracting stress to normal-yield surface in the quasi-static deformation process is retained by incorporating Eq. (7.14) with the replacement of R to sR . The explicit equation for the function ( )sRU satisfying Eq. (7.14) can be given by the identical form as Eq. (7.15), i.e.

= cot( ) 2 1( )e

e

ss

R RuRU Rπ − ⟩⟨

− (9.21)

sR can be calculated analytically through integration of Eq. (9.20)1 with Eq. (9.21)

in the viscoplastic deformation process pv ≠D 0 similarly to Eq. (7.16) as

0 01 exp= 2 ( )1 cos cos12 1 2

( ) ) (ppe

e eee

s RRR R u RRRε επππ − − −− − +−−

(9.22) under the initial condition 00 =:= pp sR Rεε .

Now, on the formulation of the viscoplastic strain rate, let the following as-sumptions be incorporated.

1. The viscoplastic strain rate is induced when the stress goes over the subloading surface.

2. There exists the limit of the dynamic-loading ratio R and viscoplastic strain rate pvD is induced infinitely as R reaches the limit value.

3. The viscoplastic strain rate has the outward-normal direction of the dynamic loading surface.

9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model 219

4. Evolution rules of all the internal variables are given by the rules for the rate-independent subloading surface model with the replacement of the plastic strain pD into the viscoplastic strain rate pvD .

Based on these postulates, let the viscoplastic strain rate be given by the following equation.

( , )=pv sC R RD N (9.23) where ( , )sC R R is the monotonically-increasing function of R fulfilling the following conditions.

= 0 for ( , )

for = ms

sRC R R

R R

R

→ ∞≤

+ (9.24)

mR signifies the limit value of the dynamic-loading ratio R . The function ( , )sC R R fulfilling Eq. (9.24) is schematically shown in Fig. 9.4. Here, the dy-

namic-loading ratio R is calculated simply by ( ) ( )/f HFσ for the initial subloading surface model, but it must be calculated by Eq. (8.9) for the extended subloading surface model. The evolution rules of the internal variables

, , , H βα s must be calculated considering the above-mentioned assumption 4. The simple equation fulfilling condition (9.24) is given as follows:

1( ) =, m

ss

R RC R R R Rμ − ⟩⟨ −

− (9.25)

R0 eR

Subloading-overstress model:1( , ) =

m

ss

R RC R R R Rμ − ⟨ ⟩−

−

1 1 nRμ − −⟨ ⟩Existing overstress model:

( , )sC R R

−

−

sR mR1 R0 eR

Subloading-overstress model:1( , ) =

m

ss

R RC R R R Rμ − ⟨ ⟩−

−

1 1 nRμ − −⟨ ⟩Existing overstress model:

( , )sC R R

−

−

sR mR1

Fig. 9.4 Function ( , )sC R R prescribing the magnitude of viscoplastic strain rate in the subloading and the existing overstress models


Substituting Eq. (9.25) into Eq. (9.23), the viscoplastic strain rate is given as

1=p

m

v sR RR R

μ − ⟩⟨ −− ND (9.26)

Furthermore, substituting Eqs. (6.29) and (9.26) into Eq. (9.1), the strain rate is given as follows:

1 1=m

sR RR R

μ −− ⟩⟨ −+ − ND E σ (9.27)

from which the stress rate is described by

1=m

sR RR R

μ − ⟩⟨ −− − EED Nσ (9.28)

The stress-strain curve predicted by Eq.(9.28) is shown in Fig. 9.5. Incorporating the tangential inelastic strain rate formulated in 7.6 into Eq. (9.27),

the strain rate and the stress rate are given as follows:

11 =

m tsR R

RR R 'τμ ξ−− ⟩⟨ − ++ − ND E σ σ (9.29)

1=1

2m

tsR GR

R R

RG

'τ

μ

ξ

− − 2− −− 1+E DNED (9.30)

whereas R must be replaced by R for the extended surface model.

σ

ε0

|| || 0≅D

|| ||→ ∞D

Rs =1

R=Rm

Quasi-static loading

Impact load increases|| ||D

：normal-yield state:

: maximum state:

( ) ( )=f HFσ

( ) ( )= mf HR Fσ

( ) ( )= sf HR F: σ

σ

ε0

|| || 0≅D

|| ||→ ∞D

Rs =1

R=RmR=Rm

Quasi-static loading

Impact load increases|| ||D

：normal-yield state:

: maximum state:

( ) ( )=f HFσ

( ) ( )= mf HR Fσ

( ) ( )= sf HR F: σ

Fig. 9.5 Stress strain curve predicted by the subloading overstress model


Chapter 10 Constitutive Equations of Metals

10 Constitutive Equations of Metals

The plasticity has highly developed through the prediction of deformation of metals up to date. The reason would be caused from the fact that, among various materials exhibiting plastic deformation, metals are used most widely as engineering materials and exhibit the simplest plastic deformation behavior without a pressure dependence, a plastic incompressibility, an independence of the third invariant of deviatoric stress and a softening. Thus, the elastoplastic constitutive equation of metals has developed to the highest level. Nevertheless, metals exhibit various particular aspects, e.g. the kinematic hardening and the stagnation of isotropic hardening in a cyclic loading. Explicit constitutive equations of metals will be delineated in this chapter, which have been formulated based on the general elastoplastic constitutive equations studied in the preceding chapters.

10.1 Isotropic and Kinematic Hardening

The yield function for the Mises yield condition is extended to incorporate

kinematic hardening by replacing 'σ to ˆ 'σ in Eq. (6.54) as follows:

3 || ||( ) =2

ˆˆf 'σσ ，ˆ ˆ ˆ= = = || ||

ˆˆ'' ''

nN N σσ (10.1)

while the subloading function ( )f σ for Eq. (10.1) is given by

3 3|| ||( ) = =2 2|| ||ˆ( )f R' '' +σ σ σ s ， = = = || ||

'' ''

nN N σσ

(10.2)

noting Eq. (8.6) or (8.9) with =β 0 .

The explicit form of the hardening function (6.54) is given as follows

(Hashiguchi and Yoshimaru, 1995)．

222 10 Constitutive Equations of Metals

0

( )peF ε

0F

01(1 )Fh+

0 1 2 2= exp( )peF F h h h' ε−1

peε0

( )peF ε

0F

01(1 )Fh+

0 1 2 2= exp( )peF F h h h' ε−1

peε

Fig. 10.1 Isotropic hardening function in the uniaxial loading process

0 1 2( ) = [1 1 exp( )]p pe eF F h hε ε+ − − , 0 1 2 2= exp( )peF F h h h' ε−

(10.3)

where 1 2, h h are the material constants. The hardening function ( )peF ε in Eq.

(10.3) increases from the initial value 0F with the plastic equivalent strain peε

and saturates when it reaches the maximum value 1 0(1 )h F+ .

The evolution rule of the back stress was given by Prager (1956) as follows:

ˆ|| ||== pp pp aa DD Nα (10.4)

where pa is the material constant having the dimension of stress. According to Eq. (10.4), the component of back stress is induced even in the direction of zero stress condition seen in the uniaxial loading process and in plane stress condition for instance. In order to avoid this inconvenience, Ziegler (1959) proposed the following equation

= || || ˆpza Dα σ (10.5)

where za is the dimensionless material constant. In Eq. (10.5) the mean

component of α is induced and thus α does not become a deviatoric tensor. The translation directions of α in Eqs. (10.4) and (10.5) are same in the deviatoric stress plane for the Mises yield surface, but they generally differ from each other in the other yield surfaces as shown in Fig. 10.2. Which rule is more pertinent physically is not clarified yet.

10.1 Isotropic and Kinematic Hardening 223

0

σ

α

σPrager :α

N

Ziegler :α

ijσ

Yield surface

0

σ

α

σPrager :α

N

Ziegler :α

ijσ

Yield surface Fig. 10.2 Comparison of Prager’s (1955) and Ziegler’s (1959) kinematic hardening rules

According to Eqs. (10.4) and (10.5), the relation of back stress vs. plastic strain is linear when pa and za are constant and thus they are called the linear-kinematic hardening rule. On the other hand, it is observed in test data that the rate of back stress decreases gradually and increases abruptly at the initiation of inverse loading. The nonlinear kinematic hardening rule taking account of these behavior was proposed by Armstrong and Fredericson (1966). The explicit equation is given as follows (Hashiguchi, 1989):

2 2, ( (|| ||= ) )3 3pr ra aF Fα αα α≡− −DN Naα αα (10.6)

where aα and ( 1)<rα are material constants. As shown in Fig. 10.3, α

translates to approach the conjugate point ( || ||)= = /r rF F '' 'α α Nσ σ σ on the

limit surface || || = r F' ασ of kinematic hardening in the deviatoric stress plane,

i.e. || ||/r F ' 'α→ σ σα in Eq. (10.6). The relation of axial back stress aα vs.

axial logarithmic plastic strain paε is given from Eq. (10.6) as follows:

= )( 2 / 3 pa a aad r dFα αα α ε± ± −

(10.7)

where the upper and lower signs signify extension and compression, respectively.

The integration of Eq. (10.7) under = const.F for sake of simplicity leads to the

following equation.


0

σ 3

σ1 σ 2

σ

αα

|| ||/ ˆˆr F ' 'α σσ

Yield surface

3/2 || || = ( )ˆ F H'σ

Kinematic hardeinglimit surface

|| || = ( )r HF' ασ

σ

0

σ 3σ 3

σ1 σ 2

σ

αα

|| ||/ ˆˆr F ' 'α σσ

Yield surface

3/2 || || = ( )ˆ F H'σ

Kinematic hardeinglimit surface

|| || = ( )r HF' ασ

σ

Fig. 10.3 Kinematic hardening rule (10.6)

0

1

1

aα

paε

2 / 3 r Fα−

2 / 3 r Fα

2 / 3 ar Fα α−

( )2 / 3 ar Fα α− +

0

1

1

aα

paε

2 / 3 r Fα−

2 / 3 r Fα

2 / 3 ar Fα α−

( )2 / 3 ar Fα α− +

Fig. 10.4 Kinematic hardening variable vs. plastic strain curve of (10.9) in uniaxial loading process

10.2 Cyclic Stagnation of Isotropic Hardening 225

00

1 2 /3ln=2 /3

ap pa a

a

r Fa r F

αα α

αα

ε ε ± −− ± −∓ (10.8)

where 0aα and 0p

aε are the initial values of aα and paε , respectively. It is

obtained from Eq. (10.8) that

00= 1 1 exp ( )2 / 3

2 / 3( ) ][ a p pa a a ar kF r F

αα

αα ε ε± − −∓ ∓ (10.9)

which is depicted in Fig. 10.4.

10.2 Cyclic Stagnation of Isotropic Hardening

The isotropic hardening of metals is induced by the equivalent plastic strain. It is observed in experiments that isotropic hardening stagnates despite of the development of the equivalent plastic strain for a certain range in the initial stage of re-yielding after reverse loading. This phenomenon remarkably influences the cyclic loading behavior in which the reverse loading is repeated. In particular, the isotropic hardening saturates finally in the cyclic loading under a constant strain amplitude. In order to describe this phenomenon the idea of the nonhardening region was proposed by Chaboche et al. (1979; see also Chaboche, 1989; Lemaitre and Chaboche, 1990) and modified by Ohno (1982) and Ohno and Kachi (1986): the isotropic hardening does not proceeds when the plastic strain exists in a certain region of the plastic strain space which expands with the cyclic loading. It is similar to the notion of the yield surface, assuming that the plastic strain is not induced when the stress lies inside it. Thereafter, the formulation that the isotropic hardening stagnates when the back stress lies in the certain region of stress space was proposed by Yoshida and Uemori (2002, 2003). However, it cannot describe the stagnation behavior of isotropic hardening of metals without the kinematic hardening. The following defects are imposed in these formulations.

1 ) The isotropic hardening is induced suddenly when the plastic strain or the back stress reaches the boundary of nonhardening region, violating the smoothness condition (Hashiguchi, 1993a, b, 1997, 2000). Thus, the smooth stress-strain curve cannot be described.

2 ) The judgment whether or not the plastic strain or the back stress reaches that boundary is required.

3 ) The numerical operation to pull back the plastic strain or back stress to that boundary so as not to go over the nonhardening region is required.


In what follows, a pertinent formulation without these deficiencies is presented for the cyclic stagnation of isotropic hardening, based on the concept of subloading surface.

Assuming that the isotropic hardening stagnates when the plastic strain pε lies inside a certain region in the plastic strain space, let the following surface, called the normal-isotropic hardening surface be introduced.

( ) =ˆ pf Kε (10.10)

where

ˆ p p −≡ε ε α (10.11)

K and α designate the size and the center, respectively, of the normal-isotropic hardening surface, the evolution rules of which will be formulated later. The

function ( )ˆ pf ε is explicitly given by

2( ) = || ||3

ˆ ˆp pf ε ε (10.12)

Further, incorporate the surface, called the sub-isotropic hardening surface, which always passes through the current plastic strain point pε and has similar shape and orientation to the normal-isotropic hardening surface (see Fig. 10.5). Then, it is mathematically expressed as

( ) =ˆ pf RKε (10.13)

where (0 1)R R≤ ≤ is the ratio of the size of sub-isotropic hardening surface to

that of the normal-isotropic hardening surface and thus it plays the role as the measure to describe the approaching degree of plastic strain to the normal-isotropic

hardening surface. Then, R is called the normal-isotropic hardening ratio. It is

calculated from the equation = ( ) /ˆ pR f Kε in terms of the known values pε ,

α and K . The material-time derivative of Eq. (10.13) leads to the consistency condition for

the sub-isotropic hardening surface:

( ) ( )t r t r =

ˆ ˆ( () )p p

pp p

f fR KR K••∂ ∂ +−

∂ ∂Dε εε ε α (10.14)

Now, for the formulation of the evolution rule of the size K of the normal-isotropic hardening surface, let it be assumed that


pijε

α

α

N

Normal- isotropichardening surface

Sub-isotropic hardening surface

( ) =pf RKε

( ) =pf Kε

pεˆ pε

pD

0 pijε

αα

αα

N

Normal- isotropichardening surface

Sub-isotropic hardening surface

( ) =pf RKε

( ) =pf Kε

pεˆ pε

pD

0

Fig. 10.5 Normal- and sub- isotropic hardening surfaces

1. The normal-isotropic hardening surface expands only if the plastic strain rate is directed outward of the sub-isotropic hardening surface,

2. The expansion rate of normal-isotropic hardening surface increases as the plastic strain approaches the normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio increases,

3. The expansion rate of normal-isotropic hardening surface has to be zero when the plastic strain lies on its center ( = 0)R in order that the normal-isotropic hardening surface begins to expand gradually, noting that = 0K

• for 0R

•<

due to the assumption 1.

Based on these assumptions, let the evolution rule for magnitude of the normal-isotropic hardening surface be given by

( ) ( ) || ||t r t r==ˆ ˆ )( ()

p pp p

p pf f

R RK C Cζ ζ• ∂ ∂∂ ∂⟩ ⟩⟨ ⟨D DNε εε ε (10.15)

where ( 1)C ≤ and ( 1)ζ ≥ are the material constants. Now, for the formulation of the evolution rule of the center α of the

normal-isotropic hardening surface, let it be assumed that

1. It translates only if the plastic strain rate is directed outward of the sub-isotropic hardening surface,


2. The translation rate of the center of the normal-isotropic hardening surface increases as the plastic strain approaches the normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio increases,

3. The translation rate of normal-isotropic hardening surface has to be zero when the plastic strain lies on its center ( = 0)R in order that the normal-isotropic hardening surface begins to translate gradually, noting that = 0α for 0R

•<

due to the assumption 1 ), 4. The direction of translation is the outward-normal of the sub-isotropic

hardening surface.

Based on these assumptions, let the following evolution rule for the center of the normal-isotropic hardening surface be given by

) || ||)= tr(tr( = ppAR RA ζζ ⟩ ⟨ ⟩⟨ NDNNND Nα (10.16)

where

( ) ( )ˆ ˆ|| ||/p p

p p

f f∂ ∂≡ ∂ ∂Nε εε ε

(10.17)

A is the material constant which will be formulated in the following.

Now, substituting Eqs. (10.15) and (10.16) into Eq. (10.14), one has

( )( ) ( ) t r)t r t r =tr(ˆˆ ˆ ( )( () )

pp ppp p

pp p

ff fR KR RA CRζ ζ•∂∂ ∂ +− ⟩⟨ ∂∂ ∂ ⟩⟨ DD ND N

εε εεε ε

(10.18)

Taking account of = 0R•

for 1=R in Eq. (10.18), it must hold that

= 1A C− (10.19)

Substituting Eq. (10.19) into Eq. (10.16), the evolution rule for the center of the normal-isotropic hardening surface is determined as

) || ||)= (1 ) (1 ) tr(tr( = pp RC CRζ ζ⟩− − ⟨ ⟩⟨ NDNNND Nα (10.20)

The normal-isotropic hardening surface expands without the translation in case of

= 1C but inversely translates without the expansion in case of = 0C .

Substituting Eqs. (10.15) and (10.20) for the evolution rules of α and K into

Eq. (10.14), we have


( ) ( )1= t r t rˆ ˆ( () )

p pp

p pf f

RR KK

• •∂ ∂− −∂ ∂⟨ ⟩Dε εε ε α

( ) ( )1 || ||)t r tr((1 )= t rˆ ˆ( ) ( )

p ppp

p p

f fC R

Kζ∂ ∂ ⟨ ⟩− −∂ ∂⟨ ⟩D DNN N

ε εε ε

( ) || ||t rˆ )(

pp

pf

R RC ζ ∂− ∂ ⟩⟨ DNεε

Then, R•

is given by

( )1 || ||= t r [1 1 (1 ) ]ˆ )(

pp

pf

R C R RKζ

• ∂− − −∂ ⟩⟨ DN

εε (10.21)

in which it holds that

( )1= t r for = 0

( )1< t r for < 1|| ||/

0 for = 1=

0 for > 1

ˆ

ˆ

)(

)(

p

p

pp

p

f RK

f RR K

R

R

•

⎧ ∂⎪ ∂⎪⎪ ∂⎪⎨ ∂⎪⎪⎪<⎪⎩

⟩⟨

⟩⟨

N

ND

εεεε (10.22)

Particularly, it is noticeable that the present formulation possesses the controlling function that the plastic strain rate is always attracted to the normal-isotropic

10

|| ||/ pR•

D

R

( )1 t rˆ )(

p

pf

K∂

∂ ⟩⟨ Nεε

10

|| ||/ pR•

D

R

( )1 t rˆ )(

p

pf

K∂

∂ ⟩⟨ Nεε

Fig. 10.6. Evolution of normal-isotropic hardening ratio


hardening surface by the incorporation of Eq. (10.21) with Eq. (10.22) (see Fig. 10.6). In addition, the judgment whether or not the plastic strain reaches the normal-isotropic hardening surface is not required therein.

Now, it is assumed that the isotropic hardening variable H evolves under the conditions:

1. The isotropic hardening is induced only if the plastic strain rate is directed outward of the sub-isotropic hardening surface,

2. The isotropic hardening rate increases as the plastic strain approaches the normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio increases,

3. The isotropic hardening rate has to be zero when the plastic strain lies on its

center ( = 0)R in order that the isotropic hardening begins to develop

gradually, noting that = 0H•

for 0R•

< due to the assumption 1, 4. The isotropic hardening variable evolves by the rule formulated in the

monotonic loading process when the plastic strain lies on the normal-isotropic

hardening surface ( = 1)R .

Then, let the following evolution rule of isotropic hardening be assumed by

extending Eq. (6.56).

2 2 )tr(|| ||)tr(= = 3 3

ppR RHυ υ• ⟨ ⟩⟨ ⟩ NNDND

(10.23)

where ( 1)υ ≥ is the material constant. Accompanying with the incorporation of the cyclic stagnation of isotropic

hardening, i.e. the adoption of Eqs. (10.23) instead of Eqs. (6.56), the rate of similarity-center in Eq. (8.17) with Eq. (10.6) and =β 0 for metals is given by

2 2 )tr(( )= 33

ˆp Fra RFR F

'c|| || υαα ⟩⟨−+ + NNND σ sαs (10.24)

Further, the plastic modulus in Eq. (8.27) is given as follows:

22 )tr( ( )tr 33ˆ[p F raM FR

Fυ

αα′ ⟨ ⟩ −+≡ NN NN σ α

( ) 1 1)( ]U RRR

c −++ σ (10.25)

On the other hand, it is insisted that the blunting of isotropic hardening in the state that the plastic strain lies inside the isotropic hardening surface is supplemented by the acceleration of kinematic hardening (Chaboche et al., 1979;


Chaboche, 1989; Ohno, 1982; Ohno and Kachi, 1986). According to this notion, the plastic modulus has to be given always from Eq. (8.27) with =β 0 for metals as

( )2 12 1( )tr 33ˆ )( ][p U RF raM F

F RRcαα

′ −− ++ +≡ NN σ α σ

(10.26)

In order that Eq. (10.26) holds for Eq. (10.23), the evolution rule (10.6) has to be

modified as follows:

2 (= ) )tr( ) || ||3 ˆ pFra F RF'υαα − ⟨ ⟩+ (1−N NN Dαα σ

2 ( ) )tr( )3 ˆ Fra F RF'υαα≡ − ⟨ ⟩+ (1−N NNa α σ (10.27)

Further, adopting Eqs. (10.23) and (10.27) instead of Eqs. (6.56) and (10.6), the evolution rule of similarity-center in Eq. (8.17) has to be modified as follows:

2 ( ) )= 3p ra F RRc|| || υαα −+ (1−+N ND σ αs

2 )tr( 3

ˆF RF' υ ⟩⟨+ NN s (10.28)

As described above, the two notions are considered as to the kinematic hardening rule during the blunting of isotropic hardening. The notion that the kinematic hardening is accelerated during the blunting of isotropic hardening is introduced by Ohno (1982, cf. also Ohano and Kachi, 1986; Chaboche, 1989) in order to avoid the deficiency that the isotropic hardening is restored suddenly when the plastic strain reaches the boundary of nonhardening region so that the plastic modulus change abruptly. However, this defect is always avoided predicting a smooth stress-strain curve in the present formulation based on the notion of the sub-isotropic hardening surface. Moreover, in the present formulation, the judgment whether or not the plastic strain reaches the normal-isotropic hardening surface is not required and the operation pulling back the plastic strain to the normal-isotropic hardening surface is not necessary, attracting always it to that surface. Further study is required as to whether or not the kinematic hardening is accelerated during the blunting of isotropic hardening.

Assuming

=p p pp• − +D ω ωε εε (10.29)


based on Eq. (4.33), the plastic strain is calculated by

( )=p p p p dt−+∫ D ω ωε ε ε (10.30)

For the infinitesimal strain, the continuum spin W can be used as the corotational spin ω .

If we adopt the Yoshida and Uemori’s (2002, 2003) idea that the isotropic hardening ceases when the back stress lies inside a certain surface, called the isotropic stagnation surface, the plastic strain pε has only to be replaced simply by the back stress α in the formations described in this section.

10.3 On Calculation of the Normal-Yield Ratio

The normal-yield ratio R can be calculated directly by = ( )/ˆR f Fσ in the initial subloading surface model. However, it has to be calculated by solving the equation of the subloading surface in the extended subloading surface model as described below.

Substituting Eq. (10.2) into Eq. (8.9) with =β 0 , the extended subloading surface is described as follows:

3 || ||= ( )2ˆR RF H' '+σ s (10.31)

i.e.

2 2 22)tr( =

3ˆR R F' '+σ s (10.32)

The normal-yield ratio R is derived from the quadratic equation (10.32) as follows:

2 2 2 2

22

2 || || || ||) )tr( tr (3

=2 || ||3

ˆ ˆ ˆ

ˆ

( )FR

F

' ' ' ' ' '

'

−

−

++σ s σ s s σ

s

(10.33)

10.4 Comparisons of Test Results

Capability of the extended subloading surface model for the prediction of deformation behavior of metals is verified by comparisons with some basic test data below (Hashiguchi and Ozaki, 2009a,b).

10.4 Comparisons of Test Results 233

The cyclic loading behavior under the stress amplitude in both positive and negative sides can be predicted to some extent by any models including even the conventional plasticity model. On the other hand, the prediction of the cyclic loading behavior under the stress amplitude in positive or negative one side, i.e. the pulsating loading inducing the so-called mechanical ratcheting effect requires a high ability for the description of plastic strain rate induced by the rate of stress inside the yield surface. Further note that we quite often encounters the pulsating loading phenomena in the boundary-value problems in engineering practice, e.g. railways, gears and plates undergoing punch-indentation. The comparison with the test data of the uniaxial deformation behavior of 1070 steel under the pulsating loading between 0MPa and 830Mpa after Jiang and Zhang (2008) is depicted in Fig. 10.7 where the material constants and the initial values are selected as follows:

Material constants:

1 2

0

Elastic moduli: = 170000MPa, = 0.3,

isotropic : = 0.4, = 170,Hardening

kinematic : = 100, = 0.3,

Evolution of normal - yield ratio : = 0.1, = 500, = 2,

Translationon of similarity - center : = 200,

se

E

h h

a r

uuR

c

α α

ν⎧⎨⎩

Stagnation of isotropic hardening : = 0.5, = 20, = 3,C ζ υ

Initial values:

0

0

0

0

0

Hardening function: = 580MPa,

Size of the normal-isotropic hardening surface : = 0 001,

Kinematic hardening variable : = k Pa,

Center of similarity :

Str

= MPa ,

: MPess a=

F

K .

0

0

0

s

The test result is simulated very closely by the present model. The movements of kinematic hardening variable α and the similarity-center s are shown in that

figure where the axial components are designated by ( )a . In addition, the

variations of normal-yield ratio R and the normal-isotropic hardening ratio R are

depicted. It is shown that the normal-yield ratio R does not increase over unity and thus the stress is automatically attracted to the normal-yield surface without


R

R

1.0

1.0

aε

as

aα

0

0 0.005 0.01 0.015

aε0.005 0.01 0.015

0 aε0.005 0.01 0.015

0.5

0.5

200

400

600

800

aσExperimentPrediction

(MPa)aσ

R

R

1.0

1.0

aε

as

aα

0

0 0.005 0.01 0.015

aε0.005 0.01 0.015

0 aε0.005 0.01 0.015

0.5

0.5

200

400

600

800


(MPa)aσ

Fig. 10.7 Prediction of zero-to-tension uniaxial loading behavior of 1070 steel (test data after (Jian and Zhang, 2007))


aε

as

aα

0 0.005 0.01 0.015

200

400

600

800


aε

aα

0 0.005 0.01 0.015

200

400

600

800


(a) 0 : Initial subloading surface model= c

(b) 0 ( const.)= =s uu

(MPa)aσ

(MPa)aσ

aε

as

aα

0 0.005 0.01 0.015

200

400

600

800


aε

aα

0 0.005 0.01 0.015

200

400

600

800


(a) 0 : Initial subloading surface model= c

(b) 0 ( const.)= =s uu

(MPa)aσ

(MPa)aσ

Fig. 10.8 Prediction of zero-to-tension uniaxial loading behavior of 1070 steel (test data after (Jian and Zhang, 2007)): (a) Initial subloading surface model, (b) u=const.

incorporation of any return-mapping algorithm. Also, the normal-isotropic

hardening ratio R does not increase over unity and thus the normal-isotropic hardening surface changes such that the plastic strain does not go out from this surface.


The simulation by the initial subloading surface model in which the center of similarity s is fixed in the origin of stress space by setting = 0c is shown in Fig. 10.8(a), while the other parameters are chosen same as those for Fig. 10.7. It is known that the subloading surface contracts and then the unloading proceeds during the decrease of stress to zero so that open hysteresis loopa are depicted predicting an unrealistically large strain accumulation. Further, the simulation in the case that the material parameter u in the evolution rule of the normal-yield ratio is taken to be constant is shown in Fig. 10.8(b). It is shown that the excessively large plastic strain is induced in the reloading processes as was described in 8.4 and thus the unrealistically large strain accumulation is predicted even in this calculation although the strain accumulation is much improved to be suppressed from the calculation by the initial subloading model in Fig. 10.8(a).

As seen in numerical experiments shown in Appendix 7, the strain accumulation is suppressed for the deformation behavior near the yield state by the extension of the material parameter u to Eq. (8.36).

The comparison with test data for the cyclic loading behavior under the increasing strain amplitudes 1, 1.5, 2, 2.5, 3%± ± ± ± ± in turn after

saturation in each amplitude is shown in Fig. 10.9. The test data are obtained for the

316L steel at 20 C after Chaboche et al. (1979). The material constants and the initial values are selected as follows:

Material constants:

1 2

0

Elastic moduli: = 170000MPa, = 0.3,

isotropic : = 0.8, = 10,Hardening

kinematic : = 100, = 0.4,

Evolution of normal - yield ratio : = 800, = 3, = 0.4,


S

s e

E

h h

a r

uu R

c

α α

ν⎧⎨⎩

tagnation of isotropic hardening : = 0.5, = 15, = 3,C ζ υ

Initial values:

0

0

0

0

0

Hardening function: = 275 MPa,

Size of the normal-isotropic hardening surface : = 0 001,

Kinematic hardening variable : = MPa ,

Center of similarity :

Str

= MPa ,

: MPess a=

F

K .

0

0

0

s

This test result is also predicted very closely. The variations of normal-yield ratio R and the normal-isotropic hardening ratio R do not increase over unity as shown in Fig. 10.10.


(b) Prediction(a) Experiment

(%)aε

(MPa)σ

3− 1− 1 2 3

250−

(MPa)σ

250

aα

(%)ε2− 0

500

500−

(b) Prediction(a) Experiment

(%)aε

(MPa)σ

3− 1− 1 2 3

250−

(MPa)σ

250

aα

(%)ε2− 0

500

500−

3− 1− 1 2 3

250−

(MPa)σ

250

aα

(%)ε2− 0

500

500−

3− 1− 1 2 3

250−

(MPa)σ

250

aα

(%)ε2− 0

500

500−

Fig. 10.9 Comparison with experiment for the cyclic loading behavior under the five levels of constant strain amplitudes in both positive and negative sides of 316L steel (test data after Chaboche et al.,1979)

R

0.0

1.0

0.5

0.1 0.2 0.30.1−0.2−0.3− 0.0 0.1 0.2 0.30.1−0.2−0.3−

0.5

1.0

R− R

0.0

1.0

0.5

0.1 0.2 0.30.1−0.2−0.3− 0.0 0.1 0.2 0.30.1−0.2−0.3−

0.5

1.0

R−R−

Fig. 10.10 Variations of R RR and R in calculation for Fig. 10.8

The prediction without the stagnation of isotropic hardening by setting = 0C

or = 0K is shown in Fig. 10.11. The isotropic hardening develops unrealistically rapidly in this calculation.


3− 1 2 3

(MPa)aσ

aα

(%)aε0

250

2− 1−

250−

500−

500

3− 1 2 3

(MPa)aσ

aα

(%)aε0

250

2− 1−

250−

500−

500

Fig. 10.11 Prediction of the cyclic loading behavior under the five levels of constant strain amplitudes in both positive and negative sides of 316 L steel (test data after Chaboche et al.,1979) without stagnation of isotropic hardening

10.5 Orthotropic Anisotropy

The kinematic hardening incorporated in the foregoing is regarded to be the induced anisotropy. On the other hand, various inherent anisotropies are induced in the manufacturing process of metals. The typical inherent anisotropy is the orthotropic anisotropy formulated by Hill (1948).

Now, consider the general yield function in the quadratic form shown as follows:

1( ) =2ij ij klijklf Cσ σ σ (10.34)

where ijklC is the fourth-order anisotropic tensor having eighty-one components

fulfilling

= = = = = = =jilkjikl klji lkij lkjiijkl ijlk klijC C C C C C C C (10.35)

by the relations ， ijijkl klC σ σ =( )ijklij klC σ σ = ijklij klC σ σ based on the

symmetry of the stress tensor = jiijσ σ . Then, the independent components

reduces to twenty-one leading to

10.5 Orthotropic Anisotropy 239

=ij klijklC σ σ

2

2

11 111111 1133 33 1112 23 31 3111 1122 11 22 11 11 12 11 1123

2233 12 2223 22312222 22 22 33 22 22 22 312212 23

2 2 2 2 2

+ 2 2 2 2

C C C C C C

C C C C C

σ σ σ σ σ σ σ σ σ σ σσ σ σ σ σ σ σ σ σ

+ + + + +

+ + + +2

2

33313333 33 3312 12 3323 33 3333 3123

12 1223 1231 121212 12 3123

2 2 2

2 2

C C C C

C C C

σ σ σ σ σ σ σσ σ σ σ σ

+ + + +

+ + +2

2323 23 312331 23

2

C Cσ σ σ+ +2313131 ) C σ+

(10.36) which is the general form of yield function in the quadratic form.

Here, assuming the plastic incompressibility, it holds that

( (2 ) / ) = ( / )

=

= =

ijpq pqklpq pqijkl

kl ijpi qj pq pqijkl ijkl p qk l

pp ppkl klpp ij ppij ijijkl kl

Cf

C C

C C C C

σ σσ σδ δσ σδ δ δ δ δ δ

σ σ σ σ

∂∂ ∂ ∂

+

+ +

2 0= =pp klklC σ

This relation must hold for any ijσ and thus one obtains

= = 0qqpp ijklC C (10.37) which leads to

1111 1122 1133

22222211 2233

33333311 3322

= 0= 0

= 0

C C CC C C

C C C

⎫+ +⎪+ + ⎬⎪+ + ⎭

(10.38)

2212 3312 3312 22121112 1112

1123 2223 3323 1123 2223 3323

2231 22311131 3331 1131 3331

= =0 ( )= =0 ( )

(= =0 )

C C C C C CC C C C C CC C C C C C

+ + +−⎫ ⎫⎪ ⎪+ + +−→⎬ ⎬+ + +⎪ ⎪−⎭ ⎭

(10.39)

The substitution of Eq. (10.39) into Eq. (10.36) gives the expression

2

2

2223 3323 111111 1133 33 1112 31 3111 1122 11 22 11 11 12 11 1123

33312233 12 2223 112222 22 22 33 22 22 31 22 312212 23

=

2 2 2 2( ) 2

2 2 2 2( )

ij klijklC

C C C C C C C

C C C C C C

σ σσ σ σ σ σ σ σ σ σ σ σ

σ σ σ σ σ σ σ σ σ+ + + + +−

+ + + +−+2

2

1112 2212 33313333 33 12 3323 33 3333 3123

12 12231212 12 23

( ) 2 2 2

2

C CC C C

C C

σ σ σ σ σ σ σσ σ σ

++ + +−

+ +2

1231 12 31

2323 233123 23 31

2

2

C

C C

σ σσ σ σ

+

+ +2313131 C σ+

(10.40)


Further, considering Eq. (10.38), one has

2 2 23333 33 22331111 2222 22 22 33 1133 3311 1122 11 22 11 + 2 2 2C C C C C Cσ σ σ σ σ σ σ σ σ+ + + +

2 2 23333 331111 2222 2211 = +C C Cσ σ σ+

2 2 21122 11 22 1122 11 1122 22( )C C Cσ σ σ σ− − + +

2 2 22233 33 2233 22 2233 3322( )C C Cσ σ σ σ− − + +

2 2 21133 33 1111 1133 33 2233( )C C Cσ σ σ σ− − + +

2 2 23333 331111 2222 221122 11 11222233 2233 1133 2233 = ( ) + ( ) ( )C C C C C C C C Cσ σ σ++ + + + + +

2 2 22233 33 1133 331122 11 22 1122( ) ( ) ( )C C Cσ σ σ σ σ σ− − − − − −

2 2 22233 33 1133 331122 11 22 1122 = ( ) ( ) ( )C C Cσ σ σ σ σ σ− − − − − −

(10.41)

Then, by setting

1 2 3

4 65

97 8

10 11 12

1513 14

2233 11331122

1112 2212 2223

3323 3331 1131

23311223 1231

313123231212

, ,

, , ,

, ,

, 2 ,

,

a a aC C C

a a aC C C

a a aC C C

a a aC C C

a a aC C C

− − −≡ ≡ ≡− − −≡ 2 ≡ 2 ≡− − −≡ 2 ≡ 2 ≡ 2

≡ 2 ≡ ≡ 2≡ ≡ ≡

Eq. (10.40) is rewritten as

2 2 21 2 33 3311 22 3 1122= ( ) ( ) ( )ij klijkl

a aaC σ σ σ σ σ σσ σ +− − −+

11 5 33 22 12334 ( ) ( )aa σ σ σ σ σ− −+ +

22 237 3311 226 ( ) ( )aa σ σ σ σ σ− −+ +

1122 33 9 22 318 ( ) ( )aa σ σ σ σ σ− −+ +

12 23 1231 1211 23 3110 a aa σ σ σ σ σ σ++ +

2 2 21213 23 15 3114 a aa σ σ σ++ + (10.42)

Equation (10.42) is the general yield function for the plastically-incompressible materials in the quadratic form.

Furthermore, assume orthotropic anisotropy. Then, if we describe the yield surface by the coordinate axes selected to the principal axes i

∗e of orthotropic


anisotropy, the yield function is independent of the sign of shear stress components in this coordinate system. Therefore, it must hold that

4 5 6 7 8 9 10 11 12= = = = = = = = = 0a a a a a a a a a

Here, replacing the symbols ia as

1 2 13 14 153, , , , ,= = = = / 2 = / 2 = / 2a a a a a aF G H L M H

used by Hill (1948), Eq. (10.42) leads to the Hill’s yield condition with orthotopic anisotropy:

( )ijf σ

2 2 222 2211 33 33 11 12 23 31

2 2 21 ( ) ( ) ( )= 2 2 2 2

F G H L M Nσ σ σ σ σ σ σ σ σ− − −+ + + + +

= ( )F H (10.43)

Here, note that = = = 1, = = = 3F G H L M N holds for isotropy and then

Eq. (10.43) reduces to || ||( ) = 3/2ijf 'σ σ which is the equivalent stress. While

Eq. (10.43) is the expression on the principal axis i∗e of orthotropic anisotropy,

it is rewritten by the following equation stipulating this fact.

( )ijf σ ∗

2 2 2 2 2 222 2211 33 33 11 12 23 31

1 ( ) ( ) ( )= 2 2 2 2

F G H L M Nσ σ σ σ σ σ σ σ σ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗− − −+ + + + +

= ( )F H (10.44)

Further, under the plane stress condition observed in the sheet metal forming it

holds that 23 3331= = = 0σσ σ∗ ∗ ∗ and thus Eq. (10.44) reduces to

22 222 2211 11 12( ) 2 ( ) 2 = 1H F F G F Lσ σ σ σ σ∗ ∗ ∗ ∗ ∗+−+ + + (10.45)

where

2 2 2 2, , ,

2 ( ) 2 ( ) 2 ( ) 2 ( )G FH LH G F L

F H F H F H F H≡ ≡ ≡ ≡

(10.46)

Here, denoting the yielding strength in the equi-two axis tension as σ and that of the pure shear as τ , it holds from Eq. (10.45) that


1/2 1/2( ) , (2 )H G Lσ τ− −≡ + ≡ (10.47)

Now, rewrite Eq. (10.45) as

22211 11

2222 12

2 21 1 1 ( ) ( 4 ) ( ) ( ) ( 4 )4 4 2 4 4

1 1 1 ( ) ( 4 ) ( ) 2 = 14 4 2

G H G H F G H G H G H F

G H G H F G H L

σ σ σ

σ σ

∗ ∗ ∗

∗ ∗

+ + + + − − + + − + +

+ + + + + + − +

which is arranged as follows:

2 2

22 2211 111 1( )( ) ( 4 )( )4 4G H G H Fσ σ σ σ∗ ∗ ∗ ∗++ + + + −

22 22211 12

1 ( )( ) 2 = 12 G H Lσ σ σ∗ ∗ ∗− − − + (10.48)

Denoting the angle measured in the counterclockwise direction from the principal

axes of anisotropy to the principal stress as α and substituting the relations

22 2211 11 ( )= , = cos2σ σ σ σ σ σ σ σ α∗ ∗ ∗ ∗ −+ + −Ⅰ ⅠⅡ Ⅱ ,

12 ( )2 = sin 2σ σσ α∗ −Ⅰ Ⅱ (10.49)

into Eq. (10.48), one has

2 2 2

2 22 2

1 1( )( ) ( 4 )( ) cos 24 4

1 1 ( )( )cos2 ( ) sin 2 = 12 2

G H G H F

G H L

σ σ σ σ α

σ σ σ σα α

++ + + + −

− − − + −

Ⅰ ⅠⅡ Ⅱ

Ⅰ ⅡⅡⅠ


2 2 22 2( ) 2 ( )cos 2 ( ) cos 2a bσ σ σ σ σ σα+ − − + −Ⅰ ⅠⅡ ⅡⅡⅠ

2 42 ( ) =LG H G H

σ σ+ −+ +

Ⅰ Ⅱ (10.50)

where

4 2, G H G H F La bG H G H

− + + −≡ ≡+ +

(10.51)


2 2 2 2 2( ) ( ) 2 ( ) cos 2( ) aσσ σ σ σ σ σ ατ+ + − − − +Ⅰ ⅠⅡ Ⅱ ⅡⅠ


2 2 2)( ) cos 2 = (2b σ σα α σ+ − (10.52)

Equation (10.52) is extended to the following equation for the plane isotropy with the material constant ( 1)m ≥ .

)| | | | = (2( )mm m mσσ σ σ σ στ+ + −Ⅰ ⅠⅡ Ⅱ (10.53)

Hill (1990) proposed the following extended orthotropic yield condition from Eqs. (10.52) and (10.53).

| | | |( )mm mσσ σ σ στ+ + −Ⅰ ⅠⅡ Ⅱ

212 2 2 2( /2) ) | | 2 ( ) ( ) cos 2 cos 2 = (2m ma bσ σ σ σ σ σ α α σ−+ + − − + −Ⅰ ⅡⅡ ⅡⅠ Ⅰ

(10.54)

Equation (10.54) involves the five material constants, i.e. the yield stress , σ τ and

the dimensionless number , , a b m . It reduces to Eq. (10.52) for = 2m and to Eq.

(10.53) for = = 0a b (or = / 4α π ). By use of Eq. (10.49), Eq. (10.54) is rewritten in the anisotropic axes as follows:

2222 2211 11 12

/2| | |( ) 4 |( )mm mσσ σ σ σ στ∗ ∗ ∗ ∗ ∗+ + − +

2 2 22 222 22 2211 12 11 11

1( /2) ) |( ) 4 | 2 ( ) ( ) = (2m ma bσ σ σ σ σ σσ σ−∗ ∗ ∗ ∗ ∗ ∗ ∗+ + + − − + −

(10.55)

Generally, the yield surface is described in the principal axes of anisotropy as follows:

( )( ) =ijf HFσ ∗ (10.56)

where

=ij ir js rsQ Qσ σ∗ (10.57)

( ) ( ) ( )( cos( , ))=ji i jij t tQ t•∗ ∗≡ e e ee (10.58)

Needless to say, Eq. (10.56) is not a general tensor expression but is merely the

expression by the components. The variation of i∗e is calculated using the

following equation with the initial value of 0i∗e .

0= ii i dt•∗∗ ∗ + ∫e e e (10.59)

where i•∗e is given by


=i i•∗ ∗ωee (10.60)

denoting the spin of principal axes of orthotropic anisotropy as ω . Here, the stress

rate ijσ• ∗ is calculated by

= = ( )ij ij rsjs rp ps rp psirQ Qσ σ σ ω σ σ ω•• ∗ ∗ − + (10.61)

for the case of the input of stress rate rsσ• from Eq. (4.33) and by

=ij rsjsirQ Qσ σ• ∗ (10.62)

after calculating the corotational stress rate rsσ from the constitutive relation for

the case of the input of the strain rate.

10.6 Representation of Isotropic Mises Yield Condition

The isotropic yield function described by Eq. (6.54) can be expressed in the

following various forms.

( ) = = = || || =3 3/2 3/2 rs rsef ' ' ' 'σσ σ σσ σⅡ

2 2 2 2 211 22 33 12 23 31= )3/2 ' ' ' ' ' 'σ σ σ σ σ σ+ + + + +2(

2 2 2 1/ 22 2 233 3311 22 22 11 12 23 31= 1 ( ) ( ) ( ) 6( )/2 σ σ σ σ σ σ σ σ σ− + − + − + + +

2 2 2= 3/2 ' ' 'σ σ σ+ +Ⅰ Ⅱ Ⅲ

2 2 2= 1 ( ) ( ) ( ) =/2 Fσ σ σ σ σ σ− + − + −Ⅱ Ⅱ Ⅲ ⅢⅠ Ⅰ (10.63)

The combined test of the tensile stress 11( )=σ σ and the distortional stress

12 ( )=τ σ for a thin wall cylinder specimen is widely adopted for metal. In this

case Eq. (10.63) is rewritten as

2 22 ( 3 ) = Fσ τ+ (10.64)

10.6 Representation of Isotropic Mises Yield Condition 245

Then, the Mises yield condition is shown by a circle of radius F in the

( , 3 )σ τ plane.

The visualization of the stress state can be realized in the space of three and less dimension. The stress state can be represented completely in the principal stress space when principal stress directions are fixed to materials and only the principal stress values change. In general, however, one must use the six-dimensional space or memorize the variation of the principal stress direction if the directions change. However, in the cases for which the number of independent variable components is less than three, such as the tension-distortion test described above and the plane stress and strain tests, the state of stress can be represented in the three and less dimensional stress space. The Ilyushin’s isotropic stress space (Ilyushin, 1963) is convenient to depict the Mises yield surface, which depends only on the deviatoric stress, as explained below.

The deviatoric stress tensor involves the five independent variables and thus the Mises yield surface in Eq. (10.63) is described by the independent components as follows:

2 2 2 211 22 11 22 12 23 31( ) = 3 3 3 )f ' ' ' ' ' ' 'σ σ σ σ σ σ σ+ + + + +3(σ

2 2 2 2

11 11 22 12 23 313 1 == 3 )2 2( ( F' ' ' ' ' 'σ σ σ σ σ σ+ + + + +3() )

and thus it can be rewritten as

2 2 2 2 2 21 2 3 4 5 =S S S S S F+ + + + (10.65)

in the five-dimensional space with the axes

32 541 11 11 22 12 23 31= (3/2) , = 3(1/2) ), = 3 , = 3 , = 3S S S S S' ' ' ' ' 'σ σ σ σ σ σ+(10.66)

Equation (10.65) exhibits the five-dimensional super-spherical surface. Further, consider the expression of the Mises yield surface for the plane stress and strain conditions in the following.

10.6.1 Plane Stress State

The plane stress state fulfilling 3 = 0jσ can be described in the three-dimensional

space 11 22 12( , , )σ σ σ and thus the Mises yield condition (10.63) is described by

the following equation.

2 2 211 11 22 22 123 = Fσ σ σ σ σ− + + (10.67)


On the other hand, Eq. (10.67) can be described in the two-dimensional principal stress plane as follows:

2 2 = Fσ σ σ σ− +Ⅰ ⅡⅠ Ⅱ (10.68)

which is the section of the Mises yield condition cut by the plane = 0σⅢ and exhibits Mises’s ellipse in the principal stress plane ( , )σ σⅡⅠ as shown in

Fig. 10.12. It holds from Eq. (1.219) 3 because of = 0σⅢ leading to =m 'σ σ− Ⅲ

that

2= cos3

)(m Fσ θ π2− +3

(10.69)

Substituting Eq. (10.69) into Eq. (1.219), one obtains

2 2 2= cos cos = sin3 3 3

22 2= cos cos = sin3 3 3

) )( (

) )( (

F F F

F F F

πσ θ π θ θ

σ θ π θ π θ

⎫2− + + + ⎪3 3 ⎪⎬

2 2 ⎪− + + −3 3 ⎪⎭

Ⅰ

Ⅱ

(10.70)

from which the coordinates of main points on the Mises’s ellipse are calculated as shown in Fig. 10.11. The thin curve shows the Hill’s orthotropic Mises yield surface in Eq. (10.45), which is rotated the principal axes of ellipse with the changes of its long and short radii from the isotropic Mises yield surface.

Next, consider the Ilyushin’s isotropic stress space in which the variables in Eq.

(10.66) are used. Here, in the present case fulfilling 3 = 0jσ leading to

54 = = 0S S the Mises yield surface is represented by the sphere in the

31 2( , , )S S S space, while it holds that

1 11 11 11 1122 22

2 11 22

11 11 22 1122 22 22

= (3/2) = (3/2) ( ) / 3 = / 2

= 3(1/2) )

= 3[(1/2) ( ) / 3 ( ) / 3] = ( 3/2)

S

S

'

' '

σ σ σ σ σ σ

σ σ

σ σ σ σ σ σ σ

⎫− + − ⎪⎪+ ⎬⎪− + + − + ⎪⎭

(10.71)

Furthermore, in the case fulfilling 12 = 0σ , the Mises yield surface is represented by the circle in the 1 2( , )S S plane (Fig. 10.13). Here, setting

10.6 Representation of Isotropic Mises Yield Condition 247

21 , 3 3

( )F F− −

( =150 )θ

, ( )F F− − F−

F ( = 60 )θ, ( )F F

0 σⅠ

σⅡ

F

F−

12 , ( )3 3F F

1 1, 3 3

( )F F−

( = 0)θ

( = 30 )θ

( =120 )θ

( =180 )θ

( = 300 )θ( = 240 )θ

( = 330 )θ

1 1, 3 3

( )F F−

( = 270 )θ

1 2, 33

( )FF ( = 90 )θ

12 , 3 3

( )F F− −( = 210 )θ

21 , 3 3

( )F F− −

( =150 )θ

, ( )F F− − F−

F ( = 60 )θ, ( )F F

0 σⅠ

σⅡ

F

F−

12 , ( )3 3F F

1 1, 3 3

( )F F−

( = 0)θ

( = 30 )θ

( =120 )θ

( =180 )θ

( = 300 )θ( = 240 )θ

( = 330 )θ

1 1, 3 3

( )F F−

( = 270 )θ

1 2, 33

( )FF ( = 90 )θ

12 , 3 3

( )F F− −( = 210 )θ

Fig. 10.12 Mises yield surface in the plane stress condition. (Thin curve describes Hill’s orthotropic Mises yield condition).

01SF

F

F−

F− φ

2S

01SF

F

F−

F− φ

2S

Fig. 10.13 Mises yield surface in plane stress state without shear stress ( 121σ =0)

1 2= cos , = sinS F S Fφ φ (10.72)

and substituting them into Eq. (10.71), it holds that

22112 2= sin , = sin

33 3( )F Fπσ φ σ φ+ (10.73)


10.6.2 Plane Strain State

If the elastic strain rate can be ignored compared with the plastic strain rate in the

plane strain state, the following relation holds by substituting 33 33= = 0pD 'λσ

into = 0rr'σ .

33 22111= ( )2

σ σ σ+ (10.74)

Then, the Mises yield surface is described from Eq. (10.63) 3 by

2 2221112 =3

2( ) F

σ σ σ− + (10.75)

which is represented by the Mohr’s circle in the plane of the normal and the shear stresses.


Chapter 11 Constitutive Equations of Soils

11 Constitutive Equations of So ils

The history of plasticity has begun with the study on deformation behavior of soils by Coulomb (1773) when he has proposed the yield condition of soils by applying the friction law of himself. Thereafter, the soil plasticity has been deprived the leading part by the metal plasticity. One of the reasons would be caused by the fact that soils exhibit various complex plastic deformation behavior, e.g., the pressure-dependence, the plastic compressibility, the dependence of the third in-variant of deviatoric stress, the softening and the rotational hardening. Explicit constitutive equations of soils will be described in this chapter, based on the elas-toplastic constitutive equations in Chapters 6-8.

11.1 Isotropic Consolidation Characteristics

The ln ln pv − linear relation ( (t r )/3p ≡ − σ : pressure) for the isotropic consolidation characteristics of soils was proposed by Hashiguchi (1974; see also 1985, 1995, 2008; Hashiguchi and Ueno, 1977), which is depicted in Fig. 11.1.

00

0

Normally - consolidated line: ln ln=

ln lnSwelling line: =

yy

yy

e

e

e

e

p pvp pv

ppvp pV

ρ

γ

⎫+⎪− + ⎪⎬

+ ⎪− ⎪+⎭

(11.1)

where 0( , )p V , 0 0( , )y yp V , ( , )yyp V are the pressures and the volumes in

the initial state, the initial yield state and the current yield state, respectively. V is

the volume in the unloaded state to the initial pressure, corresponding to the in-

termediate configuration described in 6.1. Further, ρ and γ are the material

constants prescribing the slopes of normal-consolidation and swelling lines, re-

spectively, in the (ln , ln )pv plane. (>0)ep is the material constant pre-

scribing the pressure for which the volume becomes infinite, i.e.

for ep pv → ∞ → − .

250 11 Constitutive Equations of Soils

V0yv

V

vyv

0

ρ

γ

1

1

lnv

Swelling line

Normal-consolidation line

Swelling line

ln( )ep p+y ep p+ep p+0y ep p+0 ep p+

V0yv

V

vyv

0

ρ

γ

1

1

lnv

Swelling line

Normal-consolidation line

Swelling line

ln( )ep p+y ep p+ep p+0y ep p+0 ep p+

Fig. 11.1 ln ln pv − linear relation of isotropic consolidation of soils

The logarithmic strain and its elastic and plastic components are given from Eq. (6.7) and (11.1) as follows:

0

0

ln lnln= = ln= lnln )(p yy

yy

evv v

vvVv v VV vvVV V

ε ε ε + ++ + +

0 0

0 00

ln ln= ln ln( )y y

yy

e e eee ee e

p p pp p pppp p p ppp p p

γ ρ γγ + + ++− + − − −+ ++ +

00

)(ln ln= y

y

ee

e e

p ppppp p p

γ ρ γ ++− − −+ + (11.2)

i.e.

00

0

0

)( ,ln ln=

,ln=

)( ln=

y

y

yp

y

v

v

ee

e e

eev

e

e

e

p pppp p p p

ppp p

p pp p

γε ρ γ

γε

ρ γε

+ ⎫+− − − ⎪+ +⎪⎪+ ⎪− + ⎬⎪

+ ⎪− − ⎪+⎪⎭

(11.3)

The volumetric strain rate and its elastic and plastic components are given from Eq. (11.3) as follows:

11.1 Isotropic Consolidation Characteristics 251

)= (

=

)= (

y

y

yp

y

e e

e

e

e

v

v

v

ppD pp p p

pD p p

pD p p

γ ρ γ

γ

ρ γ

••

•

•

− − −+ +

− +

− − +

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(11.4)

which show the volumetric elastic and the plastic strain rates derived exactly from

the ln ln pv − linear relation based on the multiplicative decomposition of

deformation gradient. Strictly speaking, the elastic deformation characteristic are influenced by plastic

deformation in general, but yp is not incorporated in the elastic strain and its rate in Eqs. (11.3) and (11.4) in order to avoid the elastic-plastic coupling, which would raise the complexity of formulation, whilst the coupling in soils is not so strong as in brittle geomaterials, e.g. rocks and concretes.

Adopting Eq. (11.4) in the explicit equation (5.28) for the elastic modulus tensor E in Eq. (6.29), the elastic bulk modulus K and the elastic shear modulus G are given as

3(1 2 )= , =2(1 )

ep pK G Kν

γ ν−++

(11.5)

On the other hand, the ln pe − linear relation for the isotropic consolidation has been widely adopted for constitutive equation of soils including Cam-clay models (Roscoe and Burland, 1968; Schofield and Wroth, 1968) in which the normal-consolidation and the swelling lines are given by

00

0

Normally - consolidated line: ln=

lnSwelling line: =

yyy

y

pe p

pe e p

e λ

κ

⎫⎪− −⎪⎬⎪− − ⎪⎭

(11.6)

where the material constants λ and κ are the slopes of normal-consolidation and

swelling lines, respectively, in the ( , ln pe ) plane (Fig. 11.2), where 0 0( , )p e ，

0 0( , )y yp e and ( , )y yp e are the pressure, the void ratio in the initial, the ini-

tial yield and the current yield states, respectively. In addition, e is the void ratio in

the unloaded state to the initial pressure 0p .


0e0ye

e

e

ye

ln p0

λ

κ

1

1

ln e

ypp0yp0

p

Swelling line

Swelling line

Normal-consolidationline

0e0ye

e

e

ye

ln p0

λ

κ

1

1

ln e

ypp0yp0

p

Swelling line

Swelling line


Fig. 11.2 ln pe − linear relation for isotropic consolidation of soils

However, the ln pe − linear relation has the following physical impertinence.

1. The change of void ratio induced during the same range of pressure along the swelling line is independent of a pre-consolidation pressure yp , although the plastic decrease of void ratio proceeds with the increase of yp .

2. The void ratio becomes negative if the pressure becomes large as

0 0exp( )/y yyp p e λ> or 0 exp( )/p p e λ> . 3. The relation is not given by the volume but by the void ratio. Therefore, the

relation of pressure vs. volumetric strain becomes quite complicated even if the incompressibility of soil particles is assumed.

4. The volume becomes infinite when the pressure approaches zero. 5. The relation is given by the difference of void ratio. Therefore, it fits to the

nominal strain but becomes quite complex for the logarithmic strain which is pertinent for finite strain.

Therefore, the following nominal volumetric strain has been adopted in consti-tutive equations.

, ,= = = pe

v v vv vV V V V

V V Vε ε ε− − −

(11.7)

Here, it is noteworthy that the nominal strain cannot be related to the strain rate D in the exact sense. The following approximation is used by substituting the void ratio instead of the volume into Eq. (11.7) in order to evade the above-mentioned deficiency 3.

11.2 Yield Conditions 253

0 000 0

0 0 0 0

( ) ( ) ( ), , = 1 1 1 1

y y yypev v ve e eee ee ee e

e e e ee eε ε ε

+ +− − −− −−≅≅ ≅+ + + + (11.8)


00 00ln ln1 1

yv

y

ppp pe e

λκ κε −≅ − −+ + , 0 0

ln1v

e ppe

κε ≅ − +

0 0

0 00 0 0 00ln ln ln ln=1 1 1 1

p y y yv

y y y

p p ppp p ppe e e e

κ κ κλ λε −≅− − − −+ + + +(11.9)

from which the nominal volumetric strain rate is given by

0 0( )

1 1 y

yv v

ppD p pe eκλκε

••• −≅≅ − −+ + ,

0 0( ) ( ),

1 1 p p y

yv

eev vv

ppD Dp pe eκλκε ε

••• • −≅ ≅≅ ≅− −+ + (11.10)

Equation (11.10) is adopted most widely for elastoplastic constitutive equations of soils. Nevertheless, it has the following deficiencies.

6. It is derived merely approximately from the ln pe − linear relation. 7. It cannot be adopted to describe finite deformation since it is derived based on

the definition of nominal strain, which can be adopted only for the description of an infinitesimal deformation.

8. The tangent elastic bulk modulus ( )/ evpK ε

••≡ is given by

0)(1 /= peK κ+ from Eq. (11.10). It has a crucial physical impertinence:

“The larger the void ratio, the larger is the elastic bulk modulus, i.e., the looser

the soil, the smaller is the change of void ratio to be induced”. In addition, the

tangent elastic bulk modulus K depends on the initial void ratio 0e . Conse-

quently, it is not a material constant.

Eventually, it is concluded that the ln pe − linear relation is inadequate physically and mathematically for formulation of constitutive equations for finite deformation of soils. On the other hand, the various deficiencies in the ln pe − linear relation are remedied completely in the ln ln pv − linear relation.

11.2 Yield Conditions

Various yield conditions of soils have been proposed to date. The functions ( )f σ in Eq. (6.30) can be reduced to the unique form of


( ) ( )=f pg χσ (11.11)

where

|| ||M

χ ≡η， p'≡η σ

(11.12)

M is the stress ratio || ||η in the maximum point of || ||'σ , i.e. the critical state

and is called the critical state stress ratio. The following explicit equation of M

was proposed (Hashiguchi, 2002).

sin cos( ) = (3 sin )(8 )cosc

c

Mφθ

φ θ14 63

− + 3σ

σ (11.13)

where

3 cos 6t rθ3 ≡σ τ , || ||

''

≡ στ σ (11.14)

cφ is the angle of internal friction in the critical state for the axisymmetric

compression stress state, i.e. the so-called tri-axial compression ( = / 3)θ πσ . The

section of the conical surface || ||= Mη cut by the π plane is depicted in

Fig. 11.3. It fulfills the convexity condition (cf. Eq. (A.14) in Appendix 5)

2

2

3 sin 1 1 8 ( )cos1= 0sin

)( c

c

dM Md

φθθ φ

−+ − 3 ≥14 6σ

σ (11.15)

for the whole range of cφ . On the other hand, the following simple equation has

been widely used for constitutive equations of soils (Satake, 972; Gudehus, 1973;

Argyris et al., 1973).

6 sin2)( =sin sin3 3

c

cM φθ φ θ−σ

σ (11.16)

However, Eq. (11.16) violates the convexity condition

2

21 1 1 sin( 8 sin )3= 0

6 sin2)( c

c

dM Md

φ θθ φ+ + 3 <

σσ (11.17)

for 22 01'cφ > .


0

= 45cφ

θσ

30

15

1σ 2σ

3σ

0

= 45cφ

θσ

30

15

1σ 2σ

3σ

Fig. 11.3 Section of the critical state surface of soils by π–plane (Hashiguchi, 2002)

It is premised in Eq. (11.11) that the yield surface passes through the isotropic

compression state and the null stress point. Then, the function ( )g χ must fulfill

the following conditions

( ) for = 0= 1

( ) for

g

gχ χχ χ

⎫⎬→ ∞ → ∞⎭

(11.18)

Furthermore, it is premised that the magnitude of the deviatoric stress, || ||'σ , becomes maximal at the critical state and thus it holds that

( ) ( )=1 1g g' (11.19)

(cf. Appendix 6).

Denoting the p -value in the critical state as crp , the following equation is obtained by substituting Eq. (11.11) into Eq. (6.30) at = 1χ .

= (1)crp gF/ (11.20)

The following various equations of the function ( )g χ have been proposed.


1 ) Original Cam-clay model (Schofield and Wroth, 1968)

= exp( )g χ (11.21)

= ecrp F/ (11.22)

2 ) Modified Cam-clay model (Burland, 1965; Roscoe and Burland, 1968)

2= 1g χ+ (11.23)

2=crp /F (11.24)

3 ) Hashiguchi model (Hashiguchi, 1972)

2= exp

2( )g

χ (11.25)

= ecrp F/ (11.26)

The above-mentioned three yield surfaces are shown in Fig. 11.4. Among them, only the yield surface of the modified Cam-clay model in Eq. (11.23) does not involve the corner so that the singularity of the normal of the surface is not induced. Equation (6.30) with Eqs. (11.11) and (11.23) is rewritten as

22

2

( / 2) || ||= 1

/ 2 ( /2)

p FF MF

'− + σ (11.27)

p

q

F2F

eF

eF0

Original Cam-clay model

Modified Cam-clay model

Hashiguchi (1972) model

1eM

cM

Compression

Extension

p

q

F2F

eF

eF0

Original Cam-clay model

Modified Cam-clay model

Hashiguchi (1972) model

1eM

cM

Compression

Extension

Fig. 11.4 Various yield surfaces of soils


Here, consider further the yield surface fulfilling the following conditions.

1) It involves not only positive but also negative pressure ranges. Here, note that the subloading surface is indeterminate at the null stress point in the initial subloading surface model if the normal-yield surface passes thorough at the null stress point in which the singular point of plastic modulus is induced, while this problem is not induced in the extended subloading surface model because the similarity-center of the normal-yield and the subloading surfaces is not fixed at the null stress point. The exclusion of the singularity of plastic modulus at the null stress point is of important for the engineering design of soil structures because soils near the side edges of footings, soils at the pointed ends of piles, etc. are exposed to the null or further negative stress state. In addition, the in-clusion of tensile yield strength is of importance for the engineering design of structures of natural soils such as soft rocks and cement-treated soils widely used recently, which have the tensile yield strength.

2) In the case that the anisotropy does not change, the yield surface ex-pands/contracts keeping the similarity with respect to the origin of stress space so that the yield stress increases/decreases in all directions in this space.

3) For the sake of mathematical simplicity, the yield condition is described by a

separate form consisting of the function of the stress and internal variables

( , )f βσ and the function involving the isotropic hardening variable, i.e. the

isotropic hardening function ( )F H , which describes the size of the yield

surface.

Here, the function ( , )f βσ must be a homogeneous function of the stress tensor σ in order to fulfill the above-mentioned conditions 2 ) and 3 ).

Equation (11.27) becomes the following equation through the translation to the

negative pressure range by Fξ ( )p p Fξ+→ (Hashiguchi and Mase, 2007).

22

2

( / 2) || ||1= 1/ 2 ( )/2

p F

F MF'ξ−− + σ

(11.28)

where ξ is the material constant, while it must fulfill 0 1/2ξ≤ ≤ since the

tensile yield stress is smaller than the compression stress and /ep Fξ < since the

volume does not become infinite by the elastic deformation inside the yield surface,

i.e. for p Fξ> − . The yield surface in Eq. (11.28) is depicted in Fig. 11.5

for the axisymmetric stress state, where M is given from the relation || || = 2MF/'σ = (1 2 ) 2M F/ξ− as follows:

1=1 2

M Mξ− (11.29)


cφ is described from Eqs. (11.13) and (11.29) as follows:

1 1(1 2 )3 3 = sin = sin

(1 2 )2 26 6( () )c c

cc c

M MM M

ξφξ

− − −+ + −

(11.30)

where cM and cM are the values of M and M in the axisymmmetric com-

pression stress state. q is the axial difference stress, i.e. l aq σ σ≡ − ( aσ : axial

stress， lσ : lateral stress) in Fig. 11.5.

ˆ

2MF

ξ

1

0 p

q

1

( (1=yp Fξ− ) ) (> )epFξ− −

M

p

v

ep−

Swelling line

Critical state line


0

(1/ 2 )Fξ− / 2FF

1=1 2

)(M Mξ−

ˆ

2MF

ξ

1

0 p

q

1

( (1=yp Fξ− ) ) (> )epFξ− −

M

p

v

ep−

Swelling line

Critical state line


0

(1/ 2 )Fξ− / 2FF

1=1 2

)(M Mξ−

Fig. 11.5 Yield surface of soils with tensile strength

Equation (11.28) can be transformed to the equation

22 2(1 2 )(1 ) ( ) 0=ppF F χξξξ ++ −− − (11.31)

11.3 Isotropic Hardening Function 259

where || ||M

'χ ≡ σ (11.32)

Equation (11.31) can be expressed in the separated form of the function ( , )pf χ of the stress and internal variables and the hardening function F , i.e.

( , ) =pf Fχ ,

21 ( ) for =0/( , ) = 1 ( for 0)

pppf

p pχ

χ ξχ

ξξξ

⎧ +⎪⎨ ≠−⎪⎩

(11.33)

where

, 2(1 ) (1 2 )ξ ξξ ξ ξ≡ ≡− − , 22 2p pχ χξ≡ + (11.34)

In the above, the yield surface of soils is formulated so as to fulfill the conditions

1 )-3 ) based on the modified Cam-clay model. It is difficult to derive the other yield

surface fulfilling the conditions 1 )-3 ). For instance, consider the translation of the

original Cam clay model to the negative pressure range by p p Fξ+→ .

|| ||( ) exp =( )/p F M Fp F'ξ ξ+ +σ

(11.35)

However, a separated form of the stress and internal variables and the hardening function cannot be derived from this equation. On the other hand, the translation of the yield surface to the negative pressure range by the constant value

( )y yp pC C+→ is adopted for constitutive equations for unsaturated soils (e.g. Alonso et al., 1990; Simo and Meschke, 1993; Borja, 2004). The modified Cam-clay model, for instance, is described by this translation as follows:

22

2

( / 2)1 || ||= 1/ 2 ( )/2

yp F CF MF

'− ++ σ

(11.36)

In this equation, the yield surface expands/contracts from/to the fixed point

= ( = )y ypC CIσ on the hydrostatic axis and thus it does not fulfill the condi-

tion 2 ). The incorporation of this yield condition into the subloading surface model

leads to the physical impertinence that the unloading is induced against the fact a

large plastic deformation is induced when the stress translates towards the negative

pressure direction.

11.3 Isotropic Hardening Function

The isotropic hardening/softening of soils is induced substantially by the plastic volumetric strain. Substituting (1 )=yp Fξ− into Eq. (11.3)2 in accordance with


the translation of the yield surface by Fξ to the negative pressure range based on the discussion in 11.2, one has

0

(1 ))( ln=(1 )

pv

e

e

pFpF

ξρ γεξ

− +− −− + (11.37)

Here, selecting H as

, t r=( ), , ip pv vH H D h Hε •

−≡ − ≡ − , NNσ (11.38)

so that F increases with H , the hardening function ( )HF is given from Eq. (11.37) as follows:

0 exp( ) = 11 )( ee pp HHF F ρ γ ξξ−+ −− −

(11.39)

1= 1( )epdF FF dH' ρ γ ξ+≡ −− (11.40)

While the plastic volumetric strain is dominant for the hardening/softening of soils, the deviatoric strain would induce the remarkable softening in the negative range of pressure. Then, extend the evolution rule of the hardening/softening func-tion H in Eq. (11.38) as follows:

|| || || ||, t r,= =( )ppiv p pDH h H' 'ζ ζ• − −− −− ⟨ ⟩ − ⟨ ⟩,D N NNσ (11.41)

where ζ is the material constant.

On the other hand, the following isotropic hardening rule has been used for sands, while the isotropic hardening and softening are induced by the deviatoric strain rate when the stress ratio is higher and lower, respectively, than a certain value dM (Nova, 1977; Wilde, 1977; Hashiguchi and Chen, 1998).

|| ||= || || ( )p p

d dvH D Mp''μ

•− + −D σ (11.42)

sin cos( ) (3 sin )(8 )cosd

dd

M φθφ θ

14 63 ≡− + 3

σσ

(11.43)

where dμ and dφ are the material constants. The stress paths for sands under the constant volume or undrained condition can be predicted realistically by the iso-tropic hardening rate in Eq. (11.42) with Eq.(11.43), in which the remarkable stress rise and drop are induced in loose and dense sands, respectively (see Fig. 11.6). On the other hand, these behavior can be predicted also by introducing the super-yield surface as will be described in 11.10.

11.4 Rotational Hardening 261

q

p0

q

p0Loose sands Dense sands

Critical st

ate line

Critical st

ate line

Initia l yieldsurface

Initial yieldsurface

Stress path

Stress pathq

p0

q

p0Loose sands Dense sands

Critical st

ate line

Critical st

ate line

Initia l yieldsurface

Initial yieldsurface

Stress path

Stress path

Fig. 11.6 Stress paths under constant volume or undrained condition

11.4 Rotational Hardening

The inherent anisotropy represented in the orthotropic anisotropy described in 10.1.4 cannot be ignored in metals and woods. On the other hand, the induced anisotropy is more dominant in soils since soils are assemblies of particles with weak cohesions between them and thus the rearrangement of soil particles is in-duced easily. Here, the yield surface of soils must always involve the origin of stress space but does very slightly because of the high frictional property. Note that the stress can never return to the origin undergoing the remarkable softening (contrac-tion of the yield surface) following the plastic potential flow rule, once the yield surface translates so as not to involve the origin, as illustrated on the (p, q) plane for the axisymmetric stress state in Fig.11.7. Therefore, the kinematic hardening (Prager, 1956) is not applicable to soils.

On the other hand, the anisotropy of soils with the frictional property, i.e. the pressure-dependence of yield surface can be described by the concept of the rota-tional hardening that the yield surface rotates around the origin of stress space by anisotropy. Sekiguchi and Ohta (1977) proposed the replacement of the deviatoric

q

0 p

f∂∂σ

t r : softening> 0=( )f fp

∂ ∂−∂ ∂σ

Yield surface:contractionσ

dσ

q

0 p

f∂∂σ

t r : softening> 0=( )f fp

∂ ∂−∂ ∂σ

Yield surface:contractionσ

dσ

Fig. 11.7 Inadequacy of kinematic hardening for description of anisotropy of soils


stress 'σ to the novel variable p' − βσ , where β is called the rotational hard-ening variable. Then, the yield condition in Eq. (11.28) or (11.33) is extended as follows (Hashiguchi and Mase, 2007):

22

2

( / 2) || ||1 = 1/ 2 ( /2)

p F

F M F'ξ−− + σ

(11.44)

or

( , ) =pf Fχ ,

21 ( ) for =0/( , ) = 1 ( for 0)

pppf

p pχ

χ ξχ

ξξξ

⎧ +⎪⎨ ≠−⎪⎩

(11.45)

where p' ' −≡ βσ σ (11.46)

|| ||

M'χ ≡ σ

(11.47)

sin cos3( ) =cos3(3 sin )(8 )

c

c

Mφθ

φ θ14 6

− +σσ

(11.48)

3cos3 6t rθ ≡σ τ , || ||

''

≡ σστ (11.49)

The yield surface in Eq. (11.44) or (11.45) is depicted in Fig. 11.8.

p0

q

Central axis

Compression

Extension

1 32

tan ( )aβ−

( (1=yp Fξ− ) )Fξ−

p0

q

Central axis

Compression

Extension

1 32

tan ( )aβ−

( (1=yp Fξ− ) )Fξ−

Fig. 11.8 Rotated yield surface in the (p, q) plane

11.4 Rotational Hardening 263

The evolution rule of rotational hardening tensor β is described below (Hashi-guchi and Chen, 1998; Hashiguchi, 2001). The following assumptions are adopted for the formulation of the evolution rule.

1 ) Rotation of the yield surface is induced only by the deviatoric component of the plastic strain rate independent of the mean component.

2 ) The rotation ceases when the central axis of yield surface reaches the surface, called the rotational hardening limit surface, which exhibits the conical surface having the summit at the origin of stress space. Let the rotational hardening limit surface be given by

= rMη (11.50)

where rM is the stress ratio in the rotational hardening limit surface, called the rotational hardening limit stress ratio, and let it be given by

sin (cos ) =(3 sin )(8 cos )

rr

rM

φθφ θ

14 63− + 3

σσ

(11.51)

rφ being the material constant, called the rotational hardening limit angle.

3 ) The central axis of yield surface =η β rotates towards the conjugate line

= rMβ τ on the rotational hardening limit surface, where the conjugate line is

the generating line of the rotational hardening limit surface which is observed

from the hydrostatic axis in the same direction observed from the central axis

=η β of the yield surface to the current stress (see Fig. 11.9).

4 ) In the proportional loading process, the central axis of the normal-yield surface

approaches the stress path in the proportional loading process. Further, the

magnitude of rotational rate β is larger for a larger distance from the central

axis to the current stress and thus it is the function of || ||η . This assumption is

necessary also to exclude the singularity of the rotational direction in the state

that the stress lies on the central axis of the normal-yield surface (see

Fig. 11.10).

Based on the above-mentioned assumptions, let the following evolution rule of rotational hardening be postulated.

|| |||| || )( )) (|| || ( , = = p

i rrr rH MMb bFF

''' , −− ββ βD b σσ σ ττ

(11.52)

where rb is the material constant and rM is given by the following equation in an

identical form to that of Eq. (11.13).


1σ

2σ

3σ

Conjugate generating line

Rotational hardeninglimit surface

Hydrostatic axis

Central axisYield surface

0

　

=η β

=η 0

|| || = rMη

σβτ

τ

= rMη τ

1σ

2σ

3σ

Conjugate generating line

Rotational hardeninglimit surface

Hydrostatic axis

Central axisYield surface

0

　

=η β

=η 0

|| || = rMη

σβτ

τ

= rMη τ

Fig. 11.9 Direction of rotation of yield surface (illustrated in the principal stress space)

p0

q

Compression

Singular pointof rotational direction

=aβ 0

=bβ 0aσ

bσ

Yield surface

Rotational hardening limit s

urface

Rotational hardening limit surface

Central axis

of yield surface

Extension

p0

q

Compression

Singular pointof rotational direction

=aβ 0

=bβ 0aσ

bσ

Yield surface


urface


Central axis

of yield surface

Extension

Fig. 11.10 Singularity in direction of rotation of yield surface on central axis of yield surface


sin (cos ) =(3 sin )(8 cos )

rr

rM

φθφ θ

14 63− + 3σ

σ (11.53)

The relation between the axial components of the rotational hardening variable vs. the deviatoric plastic strain in the axisymmetric stress state under = const.F for sake of simplicity is depicted in Fig. 11.11. It is identical to the kinematic hardening variable shown in Fig. 10.1.

0

123 rM

1

aβ

23 rM−

pa 'ε

3 232

( )aar rMb η β−| |

3 232

( )aar rMb η β+| |

0

123 rM

1

aβ

23 rM−

pa 'ε

3 232

( )aar rMb η β−| |

3 232

( )aar rMb η β+| |

Fig. 11.11 Relation of axial component of rotational hardening variable vs. axial plastic strain in the axisymmetric stress state

11.5 Extended Subloading Surface Model

The extended subloading surface for the normal-yield surface in Eq. (11.45) is

described by replacing σ into −≡σ σ α as follows:

( , ) =pf RFχ ,

21 for =0

( , ) =1 ( ) for 0

( )pp

pfppχ

χ ξχ

ξξξ

⎧+⎪⎪

⎨⎪ ≠−⎪⎩

(11.54)

where

22 2p pχ ξ χ≡ + (11.55)

p' '−≡ βσ σ (11.56)


|| ||

M

'χ ≡ σ，

sincos( ) =

cos(3 sin )(8 )c

cM

φθθφ

14 633− +σ

σ (11.57)

3cos 6t rθ3 ≡σ τ ， || ||

( )ij

ij

srrs

''' ' '

σ

σ στ≡ ≡στ

σ (11.58)

Because of = 0α in soils, it holds from Eq. (8.3) that

( )= (1 ) , = (1 ) = R R R R≡ − − − +−α s s s s sσ σσ (11.59)

The normal-yield and the extended subloading surfaces for the yield surface for Eq. (11.44) or (11.45) are depicted in Fig. 11.12.

p

qNormal-yield surface

Compression

Extension

(1 )Fξ−

1 32

tan ( )aβ−−

32

( )(1 )

= (1 )( )a l

a

F

F

β β

β

ξξ

− − −

− −

0

Fξ−



urface

Subloading surface

s

σ

p

qNormal-yield surface

Compression

Extension

(1 )Fξ−

1 32

tan ( )aβ−−

32

( )(1 )

= (1 )( )a l

a

F

F

β β

β

ξξ

− − −

− −

0

Fξ−



urface

Subloading surface

s

σ

Fig. 11.12 Rotated normal-yield and extended subloading surfaces in the (p, q) plane

The rotational hardening rule is given from Eqs. (11.52) with the replacement of σ to −≡σ σ α by

|| || || ||( )) )|| || ( , (= = p

ir rr rHM Mb bF F

' '' ,− −β β βD bσ σστ τ (11.60)

Further, setting = 0α , the plastic modulus is obtained from Eq. (8.27) as follows:

( ) 1 1 ;( )tr ˆ )( [pi

U RFM h HF RR

c′ −++,≡ NN σ σ σ

|| || ( , ) ( ) || ||t r( )( if H

RF' '

∂ ,−∂

βNNβ bσ σ σ

11.6 Partial Derivatives of Subloading Surface Function 267

( , )( )( )tr1 = 0

ˆˆ( ) )]i

fHR R

∂ ,− − ∂β

β bs σ s (11.61)

where the functions h and b are given by Eqs. (11.41) and (11.60).

11.6 Partial Derivatives of Subloading Surface Function

The partial derivatives of the function in Eq. (11.54) are shown below.

2 3 21 ( ) ( 2 ) for =0/( , )

= 1 21 for 02( )

pp ppf

pp

pχ

χ χ ξχ

ξξξ

−⎧ + + −∂ ⎪

⎨ ≠−∂ ⎪⎩

21 for =0

=1 for 0

( )

( )

p

ppχ

χ ξ

ξξξ

⎧−⎪

⎪⎨⎪ ≠−⎪⎩

2 2 for =0 for =0( , )

= =41 1 2 for 0 for 0

2

p ppf

p pχχ

χ χξ ξ

χχ χχξ ξξ

ξ

⎧ ⎧⎪ ⎪

∂ ⎪ ⎪⎨ ⎨∂ ⎪ ⎪ ≠≠⎪ ⎪⎩⎩

1=3

p∂ −∂

Iσ

1= =3' '∂ − ⊗

∂I I I Iσ

σ

( ) 1 1= = ( ) 32( )ij ij ij

jl jk klkl kl

ik ijilp' δσ σ

δ δ δ δ δ δσ σ∂ ∂ +

+ −∂ ∂

1 1 1= =3 3 3' '∂ +− ⊗ + ⊗ ⊗∂ I I I β I I β Iσσ

( ) 1 11= = ( ) 32 3( )ij ij ij

ijjl jk kl klkl kl

ik ijilp' 'σ βσ δ δ δ δ δ δ β δσ σ

∂ −∂ + − +∂ ∂

2

sin=

cos cos(3 sin )(8 )c

c

M φθ θφ

14 6∂ −3∂ 3− +σ σ


23 sin = =

8 cos sin( )c

c

M Mφ

θ φ−− −+ 3 14 6σ

21= ( )|| ||' '

∂ −∂

Iτ τσσ

= =(ij

rs srijijrs srsrrs

ij klkl

kl kl rs sr

' ' '''' '' ' ''

'' ' '

σ σ σσσσ σσ σ σ σ

σ σ σ στ

∂ ∂∂−

∂∂ ∂∂ ∂

1 ˆ( )2=ˆijrs srjl jk kl

rs sr

ik il '' '

' '

δ δ δ δ σσ σ

σ σ

τ+ −

11 ( )= 2ˆˆ )ij kljl jk

rs sr

ik il' '

δ δ δ δσ σ

ττ+ −

23( )t r

= 3∂∂τ ττ

= = 3 )( rs st trrs rsst tr tr st ir rj

ijjs jt jrir is itδ δ δ δ δ δτ τ τ τ τ τ τ τ τ τ τ

τ∂ + +

∂

2cos 3 cos= ( 6 )|| ||''

θ θ3∂ − 3∂

σστ τσ σ

6= = 36 6( rsnl nl rsmn mnlm lm

s n nrrsij ij ij' ' 'σ σ σ

τ ττ τ τ ττ τ τ ττ

∂∂∂ ∂∂∂ ∂ ∂

1 1= 3 ( )6 2 rs ijrjsj siripq qp

s n nr

' 'δ δ δ δ

σ στ τ τ τ+ −

1= ( )3 6 n j

pq qp

s n nin r rs ij' 'σ σ

τ τ τ τ τ τ−

3= ( cos )6 )pq qp

ijin nj' '

θσ σ

τ τ τ− 3 σ

11.6 Partial Derivatives of Subloading Surface Function 269

21 3 )cos= ( 6cos8

M'

χ θθ∂ 3+ −3+∂

σσ

ττ τσ (11.62)

cos=cos(

pq qp

pq qpij ijij

M

M

' '

' '' ' '

σ σχ χ χ θθσ σ σσ σ

∂∂∂ ∂ 3∂∂+3∂∂ ∂ ∂∂∂

σ

σ

2

cos1=cos

pq qp

pq qp

ij

ij

M

M M

' '''' '

σ σσ θθ σσ σ

∂ 3∂ ∂−∂ 3 ∂∂

σ

σ

1 3 )cos= ( 68 cos

)ij ijir rjM

θθτ τ τ τ− 3+

+ 3 σσ

( , ) ( , ) ( , )=

ˆ3

( , ) ( , ) 1 = 33

( , ) ( , ) 1tr ( ) = 33

( )

[

p p pf f p fp

p pf fp

p pf fp

''

''

' '

χ χ χ χχ

χ χ χχ

χ χ χ χχ

∂ ∂ ∂ ∂∂ ∂1− +∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂1 +− + ⊗

∂ ∂ ∂

∂ ∂ ∂ ∂1 − −− +∂ ∂ ∂ ∂

I I β I

I β II

σσ σσ σ

σ

σ σ ]

( , ) ( , ) ( , )=( rs

ij ij ijrs

p p ppf f fp

''

χ χ χ χ σσ σ σσχ

∂ ∂ ∂ ∂∂ ∂1− +3∂ ∂ ∂ ∂∂∂

( , ) ( , )=

ij rs

p ppf fp '

χ χ χσ σχ

∂ ∂ ∂∂1− +3 ∂ ∂ ∂∂

1 11( ) 32 3 rssj si ijijri rj rsδ δ δ δ δ δ β δ+ − +

( , ) ( , )= ij

rs

p pf fp '

χ χ χδ

χ σ∂ ∂ ∂1− +3 ∂ ∂ ∂

1 11( ) 32 3

)rssj si ij ijrsri rjδ δ δ δ δ δ β δ+ − +


21 11 2 tr ( ) 33

for =0( , )=

1 1 1tr ( ) 33

( )

( )

][

][

p p

pf

ppp

' '

' 'χχ

χ χ χχ

ξχχ χ χξ

ξ

∂ ∂− − + − −∂ ∂

∂∂ ∂ ∂− − +2 − −

∂ ∂

I I β I

I I β I

σ σ

σσ σ

for 0ξ

⎧⎪⎪⎪⎪⎨⎪⎪⎪

≠⎪⎩

(11.63)

( , ) ( , )=

p pf fp'

χ χ χχ

∂ ∂ ∂−∂ ∂ ∂β σ

( , ) ( , ) ( , )= = ( )( rs

rsijij ijrs rs

p p pf f f pIβ β'

' 'χ χ χ χ χσ

χ χσ σ∂ ∂ ∂∂ ∂∂ −

∂ ∂∂ ∂∂ ∂

( , ) ( , )1= ( ) =2

)sj sirj rjrsrs

p pf fp p''

χ χ χ χδ δ δ δχ χ σσ

∂ ∂∂ ∂− + −∂ ∂ ∂∂

2 for =0( , )

=

2 for 0

pf

p

p

'

'χ

χχ ξχ

χχ ξ

⎧ ∂−⎪ ∂∂ ⎪⎨∂ ∂⎪− ≠⎪ ∂⎩

βσ

σ

(11.64)

21 ( ) for =0/( , )( , ) = = 1 ( for 0)

ˆpp

pffp pχ

χ ξχ

ξξξ

⎧ +⎪⎨ ≠−⎪⎩

s

s ss

s ss

βs (11.65)

where

tr , p p'1−≡ +≡3s sIs s s (11.66)

p' ' −≡ sβss (11.67)

|| ||

M'χ ≡ss

s (11.68)

sin cos3( ) =cos3(3 sin )(8 )

c

cM

φθφ θ

14 6− +

s ss

(11.69)

11.7 Calculation of Normal-Yield Ratio 271

3cos3 6t r ˆθ ≡s sτ , || ||

''

≡sss

τ (11.70)

21 3 )cos= ( 6cos8

M'

χθθ

∂ 3+ −3+∂s

s sssss

ττ τs (11.71)

2 for =0( , )

=2 for 0

ˆfp

p

'

'χ

χχ ξ

χχ ξ

∂⎧−⎪ ∂∂ ⎪⎨∂ ∂⎪− ≠

∂⎪⎩ s

ss

s ss

ββ

ss

s

(11.72)

11.7 Calculation of Normal-Yield Ratio

The normal-yield ratio R can be calculated directly by ( )= /fR F,βσ in the initial subloading surface model but it has to be obtained by the numerical calcula-tion in the extended subloading surface model as will be described below.

First, one has

1 , (tr( )= = ) =3 m mp R R ' 'Rs 'σ− + − + +sσ sσ σ (11.73)

from Eq. (8.6). Substituting Eq. (11.73) into Eq. (11.56) and (11.57), it is obtained that

1 (tr( ) )( ) = == 3 m mR Rp ' '' 'R R s' ' σ+ ++ ++ +− β ββ sσs sσ σσ σ

(11.74)

1tr( )|| || 3=

|||| R' 'R

M M

'χ+ ++

≡βsσsσσ (11.75)

Further, substituting Eq. (11.73)-(11.75) into Eq. (11.54) of the extended subloading surface, one has the following equation and can transform it in turn.

2

2 2

( )

( = for =0) 1( )

( )1 ( () ) 2

for 0=

||||

|| ||

( )

[ ]

m m

m m

m m

m mm m m m

R' 'R

RFMRR

R' 'RR RM

RF

s

ss

ss s

σσ ξ

σ

σσ σξ ξξ

ξ

⎫+ + ⎪+

⎪− + + ⎪− + ⎪

⎪⎪⎬⎪+ ++− + +++ ⎪⎪⎪≠ ⎪⎪⎭

β

β

sσ

sσ

(11.76)


2

2 2

2) ( )||( ||( ) ( )=ˆ

for =0

( ) ( ) 2

|| ||

m mm m m m

m mm m

RRR R RFM

R' 'RRM

ss s

ss

σσ σ

ξ

σσ ξ

∗ ∗ ++++− − +

+ ++− + +

β

β

sσ

sσ

( for 0 ) = m mRRF sσ ξξξ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪≠+− ⎭

2 2

2 2

2 2

2

2

2

2

( ) ) ( )( (t r t r 2 )

( || ||) ( ) 0 = for = 0

( ) ) ( )( (t r t r)2 4

( || ||)2

m m m m

m m m m

m m m m

m m

R ' '' R'R RM

R RM RF

R ' ''M R'R R

R

s s

s s

s s

s

σ σ

σ σ ξσ σξ ξ

σξ

+ +++ + +

+ + + +

+ ++ ++ +

++

β

β

β

β

ss σσ

ss σσ

2 2( for 00[ )] = m mRM RF sσ ξξξ

⎫⎪⎪⎪⎬⎪⎪

− ≠+ ⎪− ⎭

2 2 2 2 2 2

2

2 2 2 22

2 22

2

2 )(|| || t r || ||2

( (( ) ( )2 t r t r) 2 )

)( (2 || || ) 0=

m m m m

m mm m

mm m m mm

R RM MM R R

R R R

R RR M FR F

s s '' ' '

s s' 'ss s

σ σ

σ σσ σ σ

+ + + + +

+ ++ ++ + ++ +

β ββ

s sσ σ

sσ

22 2 2

2 2

22 22

2 22

for = 0

|| ||22

( ( )|| || t r) 4 )(t r4 2

|| ||)(4 ( ) (t r 2) 2

m m m m

m m

m m mmm m

R RM MM

RRR

RRR R

s s '

s ''' '

s ss '

ξ

σ σ ξξ σξξ

σξ ξ σσ

++ +

+ ++ +

++ ++ +

β

ββ

σs σsσ

s 2

2 2 22 2( ) 2 ( ) 0= for 0

m m mmF F RM Rs sξ σ σξ ξ ξξ ξξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪

−− − − + ⎪⎪≠ ⎪⎭

2 2

2 22 2

( ) ) = (

) )2 = ( (

m mm m

mm mm

RRF F R

F FR R

s s

s s

σ σξ ξ ξξ ξσξ ξ ξσξξ ξ

+− − −⎛ ⎞⎜ ⎟⎜ ⎟− −− +⎝ ⎠

11.7 Calculation of Normal-Yield Ratio 273

2222 2 22 2 2

2

2 2

2

22

2

2

22 2

2

( )|| || t r2 || ||

)( ( ) t r 2 t r22

2 || ||( )t r2

0 for = = 0( )|| || 2 t r || ||

mm mm

m mm

m m mm

m m m

m

RM FRR R RM

RRRM

M FR RR

M

RM

ss ss ''

ss ' ''

s'

''

s

σ

σ σσ

ξσ σ σ

+++ +

++ +

+ ++

+ ++ +

β β

β

ββ

β β

s s

sσ σ

s

σ σ

222 22 22

2 22

2

22

2

22

|| |||| || 4 ( )2 t r 2

)(t r4( ) 2

|| ||4 4 4( ) ( )t r t r

4 ( )t r|| || 2 ( ) 2

mm

mm m

m mmm

mm m m

RR R

R F RRMM

RR R

F RM M

ss' '

s s ' '

ss ' '

s ' '

ξξ ξ

ξσξξ

σξ ξ ξσ

ξσσ σξ ξ ξξ

+ ++

− +− +

+ + +

+ ++− +

ββ

ββ β

β

s s

sσ

sσ

σσ

2 2 2 22 || || for 0 0=2 mm M σξ ξξσ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪≠−+⎪⎭

β

22 222

2 2

2

2

22

2

2 22

2

2

( ) || || t r2 || ||

)( ( ) ( ) t r 2 t r t r222

2 || || || ||

( )2 t r || || 0 for = = 0

|| || 2

mmmm

mm mm

m mm m

m m

m

RM FM

M

M F R M

M

s sss ''

ss ' ' ''

s '

'

s '

σσ

σσσ

σ ξσ

ξ

+++ +

++ + +

++ ++

+ +

+

β β

β β

β

β β

s s

s sσ σ

σ

σs

22 222 2

2 2 2

2 2

2

22

2

|| || ( )4 ( )t r 2

)( ( ) ( )t r t r t r2 2 2 2

|| || 2 ( ) (1 )

for0|| ||( ) =t r || || 4 2 2

m mm

mmm m

m m mm m

m m

RFM

M

F RM M

sss '

ss ' ' ''s s

' '

ξξ ξξ

ξ ξσξσσ σξ ξσξ ξξ

σσξ ξξ

− −+ +

+ ++ +

+ ++ − −

++ +

ββ

β β

β

ββ

s

s sσ σ

σσ 0ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪≠ ⎭

Solving this quadratic equation, the normal-yield ratio R is expressed as follows:

2

2

for = 0=

for 0

B AC B A

RB AC B

A

ξ

ξ

⎧ −−⎪⎪⎨

−−⎪ ≠⎪⎩

(11.77)


where

2 222

2

2 2

22

22

2

2

22 2

2

( )|| || t r2 || ||

)( ( ) ( )t r 2 t r t r222

2 || ||

( )|| || 2 t r || ||

|| || 4 ( )2 t r 2

mmmm

mm mm

mm m

m m m

mm

A M FM

B M

M F

C M

A M

s sss ''

ss ' ' ''s

''ss ' '

σσ

σσ

σ σσ

ξ ξ

++≡ + +

+≡ + +

++

≡ + + +

+ +≡ +

β β

β β

β

β β

β

s s

s sσ σ

σ σs s

22

22

2 2

2 22

2

2 22

|| ||

( )

)( ( ) ( )t r t r t r2 2 2

|| || ( ) 2

|| ||( )t r|| ||(1 ) 42 2

m

m

mmm m

mm mm

mm m

FM

B M

FM

C M

s

s

ss ' ' ''ss

' '

ξ

ξξ

ξ ξσξσσσ ξ ξξ ξ

σ σξ σξ ξξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪− −

+ +≡ +

+ −+

+≡ − + +⎭

β

β β

β

ββ

s sσ σ

σσ

⎪⎪⎪⎪⎪⎪⎪⎪

(11.78)

Explicit calculation processes:

1 ) First step (beginning of calculation):

Calculate the normal-yield ratio R by Eq. (11.77), substituting the trial value

2 6sin /3cM = φ which is the average of 2 6sin /(3 sin )c cM = φ φ +

and 2 6sin /(3 sin )c cφ φ − .

2 ) After second step:

Recalculate R by substituting the value

3

sin sincos( ) = =cos(3 sin )(8 ) (3 sin )(8 )6t r

( )c c

c c

M φ φθθφ φ

14 6 14 633− + − +

σσ τ

(11.79)

into Eq. (11.77), while the value of R obtained in the former step is used in Eq. (11.79).

11.8 Simulations of Test Results 275

3 ) Repeat the process 2 ) until R will reaches the convergence.

Note here that R is calculated by the accumulation of d

dtR R•+ with

|| ||( )= pRR U

•D in the loading process but one has to calculate R by the

above-mentioned processes in the unloading process.

11.8 Simulations of Test Results

Some simulations of test data are given below in order to show the capability of the subloading surface model to reproduce the real deformation behavior of soils (Hashiguchi and Chen, 1989). Hereafter, all stresses exhibit effective stresses, i.e. stresses excluded pore-pressure.

The simulation of the test data (after Saada and Bianchini, 1989) for Hostun sand subjected to the drained triaxial compression with a constant lateral stress, which includes the unloading-reloading process, is shown in Fig. 11.13 where the material constants and the initial values are selected as follows:

Material constants:

Yield surface (ellipsoid): = 27 ,

volumetric : = 0.008, = 0.003, = 10 kPa,isotropic

deviatoric : = 0.6, = 25 ,Hardening/softening

rotational : = 10, = 20 ,

Evolution of normal - yield rati

d d

c

e

r r

p

b

φρ γ

μ φ

φ

⎧ ⎧⎪⎪ ⎨

⎪⎨ ⎩⎪⎩

1 1o : = 1.5, = 3.8,


Elastic shear modulus : = 200 000 kPa,

mu

G

c

Initial values:

0

0

0

0

Hardening function: = 400 kPa,

Rotational hardening variable : = ,

Center of similarity : = kPa,

Stress : 100 k Pa=

F

−50−

β 0

I

I

sσ

where the equation 11( ) = (1/ 1)mU R u R − is used for the evolution rule of the

normal-yield ratio in Eq. (7.13).

The simulation of the test data (after Saada and Bianchini, 1989) for Hostun

sand subjected to the drained proportional loading with

2 3 31( ( )/( )) = 0.666= b σ σ σ σ− − ( = 19 09 )'θσ from 0 = 500 kPa− Iσ


by the true triaxial test apparatus is shown in Fig. 11.14. The material parameters

are the same as those for the above-mentioned drained triaxial compression, while

the sample was preliminarily loaded the isotropic compression from

= 100 kPa− Iσ to 500 kPa− I before the test.

|| ||dt'∫D

|| ||dt'∫D

|| ||p'σ

vε

|| ||dt'∫D

|| ||dt'∫D

|| ||p'σ

vε

Fig. 11.13 Drained behavior of Hostun sand (data from Saada and Bianchini, 1989). Meas-ured and calculated results are shown by the dashed and solid lines, respectively.


|| ||dt'∫D

|| ||dt'∫D

|| ||p'σ

vε

|| ||dt'∫D

|| ||dt'∫D

|| ||p'σ

vε

Fig. 11.14 Drained proportional loading behavior of Hostun sand (data from Saada and Bianchini, 1989). Measured and calculated results are shown by the dashed and solid lines, respectively.


|| ||dt'∫D

|| ||p'σ

|| ||dt'∫D

|| ||p'σ

Fig. 11.15 Undrained behavior of Banding sand (data from Castro, 1969). Calculated results are shown by the solid lines


The simulations of the test data (after Castro, 1969) for Banding sand subjected to the undrained triaxial compression with a constant lateral stress are shown in Fig. 11.15 where the material constants and the initial values are selected as follows:

Material constants:

Yield surface (ellipsoid): = 26, 30, 31, 32 ,

= 0.025, 0.018, 0.014, 0.010,

volumetric = 0.0067, 0.0065, 0.0060, 0.0058,

= 0, 10, 30, 80 kPa,isotropicHardening/softening

= 1.00, 0.65, 0.deviatoric

d

c

ep

φ

ργ

μ

⎧⎪⎨⎪⎩

1

1

30, 0.10,

= 40, 33, 30, 20 ,


= 0.1, 0.3, 0.5, 1.0, 33.0,Evolution of normal - yield ratio

= 0.1, 0.4, 0.5, 0,7,

Translationon of similarity - center :

d

r rb

u

m

φ

φ

⎧ ⎧⎪ ⎪⎪ ⎪

⎪⎪ ⎨⎪⎨ ⎪ ⎧⎪⎪ ⎪ ⎨⎪ ⎪ ⎪⎩⎩⎪⎪⎩

⎧⎨⎩

= 20, 18, 14, 8,

Elastic shear modulus : = 18 000, 23 000, 25 000, 35 000 kPa,G

c

Initial values:

0

0

0

0

Hardening function: = 410, 480, 520, 580 kPa,


Center of similarity : = kPa,

Stress : 67 0 k Pa=

F

.

−200, −110, −100, −80 −

β 0

I

I

sσ

where the four values correspond to the initial relative densities

= 0.27, 0.44, 0.47, 0.64rD , respectively, in this order. The simulation of the test data (after Ishihara et al., 1975) for the cyclic mobility

of loose Niigata sand 0( = 0.737)e with the constant stress amplitude 2= 0.71 kgf/cmq ± under the undrained condition is depicted in Fig. 11.16

where the material constants and the initial values are selected as follows:

Material constants:

1 1

Yield surface (ellipsoid): = 28 ,

volumetric : = 0.01, = 0.0065, = 0.05 kPa,isotropic

deviatoric: = 1.0, = 35 ,Hardening/softening


Evolution of : = 8.0, = 1.3,

T

d d

c

e

r r

p

b

muR

φρ γ

μ φ

φ

⎧ ⎧⎪⎪ ⎨

⎪⎨ ⎩⎪⎩

2

ranslationon of similarity - center : = 34,

Elastic shear modulus : = 1,800 kgf/cm ,G

c


Fig. 11.16 Cyclic mobility of loose Niigata sand (after Ishihara et al., 1975). Test data and calculated results are depicted by the dashed and the solid lines, respectively.

Initial values: 2

0

0

20

20

Hardening function: = 5 5 kgf/cm ,


Center of similarity : = 0 21kgf/cm ,

Stress : 2 1 k gf/cm ,=

F .

.

.

−

−

β 0

I

s

σ

11.9 Simple Subloading Surface Model 281

11.9 Simple Subloading Surface Model

The simulation of test data by the extended subloading surface model with the isotropic and anisotropic hardening is described above. In what follows, the simple initial subloading surface model for soils will be described below in order to exhibit differences from the other soil models.

Now, by putting

= = , =β0 0s α , = 0, = 0ep ξ , = 0dμ (11.80)

in the afore-mentioned constitutive equations, the simple yield condition is obtained as follows:

2

0

|| ||/ ,( ) 1=

exp , =( ) ,=

,, t r,t r =( ) =

6 sin2=

sin sin3 3

( )

ip

c

c

pf p

M

H dF FF FF H dH

h HH

M

'

' ρ γρ γ

φφ θ

•

⎫⎪+⎪⎪⎪≡ ⎪−− ⎬⎪−− , ⎪⎪⎪

− ⎪⎭σ

ND N

σσ

σ (11.81)

for which the plastic modulus pM is given from Eq. (7.23) as

( )t r ( )t r( )p RUM Rρ γ

− +≡ −N Nσ (11.82)

and thus the plastic strain rate is described by

t r( )( )t r ( )tr( )

p =RU

Rρ γ− +−

N NDN N

σσ

(11.83)

and thus

1 t r( )=

( )t r ( )tr( )RURρ γ

− +− +−

ND E N

N N

σσσ

(11.84)

Here, note that the subloading hardening with t r( )0 0pM → >> Nσ in Eq.

(7.29) is induced over the critical state line fulfilling t r = 0N . The partial derivatives of the yield function in Eq. (11.81) is given as follows:

2 , , , , ( )( ) 21 , = =( ) pp f pf MMM Mp M

ηη η ηη

∂∂ +∂∂

,


2, , ( ) 2= ( )pf pMMMM

η η∂ −∂

(11.85)

2

|| || 1 11 1/ || ||= = =3 3( )p pp p' '

η η∂ ∂ + +∂ ∂ I Iσ στ τσ σ (11.86)

2

sin= =

cos 8 cos(3 sin )(8 cos )c

c

M Mφθ θφ θ

14 6∂ − −3 + 3∂ − + 3σ σσ

23 sin =

sinc

c

Mφ

φ−

−14 6

(11.87)

21= ( )|| ||' '

∂ −∂ Iτ τσ σ (11.88)

2cos 3= ( cos6|| ||

)' 'θ θ3∂ − 3∂σ

στ τσ σ (11.89)

( ) , , ( )=f pf Mη∂ ∂∂ ∂σσ σ

2 1 11 2= 1 33( ) ( )p

pMM Mη η η+− + +I Iτ

2 2 23 sin 32 ( cos6|| ||sin

)( )( ) c

c

pMMM '

φη θφ

− −− − 314 6

στ τσ

2 23 sin 21= 1 ( cos63 7 sin)( ) c

cM MM M

φη η η θφ

− −− − + + 3 36

σI τ ττ

(11.90) On the other hand, the Cap model, which incorporates the Drucker-Prager

model (Drucker and Prager, 1952) for the over-consolidated state into the Cam-clay model (Roscoe and Burland, 1968; Schofield and Wroth, 1968) for the nor-mally-consolidated state, is most widely used for the prediction of soil deformation behavior. The predictions of the drained triaxial compression behavior of soils under the constant lateral stress by the subloading surface model in Eq. (11.84) and the Cap model are depicted in Fig. 11.17. Here, the curves of axial stress and volumetric strain vs. the axial strain in the loading from the heavily and the lightly over-consolidated states, i.e. points o and o' , respectively are shown in this figure.

In the loading from the lightly over-consolidated state o' , the volumetric con-traction proceeds and the axial stress increases up to the critical state. The abrupt transition from the elastic to the plastic state is predicted by the Cap model. On the other hand, the smooth behavior is predicted always by the subloading surface model as observed in experiments.


Next, consider the loading from the heavily over-consolidated state o . In the

subloading surface model, the first term ( )/t r ρ γ−− N in the plastic modulus

(11.82) decreases from the positive value to zero up to the point c on the critical

state line t r( = 0)N . On the other hand, the second term /RU is always positive.

Therefore, the plastic modulus keeps positive at the point c and thus the stress can

go up the critical state line. After going up the critical state line, ( )/t r ρ γ−− N

tends to the negative value but /RU decreases so that the plastic modulus pM

decreases to zero exhibiting the peak of stress when ( )/ (< 0)t r ρ γ−− N and

(>0)/RU cancel mutually resulting in = 0pM . Thereafter, ( )/t r ρ γ−− N

increases gradually and /RU decreases more rapidly resulting in 0pM < so that

the stress decreases toward the critical state c .

p00 0o

y

cc−

y

cc−

Elastic state

Elastoplastic state


p

= max.avD D| |/

−0 +

0

+

−

+ −

oBBpBc 0 (Critical state)

max.

pvD

min.+ −

−

0 (Critical state)B +

⎫⎬⎭

Over-consolidate state

Cap model

1m

Subloading surfaceat initial state

Normal-yield surfaceat initial state

Normal-yield surfaceat final state

0F FcF

Stress path

c

Critica

l stat

e line

Stress-strain curve

（Subloading surfacemodel：　　　　）　　　u → ∞

+

p

q

y'

o'

c'

c'

c'y'

c'

o'

Normally-consolidate state


trN t r( )Nσ

( )t r ( )( ) tr( 0)tr ( )p RUM Rρ γ

− +≡ ≥−N NN σσ t r( )p

p=MN NDσ

pMF•

trN pvD F

•U U pM t r( )Nσ

vε−vε−

aε aε

tr > 0N

0pvD >

0F• <

tr < 0N0p

vD <0F

• >qq

aε aεpM

t r ( )trρ γ

−−N Nσ

( ) ( )trRU

RNσ

pM

t r ( )trρ γ

−−N Nσ

( ) ( )trRU

RNσ

p00 0o

y

cc−c−

y

cc−c−

Elastic state

Elastoplastic state


p

= max.avD D| |/

−0 +

0

+

−

+ −

oBBpBc 0 (Critical state)

max.

pvD

min.

+ −

−

0 (Critical state)B +

⎫⎬⎭

Over-consolidate state

Cap model

1m

Subloading surfaceat initial state

Normal-yield surfaceat initial state


0F FcF

Stress path

c

Critica

l stat

e line

Stress-strain curve

（Subloading surfacemodel：　　　　）　　　u → ∞

+

p

q

y'

o'

c'

c'

c'y'

c'

o'

Normally-consolidate state


trN t r( )Nσ

( )t r ( )( ) tr( 0)tr ( )p RUM Rρ γ

− +≡ ≥−N NN σσ t r( )p

p=MN NDσ

pMF•

trN pvD F

•U U pM t r( )Nσ

vε−vε−

aε aε

tr > 0N

0pvD >

0F• <

tr < 0N0p

vD <0F

• >qq

aε aεpM

t r ( )trρ γ

−−N Nσ

( ) ( )trRU

RNσ

pM

t r ( )trρ γ

−−N Nσ

( ) ( )trRU

RNσ

Fig. 11.17 Comparison of predictions of triaxial compression behavior under constant lateral stress by the Cap model and the subloading surface model

The sign of the plastic volumetric strain rate is identical to that of t rN , whilst

the elastic volumetric strain is induced with the variation of pressure. The volu-

metric contraction is induced in the initial stage of loading. After going up the

critical state line, the plastic volumetric expansion larger than the elastic volumetric


contraction develops so that the volumetric expansion proceeds. However, reaching

the peak stress t r( ) = == 0p evDMNσ at which the subloading surface expands

most and thus t rN becomes maximum, the peak of stress and the maximum ratio

of volumetric expansion strain rate vs. axial strain rate are induced simultaneously

as has been found experimentally by Taylor (1948). Eventually, the fundamental

deformation behavior in not only normally-consolidated but also over-consolidated

states could be described concisely by the initial subloading surface model. The Cap model possesses various drawbacks compared with the subloading

surface model. In other words, the subloading surface model is fundamentally modified from the Cap model as described in the following.

1) The Cap model falls within the framework of conventional plasticity with the yield surface enclosing the purely elastic domain. Therefore, the stress-strain curve is predicted to rise up steeply (elastically) to the peak stress, and sub-sequently it decreases suddenly exhibiting a softening. It leads especially to the unrealistic prediction of softening behavior. On the other hand, the subloading surface model can describe the realistic stress-strain curve with the smooth elastic-plastic transition since the plastic strain rate develops gradually as the stress approaches the normal-yield surface, i.e. the normal-yield ratio in-creases.

2) The Cap model necessitates the judgment of whether or not the stress lies on the yield surface, i.e. a yield judgment in addition to the judgment on the di-rection of strain rate in the loading criterion as shown in Eq. (6.68). On the other hand, the yield judgment is not required in the subloading surface model since the stress lies always on the subloading surface playing the role of the loading surface as shown in Eq. (7.27).

3) The Cap model requires the operation pulling the stress back to the yield sur-face, i.e. the return-mapping algorithm since the increments of stress or strain of finite magnitudes are input in the numerical calculation. On the other hand, the subloading surface model does not require it since it possesses an automatic controlling function to attract the stress to the normal-yield surface in the loading process as was described in 7.3.

4) The cap model additionally adopts the plastic potential surface of the conical shape having a dilatancy angle lower than the Drucker-Prager yield surface leading to the nonassociativity since an excessively large plastic volumetric strain is predicted if the associated flow rule is adopted and thus the constitu-tive equation becomes complicated, including additional material parameters. On the other hand, the subloading surface model can use the associated flow rule, whereas the outward-normal N of the subloading surface in the current stress is approximately identical to the outward-normal DPN of the plastic potential surface adopted in the Drucker-Prager model as shown in Fig. 11.18.


σ

N

Drucker-Prager yield surface

q

p0

Normal-yieldsurface

Subloadingsurface

Critica

l state

line

DPN

Plastic potential surface assumedin Drucker-Prager model σ

N

Drucker-Prager yield surface

q

p0

Normal-yieldsurface

Subloadingsurface

Critica

l state

line

DPN

Plastic potential surface assumedin Drucker-Prager model

Fig. 11.18 Outward-normal of subloading surface coinciding approximately with the plastic potential surface assumed in the Drucker-Prager model

5) The cap model adopts the nonassociativity for the Drucker-Prager yield sur-face. Therefore, it is accompanied with the asymmetry of the elastoplastic stiffness modulus tensor peM . This fact engenders difficulty in the analysis of boundary value problems. On the other hand, the subloading surface model adopts the associativity leading to the symmetry of the elastoplastic stiffness modulus tensor.

6) The cap model predicts the failure surface describing the peak stresses deter-mined uniquely by the Drucker-Prager yield surface itself, independent of the loading paths, because the interior of the yield surface is assumed to be a purely elastic domain. However, the surface depicted by connecting the peak stresses depends on the loading paths and exhibits nonlinearity with the increase of pressure in real soils on the contrary to the Coulomb-Mohr failure criterion. On the other hand, the subloading surface model can describe these facts (cf. Ha-shiguchi et al., 2002).

7) The cap model requires the tension cut for the Drucker-Prager yield surface, which runs out sharply into the negative pressure range. The subloading sur-face model does not require the tension cut because it adopts the normal-yield surface passing through the vicinity of the null stress state.

8) The cap model is accompanied with the singularity of the plastic potential in the intersecting lines of the Cam-clay, the Drucker-Prager, and the tension-cut yield surfaces, which would be unrealistic and would induce the difficulty of analysis of boundary-value problems. On the other hand, the subloading sur-face model adopts a single smooth normal-yield surface. Therefore, it does not induce the singularity of the plastic modulus.

9) The cap model predicts the simultaneous occurrence of the peak stress and the maximum volumetric compression in over-consolidated clays and dense sands, in contradiction to experimental fact. On the other hand, the subloading surface model provides the realistic prediction that the peak stress and the maximum ratio of volumetric expansion strain rate vs. axial strain rate occur simultane-ously as described in the foregoing.


10) The cap model requires at least two more material constants describing the inclinations of yield and plastic potential surfaces in addition to the material constants in the Cam-clay model. On the other hand, the subloading surface model requires only material constant u in the evolution rule of the nor-mal-yield ratio despite the distinguishable high ability.

In what follows, some comparisons of the simulations of typical triaxial test data by the Cap model and the subloading surface model are shown (Hashiguchi et al., 2002).

The simulations of the test data measured by Skempton and Brown (1961) for Weald clay subjected to the drained triaxial compression with a constant lateral stress are shown in Fig. 11.19 where the material constants and the initial value are selected as follows:

0

= 0 045, = 0 002, = 0 37, = 1 2,

0 574 = 0 071 for the Drucker-Prager model,=

= 33 0 for the subloading surface model,

= 330 0 kPa,

py

. . . M .

M . , M .

u .

F .

ρ γ ν

whilst the initial stress state is 0 67 0 k Pa= .− Iσ . Here, the function yM and

pM are the inclinations of yield and the plastic potential surfaces, respectively, of

the Drucker-Prager model in the ( , || ||)p 'σ plane. The associated flow rule and

the nonassociated flow rule are abbreviated as AFR and Non-AFR, respectively, in

this figure. On the other hand, Eq. (7.17) is used for the evolution rule of the nor-

mal-yield ratio in the subloading surface model. The similar simulations for the test

data of kaolinite-silt mixtures measured by Stark et al. (1994) are shown in

Fig. 11.20 where the material constants and the initial value are selected as follows:

0

= 0 1, = 0 006, = 0 3, = 1 051,

0 528 = 0 093 for the Drucker-Prager model,=

= 35 0 for the subloading surface model,

= 6,000 0 kPa,

py

. . . M .

M . , M .

u .

F .

ρ γ ν

whilst the initial stress state is 0 1275 0 k Pa= .− Iσ . The subloading surface

model gives rise to the clearly better prediction than the Drucker-Prager model for

both the axial stress-axial strain and the volumetric strain-axial strain curves. The

curves predicted by the Drucker-Prager model are not smooth, which are formed by

the three segments, i.e. the elastic, the elastoplastic and the critical state segments,


whilst the former two form the concave curves of the ‘Eiffel-tower’ shape. Note that

the adoption of the nonassociated flow rule in the Drucker-Prager model does not

lead to the substantial improvement in simulation, whilst the subloading surface

model adopting the associated flow rule gives the realistic prediction even for the

volumetric strain. The parameter u is determined such that the stress-strain curve fit

to the gentleness in the elastic-plastic transition.

－10 －2000

50

150

100


－15

200

0.0

1.0

2.0

－10 －200 －15－5－1.0

1.5

0.5

－0.5

εv

(%)

Drucker-Prager model (AFR)Drucker-Prager model (Non-AFR)

Subloading surface modelDrucker-Prager model (Non-AFR)

Drucker-Prager model (AFR)

－5

(%)aε

(%)aε

: Test data

: Test data

|| ||

(kPa)'σ

－10 －2000

50

150

100


－15

200

0.0

1.0

2.0

－10 －200 －15－5－1.0

1.5

0.5

－0.5

εv

(%)




－5

(%)aε (%)aε

(%)aε (%)aε

: Test data

: Test data

|| ||

(kPa)'σ

Fig. 11.19 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Skempton and Brown, 1961) of Weald clay for the drained triaxial compression with the constant lateral pressure


－10 －2000

1000

3000

2000


－15

4000

0

1

3

－10 －200 －15－5－2

2

－1

εv

(%)




－5

(%)aε

(%)aε－25

－25

: Test data

: Test data

|| ||

(kPa)'σ

－10 －2000

1000

3000

2000


－15

4000

0

1

3

－10 －200 －15－5－2

2

－1

εv

(%)




－5

(%)aε (%)aε

(%)aε (%)aε－25

－25

: Test data

: Test data

|| ||

(kPa)'σ

Fig. 11.20 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Stark et al., 1994) of kaolinite-silt mixtures for the drained triaxial compression with the constant lateral pressure

The simulations of the stress paths and the stress-strain curves to the test data measured by Bishop et al. (1965) for London clay subjected to the undrained tri-axial compression are shown in Fig. 11.21 where the material constants and the initial value are selected as follows:


00

200

200

400

400

600

600

800

800

1000

1000

p (kPa)

Critical state lin

e

104553


Drucker-Prager model

p0 (kPa)

1200

1200

00

200

400

600

1000

800

−1 −2 −3 −4 −5



1200

in Drucker-Prager modelYield surface

104553

Test data

ν =0.30ν =0.45

Test data p0 (kPa)

(%)aε

|| ||

(kPa)'σ

|| ||

(kPa)'σ

00

200

200

400

400

600

600

800

800

1000

1000

p (kPa)

Critical state lin

e

104553



p0 (kPa)

1200

1200

00

200

400

600

1000

800

−1 −2 −3 −4 −5



1200

in Drucker-Prager modelYield surface

104553

Test data

ν =0.30ν =0.45

Test data p0 (kPa)

(%)aε (%)aε

|| ||

(kPa)'σ

|| ||

(kPa)'σ

Fig. 11.21 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Bishop et al., 1965) for the undrained triaxial com-pression with the constant lateral pressure

0

= 0 022, = 0 0063, = 0 82,

= 0 3 and = 0 45, = 0 62, = 0.21 for the Drucker-Prager model,

= 0 3, = 70 0 for the subloading surface model,

= 1,700 0 kPa.

y p

. . M .

. . M . M

. u .

F .

ρ γν ν

ν


|| ||

(kPa)'σ

|| ||

(kPa)'σ

00

200

100

200

300

300

p (kPa)

Critical st

ate line

50100



p0 (kPa)

400

400

00

100

300

200

−2 −4 −6 −8 −10


Drucker-Prager model400

Drucker-Prager modelYield surface of

50100

Test data

ν =0.30ν =0.43

Test data

p0 (kPa)

(%)aε

100

250

250

|| ||

(kPa)'σ

|| ||

(kPa)'σ

00

200

100

200

300

300

p (kPa)

Critical st

ate line

50100



p0 (kPa)

400

400

00

100

300

200

−2 −4 −6 −8 −10


Drucker-Prager model400

Drucker-Prager modelYield surface of

50100

Test data

ν =0.30ν =0.43

Test data

p0 (kPa)

(%)aε (%)aε

100

250

250

Fig. 11.22 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Wesley, 1990) for the undrained triaxial compression with the constant lateral pressure

The similar simulations for the test data of red clay measured by Wesley (1990) are shown in Fig. 11.22 where the material constants and the initial value are selected as follows:

11.10 Super-Yield Surface for Structured Soils in Natural Deposits 291

0

= 0 035, = 0 012, = 1 015,

= 0 3 and = 0 43, = 0 767, = 0.24 for the Drucker-Prager model,

= 0 3, = 20 0 for the subloading surface model,

= 300 0 kPa.

y p

. . M .

. . M . M

. u .

F .

ρ γν ν

ν

The test data are predicted fairly well by the subloading surface model. On the other hand, both the stress paths and the stress-strain curves predicted by the Drucker-Prager model are quite different from the test data, which are not smooth being formed by the three segments, where the Poisson’s ratio is selected two levels of = 0 30.ν and 0.45 in Fig. 11.21 and = 0 30.ν and 0.43 in Fig. 11.22.

The simple subloading surface model has been widely applied to the analyses of soil deformation behavior (e.g. Topolnicki, 1990; Kohgo et al., 1993; Asaoka et al, 1997; Noda et al., 2000; Hashiguchi et al., 2002; Nakai and Hinokio, 2004; Wongsaroj et al., 2007).

11.10 Super-Yield Surface for Structured Soils in Natural Deposits

Constitutive equations formulated to date are relevant to remolded soils. Asaoka et al. (2000, 2002) noticed that naturally deposited soils have structures and they have a strength greater than that of the remolded soils for same void ratio. To take ac-count of this fact, they incorporated the novel surface, called the super-yield sur-face, which has a similar shape and orientation to the normal-yield surface but is larger than the normal-yield surface if soils have the structure. Constitutive equa-tions based on this concept will be explained below in the revised forms.

First, let the following super-yield surface be assumed, while it is postulated that the isotropic hardening is induced only by the plastic volumetric strain, independent of the deviatoric deformation.

( )= ( )ˆf RF Hσ (11.91)

where ˆ ( 1)R ≥ is the ratio of the size of the super-yield surface to that of the

normal-yield surface and is called the structure ratio. Further, they incorporated the

super-yield ratio (0 1)R R≤ ≤ describing the ratio of the size of the subloading

surface to that of the super-yield surface, whilst the subloading surface can become larger than the normal-yield surface in structured soils. The subloading surface is described as

( ) ( ( ))= ( )= ˆf RF H RRF Hσ (11.92)

noting the normal-yield ratio ˆ(0 )R R R≤ ≤ is described by the product of

structure ratio R and super-yield ratio R as

= ˆR RR (11.93)


q

p0 RF

σ

Super-yield surface Subloading surfaceNormal-yield surface

RFF

( )= ( )ˆf RF Hσ( )= ( )f F Hσ ( )= ( )f RF Hσq

p0 RF

σ

Super-yield surface Subloading surfaceNormal-yield surface

RFF

( )= ( )ˆf RF Hσ( )= ( )f F Hσ ( )= ( )f RF Hσ

Fig. 11.23 Super-yield, normal-yield and subloading surfaces

The super-yield, the normal-yield and the subloading surfaces are shown in Fig. 11.23.

The time-differentiation of Eq. (11.92) leads to

( )t r =)( f ˆRRˆ ˆRR F R F F HR••∂

∂+ + ′σ σσ

i (11.94)

Now, assume for the upgradation ˆ( 0)R•> and the degradation ˆ( 0)R

•< of

structure as follows:

1) The upgradation ˆ( 0)R•> or degradation ˆ( 0)R

•< occurs only when the plas-

tic strain rate is induced ( )p ≠D 0 .

2) The degradation of structure ˆ( 0)R•< proceeds when the deviatoric strain rate

is induced ( )p' ≠D 0 .

3) The degradation of structure ˆ( 0)R•< proceeds when the plastic volumetric

contraction ( 0)pvD < is induced. The remolded state has the lowest grade of

structure ˆ( = 1)R , and thus the degradation occurs no more ˆ( 1)R ≥ .

4) The upgradation of structure ˆ( 0)R•> proceeds when the plastic volumetric

expansion ( 0)pvD > is induced.

Based on these assumptions, let the following evolution rule of the structure ratio

R be given as follows:


R0

1

p' ≠D 0

R0 1

0pvD <

0 1

0pvD >

dU cvU e

vU

RR0

1

p' ≠D 0

R0 1

0pvD <

0 1

0pvD >

dU cvU e

vU

R

Fig. 11.24 Functions dU dU , cvU cvU and evU evU in the evolution rule of R R

|||| ( ) ( )( )= p pp ecv vv vd ˆ ˆˆ ˆ ˆˆˆ D DR RR U UUR '

•− − +− ⟨ ⟩ ⟨ ⟩D (11.95)

where , , ecv vdˆ ˆ ˆU U U are the functions of R fulfilling the conditions (Fig. 11.24).

1( )

= 1

> 0 for

= 0 for d

Rˆˆ RUR

>⎧⎪⎨⎪⎩

, 1

( )= 1

> 0 for

= 0 for cv

Rˆˆ RUR

>⎧⎪⎨⎪⎩

,

( )> 0 for 1ev ˆ ˆˆ R RU ≥ (11.96)

which are all the monotonically-increasing functions of R . The following simple equations are assumed hereinafter.

ˆ ˆ ˆˆ( ) = ( 1)md dU R u R − , ˆ ˆ ˆˆ( ) = ( 1)c cv vU R u Rζ − , ( ) =e e

v vˆ ˆˆ ˆR RU u ξ (11.97)

where ˆ ˆ ˆ, ; , ; , ( 1)dc ev vu m u uζ ξ ≤ are the material constants.

On the other hand, let the evolution rule of the super-yield ratio R be given by the following equation which is replaced the normal-yield ratio R to the su-

per-yield ratio R in Eq. (7.13).

|| ||= ( ) for p pR U R•

≠D 0D (11.98)

where U is the monotonically increasing function of R fulfilling the conditions (Fig. 11.25):

for = 0,

= 0 for = 1,

( 0 for 1).

R

RU

R

→ ∞

< >

⎧⎪⎨⎪⎩

(11.99)


10 R

|||| ( )=p U RR

•

D

10 R

|||| ( )=p U RR

•

D

Fig. 11.25 Function U in the evolution rule of the super-yield ratio R

The following simple equation is assumed putting = 0eR in Eq. (7.15).

=( ) cot 2( )uRU Rπ (11.100)

where u is the material constant. The substitutions of Eqs. (11.38), (11.40), (11.95) and (11.98) into Eq. (11.94)

yields

( )t r =)( pv

f ˆ DRRF'∂∂

−σ σσ

|| |||| ( )|| ( ) ( )( ) pp pp ecv vv vd ˆ ˆˆ ˆ ˆˆ D D U RR RR U UU RFFR ' −+ ⟨ ⟩ − ⟨ ⟩ +− DD

(11.101) which is rewritten as

( )t r ( ) t r =pv

ˆRR F ˆ DRRF'−N Nσ σ

|| ||( )|||| ( ) ( )( ) pp pp ecv vv vd ˆ ˆˆ ˆ ˆˆ U RD DR RR U UU RFFR ' −⟨ ⟩ − ⟨ ⟩+ +− DD

(11.102)

noting the following relation due to Euler’s theorem of homogeneous function in degree-one.

( )t r( )

= =t r( ) t r( )

( )fˆf RRF

∂∂ ∂

∂ N N

σ σσ N Nσ σσσ

(11.103)


Adopting the associated flow rule (7.19) in Eq. (11.102), one has

( () )= =, tr trp

p pM Mλ D NN Nσ σ

(11.104)

where

t r t r( )( )p ˆF U UM ˆF RR'− + −≡ N σN (11.105)

setting

|||| ( ) ( ) t rt r ecv vd ˆ ˆˆ ˆˆ R RU UUU ' + ⟨ ⟩ − ⟨ ⟩−≡ NN N (11.106)

Only the softening and the hardening are predicted over and below, respectively, the critical state line in the Cam-clay model. On the other hand, the subloading

hardening t r( )( 0)>Nσ can be also predicted over the critical state line because

of the inclusion of the function U (U in case of Eq. (11.105)) in the subloading surface model without the deviatoric hardening. Furthermore, both the subloading

hardening t r( )( 0)>Nσ and the subloading softening t r( )( 0)<Nσ can be

predicted both over and below the critical state line because of the inclusion of the

functions , , c ev vdˆ ˆ ˆU U U in addition to U in the subloading super-yield surface

model (see Fig. 11.26). These facts are shown in Table 11.1. The undrained stress path in sands depicted in Fig. 11.6 and also the cyclic mobility, i.e. the

:0 0pvD F

•< >

q

p0

t r( ) 0>Nσ

t r( ) 0<Nσ

ˆStructure ratio : high,

Super - yield ratio : high

R

R

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

0UURR

<−

Loose sand

Critical st

ate line

:0 0pvD F

•> <

q

p0

t r( ) 0>Nσ

ˆStructure ratio : low,

Super - yield ratio : low

R

R

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

0UURR

− >

Dense sand

Critical st

ate line

:0 0pvD F

•> <

:0 0pvD F

•< > :0 0p

vD F•

< >

q

p0

t r( ) 0>Nσ

t r( ) 0<Nσ

ˆStructure ratio : high,

Super - yield ratio : high

R

R

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

0UURR

<−

Loose sand

Critical st

ate line

:0 0pvD F

•> <

q

p0

t r( ) 0>Nσ



R

R

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

0UURR

− >

Dense sand

Critical st

ate line

:0 0pvD F

•> <

:0 0pvD F

•< >

q

p0

t r( ) 0>Nσ



R

R

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

0UURR

− >

Dense sand

Critical st

ate line

:0 0pvD F

•> <

:0 0pvD F

•< >

Fig. 11.26 Subloading softening for a plastic volumetric expansion and subloading hardening for a plastic volumetric contraction


Table 11.1 Signs of mechanical quantities above and below the critical state line in soil models

Cam-clay Model Subloading sur-

face model Subloading superyield

surface model

tr( ),pM N t r( ),pM N t r( ),pM N

Above C.S. line

, , 0tr 0 tr 0pF•> > <N D

0< 0 or =0 or >0< 0 or =0 or >0<

C.S. line

,= , = 0tr 0 tr =0pF•N D

= 0 =0 or >0 0 or =0 or >0<

Below C.S. line

, , 0tr 0 tr 0pF•< < >N D

0> 0> 0 or =0 or >0<

butterfly-shape stress path after the decrease of pressure for the cyclic loading with the constant deviatoric stress amplitude under the undrained condition of sands could be described rigorously (Noda et al., 2007).

Based on the concept of the extended subloading surface model described in Chapter 8, the above-mentioned subloading super-yield surface model is extended so as to describe the cyclic loading behavior as follows:

Now, assume that the similarity-center s of the subloading and the super-yield surfaces moves with the plastic deformation (see Fig. 11.27). Then, the subloading surface is described by

( , ) ˆ= ( ) (= ( ))f RRF H RF Hβσ (11.107)

where ≡ −σ σ α , = (1 )R− sα (11.108)


( , ) ( , ) ( , )tr t r tr( ) ( ) ( )f f f∂ ∂ ∂− +∂ ∂ ∂

β β β ββ

σ σ σσ ασ σ

ˆˆ ˆ= RRRR F R F F HR• • •

+ + ′ (11.109)

where it holds from Eq. (11.108) that

(1 )= R R•

−−α s s (11.110)


αs

σ

σ

p0

q


Super-yield surface

Subloadingsurface

αs

σ

σσ

p0

q


Super-yield surface

Subloadingsurface

Fig. 11.27 Normal-yield, subloading and super-yield surfaces

The similarity-center has to lie inside the super-yield surface in order that the subloading and the super-yield surfaces do not intersect mutually. Then, it has to be fulfilled that

ˆ( , ) ( )f RF H≤βS (11.111)

The time-differentiation of Eq. (11.111) at the limit state that s lies just on su-per-yield surface instead of the normal-yield surface (see Fig. 8.6) leads to

( , ) ( , ) ˆˆˆ for ( , ) = ( )0t r t r ( ) ( )f f f RF HF FRR• •∂ ∂ − −+

∂ ∂≤β β βββ

ss s ss

(11.112)

Noting the relation

( , ) ˆt r = fRF

∂∂

βs ss , Eq. (11.112) becomes

( , ) ( , )( , ) 1 ˆˆt r 0t rt r ˆ( ) ( ) f ff F FRR

RF

• •∂ ∂∂ − −+ ∂ ∂∂≤β ββ ββ

s ss ss ss

ˆfor ( , ) = ( )f RF Hβs

i.e.

ˆ( , ) ( , )1t r 0t r ˆˆ( ) ])[( f f FR

FRRF

• •∂ ∂ −+ −∂ ∂

≤β H ββs s sss

ˆfor ( , ) = ( )f RF Hβs (11.113)


Then, assume the following evolution rule of the similarity-center so as to fulfill the inequality (11.113).

ˆ ( , )1= t rˆ ˆ ( )p

R

fF RF R RF

c|| ||•• ∂+ + −

∂β

D ββsσ ss (11.114)

Substituting Eq. (11.114) into Eq. (11.110), it is obtained that

ˆ ( , )1(1 )= t rˆ ˆ][ ( )p

R

fF R RRF R RF

c|| ||••• ∂ −+− + −

∂β

D ββsσ s sα

(11.115)

Further, the substitution of Eq. (11.115) into Eq. (11.109) leads to

( , )t r( )f∂

∂βσ σσ

ˆ( , ) ( , )1(1 )tr t rˆ ˆ ( ) ] [( p

R

f fF RRF R RF

c|| ||••∂ ∂+− + −− ∂ ∂

β βD ββ

σ sσ sσ

( , ) ˆ ˆˆtr =( )) fRR RRF R F FR R

• • • •∂− + ++∂

β ββσs

which, noting the relation

( , )tr ˆ( , )( , )

= = =) ) )tr( tr( tr(

( )fff FRR

∂∂ ∂

∂

βββ

N N NN N N

σ σ σσσσ σσ σ

(11.116)

becomes

ˆ)tr(

)tr(FRR N

Nσσ

ˆ ˆ ( , )1(1 )tr t rˆ ˆ)tr( ( ) ] [( p fF FRR RR

FR R RFc|| ||

•• ∂+− + −−∂

HD βN βN

sσ sσ

( , ) ˆ ˆˆtr =( )) fR RR RRF R F FR•• • •∂− + + +

∂β ββ

σs


i.e.

)tr(Nσ

ˆ ( , )1(1 )tr t rˆ ˆ ( ) ] [[ p

R

fF RRF R RF

c|| ||•• ∂+− + −−

∂H

D βN βsσ s

ˆ( , )1tr tr = 0ˆ ˆ( ) ]] [ f R R FR

FFRR R R

••• •

∂− + − − −∂

β βN βσ σs

i.e.

ˆ 1 1 (1 ) 1) trtr( ˆ()(( ) )[( pF R R RF R RR

c || ||•• •

+− − −+ ++− DNN σs sσσσ

( , ) (1 ) ( , )1 tr 0t r =ˆ ˆ( ) ( ) ])f fR

FRR FR

∂ − ∂−−∂∂

β Hβ ββ βsσ σ s (11.117)

Noting the relation

( )(1 ) = =1 1 1

( )= =

R

RR R R

⎫+ − − + − − ⎪⎬− −+ ⎪⎭

σ α s s α σsσsσ s s σσ

(11.118)

Eq. (11.117) becomes

||||ˆ 1 1) trtr( ˆ

()( )[( pRF RF RRR

c•••

+ −+ +− DNN σσ σσ

( )( , ) ( , )11 tr t r 0=ˆ ˆ( ) ( ) ])f fR

FRR FR

∂ ∂−− −∂∂

β Hβ ββ βsσ σ s

Substituting Eq. (11.38) with = 0dμ , (11.95) and (11.98) into this equation, we have the consistency condition

1 ˆ ˆˆ |||| ) trtr( ˆ

[( v vvp pp p ec

v vd D DD U UUFF R

' ⟨′ −+ −− ⟨ ⟩ ⟩−− DNN σσ

||||1 1( ) pURR

c+ −+ D σ


||||( , )1 ( )trˆ

)( ipf H

FRR'

∂− ,∂

βDβ bσ σσ

||||( ) ( , )1 ( )t r 0=ˆ

( ) ])ipR f H

FR'

− ∂− ,∂

H Dβ bs σ s

The substitution of the associated flow rule (8.25) to this equation leads to

tr) trtr( [( FFλ ′−− NNNσ

ˆ ˆ ˆ||||1 1( ) 1tr trˆ ( )ec

v vd UU U UR RR

c' ⟨− +− + ⟨ ⟩ ⟩ −− +N NN σ σ

||||( , )1 ( )tr

ˆ)( i

f HFRR

'∂ ,−

∂β

Nβ bσ σσ

||||(1 ) ( , ) ( )t r 0=ˆ

( ) ])ifR H

FR'− ∂− ,

∂H

Nβ bs σ s

from which the plastic modulus in Eq. (8.27) is derived as follows:

1 ˆ ˆ |||| tr trtr ˆ[(p cvdU U

FM F R

'′ − +≡ ⟨ ⟩− −N NNN

ˆ 1 1tr ( )ev

UURR

c⟨− −⟩ ++N σ σ

||||ˆ

( , ) ( )tr )( iRR

f HF' ∂ ,−

∂βN

β bσ σσ

)(

( , ) ( )t r1 ( ) ])iR Rf H∂ ,−+ ∂

Hβ bs σ s (11.119)

In case of =R R , ( ) = ( )U R U R for soils without a structure, i.e. ˆ = 1R

leading to ˆ ˆ ˆ = 0= = ecv vdU U U , the plastic modulus (11.119) reduces to

Eq. (8.27). The structure ratio R is calculated by the numerical integration of Eq. (11.95)

and ˆRR can be calculated by the method described in 11.7 with the replacement of

R to ˆRR . Then, the super-yield ratio R is calculated from them.

11.11 Numerical Analysis of Footing Settlement Problem 301

11.11 Numerical Analysis of Footing Settlement Problem

Numerical analysis of footing settlement problem will be shown in this section (Mase and Hashiguchi, 2009). The prediction of peak load and post-peak behavior for the footing-settlement problem on sands having the high friction and dilatancy cannot be attained in fact by the usual implicit finite element method requiring the repeated calculations of total stiffness equation which needs quite large calculation time. On the other hand, it can be attained by the explicit dynamic relaxation method in which the dynamic equilibrium equation is solved directly without solving the total stiffness equation so that the calculation time is drastically reduced. The FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions; Cundall, P. and Board M., 1988; Itasca Consulting Group, 2006) based on the explicit dynamic relaxation method is adopted in the present analysis, in which the initial subloading surface model of soils with the automatic controlling function to attract the stress to the normal-yield surface is implemented as the constitutive equation. The calcula-tion is executed by the Euler method without a calculation for convergence in this program by adopting small incremental steps so as not to influence on the calcula-tion, while this fact is examined prior to the calculation. The finite elements are composed of eight-noded cuboidal elements. Each cuboidal element is divided into the two kinds of overlays, i.e. assembly of five tetrahedral sub-elements having different directions. Then, the deviatoric variables are analyzed using individual values in each tetrahedral sub-element. On the other hand, isotropic variables are analyzed using averaging values in ten tetrahedral sub-elements in order to avoid the over-constraint problems common in finite element calculations for dilatant materials, i.e. the dilatancy locking.

Test data used for numerical simulation The test data of footing settlement phenomenon on sand layers under the plane strain condition are used for the present analysis. The sizes of the test apparatus of type A (Tatsuoka et al., 1984) and type B (Tani, 1986) have the same height 49cm and depth 40cm and the different widths 122cm and 183cm, respectively. The size of type C (Okahara et al., 1989) has the height 400cm and depth 350cm and the widths 7002cm. The footings width, denoted as B, is taken 10 cm for the types A and B and 50cm for the type C. The sand layers has been prepared carefully by the air-pluviation method for the dried Toyoura sand in order to obtain the same ho-mogeneous layers but the test data exhibit dispersion more or less test by test despite of the laborious preparation work.

Numerical analysis and comparison with test data

The finite element meshes in the present analyses for the simulations of the test data are shown in Fig. 11.28. The nodal points of soil layer contacting with the footing


and the bottom of soil bin are fixed to them, respectively. On the other hand, the nodal points at the side walls can move freely in the vertical direction. The right half of soil layer is analyzed in order to reduce the calculation time as has been done widely even for searching the localized deformation (cf. e.g. Sloan and Randolph; 1982; Pietruszczak and Niu; 1993; Stallebrass et al., 1997; Borja and Tamagnini, 1998; Siddiquee et al., 1999; de Borst and Groen, 1999; Sheng et al., 2000; Borja et al., 2003). First, the analysis of deformation caused by the gravity force was per-formed. Then, the vertical displacement of footing is given by incremental steps of

5 410 ~ 5 10 cm− −× . The material parameters in the subloading surface model are selected as

0 = 350 kPa, = 30 , = 0.001cF φ ξ ,

= 0 0015, = 0 00015. .ρ γ ,

= 0.01kPa, = 0 3ep .ν ,

= 29 , = 0 .2d dμφ ,

= 15 0u .

where Eq. (7.17) is used for the evolution rule of the normal-yield ratio in the subloading surface model. The values of material parameters listed above are used for all the following numerical calculations because Toyoura sands having the same initial void ratio 0.66 are used in these tests.

The comparisons of test and calculated results are shown in Fig. 11.29, where the prediction by Siddiquee et al. (1999) is also depicted in (c). In this figure mq is the average footing pressure, dγ is the unit dry weight, Nγ is the normalized footing pressure and S is the settlement. The qualitative trends of test results and the quantitative simulation to some extent are captured and the ultimate loads, i.e. bearing capacities are predicted well by the present analyses, although the analyses are performed for the sand with the high friction and dilatancy. Here, the post-peak behavior, i.e. the increase of load after exhibiting once the minimal value is also predicted well qualitatively. It would be provided by the adoption of the up-dated Lagrangian calculation realizing the accumulation of displacements by updating the positions of nodal points, which results in the upsurge of soils around the footing and thus the increase of footing load. However, the quantitative prediction of post-peak behavior would require the further study taking account of the tan-gential inelastic strain rate due to the stress rate tangential to the loading surface (Hashiguchi and Tsutsumi, 2001) and the gradient effect (cf. Hashiguchi and Tsutsumi, 2006) by introducing the shear-embedded model (cf. Pietruszczak and Mroz, 1981; Tanaka and kawamoto, 1988) for example, which will be described in 13.3.


C.L.

49cm

56cm5cm

(a) Type A (B: 10cm)

(c) Type C (B: 50cm)

(b) Type B (B: 10cm)

C.L.

49cm

86cm5cm40

0cm

325cm25cmC.L.

C.L.

49cm

56cm5cm




C.L.

49cm

86cm5cm40

0cm

325cm25cmC.L.

Fig. 11.28 Finite element meshes


0.00 0.05 0.10 0.150

100

200

300

(a) Type A (B: 10cm) (b) Type B (B: 10cm)


FDM by subloading surface modelTest (Tatsuoka et al., 1984): e=0.66

Test (Tatsuoka et al., 1984): e=0.66

0.00 0.05 0.10 0.150

100

200

300FDM by subloading surface modelTest (Tani, 1986): e=0.669,BC2

Test (Tani, 1986): e=0.669,BC3

0.00 0.05 0.10 0.150

100

200

300FDM by subloading surface model

FEM (Siddiquee et al., 1999)

Test (Okahara et al., 1989): e=0.66


: Average footing pressure: Unit dry weight

: Settlement: Footing width

m

d

q

SB

γ

Relative settlement S B

Relative settlement S B Relative settlement S B

Nor

mal

ized

foo

ting

pre

ssur

e N

γ=2q

m/

γ dΒ

Nor

mal

ized

foo

ting

pre

ssur

e N

γ=2q

m/

γ dΒ

Nor

mal

ized

foo

ting

pre

ssur

e N

γ=2q

m/

γ dΒ

0.00 0.05 0.10 0.150

100

200

300

(a) Type A (B: 10cm) (b) Type B (B: 10cm)


FDM by subloading surface modelTest (Tatsuoka et al., 1984): e=0.66

Test (Tatsuoka et al., 1984): e=0.66

0.00 0.05 0.10 0.150

100

200

300FDM by subloading surface modelTest (Tani, 1986): e=0.669,BC2

Test (Tani, 1986): e=0.669,BC3

0.00 0.05 0.10 0.150

100

200

300FDM by subloading surface model

FEM (Siddiquee et al., 1999)



: Average footing pressure: Unit dry weight

: Settlement: Footing width

m

d

q

SB

γ

Relative settlement S B

Relative settlement S B Relative settlement S B

Nor

mal

ized

foo

ting

pre

ssur

e N

γ=2q

m/

γ dΒ

Nor

mal

ized

foo

ting

pre

ssur

e N

γ=2q

m/

γ dΒ

Nor

mal

ized

foo

ting

pre

ssur

e N

γ=2q

m/

γ dΒ

Fig. 11.29 Comparisons of test and calculated results for footing settlement phenomenon

The displacements of nodal points from the initiation of settlement are shown in Fig. 11.30 at the settlement 11mm，15mm，80mm for Type A, B and C, respec-tively, which are the final stage of calculation. The Prantdl’s slip line solution with the triangle wedge, the logarithmic spiral zone and the passive Rankine zone is observed clearly in this figure.

On the other hand, the soils in the periphery of footing inevitably experience the null or further negative pressure since they are pulled into the vertical direction as the footing settlement proceeds (see Fig. 11.31). It causes the singularity of plastic modulus for the normal-yield surface passing through the origin of stress space at which the normal-yield and the subloading surfaces contact with each other. This defect is improved in the present model by making the normal-yield




(c) Type C ( B: 50cm)

Displacement (cm)

Displacement (cm)

Displacement (cm)

0.0～0.10.1～0.20.2～0.30.3～0.40.4～0.50.5～0.60.6～0.70.7～0.80.8～0.90.9～1.01.0～1.1

0.0～0.20.2～0.40.4～0.60.6～0.80.8～1.01.0～1.21.2～1.41.4～1.5

0.00～1.001.00～2.002.00～3.003.00～4.004.00～5.005.00～6.006.00～7.007.00～8.00

I



(c) Type C ( B: 50cm)

Displacement (cm)

Displacement (cm)

Displacement (cm)

0.0～0.10.1～0.20.2～0.30.3～0.40.4～0.50.5～0.60.6～0.70.7～0.80.8～0.90.9～1.01.0～1.1

0.0～0.20.2～0.40.4～0.60.6～0.80.8～1.01.0～1.21.2～1.41.4～1.5

0.00～1.001.00～2.002.00～3.003.00～4.004.00～5.005.00～6.006.00～7.007.00～8.00

I

Fig. 11.30 Deformed finite element meshes at final step


surface translate to the region of negative pressure as shown in Fig. 11.5, whilst the numerical difficulty can be avoided although the translation was taken quite small as 1/1000 in size of the normal-yield surface. In addition, the impertinence that the volume becomes infinite elastically is avoided by shifting the isotropic consolida-tion characteristic into the negative range of pressure as shown in Fig. 11.1. It should be emphasized that the stable analysis cannot be executed without these improvements.

−0.0001～00 ～1010～2020～3030～4040～5050～6060～

(kPa) p

I t

C.L.

−0.0001～00 ～1010～2020～3030～4040～5050～6060～

(kPa) p

I t

C.L.

Fig. 11.31 Distribution of mean pressure for type A at final step

The pertinent result for the footing-settlement problem on the sand with a high friction, one of the difficult problems in soil mechanics, is obtained in the present study as described above. Here, the peak, the subsequent reduction and the final increase of footing load are predicted well qualitatively and quantitatively to some extent. The reasons for succession are summarized as follows:

1. The subloading surface model applied in the present analysis has the advan-tages: i) It is furnished with the automatic controlling function to attract the stress to the yield surface, whilst all other elastoplastic constitutive models are required to incorporate a return-mapping algorithm to pull back the stress to the yield surface in the plastic deformation process in the normal-yield state. The distinguished advantage enables us to execute an accurate calculation by the program FLAC3D adopting the simple Euler method without a con-vergence calculation process. ii) It is not required the judgment whether or not the yield condition is fulfilled in the loading criterion. iii) It adopts the


associativity of the flow rule and thus it leads to the symmetry of elastoplasic constitutive matrix. Therefore, this model possesses the distinguishable adaptability in numerical calculation.

2. The subloading surface model of soils applied in the present analysis has the advantages: i) It is capable of describing the softening behavior and dilatancy characteristics quite realistically, predicting the simultaneous occurrence of the peak load and the highest dilatancy rate as has been found experimentally by Taylor (1948). ii) It has the full regularity since the normal-yield surface does not pass through the zero stress point and thus the subloading surface is always determined uniquely. In addition, the elastic property is improved such that the elastic bulk modulus does not become zero for the stress inside the normal-yield surface.

3. The finite difference program FLAC3D adopted in the present study is based on the explicit-relaxation method which enables us to shorten the calculation time drastically since it is not required to solve the total stiffness matrix.


Chapter 12 Corotational Rate Tensor

12 Corotational Rate Te nsor

It was studied in Chapter 4 that the material-time derivatives of state variables, e.g. stress and internal variables in elastoplasticity do not possess the objectivity and thus, instead of them, we must use their corotational derivatives. This chapter fo-cuses on the responses of simple constitutive equations introducing corotational rates with various spins including the plastic spin.

12.1 Hypoelasticity

Consider the following hypo-elastic constitutive equation in Eq. (5.22).

= (tr Gλ + 2D)I Dσ (12.1)

where the symbol L is replaced to λ . Equation (12.1) reduces to the following

equation noting that 12 21 2112= , =σ σ ω ω− and using Eqs. (2.144) and (4.33)

for the simple shear described in 2.4.2.

11 1212 12 11 22 12

22 12 12

( )2 0 1=

1 0Sym. 2G

σ σ σ σ σω ωγ

σ σ ω

• ••

•

⎡ ⎤+− − ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦+⎢ ⎥⎣ ⎦

(12.2)

12.1.1 Jaumann Rate

When the Jaumann rate is adopted for the corotational rate, Eq. (12.2) leads to the following equation by setting = Wω with Eq. (2.144).

11 1212 11 22

22 12

0 1 ( )2 =

1 0 sym.

Gγσ γ σ σ σ σ

γσ γ σ

••• •

•

••

⎡ ⎤⎢ ⎥− + − ⎡ ⎤⎢ ⎥ ⎢ ⎥

⎣ ⎦⎢ ⎥+⎢ ⎥⎣ ⎦

(12.3)

310 12 Corotational Rate Tensor

from which we have

11 12

12 11 22

22 12

=0

( =)2

= 0

G

σ γ σ

γσ σ σ γ

σ γ σ

••

•••

••

⎫− ⎪⎪⎪+ − ⎬⎪⎪+ ⎪⎭

(12.4)

Substituting

1122 =σ σ− , 11

12= σγ σ

•• (12.5)

obtained from the first and the third equations into the second equation in Eq. (12.4), yields

111112 1112 12

= Gσ σσσ σ σ• •• + (12.6)

the time-integration of which is given as

212 11 11= 2Gσ σ σ− (12.7)

Substituting this equation into the second equation of Eq. (12.4), we have

11

211

=

1 1( )G

G

σγ

σ

•

•

− −

the integration of which is given by

111cos 1 =( )Gσ γ− −

i.e. 11 22= = (1 cos ) Gσ σ γ− − (12.8)


12 = sinGσ γ (12.9)

The continuum spin W designates the instantaneous rate of rotation of the principal directions of strain rate, i.e. the instantaneous rate of rotation of the cross depicted momentarily on the material surface as described in 2.3. Therefore, if it is

12.1 Hypoelasticity 311

0.4

2.0

1.6

0.8

1.0

1.2

2.4

2.8

π/2π γ

12 /Gσ

0.0

0.4−

0.8−

Jaumann rate with W

Green-Naghdi rate with RΩ

0.4

2.0

1.6

0.8

1.0

1.2

2.4

2.8

π/2π γ

12 /Gσ

0.0

0.4−

0.8−

Jaumann rate with W


0.4

2.0

1.6

0.8

1.0

1.2

2.4

2.8

π/2π γ

12 /Gσ

0.0

0.4−

0.8−

Jaumann rate with W


0.4

2.0

1.6

0.8

1.0

1.2

2.4

2.8

π/2π γ

12 /Gσ

0.0

0.4−

0.8−

Jaumann rate with W


Fig. 12.1 Description of simple shear deformation of hypoelastic material by Jaumann rate and Green Naghdi rate (Dienes, 1979)

used in the simple shear deformation with the constant shear strain rate, i.e.

const.=γ• leading to = const.W , the material is regarded to rotate in a con-

stant velocity, while the strain rate D is also kept constant. Then, when the mate-

rial rotates by 180 , the material becomes to be subjected to the shear strain rate in the opposite direction. Eventually, the shear stress oscillates in the sine curve for the shear strain as seen in Fig. 12.1 calculated by Eq. (12.9) (cf. e.g. Dienes, 1979).

12.1.2 Green-Naghdi Rate

When one adopts the Green-Naghdi (Dienes) rate for the corotational rate with the

relative spin = RΩω , i.e. Eq. (4.43), noting

22

0 1 0 1 0 12 2= = = =,1 0 1 0 1 0cos 4

( )TRG γθ θ γθ•• • •⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥+− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦D Ω R R

(12.10)


Eq. (12.2) reads

2

11 1212 11 22

22 12

( )2 0 12=1 0cosSym. 2

Gσ σ σ σ σθ θ

θθσ σ θ

• ••

•

• •

•

⎡ ⎤+− − ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦+⎢ ⎥⎣ ⎦

(12.11)

from which we have

2

11 12

12 11 22

22 12

2 =0

2( ) =cos

2 = 0

G

σ σ θ

σ σ σ θ θθ

σ σ θ

•

• •

•

•

•

•

⎫− ⎪⎪⎪+ − ⎬⎪⎪+ ⎪⎭

(12.12)

It is obtained that

22 11=σ σ• •− ，

11

12

1= 2σθ σ

• • (12.13)

from the first and the third equations, and

212 11

21=2

d dd dσ σθ θ

(12.14)

from the first equation in Eq. (12.12). Substituting Eqs. (12.4) and (12.13) into the second equation in Eq. (12.12), we have the ordinary differential equation

2

211

11244 =

cosGd

dσ σθ θ

+ (12.15)

The roots of the characteristic equation of the second-order homogeneous linear differential equation for Eq. (12.15) are given by

2 4 0 2= = =2m m m i

± −16+ → ±

Thus, the complementary function of Eq. (12.15) is given by the following equation.

2 211 = cos2 sin2 = sin2 tan( / )A B A B A Bσ θ θ θ+ + + (12.16)

where , A B are the integral constants. Further, adding the particular solution for Eq. (12.15) itself, the general solution of Eq. (12.15) is obtained as follows:

12.2 Kinematic Hardening Material 313

2 2 211 = sin(2 tan( / )) 4 (cos2 ln cos sin2 sin )A B A B Gσ θ θ θ θ θ θ+ + + + −

(12.17)

Assuming that the initial stress is zero, Eq. (12.17) becomes

211 = 4 (cos2 ln cos sin2 sin )Gσ θ θ θ θ θ+ − (12.18)

Furthermore, substituting Eq. (12.18) into Eq. (12.14), we have

1112

1= =2 cos2 (2 2 tan ln cos tan )2

d Gdσσ θ θ θ θ θθ

− − (12.19)

These equations have been derived by Dienes (1979). The relative spin RΩ designates the mean rate of rotation of the cross depicted

on the material surface at the beginning of deformation. Therefore, it coincides with

the continuum spin W at the initial state but it decreases gradually with the shear

deformation. Then, the oscillation of shear stress observed in Jaumann rate is

not predicted if the Green-Naghdi rate is adopted as the corotational rate as seen in

Fig. 12.1 calculated by Eq. (12.19).

12.2 Kinematic Hardening Material

For the sake of simplicity, consider the response of a rigid plastic material fulfilling

= pD D and assume the linear kinematic hardening in Eq. (10.4), i.e.

2 2ˆ || ||= =3 3p p

a ah hDN Dα (12.20)

while it is set that = (2 / 3)p aa h in accordance with Dafailas (1985). Then, it holds for the simple shear that

11 12

2221

0 11= 3 1 0

ahα α

γα α

•⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(12.21)

by substituting Eq. (2.144) into Eq. (12.20). Further, substituting Eqs. (2.144), (2.164) and (4.33) into Eq. (12.21), we have


1211 12 12 11 22 12

22 12 12

( )2 0 11= 3 1 0Sym. 2ah

α α α α αω ωγ

α ωα

• ••

•

⎡ ⎤+− − ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦+⎢ ⎥⎣ ⎦

(12.22)

where

0 1= ( )

1 0z γ γ•⎡ ⎤

⎢ ⎥−⎣ ⎦ω (12.23)

2

1/2 for =( ) =

2/(4 ) for = pz γ γ

⎧⎨

+⎩

ω W

ω Ω (12.24)


1211 12 11 22

22 12

( )2 ( ) ( ) 0 11= 3 1 0Sym. 2 ( )a

z zh

z

α α α α αγ γ γ γγ

αα γ γ

• •• ••

• •

⎡ ⎤+− − ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ −⎣ ⎦+⎢ ⎥⎣ ⎦

(12.25)

from which we have

11 12

12 11 22

22 12

2 ( ) =0

1( ) ( ) = 3

( )2 = 0

a

z

hz

z

α α γ γ

α α α γγ γ

α α γ γ

• •

• ••

• •

⎫− ⎪⎪+ − ⎬⎪⎪+ ⎭

(12.26)

In addition, noting 11 22=α α− , we have

11 22 12

1112

( )= =2

1( )2 = 3 a

z

hz

' '

'

αα α γ

αα γ

⎫−⎪⎬

+ ⎪⎭

(12.27)

where ( )( ) = γd /d' . Differentiating Eq. (12.27), we have

11 1212

11 1112

2z 2 = 0

2 2 = 0

z

zz

'' ' '

'' ' '

αα α

αα α

⎫− − ⎪⎬

+ ⎪+ ⎭ (12.28)

12.2 Kinematic Hardening Material 315

which noting Eq. (12.27), becomes

1111 11

12 12 12

12z 2 2 =03 2

114 2 = 032

( )

( )

a

a

h z zz

hz z zz

''' '

'' ' '

ααα

αα α

⎫− ⎪− − ⎪

⎬⎪+ + − ⎪⎭

Then, it is obtained that

211 11 11

212 12 12

24 z = 03

14 = 03

a

a

z z hz

z zhzz z

''' '

' ''' '

αα α

αα α

⎫⎪− + −⎪⎬⎪+ +− ⎪⎭

(12.29)

12.2.1 Jaumann Rate

The substitution of Eq. (12.24) 1 into Eq. (12.29) leads to

11 11

1212

1 = 02

= 0

pa''

''

αα

αα

⎫−+ ⎪⎬⎪+ ⎭

(12.30)

from which, noting the initial condition 1211 = 0, = 0α α for = 0γ , we have

11 22

12

1= = cos )2

1= sin2

p

p

a

a

α α γ

α γ

⎫− (1− ⎪⎪⎬⎪⎪⎭

(12.31)

It is obtained from Eq. (12.31) that

11 11 22 22

12 12

1= = = = cos )211 1= = sin23 3

p

p

a

aF F

σ α ασ γ

σ γα

⎫− − (1− ⎪⎪⎬

+ + ⎪⎪⎭

(12.32)

Both 11σ and 12σ oscillates in sine curves as shown in Fig. 12.2.


0 2 4 6 8 10

3

2

1

0

4

3

2

1

00 2 4 6 8 10

0 1 2 3

3

2

1

0

−1

γ

γ γ

12α

12σ

11σ

( a ) ( b )

( c )

Green-Naghdi ratewith RΩ

Jaumann ratewith W

0 2 4 6 8 10

3

2

1

0

4

3

2

1

00 2 4 6 8 10

0 1 2 3

3

2

1

0

−1

γ

γ γ

12α

12σ

11σ

( a ) ( b )

( c )

Green-Naghdi ratewith RΩ

Jaumann ratewith W

Fig. 12.2 Description of simple shear deformation of kinematic hardening material by corotational rate with Jaumann spin and Green-Naghdi spin (Dafalias, 1985)

12.2.2 Green-Naghdi Rate

Substituting Eq. (12.24) 2 and

2 2

4= )(4

z'γ

γ−+

(12.33)

into Eq. (12.29), we have

2 2

11 11 112 2 2

2

2 2 2 2

12 12 122 2

2 2

4)(4 4 24 = 0)(2 4 4

4

4 4) )( (14 444 = 0)( 22 24

4 4

p

p

a

a

'' '

'' '

γγ αα α γ γγ

γ γγ γαα α γγ γ

⎫−⎪+− + − ⎪+ + ⎪+ ⎪⎪⎬⎪− −⎪+ ++ +− ⎪+⎪+ + ⎪⎭

12.3 Plastic Spin 317

that is to say, we obtain

11 11 1122 22

12 12 1222 22

12 16 4 0=24 4)(4

1 22 16 0=24 4)(4

p

p a

a

a h

'' '

'' '

γ αα αγ γγγγ αα αγ γγ

⎫+ + − ⎪+ ++ ⎪⎬⎪+ + −+ + ⎪+ ⎭

(12.34)

the general solution of which is derived as the following equation by the method of variable coefficients (Dafalias, 1983).

3

1 211 22

1212 2

2

211 4 4 tan 4( 4) ln= 242 4

211 4( 4)4 tan 4 1 4ln= 242 4

[ ( ) ]

[ ( ) ( )]

p

p

a

a

γ γα γ γγ γγγ γα γγ γ

−

−

⎫− − − ⎪+ + ⎪⎬⎪+− − −+ ⎪+ ⎭

(12.35)

The relation of 11 12, σ σ to 11 12, α α is given by Eq. (12.32) also in this case. A oscillation is not predicted in the simple shear deformation as shown in Fig. 12.2.

12.3 Plastic Spin

The above-mentioned Jaumann rate and the Green-Naghdi rate do not depend on the substructure of material, rather are uniquely determined only by the external appearance of the material. However, the mechanically meaningful rotation would be the spin of substructure, as known presuming the crystals of metals or the annual ring of woods, which would be the rotation of the principal direction of anisotropy (Kroner, 1960; Kratochvil, 1971). The concept of the plastic spin is proposed in order to incorporate such rotation into elastoplastic constitutive equations (Dafalias, 1983, 1985; Loret, 1983).

In what follows, in order to interpret the mechanical meaning of the plastic spin, assume the rigid plasticity and the simplest anisotropy, i.e. the traverse anisotropic material (Fig. 12.3) with the parallel line-elements of substructure having the di-rection 1e inclined /4π from the fixed base 1e in the initial state of deformation and rotates the angle ϕ in the clockwise direction with the increase of shear strain (Dafalias, 1984). Here, Eq. (6.27) is postulated for the multiplicative decomposi-tion. Then, it holds that

1 1

cos( / 4 ) sin( / 4 )= , =

sin( / 4 ) cos( / 4 )ˆ ˆ

π ϕ π ϕϕ

π ϕ π ϕ• •− −⎧ ⎫ ⎧ ⎫

⎨ ⎬ ⎨ ⎬− − −⎩ ⎭ ⎩ ⎭e e (12.36)


while =eV I (rigid plasticity) leading to =eW 0 and =p pW W . Here, refer-ring Fig. 12.1, it holds that

1tan( / 4 ) =1

π ϕ γ− + (12.37)

from which one has

222 = = tan ( / 4 )( / 4 ) )cos (1

ϕ γ γ π ϕπ ϕ γ• •

•− − − −− + (12.38)

Then it holds that

2 ]= sin ( / 4 ) = [1 cos2( / 4 ) = (1 sin 2 )2 2( )γ γγϕ π ϕ π ϕ ϕ• ••• − − − −

(12.39)

Using Eq. (12.39) along with Eq. (12.36), it is obtained that

1

sin( / 4 )= (1 sin 2 ) 2cos( / 4 )

ˆπ ϕ γϕπ ϕ

•• −⎧ ⎫−⎨ ⎬− −⎩ ⎭

e (12.40)

The substructure spin is given by the continuum spin W in the initial state of deformation but it becomes smaller than W with shear deformation. Then, let the decrease of spin from the continuum spin be called the plastic spin, and let it be

denoted by ˆ pW . Referring to the strain rate circle in Fig. 12.1 based on Fig. 2.5 in

2.3, ˆ pW is given as

0 sin 21ˆ 2 sin 2 0p ϕ

γϕ

•⎡ ⎤≡ ⎢ ⎥−⎣ ⎦

W (12.41)

Here, it holds from Eqs. (2.144) and (12.41) that

12 12

12 12

ˆ 0 cos( / 4 )

ˆ sin( / 4 )( ) 0

0 ( /2)(1 sin 2 ) cos( / 4 ) =

s ( /2)(1 sin 2 ) 0

p

p

W W

W W

π ϕπ ϕ

γ ϕ π ϕ

γ ϕ

•

•

⎡ ⎤− −⎧ ⎫⎢ ⎥ ⎨ ⎬−− − ⎩ ⎭⎢ ⎥⎣ ⎦

⎡ ⎤− −⎢ ⎥⎢ ⎥− −⎣ ⎦

in( / 4 )

sin( / 4 ) = (1 sin 2 ) 2cos( / 4 )

π ϕ

π ϕ γϕπ ϕ

•

⎧ ⎫⎨ ⎬−⎩ ⎭

−⎧ ⎫ −⎨ ⎬− −⎩ ⎭


ϕ

a aγ

1e2e

1e

2e

a

1tan γ−

2ϕ

1212( )= W Dω +

nD0

12W

/2γ•

/4π

12ˆ pW

ϕ

a aγ

1e2e

1e

2e

a

1tan γ−

2ϕ

1212( )= W Dω +

nD0

12W

/2γ•

/4π

12ˆ pW

Fig. 12.3 Substructure spin

Noting Eqs. (12.36), (12.40) and (12.41) in this equation, one obtains the fol-lowing expression for the rotation of substructure.

1 1=ˆ ˆ•

ωe e (12.42)

where ˆ p−≡ W Wω (12.43)

Next, consider the same problem by the deformation of crystals of metals (Kuroda, 1996). If the substructure does not rotate, it holds for the slip system in Fig. 12.4 that

= ( ) = ( )γ γ• •• ⊗v x n s s x n , = ( )i r r iv x n sγ• (12.44)


= =p γ•∂ ⊗∂v s nL x

, = = =p ijrij r i i j

j

vL n s s n

xγ δ γ• •∂

∂ (12.45)

12= )p γ• +⊗ ⊗(s n n sD , 1 1

2 2= = ( )( )ji

ij i j i jj i

p vvD s n n s

x xγ•∂∂ + +

∂ ∂ (12.46)

12= )p γ• ⊗ − ⊗(s n n sW , 1 1

2 2= = ( )( )ji

ij i j i jj i

p vvs n n sW x x

γ•∂∂ − −∂ ∂

(12.47)

Eqs. (12.44)-(12.47) are extended for multi slip systems as follows:

=1=

Np α αα

α

γ• ⊗∑ ( ) )(( )s nL (12.48)

=1 =1

1 )( == 2

N Np α αα αα α α

α α

γ γ• •⊗ + ⊗∑ ∑( ) ( )) )( (( ) ( ) ( )ps n n sD (12.49)

1 )(2

α αα αα ⊗≡ + ⊗) )( (( ) ( )( )p s n n s (12.50)

=1 =1

12

)(= =N N

p α αα αα α α

α α

γ γ• •⊗ − ⊗∑ ∑( ) ( )) )( (( ) ( ) ( )ws n n sW (12.51)

1 )(2

α αα αα −⊗≡ ⊗) )( (( ) ( )( ) s n n sw (12.52)

Needless to say, it holds in this formulation that ˆ= , = =p p0 W W Wω . The simple example of the plastic spin is shown above. Dafalias (1985) provided

the general mechanical interpretation of the plastic spin based on the multiplicative decomposition with the postulate of Eq. (6.27) as follows.

The velocity gradient L in Eq. (6.9) with Eq. (6.27) can be additively decomposed into the three parts, i.e. the rigid body spin, the elastic deformation rate and the plastic deformation rate as follows:

1 11=

p pe ee e − −−+ + F FL V VV Vω (12.53)

where

= =, p p pe e e e• •− −+V V V V F F Fω ωω (12.54)

Here, eV and pF are the corotational rates of eV and

pF , respectively. The strain rate D and the continuum spin W are decomposed by Eq. (12.53) as

follows:


γ.

sn

x

v

τ

τγ.γ.

sn

x

v

τ

τ

Fig. 12.4 Slip system and slip deformation (Kuroda, 1996)

11 1( )( )= ( ) = p pe e ees ss−− −+D L V V F FV V (12.55)

11 1( )( )= ( ) = p pa

e e ee aa−− −+ +W L V V F FV Vω (12.56)

Here, using Eq. (6.27) and assuming that the elastic deformation is infinitesimal

and the unloading process is not accompanied with a rotation, i.e. =e e ≅F V I

leading to 1( ),= = ee e ee a

− ≅ 0D FFVF , Eqs. (12.55) and (12.56) reduce to

= pe + DD D (12.57)

ˆ= p+W Wω (12.58) where

1ˆ ( )pp pa

−≡W F F (12.59)

Although the corotational spin ω is expressed formally as the subtraction of the

plastic spin ˆ pW from the continuum spin W in Eqs. (12.43) and (12.58), it means the rigid-body spin of substructure but, needless to say, it is impertinent to be called an elastic spin which is merely ignored in (12.56).

Substituting Eq. (12.58) and the Jaumann rate (4.41) into Eqs. (4.33), the coro-tational rate is given by

ˆ ˆ ˆ ˆ= =p p p p• +− − −( ) ( ) + −T T T T T T TW W W W W W (12.60)

Noting that the plastic spin is the skew-symmetric tensor, the following explicit

equation is proposed by Zbib and Aifantis (1989)．


ˆ = ( )2p p pρ −W D Dσ σ (12.61)

where ρ is the function of internal variables iH . The relation of the corotational rate and the Jaumann rate of Cauchy stress is

given from Eq. (12.60) with Eq. (12.61) as

ˆ ˆ= p p−+W Wσ σ σ σ

ˆ ˆ ˆ ˆˆ ( ) ( )= ρ λ − −−+ ( /2) N N N Nσ σ σ σσ σσ

i.e.

ˆ= λ+ Nσσ σ (12.62)

where 2 2ˆ ˆ ˆ(2 )ρ −( /2) −≡N N N Nσ σ σ σ σ (12.63)

Substituting Eq. (12.62) into Eq. (6.94), it follows that

ˆˆˆ ( )trtr( )ˆ = =

ˆ ˆp pM M

λλ + NNN σσ σ


ˆtr( )ˆ = pMλ Nσ , ˆtr( ) ˆ=p

pMND Nσ

(12.64)

where

ˆˆ )(tr=p pM M − NNσ (12.65)

The substitution of Eq. (12.64) into Eq. (12.62), one has

ˆt r( )= pM

+ NNσσ σ σ (12.66)

Then, the strain rate is given by

1 1ˆ ˆtr( ) tr( ) ˆ= = p

p pM M− −+ + +N

N ND E D E Nσ σσ σσ

i.e.

1 1ˆtr( ) ˆ )(= pM− −++ N

ND NE Eσ σσ (12.67)


The positive proportionality factor Λ in terms of strain rate is given by Eq.

(6.97) as it is. Then, noting = eEDσ , the Jaumann rate of Cauchy stress is given

from Eq. (6.79) and (12.62) by

ˆ( )= p Λ− − NE D D σσ

ˆ ˆt r( ) tr( )ˆ=ˆ ˆ ˆ ˆ ˆ ˆtr( ) tr( )p pM M

− −+ +

NED EDN NEED N

NEN NENσ

i.e.

ˆ ˆ )( )=

ˆ ˆ ˆtr( )[ ]

pM

+ (⊗−+

NEN ENE DNEN

σσ (12.68)

The stress and the internal variables are updated by

=• + −W Wσ σ σ σ (12.69)

ˆ ˆ|| ||=

ˆ ˆ|| ||=

p p p

p p p'

•

•

⎫+ − − −( ) ( ) ⎪⎬⎪+ − − −( ) ( ) ⎭

D W W W W

β β βD W W W Wb

aα α α (12.70)

noting Eq. (12.60). Hereinafter, limit to the Mises yield condition with the kinematic hardening.

Then, substituting Eq. (6.56) (6.85) into Eq. (6.89) into Eq. (12.61), the plastic spin reduces to the following equation given by Dafalias (1985).

ˆˆ ˆ ˆ( )= 2p ρ λ −W N Nσ σ

ˆ ˆ ( () )= =2 2|| || || || || || || ||ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆ( ) ' ' ' '' ' ' '

ρ ρλ λ −− + +σ σ σ σσ σ α ασ σσ σ σ σ

ˆ ˆˆ ˆ ) )( (= = =2 2 2|| || || ||ˆ ˆˆ ˆ

( ) p p' '' '

ρ ρ ρλ λ− − −N N D Dσ σα α α α α α

σ σ

(12.71) Then, considering the simple shear deformation and assuming the rigid plastic-

ity, the initial isotropy and the linear kinematic hardening as in 12.2, we have

11 1112 12

11 1112 12

0 1 0 1ˆ =2 2 2 1 0 1 0 ( )p ρ α α α αγ γ

α α α α

• •⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦W


111112 12 11

11 1112 12 11

0= =2 2 2 0 ( )ρ α α α α γ ρ α γα α α α α

••

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (12.72)

Substituting Eqs. (2.144) and (12.72) into Eq. (12.70)1, we have

12 1111 1211

11121112

11 12

1112

0 1 0 1 0 1 2= 3 2 2 21 0 1 0 1 0

0 1 1 0

ahα αα α ργ γ γαα αα α

α αα α

• • • • •• •

⎡ ⎤ ⎛ ⎞ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎜ ⎟+ − ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ − − −⎜ ⎟⎣ ⎦ ⎣ ⎦ ⎣ ⎦− ⎣ ⎦ ⎣ ⎦ ⎝ ⎠

⎡ ⎤− ⎢ ⎥ −−⎣ ⎦

11

0 1 2 2 1 0

γ ρ γα•

•⎛ ⎞⎡ ⎤ ⎡ ⎤⎜ ⎟−⎢ ⎥ ⎢ ⎥−⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠

11 1112 1211

11 1112 12

11 1112 1211

11 1112 12

0 1 1 = 3 2 21 0

2 2

ahα α α αγ ργ γαα α α α

α α α αργ γαα α α α

•• •

••

−⎡ ⎤ ⎡ ⎤−⎡ ⎤ + −⎢ ⎥ ⎢ ⎥⎢ ⎥ −− − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

−−⎡ ⎤ ⎡ ⎤+− ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

11 1112 1211

11 1112 12

0 1 1 = 3 1 0 ah

α α α αγ γ γρα

α α α α• • •−⎡ ⎤ ⎡ ⎤−⎡ ⎤ + −⎢ ⎥ ⎢ ⎥⎢ ⎥ − − −−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

11 11 1112

11 1111 12

1 ) ) 13 =

1 ))1 3

a

a

h

h

ρ ρα α α αγ

ρ ρα α α α

•

⎡ ⎤( (− − −1⎢ ⎥⎢ ⎥⎢ ⎥((− − − −1⎢ ⎥⎣ ⎦

(12.73)

from which we obtain

1111 12

1211 11

)=

1 )= 13 a

dd

d hd

α ρα αγ

α ρα αγ

⎫( −1 ⎪⎪⎬⎪(− −⎪⎭

(12.74)

The nonlinear differential equation (12.74) is solved numerically by Dafalias (1985). The calculation result is shown in Fig. 12.5. As seen in this figure, the non-oscillation curve is obtained by choosing the material parameter ρ


appropriately. When choosing 0.5ρ > , 11 22=σ σ− and 12σ increase mono-tonically with the increase of shear strain γ . On the other hand, the Jaumann and the Green-Naghdi rates are independent of material property and thus they would lack the physical pertinence.

The simple shear of the hypoelastic material and the kinematic hardening mate-rial exhibit the oscillatory shear stress if the Jaumann rate is adopted as described above. On the other hand, the simple shear of the isotropic elastoplastic material does not exhibit oscillatory shear stress for any corotational rate since the center of yield surface is fixed in the origin of stress space and the stress continues to lie on the yield surface in the plastic deformation process.

0 2 4 6 100

1

2

3

8

12σ

γ0

0

1

2

11σ3

2

4 6 8

2

321

1

1−

0

0

12α3

= 0.3ρ

= 0.42ρ= 0.5ρ

= 0ρ

= 1ρ

= 0ρ

=1ρ

= 0.5ρ

= 0.3ρ

= 0.42ρ

10γ

= 0ρ = 0.3ρ

= 0.42ρ

= 0.5ρ

= 1ρ

4

( a ) ( )b

( c )γ

0 2 4 6 100

1

2

3

8

12σ

γ0

0

1

2

11σ3

2

4 6 8

2

321

1

1−

0

0

12α3

= 0.3ρ

= 0.42ρ= 0.5ρ

= 0ρ

= 1ρ

= 0ρ

=1ρ

= 0.5ρ

= 0.3ρ

= 0.42ρ

10γ

= 0ρ = 0.3ρ

= 0.42ρ

= 0.5ρ

= 1ρ

4

( a ) ( )b

( c )γ

Fig. 12.5 Description of simple shear deformation of kinematic hardening material by the corotational rate with a plastic spin (Dafalias, 1985)


Chapter 13 Localization of Deformation 13 Localization of Deformation

Even if material is subjected to a homogeneous stress, the deformation concentrates in a quite narrow strip zone as the deformation becomes large and finally the material results in failure. Such a concentration of deformation is called the localization of deformation and the strip zone is called the shear band. The shear band thickness is the order of several microns in metals and ten and several times of particle radius in soils. Therefore, the large shear deformation inside the shear band is hardly reflected in the change of external appearance of the whole body. Therefore, a special care is required for the interpretation of element test data and the analysis taking accounting of the inception of shear band is indispensable when a large deformation is induced. The localization phenomenon of deformation and its pertinent analysis are described in this chapter.

13.1 Element Test

The element test of material is useful on the premise that a homogeneous deformation proceeds reflecting the constitutive property. However, the deformation becomes heterogeneous when a large deformation accompanying with

0pM <11σ

0

11dσ

11D dt 11D dt 11D dt∫

11σ

Constitutive property

Element test

0pM <11σ

0

11dσ


11σ


Element test

0pM <11σ

0

11dσ


11σ


Element test

Fig. 13.1 Stress-strain curve in the constitutive property and the element test with a localization in a softening state

328 13 Localization of Deformation the shear band is induced. On the other hand, the state of stress (rate) is almost identical outside and inside the shear band, since the equilibrium equation is fulfilled. Therefore, the strain for a given stress is measured far smaller in the element test than that predicted in the constitutive relation reflecting the mechanical property. This fact is illustratively depicted in Fig. 13.1 for the softening material. This fact must be considered when we determine the material parameters from element test data.

13.2 Gradient Theory

The material in which the stress at a certain material point is uniquely determined only by the history of the deformation gradient F at that point is called the simple material. However, when the gradient of deformation gradient F becomes large as seen inside the shear band, its history influences on the state of stress (rate). Here, there exist the limitation in the gradient so that the shear band thickness does not decrease less than a certain limitation. The limitation is regulated by the material constant, called the characteristic length. In the finite element method ignoring the influence of the gradient, the deformation concentrates in the narrow band zone corresponding to the width of one element. Then, if the finite elements are downsized aiming at obtaining an accurate solution, the shear band thickness reduces infinitely, resulting in the mesh-size dependence losing the reliability of solution, i.e. the il-posedness. The theory for the simple material ignoring the gradient is called the local theory. On the other hand, the theory taking account of the gradient is called the non-local theory. The non-local theory of elastoplastic deformation is first proposed by Aifantis (1984).

Adopting the yield condition with the isotropic hardening for sake of simplicity and introducing the gradient of the isotropic hardening variable, let the yield condition (6.30) be extended as follows:

( ) ( )=f F H⟨ ⟩σ (13.1)

where ⟨ ⟩ designates the second-order gradient, i.e.

222 =1 c⟨ ⟩ + ∇ ,

22 22

2 2 22 31x x x

∂ ∂ ∂∇ ≡ + +∂ ∂ ∂ (13.2)

2c is the material constant describing the effect of the gradient of the mechanical state. Here, for the sake of simplicity, the higher order gradient is not incorporated. The first-order gradient is not incorporated because the odd-order gradients cancel each other in opposite directions.

The material-time derivative of Eq. (13.1) is given by

2 22

( ) [ ](1 )tr = f cF H'•∂ + ∇

∂σ σσ (13.3)

13.2 Gradient Theory 329 Substituting the associated flow rule (6.48) into Eq. (13.3), one has

2 22

( ) ( )(1 )[ ], ,tr = if cHhF' λ∂ + ∇∂

Nσ σσσ (13.4)

from which we obtain

2 22 [(1 ) ]tr( )

= pc

Mλ − ∇N σ

(13.5)

2 221 ( )tr( [ ]1 )= p

c

M− − ∇+ ND E Nσσ (13.6)

2 22 (1 )[ ]trt r( ) = t r( ) t r( ) p

c

M

− ∇+ NNED N NEN σσ

2 222 2

2 (1 )[ ]tr(1 ) t r( )[ ]p

p

ccM

M

− ∇≅ + +∇ NNENσ

2 22

222

(1 ) [ ]tr= t r( ) 1

t r( ) [ ]

pp

p p

cMM cM M

− ∇+ + ∇+

NNEN

NENσ

22( ) [1 ]tr= t r( )(1 )[ ]p

p

cM

M

− ∇+ + ∇ NNEN σ (13.7)

where

2 22cϑ∇ ≡ ∇ ,

t r( )

p

pM

Mϑ ≡

+ NEN (13.8)

On the other hand, it is obtained from Eq. (13.7) that

(1 )t r( )t r( )(1 )=t r( ) t r( )

[ ]p pΛM M

− ∇− ≅∇+ +

NE DNEDNEN NEN

(13.9)

(1 )[ ]t r( )=t r( )pM

− ∇−+

NE DED E NNEN

σ (13.10)

In what follows, the above-mentioned equations are extended for the subloading surface model with the isotropic and the anisotropic hardening. Incorporating the gradient into the internal variables, Eq. (7.33) for the subloading surface is extended as (Hashiguchi and Tsutsumi, 2006):

( ) ( )=f RF H⟩, ⟩ ⟨ ⟩− ⟨ ⟨βσ α (13.11)

330 13 Localization of Deformation Considering Eq. (13.2), Eq. (13.11) leads to

2 2 22 2 22 2 2[(1 ) ] (1 ) [ (1 )]( ) [ ( )]=, c c cf RF H+ + +− ∇ ∇ ∇βασ (13.12)

The material-time derivative of Eq. (13.12) is given by

2 22 22 2[(1 ) ] [(1 ) ]( )

tr,

c cf + +− ∇ ∇∂∂

βασ σσ

22 2222 2 2

2[ ] [(1 ) ](1 )( ) [ ](1 )tr

, ccf c+− + ∇∇∂− + ∇∂

βασ ασ

22 2222 2 2

22 22

[ ] (1 )[ ](1 )( ) [ ](1 )tr(1 )[ ]

, ccf c

c+− + ∇∇∂ ++ ∇+∂ ∇

β ββ

ασ

2 22 [(1 ) ]= c R F R F

• •+ ∇ + (13.13)

The gradients of internal variables can be ignored since they are small compared to the gradient of their rates and thus Eq. (13.13) reduces approximately to

2 2 2 22 2

( ) ( ) ( )(1 )[ ] [(1 ) ]tr tr t rˆ ˆ ˆ f f fc c, , ,∂ ∂ ∂+ +− ∇ + ∇∂ ∂ ∂

ββ β ββσσ σ σασ σ

2 22 22 2[ ] [ ](1 ) (1 )= c cF R R F H'

• •+ + +∇ ∇ (13.14)

Substituting Eq. (7.13) for the evolution rule of normal-yield ratio, the consistency condition is derived from Eq. (13.14) as follows:

2 2 2 22 2

( ) ( ) ( )(1 )[ ] (1 ) [ ]tr tr trˆ ˆ ˆ f f fc c, , ,∂ ∂ ∂+− ∇ ++ ∇∂ ∂ ∂

ββ β ββσσ σ σασ σ

2 22 22 2[ ] [ ](1 ) (1 )= pc cUF R F H'

•+ + +∇ ∇D (13.15)

Further, substituting the associated flow rule (7.19) into Eq. (13.15), it is obtained that

22 2222

( ) ( ) ( ) (1 )[ ](1 )[ ]tr tr trˆ ˆ ˆ f f f cc λλ, , ,∂ ∂ ∂ ++− + ∇∇∂ ∂ ∂β β β

a bβσσ σ σσ σ

2 22 22 2[ ] [ ](1 ) (1 )= c c hUF R F' λλ+ + +∇ ∇ (13.16)

13.3 Shear-Band Embedded Model: Smeared Crack Model 331 which can be approximately given by

2 22 22 2

( ) ( ) ( )(1 ) (1 )tr tr [ ] tr [ ], , , ˆ ˆ ˆ f f fc cλ λ∂ ∂ ∂+ +− +∇ ∇∂ ∂ ∂β β βa bβσσ σ σ

σ σ

2 22( )(1 )[ ]= ' cUF RF h λ+ + ∇ (13.17)

The positive proportionality factor is derived from Eq. (13.17) as

2 222 2

2tr( ) [(1 ) ]tr( )(1 )= [ ]p p

ccM M

λ − ∇− ≅∇N Nσ σ (13.18)

where

( , )1tr + trˆ

( )( ) [ ]p fF UhMRFF R

∂′≡ + − ∂βaN bβ

σ σ . (13.19)

On the derivation of Eq. (13.18), it is assumed that the fourth-order gradient can be ignored, thereby leading to 2 22 2

2 2(1 )(1 ) 1c c+ −∇ ≅∇ . Consequently, the plastic

strain rate is given as

2 22(1 )[ ]tr( )

=pp

c

M+ ∇N ND

σ (13.20)

The shear band thickness of softening soil has been predicted adopting Eq. (13.20) by Hashiguchi and Tsutsumi (2006).

Here, it is noteworthy that we must use quite small elements with the size of several tens of shear band thickness to take the effect of the gradient into account correctly in the finite element analysis. Therefore, it is nearly impossible to apply the gradient theory to the finite element analysis of boundary value problems in engineering practice at least at present. The gradient theory is used widely for prediction of shear band thickness, size effects, etc. using fine meshes for very small specimens.

13.3 Shear-Band Embedded Model: Smeared Crack Model

Although the gradient theory is not applicable to the analysis of practical engineering problems at present, the practical model for the finite element analysis for softening materials has been proposed, as described below.

As the deformation becomes large and the shear band is formed, the plastic deformation concentrates in the shear band and thus the softening is accelerated leading to the rapid reduction of stress. As the result, inversely, the unloading leading to the elastic state occurs inside the shear band. Consequently, the

332 13 Localization of Deformation elastoplastic constitutive equation holds only inside the shear band. Then, denoting the strain rate and the plastic strain rate calculated from the external appearance by

D and pD , respectively, called the apparent strain rate and the apparent plastic strain rate, respectively, the elastoplastic constitutive equation in terms of the apparent strain rate is proposed. It is called the shear-band embedded model or smeared crack model (Pietrueszczak and Mroz, 1981; Bazant and Cedolin, 1991).

Denoting the ratio of the area of a shear band to the area of a finite element by ( 1)S , the following relations hold.

= pp SDD (13.21)

1 t r( ) = = =pp

pe e S S

M−+ + + ND D D D D E N

σσ (13.22)

Tanaka and Kawamoto (1988) proposed the simple equation of S for the plane strain condition as follows:

= ( ) /( ) = / eS w l l l w F× × (13.23)

supposing simply the square finite element ×l l with the side-length l and the shear band having the thickness w , where ( = )eF l l× is the area of the finite element.

The positive proportionality factor is expressed in terms of the apparent strain rate from Eq. (13.22) as follows:

t r( )=

tr( )pM

S

Λ+

EDN

NEM (13.24)

Then, the stress rate is given by

t r( )=

tr( )pM

S

−+

EDN NEEDNNE

σ (13.25)

Then, we have

t r( )=tr( ) tr( )

p

pM

SM +EDNN NNE

σ (13.26)

from which it is known that the stress reduction is accelerated when the shear band is induced resulting in < 1S . It is desirable to choose material parameters such that Eq. (13.22) or (13.25) fits to a measured stress-strain curve using the value of S predicted by a pertinent method, if we determine them from element test data.

13.4 Necessary Condition for Shear Band Inception 333 13.4 Necessary Condition for Shear Band Inception

Discontinuity of the velocity gradient is induced at the shear band boundary. Here, incorporate the coordinate system in which the coordinate axes 1x∗ and 2x∗ are

taken to be normal and parallel, respectively, to the shear band as shown in Fig. 13.2. The discontinuity of velocity gradient can be induced only in the 1x∗ -direction.

Therefore, only the following quantities are not zero.

211 1

1 2

1 1

, ( () )v vg gx x∗ ∗∗ ∗

∂ ∂Δ Δ≡ ≡∂ ∂ (13.27)

Therefore, the discontinuity of strain rate is given by

1 1= =2 2 ( ( ( () ) ) )j jr ri i

ijj j iri r

v x xvv vDx x x x xx

∗∗∗ ∗

∂ ∂ ∂∂∂ ∂Δ Δ + Δ Δ + Δ∂ ∂ ∂ ∂ ∂∂

1 11 1

1 1

1 1 = = ( ) ( )2 2 ( () )ji jij i

j i

vx xv g gx x x x

• •∗ ∗

∗ ∗∗ ∗∂∂ ∂∂Δ + Δ +∂ ∂ ∂ ∂ n e n e

1 11 = ( )2

jij ig gn n∗ ∗+ (13.28)

where n is the unit vector in the direction normal to the shear band, i.e. the 1x∗ -direction.

On the other hand, the discontinuity in the rate of traction vector nt applied to

the discontinuity surface of velocity gradient, having the direction vector n , is described by

1 1 11

1 12 2

1 1

2 2

1 (= = = )2

1 (= = = )2

p p k llj j jj kj kl kl j kl

p p k llj j jj kj kl kl j kl

e en

e en

g gn nt n C D n C n

g gn nt n C D n C n

σ

σ

•

•

∗ ∗

∗ ∗

⎫+Δ Δ Δ ⎪⎪⎬⎪+Δ Δ Δ⎪⎭

(13.29)

Noting

1 1 1 1

2 2111 1 1 1l 2

2 111 1 1

1 1 1

11 12 1 1

11 12 11 12

1 1( = ( ))2 2

1 = ( )2

1 = (2

p p pk l k lj jl ljk kj kl j kl j kl

p p p pj j j jl l k kj l j l j k j k

p p p pj j jl l lj l j l j l j l

e e e

e e e e

e e e e

g g g gn n n nn nC n C C

g g g gn n n nn n n nC C C C

g g g gn n nn n nC C C C

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗

+ +

+ + +

+ + + 21

211 111 12

)

= ( )

jl

p pj jl lj l j l

e e

n n

g gn nn nC C

∗

∗ ∗+

334 13 Localization of Deformation Eq. (13.29) is expressed as

1 11 11

2 21 12

11 1 2 11 12

21 2 2 21 22

= =

p p p pj j j jk kl lj l j k j l j k

p p p pj j j jk kl lj l j k j l j k

e e e en

e e e en

g gC n n C n n C n n C n nt

g gC n n C n n C n n C n nt

•

•

∗ ∗

∗ ∗

⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤Δ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪Δ ⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭⎩ ⎭

That is to say,

11111 12

1122221 12

= ( = , = )

j

i ijn

n n

n

gA Att A g

A A gt

•• •

•

∗∗∗

∗

⎧ ⎫ ⎧ ⎫Δ ⎡ ⎤⎪ ⎪ ⎪ ⎪ Δ Δ⎨ ⎬ ⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪ ⎪ ⎪Δ ⎩ ⎭⎩ ⎭

gt A (13.30)

where ijA is given by the following equation and called the acoustic tensor.

, p psrij rijs

e eA C n n≡ ≡A nC n (13.31)

Here, noting that the traction rate vector must be continuous, i.e. =nΔt 0 and thus it must hold from Eq. (13.30) that

1111 12

1122221 1

0= ( = 0, = )

0j

ij

gA AA g

A A g∗∗

∗

∗

⎧ ⎫⎡ ⎤ ⎧ ⎫⎪ ⎪⎨ ⎬ ⎨ ⎬⎢ ⎥

⎩ ⎭⎣ ⎦ ⎪ ⎪⎩ ⎭

gA 0 (13.32)

In order that Eq. (13.32) has a solution other than the non-trivial solution 1 =∗g 0 ,

i.e. that the discontinuity of velocity gradient is induced, it must hold that

det = 0A (13.33)

The eigenvalue of the acoustic tensor A becomes zero as known from Eq. (1.127) when (13.33) holds. The search for the occurrence of n fulfilling Eq. (13.33), i.e. the inception of the shear band and its direction, is called the eigenvalue analysis. Equation (13.33) is given explicitly as (Hashiguchi and Protasov, 2004)

1 1 2 2det( ) = det( )p pps sijs ijs

e ee C n n C n n+n nC

1 1 1 1 1 2 1 2 1 2 21 22 22= det( )p p p pij ij ij ije e e eC n n C n n C n n C n n+ + +

1111 1112 2111 2112 1121 1122 2121 2122

1211 1212 2211 2212 1221 1222 2221 2222

1 1 1 1 1 11 2 1 22 2 2 22 2

1 1 1 1 1 11 2 1 22 2 2 22 2

=

p p p p p p p p

p p p p p p p p

e e e e e e e e

e e e e e e e e

C n n C n n C n n C n n C n n C n n C n n C n n

C n n C n n C n n C n n C n n C n n C n n C n n

+ + + + + +

+ + + + + +

2 2 2 21111 1112 2111 2112 1121 1122 2121 2122

2 2 2 21211 1212 2211 2212 1221 1222 2221 2222

1 11 2 2 1 2 2

1 11 2 2 1 2 2

( ) ( ) =

( ) ( )

p p p p p p p p

p p p p p p p p

e e e e e e e e

e e e e e e e e

C n C C n n C n C n C C n n C n


+ + + + + +

+ + + + + +

13.4 Necessary Condition for Shear Band Inception 335

2 2 2 21111 1112 2111 2112 1221 1222 2221 2222

2 2 2 21121 1122 2121 2122 1211 1212 2211 2212

1 11 2 2 1 2 2

1 11 2 2 1 2 2

= ( ) ( )

( ) ( )

p p p p p p p p

p p p p p p p p

e e e e e e e e

e e e e e e e e



+ + + + + +

− + + + + + +

4

1111 1221 1121 1211 1= ( )p p p pe e e eC C C C n−

1111 1222 1111 2221 1112 1221 2111 1221( p p p p p p p pe e e e e e e eC C C C +C C C C+ + +　

31121 1212 1121 2211 1122 1211 2121 1211 1 2)p p p p p p p pe e e e e e e eC C C C C C C C n n− − − −

2 21111 2222 1221 2112 1121 1211 1211 2122 1 2

p p p p p p p pe e e e e e e eC C C C C C C C n n + ( + − − )

1122 2222 2111 2222 1222 2112 2221 2112( p p p p p p p pe e e e e e e eC C C C C C C C+ + + +

3

1122 2212 2121 2212 2122 1212 2122 2211 1 2)p p p p p p p pe e e e e e e eC C C C C C C C n n− − − −

4

2112 2222 2122 2212 2)p p p pe e e eC C C C n + ( −

4

1111 1212 11122

1= ( )p p pe e eC C C n−

31111 1222 1111 2212 1112 1122 1122 1211 1 2( )p p p p p p p pe e e e e e e eC C C C C C C C n n+ + − −　

2 2

1111 2222 1212 1212 1112 1211 1211 1222 1 2p p p p p p p pe e e e e e e eC C C C C C C C n n + ( + − − )

3

1122 2222 12111 2222 1122 1222 1222 1122 1 2( )p p p p p p p pe e e e e e e eC C C C C C C C n n+ + − −

4

1212 2222 12222

2)p p pe e eC C C n + ( −

which reduces to

2 3 42341 21 1 2 1 2 1 2 23 4 5det = = 0a n a n n a n n a n n a n+ + + +A (13.34)

where

1111 1221 1121

1111 1222 1111 2221 1121 2211 1122 1211

1111 2222 1221 2112 1121 1211 1211 2122

1122 2222 2111 2222 1122 2212

21

2

3

4

,

,

,

p p p

p p p p p p p p

p p p p p p p p

p p p p p

e e e

e e e e e e e e

e e e e e e e e

e e e e e e

a C C C

a C C C C C C C C

a C C C C C C C C

a C C C C C C

≡ −

≡ + − −

≡ + − −

≡ + −

　

2122 2211

2112 2222 21222

5

,p p p

p p p

e e

e e e

C C

a C C C

⎫⎪⎪⎪⎬⎪− ⎪⎪≡ − ⎭

(13.35)

Setting

1 2= cos , = sinn nθ θ (13.36)

336 13 Localization of Deformation

1x∗2x∗

1x

2x

0

0∗

n

1e

2e

v 1x∗2x∗

1x

2x

0

0∗

n

1e

2e

v

Fig. 13.2 Discontinuity of velocity gradient induced in the direction normal to the shear band

Eq. (13.34) is rewritten as 234

5 4 3 2 0( ) = tan tan tan tan = 0g a a a a aθ θ θ θ θ+ + + + (13.37)

which, noting the symmetry ( ) = ( )g gθ θ− , leads to 24

5 3 0( ) = tan tan = 0g a a aθ θ θ+ + (13.38)

There exists the possibility that a shear band occurs in the direction θ fulfilling Eq. (13.38). Here, note that Eq. (13.33) is the only necessary condition for the inception of the shear band.

We searched above the discontinuity of the velocity gradient in the direction normal to the shear band, while the traction rate vector must be continuous in that direction. Inversely, on the other hand, the search for the discontinuity of the normal stress rate component applied to the surface normal to the shear band, i.e. 22 0σ• ∗Δ ≠ , while the normal strain rate component in the direction parallel to the shear band must be zero, i.e. 22 0=ε•∗Δ , is called the compliance method (cf. Mandel, 1964). Here, note that the normal strain rate component in the direction normal to the shear band can be discontinuous, i.e. 11 0ε•∗Δ ≠ in dilatable materials. It was applied to the prediction of the direction of shear band formation in soils (Vermeer, 1982).


Chapter 14 Numerical Calculation Numerical Calculatio n

Elastoplastic deformation has to be analyzed generally by numerical calculations with finite incremental steps since the constitutive equation is rate nonlinear. In particular, the return-mapping algorithm to pull back the stress to the yield surface must be incorporated into the computer program adopting the conventional elastoplastic constitutive models. On the other hand, the subloading surface model is furnished with the distinguished advantage for the numerical calculation with the automatic controlling function to attract the stress to the normal-yield surface in the plastic deformation process and thus it does not require to incorporate the convergence computer algorithm such as the return mapping in the normal-yield state. Nevertheless, it requires to incorporate the return-mapping algorithm in the subyield state since the subyield state is out of the stress-controlling function. Basic equations for the return mapping method extended to the subloading surface model is described in this chapter.

14.1 Numerical Ability of Subloading Surface Model

The stress controlling function of the subloading surface model is described in Chapter 7. This fact will be qualitatively shown below in the numarical calculation by the concise examination for the response of the uniaxial loading behavior, adopting the simplest version, i.e. the initial subloading surface model for the isotropic Mises material with the evolution rule of the normal-yield ratio

=( ) cot( || ||/2) puRU Rπ D in Eq. (7.13) with Eq. (7.15) with = 0eR . The

response of the conventional elastoplastic constitutive model is also shown for the comparison.

The relations of the axial stress aσ and the normal-yield ratio R versus the

axial strain aε are depicted in Fig. 14.1. The responses for the linear isotropic

hardening 0= epcF F h ε+ ( ch : material constant) are depicted in Fig. 14.1(a)

and those for the nonlinear isotropic hardening in Eq. (10.3) are shown in Fig. 14.1(b). The two levels of axial strain increment and 0 .0055 = 0.0006 adε are

input for numarical calculations. Here, any special stress controlling algorithm to pull it back to the yield surface is not introduced. The material parameters are chosen as follows:

338 Numerical Calculation

aε0 0.02 0.04 0.06 0.08

1.0

0.50

1.50

2.0

R

0

200

400

600

800

1000

(MPa) aσ

(a) Linear isotropic hardening

Exact curve of conventional elastoplasticitCalculated by the conventional elastoplast

Calculated by the subloading surface mod

55= 0.00adε

0.0006=adε0.0055=adε

0.0006=adε

550.00

0.0006=adε

0.0006=adε

0.0055

aε0 0.02 0.04 0.06 0.08

1.0

0.50

1.50

2.0

R

0

200

400

600

800

1000

(MPa) aσ

(a) Linear isotropic hardening

Exact curve of conventional elastoplasticitCalculated by the conventional elastoplast

Calculated by the subloading surface mod

55= 0.00adε

0.0006=adε0.0055=adε

0.0006=adε

550.00

0.0006=adε

0.0006=adε

0.0055

Fig. 14.1 Numerical accuracies of the conventional elastoplastic and the subloading surface model: Uniaxial loading behavior of Mises material with isotropic hardening.

14.1 Numerical Ability of Subloading Surface Model 339

0.0006

0 .0055

aε0 0.02 0.04 0.06 0.08

1.0

0.50

1.50

2.0

R

0

200

400

600

800

1000

(MPa) aσ

(b) Nonlinear isotropic hardening

0 .0055

0.0006=adε

0 .0055

0 .0055

0.0006=adε

0.0006=adε

0.0006

0 .0055

aε0 0.02 0.04 0.06 0.08

1.0

0.50

1.50

2.0

R

0

200

400

600

800

1000

(MPa) aσ

(b) Nonlinear isotropic hardening

0 .0055

0.0006=adε

0 .0055

0 .0055

0.0006=adε

0.0006=adε

Fig 14.1b. (continued)

340 Numerical Calculation Material constants:

1 2

Youg's modulus: = 100000MPa,

Linar isotropic : = 7000,Hardening

Nonlinar isotropic : = 0.8, = 50,

Evolution of normal - yield ratio : = 200.

c

E

h

h h

u

⎧⎨⎩

Initial values:

0

0

Hardening function: = 500MPa,

Stress : MPa=

F

0σ

The nonsmooth curves bent at the yield stress are expressed by the conventional model. Moreover, the stress deviates from the exact curve of concventional elastoplasticity. The deviation becomes large with the increases in the nonlinearity of hardening and in the input strain increment. On the other hand, the stress is automatically attracted to the normal-yield surface in the subloading subloading

surface model even for the quite large strain increment = 0.0055 (0.55%)adε .

The zigzag lines tracing the exact curve are calculated such that the stress rises up when the it lies below the normal-yield surface but it drops down immediately if it goes over the normal-yield surface, obeying the evolution rule of normal-yield ratio

in Eq. (7.13) with Eq. (7.14), i.e. 0 for <0RR• > and 0 for >0RR

• < . The amplitude of zigzag decreases gradulally in the monotonic loading process, while, needless to say, the amplitude is smaller for a smaller input increment of strain. Eventually, the subloading surface model posseses the distinguished high ability for numerical calculation as verified also quantitatively in these concrete examples, which has not been attained in any other elastoplastic constitutive equations incuding the multi, the two, the infinite, nonlinear-kinematic hardening and the bounding surface models delineated in Chapters 7 and 8.

14.2 Return-Mapping Formulation for Subloading Surface Model

Elastoplastic constitutive equation is described by simultaneous equations of stress rate components and strain rate components which do not fulfill the complete integrability condition and thus elastoplastic deformation analysis is performed by numerical calculations with finite steps. Therefore, an error is accumulated in the calculation using the Euler method (Yamada et al., 1968) without the convergence calculation if we adopt the conventional elastoplastic constitutive models in which the interior of yield surface is assumed to be the purely-elastic domain. Then, various return-mapping algorithms, e.g. the mean normal method (Rice and Tracey,

14.2 Return-Mapping Formulation for Subloading Surface Model 341 1973; Pillinger et al., 1986) and the radial return method (Krieg and Krieg, 1977) with the correction algorithm so as to pull the stress back to the yield surface, have been proposed to date. However, they are intended for the conventional elastoplasticity premised on the assumption that the interior of yield surface is an elastic domain.

The subloading surface model is furnished with the controlling function to attract the stress to the normal-yield surface automatically as described in 14.1 and thus it has no need to incorporate the return-mapping algorithm to pull back the stress to the normal-yield surface. However, the subyield state before the stress reaches the normal-yield surface is out of this controlling function and thus the calculation error is accumulated in cyclic loading process in which the subyield state ( 1)R < is repeated. An improvement in numerical calculation so as to reflect exactly the evolution of subloading surface would be required in the subyield state. The return-mapping formulation for the subloading surface model will be given in this section, while the Jaumann rate is adopted for the corotational rate for sake of simplicity.

As known from Eq. (7.26), the stress increment in elastoplastic constitutive equation is given by subtracting the plastic relaxation stress increment due to plastic strain rate from the elastic stress increment calculated presuming that the input strain increment is elastic. Here, the stress rate is related to the plastic strain rate by the consistency condition (7.18) for the subloading surface (7.6).

Suppose that the stress nσ in the step n is calculated already by giving n-time inputs of

the strain increment. Then, for the initiation of calculation in the n 1+ step, first calculate the

elastic stress increment 1elasticn+dσ using the elastic constitutive equation for 1n+ dtD and

1n+ dtW due to the input velocity gradient increment 1n+ dtL , i.e.

1 1 1 1elastic

n n nn n n n+ + + +(( )= )d dt+DEσ σ σ σW W− (14.1)

noting Eq. (4.41). Here, 1elasticn+dσ is called the elastic predictor which is predicted as a trial

supposing the elastic deformation process. Then, the stress is updated by adding it to the stress

nσ in the last step, i.e.

1 1elastic elasticn+ n n++= dσ σ σ (14.2)

It can be stated that the calculation as the elastic deformation was correct if the subloading

surface 1elastic

n nn+( ), f −ασ β passing through 1elasticn+σ is smaller than the subloading

surface nn n( ), f −α βσ at the end of step n . Then, 1elasticn+σ is determined as the stress

at the n 1+ step. On the other hand, if 1elastic

n nn+( ), f −ασ β is larger than

nn n( ), f −α βσ , it is regarded that a plastic strain rate is necessarily induced during the

input strain increment 1n+ dtD , i.e.


11

1

Final elasticelasticn n n+1 n 1 nn n n n + ++

Final elastic(1)nn+1 +n+1

( ) or : = , =

Otherwise : ,

, ( ) ( ) p

p

ef RR F H F H≤

≠ ≠

⎫− ⎪⎬⎪⎭

D 0

D 0

α σ σσ β

σσ

(14.3) In the second case of Eq. (14.3) the plastic relaxation stress increment must be

reduced from 1elasticn+σ . Presuming that

(k) (k)1 n+1n+ , ,Hσ (k) (k) (k)

n+1 n+1 n+1, , Rβα are

already obtained repeating calculations of stress reduction by k time calculations, then let the k 1+ -th calculation be carried out. Designating the plastic strain

increment (k+1)n+1p tδD in this time, the plastic relaxation stress increment (k+1)

n+1pδσ ,

called the plastic corrector, is given by

(k+1) (k) (k+1)(k) (k+1)n+1 n+1 n+1n +1 n+1 ( )== p pd tδ− Dσ σσ σ E (14.4)

Here, it is desired that the following relation is fulfilled.

(k+1) (k+1) (k+1) (k+1)n+1 n+1 n+1 n+1( ) ( ) = 0, f R F Hˆ −σ β (14.5)

Applying the Taylor expansion to (k+1) (k+1) (k+1)n+1 n +1 n+1, , H Hα and (k+1)

n+1R in Eq.

(14.5), one has

1 1 1 1

1 11 1

(k) (k) (k) (k)(k)(k) (k) n + n + n + n + (k+1)n+1n+ n+ (k) (k) n+1

n + n +

( ) ( ), , ( ) t r t r, pf f

f ˆˆ ˆ

δ δ∂ ∂+ −∂ ∂

β βσ β σ σσ σσ α

1 1(k) (k)n+ n + (k+1)

n+1(k)n +1

( ), t r f ˆ

δ∂+∂

ββ

βσ

(k+1) (k+1)(k) (k) (k)n+1n+1 n+1 n+1 n+1 ) ( () )R F H + F H HR 'δ δ− ( +

1 1 1 1

1 11 1

(k) (k) (k) (k)(k)(k) (k) n + n + n + n + (k+1)n+1n+ n+ (k) (k) n+1

n+ n+

( ) ( ), , ( ) t r t r, pf f

f ˆˆ ˆ

δ δ∂ ∂≅ + −∂ ∂

β βσ β σ σσ σσ α

1 1

1

(k) (k)(k)n + n +n+1(k)

n +

( ), t r f ˆ

δ∂+∂

ββ

βσ

(k) (k) (k) (k) (k) (k+1)(k+1)n n n n n n1 1+1 +1 n+1 + +1 +1 +( ( ( 0) ) ) =R F H F H R F H HR 'δ δ−− − (14.6)

where from Eqs. (6.37) and (6.86) one has (k+1) (k+1)( )k( )kn n 1+1 n+1 +n+1( , = ; )p

iH th H δδ σ D (14.7)

14.2 Return-Mapping Formulation for Subloading Surface Model 343

(k+1) (k+1)( )k( )kn n 1+1 n+1 +n+1( , = )|| ||p

i tHδ δa σ Dα (14.8)

(k+1)(k+1) ( )k( )kn n 1+1 n+1 +n+1( , = )|| ||p

i tH 'δ δβ b σ D (14.9)

(k+1)(k+1) (k+1)(k) (k) n+1n n n 1+1 +1 n+ +1

) , )= = ( || || ( cot2 1( )p e

e

RR R R RtU uUR

δ δ π ⟨ − ⟩−D

(14.10) Substituting Eqs. (14.7)-(14.10) into Eq. (14.6), the consistency condition is

obtained as follows:

11 11 11

1

(k) (k)(k)(k) (k) (k+ )n + n +n+1n n++ n+ (k)

n +

( ), ( ) t r ( ), pf

f ˆ tˆ

δ∂+∂

βDσσ β Eσ

σ

1 1

1

(k) (k)(k+1)( )kn + n + ( )k

n 1n+1 +(k) n+1n +

( ), t r ( , )|| || p

if

tHˆ

δ∂−∂

a σ Dσσ α

1 1

1

(k) (k)(k+1)n + n + ( )k( )k

n 1(k) n+1 +n+1n +

( ), t r ( , ) || || p

if

tHˆ

' δ∂+∂

βbβ σ D

σ

1

11(k+ )(k) (k) (k)(k)

n n nn 1n+1 +1 ++ +( (( ) || ||) )pR F H F HRU tδ− − D

(k+1)( )k(k) (k) ( )kn n n 1+1 +1 n+1 +n+1( ( , 0;) ) =p

iR F H th H' δ− σ D (14.11)

The associated flow rule is expressed from Eq. (7.35) as

(k+1) (k) (k)(k+1)n n+1 n+11 n+1+ ( ), =p ˆ ˆ ˆλ σ βD N (14.12)

Substituting Eqs. (14.12) into Eq. (14.11), it holds that

1 1

1 11

(k) (k)(k)(k) (k) n + n + (k+1) (k) (k)n+1n+ n n+1+ n+1 n+1(k)

n +

( ), ( ) t r ( ) ( ), , f

f ˆˆ ˆtˆ

δλ∂+∂

β σσ β E σN βσσ

1 1

1

(k) (k)( )kn + n + ( )k (k+1)

n+1n+1(k) n+1n+

( ), t r ( , ) ) i

ftH

ˆδλ∂−

∂a σσ

σ α

1 1

1

(k) (k)( )k( )n + n + (k+1)k

n+1n+1(k) n+1n +

( ), ( , t r ) ) i

f|| || tH

ˆ' δλ∂+

∂β

b Nβ σσ

1(k) (k) (k) (k+1) (k)n n n n+1 n+1 +1 1+ +( ( ) () )R F H R F HU tλ δ− −

(k) (k)( ) (k+1)k(k) (k) ( )kn+1 n+1n+1n n+1 +1 n+1 n+1

( ( , 0( ); , )) =iR F H ˆ ˆh H t' λ δ− σ βσ N (14.13)

344 Numerical Calculation from which one has

1 1(k) (k) (k) (k)(k+1)n n+1 n+1n+1 + n+( )= ( , )f R F Ht ˆλ δ −σ β

1 1

1

(k) (k)( )k( ) ( )kkn n+ + n+1 n+1 n+1(k)

n +

( ), ( , t r ( ) )

[/ iˆf ˆ H∂−∂σ β σ σNEσ

1 1

1

(k) (k)( )k( )n + n+ k

n+1(k) n+1n+

( ), ( , t r ) i

fH

ˆ∂+∂

β a σσ

σ

1 1

1

(k) (k)( )n + n + k( )k

n+1(k) n+1n +

( ), t r ( , ) i

f|| ||H

ˆ'

∂−∂

βb Nβ σσ

(k)(k)(k) (k) ( ) (k) (k)k( )k n+1n+1n+1 n+1 n+1 n+1n+1 n+1( ) (( , ( ( ); ), )) ]i R F HR F H Uˆ ˆHh' ++ σ βσ N

(14.14) By use of

1 1

1

(k) (k) (k) (k)n+1 n+1 (k) (k)n + n +

n+1 n+1(k) (k) (k) (k)n + n+1 n+1 n+1

( ) ˆ( ), ˆ ˆ( )= ˆ ˆ( ) ˆt r

ˆf f∂∂

β σ β σ βNσ β σNσ

σ , , ,

(14.15)

(k)(k)n+1 (k) (k)n+1

n+1 n+1(k) (k) (k)n+1 n+1 n+1

( ) ˆ ˆ( )ˆ ˆ( ) ˆt r

( )R F H≠ σ βNσ β σN

, ,

based on Eq. (7.20), Eq. (14.14) is expressed as

1 1(k) (k) (k) (k)(k+1)n n+1 n+1n+1 + n+( )= ( , )f R F Ht ˆλ δ −σ β

(k) (k)( )k( ) ( )n+1 n+1 kk(k) (k)

n+1 n+1n+1 n+1 n+1(k) (k) (k)n+1 n+1 n+1

( , t r

( ), ( ) )( ), ( ), t r

[/ iˆf ˆˆ ˆ H

ˆ ˆ ˆ−

σ β σ σσ NβN Eσ β σN

(k) (k)n+1 n+1 ( )k( )k(k) (k)

n+1 n+1 n+1 n+1(k) (k) (k)n+1 n+1 n+1

( , t r

( ), ( ) ),

( ), t ri

ˆf ˆ ˆ Hˆ ˆ ˆ+

σ βa σσ βNσ β σN

1 1

1

(k) (k)( )n + n + k( )k

n+1(k) n+1n +

( ), t r ( , ) i

f|| ||H

ˆ'

∂−∂

βb Nβ σσ

(k)(k)(k) (k) ( ) (k) (k)k( )k n+1n+1n+1 n+1 n+1 n+1n+1 n+1( ) (( , ( ( ); ), )) ]i R F HR F H Uˆ ˆHh' ++ σ βσ N

14.2 Return-Mapping Formulation for Subloading Surface Model 345 Then, one has

(k) (k) (k) (k)(k) (k) (k)

(k+1) n+1 n+1 n+1 n+1n+1 n+1 n+1n+1

(k) (k)(k) (k)(k) (k)n+1 n+1n+1 n+1n+1 n+1

( t ( ) ) ( ) , 1 r, /=, ( ) ( ) ( )t r , p

R F Htˆ ˆˆ ˆf

ˆ ˆˆ ˆ Mλ δ

+−−

σσ β σβ Nσσ βσ NβN E

(14.16) where

( )k( )(k) (k) (k) k

n+1 n+1 n+1 n+1n+1 ( , t r ( ) ), =p

iˆ ˆ HM σ σaβN

(k) (k) ( ) (k) (k)k( )kn+1 n+1 n+1 n+1n+1 n+1( , ( ( )); , ) iR F H ˆ ˆHh'+[ σ βσ N

1 1

1

(k) (k)( ) (k)k (k)( )n + n + k

n+1n+1n+1 n+1(k)n +

( ), ( , ( ) (t r

) ) ]i

fR F H|| || UH

ˆ'∂− +

∂β

b Nβσσ

(k) (k) (k)n+1 n+1 n+1(k) (k)n+1 n+1

( ), t r( ),

ˆ ˆ ˆˆfσN β σσ β (14.17)

Then, the plastic strain rate is obtained from Eqs. (14.12) and (14.16) as follows:

1(k+ )

+1

(k) (k) (k) (k)(k) (k) (k)n+1 n+1 n+1 n+1n+1 n+1 n+1

n (k) (k)(k) (k)(k) (k)n+1 n+1n+1 n+1n+1 n+1

( t =

( ) ) ( ) , 1 r, /, ( ) ( ) ( )t r ,

pp

R F H ˆ ˆˆ ˆft ˆ ˆˆ ˆ M

δ+

−−

Dσσ β σβ N

σσ βσ NβN E

(k) (k)n+1 n+1( ), ˆ σ βN (14.18)

Here, , , , H βσ α and pε are updated from Eqs. (14.4) and (14.7)-(14.9) by

1 1(k+1)(k) (k) (k)(k+1) (k) n nnn+1 n+1 n+11 + +n+1 +n+1= ( ) )p t tδ δ− (+σ σ σσ E Dσ W W− (14.19)

(k+1) (k) (k+1)( )k( )kn n n 1+1 +1 +n+1 n+1

( , = ; )piH H th H δ+ σ D (14.20)

1 1

(k)(k)(k+1) (k) (k+1)( )k( )kn nn+ n +n n n +11 +1+1 +1 n+1 +n+1( , = )|| || )p

i ttH δ δ(++a σ D ααα α W W− (14.21)

1 1(k)(k)( )(k+1) (k) (k+1)k( )k

n n nn+ +n n +1n +11n+1+1 +1 +n+1( , = ) |||| )pi tH tδ δ(+ βββ β + b σ D W W−

(14.22)

(k+1) (k) (k+1)nn+1 n+1 1+= || ||p p p tε ε δ+ D (14.23)

where pp t|| ||ε δ≡ ∫ D (14.24)

1(k+ )+1n

pD is given explicitly by Eq. (14.18).


(k+1)n+1R has to be calculated from Eq. (7.16) as

1 n+1 nn(k+1)n+1 ( )2 1 cos= cos exp 112 2

)( ) (p p

ee e

eeR RR RR u

RRε επ ππ

− +−−− − −−

(14.25) On the other hand, (k+1)

n+1R must be calculated by the numerical integration of Eq.

(14.10), if we adopt an evolution rule of the normal-yield ratio in which the

analytical relation of R to pε does not hold. The calculation must be continued until the error of the following equation

becomes less than a certain limit as shown in Fig. 14.2.

1 n+1 nn(k+1) (k+1)n+1 n+1 ( )( ) 2 coscos exp1, 112 2

)( ) (p p

ee

eef

R Rˆ uR RRε επ π

π−−

−−− − −−[σ β

]eR+ (k+1)n+1( ) 0=F H (14.26)

which is obtained by substituting Eq. (14.25) into (14.5).

Stress

Strain0

nσ

1elasticn+dσ

(1)1n+

pδσ

1n+ dtD

= ( )( ) RF Hf ˆ , βσ

1elasticn+σ

( )21n+

pδσ( )31n+

pδσ

Stress

Strain0

nσ

1elasticn+dσ

(1)1n+

pδσ

1n+ dtD

= ( )( ) RF Hf ˆ , βσ

1elasticn+σ

( )21n+

pδσ( )31n+

pδσ

Fig. 14.2 Return-mapping process attracting the stress to the subloading surface

14.2 Return-Mapping Formulation for Subloading Surface Model 347

Basic equations of the return-mapping method are shown above for the subloading surface model which is the only pertinent unconventional model fulfilling the continuity and smoothness conditions. On the other hand, the bounding surface model with a radial-mapping of Dafalias, which has somewhat similar structures to the subloading surface model, is incapable of incorporating the return-mapping method since no loading surface passing through the current stress point is assumed and thus its evolution rule is not incorporated in the subyield state.

The return mapping method executed so as to fulfill the equation of the subloading surface is shown above for the initial subloading surface model. It can be formulated for the extended subloading surface model in the similar way. The stress is attracted to the normal-yield surface and thus an accurate calculation would be executed in normal-yield state by adopting small incremental steps even if the return-mapping algorithm is not adopted.


Chapter 15 Constitutive Equation for Friction

15 Constitutive Equatio n for Frictio n

All bodies in the natural world are exposed to friction phenomena, contacting with other bodies, except for bodies floating in a vacuum. Therefore, it is indispensable to analyze friction phenomena rigorously in addition to the deformation behavior of bodies themselves in analyses of boundary value problems. The friction phe-nomenon can be formulated as a constitutive relation in a similar form to that of the elastoplastic constitutive equation of materials. A constitutive equation for friction with the transition from the static to the kinetic friction and vice versa and the orthotropic and rotational anisotropy is described in this chapter.

15.1 History of Constitutive Equation for Friction

Formulation of the friction phenomenon as a constitutive equation was first attained for a rigid-plasticity (Seguchi et al., 1974; Fredriksson, 1976). Subsequently, it was extended to an elastoplasticity (Michalowski and Mroz, 1978; Oden and Pires, 1983a,b; Curnier, 1984; Cheng and Kikuchi, 1985; Oden and Martines, 1986; Ki-kuchi and Oden, 1988; Wriggers et al., 1990; Wriggers, 2003; Peric and Owen, 1992; Anand, 1993; Mroz and Stupkiewicz, 1998; Gearing et al., 2001) in which the elastic springs between contact surfaces is incorporated. In them the isotropic hardening is introduced to describe the test results (cf. Courtney-Pratt and Eisner, 1957) exhibiting the smooth contact traction vs. sliding displacement curve reach-ing static-friction. However, the interior of the sliding-yield surface has been as-sumed as an elastic domain. Therefore, the plastic sliding velocity induced by the rate of traction inside the sliding-yield surface is not described. Needless to say, the accumulation of plastic sliding displacement induced by the cyclic loading of contact traction within the sliding-yield surface cannot be described by these models. They might be called the conventional friction model in accordance with the classification of plastic constitutive models by Drucker (1989). On the other hand, based on the concept of the subloading surface, the subloading-friction model (Hashiguchi et al., 2005b; Ozaki et al, 2007) falling within the framework of un-conventional plasticity, called the unconventional friction model, was formulated, which describes the smooth transition from the elastic to plastic sliding state and the accumulation of sliding displacement during a cyclic loading of tangential contact

350 15 Constitutive Equation for Friction

traction. Additionally, in this model, the reduction of friction coefficient with the increase of normal contact traction observed in experiments (cf. e.g. Bay and Wanheim, 1976; Dunkin and Kim, 1996; Gearing et al., 2001) is described by incorporating the nonlinear sliding-yield surface. Besides, the decrease has not been taken into account for Coulomb sliding-yield surface, which has been adopted widely to date as a constitutive models for friction.

It is widely known that when bodies at rest begin to slide against one other, a high friction coefficient appears first, which is called the static friction. Subse-quently, a friction coefficient decreases approaching a stationary value, called the kinetic friction. Furthermore, if the sliding ceases for a while and then starts again, the friction coefficient recovers and similar behavior to that of the initial sliding is reproduced (Dokos, 1946; Rabinowicz, 1951, 1958; Howe et al., 1955; Derjaguin, et al., 1957; Brockley and Davis, 1968; Kato et al., 1972; Richardson and Noll, 1976; Horowitz and Ruina, 1989; Ferrero and Barrau, 1997; Bureau et al., 2001). The recovery of friction coefficient has been formulated using equations including the time elapsed after the stop of sliding (cf. Rabinowicz, 1951; Howe et al., 1955; Brockley and Davis, 1968; Kato et al., 1972; Horowitz and Ruina, 1989; Bureau et al., 2001). However, the inclusion of time itself leads to the loss of objectivity in constitutive equations, since the evaluation of elapsed time after the stop of sliding is accompanied with arbitrariness, as known from the state in which a sliding ve-locity varies at a slow level. On the other hand, generally speaking, the variation of material property has to be described in terms of the sliding velocity, the stress and internal variables.

The reduction of the friction coefficient from the static to kinetic friction and the recovery of the friction coefficient as described above are the fundamental char-acteristics in friction phenomena, which have been widely recognized for a long time. Difference of the static and kinetic frictions often reaches up to several tens of percent, and thus the formulation taken account of these characteristics is of im-portance for analyses of practical problems in engineering.

In addition, difference of friction coefficients is observed in mutually opposite sliding directions. That difference can be described by the rotation of a sliding-yield surface, whereas the anisotropy of soils has been described by the rotation of a yield surface, as described in 11.4. Further, the difference of the range of friction coef-ficients is observed in different sliding directions. It would be describable by the concept of orthotropy of the sliding-yield surface (Mroz and Stupkiewicz, 1994).

15.2 Decomposition of Sliding Velocity

The sliding velocity v is defined as the relative velocity of the counter body and is

expressed by the normal part nv and the tangential part tv as follows (see Fig. 15.1):

= n t+v v v (15.1)

15.2 Decomposition of Sliding Velocity 351

fn

n

f

vtv

nvtf

Counter bodyfn

n

f

vtv

nvtf

Counter body

Fig. 15.1 Contact traction f and sliding velocity v

where

,= ( ) = ( ) =

(= = ) .n n

nt

v• ⊗

⊗

− ⎫⎬− − ⎭

v v n n n n v n

v v v I n n v (15.2)

n is the unit outward-normal vector at the contact surface. nv is the normal

component of the sliding velocity, i.e.

nv •≡−n v (15.3)

where the sign of nv is selected to be plus when the counter body approaches the

relevant body. Here, it is assumed that v is additively decomposed into elastic sliding velocity

ev and plastic sliding velocity pv , i.e.

= pe +vv v (15.4)

Then, nv and tv are expressed by the elastic and the plastic parts as follows:

,=

= ,

pn n n

p

e

et t t

⎫+ ⎪⎬

+ ⎪⎭

v v v

v v v (15.5)

where

= ( ) = ( ) ,=

(= = ) ,

n

t n

ne e e e

e ee e

v• ⊗

⊗

⎫− ⎪⎬

− − ⎪⎭

v v n n n n nv

n nv v v I v (15.6)


= ( ) = ( ) = ,

( .= = )

pp p pn

p p p pt n

nv• ⊗

⊗

⎫− ⎪⎬

− − ⎪⎭

v v n n n n v n

n nv v v I v (15.7)

nev and

pnv are the elastic and the plastic part, respectively, of nv in Eq. (15.3).

The contact traction f acting on the body is expressed by the normal traction

nf and the tangential traction tf as follows:

= n t+ ff f , (15.8) where

( ) ( )= = ,

(= ) =

n n

tt n

f

f

•≡ ⊗ − ⎫⎪⎬

≡ − − ⊗ ⎪⎭

f n f n n n f n

f f f I n n f t (15.9)

whilst n and t are the unit vectors in the directions of nf and tf , i.e.

, || || || ||n

n

t

t≡ ≡f fn tf f

(15.10)

and nf and tf are the magnitudes of nf and tf , i.e.

|| || , || ||=n n ttf f•≡ − − ≡nf ff (15.11)

while the sign of nf is selected to be plus when the relevant body is compressed by the counter body. Here, note that the directions of the tangential contact traction and the tangential sliding velocity are not necessary identical in general and thus

− ≠⊗ ⊗I n n t t in the three-dimensional space. Here, it holds that

( ( )), ,= = = =

( )= =

n n

t

f ••

•

− ⎫⊗∂ ∂∂ ∂ − ⎪∂∂ ∂ ∂ ⎪⎬− ⊗∂∂ ⎪− ⊗∂ ⎪∂ ⎭

n n nf f fn nff f f

n nI ff n nIff

(15.12)

2

( ) ( )/= = =nn

nn

tt ffff

⊗ − − ⊗ +− −⊗∂ ⊗∂∂ ∂

η nnη n n n nfI If f

f (15.13)

1 == =nn

|| || || ||ff

•• •⊗ +−∂ ⊗∂ ∂ ( ) +

∂ ∂∂η η η η nn nI η nηf f

τ τ τ

(15.14) where

t

nf≡ fη (15.15)

15.2 Decomposition of Sliding Velocity 353

dtv

Bdtv

dt•f

( )tf

( )tf

( )t dt+f

dtΩfdtf

t dt+

t

A dtv

B

A

B'

A'dtv

Bdtv

dt•f

( )tf

( )tf

( )t dt+f

dtΩfdtf

t dt+

t

A dtv

B

A

B'

A'

Fig. 15.2 Objectivities of the sliding velocity v and the corotational traction rate ff

( )0=|| ||•≡ η

nητ τ (15.16)

Now, let the elastic sliding velocity be given by the following hypo-elastic rela-tion, whilst the elastic sliding velocity is usually far smaller than the plastic sliding velocity in the friction phenomenon.

1 1,= = n nn

e et t

tα αv v ff (15.17)

where and n tf f are the normal component and tangential component, respec-

tively, of f , which are related to the material-time derivative as follows:

, , = = =n nn tt t• • •− − −f f Ωf f f Ωf f f Ωf (15.18)

which is derived from

( ) ( )= = = =n n nt t nn t tt• • ••− + − + +− + −f f f ff f Ωf Ω f Ωf f Ωf f f

(15.19)

where the skew-symmetric tensor Ω is the spin describing the rigid-body rotation

of the contact surface. nα and tα are the contact elastic moduli in the normal and

the tangential directions to the contact surface. On the other hand, the sliding ve-locity v is not an absolute velocity of a point on the body surface but the relative


velocity between two points on the contact surface, independent of the rigid-body rotation of contact surface (see Fig. 15.2), and thus it can be adopted to the con-stitutive relation having the objectivity. It follows from Eq. (15.17) that

= = een t+f f f C v (15.20)

where the second-order tensor eC is the contact elastic modulus tensor given by

1

),= (

1 1 )= (

e

e

n t

n t

α α

α α−

⊗ ⊗

⊗ ⊗

⎫+ − ⎪⎬

+ − ⎪⎭

n n n nIC

nn nn IC (15.21)

15.3 Normal Sliding-Yield and Sliding-Subloading Surfaces

Assume the following sliding-yield surface with the isotropic hardening/softening and the rotational isotropy, which describes the sliding-yield condition.

( , ) =f Ff β (15.22)

where F is the isotropic hardening/softening function denoting the variation of the size of the sliding-yield surface. The anisotropy of the friction phenomenon would be substantially described by the rotation of yield surface similarly to the defor-

mation of soils as a frictional material described in 11.4. β is the vector describing the rotation of the sliding-yield surface around the origin of traction space and it does not have the normal component since it describes the anisotropy. Hereinafter,

let β be assumed to be a fixed vector leading to =β 0 for sake of simplicity. Then,

it holds that

= 0•βn (15.23)

=•β βΩ . (15.24)

Here, for simplicity it is assumed that ( , )f f β is the homogeneous function of f in degree-one. Then, it holds that

=( , ) ( , )f fs sf β f β (15.25)

( , )= ( , )

ff•

∂∂f β

f f βf (15.26)

where s is an arbitrary positive scalar quantity, whilst Eq. (15.26) is based on Euler’s homogeneous function of degree-one. Therefore, the sliding-yield surface

15.4 Evolution Rules of Sliding-Hardening Function 355

retains a similar shape and orientation with respect to the origin of contact traction

space, i.e. =f 0 for = const.β

In what follows, we assume that the interior of the sliding-yield surface is not a purely elastic domain but that the plastic sliding velocity is induced by the rate of traction inside that surface. Therefore, let the sliding-yield surface be renamed as the normal sliding-yield surface.

Then, based on the concept of the subloading surface (Hashiguchi, 1980, 1989), we introduce the sliding-subloading surface, which always passes through the current contact traction point f and retains a similar shape and orientation to the normal sliding-yield surface with respect to the origin of contact traction space, i.e.

=f 0 . Let the ratio of the size of the sliding-subloading surface to that of the normal sliding-yield surface be called the normal sliding-yield ratio, denoted by

(0 1)R R≤ ≤ , where = 0R corresponds to the null traction state ( = 0f ) as

the most elastic state, 0 1R< < to the subsliding state ( 0 f F< < ), and

= 1R to the normal sliding-yield state in which the contact traction lies on the

normal sliding-yield surface ( =f F ). Therefore, the normal sliding-yield ratio R plays the role of three-dimensional measure of the degree of approaching the normal sliding-yield state. Then, the sliding-subloading surface is described by

( , ) =f RFf β . (15.27)


= R F R F• •

• +fN (15.28) where

( , )f∂≡ ∂

f βN f

(15.29)

The direct transformation of the material-time derivative to the corotational de-rivative is verified by substituting Eq. (15.18) into Eq.(15.28), noting

( ) = 0•a Ωa for an arbitrary vector a . The general proof is given in Chapter 4. 15.4 Evolution Rules of Sliding-Harde ning F unction

15.4 Evolution Rules of Sliding-Hardening Function and Normal Sliding-Yield Ratio

Evolution rules of the isotropic hardening function and the normal sliding-yield ratio are formulated so as to reflect experimental facts.

15.4.1 Evolution Rule of Sliding-Hardening Function

The following might be stated from the results of experiments.


1) The friction coefficient first reaches the maximal value of static-friction and then decreases to the minimal stationary value of kinetic-friction if the sliding commences. Physically, this phenomenon might be interpreted to result from separations of the adhesions of surface asperities between contact bodies because of the sliding (cf. Bowden and Tabor, 1958). Then, let it be assumed that the reduction results from the contraction of the normal sliding-yield surface, i.e., plastic softening caused by the sliding.

2) The friction coefficient recovers gradually with the elapse of time and the identical behavior as the initial sliding behavior exhibiting the static friction is reproduced if sufficient time elapses after the sliding ceases. Physically, this phenomenon might be interpreted to result from the reconstructions of the adhesions of surface asperities during the elapsed time under a quite high contact pressure between edges of surface asperities. Then, let it be assumed that the recovery results from the viscoplastic hardening because of the creep phenomenon.

Taking account of these facts, let the evolution rule of the isotropic harden-ing/softening function F be postulated as follows (Hashiguchi and Ozaki, 2008):

= 1 1)( ( )p nm

k s

F FFFF

κ ξ•

− − || || −+v (15.30)

where sF and ( )k k s FF FF ≥ ≥ are the maximum and minimum values of F

for the static and kinetic frictions, respectively. Both κ and m are the material

constants influencing the decreasing rate of F because of the plastic sliding, and

both ξ and n are the material constants influencing the recovering rate of F

because of the elapse of time, whereas ξ is a function of absolute temperature in

general. The first and the second terms in Eq. (15.30) stand for the deteriorations and the reformations, respectively, of the adhesions between surface asperities. On the other hand, these phenomena have been described by the softening because of the sliding displacement and the hardening due to the time elapsed after the stop of sliding up to the present. However, the inclusion of the time itself in constitutive equations is not allowed, as described in 15.1.

15.4.2 Evolution Rule of Normal Sliding-Yield Ratio

It is observed in experiments that the tangential traction increases first elastically with plastic sliding and thereafter it gradually increases approaching the normal sliding-yield surface similarly as in the plastic deformation described in 7.2. Then, similarly to the evolution rule (7.13) of the normal-yield ratio R for the plastic deformation, we assume the evolution rule of the normal sliding-yield ratio as follows:

15.5 Relations of Contact Traction Rate and Sliding Velocity 357

R10

|| || = ( )pR U R•

v

R10

|| || = ( )pR U R•

v|| || = ( )pR U R•

v

Fig. 15.3 Function

　( )U R　( )U R for the evolution rule of the normal sliding- yield ratio R R

|| ||= ( ) for p pR U R•

≠v 0v (15.31)

where ( )U R is a monotonically decreasing function of R fulfilling the following

conditions (Fig. 15.3).

( ) for = 0,

( ) 0 for = 1,=

( ( ) 0 for 1).

R RU

R RU

R RU

⎫→ +∞⎪⎬⎪< > ⎭

(15.32)

Let the function U satisfying Eq. (15.32) be simply given by

=( ) cot 2( )uR RU π (15.33)

where u is the material constant. Equation (15.31) with Eq. (15.33) can be led to

the analytical integration of R for accumulated plastic sliding ppu dt||| |≡ ∫ v

under the initial condition 0 := pp uu 0=R R as follows:

10 0

2 cos ( )= cos exp2 2 [ ( ) ]p pR u uR uπ ππ

− −− (15.34)

15.5 Relations of Contact Traction Rate and Sliding Velocity

The substitution of Eqs. (15.30) and (15.31) into Eq. (15.28) gives rise to the con-sistency condition for the sliding-subloading surface:


1 1 || ||= )( ( ) p pm n

k s

F F FR UF Fκ ξ• − −− || || ++fN vv (15.35)

Assume that the direction of plastic sliding velocity is tangential to the contact plane and outward-normal to the curve generated by the intersection of the slid-ing-yield surface and the constant normal traction plane = const.nf , leading to the tangential associated flow rule

( )|| ||= 0, = 1pn nλ λ >t tv (15.36)

where ( ) > 0λ is a positive proportionality factor and

( )|| ||( )

n•

•

− ⊗≡− ⊗

I n n NtI n n N

(15.37)

Substituting Eq. (15.36) into Eq. (15.35), the proportionality factor λ is derived as follows:

= p

cmm

λ • −fN (15.38)

and thus

=pp

cn

mm

• −fNv t (15.39)

where

1( )p m

k

Fm FR UFκ≡ − − + (15.40)

1 ( 0)( )ncs

Fm RFξ −≡ ≥ (15.41)

Substituting Eqs. (15.20) and (15.39) into Eq. (15.4), the sliding velocity is given by

1= p

cen

mm

− • −+ fNv C tf (15.42) The positive proportionality factor in terms of the sliding velocity, denoted by

the symbol Λ , is given from Eqs. (15.42) as

=p

e c

en

mm

Λ • •• •

−+CN v

tCN (15.43)

The traction rate is derived from Eqs. (15.4), (15.20), (15.36) and (15.43) as follows:

= ( )p

e cene

n

mm

• •

• •

−−+⟨ ⟩CN v tvCf

tCN (15.44)

15.6 Loading Criterion 359

15.6 Loading Criterion

While the loading criterion for the plastic sliding velocity is similar to that for the plastic strain rate described in 6.3, it will be described below.

First, note the following facts:

1. It is necessary that

= > 0λ Λ (15.45)

in the loading (plastic sliding) process p ≠v 0 . 2. It holds that

0• ≤fN (15.46)

in the unloading (elastic sliding) process =pv 0 . Further, because of = ev v

leading to = =e e e• • •• • fvN N NC C v in this process it holds that

=p

c

en

mm

Λ •• •

−+

fNCN t

(15.47)

while it should be noted that 0cm ≥ (Eq. (15.41)). 3. The plastic modulus pm takes both signs of positive and negative, while the

first and the second terms in Eq. (15.40) are negative and positive, respectively.

On the other hand, noting that the contact elastic modulus eC is the positive

definite tensor and thus it holds that pe m• •CN N in general and postu-lating that the plastic relaxation does not proceed infinitely, let the following inequality be assumed.

0p enm • •+ >tCN (15.48)

Then, in the unloading process =pv 0 , the following inequalities hold de-pending on the sign of the plastic modulus pm , i.e. the hardening, perfectly-plastic and softening states from Eqs. (15.38) and (15.46)-(15.48).

0 and 0 when 0

or indeterminate and 0 when = 0

0 and 0 when 0

p

p

p

m

m

m

λ Λλ Λλ Λ

⎫≤ ≤ >⎪

→ −∞ ≤ ⎬⎪<≥ ≤ ⎭

(15.49)

Therefore, the sign of λ at the moment of unloading from the state 0pm ≤ is

not necessarily negative. On the other hand, Λ is negative in the unloading proc-ess. Consequently, the distinction between a loading and an unloading processes


cannot be judged by the sign of λ but can be done by that of Λ . Therefore, the loading criterion is given as follows:

,: 0

= : otherwise.

p

p

Λ ⎫≠ ⎪⎬⎪⎭

>v 0

v 0 (15.50)

or

: ,> 0

= : otherwise.

p

p

e cm• • ⎫≠ − ⎪⎬⎪⎭

v 0 CN v

v 0 (15.51)

on account of Eq. (15.48).

15.7 Sliding-Yield Surfaces

It can be stated from experiments that the friction coefficient decreases with the increase of contact pressure (cf. Bay and Wanheim, 1976; Dunkin and Kim, 1996; Gearing et al., 2001; Stupkiewicz and Mroz, 2003). Therefore, the normal slid-ing-yield surface cannot be described appropriately by the Coulomb sliding-yield surface in which the tangential contact traction is linearly related to the normal contact traction through the constant friction angle. In what follows, the slid-ing-yield surface with the nonlinear relation of tangential contact traction and normal contact traction is assumed, by which the reduction of friction coefficient with the increase of normal contact traction is described.

The closed normal sliding-yield and the sliding-subloading surfaces can be de-scribed by putting

( ) ˆ=, ( )nf f g χf β (15.52)

as follows:

ˆ ˆ= =( ) , ( )n nf g f gF RFχ χ (15.53)

where ˆ

ˆˆ , || ||

Mχ −≡≡

η η η β (15.54)

M is the material constant denoting the traction ratio η ( )= t nf f/ at the maxi-

mum point of tf . Simple examples of the function ( )g χ in the sliding-yield

function in Eq. (15.52) are as follows:

ˆ ˆ ˆ, ( ) ( )= exp( ) = exp( )g g' χχ χ χ (15.55)

15.7 Sliding-Yield Surfaces 361

2 ˆˆ ˆ ˆ, ( ) ( )= 1 = 2g g' χχ χ χ+ (15.56)

2ˆ ˆ ˆ ˆ, ( ) exp ( )= ( /2) = exp( )g g' χχ χ χ χ (15.57)

2

1 1ˆˆ , ( ) ( )= =ˆ ˆ1 2(1 )/2 /2g g' χχ χ χ− −

(15.58)

All sets of Eqs. (15.22) and (15.52) with Eqs. (15.55)-(15.58) exhibit closed sur-

faces passing through points = 0nf and =nf F at = 0tf for =β 0 . Equa-

tion (15.55) and (15.56) are based on the original Cam-clay yield surface (Schofield and Wroth, 1968) and the modified Cam-clay yield surfaces (Roscoe and Burland,

1968), respectively, for soils described in 11.2. Equation (15.57) exhibits a tear-

drop-shaped surface (Hashiguchi, 1972, 1985; Hashiguchi et al., 2005b) which is reversed from the surface of Eq. (15.55) on the axis of normal contact traction.

Equation (15.58) exhibits a parabola (Hashiguchi et al., 2005b). The normal

sliding-yield and the sliding-subloading surfaces are depicted in Fig. 15.4 for Eq. (15.57) having the teardrop shape.

00

Sliding-subloading surfacetf

nf

M1

FRFeF/eRF/

f

Normal sliding-yield surface

00

Sliding-subloading surfacetf

nf

M1

FRFeF/eRF/

f

Normal sliding-yield surface

Fig. 15.4 Teardrop shaped normal sliding-yield and sliding-subloading surfaces

It holds for Eqs. (15.13) and (15.54) 2 that

ˆ= =

nf⊗ +−∂ ∂ ⊗

∂ ∂η nη η n nI

f f (15.59)

ˆ ˆ ˆ 1 == =ˆˆ ˆˆ

nn

|| || || ||ff

•• •⊗ +−∂ ⊗∂ ∂ ( ) +

∂ ∂∂ηη η nn nη I η nηf f

τ ττ (15.60)


where ˆ

(ˆ

ˆ ˆ )0=|| ||•≡ η

nη

τ τ (15.61)

Further, it holds from Eqs. (15.12), (15.21), (15.52), (15.54), (15.59) and (15.60) that

ˆˆ / 1 = = ˆ ˆn

|| || MfM

χ•∂ ∂ ( ) +

∂ ∂η η n

f fτ τ (15.62)

1ˆ ˆ( ) = ( ) ˆ ˆnn

g gffM'χ χ •− ( ) ++ ηn nN τ τ

ˆ ˆ( ) ( )ˆ( )= ( )ˆ ˆ g gg

M M' 'χ χχ •− − +η nτ τ (15.63)

( )=( )t •∂ − ⊗∂f

I n n Nf

ˆ ˆ ˆ( ) ( ) ( )ˆ( )= ( ) ( ) =ˆ ˆ ˆ [ ]g g ggM M M' ' 'χ χ χχ• •− −⊗ − +I n n η nτ τ τ

(15.64)

= ˆn τt (15.65)

ˆ ˆ( ) ( )ˆ( ) ( = )ˆ ˆ [ ]e g ggM M' 'χ χχ• •− − +η nN C τ τ

)(n tα α• ⊗ ⊗+ − nn n nI

ˆ ˆ( ) ( )ˆ( ) (= )ˆ ˆ n tg gg

M M' 'χ χα χ α•− − +η nτ τ (15.66)

ˆ ˆ( ) ˆ( ) ( )ˆ( ) ( )= =ˆ ˆ ˆ [ ]e nn t t

g g ggM M M

' ' 'χ χ χα χ α α•• • •− − +η nN C τ τ τt

(15.67)

The substitution of Eqs. (15.21) and (15.62)-(15.67) into Eqs. (15.42) and (15.44) leads to the sliding velocity vs. contact traction rate and its inverse relation are given as follows:

1 1 )(= tnα α⊗ ⊗+ − nn n nIv f

15.7 Sliding-Yield Surfaces 363

ˆ ˆ( ) ( )ˆ( ) ( )ˆ ˆˆ

[ ]p

cg gg m

M Mm

' 'χ χχ • •− − + − +

η n fτ ττ

1 1 1(= ))(n t tα α α• +− n nf f

ˆ ˆ( ) ( )ˆ( ) ( () )) (ˆ ˆˆ

p

cg gg m

MMm

' 'χ χχ • • •− − + −+

η n f fτ ττ (15.68)

) = (n tα α⊗ ⊗+ − nn n nf I

ˆˆ ( )( )ˆ( ) ( )

ˆ( )

ˆˆˆ

][[ ]p

cn t

t

ggg mMM

gm

M

''

'

χχ αα χ

χα

•• −− +−−

+⟨ ⟩

η vnv

τττ

(= ))(n t tα α α• +− n nv v ˆ ˆ( ) ( )ˆ ))( ) ( (()

ˆ( )

ˆˆˆ

p

cn tt

t

g gg mMM

gm

M

' '

'

χ χα χ αα

χα

•••− − −+−

+⟨ ⟩

η n vv τττ

(15.69) On the other hand, the normal sliding-yield and the sliding-subloading surfaces

for the circular cone of the Coulomb friction condition are given by putting

ˆ( ) , ==, || ||f F μηβf (15.70)

as follows: ˆ ˆ, = =|| || || || Rμ μη η (15.71)

where μ is the friction coefficient and the evolution rule is given in an identical form with Eq. (15.30) as follows:

= 1 || || 1)( ( )p nm

k s

μμμ κ ξ μμ• −− + −v (15.72)

sμ and kμ signify material constants for the maximum and the minimum fric-

tion, i.e. static and kinetic friction coefficients, respectively. ( , )f f β in Eq.

(15.70) is the homogeneous function of f in degree-zero. The normal sliding-yield


0

Normal sl

iding-yield surfa

ce

nf

Sliding-subloading surface

tf

f

0

Normal sl

iding-yield surfa

ce

nf


tf

f

Fig. 15.5 Normal sliding-yield and sliding-subloading surfaces for Coulomb friction condition

and sliding-subloading surfaces in Eq. (15.71) are open surfaces having a conical

shape, whereas they expand/contact with the increase/decrease of μ and R as shown in Fig. 15.5.

It holds for Eq. (15.71) that

1)( p m

km R U

μκ μμ≡ − − + (15.73)

1 ( 0) ( )ncs

m Rμ

ξ μ−≡ ≥ (15.74)

in stead of Eqs. (15.40) and (15.41). Further, it holds from Eqs. (15.12), (15.21), (15.60) and (15.70) that

ˆ 1 = = ˆ ˆn

|| ||f

•∂ ( ) +∂η η nNf

τ τ (15.75)

15.8 Basic Mechanical Behavior of Subloading-Friction Model 365

11( () )= = = ˆˆ ˆ( )

n

t

nff• • •

∂ − −⊗ ⊗ ( ) +∂f ηI n n I n nN nf ττ τ (15.76)

1 1 ) (= =ˆ ˆ ˆ ˆn n

en n ttf f

αα αα• •• • ⊗ ⊗( ) + ( ) ++ −η ηnn nn n nCN Iτ τ τ τ

(15.77)

1= ) =ˆ ˆˆ e tn n tn nf f

αα α• • • •( +ηN tC nτ ττ (15.78)

for Coulomb friction condition. The substitution of Eqs. (15.21) and (15.75)-(15.78) into Eqs. (15.42) and

(15.44) leads to the relation of sliding velocity vs. contact traction rate as follows:

1 1 1 )(=

ˆ ˆˆ n

p

c

n t

mfmα α

• •⊗ ⊗

( ) + −+ − +

η n fnn n nIv f

τ ττ

1 () ) (1 1 1(= ))(

ˆˆˆn

p

c

n t t

mf

mα α α• • •

•+( ) −

+− +η n f f

n nf fττ

τ

(15.79)

The inverse relation of Eq. (15.79) is given as follows:

1 ) = (

ˆ ˆˆ( )n

p

cn t

n t t

n

mf

mf

ααα α α

• •

⊗ ⊗−( ) +

−+ −+

⟨ ⟩η n v

n vn n nf Iτ τ

τ

)()( )((= ))(ˆˆ ˆ

pn

ct tnn t t

t

n

mf m

f

α ααα α α α•• ••

+ −−+−+

⟨ ⟩η n vvn nv vττ τ

(15.80)

15.8 Basic Mechanical Behavior of Subloading-Friction Model

We examine below the basic response of the present friction model by numerical experiments and comparison with test data for the linear sliding phenomenon without a rigid-body rotation under a constant normal traction and with the fixed direction of tangential contact traction on the assumption of isotropy for sake of simplicity. Then, it holds that


, ,=const. const. ,= = , ,,= = =

, , .= = =

n

tn t t

ppt t t

e e

f f

v v v

• •⎫⎪⎪⎬⎪⎪⎭

f t Ω 0

t tf 0 f f

v t v t v t (15.81)

from Eqs. (15.4), (15.20) and (15.36).

15.8.1 Relation of Tangential Contact Traction Rate and Sliding Velocity

Substituting Eqs. (15.81) into Eqs. (15.68) and (15.69), the tangential sliding ve-

locity vs. contact traction rate and its inverse relation are given as follows:

1( ) /1=11

( )

(( )) mm

n

tt

kk

st

F Rg M Fv fFF FF R UR U FF

' ξχα κκ

•−

+ −−− − +− +

(15.82)

( )1

=( )1

( )

( )

m

n

tt

k

t t st

t

g Fv RFMvfgF FR U

F M

'

'

χα ξα

χακ

•− −

−− +− +⟨ ⟩ (15.83)

On the other hand, substituting Eqs. (15.81) into Eqs. (15.79) and (15.80), these

relations for Coulomb friction condition are given as follows:

11/1=1 1

( )( ) ( )

n

n

m mtt

k k

st

Rfv f

R U R U

μξ μμ μα κ κμ μμ μ

•−

+ −− − − −+ +

(15.84)

1 1=

1

( )( )

t

n

n

n

tt m t

k

st

v Rff vR U f

μξ μα αμ μκ μ

• − −−

+− − +⟨ ⟩ (15.85)

The relation of the tangential contact traction tf vs. the tangential sliding dis-placement ( )ttu v d t≡ ∫ is schematically shown in Fig. 15.6 for a high sliding velocity process in which the creep hardening of the second term in Eq. (15.82) is negligible. The relation by the conventional friction model with the sliding-yield surface enclosing an elastic domain is also shown as bold curves. In


the subloading-friction model, the softening term ( ) ( 0)1 mkF/F Rκ− − ≤ in-

creases monotonically from the negative value to zero and inversely the normal sliding-yield term ( 0)FU ≥ decreases monotonically from the infinite value to zero in the denominator of the plastic sliding velocity in second term in the bracket in Eq. (15.82). In the initial stage of sliding, the plastic modulus is positive, i.e.

0pm > so that the tangential contact traction increases but thereafter these terms cancel mutually leading to = 0pm at which the tangential contact traction reaches the peak, i.e. the static friction point p . Thereafter, the softening term decreases gradually to zero but the normal sliding-yield term decreases rapidly resulting in 0pm < so that the tangential contact traction decreases to the kinetic friction point k .

elasticelastoplastic

Subloading-friction model

Conventional friction model

= const.nf

tftf

yyp

kF

k k

0F nfo0 0tu

Initia l normal sliding-yield surfaceat static friction

Final normal sliding-yield and

subloading-sliding surfacesat kinetic friction state

Initia l subloading-sliding surface

= , 1= ( )p pp

m

k

f FR Fm Um Fκ

•+− −v t

f•

o

+

λ

+ 0

0

p k

−

−

+

0F kFF

∞+ 0

0

(Softening)

1( )m

k

FR FUF

κ +− −

⎫⎫⎬⎬⎭⎭

p

elasticelastoplastic

Subloading-friction model

Conventional friction model

= const.nf

tftf

yyp

kF

k k

0F nfo0 0tu

Initia l normal sliding-yield surfaceat static friction

Final normal sliding-yield and

subloading-sliding surfacesat kinetic friction state

Initia l subloading-sliding surface

= , 1= ( )p pp

m

k

f FR Fm Um Fκ

•+− −v t

f•

o

+

λ

+ 0

0

p k

−

−

+

0F kFF

∞+ 0

0

(Softening)

1( )m

k

FR FUF

κ +− −

⎫⎫⎬⎬⎭⎭

p

Fig. 15.6 Prediction of linear sliding behavior from the static to the kinetic friction by the conventional friction and the subloading friction models at a high sliding rate without the creep hardening

15.8.2 Numerical Experiments and Comparisons with Test Data

Numerical experiments and comparisons with test data for the subloading-friction model are shown below, whilst Eq. (15.83) with the teardrop shaped normal slid-ing-yield surface in Eq. (15.57) is adopted.


0.8

0.6

0.4

0.2

0.0

/t nf f

0.1 0.20.0(mm)tu

11.0−

5= 1.0 mm/stv − 41.0−

21.0−

31.0−

/sntf f

_

/kntf f_

/ sF F

1.0

0.8

0.6

0.4

0.2

0.0

Stationary time (s)0.1 0.20.0

(mm)tu0 200 400 600

0

0

0

0

051.0−

41.0−

21.0−

31.0−

11.0 mm/stv −=

0.8

0.6

0.4

0.2

0.0

/t nf f

0.1 0.20.0(mm)tu

11.0−

5= 1.0 mm/stv − 41.0−

21.0−

31.0−

/sntf f

_

/kntf f_

/ sF F

1.0

0.8

0.6

0.4

0.2

0.0

Stationary time (s)0.1 0.20.0

(mm)tu0 200 400 600

0

0

0

0

051.0−

41.0−

21.0−

31.0−

11.0 mm/stv −=

Fig 15.7 Variations of the friction coefficient and hardening function with tangential sliding distance and stationary time for various tangential sliding velocity ( u=100mm-1)

For numerical experiments, the material parameters and the normal contact traction are selected as follows:

0

1

= = 100MPa, = 30MPa, = 0.3,

= 1 000MPa/mm, = 1, = 10MPa/s, = 2,

= 5, 10, 50, 100 and 1 000mm ,

= = 100MPa/mm,

= 10MPa

k

tn

n

sF F F M

m n

u

f

κ ξ

α α

−

where 0F is the initial value of F . The evolution rule = ln( ) uR RU − of the

normal sliding-yield ratio is used in stead of Eq. (15.33).


0.1mm/stv =

1005010

5

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.001mm/stv =

1005010

5

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.00001mm/stv =

1005010

5

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

= 1000mmu 1−

= 1000mmu 1−

= 1000mmu 1−

0.1mm/stv =

1005010

5

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.001mm/stv =

1005010

5

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.00001mm/stv =

1005010

5

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

= 1000mmu 1−

= 1000mmu 1−

= 1000mmu 1−

0.1mm/stv =

1005010

5

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.001mm/stv =

1005010

5

0.8

0.6

0.4

0.2

0.0

/t nf f

0.05 0.10 0.15 0.20 0.250.0(mm)tu

0.00001mm/stv =

1005010

5

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

= 1000mmu 1−

= 1000mmu 1−

= 1000mmu 1−

Fig. 15.8 Influences of the material constant u in the evolution rule of normal sliding-yield surface on the relation of friction coefficient vs. tangential sliding displacement for three levels of tangential sliding velocity


Stationary time: 1s

10ξ =

0ξ =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

/t nf f

0.6

0.8

0.4

0.2

0.0

Stationary time: 200s

10ξ =

0ξ =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

/t nf f

0.6

0.8

0.4

0.2

0.0

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

Stationary time: 1s

10ξ =

0ξ =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

/t nf f

0.6

0.8

0.4

0.2

0.0


10ξ =

0ξ =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

/t nf f

0.6

0.8

0.4

0.2

0.0

/sntf f

_

/kntf f_

/sntf f

_

/kntf f_

Fig. 15.9 Influence of stationary time on recovery of friction coefficient ( 　　　　　 )0 01mm/stu .=( 　　　　　 )0 01mm/stu .=

Variations of the friction coefficient /t nf f with the tangential sliding dis-

placement tu for various tangential sliding velocities are shown in Fig. 15.7, where

and kst tf f are the values of the tangential contract traction tf calculated from

the normal sliding-yield surface for = and =s kF F F F , respectively, and the

prescribed constant value of nf . The reduction of friction coefficient becomes remarkable with the increase of tangential sliding velocity tv . The reduction of the hardening function F with the tangential sliding displacement and its recovery with the time elapsed after the stop of sliding are also shown in this figure, while almost complete recover is realized for the stationary time of 400s.

In addition, the influence of the material constant u in the evolution rule of

normal sliding-yield ratio R on the friction coefficient is shown in Fig. 15.8. The


curve of tangential contract traction vs. tangential sliding displacement becomes smoother for the smaller vale of u inducing a larger plastic sliding velocity by the rate of contract traction inside the normal sliding-yield surface, whilst the conven-tional friction theory premising that the interior of sliding yield surface is an elastic domain is realized for u → ∞ .

The influence of the stationary time on the recovery of the friction coefficient is shown in Fig. 15.9. The recovery is larger for a longer stationary time.

The accumulation of sliding displacement under the cyclic loading of tangential

contact traction of 80% of s

tf is shown in Fig. 15.10, whilst it cannot be predicted

by the conventional friction model ( u → ∞ ). It is shown that the accumulation proceeds more quickly as the sliding velocity increases.

0.00001mm/stv =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

0.8

0.4

0.2

0.0

/t nf fConventional model

80%0.6

0.1mm/stv =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

0.8

0.4

0.2

0.0

/t nf f80%

0.6 Conventional model

Present model

Present model

/sntf f

_

/sntf f

_

0.00001mm/stv =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

0.8

0.4

0.2

0.0

/t nf fConventional model

80%0.6

0.1mm/stv =

0.1 0.2 0.3 0.4 0.50.0(mm)tu

0.8

0.4

0.2

0.0

/t nf f80%

0.6 Conventional model

Present model

Present model

/sntf f

_

/sntf f

_

Fig. 15.10 Accumulation of sliding displacement under cyclic loading for two levels of sliding velocity ( u=100mm-1)


0.02 0.04 0.06 0.08 0.10.0

(mm)tu

/t nf f

0.5

0.6

0.4

0.2

0.0

0.1

0.3

ExperimentPresent theory

0.02 0.04 0.06 0.08 0.10.0

(mm)tu

/t nf f

0.5

0.6

0.4

0.2

0.0

0.1

0.3


0.02 0.04 0.06 0.08 0.10.0

(mm)tu

/t nf f

0.5

0.6

0.4

0.2

0.0

0.1

0.3

ExperimentPresent theoryExperimentPresent theory

Fig. 15.11 Reduction process of friction coefficient from the static to kinetic-friction under the infinitesimal sliding velocity (Test data after Ferrero and Barrau, 1997)

The comparison with test data for the reduction process of friction coefficient from the static- to kinetic-friction is shown in Fig. 15.11. The test curve for sliding

between roughly polished steel surfaces (Ferrero and Barrau, 1997) under the in-

finitesimal sliding velocity 0.0002 mm/stv ≤ is simulated well enough by the present model, where the material parameters are selected as follows:

0

1

= = 120MPa, = 25MPa, = 0.28,

= 3 000MPa/mm, = 2, = 0.1MPa/s, = 2,

= 1 500mm ,

= = 10GPa/mm,

= 10MPa, = 0.0002mm/s

k

tn

tn

sF F F M

m n

u

f v

κ ξ

α α

−

The comparison with test data for the recovery of friction coefficient by the stop

of sliding on the way of the reduction process from the static- to kinetic-friction is depicted in Fig. 15.12. The test curves for sliding between roughly polished steel

surfaces (Ferrero and Barrau, 1997) under the infinitesimal sliding velocity

0.0002 mm/stv ≤ and the stationary time 20s and 400s are simulated suffi-ciently well by the present model, where the material parameters are selected as

follows:


0

1

= = 120MPa, = 30MPa, = 0.28,

= 3 000MPa/mm, = 2, = 0.1MPa/s, = 2,

= 1 500mm ,

= = 100GPa/mm,

= 10MPa, = 0.0002mm/s

k

tn

tn

sF F F M

m n

u

f v

κ ξ

α α

−

1( 3000 mm )u −=

/t nf f

0.5

0.6

0.4

0.2

0.3



0.01 0.02 0.03 0.04 0.050.0(mm)tu

1( 1500 mm )u −=


/t nf f

0.5

0.6

0.4

0.2

0.3


0.01 0.02 0.03 0.04 0.050.0(mm)tu

1( 3000 mm )u −=

/t nf f

0.5

0.6

0.4

0.2

0.3



0.01 0.02 0.03 0.04 0.050.0(mm)tu

1( 1500 mm )u −=


/t nf f

0.5

0.6

0.4

0.2

0.3


0.01 0.02 0.03 0.04 0.050.0(mm)tu

1( 3000 mm )u −=

/t nf f

0.5

0.6

0.4

0.2

0.3



0.01 0.02 0.03 0.04 0.050.0(mm)tu

1( 1500 mm )u −=


/t nf f

0.5

0.6

0.4

0.2

0.3


0.01 0.02 0.03 0.04 0.050.0(mm)tu

Fig. 15.12 Recovery of friction coefficient by the stop of sliding in the reduction process from the static-to kinetic-friction under the infinitesimal sliding velocity (Test data after Ferrero and Barrau, 1997)


/maxt nf f

0.4

0.3

0.2

0.1

Stationary contact time (s)10 20 30 40 500


/maxt nf f

0.4

0.3

0.2

0.1



/maxt nf f

0.4

0.3

0.2

0.1



Fig. 15.13 Recovery process of friction coefficient with the elapsed time after the stop of sliding (Test data after Brockley and Davis, 1968)

The comparison with test data for the recovery of friction coefficient is shown in Fig. 5.13. The sliding was first given reaching the kinetic-friction and then the tangential contact traction was unloaded to zero. Further, after the stop of sliding for a while the sliding was given again. The relations of the maximum value of fric-tional coefficient max

ntf / f vs. the time elapsed after the stop of sliding are plotted in this figure. The test curve for sliding between hardened and fully-annealed coppers (Brockley and Davis, 1968) is simulated sufficiently well by the present model, where the material parameters are selected as follows:

0

1

= = 120MPa, = 200MPa, = 0.16,

= 10MPa/mm, = 2, = 230MPa/s, = 2,

= 3 000mm ,

= = 10MPa/mm,

= 138kPa, = 0.1mm/s (before stop of sliding)

k

tn

tn

sF F F M

m n

u

f v

κ ξ

α α

−

The similar comparison with test data for sliding between cast irons (Kato et al., 1972) is shown in Fig. 5.14. The test data are simulated quite well by the present model, where the material parameters are selected as follows:

0

1

= = 3 000kPa, = 50kPa, = 0.19,

= 5MPa/mm, = 2, = 20MPa/s, = 2,

= 1 000mm ,

= = 10MPa/mm,

= 33.4kPa, = 0.05mm/s (before stop of sliding)

k

tn

tn

sF F F M

m n

u

f v

κ ξ

α α

−

15.9 Extension to Orthotropic Anisotropy 375


Stationary contact time (s)

/maxt nf f

0.5

0.4

0.2

0.1

0.3

50 100 150 200 2500



/maxt nf f

0.5

0.4

0.2

0.1

0.3

50 100 150 200 2500



/maxt nf f

0.5

0.4

0.2

0.1

0.3

50 100 150 200 2500

Fig. 15.14 Recovery process of friction coefficient with the elapsed time after the stop of sliding (Test data after Kato et al., 1972)

15.9 Extension to Orthotropic Anisotropy

The difference of friction coefficients in mutually opposite sliding directions can be described by the aforementioned rotational anisotropy. However, the difference of friction coefficients in the mutually-perpendicular directions cannot be described by the rotational anisotropy. In order to extend so as to describe this difference, let the concept of orthotropy be further incorporated below.

The simple surface asperity model is illustrated in order to obtain the insight for the anisotropy in Fig. 15.15. Here, the directions in the inclination of surface asperities would lead to rotational anisotropy. In addition, the anisotropic shapes and intervals of surface asperities would lead to the orthotropic anisotropy.

Now, choosing the bases 1*e and 2*e in the directions of the maximum and

the minimum principal directions of anisotropy, respectively, and letting 3*e

coincide with n to make the right-hand coordinate system 21 3( , , )∗ ∗ ∗e e e , it can

be written as

21 32 31

1 2 331 2

=

=

f ff

β β β

∗ ∗∗ ∗ ∗∗ ⎫+ + ⎪⎬∗∗ ∗ ∗∗ ∗+ + ⎪⎭

f

βe e e

ee e (15.86)

while the spin Ω of the base 21 3( , , )∗ ∗ ∗e e e coinciding with that of the contact

surface and thus it holds that


** , =r r i i•• ∗ ∗≡ ⊗eΩ e e eΩ (15.87)

Equation (15.86) is rewritten by 21 31 2= = =, , ,nt tf f f ff f∗ ∗∗ ∗ ∗ ∗−

21 1 2= =, ,t tβ β β β∗ ∗∗ ∗ 3 0=β ∗ as follows:

21 31 2

1 21 2

=

=

n

t t

t t ff f

β β

∗ ∗∗ ∗ ∗⎫+ − ⎪⎬∗∗ ∗ ∗+ ⎪⎭

f

βe e e

ee (15.88)

2∗e

1∗e

2∗e

1∗e

Fig. 15.15 Surface asperity model suggesting the rotational and the orthopic anisotropy

Invoking the orthotropic anisotropy proposed by Mroz and Stupkiewicz (1994), let Eq. (15.52) with Eq. (15.53) taking account of the rotational anisotropy be extended as follows:

, ˆ( ) = ( )nf f g χ∗f β (15.89)

ˆ ˆ= =( ) , ( )n nf g f gF RFχ χ∗ ∗ (15.90)


where

2 2 2121 1 2

21

ˆ ˆˆ ˆ ˆ ˆˆ , ,MM

η ηχ χ χ χχ ∗ ∗∗ ∗ ∗ ∗∗ + ≡ ≡≡ (15.91)

1M and 2M are the material constants standing for the values of M in the

maximum and the minimum principal directions of anisotropy, respectively. The sliding-yield surface having the rotational and the orthotropic anisotropy is depicted in Fig. 15.16.

v 2v∗

1∗e

1β ∗2β ∗

nf

tf

3 =∗e n

f1f∗

0

2f∗

f

1nf M1nf M

2nf M

2nf M

0

Rotation

1v∗

2∗e

Normal sliding - yield surface


2f ∗

1f ∗

βv 2v∗

1∗e

1β ∗2β ∗

nf

tf

3 =∗e n

f1f∗

0

2f∗

f

1nf M1nf M

2nf M

2nf M

0

Rotation

1v∗

2∗e



2f ∗

1f ∗

βv 2v∗

1∗e

1β ∗2β ∗

nf

tf

3 =∗e n

f1f∗

0

2f∗

f

1nf M1nf M

2nf M

2nf M

0

Rotation

1v∗

2∗e



2f ∗

1f ∗

β

Fig. 15.16 Sliding-yield surface with the rotational and the orthotropic anisotropy


The partial derivatives for Eq. (15.89) are given as

2

1

1 2

ii i i

ii i

ii iii i

i

i ii

ii i

i ii i i

ˆ ( ) // 1= =

ˆ ( )/ /== =

ˆ 1 ˆˆ= 2 =ˆ ˆ2

ˆˆ ˆ ˆ ˆ 11= = =ˆ ˆ

ˆˆˆ ˆ ˆ= ˆ ˆ

n

n

n

n n nn

nn

n n

t

t t

t t

t t

f Mff f f M

f ff Mf f ff M

ff f f M M

f f

χ β

χ β χ

χ χ ζχ χ

χ χ χ χ ζχ χ

χχ χ χχ χ

∗ ∗∗ −∂∂∗ ∗∂ ∂

∗ ∗∗∗ ∗−−∂∂ −∂ ∂

∗∂ ∗∗∗ ∗∂

∗∗ ∗∗ ∗ ∂∂ ∂∗ ∗∗ ∗∂∂ ∂

∗∗ ∗ ∗ ∂∂∂ ∂ ∂+∗ ∗∂ ∂∂ ∂2

21 211 ˆ ˆ= ( )

n nf fχ χχζ ζ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪

∗ ⎪∗∗ ∗∗− + ⎪∂ ⎪⎭

(15.92)

where

ii

ˆˆˆχ

ζ χ∗

≡ ∗ (15.93)

Subscript i takes 1 or 2 and is not summed even when it is repeated. It holds from Eqs. (15.21) and (15.92) that

212 21 1 21 21

1 221

ˆ ˆˆ ˆ ˆˆ 1 ˆ ˆ )(= = n nt t f ff f M M

χ χ χχ ζ ζ χχζ ζ∗ ∗∗ ∗ ∗∗ ∂ ∂ ∂∂ ∗∗ ∗ ∗∗ ∗∗ ∗++ − + +∗ ∗ ∂∂ ∂ ∂f

e ee en n

(15.94)

ˆ ˆ( )( ) ˆˆ ( )= = ( )n nn

gf f gg f 'χ χχχ∗ ∗∂ ∂ ∂∗∗ +∂ ∂ ∂

Nf f f

2121 21 21

1 2

ˆ ˆ1 ˆ ˆˆˆ ( ) )= (( ) nn

gg ff M M

'ζ ζχ χχ χζ ζ

∗ ∗ ∗ ∗∗ ∗∗ ∗∗∗ +− + + + een n

2121 21 21

1 2

ˆ ˆˆ ˆˆˆ ( ) )= (( ) g g

M M'

ζ ζχ χχ χζ ζ∗ ∗ ∗ ∗∗ ∗∗ ∗∗∗ +− + + +e nen

1 21 2 21 21

1 2

ˆ ˆˆ ˆ ˆˆˆ( ) = ( ( ) )( )( )g gg

M M' '

ζ ζχ χχ χχζ ζ∗ ∗∗ ∗ ∗∗ ∗ ∗∗∗∗−+ − + ne e

(15.95)


( () )=•− −⊗ ⊗I INn n n n

1 21 2 21 21

1 2

ˆ ˆˆ ˆ ˆˆˆ ( ) (( ) )( )( ) ][g gg

M M' '

ζ ζχ χχ χχζ ζ•∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗∗∗− −+ + ne e

1 21 2

21

ˆ ˆˆ( )= ( )g

MM'

ζ ζχ ∗ ∗∗ ∗∗ +e e (15.96)

1 21 2

1 2

2 21 2

1 2

ˆ ˆ

=ˆ ˆ

( ) ( )n

M M

MM

ζ ζ

ζ ζ

∗ ∗∗ ∗+

∗ ∗+

t

e e

(15.97)

1 21 2 21 21

1 2

ˆ ˆˆ ˆ ˆˆˆ= ( ) (( ) )( )( )[ ]e g gg

M M' '

ζ ζχ χχ χχζ ζ•∗ ∗∗ ∗ ∗∗ ∗ ∗∗∗∗− −+ +CN ne e

3 3 1 1 2 2 ( )n tα α• ⊗ ⊗ ⊗∗ ∗ ∗ ∗ ∗ ∗+ +e e e e e e

1 221 21 12

1 2

ˆ ˆ ˆ ˆˆˆˆ (( ) )( )= ( )( ) nt gggM M

''ζ ζ χχ χα χα χ ζ ζ

∗ ∗ ∗∗ ∗∗∗∗ −∗ ∗∗ − ++ ne e

(15.98)

1 21 2

1 2

ˆ ˆˆ= ( ) ( )[e

tn gM M

'ζ ζα χ• •

∗ ∗∗ ∗∗ +N tC e e

21 21ˆˆˆ (( ) )( ) ]n gg ' χχ χα χζ ζ ∗∗ ∗∗∗∗− − + n

1 21 2

1 2

2 21 2

1 2

ˆ ˆ

ˆ ˆ( ) ( )

M M

MM

ζ ζ

ζ ζ•

∗ ∗∗ ∗+

∗ ∗+

e e

2 21 2

1 2

ˆ ˆˆ= ( ) ( ) ( )t g

M M'

ζ ζχα∗ ∗∗ + (15.99)


Substituting Eqs. (15.21) and (15.95)-(15.99) into Eqs. (15.42) and (15.44), we obtain the sliding velocity vs. contact traction rate and its inverse relation as follows:

1 221

1 1 )( (= ) nn tf f fα α⊗ ⊗ ∗ ∗+ − −+v In n n n e e n

1 221 211 2

1 2

1 221

ˆ ˆ ˆˆ ˆ ˆ ˆ ( )( ) ( )( )

( )

( ) ][

p

nc

gggM M

f f f m

m

''ζ ζ χχ χχχ ζ ζ

•

∗ ∗ ∗∗ ∗∗∗ ∗− −∗ ∗∗ ++

∗ ∗ −+ −+

ne e

e e n

1 221

1 1(= ) nntf f fαα ∗ ∗ −+ ne e

1 221

1 2

21 21

ˆ ˆˆ( )

ˆ ˆˆˆ (( ) ) ( )

( )

p

nc

g f fM M

g mg fm

'

'

ζ ζχ

χ χχχ ζ ζ

∗ ∗∗ +

∗∗ ∗∗∗− −∗+ ++

1 21 2

1 2

2 21 2

1 2

ˆ ˆ

ˆ ˆ

( ) ( )

M M

M M

ζ ζ

ζ ζ

∗ ∗∗ ∗+

∗ ∗+

e e

(15.100)

11 1 22 212 ( = ) [ nn t vv vα α⊗ ⊗ ⊗∗ ∗ ∗ ∗ ∗ ∗+ ++ −f n n ne ee e e e

1 21 2

1 2

1 22 211 21

2 21 2

1 2

ˆ ˆ ˆˆˆ ( )( )( )

ˆ ˆ ( ) ( )

ˆ ˆˆ( )

( )

( ) ( )

p

n

nt

c

t

gggMM

vv v m

gmMM

''

'

ζ ζ χχαα χ

χχζ ζ

ζ ζχα

•

∗ ∗ ∗∗∗ ∗∗ − −+

∗ ∗∗∗ ∗∗ + −+ −−

∗ ∗∗+ +⟨ ⟩

e e

e en n


1 21 2

1 2

2 21 2

1 2

ˆ ˆ

ˆ ˆ( ) ( )

]M M

M M

ζ ζ

ζ ζ

∗ ∗∗ ∗+

∗ ∗+

e e

1 21 2(= ) nnt vv vαα ∗ ∗+ − ne e

1 21 2

1 2

21 21

2 21 2

1 2

ˆ ˆˆ( )

ˆ ˆˆˆ ( ( ) )( )

ˆ ˆˆ( )

( )

( ) ( )

p

n

t

cnt

t

g v vMM

gg v m

gmM M

'

'

'

ζ ζα χ

χχ χα χζ ζαζ ζχα

∗∗∗ +

∗∗ ∗∗∗∗ − −+ +− ∗ ∗∗+ +

⟨ ⟩

1 21 2

1 2

2 21 2

1 2

ˆ ˆ

ˆ ˆ( ) ( )

M M

M M

ζ ζ

ζ ζ

∗ ∗∗ ∗+

∗ ∗+

e e

(15.101)

Equation (15.70) with Eq. (15.71) for the Coulomb friction condition with rota-tional anisotropy can be extended to the orthotropic anisotropy as follows:

ˆ( ) , ==, cf Fχ μ∗βf (15.102)

ˆ ˆ, = =c c Rχ χμ μ∗ ∗ (15.103)

where

2 2 1 221 1 2

21

ˆ ˆˆ ˆ ˆ ˆ ˆ, , c cc c cC Cη ηχ χ χ χ χ∗ ∗∗ ∗ ∗ ∗ ∗+ ≡ ≡≡ (15.104)

1 = 1C and 2 ( 1)C ≤ is the material constant, whereas the 1 -∗e direction is

chosen for the long axis of ellipsoid in the section of sliding-yield surface so that μ

designates the friction coefficient in the 1 -∗e direction. The partial derivatives for

Eq. (15.102) are given as follows:


1

12

1

ii i

ii

ii ii i

i

iii

i i i

iii ii

ˆ ( ) // 1= =

ˆ ˆ( )/ /== =

ˆ 1 ˆˆ= 2 =ˆ ˆ2

ˆˆ ˆˆˆ 11= = =ˆˆ

ˆ ˆ=

ˆ

n

nj

n

n n nn

c

c

cc

nnc

c c

n

c

c c

ccc

cc c

c

c

t

t t

t t

t t

f f Cf Cf f

f ff Cf f f fC

f Cf Cf f

f

χ β

χ χβ

χ χ ζχ χ

χ χχχ ζχχ

χ χχ

∗∗ ∗−∂ ∂∗ ∗∂ ∂

∗ ∗∗ ∗ ∗−−∂ ∂ −∂ ∂

∗∂ ∗∗∗ ∗∂

∗∗ ∗∗∗ ∂∂∂∗ ∗ ∗∗∂ ∂∂

∗ ∗ ∂∂ ∂∗∂ ∂

211 1 2 2

2

ˆˆ ˆ 1 ˆ ˆ=ˆ

( )c

n n n

ccc cc c

cf f f

χχ χ χ χζ ζχ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪

∗∗ ⎪∗ ∂∂ ∗ ∗∗ ∗+ − + ⎪∗∂ ∂∂ ⎪⎭

(15.105)

Further, it holds from Eqs. (15.21) and (15.105) that

1 221 1 21

1 2

ˆ ˆˆ 1 ˆˆ )(== n

c ccc c ccf CC

χ ζ ζ χ χζζ∗ ∗∗∂ ∗ ∗∗ ∗∗∗ ++ +

∂Nf

ee n (15.106)

1 221 1 21

1 2

ˆ ˆ1 ˆˆ )( ( () )= n

c cc c ccf CC

ζ ζ χ χζζ• •∗ ∗ ∗ ∗∗ ∗∗∗ +− − + +⊗ ⊗I IN een n n n n

2121

1 2

ˆˆ1= ( )n

ccf CC

ζζ ∗∗ ∗∗ + ee (15.107)

211 2

1 2

2 221

1 2

ˆˆ

=ˆˆ

( ) ( )

cc

n

cc

CC

CC

ζζ

ζζ

∗∗∗ ∗+

∗∗+

te e

(15.108)

1 221 1 21

1 2

ˆ ˆ1 ˆˆ )(= n

c cec c ccf CC

ζ ζ χ χζζ•∗ ∗ ∗ ∗∗ ∗∗∗ ++ +CN nee

3 3 1 1 2 2 ( )n tα α• ⊗ ⊗ ⊗∗ ∗ ∗ ∗ ∗ ∗+ +e e e e e e


1 21 1 21

1 2

ˆ ˆ1 ˆˆ )(= ( ) n

c cn c c ct cf CC

ζ ζ χ χα α ζζ∗ ∗ ∗ ∗ ∗∗∗ +++ ne (15.109)

1 21 1 21

1 2

ˆ ˆ1 ˆˆ )(= ( ) n

c ce n c c ct cn f CCζ ζ χ χα α ζζ• •

∗ ∗ ∗ ∗ ∗∗∗ +++N tC ne

211 2

1 2

2 221

1 2

ˆˆ

ˆˆ( ) ( )

cc

cc

CC

CC

ζζ

ζζ•

∗∗∗ ∗+

∗∗+

e e

2 221

1 2

ˆˆ= ( ) ( )

n

t ccf CCα ζζ ∗∗

+ (15.110)

Substituting Eqs. (15.21) and (15.106)-(15.110) into Eqs. (15.42) and (15.44), we obtain the sliding velocity vs. contact traction rate and its inverse relation as follows:

1 221

1 1 )( (= ) nn tf f fα α⊗ ⊗ ∗ ∗+ − −+v In n n n e e n

1 221 1 21

1 2

1 221

ˆ ˆ1 ˆˆ )(

( )

n

p

n

cc cc cc

c

f CC

f f f m

m

ζ ζ χ χζζ

•

∗ ∗ ∗ ∗∗ ∗∗∗ ++ +

∗ ∗ −+ −+

ee n

e e n

211 2

1 2

2 221

1 2

ˆˆ

ˆˆ( ) ( )

cc

cc

CC

CC

ζζ

ζζ

∗∗∗ ∗+

∗∗+

e e

1 221

1 1(= ) nntf f fαα ∗ ∗ −+e e n

1 221 1 211 2

ˆ ˆ1 ˆˆ )( n

p

nc c c

c c ccf f mff CC

m

ζ ζ χ χζζ∗ ∗

∗ ∗ ∗∗ + −+ −+


211 2

1 2

2 221

1 2

ˆˆ

ˆˆ( ) ( )

cc

cc

CC

CC

ζζ

ζζ

∗∗∗ ∗+

∗∗+

e e (15.111)

11 1 22 212 ( = ) ( nn t vv vα α⊗ ⊗ ⊗∗ ∗ ∗ ∗ ∗ ∗+ ++ −f n n ne ee e e e

1 21 211

21

1 221

2 221

1 2

ˆ ˆ1 ˆˆ )(

( )

ˆˆ

)(

( ) ( )

n

p

n

n

c cn c c ct c

c

t cc

f CCvv v m

m f CC

ζ ζ χ χα α ζζ

α ζζ

•

∗ ∗ ∗ ∗∗∗ ∗+++∗ ∗+ − −

−∗∗

+ +

⟨ ⟩ne

e e n

211 2

1 2

2 221

1 2

ˆˆ

ˆˆ( ) ( )

cc

cc

CC

CC

ζζ

ζζ

∗∗∗ ∗+

∗∗+

e e

1 21 2(= ) nnt vv vαα ∗ ∗+ − ne e

1 21 1 21221

2 21 2

1 2

ˆˆ1 ˆˆ )(

ˆ ˆˆ( )

( )

( ) ( )

n

p

nc cc n c c ct c

t

v mv vf CC

gmM M

'

ζζ χ χαα ζζ

ζ ζχα

∗∗ ∗ ∗∗ ∗+− −+−

∗ ∗∗+ +

⟨ ⟩

211 2

1 2

2 221

1 2

ˆˆ

ˆˆ( ) ( )

cc

cc

CC

CC

ζζ

ζζ

∗∗∗ ∗+

∗∗+

e e (15.112)

The calculation for sliding with the orthotropic anisotropy must be performed in the coordinate system with the principal axes of orthotropy, i.e. 21( , , )∗ ∗e e n .

We examine below the basic response of the present friction model by numerical experiments and comparison with test data for the linear sliding phenomenon without a normal sliding velocity leading to


=nv 0 (15.113)

The traction rate vs. sliding velocity relation is given by substituting Eq. (15.113) into Eq. (15.101) as

1 21 2(= )t vvα ∗ ∗+f e e

1 21 2 1 21 21 21 2

2 2 2 21 2 1 2

1 12 2

ˆ ˆˆ ˆˆ( )

ˆ ˆ ˆ ˆˆ( )

( )

( ) ( ) ( ) ( )

p

ct

t

t

g mv vM MMM

gmM M M M

'

'

ζ ζζ ζα χα

ζ ζ ζ ζχα

∗ ∗∗∗ ∗ ∗∗ − ++−

∗ ∗ ∗ ∗∗+ + +⟨ ⟩

e e (15.114)

while it holds that = const.nf The traction rate vs. sliding velocity relation is given by substituting Eq. (15.113)

into Eq. (15.112) as

1 21 2(= )t vvα ∗ ∗+f e e

2121 1 21 21 221

2 2 2 21 2 21

1 1 22

ˆˆˆˆ1

ˆˆ ˆ ˆˆ( )

( )

( ) ( ) ( ) ( )

n

p

ccc cct

cct

mv vf CCCC

gmC CCC

'

ζζζζα

ζζ ζ ζχα

∗∗∗∗ ∗ ∗− ++−

∗∗∗ ∗∗+ + +⟨ ⟩

e e (15.115)

where pm and cm are given by Eqs. (15.73) and (15.74). The influence of the rotational hardening is shown by the numerical experiments

using Eq. (15.115) for the Coulomb friction condition in Fig. 15.17, where the material parameters are selected as follows:

0

1

= = 0.2, = 0.2, = 0.19,

= 10/mm, = 1, = 0.1/s, = 1,

= 1 000mm ,

= = 1000MPa/mm,

= 10MPa.

k

tn

n

s M

m n

u

f

μμ μκ ξ

α α

−

assuming 2 32 3= = 0, = = 0t tv vβ β . It is shown that the difference of

friction coefficients in the opposite sliding directions increases with the rotational anisotropy.


-6

-4

-2

0

2

4

6

-0.2 -0.1 0 0.1 0.2-6

-4

-2

0

2

4

6

-0.2 -0.1 0 0.1 0.2

sμ

sμ sμ

sμ

kμ

kμ

kμ

kμ

1 0.05tβ = 1 0.1tβ =

1 0.0001tv = ±

1

n

tf

f1

n

tff

1 (mm)tu 1 (mm)tu

1 0.001tv = ±

1 0.01tv = ± 1 0.1tv = ±

-6

-4

-2

0

2

4

6

-0.2 -0.1 0 0.1 0.2-6

-4

-2

0

2

4

6

-0.2 -0.1 0 0.1 0.2-6

-4

-2

0

2

4

6

-0.2 -0.1 0 0.1 0.2-6

-4

-2

0

2

4

6

-0.2 -0.1 0 0.1 0.2

sμ

sμ sμ

sμ

kμ

kμ

kμ

kμ

1 0.05tβ = 1 0.1tβ =

1 0.0001tv = ±

1

n

tf

f1

n

tff

1 (mm)tu 1 (mm)tu

1 0.001tv = ±

1 0.01tv = ± 1 0.1tv = ±

Fig. 15.17 Influence of rotational anisotropy on relation of tangential traction and dis-placement


Appendixes

Appendix 1: Projection of Area

Consider the projection of the area having the unit normal vector n onto the surface having the normal vector m .

Now, suppose the plane ( abcd in Fig. A.1) which contains the unit normal vectors m and n . Then, consider the line ef obtained by cutting the area having the unit normal vector n by this plane. Further, divide the area having the unit normal vector n to the narrow bands perpendicular to this line and their projections onto the surface having the normal vector m . The lengths of projected bands are same as the those of the original bands but the widths become to the ones multiplied by the scalar product of the unit normal vectors, i.e. •m n . Eventually, the projected area da is related to the original area da as follows:

da = da•m n (A.1)

a

e

n

m

db

db•n m

b

c

d

f

a

e

n

m

db

db•n m

b

c

d

f

Fig. A.1 Projection of area

388 Appendixes

Appendix 2: Proof of ( / ) / = 0jjA xF J ∂∂

2

( / )1= jA

j jA

jAjj

Fx xX JJ J xx XJx

∂ ∂ ∂ ∂∂ ∂− ∂ ∂∂∂

1 32

1 322

( / )1 = PQR

j Rj QPAPQR

jAj RQP

x x xX X Xx xX x x x

xx X X X XJ

εε

∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ −∂ ∂ ∂ ∂ ∂∂22 2

1 322

321

1 321 = ( PQRA AA RQP

xx x x x xx x xX X XJ X X X

ε∂∂ ∂ ∂ ∂ ∂+ +∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂)

22 233 11 22 3 1 2

321

1 32( )PQRA A AP QR RQ P R QP

xx x xx xxx x x x xX x X X X X x X xX X XX X X

ε ∂∂ ∂∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂− + +∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂∂ ∂ ∂∂ ∂

22 23 3 31 1 12 2 2

2321

1 321 = (PQRA AA R R RQ Q QP P P

xx xx x xx x x x x xx x xX X XJ X X X X X X X X X

ε ∂∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂+ +∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ )

22 233 11 22 3 1 2

321

1 32(A A AP QR RQ P R QP

xx x xx xxx x x x xx X X X X x X x XX X XX X X

∂∂ ∂∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂− + +∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂∂ ∂ ∂∂ ∂ )

22 23 31 12 2

21 321 = PQR

A AA R R RQ Q QP P P

xx xx xx x x xX X XJ X X X X X X X X X

ε ∂∂ ∂∂ ∂∂ ∂ ∂ ∂+ +∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ )

22 23 12 3 1 21 32 = 0(

A A AP QR RQ P R QP

xx xxx x x x xX X X X X XX X XX X X

∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂− + +∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ )

(A.2)

Appendix 3: Euler’s Theorem for Homogeneous Function

The homogeneous function of degree n is defined to fulfill the relation

1 12 2( , , , ) = ( , , , )nm mf ax ax ax a f x x x••• ••• (A.3)

assuming the variables 1 2, , , mx x x••• and letting a denote an arbitrary scalar

constant. Then, the homogeneous function can be given by the polynomial

expression:

1 21 2 21

1=

( , , , ) = nm

i i i

mi

snn nf x x x x x x••• •• • (A.4)

Appendix 4: Normal Vector of Surface 389

where s is the number of terms, provided to fulfill

1=

= for each m

ij

j

n n i (A.5)

Eq. (A.4) leads to

1 2 1 21 22 21 1

1 1 1 1= = = =

1( , , , )= =j n n

j

m m ii

m mi

i i ii imj j j

jj i j

n n nn n n nissf x x x

n nx x x x x x x x xx

••• −•• • •• • •• •

∂∂

Then, it holds that

1 21 2

1=

( , , , )= ( , , , )

mm

j mjj

f x x xnx f x x x

x

••••••

∂∂∑ (A.6)

which is called the Euler’s theorem for homogeneous function.

For the simple example ( = 3, = 4, = 3m n s ):

3 24( , , ) =f x y z x x y x yzα β γ+ +

Eq. (A.6) is confirmed as follows:

2 33 2 2(4 2 ) ( 4= =f f f

x y z x x y xyz x x x z y x y z fy zxα β γ β γ γ •

∂ ∂ ∂+ + + 3 + + + ) +∂ ∂ ∂

Eq. (A.6) leads to Eq. (6.32) for the yield function ( = 1)n .

Appendix 4: Normal Vector of Surface

The quantity ( )tr( / ) f d∂∂ σ σ σ is regarded as the scalar product of the vectors

( )/f ∂∂ σ σ and dσ in the nine-dimensional space. Here, it holds that

> 0 : is directed outward of yield surface( )

tr = 0 : is directed tangential to yield surface

< 0 : is directed inward of yield surface

( )d

f d d

d

∂∂

(A.7)

Therefore, ( )/f ∂∂ σ σ is interpreted to be the vector designating the outward-normal of the yield surface. This fact holds also for general scalar function

( )f T of arbitrary tensor T .

390 Appendixes

Appendix 5: Convexity of Two-Dimensional Curve

When the curve is described by the polar coordinates ( , )r θ as shown in Fig. A.1,

the following relation holds

tan = rddr

θα (A.8)

where α is the angle measured from the radius vector to the tangent line in the

anti-clockwise direction. Eq. (A.8) is rewritten as

cot = r'rα , (A.9)

where ( )' designates the first order differentiation with respect to θ .

0

Curve

Tangent line

α

r

x

y

R

θΘ

dθ

rdθ

dr

π/2−α

( / 2 )Θ θ π α− − −

dr

rdθ

α

0

Curve

Tangent line

α

r

x

y

R

θΘ

dθ

rdθ

dr

π/2−α

( / 2 )Θ θ π α− − −

dr

rdθ

α

Fig. A.2 Curve in the polar coordinate (r, θ )

The equation of the tangent line at ( , )r θ of the curve = ( )r r θ is described

by the following equation by using the current coordinates ( , )R Θ on the tangent

line.

cos[ ( / 2 )] = cos( / 2 )R rΘ θ π α π α− − − −

Appendix 6: Derivation of Eq. (11.19) 391

which is rewritten as 1 1sin( ) = sin =

sincos( )

1 1 1= cos( ) cot sin( )

R rrR

r rR

Θ θ α α αΘ θ α

Θ θ α Θ θ

− − − → −− −

→ − − −

Substituting Eq. (A.9) to this equation and noting 2(1/ ) =r r /r' '− , one has the

relation 1 1 1= cos( ) sin( )( )'r rR

Θ θ Θ θ− + − , (A.10)

Equation (A.10) is rewritten by the Taylor expansion as

21 1 1 1 1= cos sin = 1( ( () ) ) 2( )( ) ( )'

r r rR ϑ ϑ ϑθ θ θΘ • • •+ − +

31 1( ) 6

( ) ( )'r ϑ ϑθ • • •+ − +

21 1 1 1=( ( () ) )2

( )'r r rϑ ϑθ θ θ • • •+ − + (A.11)

where ϑ Θ − θ≡ . On the other hand, the radius ( )r Θ for =Θ θ ϑ+ on the

curve is described by

21 1 1 1 1=( ( ( () ) ) )2

( ) ( )'r r r r

"ϑ ϑθ θ θΘ • • •+ + + . (A.12)

Eqs. (A.11) and (A.12) lead to

211 1 11=( () )( ) 2 ( )( ) r rR r

" ϑΘ Θ θθ • • •− + + . (A.13)

In order that the curve is convex ( ( ) ( )r RΘ Θ≤ ), the following inequality must hold from Eq. (A.13).

1 1 > 0( )r r "+ (A.14)

Appendix 6: Derivation of Eq. (11.19)

Differentiation of Eq. (11.11) under the condition ( const.) =f σ leads to

|| ||( ) ( )|| ||

( )g dp pg dp dp '' 'χ χχ χ ∂ ∂+ +

∂ ∂σσ

2

|| || 1 || ||( )( ) = 0= ( )gg p pd dp dp pM M' ''χ χ+ − +σ σ

392 Appendixes

from it holds that || ||

( ) ( )|| || ( )= =

( )1 ( )( )

g gp gd M Mdp gg

M

'''

''

χ χ χ χχχ

−−

σσ

(A.15)

Considering || || / 0d dp='σ at =1χ in Eq. (A.15), one has Eq. (11.19).

Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State

The numerical experiments of small cyclic tension uniaxial loading behavior near yield state of 1070 steel are depicted in Fig. A3 for u in Eq. (8.36) and Fig. A4

aε

as

aα

0 0.005 0.01 0.015

200

400

600

800

(MPa)aσ

R

1.0

0 aε0.005 0.01 0.015

0.5

aε

as

aα

0 0.005 0.01 0.015

200

400

600

800

(MPa)aσ

R

1.0

0 aε0.005 0.01 0.015

0.5

Fig. A.3 Numerical experiment of small cyclic tension uniaxial loading behavior near yield state of 1070 steel

Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State 393

aε

as

aα

0 0.005 0.01 0.015

200

400

600

800

(MPa)aσ

R

1.0

0 aε0.005 0.01 0.015

0.5

aε

as

aα

0 0.005 0.01 0.015

200

400

600

800

(MPa)aσ

R

1.0

0 aε0.005 0.01 0.015

0.5

Fig. A.4 Numerical experiment of small cyclic tension uniaxial loading behavior near yield state of 1070 steel for u = const.

for const.=u , respectively, while the same values of material parameters for Fig. 10. 7 are used (Hashiguchi and T. Ozaki, 2009). It is also here shown that the strain accumulation is suppressed by the extension of the material parameter u .

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Index

A

Accumulated plastic strain 146 Acoustic tensor 334 Additive decomposition of strain rate 142 Admissible field kinematically 108 statically 108 Admissible transformation 59 Almansi (Euler) strain 64, 82, 85, 113 Alternating symbol 2 Angular momentum 102 Anisotropy 156 kinematic 157, 186, 197, 222, 313, 324,

329 orthotropic 238 rotational 157, 197, 261 rotational for friction 354 sliding-yield surface 354, 375 traverse 317 yield surface 157, 186, 197, 222, 261, 313,

324, 329 Anti-symmetric tensor 23 Associated flow rule (Associativity) 145,

151, 158, 179, 203, 219

Drucker’s interpretation 151 plastic relaxation work rate 152 Prager’s interpretation 151 Associative law of vector 8 Axial vector 25

B

Back stress 157 Bauschinger effect 157 Bingham model 211 Biot strain tensor 67 stress tensor 125 Body force 102 Bounding surface 184 model 184 surface model with radial mapping 184 Bulk modulus 132, 251

C

Caley-Hamilton theorem 35 Cap model 282 Cam-clay model 282 modified 256 original 256 Cartesian summation convention 1 decomposition 24 Cauchy ’s fundamental theorem 103 ’s law of motion 106 ’s stress principle 103 stress 103, 114, 128 elastic material 130 Cauchy-Green deformation tensor 63

408 Index

Characteristic direction 26 equation 26 length 328 value 26 vector 26 Circle of relative velocity 80 Coaxility 35, 160 Cofactor 4 Commutative law of vector 8 Compliance method 336 Component of tensor 17, 20 vecto 10 Cofactor 4 Configuration current 57 initial 57 intermediate 136 reference 57 relative 58 Conservation law of angular momentum 102 mass 101 momentum 101 Consistency condition for conventional plasticity 143 extended subloading surface model 203 friction 357 initial subloading surface model 179 subloadong-superyield surface model 294,

299 Consolidation ln pe − linear relation 251 isotropic 249 ln ln pv − linear relation 249 normal 249 Contact elastic modulus 353 normal traction 352 tangential traction 352 traction 352 Continuity condition 171

Continuity equation 101 Continuum spin 71, 114, 118, 310, 313 Contraction of tensor 16 Convected stress 105 stress rate 123 Convective term 51 Conventional plasticity mode 135, 142 friction model 349 Convexity condition 254, 390 Convexity of yield surface 151, 254 Coordinate system 11 transformation 11, 15 Corner theory 160 Corotational rate 118 derivative 114, 119 Green-Naghdi (Dienes) 119, 122, 311,

316 Jaumann 107, 118, 122, 129, 168, 309,

315, 322 rate 110, 114, 118, 119, 309, 321 stress rate 122, 123, 309, 311, 315, 316 Cosserat elastic material 133 Cotter-Rivlin stress rate 123 Coulomb friction (sliding-yield) condition (surface)

350, 354, 360, 381 -Mohr failure criterion 285 Couple stress 133 Creep model 214 hardening of friction 356 Critical state 254 Cross product 9 Curl (rotation) of tensor field 52 Current configuration 57 Cyclic loading 170, 191 mobility 279, 295 plasticity m1odel 191 stagnation of isotropic hardening 225

Index 409

D

Definition of tensor 14 vector 8 Deformation 57 gradient 59, 63, 136 of crystal 319 rate tensor 71 relative gradient 63 theory 161 Derivative material-time 50 spatial-time (local) 50 Description Eulerian 58 Lagrangian 57 material 57 relative 50, 58 spatial 50, 58 total Lagrangian 58 updated Lagrangian 58 Determinant 3 Vandermonde’s 39 Deviatoric part 24 invariant 30 plane (π -plane) 41 projection tensor 162 tangential stress rate 162 tangential projection tensor 163 Dienes rate 119, 123, 311, 316 Differential calculi of tensor 48 Dilatancy locking 301 Direct (Symbolic) notation (description) 17 Direction cosine 10 Discontinuity of velocity gradient 333 Displacement 65 Dissipation energy 127 Distributive law of vector 8 Divergence of tensor field 51 Divergence theorem 53 Drucker’s postulate for stress cycle 151 classification of plasticity model 135

Drucker-Prager model 282 Dummy index 1 Dyad (tensor or cross product) 19 Dynamic loading ratio 218 surface 213 E

Eddington’s epsiron 2 Eigen direction 26 projection 32 value 26, 31, 33 vector 26, 32, 33 Einstein’s summation convention 1 Elastic bulk modulus 132 constitutive equation 127 deformation gradient 136 material 127 modulus 131 predictor 341 shear modulus 131 strain energy 127 stress rate 153 volumetric strain 138 Elastic-plastic transition 182 Elastoplastic modulus tensor 145 Element test 327

ln pe − linear relation 252 Equilibrium equation 105 incremental(rate)-type 106 moment 107 Equivalent plastic strain 146 stress 146 viscoplastic strain 213 Euler ’s first law of motion 102 ’s second law of motion 102 ’s theorem for homogeneous function 142,

179, 202, 388 Eulerian description 58

410 Index

spin tensor 73, 90 strain tensor 65, 82, 85, 113 triad 61, 73, 86 Extended subloading surface model 192,

196, 265, 296 F

Failure surface 285 Finite strain theory 165 First Piola-Kirchhoff stress 103, 114, 127 Flow rule associated 145, 151, 158, 179, 203, 219 friction 358 nonassociated 284 plastic 143 viscoplastic 219 Footing settlement analysis 301 Frame-indiffrence 111 Functional determinant 59 G

Gauss’ s divergence theorem 53 Gradient of tensor field 51 Gradient theory 328 Green elastic material 127 strain 65, 82, 85, 113, 165 Green-Naghdi rate 119, 123, 311, 316

H

Hamilton operator 51 Hardening 149, 180 anisotropic 157, 186 function 142, 222, 259 isotropic 142, 180, 222, 259 kinematic 157, 186, 222, 313 nonlinear-kinematic 195, 223 rotational 157, 186, 261 variable 142, 157 Hencky deformation theory 161 strain 69 Hooke’s law 131

Hyperelasticity 127 Hyperelastic-plasticity 165 Hypoelasticity 131, 142, 309 Hypoplasticity 133, 160 Hysteresis loop 195 I

Identity tensor fourth-order 21 second-order 21 Il-posedness 328 Ilyushin’s isotropic stress space 245 postulate of strain cycle 152 Impact load 211, 215, 220 Infinite surface model 195 Infinitesimal strain 65 Initial configuration 57 Initial subloading surface model 174, 281 Inner product 8 Intermediate configuration 136 Internal variable 143 Intersection of yield surfaces 160 Invariant 30 Inverse loading 206 relation 144, 168, 180, 205, 216 tensor 22 Isotropic consolidation 249 function 37 hardening (variable) 142, 222, 259 material 156 scalar-valued tensor function 28 tensor-valued tensor function 37

J

Jacobian 59 Jaumann rate 118, 309, 315 of Almansi strain tensor 78 of Cauchy stress 107, 122, 129, 168, 322 of Kirchhoff stress 123, 168 J2-deformation theory 161

Index 411

K

Kinematic hardening 157 linear 222, 313 nonlinear 195, 223 Prager 222 Ziegler 222 Kimatically admissible velocity field 108 Kinetic friction 349, 356 Kirchhoff stress 103, 114, 128 Kronecker’ delta 1

L

Lagrangian description 57 spin tensor 72, 93 strain 65, 82, 85, 113, 165 triad 61, 72, 87 Lame coefficients 131 Laplacian (Laplace operator) 52 Lee decomposition 136 Leftt Cauchy-Green deformation tensor 63, 114 polar decomposition 37, 60 relative Cauchy-Green deformation tensor

64 stretch tensor 61 Lie derivative 123, 167 Linear kinematic hardening rule 222, 313 transformation 16 ln ln pv − linear relation 249 Local form 106 Local theory 328 Local time derivative 50 Loading criterion for plastic sliding velocity 359 plastic strain rate 148, 180 Localization 327 Lode’s variable 44 Logarithmic strain 69, 79 volumetric strain 69, 139, 249

M

Macauley bracket 178 Magnitude of tensor 22 vector 8, 10 Masing rule 192, 206 Material description 57 frame-indifference 111 spin 118 -time-derivative 50, 55 Matrix 2 Maxwell model 211 Mean (Spherical) part 24 Mechanical ratcheting effect 193, 195, 233 Mesh size dependence (sensitivity) 328 Metal 146, 161, 221 Mises ellipse 246 yield condition 146, 244 Modified Cam-clay model 256 Modified overstress model 216 Mohr’s circle 45 Momentum 101 Motion 57 Multi surface (Mroz) model 182, 191 Multiplicative (Lee) decomposition of

deformation gradient 135

N

Nabra 51 Nanson’s formula 98 Natural strain 79 Navier’s equation 130 Negative transformation 59 Nominal strain 79 stress 103, 114, 128 stress rate 107 stress vector 103 Nonassociated flow rule (Nonassociativity)

284

412 Index

Nonhardening region 225 Nonlinear kinematic hardening model 182, 195 rule 223 Non-local theory 328 Non-proportional loading 160 Non-singular tensor 22 Non-steady term 50 Normal component 20 Normal isotropic hardening ratio 226, 229 surface 226 Normal stress rate 160, 162 Normal vector of surface 389 Normal sliding-yield ratio 355 surface 354 Normal-yield ratio 175, 197, 232, 271 surface 174, 197 Normality rule 145, 151 Normalized orthonormal base 9 Notation of tensor 17 Numerical calculation 340 O

Objectivity 111 Objective stress rate tensor 122 tensor 15 transformation 15 Octahedral plane 147 shear stress 147 Oldroyd stress rate 123 Original Cam-clay model 256 Orthogonal tensor 18 Orthotropic anisotropy 238 Orthotropic anisotropy of friction 375 Overstress model 211

P

Parallelogram law 8 Partial differential calculi 47

Perfectly-plastic material 149, 172 Permutation symbol 1 Perzyna’s over stress model 213 Piola-Kirchhoff stress 103 π -plane 42 Plane strain 248 stress 245 Plastic compressibility 249 corrector 342 deformation gradient 136 flow rule 143, 145, 151, 179, 203 modulus 144, 145, 179, 204, 230, 266,

281, 295, 300, 322 potential 145, 153 relaxation modulus tensor 144 relaxation stress rate 145, 153 shakedown 193 sliding flow rule 358 spin 317 strain rate 142, 144, 179, 203 volumetric strain 138 Poisson’s ratio 132 Polar decomposition 36, 60, 141 spin 72, 88, 311 Pole 47, 81 Positive definite tensor 35 proportionality factor 143-145, 159, 179,

180, 203 Potential energy 127 Prager’s continuity condition 172 kinematic hardening rule 222 overstress model 212 Prandtl-Reuss equation 147 Pressure-dependence 249 Principal direction 26 invariant 30 space 40 stretch 61

Index 413

value 26 vector 26 Principle of maximum plastic work 155 objectivity 111, 141, 354 material-frame indifference 111 Product law of determinant 5 Projection of area 387 Projection tensor deviatoric 162 deviatoric-tangential 163 Pull-back 123, 165 Pulsating loading 233 Push-forward 123 Q

Quasi-static deformation 215 Quotient law 15 R

Ratcheting effect 193, 195, 233 Rate-dependence 211 Rate of area 99 extension 76 normal vector of surface 99 shear strain 76 volume 99 Rate-type equilibrium equation 106 virtual work principle 109 Reference configuration 57 Relative configuration 58 deformation gradient 63 description 58 left Cauchy-Green tensor 64 right Cauchy-Green tensor 64 spin 72, 88, 311, 313 Representation theorem 40 Return-mapping 340 Reynolds’ transportation theorem 56 Right Cauchy-Green deformation tensor 63, 114

polar decomposition 37, 60 relative Caucgy-Green deformation tensor

64 stretch tensor 61 Rigid-body rotation 111, 112 Rigid-plasticity 313, 317 Rotation (curl) of tensor field 52 Rotation of triad 61, 90, 93 Rotation rate tensor 71 Rotational hardening (variable) 157, 261 Rotational anisotropy of friction 354 Rotational strain tensor 133

S

Second Piola-Kirchhoff stress 104, 114, 128

Scalar product (inner product) 8 triple product 9 Shakedown 193 Shear band 327 band inception 333 band thickness 331 -band embedded model 331 component 20 modulus 131, 251 strain rate 76 Similar tensor 28 Similarity-center 174, 197 surface 207 yield ratio 207 Similarity-ratio 174 Simple shear 84, 309 Single surface model 195 Skew-symmetric tensor 23, 33 Sliding- hardening function 355 subloading surface 354 velocity 350 yield condition (surface) 354, 360 Smeared crack model 331 Smoothness condition 172 Softening 149, 180

414 Index

Softening of sliding-yield surface 356 Soil 249 Spatial description 58 Spectral representation 31 Spherical part 24 Spin continuum 71, 114, 118, 310, 313 Eulerian 72, 90 Lagrangian 72, 93 of base 113 polar (relative) 72, 88, 311, 313 tensor 26 vector 26, 75 Stagnation of isotropic hardening 225 Static friction 350, 356 Statically-admissible stress field 108 Steady term 51 Strain 64 Almansi (Eulerian) 65, 82, 85, 113 Boit 67 cycle 153 energy function 127 Green (Lagrangian) 65, 82, 85, 113, 165 Hencky (logarithmic) 69 infinitesimal 64 logarithmic (natural) 69, 79 rate 71 space 150 tensor 64 Strain rate 70, 114 elastic 142 plastic 142, 144, 179, 203 viscoplastic 213, 219 Stress-controlling function 182 Stress cycle 151 Stress tensor 102 Biot 125 Cauhy 103, 114, 128 convected 105 Kirchhoff 103, 114, 178 first Piola-kirchhoff (nominal) 103, 114,

127, 219 nominal 103, 114, 128 second Piola-kirchhoff 104, 114, 128

Stress rate tensor 122 Convected (Cotter-Rivlin) 123 Jaumann of Cauchy stress 107, 122, 129,

168, 322 Jaumann rate of Kirhhoff stess 123, 168 Green-Naghdi (Dienes) 119, 123, 311, 316 nominal 107 Oldroyd 122 Truesdell 123 Truesdell rate of Kirhhoff stress 123, 129,

168 Stress vector (Traction) 101, 103, 352 Stretch left 61 principal 61 right 61 Stretching 71 Structure ratio 291 Subloading surface model 174, 196 normal-yield ratio 175, 197 normal-yield surface 174, 197 subloading surface 174, 197 stress-controlling function 182 Subloading overstress model 217 dynamic-loading ratio 218 dynamic-loading surface 218 subloading ratio 218 Subloading-friction model 350 sliding-hardening function 355 normal sliding-yield ratio 355, 356 sliding-subloading surface 355 normal sliding-yield surface 354 Sub-isotropic hardening stagnation surface

model 225 normal-isotropic hardening ratio 226 normal-isotropic hardening surface 225 sub-isotropic hardening surface 226 Subloading-superyield surface model 291 substructure ratio 291 super-yield ratio 291 super-yield surface 291 Summation convention 1 Superposition of rigid-body rotation 112 Surface element 97

Index 415

Symbolic notation (description) 17 Symmetric tensor 23 Skew-symmetric tensor 23, 33 Sylvester’s formula 33 Symmetry of Cauchy stress 108

T

Tangent (stiffness) modulus 131 Tangential associated flow rule of friction 358 contact traction 352 inelastic strain rate 159, 187, 208, 220 stress rate 160, 162 velocity 351 Tension cut of yield surface 285 Tensor 14 acoustic 334 anti-symmetric 23, 33 Cartesian decomposition 24 characteristic equation 26, 29-31 coaxiality 35, 160 component 20 component notation 17 component notation with base 17 contraction 16 definition 14 deviatoric part 24 direct notation 17 eigenprojection 32 eigenvalue 26, 31, 33 eigenvector 26, 32, 33 identity 21 invariant 30 inverse 22 magnitude 22 matrix notation 17 mean (spherical) part 24 non-singular 22 normal component 20 notation 17 objective (transformation) 15 orthogonal 18 partial derivative 47 positive definite 35

principal direction 26 principal invariant 30 principal value 26 product (cross product, dyad) 19 polar decomposition 36, 60, 141 representation in principal space 40 shear component 20 similar 28 skew-symmetric 23, 33 similar 28 spectral representation 31 spin 26, 70, 72, 88, 113, 311 strain 64, 65, 67, 69, 79 strain rate 70, 114, 142, 179, 203, 213, 219 stress 102-105 stress rate 122-125 symbolic notation 17 symmetric 23 time-derivative 50 trace 20 transpose 21 triple decomposition 24 two-dimensional state 44 two-point 113, 136 unit 22 zero 21 Time-dependence 211 Time derivative local (spatial-time) 50 material-time 50 non-steady (local time derivative) term 50 steady (covective) term 51 Total Lagrangian description 58 Trace 20 Traction (Stress vector) 101, 352 Triad Eulerian 61, 93 Lagrangian 61, 72, 87 Transportation theorem 56 Transpose 3, 21 Traverse anisotropy 317 Triple decomposition 24 Truesdell stress rate 123 Truesdell rate of Kirchhoff stress 123, 129,

168

416 Index

Two-dimensional state 44 Two-point tensor 113 Two surface model 182, 194 U

Unconventional (elasto)plasticity 135, 171 friction model 349 Uniaxial loading 81, 233, 337 Uniqueness condition 171 Uniqueness of solution 171 Unit tensor 21 vector 8 Updated Lagrangian description 58

V

Vandermonde’s determinant 39 Vector 8 associative law 8 axial 25 component description 9 commutative law 8 definition 8 direction 8, 10 distributive law 8 eigen (principal) 26 equivalence 8 magnitude 8 parallelogram law 8 product 9 scalar product 8 scalar triple product 9 spin 75

unit 8 zero 8 Velocity gradient 70, 336 Vertex of yield surface 160 Virtual work principle 108 Viscoelastic model 211 Viscoplastic coefficient 213 model 212 strain rate 213 Volume element 97 Volumetric strain (rate) 79, 140, 249 Vorticity 75 W

Work Conjugacy 124 conjugate pair 125 hardening 147 rate 124, 152 Y

Yield condition (surface) 142, 146, 150, 152,

238, 253 condition (surface) for friction 360, 363,

376, 381 Young’s modulus 132 Z

Ziegler’s kinematic hardening rule 222 Zero tensor 21 vector 8

elastoplasticity theory - friedrich pfeiffer, peter wriggers.pdf

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