IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials

SOLID MECHANICS AND ITS APPLICATIONS

Volume 114

Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is the first year graduate student. Some texts are mono­graphs defining the current state of the field; others are accessible to final year under­graduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

IUTAM Symposium on

Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials Proceedings of the IUTAM Symposium held in Marrakech, Morocco, 20-25 October 2002

Edited by

S.AHZI University Louis Pasteur, Strassbourg, France

M. CHERKAOUI University of Metz, Metz, France

M.A. KHALEEL Hydrogen and Industrial Transportation Program, NorthWest Pacific National Laboratory, WA, U.S.A.

H.M. ZBIB Washington State University, Pullman, WA, U.S.A.

M.A. ZIKRY North Carolina State University, Raleigh, NC, U.S.A.

and

B. LAMA TINA Materials Research Program Director, Army Research Office, Raleigh, NC, U.S.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.l.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6529-2 ISBN 978-94-017-0483-0 (eBook) DOl 10.1007/978-94-017-0483-0

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© 2004 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 Softcover reprint of the hardcover 1 st edition 2004

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Contents

Preface

New developments in the brittle to ductile transitions of fracture in intrinsically brittle crystals and polycrystals A.S.Argon

Dislocation-based length-scales in crystal plasticity:

xi

experiments and modeling - S. Nemat-Nasser 5

Application of a variational self-consistent procedure to the prediction of deformation textures in polycrystals P. Gilormini, Y. Liu, P. Ponte Castaneda 17

What about the yield transformation surface determination (austenite -+ martensite) with the measurement of austenite and martensite lattice parameters for some shape memory alloys? - C. Lexcellent, P. Blanc and C. Bouvet 25

Micro to macroscopic deformation behavior of amorphous polymer with slightly heterogeneous distribution of molecular chains - Y.Tomita and M Uchida 33

Cross slip viewed at the nano - and micrometer scale. ~ T. Leffers and OB. Pedersen 41

A multi scale micromechanics approach to describe environmental effects on surface crack initiation under cyclic loading - E.P. Busso, G. Cailletaud and S. Quilici 49

Atomic - scale modeling of dislocation behaviour under stress D.J Bacon and Y.N Osetsky 59

v

VI

Coalescence and evolution of nanoscale islands during polycrystalline thin film growth. - M 0. Bloomfield, YH 1m, H Huang and TS. Cale 67

On pIc band propagation velocity under stress controlled tests in aluminium alloys- M Abbadi, D. Thevenet, P. Hahner and A. Zeghloul 75

Effect of some parameters on the elastoplastic behavior of green sand - R. Ami Saada 83

Experimental investigations of size effects in thin copper foils - G. Simons, Ch. Weipert, J Dual, and J Villain 89

Plastic response of thin films due to thermal cycling-L. Nicola, E. van der Giessen and A. Needleman 97

Measurement of the stress intensity factor, k1, for copper by a digital image correlation method - S. M'guil, C. Husson and S. Ahzi 105

Homogeneization of viscoplastic materials - A. Molinari, S. Mercier 113

Collective dislocation behavior in single crystalline aluminum under indentation - Y Shibutani, A. Koyama and T Tsuru 125

Multiscale modeling of texture gradient effects on localization in fcc polycrystals- K W. Neale, K Inal and P.D. Wu 133

Multiaxial plastic fatigue behavior with multiscale modeling -A. Abdul-latif, K Saanouni and J Ph. Ding/i 141

Damage, opening and sliding of grain boundaries -G. Cailletaud, 0. Diard, A. Musienko 149

Gradients of hardening in non local dislocation based plasticity - G.z. Voyiadjis and R.J Dorgan 157

VB

Determination of the material intrinsic length scale of gradient plasticity theory - G.z. Voyiadjis and R.A. AI-Rub 167

Computer simulation of contact force distribution in random granular packings - A.H W Ngan 175

Three - dimensional structures of the geometrically necessary dislocations generated from non - uniformities in metal microstructures - T. Ohashi 183

Simulation of texture evolution in equal channel angular extrusion of copper using a new flow field L.S Toth, R. Massion, L. Germain and se. Baik 191

Initial energy dissipation mechanism at crack tip on the ductile to brittle transition - J W Kysar 199

Constitutive modeling of viscoelastic unloading of glassy polymers - Y. Remond 207

On the constitutive theories of power - law materials containing voids - e. Y. Hsu, B.J Lee and ME. Mear 217

Objective quantification of the ductility within the coupling elasticity - damage behavior: formulation - H Bouabid, Se. D'Ouazzane, M El Kortib and o.fassi-Fehri 227

Discrete dislocation predictions for single crystal hardening tension vs bending- A.A. Benzerga and A. Needleman 235

On plasticity and damage evolution during sheet metal forming- e. Husson, e. Poizat, N Bahlouh S Ahzi, T. Courtin and L. Merle 243

Modeling of thermo-electro-elastic effective behaviors of piezoelectric composite mediums and analysis of reinforcement orientation effects - N F akri, L. Azrar and L. EI Bakkali 25 I

Investigations in size dependent torsions and fractures -P. Tong, D.e.e. Lam, F. Yang and J Wang 259

V11l

Influence of microstructural parameters on shape memory alloys behavior - C. Niclaeys, T Ben Zineb and E. Patoor 267

Investigation of ridging in ferritic stainless steel using crystal plasticity finite element method - HJ. Shin, J.K. An and D.N Lee 275

Grain boundary effects and failure evolution in polycrystalline materials - W.M Ashmawi and MA. Zikry 283

The influence of an heterogeneous dispersion on the failure behaviour of metal-matrix composites: micromechanical approach - K. Derrien and D. Baptiste 291

A cohesive segments approach for dynamic crack growth -J.J. C. Remmers and R. de Borst 299

A linear model of processing path in cubic-orthotropic system - D.S. Li and H Garmestani 307

Taylor theory with microscopic slip transfer conditions. -B.L. Adams, B.S. Dasher, R. Merrill, J. Basinger and D.S. Li 315

Dynamics of nanostructure formation during thin film deposition. - D. Walgraef 325

Prediction of damage in randomly oriented short-fibre composites by means of a mechanistic approach-B. Nghiep Nguyen and MA. Khaleel 333

Nonsteady plain - strain ideal plastic flow considering elastic dead zone - W.Lee, K. Chung, TJ. Kang and J.R. Youn 343

Multiscale modeling of non - linear behaviour of heterogeneous materials: comparison of recent homogeneisation methods - P. Kanoute, J.L. Chaboche 351

IX

Thermomechanical behaviour of shape memory alloy taylor's model- M 0. Bensafah, L. Boufmane and A. Hihi 359

Multiscale analysis of dynamic deformation in monocrystals­MA. Shehadeh, HM Zbib, TDiaz de fa Rubia and V Bulatov 367

Micro/meso-modeling of polymeric composites with damage evolution - F. Ellyin, Z. Xia and Y Zhang 379

An alternative approach for heterogeneous material behaviour modelling. - 0. Bouaziz and P. Buessler 389

On anisotropic formulations of the elastic law within multiplicative inelasticity - C. Sansour 397

Deep drawing process ofthe aisi 304 stainless steel cup: interaction between design tools and kinetic of plastic strain induced martensite - Z. Tourki and M Cherkaoui 405

Modeling and simulation of dynamic plasticity and failure in ductile metals - L. Campagne, L. Daridon and S. Ahzi 413

Effects of polymeric additives on the morphology and the structure of the calcium carbonate material- A. Jada 421

Preface

The papers in this proceeding are a collection of the works presented at the IUTAM symposium-Marrakech 2002 (October 20-25) which brought together scientists from various countries. These papers cover contemporary topics in multiscale modeling and characterization of materials behavior of engineering materials. They were selected to focus on topics related to deformation and failure in metals, alloys, intermetallics and polymers including: experimental techniques, deformation and failure mechanisms, dislocation-based modelling, microscopic-macroscopic averaging schemes, application to forming processes and to phase transformation, localization and failure phenomena, and computational advances. Key areas that are covered by some of the papers include modeling of material deformation at various scales. At the atomistic scale, results from MD simulations pertaining to deformation mechanisms in nano-crystalline materials as well as dislocation-defect interactions are presented. Advances in modeling of deformation in metals using discrete dislocation analyses are also presented, providing an insight into this emerging scientific technique that can be used to model deformation at the microscale. These papers address current engineering problems, including deformation of thin fIlms, dislocation behavior and strength during nanoindentation, strength in metal matrix composites, dislocation-crack interaction, development of textures in polycrystals, and problems involving twining and shape memory behavior.

On Behalf of the organizing committee, I would like to thank Professor P. Germain for his support and help to organize this symposium and I acknowledge the support from our sponsors: the International Union of Theoretical and Applied Mechanics; the Moroccan State Secretary for ScientifIc Research; the PacifIc Northwest National Laboratory, WA, USA; the European Research Office of the US Army; the University Cadi Ayyad, FSSM, Marrakech, Morocco; the University of Metz, UFR MIM, Metz, France and the University Louis Pasteur at Strasbourg, UFR IPST, France

SaidAhzi

xi

Multiscale Modeling and

Characterization of Elastic-Inelastic Behavior of Engineering Materials

Proceeding of IUT AM

SymposiumOctober

20-25, 2002 Marrakech, Morocco

Sponsors • International Union of Theoretical and Applied

Mechanics • Moroccan State Secretary for Scientific Research • Pacific Northwest National Laboratory, WA, USA • European Research Office of the US Army • University Cadi Ayyad, FSSM, Marrakech, Morocco • University ofMetz, UFR MIM, Metz, France • University Louis Pasteur at Strasbourg, UFR IPST,

France

Organizing Committee

S. Ahzi (Chair), M. Cherkaoui , M.A. Khaleel, H.M. Zbib, M.A. Zikry

International Scientific Committee of the Symposium

S. Ahzi (France, Chair), M. Cherkaoui (France), H. Zhib (USA), E.C. Aifantis (Greece), E.P. Busso (UK), P. Germain (France), D.

xiii

xiv

McDowell (USA), P. McHugh, (Ireland), c. Miehe, (Germany), S. Nemat-Nasser (USA), K. Neale, (Canada), J. Salem;on, (France)

Local Organizing Committee

M. Hasnaoui, A. Oueriagli, O. Oussouaddi, K. Naciri, A. Hihi, M. Ben-Salah

NEW DEVELOPMENTS IN THE BRITTLE TO DUCTILE TRANSITIONS OF FRACTURE IN INTRINSICALLY BRITTLE CRYSTALS AND POLYCRYSTALS

ALI S. Argon

Massachusetts Institute of Technology Cambridge, MA 02139, USA Email: argon@mit.edu

KeyWords: brittleness, ductility, fracture transitions, silicon, Fe-3% Si Alloy

1. INTRODUCTION

We will briefly review some recent developments in the study of the brittle to ductile transitions in fracture in some intrinsically brittle crystalline solids both on the experimental and theoretical level or with computer modeling.

2. CRACK TIP PROCESSES OF DISLOCATION EMISSION

The fracture transitions in brittle solids have been considered on a fundamental level as bifurcation phenomena at the tips of atomically sharp cracks in material with inadequate supplies of mobile dislocations, requiring emission of dislocations from the crack tip as a first necessary step for a fracture transition. In a' trend setting paper Rice and coworkers [1] considered the energy barriers to the emission of dislocations in a 2-D setting from crack tips under Mode III and mixed Mode I and Mode II loading to describe the character of the key phenomena that consisted of the formation of only a half of a dislocation core that was sufficient to initiate subsequent rapid multiplication of dislocations to initiate plasticity. Such energy barriers were determined further in 3-D for all modes of loading by Xu, et al. [2,3] clarifying the importance of crack front cleavage ledges to "catalyse" the key dislocation emission processes. These modeling studies

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 1·4. ©2004 Kluwer Academic Publishers.

2 A. S.Argon

demonstrated very good agreement with experiments but did not fully answer whether the actual transition was governed by the essential steps of dislocation emission from crack tips or by the mobility of dislocations away from the crack tip. These considerations were answered by means of definitive experiments on Si and Fe-3% Si single crystals performed by Argon and co-workers as discussed below.

3. EXPERIMENTAL STUDIES OF FRACTURE TRANSITIONS IN Si and Fe-3% Si CRYSTALS

Silicon single crystals are ideal for fundamental experimental studies on the fracture transition [4-6]. New experiments were conducted on the brittle-to­ductile (B-D) transition of fracture on special double cantilever beam specimens of dislocation-free Si single crystals in which brittle cracks were propagated up a temperature gradient at different velocities on {110} cleavage planes until cracks were arrested by dislocation emission from the crack tip to result in ductile behavior [7,8]. Similar experiments were also conducted on large Fe-3% Si single crystals [9].

The overall description of the arrest phenomena in fully quantitative detail with excellent connections to experimental findings of both recent research and earlier research of others was described by Argon [10].

4. ROLE OF GRAIN BOUNDARIES IN THE FRACTURE TRANSITIONS OF POLYCRYSTALS

In early pioneering experiments Hahn et al. [11] had demonstrated that as long as microcracks formed by slip incompatibilities inside grains could be contained into individual grains by grain boundaries a transition from ductile to brittle behavior could be averted, indicating that the cleavage crack arrest functions of grain boundaries were crucial.

The crack arrest behavior of individual grain boundaries was studied in detail by Qiao and Argon [2] in bi-crystals extracted from a large Fe-3% Si alloy ingot containing very large grains. The increments in the fracture toughness, AK, of individual grain boundaries in such bi-crystals were measured and related to the angles of tilt and twist across the boundaries. SEM studies of fracture surfaces indicated that grain boundaries force impinging cleavage cracks to break-up at the grain boundary and enter into the adjacent grain in a staircase manner on the cleavage planes of the adjoining grain. The additional cleavage fracture surface that needs to be created in the process and the associated ductile tearing of the portions of the grain boundary islands left behind are responsible for the increment in

New developments in the brittle to ductile transition of fracture in 3 intrinsically brittle crystal and polycrystals.

toughness. A simple geometrical model was quite successful in accounting for the dependence of the measured I:lK, on the misfit angles of tilt and twist across the boundaries [12,13].

5. PERCOLATION OF CLEAVAGE FRACTURE ACROSS A FIELD OF BRITTLE GRAINS

In steels in the lower shelf region, below the fracture transition temperature, the work of fracture is influenced directly by the same geometrical processes that govern the toughness increments of individual grain boundaries in polycrystals. The overall lower shelf toughness however, also depends strongly on the many necessary local plastic tearing events when cleavage cracks propagate across a field of brittle grains. These processes were studied in detail experimentally in very coarse grained Fe-2% Si alloy by Qiao and Argon [14] to map out the percolation path of the fracture front. A model based on the crack tip weight-function approach and utilizing the geometrical aspects of resistance of cracking across grain boundaries derived from the bi-crystal experiments referred to in Section 3 above gave excellent agreement with the experimental findings [14].

6. DISCUSSION

While the fundamental experiments and models of the brittle-to-ductile transition process described briefly above have clarified the mechanisms considerably many important considerations based on the statistical aspects of the fracture instabilities still require attention on an operational level [15].

7. ACKNOWLEDGEMENT

The research of A.S. Argon on the B-D transitions has been supported by the ONR under grants N00014-92-3-4022 and NOOOI4-96-1-0629, as well as by the NSF under grant DMR-9906613.

8. REFERENCES

1. Rice, J.R., Beltz, G.B. and Sun, Y. (1992), in "Topics in Fracture and Fatigue", edited by AS. Argon, Springer: New York, pp. 1-58.

2. Xu, G., Argon, AS. and Ortiz, M. (1995), Phil. Mag., 72, 415-451.

3. Xu, G., Argon, AS. and Ortiz, M. (1997), Phil. Mag., 75, 341-367.

4 A. S. Argon

4. St. John, C. (1975), Phil. Mag., 32, 1193-1212. 5. Brede, M. and Haasen, P. (l988), Acta Metall., 36, 2003-2018. 6. Hirsch, P.B., Roberts, S.G., Samuels, 1. and Warren, P. (1989),

in "Advances in Fracture Research", edited by Salama, K., et aI., Pergamon Press: Oxford, Vol. 1, pp. 139-158.

7. GaIly, B.1. and Argon, A.S. (2001), Phil. Mag., 81, 699-740. 8. Argon, A.S. and Gally, B.J. (2001), Scripta Mater., 45, 1287-

1294. 9. Qiao, Y. and Argon, A.S. (2003), Mech. Materials, in the press. 10. Argon, A.S. (2001), J. Eng. Mater. Technol., 123, 1-11. 11. Hahn, G.T., Averbach, B.L., Owen, W.S. and Cohen, M.

(1959) in "Fracture", edited by Averbach, B.L. et aI., MIT Press: Cambridge, MA. pp.91-116.

12. Qiao, Y. and Argon, A.S. (2003), Mech. Materials, 35, 313-332.

13. Argon, A.S. and Qiao, Y. (2002), Phil. Mag., 82, 3333-3348 14. Qiao, Y. and Argon, A.S. (2003), Mech. Materials, 35, 129-

154. 15. Knott, 1.F. (2002), Phil. Mag., 82, 3455-3470.

DISLOCATION-BASED LENGTH-SCALES IN CRYSTAL PLASTICITY: EXPERIMENTS AND MODELING

Sia Nemat-Nasser

Center of Excellence for Advanced Materials University of California San Diego

9500 Gilman Drive, La Jolla, CA 92093-0416

Abstract: Considered is a dislocation-based plasticity model that includes both temperature- and strain-rate effects, and heavily draws from a body of experimental data on various metals over broad ranges of strain rates, from quasi-static to 104/s and greater, and temperatures from 77 to 1,300K and greater. In this model, the role of the strain gradient is embedded in the nature of the dislocations, their density and distribution, and the manner by which they produce slip in crystal plasticity and affect the overall flow stress. The model includes length scales that are directly related to the dislocation densities and hence change with temperature and the strain-rate histories. The model can be used to calculate the force-deformation relations at micron to continuum dimensions. For plastic deformation of small polycrystalline samples involving only a few grains, geometric and textural incompatibilities will most likely manifest themselves through a size effect, and may affect the overall materials' resistance to deformation (flow stress). This size effect is distinguished from the length scales in plasticity, and the size effect is viewed as a problem-dependent phenomenon. For a few interacting crystals, the proposed model of slip-induced crystal plasticity should adequately account for any such size effects

Keywords: plasticity, length-scale, size effect, temperature, strain-rate

1 INTRODUCTION

Plastic deformation of a broad class of metals occurs by the motion of dislocations. The structure of dislocations, their density and distribution, as well as their interaction with each other and with the solute atoms and other defects lie at the foundation of slip-induced crystal plasticity. Collectively, all these affect the motion of dislocations and, hence, the resulting' plastic deformation. The collective resistance to the dislocation motion defines the flow stress of the material at the continuum crystal scale

5

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization oj Elastic-Inelastic Behavior oj Engineering Materials, 5-16. ©2004 Kluwer Academic Publishers.

6 S. Nemat Nasser

and the interactive resistance to the deformation of the crystals within a polycrystalline metal defines the overall flow stress of the metal. When there are few grains, geometric and textural incompatibilities will most likely manifest themselves through a size effect, and may affect the overall materials' resistance to deformation (flow stress) in a major way. This size effect should be distinguished from the length scales in plasticity. Dislocation-based metal plasticity naturally involves length scales associated with the densities of dislocations (mobile and tota/), whether or not there is also a size effect. While these length scales naturally enter plasticity constitutive relations, the size effect is a problem-dependent phenomenon that must be examined in each case using a relevant material­specific dislocation-based crystal plasticity model. For a few interacting crystals, the slip-induced crystal plasticity should be used and would adequately account for any such size effects.

To illustrate the size effect on the response of crystalline metals, consider a bicrystal deforming in simple shear, each crystal having only a single slip plane. Some hcp crystals, e.g., certain crystalline ice, easily slip on their basal plane but hardly on other potential slip planes. Consider the ideal cases shown in Figures 1 a and 1 b. In Figure 1 a, the two grains in the bicrystal have parallel slip planes, shown by dotted lines, whereas in Figure 1 b, these planes are normal to one another. If under an applied shear stress, 1'0 , the engineering shear strain in the first case is 2ro' then under the same shear stress, the engineering shear strain in the second case would be roo As is shown in Figure Ie, the second bicrystal will be twice as stiff as the first one, simply due to textural incompatibility.

21. -~_T

I

a

y, _ T

"2 To I

b ' C

Iro~ L..-__ --L-__

Y. 2y.

Figure 1. A bicrystal with parallel (aj, and normal (bj, slip planes; and the resulting

response (cj

2 DISLOCATIONS AND LENGTH SCALES

Over the past several years, I have examined an approach to plasticity, based on dislocation models of plasticity that include both temperature- and strain-rate effects, and heavily draws from a body of experimental data on various metals over broad ranges of strain rates, from quasi-static to 104/s and greater, and temperatures from 77 to 1,300K and

Dislocation-based length-scales in crystal plasticity: experiments and 7 modeling

greater. In this approach, the role of the strain gradient is embedded in the nature of the dislocations, their density and distribution, and the manner by which they produce slip in crystal plasticity and affect the overall flow stress. Here the length scales are directly related to the dislocation densities and hence change with temperature and the strain-rate histories. The model can be used to calculate the force-deformation relations at micron to continuum dimensions.

As is well known, the core of a dislocation involves only a few lattices within which immense lattice distortion and strain gradient are present. Since this strain gradient dies inversely with the square of the distance from the dislocation line [1], each dislocation line carries with it a concentrated strain gradient field at the nano-scale, and hence large strain gradients exist within each cluster and across various clusters of dislocations on each slip plane of each crystal in a polycrystalline metals.

3 RESISTANCE TO DISLOCATION MOTION

The motion of the dislocations on a slip plane in a slip direction is resisted by short-range and long-range barriers. In addition, the dislocations must overcome any drag forces that may act on them as they move from one set of short-range barriers to the next. The dislocations can overcome their short-range barriers partly by their thermal activation, and partly by the action of the net shear stress due to the externally applied forces.

The average dislocation velocity v can be expressed in terms of the density of the short-range barriers that the dislocations must overcome in their motion, and their average activation energy, LtG. For fcc and some hcp metals, it is usually the total dislocations which intersect the slip plane that are the barriers to the motion of mobile dislocations lying on the slip plane. For bcc metals on the other hand, the lattice resistance itself (the Peierls stress) may be the dominating hindrance to the dislocation motion. To estimate the average dislocation velocity, we now divide the average spacing of the short-range barriers, Is> by the sum ofthe average waiting time to cross the barrier, tw, and the running time, tro to move between the barriers [2],

(1)

The waiting time is estimated from the dislocation's rate of success in overcoming its short-range barriers. Using classical statistical arguments, it can be shown that

(2)

wheremo is the total attempt frequency (which depends on the dislocation

core structure), T is temperature, and k is Boltzmann's constant. The running time may be estimated using a linear drag model, in which the average running velocity, v" is related linearly to the net driving force,

(t' - t'a)b, by

8 S. Nemat Nasser

(3)

where D is the drag coefficient, T is the resolved shear stress in the slip direction, and Ta (the stress due to the long-range barriers) is the resistance imposed on the moving dislocations by the elastic stress field of all dislocations and defects. The slip rate, f , is given by t = b Pm v, where Pm is the density of mobile dislocations. Combining (1) to (3), obtain

( J[ O()~I . . lis TD Is 0 • b y=yo -2 exri.,.dJ/ KI)+~ - ,TD =OJoDfo/b, Yo =-OJo' (4)

1m T Ta 10 10

where 10 is some convenient reference length and to is a corresponding

reference strain rate, and 1m = Pm -1/2 is the average spacing of the mobile dislocations. For bcc crystals, for example, one may use the lattice spacing for 10 , and for fcc crystals it may be more appropriate to use the average spacing of the total dislocations at some reference state. In (4), Is and 1m are viewed as natural length scales that characterize the microstructure and dislocation activities, evolving with the temperature and deformation histories. They thus require evolutionary constitutive descriptions.

The resistance of the long-range barriers, Ta , is often referred to as the athermal component of the slip resistance. Being due to the elastic field of the dislocations and defects, its dependence on temperature is through the temperature dependence of the elastic moduli, especially the shear modulus, Ji(T) , and the temperature-history dependence of the microstructure, e.g., dislocation density. This gives,

Ta = f(p,,···)Ji(T)/ Jio (5)

where p, is the average total dislocation density, the dots stand for parameters associated with other defects and impurities that help to create an elastic stress field, and Jio is a reference value of the shear modulus. Set

I, = p,-1/2 and from (5) obtain,

To = f(/" ... )Ji(T)/ Jio' (6)

In general, the dependence of Ta on the length parameter I, is nonlinear.

Often it is assumed thatf r:::, ko II" following Taylor's model of dislocation plasticity [3, 4]. It is necessary to develop a constitutive relation for the evolution of It in terms of the deformation and temperature histories.

Dislocation-based length-scales in crystal plasticity: experiments and 9 modeling

The drag effect becomes significant especially at high temperatures. From

(4), for suitably large T, and with ~(~»> 1, obtain T - To 10

T R! To + Td, Td = Eo (~: J f, Eo = D( ~J (7)

In some applications, the drag effects may be neglected. Then, setting D = 0 in (4), we obtain,

f = fo (~:; }xP( -L1G / kT). (8)

4 ACTIVATION ENERGY

Consider now a typical slip system and examine the following relation for the activation energy of the dislocations:

L1G=G [l_(T-Ta)P]q f=~=f 10 of' bAI 0 I '

, Go T =--

o bAl' o

(9)

Here, Go is the total short-range barrier's energy, f is the resolve shear stress above which a dislocation can glide over its short-range barriers without the thermal assistance, and A, p, and q define the structure of the short-range barriers, with V* = bAI being the corresponding activation volume and Va * = bAlo its reference value. In (9), I is another length scale that characterizes the distribution and structure of the short-range barriers.

Equation (9) has been obtained empirically by the author and coworkers, but it has a long history going back to Ono [5] and Kocks et al. [6]. Ono suggests 0 < p ::;; land 1::;; q ::;; 2 for most energy barrier profiles. We have now extensive experimental data for tantalum and its alloys, molybdenum, niobium, vanadium, various types of titanium, stainless steel, DH-36 steel, HSLA65 steel and OFHC copper. All these data support Ono's suggestion.

Equations (4), (6), and (9) now define the slip rate in terms of the resolved shear stress, T, temperature, T, and four length parameters, 1m, Is, 1/, and I. These length scales directly relate to the physics of dislocation­induced plastic deformation of metals, and their evolution with deformation and temperature variation can be modeled based on experimental results and physical arguments.

10 S. Nemat Nasser

5 SINGLE CRYSTALS

Fcc crystals have a total of 12 slip systems. There are four slip planes, the {Ill }-family. Each plane has three slip directions, the <llO>-family. On each slip plane, one direction can be expressed as a linear combination of the two other directions. Additionally, one of the four unit vectors normal to the slip planes can be expressed in terms of the other three. Hence, in general only six independent slip systems can be identified. Also, since the tensor s ® n has zero trace, there are actually a maximum of only five independent slip systems for any crystal structure, even though there may exist many potential slip systems, e.g., 48 in tantalum [7]. The number of independent slip systems may be less than five if there are other constraints, e.g., in some hcp crystals. With these comments in mind, we identifY the N slip systems of a crystal by sa ® na , a = 1,2, ... , N, and define the slip­

induced plastic distortion of the crystal by

N

[J' = Ifasa ®na , (10) a=l

where r is now defined by an equation similar to (4)1.

6 DISLOCATION-BASED CONTINUUM APPROACH TO POLYCRYSTALS

The formulation presented above has been successfully used to model the observed thermo-mechanical response of a number of polycrystalline metals over broad ranges of temperatures, strains, and strain rates. For polycrystals, T and f are the effective von Mises stress and strain

rate, respectively defined by

(3 )112

T = "2 (T~ (T~ ,

(11)

where (T~, i, j = 1,2,3, are the rectangular Cartesian components of the

deviatoric part of the true stress tensor, and D: are the components of the

deviatoric part of the plastic deformation rate tensor. In (11), repeated indices are summed over 1, 2, and 3. For uniaxial tests, T is the axial stress and f is the axial strain rate. To illustrate the use of model res~lts in three­dimensional settings, consider a plasticity model in which the Jaumann rate

'" of true stress, (T ij , is related to the elastic deformation rate tensor by

Dislocation-based length-scales in crystal plasticity: experiments and 11 modeling

a a ai} =Ci}kJ (DkJ -Dt), ai} =&i} -W;ka.y +a;kw.y,

(12)

where Ci}kJ is the instantaneous elasticity tensor, Di} and wi} are the

deformation rate and spin tensors. We now consider the simplest model for the deviatoric plastic deformation rate, D% ' as follows:

(13)

Since i is positive, r is monotonically increasing and may be used as a time parameter.

7 EXPERIMENTAL VERIFICATION

As commented before, the author and coworkers have been characterizing a number of metals over broad ranges of temperatures and strain rates. The formulation given in the present work was originally suggested by the experimental data obtained from direct measurement of the strain hardening, the strain rate effect, and the thermal softening in tantalum and tantalum-tungsten alloys [8] and subsequently applied to model experimental data on OFHC copper [9] and many other metals.

Using recovery Hopkinson techniques [10], it has become possible to obtain isothermal stress-strain relations over a broad ranges of temperatures and strain rates. In addition, techniques have been developed to implement strain-rate jumps in high strain-rate tests [II]. From such data, the stress that the sample can carry (the flow stress) can be obtained as a function of temperature for various strains at a fixed strain rate, as shown in Figure 2a for commercially pure vanadium. Since at high strain rates, adiabatic heating increases the sample temperature, this temperature increase is calculated from

L1T ~ r !Z.~dr, pCv

(14)

where p is the mass density of the metal, C v is the heat capacity at constant

volume (generally a function of temperature and pressure), and 1] is the

fraction of plastic work actually converted to heat. While 1] < I is generally

expected, our experimental results show that 1] ~ 1, at high strain rates and

for suitably large plastic strains [12]. The temperature correction given by (14) is incorporated in Figure 2a. The data suggest that the flow stress is essentially temperature independent at sufficiently high temperatures. The

12 S. Nemat Nasser

limiting values of the flow stress may thus be estimated and plotted in terms of the strain, as shown in Figure 2b. Subtracting these from the stress­temperature results of Figure 2a, arrive at the results shown in Figure 2c. To interpret these experimental data, neglect for the time being the drag effects, and combine (8) and (9), arriving at

r*=r-r =r [1-[- kT InL)l/Q]IIP a 0 G' ,

o Yr

(15)

Our experimental results give p = 2/3 and q = 2 for many metals (see Table 1.), in line with Ono's conclusion [5]. In addition, from Figure 2b, note the following empirical result:

(16)

where n = 115, r! and r~ are material constants.

1000,--------------, ~,-------------,

Vanadium, 2,500 S·1 .. BOO

600

400 ~

• strain=O.05 • strain=O.1 a strain"'O.2 ~ strain"'O.3 s strain=O.4

• strain=O.45

• Experiman1s --200 ~~ __ ~ ___ ~~~_---.J

~L-__________________ ~

o 200 400 600 800 1000 000 0.10

Temperature, (K)

(a)

700,--------------, Vanadium, 2,500 s"

200 400 600

• strain-a.OS .. strain=O.1 ~ strBin=O.2 " strain=O.3 • etrein=O,4 + strain=0.45

800 .100 L..... ____________ -'

Temperature, (K)

(c)

'al 0.30 'AO 0.50 True Strain

(b)

Figure 2. Flow stress of vanadium at 2,500/s strain rate: (a) as afunction of temperature, (b) the athermal part as a function of strain, and (c) the thermal part as a function of temperature; from Nemat-Nasser and Guo [13 J.

Dislocation-based length-scales in crystal plasticity: experiments and 13 modeling

The fmal results for vanadium is independently checked for an 8,000/s strain rate in Figure 3.

700 r-------------------------------------~

600

500 ., a. ~400 '" ., ~ en 300 ., 2 I-

200

100

Vanadium, 8.000 s"

.. --:::._-----~--------------------==------~=::__-----LI1T ; 296K

o ~----------------------__ ~ ____ ~~ __ _J

0.00 0 ,05 0.10 0.15 0.20 0.25 0.30 0.35 OAO 0,45

True Strain

Figure. 3. Comparison of model predictions with experimental results for indicated initial temperatures and indicated strain rate

Table 1 summarizes the material parameters for tantalum and other bcc

metals that we have experimentally characterized. The constant term,.!, is

small for these materials and may be set equal to zero.

Table 1. Model parameters for indicated metal

p q f~ , n f o , YT k/Go

MPa MPa 107 S'l lO'5K"1

Ta 213 2 473 1/5 1,100 54.6 8.62 Va 2/3 2 305 1/5 1,050 0.358 12.72 Nb 1/2 2 440 114 1,680 0.35 12.4 Mo 213 2 720 1/4 2,450 1.45 8.62

For many metals, the lattice resistance to dislocation motion may not entail as much energy as the resistance due to the dislocation forests which intersect the slip plane. Most fcc metals are of this kind, as are some hcp and others. Since the dislocations are the dominating short-range barriers, their evolution must be considered and modeled. As a starting point, we note that the density of dislocations decreases with increasing temperature, and it

14 s. Nemat Nasser

increases with further plastic deformation. Thus length scales Is and I change with temperature and plastic deformation.

The quantity r is a monotonically increasing parameter and may be used as

a load parameter. Following this and guided by a vast body of experimental data, we let the variation of the length scales in (15) have an evolution defined by

10 ~ 10 = f(r,T) > 0, f(O,1~) = 1, of ~ 0, of:s; o. (17) I Is or oT

The functionf can now be established empirically for a given metal based on experimental results. My coworkers and I have found the following expression to be suitable for a variety of metals, including OFHC copper [9], AI-6XN stainless steel [14], and several Ti-6AI-4V alloys obtained through different processing [15],

f = (1 + a(T)rm), aCT) = ao[I- (T ITm )2], (18)

where Tm is the melting temperature of the metal (about I,350K for copper) and ao is a constant that depends on the initial state of the material, i.e., whether annealed or otherwise. Figures 4a and 4b illustrate this, and Table 2 provides the values of constitutive parameters for indicated metals.

OOD Solid Cun.: Model Pl1IdictiDns Dahed Cuv.: ExperimenIB

OFHC Copper, 4,OOOIs

7tID

OlIO

.. ~ -50D

~ .... .... .,-., ..'"

~ 3DD >-

"'" 100

D.2 •• D .• D .• 1.2

True strain

(a)

Dislocation-based length-scales in crystal plasticity: experiments and 15 modeling

BOO Solid Curves: Model Predictions Dashed Curves. Experiments

OFHC Copper, 4,00015

500

400 ---'" --a. --~ -~

i': 300

1i5

" 2 .... 200

100

02 OA 0.6 0.8 1.2

True Strain

(b)

Figure 4, Comparison between experimental and model results for OFHC copper at indicated initial temperatures and at a strain rate of 4,000/s, for the as-received (a) and annealed (b) material; from Nemat-Nasser and Li [9J

Table 2, Model parameters for indicated metal

p q /rIGo,K To' MPa ,0 1

Y, ,s OFHC Cu, .s 2/3 2 4.9x 10- 400 2xlO received OFHC Cu, 2/3 2 4.9xI0- 46 2x10'" annealed AL·6XN 213 2 66x 10- 630 2x10'"

DH36 213 2 6.6xIO' 1,500 2xlO

Ti·1 I 2 6.2x lO' 1560 L3xlO'"

Ti·2' I 2 6.2xlO- 1900 L3xIO'"

Ti·3' I 2 6.2x 10- 1620 L3xlO'"

1 CommercIal 2 Rapidly solidified Ti-6AI-4V powder is milled and then HIPed [15] 3 Rapidly solidified Ti-6AI-4 V powder is HIPed [15] • NA stands for "not available"

8 REFERENCES

Qo m r;, MPa n

1.8 112 220 0.3

20 112 220 0.3

5 112 900 0.35

0 - 750 0.25

2.4 I 685 0_05

2.4 I 710 0.03

2.4 I 680 0_04

[1] J. Hirth and J. Lothe, Theory of dislocations. 2nd Ed., John Wiley & Sons, New York, 1992.

[2] G. Regazzoni, U.F. Knocks, and P_S. Follansbee, "Dislocation kinetics at high strain rates," Acta Metall., vol. 35, pp. 2865-2875,1987.

[3] G. I. Taylor, "The mechanism of plastic deformation of crystals -I, II," Proc. R. Soc. Lond. A., vol. 145, pp. 362-387, pp. 388-404, 1934.

[4} G. I. Taylor, "Plastic strain in metals," J. Jnst. Metals, vol. 62, pp. 307-324, 1938. [5] K. Ono., "Temperature dependence of dispersed barrier hardening," J. Appl. Phys.,

vol. 39, pp.1803-1806, 1968.

16 S. Nemat Nasser

[6] U. F. Kocks, A. S. Argon, and M. F. Ashby, "Thermodynamics and kinetics of slip," Progress in Materials Science, vol. 19, pp. 1-271, 1975.

[7] S. Nemat-Nasser, T. Okinaka, and L. Ni, "A Physically-based constitutive model for bcc crystals with application to polycrystalline tantalum," J. Meeh. Phys. So/ids, vol. 46, no. 6, pp. 1009-1038,1998.

[8] S. Nemat-Nasser and J. B. Isaacs, "Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta­W alloys," Acta Mater., vol. 45, pp. 907-919, 1997.

[9] S. Nemat-Nasser and Y. L. Li, "Flow stress of fcc polycrystals with application to OFHC Cu," Acta Mater., vol. 46, pp. 565-577, 1998.

[10] S. Nemat-Nasser, J. B. Isaacs, and 1. E. Starrett, "Hopkinson techniques for dynamic recovery experiments," Proe. R. Soc. Lond. A., vol. 435, pp. 371-391, 1991.

[II] S. Nemat-Nasser, Y. F. Li, and J. B. Isaacs, "Experimental/computational evaluation of flow stress at high strain rates with application to adiabatic shear banding," Meeh. Mat., vol. 17, pp. II 1-134, 1994.

[12] R. Kapoor and S. Nemat-Nasser, "Determination of temperature rise during high strain rate deformation," Meeh. Mat., vol. 27, pp. 1-12, 1998.

[13] S. Nemat-Nasser and W. G. Guo, "High-strain-rate response of commercially pure vanadium," Meeh. Mat., vol. 32, no. 4, pp. 243-260, 2000.

[14] S. Nemat-Nasser, W. G. Guo, and D. P. Kihl, "Thermomechanical response of AL-6XN stainless steel over a wide range of strain rates and temperatures," J. Meeh. Phys. Solids, vol. 49, pp. 1823-1846,2001.

[15] S. Nemat-Nasser, W. G. Guo, V. Nesterenko, S. S. Indrakanti, and Y. Gu, "Dynamic response of conventional and hot isostatically pressed Ti-6AI-4V alloys: experiments and modeling," Meeh. Mat., vol. 33, no. 8, pp. 425-439, 2001.

APPLICATION OF A VARIATIONAL SELF ·CONSISTENT PROCEDURE TO THE PREDICTION OF DEFORMATION TEXTURESINPOLYCRYSTALS

Pierre Gilormini1), Yi Liu2) , and Pedro Ponte Castafieda2)

1) Lahoratoire de Mecanique et Technologie,

ENS de Cachan-CNRS-Universite Paris 6, 94235 Cachan, France

Email: Pierre.Gilormini@lml.ens-cachan.fr

2) Mechanical Engineering and Applied Mechanics,

University of Pennsylvania, Philadelphia. PA 19104-6315, USA

Email: yiliu2@seas.upenn.edu, ponte@seas.upenn.edu

Abstract A fundamental problem in the mechanics of materials is the computation of the macroscopic response of polycrystalline aggregates from the properties of their constituent single-crystal grains and the microstructure. In this paper, the nonlin­ear homogenization method of deBotton and Ponte Castaneda is used to compute "variational" self-consistent estimates for the effective behavior of pOlycrystals. Earlier papers have detailed the "instantaneous" mechanical response of poly­crystals, but the present study focusses on the evolution the crystallographic texture predicted by this procedure.

Keywords: polycrystals, textures, variational procedure, self-consistent model

1. INTRODUCTION

Forming processes involve large finite strains that are known to induce strong anisotropy of the mechanical properties of polycrystalline metals, especially when their crystal structure is hexagonal. This is due to the reorientation of the crystalline axes induced by plastic strain, and to the highly anisotropic behavior of hexagonal crystals. The prediction of the evolution of the so-called deformation texture, and of the corresponding anisotropy, has been performed for several decades by means of various models, as reported for instance in [1].

The aim of this paper, which reports on a preliminary study (more extensive results will be presented in [2]), is to show that a method that has been pro-

17

S. Ahzi et at. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 17-24. ©2004 Kluwer Academic Publishers.

18 Pierre Gilormini, Yi Liu and Pedro Ponte Castaneda

posed recently for computing the overall properties of polycrystals ([3], [4], [5]), is also able to predict deformation textures with interesting features. A comparison will be made with the results of the most popular model that is presently used ([6], [7]), and with some experimental textures measured on titanium polycrystals obtained by [8].

2. VARIATIONAL PROCEDURE

The variational procedure that is used in this study follows from the original paper by [9], on nonlinear composites. Applications to polycrystals with fixed textures have already been presented in several papers, including [3], for face­centered cubic crystals, [4], for other cubic structures, and [5], for hexagonal crystals. Consequently, the details of the procedure will not be repeated here, and only the essential equations will be recalled, as they apply specifically to polycrystals.

The approach applies to crystals where the behavior of the slip systems is governed by a convex potential. This includes viscoplastic crystals, for which the standard example of the potential obeys a power law:

(I (g)l) ~ (g) _ m (g). T(s)

'P(s) - 1 + m TO(s) 'Yo ----cg) TO(s)

(1)

where the slip rate sensitivity m is between 0 and 1, T(~} = IL~~~ : u (with

,,(g) = 1 (n(g) @ meg) + meg) @ neg») meg) and neg) being unit vectors ""(s) 2 (s) (s) (s) (s)' (s) (s) parallel to the slip direction and normal to the slip plane, respectively) is the resolved shear stress on the system s of grain 9 through the stress tensor u, Tcif1) is a reference shear stress and 'Yo a reference slip rate (assumed the same for all systems for convenience). Such a power-law potential is assumed to hold in this work for each slip system in the polycrystal, although this is not a limitation of the approach ([5], for instance). Moreover, m will be assumed the same for all systems, which is not required by the approach either (an example with a non-uniform m is also given in [5]).

Basically, the variational procedure introduces a linear comparison poly­crystal in a systematic manner, which involves the same slip systems as in the nonlinear polycrystal, but obeying quadratic potentials:

~(g) _ 1 ((g»)2 'P(s) - 2.,.,(g) T(s)

"(s)

(2)

A complete equivalence between the nonlinear and comparison polycrystals

is obtained when the stiffness 'f/~;? of each system is allowed to vary not only from system to system in any grain, and from grain to grain, but even inside

Application of a Variational Self-Consistent Procedure 19

each grain, provided that a maximization problem can be solved to optimize

the choice of the 77~!? field. In practical applications, 77~!? is assumed uniform in each grain, which leads to an underestimate of the effective stress potential cp(E) of the polycrystal:

q,(~) ~ max [~~: M: ~ - L w(g) L v:~g)l (3) (g»o 2 ( )

1/(0) _ 9 s

where w(g) is the volume fraction of grains with crystallographic orientation g

and where Yc.~» is given by:

(g) • v:(g) _ 1 - m 7"0(8)'Y0

(8) - 1+m -2- ( 7"ci(~) ) ~

--w-:-77(8) 'Yo (4)

If M is evaluated in the linear comparison polycrystal by the self-consistent model, equation (3) gives a self consistent estimate of the response of the nonlinear polycrystal. The computation of M involves the compliance of each grain in the linear comparison polycrystal:

M(g) = ~ _1_ ,,(g) ® ,,(g) L.J (g) r(s) r(s)

8 77(8)

(5)

The nonlinear self-consistent estimate that will be obtained with this procedure will be in agreement with the bounds that can be defined on the nonlinear polycrystal by using lower bounds of M in equation (3).

This brief outline of the method shows that a large optimization problem has to be solved to apply the variational procedure, as shown in equation (3): the number of optimization variables is equal to the total number of slip sys­tems available in the polycrystal. Simple algorithms were feasible as long as transversely isotropic polycrystals were considered, without texture evolution. The Powell method was used in [4], and [5], for instance, because a small number of crystallographic orientations was sufficient. No partial derivatives were required, but the large numbers of grains (and, consequently, variables) that are involved in texture predictions could not be afforded. The present work makes use of a more efficient method, that is more suitable for the large scale of the optimization problem considered, namely the modification of the BFGS method proposed by [10]. This requires the partial derivatives of the function to optimize, which were computed from analytical formulae derived from equation (3) by using symbolic programming.

After the linear comparison polycrystal has been obtained through the op­timization procedure, there remains to allow the microstructure evolve before repeating the process at the next time increment. As explained in [11], the tex­ture change is computed from the slip rates obtained for each crystal orientation

20 Pierre Gilormini, Yz Liu and Pedro Ponte Castaneda

in the linear comparison polycrystal, which lead to the plastic spin and, finally, to the rotation rate of the crystallographic axes of the grains.

3. MATERIAL AND LOADING CONDITIONS CONSIDERED

The above procedure has been applied to the titanium polycrystal described in [8]. Twenty-four slip systems were considered, belonging to 4 families of systems: 3 {OOOI }(1120) systems for basal slip, 3 {loIO}{1120} for prismatic slip, 12 {lOIT}{1123} for first-order pyramidal (c+a) slip, and 6 {1122}(1123) for second-order pyramidal (c + a) slip. A slip-rate sensitivity of m = 0.16 has been used on all systems, as well as a reference slip rate of 1-'0 = 10-3 per second. Different reference shear stresses were taken into account: the initial values were Tci(;) = 8.2 MPa for basal and prismatic slip, and Tci(;) = 82 MPa

for pyramidal slip, with a hardening law that increased all Tcif;) values with the same rate

[ T(g) ( . ) n]

i(g) = ho ""' 1 - O(r) ~ 11-'(g) 1 O(s) L..t T () I' (g) 1 (r)

r Ir I(r)

(6)

with ho = 12 MPa, n = 0.1, and TI(r) = 18 MPa for basal and prismatic systems and 180 MPa for pyramidal systems in the above summation. Since all reference shear stresses in a grain are assumed to change with the same amplitudes, there results a decrease in the local contrast from the initial value of 10, i.e. in the ratio between the "hardest" and "softest" systems in each grain. It should be noted that, because of the lack of direct measurements on single crystals, these values were deduced by [8], by fitting the predictions of their finite element model to experimental measurements obtained with titanium polycrystals. As a result, these data are probably not the best possible for getting good agreement between experiments and other models. They will nevertheless be used in this study as a reasonable starting set of values that allows some preliminary comparisons between models. Therefore, only qualitative comparisons with experimental results will be possible. More quantitative analyzes would require new fittings (one for each model, actually), and are presently underway.

The texture is not isotropic initially, as shown in figure 3, where orthotropic symmetry has been applied to the same set of 729 orientations as used by [8], to simulate their experimental pole figures. The simulations presented below use the same data file and symmetrization and, consequently, 24 x 729 = 17,496 variables are involved in the maximization procedure of the self-consistent variational model.

Uniaxial compression has been applied to the above initial texture (other loading types will be reported in [2]). A reduction in height f::l.H / Ho of 63% (i.e., an axial strain of -1) was prescribed along the axis normal to figures 3

Application of a Variational Self-Consistent Procedure 21

Figure 1. {OOOI} and {lOll} polefigures of the initial texture.

Dotted areas are below the lowest level shown. Equal area projection is used.

and 2, with a constant axial strain rate iI / H of 10-3 per second, keeping all lateral sides stress-free. This corresponds to an experiment of [8], with friction neglected in our simulations.

4. RESULTS In the present paper, which reports a preliminary study, the results obtained

with the variational procedure are compared only to those given by the tangent model of [6], and [7]. This approach applies the self-consistent model to a tan­gent approximation of the nonlinear behavior of the slip systems, and is widely used to simulate the texture evolution in metals as well as in minerals (many examples are presented in [1], for instance). The VPSC5 program developed by R. Lebensohn and C. Tome has been used in the simulations that are presented here.

Figure 4 shows the pole figures predicted by the variational procedure and by the tangent model after the uniaxial compression has been applied during 100 equal time increments. It can be observed that the two models used with the same data do predict different results. The tangent model predicts a {1 011 } pole figure with four groups of high intensity areas, including two located along the axis 2. This is absent from the predictions of the variational procedure as well as from the experimental pole figures shown in [8] (and from the finite element simulations in the same reference). A closer analysis of the differences between the two models can be performed by looking at the relative system activity, as defined in [7]. Figure 4 shows that the activity of the prismatic systems, which decreases when compression proceeds, is larger in the tangent model, while that of the basal systems (which increases) is smaller than in the variational model. The latter also predicts a significant activity of the pyramidal systems, that is

22 Pierre Gilormini, Yi Liu and Pedro Ponte Castaneda

Figure 2. Pole figures 0/ the textures predicted by the variational procedure (above)

and the tangent model (below) after a uniaxial compression ~H/Ho 0/63%.

almost absent in the results for the tangent model. The activities predicted by the (uniform strain) Taylor model are also shown in the figure, and it may be observed that they are very different from what the two models considered in this paper suggest. The pyramidal activity, for instance, is very large in the Taylor model, which is due to a complete lack of grains interaction: the model requires all grains to deform equally and consequently pyramidal systems are activated in most grains. The self-consistent model, on the opposite, allows each grain to deform differently, with a larger contribution of the soft systems. Another quantity related to this result is the average number of (significantly) active systems: the value computed from the Taylor model was about 10, whereas the variational and tangent models used respectively about 5 and 3 systems per grain on average (recall that prismatic and basal systems amount to a total of 6).

Application of a Variational Self-Consistent Procedure

., 1l .s: .tj .. ] 0.6

~ ~~:'~~.~~~flL ... ~::=~:=:::~ .... _=: .2 0.4 ,., .. ' ~ ......... """"'rangent -&. 0.2 ................... >,,··"::::~·······--··'-······Tayior············

.............

°0~~0~.2~~0~.4~~0~.6~~07.8~~ E

.................. Taylor

tan e °0~~0~.2~-0~.4~~0.6~~0.~8~~

E

Figure 3. System activities predicted in simulations of uniaxial compression.

Prismatic (solid lines) and basal (broken lines) systems on the left.

First-order (solid lines) and second-order (broken lines) pyramidal systems on the right.

s. CONCLUSION

23

It has been demonstrated that the variational procedure, combined with the self-consistent model, can be applied to the computation of the texture evolution induced by the deformation of polycrystals.

For titanium polycrystals, comparison with another, widely used, model has shown that some differences in the predictions are generated. Moreover, qualitative agreement with experimental pole figures has also been found. Now that computational issues have been addressed, more quantitative comparisons with experimental results are possible.

Acknowledgments

R. Lebensohn (IFIR, Argentina) and C. Tome (LANL, USA) are gratefully acknowledged for making some of their programs available. We are grateful to L. Anand (MIT, USA) for kindly providing the initial texture data file and valuable discussions.

References

[1] U. F. Kocks, C. N. Tome, andH. R. Wenk, Texture and Anisotropy. Cambridge University Press, Cambridge, 1998.

[2] Y. Liu, P. Gilonnini, and P. Ponte Castafieda, Variational self-consistent estimates for texture evolution in hexagonal polycrystals, submitted for publication, 2002.

[3] G. de Botton and P. Ponte Castaiieda, Variational estimates for the creep behavior of polycrystals, Proceedings of the Royal Society of London A, vol. 448, pp. 121-142, 1995.

[4] M. V. Nebozhyn, P. Gilorrnini, and P. Ponte Castaiieda, Variational self-consistent esti­mates for cubic viscoplastic polycrystals: the effect of grain anisotropy and shape, Journal of the Mechanics and Physics of Solids, vol. 49, pp. 313-340, 2001.

24 Pierre Gilormini, Yi Liu and Pedro Ponte Castaneda

[5] P. Gilonnini, M. V. Nebozhyn, and P. Ponte Castaneda, Accurate estimates for the creep behavior of hexagonal polycrystals, Acta Metallurgica, vol. 49, pp. 329-337, 2001.

[6] A. Molinari, G. R. Canova, and S. Ahzi, A self-consistent approach to the large deforma­tion polycrystal viscoelasticity, Acta Metallurgica, vol. 35, pp. 2983-2994, 1987.

[7] R. A. Lebensohn and C. N. Tome, A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys, Acta Metallurgica, vol. 41, pp. 2611-2624, 1993.

[8] S. Balasubramanian and L. Anand, Plasticity of initially textured hexagonal polycrystals at high homologous temperatures: application to titanium, Acta Materialia, vol. 50, pp.133-148,2002.

[9] P. Ponte Castaneda, The effective mechanical properties of nonlinear isotropic composites, Journal of the Mechanics and Physics of Solids, vol. 39, pp. 45-71, 1991.

[10] D. C. Liu and I. Nocedal, On the limited memory bfgs method large scale optimization, Mathematical Programming, vol. 45, pp. 503-528,1989.

[11] P. Ponte Castaneda, Nonlinear polycrystals with microstructure evolution, In E. Inan and K.Z. Markov, editors, Continuum Models and Discrete Systems. Proceedings of the 9th International Symposium, pp. 228-235. World Scientific Publishing Co., 1999.

WHAT ABOUT THE YIELD TRANSFORMATION SURFACE DETERMINATION (AUSTENITE -+ MARTENSITE) WITH THE MEASUREMENT OF AUSTENITE AND MARTENSITE LATTICE P~ETERSFORSOMESHAPEMEMORY ALLOYS?

C. Lexcellent, P. Blanc and C.Bouvet

Laboratoire de Mecanique Appliquee R. Chaieat. UMR 6604 CNRS-Universite de Franche-Comte. 24 rue de I'Epitaphe 25000 Besam;on (France). christian.lexcellent@univ-fcomteJr

Abstract: Like in the plasticity theory, the prediction of phase transfonnation yield

surfaces constitutes a key point in the modeling of polycrystalline shape

memory alloys thermomechanical behavior. Generally in some micro-macro

integration, the nature of the interface between austenite and twinned or

untwinned martensite under stress free state and the choice of correspondance

variants (CV) or habit plane variant (HPV) are determining for the explicit

expression of the yield criterion. If the prediction of some copper-based alloys

(interface between austenite and one single variant of martensite) and the Cu­

AI-Ni for cubic to orthorhombic phase transformation (interface between

austenite and twinned martensite) is fairly good, the prediction is not efficient

for the important case of Ti-Ni (interface between austenite and twinned

martensite with stress free state). The usual hypothesis consisting in neglecting

the effect of stress on the interface geometrical configuration must be

revisited.

Key words: shape memory alloys, phase transfonnation, interface, yield surface.

25

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 25-32. il:l2004 Kluwer Academic Publishers.

26 C Lexcellent, P. Blanc and CBouvet

1. INTRODUCTION

Shape memory alloys (SMA) are materials undergoing a first order structural phase transformation from a high symmetry parent phase (called Austenite) to a lower symmetry product phase (called Martensite).

Understanding the deformation behavior of SMA is predicted upon knowing the various microstructures which form during the transition between the phases under various loading conditions [1].

At first, a classical theoretical analysis of the austenite-martensite transformation called WLR [2] was presented. It predicts the habit plane, orientation relationships and macroscopic distortions only from the knowledge of the crystal structures and lattice parameters of the parent and the product phases. Hence a martensite variant is identified by its own shape strain and habit plane normal vectors.

More recently, the theory (called Crystallographic Theory of Martensite : CTM) used to construct microstructures is a geometrically non linear theory of martensitic transformation performed by Ball and James [3,4]. These authors formulate a free energy function that would produce the A-M interface by energy minimization and relate it to crystal structure.

In general, as said Bhattacharya [5], a single variant of martensite cannot have a coherent interface with the austenite. However a region consisting of fine twins of two martensitic variants can form a coherent interface with the austenite: Ni-Ti, Cu-AI-Ni (cubic ~ orthorhombic) ... As shown by Hanes [6] a number of alloys such as Cu-AI-Ni (cubic ~ monoclinic) Cu-Zn, Cu­Zn-Al, Cu-Zn-Ga to name of few [7] exhibit an undeformed interface between austenite and a single variant of martensite nevertheless.

The CTM gives mathematical tools to predict untwinned or twinned martensite at the interface with the austenite phase.

But the different possible configurations at the interface A-M are predicted for stress free state. This theory neglects any elastic strains and assumes that the martensite have stretches !L and the austenite is unstrained, even in the presence of stresses [8, 9, 10].

Even to analyse the mechanical behavior of SMA, the crystallographical microstructure is viewed as stress free. It means that exact compatibility of the transformation strains at zero stress is adopted.

The main question of the present paper consists in the workability of this important hypothesis. In this order, a comparison between theoretical predictions of the initiation surface of phase transformation and experimental points for biaxial proportional loadings (tension (compression)-torsion, bicompression tests) is done.

Two microstructural configurations are investigated, (i) the coherent interface between austenite and one single variant of martensite exhibited by Cu-AI-Be and Cu-AI-Zn for special composition, (ii) the transformed region

What about the yield Transformation surface determination ... 27

consists of parallel bands containing alternately two different variants of martensite exhibited for instance by Ni-Ti and Cu-AI-Ni (cubic ~ orthorhombic).

2. AUSTENITE-MARTENSITE POSSIBLE INTERFACES

2.1 Cubic to monoclinic phase transformations

Some copper based alloys and nitinols exhibit a phase transformation between a cubic parent phase A (lattice parameter a,,) and a product phase M which is monoclinic (lattice parameters a, b, c and e, angle between the edges with lengths a and c).

!­The strain gradient tensor.E and .!l =!:T!: can be written as :dXO(A)~dX(M)

!: = (: :~~::] .!l = (a,~:,o a't p~] a~l o 13 (frame of A) (frame of A) (2)

where the transformation stretches are

Fza b Fzc Fzc a=-- , 13=- and y=-- (MI8R)ory=-- (6M) [II]

a o a o 9ao 3ao

In addition, the eigenvalues AI, A2 and A3 of.!l are

2 2_ 12 22 22 2 2 a + y +" (a - y ) + 4a y cos e (3) "'I = 13, A2/3 =

2

With the usual values of a, \3, y obtained with lattice parameters X Ray measurements for these alloys, one obtains Al < A2 < A3 .

Following Ball and James [3,4] an interface exist between the austenite and a single variant of martensite called Mi (i = 1...12) if and only if the symmetric matrix Ii has an eigenvalue less than one (called AI), an eigenvalue greater than one (called 1...3) and the third equal to one (1...2)'

!J2 has an eigenvalue 1...2 = 1 if and only if, the monoclinic angle e and the transformation streches a and y are related by

2 2 2 (1 - a. )(1 - Y ) (4) cos e=~-....:...:..-~

2 2 a. y

28 C. Lexcellent. P. Blanc and C.Bouvet

For instance, we confmn that the situation is fulfilled for some copper based alloys (Cu-AI-Be, Cu-Zn-AI and Cu-AI-Ga) [6, 11, 12] but not for Ni­Ti [1] alloys (equation 4 and ~ values around 0,93).

2.1.1 Austenite - single variant of martensite microstructure

The precedent discussion means that the "habit plane equation" with the monoclinic variant !Ii (i = 1...12)

R. U. -l=b®m (5) -1-1 -

has resolved values for Ri (rotation matrice), iii: habit plane normal and b: the shape strain vector.

Besides, with the eingenvalues from eqn (3), it can be shown that the shape strain vector b and the habit plane normal m can be obtained [6, 11, 12]. At last for completeness, as recalled before, there is twelve distincts

variants U1 ... UI2, and 24 distinct possible couples (bk,mk) in this type of

phase transformation.

2.1.2 Austenite-twinned martensite microstructure

The austenite-twinned martensite microstructure consists of two adjacent regions in one of which the austenite phase is present and the other contains parallel bands of alternating layers of two martensite variants. Following Ball and James [3,4] with the martensite variant pair (i, j) the compatibility equations are (i) the twinning equation between the martensite variants i and j itself

R .. U· - U· =a®ii (6) -IJ -1 -J

(ii) and an equation called the habit plane equation which can be written as

R,,(AR .. U· + (l-A)U,) -1 = b ® iii (7) -IJ -IJ-l -J

The algorithm to fmd the solutions is given in the paper [1]. In a first step, the resolution of the twinning equation (6) delivers the

pairs (i, j) which are compatible. They are often called the correspondence variants CV s. For Ni-Ti, there are 132 correspondence variant pairs.

Secondly, the resolution of the habit plane equation permits to choose among the CV s pairs, the so-called habit plane variant (HPV) which is a

compound twin of AUi and (l-A)Uj . At last for Ni-Ti, 192 different

couples of (bik ,mjk) are obtained (with i,j =1 to 12, i;e j, k=l to 8).

If the twinning equation (6) is substituted in the habit plane equation (7), then the habit plane equation becomes

~ij 01 j + A a ® ii) -1 = b ® iii (8)

What about the yield Transformation surface determination ...

Let the symmetric matrix hii/..) defined by

~ij(A) = (llj + A a ® Ii)T (Qj + Aa ® Ii)

29

(9)

With the knowledge of el and e3 ' the eigenvectors corresponding to the

eigenvalues Al and A3 of ~ij (A) (AI ~ A2 = 1 ~ A3), b and in can be

obtained [1]. For Ni-Ti, the second eigenvalue of ~ij is exactly equal to 1

according to our calculations.

2.2 Cubic to orthorhombic transformation (examination of Cu-13.95 AI-3.93 Ni wt. (%))

In a similar way, 112 = eeE) can be given [8] with the eigenvalues :

Al = ~2 ,A2 = l , A3 = a 2 where a = 1.0619, 13 = 0.9178 and y = 1.023

(obtained for Cu-14.2 AI-4.3 Ni (wt. %) [13]). Evidently /..2 is different from 1. However, the calculation of eigenvalues

of gij (eq. 9) delivers the second root equal exactly to 1 and confirm the

presence of an interface between austenite and twinned martensite. Thus, we have 96 A-M possible austenite-martensite interfaces in this

alloy i.e. 96 different couples (bk , m k ) .

3. PHASE TRANSFORMATION SURFACE (AUSTENITE --+ MARTENSITE)

3.1 Theoretical considerations

Let have a biaxial loading

Q = 0'1 el ® el + 0'2 e2 ® e2 (10)

The predicted phase transformation surface must be at least convex in the stress space (O'b 0'2) and also must take account of the general asymmetry between tension and compression observed in SMA and at last must fitted the experimental yield points obtained for proportional biaxial loading as tension( compression)-torsion or bi-compression.

The process of homogenisation in order to obtain the yield phase transformation surface for a polycrystalline alloy is described in [12].

There is also a prediction of these surfaces at the macroscopic scale. As in the classical plasticity theory, a combination of the second (J2) and the

30 C. Lexcellent, P. Blanc and C.Bouvet

third (J3) invariants of the deviatoric stress tensor dev Q is used [12, 14, 15, 16]. This theory is called phenomenological one.

3.2 Phase transformation surface prediction

For the two copper-based alloys investigated (Cu-23.73 Zn - 9.4 Al (at. %) and Cu-24.2 Al - 2.95 Be (at. %) where an "austenite-single variant martensite" interface is predicted with stress free state, a good agreement is obtained between prediction and experiments (figure 1 and 2) .

.. 0, 2 cr/tl. CwA'B~fHJIycryllll

(M9",

Q ("P~

(M'. '" .,

~ ... .,

Figure I : Yield surface of Cu·Zn-AI polycrystal in the Figure 2 : Yield surface of Cu-AI-Be polycrystal in the

space (<>1.<>2) (austenite -+ martensite) (e experimental space (<>10<>2) (austenite -+ martensite) (e experimental

points micro-macro simulation points micro-macro simulation;

phenomenological simulation). phenomenological simulation).

For the Cu-13.95 Al - 3.93 Ni (wt. %) which is a seat of a cubic to orthorhombic transformation, remember that an interface between austenite and twinned martensite is predicted with stress free state. The micro-macro prediction (fig. 3) seems consistent with the two precedent shape yield curves obtained for the Cu-Zn-AI and Cu-AI-Be.

In particular, the established asymmetry in tension-compression for this alloys [8, 10] is taken into account in the prediction. -,,"

c. I"'~"", 'CI'istal

----- '\ '" I{){)~ "" ,'" / ) '" / /

(Mpo) ,

r / \ /

. ..,

....... t----cr, t- .... Mpo)

.,,,, .1~ -'00 -$ :iO 100 ISO

Figure 3 : Phase transformation surface of Cu-AI­

Ni polycrystal in the space (<>1. <>2)

(austenite -+ martensite).

Figure 4 : Phase transformation surface of Ni-Ti

polycrystal in the space (<>,. <>2) (austenite -+ twinned

martensite) (_ experimental points ; - micro-macro

simulation. _ phenomenological simulation)

What about the yield Transformation surface determination ... 31

The prediction obtained of Ti-49.75 Ni (at. %) with an austenite-twinned martensite interface with stress free state is more questionable (fig. 4).

For instance, the dissymetry between tension and compression which has been measured on this alloy [14,17] is not predicted.

This gap between prediction and real material behavior for Ni-Ti IS

confirmed by the work ofPatoor et al [18]. umbe<ofgrains - IOOO

At first, one has to note for Cu N;.Tipolycri".'" a,(Mpa)

based alloys their micro-macro fro - J70 Mpo ..

approach [18] and hour investigations give nearly the same yield phase transformation surface.

Secondly, in spite of the defintion of a new interaction matrix considering the existance of two C.Vs inside each HPV , they also failed in Figure S : Phase transfonnation surface of Ni-Ti

their prediction [18]. polycrystal in the space (<1,. <1,) (austenite ..... martensite as if it was single martensite)

With a slighty different point of (. experimental points; - micro-macro simulation, view, we do the same investigation phenomenological simulation)

with the "double" Hadamard condition and the obtained results are not efficient. At last, one has to say that the Patoor calculation [18] accouTrrn' for the interaction between the grains. Our interpretation of this strange Ni-Ti behavior is the following. It seems that the interface configuration (austenite-twinned martensite All i + (1 - A) 1I j) predicted with stress free state is not the same under

stresses. For instance, a calculation as if the interface was between the austenite and a single variant of martensite !L gives good yield surface prediction (fig. 5), but it seems purely heuristic !

3.3 Conclusion

On one part, the crystallographic theory of martensite performed by Ball and James [3,4], Bhattacharya [5], Hane [6], Shield [8] permit to determine the nature of the interface i.e. between austenite and a single variant of martensite or between austenite and a twinned martensite, with stress free state. The determination of the habit plane normal m and the shape strain vector b permit to obtain the yield phase transformation criterion.

The yield surface prediction is efficient for some Cu-Zn-AI and Cu-AI­Be (austenite/untwinned martensite) and Cu-AI-Ni (austenite/twinned martensite) but not for Ni-Ti. For this alloy, the hypothesis of neglecting stress effect on the geometrical interface configuration between austenite and martensite, must be revisited.

32 C. Lexcellent, P. Blanc and C.Bouvet

REFERENCES

[1] KF. Hane, T.W. Shield, "Microstructure in the cubic to monoclinic transition in Titanium-Nickel shape memory alloys", Acta Mater., 47, 9, pp. 2603-2617,1999.

[2] M.S. Wechsler, D.S. Lieberman and T.A. Read, "On the theory of the formation of martensite",

Trans. AlME 197, pp. 1503-1515, 1953.

[3] J.M. Ball and R.D. James, "Fine phase mixtures as minimizers of energy", Archs Ration Mech.

Analysis, 100, pp. 13-50, 1987.

[4] J.M. Ball and R.D. James, "Proposed experimental tests of a theory of fine microstructure, and the

two-well problem", Phil. Trans. R. Soc. Lond A, 338-389, 1992.

[5] K Bhattacharya, "Wedge-like microstructures in martensite", Acta Metall. Mater., 39 (10), pp.

2431-2444,1991.

[6] K.F. Hane, "Bulk and thin film microstructures in untwinned martensite", J. Mech. Phys. Solids,

47,pp. 1917-1939, 1999.

[7] H. Funakubo (Ed.), "Shape Memory Alloys", Gordon and Breach, New York, 1987.

[8] T.W. Shield, "Orientation dependence of the pseudoelastic behavior of single crystals of Cu-AI-Ni

in tension", 1. Mech. Phys. Solids, 43, pp. 869-895, 1995.

[9] 1. Ball, C. Chu and R.D. James, ''Hysteresis during stress-induced variant rearrangement", J. de

Physique IV, C8:245--2S1, 1995.

[10] S. Stupkiewicz, H. Petryk, "Modelling of laminated microstructures in stress-induced martensite

transformations", J. Mech. Phys. Solids, Vol SO, pp. 2329-2357, 2002.

[11] R. D. James and K F. Hane, "Martensitic transformations and shape memory materials", Acta

mater., 48, pp. 197-222,2000.

[12] C. Lexcellent, A. Vivet, C. Bouvet, S. Calloch and P. Blanc, "Experimental and numerical

determination of the initial surface of phase transformation under biaxial loading in some

polycrystalline shape memory alloys", J. Mech. Phys. Solids, vol 50, pp 2717-273S, 2002.

[13] K Otsuka, K Shimizu, "Morphology and crystallography of thermoelastic Cu-AI-Ni martensite

analyzed by the phenomenological theory", Trans. J.I.M., IS, pp. 103-108, 1975.

[14] B. Raniecki, K Tanaka, A. Ziolkowski, "Testing and modeling of Ni-Ti SMA at complex state­

Selected results of Polish-Japanese Research Cooperation", Material Science Research

International Special Technical Publication, 2, pp. 327-334,2001.

[IS] Gillet Y., Patoor E. ,and M. Berveiller, "Calculation of pseudoelastic elements using a non­

symetrical thermomechanical transformation criterion and associated rule", Journal of intelligent

material systems and structures, 9, 366-378, 1998.

[16] B. Raniecki and Ch. Lexcellent, "Tbermodynamics of isotropic pseudoelasticity in shape memory

alloys", Eur. J. Mech., A/Solids, 17, nO 2, 185-20S, 1998.

[17] L. Orgeas, D. Favier, "Stress-induced martensitic transformation of a Ni-Ti alloy in isothermal

shear, tension and compression", Acta Materialia, 46 (IS), pp. 5579-5591, 1998.

[18] E. Patoor, C. Niclaeys, S. Arbab Chirani and T. Ben Zineb,' "Influence of microstructural

parameters on shape memory alloys behaviOr", IUT AM Symposium on Mechanics of Martensite

Phase Transformation in Solids, pp. 131-138, Q.P. Sun (Ed.), Kluwer Academic Publishers, 2002.

MICRO- TO MACROSCOPIC DEFORMATION BEHAVIOR OF AMORPHOUS POLYMER WITH SLIGHTLY HETEROGENEOUS DISTRIBUTION OF MOLECULAR CHAINS

Yoshihiro Tomital ) and Makoto Uchida2)

1),2) Graduate Shool of Science and Technology, Kobe University Nada Kobe, Japan 657-8501 E-mail: tomita@mech.kobe-u.acjp

Abstract: The micro- to macroscopic defonnation behavior of the polymer under macroscopically unifonn tension and shearing, and surface defonnation of the plane strain polymer block under compression were investigated by means of computational simulation with the nonaffine molecular chain network model with slightly heterogeneous distribution of the molecular chain, in other words, the distribution of the initial strength of the polymer. The results clarified the onset of microscopic shear bands emanating from the slightly weak points and their evolution, interaction and percolation. The interaction of weak points and the evolution of surface undulation under compression have been demonstrated.

Key words: Amorphous Polymer, Distribution ofInitial Shear Strength, Microscopic Shear Band, Macroscopic Shear Band, Molecular Chain Network Theory, Computational Simulation

1. INTRODUCTION

The plastic flow in an amorphous polymer that deforms due to the onset and growth of shear bands is initiated at a stress level lower than the macroscopic yielding point. Such shear bands in an amorphous polymer are oriented along a direction very close to the direction of maximum shear stress. Subsequently, many shear bands form with the increase of deformation. Beyond the macroscopic yield point, such localized shear

33

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 33-40. ©2004 Kluwer Academic Publishers.

34 Yoshihiro TOMITA and Makoto UCHIDA

bands are transformed into the macroscopic deformation. These characteristic deformation behaviors are deeply related to the highly heterogeneous microstructure of an amorphous polymer. Nevertheless, the discussions associated with the heterogeneity of the microscopic structure are very restricted [1-3]. The distribution of strength with double peaks in low and high strength regions was introduced and the nonlinear deformation behavior before macroscopic yielding and reverse loading processes and other mechanical characteristics were clarified [1]. An experimental investigation by Atomic Force Microscope (AFM) observation [2] and large scale Molecular Dynamic (MD) simulation [3] suggested the heterogeneous distribution of the molecular chains.

In this paper, the characteristic deformation behavior of an amorphous polymer is numerically specified by employing a nonaffine molecular chain network model [4] and finite element simulation of the amorphous polymer with a slightly heterogeneous chain distribution which was replaced by the heterogeneous initial shear strength. Currently the information associated with the concrete distributions of chain density or initial strength is not available; therefore, we will employ the normal distribution of the initial shear strength of a polymer. The micro- to macroscopic deformation behavior of a unit cell under uniform tension and shearing, and the evolution of undulation of a stress-free surface of blocks under plane strain compression will be discussed.

2. CONSTITUTIVE EQUATION

The complete constitutive equation for a polymer employed in this investigation is given in references [4], thus we provide a brief explanation of the constitutive equation here. The total strain rate is assumed to be decomposed into elastic strain rate and plastic strain rate. Elastic strain rate is expressed by Hooke's law and plastic strain rate is modeled using a nonaffine eight-chain model. The fmal constitutive equation that relates the rate of Kirchhoff stress S ij and strain rate & IcJ becomes

Sij = Lijkl&1cJ - p~, Lijkl = n;kl - Fijkl' FijlcJ =~(O"ikOjl +O"i/Ojk +O"jlOik +O"jkOi/)

(1)

where Dijkl is the elastic stiffness tensor and 0" ij is the Cauchy stress. The shear strain rate yP in Eq. (1) is given as [5]

i' = i. ex{ ( -A;X'-(f)"'}]- (2)

where Yo and A are constants, T is the absolute temperature, 'i is the

Micro- to macroscopic deformation behavior of amorphous polymer with 35 slightly heterogeneous distribution of molecular chains

applied shear stress, s = S + ap indicates shear strength [6], s is the shear strength which changes with plastic strain from the athermal shear strength So = 0.077 f.J /(1- v) to a stable value Sss' p is the pressure, a is a pressure- dependent coefficient, f.J is the elastic shear modulus and v is Poisson's ratio. Since s depends on the temperature and strain rate, the evolution equation of s can be expressed as s=h{I-(s/s,,)}rP , where h is the rate of resistance with respect to plastic strain. Furthermore, Blj in Eq. (I) is the back-stress tensor of which principal components are expressed by employing the eight-chain model [7] as:

B. =!..cRffi v/ -}} L-l(~), '3 A, ffi

1 2 1 2 2 2 (3) L(x)=cothx--, A, =-(V1 +V2 +V3 ), x 3

where V; is the principal plastic stretch, N is the average number of segments in a single chain, cR = nkT is a constant, n is the number of chains per unit volume, k is Boltzmann's constant, and L is the Langevin function. In the nonaffme eight-chain model [4], the change in the number of entangled points, in other words, the average number of segments N, may change depending on the temperature change and distortion of ~ which represents the local deformation of a polymeric material. The simplest version of the expression of the number of entangled points is m = mr exp{- c(l- ~)}, where mr is the number of entangled points at reference state and c is a material constant [4].

3. COMPUTATIONAL MODEL

Here, we evaluate in detail the characteristics of microscopic deformation to macroscopic deformation of an amorphous polymer with heterogeneously distributed molecular chains. Quite localized deformation, in the form of microscopic shear bands, is observed in the initial stage of deformation and subsequently these bands develop and unify to macroscopic shear bands. The detailed characteristic feature of these deformation behaviors is clarified by careful observation [2] using an Atomic Force Microscope (AFM) under macroscopically uniform tension and shearing. Furthermore, large- scale Molecular Dynamic (MD) simulation [3] suggested the heterogeneous distribution of the molecular chains. Here, we restrict our investigations to the effect of heterogeneity of the distribution of the initial strength for the micro- to macroscopic deformation behavior of an amorphous polymer and will employ the normal distribution of the initial strength ofthe polymer.

Two models shown in Figs.! (a) and (b) are prepared for the investigation of plane strain micro- to macroscopic deformation behavior. Figure 1 (a) illustrates the plane strain computational unit cell model for macroscopically

36 Yoshihiro TOMITA and Makoto UCHIDA

uniform deformation in which heterogeneous distribution of initial shear strength of polymer So is assumed. The distribution of So is assumed instead of the chain density distribution. Figure I (c) indicates the normal distribution of So with the mean value sOm' The distribution of So is specified such that depending on the total number of finite elements in a unit cell, a specific value of So is allocated to a square element and the number of elements with specific values of So has normal distribution as indicated in Fig.l (c). Figure I (b) indicates the computational model for investigation of the surface morphology under plane strain uniform compression. To capture the more local deformation near the surface of the polymer, finer meshes are used. The macroscopically homogeneous deformations shown in Fig. I (a) and uniform horizontal compression to (b) are applied. In all cases, except for simple shear deformation, shear free conditions are set at the surface of the boundary.

Here, we discuss the effects of the distribution of So on such macroscopic deformation behavior as the average stress-strain relationship and on microscopic deformation behavior such as onset and propagation of the microscopic shear bands, and on the evolution of microscopic shear bands and their unifications. We define the average strain rates E"E2 or stress rates £" £2 with respect to the coordinate directions X"X2 , respectively, and macroscopIC equivalent stress and strain are defined as

So [MPo]

139.3 97.3 55.3

(a) Plane Strain Model for Macroscopically Uniform DeformatIon

(b) Plane Strain Model for Evoluation of Surface MorpholollY Under CompressIon

Fig.] Simulation Model

!'l c: u E u t;j ... o

.8 j E Z

55.3 97.3 139.3 So (MP.j

(c) Normal Distribution of So with mean valueS ... =97.3MPa

Ie = (3I;'I; /2)112 and Ee = (2E;E; /3)1/2 for the case of macroscopically uniform deformation. In Fig.l(a), a macroscopically homogeneous strain rate Ee = Eo = 10-5 / s is applied and Ii / La = 10-5 / s for (b). Since the strain rate is sufficiently low, here we disregard the heat generation due to irreversible works. The material parameters for the polymer employed are Em / sOm = 23.7, sss / sOm = 0.79, h / sOm = 5.15, ASOm /1'0 = 78.6, a = 0.08,

Micro- to macroscopic deformation behavior of amorphous polymer with 37 slightly heterogeneous distribution of molecular chains

r 0 = 2.0 X lOIS / s, SOm = 97 MPa, To = 296K, mo = 7.83 x 1026 and c = 0.33, which were the modified versions of those for the affine eight-chain model [7].

4. RESUL TS AND DISCUSSION

We will first discuss the micro- to macroscopic deformation behavior under uniform tension. Figure 2 (a) shows the macroscopic stress to strain relations for tension. The effects of the microscopically distributed feature of So are substantial in the early stage of deformation. The yielding due to heterogeneous distribution of So causes the nonlinear response prior to the macroscopic yielding and suppresses the corresponding macroscopic yielding stress. Furthermore, continuous occurrence of yielding at different positions on the polymer results in the rather moderate change of average stress with deformation as compared with the homogeneous case. However, the critical stretch is assumed to be identical in this investigation, therefore, the resistance of the deformation in the later stage asymptotically approaches that of the homogeneous case.

The localized microscopic shear bands, which connect the microscopically weak regions, appear at almost 45 degrees with respect to the tension

1.00-----..---~-,--, ,,-1101lI0_ ~ ---- li<t<ro .........

(..,f (7 DEJBJEJ

(I) (2) (3) (4)

I..... (2) (3) (4)

(1

0.00 0.40 E"'l

DDDD'~~ (5) (6) (7) (8) 0.80

(a) Equivalent Stress - Strain Rerations (b) Distribution of Equivalent Strain Rate

Fig.2 Macroscopically Uniform Tension

1.00-----.,....-- ---,

o.s

A - 110lII0I''-'' ~ - - - - Htlc:roaCBeOW

"'l8" DDDD';~ (I) (2) (3) (4) O.

I..... (2),;.(3",)(,",,-4)_--1

(1)

0.00 0.40 £"'1 0.80

(0) Equivolenl Stress - Strain ReranODS

000// (5) (6) (7) (8)

(b) Distribution of Equivalent Strain Role

Fig. 3 Macroscopically Uniform Shearing

38 Yashihiro TOMITA and Makata UCHIDA

direction, and with the increase of deformation, their percolation and intensification and propagation along the normal direction to the shear bands are observed accompanied by the rotation of shear bands. The additional macroscopic deformation causes the attainment of the critical stretch value for the specific shear bands, which yields the macroscopic increase in the resistance of deformation as observed. Subsequently, the increase of the number of shear bands in which the maximum stretch attains the critical value results in the significant increase of the resistance of deformation.

Similar behavior to that observed in the tension case can be seen in shearing cases except for the shear band direction which depends on the principal shear direction. Figure 3 indicates the corresponding results for shearing of the unit cells. Since the maximum shear directions are the horizontal and vertical directions for the case of shearing, therefore, the corresponding directional shear bands evolve and propagate. With regard to the effect of the heterogeneous distribution of the shear strength on the energy consumption for the deformation, generally a higher energy is required for the heterogeneous case.

I.sr----------r------~ .. .I ~

II:>

1.0

0.0

--Homogeneous - - - -HClcrO,gcnc:ou.s

0.2 "/Lo (a) ominal Stress - Strnin Relations

0.4 /

--

DD[Jm~EJ~L--...I (I) (2) (3) (4) (5) (6) (7) (8)

(b) Distribution of Equivalent Strnio Rate

(S)

FigA Evaluation o/Surface Undulation under Plane Strain Compression

Ceq

1.0e-4

0.0

So far, the discussion has been restricted to the heterogeneous distribution of the initial shear strength, however, the heterogeneous distribution of molecular chains may affect the number of entanglements, in other words, the average number of segments for a chain, which could affect the results. This is the subject of future work.

Micro- to macroscopic deformation behavior of amorphous polymer with 39 slightly heterogeneous distribution of molecular chains

Figures 4 (a) and (b) respectively indicate the average stress-strain relation, and strain rate distribution and the evolution of surface undulation for the plane strain blocks due to inhomogeneous deformation caused by the distribution of So under macroscopically uniform compression. The significant increase in the undulation starts at the specific deformation where micro shear bands connecting weak points developed, and subsequently, the rate of increase in undulation decreases as opposed to the case of metallic materials. This is due to the characteristic feature of the polymeric materials, which is caused by the orientation hardening. More precise observation clarified that in the initial stage of deformation, onset of the long wavelength undulation is predominant and sharpened with further deformation. Subsequently, due to the orientation hardening, the evolution of long wavelength undulation is suppressed. With further deformation, onset, propagation and percolation of microscopic shear bands cause the additional short wavelength surface undulation which can be clearly seen in the magnified figure of the surface of the block in Fig. 4 (b). These characteristic deformation behaviors qualitatively reproduce the main features observed in AFM observation of the surface of PC [2].

5. CONCLUSION

The series of computational simulations of the deformation behavior of an amorphous polymer with heterogeneous distribution of the shear strength clarified the characteristic features of the micro- to macroscopic deformation behavior. The results are summarized as follows. (1) The distribution of the shear strength causes the onset of micro shear

bands connecting the weak points, which causes the softening of the macroscopic stress and strain relations, and nonlinear response before the start of macroscopic yielding.

(2) A nearly steady-state deformation in which the resistance of deformation is almost constant is attained during the propagation of shear bands with respect to the normal direction of the micro shear bands and percolation of new shear bands. Termination of the propagation results in the significant upward trend in stress and strain relations.

(3) The undulation caused by the nonuniform deformation due to the distribution of the initial shear strength manifests different features depending on the stage of deformation. The wavelength of the undulation decreases as the deformation proceeds which is caused by the onset, propagation and percolation of shear bands.

(4) In macroscopically homogeneous deformation as well as heterogeneous deformation, the deformation resistance increases with the introduction

40 Yoshihiro TOMITA and Makoto UCHIDA

of heterogeneity of the initial shear strength and therefore, energy consumption for the deformation increases as well.

ACKNOWLEDGMENT

Financial support from the Ministry of Education, Culture, Sports, Science and Technology of Japan is gratefully acknowledged.

REFERENCES

[1] M. C. Boyce and C. Chui, "Effect of heterogeneities and localization of polymer deformation and recovery", T., De Borst, R and Van der Giessen, E., eds. Proc. IUTAM Symposium on Materials Instabilities, pp. 269-285, 1998, John Wiley & Sons Ltd.

[2] Y. Kashu, T. Adachi and Y. Tomita, "AFM observation of microscopic behavior of glassy polymer with application to understanding of macroscopic behavior", Eds. Abe, T and Tsuta, 1; Proc. AEPA'96Pergamon, pp. 501-505,1996.

[3] T. Itoh, K. Yashiro and Y. Tomita, "Molecular dynamic study on deformation of molecular chains in amorphous polymer", JSMS 7 th Symposium on Molecular Dynamics, pp. 50-55, 2002.

[4] Y. Tomita, T. Adachi and S. Tanaka, "Modelling and Application of Constitutive Equation for Glassy Polymer Based on Nonaffine Network Theory", Eur. J. Mech. AlSolids, vol. 16, pp. 745-755,1997

[5] A. S. Argon, "A theory for the low-temperature plastic deformation of glassy polymers", Phil. Mag. Vol. 28, pp. 839-865,1973.

[6] M. C. Boyce, D. M. Parks and A. S. Argon, "Large inelastic deformation of glassy polymers, Part I: rate dependent constitutive model", J. Mech. Mater. , vol. 7, pp. 15-33, 1988.

[7] E. M. Arruda and M. C. Boyce, "A three-dimensional constitutive model for large stretch behavior of rubber materials", J. Mech. Phys. Solids, vol. 41, pp. 389-412, 1993.

CROSS SLIP VIEWED AT THE NANO- AND MICROMETER SCALE

T. Leffers and O.B. Pedersen

Materials Research Department Rise National Laboratory DK-4000 Roskilde, Denmark E-mail: torben.leffers@J"isoe.dk

Abstract: The results of recent work on atomic-scale modelling of cross slip of non­jogged and jogged screw dislocations in copper are summarized - with special emphasis on the activation energy for cross slip. The results are compared with observations of the texture transition in brass and with microstructural observations on cyclically defonned copper. The indication is that cross slip plays a governing role in texture fonnation and in cyclic defonnationlfatigue.

Key words: cross slip, atomistic modelling, texture transition, fatigue

1. INTRODUCTION

The concept of cross slip was introduced on the basis of slip-line observations on polished surfaces [1,2]. Cross slip means that a dislocation leaves its original slip plane and continues slip on another, crossing slip plane. Only screw dislocations can cross slip: as opposed to other dislocations they do not have their slip plane fixed geometrically (as the plane containing Burgers vector and line vector). It is generally accepted that cross slip is a thermally activated process, e.g. [3]: the screw dislocations are dissociated in their original slip plane, and constriction and redissociation in a new slip plane requires energy.

It is also widely accepted, in agreement with early suggestions, that cross slip plays a decisive role in various processes during plastic deformation (monotonic and cyclic). Seeger [4] suggested that the transition to stage III in the plastic deformation of fcc single crystals is governed by cross slip. Mott [5] suggested that cross slip is the essential fatigue process. Smallman

41

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 41-48. ©2004 Kluwer Academic Publishers.

42 T. Leffers and o.B. Pedersen

and Green [6] and Dillamore and Roberts [7] suggested that the fcc rolling­texture transition is governed by cross slip.

Since cross slip involves rearrangements in the dislocation core, attempts to describe cross slip theoretically on the basis of a continuum approach, which was the only possibility up to the nineteennineties, were of questionable value. Since the late nineteennineties the present authors have been involved in atomistic modelling of cross slip in copper in collaboration with Institute of Physics at the Technical University of Denmark. In the present work we recapitulate the results of this atomistic modelling - and the limitations we are up against. And we describe some applications of the modelling results - on the fcc rolling-texture transition and on cyclic plasticity/fatigue. Because of the limited space available we concentrate on the description of the atomistic modelling organized in such a way that it leads up to the applications, particularly so for the fcc texture transition. For the actual applications we largely refer to references available in open literature.

2. ATOMISTIC MODELLING OF CROSS SLIP

2.1 Modelling procedure

The interatomic potential used in the atomistic modelling is derived from "Effective Medium Theory" (e.g. [8]). It reproduces the elastic properties of copper quite well, but it gives a stacking fault energy (31 mJm-2) which is somewhat lower than the normally quoted value of ~50 mJm-2•

Molecular dynamics (e.g. [9]) is the theoretically ideal procedure for the modelling of thermally activated processes. However, the capacity of presently available computers is normally insufficient to do molecular­dynamics modelling of complex processes like cross slip. Therefore, we have mainly used an alternative procedure, the "nudged elastic band" [10]. With this procedure we define the initial and the final configuration, and the computer finds the lowest-energy path in configuration space (strictly speaking the computer fmds a low-energy path but in practice it is the lowest-energy path). Vegge et al. [11] have provided an easily understandable explanation of the nudged-elastic-band procedure. During the path in configuration space the computer keeps track of the energy levels, and therefore we can extract the activation energy.

2.2 Cross slip of non-jogged dislocations

Rasmussen et al. [12] determined the activation energy for cross slip of a single non-jogged screw dislocation in copper: -3eV. This activation energy is prohibitively high for cross slip at room temperature. Based on atomistic

Cross slip viewed at the nano- and micrometer scale 43

modelling of cross slip in nickel (with different procedures) Rao et al. [13] suggested a substantially smaller activation energy for cross slip of non­jogged screw dislocations in copper (-1.15eV), but this "translation" from nickel to copper has been questioned by Rasmussen [14] and Vegge [15].

Cross slip may be assisted by stress, for instance the attractive stress between the two dislocations in a screw-dislocation dipole with sufficiently low dipole height. This was investigated by Rasmussen et al. [16]. They found an approximately linear relation between the inverse dipole height and the activation energy for dipole annihilation by cross slip 1. For dipole heights less than five or six {Ill} interplanar distances (depending on the configuration) the activation energy is zero, i.e. the two dislocations annihilate by spontaneous cross slip.

For screw-dislocation dipoles with dipole height just above the critical dipole height for spontaneous annihilation - six {Ill} interplanar distances - the activation energy is so low that the annihilation process can be modelled by molecular dynamics (Vegge et al. [17]). This allowed us to detennine the preexponential P (with dimension m-ls- l) in the equation

F = P exp(-ElkT) (1)

where F is the cross-slip frequency per metre per second, E is the activation energy, k is Boltzmann's constant and T is the temperature. The resulting P value was 2·1023m- ls-1 or 5·1013b-ls-1 where b is the Burgers vector. This is the first example of such a detennination of the preexponential for a complex process like cross slip/dipole annihilation. One notices that P as referred to the Burgers vector is of the same order of magnitude as the Debye frequency as one would expect intuitively.

2.3 Cross slip of jogged dislocations

As described in 2.2 the activation energy for cross slip of a single non­jogged screw dislocation as derived from atomistic modelling, -3eV, is prohibitively high for cross slip at room temperature. A jog on the screw dislocation would provide a ready-made constriction and thus reduce the activation energy for cross slip. Cross slip of a jogged screw dislocation in copper has been modelled by Vegge et al. [18]. They found activation energies of O.86eV or O.87eV depending on the jog configuration. This dramatic reduction of the activation energy for cross slip means that the theoretical activation energy for cross slip in copper is brought to a level which allows cross slip at room temperature. It also means that the theoretical activation energy is brought to agreement with the experimental activation energy for cross slip in copper detennined by Bonneville et al. [19]: 1.15eV±0.37eV. However, there is serious disagreement between the theoretical and the experimental values for the activation volume as to be described later.

44 T Leffers and o.B. Pedersen

For detailed comparison between the activation energy for cross slip of jogged screw dislocations and the experimental activation energy for the fcc rolling-texture transition (3.1) we need to know the stress dependence of the theoretical activation energy for cross slip of jogged screw dislocations in a polycrystalline material. Such a stress dependence is implicit in the annihilation of non-jogged screw-dislocation dipoles as described in 2.2, e.g. Figs. 1 and 2. The obvious solution is then to repeat the computer experiments by Rasmussen et al. [16] on annihilation of non-jogged dipoles for jogged dipoles. However, when we attempted to do so, the computer found an alternative annihilation process, "stress-aided jog migration", with very low activation energy (Vegge et al. [20]). For jogged screw-dislocation dipoles stress-aided jog migration (with an activation energy of ~lOmeV) gradually reduced the dipole height to eleven {Ill} interplanar distances, and then spontaneous cross slip (with zero activation energy) finished the annihilation. Stress-aided jog migration requires free supply of new jogs of a specific type which, in the computer, is ensured via the periodic boundary conditions applied. In the real world free supply of jogs requires bulk diffusion which we consider to be unrealistic in connection with monotonic plastic deformation leading to deformation texture. In connection with 3.1 we therefore take stress-aided jog migration to be an artifact introduced by the structure of our computer program, and consequently we alternatively assume that in reality (without stress-aided jog migration) there is an approximately linear relation between inverse dipole height and the activation energy for annihilation of jogged screw-dislocation dipoles by cross slip similar to the relation for non-jogged dipoles depicted in [16] (cf. 2.2).

However, so far we have not been clever or imaginative enough to establish the modelling conditions for annihilation of jogged screw­dislocation dipoles without stress-aided jog migration. Therefore, we must base the linear relation on the two points we know: for infinite dipole height (for an individual jogged screw dislocation) the activation energy is 0.86eV or 0.87eV, and for a dipole height of eleven {11I} interplanar distances the activation energy is zero. The resulting linear relation is shown in Figure 1.

We now return to the question of the stress dependence of the activation energy for cross slip of jogged screw dislocations. There is no doubt that the reduction in activation energy with decreasing dipole height, e.g. Figure 1, is caused by the stress field from one of the two dislocations in the dipole at the position of the other. Simplistically we use the isotropic expression for the shear stress caused by an undissociated screw dislocation at a distance r

-r= ~b/21tr (2)

Cross slip viewed at the nano- and micrometer scale 45

Applied Stres. (MPa)

SOli lQOO 1500 2000.2500 3000 j i ··-~r·

Inlle ..... DiPOle Heigh! (nm")

Figure 1, The suggested linear relation between the activation energy for cross slip of a jogged screw dislocation in dipole configuration and inverse dipole height - with an added

alternative abscissa axis in terms of applied normal stress,

where ).I. is the shear modulus and b is the Burgers vector. If we take r to be equal to the dipole height, we can use equation (2) to convert the inverse dipole height in Figure 1 to a shear stress. However, for our application in 3.1 we need to convert the shear stress to a normal stress. We do this conversion with reference to heavy plastic deformation where we assume that the role of cross slip is to help screw dislocations to by pass obstacles -as illustrated by the very ftrst references to cross slip based on slip-line observations as quoted in section 1. First we multiply by the Sachs m factor 2.22. Then we mUltiply by a factor slightly above unity (taken to be 1.36 in order to get a simple total conversion factor of 3) with reference to the possibility that other stress components than the shear stress in the cross-slip plane (with a higher m factor) may be important for stress-assisted cross slip. Alternatively we might just have multiplied by the Taylor m factor 3.08 with almost the same result. We can thus add an alternative abscissa axis with units of normal stress as shown in Figure 1. For further details we refer to [21].

The activation volume is deftned as dEld't, E being the activation energy and 't being the shear stress. We can thus derive the activation volume for cross slip of jogged screw dislocations from Figure 1 as the slope divided by -3. The resulting activation volume is -IOb3 where b is the Burgers vector. With similar arguments Rasmussen et al. [16] derived an activation volume of approximately the same magnitude for cross slip of non-jogged screw dislocations. Rao et al. [13] quote an activation volume of20b3 for cross slip in copper derived from their atomistic modelling of nickel. These theoretical activation volumes are very different from the experimental activation volume for cross slip of 250b3 or even more quoted by Bonneville et al. [19]. We shall not go into a discussion of this difference. For such a discussion we can refer to PUschl [22].

46 T. Leffers and G.B. Pedersen

The indirect way we have had to pursue in order to get to the stress dependence of the activation energy for cross slip of jogged screw dislocations illustrates the present limitation in atomistic modelling: even though we are basically in a position to model processes at the atomic level, there are situations which we cannot yet model- because we are not clever enough and/or because our computers are not big enough. Apparently the application of a stress on a single jogged screw dislocation would be a more direct way to the stress-dependence of the activation energy, but we did not manage to find a workable formulation of this problem either.

3. APPLICATION OF MODELLING RESULTS

3.1 The fcc texture transition

For more than 50 years the existence of two types of rolling texture in fcc metals and alloys - the fcc texture transition - has puzzled the texture community. As mentioned in section 1 Smallman and Green [6] and Dillamore and Roberts [7] already in 1964, on the basis of circumstantial evidence, suggested that the texture transition is governed by cross slip, the copper-type texture and the brass-type texture being favoured by high and low cross-slip frequency, respectively.

Weare now in a position to compare the experimental activation energy for the texture transition with the theoretical activation energy for cross slip as derived above. This is described in details in open literature [21,23,24], and we can therefore restrict ourselves to quote the final result. From Figure 1 we can derive the relevant theoretical activation energy for cross slip in "computer copper" (with a stacking fault energy similar to that of Cu-5%Zn): 0.73-0.83eV. The experimental activation energy for the texture transition in Cu-5%Zn is 0.70eV ± O.lOeV. Thus, we have solid quantitative support for the idea that the texture transition is governed by cross slip (via some "catalytic" effect rather than the direct effect suggested in [6,7]).

3.2 Cyclic hardening and fatigue

Microstructural modelling and characterization [25,26] reveal the successive processes whereby cyclic straining induces persistent slip bands (PSBs) in metals like copper at temperatures down to 4K. The essential processes are dislocation dynamics (DD) and stress-aided cross-slip [5], since point defect processes cannot be activated at 4K.

We first ask whether equation (1) accounts for storage of edge dipole loops (EDLs) in cyclic straining, and we find that heights of EDLs are limited only by EDL flipping (to appear). Thus cyclic hardening proceeds to the fatigue endurance limit controlled [26] by avalanches of EDL flipping.

Cross slip viewed at the nano- and micrometer scale 47

The avalanches account for PSB nucleation, point defect production, surface damage and ultimately for the fatigue crack.

The wall spacing d of PSBs is proportional to the critical annihilation height he, defined so that screw dipoles narrower than he annihilate before stress reversal [25-27]. Spontaneous annihilation [16,18] accounts for PSBs at 4K. Brown [27J suggests that stress-aided jog migration along screw­oriented dipoles determines he above 150K. An activation energy of about O.leV accounts for observed d-values [27], but the atomistic jog migration energy [20J is only O.OleV. Nevertheless, the models [26,27J are compatible and together they account for both d and surface damage (to appear).

ACKNOWLEDGEMENTS

The present paper relies very heavily on the results from our collaboration with T. Rasmussen, T. Vegge and K.W. Jacobsen. The work was supported by the Engineering Science Centre for Structural Characterization and Modelling of Materials.

REFERENCES

[1] R. Maddin, C.H. Mathewson and W.R. Hibbard, "Unpredicted Cross-Slip in Single

Crystals of Alpha Brass", Metals Transactions, vol. 175, pp. 86-99, 1948.

[2] G.J. Ogilvie and W. Boas, ibid., pp. 102-104. [3] H. Wolf, "Die Aktivierungsenergie fUr die Quergleitung aufgespaltener

Schraubenversetzungen", Zeitschriftfor Naturforschung, vol. 15a, pp. 180-193, 1960.

[4] A. Seeger, "The Mechanism of Glide and Work Hardening in Face-Centered Cubic and Hexagonal Close-Packed Metals", in Dislocations and Mechanical Properties of Crystals, New York, John Wiley and Sons, pp. 243-329, 1957.

[5] N.F. Mott, "A Theory for the Origin of Fatigue Cracks", Acta Metallurgica, vol. 6, pp.

195-197,1958. [6] R.E. Smallman and D. Green, "The Dependence of Rolling Texture on Stacking Fault

Energy", Acta Metallurgica, vol. 12, pp. 145-154, 1964. [7] LL. Dillamore and W.T. Roberts, "Rolling Textures in F.C.C. and B.c.c. Metals", Acta

Metallurgica, vol. 12, pp. 281-293,1964. [8] K.W. Jacobsen, J.K. N0rskov and MJ. Puska, "Interatomic Interactions in the Effective­

Medium Theory", Physical Review B, vol. 35, pp. 7423-7442,1987.

[9] J.B. Gibson, A.N. Goland, M. Milgram and G.H. Vineyard, "Dynamics of Radiation

Damage", Physical Review, vol. 120, pp. 1229-1253, 1960. [10] H. Jonsson, G. Mills and K.W. Jacobsen, ''Nudged Elastic Band Method for Finding

Minimum Energy Paths of Transitions", in Classical and Quantum Dynamics in

Condensed Phase Simulations, Singapore, World Scientific, pp. 385-404, 1998.

48 T. LefJers and OB. Pedersen

[II] T. Vegge, T. Leffers, O.B. Pedersen and K.W. Jacobsen, "Atomistic Simulations of Jog

Migration on Extended Screw Dislocations", Materials Science and Engineering A, vol.

319-321, pp. 119-123,2001.

[12] T. Rasmussen, K.W. Jacobsen, T. Leffers, O.B. Pedersen, S.G. Srinivasan and H. Jonsson, "Atomistic Determination of Cross-Slip Pathway and Energitics", Physical

Review Letters, vol. 79, pp. 3676-3679, 1997.

[13] S. Rao, T.A. Parthasarathy and C. Woodward, "Atomistic Simulation of Cross-Slip Processes in Model FCC Structures", Philosophical Magazine A, vol. 79, pp. 1167-1192,

1999. [14] T. Rasmussen, "Comment on 'Atomistic Simulation of Cross-Slip Processes in Model

FCC Structures''', Philosophical Magazine A, vol. 80, pp. 1291-1292,2000.

[15] T. Vegge, "Atomistic Simulations of Screw Dislocation Cross Slip in Copper and

Nickel", Materials Science and Engineering A, vol. 309-310, pp. 113-116,2001.

[16] T. Rasmussen, T. Vegge, T. Leffers, O.B. Pedersen and K.W. Jacobsen, "Simulation of

Structure and Annihilation of Screw Dislocation Dipoles", Philosophical Magazine A,

vol. 80, pp. 1273-1290,2000.

[17] T. Vegge, T. Rasmussen, T. Leffers, O.B. Pedersen and K.W. Jacobsen, "Determination

ofthe Rate of Cross Slip of Screw Dislocations", Physical Review Letters, vol. 85, pp.

3866-3869,2000.

[18] T. Vegge, T. Rasmussen, T. Leffers, O.B. Pedersen and K.W. Jacobsen, "Atomistic

Simulations of Cross-Slip of Jogged Screw Dislocations in Copper", Philosophical

Magazine Letters, vol. 81, pp. 137-144,2001.

[19] J. Bonneville, B. Escaig and J.L. Martin, "A Study of Cross-Slip Activation Parameters in Pure Copper", Acta Metallurgica, vol. 36, pp. 1989-2002, 1988.

[20] T. Vegge, O.B. Pedersen, T. Leffers and K.W. Jacobsen, "Atomic-Scale Modeling of the

Annihilation of Jogged Screw Dislocation Dipoles", Materials Research Society

Symposia Proceedings, vol. 578, pp. 217-222, 2000. [21] T. Leffers and O.B. Pedersen, "The Activation Energy for the FCC Rolling Texture

Transition and the Activation Energy for Cross Slip", Rise-R-1308 (EN). Available

electronically via http://www.risoe.dklrispubI!AFM/ris-r-1308.htm

[22] W. Puschl, "Models for Dislocation Cross-Slip in Close-Packed Crystal Structures: a

Critical Review", Progress in Materials Science, vol. 47, pp. 415-461, 2002.

[23] T. Leffers and O.B. Pedersen, "The Activation Energy for the FCC Rolling-Texture

Transition as Related to the Activation Energy for Cross Slip", Scripta Metallurgica, vol. 46, pp. 741-746, 2002.

[24] T. Leffers and O.B. Pedersen, "Can We Relate the FCC Rolling Texture Transition to

Cross Slip?", Materials Science Forum, vol. 408-412, pp. 365-370,2002.

[25] O.B. Pedersen and A.T. Winter, "Cyclic Hardening and Slip Localization in Single Slip

Oriented Copper Crystals", Physica status solidi (a)vo1.149, pp. 281-296,1995.

[26] O.B. Pedersen, "A nanotheory ofthe intense slip localization causing metal fatigue",

Zeitschriftfor Metallkunde, vo1.93, pp. 790-798,2002.

[27] L.B. Brown,"A dipole model for the cross-slip of screw dislocations in fcc metals",

Philosophical Magazine A, vo1.82, pp.l691-1711, 2002.

A MULTISCALE MICROMECHANICS APPROACH

TO DESCRIBE ENVIRONMENTAL EFFECTS ON SURFACE CRACK INITIATION UNDER CYCLIC LOADING

E.P. Bussol ), G. Cailletaud2), and S.Quilici2)

l) Department of Mechanical Engineering

Imperial College London, United Kingdom

2) Centre des Materiaux

Ecole des Mines de Paris,Evry, France

Abstract In this work, a multiscale mechanistic approach is employed to study the effects of casting-related porosities and environment on the initiation of surface cracks in single crystal superalloys under predominantly cyclic loading conditions. At the scale of the porosity (i.e. mesoscale), a micromechanics-based probabilistic formulation is relied upon to describe the initiation and growth of fatigue cracks from spherical defects. The effects of loading at the scale of the component (i.e. macroscale) and of the interaction of the void with a free surface on the mesoscopic stress variations within a loading cycle are quantified from detailed finite element analyses of a representative material volume element. The effect of a reduction in the volume fraction of the " precipitate phase due to surface oxidation on the time-dependent notch stresses is analysed numerically using a crystallographic formulation for the single crystal which depends explicitly on the precipitate volume fraction at the microscale. The framework is then used to investigate the fatigue behaviour of a typical notched tensile bar.

Keywords: Surface cracks, superalloy single crystals, oxidation effects, porosity-induced damage, multi-scale model

1. INTRODUCTION

In high temperature single crystal components, fatigue cracks are often linked to the presence of defects within the material, as they act as stress concentrators leading to a magnification of the local stress and strain fields from those expected in a defect-free material. Furthermore, the position of these defects relative to a free surface of the material is important since embedded defects which are very

49

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 49-57. ©2004 Kluwer Academic Publishers.

50 E. P. Busso, G. Cailletaud and S. Quilici

close to a free surface can lead to highly localised deformation which favours the nucleation of surface cracks [1][2][3]. In most single crystal superalloys, the dominant mechanism of fatigue crack growth is known to be the linking up of microcracks which nucleate at pre-existing 10 to 20 J.l.m casting-related porosities or voids. The rate at which these microcracks grow is generally controlled by thermally induced stresses and, for cracks exposed to an oxidising environment, by the local microstructural degradation around the crack faces caused by oxidation [4]-[7].

In this work, a multiscale mechanistic approach is employed to study the effects of environment and microstructure on the local stress and strain fields around voids. The behaviour of the single crystal superalloy is described by a multi-scale rate-dependent crystallographic theory, which can also account for the reduction in the volume fraction of the " precipitate phase caused by sur­face oxidation. This is achieved by expressing the model hardening parameters explicitly in terms of the volume fraction of the precipitate phase at the mi­croscale [1][8]. At the mesoscale, a statistical-based formulation is relied upon to describe the growth of fatigue cracks from initially spherical voids. Finite element (FE) analyses on a representative material volume element containing a single void subjected to different loading conditions are then performed and the results used to quantify the local stress variations around the defect within each loading cycle. The numerically predicted local stress ranges are combined with fatigue data on smooth specimens to calibrate a probabilistic formulation for fatigue crack initiation, which is then used to study the behaviour of a single crystal notched bar.

2. MULTISCALE PROBABILISTIC MODEL The relevant length scales which define the interaction between a single void

with a free surface are indicated in Figure 1. In general, the location of the closest free surface to the void depends on the relative position of the void. Here, the free surface is that of the root of a notch in a single crystal specimen subjected to an axial load parallel to the [001] crystallographic axis. The defect, a spherical void of radius R, is located at a distance L from the free surface, and its position with respect to a reference system is given by a generic vector, x. Note that due to the fact that voids in superalloy single crystals are widely spaced, interactions between individual voids are not considered. Therefore, for a given single crystal microstructure, the ratio L / R uniquely defines the length scale of the problem [3]. I should also be noted that the presence of the void affects the stresses at the microscale, viz. T in Figure 1, but not those at the macroscale, viz. ~. Thus ~ represents the macroscopic stresses in a flawless component.

As defects act as stress concentrators, especially in the vicinity of a free surface, then a general relation between the mesoscopic stress T around a void

A Multiscale Micromechanics Approach to describe Environenmental effects51 on Suiface Initiation under Cyclic Loading

t F MacmE(x)

Macro I: .......... 1 ........... , .... / .. ,,! IR~.IJ oid !

: : . : ~ i ........ : :

""' ... LM~·:r.r····· · ···:

I:(x) ! Macroscopic scale Mesoscopic scale

Figure 1. Relevant length scales in the multiscale modelling framework

and the macroscopic stress ~ can be expressed as,

(1)

where A is a fourth order tensor operator which depends on the ratio Rj L. As it will be discussed later in the text, the characterization of such an operator will be done through FE computations of a representative volume element of the single crystal containing a single void.

In the general multiscale formulation, it is assumed that the local stress T constitutes the dominant driving force for the growth of microcracks from the surface of the voids and that the initiation process happens relatively much faster. A simple fatigue life relation can be expressed in terms of the greatest variation, within a loading cycle, of the maximum principal stress anywhere around the vicinity of the void. Let this variation be expressed in terms of the maximum (T1 max) and minimum (Tl min) values in a cycle so that LlTl = Tl max - Tl min· Then

(LlTl) -m

Nf = To (2)

where m is a material parameter, and To a parameter which depends both on temperature and microstructure.

A probabilistic approach must be considered to account for the statistical variations associated with both void size and location. To estimate the prob­ability that component failure will occur after N cycles, Pf' then a suitable probability function needs to be defined. Following the weakest link assump­tion used in brittle failure [9], it will be assumed that it remains applicable to the growth of microcracks from pre-existing defects of very low volume fraction so that no significant interaction exists between defects. Then

Pf(N) = 1 - exp [- f ( (Xl fv(R) dR) dV(X)] (3) iv iRc(N,x)

52 E. P. Busso, G. Cailletaud and S. Quilici

where the integration is over the component volume, V, and the defect radius R. Here, fv{R) denotes the probability density function for the defect size distribution measured experimentally, and Rc is a critical defect size at cycle N and at a generic material point of coordinate x.

To determine the probability that failure will occur at a specified number of cycles, N = Nt, requires first that the mesoscopic stress range corresponding to Nt be obtained from the fatigue life relation, Eq. 2. Thus let ~Tlc = ~Tl when Nt = N. Then, at each point of known coordinate x, free-surface dis­tance L, and macro stress range ~E, the corresponding critical defect size Rc = R{ Nt, x) can be extracted from the largest variation of Eq. 1 within a cycle. With Rc now determined at all material points, the failure probability can then be found by integrating Eq. 3.

3. MODEL CALmRATION

3.1 Defect probability density function A Gaussian distribution was chosen to describe the measured statistical vari­

ation of void sizes in the Ni-base superalloys. The probability distribution function (PDF) for the defect radius in Eq. 3 is defined as,

[ ( R - b2)2] fv{R) = b1 exp - ~

0.05 r-------:--------,

Q.04

~ 0.03

0,02

0.01

_ .. -Data __ •

ooo~~~~~-k~~~~

o 4 12 16 20

o.r.ct ..diu~ R "'m)

Figure 2. Experimentally measured defect size distribution

(4)

where bl, b2 and b3 are fitting parameters. A comparison between the measured PDF of defect size versus that predicted by Eq. 4 with b1 = 4.59 X 10-2, ~ = 7.3 p.m, and b3 = 4.3 p.m is shown in Figure 2.

3.2 Macroscopic-mesoscopic stress link

A typical FE mesh of a representative volume element containing a single void (with L / R = 2) used for the void-free surface interaction studies is shown in Figure 3. Cyclic strain histories were applied along the X3 direction at a rate of 10-3 lis at 950°C, with both a zero (-10,+10) and a positive (0,+210) mean cyclic strain. Geometries with several L / R ratios were analised, ranging from o (void open to the free surface) to 00 (defect-free or macroscopic case). FE

A Multiscale Micromechanics Approach to describe Environenmental effects53 on Surface Initiation under Cyclic Loading

Table 1. Maximum values of component of the local stress T, total ff and inelastic fin strains along the tensile (i.e. [001» loading direction for different defect geometries

0.0 1.1 1.4 2.0

345

0.08 0.64

calculations were performed with the commercial FE code ZEBULON [10] using the crystallographic single-crystal model identified for CMSX4 at 950°C (see [1] for details).

r U •• -

.­-Figure 3. Typical FE mesh for UR=2 containing 4304 elements and 58239 degrees of freedom

Typical results of the FE calculations of representative volume elements with different L I R ratios are summarized in Table 1 and Figure 4. The stresses in Table 1 are the maximum values identified during a steady state cycle. The most highly stressed location was generally found near the void surface on the free surface side, indicated by the arrow A in Figure 3. Also shown in Table 1 are the corresponding values of inelastic and total strains at these locations.

The calculations revealed that, after a few cycles, similar local stress ampli­tudes are obtained irrespective of the mean cycle strain. Table 1 also shows that significant stress concentration is mainly experienced by defects located close to the surface (LI R < 1.4) and that open defects (LI R = 0) behave similarly as defects situated far from the surface (LI R = 00). From these observations, the influence of open defects can be neglected, and the integration upper bound in Eq. 3 might be replaced by L leaving

Pf(N) = 1 - exp [- r (rL(X) Iv(R)dR) dV(X)] (5) iv iRc(N,x)

Based on the above numerical results, an approximate analytical expression to Eq. 1 was found for the largest variation of the maximum principal stresses at

54 E. P. Busso, G. Cailletaud and S. Quilici

the mesoscopic level within a cycle. Here,

(6)

Optimum values for Eq. 6 parameters to fit the FE results were found to be C1 = 0.175 MPa(l-C2), C2 = 1.2, and C3 = -0.65, C4 = 3.9. A comparison between the numerical results and Eq. 6 predictions is shown in Figure 4.

110{) UR FE Model 1.1 . lA 2.0

"l 4.0 . ~

.. 00

'" ~ 200

100 200 300 ... A172 [MPa)

Figure 4. Comparison between the macro-meso stress relations

3.3 Fatigue life relation parameters

The calibration ofEq. 2's parameters was done by solving an inverse prob­lem using the multiscale probabilistic model in conjunction with experimental fatigue data. The fatigue life relation parameters from Eq. 2 were identified so that the results predicted by the model fit fatigue data obtained from smooth specimens of SC16 at 950°C [2]. Note that even though the fatigue tests were carried out on a different superalloy from the one of interest, viz. CMSX4, it is safe to assume that both materials exhibit a similar crack initiation be­haviour due to their similar microstructures. The macroscopic stress amplitude vs. number of cycles to failure curve predicted by the multiscale model for a probability of failure of 0.5 was compared with the data, see Figure 5. When the effects of surface oxidation were not considered, consistent predictions were obtained using To = 1768 MPa, and m = 8.7 in Eq. 2.

Figure 5.

41:,/2

10000 '--~~"""'-~~~""--'Oalac-r-"""""'II"" Model-

[MPa) 1000

,OO~ ____ ~ ______ ~ ____ ~ 100 1000 10000 100000

Number of cycle. to failure Identification of fatigue life relation parameters CEq. 2) with surface oxidation

A Multiscale Micromechanics Approach to describe Environenmental effect~5 on Surface Initiation under Cyclic Loading

4. ANALYSIS OF A NOTCHED SPECIMEN The approach required to obtain probabilities of failure of a structural compo­

nent with the proposed multiscale probabilistic model involves the calculation of the range of near-void stresses during the stabilised cyclic response of the component. This approach will now be applied to a typical CMSX4 notched fatigue specimen loaded axially along the [001] crystal direction. The speci­men has a 26 mm long and 9 mm diameter gauge length, and a 1 mm notch radius. As in the macro-meso stress calculations, the single crystal model to be used to describe the deformation behaviour of the component was that for CMSX4 [1]. The FE mesh ofthe 1I8th of the specimen required to be modelled, due to symmetry, consisted of 690 quadratic elements. The FE analyses were performed under an imposed displacement at its ends of ±O.O 1613 mm at a rate of 1.08 x 10-2 mmlsec at 950°C.

To investigate the interaction between environmental effects and the local conditions for crack initiation and growth in the notched specimen, PI pre­dictions were obtained under vacuum and an oxidising environment. The for­mation of the oxide scale was not explicitly included in the FE analysis as its thickness can be considered to be negligibly small when compared to the notch radius. Therefore, only Al diffusion and the subsequent growth of the pure "I region next to the oxide was modelled. In this case, the only parameters which need to be prescribed in the FE mass diffusion analysis are the surface diffusivity and the AI flux boundary condition, given by

(7)

where kp is the parabolic rate constant, t, the time, ell' the Al concentration at the metal-oxide interface (i.e. "I region-oxide) and VAl the partial molar volume of Al (see [1] for details). To solve the overall oxidation-deformation problem, sequentially coupled Al mass diffusion and stress analyses were performed. During the first stage of the analysis, the nodal Al concentrations were calculated and stored as a function of time. The single crystal model parameters at each integration point were then chosen based on the current value of the local Al concentration. If the concentration was above the upper solubility limit of Al in "I, the material properties were assigned to be those of ("( + "I'), otherwise they were defined as those of pure "I (see[1]). Here, a 1000 hour exposure to 950°C was simulated prior to cycling to introduce surface oxidation.

The different steps required to obtain the probabilities of failure are:

• FE calculation of several loading cycles up to a stabilized response.

• Post-processing the FE results of the stabilized cycle to evaluate, at each material point, the largest range of the macroscopic maximum principal stress ~El within the cycle.

56 E. P. Bussa, G. Cailletaud and S. Quilici

(a) Mises (vacuum) (b) .!' (vacuum) (e) .!' (air) Figure 6. Contours of (a) Mises stress and (b) accumulated inelastic strain with no surface oxidation (vacuum) and of (c) accumulated inelastic strain with surface oxidation (air) at the end of the second cycle

• Generation of a 2D surface mesh and transfer of results from the 3D FE analysis, as it was found that surface failure probabilities are non­negligible within a void diameter from the surface (see [2]).

• Application of the multiscale probabilistic model on the surface mesh.

As previously discussed, the value of the fatigue life parameter To (see Eq. 2) depends on the local phase composition and temperature. Thus, for material points in the pure ,-phase region, a value of To = 1167 MPa was used in Eq. 2. This value is consistent with the predicted decrease in the original flow stress of the (, + ,') microstructure at 950°C and a strain rate of 10-3 sec. as a result of the Al loss to form the surface oxide [8].

Figure 6 shows contours of the (a) Mises stress and (b) accumulated inelas­tic strain without surface oxidation, and (c) accumulated inelastic strain with surface oxidation at the end of the second cycle. A large increase in the accu­mulated inelastic strain due to surface oxidation can be seen in Figure 6, i.e. from 3.1 x 10-6 to 2.2x 10-3 , due to the weaker material in the approximately 100 /-Lm thick pure, layer on the specimen surface. It was also found that, when surface oxidation is accounted for, the maximum Mises stress decreases from 674 to 557 MPa.

The predicted contours of the probabilities of failure in 20 cycles (a) with and (b) without surface oxidation are shown in Figures 7(a) and (b), respectively. It can be seen that fatigue cracks are predicted to develop at the < 100 > locations just above the notch root symmetry plane, in agreement with the experimental observations of [2]. Figure 7(c) shows the predicted effect of surface oxidation on the probability of failure vs. number of cycles. It can be seen that the probability of failure only becomes significant after about 50 cycles for the unoxidised case, and that surface oxidation actually decreases the fatigue life of the specimen to about 5 cycles. This trend is consistent with measured effects of oxidation on crack growth rates on CMSX4 at lOOO°C reported by Rieck [11].

A Multiscale Micromechanics Approach to describe Environenmental effects57 on Surface Initiation under Cyclic Loading

i 1.0 rr---~:::------'

! D .•

I Pt ... ZOt)l •• ~ CI.'S ... ...... 0..0 "' .• 1"'10~ "-"ZOf:J~ L. O~ .. a..n...oo " ...... lD

W ~ '" Figure 7. Contours of the probability of failure occurring in 20 cycles (a) with (b) and without surface oxidation, and (c) effect of surface oxidation on the probability of failure VS. number of cycles

5. CONCLUSIONS A multi scale probabilistic framework has been presented to predict the fail­

ure probability by fatigue cracks growing from porosity-related voids in sin­gle crystal superalloy components. The overall findings confirm experimental observations that surface oxidation has a detrimental effect on the growth be­haviour of surface cracks by lowering the local material flow stress more than the stresses responsible for microcrack growth.

Acknowledgments

Financial support for this work by the European Union through grant BRPR­CT970428 and the EPSRC (UK) through grant GRlN12312 are gratefully ac­knowledged. The authors are very grateful to Dr. P. Boudibi (Ecole des Mines) for the provison of experimental data and Dr. L. Zhao (IC) with his help with the FE modelling work.

References

[l] Dumoulin, S., Busso, E.P., O'Dowd, N.P. and Allen, D .. Philosophical Magazine. In press.

[2] Boubidi, P., PhD Thesis, Ecole des Mines de Paris, France, Dec. 2000.

[3] Busso, E. P., O'Dowd, N. P., and Dennis, R. J., 2oo1a, Proc. Fifth /uTAM Symp. on Creep in Structures, S. Murakami and S. Ohno (eds.), pp. 41-50.

[4] Martinez-Esfiaola, J. M., Martin-Meizoso, A., Affeldt, E. E., Bennett, A. and Fuentes, M., 1997, Fatigue Fract. Engng. Mater. Struct., V. 20,771.

[5] Andrieu, E., and Pineau, A., 1999, J. de Physique IV, V. 9, 3.

[6] Nusier, S. Q., Newaz, G. M., 1998, Engng Fract. Mech., 60, 577.

[7] Sfar, K., Aktaa, J., Munz, D., 2002, Mat. Sc. Eng. A, 333,351.

[8] Busso, E. P., Meissonnier, F., and O'Dowd, N.P., 2000, J. Mechanics Physics Sol., V. 48, 2333.

[9] Beremin, F.M., Metall. Trans. A, 14, p. 2277.

[10] Besson, J., Le Riche, R., Foerch, R. and Cailletaud, G., R.Eur.Elem.Fin. (1998),7,567-

588.

[11] Rieck, T., PhD Thesis, Technischen Hochschule Aachen, Germany, 1999.

ATOMIC-SCALE MODELLING OF DISLOCATION HEHA VIOUR UNDER STRESS

D.J.Bacon and Yu.N.Osetsky

Department of Engineering The University of Liverpool Brownlow Hill Liverpool L693GB, UK e-mail: djbacon@liv.ac.uk

Abstract: Computer simulation is used to investigate glide of an edge dislocation in (I-iron over large distances on the atomic scale and in the presence of obstacles in the form of either voids or coherent copper precipitates. Strength characteristics and dislocation configuration information are obtained, and atomic-scale mechanisms associated with strengthening due to these obstacles are identified. The role of a dislocation-induced phase transformation in the larger copper precipitates is revealed. The results are compared with those obtained from models based on continuum treatments.

Keywords: computer simulation, dislocation line tension, precipitates, voids, yield strength

1. INTRODUCTION

Multiscale materials modelling requires coupling between the scales and one of the most important lies at the interface between the atomic and continuum levels. Continuum elasticity theory can describe the role of dislocations in the mechanical response of materials, but there are obvious problems in using it to treat phenomena that are controlled by atomic-scale mechanisms, such as the interaction of gliding dislocations with obstacles. In principle, such processes can be studied via atomic modelling techniques and the information then used to validate the continuum approach. This requires the simulation of large enough length and time scales in the atomic modelling, and to this end we have developed a model [1] based on an approach proposed in [2]. It models a crystal containing an initially straight edge dislocation and allows: simulation of independent application of applied stress or strain; calculation of the resultant strain or stress and crystal energy; realistic dislocation density and spacing between obstacles; and glide at either zero or non-zero temperature, T. In this paper we present results on the interaction between a moving dislocation and either voids or coherent copper precipitates in iron. The results are compared with previous continuum treatments and conclusions are drawn on the importance of effects at the atomic scale. A fuller description is to be published elsewhere [1].

59

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 59-66. ©2004 Kluwer Academic Publishers.

60 D.J.Bacon and Yu.N. Osetsky

2. MODEL

The simulated crystal is sketched in fig. 1 , where the edge dislocation lying along the z axis with b along the x axis is created in a region of mobile atoms (denoted as A) with x, y, z dimensions Lb, H, L. (For the study of u­iron considered, Burgers vector b == Y2[111], y is [110] and z is [112].) The model has periodic boundary conditions applied along x (regions P in the Figure) as well as z, i.e. it simulates a periodic array of dislocations, but with large spacing. Regions Band F are blocks of crystal where individual atoms are immobile: F is fixed but B is moveable. For molecular statics (MS) simulation, T = OK, shear strain is applied by incremental displacement of B in the x direction. The potential energy is minimized at each step and the applied shear stress calculated as O'xy = F/L~, where F is the x component of the total force from all atoms in A to block B. For molecular dynamics (MD) modelling, T > OK, B can be moved either at a chosen velocity (applied strain rate) and stress determined as above, or under a constant force, i.e. applied stress, and treated as a particle in the MD loop with strain measured from its displacement [1]. Only results for the MS study are presented in this paper.

The near-spherical obstacle with its equator in the glide plane y = 0 was either a void of vacancies or a copper precipitate. The periodic boundary condition along z results in the infinitely long dislocation encountering a periodic row of obstacles of spacing L. Voids and Cu-precipitates of diameter, D, from 0.9 to 4.0 nm, i.e. from 27 to 2900 vacancies or Cu atoms, and with spacing, L, equal to either 41.4nm (= 167b) or 83nm (== 334b). The precipitates were created with bcc structure coherent with the iron matrix, as found experimentally for small precipitates [3]. Crystals containing from

(110)

IlIz;L 11111 z x

Fig. I. Schematic presentation a/Simulated crystallite.

Atomic-scale modelling of dislocation behaviour under stress 61

about 2 to 4 million mobile atoms were employed. Different sizes (H up to 19.9nm and Lb up to 59.6nm) were used in order to check the sensitivity of the results to size. The interatomic potentials used were derived in [4,5].

3. RESULTS

3.1. Dislocation void interaction

As an example, the stress-strain curve for a dislocation cutting through the row of 2nm diameter voids (339 vacancies each) and spacing 41.4nm is presented in fig.2(a). Regions I-IV correspond to: I dislocation glide to where it starts to intersect the void; II dislocation attraction into the void and the creation of a step of length b on the entry surface; III dislocation bowing between the voids under increasing applied strain until it is released at the critical resolved shear stress, 'te, of 207MPa; and IV glide of a dislocation containing a superjog, because on breaking away, the dislocation climbs to absorb a few vacancies, i.e. the void volume is reduced. The configuration of the dislocation in its slip plane just before it leaves the void is presented in fig.3. The core has a preference for low-index directions, [111] being the screw orientation as the line leaves the void.

Simulations of models with different void spacing, L, show that 'te has an inverse dependence on L, as predicted by elasticity theory of dislocations for strengthening by localised obstacles (e.g. [6]). The results for 'te versus D (log scale) are presented in fig.4 for L = 41.4nm and 83nm. Both plots are close to linear. The mechanism of dislocation-void interaction is void-size dependent. For small voids the minimum angle, <p, between the two dislocation segments

to D. :E iii If)

~ "-to CD

..r::. If)

200 II IV II III IV 200

150 150

100 100

50 50

0 0

[i] -50 -50

0.0 0.2 0.4 0.6 0.8 1.0, 1.2 0.0 0.2 0.4 0.6 0.8

shear strain, % shear strain, %

Fig-2- Stress versus applied strain for an edge dislocation gliding through a row of (a) voids and (b) Cu precipitates. (D = 2nm, L = 41Anm, strain increment'" 1(J4)

62 D.JBacon and Yu.N Osetsky

emerging from a void at 'tc decreases with increasing D and becomes zero for D = 2nm, see fig.3, and remains zero for larger voids. This is similar to the critical shape in the Orowan process, but without creation of an Orowan loop. Further increase of'tc for larger voids is reflected only in the increase of the length and spacing of the two parallel segments of screw dislocation.

3.2. Dislocation-precipitate interaction

The stress-strain curve for the row of Cu precipitates with D = 2nm and L = 41.4nm is presented in fig.5 and can be compared with that for voids of the same D and L in fig.2(b). The precipitate is a weaker obstacle, for 'tc (= 123MPa) is significantly lower. Data for 'tc for the row of precipitates with L=41.4nm are plotted against D in fig.4. Although 'tc again increases with

20

10

o

-10

-20

-40 -30 -20 -10 o 10 20 30 40

Fig. 3. Projection on the glide plane of atoms in the dislocation core at Tcfor the void simulation offig.2(a}. (Length unit = a)

In(D), the fit is not as good as for voids. The data suggest different gradients for precipitates with small and large D, and the simulations do indeed show that the interaction mechanism for a coherent Cu precipitate is dependent on D [1]. Small precipitates are simply sheared, with a breaking angle <p > O. Larger precipitates exhibit a structural change associated with a bee-fcc transformation: this provokes absorption of both vacancies and interstitials from the dislocation line, with a balance in favour of vacancies. The influence of this dislocation-induced transformation is such that for

Atomic-scale modelling of dislocation behaviour under stress

300 <> void (L=41.4nm) 0 void (L=83nm)

250 0 precipitate (L=41.4nm)

200

(11

a.. 150 ::E

"..,

100

50

5 10 0, b

15 20

Fig. 4. Tc versus D. (Diamonds: voids L = 41.4nm; squares: voids L = 83nm; circles: Cu orecioitates L = 41.4nm)

63

precipitates with D ~ 3run the critical line shape becomes Orowan-like (<p = 0), but, as with voids, no Orowan loop is created.

4. DISCUSSION

The simulations show that strengthening due to small voids and precipitates is significantly different, whereas for the larger obstacles it is similar. This is apparent from the 'tc values presented in figs. 2 and 4, and also from the critical line shapes for obstacles with L = 41.4run and D = lrun or 4 run plotted in figs.5(a) and (b), respectively. The critical angle <p is close to zero for voids and the larger precipitate, but the line is only slightly bent (<p = 136°) by the Inm precipitate. Climb is not apparent in these projections, but occurs for both voids and precipitates when the dislocation adopts a shape with critical angle <p ~ 0 at 'tc, although the climb processes are not identical. Superjog production and glide appears to occur more easily than mutual annihilation of the two screw segments by glide, presumably due to the high Peierls stress of the screw.

The 'tc values for void and precipitate strengthening of fig.4 are re­compared in fig.6, where 'tc, in units of GblL is plotted against twice the harmonic mean, (D-1+L-1rt. in units ofb. (Here G = 62.5GPa is the effective shear modulus for a dislocation of the < 111> {I TO} glide system in Fe [7] .) The harmonic mean of D and L was found by Bacon et al. [8] to give an excellent correlation for the Orowan stress of a row of impenetrable obstacles in a computer simulation based on elasticity theory in which the self-stress of a flexible dislocation was included explicitly. The harmonic mean becomes D when D « Land L when L «D. This recognises that the

64

60

40

20

o

·20

0 l I

\D=1nmJ I j f

V '\

\

~,

\ I I!reclpitate I ; I void I

..w ·20 o 20 40 60

D.J.Bacon and Yu.N.Osetsky

b

IO=4nmJ

-40 ·20

Fig.5. Projection on the glide plane of atoms in the dislocation core at "clor (a) J nm and (b) 4nm obstacl(:!s.

critical Orowan line shape is achieved when 'tc can draw out a dipole of spacing D (energy ex: In(D)) when D « L and spacing L (energy ex: In(L)) whenL«D.

The dashed line in fig.6 is the fit obtained when ro, the dislocation core cutoff radius used as a unit of length for D and L, is set equal to b [8]. The solid line was obtained in a separate continuum simulation of a dislocation passing through a row of voids [9], in which the boundary condition for a dislocation at a void surface was treated to allow for the effect of creation of a surface step. A range of surface energy values was considered and the line in fig.6 is for the highest value thought reasonable. Thus, the continuum simulations give a critical stress

'tc = Gb [In(D-1 + L-1 t + BJ, (1) 2nL

where B = 0.7 for impenetrable obstacles and 1.52 for voids. The agreement between the void data of the atomic-level modelling and

the continuum treatment is rather surprising, particularly since the latter contained several approximations. The reason is that the dislocation segments that emerge from a void at 'tc are almost parallel, thereby matching the Orowan critical shape. They are screw in character, so 'tc acting over the length L draws out a screw dipole of spacing D, the line tension of which is proportional to In(D). The smallest voids do not fit this model quite so well because the angle <p in the critical condition is greater than zero.

The smaller precipitates of copper modelled in the present research are sheared by dislocations with critical angle <p > 0, and so their 'tc values fall

Atomic-scale modelling of dislocation behaviour under stress 65

below the correlation of eq.(l). For the larger precipitates, however, when the dislocation induces a phase change in the unstable copper, 'tc approaches the value for voids. The importance of the bcc => fcc transformation as a strengthening mechanism was recognized in the atomic modelling of Harry and Bacon [10], who studied the interaction energy of a screw dislocation with a Cu precipitate at OK. It was also observed in the study by Hu et al. [11] using an edge dislocation model similar to that used here for precipitates with D = 2.8nm and L = 8.6nm. The critical line shape corresponded to <p = 0 and, although the precipitate and dislocation structure after the interaction was not described, it is possible to conclude from the published figures that the dislocation had a positive jog and left a few interstitials inside the precipitate.

The present work contributes to a framework for multiscale modelling of mechanical properties, in which the characteristics of dislocation-obstacle interaction required for continuum dislocation dynamics (DD) simulations, e.g. maximum obstacle force, critical breaking angle, etc., are either obtained or validated by atomic-level modelling. The results reveal that the atomic mechanisms are often impossible to predict. Thus, voids appear as strong obstacles with small critical breaking angle <p, whereas Cu precipitates are weaker obstacles when small, e.g. they are overcome with large <p, and become strong obstacles as they grow. Furthermore, the underlying processes of dislocation climb at breakaway are atomic in nature.

The good fit between the correlations for Orowan and void strengthening deduced in [8,9] and 'tc obtained here demonstrate how successful DD

0.8.-------------------, 0 .7 ',..,= ~l!n(O·' +L·'r'+1.52) ~

0.6

....J ::0 0.5 C)

"u 0.4 ....

0.3

0.2

0.1

0.0

.. ' <>

.. , ... .g ..... \" .

<> .......... ) <Sl •......•.. , = Gb)ln(O·'+L·'r'+O.7)

. .... 0'... 0 ""'- 2nl!

.. ,

Diameter D (nm): 0.91 .0

o

1.5 2.0 3.0 4.0

simulation can be when dislocation self-stress is included. This effect is not

Fig. 6. Tc versus (D' I +L· I)". Lines show empirical correlations found by continuum simulation for impenetrable obstacles {B] (dashed line) and voids {9] (solid line).

66 D.JBacon and Yu.N.Osetsky

described in the line tension approximation. Russell and Brown [12] first estimated the value of te for precipitates of Cu in Fe, using the premise that the precipitates have a lower elastic modulus, i.e. lower dislocation energy, than the matrix. This enables <p, and hence t e, to be estimated in the constant line tension approximation:

3

t = Gb [COs(~)] if <p ::;; 50° . t = Gb [cOS(<p)]2 if ~ ~ 50° (2) e L 2' 2 ' e L 2' 2

However, the angle <p in this approximation should not be confused with the critical angle found in atomic modelling. For example, if te in eq.(2) for strong obstacles is set equal to the accurate value given by eq.(1) with B = 1.52, <p is found to lie in the range ~90-110°, rather than close to zero. In other words, self-interactions reduce te to values it would have if voids and large Cu precipitates were weak obstacles in the line tension model. Hence, the Russell-Brown formula overestimates the strength if the true angles are used. This does not mean that such obstacles can be treated in the line­tension framework by simply choosing a large <p for breakaway, because then the shape of the bowing line would be in error, and this would affect the statistics of dislocation-obstacle interaction. Hence, care has to be exercised in selecting the approximations in continuum DD modelling.

ACKNOWLEDGEMENT

This research was supported by the UK Engineering and Physical Sciences Research Council.

REFERENCES

1. Osetsky Yu.N., Bacon D.l, 2002, Modelling Simul. Mater. Sci. Eng., submitted. 2. Baskes M.I. and Daw M.S., 1989, in Fourth Int. Conf. On the Effects of Hydrogen on the

Behaviour of Materials (Jackson Lake Lodge, Moran, WY) eds. N.Moody and A.Thompson (The Minerals, Metals and Materials Society, Warrendale, PA).

3. Othen P.l, Jenkins M.L. and Smith G.D.W., 1994, Phil. Mag. A 70, 1. 4. Ackland GJ., Bacon DJ., Calder A.F. and Harry T., 1997, Phi/os.Mag. A, 75,713. 5. Ackland, G.l, Tichy, G., Vitek, V. and Finnis, M.V., 1987, Phi/os. Mag. A, 56,735. 6. Hull D. and Bacon DJ., 2001, Introduction to Dislocations, 4th edition (Butterworth­

Heinemann, Oxford) 7. Bacon D.l, 1985, in Fundamentals of Deformation and Fracture (eds B.A. Bilby, K.J.

Miller and lR. Willis), Cambridge University Press, 1985, p.401. 8. Bacon D.l, Kocks U.F. and Scattergood R.O., 1973, Phi/os.Mag., 28, 1241. 9. Scattergood R.O., Bacon D.J., 1982, Acta Metall., 30, 1665. 10. Harry T. and Bacon DJ., 2002a, Acta Mat., 50, 195; 2002b, Ibid, 209. 11. Hu S.Y., Shmauder S. and Chen L.Q., 2000, Phys. Stat. Sol. (b), 220, 845. 12. Russell K.C., Brown L.M., 1972, Acta. Metall., 20,969.

COALESCENCE OF NANOSCALE ISLANDS DURING POL YCRYSTALLINE THIN FILM GROWTH

Max O. Bloomfield+, Yeon Ho Im+, Hanchen Huang* and Timothy S. Cale+ +Focus Center - New York, Rensselaer: Interconnections for Gigascale Integration *Department of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute, Troy NY 12180-3590 USA

Abstract: We describe a simulation framework designed to track individual grains in a material during simulations of their formation and processing. The framework employs a "grain continuum" model of films [Cale et a!., Compo Mat. Sci. 23 (3), (2002)] and can be used to complement discrete atomistic simulations, and link their results to continuum simulations. We demonstrate the use of multiple-levelset methods to track islands nucleated on substrates, during growth and impingement to form polycrystalline films. We briefly discuss how this simulation tool might be used in an integrated multi scale process simulation environment [Bloomfield et a!., Electrochem. Soc. PV2001-24, pp.77-84, 2001] to establish a link from atomistic simulations upward to feature, pattern, and reactor scale simulations.

Key words: grain structure, computer simulation, level set methods, multiscale

1. INTRODUCTION

Thin films are critical to the performance of many products. For example, progress in IC performance has been largely driven by the ability to fabricate smaller devices and circuits. Process engineers have worked towards understanding thin film deposition processes, as well as properties of the deposited films; e.g., thickness uniformity, adhesion to the substrate, roughness, and texture. As film thicknesses and the sizes of the features onto which the films are deposited continue to decrease in microelectronics applications, the underlying microstructure of these, often polycrystalline,

67

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 67-74. ©2004 Kluwer Academic Publishers.

68 Max 0. Bloomfield, Yeon Ho 1m, Hanchen Huang and Timothy S. Cale

films has become increasingly important. A recent study of copper electroplating baths reported copper grain sizes ranging from 0.1 to 0.25 mm for a widely used electrolyte [1]. With the state of the art in copper interconnect technologies for ICs reaching this size scale [2], the grain structures of these films can have a large impact on reliability [3], manufacturability[4], and performance [5] of those circuits.

In order to improve understanding of grain structure development during deposition, and evolution during later processing, we have developed a simulation environment that we call the finite element based levelset evolver, or FEBLE. This simulator uses multiple levelsets to implement a grain-based method in 3D. We have chosen this approach in response to a perceived need for a tool that can investigate grain structures in 3D on length scales not practical using discrete atomistic simulations, and to bridge the gap between such discrete simulations and continuum representations. Because FEBLE represents complex structures in 3D, it can be used to investigate phenomena that may not be quantitatively the same when simulated in 2D, such as shadowing, motion due to grain boundary energy gradients, and diffusion along many realistic types of grain boundary networks found in interconnects.

In the following section, we discuss the use of FEBLE in an integrated multi scale process simulation context. The multi scale aspects of this work include methodologies for converting discrete atomistic information, such as might result from kinetic lattice Monte Carlo (KLMC) simulations, to what we call a "grain continuum" representation, as well as methodologies for connecting reactor scale and pattern scale simulations to grain scale simulation through boundary conditions. Finally, we summarize an example of deposition of a grain-like structure, based on a kinetically limited electroless deposition (ELD) process.

2. SIMULATION METHOD

We use FEBLE to track the evolution of an initial nucleation. FEBLE is a grain-continuum code, which uses a grain-based levelset method, similar to those in Ref. 6. Levelset methods are based upon representing an interface implicitly, as a contour or "levelset" in a scalar field, and reducing the explicit motion of the interface to the time evolution of the scalar field under an ordinary differential equation [7,8]. The computational overhead of representing a field in n dimensions to track an (n-l )-dimensional interface is made up for by not having to deal explicitly with several geometrically troublesome issues, such as topological changes in the structure.

During levelset calculations, the scalar field variable, f/J, is almost always chosen to be initialized to the signed distance from the interface [7,8]. This

Coalescence of nanoscale islands during polycrystalline thin film growth 69

choice of field has many desirable properties, including that it has its zero contour along the interface, that its contours are equally spaced, and that the various derivatives of the field can be equated to physical quantities, such as curvature and surface normal. The ~field is often represented on a tensor­product grid [8], and the levelset evolution equation (which is of the Hamilton-Jacobi type), given by

(1)

is solved using finite difference approximations to the gradient (which must be of the upwind variety [8]) and explicit time-stepping.

Although we use explicit stepping in time, we represent <p-fields on an unstructured mesh. Barth and Sethian [9] formulated a solution technique of equation (1) for frrst-order, finite element representations of the scalar field on unstructured meshes. PEBLE uses this strearnline-upwinding Petrov­Galerkin (SUPG) formulation by Barth and Sethian [9] to solve the Hamilton-Jacobi equation for simplex meshes, which they demonstrate for triangular meshes, on a tetrahedral volume mesh. We have chosen the finite element based approach to levelset evolution for a variety of reasons, including to enable the use of an unstructured mesh, which is excellent for automatic local refinement.

Because we are very interested in the grain boundaries that form when grains or islands collide, we use multiple levelsets for each simulation, associating one levelset with each distinct material or grain orientation. This one-to-one mapping between levelsets and distinct materials preserves material interfaces after collision of neighboring domains identified with distinct materials.

Grain continuum codes such as PEBLE can be used in a multi scale context to establish a link between grain scale simulation and larger scale simulations, such as to a reactor scale code. Gobbert and co-workers [12,13] and Bloomfield et al. [14] have successfully linked feature scale simulations for chemical vapor deposition and electrochemical deposition respectively, by homogenizing information [15] from the feature scale and using it as a boundary condition for the reactor scale. Essentially, PEBLE provides a generalized three-dimensional multiple material moving algorithm, and uses a user-supplied process model to calculate the growth rates of the various interfaces in the 3D structure. By using an appropriate process model with PEBLE, the same method as in Refs. 12-15 can be applied to link reactor scale and grain scale simulations [16].

70 Max 0. Bloomfield, Yeon Ho 1m, Hanchen Huang and Timothy S. Cale

3. RESULTS

Atomistic simulations that use discrete representations of the atoms making up grain structures are the desired starting point for many grain continuum simulations (FEBLE). We have developed some initial "encapsulation" methods [17] for converting from discrete to continuum representations, but additional work is proceeding. Figure 1 shows results of small Monte Carlo atomistic simulation and the resulting encapsulation made using a brute force method followed by a volume conserving smoothing process. For larger data sets, it may not be practical nor desirable to encapsulate on an atomic scale, in which case a suitable homogenization must be employed.

~ .. .. " .- -I, ._-'_. _

' .. ~ -- ..... -'., . ... ~

Figure 1: (left) data from a small kinetic lattice Monte Carlo (KLMC) atomistic simulation of physical vapor deposition showing texture competition between pyramidal <100> islands and puddle <111> deposition. (right) "Encapsulated" grain-continuum representation of (left).

For many process or grain evolution models, atomistic simulation may be needed to calculate accurate growth rates and other parameters of interest, such as diffusion coefficients or stress states, for the grain continuum representation. In this case, both "re-atomation" continuum-to-discrete and encapsulating discrete-to-continuum methods may be required. Again, for large simulations, we might want to re-atomate only a small subdomain in the neighborhood of the free surface or of a grain boundary. These requirements underscore the need for flexible encapsulation and re­atomation techniques in an integrated multi scale process simulation context.

As a simple example of how FEBLE can be used to simulate a thin film deposition in an IMPS environment, we have chosen electroless deposition (ELD) of copper in a kinetically limited regime. We deliberately chose a kinetically limited process as an example in order to remove questions of transport on the grain scale from the calculation and highlight the 3D shape evolution and material interaction. This is almost the simplest process model possible, because it yields bimodal velocities for each material: the average deposition rate for the free surface, and zero for subsurface material boundaries, such as where a grain meets a substrate, or a grain boundary has formed.

Tseng et al. [18,19] published data on a HN03-CuCI2 bath for copper deposition, which includes several SEM micrographs of copper islands

Coalescence ofnanoscale islands duringpolycrystalline thinfilm growth 71

before coalescence into a blanket film. Visual inspection indicates that most of the grains are approximately within a factor of two in radius, indicating an initial period of nucleation followed by nucleation free growth, and approximately spherical in shape, indicating transport of reactant to the surface is not limiting. We have modeled these aspects of the system by assuming an initial set of hemispherical nuclei, approximately normally distributed around a central size. These initial nuclei are placed on the substrate randomly, such that no two islands impinge on each other in the initial condition. After the initial nucleation stage, the islands are allowed to grow without further nucleation until a blanket film forms. Also, although we consider each island to be distinct and do not allow them to merge with other islands without a boundary upon collision, this particular simulation does not go so far as to associate a specified grain orientation with each island. Thus the kinetics at the free surface is the same on each island, giving isotropic evolution of the free surfaces. The micrographs from Ref. 18 show that many of the resulting grains are close to spherical, and thus we feel that this essentially isotropic model is a reasonable starting point.

We simulated deposition on two substrates, a slightly rough (5 nm RMS roughness) substrate, and the more complicated long, thin trench. In the first case we have specified natural, upwind boundary conditions at the domain sidewalls, but for the trench, we have specified a periodic boundary conditions at the sides of the domain perpendicular to the substrate, for both substrate and growing islands, in order to create a repeating array of infinitely long trenches.

Setting a periodic boundary condition in a particular direction requires that the volume mesh be "matched" in that direction, i.e., that every mesh node that falls on the periodic boundary have a corresponding mesh node on the periodic dual of that boundary. We used MeshSim [20] to construct on a unit cube a tetrahedral mesh that matches in the x and y directions. Using a periodic boundary condition places an implicit condition on the simulation that the initial condition, including the substrate, be periodic as well. Other possible boundary conditions include reflective domain sidewalls and statistical models derived from other simulations.

Figure 4 shows the evolution of our set of islands, using the ELD process model. A small (10%) noise term has been introduced to the surface velocity to mimic local fluctuations in deposition rate. As the islands grow, they begin to impinge on one another and form grain boundaries. Each island has been considered to have a distinct orientation. In Figure 5a, the resulting blanket film has been rotated and the substrate has been made transparent to expose the grains boundaries at the substrate-film interface.

72 Max 0. Bloomfield, Yeon Ho 1m, Hanchen Huang and Timothy S. Cale

Figure 4: An evolving set of islands growing and coalescing into a blanket film under kinetically limited electroless deposition with a 10% noise term.

Figure 5b shows a single grain from this simulation, with each grain boundary colored to indicate the particular grain with which it is in contact. The line along which any two of these colors meet should be regarded as a triple line or the intersection of a grain boundary and the free surface. This isolated grain has a surface mesh and is ready for further calculations, including grain property analysis or further evolution

Figure 5: The blanket film deposited in Figure 4. In (left), the substrate has been made transparent and the structure rotated to expose the grains boundaries at the substrate-film interface. In (right), a single grain has been pulled out and colored to indicate the different grain boundaries. Each color indicates a boundary with a distinct grain, and the seams between grains indicate triple lines

Figure 6 shows a section of a similar evolution a periodic array of infinitely long, aspect ratio 1.3 trenches. In our simple process model, the initial placement of islands is made without effort to account for the edges of the trench or any effect they may have on nucleation density. The islands are allowed to grow, impinge, and form grain boundaries. No noise term has been included in this version of the process model, making for very regular, smooth shapes

Coalescence of nanoscale islands during polycrystalline thin film growth 73

Figure 6: A section of a periodic array of infinitely long, aspect ratio 1.3 trenches evolved from islands to full coverage. In this figure, the array has been cut perpendicular to delineate the shape of the trenches. The initial distribution of islands is as noted in the text.

4. CONCLUSIONS

By using a grain-continuum based approach, in particular, a grain-based multiple level set representation, we can simulate grain structure formation during deposition, starting at the island stage and allowing the system to grow through coalescence to a blanket film. We have identified each grain with a different levelset and, the grains remain distinct after coalescence. Grain boundaries can be located and classified as to their shape and what grains they bound. We chose a finite element based level set method to allow for local spatial refinement of the underlying unstructured mesh.

The FEBLE software can be used with a variety of process or evolution models. As such, it should be regarded as a tool for investigating the impact of particular phenomena on grain structure, and the microstructural ramifications of various models. The sample models shown in this article are decidedly simple, but have been chosen to show some of the basic categories of problem that FEBLE can be used to investigate. We expect to add, refine, and calibrate additional models of deposition and evolution to the available simulation options.

FEBLE gives an opportunity for multi scale simulation, and linking process setpoints to changes in grain structure. The methodologies for linking reactor scale simulations to process models on this scale already exist [12, 13, 17, 21], and it is simply a matter of implementing them in an appropriate environment.

We believe grain-continuum codes such as FEBLE can complement discrete atomistic methods such as kinetic lattice Monte Carlo techniques by extending the length scales accessible to computation. Techniques still need to be developed to move efficiently from discrete to continuum and vice versa, in order to automate using atomistic inputs to codes such as FEBLE. For systems for which models of nucleation or island formation already exist for pre-coalescence, FEBLE can be used to follow the system through coalescence to the blanket film stage.

74 Max 0. Bloomfield, Yeon Ho 1m, Hanchen Huang and Timothy S. Cale

REFERENCES

1. 1. C. Seah, S. Mridha, and L. Chan, J. Vac. Sci. Technol. B 17(5), 2362 (1999). 2. P. Besser, E. Zschech, W. Blum, D. Winter, R. Ortega, S. Rose, M. Herrick, M. Gall, S.

Thrasher, M. Tiner, B. Baker, G. Braeckelmann, et aI., J. of Electron. Mat. 30(4), 320 (2001).

3. A. Fischer, A. von Glashow, A. Huot, and R. Schwarzer, in Advanced Metallization Conference 1999 (AMC 1999), edited by M. Gross, T. Gessner, N. Kobayashi, and Y. Yasuda (Mater.Res. Soc., Warrendale, PA, USA, 1999), pp. 137-141.

4. V. Dubin, C. Thomas, N. Baxter, C. Block, V. Chikarmane, P. McGregor, D. Jentz, K. Hong, S. Hearne, C. Zhi, D. Zierath, B. Miner, et aI., in Proc. IEEE 2001 Inti. Interconnect Tech. Can! (IEEE, Piscataway. NJ, 2001), pp. 271-273.

5. Y. Morand, Microelectron. Eng. 50(1-4), 391 (2000). 6. G. Russo and P. Smereka, SIAM J. Sci. Comput. 21(6),2073 (2000). 7. S. Osher and J. Sethian, J. Camp. Phys 79, 12 (1988). 8. J.A. Sethian, Level Set Methods and Fast Marching Methods, no. 3 in Cambridge

Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, UK, 1999), 2nd ed.

9. T J. Barth and J.A. Sethian, J. Camp. Phys 145 (1), I (1998). 1O.D.F. Richards, M.D. Bloomfield, S. Sen, and T.S. Cale, J. Vac. Sci. Technol. A 19 (4),

1630 (200 I). II.M.O. Bloomfield, D.F. Richards, and T.S. Cale, submitted to Philosophical Magazine A. 12.M.K. Gobbert, C.A. Ringhofer, and T.S. Cale, J. Electrochem. Soc. 143 (8),2624 (1996). 13.M.K. Gobbert, T.P. Merchant, L.J. Borucki, and T.S. Cale, J. Electrochem. Soc. 144 (II),

3945 (1997). 14.M.O. Bloomfield, K.E. Jansen, and T.S. Cale, in Morphological Evolution in

Electrodeposition and Electrochemical Processing in ULSI Fabrication IV, edited by P. e. Allongue, P.e. Andricacos, F. Argoul, D. P. Barkey, J. C. Bradley, K. Kondo, P. e. Searson, C. Reidsma-Simpson, J. L. Stickney, and G. M. Oleszek, vol. PV 2001-8, Electrochem. Soc. 2001.

15.M.K. Gobbert and C.A. Ringhofer, SIAM J. Appl. Math 58 (3), 737 (1998). 16.M.O. Bloomfield, S. Sen, and T.S. Cale, in Thin Film Materials, Processes, and Reliability

in Microelectronics, edited by G. S. Mathad, M. Yang, M. Engelhardt, H.S. Rathore, B.C. Baker, and R.L. Opila, vol. PV 2001-24, Electrochem. Soc., pp. 77-84, 2001.

17. T.S. Cale, M.D. Bloomfield, D.F. Richards, K.E. Jansen, J.A. Tichy, and M.K. Gobbert, Camp. Mat. Sci. 23,3 (2002).

18. W.-T. Tseng, e.-H. Lo, and S.-C. Lee, J Electrochem. Soc. 148 (5), C327 (2001). 19. W.-T. Tseng, C.-H. Lo, and S.-C. Lee, J. Electrochem. Soc. 148 (5), C333 (2001). 20.MeshSim 3.1 from Simmetrix, Inc., http://www.simmetrix.com. 21.T.P. Merchant, M.K. Gobbert, T.S. Cale, and L.J. Borucki, Thin Solid Films 365 (2), 368

(2000).

ON PLC BAND PROPAGATION VELOCITY UNDER STRESS-CONTROLLED TESTS IN ALUMINUM ALLOYS

Mohammed Abbadi!), David Thevenet2), Peter Hahner!), and Abderrahim Zeghloue)

J) DG-Joint Research Centre, European Commission, Institute/or Energy Postbus 2 NL-1755 ZG Petten, The Netherlands E-mail: abbadi@frc.nl 2) Mechanics o/Naval and Offshore Structures Laboratory, ENSIETA, 2 rue Fram;:ois Verny F-29806 Brest Cedex 9, France 3) Laboratoire de Physique et de Mecanique des Materiaux, URA CNRS 1215 ISGMP, Universite de Metz, ne du Saulcy F-57045 Metz Cedex 01, France

Abstract: Plastic flow of solid solutions, particularly lightweight alloys, is unstable within a certain regime of temperature, ageing and loading rates. This instability of strain rate softening type is associated with dynamic strain ageing due to the interaction between mobile dislocations and clouds of impurities and is characterized by the appearance of serrations (respectively strain bursts) when testing is performed at constant strain rate (respectively constant stress rate). Each serration or strain burst corresponds to the localization of deformation in a band which may propagate along the tensile specimen. To investigate the effect of testing conditions on band propagation velocity, tensile tests were carried out with a soft machine (constant stress rate) on two aluminium alloys of the 5000 and 7000 series. It is important to note that a decrease of band propagation velocity with increasing stress rate was observed for different temperatures and ageing. This trend is well predicted by a recent model proposed by Hahner.

Key words: plastic instability, dynamic strain ageing, band propagation velocity, soft tensile machine, aluminum alloy.

75

S. Ahzi et al. (eds.}. Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 75-82. ©2004 Kluwer Academic Publishers.

76 Mohammed Abbadi, David Thevenet, Peter Hahner and Abderrahim Zeghloul

1. INTRODUCTION

Since it was brought to the fore, the PLC effect has continued to attract scientific interest, according to its manifestation in many materials, in particular lightweight alloys. So, it has given rise to numerous theoretical and experimental studies. Microscopically, PLC effect is associated with dynamic strain ageing (DSA) due to interactions between mobile dislocations and diffusive alloying atoms in the material [1,2]. From a macroscopic point of view, this phenomenon is characterized by the appearance of stress serrations or strain steps [3,4] on the stress-strain curves according to whether tensile tests are performed at constant strain or stress rates. Most of the previous studies devoted to the PLC effect were based on hard machine results (constant strain rate tests). In this kind of tests, the band propagation arises in continuous (type A bands) or discontinuous (B and C types) way and the measurement of the deformation band characteristics such as band propagation velocity requires more sophisticated techniques, such as the laser scanning extensometry [5]. However, in spite of considerable experimental work, some characteristics remain controversial especially for the results related to the band propagation velocity variations with applied stress rate.

In the present work, we investigate the effect of ageing, temperature and stress rate on the band propagation velocity in two aluminium-based alloys of 5000 and 7000 series, respectively. Finally, the experimental finding will be compared to a recent theoretical model proposed by one of us [6,7].

2. EXPERIMENTAL PROCEDURE

The materials investigated were commercial aluminium alloys, type 5182 o and 7475, provided by the research centre of Pechiney (Voreppe) and containing 4.65% Mg, 0.37% Mn, 0.25% Fe, 0.2% Si, less than 0.1 % Cu and 5.81% Zn, 2.08% Mg, 1.49% Cu, 0.40% Si, 0.21% Cr, 0.08% Fe, 0.02% Ti, 0.01% Mn, 0.004% Zr (wt.%), respectively. Flat samples of 30.1 mm gage length, 5 mm width and 1.2 mm thickness were cut from the sheet with the tensile axis chosen in the rolling direction. Furthermore, The specimens of 7475 alloy were heat treated at 475DC for 1 h and quenched into water at room temperature. Mg and Zn are here in supersaturated solid solution in the alloy. Then, they were aged at various temperatures Tageing (from 20DC to 200DC) for various times tageing (from 10 min to 2 h) and cooled in air [4,8]. Immediately after heat treatment, tensile tests were performed with a soft machine (constant stress rate). To soften the machine, a spring of low stiffness (20 N mm·1) was inserted between the specimen and the load cell. A highly accurate Sandner axial extensometer (clip-on type with 30.1 mm gage

On PLe band propagation velocity under stress-controlled tests in 77 aluminum alloys

length) was attached to the extremities of the gage length. A data acquisition system using an IEEE card was used to record and store load, time and extension for post-processing of the data.

Tensile tests were performed at different stress rates between 2.5xlO-1

and 3.2xlO+1 MPa sol at Tageing= 20°C in the case of 7475 alloy and within a temperature range from 0 to 1600C at constant stress rate varied from 2xlO-3

and 42.57 MPa sol in the case of 5182 0 alloy.

3. RESULTS AND DISCUSSION

For both materials investigated, the band propagation velocity is found to increase with increasing strain until a maximum value of saturation for different temperatures, ageing conditions and stress rates. This trend is in agreement with other experimental results obtained on a soft tensile machine [9,10]. To investigate the dependence of the band propagation velocity Vb on stress rate cT for different temperature T, ageing temperature Tageing and ageing time tageing, we determined the average value of Vb in the domain of strains greater than 5%. Indeed, beyond this value, Vb vary slightly with increasing strain. Fig. 1 represents the variations of Vb with stress rate for different tageing in the case of 7475 alloy and Fig. 2 shows the dependence of Vb on stress rate for different temperatures in the case of 51820 alloy. It is important to note that Vb decreases when cT increases.

Various non-local approaches of the literature [11-14], which attempted to model PLC band propagation while including spatial variables in their constitutive laws in terms of second-order gradients in space, predict completely different tendencies. When one tries to compare the theoretical prediction and experimental variations of the band propagation velocity Vb with stress rate cT, it turns out that the experimental results are in quantitative agreement with results by Hahner (13,14]. Fig. 1 includes the prediction of the model for different ageing time and Fig. 2 includes two example results from the model at low and high temperatures, respectively.

Based on a gradient-dependent generalization of Penning's phenomenological PLC model [15], Hahner proposed a model which predicts the evolution of different band characteristics [6,16], in particular the band propagation velocity evolution. To this end, he started from the idea that ageing occurs while glide dislocations are temporarily arrested at localized forest dislocations that are overcome with the aid of thermal activation. This is expressed by an arrhenius law for the plastic strain rate E,t with an effective Gibbs' free enthalpy, G = Go + I1G, where Go is the activation enthalpy in the absence of any ageing and I1G is the extra contribution due to ageing.

78 Mohammed Abbadi, David Thevenet, Peter Hahner and A bderrahim Zeghloul

e=vQexp - 0 +~ =1'/ Qexp-g f . [ G + ~G CT ] [ ]

kT So (1)

Here v is the attempt frequency, n is the elementary strain associated with a single activation step, k is the Boltzmann constant, and T is absolute temperature. The effective stress which assists thermal activation, CTeff = CText - CT int' is given by the external stress CText (flow stress) minus the internal stress CT int (back stress affected by strain hardening). Finally, So = 8CText Ilne le •• w is the instantaneous strain-rate sensitivity of the flow stress, in the absence of any changes in hardening state (e) and ageing state (~G). The generalized driving force f and the reduced additional enthalpy g have been defined by:

f = '!... exp[-Go] exp[CTeff ] and 1'/ k T So

~G g=­

kT (2)

where 11 expresses the ageing rate (ex: solute mobility) and g is subject to the competition of two counteracting effects and considered as a dynamical variable of the form:

g=1J(g-gJ-~ g+Dg" (3)

where the first term represents the ageing due to the accumulation of solute clouds, the second the release of solute clouds due to dislocation unpinning and the third the spatial coupling accounting for long-range dislocation interactions.

Using a non-linear analysis and after suitable approximations, Hahner obtained the band propagation velocity Vb at constant stress rate as [17]:

(4)

D is the diffusion coefficient and go is the ageing during the loading time. To compare the model's predictions to the experimental results of Fig. 1,

the fitting of the experimental data was realized according to the following form:

On PLC band propagation velocity under stress-controlled tests in 79 aluminum alloys

(5)

with PI oc.fi:, P2 oc g", and g", representing the saturation enthalpy. The proportionality constants for the two parameters were fixed by best fits of the 120 min data to Ph P2 and g", according to Eq. (5). This result is demonstrated by the lowermost solid line in Fig. 1. The other data for 60, 30 and 10 min ageing time, respectively, were then approximated by one­parameter fits to g", using the same proportionality constants. The values of g '" deducted from the data fitting are summarized in the inset of Fig. 1. One observes a good correlation between experimental results and theoretical predictions with a slight deviation for the uppermost curve corresponding to the shortest ageing time of 10 min. The depletion of the experimental results related to the specimens with 10 min pre-deformation ageing treatment may be explained by the fact that these specimens underwent substantial additional precipitation (formation of Guinier-Preston zones) during tensile deformation and which becomes more appreciable at low stress rates. Thus, the saturation enthalpy of DSA tends to lower. This is why g", decreases with increasing pre-deformation ageing time (see Fig. 1).

The advantage of the present model consists in the fact that with the same relationship of Vb (Eq. 4) and suitable approximations, one obtains the evolution of Vb as a function of stress rate for different testing temperatures. Starting from Eq. (4) and using the fitting expressed by Eq. (5) with:

where Q relates to thermally activated migration of solute atoms. After neglecting the second term with g", of Eq. (5), Hahner obtained results in qualitative agreement with the experimental data. One should be satisfied with the prediction of the model if we look at the scattering of the experimental results.

80 Mohammed Abbadi, David Thevenet, Peter Hahner and Abderrahim Zeghloul

600

550 • . . •

~ 500 4 4 4

'" 450

1 400

.£ 350

(.) 300 0 250 ag!ling time at 20"C '0 • 10 min , g.',,46 > 200 '"C • 30 min, g.'038

~ 150 ... 60 min , g~" 32 ~ 100

* 120 min, g .,26 50

0 0.1 10

Stress rate [MPa/S]

Figure 1. Stress-rate dependence o/band propagation velocity Vb/or different ageing time at 20"(; (symbols) compared to Hahner's theoretical results (solid lines)

700

600

500 ,........,

'" 400 ] ........ 300 ~

200

100

0

o Ii

* * * * * *

Experimental results

* T=O"C 0 T=4O"C • ~OC 'V T=80"C

Hiihner's model - - - - Low temperatures -- High temperatures

0.01 0.1 1

cr [MPa/s]

* , * ",**

't ~, .........

10 100

Figure 2. Stress-rate dependence o/band propagation velocity Vb/or different testing temperatures (symbols) compared to Hahner's model results (dashed and solid lines)

Fig. 2 shows the experimental results of Vb as a function of stress rate for different testing temperatures, namely 0, 40, 60 and 80°C (symbols) and

On PLC band propagation velocity under stress-controlled tests in 81 aluminum alloys

theoretical results at low (O°C dashed line) and high (80°C solid line) temperature.

4. CONCLUSIONS

This study was based on results obtained with a tensile soft machine to investigate the effect of testing conditions, such as temperature, ageing and stress rate, on the band propagation velocity. For this, two aluminium-based alloys of the 5000 and 7000 series, respectively, were investigated. The decrease of band propagation velocity with increasing stress rate is observed in both materials for various testing conditions.

The most prominent result of the present work resides in the fact that a recent model achieved by Hahner predicts quantitatively the decrease of band propagation velocity when stress rate increases for different temperatures and ageing.

ACKNOWLEDGEMENTS

The authors want to express their gratitude to Pechiney company for their support in supplying sheets of aluminum alloys.

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82 Mohammed Abbadi, David Thevenet, Peter Hahner and Abderrahim Zeghloul

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1177-1182,1993. [13] P. Hahner, "Modelling the spatio-temporal aspects of the Portevin-Le Chatelier effect",

Mater. Sci. Eng. A, vol. 164, pp. 23-34, 1993. [14] P. Hahner, "Modelling of propagative plastic instabilities", Scripta Metallurgica et

Materialia, ViewpointSet21, vol. 29,pp.1171-1176, 1993. [IS] P. Penning, "Mathematics of the Portevin-Le Chatelier effect", Acta Metall., vol. 20, pp.

1169-1175, 1972. [16] M. Abbadi, P. Hahner, and A. Zeghloul, "On the characteristics ofPortevin-Le Chatelier

bands in aluminum alloy 5182 under stress-controlled and strain controlled tensile testing", Mater. Sci. Eng. A, vol. 337, pp. 194-201,2002.

[17] P. Hahner, "On the velocity selection of propagating Portevin-Le Chatelier deformation bands during constant strain-rate and constant stress-rate testing", unpublished.

EFFECT OF SOME PARAMETERS ON THE ELASTOPLASTIC BEHAVIOR OF GREEN SAND

R. Ami Saada,

Laboratoire de Mecanique (LaM) Universite de Marne La Vallee Cite Descartes, 5, Boulevard Descates, 77454 Marne La vallee cedex 2, France

Abstract: The behavior of the casting sands at high temperatures (up to 600°C) is experimentally presented. Based on a parametric study, the predictions are well discussed and compared with the experimental results. Triaxial compression tests at different temperatures (20 to 600°C) were performed in foundry sand (Ami Saada et ai., 1996, Ami Saada 1997, Ami Saada et ai., 1999). They have shown that the material strength is governed by the presence of bentonite and water. For the green sand at the temperature range from 20 to 300°C, Three elastoplastic models are considered in this study. Especially, the effects of some parameters on the mechanical behavior of the casting sands are investigated, such as preconsolidation pressure Ppo.

Key words: Thermal consolidation, Therrnoelastoplasticity, Green sand, Core sand, Casting process, Triaxial compression tests, Uniaxial compression tests.

1 INTRODUCTION

During the industrial casting process, the rupture of molds appears often. These laters are in general, made from clay sand (silica sands with some percents of bentonite, mineral carbon and water) called casting sands or made from sand and resin called core sands. So, for finding the solutions to this rupture problem, some authors (Alexandre et al. 1990; Bellet & Chenot 1993;) have developed simulation tools for the casting process by taking into account only the thermal transfer and the change of phase in the liquid metal. However, the mechanical behavior of foundry sands modifies considerably the economical losses (Williams, 1967). In order to introduce the mechanical behavior of molds in the simulation of the casting process as a first step, the results obtained by triaxial compressions tests are analysed at

83

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 83-88. ©2004 Kluwer Academic Publishers.

84 R. AmiSaada

different temperatures in the green sands. In the second step, the effect of some parameters involved in the adopted elastoplastic models initially employed by the authors (Ami Saada et al. 1996) are analysed for the green sands and for the temperature range from 20 to 600°C.

2 EXPERIMENTAL 2.1 Material

Triaxial compression tests at different temperatures are perfonned on the green sand composed by silica (98.8%), bentonite with few percent of mineral carbon and water.The sample is elaborated by compacting green sand in a closed container, obtaining therefore a cylinder of 35 mm diameter and 60 mm height. The specific density of the grain is taken equal to that of silica 1s = 2.65 Kg dm-3• The initial density of the specimen is 1d = 1.6 Kg dm-3 and the initial void ratio eo' defined as the volume ratio of the void to the solid phase (eo = vjvJ, is equal to 0.64.

2.2 Discussion

The principal observations obtained by the triaxial compression tests perfonned for the green sand at different temperatures and under three strain rates values (&=10-2S-1, &=10-3S-1 and &=10-4S-1) (Ami Saada et al. [1996]) show that : • The casting sands exhibit no strain rate effect for the temperatures

ranging from 20 up to 600°C. • The strength of this material is governed by the presence of bentonite and

water. In this case, it increases when the bentonite is at temperature range (20 to 600 0c). This interval may also be divided into three zones: zone 1 (20 to 200°C) where the strength increases due to the evaporation of water; zone 2 (200°C to 400°C) where strength is constant, zone 3 (400°C to 600°C) where strength increases due to the evaporation of water originally adsorbed by the bentonite.

3 ADOPTED MODELS 3.1 Expression of yield surfaces and plastic potentials for the elastoplasctics models

The mechanical behavior of casting sands is defined by elastoplastic models adopted the critical state assumption (Roscoe et al 1968). These models were used by the authors (Ami Saada et aI., 1996) for the associated or non associated laws. The expressions of the yield surfaces and plastic potentials of the elatoplastic models are based on the critical state principle. They consider implicitly the temperature, through the model parameters,

Effect of some parameters on the elastoplastic behavior of green sand. 85

while the cohesion strength of green sand is considered as explicit governing parameter. The yield surface and potential functions of the models are detailed here:

3.1.1 Basic Cam Clay Model (CC)

f(p P) q +Log( P - PeO ) ,q,8v M (p - P eO) P P - P eO

3.1.2 Modified Cam Clay Model (MCC)

f(P,q,8e) 2 q

2( )+(p - PeO)-(p P - PeO) M P- PeO

3.1.3 Revised Modified Cam Clay Model (RMCC)

l f 1(p,q,8e) 2( )+(p-Peo)-(Pp-Peo)

M p- Peo

f ( P) - M (p p - P eO) 8~ 2 p,q,8q -q 2 p

a+8q

3.1.4 Hujeux Model

f( P P) q +Log( p-Peo)~ p,q,8v,8q M (p - P eO) P P - P eO a + 8~

g(p,q) q +Log( P - Peo) M(p - PeO) Pa - PeO

with

p P = P pO exp(f3 8e) and

13=((1 + eo)/(J. - k))

where M, P eO' J., k, P pO and a are the governing parameters.

4 MODELS PARAMETERS

(1)

(2)

(3)

(4)

(5)

(6)

(7)

The results obtained by using the elastoplastic models described before are reported in figure 1.

86

2.25 ---- Experimental curve (I) Basic Cam Clay Model

1.89 (2) Revised Modified Cam Clay Model

~ (2) c... :E 1.54

0.83 " " " " " " " \ 0.47 L---L~_~'--_---'~~_~_--'

0.00 0.05 0.10 0.15 0.20 0.25 0.30 Axial Strain

R. AmiSaada

Figure 1. Results of drained triaxial lest at T = 20 °C and 0'3'" 0.47 MPa. Comparison with theoritical prediclion using basic Cam Clay Model and Revised Modified Cam Clay Model

They show that the basic Cam Clay Model based on the associated flow rule can not describe correctly the casting sand behavior (figure 1).

\ PpO

Yield surW:e 2 I'R.M.CC. only)

(8)

Figure 2. Yield Surfaces in q-p plane : (a) Basic Cam Clay Model (b) Revised Modified Cam Clay Model

(b)

This may be explained by the non uniqueness of the normality of the increment of plastic strain, as it is the case of the result obtained under the triaxial compression test at 20°C with the lateral pressure (0'3) is equal to the preconsolidation pressure Ppo equal to 0.47 MPa. Actually, in figure 2 the

Effect of some parameters on the elastoplastic behavior of green sand. 87

yield surfaces of basic Cam Clay Model (CC) and of Revised Modified Cam Clay Model (RMCC) are plotted.

They show that for the point obtained by the intersection between the yield surface and the p axis represented by Ppo , there is an infinity of normal plastic strain for the CC Model, but only one normal plastic strain for the. Hence, the Revised Modified Cam Clay model describes appropriately the casting sand behavior (figure 1).

Figures 3 and 4 compare the experimental results with predictions in the case of triaxial compression tests for two distinguished (0.6 and 0.8 MPa) lateral pressures and for 300°C. It is obvious that the Hujeux model and RMCC model based on two yield surfaces, are in good agreement with the experimental results.

o ~--------~--------~"-------"--~ 0.00 0.05 0.10 O.IS

Axial Strain

Figure.}. Results of drained triaxial test at T:: 300 "C and at various lateral J'tesures«l) (13 = 0.6 MPa, (2}. ClJ=O.8MPa). Comparison with theoriticalpred.iction us.mg.Revised Modified Cam Clay Model. .

5 CONCLUSION

By taking into account the effect of some model parameters, especially the preconsolidation pressure Ppo this study point out that the elastoplastic models based on the associated flow rule do not predict correctly the casting sands behavior. In the case non associated models, the theoretical results describe fairly well the experimental results. In general, elastoplastic models that consider two yields surfaces, give an appropriate of casting sands.

88 R. Ami Saada

~

" ---- Experimental curve Hujeux Model

'" 3

c... , 2: i ~2L ____ ! I

vi' I

'" 2 ~----------------- 1 ) " !:: </)

-;; .i< ..0::

0 I O.UD 0.05 O. iO O. i 5

Axial Strain

Figure 4. Results of dramed triaxial test at T '" 300°C and at various lateral presurcs « 1) 03 = 0.6 MPa, (2). O"J~!l MPa). Comparison with theoritical prediction using Hujeux Model.

REFERENCES

Alexandre P. et a1., Proe. of the 7 th Eng. Found., Davos, pp. 30-38, (1990),

Ami Saada R., High temperature behavior of core sands: Experimental contribution; Proc. of Plasticity '97, The Sixth International Symposium on Plasticity, Juneau (USA), 291-292, July, 1997.

Ami Saada R., Bonnet G., Therrnomechanical Behavior of Core Sands Under Triaxial Sollicitations, The Seventh International Symposium on Plasticity, Cancun, 537-540, January, 1999.

Ami Saada R., Bonnet G, Bouvard D., Therrnomecanical Behavior of Casting Sands: Experiments and Elastoplastic Modeling,. Int. J. of Plasticity 12, N° 3, 273-294, 1996.

Bellet M., & Chenot, J.L., Compo Mec. Pub. & Els. Ap. Sci., Chap. 13, pp. 287-316, (1993).

Hujeux lC., Calcul Numerique de Problemes de Consolidations Elastoplastiques, These de Docteur Ingenieur, Ecole Centrale Paris, 1979.

Ota H. et aI., Foundryrnan, Vol. 82, N° 6, pp. 278-283, (1989).

Roscoe K.H., Schofield A. ,and Wroth C., On the Yielding Of Soils, Geotechnique, 9, 22-53, 1968.

Williams D.C., What can the foundryrnan do to control the Shear Strength of Squeeze Molded Sands Mixtures 2 -Not Much, Transaction of A.F.S., pp. 80-86, 1967

EXPERIMENTAL INVESTIGATIONS OF SIZE EFFECTS IN THIN COPPER FOILS

Gerd Simons!), Christina Weipperf), Jiirg Dual l ), and Jiirgen Villain2)

J) Institute of Mechanical Systems ETHZurich 8092 Zurich, Switzerland E-mail: simons@imes.mavt.ethz.ch 2) University of Applied Sciences Augsburg 86161 Augsburg, Germany

Abstract: This work deals with the characterization of the deformation behavior of thin copper foils with the goal of investigating size effects. Tensile tests are performed with specimens, which possess a comparable microstructure, a constant thickness/width and widthJlength ratio whereas the thickness varies from 10 to 250 Ilm. Results show a transition from ductile to a macroscopically "brittle" behavior in the range of about 20 Ilm.

Key words: size effects, thin copper foils, microstructure, tensile testing, fracture surface

1. INTRODUCTION

In the last years there has been a growing interest in the use of copper for applications in very small dimensions, e.g. the current feature size of copper configurations on microprocessors is in the range of 0.2 .... m [1]. The ability of manufacturing such small structures of copper and their wider application led to a number of experimental studies on its mechanical behavior, e.g. for thin films [2-4], wires [5, 6], or foils [7-11]. Anwander et al. [7] and Hadrboletz et al. [8, 9] investigated the fatigue and fracture properties of thin copper foils (electrodeposited and rolled) by testing thin stripes of various thicknesses (thickness ranging from 9 to 250 ~), but relatively large lateral dimensions (20x20 mm and 10x40 mm). Their results show a decrease in fracture strain and ultimate strength in tensile testing with decreasing thickness.

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90 Gerd Simons, Christina Weippert, Jiirg Dual and Jiirgen Villain

The effect of the size of a specimen on its mechanical behavior ("size effect") was demonstrated only on certain mechanical properties. For plastic behavior, a large number of size effects have been reported, many of them explained by the so called "strain gradient plasticity" [5, 12-20]. When comparing small Ni specimens with bulk Ni [21-23] no influence was found on Young's modulus (an elastic property).

Due to the complexity of material behavior - particularly in the interaction of microstructure (e.g. grain size), the specimen's loading and the specimen's overall dimensions - there is still a strong need for better understanding of effects of size in materials based on a large variety of experimental investigations. Tensile testing with a predominantly uniaxial stress field is one of the key experiments.

2. EXPERIMENTAL CONSIDERATIONS

The goal of this study is to perform experiments in order to isolate the influence of dimensional parameters in a predominantly uniaxial stress field on the mechanical behavior of thin copper foils.

2.1 Methodology

In order to reach the above mentioned goal a suitable testing method as well as comparable specimens have to be chosen. The basic idea of this work is to perform tensile tests with a constant strain rate, where the specimens have a varying thickness but possess a comparable microstructure and a linearly scaled geometry.

The authors perform tensile tests in order to eliminate the influence of external strain gradients as they scale with the number of geometrically necessary dislocations which themselves are assumed to influence the hardening behavior [5]. The strain rate is kept constant to cancel out the influence of viscous material effects.

Table 1. Dimensions a/tested Cujoi/ specimens Thickness [~m] Width [~m] 10 200 20 400 34 680 100 250

2000 5000

Gauge length [mm] 2 4 6.8 20 50

In order to have geometrically similar flow conditions the specimens are designed as follows: The specimens have standard tensile test specimen geometry with a testing region whose dimensions scale with the thickness:

Experimental Investigations o/Size Effects in Thin Copper Foils 91

The length is 200 times the thickness of the specimen and the width 20 times (see Table 1 and Figure 1). The large plates at the end are required for handling purposes.

.. tI Figure 1. Layout for 10, 20 and 34 pm thick copper specimens where the testing region has

the dimensions given in Table 1 (true size).

The microstructure of a specimen plays a crucial role in its mechanical behavior. Therefore, it is mandatory for studying size effects that only specimens with a similar microstructure are compared. For polycrystals the influence of many microstructural parameters is understood to a certain extent: - Texture: The influence of the texture is twofold: On the one hand, the

type of texture influences the yield strength as can be seen from the model of Bishop and Hill (e.g. in [24]). On the other hand, differences in the orientation of a specimen relative to the orientation of its grains (e.g. whether the specimen's axis lies in rolling or transversal direction of a rolled foil) modify - amongst other properties - Young's modulus in the range of ±20 % as reported in [25].

- Grain size: Its major influence is on the yield strength as described by the Hall-Petch relation which states that the yield strength cry scales with the inverse square root of the grain size [12]:

1 u cx;-YJd

- Grain boundaries: They act as a barrier for dislocations and therefore play an important role for the deformation process [26], especially in the presence of impurities as e.g. hydrogen in copper.

The influences of other factors, e.g. the grain size distribution, the grain geometry, the internal stress state, and the dislocation density, are more complex to interpret.

Out of these considerations, the authors decided to investigate specimens of varying thickness which have a comparable microstructure: They should have the same texture, grain size, and hardness (as an indicator for plastic properties). The specimen's axis is parallel to the rolling direction of the foil.

92 Gerd Simons, Christina Weippert, Jurg Dual and Jurgen Villain

2.2 Experimental Setup

Two types of experimental setups are used for performing the tensile tests: one for small and another for larger specimens.

The smaller specimens (thickness <= 34 !Jlll) are tested based on a setup of Mazza [21]: The specimen is fixed at the lower end to a weight on a precision balance and then the upper part of the specimen is translated vertically, the resulting reduction in weight is the force acting on the specimen. The strain is measured optically (least square template matching algorithm, deformation resolution 20 nm, see [27]).

The larger specimens are tested with a commercial tensile test machine (Zwick 1445) with a ION and 200 N load cell and a specially designed clamping apparatus to allow clamping free of initial tension.

Further details on the setup can be found in [11].

2.3 Fabrication of Specimens

The specimens are made by wet etching. Whereas the thicker foils (thickness> 34 J..Lm) can be produced by conventional wet etching, the thinner foils have to be fixed to a substrate for stability reasons before processing. Therefore, they are glued on a silicon wafer, afterwards the foils can be patterned by standard photolithography (spin-coating, exposing, developing). The etching of the copper foils is performed in a 50°C sodium persulfate (Na2S20g) solution for several seconds to a few minutes depending on the thickness of the foil; this is the only difference in the fabrication process of foils of varying thickness. Finally, the foil is removed from the silicon wafer by putting it into a solvent. With this procedure beams of 50 !Jlll width can be produced. The under etching is negligible in comparison with the width, which is 20 times the thickness.

3. RESULTS

First, results for the microstructure of the tested specimens are presented. Afterwards, tensile test results are given concluding with an interpretation of the former.

3.1 Microstructure

The texture of the Cu foils was measured by X-ray diffraction. The tested foils show a preferred orientation of (100)[001], a cube texture. For determining the grain size two methods were applied:

Experimental Investigations a/Size Effects in Thin Copper Foils

Analysis of standard micrographs produced by polishing and etching (Figure 2, left image) Cross section milled by a focused ion beam (Fm) (Figure 2, right image)

93

Figure 2. Determining grain size for a 20 J.Il1I thick copper foil: over etched micrograph (left) and FIB milled cross section (right).

Both methods reveal a grain size in the order of 1 J.Ul1 for the various foil thicknesses.

3.2 Tensile Test

In Figure 3 on the left the results of a tensile test for a 10 and 34 f.1m thick foil are presented. Whereas the 34 f.1m thick foil still shows a typical stress strain relationship for ductile materials, the 10 J.Ul1 thick foil has a stress strain behavior which is typical for brittle materials (only elastic part). The right graph of Figure 3 displays the relationship between foil thickness and fracture strain (circle, left vertical axis) and ultimate strength (asterisk, right vertical axis).

u

Figure 3. Tensile test results (strain rate 2.5·UIss·'): stress-strain curve/or a 10 and 34 J.Il1I thickfoil (left image) andfracture strain (circle. left vertical axis) and ultimate strength

(asterisk. right vertical axis) vs.foil thickness (right image).

94 Gerd Simons, Christina Weippert, Jiirg Dual and Jurgen Villain

The main difference between foils of varying thickness lies in their plastic behavior, foils thinner than 100 !Jll1 show a strong decrease in fracture strain. For foils thinner than 20 /-lm hardly any plastic deformation is measurable macroscopically. There is as well an increase of ultimate strength with decreasing foil thickness. The initially relatively flat surface (Ra == 0.4 /-lm on the top surface) is roughened in the vicinity of the fracture surface largely.

3.3 Interpretation

The decrease of fracture strain with decreasing foil thickness is confirmed by SEM images of the faces of rupture (Figure 4). Foils thinner than 20 !Jll1 show quite a flat fracture surface with a knife edge rupture, whereas the thicker foils have voids and dimples which results in a much larger surface area. This indicates a change in the deformation behavior as the formation of new surfaces requires energy. This could be explained by the fact that there are more grains in the thicker foils and hence, the area of grain boundaries and the amount of dislocations is larger in thicker foils. This would mean an increase in number of dislocations which can glide as well as an increase of grain boundaries which are able to slide. As a consequence, the elongation of rupture could increase and the ultimate strength decrease. Additionally, there could be a difference in stress state as in the thinner foils it is only a few grains which carry the load. Further tests have to be performed to confirm the above mentioned hypotheses.

Figure 4. Faces of rupture ofa 10 J1m (left) and a 34J1m (right) thickfoil.

4. CONCLUSIONS AND OUTLOOK

Tensile tests of copper foils of varying thickness with a comparable microstructure and geometrically scaled specimens reveal a decrease in fracture strain which is due to a change in the deformation behavior.

Experimental Investigations of Size Effects in Thin Copper Foils 95

The most relevant factor for this finding seems to be the ratio of grain size to foil thickness. Further studies with various grain sizes as well as the analysis of cross sections close to the fracture faces, which could be made e.g. by FIB milling, would bring further insight.

ACKNOWLEDGEMENTS

The authors want to express their gratitude to P. Gasser, EMPA Diibendorf, for carrying out the FIB investigations, to A. Wahlen and S. Stahel, Institute of Virtual Production, ETH Zurich, for invaluable help in performing the texture measurements and to B. Brabetz, Siemens ZT Munich, for etching the large test specimens.

REFERENCES

[1] "Back to the Future: Copper Comes of Age", IBM Research Magazine, vol. 35, 1997.

[2] S. P. Baker, R. M. Keller, A. Kretschmann, and E. Arzt, "Deformation mechanisms in

thin Cu films", in Materials Reliability in Microelectronics VIII. Symposium. 13-16

April 1998; San Francisco, CA, USA, vol. 516, Mat. Res. Soc. Symp. Proc., J. C.

Bravman, T. N. Marieb, J. R. Lloyd, and M. A. Korhonen, Eds., 1998, pp. 287-298. [3] R. M. Keller, S. P. Baker, and E. Arzt, "Quantitative analysis of strengthening

mechanisms in thin Cu films: Effects offilm thickness, grain size, and passivation",

Journal of Materials Research, vol. 13, pp. 1307-1317, 1998.

[4] D. T. Read, "Tension-tension fatigue of copper thin films", International Journal of

Fatigue, vol. 20, pp. 203-209, 1998.

[5] N. A. Fleck, G. M. Muller, M. F. Ashby, and J. W. Hutchinson, "Strain Gradient

Plasticity - Theory and Experiment", Acta Metallurgica Et Materialia, vol. 42, pp. 475-

487, 1994. [6] R. Hofbeck, K. Hausmann, B. Ilschner, and H. U. Kunzi, "Fatigue of Very Thin Copper

and Gold Wires", Scripta MetaUurgica, vol. 20, pp. 1601-1605,1986. [7] M. Anwander, A. Hadrboletz, B. Weiss, and B. Zagar, "Thermal and mechanical

properties of micro materials using laser optical strain sensors", Proceedings of the SPIE The International Society for Optical Engineering, vol. 3897, pp. 404-413, 1999.

[8] A. Hadrboletz, G. Khatibi, and B. Weiss, "The "Size-Effect" on the Fatigue and Fracture

Properties of Thin Metallic Foils", presented at Euromat 99, October 1999, Munich,

1999.

[9] A. Hadrboletz, B. Weiss, and G. Khatibi, "Fatigue and fracture properties of thin

metallic foils", International Journal of Fracture, vol. 109, pp. 69-89, 200!.

[10] M. ludelewicz, "Cyclic Deformation of 100-Mu-M Thin Polycrystalline Copper Foils",

Scripta Metallurgica Et Materialia, vol. 29, pp. 1463-1466, 1993.

[II] G. Simons, C. Weippert, 1. Dual, and 1. Villain, "Investigating Size Effects on

Mechanical Properties: Preliminary Work and Results for Thin Copper Foils", presented

at Materialsweek, International Congress Centre Munich, 2001.

96 Gerd Simons, Christina Weippert, Jiirg Dual and Jiirgen Villain

[12] E. Arzt, "Overview no. 130 - Size effects in materials due to microstructural and

dimensional constraints: A comparative review", Acta Materialia, vol. 46, pp. 5611-

5626,1998.

[13] M. R. Begley and J. W. Hutchinson, "The mechanics of size-dependent indentation",

Journal of the Mechanics and Physics of Solids, vol. 46, pp. 2049-2068, 1998. [14] N. A. Fleck and J. W. Hutchinson, "A reformulation of strain gradient plasticity",

Journal of the Mechanics and Physics of Solids, vol. 49, pp. 2245-2271,2001. [IS] H. Gao, Y. Huang, W. D. Nix, and J. W. Hutchinson, "Mechanism-based strain gradient

plasticity - 1. Theory", Journal of the Mechanics and Physics of Solids, vol. 47, pp.

1239-1263, 1999. [16] J. W. Hutchinson, "Plasticity at the micron scale", International Journal of Solids and

Structures, vol. 37, pp. 225-238,2000.

[17] 1. G. Sevillano, "Intrinsic and extrinsic size effects in plasticity by dislocation glide", in

Multiscale Modeling of Materials - 2000. Symposium. 27 Nov.-J Dec. 2000; Boston,

MA, USA, vol. 653, Mater. Res. Soc. Symp. Proc., L. P. Kubin, R. L. Selinger, J. L. Bassani, and K. Cho, Eds., 2001.

[18] J. G. Sevillano, 1. O. Arizcorreta, and L. P. Kubin, "Intrinsic size effects in plasticity by

dislocation glide", Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, vol. 309, pp. 393-405,2001.

[19] J. Villain and O. S. Briiller, "Influence of Specimen Dimensions on Creep Behaviour of

Solder Materials", presented at MicroMat 2000, Berlin, Germany, 2000.

[20] J. Villain, C. Weippert, G. Simons, and J. Dual, "Size Effects and Mechanical

Properties: Results for Thin Copper Foils", presented at Materialsweek, International

Congress Centre Munich, 2002.

[21] E. Mazza, S. Abel, and 1. Dual, "Experimental determination of mechanical properties of Ni and Ni-Fe microbars", Microsystem-Technologies, vol. 2, pp. 197-202,1996.

[22] J. R. Davis and ASM International Handbook Committee, Properties and selection: Nonferrous alloys and special-purpose materials, vol. 2, 10th ed. Materials Park, OH: ASM International, 1990.

[23] K. E. Volk and R. Ergang, Nickel und Nickellegierungen Eigenschaften und Verhalten. Berlin etc.: Springer, 1970.

[24] W. F. Hosford, The mechanics of crystals and textured polycrystals. New York etc.: Oxford University Press, 1993.

[25] G. Wassermann and J. Grewen, Texturen metallischer WerkstofJe, 2nd. ed. Berlin etc.:

Springer, 1962.

[26] 1. P. Hirth, "The influence of grain boundaries on mechanical properties", Metallurgical­

Transactions-A-(Physical-Metallurgy-and-Materials-Science), vol. 3, pp. 3047-3067,

1972.

[27] G. Danuser and E. Mazza, "Observing deformations of20 nanometer with a low

numerical aperture light microscope", Proceedings of the SPIE The International Society

for Optical Engineering, vol. 2782, pp. 180-191, 1996.

PLASTIC RESPONSE OF THIN FILMS DUE TO THERMAL CYCLING

Lucia Nicola 1), Erik Van der Giessen 1) and Alan Needleman2)

1) The Netherlands Institute for Metals Research/Dept. of Applied Physics

University ofGroningen. Nyenborgh 4. 9747 AG Groningen. The Netherlands

2) Division of Engineering

Brown University. Providence. RI02912. USA

I.i.nicola@phys.rug.nl

Abstract Discrete dislocation simulations of thin films on semi-infinite substrates un-

der cyclic thermal loading are presented. The thin film is modelled as a two­dimensional single crystal under plane strain conditions. Dislocations of edge character can be generated from initially present sources and glide in the film on a given set of slip systems. At each time step of the simulation, the stress field in the film is calculated through the solution of a boundary value problem, taking into account the long-range stress contribution of the current dislocation structure. The numerical results show a clear size effect in the plastic behaviour of two films with thicknesses of O.25pm and O.5pm. The mechanical response of the two films during the cyclic thermal loading is analysed, with an emphasis on the evolution of the dislocation structure.

Keywords: Thin films, thermal cycling, discrete dislocation plasticity

1. INTRODUCTION

Thin films are extensively used in the electronics industry; metallic layers deposited on a substrate are at the basis of most microelectronic devices such as integrated circuits and transistors. Thin metallic films are also used as pro­tective coatings for other materials, for example to improve wear resistance or to affect heat conduction. Thin films are primarily chosen for their elec­tronic, magnetic, optical or corrosion-resistance properties, while mechanical integrity is generally not the main design consideration. However, thin films can fail mechanically during processing or while in service, mainly when the film and substrate undergo thermal cycling. High thermal stresses build up in the film because of the thermal mismatch with the substrate. Even if the plastic deformation caused by thermal stress is not severe enough to cause film failure, it can strongly affect the film performance.

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s. Ahzi et al. (eds.). Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials. 97-104. ©2004 Kluwer Academic Publishers.

98 L. Nicola, E. Van der Giessen and A. Needleman

High stress levels can be attained in thin films (thickness ranging from a few nanometers to several tens of micrometers), with the stress levels increasing with decreasing film thickness. This size effect has been shown experimentally by monitoring the average strain in the film during thermal cycling, by means of X -ray diffraction [1, 2] and by wafer curvature techniques [1, 3]. The theoretical prediction of this size effect has proven to be a challenge. Classical continuum plasticity theory does not include a material length scale and therefore does not predict a size effect. Non-local phenomenological continuum theories (e.g. [4,5,6]) have been proposed that do incorporate a length scale and thus can, at least in principle, predict the size effect in thin films. Single dislocation models [7, 8] suggest that dislocation nucleation and motion are affected by film thickness and this give rise to a size effect. We have presented a dislocation plasticity simulation in [9] which involves the nucleation and motion of many dislocations. Our analysis indicates that the local concentration of stresses caused by dislocation pile-ups at the film-substrate interface plays an important role in the observed size effect. The present paper builds on [9] and considers thin films subject to thermal cycling.

2. MODEL

The thin film is modelled as a two-dimensional single-crystal of thickness h, perfectly bonded to an infinitely large elastic substrate. Plasticity in the film is described by the evolution of edge dislocations in an otherwise elastic contin­uum. Dislocation glide takes place on a set of three slip systems, characterised by the angle C\l(Il) (p = 1,2,3) with the interface. The interface between film and substrate acts as an impenetrable barrier for the dislocations, so that the sub­strate remains elastic. While changing the temperature of the film-substrate system, the evolution of the stress state in the film is calculated, assuming plane strain conditions. The thermal stress is solely caused by the difference in the thermal expansion coefficient of the film (ar = 23.2 x 10-61K) and the substrate (as = 4.2 x 1O-61K), which are representative of an aluminium film on a silicon substrate. The difference in the elastic properties is neglected as well as the elastic anisotropy of the single crystal. Both materials follow the thermoelastic relation

(1)

where t!T is the temperature change relative to the undeformed state, a = a.r - as, C = 76GPa is the bulk modulus and 11 = 26GPa is the shear modulus; representative values for aluminium.

The film-substrate system is taken to be periodic, with periodicity w (see Fig. la). To simplify the analysis, we decompose the problem into two linearly additive parts: (i) the substrate and the film are taken to have the same coefficient of thermal expansion a = as; and (ii) the substrate does not expand, i.e. as = 0, while the film has a coefficient of thermal expansion a = ar - as. The solution

Plastic response of thin films due to thermal cycling 99

W X2 ~-----------7

(a)

+

(b)

Figure 1. (a) Geometry of the film-substrate problem. (b) Decomposition of the unit-cell problem into a thenno-elastic expansion problem and a plastic relaxation problem.

to problem (i) is uniform, stress-free expansion and is not relevant for the further stress analysis. The solution to problem (ii) accounts for the build up of thermal stress in the film and is solved using the discrete dislocation plasticity methodology introduced in [10]. This is an incremental method in which at each step the dislocation positions are updated and the new elastic fields are calculated. These fields are obtained as a linear combination of two contributions: the sum of the long-range, singular stress fields of all dislocations in the film as if they were in an infinite medium; and the stress field obtained as solution of the linear boundary value problem which enforces the boundary conditions. The latter, finite-body fields are regular and are obtained here by a finite element method. Beside periodicity, stress-free surface conditions are prescribed on the unit cell. A complete description of the problem formulation and solution method are given in [9].

The evolution of the dislocation structure is determined by a set of consti­tutive rules, which govern dislocation nucleation, glide and annihilation. The constitutive equations are based on the Peach-Koehler force acting on disloca­tion I, which is given by

(2)

100 L. Nicola, E. Van der Giessen and A. Needleman

where cr~p is the singular stress field caused by dislocation J and aij is the

finite-body correction field; b}l) is Burgers vector (of magnitude b = 0.25nm)

and n~1) is slip plane normal. The glide velocity v{J) of dislocation I is taken to be proportional to the Peach­

Koehler force so that v(1) = /(1) / B, where B is the drag coefficient. We ignore the temperature dependence of B and use the representative value B = 1O-4Pas for aluminium.

Dislocations are nucleated at a point source, which is regarded as a two­dimensional representation of a Frank-Read source. The source generates a dipole when the Peach-Koehler force on it exceeds the critical value 'tnucb for a time span tnuc . The values of the nucleation strength'tnuc are chosen randomly from a Gaussian distribution with average value 25MPa with a standard devi­ation of 5MPa. The nucleation time is taken to be tnuc = IOns for all sources. The newly created dipole is taken to have a size Lnuc such that it will not col­lapse onto itself under a resolved stress equal to 'tnuc . The mean dipole size corresponding to the average nucleation strength is Lnuc = 250b = 62.5nm. Dislocations annihilation occurs when two dislocations with opposite Burgers vectors are closer than Le = 6b.

3. RESULTS

Results are presented for two films, one of thickness h = 0.25,um and a thicker one with h = 0.5,um. The film material contains three slip systems, with slip plane orientations: cp{l) = 0°; cp(2) = 60°; cp(3) = 120°. A random distribution of sources is placed on the slip planes. The density of sources in both films is taken to be 60/,um2, corresponding to an average source spacing of 0.13,um = 520b. Assuming an initial temperature of 600K, the film-substrate system is first cooled to 4OOK, then heated up to 600K and again cooled to 400K. For the interpretation of the straining direction induced in the film, it is important to note that we effectively deal with a film with thermal expansion coefficient ar - as > 0 constrained by a non-expanding substrate; i.e., the film is stretched during cooling and compressed during heating. The temperature varies linearly with time at a rate of 40 x 106Kjs. This very high rate is used to reduce the computing time.

At the beginning of the simulation, the film is stress free and dislocation free. When cooling starts, the film first behaves elastically, and a homogeneous tensile stress builds up in the film. Plasticity starts by nucleation from the weakest source. After a dipole has been nucleated, the dislocations move in opposite directions to produce plastic deformation and relax the stresses in the film. If they do not meet other dislocations on their path, one dislocation reaches the interface while the other moves out of the film through the free surface (leaving a step at the surface). With continued cooling, more sources activate, partly because of the influence of previously nucleated dislocations.

Plastic response of thin films due to thermal cycling 101

The dislocation distribution for the O.25,um film at the end of the first cooling stage (T = 400K) is shown in Fig. 2a, together with the distribution of all, the normal stress parallel to the interface. The latter is normalized by the elastic stress O"n that would be in the film without plastic relaxation; O"n =MPa for tlT = 200K. Many dislocations pile up at the impenetrable interface, where they form a hard and highly stressed boundary layer. Stresses in the rest of the film are more relaxed (see [9] for more details). If the temperature is kept constant at 400K, no significant evolution of the dislocation structure is observed, indicating that the dislocation structure is close to equilibrium.

During heating, when the film straining changes sign, the direction of dis­location motion is reversed. The high back stress built up during cooling acts to enhance dislocation motion, so that reverse plastic deformation occurs. The dislocations that were forming pile-ups at the interface progressively reach the free surface and leave the film. At the end of the heating process (see Fig. 2b) only a few dislocations are left in the film. A few of those dislocations have signs opposite to the signs of the dislocations nucleated during cooling; the opposite­signed dislocations were nucleated during heating when the mean stress state became sufficiently compressive. The average O"ll stress (in absolute value) reached after heating is much lower than that after cooling (note that the stress

E 0.2 aliI an ;;:

0 (a)

8.00 ~ 2.67 ~ -0.2

·2.67 .0 ·8.00 1;l -0.4

0 0.5 1.5 2

E 0.2

01/ On ~ 0

2,00 ., (b) 0.67 g ·0.2

-0,67 VI

·2.00 .0 ~ -0.4

0 0.5 1.5 2

E 0.2

0"1" cr. 2 0

8.00 ., (C) 267 g -0.2

-2.67 VI

·8.00 .0 :::> -0.4 "'

0 0.5 1.5 2

Figure 2. Dislocation distribution and in-plane stress, normalised by the elastic stress an, in the film with thickness h = O.2SJlIl1: (a) at 400K after the first cooling, (b) at 600K after heating and (c) at 400K after the second cooling.

102 L. Nicola, E. Van der Giessen and A. Needleman

range in Fig. 2b is smaller than in Figs. 2a and 2c) but is not zero as it is prior to the first cooling. Thus, the response is not reversible, as is indeed seen in experiments. After the second cooling (Fig. 2c) the stress state in the film is very similar to the one obtained after the first cooling, with small differences in the dislocation structure.

The evolution of the average in-plane stress in the two films during the imposed thermal history is shown in Fig. 3. Comparison between Fig. 3a and b for h = O.25pm and O.5pm gives evidence of a quite pronounced size effect. During the first cooling cycle, the O.25pm film hardens much more than the film with h = O.5,um, as discussed in more detail in [9]. In the O.25pm film, the high stress level pushes the dislocations in the pile-ups close together. This results in large back stresses which cause reverse plasticity in the early stages of the subsequent heating process.

In the h = O.5pm film the back stress associated with the pile-ups is lower. As a consequence, this thicker film unloads elastically almost until the stress changes sign. A compressive stress builds up in the film at around 430K, leading to the dislocation distribution at 600K shown in Fig. 4. The dislocation density after unloading is greater in the O.5pm film than in the O.25pm film. (Fig. 4 versus Fig. 2b).

The dislocation structure at 600K strongly influences the material response during the last cooling sequence. The dislocation density in the thin h = O.25,um film is very low, Fig. 2b, and most of the dislocations have been generated during the first cooling cycle. This situation is very similar to the initial condition, when the film was dislocation free. Therefore the response of the O.25pm film during first and second cooling differs only in the initial stages (until 550K), see Fig. 3a.

<<rll>JMPaJ 160

450 500 (a)

h=O.5llm 120

550 600 T[KJ

Figure 3. Average in-plane stress in the film versus imposed temperature for film thicknesses: (a) h = O.25JllIl and (b) h = O.5JllIl.

Plastic response of thin films due to thermal cycling

2.00 0.61

-0.61 ·2.00

0.4

..s 0.2 I.::

o !! ~ -0.2 .2 5: -0.4

o 0.5

103

1.5 2

Figure 4. Dislocation distribution and in-plane stress nonnalised by the elastic stress Gn in the film with thickness h = O.5,um after cooling to 400K and re-heating to 600K.

During re-cooling of the O.5J.lIIl film, Fig. 3b, the dislocations that were nucleated during heating are already available to move and relax the stress, giving rise to a difference in initial plasticity compared to the first cooling. Subsequently, around 500K, relaxation becomes mainly nucleation controlled, and the stress level reaches and overtakes the level in the first cooling.

4. CONCLUSIONS

Two-dimensional discrete dislocation simulations of single crystal thin films under thermal cycling have been carried out. The results show a distinct size effect in the relaxation behaviour of films of thickness h = O.25J.lIIl and O.5J.lIIl.

• For both values of the film thickness, the high long-range back stress accumulated inside the films during the first cooling process induces early reversed plasticity during subsequent heating.

• When heating starts, de-stressing is elastic over a range that is almost equal to two times the initial elastic range, with an average yield stress of 40MPa in both films, predicting that hardening in thin films is essentially of kinematic character.

• Plastic relaxation during heating takes place mainly by the movement of dislocations already present in the films. In the O.5J.lIIl film, there is a small contribution to relaxation by nucleation of new dipoles at the end of the cooling process, when the mean stress becomes sufficiently compressive to activate the sources.

• Upon re-cooling, the dislocation density in the O.25J.lIIl film is so low, that the plastic behaviour during the first and second cooling cycles are very similar. However, the dislocations nucleated during heating in the thicker h = O.5J.lIIl film influence the hardening of that film during the second cooling.

104 L. Nicola, E. Van der Giessen and A. Needleman

s. ACKNOWLEDGEMENTS This research was carried out under project number MS97007 in the frame­

work of the Strategic Research Program of the Netherlands Institute for Metals Research in the Netherlands (www.nimr.nl). A.N. is pleased to acknowledge support from the Materials Research Science and Engineering Center on On Micro-and-Nano-Mechanics of Electronic and Structural Materials at Brown University (NSF Grant DMR-0079964).

References [1] O.S. Leung, A. Munkholm, S. Brennan, and W.D. Nix, "A search for strain gradients in

gold thin films on substrates using x-ray diffraction", 1. Appl. Phys., vol. 88, pp. 1389-1396, 2000.

[2] M. Hommel and O. Kraft, "Deformation behavior of thin copper film on deformable substrates", Acta Mater., vol. 49, pp. 3935-3947, 2002.

[3] R. Venkatraman and J.C. Bravman, "Separation of film thickness and grain boundary strengthening effects in AI thin films on Si", J. Mat. Res., vol. 7, pp. 2040-2048, 1992.

[4] N.A. Fleck and J.W. Hutchinson, "Strain gradient plasticity", Adv. Appl. Mech., vol. 33, pp.295-361,1997.

[5] N.A. Fleck and J.W. Hutchinson, "A reformulation of strain gradient plasticity", 1. Mech. Phys. Solids, vol. 49, 2245-2271, 2001.

[6] M.E. Gurtin, "A gradient theory of single-crystal viscoplasticity that accounts for geomet­rically necessary dislocations", 1. Mech. Phys. Solids, vol. 50, pp. 5-32, 2002.

[7] L.B. Freund, "The stability of dislocations threading a strained layer on a substrate", J. Appl. Mech., vol. 54, pp. 553-557,1987.

[8] W.D. Nix, "Yielding and strain hardening of thin metal films on substrates", Scripta Mater., vol. 39, 545-554, 1998.

[9] L. Nicola, E. Van der Giessen, and A. Needleman, "A discrete dislocation analysis of size effects in thin films", (submitted).

[10) E. Van der Giessen and A. Needleman, "Discrete dislocation plasticity: a simple planar model", A. Simul. Mater. Sci. Eng., vol. 3, pp. 689-735, 1995.

MEASUREMENT OF THE STRESS INTENSITY FACTOR, Kl, FOR COPPER BY A DIGITAL IMAGE CORRELATION METHOD

Siham M'Guil, Christophe Husson and Said Ahzi University Louis Pasteur - IMFS - UMR 75071CNRS-2 Rue Boussingault, 67000 Strasbourg, France.

Abstract A digital image correlation method is presented in this paper. This non-contact technique can be used in many applications to measure the displacement and strain fields. Like the speckle method, it uses the digital image correlation principle but it is much simpler to use. In fact, a set of pixels, called pattern, in an initial image is directly compared to the pixels of the final image. The accuracy of displacement measurement could reach 1/60th of a pixel. Treatment of two images with large strain (= 100 %) is possible. Very accurate cartography of strain field is obtained with this method. The covered field can range from few square millimetres to few square metres. The measurement of the stress intensity factor KI for thin sheets of copper is given as example in this paper.

Keywords: measurement of displacement and strain field, digital image correlation method, stress intensity facto

1 INTRODUCTION

The measurement of displacement fields with accuracy and speed of treatment is important for the characterization of the mechanical response of materials. Experimental methods such au the one based on photo-elasticity have been used to determine various fracture mechanics parameters. However, these methods have some limitations. For example, the experimental determination of the stress intensity factors in opaque materials is difficult with the photo-elasticity method. Irwin [1] first suggested the use

105

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106 Siham M'Gui/, Christophe Husson and Said Ahzi

of strain gages to determine the stress intensity factors near the tip of a crack. Measuring strain in materials with strains gages is practical but errors are introduced from the physical nature of the strain gage. The error is acceptable when the gage is placed on isotropic materials with relatively uniform stress distributions, but increases significantly when the gage is used to measure strains at discontinuities such as holes, notches and crack tips.

Different methods of strain gages technique have been proposed [2, 3, 4, 5] to measure the stress intensity factor, Kl. However, in the case of thin sheet metals (thickness < 0.5 mm) the local influence of strain gages is clearly shown [6, 7].

The presented method uses the technical of the direct digital images correlation. This method compares directly the pixels of an initial image with those of an image obtained after distortions of the specimen. An advantage of this technique is the abandonment of the classical marking. A completely random pattern, speckle type that is spied on the specimen with a simple bomb of paint is sufficient to obtain an optimal measure of a displacement field and then a distortions field. This non-contact technique has in the same time a very large strain range measurement and a good accuracy of the measurements. This method is used here for the determination of the stress intensity factor thin copper sheets.

2 PRINCIPLE OF THE CORRELATION ALGORITHM

The purpose of a correlation technique is to determine the displacement fields for plane surface between two states: an initial and a final one. Figure 1 shows an example of two images taken for a tensile test of an initially notched specimen with speckle aspect. The discrete function which represent the grey level is denoted by f(x,y), for the initial image and by fI'(x*,y*) for the final image.

Initial state: Final state : Grey level f(x,y) Grey level f'I'(x*,y*)

Figure 1. Tensile test for an initially notched specimen with speckle aspect.

The grey level for a pixel in the initial image f(x,y) becomes fI'(x*;y*) in the final image according to the following relation:

r" (x·, y") = r" (x +u(x, y), y + vex, y)) (1)

Measurement of the stress intensity factor Kl 107

u and v represent the displacement field for one pattern. The research of this last one is done by correlation between one pattern of the initial image and its correspondent in the deformed image. This operation is done for the entire pattern in order to obtain the total displacement field.

2.1 Basic elements for the correlation method

The four elements for direct correlation method are: a mathematical defmition of the displacement field for the pattern, a correlation criterion , an interpolation of the image in grey level to reach a precision sub pixel and a mathematical solution for the determination of the distortions terms and elongations for a pattern. Usually, the displacement field for a pattern is taken homogenous and bilinear in x and y, [8, 9] :

{U(X, y) = au x x + bu x y + cu x X x y + du

v(x,y) = av x x + bvxy+cvxx x y+dv (2)

Relation (2) contains the translation terms of a rigid body (du and dv); the elongation terms (au, av and bu, by) and the shearing terms (cu et cv). The displacement components u(x, y) and vex, y) are determined by a mathematical correlation of f(x, y) and (x·, y.). The most used correlation coefficients are the least square and crossed coefficient:

J f(x, y)f· (x· , y. )dxdy

C2=1--r===~~~==========~== J f(x,y)2 dxdy J r*(x·,y*)2dxdy

(3)

~ AM

C1 = I (f(x,y)-r*(x·,y·)fdxdy AM

(4)

These are minimized for the research of displacement field between a pattern of the initial image and its correspondent in the final image. For the grey level, the classic interpolation is the first order bilinear interpolation or the interpolation by splines bi-cubiques.

2.2 Correlation algorithm

A schematic of an initial pattern and its correspondent in the final image is shown in figure 2. To obtain the displacement filed for one pattern centred in P, four correlation calculations are done on 4 patterns centred in A, B, C and D as shown in figure 2, [8, 9].

108 Siham M'Guil, Christophe Husson and Sai'd Ahzi

0, 0* initial pattern I{x,y) ~ x,x

A V

f'(x·,y")

0 :--"Ic ,,----'-

" 'B v A'"') I-.

1"·-- --+-1 f"'p· If. ... c·

defunned pattern

0

y,y" IJ. M = surfiu:e .fthe pattern in the initial image

Figure 2. Evolution of a pattern from the initial to the final image.

We assume that their distortion field are identical to the one of the pattern cantered in P. The calculations will determine the rigid body displacement of the points A, a, C, and D which are used to obtain all the components of the distortion field of the pattern centred in P. The exact solution from numeric viewpoint will be detennined with the help of an iterative process: the rigid body displacement of the points A, a, C, and D will be searched during the iteration 'i' with the help of the components of the distortion field found at the previous iteration '(i - 1)'.

2.3 Determination of the distortion field

The following calculations are given for one pattern cantered in P (figure 2). The main distortion for a pattern and its main axes of distortions are calculated four times and averaged by considering the triangles ABC, BAC,

CaD and DAC. For the triangle ABD, the variables (Xl' Yl)' (X;, y;) and

(x2' Y2)' (x;, y;) are defined by:

(5)

The variables (x\'YI), (x: ,y:) and(x2' Y2)' (x;,y;) are obtained by the same manner for the remaining triangles by circular permutations.

The two dimensional deformation gradient F has for components:

XI 'Y2 -x2 'YI • • XI 'Y2 -X2 ,y\

(6)

XI 'Y2 -x 2 'YI

Measurement of the stress intensity factor Kl 109

The Green - Lagrange strain tensor is given by :

1 T [ ] [Ell E=-( FF-I); E = 2 E21

With these distortion components, the principal distortions are given by :

{EJ = In(Ell + E22 +~(Ell -E2S + (2E12 )2 +1)

En = In(Ell + E22 - ~(Ell - E22)2 + (2E,J2 + 1)

3 STRESS INTENSITY FACTOR Kl

(8)

The fracture mechanic theory allows the study of macroscopic cracks. The magnitude of the elastic stress field can be described by a single parameter Kl, the stress intensity factor. In this paper, only the mode I opening is considered (see figure 3).

It has been show clearly that the stresses associated with a single-ended 2D cracks can be represented by an asymptotic expansion of the corresponding stress function [9] (see figure 4).

y

Crack

Figure 3. Mode I - opening mode. Figure 4. Stress components near the crack tip.

The stress distribution near the crack tip can be approximated as :

110 Siham M'Guil, Christophe Husson and Said Ahzi

. 9 . 39 I-sm-sm-

2 2 . 9 . 39

I+sm-sm-2 2

. 9 39 sm-cos-

2 2

(9)

Here, rand 9 are the polar co-ordinates (figure 4). Plane stress assumption is employed for the measured strain and if we denote by E and v the Young's modulus and Poisson ratio, respectively, the elastic strain components are given by

(10)

3.1 Measure of Kl by two strain gage technique

Whei and Zhao [2] proposed a two-strain gage technique for determining mode I stress intensity factor. They show experimentally and numerically

that the terms with r-I/ 2 and rl/2 in the eigenfunction expansion of the strains can describe the crack tip strain distribution with sufficient accuracy. A set of two linear equations can be obtained to determine the stress intensity factor KI using only two strain-gages. For, e = rc/2, two strain

gages readings Eyyl and Eyy2 and the corresponding position rl and r2 to the

crack tip are needed. The authors gave the following relation:

( V-3) -112 A(9V-3) 112 E =a -- xr: +.., -- xr: yyl 2.J2E I 2.J2E I

( V-3) -112 (9v-3) 112 Cyy2 = a 2.J2E x r2 + ~ 2.J2E x Ii

(11)

The coefficient a is directly related to the stress intensity factor as follows:

(12)

Measurement of the stress intensity factor Kl III

3.2 Strain gage position

The precise locations of the two strain gages are important. They should not be to close to the crack tip where the plastic-zone prevails in addition to the high strain gradients. Also, they should not be too far from the crack tip because the lower strain level would reduce the measurement accuracy. It is suggested that the location of two strain gages, such as rl and r2 should

follow the inequalities:

(13)

where ry is the plastic zone radius and Llr is the difference between rl and r2 • These values will help us choose the position of the measured points.

4 EXPERIMENTAL MEASUREMENT OF Kl

The correlation technique requires a numeric camera connected to an image analyser. The images must have the most random possible aspect so that each pattern in the image is different to the other. The taking of views is done with a focal distance of 200 mm. The tests were performed on a electro-mechanical tensile machine Deltalab DN30 at room temperature on initially notched specimen. The specimen have the dimension L (length) = 200 mm, I (width) = 24 mm and e (thickness) = 0.6 mm. Several images have been taken for different loads. Following incremental maps these images show the deformation in front of the notch at different loading increments during the tensile test (figure 5) :

,%

Load = 7.29 kN Load =7.25 kN Load =6.86 kN Load =2.81 kN Figure 5. Evolution at the cracking and plastic zone.

It%

The longitudinal major strains versus the distance from the crack tip are plotted in figure 6.

Table 1. Experimental Kl for different loads. Load KN 7.02 7.19 7.29 7.33 7.25 6.89 2.81 Kl MPa.m1tl 8.71 11.27 34.65 62.75 85.35 237.07 161.44

112 Siham M'Gui/, Christophe Husson and Said Ahzi

The transversal strains Eyy\ and Eyy2 for two points A and B are

substituted in equation 11, then equation 12 is used to calculate Kl (Table 1 and Fig. 7).

10 Major strain (%)

8 --Load=281kN --Load = 7'25 kN

6 -6- Load = 7:29 kN

4

2 o~~~~~~~~~ o 100 200 Pixels 300

Figure 6. Measured longitudinal strain e xx at crack tip.

5 CONCLUSION

70 60 50 40 30 20 10

Load = 7,28 kN

0+--------------, o Position of the measured points 120

Figure 7. The variation ofKl at different measured points for the specimen.

This correlation technique is quite handy to use. The coloured pattern applied on the sample surface is easy to create. The optics is much simpler than that required by the speckle analysis technique. The same algorithm and experimental set-up may evaluate both small strains (5.10-5) and large strains. This technique can be extended to determine the stress intensity factor Kl for crack bodies with irregular geometry and complex loading. A Kl factor distribution on the specimen is going to be determined.

6 REFERENCES

[1] Irwin G.R., Analysis of stresses and strains near the end ofa crack traversing a plate. I. Appl. Mech. 1957; 24:361-364

[2] Wei I. & Zhao J.H., A two-strain-gage technique for determining mode I stress­intensity factor. Theor. and Appl. Fract. Mech. 1997; 28:135-140

[3] Parnas L., Bilir O.G. & Tezcan E., Strain gage methods for measurement of opening mode stress intensity factor. Engng. Fract. Mech. 1996; 55;3:485-492

[4] Kunag I.H. & Chen L.S., A single strain gage method for Kl measurement. Eng. Fract. Mech. 1995; 51;5:871-878

[5] Dally J.W. & Sanford R.I., Strain-gage methods for measuring the opening-mode stress intensity factor Kl. Exp Mech. 1987; 127:381-388

[6] Husson Ch., Bahlouli N., M'Guil S. & Ahzi S., Mechanical caracterisation of thin sheet metal by a digital image correlation method. J. Phys. IV France; 12:393-400

[7] Younis N.T. & Mize I., Discrete averaging effects of a strain gage at a crack tip. Engng. Fract. Mech. 1996; 55;1:147-153

[8] M'Guil S., Brunet M. & Morestin F., Comparison between experimental and theoretical Forming Limit Diagrams for aluminum sheets.Numiform'98;Pays-Bas, 1998

[9] Williams M.L., On the stress distribution at the base of a stationary crack. J. Appl. Mech. 1957; 24:109-114

HOMOGENEIZATION OF VISCOPLASTIC MATERIALS

Alain Molinari and Sebastien Mercier

Laboratoire de Physique et Mecanique des Matenaux UMR CNRS 7554, ISGMP Universire de Metz, lIe du Sauley, 57045 Metz, FRANCE

Abstract The approximate solution of the non-linear inclusion problem, Molinari, Canova, Ahzi (1987), Molinari (1997) is used to define various averaging schemes for viscoplastic heterogeneous materials, among which the tangent self-consistent model and the non-linear Mori-Tanaka model.

Keywords: Non-linear inclusion, viscoplasticity, averaging, self-consistent tangent model, Mori-Tanaka model.

1. INTRODUCTION

The Eshelby's solution (1957) of the inclusion problem for linear elastic materials was published in one of the most famous paper in mechanics. This work had a tremendous impact in micro-mechanics since it provided a way to investigate (through an appropriate averaging scheme) the strain distribution in an heterogeneous elastic body and finally to estimate the overall properties of the material. The structure of the solution of the Eshelby's linear inclusion problem is preserved in the tangent approach of the corresponding non-linear problem proposed by Molinari et al (1987), see also Molinari (1997). This approximate solution has been compared to a detailed Finite Element (FE) analysis, Molinari et al (2000), Molinari et al (2002). Calculations have been conducted for various values of:

-strain rate sensitivities of inclusion and matrix, -contrast between flow stress resistances of inclusion and matrix, -inclusion aspect ratio.

Different loading paths have also been considered. In all cases, a fairly good agreement has been obtained between the analytical solution (tangent model) and Finite Element calculations as far as the prediction of the average strain rate within

113

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 113-124. ©2004 Kluwer Academic Publishers.

114 Alain Molinari and Sebastien Mercier

the inclusion is concerned. FE calculations were those of Gilormini and Germain (1987), Gilormini and Michel (1998), Molinari et al (2000), Molinari et at (2002 ).

An essential aspect of the non-linear inclusion problem is the fact that the strain distribution within the inclusion is non-uniform. The tangent model does not provide any information on strain heterogeneity. But information on the strain distribution is provided by FE calculations showing that the strain heterogeneity increases in the inclusion for small values of the strain rate sensitivity and for a large contrast between the flow stress resistances. From these results, a large range of material properties is found for which the strain is almost uniform within the inclusion. In these cases the tangent model provides a good approximate solution by analytical means of the non-linear Eshelby problem.

For linear elastic materials, knowing the average strain per phase is a sufficient information for calculating the effective elastic moduli. It is not so for non-linear material behaviors. This is due to the fact that phase moduli are now strain dependent for a non-linear elastic material and strain-rate dependent for a viscopiastic material. If strain is non-uniform within a phase, so are the phase­moduli for a non-linear elastic material. Intra-phase moduli heterogeneities are difficult to account for in averaging schemes. Recent efforts have been devoted to the evaluation of per-phase second order moments of strains. Approaches are based on linearization of the local material response and use of energetic methods developed by Kreher (1990) in linear elasticity and thermo-elasticity. Application of these techniques to materials with non-linear response led to some improvements on the estimate of the overall response, but the theoretical background remains to be clarified. Bounding techniques provide an exception; new bounds have been derived by Castaneda (1992) within a rigorous mathematical framework. Improved upper bounds have been obtained for rigid viscoplastic materials but no new information is provided on lower bounds. In addition these bounding methods cannot be presently used for elastic-viscoplastic materials.

As for most of the averaging techniques used for non-linear materials, we shall not account here for the per-phase strain heterogeneity. We shall elaborate different classes of averaging schemes based on the approach of the non-linear Eshelby problem proposed by Molinari et at (1987). Rigid viscoplastic materials are considered.

Notation: Vectors and second order tensors are underlined by a single bar, fourth order

tensors are underlined by a double bar: ;i. Cartesian notations are used and the convention of summation on the repeated indices is

adopted. The double contracted products between second order tensors or between fourth and second order tensors are, respectively, defined as:

g:lJ. = aijbji (;i:g)ij = Aijklalk·

Homogeneization of visco plastic materials 115

The fourth-order unit tensor K operating on symmetric traceless second-order tensors is

defmed by

(1)

2. THE NON-LINEAR ESHELBY INCLUSION PROBLEM

A non-linear viscous material is considered. Developments would be similar for non-linear elasticity. The deviatoric Cauchy stress tensor :s. is assumed to be related to the strain rate tensor d by differentiating the potential g :

s=o g @ (2) - 0 d

Incompressibility is assumed: tr(4J = O.

An ellipsoidal inclusion I is embedded in an infinite matrix. The material

properties in the inclusion are defined by the stress potential gl. The material is

assumed uniform in I; that means that the potential g I is the same for each

material particle in I. The stress potential in the matrix, denoted by g O , is also

uniform in the matrix. The overall strain rate D is applied at infinity, with the constraint tr(D) = 0 . The problem is to estimate the strain rate within the inclusion

I in terms of D. The approach of this problem by Molinari et al (1987), Molinari (1997) is

briefly presented. In the Figure 1, a schematic representation of the non-linear response of the matrix is shown. The idea is to approximate this non-linear response by a first order Taylor development at D (overall strain rate). Therefore

the non-linear law :s. = (8g 0 /84)(4) is approximated in the whole matrix by the

linearized law :

4. ~ §. = ;t(Q):4. +§.o(D) (3)

82 ° where A,g (D) = --g-(D), is the viscoplastic tangent modulus of the matrix

= - 84. 84. -8 0

defined at ll, and §.\Q) = :a (!2) - 4'g (D) : Q. Note that at D the linearized

law restitutes exactly the value of the stress at infinity s..

116

&. = ~,. (lL) : d.. + &.0 (lL)

First order

Alain Molinari and Sebastien Mercier

&.=£L(d..) ild..

Matrix non-

Figure 1. Linearization of the matrix response

To the original non-linear inclusion problem (Pb 1), which has no general solution, we have substituted (pb m, where the matrix non-linear response is replaced by the linearized law (3). It turns out that (pb m has an exact solution which can be obtained with use of the Green function technique. This solution (referred to as the tangent model) can be presented in the form of the following interaction law:

l-~=(~t(Q)-(~lg(D)r): (AI -D) (4)

where d/ and i are the strain rate and the deviatoric stress in the inclusion I.

The tensor P Ig • d fi d b nlg - 1 IT,g Ttg T tg T tg ) 'th IS e me y: rijkJ --4"\ ijkJ + jikJ + ijlk + jill' Wl

T:t = II G:t,jl (;I - .i)d.i. The Green functions G:: are associated to the infmite

space with moduli ~tg •

The value of the strain rate d..I in the inclusion is solution ofEq. (4) where I2 is given and where the stresses are related to the strain rates by the laws:

8 I 0 / = l(d I ) and S = ag (ll). When a linear material response is considered, - 84 - - ad.. the interaction law restitutes the classical result obtained by Eshelby (1957).

Comparison with finite element calculations, Molinari et al (2000-2002), have shown that the tangent model restitutes with a good precision the value of the average strain rate in the inclusion. But, as said before, the strain rate distribution within I cannot be described in this framework. Nevertheless, there is a large range of strain rate sensitivities and of material property contrasts for which the strain rate is nearly homogeneous within I .

It is worth to compare the tangent model with other well known approaches. In the secant model, the matrix response is linearized with use of the secant modulus (which is actually not uniquely dermed) so as:

4~! = 4'@:4 with ~ = ~s(Q) :D (5) The interaction law predicted by this model is stiffer than those obtained for the tangent model, Molinari and Toth (1994).

Homogeneization of visco plastic materials 117

The incremental model was also a frequently used approach. This model is based on an incremental linearization of the material response. Considering the time derivation of (2), a linear relationship is obtained between the stress increment and the strain rate increment:

with a/g (d) - 8 2 g ( d) = - 84 84 \!!.

(6)

At each time increment, a linear Eshelby inclusion problem is solved, by assuming that the matrix has uniform incremental tangent moduli. This approximation leads to a cumulated error which increases with time integration. Indeed, the results of the incremental model are shown to be identical to the secant formulation in certain cases, see Molinari (1997). Therefore this model is also too stiff. This is the reason why averaging schemes based on an incremental formulation, Hill (1965), Hutchinson (1976) are overestimating the overall material response, see Molinari et al (1997). Averaging schemes based on the tangent model (which softens the inclusion-matrix interaction law) are aimed at providing softer stress­strain responses than incremental schemes.

3. GENERAL LOCALIZATION LAWS

For the purpose of predicting crystallographic texture evolution in polycrystalline materials it is worth to express the inclusion problem in a more general form involving rotation rates. Let us consider an ellipsoidal inclusion

Ie embedded in an infinite matrix with affme response: 4 ~ ~ = 4° : 4 + §..O . For

a given velocity gradient L· applied at the remote boundaries of the matrix, the

velocity gradient ZC within the inclusion is found to be, Molinari et al (1987), Molinari (1999):

r =L.0 +t :(l-;t :4c -t) (7)

with T,,~j = t, G~i.jk (!! -!!' ) dx' , 02i being the Green functions associated to the

infinite medium with uniform modulus ;t. Taking the symmetric and anti­

symmetric parts of (7), it follows that:

4c = DO -t :(l-l :4c _§..O) (8)

(9)

118 Alain Molinari and Sebastien Mercier

with and

(2. = ~ (L· - Lor). The tensor t: is defmed in terms of the Green functions a2i

and £.0 is given by: B8kl = ~ (rJkl - rj~kl + rJIk - rj~lk). From the localization law (8), the following interaction law is obtained which is similar to (4):

l-t =(l-t-1): (4c -Do)

• with s.. being the stress at the remote boundary of the matrix given by:

t =l:Do +~o.

4. AVERAGING

The material is assumed to be constituted by N phases with uniform material properties. This material is subjected at remote boundaries to the overall velocity gradient L . Averaging schemes are defined by suitable choices of the

homogeneous reference medium with affine response defmed by (Ao and s..0 being uniform):

4.. ~ §.=l :4..+~o (10)

Each phase (c) is assumed to be represented by an ellipsoidal inclusion

Ie embedded in the reference medium to which is applied the velocity gradient L·

at infinity. For each phase (c) the intra-phase velocity gradient I C is obtained from

the solution of the inclusion problem (I C embedded in the reference medium).

Finally the velocity gradient L· is determined so as to satisfy the consistency condition:

(11)

where (.) represents volume averaging on the whole aggregate.

From (11) we have (d c) = D. and (0/) = (2 , and from (8) and (9) the following

results are obtained:

(12)

(13)

Homogeneization of visco plastic materials 119

As said before the choice of the homogeneous reference medium determines the type of averaging model. This choice has to be made so as to account for the internal structure of the heterogeneous material under consideration, as discussed in following sections.

4.1 Self consistent tangent model

When each phase element is surrounded by elements of all other phases in a rather disordered way (as for instance in most of polycrystalline aggregates), self­consistent modelling is an adequate averaging scheme. In such model, the reference medium is supposed to have the effective properties of the macroscopic aggregate. In other words, the surrounding of any phase element is replaced by an infinite homogeneous matrix having the 'averaged material properties' of the aggregate. Frequently this reference medium is referred to as the Homogeneous Equivalent Medium (HEM).

A class of self-consistent model is defmed here based on the characterization of the macroscopic tangent modulus from the self-consistent incremental model of Hutchinson (1976). Another class of self-consistent models can be also developed for materials with a positive homogeneous potential, see Canova et al (1992) and Lebensohn and Tome (1993), see also Molinari (1999).

The non-linear inclusion problem is approached by using a tangent linearization

of the HEM, of the form (10) with l being replaced by the effective tangent

modulus !t (D) of the aggregate ( i.e. of the HEM) evaluated for the value D. of

the macroscopic strain rate. s.0 is defmed as:

s..0(Q) = s.. -lg(D) :D (14)

where S. is the macroscopic deviatoric stress in the aggregate generated by D.. Considering that (s.) = s., the expressions (12) and (13) reduce to:

• ( I -1 )-1 ( I -1 ) Q = §,.g : §,.g :@ (15)

When f,g and t g are phase independent, a simple result is obtained:

D. 0 =D. 0° =0 (16) To complete the set of equations, we have to evaluate the effective tangent

modulus 19. This evaluation can be provided by the self consistent incremental

approach of Hutchinson (1976), from which the following relationship can be obtained, see Appendix A:

4'g =(lg: [K +f/g: ~,g -4,g)r}( [K +E.'g: ~,g -4,g)rr (17)

120 Alain Molinari and Sebastien Mercier

where g,g (4) = (ii g / 84.f)4)(4) is the local tangent modulus and K is the unit

identity tensor defined by (1). In the thermoelastic analogy used by Masson and

Zaoui (1999) to elaborate the 'affine approach' the expression (17) for A,g is

obtained, which indeed can be obtained directly from the Hutchinson's incremental approach.

By combining the localization law (8) with (14), results can be presented in terms of the following interaction law between the phase (c) and the HEM:

19 :(4C _D)_!g-l :(4c _Do)=l_§. (18)

with A'g and f:: ,g being evaluated at D. The intra-phase stress is given by

s..C = agC / ad(dC ), n° is given by (15). Thus by considering the consistency N

condition S. = L Ic:i C where Ic is the volume fraction of phase (c), we end up c=1

with a system on N (number of phases) non-linear equations for the unknowns d C

( c = 1, ... N). This system can be efficiently solved with use of a Newton-Raphson method.

For rigid viscoplastic materials, it is shown in Appendix B that the self­consistent tangent model based on the approximate solution (18) of the non-linear inclusion problem, contains as a particular case the affine approach of Masson and Zaoui (1999), see Appendix B for further comments.

Note that the flexibility of the present modeling relies on the possibility left to evaluate the macroscopic tangent modulus by other means than (17) (as for instance in Canova et al (1992) and Lebensohn and Tome (1993», thereby defining other classes of self-consistent models.

4.2 Non-linear Mori-Tanaka models

We assume here that the material is constituted by a matrix (phase (1» in which ellipsoidal inclusions of phases (2) .... (N) are embedded. The reference material is now chosen as having the mechanical properties of phase (1). Each phase (c) (c=1

. .. N) is represented by an ellipsoid r embedded in the reference material

(matrix). The macroscopic velocity gradient ~o is applied at infmity on the

reference medium. The non-linear response of the reference material is approximated by the following affine law:

4. ~~=l :4.+t (19)

with AO being the tangent modulus of the matrix material evaluated for the value

D· of the strain rate:

Homogeneization of visco plastic materials

02 (I)

AO =-g-fDo) = oDOD~

121

(20)

(21)

DO is related to the macroscopic strain rate D applied to the composite material

by the relationship (12):

D" = (t-I(Do)f :(t-l (!i):4)-(t- l (!i)f : (l(Do): D+~o(!i)-(§.(4»))

(22)

The strain rates in phases c=l...N are given as solutions of the set of non-linear equations (8):

o (c)

4c =Do -t(Q°): ( !4 (4C)-l(Do) :4c _§.0(Q0» (23)

Note that 4c = DO for c=1.

122 Alain Molinari and Sebastien Mercier

Appendix A: Incremental self-consistent approach

An incremental self-consistent model for viscoplastic materials with local constitutive law of the type (2) has been formulated by Hutchinson (1976). Considering an increment 0 D of the macroscopic strain rate, the resulting increments of local strain rate, deviatoric stress and macroscopic stress, respectively denoted by 04 , O§. and o§.., are related by the following relationships:

0§.=glg(4):04,0§..=lg(D):oD (AI)

with the local tangent modulus being defined in (6). To evaluate the local strain rates, we proceed as usual by considering that an

element of phase (c) is represented by an ellipsoidal inclusion Ie embedded in the

HEM with linear response oD ~ O§.. = 19 (D): oD. The strain rate oDo is

applied at the remote boundaries of the HEM. The strain rate in r is given by the solution of the linear inclusion problem by Eshelby (1957):

04c = S : oDo with ~c = [ K + t:"g : (gtg(C) -lg) r (A2)

o DO can be evaluated in terms of 0 D from the consistency condition

(84)=8D:

8Do=(~t:oD (A3)

Finally, the following localization law is obtained:

04c = ~c : ((}.r : oD (A4)

Combining this result with (AI) and considering the consistency condition (8~) = 8§.., the relationship (17) follows:

4,g = (g,g :~): (~t (AS)

Two relationships, which will be useful, are now proved.

8D· =(e-1f : (t:,Ig-1 :04) (A6)

(t::g-I)=(~-I :~) or (t::g-I)=(~T :!!.g-I) (A7)

where (.)' denotes the transpose of operator (.). From (A2) we have

o D· = 04c + {!g : (lg(C) -dig) : 04c which upon left multiplication by pg-I and space averaging leads to (A6). Substitution of (A2) into (A6) provides (A 7).

The second result in (A 7) comes from the fact that (~tg r = eg •

Homogeneization of visco plastic materials 123

Appendix B : Link between the self-consistent model of section 4.1 and the self consistent affine model

Note that linearization schemes can be also based on the local tangent linearization proposed by Molinari et al (1987). This local tangent linearization provides a thermoelastic analogy, Castaneda (1996). Two general formulations of the locally linearized problem can be found in the literature. The first is presented in terms of an integral equation (LSD equation), Molinari et al (1987), Molinari (1999). The other, based on the thermoelastic analogy, was formulated by Masson and Zaoui (1999), and called the 'affine model'. In the latter model, localization tensors have to be eventually estimated per phase by referring to the linearized Eshelby inclusion problem. Thus it is shown here that the 'affine model' enters into the framework of the tangent approach based on the solution of the non-linear Eshelby problem, providing that the estimation of the macroscopic tangent modulus is given by the incremental formulation of Hutchinson (1976). The affine model is based on the following linearization of the local non-linear response, Molinari et al (1987):

4' ~ ~tg«(fJ:4' +§:o(4) (Bl)

where 4 is the local strain rate field resulting from the application of Q, ~tg (4)

is the local tangent modulus defined by (6) and

l (4:J = ~~ (4:,,) - ~/g «(fJ: 4 = §: - ~tg (4:,,): 4· Using this expression together with

(14) to eliminate §. -!i in (18), the local strain rate can be given as:

4c =!}.c :D* _!}.c :!:,'g :(l-t) (B2)

-I with A C defined in (A2). After left multiplication by e.tg ,(B2) can be written

as:

f g -I :4c =(~ct :fg -

I :D* -{st :(§:o -t) (B3)

Considering volume average of (B3) and using results (A6) and (A 7) it foIIows that:

(B4)

This is precisely the relationship relating the micro-field §.D to the macroscopic

quantity tiD that is used in the affine formulation. Therefore it can be concluded that for rigid viscoplasticity, the affine formulation belongs to the class of averaging models based on the approximate solution of the non-linear inclusion problem, Molinari et al (1987), Molinari (1997). Results obtained with the self­consistent tangent model, Molinari (1999), defined in section 4.1, are identical to those of the affine model.

124 Alain Molinari and Sebastien Mercier

REFERENCES

Canova, G.R., Wenk, H.R. and A. Molinari: Defonnation Modelling of Multiphase Polycrystals : case of a quartz-mica Aggregate, Acta Met., (1992), 1519-1530. Castaneda P.P., New variational-principles in plasticity and their implication to composite materials, J. Mech. Phys. Solids, 40 (1992),1757-1788. Eshelby, J.D. : The detennination of the elastic field of an ellipsoidal inclusion,and related problems, Proc. Roy. Soc. London, A241 (1957),376. Gilonnini, P., and Y. Gennain: Int. J. Solids Struct., 23 (1987), 413. Gilonnini P, and Michel JC, Finite element solution of a spherical inhomogeneity in an infinite power-law viscous matrix, Eur. 1. Mech. A-Solids, 17, (1998), 725-740. Hill, R. : Continuum micro-mechanics ofe1astoplastic polycrystals, J. Mech. Phys. Solids, 13 (1965), 89. Hutchinson, J.W. : Bounds and Self-Consistent Estimate for Creep of Poly crystalline Materials, Proc. Roy. Soc., A348 (1976),101. Kreher W.: residual stresses and stored elastic energy of composites and polycrystals, 1. Mech. Phys. Solids, 38 (1990),115-128. Lebensohn, R.A. and C.N. Tome: A self-consistent anisotropic approach for the simulation of plastic defonnation and texture development of polycrystals : Application to zirconium alloys, Acta Metall. et Mater., 41 (1993),2611-2624. Masson R, Zaoui A : Self-consistent estimates for the rate-dependent elastoplastic behaviour of polycrystalline materials, 1. Mech. Phys. Solids, 47 (1999),1543-1568 Molinari, A., G.R. Canova and S. Ahzi : A self-consistent approach of the large defonnation polycrystal viscoplasticity , Acta Metall., 35 (1987),2983-2994. Molinari, A., and L. Toth : Tuning a self-consistent viscoplastic model by finite element results I : Modeling, Acta Metall. Mater., 42 (1994), 2453-2458. Molinari, A. : Self consistent modelling of plastic and viscoplastic polycrystalline materials, CISM lecture notes, (Ed. C. Teodosiu), Springer Verlag (1997), ppI73-246. Molinari A.: Extensions of the self-consistent tangent model, Modelling and Simulation in Materials Science and Engineering, 7, (1999), 683-697. Molinari, A., F. EI-Houdaigui and 1.S. Toth: Comparison of the tangent model predictions to finite element results for the solution of the inclusion problem in viscoplasticity , ZAMM, Z. Angew. Math. Mech., 80 (2000),21-24. Molinari A.: Averaging models for heterogeneous viscoplastic and elastic visco-plastic materials, ASME J. of Engineering Materials and Technology, 124, (2002), 62-70.

COLLECTIVE DISLOCATION BEHAVIOR IN SINGLE CRYSTALLINE ALUMINUM UNDER INDENTATION

Yoji Shibutani, Atushiro Koyama and Tomohito Tsuru

Dept. of Mechanical Eng. and Systems,

Frontier Research Center, Osaka Univ.

2-1, Yamadaoka, Suita, Osaka 565-0871, JAPAN

1. INTRODUCTION Recent mesoscale experiments resulting in scale-dependency on the

mechanical deformation have yielded the strain-gradient plasticity [1] and furthermore motivated the linkage between the discrete defects dynamics methodology and the continuous plasticity studies [2]. Especially,

nanoindentation has been recognized as the most appropriate material testing to quantify the characteristic length [3]. Taking advantage of the controllable

j.l.N-level indent load and the nanometer-level displacement resolution, it can accurately monitor the mechanical response of the extremely localized stress and strain field. The reason of increase of microhardness observed in the ductile materials has been thought to be due to collective dislocation behavior extending under the indentation [4]. In fact, the density of the geometrically-necessary dislocation (GN dislocation) emitted from the surface is related to the strain gradient by compatibility requirements [5] and one can easily imagine the high density region of dislocation just beneath the indentation [6]. However, no one still refers the physical process of how the aggregate of dislocations dynamically evolves under the nonuniform stress

distribution and leads to the scale-dependent hardening mechanism which may, in principle, be based on the mobility of the dislocations.

The present paper is focused on the dislocations emission and their

interaction under the nanoindentation in the single crystalline aluminum by

the experiment and the molecular dynamics simulations. The early stage

125

S. Ahzi et al. (eds.). Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 125-132. ©2004 Kluwer Academic Publishers.

126 Yoji Shibutani, Atushiro Koyama and Tomohito Tsuru

when the first dislocations burst at the threshold of the operating shear stress

and interact among them in the vicinity of the indented surface is carefully

investigated. 2. NANOINDENTATION

A single crystalline aluminum with (100) crystallographic plane is

supplied for the indentation test using a commercial NanoIndenter

XP/DCM™ (MTS Corporation). The surface of the sample with size of

1 Omm X 10mm X 2mm is sufficiently electropolished and the final surface

roughness Rmax measured by the atomic force microscopy is almost 10nm. Relations between indent load and maximum depth of the indenter tip are

obtained from the results of the load-controlled experiments at RT, as shown

in Figurel. At the minimum indent load of lOflN which is, in fact, overshot

to about 11 ".IN, almost reversible hysteresis curve is depicted by shifting the unexpected creep-like slide at the maximum load. It should be interpreted as

a pure elastic deformation. One can see the irreversible irregularity [7] in the

loading curves at the more than 20flN. The first discontinuity observed in the

Figure l(b) is the threshold of the homogeneous dislocation nucleation [8]

50

Depth h I nm I

(b) P=50",

100 150 200

Depth h (nm)

Figure 1. Relations between indent load and depth of indenter tip, ranging from 10",N

to 500",N.

250

Collective dislocation behavior in single crystalline aluminum under 127 indentation.

under the nanometer-scale indentation. The snap-through type bifurcation to

the following loading stage is likely to be caused by the instability of the

deformation-induced internal structure due to the burst of the dislocations. It looks like the snap-through buckling obtained in the load-controlled

compression test of the spring washer. The first flat portion seems to be the

largest among the followings even though considering scatter of the samples.

Moreover, the discontinuities become smooth as the load. The width of the

plateau is physically related to the mobility of the dislocation. Thus, these

results suggest that the deformation at the early stage is governed mainly by the homogenous nucleation of the dislocations in the defect-free field and

up-sizing effect intends to promote the interaction with the pre-existing

dislocations.

Fitting the elastic response in Figure lea) to the analytic elastic solution

of the semi-infinite isotropic material indented by the sphere indenter [9], the

radius of the assumed round indenter tip might be roughly 50nm. The

maximum shear stress distribution at the first dislocation burst is shown in

Figure 2 and the critical shear stress necessary for the dislocation nucleation

can be estimated to be 4.8GPa. It roughly corresponds to one-fourth of the

shear modulus, J.1l4, where J1. is the shear modulus of about 20GPa in this

case. According to the estimation based on the free energy balance consisting

of the line strain energy of dislocation, the external work and the formation

energy of stacking fault [8], it can be reduced to about one-twentieth of the

shear modulus, J.1l20, which value is likely to be much more acceptable.

1 5

4 N

3

2

40

501 ~ - 0 0 10 20 30 40 [GPa]

r [nm]

Figure 2. Maximum shearing stress obtained by the Sneddon's elastic solution [9].

128 Yoji Shibutani, Atushiro Koyama and Tomohito Tsuru

3. MOLECULAR DYNAMICS SIMULATIONS 3.1 Dislocation Emission

The molecular dynamics simulations with atoms more than one million are performed for this indentation problem. Two types of the indenter, the

sphere punch and the Berkovich type, are atomistically modeled as the rigid carbon-based structure. The interaction between the indenter and the

aluminum substrate is determined by the simple Morse type potential. The aluminum substrate is constructed by the Finnis and Sinclair type

many-body potential, by which the stacking fault energy is much

u:t:lderestimated to the experimental data.

Figures 3 and 4 indicate the total potential energy per atom around the first dislocation burst and the indent load as the reacted force from the Al

substrate to the rigid indenter, respectively. Since the initial relaxation of Al atomic model before loading has not been fully taken here, you see the first

unexpected decrease of the potential energy curve even during loading. The slight change at 7ps of the curves by the spherical punch corresponds to the small flat portion in the indent load curves, which implies the instability like Figure l(b).

Internal structural changes at this unstable stage are visualized in Figures 5. Only the atoms with higher energy than -3.22eV are drawn in order to figure out the defect nucleation.

E -3.16 [ 6

£ Z co ::t :> ~ 0..

-3.18 Radius of spherical punc . -0 4 . " ~ r=50nm .2

c: c:: .,

III r=30nm -0

"iii - 3.20 1 .5 2 "::l

Berkovich c: '" ~ 0 "'-

- 3.22 - .. - 0 ---0 5 10 0 5 10

Time t Ipsl Time llpsl

Figure 3. Time evolution of potential energy per atom. Figure 4. Time evolution of indent load.

Collective dislocation behavior in single crystalline aluminum under 129 indentation.

/ /

/

/

/ /

(a) By Berkovich punch (b) By spherical punch

Figure 5. Dislocation bursts under the indentation by two kinds of punch.

3.2 Interaction Between Emitted Dislocations To understand crystallographic events being happened by intersection

between dislocations emitted from the surface, we take two dissociated partial dislocations which are gliding on (11 I) and (1 11) planes, respectively, as shown in Figure 6. One can see a leading partial dislocation

POOl

I)

Figure 6. A stair-rod dislocation at the line of intersection 0 f the two glide planes of

(III) and (II!).

130 Yoji Shibutani, Atushiro Koyama and Tomohito Tsuru

(LPD) followed by the trailing partial one (TPD). The stacking fault region

between the two partial dislocations seems to be large due to the low

estimated stacking fault energy. Some immobilization mechanism by the reaction between the LPDs with different glide planes have been proposed

and the typicallow-fault-energy barrier is the Lomer-Cottrell barrier which is thought to be related to the macroscopic hardening. All of the possible

reactions of the stair-rod dislocations, which are defined as the interaction between the different two glide planes [10], are summarized in Table 1.

Table 1. Reactions between possible LPDs. ii== a/6

a[121] a [112] Ii [2 I I] Ii [In) Ii [In) a(211]

a [21 I) x 4 3 4 X X

a [I 12] I.- 'x ,";'" .);'" X 4 1 2 X

, . C' ~

il[121] X }:,

X X 2 1 4

a[21 1] 4 x X X 4 3

a [I 12] 1 2 X X X 4

a [1211 2 1 4 X X X

The numbers of 1, 2, 3 and 4 in Table 1 should be stable, being based on the

b2criterion. They are called the Lomer-Cottrell barriers. Using the Thompson notation [10], these reactions are rewritten in Table 2 and the smaller number

is energetically favorable.

Table 2, Thompson notation and classification of the Lomer-Cottrell barrier in Table 1.

Number Stair-rod: Thompson's natation: bJ2(xa 2/36) b;+b;

in Tablel bJ b2+b3=b1 (xa 2 /36)

1 a (110) oB+By ==oy 2 12

2 ii (200) oB+yA==oy/BA 4 12

3 a (220) oC+Dy==oD/Cy 8 12

4 a (310) OB+yD==oy/BD 10 12

Collective dislocation behavior in single crystalline aluminum under 131 indentation.

Consider the possible reactions of the stair-rod dislocation obtained by the MD simulations in Table 1. First, in order to determine the Burgers vector of the

partial dislocation, the mobility of the dislocation

core at which another glide plane is crossing to the plane with the present partial dislocation has been traced (see Figures 7 and 8).

It is found in the

case that both

dislocations

present

perfect before

dissociation are gliding in

r

,~:: (010)

·3.226 leV]

·3.24S leVI

Figure 7. Dislocation core crossing two gliding planes.

the [011] direction and then the possible reactions seem likely to be the

combinations grayed in Tablel. They are all energetically unfavorable, that is, unstable than the two LPD reactants. One can easily imagine the subsequent collapse of the stair-rod dislocations after coming the TRDs. It is, therefore, expected that the emitted dislocations move far into the bulk without any pile-up due to their immobilization in the vicinity of the indented surface. Further discussion using the larger scale modeling is

crucial when considering the intersection between the perfect dislocations gliding into the deeper bulk. 4. CONCLUDING REMARKS

Collective dislocation behavior observed in the nonuniform stress distribution under the indentation of the single crystalline aluminum is investigated using the nanoindentation tests and the molecular dynamics simulations. The continuous snap-through type instability is acknowledged

from the relations between the indent load and the maximum indent depth

and it could be conjugated with the homogeneous dislocation nucleation.

The maximum shear stress of the dislocation emission can be estimated to be roughly one-twentieth of the shear modulus. The large-scaled molecular

dynamics simulations being focused on the intersection of the emitted partial

dislocations give a fact that the high-energy-fault stair-rod dislocations are

132 Yoji Shibutani, Atushiro Koyama and Tomohito Tsuru

Figure 8. Detennination of the Burgers vector from the moving direction of the dislocation core.

nucleated and disappeared by the trailing partial dislocations. Therefore, the

emitted dislocations can expand into the farther bulk without immobilization

in the vicinity of the indented surface.

ACKNOWLEDGEMENTS This work was supported by the JSPS Grands-in-Aid for Scientific

Research «B)(2) 13450047).

REFERENCES [1] N.A.Fleck, GM.Muller, M.F.Ashby and J.W.Hutchinson, Acta metall. mater, 42 (1994),

475.

[2] E.van der Giessen and A.Needleman, Modelling Simul. mater. Sci. Eng., 3 (1995), 689.

[3] H.Gao, Y.huang, W.D.Nix and J.W.Hutchinson, J. Mech. Phys. Solids, 47 (1999),1239.

[4] Q.Ma and D.R.Clarke, J. Mater. Res., 10 (1995), 399.

[5] M.F.Ashby, Phil. Mag., 21 (1970), 399.

[6] M.C.Fivel, C.F.Robertson, GR.Canova and L.Boulanger, Acta mater., 46 (1998), 6183.

[7] S.suresh, T.-GNieh and B.w.Choi, Scripta Materialia, 41 (1999), 951.

[8] T.A.Michalske and J.E.Houston, Acta mater., 46 (1998), 391.

[9] I.N.Sneddon, Int. J. Engng. Sci., 3 (1965), 47.

[10] J.P.Hirth and J .Lothe, Theory of Dislocations, John Wiley and Sons, New York, 1982.

MULTISCALE MODELLING OF TEXTURE GRADIENT EFFECTS ON LOCALIZATION IN FCC POLYCRYSTALS

Kenneth W. Nealel), Kaan Inall ), and Pei-Dong Wu2)

I) Faculty of Engineering, University of Sherbrooke Sherbrooke, Quebec. Canada, JIK 2RI E-mail: Kenneth.Neale@Usherbrooke.ca 1) Alcan International Limited, Kingston R&D Centre Kingston, Ontario, Canada, K7L 5L9

Abstract: The effects of through-thickness texture gradients on instabilities and localized deformation in FCC polycrystals are investigated. In-house finite element analyses based on a rate-dependent crystal plasticity model have been developed to simulate large strain behaviour for sheet specimens subjected to plane strain tension. Modelling of the polycrystalline aggregates is carried out at various scales, and predictions of localized deformation are compared against each other.

Key words: multiscale modelling, crystal plasticity, texture gradients, instabilities, localized deformation

1. INTRODUCTION

The mechanical properties of a polycrystalline metal depend on many attributes of its microstructure; consequently, considerable efforts have been devoted to the modelling of micromechanical phenomena. These studies indicate that, among the factors which result in the plastic deformation of single crystals and polycrystals, crystallographic slip occurring by the migration across the slip planes of atomic defects, termed dislocations, is the dominant one. Crystallographic slip induces lattice rotations, which result in a non-random distribution of the crystal orientations in polycrystals.

133

s. Ahzi et al. (eds.), Multiscale Modeling and Characterization o/Elastic-Inelastic Behavior o/Engineering Materials, 133-140. ©2004 Kluwer Academic Publishers.

134 Kenneth W. Neale, Kaan Inal, and Pei-Dong Wu

Textures are macroscopic averages of such non-random orientations. These textures have profound effects on the mechanical and the thennal properties of metals.

To model processes such as texture evolution and its influence on defonnation-induced anisotropy, models based on crystal plasticity have been employed in numerous studies [e.g. 1-4]. In these studies it has been assumed that the textures employed are representative of the entire volume. However, most forming processes do not produce materials with unifonn spatial distributions of texture. Indeed, without sufficient care in the fonning process, significant texture gradients can develop (e.g., surface-to-mid-plane texture gradients in rolled materials, surface-to-core gradients in drawn wires). Thus, from a theoretical or practical point of view, it is important to investigate these texture gradient effects on plastic defonnation properties.

In this paper, the effects of through-thickness texture gradients on instabilities and localized defonnation have been investigated for an aluminium alloy under the assumption of plane strain tension. Textures measured at various locations through the thickness of the aluminium alloy are employed in the simulations. Modelling of the polycrystalline aggregates is carried out at various scales. In the first set of simulations, the Taylor theory of crystal plasticity is adapted to model the behaviour of the polycrystalline aggregate [e.g., 5-7). For the second set of simulations, an element of the finite element mesh is considered to represent a single crystal within the polycrystalline aggregate [e.g., 8, 9]. Henceforth, these two models will be referred to as the Taylor model and FE/grain model respectively.

2. CONSTITUTIVE MODEL

A rigorous framework for the kinematics of the finite plastic defonnation of a crystal has been finnly established for some time. This basic fonnulation has been incorporated into a rate-dependent description of crystal plasticity constitutive relations [2]. Within a FCC crystal, plastic defonnation occurs by crystallographic slip on 12 distinct slip systems. In the rate-sensitive crystal plasticity model employed, the elastic constitutive equation for each crystal is specified by:

\1 .

(j = LD - (j0 - attD (1) \1

where (j is the Jaumann rate of Cauchy stress, D represents the strain-rate

tensor and L is the tensor of elastic moduli. The tenn (j0 is a viscoplastic type stress rate that is determined by the slip rates on the 12 slip systems of a

Multiscale modeling of the effects of texture gradients on instabilities and 135 localized deformation

FCC crystal. The slip rates are assumed to be governed by the following power law expression:

Y.. !"(a) r (a) = r (0) sgn !"(a) -­

g(a)

(2)

where r (0) is a reference shear rate taken to be the same for all the slip

systems, !"(a) is the resolved shear stress on slip system a (a ranging from

1 to 12), g(a) is its hardness, and m is the strain-rate sensitivity index. Based

on measurements of strain hardening of single crystals of aluminium alloys by Chang and Asaro [10], the following slip hardening rate is employed:

h,=h,-.{iI,-h,) sech' {( ~ =~;)r. } (3)

where Ito and hs are the system's initial and asymptotic hardening rates, and

r a is the accumulated slip.

Two different models are employed to obtain the response of a polycrystal comprised of many grains. In the Taylor model, the material response is obtained by invoking the Taylor assumption. Accordingly, at a material point representing a polycrystal of N grains, the deformation in each grain is taken to be identical to the macroscopic deformation of the continuum. Furthermore, the macroscopic values of all quantities, such as stresses, stress-rates and elastic moduli, are obtained by averaging their respective values over the total number of grains at the particular material point. In the FE/grain model an element of the finite element mesh represents a single crystal, and the material response is described by the single crystal constitutive model. This approach enforces eqUilibrium and compatibility between grains throughout the polycrystalline aggregate in the weak finite element sense.

3. PROBLEM FORMULATION

A thin, orthotropic sheet specimen submitted to plane strain tension is modelled (Figures la-b), where Xl and X3 represent the rolling and normal directions of the sheet, respectively. The analyses assume no initial geometric imperfection. Localized deformation occurs as a result of "built­in" boundary conditions applied at the ends (Xl= ±Lo). The boundary conditions are

U3 = 0 along Xl = ± Lo

136 Kenneth W Neale, Kaan Inal, and Pei-Dong Wu

UI = V (applied velocity) alongX\ = Lo

ul = -V (applied velocity) alongX\ = -Lo

r

(b)

-L.

Figures la-b. Initial meshes used in the simulations.

(4)

Two different finite element meshes are employed in the simulations (Figure J). For the Taylor model simulations, a non-homogeneous mesh with 40 x 56 elements is employed (Figure Ja); while a homogeneous mesh with 56 x 168 elements is employed for the FE/grain model simulations (Figure J b).

The finite element analyses incorporate certain parallel computing algorithms so that simulations could be executed with models containing sufficiently large numbers of elements. The parallel algorithms used in our simulations are designed to distribute data (on the microscopic level) and the finite element meshes over the processors of an ffiM SP3 supercomputer [11].

Multiscale modeling of the effects of texture gradients on instabilities and 137 localized deformation

4. RESULTS AND DISCUSSION

A set of discretized orientations of approximately 400 grains, measured at 7 different locations through the thickness of an aluminium alloy is employed in the simulations (Figures 2a-g).

a b c

~

d e f

Figures 2a-g. Initial textures of the aluminum alloy represented by {J J J} stereographic pole figures from the surface towards the centre of the sheet.

The values of the material properties used in the simulations are

To=95 MPa, ho/ro=1.2, T./To=1.16, h./ro =0 (5)

The slip system reference plastic shearing rate Yo and the slip rate sensitivity parameter m are taken as Yo = 0.001s·1 , and m=0.002,

138 Kenneth W Neale, Kaan Inal, and Pei-Dong Wu

respectively, with the crystal elastic constants taken as C,,=206 GPa, C'2=118 GPa and C44 =54 GPa.

A quantitative representation of shear band development as predicted with the Taylor model is presented in Figure 3 where contours of true strain (in the rolling direction) are plotted versus the normalized elongation. It can be seen that, at an elongation of U1Lo = 0.065, a shear band passing through the centre of the specimen has already developed (Figure 3a). With further stretching (u/Lo = 0.07), even though the strain has begun to concentrate in this well defined shear band, a second shear band has developed perpendicular to the first one (Figure 3b). Figure 3c shows the fully developed shear bands at u/Lo = 0.09. Note that, although there are two fully developed shear bands intersecting at the centre of the specimen, the primary (first formed) shear band is sharper and wider than the secondary shear band.

(a) UILo = 0.065

(b) UlLo = 0.07

~!~~R:~

~ . .

(c) UILo = 0.09

Figure 3 True strain contours at various elongation levels.

EPS 0.26 021 016 011 006

Figure 4 presents contour plots of true strain (in the rolling direction) at an elongation of u/Lo = 0.065 obtained from the FE/grain model. It can be seen that the strain pattern is non-uniform throughout the specimen. Contrary to the simulations with the Taylor model, where two shear bands were predicted, the FE/grain model has predicted only a single shear band. Furthermore, the sharpness of the band is non-uniform; sharper intensities are predicted towards the edges of the specimen.

Multiscale modeling of the effects of texture gradients on instabilities and 139 localized deformation

"I' . , . \.. I ... \. \

. /' :~ ~ I "

\. • • \_~ ' . "/ J .'

EPS 02fl 021 018 Oil ODll

Figure 4 True strain at U/Lo=O.065 predicted with the FE/grain model.

5, CONCLUSIONS

Plane strain crystal plasticity based finite element models have been developed to simulate the effects of through-thickness texture gradients on instabilities and localized deformation in an FCC polycrystal. The response of a thin sheet aluminum alloy was investigated where a set of discretized orientations of approximately 400 grains, measured at 7 different locations through the thickness of the sheet, was employed in the analyses. Simulations with the Taylor model predicted two shear bands intersecting at the centre of the specimen. Note that these shear bands did not occur simultaneously, and that they do not have the same intensity. This pattern is due to the existing through-thickness texture gradients. Previous studies [5, 6] have indicated that, when a single layer of texture was employed in the simulations (no texture gradients), multiple shear bands occurred simultaneously with the same intensities. It should also be mentioned that simulations of plane strain tension, where only a single layer of the initial textures (Figures 2 a-g) was employed (no texture gradients), always predicted a single shear band.

Contrary to the simulation with the Taylor model, the FE/grain model has predicted a single shear band. Even though the FE/grain model provides a better understanding of the formation of localized deformation (since it can account for grain interactions), it should be noted that the predictions of shear bands with this model can be strongly sensitive to the initial spatial distribution of the individual grain orientations.

ACKNOWLEDGEMENTS

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and a grant from Alcan International Limited (Kingston Research and Development Centre).

140 Kenneth W. Neale, Kaan Inal, and Pei-Dong Wu

REFERENCES

[I] D. Peirce, R. J. Asaro, and A. Needleman, "Material rate dependence and localized deformation in crystalline solids", Acta Metallurgica, vol. 31, pp. 1951-1976, 1983.

[2] R. J. Asaro, and A. Needleman, "Texture development and strain hardening in rate dependent polycrystals", Acta Metallurgica, vol. 33, pp. 923-953, 1985.

[3] P. D. Wu, K. W. Neale, and E. Van Der Giessen, "On crystal plasticity FLD analysis", Proceedings of the /Wyal Society of London, vol. 453, pp. 1831-1848, 1997.

[4] K. Inal, P. D. Wu, and K. W. Neale, "Simulation of earing in textured aluminum sheets", International Journal of Plasticity, vol. 16, pp. 635-648, 2000.

[5] K. Inal, P. D. Wu, and K. W. Neale, "Instability and localized deformation in

polycrystalline solids under plane strain tension", International Journal of Solids and Structures, vol. 39, pp. 983-1002,2002.

[6] K. Inal, P. D. Wu, and K. W. Neale, "Finite element analysis oflocalization in FCC

polycrystalline sheets under plane stress tension", International Journal of Solids and Structures, vol. 39, pp. 3469-3486, 2002.

[7] K. Inal, P. D. Wu, and K. W. Neale, "Large strain behaviour of aluminium sheets

subjected to in-plane simple shear", Modelling and Simulation in Materials Science and Engineering, vol. 10, pp. 237-252, 2002.

[8] L. Anand, and S. R. Kalidindi, "The process of shear band formation in plane strain

compression of FCC metals: effects of crystallographic texture", Mechanics of Materials, vol. 17, pp. 223-243,1994.

[9] P. D. Wu, K. Inal, K. W. Neale, L. D. Kenny, M. Jain, and S. R. MacEwen, " Large

strain behaviour of very thin aluminium sheets under planar simple shear", Journal de Physique IV, vol. II, pp. 229-236, 2001.

[10] Y. W. Chang, and R. J. Asaro, " An experimental study of shear localization in

aluminum-copper single crystals", Acta Metal/urgica, vol. 29, pp. 241-254, 1981. [11] K. Inal, K. W. Neale, and P. D. Wu, "Parallel finite element algorithms for the analysis

of multiscale plasticity problems", in Application of High-Performance Computing in Engineering VII - 2002. Conference. 23 Sep.-25 Sep. 2002; Bologna, ITALY, C. A.

Brebbia, P. Meli, and A. Zanasi, Eds., 2002

MULTIAXIAL PLASTIC FATIGUE BEHAVIOR WITH MUL TISCALE MODELING

A. ABDUL-LATIF l), K. SAANOUNf & J. Ph. DINGLI2)

I) ERBEM, lUT de Tremblay, 93290 Tremblay-en-France, France E-mail: aabdul@iu2t.univ-paris8.fr 2) GSMlLASMlS Universite de Technologie de Troyes-B.P. 2060, Troyes cedex, 10010 Troyes, France

Abstract: With a small strain assumption, a multiscale model of damage initiation in Low-Cycle Fatigue (LCF) is proposed to describe the cyclic damaged behavior of polycrystal under different cyclic loading paths (simples and complexes). The evolution of the internal variables on the Crystallographic Slip System (CSS) are recorded under these loading paths up to final fracture of the Representative Volume Element (RVE). It is well recognized that the model can appropriately reproduce the macroscopic damaged responses of polycrystals in plastic fatigue.

Key words: multiscale model, multiaxial cyclic loading, damage, low-cycle fatigue

1 INTRODUCTION

The modeling of fatigue life has been first developed using the macroscopic approach (Browen & Miller 73, Garud 81, Krempl 81) or phenomenological one (Lemaitre & Chaboche 85). Using the multiscale approach, some attempts have been recently conducted to describe the damaged-inelastic cyclic behavior of polycrystals under simple and complex loading paths (Abdul-Latif & Saanouni 94, 96, 97, Saanouni & Abdul-Latif 96, Abdul-Latif 99, Abdul-Latif et al. 99). The Micro-Crack (MC) initiation criterion is defined as an amount of accumulated slip on the CSS. Adopting the isotropic damage behavior of polycrystals, the model can correctly predict the fatigue live in LCF.

In this work, a multiscale model of damaged-inelastic behavior of polycrystals is proposed. It can appropriately reproduce the damaged behavior under different cyclic loading paths. The effect of the model

141

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 141-148. ©2004 Kluwer Academic Publishers.

142 A. ABDUL-LATIF, K. SAANOUNI and J. Ph. DINGLI

parameters on the fatigue life is also investigated. Some macroscopic, mesoscopic and microscopic predicted responses are recorded and discussed.

2 MULTISCALE MODEL

We limit ourselves to a shot description of the main features of the proposed model. It is expressed in the framework of the self-consistent approach of time dependent plasticity for a small strain theory. Based on the slip theory, the model of the elasto-inelastic behavior of polycrystals which has been recently developed (Abdul-Latif et al. 02) is now coupled with damage (Table 1). The resolved shear stresses 'to are determined for all slip systems as a function of the granular stress Qg by means of the Schmid factor matrix (orientation tensor) m" (Eq. 1). It is assumed that the inelastic deformation of single crystal is defined by the slip processes neglecting the other mechanisms like twinning, grain boundary sliding, etc. The slip rate can be determined as long as the shear stress and the hardening variables are known

(Eq. 4). The parameter k; is the initial value of the critical resolved shear stress for each ess. Tow internal state variables are used at the ess level representing the intragranular isotropic hardening (describing the expansion of the elastic domain of the system s) (q", R") (Eq. 2b & 3b) and intragranular damage (dS, YO) (Eq. 5 & 3a). For a system s, the internal variable dS

represents all the microdefects localized on the system, i.e., vacancy dipoles and extrusion, or interstitial dipoles and intrusion which are due to the dislocation motions. On the other hand, the modeling of the spatial localization is not a trivial task. This difficult problem is not explicitly treated in the present work, but one can indicate hereafter an approximate numerical method to model this phenomenon using the finite element method. Fig.l shows two zones: the first one is a thin layer located at the free surface having constitutive equations coupled with damage, while the second zone represents the remaining part of the specimen having constitutive equations without damage. This allows the localization of the fatigue damage at the finite element zone 1 (near the free surface). The model is then devoted to describe the damage behavior inside the thin layer.

~tt~~r Zone without damage

Zone with damage

~ ~ Thin layer width

Fig. J Schematic representation of the quarter of a round tensile bar.

A physically based model for dynamic failure in ductile metals 143

At the microscopic level '[s =Qg :ms

RS =Qs~l_ds tHrsqr~l-dr (2a); r=1

(1)

qS = j.s (1- bSqs) (2b) ~

RS - S yS =_ys = q

2(1- dS) - RS r:--:; (3a); RS = ; <is =qS"I_dS ~

(4a); fs=ll"sl_Rs_k~

SS r SS+1 j dS = j.s(Ys J () H(AS - r~) t [yr J () H(X - r~) Ss (l_dS)WS r=l SS (l_dr)WS

At the mesoscopic level 2A@g -~) + Btr@g - i;)! - a (~g -~) = (~g - g) A= -(S.u+ 3A) (7a); B= (6J.1+A.)(3A.+SJ.1)

2.u(14.u + 9.1.) J.1( 44SJ.1 2 + 456.uA + 1 OSA. 2 )

• n • dg = Ids

s=l

with yS = j. SsignerS)

At the macroscopic level • . e . in E=E +E - - - (12);

(14);

(3b)

(4b)

(5)

(6)

(7b)

(S)

(9)

(10)

(11)

(13)

(15)

Table 1: Complete set of the constitutive equations of the developed model

144 A. ABDUL-LATIF, K. SAANOUNI and J. Ph. DINGLI

The isotropic hardening is defined by its modulus Q", its non-linearity coefficient b" and the hardening interaction matrix HIS for FCC polycrystals.

The value of pseudo-multiplier i" for each slip system is a power function of the distance to the yield point as defined by (Eq. 4a). The parameters K"

and .z: characterize the viscous sensitivity of the material. Moreover, s, S~

and w" are the intragranular damage material constants; 'Y~ represents some

threshold of the accumulated slip AS. DIS is the damage interaction matrix

(Eq. 5). The notation H (A" - 'Y~ ) is the Heaviside function. dg represents the

granular damage (Eq. 8). It is assumed that the granular fracture takes place

when 1 ~ d g ~ d~r ~ O. The granular inelastic strain rate is deduced as the

sum of the contribution from all activated slip systems (Eq. 10). Throughout this paper, the index s e {1,2,3, .... ,n} is associated with the slip system rank

with n being the maximum number of octahedral systems in the grain (n= 12 for FCC materials). The index ge p,2,3, .... ,Ng } describes the grain rank

with Ng being the maximum number of grains contained in the aggregate. The elastic behavior (assumed to be uniform, isotropic and compressible) is

determined at the granular level ~: being considered as an internal state

variable and its associated thermodynamic force is g,g (Eq. 9). ! is the second order unit tensor. A and J..I. are the classical Lame's constants of the grain. It is worth intriguing that the intra and intergranular kinematic hardening effects are naturally and globally described by the scalar parameter a. (simplified interaction law (Eq. 6», which is independent on the strain history and is capable to satisfy the self-consistency condition at each instant. For further details about this parameter effect, the reader is to refer to (Abdul-Latif et al. 02, Dingli et al 2000). Q:B and ~ are the granular and macroscopic Cauchy stress rate tensors, respectively. '§.g and.s. are respectively the deviatoric parts of the granular and macroscopic stress tensors. A and B are constants depending on the granular Lame's coefficients. The overall elastic and inelastic strain rates are calculated by averaging procedures depending on the granular rates (Eqs. 12 & 13).

3 APPLICATION

The main goal of this paragraph is to qualitatively describe the overall damaged-inelastic behavior of polycrystals in multiaxial LCF. The identification of the model is performed by applying the following steps: choice of the microstructure (aggregate of grain composition) and determination of the model constants.

A physically based model for dynamic failure in ductile metals 145

In order to minimize the model complexity and the number of the model constants, all the grains as well as the CSSs are assumed to have the same material properties. Hence, all the grains have the same constants (A. and Il)

and all the slip systems have also the same constants (ZS, KS, k~, QS, bS, s,

S~, wS and y~). The microstructure of the aggregate of grains can be determined by using the Euler angles giving the orientation of each grain. All the numerical simulations are conducted by employing an aggregate (RVE) of 200 grains. The identified model parameters are summarized in Table 2. Note that the damage parameters are chosen in such a manner that the fatigue life has to be quit short for saving the calculation time. Different cyclic loading paths are utilized in this investigation, namely: uniaxial tension-compression (TC), uniaxial torsion-torsion (TT) and biaxial tension­torsion with various out-of-phase angles: ~=O° (TTO), ~=30° (TT30), ~=45° (TT45), ~=60° (TT60) and ~=90° (TT90).

A. (MPa) J.1 (MPa) CL k,,(l\IPa) z K Q(MPa) b h1 h2 h:3 h4 h5 he 156900 80830 5x1o-7 200 15 31 298 9.9 1 1.2 1.1 1.16 1.17 3.27

Y~ s! S· vi' d1 ~ da d4 ds de 1.5 4 0.9 1.2 1.5 1.1 2.46

Table 2: Identified parameters of the model

The influence of the loading paths on the fatigue lives and its effect on the cyclic hardening which is governed by the slip system multiplication, is obviously pointed out in Table 3.

f'UrtJs'ci IIlJrb3rci f'UrtJs'ci IIlJrb3rci N.mJerci Estirralal

Typed ~max pastifial cssWth cssWth gainWth gainYtith

fatigJe life

1ca:l1ll JDh rra:ro (M='a) (1'l.I'TtJer ci css ds>O ds>O.6 D,;pO D,;pO.6

cydes)

lC 676 340 226 19 187 19 3)1

TT Em 254 188 24 171 24 300

no 649 3)4 203 34 183 34 418

TT45 712 461 143 10 125 10 129

TlOO fJJl 625 88 8 78 8 73

nro ~ 768 42 5 39 5 46

Table 3: Influence of the loading path on the fatigue lives

146

1000

900

800

700

" 600 .. ! i 500

~ 400

300

200

100

0

0 0.5

A. ABDUL-LATIF, K. SAANOUNI and J. Ph. DINGLI

1.5 2.5 3.5 4 4.5

Macro .... ccumulated Inelast~ strain

n90 ~n60

~n45

- no ~n

-TC_

5.5

Figure 2 Predicted evolution of the overall maximum stress up to the final fracture of the RVE under different loading paths.

1000

900 Loading ty:~:S~~.-- -+--~ ~ ..... -~ ~.-~ ... --.-~. 0.9

800 /"v 0.8

~ 700 ,'~

~ 600 .I e ~ 500

t>

.lI 400 ~

<.:> 300

200

100

0.5 1.5

Granular accumulated inelastic s!faln

0.7

0 .6 .. .., 0.5

0.4

0.3

0.2

0.1

2.5

Figure 3 Predicted evolution of the maximum granular stress and granular damage in TT90 up to the final granular fracture

The ability of the model in describing the multiaxial behavior of polycrystals is examined by using the different out-of-phase loading paths. This capability is tested to determine whether the model can correctly describe in a natural manner the multiaxiality effect on the hardening behavior and on the fatigue life. All these tests are carried out under strain-controlled condition with the maximum von-Mises equivalent plastic strain being maintained constant at 0,5% during the test. The obtained results are

A physically based model for dynamic failure in ductile metals 147

recorded in Table 3 in which one can observe that the fatigue life decreases remarkably with increasing ~. At each grain, only one damaged slip system is systematically investigated whatever the cyclic loading path is giving us a "localization phenomenon". Therefore, the number of the damaged grains is equal to the number of the damaged slip systems (Eq. 8). An examination of the Table 3 reveals that dg

and dS decrease with the increasing of the applied loading complexity. According to the localization nature of damage, hence the smallest fatigue life (highest damaging case) takes place in TT90° (NF 46 cycles). This is due to the multiplicity of the activated slip systems induced by the change of the loading direction in the course of cycling (Fig. 2).

140·'

Loading type: TT90

120

100

20 / o~----~~~-r----~r-----~--~-r-----+

o 0.5 1.5

Accumulated slip 2 2.5 3

0.9

0.8

0.7

0.6

0.5 ..s

0.4

0.3

0.2

0.1

0

Figure 4 Predicted evolution of the intragranular isotropic hardening and intragranular damage in TT90 up to the final fracture

To study the internal variable evolution at TT90 up to the final granular fracture, a damaged grain is selected beforehand. The maximum granular stress and granular damage evolutions of the grain are recorded during the cycling (Fig. 3). When the granular damage reaches a considerable value (dg

> 0.3), the corresponding stress decreases rapidly and when dg is equal to one giving consequently the final fracture of the grain. The variation of the intragranular internal state variables (RS and dS) are illustrated in Fig. 4 for a favorably oriented system in TT90. This leads to the fact that such a system can be well plastified and then damaged. Hence, at the CSS level, RS evolves (for a positive hardening) up to a maximum

148 A. ABDUL-LATIF. K. SAANOUNI and J. Ph. DINGLI

value. This force tends progressively to zero with increasing dS leading as consequence to a final fracture of the slip system.

4 CONCLUSION

The proposed model aims to predict the LCF behavior of the RVE under different uniaxial and multiaxial loading situations for FCC polycrystalline structure. Qualitatively, it reproduces successfully the inelastic fatigue behavior of polycrystals. This model can naturally describe the overall cyclic stress-strain evolution using a local damaged constitutive equation on the crystallographic slip system level. It gives a fairly well agreement with the experimental observations particularly for non­proportionality effect of the loading paths on the plastic fatigue life.

REFERENCES

Browen, M. W. and Miller, K. J. (1973). "A Theory for Fatigue under Multiaxial Stress-Strain Conditions," Proc. Inst. Mech. Engineers, 187,745.

Garud, Y. S. (1981). "Multiaxial Fatigue: A Survey of the State of the Art, "J. Testing and Evaluation,9,165.

Krempl, E. (1974). ''The Influence of State of Stress on Low-Cycle Fatigue of Structural Materials: A literature Survey and Interpretive Report," ASTM STP 549, ASTM, Philadelphia, pp. 46.

Lemaitre, J. and Chaboche, J. L. (1985). "Mecanique des Materiaux Solides," Dunod, Bordas, Paris.

Abdul-Latif, A. and Saanouni, K. (1994). "Damaged Anelastic Behavior of FCC Poly­crystalline Metals with Micromechanical Approach," Int. J. Damage Mech., 3, 237.

Abdul-Latif, A. and Saanouni, K. (1996). "Micromechanical Modeling of Low Cycle Fatigue under Complex Loadings - Part ll. Applications," Int. J. Plasticity, 12, 1123.

Abdul-Latif, A. and Saanouni, K. (1997). "Effect of some Parameters on the Plastic Fatigue Behavior with Micromechanical Approach," Int. J. Damage Mech., 6, 433.

Abdul-Latif, A., Ferney, V., and Saanouni, K. (1999). "Fatigue Damage of Waspaloy under Complex Loading," ASME, J. Engeg. Mat. Tech., 121, 278.

Abdul-Latif, A. (1999). "Unilateral Effect in Plastic Fatigue with Micromechanical Approach," Int. J. Damage Mech., 8, 316.

Saanouni, K., Abdul-Latif, A. (1996). "Micromechanical Modeling of Low Cyclic Fatigue under Complex Loadings-part I. Applications," Int. J. Plasticity, 12, 1111.

Abdul-Latif, A., Dingli, J. Ph., and Saanouni, K. (2002). "Elastic-Inelastic Self-Consistent Model for Polycrystals," J. of Applied Mechanics, 69, 309.

Dingli, J. P., Abdul-Latif, A., and Saanouni, K. (2000). "Predictions of the Complex Cyclic Behavior of Poly crystals Using a New Self-consistent Modeling," Int. J. Plasticity, 16,411.

DAMAGE, OPENING AND SLIDING OF GRAIN BOUNDARIES

A. Musienko and G.Cailletaud Centre des Materiaux de l'Ecole des Mines de Paris, BP 87, 91003 Evry, France

O.Diard Electricite de France, Centre des Renardieres

77818 Moret-sur-Loing, France

Abstract This paper presents an approach to the modeling of damage, opening and

sliding of the grain boundaries in zircaloy submitted to stress corrosion cracking. Grain boundaries are seen as a continuous material. The grains are modeled by a model of crystal viscoplasticity. Environement is taken into account by means of weak coupling between diffusion and mechanics. 2D and 3D finite element computations for polycrystalline aggregates are presented.

Keywords: Stress corrosion cracking, Intergranular failure, Grain boundaries, Zircaloy, Crystal plasticity.

1. INTRODUCTION

In the nuclear pressure water reactors, uranium oxide is contained by long tubes made of zircaloy, which are submitted to aggressive loading conditions, involving thermomechanical fatigue and stress corrosion cracking. A mechan­ical multiscale analysis has already been made Diard et aI., 2003. Since ura­nium pellets break during operation, they indent the tube, and a high stress concentration follows, on a very small area (several grains). Iodine is shown to have also an effect on the failure process, which is first intergranular then trans granular (see for instance Fregonese et aI., 1999). It is adsorbed at the grain boundary (GB), and dramatically increases intergranular damage rate. The purpose of the paper is then to model environment assisted intergranular damage.

A lot of papers dealing with intergranular damage can be found in the lit­erature. Initial studies were concerned with metallurgical aspects Raj and

149

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 149-156. ©2004 Kluwer Academic Publishers.

150 A. Musienko, G. Cailletaud and 0. Diard

Ashby, 1975, then authors's goal was more and more to deliver quantitative stress and strain levels and a realistic view on the local mechanisms related to cavity growth Rice, 1981; Tvergaard, 1984. A series of mechanical mod­els have been proposed in the last twenty years, after the so called cohesive zone model Needleman, 1987. As pointed out in a recent review Chaboche et al., 2001, this class of model Tvergaard, 1990; Allix and Ladeveze, 1992 can be applied to any type of interface, and is intermediate between continuum damage mechanics and fracture mechanics. So, the behavior of the boundary layer is described by means of a normal and a tangential displacement, and the corresponding forces. Damage evolution produces a transition between the initial state for which the GB is elasto(visco)plastic (damage equal zero) and is broken (damage equal to 1).

The choice made in this paper consists in accepting the continuous formal­ism of damage mechanics Lemaitre, 1996, written in terms of stress and strain, keeping in mind that GB is a highly anisotropic area. It will be modeled in a finite element (FE) mesh by classical elements, with a specific local behavior.

Coupled computations are performed, by introducing a weak coupling be­tween mechanics and environment (represented by diffusion at the GB).

In what follows, the model is presented (section 2), then the coupling algo­rithm is briefly described (section 3). The numerical implementation is per­formed in the FE code ZSeT/zeBuLoN, which allows easy material model development Besson et al., 1998. Typical results in 2D and 3D FE meshes are finally discussed in the last part (section 4).

2. DAMAGE, OPENING AND SLIDING MODEL

The GB is supposed to be a regular surface. The constitutive equations of the GB will be expressed in a local framework, with the axis Xl normal to the plane, and axes X2 and X3 in the plane of the GB. The constitutive equations are anisotropic, since GB material should resist to normal stress, applied in 1 direction, but not in 2 or 3. The material must be very stiff in the normal direction, so that the grains remain pasted in absence of damage. The moduli must be low in the transverse direction, so that the boundaries do not carry any load and do not perturb grain equilibrium. As well, GB should accept in-plane shear (12 and 13), but not the 23 shear. These ideas are actually consistent with the "springs" of Onck and van der Giessen, 1997. Such an anisotropy lead to the special elastic tensor form:

~ = diag(E, 'fl, 'fl, J.L, 'fl, J.L) (1) ~

where the terms 'fl remain small. Their effect will be neglected in the follow­ing, to let GB just follow grain behavior for 22, 33 and 23 stress components. Having this anisotropy in mind, the scheme of damage mechanics must be

Damage, opening and sliding of grain boundaries 151

adapted. Introducing only one variable D figuring scalar damage, the classical approach Lemaitre, 1996 proposes a modification of the elasticity related part of the free energy, so that:

1 p'IjJe = "2(1 - D)foe : 1- : foe (2)

The variable Y energetically conjugated with D is obtained by taking the op­posite of the partial derivative of p'IjJe with respect to D. Using the previ­ously defined elastic tensor, and preserving only the three predominant terms, it comes:

(3)

As expected, GB opening a and GB sliding I will respectively correspond to (/ n and T, and D will correspond to Y. In the following, a new variable (/ d will be considered instead of Y:

(4)

One can easily see, that

(5)

In this variable, which is derivated from Y, the coefficient (3 characterizes the ratio E / fL. For a zero value of (3, damage will be driven by the normal stress only.

The evolution equations for D, a and I are defined in the framework of a viscoplastic multipotential approach, since all the phenomena are time depen­dent. To get correct constitutive equations, respecting the second principle of thermodynamics, it is enough to chose convex potentials for all 3 variables. These potentials should be dependent on the associated stresses. Norton-like viscous potentials, with a threshold are chosen. The potential driving the nor­mal flow depends on the positive part of (/n, and shear depends on the absolute value OfT:

(6)

(7)

(8)

152 A. Musienko, G. Cailletaud and 0. Diard

Derivation with respect to the conjugate stresses will immediately give us the flow rules:

i: = oFn= / < (In > /(1- D) - Rn)nn (1- D)-ln® n = 8N (9) "'I"t of!. \ Kn - - ~

~ = 0:; = (Irl/(l t) -Rt ) nt (1- D)-l{n ®!} = 1'1' (10)

One can mention, that with Rd == 0, nd - 1 == r and K'd ~ == AT the last equation becomes classical Kachanov-Rabotnov's law for creep damage:

b = (:tr (1 - D)-k (12)

In the described model, inelastic flow can start without any damage, provided the corresponding stress becomes higher than the threshold. Opening and shear are not coupled as long as non damage exists. On the other hand, either a posi­tive normal stress or a shear stress can develop damage, and have an influence on sliding and opening rate.

3. COUPLED COMPUTATION PRINCIPLE

The method of weak: coupling was used to take into account the environ­mental effect. Iodine adsorption is simulated by the solution of a diffusional problem in which the diffusion coefficient strongly depends on damage. Con­sequently, iodine propagation follows damage front in the GB. From a nu­merical point of view, the coupling is obtained by running two simultaneous problems, which are coupled at each time step. For each integration step, dif­fusion problem is solved first. Iodine concentration is then exported to the mechanical problem to compute concentration depdendent parameters. With these newly obtained parameters, the mechanical problem is solved. It provides stress, strain and damage fields. Damage is then exported to the diffusion prob­lem, to compute damage depdendent parameters, and recompute the same time step. For each time step, a fixed point algorithm is then introduced. At least two loops are performed for each time step (diffusion-mechanics-diffusion­mechanics).

4. RESULTS

Special finite element meshes with GB elements were created. Figures 1 and 2 show the examples of2D and 3D finite element meshes. As shown in Fig.l.

Damage, opening and sliding of grain boundaries 153

GB are represented by quadrilateral elements. Linear or quadratic elements are accepted. Triangles are introduced at triple points. For the case of 3D, GB -which are volumic elements now - create a quite complex structure, as seen in Fig.2. Elements in the current GB are bricks, but prisms and thetrahedra are also needed to connect the joints at the triple lines and multiple points. Careful examination of all the geometrical situations has been made, and a specific code, written in C++, is available for any type of mesh generation.

Figure 1. 2D finite element mesh with grain boundaries

a b

Figure 2. 3D finite element mesh with grain boundaries, (a) - outside view of the 20-grain aggregate, (b) - GB only for the same aggregate

Number of computations were made on 2D aggregate (Fig.}) first. It was an aggregate with 50 grains, 9834 nodes and 4553 quadratic finite elements. For this aggregate, tension in vertical direction till 5% was applied. Purely mechanical computation was made with isotropic elastic grains (E = 99300

154 A. Musienko, G. Cailletaud and 0. Diard

MPa, v=O.37) and the following parameters for damage model (Kn = K t = 400MPa· sl/nn; nn = nt =5.1; A=lOOOMPa; r=2; k=3; Rn = 260MPa; (3 = 0; orthotropic elasticity (MPa) : A 3333=1, A 2222=1, A n22=O, A 2233=O, A 3311 ==O, A 2323==1, Anu = 100 + 99300· (1 - D), Al212 = 100 + 72400· (1 - D), A3131 = 100 + 72400 . (1 - D))

a b

Figure 3. 2D aggregate computational results, (a) - damage distribution for purely mechan-ical computation, (b) - damage distribution for coupled computation

a b

Figure 4. 3D results for diffusion, (a) - outside view, (b) - grain boundaries only

One can see (FigJa), that for purely mechanical case damage appears for all the grain boundaries, normal to the tension direction, everywhere in the aggregate. To represent progressive crack propagation, it is then necessary to consider the diffusion problem.

Diffusion properties were described with a scalar diffusion coefficient, which was chosen much higher for the grain boundaries, than for the grain core (l05 and 1O-10m-2s-1 respectively). Mechanical properties were taken dependent on the iodine concentration (C) : A = 100 + 900 . exp( -100 . C).

Damage, opening and sliding of grain boundaries 155

a b

Figure 5. 3D results for coupled computation, (a) - grain boundaries FE mesh, (b) - damage distribution

Mechanical tension for this coupled computation was applied in vertical direc­tion, and the iodine concentration equal to 1 was applied on the left side of the square (Fig. 1 ). Typical result (Fig.3b) of inelastic strain intensity distribution shows now the progressive intergranular failure propagation from the left to the right.

Similar type of computations was done on 3D aggregate (Fig.2) - 20 grains, 4763 nodes, 14934 linear finite elements. Pure diffusion computation was done first. Diffusion coefficients for the grains and grain boundaries were the same, as for 2D case. Iodine concentration equal to 1 was applied on the bottom of the cube (see Fig.4). In the results one can see the iodine propagation through the grain boundaries (Fig.4a). It is interesting to remark the faster diffusion in the triple joints of grain boundaries, rather then in the grain bound­ary planes (Fig.4b). Coupled diffusion-mechanical computation was then per­formed. Material parameters were the same as for 2D case. Iodine concentra­tion was applied on the face x=O of the cube, and mechanical tension in the direction z was performed. One can see (Fig.5b) progressive grain boundary damage propagation due to iodine diffusion.

5. Concluding remarks

All the computations made up to now were qualitative only. The initial purpose of the study was to show the feasability of the computation. More realistic computations and detailed analysis of the numerical results are now underway.

156 A. Musienko, G. Cailletaud and 0. Diard

REFERENCES

Allix, O. and Ladeveze, P. (1992). Interlaminar interface modeling for the prediction oflaminate delamination. J. Compos. Struct., 22:235-242.

Besson, J., Le Riche, R., Foerch, R., and Cailletaud, G. (1998). Object-oriented programming applied to the finite element method. part ii. application to material behaviors. Revue Eu­ropeenne des Elements Finis, 7(5):567-588.

Chaboche, J.-L., Feyel, F., and Monerie, Y. (2001). Interface debonding models: a viscous reg­ularization with a limited rate dependency. Int. J. Solids Structures, 38:3127-3160.

Diard, 0., Leclercq, S., Rousselier, G., and Cailletaud, G. (2003). Modeling of iodine-induced stress corrosion cracking in a zirconium alloy. chemical-mechanical coupled formulation at the granular level, application to intergranular damage modeling. submitted.

Fregonese, M., Lefebvre, F., Lemaignan, C., and Magnin, T. (1999). Influence of recoil­implanted and thermally released iodine on i-scc of zircaloy-4 in pci-conditions: chemical aspects. J. Nucl. Mat., 265:245-254.

Lemaitre, J. (1996). A course of Damage Mechanics. Springer Verlag.

Needleman, A. (1987). A continuum model for void nucleation by inclusion debonding. J. of Applied Mechanics, 54:525-531.

Onck, P. and van der Giessen, E. (1997). Microstructurally-based modelling of intergranular creep fracture using grain elements. Mech. of Materials, 26: I 09-126.

Raj, R. and Ashby, M. (1975). Intergranular fracture at elevated temperature. Acta Metal!., 23:653-666.

Rice, J. (1981). Constraints of the diffusive cavitation of isolated grain boundary facets in creep­ing polycrystals. Acta Metallurgica, 29:675-681.

Tvergaard, V. (1984). On the creep constrained diffusive cavitation of grain boundary facets. J. Mech. Phys. Sol., 32(5):373-393.

Tvergaard, V. (1990). Effect of fiber debonding in a whisker-reinforced metal. Material Science and Engineering, 125:203-213.

GRADIENTS OF HARDENING IN NONLOCAL DISLOCATION BASED PLASTICITY

George Z. Voyiadjis & Robert J. Dorgan Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, LA 70803 USA

Abstract: In this work, a thermodynamically consistent framework for a gradient enhanced plasticity model is given to introduce material length scales through the second order gradients of both the kinematic and isotropic hardening variables. In order to give a micromechanical basis for the gradient enhanced continuum model, the evolution equations of the internal state variables derived through the gradient theory are compared to the evolution equations based on dislocation theories involving mobile and immobile dislocations, and the gradient coefficients are defined using material parameters from thes dislocation theories.

Key words: Gradient; dislocation; plasticity

1. INTRODUCTION

Microstructural characteristic lengths are significant in the analysis of the material at a scale where the microstructure characteristic length is greater than the required resolution length, or where the size of the representative volume element is significant compared to the specimen size. Local models do not include an internal length scale, and thus element size becomes the length scale. Because no length scale is involved, local numerical simulations suffer from pathological mesh dependence. Enhanced continuum models, including but not limited to nonlocal theories, introduce microstructural characteristic lengths in an effort to remove this mesh dependence. Additionally, they attempt to account for long-range microstructural interaction.

157

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 157-165. ©2004 Kilmer Academic Publishers.

158 George Z. Voyiadjis & Robert J. Dorgan

In this work, the gradient dependent evolution equations introduced in Bammann & Aifantis1981 ;1982 and Aifantis1982;1983;1984a,b, known as the Walgraef-Aifantis diffusion equations, are used in conjunction with averaged macroscopic equations for work hardening of crystalline materials in the slip-dominated plastic regime, i.e., low temperatures and quasi-static conditions. The equations for hardening are then compared with the gradient enhanced formulation similar to that presented in Voyiadjis & Dorgan2002 to give a micro-structural basis for the gradient dependent formulation, thus attempting to bridge the length scales by defining gradient coefficients through this dislocation theory.

2. CONSISTENT, GRADIENT ENHANCED FORMULATIONS

In order to introduce long-range microstructural interaction, the stress response at a material point is assumed to depend on the state of its neighborhood in addition to the state of the point itself (Figure 1). The use of nonlocal continua theory is made in order to achieve that, and a gradient continuum enhancement is used as a special case of the general concept for nonlocal continua.

Kroner1967 and Eringen & Edelen1972 incorporated nonlocal terms through an integral equation for the elastic material models where an integral expression was used to express the average strain at a material point. In a similar fashion, at a material point x, a nonlocal measure A of a local internal state variable A is defined here as the weighted average of its local counterpart over a surrounding volume Vat a small distance I~I:$; Lc from x. This integral equation is the basis for standard nonlocal theories (Baiane984;

Pijaudier-Cabot & Baiant1987; BaZant & Pijaudier-Cabot1988). However,

....... ....... ..... ~"

/ .. ~ " ~ \ 0) -\-

\ 0 w 'r \~ . I ,., . r~' l~ __ ~_-

\ ", - i. . o Ow .1.

" ~ .jI

Figure 1. Nonlocal behavior incorporating microstructure (dislocation) interaction

Nonlocal dislocation based plasticity 159

following the work of Miihlhaus & Aifantis l991 and Vardoulakis & Aifantis l99 \ this integral equation for the nonlocal measure is approximated by a the following partial differential equation:

(1)

where a is a constant proportional to a length squared and weights each component of the gradient term identically.

In this work, nonlocal measures will be used for the two plasticity internal state variables considered: isotropic hardening denoted by the scalar p and kinematic hardening denoted by the tensor ay. The first type of hardening is modeled by the expansion or contraction of the yield surface in the stress space while maintaining its shape and having a fixed center, while the second type is modeled by the movement and distortion of the yield surface. Note that the subscripted letters after the kinematic hardening variable indicates the tensorial nature of the variable.

With regard to using gradients to describe the non-local behavior of the material, the nonlocal measures ]5 and aij (which are not necessarily internal state variables) are used to characterize the nonlocal isotropic and kinematic hardening, respectively, and are given such that:

- n2 P == p+c2 v P (2)

As in equation (1), the coefficients Cl and C2 introduce length scales and are defined as a constant proportional to a length squared.

The thermoelastic Helmoltz free energy is now expressed in terms of the independent internal state variables as:

(3)

Using the same procedure given in Voyiadjis & Dorgan (2001), the definitions for the local thermodynamic conjugate forces are derived as:

(4)

where X;j and R are defined as the thermodynamic conjugate forces corresponding to the flux variables aiJ and p, respectively.

Nonlocal measures of the thermodynamic forces are assumed to have the same form as that given by Eqs. (4). Making the assumption that the coefficients kl and k2 are spatially constant, i.e. \12 kl == \1 2 k2 = 0, the nonlocal measures of the conjugate forces are defined as follows:

160 George Z. Voyiadjis & Robert J. Dorgan

(5)

These nonlocal measures are introduced into the plastic potential and yield function. In order to be consistent and satisfy the generalized normality rule of thermodynamics, the plastic potential function, F, is defined as:

k - - k-2 F = f+....l.x.x .. +-...i.R

2 lj lj 2 (6)

In this equation, the constants k3 and k4 are used to adjust the units of the equatign, andfis a yield function of Von Mises type:

(7)

where sij is the deviatoric components of the Cauchy stress tensor. From the generalized normality rule of thermodynamics for the case when F ~ 0, the following evolution equations are obtained:

(8)

. . of '(" - ) a .. =-A-=A n .. -k3X. ; lj ax. IJ lj

lj

where fly is the normal to the yield surface. The evolution equations for the thermodynamic conjugate forces are obtained through the time derivative of Eqs. (4).

3. IDENTIFICATION OF MODEL PARAMETERS

A number of researchers have described dislocation densities through gradients (Nye1953, Cottrell1964, Ashby1970, Menzel & Steinmann20oo, Shizawa et a1.2001). The storage of dislocations is responsible for material work hardening. With an increase in immobile dislocation density, mobile dislocations begin to have more interactions with the immobile dislocations such that movement becomes more difficult and the stress required to produce additional plastic deformation increases, i.e. the material hardens. Alternatively, with an increase in mobile dislocation density, strain gradients

Nonlocal dislocation based plasticity 161

are induced from which hardening occurs. For example, as an indenter size decreases, the strain gradients increase, and so does the density of mobile dislocations.

Before giving the definitions to be used for the isotropic and kinematic hardening terms, we will begin with the definition for the evolution of the plastic strain in macroscopic plasticity theory which is dependent on the density of the mobile dislocations as follows (Aifantis & HirthI985):

(10)

In relating the plastic strain at the macro scale to the plastic shear strain at the micro scale, an average form of the Schmidts orientation tensor, Mi}' is used since plasticity at the macroscale incorporates a number of differently oriented grains into each continuum point (AifantisI987; Hammann & AifantisI987). Similar to using an average Schmidts tensor, average values are also used for the Burgers vector magnitude, b; average trapped dislocation free path, I; and the average dislocation glide velocity, i.

The evolution of mobile and immobile dislocation densities is given by the reaction-diffusion equations initially proposed by Bammann & Aifantis 1981 ;1982 and Aifantis I982;1983;1984a.b and further developed by Walgraef & AifantisI985.1988. In this model, spatial gradients are introduced into the evolution equations of both the mobile and immobile dislocations. When Walgraef-Aifantis model is generalized to 3D, the evolution equations are written as:

(11)

(12)

where Di and Dm are diffusion-like coefficients for the mobile and immobile dislocations, f(pi ,pm) denotes the interaction between mobile and immobile dislocations, and g(pi) denotes the production/annihilation of immobile dislocations.

3.1 Isotropic Hardening

The isotropic hardening variable in macro-plasticity is a scalar quantity and its evolution is normally expressed in terms of the evolution of the equivalent plastic strain:

162 George Z. Voyiadjis & Robert J Dorgan

(13)

which, after substitution of the evolution equation for the mobile dislocation density given by Eq. (12), is rewritten as:

(14)

From this equation, the nonlocal dependence of the evolution of the isotropic hardening can be seen through the gradient term. The evolution equation derived through the nonlocal theory in Section 2 is compared to the dislocation based evolution equation in order to obtain the following relations:

(15)

(16)

From this last relationship, we assume here that the isotropic hardening is equal to the mobile dislocation density. Based on this assumption, the coefficients in the gradient theory are defined based on a micromechanical characterization as follows:

k -Z-I 2 - , (17)

As defined here, it is seen that the coefficients k4 and C2 are not constant. In fact, both terms vary with the dislocation glide velocity, and the coefficient k4 is dependent on the interaction between mobile and immobile dislocations.

3.2 Kinematic Hardening

The model introduced by Bammann & Aifantis l982 will be used here to define the backstress. In this model, the continuum is considered to be composed of three states: the perfect lattice state, the mobile dislocation

Nonlocal dislocation based plasticity 163

state, and the immobile dislocation state. The total stress is therefore additively composed of the mobile and immobile dislocation stresses and the lattice stresses. The same form for the stresses supported by the mobile and immobile dislocation states defined by Bammann & Aifantis1982 are used:

xm m MM ij =-tr P ij (18)

where trM and tr i are constants and M ij is the symmetric macroscopic Schmidts orientation tensor. The sum of the stresses due to the immobile and mobile dislocations acts as the backstress. Again assuming that the slip system remains unchanged with time, the evolution equation for the backstress can thus be obtained as:

X xm Xi (m . mi' i)M ij = ij + ij = - tr P +tr P ij (19)

Using the Walgraef-Aifantis evolution equations for the dislocation densities, the evolution equation for the kinematic hardening is given in terms of the dislocation densities and the gradients of the dislocation densities as:

Xij =_[trMDmV 2pM +triDiV 2pi +(trM -tri)f(pi,pM)+trig(pi)]Mif

(20)

A relationship between this equation and the evolution equation for the kinematic hardening derived through the nonlocal theory in Section 2 cannot directly be determined. Thus, we assume that the Laplacian of the constants trM and tr i is zero and that V2 M if can be neglected, and define the nonlocal measure as:

(21)

This measure is substituted into the evolution equation derived through the nonlocal theory in Section 2, along with the evolution equation for the plastic strain and the evolution equation for the Lagrange multiplier as defmed for the isotropic hardening in the previous section. Substitution of the evolution equation for the mobile and immobile dislocation densities given by the Walgraef-Aifantis diffusion equations gives:

164 George Z. Voyiadjis & Robert J. Dorgan

Xij=k)v2pm(Dmbl + k3b1 f(/,pm)c)1l'm ~!MpqMpq )Mij

+k)V2/ (Isbl f(pi ,pM)C)1l'i ~!M pqM pq )Mij

+k)pM (bi + k3b1 f(pi ,pm)1l'M ~!M pqMpq )Mij

+k)/ (k3b1 f(/,pm)1l'm ~!M pqM pq )Mij

+k) (blf(pi ,pm»)Mij

(22)

This equation is then compared to the dislocation based evolution equation in order to defme the coefficients in the gradient theory based on a micromechanical characterization as follows:

(23)

(24)

Note that the coefficients C] and k3 vary with dislocation glide velocity, the mobile and immobile dislocation densities, f(/ ,pM), and g(pi).

4. CONCLUSIONS AND FUTURE WORK

A thermodynamically consistent gradient enhanced approach to plasticity is formulated in this paper. The proposed capability of the model is to simulate properly size dependent behavior of the materials together with localization problems through the incorporation of a internal material length scale. We have shown a dependence of the materials parameters on the microstructure and the respective evolution of the microstructure through the dislocation densities, thus attempting to bridge the length scales between the macro (continuum) and micro (crystal) levels of the material.

Nonlocal dislocation based plasticity 165

ACKNOWLEDGEMENTS

The authors acknowledge their appreciation to Dr. DJ. Bammann of Sandia National Laboratories in California for the numerous discussions on dislocation based plasticity. The first author acknowledges his appreciation on the discussions on this subject he had with Dr. E.C. Aifantis during his visit at the Aristotle University of Thessaloniki in Greece under the REVISA project. The support of Robert Dorgan by the Board of Regents Fellowship is gratefully acknowledged.

REFERENCES

Aifantis, E.C., In S. N. Atluri and J. E. Fitzerald (005.), NSF Workshop on Mechanics and of Damage and Fracture, Georgia Tech., Atlanta, 1-12 (1982).

Aifantis, E.C., In G. C. Sih and J. W. Provan (eds.) Defects, Fracture, and Fatigue (Proceedings of International Symposium held in May 1982, Mont Gabriel, Canada), Maetinus-Nijhoff, The Hague, 75-84 (1983).

Aifantis, E.C., J. Eng. Mater. Technol. (Transactions of ASME), 106: 326-330 (1984a). Aifantis, E.C., Int. 1. Engng. Sci, 22: 961-968 (1984b). Aifantis, E.C. and Hirth, J.P. (eds.), The Mechanics of Dislocations, ASM, Metals Park, 1985. Aifantis, E.C., Int. 1. Plast., 3: 211-247 (1987). Aifantis, E.C., J. Eng. Mat. Tech., 121: 189-202 (1999). Bammann, 0.1. and Aifantis, E.C., In A.P.S. Selvadurai (ed.), Mechanics of Structured

Media, Preceedings of the International Symposium on the Mechanical Behaviour of Structured Media, Ottawa, 79-91 (1981).

Bammann, D.J. and Aifantis, E.C., Acta Mech., 45: 91-121 (1982). Bammann, 0.1. and Aifantis, E.C., Acta Mech., 69: 97-117 (1987). Fleck, N.A. and Hutchinson, J.W., J. Mech. Phys. Solids, 41(12); 1825-1857 (1993). Menzel, A. and Steinmann, P., 1. Mech. Phys. Solids, 48: 1777-1796 (2000). Miihlhaus, H.B. and Aifantis, E.C., Int. 1. of Solid and Structures, 28: 845-857 (1991). Vardoulakis, I. and Aifantis, E.C., Ingenieur- Archive, 59: 197-208 (1989). Walgraef, D. and Aifantis, E.C., J. Appl. Phys., 58: 668-691 (1995). Voyiadjis, G.Z. and Deliktas, B., Mech. Research Communications J., 27(3); 1 (2000). Voyiadjis, G.Z., Deliktas, B., Aifantis, E.C., 1. Eng. Mech., 127(7); 636 (2001). Voyiadjis, G.Z., Dorgan, RJ. and Dorroh, J.R, Proc. of the 2001 Energy Sources Tech.

Conference & Exhibition, ASME Publishing Company, Houston, Texas (2001). Voyiadjis, G.Z. and Dorgan, R1., Arch. ofMech., 53(4-5): 565-597 (2001). Walgraef, D. and Aifantis, E.C., 1. Appl. Phy., 58: 688-691 (1985). Walgraef, D. and Aifantis, E.C., Res Mech., 23: 161-195 (1988). Zbib, H.M. and Aifantis, E.C., Res Mechanica, 23: 261-305 (1988).

DETERMINATION OF THE MATERIAL INTRINSIC LENGTH SCALE OF GRADIENT PLASTICITY THEORY

George z. Voyiadjis and Rashid Abu AI-Rub

Department of Civil and Environmental Engineering Louisiana State University Baton Rouge,Louisiana 70803, USA E-mail: voyiadjis@eng.lsu.edu

Abstract: The enhanced strain-gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. To assess the size effects it is indispensable to incorporate an intrinsic material length parameter into the constitutive equations. However, the full utility of gradient­type theories hinges on one's ability to determine the constitutive length-scale parameter. The classical continuum plasticity is unable to predict properly the evolution of the material flow stress since the local deformation gradients at a given material point are not accounted for. The gradient-based flow stress is commonly assumed to rely on a mixed type of dislocations: statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNOs). In this work a micromechanical model to assess the coupling between SSDs and GNDs, which is based on the Taylor's hardening law, is used to identify the deformation-gradient-related intrinsic length-scale parameter in terms of measurable microstructural physical parameters. This work also presents a method for identifying the length-scale parameter from micro-indentation tests.

Key words: Gradient plasticity; Size effects; Intrinsic material length-scale; Geometrically necessary dislocations; Micro-hardness.

1. INTRODUCTION

In the last ten years a number of authors have physically argued that the size dependence of the material mechanical properties results from an

167

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 167-174. ©2004 Kluwer Academic Publishers.

168 George Voyiadjis and Rashid Abu AI-Rub

increase in strain gradients inherent in small localized zones leads to geometrically necessary dislocations that cause additional hardening [1]. A number of gradient-enhanced theories have been proposed to address the size effects through incorporation of intrinsic length-scale measures in the constitutive equations, mostly based on continuum mechanics concepts [2,3].

Although there has been a tremendous theoretical work to understand the physical role of the gradient theory, this research area is still in a critical state with numerous controversies. This is due to some extent to the difficulty in calibration of the different material properties associated with the gradient-dependent models, which is impossible for certain cases. From dimensional consideration, in gradient-type plasticity theories, length scales are introduced through the coefficients of spatial gradients of one or more internal variables. Thus, the full utility of gradient-based models hinges on one's ability to determine the constitutive length parameter that scales the gradient effects. The work we report here aims at remedying this situation.

However, it is believed that the calibration of the constitutive coefficients of a gradient-dependent model should not only be based on stress-strain behavior obtained from macroscopic mechanical tests, but should also draw information from micromechanical gradient-dominant tests, such as micro­indentation and nano-indentation tests [4]. Nix and Gao [5] estimated the material length scale parameter l from the micro-indentation experiments of McElhaney et al. [6] to be l = 12,um for annealed single crystal copper and l = 5.84,um for cold worked polycrystalline copper. By fitting micro­indentation hardness data, Begley and Hutchinson [7] have estimated that the material length-scale associated with the stretch gradients ranges from 0.25 to 0.5,um, while the material lengths associated with rotation gradients are on the order of 4,um.

Recently, Voyiadjis and Abu AI-Rub [8] developed a general thermodynamic framework for the analysis of heterogeneous media. They showed that the variety of plasticity and damage phenomena at small-scale level dictate the necessity of more than one length parameter in the gradient description. They expressed these material length-scales in terms of macroscopic measurable material parameters. However, this work concerns with the identification of the material intrinsic length-scale parameter l for gradient isotropic hardening plasticity. This can be effectively done through establishing a bridge between the plasticity at the micromechanical scale with the plasticity at the macromechanical scale. This bridge is characterized by the gradient plasticity theories. This constitutive framework yields expressions for the deformation-gradient-related intrinsic length-scale parameter in terms of measurable microstructural physical parameters. Moreover, we present a method for identifying the material intrinsic length parameter from micro- and nano-indentation tests using conical or pyramidal indenters.

Determination of the material intrinsic length of gradient plasticity theory 169

2. BRIDGING OF LENGTH SCALES

Many researchers tend to write the non-local weak form of the conventional effective plastic strain, p, in terms of its local counterpart, p,

and high-order gradient terms. The following modular generalization of p can then be defined [2] by:

(1)

where e is a length parameter that is required for dimensional consistency

and whose physical interpretation will be discussed in detail later in this

paper. TJ is the measure of the effective plastic strain gradient of any order.

The superimposed hat denotes the spatial non-local operator. r is a constant which can be interpreted as a material parameter.

The critical shear stress that is required to untangle the interactive dislocations and to induce a significant plastic deformation is defined as the Taylor flow stress:

r = [r% + rg J P (2)

with rs and rG are given by the Taylor's hardening laws related to the SSD density, Ps, and GND density, PG' respectively, as follows:

rs =asGbs.[p; (3)

rG = aGGbG/P: (4)

where bs and bG are the magnitudes of the Burgers vectors associated with SSDs and GNDs, respectively, G is the shear modulus, and as and a G are statistical coefficients which account for the deviation from regular spatial arrangements of the SSD and GND populations, respectively. P is a constant which is interpreted as a material parameter similar to that of r .

Expressing Eq. (2) in terms of Eqs. (3) and (4) yields a general expression for the overall flow stress in terms of an equivalent total dislocation density, PT' such that:

r::-. [ P/2 ( 2 2 / 2 2 )P/2 J2/ P r = asGbs'V PT WIth PT = Ps + aGbGPG asbs (5)

Eq. (5) is expressed at the microscale; however, plasticity is the macroscopic outcome from the combination of many dislocation elementary properties at the micro and mesoscopic scales. One can then write the flow stress at the macroscopic scale using a power law (0" = kjJl/m ) and Eq. (1) as follows:

O"=k[pr+{TTJrJmr (6)

where r , k , and m are material constants. It is important to emphasize that the non-local effects associated with the presence of local deformation

170 George Voyiadjis and Rashid Abu AI-Rub

gradients at a given material point are incorporated into Eq. (5) through the GNDs density, PG' and Eq. (6) through the strain-gradient, 11.

Generally, it is assumed that the total dislocation density, PT' represents the total coupling between two types of dislocations which playa significant role in the hardening mechanism. Material deformation enhances dislocation formation, dislocation motion, and dislocation storage. Dislocation storage causes material hardening. Stored dislocations generated by trapping each other in a random way are referred to as statistically-stored dislocations (SSDs), while stored dislocations required for compatible deformation within the polycrystal are called geometrically-necessary dislocations (GNDs). Their presence causes additional storage of defects and increases the deformation resistance by acting as obstacles to the SSDs [3]. The SSDs are created by homogenous strain and are related to the plastic strain, while the GNDs are related to the curvature of the crystal lattice or to the strain gradients. Plastic strain gradients appear either because of geometry of loading or because of inhomogeneous deformation in the material. Hence, GNDs are required to account for the permanent shape change. The non­local effective plastic strain in Eq. (1) is intended to measure the total dislocation density that accounts for both: dislocations that are statistically stored and geometrically necessary dislocations induced by the strain gradients.

The gradient in the plastic strain field is accommodated by the GND density, PG' so that the effective strain gradient 11 that appears in Eq. (1) can be defined as follows [3]:

11 = PG bGlF (7) where r is the Nye factor.

The plastic shear strain, yP, as a function of the SSD density, Ps' is defined as [9]:

(8)

where Ls is the mean spacing between SSDs which is usually in the order of submicron. The plastic strain in the macroscopic plasticity theory is defined in terms of the plastic shear strain and an orientation tensor as follows [9]:

6; = yPMij (9)

where Mij is the symmetric Schmidt's orientation tensor. The flow stress u is the conjugate of the effective plastic strain variable

p in macro-plasticity. For proportional, monotonically increasing plasticity, p is defmed as:

p = 26;6;/3 (10)

Hence, utilizing Eqs. (8) and (9) in Eq. (10) one can write p as a function of SSDs as follows:

(11)

Determination of the material intrinsic length of gradient plasticity theory 171

where M =~2MijMij/3 can be interpreted as the Schmidt's orientation

factor, usually taken equal to 1/2. Substituting Pa and Ps from Eqs. (7) and (11), respectively, into

(J = .fj't' , where 't' is given by Eq. (5) yields the following expression for the flow stress:

J [ ( - P/2 JI/P (J = asG'IJ 3bs / LsM pP/2 + a~baL.M r / a~bs ) "P/2 (12)

Comparing Eq. (12) with Eq. (6) yields the following relations:

y=p, m=2, k=asGJ 3b~, £=(aa/as)2(ba/bs )LsMr (13) LsM

Substituting LsM from Eq. (13)3 into Eq. (13)4 one can express the

material intrinsic length parameter .e by:

£ = 3a~bar( ~ J (14)

We note that the above equation implies that the length-scale parameter may vary with the strain-rate and temperature for a given material for the

case k = k(P,T) , where p = 2&t it /3. However, for most metals, the flow

stress increases with the strain rate and decreases with temperature increase. Thus causing the intrinsic material length-scale to decrease with increasing strain-rates and to increase with temperature decrease. However, opposite behavior is concluded for the gradient term " .

3. IDENTIFICATION FROM MICRO-HARDNESS TESTS

It is well-known by now that indentation tests at scales on the order of one micron have shown that measured hardness increases significantly with decreasing indent size. This has been attributed to the evolution of GNDs associated with gradients. Consider the indentation by a rigid cone, as shown schematically in Figure 1. GND density can then be defined by [5]:

Pa = 3 tan 2 (}/2bah (15) where h is the indentation depth, and (} is the angle between the surface of the conical indenter and the plane of the surface. This angle is related to the indentation depth h and the radius of the contact area of the indentation a through the following relation, tan (} = h/ a . Both h and a are measured in

the unloaded configuration and characterized as the residual values after unloading.

172 George Voyiadjis and Rashid Abu AI-Rub

The mapping from the hardness-indentation depth curve (H - h curve, where H is the hardness) to the tensile stress-plastic strain curve «(J - p) is defined by [10]:

H = Ka , p = c ( hi a) = c tan 0 (16)

where K is the Tabor's factor of K = 2.8 [10] and c is a material constant on the order of c = 1 [11].

The substitution ofEq. (5) into Eq. (16) with (J = 13. and Eq. (13)4 into Eq. (11), yield the following expressions for hardness, H, and the SSD density, Ps' respectively, as follows:

r:; [ PI2 ( I )p PI2 Jl/P H=....;3KasGbs Ps + aGbG asbs PG (17)

Ps = cra~bG tan 0 / Rb~a~ (18)

Moreover, we can define the macro-hardness Ho as the hardness that would arise from SSDs alone in the absence of strain gradients, such that [5]:

Ho =J)KTs =J)KasGbs'[p; (19)

With these relations we can now write the micro-hardness for the conical! pyramidal indenter using Eqs. (16) - (19) as follows:

(H/Ha)P =1+{h*/h)P!2 with h'=(R and (=3tan 3 0/2cr (20)

where h' is a material specific parameter that characterizes the depth dependence of the hardness and depends on the indenter geometry as well as on the plastic flow. Eq. (20)2 shows that h* is a linear function of the length­scale parameter R. Thus, h' is a crucial parameter that characterizes the indentation size effects and its accurate experimental measure gives a reasonable value for the length-scale parameter R obtained by using Eqs. (20)z and (20)3, We note that if f3 = 2 in Eq. (20), one retains the relation originally proposed in [5]. Moreover, substituting Eq. (18) into Eq. (19) along with Eq.(14), one can obtain a simple relation to predict the macro­hardness Ho as:

(21)

where k can be obtained from Eq. (13)3, The characteristic form for the depth dependence of the hardness

presented by Eqs. (20) gives a straight line when the data are plotted as (H / Hal versus h-PI2 , the intercept of which is I and the slope is h ,P!2 . The length-scale parameter R = h' / ( can then be calculated using Eq. (20)2, where ( is determined in terms of the shape of the conical indenter (i.e. tanO) and the material properties (i.e. r and c) which are known. Therefore, by using Eq. (20) to fit the hardness experimental data obtained from indentation tests, one can simply compute the intrinsic length-scale parameter that characterizes the size effects.

Figure I shows that Eq. (20)1 fits the hardness experimental data [4] very well for f3 = 2. From the slope of the solid line, one obtains h * = 1.538,um

Determination of the material intrinsic length of gradient plasticity theory 173

for 111 single crystal annealed Cu and h' = 0.444,um for the cold-worked polycrystal Cu. Using Eqs. (20h and (20h for tan () = 0.358, r = 2, c = I we obtain ,= 0.268 and f = 5.74,um for III single crystal annealed Cu and f = 1.66,um for the cold-worked polycrystal Cu. A value of f = 5.84,um has been reported in [5] for III single crystal annealed Cu and f = l2,um for the cold-worked polycrystal Cu.

One can note that e for the cold worked sample is smaller than the value for the annealed sample, indicating that spacing between statistically stored dislocations is reduced in the hardened-worked material. Apparently, numerical experiments of the indentation problem using finite element method are required to verify those findings.

12.0 -r-----------------------,

10.0

8.0

4.0

2.0

0.0

Experimental Data (McElhaney et al. 1997)

6 111 Single Crystal Cu

o Cold-Worked Polycrystal Cu

1.0 2.0 3.0 4.0

h·1!I2 (I'm·1I/2)

5.0 6.0 7.0 8.0

Figure 1. Comparison of the experimental results and the prediction of Eq. (20) J to determine the intrinsic material length scale for copper.

4. CONCLUSIONS

This work uses the gradient theory to bring the microstructural (described by the Taylor's hardening law) and continuum (described by the strain­hardening power law) descriptions of plasticity closer together in order to identify the material intrinsic length-scale parameter f. As a result f is defmed in terms of the average distance between statistically stored dislocations Ls; the Nye factor r; the Schmidt's orientation factor M; the Burgers vector b; and the empirical constant a .

174 George Voyiadjis and Rashid Abu AI-Rub

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support by the Air Force Institute of Technology, at Wright Patterson Air Force Base, Ohio.

REFERENCES

[1) Stelmashenko, N.A., Walls, M.G., Brown, L.M., and Milman, Y.V. (1993).

"Microindentation on W and Mo oriented single crystals: An STM study," Acta

Metallurgica et Materialia, 41, 2855-2865.

[2) Fleck, N.A. and Hutchinson, J.W. (1997). "Strain gradient plasticity," Adv. App!.

Mech., 33, 295-361.

[3) Gao, H., Huang, Y., and Nix, W.D. (1999). "Modeling plasticity at the micrometer

scale," Naturwissenschaften 86, 507 -515.

[4) Poole, W.1., Ashby, M.F., and Fleck, N.A. (1996). "Micro-hardness of annealed and

work-hardened copper polycrystals," Scripta Materialia, 34, 559-564.

[5) Nix, W.D. and Gao, H. (1998). "Indentation size effects in crystalline materials: A law

for strain gradient plasticity," J. Mech. Phys. Solids, 46, 411-425.

[6) McElhaney, K.W., Valssak, J.J., and Nix, W.D. (1998). "Determination of indenter tip

geometry and indentation contact area for depth sensing indentation experiments," 1. Mater. Res., 13, 1300-1306.

[7) Begley, M.R. and Hutchinson, 1.W. (1998). "The mechanics of size-dependent

indentation," 1. Mech. Phys. Solids, 46, 2049-2068.

[8) Voyiadjis, G.Z. and Abu AI-Rub, R.K. (2002). "Thermodynamic formulations for non­

local coupling of visco plasticity and anisotropic viscodamage for dynamic localization

problems using gradient theory," Int. J. Plasticity (submitted for publication).

[9) Bammann, D.J. and Aifantis, E.C. (1982). "On a proposal for a continuum with microstructure," Acta Mech., 45, 91-121.

[10) Tabor, D. (1951). The hardness of metals, Clarendon Press, Oxford.

[II) Xue, Z., Huang, Y., Hwang, K.C., and Li, M. (2002). "The influence of indenter tip

radius on the micro-indentation hardness," ASME 1. Eng. Mater. Techno!., 124, 371-

379.

COMPUTER SIMULATION OF CONTACT

FORCE DISTRIBUTION IN RANDOM

GRANULAR PAC KINGS

A.H.W. Ngan

Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China

Abstract Discrete element simulations show that the contact force distribution in a stressed granular packing can be described by a single, lumped parameter known as the "mechanical temperature". The simulated results suggest that equilibrium is governed by a free energy functional containing an energy and an entropy component. Similar to the role of the conventional thermal temperature, the mechanical temperature controls the relative importance between energy and entropy.

Keywords: Discrete element simulation, granular materials, statistical physics

1 INTRODUCTION

Many engineering materials are randomly structured either by design or otherwise. Examples include amorphous solids or polymers, or macroscopically disordered materials such as foam materials or random grain piles. Because of structural disorderness, the internal force distribution in these materials due to external loadings will not be uniform, and a satisfactory description should involve concepts from statistical physics. In this work, discrete element simulations were performed to model the internal forces of random granular packings. Granular packings are used here as a prototype for random materials because a wide body of literature has been established through intensive investigations in the past decade [1-13]. It is

175

S. Ahzi et al. (eds.). Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 175-182. ©2004 Kluwer Academic Publishers.

176 A.HW. Ngan

expected that the concepts developed in this work should also be valid for other random materials such as open cell foam materials.

2 COMPUTER SIMULATION

Computer simulations were performed on elastic grains contained in a box, with compressive load applied from the top through a piston. Both 2-D and 3-D simulations were performed. The 2-D simulations were performed on 1 x 104 elastic grains, and the grain sizes distributed approximately uniformly throughout a range of ± 10 % of the mean value to prevent crystallization. For 3-D, 5x104 grains were simulated, and the grain size was uniform since it was observed that the structure did not crystallize easily. The Young's modulus for 3-D was selected to be 200 GPa, and Poisson's ratio 0.3. Only elastic Hertzian contact forces [14] were assumed, and friction was not included in the simulations.

10'

~ .~ 10' \ ............................ . (l)

o ~ ~ 10-2 ............ ----.. ------

.0 e a..

10" .---- .. -.. -------

o 2 3

--Model curve: "20 = 0.2

--Model curve: "20 = 0.7 o Load = 0.0001 unit /', Load = 0.001 unit o Load = 0.01 unit 'V Load" 0.1 unit o Load" 1 unit o Load = 10 units

4 5 6

Normalised Contact Force

Figure 1; Computer simulation results of contact force distribution in 2-D under hydrostatic load. 1 unit ofload = 2 x 10-3 Ed, E = Young's modulus, J = average grain size.

Fig. 1 shows the simulated results for 2-D. Here, the forces are normalized by the mean force value. It can be seen that the normalized force distribution is invariant with respect to the applied hydrostatic load over a

Computer simulation of contact force distribution in random granular 177 packings

four orders-of-magnitude change of load up to about 1 unit. Beyond 1 unit of load, the force distribution changed to another form with a smaller variance. Accompanying this change was an observed drastic change in the average number of contacts per grain, which remained roughly constant at the rigid­grain limit [15] of 4 when the load was smaller than about 1 unit, but increased to larger than 5 when the load was 10 units. The higher coordination at large loads corresponds to a more regular arrangement or increased degree of crystallinity of the packing. The 3-D simulation results are shown in fig. 2. Here, it can be seen that the force distributions under applied pressure from 0.001 to 1 GPa are all invariant with respect to load.

10' 3D Unifonn Compaction Grain size'" 3 nm, no cohesion

10· -,I ............ ; ..... .. ..... , ............ .. ; ........... . ; ........ . --Model curve: K30 = 0.7

6 Load = 0.001 GPa z. "V Load = 0,01 GPa .~ 10" + ...... · .. ·~ .. ·········F".""· .. ·i···· .. · ·····~· .. ···· .. · o Load = 0,1 GPa Q)

o / Load = 1 GPa

~ ii 10·' + .......... . ; .. .. ..... ... i ............. i .... <iI~ i!: e

11.

10 .. + .. · ...... ·; .. · .. · ...... ·i ............ ·; .. · .... · .... ·i· .. -:-; 'i~ .. ... ..... -............ .

¢. C. I~

10~+-~+-~r-~~~~-T~~~I~~~~~~'~_' -~'~~:~_· +f_~6~ o 2 3 4 5 6 7 8 9 10

Normalised Contact Force

Figure 2: Computer simulation results of contact force distribution in 3-D under hydrostatic load.

3 THEORY

A force distribution Pif) enables the definition of an entropy in the statistical sense:

178 A.H.W. Ngan

00

S = - fp(f) In [P(f)]df . (1) o

Bagi [5] and Evesque [6] have argued that in a structurally random packing, the entropy in eqn. (1) should attain maximum value. When this entropy is maximized subj ect to the constraints

00 00

f jP(f)df = 1 = constant, and f P(f)df = 1 , (2) o 0

the force distribution would be the Maxwell-Boltzmann (MB) distribution

P(f) = exp( - f / 1) . The simulation results shown in figs. 1 & 2 above are

clearly not MB. On the other hand, a Hertzian contact force fbetween two grains will be associated with a work done W(f). With a force distribution P(j), this will be associated with an energy functional

00

E = JP(f)W(f)df, (3) o

which, when minimized subject to the same constraints in eqn. (2), will yield

(4)

where A;'s are Lagrange multipliers. Since eqn. (4) does not involve P(f), the solution to it would be the value all contact forces should adopt. In other words, minimization of the energy functional alone always yields the delta function for P(f) and not a distribution.

The observed force distributions in figs. 1 & 2 are intermediate between the two extreme cases of pure energy and pure entropy. In other words, equilibrium is given by a compromise between energy minimization and entropy maximization. This has the same spirit as minimization of the Helmholtz or Gibbs free energy in a thermal system, but nevertheless cannot be embodied by the existing definitions of these free energies because the current mechanical problem is athermal. To express the minimization principle in the purely mechanical case, one would need to introduce a mechanical analogue of temperature, e, so that the analogue of the thermal free energy is

F=E-fE. (5)

Computer simulation of contact force distribution in random granular 179 packings

E here is the strain energy functional in eqn. (3), and S the entropy defined in eqn. (1). Bhas the same dimension as E, and is analogous to the product kT, the Boltzmann constant times the absolute temperature, in the thermal case. B = 0 means that minimization of F is equivalent to minimization of E, which as we have seen above will yield the perfect crystallinity behaviour. On the other hand, B ~ 00 corresponds to the MB behaviour. Hence, B is a signature for the structural randomness of the granular packing.

For a fixed B, the equilibrium force distribution can be obtained by minimizing F subject to the constraints in eqn. (2). This yields

P(f) = A exp[- ~(W(f) - ,1/)], (6)

where A and A are normalization constants which make Pif) satisfy (2). In a 2-D granular packing, the contact force between two grains is given by

f = rrEra 2 12R, where Er is the reduced modulus, R the radius of the

cylindrical grains, and a the radius of the contact zone [14]. The work done

by f is W(f) = - If' dr da, where r = 2~ R2 - a 2 is the distance between o da

the grain centers. Wif) can be shown to be given by

where the simplification at the end is accurate whenflErR is small compared to unity. With eqn. (7), Pif) in eqn. (6) would adopt a Gaussian form

(8)

() - - .&: 2]2 1.. d

where / = / / /' / being the mean loree, K = --. - IS an Inverse an 3rrEr B

dimensionless measure of the mechanical temperature, and A and 10 are normalization constants. Similarly, for a 3-D granular packing, the force distribution can be shown to be

P(f) = A expl- K((J) 5/3 - A(J) )j, (9)

180 A.H.W Ngan

10'

10·

i:!" -iii c Q)

Cl 10"

~ :0 m .0 e 10-' a..

10->

10 0.4 , 0.2 I( = 0.01

10~

0 2 3 4 5 6 7 8 9 10

Normalised contact force

Figure 3: 20 equilibrium force distribution at different "mechanical" temperatures. Kis an inverse measure of the mechanical temperature o (see text).

10 '

10·

i:!" -iii c Q)

Cl 10"

~ :0 m .0 e 10-' a..

10->

10 0.4 0.2 I( = 0.01

10~

0 2 3 4 5 6 7 8 9 10

Normalised contact force

Figure 4: 30 equilibrium force distribution at different "mechanical" temperatures. Kis an inverse measure of the mechanical temperature o (see text).

Computer simulation of contact force distribution in random granular 181 packings

where K = ~( 3R J2/3 j5/3 . .!.., and A and /I.. are normalization constants.

5R 8E, ()

Figs. 3 & 4 show the equilibrium Pif ) as at different mechanical temperatures for 2-D and 3-D respectively.

In the 2-D simulated results in fig. 1, the probability curves from 0.0001 to I unit of load can be fitted accurately by eqn. (8) with K= 0.2. The curve at 10 units of load can be fitted accurately by K= 0.7. The good fit in both cases indicates the validity of the theory above, namely, the equilibrium distribution corresponds to minimization of F = E - as at constant g. The fitted results also indicate that K is constant over a four orders-of-magnitude change in the applied load up to about 1 unit, but starts to decrease when the load becomes larger. The 3-D simulation results shown in fig. 2 under applied pressure from 0.001 to 1 GPa can all be accurately fitted by eqn. (9) with the K3D parameter chosen to be 0.7.

4 CONCLUSIONS

In stressed granular packings, the internal force distribution due to external loading is not uniform. The internal force is associated with a strain energy functional and the distribution of forces enables the definition of an an entropy functional. Computer simulation results of the force distribution show that equilibrium is governed by a compromise between the energy and entropy, i.e. the equilibrium force distribution is given by minimization of a free energy functional which is a mixture of energy and entropy. The mixity between energy and entropy is controlled by a parameter known as the "mechanical temperature", which is an increasing function of the applied stress.

ACKNOWLEDGMENT

This work is supported by a research grant from the University Research Committee of the University of Hong Kong (Project number: 10204222.16180.14500.323.01)

REFERENCES

[1] S.F. Edwards, in "Granular Matter - An Interdisciplinary Approach", ed. A. Mehta, (Springer-Verlag: New York), (1994), Chp. 4. [2] C.H. Liu et ai, Science, 269, 513 (1995). [3] F. Radjai et ai, Phys. Rev. Lett., 77, 274 (1996). [4] D.M. Mueth, H.M. Jaeger and S.R. Nagel, Phys. Rev. E, 57, 3164 (1998). [5] K. Bagi, in "Powders and Grains 97", ed. R.P. Behringer and J.T. Jenkins, (A.A. Balkema, Rotterdam, Netherlands), p. 251 (1997).

182 A.HWNgan

[6] P. Evesque, in "Powders and Grains 2001", ed. Y. Kishino, (A.A. Ba1kema, Lisse, Netherlands), p. 153 (2001). [7] N.A. Makse, D.L. Johnson and L.M. Schwartz, Phys. Rev. Lett., 84,4160 (2000). [8] M.L. Nguyen and S.N. Coppersmith, Phys. Rev. E, 62, 5248 (2000). [9] C.S.O'Hem et ai, Phys. Rev. Lett., 86, 111 (2001). [10] S.N. Coppersmith et ai, Phys. Rev. E., 53, 4673 (1996). [11] J.E.S. Soco1ar, Phys. Rev. E, 57, 3204 (1998). [12] P. C1audin et ai, Phys. Rev. E, 57,4441 (1998). [13] O. Narayan, Phys. Rev. E, 63, 010301(R) (2000). [14] K.L. Johnson, "Contact Mechanics", Cambridge University Press, (1985), p. 101. [15] C.F. Moukarze1, in "Rigidity Theory and Applications", ed. M.F. Thorpe and P.M. Duxbury, (K1uwer AcademiclPlenum, New York), p. 125 (1999).

THREE-DIMENSIONAL STRUCTURES OF THE GEOMETRICALLY NECESSARY DISLOCATIONS GENERATED FROM NON-UNIFORMITIES IN METAL MICROSTRUCTURES

Tetsuya Ohashi

Kitami Institute of Technology, Koencho 165, Kitami, 090-8507, Japan E-mail: ohashi@newton.mech.kitami-it.acJp

Abstract: Slip defonnation in microstructures of f.c.c. type metals are analyzed by a finite element technique and the density distribution of the geometrically necessary dislocations is evaluated. Results show development of wall like structure of dislocations in some of single crystals and also dislocation half loops within grains and their pile up at grain boundaries in mUltiple crystal models.

Key words: single and multiple crystals, microstructure, crystal plasticity analysis.

1. INTRODUCTION

Development of dislocation structures during deformation has long been studied. Two types of dislocation densities can be evaluated; the statistically stored (SS) and the geometrically necessary (GN) dislocations (Ashby, 1970). Density increment of the SS dislocations is related to the increment of plastic shear strain and the mean free path of moving dislocations, while the density of the GN ones is related to the spatial gradient of the plastic shear strain on slip systems. Scale dependent characteristics of the GN dislocations have been attracting much attention in the research field of solid mechanics and some models for scale dependent crystal plasticity constitutive laws were proposed (for example, Fleck, et al.,

183

S. Ahzi el al. (eds.), Mulliscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 183-190. ©2004 Kluwer Academic Publishers.

184 Three dimensional structures of the geometrically necessary dislocations generated from non-uniformities in metals microstructures.

1994). On the other hand, the structure of the aggregate of the GN dislocations is less studied, although some typical structures such as the Orowan loops formed around precipitates or piled up dislocations at grain boundaries result from non-uniform deformation and thus, be understood as the ones made up of the GN dislocations.

In the present paper, we analyze slip deformation in single and multiple crystals of the face centered cubic type metals and evaluate edge and screw components of the GN dislocations. Their density norm and direction vector of the dislocation line segments are also deduced from the edge and screw components. After this process, we can reconstruct images for three-dimensional structures of the GN dislocations in deformed microstructures.

2. BASIC EQUATIONS

Slip deformation is supposed to take place on {Ill} crystal plane and in < 11 0> crystal direction. The activation condition of the slip system n is supposed to be given by the Schmid law;

P en) = B(n) p(n)' = B'(n) ij (Y ij 'ij (Y ij , (n = 1, .. ·,12), (1)

and,

(2)

where, (Y ij and B(n) denote the stress and the critical resolved shear stress on

the slip system n, respectively. The slip plane normal v}nl and the slip

direction bin) define the Schmid tensor ~in). Quantities with dot indicate

increments of them. Increment of the critical resolved shear stress is written as follows;

(3) m

Here, t and i(m) denote the increments of temperature and the plastic

shear strain on slip system m, respectively. If the deformation is small and rotation of the crystal orientation is neglected, the constitutive equation is written as follows (Hill, 1966, Ohashi, 1987, 1994),

(4a)

Tetsuya Ohashi 185

where,

(4b)

and,

(4c) m

S;kl' a, and Dkl denote elastic compliance, thermal expansion coefficient

and the Kronecker's delta, respectively. Summation is made over the active slip systems.

Let us suppose that the critical resolved shear stress is a function of the Bailey-Hirsch type and given by the following equation (Ohashi, 1987, 1994);

(5)

where, ()o (T) denotes the lattice friction term, which is, in general,

dependent on temperature, and p~m) denotes the dislocation density that

accumulate on the slip system m. Reaction between dislocations on slip systems nand m defines the magnitude of the interaction matrix n(nm). In the present study, we choose parameters to express pseud-isotropic hardening character for every slip system.

The dislocation density on the slip system n is given by the following equation;

(6)

where, p~n) and IIp~n)11 denote the densities of the SS and GN dislocations,

respectively. Increment of the SS dislocations is given as follows;

.(n) _ C"( .(%) Ps - bL(n) , (7)

where, L(n) is the mean free path of dislocations on slip system n and, in this paper, we use the modified Seeger's model for it;

186 Three dimensional structures of the geometrically necessary dislocations generated from non-uniformities in metals microstructures.

1 L(n)

L'" - ~ r'O' _ (y~ _ AI 4;') {

single slip for , (8)

multiple slip

where, A is a material constant and yO denotes the plastic shear strain when multiple slip start.

The edge and screw components of the geometrically necessary dislocations are obtained from the strain gradients (Ohashi, 1997);

10, .. }n) (n) I

PG •edge = -I8e' lo",(n)

(n) _ I

PG,screw - be( (9)

Here, ~ and , denote directions parallel and perpendicular to the slip direction on the slip plane, respectively. Norm of two components defines the scalar density for the GN dislocations,

(10)

Evaluation of the edge and screw components for the GN dislocations enables one to calculate the tangent vector I(n) of the dislocation line segments (Ohashi, 1999);

I(n) - _1_( (0) • b(o) + p(o) • b(n) X v(n») (11) - Ilp~n) II PG,screw G,edge •

Data for GN dislocations are obtained for each finite element and then, we can draw line segments of dislocations in three-dimensional space. We will draw one line segment at the center of each element. Direction of the line segment is given by eq. (11) and its length and thickness is determined by

the density norm Ilpb)ll. Numerical parameters s and g in the equation (6) are introduced to

control the complexity of the simulation. In the present paper, we suppose s= 1 and g=0 for simplicity and the strain hardening coefficients in equation (3) are given by the following equation;

o q =-() (T) aT 0 ,

(12a)

%tsuya Ohashi

(12b)

3. RESULTS AND DISCUSSION

3.1 Single crystals with non-uniform initial dislocation densities

187

Let us examine tensile deformation of single crystal bars where distribution of the initial dislocation densities is not uniform. Figure 1 shows the geometry of the specimen employed for the analysis. The specimen is divided into 8x30x8 finite elements of composite type with eight nodes and the top surface is subjected to a uniform displacement in y direction, while the bottom surface is fixed. Initial dislocation densities on twelve slip systems in each elements are decided by a normally distributed deviates with the mean value Po = 109 m-2• Crystal orientation of specimens #1 - #3 is the same and positioned so as that the Schmid factor of the primary slip system is at the maximum value of 0.5. Slip plane and slip direction for the primary system in the specimens #1 - #3 are schematically shown in Figure 1. While, the orientation of the specimen #4 is very close to double slip orientation. The standard deviations of the initial dislocation densities in the specimens #1 - #3 are 0, O.lpo, and 0.25po, respectively. The standard deviation for the specimen #4 is 0.1 Po.

Figure 2 shows numerical results for the load-elongation curve. In specimens #2 - #4, slip on a secondary (conjugate) slip system superimpose after some amount of slip on the primary one and this causes decrease in the mean free path of the dislocations as shown in eq. (8) and result in the onset of the deformation stage II. Duration of the stage I depends on the crystal orientation and the magnitude of the non-uniformity of the initial dislocation density. Figure 3 compares density distributions of the GN and SS dislocations on the primary slip system in the specimen #3 when the average tensile strain is 4.625%. Walls of GN dislocations, which consist mainly of edge type dislocations, develop in the direction perpendicular to the slip plane and extend as wide as the width of the whole specimen. On the other hand, the distribution of the SS dislocations remains to be random, although the density is one order higher than that of GNDs. Tri-axial stress field, which accompanies to non-uniform slip is supposed to be responsible to the generation of the long-range structure of GNDs.

3.2 A multi-crystal plate of copper

Figure 4(a) shows the multi-crystal model we employed in this study.

188 Three dimensional structures of the geometrically necessary dislocations generated from non-uniformities in metals microstructures.

y ..

42

Figure 1 Geometry of the single crystal specimen employed in this study. Dimensions are given in unit off,Lm.

lOll m ·2

3,22

2,51

1.80

1,09

0,38

YL x

14 0:1 12 c... ~ vi 10 Vl Q)

8 !:: '" 0:1 6 c::

E 4 a :z

2

0 0 0.02 0.04 0.06 0.08 0.1 0.12

Nominal strain

Figure 2 Load-elongation curves calculated for single crystal specimens #1 - #4. The initial dislocation density for the specimens #2 - #4 is not uniform and given by normally distributed deviates. See text for details.

10 12 m·2

4 .S9

3.70

2,82

1.93

1.05

(8) (b)

Figure 3 Cross sectional views for the distribution of the geometrically necessary dislocations (a), and statistically stored ones (b) in the single crystal specimen #3, when the mean tensile strain is 4.625 %.

Thtsuya Ohashi

~ ~

}. z x (a)

.. ,/\ ~ (b)

15

Figure 4(a) Geometry of the multi-crystal model. Dimensions are given in f.lm. (b) Crystal orientation of grains 1-6.

, I

!

~ (a)

/

I I

189

r

~ . . ., /

4

2

o

0.061 0.068 x I 0-2

Figure 5(a)-(c): Distribution of plastic shear strain on the primary slip system when the average tensile strain is 5.3, 6.1, and 6.8 xl0-4, respectively. (d)-(f) Density distribution of the geometrically necessary dislocations which correspond to the primary slip shown in (a)-(c).

Figure 6 Dislocation segments in a thin foil in the grain 1. The foil is parallel to the primary slip plane and its thickness is 0.4 J.Illl. View direction of this figure and position of the foil is illustrated in Figure 4(a).

190 Three dimensional strnctures of the geometrically necessary dislocations generated from non-uniformities in metals microstrnctures.

The model is made from six copper crystal grains and their orientations are determined by random numbers and exhibited in Figure 4(b). All grain boundary planes are flat and positioned perpendicular to the x-y plane. The specimen is divided into 4864 finite elements and uniform tensile displacement is given to the top and bottom surfaces.

Figure 5(a)-(c) show evolution of the plastic shear strain on the primary slip systems at three stages of deformation. The first plastic slip takes place in the grain I near a grain boundary between the grains I and 2, although the slip at the interior of the grain I starts immediately after it and grows faster. Slip deformation in the grain I induces slip deformation in the grains 2, 3 and 4, which start from grain boundary triple junctions. Figure 5( d)-( f) show distribution of GN dislocations. . Rather uniform accumulation of GN dislocations in the grain I is observed first, and then the density near grain boundaries gradually builds up. To examine the structure of GN dislocations in more detail, we cut out a foil from the grain I and observe the structure. The foil is schematically illustrated as a platelet in the grain I in Figure 4(a). The foil is parallel to the slip plane and its thickness is 0.4~. Figure 6 shows the line segments of GN dislocations, which are positioned within the volume of the foil. Half loop shaped structure of dislocations is observed to expand from specimen surface and the grain boundary pile up of dislocations is also observed.

4. SUMMARY

We analyzed plastic slip deformation in FCC type single- and multi­crystals and three-dimensional structure of the geometrically necessary dislocations were evaluated. Results for single crystals showed a development of wall shaped structure of GN dislocations and fairly random distribution of SS dislocations. GN dislocations in a multi-crystal model were depicted to emerge from specimen surface and grow in the shape of half-loops before they pile-up at grain boundaries.

References

Ashby, M.F., 1970, Phil. Mag. 21, 399.

Fleck, N.A., et al., 1994, Acta metall. mater., 42, 475.

Hill, R. 1966, J. Mech. Phys. Sol., 14,95-102. Ohashi, T., 1987, Trans. Japan Inst. Met. 28, 906.

Ohashi, T., 1994, Phil. Mag., A70, 793.

Ohashi, T., 1997, Phil. Mag. Lett., 75, 51. Ohashi, T., 1999, J. Phys. IV France, 9, Pr9-279.

SIMULATION OF TEXTURE EVOLUTION IN EQUAL CHANNEL ANGULAR EXTRUSION OF COPPER USING A NEW FLOW FIELD

Laszlo S. Totht, Roxane Massiont, Lionel Gennain1, Seung.C. Baik2

1 Laboratoire de Physique et Mecanique des Materiaux, Universite de Metz, lIe du Saulcy 57045 Metz. France, laszlo(ij)lpmm.univ-metz.{r 2lnstitut! WerkstofJkunde u. Werkstofftechnik, TU Claus thaI, Agricolastr. 6

38678 Clausthal-ZelierJeld, Germany,

Abstract: The deformation textures of OFHC copper that develop in r;qual ~hannel Angular r;xtrusion (ECAE) from one to 12 passes are examined in route A. They are also simulated with the help of the viscoplastic Taylor and Self Consistent models, first using the classical discontinuous shear approach, then a new analytical flow line model. The latter one is calibrated with finite element calculations of the ECAE test. It is shown that the new model describes the evolution of the texture better than the discontinuous shear approach.

1. INTRODUCTION

The evolution of the microstructure during severe plastic defonnation is the subject of intensive investigations recently, due to the unique physical and mechanical properties of severely defonned materials [1]. Equal !;;hannel Angular Extrusion (ECAE), where ultra-fine grain sizes (~ 200 nm) and very large strains can be obtained, receives special attention.

These last years, two models (the Taylor and the Self Consistent models of polycrystal viscoplasticity) have been used to predict the texture evolution in ECAE defonnation [2-7].

Using texture measurements, it will be shown first in the present work that the discontinuous approach gives only a first approximation of the textures. Then a new, more precise flow field is proposed which uses an

191

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 191-198. ©2004 Kluwer Academic Publishers.

192 Laszlo S. Toth, Roxane Massion, Lionel Germain, Seung. C. Baik

analytical flow function. The texture predictions improve significantly with the new flow line description of ECAE deformation.

2. COMPARISON OF ECAE TEXTURES TO TEXTURES OF SIMPLE SHEAR

As the shape change of a material element leaving the deformation zone of the die is exactly that of produced by simple shear (see Fig. I), it is legitimate to compare the ECAE textures with simple shear textures (to save space here only the f/JJ. = 0° sections of the ODFs are shown).

y y

a) b)

x x z

Figure I. a: The ECAE test. b: Strain mode in the die.

The similarity between the textures of the two different tests is quite striking (Fig. 2).

ECAE, one p.,. Simple shear, y ~ 2 100

ECAE, 1""0 pa .... Simple shear, y ~ 5.5 ,.0 o

$0 +P=HI::-_~---..L....I.--1

Figure 2. Comparison of ECAE textures and simple shear textures of copper.

It is also interesting to examine the "tilts" of the texture components of ECAE deformation in the f/JJ. = 45° section. In contrast to simple shear, the

Simulation of texture evolution in equal channel angular extrusion 193 of copper using a new flow field

ideal components are clearly rotated away from the ideal positions in the direction of the rigid body rotation in Fig. 3b.

a) b) o· 180'

o' ,----------------------, ;

A I" c A2* ......... " ...... .... " .. .. ............... , ................ ,""' . ................ ,

A

w·~~~ __ ~~~~~~~ w· ~---------------------J ~ L.J d2" AI· c

Figure 3. a: Key figure for ideal orientalions for ¢2=O°. b: The ¢l= 45° section of the ODF ofcopper after one pass in the ECAE die. Bold poinls are the ideal positions ofsimple

shear textures. The arrow indicates the average lauice rolalion in case of simple shear.

3. THE DISCONTINUOUS SHEAR MODEL

The velocity gradient of simple shear (L ') in the plane of discontinuity and in the x-y reference system of Fig. I (L) are the following:

-f o o

with f positive. L is illustrated in Fig. 1 b.

-1

-1 (1)

o

Fig. 4 displays the textures obtained by the Taylor model after one pass and after two passes. By comparing this prediction to the experimental textures (Fig. 2 and Fig. 3b), one can see that there are significant differences. First of all, the Al * component is absent after one pass in the simulation. The A2* component is not strong enough, and seems also to be too close to the C component. After the second pass, the Al * component is present but in an excessive rotated position with respect to the experience.

One pass:

Two passe.!:

I_,J: :+:J ~L_i ~i ~E:llJJlLI

Figure 4. Two sections of the ODFs obtained by the Taylor approach using the discontinuous shear model after one and two passes in the ECAE die.

194 Laszlo S. Toth, Roxane Massion, Lionel Germain, Seung.c. Baik

The results obtained with the SC model are better than those of the Taylor model (Fig. 5). The Al * component is present already after the first pass and the rotated cube component is predicted in its correct position. The A I * and C components, however, are in too much rotated positions.

,~ __ ~ ______ ~ __ ~'OO o~o ______ ~ ______ ~'00

Two passes:

Figure 5. Two sections of the ODFs obtained by Self Consistent approach with the discontinuous shear model after one and two passes in the ECAE die.

4. FLOW LINE MODEL

In order to avoid the discontinuity of the deformation process in the classical approach described above, the following flow function is proposed to better approximate the material flow in the ECAE die:

(2)

In this expression d is the diameter of the die, Xo defines the incoming (and outgoing) position of the flow line, and n is a parameter (see Fig. 6). This function is only valid in the working part of the die, i.e. between the lines defined by y = d and x = d. At these positions, the flow lines are perfectly parallel to the compression direction and the outgoing flow direction, respectively. The n parameter describes the possible shapes of the flow lines.

The velocity field corresponding to the flow function ¢ is given by:

( ),,-1 ()"-1 d-x d-y Vx =Vo -- , Vy =-Vo -- ,

d -Xo d -Xo (3)

Simulation of texture evolution in equal channel angular extrusion 195 of copper using a new flow field

with Vo the incoming velocity of the material.

y

Xu

~ .... \

a\

d

n=4

d

z ~--------------------~-. X

Figure 6. Description of the flow field by flow lines . .. n .. indicates the value of the exponent in the proposed flow jUnction determined by finite elements calculations.

The velocity gradient field can be obtained by simple partial derivation of (3), after fully expressing Vx and Vy as a function of the coordinates x and y (by using Eq. (2) in (3)). One obtains:

8v ()( )n-l ( )n-l ( )1-2n Lxx = _x = -vo 1- n d - x d - y d - Xo , ox Lyy = -Lxx,

8v ()( )n ( )n-2 ( )1-2n Lxy = a; = Vo 1- n d - x d - y d - Xo , (4)

8v y ()( )n ( )n-2 ( )1-2n Lyx =-=-vo I-n d - Y d -x d -xo . ox

The obtained velocity gradient field describes compression along axis y, tension in direction x and shear on both the y and x planes. It gives also a large rigid body rotation, maximum at the symmetry plane of the flow (at 45°). Although (4) is almost of the same nature as that of corresponding to the discontinuous shear model (see Fig. Ib), it differs from it by its dependence on the position along the flow line.

The strain rate tensor is the symmetrical part of the velocity gradient:

196 Laszlo S. Toth, Roxane Massion, Lionel Germain, Seung.c. Baik

( ) ( )n-l ( )n-l ( )1-2n ixx = -vo 1- n d - x d - y d - Xo ,

. () ( )n-l ( )n-l ( )1-2n b'y'y=vo I-n d-x d-y d-xo ,

ixy =~vo(1-n)(d _xo)'-2n [(d -xf (d - yr2 -(d -yf (d -xr2] .

(5)

The equivalent strain rate in the sense of von Mises is:

- 1 ()1-2, ( )'-2 ( ),-2 [( )2 ( )2 ] dE 0= .Jjvo(n-l) d-xo d-x d-y d-x + d-y .

(6)

Eq. (6) can be integrated along a flow line, it gives:

(7)

This formula of the total accumulated equivalent strain is interesting. As it is independent of the d and Xo parameters, it- means that the total strain is the same in the whole cross section of the die, as long as the n parameter is kept constant.

In order to identify the value of the n exponent of the flow field, results of finite element calculations were used. Details about the finite element simulations are available in [6].

0,025 ..--~------

0,02

0,Q15

~ 0,01

~ 0,005

~ -0,005

~ ·0,01

·0,015

10 20 30 40 50 60 70 80

angular position on the flow line

Figure 7. Comparison of the strain rate components obtained from finite elements calculations (symbols) with that of the flow line

model (continuous line) for one flow line identified by n=4.

90

Simulation of texture evolution in equal channel angular extrusion 197 of copper using a new flow field

The analytical flow line model gives a velocity field which is in good agreement with the finite element results; an example is shown in Fig. 7. n values between 4 and 9 were identified, depending on the position of the flow line within the die.

5. TEXTURES PREDICTED BY THE FLOW LINE MODEL

Deformation texture development was predicted using the present flow line model by employing the Taylor and SC polycrystal models. The initial texture was placed on a flow line at an infinitesimal point and subjected to the strain field defined by the velocity gradient in Eq. (4). The results obtained for an n value of 8 are presented in Fig. 8:

Taylor model Self Consistent model

Pass one: Jtm~I[I'j o l~

]! ] -!:[J~~o 0--""-----1' I Pass two:

Figure 8. Textures predicted by the flow line approach with n =8.

Both the Taylor and SC models lead to textures which are in much better agreement with the experiment (Fig. 2), when they are compared to the textures of the discontinuous shear approach (Figs. 4-5). Even the Taylor model reproduces the Ai * component in the very first pass, which was absent in the simulation carried out by the classical approach (compare Fig. 8 with Fig. 4). Another important result of the flow line model is that the rotated positions of the ideal components are well reproduced. The SC model gives similar results with the difference that the fiber nature of the Al * component in the first pass is also well reproduced. The main feature of the experimental textures is that the C and A2* components are the major ones in the first pass, while it is the C which is the strongest component of the texture in the second pass. These relative intensities of the predicted texture components are reproduced for both polycrystal models employed.

198 Laszlo S. Toth, Roxane Massion, Lionel Germain, Seung.c. Baik

6. CONCLUSIONS

The strain field in ECAE testing of metals was investigated in the present paper. The evolution of crystallographic texture was used as an indicator of the strain mode that a polycrystal experiences during passing through the die. The classical discontinuous shear approach was examined and a new analytical flow line field has also been proposed. The latter one was tuned with finite element results. From the results obtained, the following conclusions can be drawn:

1. ODFs ofECAE textures are similar to ODFs of simple shear, however, with dpposite tilts of the ideal components. 2. The classical discontinuous approach gives textures that are about 10°

rotated with respect to measured ones.3. A new flow line description in 90° ECAE testing has been proposed with one parameter only. 4. The proposed flow line gives a varying deformation field along the flow line in good agreement with fmite element calculations. 5. The Taylor and Self Consistent models implemented into the flow lines give textures in reasonable agreement with experiments up to two passes.

ACKNOWLEDGEMENT

L.S. Toth acknowledges helpful discussions on the subject with Profs. A. Eberhardt and A. Molinari (University of Metz).

REFERENCES [1] Segal V.M., Mat. Sci. Eng., 1995, A197, 157. [2] Toth L.S., Kopacz I., Zehetbauer M., Alexandrov I.V., Proc. TIIERMEC-2000, Eds.

Chandra T., Higashi K., Suryanarayana C., Tome C., THERMEC-2000, Las Vegas, USA, Dec. 2000, on CD.

[3] Kopacz I., Zehetbauer M., Toth L.S., Alexandrov 1.V., Proc. 22ru1 Riso Int. Symp. on Mechanical Science: Science of Metastable and Nanocrystalline Alloys Structure, Properties and Modelling, Eds. Dinesen A.R., Eldrup M., Juul Jensen D., Linderoth S., Pedersen T.B., Pryds N.H., Schroder Pedersen A., Wert J.A., Roskilde, Denmark, Sept. 2001,295.

[4] Agnew S.R., Kocks U.F., Hartwig K.T., Weertman J.R., Proc. 19th Riso. Int. Symp. Mat. Sci., Denmark, 1998,201.

[5] Agnew S.R., Proc. of ICOTOMI2, the 12th Int. Conf. on Textures of Materia is, Ed. Szpunar J.A., Montreal, August 9-13, 1999,575.

[6] Baik S.C, Estrin Y., Kim H.S., Hellmig R.J., .. Dislocation Density Based Modeling of Deformation Behaviour of Aluminium under Equal Channel Angular Pressing", to be published.

[7] Gholinia A., Bate P., Prangnell P.B., Acta Mater., 2002, 50, 2121.

INITIAL ENERGY DISSIPATION MECHANISM AT CRACK TIP ON THE DUCTILE TO BRITTLE TRANSITION

Jeffrey W. Kysar Department of Mechanical Engineering Columbia University, New York, NY 10027, USA

Abstract: The objective of this study is to investigate energy dissipation mechanisms that operate at different length scales during fracture in ductile materials. A dimensional analysis is performed to identify the sets of dimensionless parameters which contribute to energy dissipation via dislocation-mediated plastic deformation at a crack tip. However rather than use phenomenological variables such as yield stress and hardening modulus in the analysis, physical variables such as dislocation density, Burgers vector and Peierls stress are used. It is then shown via elementary arguments that the resulting dimensionless parameters can be interpreted in terms of competitions between various energy dissipation mechanisms at different length scales, for example between dislocation nucleation from a crack tip and dislocation nucleation from a Frank-Read dislocation source in the material close to the crack tip. Criteria are established which are used to determine the initial, and perhaps dominant, energy dissipation mechanism at a crack tip.

Keywords: Fracture, dislocation, crack tip dislocation nucleation, Frank-Read dislocation

source, ductile to brittle transition

1. INTRODUCTION

The brittle to ductile transition of a material in the presence of a crack is of great interest. Two distinct approaches to understanding this transition are commonly invoked. In one approach, dislocation mobility plays the key role. In the other approach, the competition between crack tip dislocation nucleation and cleavage plays the key role. The two approaches are reconciled in the present study. This is accomplished by considering the

199

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 199-206. ©2004 Kluwer Academic Publishers.

200 Jeffrey W. Kysar

competitions that exist between various energy dissipation mechanisms which contribute to macroscopically measured fracture toughness. A dimensional analysis is performed and it is shown that the resulting dimensionless parameters can be interpreted in terms of competitions between various energy dissipation mechanisms such as cleavage, crack tip dislocation nucleation, and also dislocation nucleation from a Frank-Read dislocation source. Therefore materials can be classified into three groups. The first two groups are the well-known intrinsic brittle and ductile behaviour. The third group is designated to have extrinsic ductile behaviour for which Frank-Read source dislocation nucleation is the initial energy dissipation mechanism. It is shown that a material is predicted to exhibit extrinsic ductility if the dimensionless product of the Burgers vector and square root of dislocation density is within a certain range. The lower limit of the range is determined by a dimensionless parameter characterizing dislocation mobility. The upper limit is determined by dimensionless parameters that govern either cleavage or crack tip dislocation nucleation.

2. DIMENSIONAL ANALYSIS

The fracture criterion that is commonly used for materials in which a significant amount of plastic deformation occurs is G = 2r. + r p' where G is the applied energy release rate available to effect fracture, r. is the free energy of the newly created surface, and r p is the energy dissipated through various irreversible processes in the near crack tip region. The magnitude of r p often exceeds r. by orders of magnitude. Nevertheless Rice (1965) showed that r p = r p (r.), so the energetic contribution of the newly created surfaces can not be neglected.

Let us consider in a dimensional analysis the effect that other variables can have on r p' We assume that an atomically sharp crack exists in an elastic-plastic material within which plastic deformation occurs via the creation and motion of dislocations on discrete slip planes and in discrete directions at a critical resolved shear stress. In addition to r., other important variables include: P the elastic shear modulus, Pi} the so-called Schmid factor that contains information about orientations of the plastic slip systems, and G'max the maximum theoretical tensile stress that the material can support in the absence of any defect. We also need to include information about the yield stress in the dimensional analysis. However rather than use yield stress itself, which is a phenomenological and ill­defined variable, we instead invoke the physics-based variables which determine macroscopic yield stress: b is Burgers vector, P dul is density of mobile dislocations, and T p is the Peierls stress. Performing a

Initial energy dissipation mechanism at crack tip on the ductile to brittle 201 transition

straightforward analysis to obtain the dimensionless groups which determine r p yields

(1)

We now explore the physical significance of each dimensionless group. The term r./ ph, first derived by Armstrong (1966) and Rice and Thomson (1974), expresses the competition between cleavage at a crack tip and crack tip dislocation nucleation. Its physical significance is more clearly illustrated if it is rescaled so the numerator and denominator have units of energy per unit length

2r.b 1 ph2/5 < . (2)

The numerator of Eq.(2) can be interpreted as the activation energy per unit length necessary to propagate a crack in the absence of any plastic deformation, because cleavage can not occur in increments of less than one atomic spacing, or approximately b. The denominator of Eq.(2) scales as the two-dimensional activation energy per unit length of a dislocation that is emitted from a crack tip (Li, 1986; SchOck and Piischl, 1991; Rice and Beltz, 1994). Therefore the inequality in Eq.(2) represents the condition where the activation energy to provoke crack extension is smaller than the activation energy to emit a dislocation from a crack tip. The ideas of crack tip dislocation nucleation were clarified and expanded upon by Rice (1992) with an analysis that invoked the Peierls relationship between slip and shear stress. The result of this analysis is that the applied energy release rate necessary for crack tip dislocation nucleation is proportional to the unstable stacking energy, ru.' However it is important to note that ru. scales with ph, so that the ratio r s / ph maintains its interpretation as the energetic competition between cleavage and crack tip dislocation nucleation.

The next dimensionless quantity, T p / II, determines the mobility of dislocations in a material. Since dislocation motion is thermally activated, the effective value of T p / II is a function of temperature.

The third dimensionless quantity can be rewritten as

.. I. 1/2 f-lUPdi.l (3)

202 Jeffrey W Kysar

The numerator of Eq.(3) can be interpreted physically as scaling with r FR = (l/2)f1h/ L, where r FR is the resolved shear stress necessary to activate a Frank-Read (FR) dislocation source for a pure edge dislocation, and L is considered to be the mean distance between impediments to dislocation motion. For pure single crystals, the distance between impediments scales as L ~ P ~i~2, which corresponds to the mean distance between dislocations. Thus, the numerator of Eq.(3) corresponds to the resolved shear stress at which dislocations are multiplied via FR sources, and hence with the macroscopically measured yield stress (Taylor, 1934). However a necessary requirement for a FR source to be active is that the dislocation spanning the impediments must be mobile. Hence, rp < (l/2)f1h/ L is a necessary condition for plastic deformation to occur.

Another possible physical interpretation for the third ratio in Eq.(1), this time in terms of activation energy per unit length is

3 "J.,3 1/2 f<U Pdis[ (4)

As will be shown in Section 3, the numerator of Eq.(4) scales as the applied energy release rate necessary to activate a FR source in the vicinity of a crack tip. The denominator scales as the activation energy necessary per unit length for crack tip dislocation nucleation. Therefore the ratio in Eq.(4) represents the energetic competition between dislocation nucleation at a FR source near a crack tip and dislocation nucleation from a crack tip.

The fourth ratio on the right hand side of Eq.(l) can be rewritten (J" max / E , where E is the elastic stiffness of the material. This ratio plays an important role because it determines when a crack tip will cleave in the presence of dislocations which act to shield the crack tip from the far-field loading. Finally, the Schmid factor, f.Jij' is dimensionless and can be combined with any of the other dimensionless groups in Eq.(1).

Thus we see that the dimensionless groups obtained via a simple dimensional analysis can be interpreted in terms of energetic competitions between the energy dissipation mechanisms of cleavage, crack tip dislocation nucleation and also FR dislocation nucleation

3. COMPETITION BETWEEN CRACK TIP AND FRANK-READ DISLOCATION NUCLEATION

The ratio in Eq.(4) is now derived from the viewpoint of dislocation mechanics near a crack tip. The derivation is elementary and necessarily approximate, but nevertheless suffices to reveal the physical significance of

Initial energy dissipation mechanism at crack tip on the ductile to brittle 203 transition

the ratio. The goal is to calculate the applied energy release rate of a crack at which the FR sources nearest the tip will be activated. It is not possible to derme a priori the precise position of each source. Nevertheless, it is well­accepted that the distance between dislocations, and hence between FR dislocation sources, scales as L ~ p~~2 (e.g. McClintock and Argon, 1966; Ashby and Embury, 1985). Hence we assume that a FR dislocation source exists at a radial distance r ~ p ~~2 from a crack tip. The source consists of an edge dislocation that is parallel to the crack front which spans two impediments. The spacing between the two impediments scales as L ~ P ~~2 , which implies that the macroscopic yield stress of the material scales with )Jbpi;/ (Taylor, 1934). We also assume that the material is fully annealed so that the Burgers vectors of all other pre-existing dislocations do not have any preferred orientation. Thus the net stress due to all other surrounding dislocations at any particular position is, on average, zero (Lawn, 1993). Furthermore the net image stress on all dislocations is, on average, zero.

Therefore the dislocation of the FR source interacts only with the stress field of the crack tip and with its own image force. The interactions can be quantified in terms of the Peach-Koehler (PK) force, f , defined as f = ro , where .. is the resolved shear stress on the dislocation. The FR source is activated once the sum of the PK forces from the crack tip and image force equals .. FRb • Finally for simplicity we assume that the dislocation exists on a slip plane which intersects the crack tip (this assumption is not necessary, and does not affect the order of magnitude of the result). Rice and Thomson (1974) discussed the PK forces on a dislocation due to a crack tip and its image force under the same conditions assumed here. Setting the sum of the PK force from the crack tip and from the image force equal to .. FRb yields

(5)

where K[ is the Mode-I stress intensity factor, v is Poisson's ratio and r is the radial position of the dislocation. The first term on the left side of Eq.(5) is from the singular stress field of the crack tip and the second term represents the image force. It should be noted that the trigonometric functions normally associated with the crack tip term are approximated as unity, which does not affect the order of magnitude of the result. Substituting r ~ p~~2 and L ~ p~~2 while assuming that v = 1/3 yields K[ ~ 3)JbpJ~, which represents the order of magnitude of the applied stress intensity factor at which the FR sources nearest the crack tip are expected to be activated. This corresponds to the applied energy release rate of

(6)

204 Jeffrey W. Kysar

which has units of energy per unit area. Using this result it is possible to construct a new dimensionless grouping by taking the ratio of G FR with the quantity 2ys' and multiplying the resulting numerator and denominator by b so that they both have units of energy per unit length to obtain

3,,1.3 1/2 --=f'U,----,P,-"d=-isl < 0(1) .

2Ysb (7)

The numerator is then interpreted as the activation energy per unit length along a crack front to activate a FR source. Therefore the inequality in Eq.(7) represents the energetic competition between cleavage and the nucleation of a dislocation at a FR source, with FR dislocation nucleation energetically preferred when the ratio is less than order unity.

Likewise we can express the energetic competition between dislocation nucleation at a FR source and dislocation nucleation at a crack tip by taking the ratio ofEq.(7) with Eq.(2) to yield

(8)

Again both the numerator and denominator are interpreted as activation energies per unit length along the crack front from, respectively, a FR source and the crack tip, with dislocation nucleation from a FR source preferred when the ratio is less than order unity. Thus we accomplish our goal of elucidating the physical significance of the ratio in Eq.(4) that was obtained via dimensional analysis.

An additional necessary condition for a FR source to operate is that Tp < (1/2)pbP~;I' Ashby and Embury (1985) addressed this by requiring that T p be less than the resolved elastic shear stress which is induced onto the dislocation of a FR source from a crack tip stress field. They too assumed that the FR source position scales as r ~ P :n~2 from the crack tip and approximated the trigonometric terms as unity. Ashby and Embury (1985) then obtained (apart from slightly different numerical factor) that p~2 < K: /8!!,,; , which gives a bound on the dislocation density, at a given K[ for which the dislocation nearest the crack tip is mobile. We choose the value of K[ at which cleavage fracture occurs and rewrite the inequality as

(9)

Initial energy dissipation mechanism at crack tip on the ductile to brittle 205 transition

which expresses the condition for mobility of a FR dislocation prior to cleavage fracture. Taking the ratio of Eq.(9) and Eq.(2) yields a condition for mobility of a FR dislocation prior to crack tip dislocation nucleation as

3 2b 1/2 f1. Pdis! > 0(1). 40JrT!

(10)

4. DISCUSSION

It is now possible to determine the initial energy dissipation mechanism that is activated at a crack tip by using the various inequalities that have been derived. We first express the conditions for which FR sources are the initial mechanism. Materials that satisfy this condition will be referred to as possessing extrinsic ductility, since the initial energy dissipation is extrinsic to the crack tip. We combine Eq.(7) and Eq.(9) to obtain the condition which ensures that FR sources are activated prior to cleavage given as

(11)

Similarly we combine Eq. (8) and Eq.(lO) to obtain the condition which ensures that FR sources are activated prior to cleavage given as

40Jr Tp 1/2 I [ [ 211 ( ) o -3- -;;z < bpdisl < 0 15 . (12)

Therefore, the dimensionless parameter bp~;1 of a material that exhibits extrinsic ductility satisfies the conditions in both Eq.(ll) and Eq.(12). These conditions imply that dislocation nucleation from a FR source is the initial energy dissipation mechanism, irrespective of the competition between cleavage and crack tip dislocation nucleation.

The condition for an intrinsically brittle material is that cleavage occurs prior to dislocation nucleation either from a crack tip or from a FR source. From Eq.(2), such a material must satisfy r) f1.b < 0.1; also bp~;1 must lie outside the range defined in Eq.(ll). Likewise the conditions for an intrinsically ductile material are, from Eq.(2) that r, I f1.b > 0.1, and that bp~;1 must lie outside the range defined in Eq.(12).

206 Jeffrey W. Kysar

The analysis, though elementary, determines the initial energy dissipation mechanism that is activated at a crack tip. It should be emphasized that the initial mechanism is not necessarily the dominant mechanism. Nevertheless it is likely that the transition between energy dissipation mechanisms does signal a change in the overall brittle and ductile response of a material.

The topics presented herein are explored in much greater detail in Kysar (2002) where the predictions are shown to compare favourably with documented material behaviour.

5. ACKNOWLEDGEMENTS

This work was supported by the National Science Foundation under the Faculty Early Career Development Program with grant CMS-0134226.

6. REFERENCES

Armstrong, R. (1966). "Cleavage crack propagation within crystals by the Griffith mechanism versus a dislocation mechanism." Mater. Sci. Engng., 1, 251-256.

Ashby, M. F., and Embury, 1. D. (1985). "The influence of dislocation density on the ductile­brittle transition in BCC metals." Scripta metall., 19,557-562.

Lawn, B. R. (1993). Fracture of brittle solids, Cambridge University Press, Cambridge. Kysar, 1. W. (2002) "Energy dissipation mechanisms in ductile fracture." Submitted to J.

Mech. Phys. Solids Li,1. C. M. (1986). "Computer simulations of dislocations emitted from a crack." Scripta

metall.,20,1477-1482. McClintock, F. A., and Argon, A. S. (1966). Mechanical Behavior of Materials, Addison­

Wesley, Reading, Massachusetts. Rice, 1. R. (1965) "An examination of the fracture mechanics energy balance from the point

of view of continuum mechanics." Proceedings of the 1st International Conference on Fracture, Sendai, (eds. T. Yokobori, T. Kawasaki, and 1. L. Swedlow), Japanese Society for Strength and Fracture of Materials, 1,309-340.

Rice, 1. R. (1992). "Dislocation nucleation from a crack tip: An analysis based on the Peierls concept." J. Mech. Phys. Solids, 40,239-271.

Rice, 1. R., and Beltz, G. E. (1994). "The activation energy for dislocation nucleation at a crack." J. Mech. Phys. Solids, 42, 333-360.

Rice, 1. R., and Thomson, R. (1974). "Ductile versus brittle behaviour of crystals." Phil. Mag., 29, 73-97.

Schock, G., and PUschl, W. (1991). "The formation of dislocation loops at crack tips in 3 dimensions." Philosophical Magazine A, 64, 931-949.

Taylor, G. 1. (1934). "The mechanism of plastic deformation of crystals. Part 1.--Theoretical." Proc. R. Soc. Land. A, 145,362-387.

CONSTITUTIVE MODELING OF VISCOELASTIC UNLOADING OF GLASSY POLYMERS

YvesREMOND

Universite Louis Pasteur Institut de mecanique desfluides et des solides - UMR 7507 ULPICNRS 2 rue Boussingault 67000 Strasbourg remond@imfs·u-strasbgfr

ABSTRACT: The simulation by traditional uni-dimensional rheological models of viscoelastic unloading to zero stress after tensile testing of polyethylene and its composites is poor. The models significantly underestimate recovery rates, even with small amounts of strain. The use of a finite number of relaxation times does not sufficiently increase recovery rates during unloading when models are generated from the responses of materials under load. Similar results and observations are obtained using rate jumps in loading and unloading. 3D models developed using local state methods require that an additional recovery potential be used. A simple 2D model is proposed here which takes into account the differences in local behaviour seen during loading and unloading, thus justifying the existence of this potential. The similar situation that exists for composites means that the phenomenon must not be confused with material damage.

Key words: Viscoelasticity, unloading, constitutive equations, polyethylene, polypropylene

207

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 207-216. ©2004 Kluwer Academic Publishers.

208 Yves Remond

1. INTRODUCTION

Modelling the behaviour of polymeric or composite structures using a polyethylene matrix requires that the model used is satisfactory for the viscoelastic and visco-plastic behaviour of these materials, both during loading and unloading. Experimenters have long known that for many polymers viscoelastic unloading after tension is accompanied by a viscosity effect that is entirely unlike that exhibited during tension. The intention in this study was to consider this question in full, and to evaluate the characteristics of the various models for behaviour that are currently available (and if appropriate improve them) so that these phenomena are adequately handled by models. Many authors have provided significant information on the microscopic aspects of deformation in semi-crystalline materials (Argon, Drozdov, Ahzi, Berstrom, Boyce etc.), others on viscoelasticity (Salen~on, Shapery, etc.) and phenomenological viscoelastic and/or visco-plastic models of polymers and composites (Chaboche, Maire, Chambaudet etc.), and others on polyethylene (G'seIl, Kichenin, Bellouettar) or the numerical aspects associated with such models (Cognard, Stehly).

2. EXPERIMENTAL OBSERVATIONS FOR PE, PP

In order to examine fully the viscoelastic behaviours that occur during small deformations of polyethylene, several types of experiments were carried out at ambient temperature, using small deformations of standard low or medium density polyethylene, as well as with PE with 20% by volume short fibre loading. The following experiments were undertaken:

experiments with monotonic tension at various strain rates between 5. 10-3 and 5.10-6 and between 0.1 and 1. Experiments using monotonic tension with jumps (both increasing and decreasing) in strain rate tensile tests as described above, followed by unloading at different strain rates, followed by a recovery phase.

Test pieces used were injection moulded and used in accordance with usual experimental standards. In the case of GF-PE composite, the average length of the fibres was O.4mm, which, because of the injection flow, were oriented in the longitudinal direction of the test piece. The properties of the composite obtained were, naturally, anisotropic. Only the longitudinal properties, however, were of interest in this study. The test pieces were subjected to strains of the order of 7 to 8%. At this level of deformation, unloading

Constitutive modeling of viscoelastic unloading of glassy polymers 209

followed by recovery over several days results in full reversibility in deformation. The situation was, therefore fully viscoelastic, and did not enter the viscoplastic region, (despite the difficulties that these definitions can cause in polymers).

The results are shown in figure 1, 2 and 3. Figure 1 shows the effects of sudden changes in strain rate on medium density PE. Where sudden changes are induced, the graph shows a standard discontinuity of gradient, and asymptotically approaches the monotonic curve for the final deformation rate. This result is also well illustrated in figure 2 where the differences in the load path as a function of rate can be seen in addition. Note that these differences are much less pronounced in standard creep or relaxation studies, since workers in these fields are more interested in longer term asymptotic effects, whilst the effects that we are discussing here are exhibited over the short term.

Jump from 0.03 to 0.0003 mmlmn

2O -.----......,....--~---~_._.,

15

10

5

O~-~-~--r--,--~

o 0.02 Q04 0.00 Q08 0.1

Jump from 0.03 to 0.3 mmlmn

o 0,02 0,04 0,06 0,08 ,1

Figure 1 " Polypropylene strain rate jumps

20

{ " J I.

... ... .. ..

,.1 Experiments

O,D:l 0,011 0,05 O,ot 0.1 -Figures 2, 3 " Viscoelastic behavior of different PE, PP

210 Yves Remond

3. VISCOELASTIC MODELING

In order to evaluate existing models, and to estimate how adequate they were in simulating behaviour during unloading, standard rheological models and their generalisations with a finite number of relaxation times were used. These were followed by non-linear internal-variable models based on Zener­Weber models, and finally models using dissipation pseudo-potentials constructed using the local state method were examined.

The first two approaches were carried out using the universal Zener-Weber form, by making: 'P = 'P(e,p) the free energy, where p represents an internal variable of the uni-dimensional viscoelastic system. In addition, <l> = <l>(p,P) the dissipation potential.

The equations of state are then expressed as cr = 8'P/fJs and P = - 8'P/ap And the complementary dissipation equation as p = aq,/8P The equation for the behaviour then assumes the following form:

cT = a(s,a)& + P(s, a)

a2'1' p(s,a)=-asap

The functions a(cr,e) and ~(cr,e) are independent of the strain rate and may assume a range of values. The table below shows the principal values of these functions corresponding to the basic models, through to those used for the non-linear Weber-Zener model. The models referred to as "generalised" which are associated with a large number of relaxation times do not significantly alter the results presented later (figure 4).

Models n(o E) ]to,E) Maxwell E -01'11 Zener E1Elll -( El + E2)0[1l Soko1ovskii (1948) E Oif o <cry

-sgn(o).F(lcr - cry!) otherwise Malvern (1951) E o if 0 d{e)

-kF(Jo - f{E)J) otherwise Kunundjanov (1967) E -k«o - f{e»/a)" Weber (1970) n(o,E) 13(0 e) Critescu (1972) n(o,E) -k(e)F(o - f{E» Weber (1996) n(o,E) -k(E)sgn(O - f{e» .F(o - f{e»

Constitutive modeling of viscoelastic unloading of glassy polymers 211

Finally, several variants of more complex dissipation potential models were examined. Their utility will be discussed later. These were based on zero­threshold viscoplastic type models (and so can be used as viscoelastic models) with isotropic and kinematic two-point work-hardening, which enables the dip that appears on the first load in certain polymers to be simulated, with this effect playing no part in these studies.

The thermodynamic potential selected is given by:

where the equations of state are: a = KEe, X\ = 2/3C\a\, X2 = 2/3C2a2, R=h'(P)

The dissipation potential is expressed as: <1>* = <1>*\ + <1>*2

With <I>*\(a, XI,X2, R, p) = Kln+ 1< (J2(a-X I -X2) -R(P) - Ro)1K > n+1

and R(p) = Q\ (1- exp(bl(p» + Qz( 1 - exp(bz(p», and QI>O et Qz< 0

With <1>*2 the annealing potential selected for the recovery of the strain induced. This allows us to adequately describe the viscoelastic behaviour with rate jumps and unloading.

Comparison of viscoelastic models for unloading with strain rate jumps :

a- Simple 10 rheological Maxwell and Zener-type models and their generalisations. No difficulty in simulating monotonic non-linear viscoelastic loading.

b- Simple 10 rheological Maxwell and Zener-type models and their generalisations. The viscosity introduced in order to generate non­linearity is frequently too high and rate jumps are poorly simulated. Hardening (or softening) is also over estimated, due to strain rates that are too large (or small).

c- Simple 10 rheological Maxwell and Zener-type models and their generalisations. The viscosity introduced in order to produce non­linearity is too large to correctly simulate unloading; strain recovery is too small.

d- Simple 10 rheological Maxwell and Zener-type models and their generalisations. The estimation of viscosity parameters from the two loading curves obtained at different strain rates show that these seem to decrease significantly with the rate. This effect cannot be

212 Yves Remond

explained at present by a microscopic approach involving changes in levels of crystallinity etc.

e- Simple ID rheological Maxwell and Zener-type models and their generalisations. Regardless of loading rate, viscosity parameters estimated do not result in the unloading curve being reproduced.

f- The above results are similar when rheological models with a fmite number of relaxation times are used.

g- Non-linear internal-variable Zener-Weber type viscoelastic models. There is no difficulty in simulating viscoelastic loading.

h- Non-linear internal-variable Zener-Weber type viscoelastic models. Parameters can be chosen that adequately simulate strain-rate jumps.

i- Non-linear internal variable Zener-Weber type viscoelastic models. Parameters estimated during loading, and compatible with the various rates used do not result in satisfactory unloading, except where there is setting of specific characteristic as unloading parameters, which is an artificial situation.

j- Non-linear internal variable Zener-Weber type viscoelastic models. Non-linear functions 0.(0',&) and P(O',&) can be identified such that loading and unloading at given rate are correctly predicted.

k- Non-linear internal variable Zener-Weber type viscoelastic models. The functions 0.(0',&) and P(O',&) identified during loading from varied rates and those which simulate unloading correctly are not the same.

1- Modeles viscoelastiques non lineaires a variables internes de type Zener-Weber. Les fonctions 0.(0',&) et P(O',&) permettant de simuler des chargements Ii des vitesses diff6rentes, ne permettent pas d'obtenir une bonne simulation du dechargement Ii une vitesse quelconque sans parametrage complementaire du dechargement.

m- 3D dissipation potential models with continuous-spectrum relaxation times. There is no problem in simulating loading at constant rates and with rate jumps, but unloading is always underestimated.

n- 3D dissipation potential models with continuous-spectrum relaxation times. Only by using an additional specific dissipation pseudo­potential for recovery can unloading be modelled.

0- 2D rheological models with uni-lateral effect. Rate jumps and unloading can be modelled if the longitudinal component is not too small (Zener-Weber). The transverse viscosity component is, for the time being, only hypothetical, with no microscopic experimental basis. It does, however, explain the requirement for an additional potential that is independent of the local state method.

Constitutive modeling of viscoelastic unloading of glassy polymers 213

Models Experimen

112<

a,

Figure 4 .' Comparison of unloading and jump viscoelastic models

214 Yves Remond

4. MICROSCOPIC MECHANISMS

It was seen earlier that classical linear or non-linear rheological models are not capable of reproducing the unloading behaviour of the materials examined. Similarly, dissipation equations for behaviour based on the existence of a dissipation potential require that a recovery term be introduced in order to obtain satisfactory results. The experimental results obtained show that viscosity estimated on loading is much too great, and prevents sufficient recovery from taking place. In order to overcome this difficulty, a uni-Iateral approach is proposed which allows the load viscosity to be regarded as the sum of an intrinsic viscosity <11m> (which would naturally involve a continuous relaxation time spectrum), and a uni-Iateral viscosity <11conf>, due to the restriction of movement of certain macromolecular chains. These chains are associated with specific conformations (for example, loops), and the latter viscosity would be greater than the average viscosity in order to result in a uni-Iateral effect. During loading, both types of viscosity would exist and be active, but on unloading, only the average viscosity would exert an effect, which would allow materials to recovery more rapidly after deformation. This hypothesis is based on the widely accepted, but as yet not-proven concept that amorphous zones contribute more to deformation than crystalline zones, for small amounts of strain.

For a >0 and da/dt >0,

For a >0 and da/dt <0,

The rheological diagram below illustrates the resulting behaviour, and indicates the need to explain in 2D the uni-dimensional behaviour observed during tensile testing with a return to zero load. During load, the movement of some macro-molecular chains can be hampered by other configurations in the amorphous phase. On the other hand, the return of these same chains to their initial configuration during unloading is easier if <11conf> > <11m>.

The value of <11conf> can be estimated simply from the difference between viscoelastic behaviour during loading and unloading. Very similar behaviour is also found in polypropylene and a number of other semi-crystalline polymers.

Constitutive modeling of viscoelastic unloading of glassy polymers 215

Figure 6 .' 2D Viscoelastic model displaying the uni-lateral unloading behavior

5. CONCLUSIONS

Semi-crystalline thennoplastic polymers such as polyethylene, polypropylene and their composites exhibit complex viscoelastic behaviour during unloading, even for small amounts of strain at ambient temperatures. Analysis of uni-dimensional mechanical tests show that the return to zero load occurs with relaxation times that are very different from those that result during loading. Most equations for behaviour do not take this difference into account, and as a result, do not allow adequate simulation of unloading, which presents difficulties for the simulation of structural behaviour. . The Local State method only works if an additional specific dissipation potential is constructed to allow for recovery. This leads to the possibility that the macro-molecular microscopic mechanisms involved are themselves different from those proposed by various authors to distinguish between tension and compression. A local elementary interpretation is proposed here which allows this to be taken into account in a simple way. In composites. with matrices reinforced by short fibres, the reinforcement largely prevents very long-tenn viscoelastic defonnation from taking place, and the phenomenon is less noticeable. It is still present however, and must not be combined with or confused with indications of damage. Finally, this study also reveals the need to test the validity of viscoelastic models by the use of rate jumps in addition to traditional tensile and creep/relaxation testing. This seems to provide a way of discriminating between the adequacy of the models.

216 Yves Remond

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ON THE CONSTITUTIVE THEORY OF POWER­LAW MATERIALS CONTAINING VOIDS

c. Y. Hsu1), B. J. Lee2), and M. E. Mea2)

J) Department of Hydraulic Engineering 1) Department of Civil Engineering Feng Chia University, Taichung, Taiwan, ROC E-mail: bjlee@fcu.edu.tw 3) Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin, Austin, TX78712, USA

Abstract: In the analysis of non-linear porous solids, it is commonplace to employ a spherical unit cell owing to the simplicity it affords. The macroscopic constitutive response is then predicted based upon either uniform traction or linear displacement/velocity boundary conditions applied on the outer surface of the cell. In this investigation we carry out a careful computational investigation of the effect of these two types of boundary conditions upon the predicted macroscopic response, and in particular, we explore the sensitivity of the predicted response to the macroscopic stress state and the degree of matrix non-linearity. In addition, we contrast the accurate numerical results obtained here with various approximate constitutive models in order to provide additional insight into the predictive capabilities of these models.

Key words: non-linear porous solids, voids, micromechanics

1. INTRODUCTION

In the development of constitutive relations for the inelastic defor­mation of porous solids, the ''unit cell" which is most commonly adopted is that in the form of a thick-walled spherical shell (see Figure 1). The choice of this unit cell is driven primarily by the relative simplicity it affords for both analytical and numerical treatment, but it can be argued that it is a reasonable choice for situations in which the voids are roughly spherical and near uniformly distributed. Indeed, the thick-walled spherical shell can be considered to be a simplification of the unit cell which would pertain exactly to an idealized porous solid for which perfectly spherical voids are spaced in

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218 C. Y. Hsu, B. J. Lee and M E. Mear

a uniform cubic array. However, while the boundary conditions for the "exact" unit cell representing such an array of voids (viz. a cube of matrix material containing a spherical void) can be precisely discerned, the boundary conditions appropriate for the "approximate" spherical unit cell cannot be clearly identified. To retain the simplicity afforded by this unit cell, as well as to guarantee certain features of the connection between the macroscopic stress and strain/strain-rate, attention is restricted to two types of boundary conditions: uniform traction boundary conditions and linear displacement/velocity boundary conditions (e.g. [1-6]). While some arguments have been put forth in favor of the linear displacement/velocity boundary conditions (see [7] and references therein), there seems to be no a priori guarantee as to which of the boundary conditions will yield a better estimate for the constitutive response of the (idealized) porous solid under any particular combination of matrix non-linearity and macroscopic loading.

The purpose of this investigation is to perform a careful computational investigation of the effect of the two types of boundary conditions upon the predicted macroscopic response, and in particular, to elucidate the sensitivity which the (difference in) response has to material non-linearity and macro­scopic stress state. In addition, we contrast our accurate numerical results with the approximate constitutive models given by [1-6] in order to provide additional insight into the predictive capabilities of these models. We remark that the development of various simple models such as [1-6] is based upon (approximate) solutions for the axisymmetric deformation of the unit cell, and in the current investigation we also restrict attention to such loading.

S,Er;

+

Figure 1. Schematic showing spherical unit cell.

2. STRUCTURE OF CONSTITUTIVE RELATIONS

Consider a unit cell in the form of a thick-walled spherical shell as shown schematically in Figure 1. The void volume fraction, defined simply as the volume occupied by the pore divided by the total volume of the cell, is

On the constitutive theory of power-law materials containing voids 219

given by c = (alb) 3 where a and b are the inner and outer radius of the shell, respectively. The matrix material comprising the cell is assumed to be isotropic, incompressible and to deform under conditions of steady-state power-law creep. Specifically, the multi-axial relation between the local, or microscopic stress lJ and the local strain-rate i is taken as

. 3e. (ae )n-1 , li=-- -- u 2u. a.

(1)

where u' is the stress deviator, a. = (3u' : u' / 2)1/2 is the effective stress, i. and a. are a reference strain-rate and stress, respectively, and n is the hardening exponent. For future reference we note that i = alfl/ au and u' = ai/J / ai where the strain-rate potential lfI(lJ) and the stress deviator potential I/J(i) are given by

1 n+l

IfI(U) = i.a. (!!.L)n+, I/J(i) = nip'. (~.)-;; (2) n+1 a. n+l c.

in which Be = (2i : i / 3)1/2 is the local effective strain-rate given in terms of the strain-rate deviator (which, since the matrix material is incompressible, is identical to the strain-rate).

Toward characterizing the overall, or macroscopic response of the porous solid we adopt the usual stress and strain-rate quantities defined in terms of the local fields by (see [8])

I=(U)=~ ft®xdS, (3) V s.

t = (i) = _1_ f[v®n+n®v]dS. 2V s.

(4)

In these relations, the operator (-) denotes the average over the volume V of the unit cell, t and v are the traction and velocity on the outer surface S. of the unit cell, n is the unit outward normal to S., and x is the position vector.

The connection between the macroscopic stress and strain-rate (i.e. the macroscopic constitutive equation) is sought based upon the two types of boundary conditions mentioned earlier. For the case of uniform traction boundary conditions, the macroscopic stress I is specified and the traction t = I . n is applied to the outer boundary of the cell. On the other hand, for the case of linear velocity boundary conditions it is the macroscopic strain­rate t which is specified, and the velocity v = t . x is prescribed on the outer boundary of the cell. For either type of boundary condition, the macroscopic constitutive response of the porous solid can be expressed in the form [8,9]

I _ acJ>(E) - at ' (5)

220 C. Y. Hsu, B. J. Lee and M. E. Mear

where tPCE) and 'l'(I) are a macroscopic stress and strain-rate potential, respectively, which are related to the local fields by

tP(E) = (¢(e)) , 'l'(I) = (/f/«(f») . (6)

It is important to note that while the form of these relations applies to either uniform traction or linear velocity boundary conditions, the macroscopic potentials associated with one type of boundary condition will, in general, differ from that associated with the other (also see [8]). In what follows we will introduce notation to clearly distinguish the two cases.

We now restrict attention to axisymmetric deformation of the unit cell. Specifically, with {x"x1,xJ being a cartesian coordinate system for which the XJ axis is directed along the axis of symmetry (Figure 1), the loading is taken such that the only non-zero components of the macroscopic stress and strain-rate satisfy {Ell =E22 =T,E33 =S>T} and {Ell =E22 =ET ,

E33 = Es > ET } • The macroscopic effective and mean stress and the macroscopic effective and mean strain-rate (which are defined analogously to their microscopic counterparts) are then given by

1 E. =S-T, EM =-(2T+S) (7)

3 and

(8)

As measures of the stress and strain-rate states which will prove useful below, we introduce the stress triaxiality X = EM/E. and the strain-rate triaxiality Y = Em/E. and note that

X= 2T+S 3(S-T) ,

(9)

Now, for the axisymmetric loading considered here, the macroscopic potentials (6) can be expressed in the form (see [11,12])

( )

"+1 eo- E 'l'(I) = (1 + p)-'-' -' = (1 + p)/f/(I)

n+ 1 0-. (10)

and

. nco- (E )":1 . tP(E) = (1 + K)-' -' ~ = (1 + K)¢(E)

n+ 1 liD (11)

in which p = p(X,c,n) and K = K(Y,c,n) are functions of the void volume fraction c, the hardening exponent n, and the stress or strain-rate triaxiality as indicated, but are independent of the magnitude of the stress and strain-rate quantities. An expression for the strain-rate follows from (5) and (10) as

. . [ 3 I' 1 ](E,)" E = lip (1+h)2~+3g I -;: (12)

in which I is the second order identity tensor and

On the constitutive theory of power-law materials containing voids 221

1 8p g= n+18X' h=p-Xg. (13)

Similarly, an expression for the stress follows from (5) and (11) as

I=ao[{l+,u)3.~' +!11I](~.)-; (14) 3 E. 3 Eo

with n 8lC

11= n+18Y' ,u=lC-Y11· (15)

We further note that, as follows directly from (12) and (14), there exists the connections

1+,u= 1 11= 3X y=!-g- (16) (l+h)l/' , (1 + h)l/. ' 3 (1+h)

and

l+h= __ I_ (l +,u)"

3Y g = (1+,uY ,

X=!_11_. 3 (1+,u)

(17)

In the next section we present a numerical procedure for use in determining the unknown scalar quantities which appear in these constitutive relations. As mentioned previously, it is important to note that the predicted constitutive response depends upon the type of boundary conditions employed. To clearly distinguish between the constitutive response associated with uniform traction boundary conditions and that associated with linear velocity boundary conditions, we follow Nemat-Nasser and Hori [8] and introduce the following notation. Quantities associated with prescribed tractions will be denoted with a superscript I whereas those associated with prescribed velocities will be denoted with a superscript i. . For example, the quantities {g,h} corresronding to uniform traction boundary conditions will be denoted {g ~ , h ~ while those corresponding to linear velocity boundary conditions will be denoted {gE ,hE}.

3. NUMERICAL SOLUTION PROCEDURE

To solve the requisite boundary value problems for the unit cell, we adopt a Ritz procedure based upon Hill's minimum principle [10] for the velocity. In the case in which uniform tractions t = I· n are applied on S. (and, of course, the surface of the void is traction free) the functional is given by

(18) J'", s.

where Vm is the volume occupied by the matrix material. On the other hand, in the case in which linear velocities v = E . x are applied on S. , the functional assumes the reduced form

222 C. Y. Hsu, B. J. Lee and M E. Mear

QE(V) = fiP(li)dV. (19) y.

In the numerical scheme, a spherical coordinate system {r,f),tp} is adopted for which the only non-zero physical components of the (axisymmetric, divergence free) velocity field are {v r' V 8}' and the trial velocity field is taken in the form (see [11-13])

v, = ;2 [A+ .~.~(k+l)l!(COSO)Fk(P)]' (20)

VB = ; [k~.~.B(COSf))F~/P)] (21)

where P = ria, the l! are Legendre polynomials and L M

F.(p) = L B"p' + L c .. pocm+l ) • (22) ' .. 0,1.... ..0.1 •...

We remark that for the case of linear velocity boundary conditions, it is necessary to satisfy the constraint on the velocity dictated by the boundary data, and here the method of Lagrange multipliers was employed for this purpose.

The constants {A,B,I'Cbo } are determined using the Newton-Raphson method, and for all the results to be presented below, 33 coefficients corresponding to K = 8 , L = 2 and M = 4 were utilized. Convergence studies which have been performed indicate that the results obtained using this set of coefficients are highly accurate.

Once the coefficients have been determined, the quantities {p,g,h,K,,u,l1} appearing in the macroscopic constitutive relations are readily determined. We discuss separately the cases of uniform traction and linear velocity boundary conditions.

Consider first uniform traction boundary conditions for which the stress components {S,T} (and so the triaxiality X) are specified. The quantity p~ is determined from (see [11-13])

(23)

in which Q:,o is the stationary value of the functional (18), and g~ (which, as can be seen from (12), has a direct correspondence to the macroscopic dilatation-rate) is given in terms of the single coefficient A as

~ 3 ,13 1 (S-T}O'A g=c -:--- . &. u.

(24)

From (13) we then find h~=p~-Xg~ (25)

and using (15) and (16), the quantities {K~,,u~,7]~,y} readily follow. Consider next linear velocity boundary conditions for which the strain­

rates {Es' ET } (and so the strain-rate triaxiality Y) are specified. In this case K E is determined from

On the constitutive theory of power-law materials containing voids 223

r.l' l+Kt =~ (26)

Vt/J(E)

in which Q;'in is the stationary value of the functional (19). Toward determining the remaining constitutive functions, we first note that Ee = (3(0").' (0") l2yn in which 0" is related to the local strain-rate field by (1), hence the macroscopic effective stress can be readily calculated in terms of the local strain-rate field. Once E e has been evaluated, we find from (14) that

1

E Ee [2 . . ]--;; 1+,u =- -. (Es-Er)

a 0 3&0 (27)

and from (IS) that . 1 . .

1]E =_(K E _,uE). Y

(28)

With these quantities in hand, all other quantities of interest immediately follow. In particular, we have

E 3Y hE = 1 . 9 = (1+,uE)" , (1+,uE)" (29)

and we note that the stress triaxiality (corresponding to the prescribed strain­rate triaxiality) is given by

1 1]" X =3 (1+ ,uE) (30)

4. SELECTED RESULTS

Attention is now focused on the dilatation-rate and the change in effective strain-rate induced by the presence of the porosity. From (12) we find that the dilatation-rate and the change in effective strain-rate (i.e. the difference between the effective strain-rate for the porous solid and that for a fully dense solid) are given by

E •• = g(X,c.n) E;, !lE, = h(X,c.n)E; (31)

in which

E; = e. (E,I aJ" (32)

is the effective strain-rate in the absense of porosity. For any given stress triaxiality X and effective stress E" the ratios 9 E / 9 r. and hE / h r. serve as measures of the discrepancy between the macroscopic strain-rates predicted by use of linear velocity boundary conditions and those predicted by use of uniform traction boundary conditions. A (representative) set of results for gE/gr. and hE/hr. are displayed in Figure 2 as a function of X. The results shown are for the void volume fraction c=O.OS and the matrix hardening exponents n={3,S,1O}. Not

224 C. Y. Hsu, B. J Lee and M. E. Mear

surprisingly, it is found that the change in macroscopic strain-rate associated with uniform traction boundary conditions exceeds that for the (more constrained) case of linear velocity boundary conditions. Of key interest is the degree to which the two predictions differ. As can be seen from the figure, the difference between the predictions becomes progressively larger with increasing matrix non-linearity, and this difference can be very substantial depending upon the hardening exponent and the stress triaxiality. For example, with n=IO and X=4/3 (Le. SIT=2) we find that gE /gL '" 0.41 and hE /h L '" 0.43.

1.0. lD ,--------------,

0.0 ,-

,-

/

gF. 0.." /

/

7" /

0..4 . _.

I c .. c.os

"- 5

0.2 "-5 H" 10

0..0. 0..0. 2.0. 4.0.

x X Figure 2. Results for 9 E / gLand hE / h I. as a function of stress triaxiality.

Cases shown are for c=O.05 and n={3,5,JO}.

As mentioned previously, various approximate constitutive models for porous solids have been developed based upon analysis of a spherical unit cell. We now consider the analytical models devised by Gurson [I], Cocks [2], Duva and Crow [3], Sofronis and McMeeking [4], Haghi and Anand [5], and Michel and Suquet [6], and we contrast the predictions of these models with our numerical findings. We remark that Gurson [I] and Cocks [2] utilized linear velocity boundary conditions in the development of their models whereas the others adopted uniform traction boundary conditions.

Here we limit attention to the single case for which c=O.05 and n=IO. Results for the normalized macroscopic dilatation-rate EufE: = 9 and the normalized change in macroscopic effective strain-rate 11 E/E; = h are shown in Figure 3 as a function of X. For this particular void volume fraction and hardening exponent, the predictions of the models [2-6] essentially all lie between the numerical results obtained for linear velocity boundary conditions and those obtained for uniform traction boundary conditions. On the other hand, the predictions of Gurson's model are strictly less than the numerical results obtained for the (highly constrained) case of

On the constitutive theory of power-law materials containing voids 225

linear velocity boundary conditions. It should be noted that for a given triaxiality X, there are significant differences between the predictions of the models both relative to each other and relative to the numerical results.

M .5.0

'.0 '.0

t,. 3.0

3.0

~ --.-.-E. ~ ~.O

. "0

1.0 1.. ... /D

0.0 0.0 0.. 0.. 0.8 1.2 1 .• 2.~ 0.0 0.. ... 1.2 1 .• 2.(

X X

-e-- G"'''''~I --.- Cocks PI

--v- O,,,"Urowtl},MkhellSuquetlll

___ Set"",11 &McM .. Wng 141

~ H·!t>l&A ... ·~1

L .... r .. locftyB.C.

Ul1lbrm hdon B.C.

Figure 3. Results for EuIE: and t. E,IE: based upon linear velocity and uniform traction boundary condition, along with predictions of the models [1-6].

Case shown is for c=O. 05 and n = 10.

Finally, we note that while the selected results presented here serve to illustrate the nature and intent of the present investigation, they do not allow a full characterization of either the effect of boundary conditions or the performance of the analytical models. An extensive set of results (for a wide range of void volume fractions, hardening exponents and stress triaxialties) will be presented and discussed in a separate article.

ACKNOWLEDGEMENT: B. 1. Lee wishes to thank the financial support from the National

Science Council ofR.O.C. under Grant NSC 91-2625-Z-035-004.

REFERENCES

[I] A. L. Gurson, "Continuum theory of ductile rupture by void nucleation and growth: Part

I-Yield criteria and flow rules for porous ductile media", J. Eng. Materials Tech., vol.

99, p. 2, 1977.

226 C. Y. Hsu, B. J. Lee and M E. Mear

[2] A. C. F. Cocks, "Inelastic deformation of porous materials", J. Mech. Phys. Solids, vol. 17, p. 693, 1989.

[3] J. M. Duva and P. D. Crow, "The densification of powders by power-law creep during hot isostatic pressing", Acta. Metall. Mater., vol. 40, p. 31, 1992.

[4] P. Sofronis and R. M. McMeeking, "Creep of power-law material containing spherical voids", J. Appl. Mech., vol. 59, p. 88, 1992.

[5] M. Haghi and L. Anand, "A constitutive model for isotropic, porous, elastic-viscoplastic metals", Mech. Mater., vol. 13, p. 37,1992.

[6] J. C. Michel and P. Suquet, "The constitutive law of nonlinear viscous and porous materials", J. Mech. Phys. Solids, vol. 40, p. 783, 1992.

[7] A. A. Benzerga and J. Besson, "Plastic potentials for anisotropic porous solids", Eur. J. Mech. A/Solids, vol. 20, p. 397,2001.

[8] S. Nemat-Nasser and M. Hori, Micromechanics: overall properties of heterogeneous materials, Elsevier Science, 1993.

[9] J. W. Hutchinson, Micromechanics of damage in deformation and fracture, Technical University of Denmark, Denmark, 1987.

[10] R. Hill, "New horizons in the mechanics of solids", J. Mech. Phys. Solids, vol. 5, p. 66, 1956.

[11] J. M. Duva, "A constitutive description of nonlinear materials containing voids", Mech. Mater., vol. 5, p. 137, 1986.

[12] B. J. Lee and M. E. Mear, "Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids", J. Mech. Phys. Solids, vol. 40, p. 1805, 1992.

[13] K. C. Yee and M. E. Mear, "Effect of void shape on the macroscopic response of non­linear porous solids", Int. J. of Plasticity, vol. 12, p. 45, 1996.

OBJECTIVE QUANTIFICATION OF THE DUCTILITY WITHIN THE COUPLING ELASTICITY-DAMAGE BEHAVIOR: FORMULATION

H. Bouabid*, S. Charif-D'Ouazzane*, M EI Kortib** and O. fassi-Fehri***

* Laboratoire Modelisation et Calcul en Mecanique ENlM B.P. 753 RABAT - MOROCCO ** CRR-Batiment, Laboratoire Public d'Essais et d'Etude 25 Rue d 'Azilal Casablanca, Morocco *** Laboratoire de Mecanique et Materiaux Faculte des Sciences B.P. 1014 Rabat - Morocco

Abstract

Keywords

For the inelastic behavior (elasticity-damage coupling), the ductility is quantified by the ductility factor expressed as the ratio of the limit constraint to the elastic constraint. This ratio is then explicitly developed to be found written only by the damage coefficient as a correcting term which is added to linear elasticity to take into account the state of deterioration occurring in the material.

Ductility; damage coefficient; behavior; elasticity-damage coupling.

1. INTRODUCTION

The ductility of a material represents its aptitude to bear deformation under external loads without breaking. In fact, it determines the level of cohesion between the particles of the material to be enough linked one to another. But, its quantification is quite difficult particularly for the heterogeneous materials presenting non linear behavior. The mechanical behavior of a material is never purely elastic and linear. In fact, many intrinsic phenomena occur and render it to a macroscopic scale non linear. The amplitude of this non linearity informs whether the material is fragile or ductile. The quantification of the ductility is quite difficult, particularly for the heterogeneous materials: composite, concrete, compressed earth block, etc. This notion is commonly used to reflect the

227

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228 H Bouabid, S. Charif-D 'Ouazzane, MEl Kortib and O. fassi-Fehri

state of the damage (ultimate stage). However, this state of damage is just a part of the behavior of the material. Nevertheless, the interpretation of ductility must be done with respect to either the deformation and the rigidity. For concrete, the ductility is measured by the ratio of the limit strain to the elastic strain, with the hypothesis that de difference between these strains is simply the plastic strain 1.

According to Taerwe 2, the ductility could also be associated with the ability of a material to form a plastic swivel. However the ductility is not always correlated to the plasticity. It is attributed to many different phenomena, such as viscosity, damage, etc. Chanvillard 3 pointed out that the non fragility of the reinforced concrete is necessary to ensure a multi-cracks. He added that the non fragility informs about the capacity of a cross section to resist to a stress post fissure greater than the stress which caused the crack. The importance of determining the ductility is also due to the fact that it is directly related to the deformation of the material which is admitted to be a source of the crack. Indeed, the crack is consecutive to an overflow of the strain or to a limitation of the strain which then induces stresses that cause fissures 4.

In a proposition to the EUROCODE of a simplified model to determine the strength, Delmotte and al.5 had integrated the limit strains of both the block and the mortar. Another phenomenon of great importance, that is related to strain, is the interface block-mortar which is known to be the weakest element of the masonry. A better understanding of the deformation and consequently the ductility of each of the block and mortar is fundamental to know more about the interface and then to optimise its strength.

2. FORMULATION

2.1. Basic Formula

The ductility or the fragility is herein described by the factor !e which gives the ratio of the limit strain E lim to the elastic one E e' The latter is the strain due the stress limit O'lim (Rc) as if the material had an elastic behavior. In

addition, O'e represents the stress corresponding to the strain limit E lim as if the material had an elastic behavior (Fig. I).

(1)

Objective quantification of the ductility within the coupling elasticity- 229 damage behaviour: formulation.

The different parameters are depicted in the figure 1.

cr (MPa)

cr ...................................................... . e

E (%)

Figure 1 : particular strains and stresses

The behavior using the elastic damage coupling is written as below 6 :

cr=E(l-(~)S)E ER

(2)

where E denote the macroscopic instant strain, E is the Young modulus, ER

is the damage strain - corresponding to cr = 0 - and s the damage coefficient.

The part D=(~)S is the correcting term called damage factor. ER

2.2. Ductility Factor

Using the law in the equation 2, one could write for E = E lirn :

(3)

230 H. Bouabid, S. Charij-D'Ouazzane, MEl Kortib and o.fassi-Fehri

Besides, we know that the strain limit ELim corresponds to the strength O'lim

(Rc) which is the maximum of the stress that is :

Consequently, it comes from equations 3 and 4 that:

or also:

Replacing the first term of the equality 6 into the equation 3, we obtain:

S O'lim == E-1-E lim +s

In addition for a supposed elastic material the stress O'lim is :

(4)

(5)

(6)

(7)

(8)

when equalizing the equations 7 and 8 we deduce for the factor r& in equation 1:

(9)

The equation 9 shows that the factor of ductility is function only of a parameter characterizing the damage of the material which is simply the coefficient of damage s. Throughout this coefficient, the damage is

introduced by a correcting term 1 which is always positive. In deed, the s

limit strain is always superior than the elastic strain which represents the approximate value of the strain for elastic behavior. For a perfect fragile material the factor r& is equal to 1. It means in this case that the correcting term is vanishing and the strain E lim is equal to the elastic

strain Ee'

Objective quantification of the ductility within the coupling elasticity- 231 damage behaviour: formulation.

However for a ductile material, which is more real, the factor of ductility is greater than 1. It is maximal when the damage coefficient is minimal. For instance, when the damage factor varies in a square root - s is equal to 0.5 -the strain limit is three times the elastic strain. The evolution of the factor of ductility is shown in the figure 2 where s varies from 0.5 to 10.

3

2.8

2.6

2.4 ~

~

~ 2.2

"C 2 -8

Ductile Fragile

i 1.8

&l u. 1.6

1.4

1.2

2 3 4 5 6 7 8 9 10 Coefficient du dommage s

Figure 2 : Evolution of the factor of ductility versus damage coefficient

The figure 2 shows that the zone of fragility - or the approximation in this zone - is relatively more extended compared to the one of the ductility. The factor of the ductility varies sharply with the coefficient of damage in small values ofs.

2.3. Strength Quantification

On the other side we clearly know that the presence of intrinsic phenomenon induces a loss of the rigidity accompanied with a decrease of fragility. Then by analogy with the deformation, we will quantify this loss of rigidity. We write fa representing the ration of the stress limit O"lim to the elastic stress

0". corresponding to the strain limit clim as :

(10)

232 H. Bouabid, S. Charif-D 'Ouazzane, MEl Kortib and 0. fassi-Fehri

From the figure 1, we could write:

(11)

from equations 1, 10 et 11, we deduce that :

(12)

The relationship pointed in equation 12 is simply a property of geometry. We also find in this expression an evident result that means a material which is more ductile is a material less fragile and vice-versa. But the objective conclusion to make is that the ductility which normally appears with an increase in 'deformability' induce a decrease in the rigidity in the same proportion. In other words, what is gained in the rigidity is lost in the same proportion in the deformation and vice-versa.

Aknowledgement

Research partially sponsored by the Program PROT ARS (P2-T3/31) of the Government of Morocco

3. CONCLUSION

With the formulation above we are able to quantify objectively the ductility of a material with an elastic-damage coupling behavior. Representing a ratio of strains, the factor of ductility is found simply depending of the coefficient of damage. The latter was introduced to correct the elastic behavior of the material. This ratio will be of a simple and a standard use in engineering.

This formulation was successfully applied to the compressed earth block and now is under validation for concrete.

4. REFERENCES

1 NOUARI L., AZIZI A., TOUGUY M. Sur la ductilite des betons a hautes performances

Rev. Mar. De Gen. Civ., N° 72, Dec. 1997, pp: 7-11.

2TAERWEL. Brittlness versus ductility o/high strength concrete

Int Jour. ofStruct Eng., N° 4, 1991.

3 CHANVILLARD G.

Objective quantification of the ductility within the coupling elasticity- 233 damage behaviour: formulation.

Caracterisation des performances d'un beton renforce de fibres a partir d'un essai deflexion. Partie 2,' Identijication d'une loi de comportement intrinseque en traction.

Mat. and Struct. Journal, Vol. 32, Oct. 1999, pp : 601 - 605

4 MATTONE R., P ASERO G. Block interlocking as a means to improve the behavior of stabilized earth masonry in seismic areas.

Int. Con. On Earthquake Eng. Vol 2, Oct. 95, pp: 794-800.

5 DELMOTTE P., LUGEZ J., MERLET J. D. Resistance des ma~onnerie sous charges verticales. Proposition d'un modele simplijie de calcul.

Cahier CSTB, Paris, Jan.-Fev. 92.

6 HAKIMI A., FASSI-FEHRI 0., BOUABID H., CHARIF D'OUAZZANE S., ELKORTBIM.

Non linear behaviour of the compressed earth block by elasticity­dammage coupling

Mat. and Struct. Journal, Vo1.32, Aug./Sep. 1999, pp : 539-545

7 BOUABIDH. Contribution a I' etude du comportement mecanique du bloc de terre comprimee et du mortier de terre stabilises - proposition d'un Optimum Technico-Economique

These de Doctorat d'Etat (PhD Thesis), Jul. 2000, Fac. ofSc. Rabat.

DISCRETE DISLOCATION PREDICTIONS FOR SINGLE CRYSTAL HARDENING: TENSION VS BENDING

Ahmed Amine Benzerga and Alan Needleman Division of Engineering, Brown University, Providence, RI02912, USA

benzerga@engin.brown.edu

Yves Brechet L. T.P. C.M., 1130 Rue de la Piscine, BP 75, Domaine Universitaire, 38402

Saint Martin D'Heres Cedex, Prance

Erik Van der G iessen The Netherlands Institute for Metals Research/Dept. of Applied Physics

University of Groningen, Nyenborgh 4, 9747 AG Groningen, The Netherlands

Abstract Two boundary value problems are solved for a planar single crystal strip: ten­sion and bending. Plastic flow arises from the motion of discrete dislocations, which are modeled as line defects in a linear elastic medium. Two sets of con­stitutive rules for sources and obstacles are used: (i) rules that only account for a static set of initial point sources and obstacles; (ii) rules that, in addition, account for the dynamic creation (and possible destruction) of dislocation junctions that can act as sources or obstacles. In tension, the overall stress-strain response is essentially ideally plastic when rule set (i) is employed while a two-stage hard­ening behavior, with a high hardening second stage, occurs when the number of sources and obstacles evolves dynamically. No major difference between the predictions of the two sets of constitutive rules is found in bending where the density of geometrically necessary dislocations dominates.

Keywords: Constitutive behavior, dislocations, metallic materials.

1. INTRODUCTION Two-dimensional calculations are always less computationally intensive than

corresponding three-dimensional calculations. However, there often are as-

235

S. Ahzi et al. (eds.). Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials. 235-242. ©2004 Kluwer Academic Publishers.

236 Ahmed Amine Benzerga and Alan Needleman

pects of the physical process that are inherently three dimensional. This is cer­tainly the case for discrete dislocation plasticity. Therefore, in [1] we extend the current two-dimensional dislocation plasticity formulation to account for some key aspects of the three dimensional physics, such as junction formation and line tension.

The main effect of the three-dimensional physical processes considered here is that the number of dislocation sources and obstacles evolves dynamically. We consider the implications of this dynamical evolution in two boundary value problems: (i) tension of a single crystal strip; and (ii) bending of the same strip, both in plane strain. The difference between these two problems is that bending induces deformation gradients that require dislocation accumula­tion. As a consequence, the dislocation density is dominated by dislocations that are geometrically necessary in the sense of Nye [2] and Ashby [3]. On the other hand, no such dislocation accumulation is required as a consequence of the imposed deformation in tension. In this case, statistical dislocations are rather increasingly stored, consistent with the physics of work-hardening.

2. DISCRETE DISLOCATION PLASTICITY The dislocations are modeled as line defects in an isotropic linear elastic

continuum and are restricted to gliding in their slip plane. The stress and dis­placement fields associated with the dislocations are singular and not compat­ible with the imposed boundary conditions, but directly account for the long­range interactions between dislocations. To enforce the boundary conditions, image fields are introduced that are smooth and governed by equations that can be solved by the finite element method. Superposition is then used to obtain the actual fields [4].

In a two-dimensional plane strain analysis, dislocation loops are modeled as edge dipoles in the plane of deformation. Short-range interactions enter through a set of constitutive rules. The latter are needed for: (i) dislocation glide, (ii) dislocation nucleation, (iii) dislocation annihilation and (iv) disloca­tion interaction with obstacles. The motion of dislocation i is determined by the Peach-Koehler force, p = Tib, with Ti being the resolved shear stress. Dislocation glide is assumed to be drag controlled so that for each dislocation B vi = Tib, where vi is the glide velocity.

In previous analyses, e.g. [5, 6], dislocation nucleation has been taken to occur at a fixed number of initial Frank-Read type sources. These sources generate a dislocation dipole when the magnitude of the Peach-Koehler force at source location i, exceeds a critical value T~rcb for a prescribed time tsrc

(b is the Burgers vector length). Dislocation annihilation occurs when oppo­site signed dislocations approach each other within an annihilation distance Le. In addition, a distribution of point obstacles is specified to be initially present. Dislocations can get pinned at such obstacles and are released once the resolved shear stress attains the obstacle strength Tobs. A key aspect of this

Discrete dislocation predictions for single crystal hardening: tension vs 237 bending

formulation is that the density of dislocation sources and obstacles is fixed. As shown in [1], this significantly restricts the hardening behavior that evolves.

In plane strain, the formation of a junction is taken to occur when two dislo­cations gliding on intersecting slip planes approach within a specified distance d* from the intersection point of the slip planes. Such a junction is much stronger than a dipole that may form due to elastic interactions. Junction for­mation is taken to be independent of the signs of the Burgers vectors of the dislocations comprising it. When a junction forms, dislocations gliding on ei­ther of the two slip planes are kept at a distance greater than or equal to d* from the core of the junction.

A junction I can be destroyed if the Peach-Koehler force acting at its lo­cation exceeds a value T6rkb. With S~st denoting the distance to the nearest junction, the breaking stress for junction I is specified by (see [1])

I p,b Tbrk = (3 Sl

nst (1)

where p, is the shear modulus and (3 a factor which reflects the strength of the junction. From eq. (1), the closer the junctions are, the harder they are, which is consistent with the results in [7, 8] (also see [9]). When the junction is destroyed, the dislocations forming the junction, as well as those pinned at the junction, are released and free to glide on their respective slip planes. The mechanism for junction destruction that motivates eq. (1) is described in [1].

This same mechanism is invoked for source activation, depending on the junction's three-dimensional structure and orientation. Since these details can­not be captured in two dimensions, whether or not a junction acts as a source is taken to be a statistical event and is decided at the creation of the junction. The probability that a junction acts as an anchoring point for a new source has the prescribed value p. Source operation is a cooperative process involving two junctions, as discussed in [1].

The criterion used to operate a source is similar to the criterion used to destroy a junction. In plane strain, a dislocation dipole is nucleated at source I if the value of the Peach-Koehler force at the junction exceeds the value T~ue b,

during a time t~ue' with

I p,b Tnue = (3 Sl '

nst

t l J-l S~st t nue = 'Y bl ITI I sre (2)

where TI is the current resolved shear stress at the location of junction I, 'Y is a parameter that controls the multiplication rate and tsre is the nucleation time for initial point sources. A justification of the expression for t~ue is given in [1]. The distance, L~ue' between the freshly nucleated dislocations reflects the size of the new loop. Based on three-dimensional calculations [10], L~ue is taken to be in the range one to three times S~st. In evaluating S~st all junctions are considered. Since the dislocations comprising the source are presumed to

238 Ahmed Amine Benzerga and Alan Needleman

have cross-slipped [1], dislocations on the original slip plane can pass by the source without destroying it.

3. NUMERICAL RESULTS A planar crystal, with the XI-X2 being the plane of deformation, and having

dimensions l = 12/-tm by h = 4/-trn is considered. The crystal has two slip systems, symmetrically oriented with respect to the xl-direction by CPo = 35.25° and b.cp = 109.5° (an fcc orientation), see Fig. 1. The spacing of potentially active slip planes is d = 100b. The calculations are carried out assuming infinitesimal strains and rotations.

tX2

Xl-a,E - I""'. M, ()

Figure 1. Geometry of a fcc crystal oriented for double slip under [110] tension (0", €) or bending (M, 8) with t:J.cp = 109.5° and cpo = 35.25° referring to the angle between the xl-axis and the [112] direction.

1\\'0 plane strain loading conditions are analyzed: (i) tension and (ii) bend­ing. In both cases, the surfaces at X2 = ±h/2 are traction free and the shear stress vanishes at Xl = ±l/2. For tension a uniform displacement Ul = ±U /2 is prescribed along Xl = ±l/2, while for bending a rotation () is prescribed along Xl = ±l/2, with the axial force vanishing. The numerical procedure used to assure vanishing of the axial force in bending is described in [5]. The loading rates are € = U /l = 1678-1 for tension and iJ = 103 8-1 for bending.

Material parameters are used which are representative of aluminum with v = 0.3, E = 70 GPa and b = 0.25 nrn. Initial sources having an average nucleation strength of Tsrc = 50MPa with a standard deviation of 15 MPa are distributed randomly throughout the specimen with a density Po = 12/-trn-2•

Furthermore, tsrc = 10 ns and Le = 1.5 nrn. Dislocation obstacles of uniform strength Tobs = 150 MPa are also distributed randomly with a density Pobs = 18 j.tm-2 • The drag coefficient is B = 10-4 Pa s.

Emphasis in the following is on the calculations allowing for the dynamic creation/destruction of junctions, but results with only static sources and obsta­cles are shown for comparison. The calculations allowing for dynamic junc­tion formation use a critical distance d* = 2Le and a source probability of p = 0.05. Also, {3 = 1 in eqs. (1) and (2), following [8], and'Y = 0.1 in eq. (2).

Discrete dislocation predictions for single crystal hardening: tension vs 239 bending

3.1 Tension

The response in tension using the two sets of constitutive rules is compared in Fig. 2(a). When the dynamic formation of junctions is not accounted for, the response is perfectly-plastic. In contrast, the dynamic source and obstacle creation leads to a two-stage hardening.

(a) 7 (b) 2.5

6

~ 2.0 5

'" N 0 I 1.5 .... 4 S x '-' 0

kl 3 C ----

.... 1.0 b "'-

2 0.5

0 0.0 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06

Figure 2. Response in tension with constitutive rules allowing for the dynamic creation and destruction of junctions. The calculation with only static initial sources and obstacles (SI) is shown for comparison. (a) Stress versus strain. (b) Dislocation densities versus strain; for the SI results, only the total dislocation density is shown.

There is a relatively high initial transient hardening rate for E = U / I < 0.5 X 10-2 due to the formation of pile-ups on a few slip planes where the first junctions form. For E E [0.5 X 10-2, 1. 75 x 10-2] an easy-glide stage is identified where the flow strength remains roughly constant. This is due to the fact that (i) the slip is initially not symmetric due to static sources of unequal strengths, so that only a few junctions form; (ii) most new junctions are obstacles (recall that p = 0.05) which keep being destroyed because of local stress concentrations. Because the dynamically created sources cannot be destroyed but the dynamically created obstacles can, the number of sources at higher strains (E > 0.015) becomes significantly greater than the number of obstacles. As a consequence, a stage with a relatively high hardening rate is attained with a continuous production of sources which further promotes dislocation multiplication, see Fig. 2(b). Although::;:; 2/3 of the dislocations are mobile (i.e. with vi i- 0), as seen in Fig. 2(b), their effective glide distance is significantly reduced since most of them are caught in pile-ups near the junctions that have formed.

The dislocation structure at a strain of E = 0.04 in Figure 3(a), shows re­gions that ate relatively free of dislocations separated by dense walls. By way of contrast, such a patterning is absent when the dynamic creation of junctions is not accounted for, see Fig. 3(b). This incipient organization appears at the micron scale, which is indeed the observed scale of dislocation substructures.

240 Ahmed Amine Benzerga and Alan Needleman

(a)

(b)

'":1 '"

500nm

t--I

Figure 3. Dislocation patterns in [1lOJ tension at € = 0.04 (a) with constitutive rules allowing for dynamic junction creation; (b) with only static initial sources and obstacles.

3.2 Bending

The moment (M) versus rotation «() response in bending is shown in Fig. 4(a) with Mref = 1/6 fsrch 2 • In bending, dynamic junction formation has little ef­fect on the overall response, at least up to () = 0.06. In addition, the total dislocation density is only slightly higher with dynamic junction formation than with the old static rules SI, see Fig. 4(b).

The sensitivity of the behavior to dynamic junction formation is related to the character of the dislocation density in bending compared to tension. The concept of geometrically necessary dislocations (GNDs) was introduced by Nye [2] and Ashby [3] to account for modes of plastic deformation where an internal accumulation of a density of dislocations is required to accommo­date the gradients of plastic strain induced by the deformation. The density of GNDs in bending is given by [2, 3],

PG = 2()P = ~ [2() _ M] lbl bl I D

(3)

where ()P is the plastic rotation, bl is the projected Burgers vector length b cos <p

in the Xl direction, and D = EH3 /[12(1-1/2)] is the elastic bending stiffness. In bending, the dislocation density that evolves is largely "geometrically

necessary." For example, at () = 0.06, GNDs constitute 86% of the total dislo­cation density for the calculation with only static initial sources and and 80% of the total density when the constitutive rules for dynamic junction forma­tion are used. On the other hand, in tension, the imposed average deformation can be accommodated by a uniform deformation and the dislocation density

Discrete dislocation predictions for single crystal hardening: tension vs 241 bending

(a) 3 (b) 0.6 Dynamic Junctions -- Dynamic Junctions --

2.5 SI - - - - _. 0.5 SI - - - - _.

2 ~ 0.4 '" 'll I

:i s 1.5 '-' 0.3 0 ....... a ~ ,...,

<:>. 0.2

0.5 0.1 sources

0 0.0 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06

() ()

Figure 4. Bending of planar single crystal using the constitutive rules for the dynamic creation and destruction of junctions. Results with only static initial sources and obstacles (SI) shown for comparison. (a) Moment versus imposed rotation. (b) Dislocation densities versus imposed rotation. The evolution of the source density is also shown for the calculation using the constitutive rules for the dynamic creation and destruction of junctions.

is referred to as statistical. Thus, when the dislocation density is dominated by geometrically necessary dislocations, the dynamic creation of junctions has a relatively small effect on the overall response. It is possible, however, that dynamic junction formation plays a more important role for larger specimens or, for a specimen of the size analyzed here, at larger deformations.

4. CONCLUSIONS

Discrete dislocation analyses have been carried out for two boundary value problems for a micron size planar strip of a single crystal: tension and bending.

• There is no simple connection between the stress-strain response in ten­sion and the bending moment-rotation response because the correspond­ing hardening mechanisms have different origins and are associated with different dislocation structures.

• In tension, constitutive rules that allow for dynamically evolving junc­tion formation lead to a two-stage hardening associated with the refine­ment of a dislocation structure consisting of cells and walls.

• In the bending calculation, the dislocation structure that forms is domi­nated by geometrically necessary dislocations and the constitutive rules allowing for dynamically evolving junction formation have little effect on the moment-rotation response.

242 Ahmed Amine Benzerga and Alan Needleman

ACKNOWLEDGMENTS

Support from the Materials Research Science and Engineering Center on On Micro-and-Nano-Mechanics of Electronic and Structural Materials at Brown University (NSF Grant DMR-0079964) is acknowledged.

References

[1] A. A. Benzerga, Y. Brechet, A. Needleman, and E. Van der Giessen. Incorporating 3D Mechanisms into 2D Dislocation Dynamics. in preparation.

[2] J. F. Nye. Some geometrical relations in dislocated crystals. Acta metall., 1:153-162, 1953.

[3] M. F. Ashby. The deformation of plastically non-homogeneous materials. Phil. Mag., 21:399-424,1970.

[4] E. Van der Giessen and A. Needleman. Discrete dislocation plasticity: a simple planar model. Modelling Simul. Mater. Sci. Eng., 3:689-735, 1995.

[5] H. H. M. Cleveringa, E. Van der Giessen, and A. Needleman. A Discrete Dislocation Analysis of Bending. Int. 1. Plasticity, 15:837-868, 1999.

[6] A. A. Benzerga, S. S. Hong, K.-S. Kim, A. Needleman, and E. Van der Giessen. Smaller is Softer: A Discrete Dislocation Analysis of an Inverse Size Effect in a Cast Aluminum Alloy. Acta mater., 49:3071-3083, 2001.

[7] G. Saada. Sur Ie durcissement dii a la recombinaison des dislocations. Acta metall., 8:841, 1960.

[8] V. B. Shenoy, R. V. Kukta, and R. Phillips. Mesoscopic Analysis of Structure and Strength of Dislocation Junctions in fcc Metals. Phys. Rev. Lett., 84:1491-1494, 2000.

[9] A. J. E. Foreman. The Bowing of a Dislocation Segment. Phil. Mag., 15: 1011-1021,1967.

[10] A. Moulin, M. Condat, and L. P. Kubin. Simulation of Frank-Read Sources in Silicon. Acta mater., 45:2339-2348, 1997.

ON PLASTICITY AND DAMAGE EVOLUTION DURING SHEET METAL FORMING

Christophe Hussonl ,3), Christophe Poizaf), Nadia Bahlouli l), Sai'd Ahzi l),

Thierry Courtin3), and Laurent Merle3)

I) University Louis Pasteur -lMFS - UMR 75071CNRS - 67100 Strasbourg, France.

2) Fraunhofer IWM - Wohlerstrasse 11, D-79108, Freiburg i. Br., Germany.

J) FCl/ Corporate Research Center - 72000 La Fertfj-Bemard, France.

Abstract Based on experimental considerations, the proposed damage law considers that the growth of micro-voids results in a non-linear damage accumulation with plastic deformation. This paper presents a non-linear Continuum Damage Mechanics (C.D.M.) model for the prediction of ductile plastic damage analysis under large deformations such as in the case of metal forming. The dependence of the damage on temperature, strain rate and microstructure are discussed.

Keywords: ductile damage; plasticity; C.D.M.; metalforming

1. INTRODUCTION

In industrial processes, the numerical simulation of manufacturing operations is particularly attractive because the trial-and-error procedure is more expensive. During sheet metal forming the material is subjected to large elasto-plastic deformations. Ductile failure in metals is caused by the nucleation, growth, and coalescence of voids initiated by dislocation pile­ups, or imperfection particles. The damage law plays an important role for process modelling, process design and process control. In order to correctly simulate metal forming processes, the modelling of the damage evolution can be described by means of Continuum Damage Mechanics (C.D.M.). General considerations of C.D.M. are based on the general framework of thermodynamics of irreversible processes (Kestin, 1966; Kestin and Rice, 1970; Rice, 1971), internal state variable theory (Coleman and Gurtin, 1967), relevant physical considerations such the kinetic laws for damage

243

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 243-250. ©2004 Kluwer Academic Publishers.

244 Christophe Husson, Christophe Poizat, Nadia Bahloul, Sai"d Ahzi, Thierry Courtin, and Laurent Merle

evolution, the defmition of the damage variables ... An acceptable damage model for metals should be able to correctly describe the behavior of the material and its properties such as strain hardening strain rate dependence and temperature dependence.

In this study, we first propose a non-linear quasi-static damage evolution law based on the C.D.M. approach. Then we extend this proposed model to the dynamic case. In the last section of this paper, we discuss how the proposed quasi-static to dynamic damage evolution law can be used to simulate the blanking process.

2. MOTIVATION

To predict the failure during manufacturing process, the plasticity model for the flow stress can combined with failure criterion. The flow stress can be formally written by mean of the following expression:

a=u(p,p, T, ... ) (1)

where cr is the flow (effective) stress, p is the effective accumulated plastic strain, p is the effective accumulated plastic strain rate. This model can't predict damage softening because of the parameter D does not affect then constitutive material model. Thus, in the originally model, damage is derived in a post-processing procedure from the following cumulative damage law:

D=v(p,p, T , ... ) (2)

In the present paper, we proposed to insert the quasi-static to dynamic damage evolution law developed in the Johnson and Cook material model:

a=(I-D)Xu(p,p, T, ... ) (3)

where D is the proposed nonlinear isotropic damage evolution law based C.D.M. approach.

On plasticity and damage evolution during sheet metal forming 245

3. A NONLINEAR C.D.M. APPROACH

3.1 Nonlinear C.D.M. model

The expression of the damage dissipation potential, FD, proposed by Bonora [1] is rewritten in the following expression form to develop our model:

(4)

where So is the initial damage energy strength, Y is the damage energy release rate (associated variable to the internal damage variable D), k is the damage exponent characteristic of the material, h is a material exponent, and Do is the initial amount of damage. The damage evolution kinetic law is

(5)

where j.. is the multiplier factor; <p (p - Pth) is the Heaviside function, and

Pth is the onset strain for damage nucleation. Using the definitions of the

plastic multiplier i = p (1- D) [3] and the internal variable associated to

damage Y=(-a;q.f(aH/aeq ))/(2E.(1-D)); and substituting Eq.(4) into

Eq.(5), we get

(6)

where k gives the degree of non-linearity of damage evolution, E is the

Young modulus, f(aH/aeq) is the triaxiality factor where (Jeq is the von

Mises equivalent stress and aH is the hydrostatic stress.

3.2 Quasistatic behavior

For polycrystalline Face Centered Cubic (F.C.C.) metals, the effective equivalent von Mises stress is considered as a function of the accumulated

246 Christophe Husson, Christophe Poizat, Nadia Bahloul, Sai'd Ahzi, Thierry Courtin, and Laurent Merle

plastic strain : monotonously increasing stress function with a maximum value at a finite "strain". Using the Kovacs and Voros power law [2], we can write

(7)

where Pm is the strain corresponding to the maximum stress, p and 0"0 are fitting parameters.

Eq.(6) can be integrated between the initial condition 0 = Do and the critical condition 0 = Ocr. where Ocr is the critical value of damage at failure. At the initial amount of damage Do, the intrinsic value of strain is Pth; and at the rupture damage Ocr, the intrinsic value of strain will be Pcr. According to the general form of damage dissipation potential, and integrating Eq.(6), we obtain the analytical form of the damage law

In the case of uniaxial loading (the triaxiality factorJ= 1) and integrating Eq.(6) firstly between the initial state and the critical state, and secondly between the initial state and at any state, we get the general damage evolution law [3] :

k

f:iliiM 0, +~" + -m(::J)J dp

fPer ~[(Jo + ~o P (1 _ In (l))]2 dp JPth (p) Pm

(9)

3.3 Results of the quasistatic case

A preliminary parameters study (Husson et aI., 2003) has shown that the best results for damage evolution are obtained for the parameter h=O, see Eq.(9). The retained expression of 0 for the following analyses is for h = O.

For this case, the damage evolution is shown in Figure 1 for an pure copper (O.F.H.C.) and for an aluminium alloy (AU 4 GT4). The predicted results are in very good agreement with the experiments data proposed by Lemaitre and Dufailly [4].

On plasticity and damage evolution during sheet metal forming 247

0.9 0.25

0.8

0.7 0.2

Q 0.6

!,i O.S

J 0.4

0.3

Q

!,i O.IS

J 0.1

0.2 O.OS X

0.1

0.2 0.4 0.6 0.8 I.l 0.1 0.2 0.3

strain strain

(a) Damage evolution in OFHC Cu. (b) Damage evolution in AU 4 GT4 alloy. Figure 1. Comparison between experimental data [4] and the proposed model.

3.4 Dynamic behavior

In this part, we develop a constitutive model for the macroscopic damage using microstructural point of view. Strain rate and temperature dependence are included to make this model valid for a wide range of strain rates, from quasi-static to dynamic. Like in the M.T.S. model [5], the effective equivalent von Mises stress is modeled as the sum of two components: one temperature and strain rate dependant 0' , and one athermal Oath :

0' eq / (1 - D) = a (p, p, T, microstructure) + 0' ath (microstructure) . (10)

The athermal stress Oath is due to the strengthening mechanisms and grain size effect. The grain size influence on the flow stress at low homologous temperatures is considered in terms of the Hall-Petch relation:

( . ) k d'I12 O'ath mIcrostructure = O'sm + HP' = constant. (11)

In the above equation, O'sm is the contribution from strengthening

mechanisms (including lattice, solution and precipitation effects; O'sm = 20

MPa for pure copper), d is the mean grain diameter and kHP is the Hall­

Petch constant (kHP = 4.6 MPa.mml/2 and d = 5.10-2 mm for OFHC copper).

The thermal stress a is mainly induced by the interaction between dislocations and obstacles with a density independence on strain [6]. This component can be written in the following form :

a(p, p, T, microstructure) = O'kv (p). R (p, T). S(p, T). (12)

248 Christophe Husson, Christophe Poizat, Nadia Bahloul, Sai"d Ahzi, Thierry Courtin, and Laurent Merle

In the previous equation, O'Jev (p) is the quasi-static component proposed by

Kovacs et al. [3]. The term R(p, T) represents the ratio of hardening

coefficients due to dislocation generation at the temperature T and the strain

rate p to its maximum value Hmax. R(p, T) is expressed as following

The above propose form for R(p, T) is a modification of the work proposed by Maudlin et al. [7] to include temperature effects in a way similar to the Johnson-Cook model [8], m is a softening exponent, T}llf is a reference temperature and T melt is the melting temperature. The term S\I), T) is the microstructural evolution component defmed by :

(14)

where ka is the Boltzmann constant, b is the magnitude of the Burgers vector, go is a normalized activation energy, Po is a constant, I! is the shear modulus, and 0 1 and O2 are constants (resp. 2/3 & 1) that characterize the shape of the obstacle profile. The parameters involved in Eq.(14) are defined in the work of Kocks et al. [5] for OFHC copper. The proposed model is used to predict stress-strain curves for pure copper deformed under uniaxial load in the quasi-static case and dynamic case. The comparison between experimental data [9] and proposed model is presented in Figure 2 for different strain rates. The parameters used are resumed in Table 1.

According to the form of kinetic damage evolution law - Eq.(6) - , in which we insert the stress-strain constitutive model including T, £ and the microstructure - Eq.(lO) - , we obtain the quasi-static to dynamic damage evolution law in polycrystalline F.C.C. metals (Figure 3) :

0=00 ,

k

Per <p O=Ocr'

On plasticity and damage evolution during sheet metal forming 249

400 400

350 3S0

300 300

~ 2S0 .. P- 250

6 200

'" 6 200

'" '" '" ,g ISO

'" ~ ISO

100 100

SO SO

n, 0,. n SI 0.2 0.4 0 .• 0.8

strain strain

(a) T = 298 K, E = 0.0004 /s (b) T = 298 K, E = 6000 /s

100,-------------, 250

!50 200

wo .. ISO

P- ISO

6 '"

100 '" 100 ,g '"

so so

0.2 0.4 0.6 O.K 0.2 0.4 0 .• 0.8

strain strain

(c)T=542K, E =0.1Is (d) T = 542 K, E = 5200 /s

Figure 2. Comparison between experimental data [9] and proposed model for OFHC Cu.

Experiments: (a) & (c) : isothermal compression; (b) & (d) : adiabatic compression.

0.9 0.9

0.8 0.8

0.7 0.7

Cl 0 .• 0 .•

f 0.5

0.4

0.3

Cl 0.5

,i 0.4 OIl

~ 0.3

0.2 '0 0.2

0.1 0.1

0.2 0.4 0.6 0.8 1.2 0.2 0.4 0 .• 0.'

strain strain

(a) Quasistatic case:T = 298 K, E = 4.10.03 /s (b) Dynamic case:T = 500 K, E = 1O+03 /s

Experiment: D cr = 0.85, Ecr = 1.04 Hypothesis: D cr =0.85, Ecr = I

Figure 3. Damage evolution in OFHC Cu : comparison with experiments [4].

For the dynamic case, no experimental data are available for comparison

250 Christophe Husson, Christophe Poizat, Nadia Bahloul, Sai"d Ahzi, Thierry Courtin, and Laurent Merle

Table 1. Model parameters for OFHC Cu

<I{) al a2 Hmax Tref Tmel! m ksfb3 go k MPa MPa MPa MPa K K MPaIK

2370.7 8.295 3.506 2370.7 300.15 1356 0.8 0.823 1.6 1.17

4. CONCLUSION

We proposed a model for plasticity and damage evolution in metals. The comparison between our predicted results and the experiments show a good agreement. Our motivation for this work is the simulation of blanking process at high stroke rates. But the materials model used in this code like the Johnson-Cook model are not sufficient to describe blanking correctly. To improve the modelling, a constitutive damage model is therefore developed. It is based on a microstructural approach and integrates T, f; and grain size effects. The model is valid for a broad range ofT, e and f;.

5. REFERENCES

[1] Bonora N. A nonlinear CDM model for ductile failure. Engng. Fract. Mech., Vol. 58,

No. 1/2, pp. 11-28 (1997).

[2] Kovacs I. and Voros G., On the mathematical description of the tensile stress-strain

curves of poly crystalline face centered cubic metals. Int. J. of Plasticity, Vol. 12, No.1,

pp. 35-43 (1996).

[3] Husson C., Ahzi S. and Bahlouli N., Damage evolution law in polycristalline FCC

metals .. a nonlinear isotropic CD.M approach. Int. To be Submitted (2003).

[4] Lemaitre J. and Dufailly J., Damage measurements. Engng. Fract. Mech., 28 (5/6), pp.

643-661 (1987).

[5] Follansbee P.S. and Kocks V.F., A constitutive description of copper based on the

deformation of copper based on the use of the mechanical threshold stress as an

internal state variable. Acta metal. 36(1), pp. 81-93 (1988).

[6] van Liempt P., Onink M. and Bodin A., Modeling the influence of dynamic's strain

ageing on deformation behavior. Adv. Engng. Mat., Vol. 4, No.4, pp. 225-232 (2002).

[7] Maudlin P.J., Davidson R.F. and Henninger R.J., Implementation and assessment of the

Mechanical-Threshold-Stress model using the EPIC2 and PINON computer codes. Los

Alamos National Laboratory, LA-I 1895-MS (1990).

[8] Johnson G.R. and Cook W.H., A constitutive model and data for metals subjected to

large strains, high strain rates and high temperatures. i h Int'I Symp. Ballistics,

Netherlans, pp. 541-547 (1983).

[9] Tanner A.B., McGinty R.D. and McDowell D.L., Modeling temperature and strain rate

history effects in OFHC Cu. Int. J. of Plasticity, Vol. 15, pp. 575-603 (1999).

MODELING OF THERMO-ELECTRO-ELASTIC EFFECTIVE BEHAVIORS OF PIEZOELECTRIC COMPOSITE MEDIUMS AND ANALYSIS OF REINFORCEMENT ORIENTATION EFFECTS

N. Fakri, L. Azrar and L. El Bakkali*

Equipe MMPM, Faculte des Sciences et Techniques de Tanger, Universite Abdelmalek Essaadi; BP 416 Tanger; Morocco * Faculte des Sciences de Tetouan, Universite Abdelmalek Essaadi, Tetouan; Morocco

Abstract: In this paper a thermo-electro-elastic modeling for piezoelectric inclusions in an infinite non-piezoelectric matrix is proposed. Extension of the heterogeneous inclusion problem of Eshelby for elastic to electro­elastic behavior is formulated in terms of four interaction tensors. These tensors are basically used to derive the self-consistent model and Mori­Tanaka approaches for ellipsoidal piezoelectric inclusions. Solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective thermo-electro-elastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. The influence of the pooling direction effect on the thermo-electro-elastic coefficients is studied and several numerical tests of Ceramic/Epoxy composites are investigated.

Key words: micromechanical models, interaction tensors, concentration tensors, heterogeneous inclusion, piezoelectric material, thermo-electro-elastic.

1 INTRODUCTION

Smart materials such as piezoelectric composites have a large impact in several technologies and constitute an important branch of the recently emerging technologies of modem engineering materials. The thermo-piezo­electric behavior of composites such as, ceramic-polymer, is a good example of coupled behavior which is strongly used in transducers, sensors and actuators. Micro-mechanical models focusing on the interaction between inclusions and surrounding matrix are powerful tools to predict the effective properties. Many investigations dealing with micromechanical analysis have

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252 NFakri, L. Azrar and L. EI Bakkali

been done by many authors. The homogenization problem of micro­heterogeneous piezoelectric mediums has been studied by Benveniste (92), Dunn and Taya (93) and Dunn (94,95) based on the theory of heterogeneous inclusion problem of Eshelby (57). Effective field method has been used by Levin et al. (99) for thermo-electro-elastic properties. Optimal electro­mechanical properties are numerically predicted in accordance with inclusion orientations and concentrations in a previous work Fakri et al. (2003).

The main purpose of this paper is the prediction of effective thermal and piezoelectric properties of composite materials containing spatially oriented piezoelectric reinforcements. The extension to electro-elastic behavior of the heterogeneous elastic inclusion problem of Eshelby is formulated in terms of four interaction tensors Til. These tensors are basically used to derive the I-site self-consistent model and Mori-Tanaka's approache for ellipsoidal piezoelectric inclusions. Explicit expressions of the concentration tensors are presented and can be computed using the numerical values of Til. Based on the obtained concentration tensors, effective thermo-electro-elastic properties are investigated for various polling directions and fibers concentrations.

2 MATHEMATICAL FORMULATION 2.1 Constitutive equations

Let us consider a homogeneous elastic piezoelectric material. The stationary linear responses of such linear and transversely isotropic medium satisfy the following equations of thermo-electro-elasticity where only the electric and elastic fields are coupled. The temperature e is considered to be constant in the medium:

O'ij = Cijmn Emn - enij.En- f3ij e Di = eimn Emn + Kin. En - Xi e (1)

where 0' and E are the stress and strain tensors, E and D the electric field and electric displacement vectors, C is the tensor of elastic moduli, K is the dielectric tensor (primittivity), 'e' is the tensor of piezoelectric constants characterizing coupled electroelastic effect, f3 is the tensor of temperature stress coefficients, X is the vector of pyroelectric constants, and e denotes the temperature change from a reference temperature. Let us assume that the temperature field, obtained by solving uncoupled problem of heat­conduction, is not affected by mechanical and electrical fields. Additionally, strain and electric fields derive respectively from elastic displacement u and electric potential <1>:

Emn = -} (um,n+ un,m) En = - <I>,n (2)

Modeling of thermo-electro-elastic effective behaviors of piezoelectric 253 composite mediums and analysis of reinforcement orientation effects

In the absence of body force and electric charge, the stress and electric displacement fields verify the following divergence equations.

cr· = 0 Ij,1 (3)

2.2 Local shorthand expressions of constitutive equations

The constitutive equations of the considered piezoelectric medium can be written as follows:

(4,5)

where F represents the inverse electroelastic constants and can be derived from the electroelastic constants by 'inversion'; FMniJ = L·1MniJ

The tensorial quantities in the last expressions are represented by the following matrices:

L iJMn = [ C;nm tl ZMn =[ Emn ]; L. = [ cr ij ] n.=[~ij] ern; , einm -Kin -En 1J D i lJ Xi

F MoiJ = [ &ru.ij t 1 (6) ginm A Mn = [<X1t: ]

gnij -llni where the (9x9) 'matrix' L (F) and the (9xl) 'vector' IT (A) have to be considered as linear operators, which transform the tensor-vector pair [E, E] into another tensor-vector pair [cr, D] and vice-versa. S = (C + e1K·1 erl, TJ = (K + e C- I el rl, gl = C- I el TJ, g = K-1 e S ZMn derive from elastic displacement-electric potential fields UM by means of the following differential relationship:

(7,8)

In equations (6), the (9x9) matrices L and F are symmetric. Additionally, thermal and pyroelectric tensors AM", ITiJ are related to the elctroelastic compliance tensors FMniJ as follows:

(9-a,b)

The homogenization methods will be used to solve microheterogeneous­macrohomogeneous scale transition problem.

254 N.Fakri. L. Azrar and L. El Bakkali

2.3 Effective coefficients formulation Consider a heterogeneous medium consisting of a composite

material of N homogeneous perfectly bonded phases. Equations (4) and (5) can be expressed in a short form for each phase (p=l~N) as:

(lO-a,b)

For each phase, the relations (9-a) and (9-b) can be rewritten as:

(ll-a,b)

where Fp, Lp, , rrp and Ap are the thermoelectroelastic behaviour constants of the pth phase. Thermoelectroelastic effective behavior of the composite medium is described by the global constitutive equations:

(12-a,b)

where, Feff, Leff , A eff and rreff are the effective thermoelectroelastic constants

of equivalent homogeneous medium and Z, f, e and the macroscopic homogeneous fields. Let us recall that the effective coefficients defined above verify:

Feff = (Uff)"' and rreff = U ff N ff (13)

Assuming that the considered composite medium verifies the Hill hypothesis: the medium is considered free of volume forces, submitted at low perturbations and the heterogeneities have a smaller size than medium size. The global volume V of composite medium can be considered as representative volume element attributed to Hill and the volume average of quantities over the entire volume V of composite are considered. The global stress-electric displacement (~) and strain-electric fields (Z) are related to local fields by means of average relations of Hill-Mandel. For the composite material, the volume-averaged thermoelectroelastic fields can then be expressed as:

- N . - N . L =I f'~i and Z=I f'Zi (14,IS)

i=! i=l

where the over bar denotes the volume average of quantity. The subscript 'i' denotes the fh phase and fi is the corresponding concentration. ~i and Zi are uniform fields in ith phase. In order to make the transition since the micro­structural scale of material to the global scale, two localization relationships between global and local quantities are introduced:

i - . i - . Zi = A Z + al e and Li = B L + b l e (16,17)

where Ai , ai, Bi and bi are concentration factors verifying:

Modeling of thermo-electro-elastic effective behaviors of piezoelectric 255 composite mediums and analysis of reinforcement orientation effects

t fi Ai=t fiBi=I and t fiai=t fibi=O (18) i=l i=l i=l i=l

In which I is the (9x9) identity matrix. The localization tensors Ai and Bi depend on the considered micro-mechanical approximation and will be expressed by means of so called 'interaction' tensors TIl Fakri et al.(2003). The computation of the interaction tensors, TIl, is obtained by Fourier's transforms of the associated Green's functions. The explicit formulation and numerical details are presented in Fakri et al. (2003). The localization and interaction tensors A and TIl are basically used to derive the micro­mechanical models used in this analysis.

2.3.1 The Mori-Tanaka mean field approach The Mori-Tanaka's approach consists in considering that the

interactions among the matrix and inclusions are taken into account in term of inclusions concentration in the matrix. The corresponding concentration tensor AMT is then given by the solution for a single inclusion embedded in an infinite matrix in the same manner as heterogeneous inclusion problem of Eshelby.

AMT = [I + L TIl (LI_Lm)rl (19) Vi

In which f m as matrix concentration and Vi is the volume of the inclusion

2.3.2 The Self-Consistent method The self-consistent method consists in considering one single

heterogeneity (inclusion) embedded in an homogeneous medium (matrix) with effective electroelastic moduli L eff UMn not yet known and taking into account the equivalent behavior of neighboring medium of inclusions. Under these conditions, the expression of the concentration tensors, ASc:

Ase = [I + _1_ TIl (LI_Lerr)rl (20) VI

where Ase depends on Leff not yet known. This leads to an implicit algebraic problem which is numerically solved by an iterative procedure. In the other hand, substituting (16,17) into (14, 15) and considering (12a-b), one can obtain the effective thermoelectroelastic constants of the considered composite mediums as the following:

LetT= t f LI AI FetT= t f FI BI n etT = t f AI nl (2la-b,22) i=l i=l i=l

Where the effective thermal properties netT can be obtained by relationships analogous to Levin's relations in thermoe1asticity see Dunn (95).

256 N.Fakri, L. Azrar and L. El Bakkali

3 NUMERICAL RESULTS

In this work, "Ceramic-Polymer" composites are considered. The first phase, matrix, is constituted of spherical inclusions of isotropic polymer. The second phase, reinforcement, is a transversely isotropic ellipsoidal inclusion of ceramic with semi-axes a, b and c. The global coordinate system related to matrix is (X"X2,X3) and the third semi-axe 'c' of inclusion is respected with the polling direction X 3 of coordinate system (X I, X 2, X 3) related to piezo­inclusions (figure 1). Namely, a spatial orientation of inclusion in generally anisotropic medium can be described by three Euler angles e, </I and ffi which are the rotations existing between local and global coordinate systems.

e Q 0 0 G G 0

}-o1 ~ X1

'------~

c=::J matrix cs:::::::::::J inclusion

Figure I: Scheme of spatial distribution of inclusions for which the third half axe 'c' is aligned with polling and global direction X3.

The micromechanical approaches presented above are applied for different composites to predict effective thermoelectroelastic moduli LeffiJMn, Aeff and neff. The thermoelectroelastic moduli of the constituents used in our investigation are obtained from the paper of Levin et ai. (99). For the sake of brevity only some tests are presented. Figure 2 represents the effective piezoelectric coefficient d33 (d3j=e3i c·1ij) of the composite BaTi03/Epoxy with respect to concentration of BaTi03. It can be seen that the evolution of this coefficient with ceramic concentration is affected by the orientation of polling direction and the predictions of the two micro-mechanical models SC and MT differ for high concentrations and particularly for the polling direction (</I =60°). For (</1=0°), the two models coincide perfectly and a small discrepancy is shown for (</1=30°). The predicted value of d33 is increasing with respect to inclusion concentrations. At (</1=60°), the prediction given by MT's model is near zero up to 95% of inclusions and will completely vanish at (</1=90°). In fact, there is an exchange between d33 and d22 and d22 is increasing from zero to the same value of d33 at (",=0°). Figure 3 shows the evolution of the effective hydrostatic coefficient dh=d33+ d31+ d32 for the composite BaTiOJEpoxy with the concentration of BaTi03. It can be seen that the highest value of db is obtained at '" = 30° and the concentration around 50%. The evolution of the effective thermal expansion U33 of the composite with BaTi03 concentration is shown in figure 4

Modeling of thermo-electro-elastic effective behaviors of piezoelectric 257 composite mediums and analysis of reinforcement orientation effects

:IJ .....-" •• -.,_..---u ~ u U ~ u ~ ~ u u ~

~d.e..TOI

. . " ~ ~ a ~ ~ @ D ~ U ~

_dBoToCl,

Fig.2: variation of d)) with respect to Fig.3: variation of dh with respect to concentration of BaTiO) into epoxy matrix concentration of BaTiO) into epoxy matrix for ~ = 0°,30°,60°. for ~ = 0°, 30°,60°.

'" ....

---Ml'

......... . .... sc

..

. . u u ~ U ~ u ~ ~ ~ u w

BaT.o1'tCII16n8 ftadIol

..

+--..---~~~~...,.-~....,....~-l ." ~ ~ ~ ~ u u v ~ ~ ~

.. T'O'''' ...... --.

Fig.4: variation of a)) with respect to Fig.5: variation of 1t) with respect to concentration of BaTiO) into epoxy matrix concentration of BaTiO) into epoxy matrix for ~ = 0°, 30°, 60°, 90° for ~ = 0°, 30°, 60°, 90°

It can be seen that for fiber reinforced composite, the two micro-mechanical models used in the computation, lead to the same results when the fibers long semi-axe lay to the polling direction and the value of u)) decrease strongly when the concentration of the piezoelectric reinforcements increase. The two micro-mechanical models lead to different results at the high concentrations for the fiber orientations since <I> = 30° to <1>=90°. Namely, at the orientation <I> = 90° an increase of the thermal coefficient is observed at the values of the reinforcement concentration lower than 1 0%; after this concentration it decreases quasi-linearly to the full material value. In figure 5, the predictions of pyroelectric coefficient 1t3 given by the Self-consistent and Mori-Tanaka's models are the same. We can observe a decrease of the pyroelectric coefficient values for all BaTi03 reinforcement concentrations. The curves for different piezoelectric polling directions can show a big increase of the pyroelectric coefficient immediately after introducing a little quantity of piezoelectric reinforcement near to I %. After this concentration the pyroelectric coefficient 1t3 decreases strongly for concentrations lowest than 25% and becomes quasi-constant.

258 NFakri, L. Azrar and L. El Bakkali

4 CONCLUSION

In this work, a thermo-electro-elastic modeling of composite materials behavior is proposed based on heterogeneous inclusion problem. Effective thermoelectroelastic coefficients are formulated using the concentration tensors. An efficient numerical procedure is developed for numerical predictions of thermoelectroelastic coefficients and a large part of heterogeneous piezoelectric problems may be studied. It can be seen that for piezoelectric and thermal effective coefficients, the SC and MT models, lead to the nearly same results when the fibers long semi-axe direction, which is also the polling direction, lay with third axe of global referential of the matrix. When the polling direction changes, the two models generally differ for the high concentrations of piezoelectric reinforcements, except for diagonal coefficients as pyroelectric constant 1t3 for which the different micromechanical models predictions are the same for each fibers orientation. It can be concluded that the effective thermo-electro-elastic coefficients are strongly affected by reinforcement spatial orientation, concentration and shape.

REFERENCES

Benveniste Y., The determination of the elastic and electric fields in a piezoelectric inhomogeneity. Journal of Applied Physics. 1992; 72(3): 1086-1095.

Dunn M. L. and Taya M., Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Struct. 1993; 30(2): 161-175

Dunn.M.L., Electroelastic Green's functions for transversely isotropic piezoelectric media and their application to the solution of inclusion and inhomogeneity problems. 1994; International Journal Engineering. Sciences. 32(1): 119-131

Dunn M.L. A theoretical framework for the analysis of thermoelectroelastic heterogeneous media with applications. J. ofIntelligent Material Syst. and Struct. March 1995; 6: 225-265

Eshe1by J.D., Proceeding Royal Society London. 1957; A241: 376

Levin V.M., Rakovskaja M.I., Kreher W.S., The effective thermoelectroelastic properties of microinhomogeneous materials, lnt. Journal of Solids and structures. 1999; 36: 2683-2705

Fakri N; Azrar L. and EI Bakkali L. Electroelastic behaviour modeling of piezoelectric composite materials containing spatially oriented reinforcements, International Journal of Solids and Structures.(40) 2003; 361-384.

INVESTIGATIONS IN SIZE DEPENDENT TORSIONS AND FRACTURES

1) Department o/Mechanical Engineering, The Hong Kong University o/Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 2) Institute 0/ Computational Engineering and Science, Southwest Jiaotong University, Chengdu 610031, Sichuan, China

Abstract: Size effects of fine structures due to stain gradients were exhibited experimentally and theoretically in elastic bending of micrometer and nanometer size beams. The Saint-V enant torsion theory of cylinders are extended to explore the relationship between strain gradients and size dependent phenomena. The prediction of the extended torsion theory is in good agreement with experimental data. Furthermore, a higher order J-integral criterion for fracture, which accounts for strain gradients, and its application are reported.

Key words: strain gradients, size dependence, elasticity, torsion, fracture, J integral

1. INTRODUCTION

Structures in engineering are pushing from macro scale to micrometer and nanometer scales [1]. Microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) are structures used to probe surfaces, study cells and neurons, move microliters of fluids and divert photons in optoelectronics [2-5]. These structures are microns to nanometers in size, and usually bend and/or twist elastically when loaded. The conventional elasticity has been used to predict their deformation behaviors, but it is unclear whether the conventional elasticity is applicable a priori to structures of micron and nanometer scales.

The classical strain-based isotropic elasticity is a geometry independent theory, and has been used indiscriminately to predict the elastic deformation behaviors of large and small (micron-scaled and nanometer-scaled)

259

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 259-266. ©2004 Kluwer Academic Publishers.

260 P. Tong, D. C. C. Lam, F. Yang, and J Wang

structures. Elastic theories accounting for size effects in material deformation were proposed in 1960s [6-10]. Fleck and Hutchinson [11] reformulated Mindlin's theory [10] and renamed it as the strain gradient theory. Recently, Lam et al. [12] modified Fleck and Hutchinson's strain gradient theory [11] with a different decomposition of the second order strain gradient tensor. The reformulation permitted application of the equilibrium equation for moment distribution [13] to the strain gradient theory. This gives rise to a modified version of strain gradient theory, which accounted for the full set of higher order strain gradients and higher order equilibrium relation.

Experimentally, elastic bending stiffness of carbon nanotubes has been reported to be dependent on the size of the nanotubes [14]. The effective elastic modulus determined using conventional elasticity was found to increase from 0.1 TPa to 1 TPa when the tube size is decreased. Recently, Lam et al. [12, 15] conducted a series of tensile and bending tests of micron­sized epoxy beams and develop a strain gradient based bending theory, which exhibits the significant contribution to size dependence from strain gradients. The experimental results shown in Figs. 1 and 2 indicate Young's modulus is practically identical for beams of all sizes while the normalized bending rigidity varies linearly as the inverse of the square of beam thickness. This is in good agreement with the prediction of the strain gradient based bending theory.

b D.H F?9=<====lo17-~ ; \, ;" 'V1 'a:G LLI ~ 0.2 . . ' "';' _.- ~~ ~ . _.- : .. c .".

1 ~ O.IS

0.1 '----'-_'-i' --'-_-'-----'-----'

0.5

O L-~~ __ '__~~~ ow. ~ W 100 IW

Thiekne .. Uim) Thickness Uim)

Figure 1. D 'and Do vs beam thickness (E of Figure 2. Elastic module vs beams tension tests is used to computed Do). thickness.

In this paper, the strain gradient based Saint-Venent torsion theory is developed. The theoretic prediction for torsion of fine cylinders is compared with experimental data. A higher order J-integral criterion for fracture accounting for strain gradient effects and its applications are reported.

Investigation in size dependent torsions ans fractures 261

2. SIANT-VENANT TORSION IN ELASTICITY

The problems of Saint-Venant torsion are described by the equilibrium equation in the cross section of a cylinder bar and the traction free boundary conditions on the lateral circumference surface. For a cylindrical shaft along

the xI-axis in the rectangular Cartesian coordinates x"x2'x3 ' following

Saint-Venant's approach, we assume that, as the shaft twists, the plane cross sections warp, but rotates as a rigid body that

ul == aw(x2 ,x3 ), u2 == -ax\~, u3 == ax\x2

where u's are the displacement components in the x\,x2,x3 -directions, w is

the warping function, which is a function of X2, X3 only, and a, a small

constant «<1), is the angle of twist per unit length along the shaft. The strains and the strain gradients can be derived according to the modified strain gradient theory [12]. We can show that the equilibrium equation in terms of the warping function w is

V2(W-&2V2W)=0, (1) where V2 is the two-dimensional Laplace operator. The boundary conditions are

a 2 2 2 a2 w 2 a 1 Ow -(w-& V W-& -)+& -(--)=nx -n x f) \a2 If) a \2 21' n 'S 'Sp'S

(2) 2 2 2 a2 w 1 Ow

& V W-&, (-+--)=0, as2 pan where p, n and s are the curvature and the local normal and tangential coordinates along the lateral circumference, n's are the components of a unit outer normal vector, and 8 and 81 are constants associated with the material length scale parameters l's defmed by Lam et al. [12] as follows,

2 8/12 I; 2 2/12 I; & =-+- & =-+-

15 4' , 3 2 (3)

The torque of torsion is given as

T = L (0'23 XI - O'ux2)dA + 31; Apa (4)

where f.1. is shear modulus, A is area of the cross section, and the stresses are O'u = 0'31 = pa(w'l -x2), 0'23 = 0'32 = pa(w'2 +xl) (5)

The torsion stiffuess of cylinders is higher than that predicted by the conventional Saint-Venant theory when strain gradients are active. For different cross sections, the normalized torsion rigidities (the torque normalized by the conventional torque solution To) are plotted in Fig. 3. Of the cases considered, triangular bars have the strongest size stiffening. Comparison with experimental data [16] shown in Fig. 4 indicates the prediction of the strain gradient torsion theory is in good agreement with the experimental results.

262 P. Tong, D. C. C. Lam, F. Yang, and J Wang

- ci=lar 6 ... triangular

- - square

I 0

• • rectangular (bI. - 3)

0.1 0.2

. . - :-

0.3 0.4

Nom-oalized lergth scale parameter 0.5

Figure 3. Normalized torsion rigidity (TIT aJ vs. normalized length parameter (l2t2r for circular, 12th for triangular and l2t2a for square and rectangular microbars) with II = 12 .

die W ~ ~ ~ W M W ~ 100

1.2 ~-"1 --'---,,--r-. . . : .

l.l ~ ; .,

1.05 ...... ': . .. . ~.:.~::~ . .. , .

Figure 4. Lakes' rod torsion experimental data [16] fitted by strain gradient torsion theory, d and c are diameter and average cell size, respectively.

3. HIGHER ORDER J INTEGRAL FOR FRACTURE

3.1 Energy Release Rate and Higher Order J-Integral

The classical Griffith energy criterion for crack propagation is, G>~ 00

where the energy release rate G is defmed as

G = _d%da ' (7)

n is the potential energy, and a and B are the crack length and width

A , 1 ' - " - ·· _ ;" - ' "' .

" "

\

.- .. -..... .

Figure 5. A crack and its J-integral path r.

Investigation in size dependent torsions ans fractures 263

respectively. The potential energy accounts for the contribution of strain gradients when they are active. Following Griffith's concept of energy release rate, we have derived a path independent contour integral including the contribution of strain gradients, named as higher order J-integral, instead of calculating the energy release rate. The higher order J-integral is

Jh = r[WdY - (tk Ouk + qk O()k + r oGn )dS], (8) !- ax ax ax

where W is the strain energy density, which depends on the classical strains and the strain gradients, Uk, ()k and Gn are displacements, rotations and outer normal strain, respectively, and tk, qk and r are the tractions of force, couple and double force, which are the work conjugates to Uk> ()k and Gn' The integral path r is an arbitrary opened contour from one crack surface to the other as shown in Fig. 5. The criterion for crack propagation in Eq. (6) becomes

J h ~ J c = Gc ' (9)The path independence of the higher

order J-integral can be proved by using virtual work principle.

3.2 Application of Higher Order J-integral to Elastic Fracture

Figure 6. Schematic of the mesh used in finite element analysis and contours used

. ~

l.t "--'---'--- '!. --'!, --,

0.9

0.8

0,7

0.6

0

: 0

. .. ..... -: ........ ;. . '

o : lI:

... .... : .... .. . : ........ " .... : ..... -o J IG

C,HO c .. ... ... R • • • ••• _ ; _ .

• J IG e,O c:

i i i

10 1 S 20 2>

htL

Figure 7. Conventional Jc and high order Jjc• normalized by Go varied with normalized thickness.

Bending of micron-sized cantilever beams exhibits strain gradient stiffening when the beam thickness is decreased [12, 15]. If a beam is only partially attached to a substrate, a crack is formed between the beam and the substrate. The bending stiffness of the beam on the substrate is stiffened by

264 P. Tong, D. C. C. Lam, F. Yang, andJ Wang

train gradients when the beam thickness is near the magnitude of the higher order length scale parameters. Since the energetic contribution from strain gradients is significant, Jh, which includes the strain gradient energetic contribution, should be used in place of the conventional J-integral. For the linear elastic beam on a rigid substrate loaded to the critical load Pc as shown in Fig. 6, the J-integral on a contour around the crack gives the strain energy release rate Gc• If the beam thickness is significantly larger than the higher order length scale parameters, the conventional and higher order J-integrals can both be used to determine Gc• The quantity Gc is a material constant when fracture is elastic. Since the external work needed for fracturing an interface equals to Gc, the integral of the load-deflection curve to criticality is constant for all beam thicknesses. We conducted a conventional FEM analysis to calculate the displacement field under a critical load Pc approximately. Both J and Jh of the conventional and higher order J-integrals were obtained from the approximate displacement field through the principle of the energy balance and identified the integrals as Jc and Jhc, respectively. The ratios of critical Jc and Jhc to Gc are plotted in Fig. 7. The plot reveals that the ratio Jhc /Gc remains constant and is independent of thickness while Jc /Gc decreases with thickness. This indicates that Jhc properly accounts for the strain gradient contribution to the deformation energy, and is consistent with the energy equivalence notion of the critical higher order J-integral Jhc

and the enery release rate Gc for elastic fracture. In contrast, while the conventional Jc is constant and is equivalent to Jhc when hll is large, the exclusion of the strain gradient energetic contribution leads to artificial size dependence when the beam is thin (Fig. 7).

3.3 Discussion

The deformation energy release rate G and J-integral J for fine structures developed in this study extended the energetic consideration to include the contribution from strain gradients. In structures in which strain gradients are negligible, terms involving strain gradient do not need to be considered and G and J reduce to the expressions of the conventional theory. For elastic fracture, the J-integral determined at critical fracture load is J c and equals to the critical strain energy release rate Gc, which is a material parameter. However, strain gradients can not only elastically stiffen, but also harden materials. While purely elastic deformation does not dissipate extra energy, extra irrecoverable energy is expended toward the creation of extra dislocations in plastic strain gradient hardening. Because of this plastic strain gradient-based energy dissipation mechanism, the assumption of strain energy release rate being independent of beam thickness (i.e., independent of strain gradients) is valid only if the plastic strain gradient contribution is small. When plastic strain gradient hardening is non-negligible, it will shield the crack and increase Gc• As a result, Gc becomes a geometry-dependent

Investigation in size dependent torsions ans fractures 265

parameter if plastic strain gradient hardening is significant and is a function of geometry.

From materials perspective, engineers may seek out materials and design geometries to take advantage of the toughening provided by plastic strain gradients. In the design of thin film structures, the film material maybe selected on the basis of the magnitudes of the higher order material length scale parameters for its higher order strain gradient toughening enhancements, in conjunction with other material selection criteria. From this perspective, the developments of the higher order G and J open new degrees of freedom, namely the scale of the structure and the higher order length scale parameters, in the design of thin film structures in microelectronics, MEMS and NEMS.

4. CONCLUSIONS

A strain gradient based Saint Venent torsion theory for fine cylinders has been developed. Comparatively, similar to the strain gradient based bending theory, the prediction of the developed theory agrees with available experimental data very well. The agreement is a new attestation to that size effects is associated with activity of strain gradients.

Conventional J-integral was developed on the basis of conventional elasticity without accounting for the contributions of higher order stresses and strain gradients. It underestimates the released energy for propagation and growth of a crack when the fractured structure is at micron or nanometer scales. The error of underestimation is larger when the structure is finer. A new strain energy release rate and a higher order J-integral have been developed to correct the underestimation of the released energy.

The higher order critical J-integral remains to be a constant at the critical applied load for beams of different sizes. The numerical study of the interfacial crack between a thin film and a rigid substrate is in good agreement with the conclusion.

ACKNOWLEDGEMENTS

This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, People's Republic of China. F. Yang and J. Wang acknowledge the support of the Fund of Southwest Jiaotong University, People's Republic of China.

266 P. Tong, D. C. C. Lam, F. Yang, and J Wang

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[I] T. R. Albrecht, S. Akamine, T. E. Carver and C. F. Quate, "Microfabrication of

cantilever styli for the atomic force microscope", Journal of Vacuum Science & Technology A-Vacuum Surfaces & Films,~, pp.3386-3396, 1990.

[2] R. Bashir, A. Gupta, G. W. Neudeck, M. McElfresh, and R. Gomez, "On the design of

piezoresistive silicon cantilevers with stress concentration regions for scanning probe microscopy applications", J. ofMicromech. & Microengineering, 10, pp.483-491, 2000.

[3] D. W. Carr and H. G. Craighead, "Fabrication ofnanoelectromechanical systems in

single crystal silicon using silicon on insulator substrates and electron beam

lithography", Journal of Vacuum Science & Technology B, 15, pp. 2760-2763, 1997. [4] H. G. Craighead, ''Nanoelectromechanical systems", Science, 290, pp.1532-1535, 2000.

[5] E. Manias, 1. Chen, N. Fang and X. Zhang, ''Polymeric micromechanical components

with tunable stifthess", Applied Physics Letters, vol. 79, pp. 1700-1702,2001.

[6] R. A. Toupin, "Elastic materials with couple stresses", Arch. Ration. Mech. Anal., 11, pp. 385-414, 1962.

[7] R. D. Mindlin and H. F. Tiersten, "Effects of couple-stresses in linear elasticity", Arch. Ration. Mech. Anal., 11.. pp. 415-448,1962.

[8] W. T .. Koiter, "Couple stresses in the theory of elasticity. I and II.", Proc. K. Ned. Akad. Wet. (B), vol. 67, pp. 17-44, 1964.

[9] R. D. Mindlin, "Micro-structure in linear elasticity", Arch. Ration. Mech. Anal., vol. 16,

pp. 51-78,1964. [10] R. D. Mindlin, "Second gradient of strain and surface tension in linear elasticity", Int. J.

Solids Struct., vol. I, pp. 417-438,1965.

[II] N. A. Fleck and J. W. Hutchinson, "Strain gradient plasticity", In Advances in Applied Mechanics, 33, Hutchinson and T. Wu, Eds., Academic Press, NY, pp.295-361, 1997.

[12] D. C. C. Lam, F. Yang, A.C.M. Chong, 1. Wang and P. Tong, "Experiments and theory in strain gradient elasticity", submitted, 2002.

[13] F. Yang, A. C. M. Chong, D. C. C. Lam and P. Tong, "Couple stress Based Strain

gradient theory for elasticity", Int. J. Solids & Struct., vol. 39, pp. 2731-2743,2002.

[14] E.W. Wong, P.E. Sheehan and C. M. Lieber, ''Nanobeam mechanics: Elasticity, strength and toughness ofnanorods and nanotubes", Science, vol. 277, pp. 1971-1975, 1997.

[15] D. C. C. Lam, F. Yang, P. Tong, A. C. M. Chong and 1. Wang, "Size dependence in

nanometer- and micron-scaled structures", Chinese National Conference for Solid

Mechanics, Dalian, China, August, 2002. [16] R. S. Lakes, "Experimental microelasticity of two porous solids", Int. 1. Solids &

Struct., vol. 22, pp. 55-63, 1986.

INFLUENCE OF MICROSTRUCTURAL PARAMETERS ON SHAPE MEMORY ALLOYS BEHAVIOR

C. Niclaeys, T. Ben Zineb, E. Patoor LPMM-ISGMP UMR CNRS 7554

ENSAM Metz, 4 rue Augustin Fresnel

57078 Metz, France

Abstract: Modeling the functional behavior of material undergoing a martensitic phase transformation like in shape memory alloys is a challenge for the development of industrial applications using these materials. Multiscale modeling concepts developed in mechanics of materials are well adapted to solve this problem. Using a crystallographical description for the single-crystal behavior law, two examples of how the internal stresses influence the transformation in thin films and bulk materials are presented. Scale transition is performed with the self­consistent approximation in the bulk and with FEM in thin film.

Key words: Martensite, Thin film, Micromechanics, Scale transition, Finite Element

1. INTRODUCTION

Shape memory alloys playa large role in the development of intelligent systems, many applications in this field are related to micro-systems: mini­pumps installed in the human body deliver micro-doses of cortisone to calm chronic pains [I]. Numerical tools developed these last years for dimensioning macro-components cannot be straightforward applied for micro-systems. In micro-systems, scale transition scheme must take into account that grain size is of the same order of magnitude as the component itself so the notion of equivalent homogeneous material is no longer valid. In addition, surface effects are now very important. The Finite Elements

267

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 267-274. ©2004 Kluwer Academic Publishers.

268 C. Niclaeys, T. Ben Zineb, E. Patoor

Method (FEM) is well adapted to deal with grain shape, grain size and also the relative position of each grain with respect to the film surface.

In this paper, a single crystal constitutive law for the superelastic behavior of SMA is used. This law is implemented into the FEM code ABAQUS via the UMAT (Users Material) routine. Results obtained for tensile test on a multicrystalline film are discussed and compared with those obtained for bulk material with a self-consistent approach.

2. SINGLE CRYSTAL BEHAVIOR LAW

The single crystal behavior law is based on the description of the reversible martensitic transformation during a mechanical or thermal loading. Total strain results from the transformation strain ED and from the elastic strain Ee [2]. When many variants are activated, each variant contributes to the total transformation strain with respect to its volume fraction f'. The total strain rate is given by:

de·· - de~ + det = de~ + '" s~dfn 1) - 1) 1) 1) £.J 1) (1) n

Strain evolution is obtained from the definition of a thermodynamical potential, function of the control parameters (applied stress cr, temperature T) and of the set of internal variables f'. For an unit reference volume V of parent phase, the complementary free energy '¥(cr, T, f) composed by elastic and chemical contributions is expressed by [3]:

1 (2) +cr .. "'E~rn --"'HnmfDfm

1J £.J I) 2 £.J n n,m

In relation (2), To denotes the thermodynamical equilibrium temperature, B is the coefficient for the chemical energy, S is the fourth order stiffhess tensor and II"'" is the interaction matrix which described the compatibility or the incompatibility between two variants [4, 5]. In this paper, we restrict our objective to describe the loading sequence, so we can neglect the hysteresis phenomena. In this case, no dissipation takes place and the transformation over a variant n occurs when the thermodynamical force (O'¥/Of D ) is equal to zero. So we obtain:

Influence of microstructural parameters on shape memory alloys behavior

m

The local behavior law is obtained from equation (1) and (3):

269

(3)

dEij ::: Sijkl dcrld + L Eij ~)Hnm r 1 (E~ dcrld - BdT) (4) n m

For a superelastic loading, we assume the temperature is kept constant, so we obtained:

(5) n,m

The implementation of this law in the ABAQUS code is realized thanks to the routine UMAT (Users Material) [6]. This routine allows to actualize stresses, internal variables and the tangent modulus necessary to the determination of the rigidity matrix for each element.

400

Analytic modeling ~300+-----------~=-~=-----------~--~

~ ~ 200+-------~~~~~~~::==_I~----~ ~ CIl ABA US simulation 100+-~------------------~~--~~~~

O+-----.-----.----.----~----~----~

0,00 0,01 0,02 0,03 0,04 0,05 0,06

Strain

Figure I. Comparison between the ABAQUS simulation and the analytic model

To verify the implementation of the single crystal behavior law (5) in the ABAQUS code, a tensile test a single cubic element is performed. We compare the FEM result obtained with the analytic ones (figure 1). The small discrepancy between the two curves comes from difference in the loading condition imposed in the two approaches. A pure tensile test is described with the analytical model but in the ABAQUS code a pseudo tensile test is applied. From this result we consider we have a successful implementation of the SMA single crystal behavior in the ABAQUS code. We applied this UMA T routine to analyze the response of a SMA film. We consider a tensile loading applied along the longitudinal axis of a SMA multicrystalline plate.

270 C. Niclaeys, T. Ben Zineb, E. Patoor

3. APPLICATION TO A MULTICRYSTALINE FILM

3.1 Geometrical description and boundary conditions

We consider a plate structure composed with six grains having one grain in the thickness (figure 2). A 10x5xO.l mm CuAIBe superelastic SMA plate is meshed. Each grain are described with 144 isoparametric tridimensionnal elements [6). Euler angles and the Schmid factor (Rll ) in the longitudinal direction are given in table 1.

Table 1. Crystallographical orientation (Euler angles cpl, cp, cp2) and Schmid factor for a I . d· I ·1 t fi h . ongltu rna tens) e tes or eac gram.

cpl cp cp2 RIl

Grain # 1 351.9 21.3 212.1 0.49

Grain # 2 173.4 104.5 34.5 0.43

Grain # 3 88.6 55.6 327.6 0.4

Grain#4 210.3 91.9 131.8 0.21

Grain #5 21.6 127.7 56.96 0.33

Grain # 6 234.4 148.3 268.5 0.28

5 1 I\~ 2 § U on

~I 3

2 4

lOmm Figure 2. Representation of the geometry adopted for the thin film

From data in table 1 we observed that grain # 1 is the best oriented with respect to the longitudinal loading direction. In the opposite grain # 4 has the smallest Schmid factor. We simulated a tensile test along the longitudinal direction 1. Boundary conditions, loading and meshing are represented in figure 3. We imposed the following boundary conditions: - All the nodes of the left face are blocked in direction 1, a node of this

face is blocked in directions 2 and 3. - All the nodes of the right face are submit to a displacement according to

direction 1.

Influence of microstructural parameters on shape memory alloys behavior

271

Figure 3. Representation of boundary conditions applied and the meshing of the plate

3.2 Martensite volume fraction

This model allows to detennine the evolution of the martensite volume fraction during the loading sequence. The ftrst variant of martensite appears in grain # 1. The last grain where the transformation takes place is logically grain # 4. This sequence is in good agreement with experimental features.

A very large intergranular heterogeneity is observed for the volume fraction of martensite (ftgure 4). When grain # 1 is almost fully martensitic (92 % martensite), grain # 4 is still almost austenitic (only 4 % martensite) and the global average volume fraction for the structure reaches 50%. It is also important to notice the transformation occurs with great difftculties around the grain boundaries. This can be related to the stress fteld, which is disturbed by the neighboring grains. To better illustrated how the transformation proceeds, the mean value of the volume fraction of martensite induced inside each grain is computed.

Figure 4. Mapping of the volume fraction when grain # I is quite fully martensitic

The six curves obtained are shown on ftgure 5. We observe that the transformation sequence is the following: grain # 1, 2, 3, 5, 6 and ftnally

272 C. Niclaeys, T. Ben Zineb, E. Patoor

grain # 4. This is in close relation with the Schmid factor listed for each grain in table 1. Some deviation to this sequence must be notice: at the end of the loading sequence, the volume fraction is larger in grain # 5 than in grain # 3 whereas the Schmid factor is larger in grain # 3. This must be related to the influence of the intergranular stress field.

Volume fraction

o o

:k:: o +""=-------, o

o

o Chronological , ________________________________________ -.. time

Figure 5. Evolution for each grain of the volume fraction during the loading sequence

3.3 Thin fIlm effect

This heterogeneity in volume fraction is also observed on the mechanical behavior. On figure 6, stress-strain curves for each grain are plotted and compared with the macroscopic response. We observed than grain # 1 first easily transforms at the beginning of the loading and its transformation strain saturates as the transformation progresses in the adjacent grains. This transition is related to a strong inflexion on the macroscopic curve. We also observe that for the other grains, stress-strain curves have a more complex shape that can be related to the formation of several different variants of martensite inside these grains.

Influence of microstructural parameters on shape memory alloys behavior

120

.-.. 100 ~

g., :t 80 --.... 60 '" '" ~ 40 ... -fJ}

20

0

0,00 0,02 0,04 0,06 0,08

Transformation strain 11

0,10

Figure 6. FEM determination of stress-induced curves grains in thin film

273

0,12

The strong influence on the surrounding on the grain individual behavior can be underline if we compare the behavior exhibit by a grain inside a thin film with the stress-strain curve determined for a grain having the same crystallographic orientation and the same elastic and phase change properties but embedded in a bulk polycrystalline structure. Curves; presented in figure 7 are computed using the self consistent scheme developed to describe the superelastic behavior of SMA [3, 7]. We consider an aggregate composed with 106 spherical grains with an isotropic crystallographical texture and the constitutive behavior defined in equation (9). The longitudinal direction of the thin film is kept as the tension loading axis for the bulk material. So every parameters are kept constant, only the internal stress field which is directly associated to how strain incompatibilities are developed during the phase transformation can change.

500 450

..-.. 400 '" Grain # 2 ~ 350

300 '-'

250 ---l

'" 200 ~ -i '" ~ 150 .tl fJ} 100

l 50 0 0.00 0.01 0 ,02 0,03 0,04 0 ,05 0,06

Transfonnation strain 11

Figure 7. Stress-strain curves computed using a self-consistent code for grains having the same crystallographic orientation as in table 1 but embedded in a bulk material

274 C. Niclaeys, T. Ben Zineb, E. Patoor

Comparison between the stress-strain curves in figures 6 and 7 undoubtedly establish the leading influence of the internal tress field on the way the phase transformation proceeds in a given grain. If we consider, inside the bulk material, a grain having the same well-oriented crystallographic direction as grain # 1, we observed that this grain is only partially transformed and this despite a stress level now four time higher than in the thin film. In figure 6, the transformation strain for grain # 1 reaches 10% at 100 MPa and in figure 7, at the same test temperature, this transformation strain is now lower than 6% but the stress exceeds 400 MPa. Similar observations can be made on other grains like grain # 2 for instance.

4. CONCLUSION

In this work a crystallographical description of the single crystal behavior law for superelasticity in SMA was successfully implemented in the finite element code ABAQUS. This approach is well adapted to compute the mechanical response of thin film. Crystallographical orientation, grain shape, grain size and surface effect are accounted in that way. Experimental validation is under way using EBSD and X-ray diffraction techniques. We also made a comparison with a self-consistent simulation to established the major influence of the surrounding on the way the transformation proceed in an austenitic crystal for a given crystallographic orientation.

REFERENCES

1. Eiji Makino, Takashi Mitsuya and Takayuki Shibata, "Fabrication of TiNi shape memory micropump", Sensors and Actuators A: Physical, Volume 88, Issue 3, 5 January 2001, Pages 256-262.

2. Patoor E., Eberhardt A., Berveiller M., "Thermomechanical behaviour of shape memory alloys" Archives of Mechanics, Vol. 40, pp. 775-794,1988.

3. Patoor E., Eberhardt A., Berveiller M., "Micromechanical modelling of the superelastic behavior", Journal de Physique IV, Vol. 6, pp. CI-277-292, 1996.

4. Siredey N., Patoor E., Berveiller M., Eberhardt A., "Constitutive equations for polycrystaIline thermoelastic shape memory alloys. Part I. Intragranulaire interactions and behavior of the grain", International Journal of Solids and Structures, Vol. 36, pp. 4289-4315,1999.

5. Niclaeys C, Ben Zineb T., Arbab Chirani S, Patoor E., "Determination of the interaction energy in the martensitic state", International Journal of Plasticity, Vol. 18, pp. 1619-1647, 2002.

6. Niclaeys C, Thesis, Metz University, "Comportement des monocristaux en AMF. Application au comportement des polycristaux", 2002.

7. Entemeyer D., Patoor E., Eberhardt A., Berveiller M., "Strain rate sensitivity in superelasticity", International Journal of Plasticity, Vol. 16, pp. 1269-1288,2000.

INVESTIGATION OF RIDGING IN FERRITIC STAINLESS STEEL USING CRYSTAL PLASTICITY FINITE ELEMENT METHOD

Hyung-Joon Shin\), Joong-Kyu A1I?l, and Dong Nyung Lee3)

1) Center for Science in Nanometer Scale, ISRC, Seoul National University Seoul 151-742, Korea E-mail: hjshin@Csns.snu.ac.kr 2) Metal&Ceramic Research Group, LG Cable Ltd. Anyang-si, Kyung-ki 431-080, Korea E-mail: seifert@dari.cable.lg.co.kr 3) School of Materials Science and Engineering, Seoul National University SeouI151-742,Korea E-mail: dnlee@Snu.ac.kr

Abstract: The ridging problem in ferritic stainless steel is well lmown and unsolved for more than two decades. Ferritic stainless steel (FSS) sheets exhibit ridging parallel to the rolling direction when subjected to tension or deep drawing. The origin of ridging has not been clearly explained yet. Most models suggested before are too simplified and underestimate the influence of neighboring grains. In this study, we simulate the ridging phenomenon using the crystal plasticity finite element method (CPFEM). We test the previous models with CPFEM and investigate the relations between orientations and ridging by simulating a more realistic case using EBSD results.

Key words: crystal plasticity, ridging, ferritic stainless steel, finite element method, texture

1. INTRODUCTION

Ridging or roping is undesirable surface corrugation ofFSS sheets. When FSS sheet is pulled or deep drawn, it shows undulations, with peaks on one side of the sheet coinciding with valleys on the other side without changes of the thickness. The columnar structure of the FSS slab has NO//<100> orientation, which is hard to recrystallize [1]. In addition, there is no transformation such as a. to y or y to (l phase during all processes. Therefore,

275

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 275-282. ©2004 Kluwer Academic Publishers.

276 H.-J Shin, J-K. An and D.N Lee

long grain colonies with similar orientations originated from the columnar structure of the slab exist even after cold rolling and annealing. Many people agree that ridging originate from different plastic anisotropies of these colonies. At present there exist three essential concepts [2-4] associating the ridging phenomenon with anisotropy of plastic deformation. Though they give good physical pictures of ridging, they are rather simplified models, in which interactions of neighboring grains are underestimated. The objective of this study is to test the previous models more quantitatively using CPFEM, which is a most useful tool taking anisotropic behavior of materials into account and to simulate ridging of a FSS sheet using the orientation distribution through the thickness measured by EBSD.

2. EXPERIMENTAL AND NUMERICAL PROCEDURES

In experiment, a STS409L sample obtained from the columnar structure zone of the slab was hot rolled, cold rolled and annealed. The details of process conditions are given in Table 1. The material used was STS409L containing 0.008% C, 0.56% Si, 0.25% Mn, 11.4% Cr, 0.23% Ti and 0.009% N. The ridging characteristics were determined after 15% tensile elongation. The local textures of the sheet were measured by SEM-EBSD from RD and ND.

In order to investigate the ridging phenomena, we simulated 20% tensile straining of the FSS sheet. The crystal plasticity description was implemented employing the method of Kalidindi et al [5). The CPFEM is one of the powerful simulation tools. The crystal plasticity provides a micromechanical model for slip dominated plastic flow and serves as a constitutive theory. The finite element method offers a numerical means to solve partial differential equations, such as the field equations of elasticity and plasticity. For a rate-sensitive slip, the usual power law is used to relate the plastic shearing rate on the (l'th slip system, ya, to the resolved shear stress, .a, as can be seen in Eq. 1 [6].

Table J Process conditions STS409L

Reheating Temp. rOC1 1200 Finishing Temp. rOC] 930 Pass Schedule [rnrn] 25-> 17-> 12->8->5.2->3.5 HR Annealing Temp. rOC] 930 CR Annealing Temp. [0C] 930

Investigation of Ridging in Ferritic Stainless Steel Using Crystal Plasticity Finite Element Method

. <l _. ~I~I(~-l) Y -Yo <l <l

g g

277

(1)

where g<l and m represent slip system resistance parameter and rate sensitivity of slip, respectively. The self and latent hardening can be readily accounted for by suitable evolution of g<l values in the constitutive law of Eq. 1.

(2)

The b<lf3 is nxn hardening matrix, where n is the total number of slip system. It describes the rate of increase of the deformation resistance on slip system a due to shearing on slip system 13. The several simple phenomenological forms for the hardening matrix, b<l(3, have been suggested so far. Peirce et al. [7] used the following simple form for the hardening law.

(3)

with h f3 denoting the self hardening rate and parameter q representing the latent hardening parameter- h f3 can be obtained by Eq. 4.

(4)

where 110, a, and gsat are slip system hardening parameters. The material

Table 2. Materialparameter used/or simulation

Elastic constant [GPa] 11 = 80.69

A. = 111.44

Strain rate sensitivity m=0.02

Initial value of slip resistance parameter [MPa] go= 110

Saturation value of slip resistance parameter [MPa] gsat= 252

Reference shear strain rate Yo = 0.0001

q = 1.4

Hardening parameter a= 2.24

ho = 1.7 [GPa]

parameters used for simulations are given 10 Table 2. The 24 shp systems (12{llO}<1l1> and 12{112}<111» were assumed to be active during

278 H.-J. Shin, J.-K. An and D.N Lee

deformation. A finite element mesh was formed by ABAQUS-C3D8R element [8]. Each element represents one orientation.

3. RESULTS AND DISCUSSION

3.1 Simulation of Previous Models

Chao attributes ridging to different plastic strain ratios between the ND//<lll> and ND//<100> components as shown in Fig. 1a [2]. The most serious problem in his model is that the deformed outline of the surface is far from that of real materials, which shows undulations. The peaks or valleys are symmetric with respect to the center plane. The calculated result is similar to his prediction (Fig. 1 b). The normal strains, ETD and END, assumed by him results in the symmetric deformed surface. However, the calculated result shows that not only the normal strains but also the shear strain YNT (N=ND; T=TD) plays a role in ridging. T~e {111}<110> components buckle during deformation. When a (111)[011] crystal is pulled along rolling direction, (211)[111] and (211)[111] slip systems are most activated, which are asymmetric with respect to the RD. Therefore, YNT is generated and undulations occur macroscopically in Chao's model too.

Takechi focused on different shear strains between RD//<110> fibers. As mentioned above, the {Ill }<11 0> components result in shear deformation when pulled along the rolling direction. Though his model is very simple, he considered crystal plasticity, so the result calculated in this study is in good agreement with his prediction (Fig 2). The shear direction varies with respect to the RD or the ND.

According to Wright, there occurs the compatibility problem after deformation due to different plastic strain ratios between the {Ill} < 112> matrix and the {OOl }<11O> band. The {001 }<110> orientation gives rise to lower r-values than the {Ill} < 112> orientation. In order to satisfy the compatibility condition, he claimed that the {001 }<11O> band should buckle along the sheet normal direction. Fig. 3 shows the CPFEM results for Wright's model. Instead of buckling of the {OOl }<110> band, it shrinks only along the thickness direction on the both sides (Fig. 3a, b). This discrepancy comes from his strict assumption. The model implies that distortion is limited to the {OOl}<llO> band, i.e., AB and CD lines in~. 3c must remain straight after deformation. If the lateral side of sheet, AB or CD, is free to move, the compatibility is compensated by a little contraction of matrix.

Investigation of Ridging in Ferritic Stainless Steel Using Crystal 279 Plasticity Finite Element Method

.-111 1 1 - -

(001)[110] (111)[011] (001)[110] (111)]011] (001)[110]

(a) (b)

Figure 1. (a) Schematic diagram explaining Chao's model and (b) result simulated by CPFEM.

1I11)[01l ]lIi1)(011] (111)(011] (111)[0111 (111)(0111

(a) (b)

Figure 2. (a) Schematic diagram explaining Takechi 's model and (b) result simulated by

CPFEM.

NO

RO~TO

NO

R~TO

(111)[1]2) (001)[1 (0]

ero C -1..40&01

-UIE-/tl

_,~t

_1"~

_'~1

.,~

.I,:z!,(tl

...-... (a) upper side .T..IoII!!~

4"l!o02 ......... ~-... ......,

A TO

~AO

(c) Ern distribution

(b) lower side Figure 3. Result simulated by Wright's model.

D

B

280 H.-J. Shin, J.-K An and DN Lee

3.2 The Effect of Colonies

In order to investigate the effects of different colonies, we simulate the tensile deformation of sheet having {OO I }<11 0>, {111 }<11 0> and {112}<110> colonies in the middle of textured matrix. Fig. 4 shows the initial texture of matrix, which was calculated using CPFEM for 50% plane strain compression of a randomly oriented bcc metal.

{OO I } < II 0> colonies shrink more in thickness direction than matrix, as a result, the plate shows ridges at the surface layer (Fig. 5b). It could be thought that it's a kind of extended concept of Wright's model. There is only one layer considered in Wright's model, while the {001}<1l0> colonies exist in the middle of the plate in this case. From this result, we could also know that Wright's prediction is far from behavior of real materials. The compatibility between colonies and matrix is compensated for not by buckling of {001}<110> colony, but by deformation of colony and matrix under restriction. Therefore, the ridging occurs owing to different plastic strain ratio between colony and matrix and it is not in the form of undulation.

When {111 }<110> colonies exist in the plate, there occur undulations (Fig. 5d). It is similar to Takechi's model. In this simulation, the interactions between matrix and colonies are taken into account. The degree of ridging is less than that of Takechi's model due to interaction of surrounding grains. The adjacent colonies with different rolling directions show different shear direction after deformation, which results in distinct undulations. If the shear directions of the adjacent colonies were the same, the distance between ridges might be wider.

{112}<1l0> is one of the common orientations among a-fiber. The deformed shape is similar to the case of {IlI}<llO> colony (Fig. Sf). The undulations occurs at the position of {II2}<11O> colonies. The undulations occUr, because the {112}<1l0> colonies bring about shear strain, 1TN, during tensile deformation. The difference between effects of the {Ill} < II 0> and { 1I2} < II 0> colonies is that contraction along the transverse direction of the {1l2}<1l0> colonies is smaller than that of the {Ill}<IlO> colonies, which comes from the different r-values between these components.

MAX.- 7.7

Figure 4. ODF (rp2=45j a/initial orientations used/or simulation.

Investigation of Ridging in Ferritic Stainless Steel Using Crystal 281 Plasticity Finite Element Method

Figure 5. Initial and deformed meshes of (a), (b) {OOI} < 1 10>; (c), (d) {111}<I1O>; and (e), (f) {1l2}<llO> colonies.

3.3 Simulation Based on the EBSD Data

The Fig. 6 shows the EBSD mapping of STS409L, where the darkest and the brightest fields represent <Ill> and <110> orientation, respectively. Its texture can be approximated by {1l1}<1l0>. An about 600 f..I1Il wide colony of {OOl}<l1O> is found in the center layer. The measured ridging height was about 55).1m.

Ridging of STS409L sheet is calculated here using CPFEM based on EBSD data. It is desirable that morphologies and orientations of grains that are identical to the measured data are used for CPFEM calculation. However, it is impractical to measure the shapes and orientations of all grains in the specimen from its two-dimensional sections. The EBSD data indicate that

282 H-J Shin, J-K. An and D.N Lee

(a) normal direction (b) rolling direction Figure 6. EBSD mappingfrom rolling direction ofSTS409L.

STS409L specimen has the y-fiber texture as the main component and some colonies. To simplify the problem, the specimen is modeled as y-fiber textured sheet embedded by some colonies measured by EBSD. Fig. 7 shows the initial and deformed meshes of the specimen, which indicates that ridging is well simulated by CPFEM. The low plastic strain ratio of the {OOl }<110> colonies and different shear deformations of the {Ill }<110> or {l12}<11O> colonies give rise to ridging. If orientations are randomly distributed, plastic anisotropy of each grain can be weakened and compensated by neighboring grains. But if they are in the form of colonies, their anisotropies cannot be neglected. They bring about macroscopic corrugations in the end. Therefore, the colonies should be eliminated for the good surface quality of FSS.

7'.

(a)L~ 11111111.,

1111 11''''

Figure 7. (a) Initial and (b) deformed meshes of specimen having various colonies.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the fmancial support by POSCO and Texture Control Laboratory (NRL), Seoul National University.

REFERENCES

[1] N. Tsuji, K. Tsuzaki and T. Maki: ISIJ int. Vol. 33 (1967), p. 783. [2] H. Chao: Trans. ASM Vol. 60 (1967), p. 33.

[3] H. Takechi, H. Kata, T. Sunami and T. Nakayama: Trans. nM. Vol. 78 (1967), p. 233 .

[4] R.N. Wright: Metall. Trans. Vol. 3 (1972), p. 83.

[5] S.R. Kalidindi, C.N. Bronkhorst and L. Anand: J. Mech. Phys. Sol. Vol. 40 (1989), p. 547.

[6] J.W. Hutchinson: Proc. R. Soc. Lond. Vol. A348 (1976), p. 101.

[7] D. Pierce, R.J. Asaro and A. Needleman: Acta meta)). Vol. 30 (1982), p. 1087.

[8] ABAQUS, Reference Manuals, Version 5.8, (1998) Hibbitt, Karlsson & Sorrensen, Inc

GRAIN BOUNDARY EFFECTS AND FAILURE EVOLUTION IN POLYCRYSTALLINE MATERIALS

W. M. Ashmawi I) and M. A. Zikryl)

J) Department of Mechanical and Aerospace Engineering North Carolina State University

Raleigh, North Carolina 27695-7910, U.S.A.

Email: w.ashmawi@ncsu.edu

Abstract: Dislocation density based multiple-slip constitutive formulations and specialized computational schemes are introduced to account for large-strain ductile deformation modes in polycrystalline aggregates. Furthermore, new kinematically based interfacial grain boundary regions and formulations are introduced to account for dislocation density transmission, absorption, and pile-ups that may occur due to grain boundary misorientations and properties.

Key words: polycrystalline materials, grain boundary, dislocation density, crystal plasticity

1. INTRODUCTION

Grain boundary (GB) structure, orientation, and distribution are essential microstructural features that characterize the initiation and evolution of failure modes in crystalline metals, alloys, and intermetallics. Physically­based constitutive descriptions are needed that can account for dominant physical mechanisms that may occur at different physical scales. The primary purpose of this study is the introduction of an inelastic dislocation density based multiple-slip crystalline constitutive formulation that can be used to obtain a detailed understanding and accurate prediction of interrelated local material mechanisms that control and affect global deformation modes in f.c.c. polycrystalline aggregates with random GB orientations and distributions. In this formulation, the length scale between multiple-slip crystalline formulations and dislocation densities is bridged by

283

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 283-290. ©2004 Kluwer Academic Publishers.

284 W M Ashmawi and M. A. Zikry

coupling evolutionary equations for the mobile and immobile dislocation densities, through the temperature dependent flow stress and slip rates on each slip system, to a multiple-slip rate-dependent crystal plasticity formulation. The derivation of these evolutionary equations are based on accepted physical relations, and generally account for thermally activated dislocation activities such as generation, interaction, and annihilation that are generally representative of the dislocation structures in cubic crystalline metals (see for example [I)).

Most polycrystalline formulations generally do not account for GB effects such as dislocation density and slip transmission, blockage, and absorption. These effects could result due to GB orientation, structure, or interfacial stress mismatches. In this study, GB effects are accounted for by the introduction of interfacial regions that are used to track slip and dislocation density transmissions and intersections. These accurate representations of overall polycrystalline aggregate behavior are needed for the prediction of failure initiation due to GBs, sub-grains, and cell-walls.

2. MULTIPLE-SLIP RATE-DEPENDENT CRYSTAL PLASTICITY FORMULATION

The crystal plasticity constitutive framework used in this study is based on the formulation developed in Kameda and Zikry [2] and Ashmawi and Zikry [3]. In that formulation, it has been assumed that the velocity gradient can be decomposed into a symmetric part, the deformation rate tensor, Dij

and an anti-symmetric part, the spin tensor, W;j' It is further assumed that

the total deformation rate tensor and the total spin tensor can be then additively decomposed into elastic and plastic components. The inelastic parts are defined in terms of the crystallographic slip rates as

D~ = p(a)y' (a), '1 lJ

wP = W(a)y' (a) lJ IJ '

(la-b)

where a is summed over all slip systems. The superscript p denotes

the plastic part, and the tensors E:;a) and w~a) are defined in terms of the

unit normals and the unit slip vectors. For a rate-dependent inelastic formulation, the constitutive description on each slip system can be characterized by a power law relation

Grain boundary effects and failure evolution in polycrystalline materials 285

Y·(a) =y.(a)[~l[J:jl;-1 ref r(a) r(a) no sum on a ,

ref ref

(2)

where r!:) is the reference shear strain rate which corresponds to a

reference shear stress, r!:) and m is the rate sensitivity parameter.

The goal of this study is to perform experiments in order to isolate the influence of dimensional parameters in a predominantly uniaxial stress field on the mechanical behavior of thin copper foils.

3. LOCAL DISLOCATION DENSITY STRUCTURE

To gain a more fundamental understanding of dislocation motion, interaction, and transmission on material failure modes, the crystal plasticity constitutive formulation is coupled to internal variables that account for a local description of the dislocation structure in each crystal. Specifically, we have used the mobile and the immobile dislocation densities as the internal variables in our constitutive formulation. In inelastic deformations of ductile metals, the characteristics of the microstructure are governed by the mechanisms of dislocation production and dynamic recovery. As the material is strained, immobile dislocations are stored in each crystal, and these dislocations act as obstacles for evolving mobile dislocations. Therefore, the immobile and mobile dislocation densities can be coupled, due to the continuous immobilization of mobile dislocations. The reference

stress, on each slip system, can be given as a function of P:':) , the immobile

dislocation density. The reference stress that is used here is a modification of widely used classical forms (see for example [4]) that relate the reference stress to a square-root dependence on the immobile dislocation density as

(3)

where G is the shear modulus, b is the magnitude of the Burgers vector, T;a)

is the static yield stress, and the coefficients, a~ (; = 1,12) are interaction

coefficients, and generally have a magnitude of unity.

286 W. M Ashmawi and M A. Zikry

Now consider a given state for a deformed material, which has a

dislocation structure of total dislocation density, p(a). This total dislocation

density is assumed to be additively decomposed, into a mobile dislocation

density, p~Q> , and an immobile dislocation density Pi<;>' Furthermore, we

have assumed that during an increment of strain, an immobile dislocation density rate is generated and an immobile dislocation density rate is annihilated. The balance between dislocation generation and annihilation equations is the basis for the evolution of mobile and immobile dislocation densities as a function of strain. Based on these arguments, it can be shown (see [5] for a detailed presentation) that the coupled set of nonlinear evolutionary equations of mobile and immobile dislocation densities can then be given by

dp~a> = y.(Q>(gsou, (Pi<;»_ gminte, exp(-.!!...)- gimmob Cp<Q>] (4) dt b2 p~a> b2 kT b "P ,m ,

dp(a) .( > (g . H g. ob JP(Q> H (Q») _'m_ = y Q ~exp(--)+-1!!!!!!- p. -g exp(--)p. (5) dt b2 kT b 1m recov kT 1m ,

where gsour is a coefficient pertaining to an increase in the mobile dislocation density due to dislocation sources, gminter is a coefficient related to the trapping of mobile dislocations due to forest intersections, cross-slip around obstacles, or dislocation interactions, grecov is a coefficient related to the rearrangement and annihilation of immobile dislocations, gimmob is a coefficient related to the immobilization of mobile dislocations, H is the activation enthalpy, k is Boltzmann's constant, and T is the temperature. As these evolutionary equations indicate, the dislocation activities related to recovery and trapping are coupled to thermal activation.

4. NUMERICAL TECHNIQUE

The total deformation rate tensor, Dij' and the plastic deformation rate

tensor, I{ , are needed to update the stress state of the crystalline material.

The numerical method used here is one developed by Zikry [6], for rate­dependent crystalline plasticity formulations. An implicit finite-element method is used to obtain the total deformation rate tensor, Dij' To overcome

numerical instabilities associated with stiffness, a hybrid explicit-implicit

Grain boundary effects and failure evolution in polycrystalline materials 287

method is used to obtain the plastic deformation rate tensor, Dt . This hybrid

numerical scheme is also used to update the evolutionary equations for the mobile and immobile densities.

5. GRAIN BOUNDARY INTERFACIAL REGIONS

It is clear that GBs playa considerable role in controlling the mechanical and physical properties and response of polycrystalline aggregates, which in combination with other factors influence material flow and fracture. As indicated by Miller [7], most existing models treat GBs as either one­dimensional rigid walls, or only as interfacial quantities which are not accurately representative of GB morphology, structure and interfacial mismatches that may occur due to stress and strain gradients. In this study, the GB region is modeled as an interfacial region with structure and properties that are different from the bulk grain regions. This layer is assumed to be a crystalline region that has a specific orientation for its crystallographic planes. Special kinematic schemes are introduced that account for slip transmission and impedance at the GB region. These schemes are based on the identification and tracking of rotating slip systems in the interfacial region, as strain evolves, such that slip and dislocation density compatibilities and incompatibilities are used to delineate regions of transmission and pile-ups; for a detailed presentation, see Ashmawi and Zikry [3].

6. RESUL TS AND DISCUSSION

A polycrystalline aggregate was simulated to illustrate the effects of the presence of GB interfacial regions. The material properties that are used here are representative of polycrystalline copper [5]. Grain bulk and GB properties are given in Table (1). The initial mobile and immobile dislocation densities within GB interfacial regions were varied randomly as a function of GB misorientation (for further details, see [3]). Using the method outlined in Zikry and Kao [5], the saturated immobile dislocation

density, Pi"'''' was calculated as 1014 m-2 and the saturated mobile dislocation

density, .oms, was calculated as 4.3x1013 m-2. Using these values, the initial

coefficient values and the enthalpy, needed for the evolution of the immobile and mobile density Eqs. (4-5), are calculated as

gminter = 5.53, greeov = 6.67, gimmob = 0.0127, gsour = 2.76xl0-5,

288 W. M Ashmawi and M A. Zikry

3 0

H/k=3.289xl0 K. (6)

All twelve-slip systems were assumed to be potentially active in each grain and GB region. Random low angle GB orientations were used with misorientations not exceeding 12° between adjacent grains. A representative aggregate size was determined by modeling the response of aggregates with different numbers of grains. This aggregate was subjected to an axial strain rate of 10-3 S-l by applying a displacement along the [001] direction. This

resulted in a plane strain deformation of the aggregate (symmetry boundary conditions were applied). Based on a convergence analysis, a minimum of 1800 four node quadrilateral elements were used for the different analyses for this study.

The effects of the GB interfacial region on the evolution of the total dislocation density can be clearly seen in the contours shown in Fig. (la). These contours correspond to a nominal strain of 5%. Dislocation densities,

corresponding to slip system(11 T)[Oll], which is one of the more active

slip systems, have localized and accumulated within some of the grains and at the GB regions. This distribution indicates that slip activity in the neighborhood of these GB regions may result either in slip transmission or blockage. Furthermore, this accumulation will lead to a buildup of normal stresses [2]. If GB effects had been ignored, these dislocation density patterns would not have evolved. It can also be seen from Fig. (1 b) that accumulated plastic strains, which are local deformation bands associated with all active slip systems, are concentrated in regions corresponding with high dislocation density activity.

7. CONCLUSIONS

A multiple-slip crystal plasticity constitutive formulation that is coupled to the temperature dependent evolution of mobile and immobile dislocation densities has been developed for a detailed understanding and prediction of the deformation modes of polycrysta1line aggregates with GB effects. The predictive capabilities and accuracy of the constitutive formulation and the specialized finite-element computational scheme have been used to investigate the effects of random GB orientations on material mechanisms in polycrystalline copper. The overall response of polycrystalline aggregates has been shown to be directly related to GB orientation, distribution, and structure. In future investigations, the constitutive formulation and the computational schemes will be used to investigate the effects of dislocation

Grain boundary effects and failure evolution in polycrystalline materials 289

motion, interaction, and transmission on the growth and the interaction of void clusters in crystalline materials separated by random GBs.

Table 1. Properties of grains and GB interfacial regions

Young's modulus, E

Static yield stress, (Y y

Poisson's ratio, V

Rate sensitivity parameter, m

Reference strain rate, r ref

Criti cal strain rate, r critical

Burgers vector, b

Initial immobile dislocation density, p~':

I .. I b'l d' I . d . pea) mba mo 1 e IS ocatlOn enSlty, rna

ACKNOWLEDGEMENTS

Grain Bulk

110 GPa

110 MPa

0.30

0.005

0.001 S·1

1104 S·1

. 3.0 X 10.10 m

GB Interfacial Region

110 GPa

330MPa

0.30

0.005

0.001 S·1

104 S·l

3.0 x 1O·lO m

Varies as a function of GB

orientation, 1010 - 1012 m·2

Varies as a function of GB

orientation, 105 - 107 m·2

The computations were performed at the North Carolina Supercomputing Center. The assistance of the staff at NCSC is deeply appreciated.

REFERENCES

[1] B. Bay, N. Hansen, D. A. Hughes, and D. Kuhlmann-Wilsdorf, "Evolution of f.c.c.

deformation structures in polyslip", Acta Metallurgica et Materialia, voL 40, pp. 205-

219,1992.

[2] T. Kameda and M. A. Zikry, "Inelastic three dimensional high strain-rate dislocation

density based analysis of grain-boundary effects and failure modes in ordered

intermetallics", Mechanics of Materials, voL 28(1), pp. 93-102, 1998.

[3] W. M. Ashmawi and M. A. Zikry, "Effects of grain boundaries and dislocation density

evolution on large strain deformation modes in fcc crystalline materials", Journal of

Computer-Aided Materials Design, voL 7, pp. 55-62,2000.

290 W M. Ashmawi and M. A. Zikry

[4] H. Mughrabi, "A 2-parameter description of heterogeneous dislocation distributions in

deformed metal crystals", Materials Science and Engineering, vol. 85(1-2), pp. 15-31,

1987.

[5] M. A. Zikry and M. Kao, "Inelastic microstructural failure mechanisms in crystalline

materials with high angle grain boundaries", Journal of the Mechanics and Physics of

Solids, vol. 44, pp. 1765-1798, 1996.

[6] M. A. Zikry, "An accurate and stable algorithm for high strain-rate finite strain

plasticity", Computers and Structures, vol. 50, pp. 337-350, 1994.

[7] G. R. MiJler, "The behavior of a crack near a low-angle grain-boundary", International

Journal of Fracture, vol. 31 (2), pp. 143-150, 1986.

022301 o •• M19 020481 o 404iO 01_ O.3M90

0111648 o 3321U 0150218 02H92

o 132'Oe O.2!1OVJ 0 .11388 0224a. o Qilse. O.lhg,,5. 0 .01748 O.l!12H

o 0592~ 0.11897

0.04105 ooaoga 0 .02235 004499

000<185 000Il00

(a) (b)

Figure J. (a) Total dislocation density at a nominal strain of,5% for aggregate with GB

interfacial region for slip system (11 T) [ 011] ; (b) total accumulated plastic strain at a

nominal strain of 5%

THE INFLUENCE OF AN HETEROGENEOUS DISPERSION ON THE FAILURE BEHAVIOUR OF METAL-MATRIX COMPOSITES: MICROMECHANICAL APPROACH

K.Derrien, D.Baptiste

LM3 CNRS ESA 8006.

EN SAM Paris.

151 Bd de I'H6pjta~ 75013 Pari.

Abstract: We use an homogenisation method in order to predict the failure behaviour of metal-matrix composites. The main damage mechanism is particle failure. We study the influence of an heterogeneous dispersion of the reinforcement on the damage development and the failure strain of composites which contain locally a higher volume fraction of reinforcement. We compare experimental and theoretical results.

Key words: Metal-matrix composites (MMCs) - Fracture, Damage- Heterogeneous dispersion, Homogenisation method -

1. INTRODUCTION

Metal-matrix Composites (MMCs) have been developed in order to combine of the following properties : rigidity of the reinforcement, with the ductility and fracture toughness of the matrix. They constitute a very attractive range of materials particularly for the construction of aeronautical structures looking for performance improvement as well as structure integrity. Our objective is to predict the tensile behavior, the damage and the failure of these composites as a function of their microstructure. We are interesting in aluminum alloys X2080 and X2124 reinforced by different volume fraction of silicon carbide particles. The materials were made by a powder blending and extruded route.

291

s. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 291-298. ©2004 Kluwer Academic Publishers.

292 K.Derrien, D.Baptiste

In previous publications (1-2), we have presented multi-scale modeling of the damaged plastic behavior and failure of AIISiCp composites. Here, we study the influence of an heterogeneous dispersion of the reinforcement on the damage development and the failure strain of composites which contain locally a higher volume fraction of reinforcement. We consider that a composite which contains locally a higher volume fraction of reinforcement results on the association of two homogeneous micro-composites. Each one is modelled by Mori-Tanaka's model; we choose the homogenization by the self consistent model for the blend of these micro-composites.

2. FAILURE MODELLING OF METAL MATRIX COMPOSITE

We use a micromechanical approach based on Mori-Tanaka's model (3) in order to connect the microstructure and the macroscopic properties of the material. The damage mechanisms are identified at the reinforcement scale, by in-situ tensile tests inside a scanning electronic microscope, and they are modeled and integrated in our model at the micro scale. Particle cracking is the principal source of damage and we use a Weibull law in order to define the particle fracture (4). The statistical development of damage is introduced in the elasto-plastic behavior law (5). Our hypothesis is that failure of the specimen takes place by linking of the micro-cracks initiated in the matrix from the broken particles. The first stage of a failure criteria defmition is to determine the range of stress and strain fields close to the SiCp broken particles, in accordance with the composition of the elementary representative element and the macroscopic plastic strain. At the second stage we determine the growth rate of cavities initiated on the precipitates. The linking of the micro-cracks depends on the value of the growth rate and on the distance from the crack where this growth rate is attained. The strain distribution near the crack is obtained from the Hutchinson (6), Rice and Rosengren (7) solution. This original solution is not suitable for porous materials because the yield criterion used is a Von Mises criterion. In our case we want to take into account the local porosity owing to the pulling-out of the precipitates. So we have modified the original theory (1) , following the same procedure used by Li and Pan (8). The growth rate of the cavities associated with the precipitates and therefore with their coalescence is a decreasing function of the distance to the broken particles. Ligament failure arises when the growth rate becomes critical on a distance at least equal to half of the distance between two particles. The volume fraction matches the average half inter-particle distance, it depends on the R ray of the particles and f the total volume fraction of reinforcement. Departing from the failure

The influence of an heterogeneous dispersion on the failure behavior of 293 metal-matrix composites: Micromechanical approach

strain of a given composite, this study allows us to forecast the failure strain of composites based on their particle content.

3. THE INFLUENCE OF AN HETEROGENEOUS DISPERSION OF THE REINFORCEMENT

Using an homogenisation method, we assume that the distribution of the reinforcement is homogeneous. In reality, a composite is rarely homogeneous and locally we can observe a high concentration of particles that will leads to a dispersion of failure strain. We shall study the influence of a heterogeneous dispersion using two steps in our homogenisation technique.

3.1 Method employed:

In any method of processing of a heterogeneous material by a method of homogenization, the first stage of the processing, " stage of representation " consists of a definition of the phases constituting material as well as the description of their space distribution and their mechanical behavior. In this stage it is initially necessary to choose a level characteristic of heterogeneity and to determine the nature of the parameters allowing the description of the corresponding phases. It is this step which we followed until now by supposing that the law of behavior of our composite was entirely determined by the knowledge of the laws of behavior of the matrix and the silicon carbide particles on the one hand, of the respective volume fractions of each one of these phases on the other hand. In order to integrate a local volume fraction of reinforcements, we will take again this step but by now considering two levels of heterogeneity and thus two successive homogenizations. The first level of heterogeneity remains the reinforcement, the second level represents on a scale "meso" the heterogeneity of distribution in the composite.

We consider that the association of these identical elementary volumes constitute what we can call a " microcomposite ". The composite resulting from the association of " microcomposites " of different composition, the logical continuation of our step will thus consist of a mixture of these " microcomposites ". For the mixture of these microcomposites, we choose a model of homogenization naturally in order to account for the interaction of these microcomposites between them. Indeed, these microcomposites does not move independently the ones of the others, when a fraction of the composite is more damaged, the others undergoes the by-effect of it and must support a greater part of the load applied.

294 K.Derrien, D.Baptiste

Whereas for the first homogenization, the model of Mori and Tanaka had seemed to us adapted, we choose the self-consistent model for the phase of setting in mixture. With such a model, it is not possible to hold account of a precise space distribution of the heterogeneous zones, the morphological reasons for the mixture are perfectly disordered points without dimension, it is necessary thus to have for the spirit that in our processing of the problem.

3.2 Calculation:

We consider a composite which results on the association of two phases: the matrix and the reinforcement phase (silicon carbide for example). If the distribution of these two phases is homogeneous (figure 1) the law behaviour of the composite is entirely determined by the knowledge of the behaviour law of the two phases and the geometry and volume fraction of the reinforcement phase. The characteristic level of the heterogeneity is that of the reinforcement, the homogenisation of phases is made using Mori and Tanaka's model.

If we attribute to the operator 0 the signification 'homogenisation by Mori and Tanaka' we can write:

Lho =~(J;,Li) i=l,n fi volume fraction of phase i, Li rigidity tensor of phase i Lho : rigidity tensor of the homogeneous composite If the phases are not evenly distributed (figure 2), we consider that the

composite results on the association of m micro-composites ct, each one representing one fraction tim of global composite. Each one of these 'micro­composite' is characterized by the volume fraction fji of reinforcement they contain. The reinforcement distribution on a 'micro-composite' is homogeneous.

O· rac' . . . @' . .-. :.': : . ~: .' ... ~ . . . .

C2

Fig I Fig 2

If we have n different phase in a micro-composite,j=l,n

Lf = ~V},J!}) ~V},J!})= If}.I!}: A5 (1) } } j

A; = r,{~f;.r, r (2), r} = I +Sr} : L~ -1 (err:.t (3)

L~ : rigidity tensor of the micro composite i

The influence of an heterogeneous dispersion on the failure behavior of 295 metal-matrix composites: Micromechanical approach

Ii: volume fraction of phase j in micro-composite i Lj : rigidity tensor of phase j in micro-composite i Sr; Eshelby tensor of the phase j in micro-composite i The logical step consists of a blend of these micro-composites. We

attribute to the operator V the signification 'homogenization by the self consistent model'

Lhe : rigidity tensor of the heterogeneous composite

Lhe = v(rt ,Lj) ,v(rjm ,Lj )= L it Lj : V + srt : the : (Li - the )j1 (4) , I

J;m: volume fraction of micro-composite i in global composite L~: rigidity tensor of the micro composite i (equation 1,2,3) Sr;m : Eshelby tensor of the micro composite i in global composite As a result, the estimation of the composite stiffness tensor can be written

Lhe = Y(J: m , ~(/; ,Ej ))

The model is extended to the elastoplastic behavior using the concept of the secant moduli, the composite stress-strain curve is determined step by step by varying progressively the matrix secant modulus.

In order to predict the tensile behavior of composite material including the damage effect, we calculate the stress (J~ induced in the SiC particles and determine their fracture's probability PJ at each step of the applied macroscopic load.

(J~ = AtAC:L, AtAC = [1 +(Sr;m:Lh~-~(L~ - Lhe)r (J~ = A~MT:(J~, AtT = T): LI;T) The failure criterion is cal wated £ reach i micro-composite. The

composite failure arises when the failure criterion is reached in one of the micro-composite.

3.3 Results obtained:

We have plotted the behavior law of two composites reinforced by 15% of silicon carbide particles. The distribution of the reinforcement is homogeneous in the case of the first material, the second material contains locally a high volume fraction (0.35) of reinforcement. The stress-strain curves are stopped when the failure criterion defined above is reached.

If the two macroscopic tensile curves are superposed (figure 3), the composite which contains clusters will fail prematurely on account of the presence of the particle cluster. The damage is less significant with the cluster ( figure 4) but this damage is more significant on account ofthe small distance between particles in the clusters.

296

f .. " ... , ~ .. u

! ,~

D"' ... I!'WIII.IttI. : I l'IIa(tll_.t .110 ... 1' trutin fIIllif nl.r ... «mnI'

Figure 4 : The dmnllge Ilccumuludon, Ilfuncdon of the {ocm fraction of the reinforcement

K.Derrien, D.Baptiste

T,ltll. Ittl ' let ., ••••• r.e •• , , •• ,.tllt •• , I., . , ...... 111. _11 1111

t . ... I.' p.ul,h', thl!".

r7 ...... ~1t. ~" •• t .. n !=',n ll.

Figure 3 : Tensile curve of two composites:

one is homogeneous the other is heterogeneous

We must however relatives these results: at the time of the development of our criterion of rupture we supposed that the rupture of the ligament of matrix between two broken particles conditioned the rupture of the composite, if the zones locally very rich in reinforcements do not represent that a negligible part of the structure we cannot say if the rupture of the V.e.r will be sufficient to involve the instability of the whole composite. Moreover the small quantity of particles broken in the zones rich in reinforcements the probability decreases of finding two particles broken close

3.4 Second application :

We have studied a composite which has been deliberately manufactured for heterogeneity (figure 5) by mixing an aluminum's powder with a composite's powder. We can represent this material as shown in figure 6.

We have modeled the behavior law of two composites including 6% of particles. The first composite is homogeneous, the second contains areas with 25% of particles and areas without

The influence of an heterogeneous dispersion on the failure behavior of 297 metal-matrix composites: Micromechanical approach

particles. A comparison between experimental and theoretical results is plotted on figure n07.

Figure 5 : heterogeneous composite

Matrix reinlorc~d with 25% o/particles

Matrix only

Figure 6 : heterogeneous composite, an outline

In the case of the heterogeneous material, the area which is highly reinforced controls the composite's failure.

We will moderate all the same the results obtained: we supposed that the rupture of a zone containing a significant proportion of reinforcements conditioned the rupture of the whole composite, this assumption reflect our experimental results well but are in contradiction with results published by Weichert &al (9). In a study aiming at optimizing the choice of new materials intended to be forged cold, these authors calculate by finite elements lengthening with steel rupture reinforced by hard particles distributed according to various configurations. The studied configurations are of homogeneous type or cluster as in our case. Their results are in agreement with the ours with regard to the rigidity of material and the fact that the damage occurs in the zones rich in reinforcements. On the other hand their study would tend to show that once the cracks initiated in the clusters, they would be stopped by the small islands of matrix alone and lengthening with rupture of material would be some increased.

Figure 7: The influence of an heterogeneous dispersion, comparison between experimental ant theoretical results

298 K.Derrien, D.Baptiste

4. CONCLUSION

The failure strength of a composite material is released to the defects present in the material and the heterogeneous distribution of the reinforcement.We have demonstrated that the particle clusters in a material could control the failure strain. Particles cracking is less important in the clusters, however the proximity of the fissures then created is damageable for the material. These results are in accordance with previous experimental observations. Nevertheless they overestimate the failure strain when there is a high proportion of reinforcement The highest the number of broken particles the highest the probability to find two immediate broken particles. Therefore we must introduce a correcting coefficient measuring the probability to find two neighboring broken particles.

5. REFERENCES:

I) Derrien K, Baptiste D, Guedra-Degeorges D, Foulquier 1. Multiscale modeling of the

damaged plastic behaviour of AIISiCp composites, International Journal of Plasticity, nOl5

(1999) p 667-685

2) Derrien K, Baptiste D, Guedra-Degeorges D. Prediction of damaged behaviour and failure

of a metal matrix composite using a multi-scale approach, Damage Mechanics in Engineering

Materials, 1997, Publishers: G.Z.Voyiadjis, 1.W.Ju and 1.L.Chaboche Elsevier Science

3) Mori T , Tanaka K. Average stress in matrix and average elastic energy of materials with

misfitting inclusions, Acta Metall, vol 21, 1973, pp571-574

4) Mochida T, Taya M, Lloyd D.l Fracture of particles in a particle/metal matrix composite

under plastic straining and its effect on the Young's modulus of the composite, Materials

Transactions, TIM, vo132, nOlO, 1991, pp 931-942

5) Qiu Y.P.,Weng G.1. A Theory of plasticity for porous materials and particle-reinforced

composites, Journal of Applied Mechanics, vo159, June 1992, pp 261-268

6) Hutchinson 1.W. Singular behaviour at the end of a tensile crack in a hardening material,

J.Mech Phys Solids, vol 16, 1968,pp 13-31

7) Rice lR., Rosengren G.F. Plane strain deformation near a crack in a power law hardening

materials. J.Mech.Phys.Solids 1968, vol 16, pp 1-13

8) Li F.Z, Pan J. Plane-Strain Crack Tip fields for pressure sensitive dilatant materials.

Journal of Applied Mechanics .voI57, march 1990, p 40-60

9) Berns H, Broeckman C, Weichert D. The effect of coarse second phase particles on the

creep behaviour ofard metallic alloysKey Engineering Materials, vols 118-119, 1996

A COHESIVE SEGMENTS APPROACH FOR DYNAMIC CRACK GROWTH

Joris J.C. Remmers and Rene de Borst Faculty 0/ Aerospace EngineeringlKoiter Institute Delft,

Delft University o/Technology, PO Box 5058 Delft, The Netherlands

J.J.C.Remmers@LR.TUDelft.nl

Alan Needleman Division 0/ Engineering, Brown University. Providence. RI02912, USA

Abstract In the cohesive segments method. a crack is represented by a set of overlapping cohesive segments which are inserted into finite elements as discontinuities in the displacement field using the partition-of-unity property of shape functions. The evolution of decohesion of the segments is governed by a relation between the displacement jump and traction across the segment. The formulation permits both crack nucleation and discontinuous crack growth to be modelled. Here. the cohesive segments formulation for dynamic crack growth is presented and application of the methodology is illustrated in two numerical examples.

Keywords: Cohesive segments method. fast crack growth. explicit transient analysis

1. Introduction The cohesive approach to fracture pioneered by Barenblatt [1], Dugdale [2]

and Hillerborg et al. [3] provides a unified framework for addressing a broad range of fracture issues. The development of complex fracture processes, in­cluding crack nucleation and crack branching, can be accounted for, see e.g. [4, 5]. In the cohesive framework, constitutive relations are specified inde­pendently for the bulk material and for one or more cohesive surfaces. The cohesive constitutive relation embodies the failure characteristics of the mate­rial and characterises the separation process. The bulk and cohesive constitutive relations together with appropriate balance laws and boundary (and initial) con­ditions completely specify the problem. Fracture, if it takes place, emerges as a natural outcome of the deformation process.

Although the cohesive surface methodology has proven useful in addressing a broad range of fracture issues, some fundamental numerical problems remain. In particular, there is a need for the development of a numerical method that

299

S. Ahzi et al. (eds.). Multiscale Modeling and Characterization of Elastic-In elastic Behavior of Engineering Materials, 299-306. ~2004 Kluwer Academic Publishers.

300 ].J.e. Remmers, R. de Borst and A. Needleman

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I~ I ,

" , ii

::r--.

Figure 1. Domain n crossed by two discontinuities, r d,l and r d,2 (dashed lines).

allows for crack growth along arbitrary directions to be modelled in a mesh independent way. We have recently developed a cohesive segments method, Remmers et al. [6], that appears promising in this regard. A crack is modelled as a collection of overlapping cohesive segments, which are incorporated as discontinuities in the displacement fields by exploiting the partition-of-unity property of finite element shape functions, see [7, 8, 9]. The method, in princi­ple, allows for complex cracking behaviour such as crack branching or discon­tinuous crack growth.

In [6], quasi-static crack growth was considered. Here, we formulate the co­hesive segments method for dynamic crack growth and illustrate its application in two example problems.

2. Kinematic relations The cohesive segments approach allows for the presence of multiple cracks

in a domain. Consider the domain 0 with boundary r as shown in Figure 1. It contains m discontinuities r d,j, j = 1.. m. Each discontinuity splits the domain into two parts, which are denoted OJ and OJ. The displacement field in the domain consists of a continuous regular field fi and m additional continuous displacement fields Uj:

m

u(x, t) = fi(x, t) + L llr d,i (x)Uj (x, t) , (1) j=l

where x denotes the position of a material point in the domain, t is time and llr d,i (x) are Heaviside step functions, which are equal to 1 when x E OJ and o otherwise. The corresponding strain field for a small strain formulation is equal to:

m

E(X, t) = VSfi(x, t) + Lllrd,j(X)VSUj(x, t), (2) j=l

where superscript s denotes the symmetric part of the differential operator. The strain field is unbounded at the discontinuities, with the magnitude of the displacement jump given by:

x on rd,j. (3)

A Cohesive Segments Approach for Dynamic Crack Growth 301

3. Linear momentum balance

The linear momentum balance without body forces can be written in weak form as: ! 7]' (pii) dO + ! 7]' (V'·O")dO = 0, (4)

n n where 7] is a variational displacement field, p is the density of the material, ii denotes the the second derivative in time of the displacement field and 0" is the Cauchy stress in the bulk material. Following a Bubnov-Galerkin approach, the admissible variation 7] must be of the same form as the displacement field, Eq. (1):

m

7] = fJ + Lf-ird,jiij' (5) j=l

After applying Gauss' theorem, eliminating some of the Heaviside functions by changing the integration domain and incorporating the boundary conditions at the external boundary and at the discontinuities, the momentum balance, Eq. (4), can be written as:

where t are the prescribed tractions on boundary r t and tj are the tractions on discontinuity r d,j'

4. Finite element discretisation

The finite element formulation rests on exploiting the partition-of-unity prop­erty of finite element shape functions [7]. A field that is discretised with standard finite element shape functions can be enhanced with a number of additional base functions which are supported by additional degrees of freedom. In this case, the Heaviside functions can be considered as enhanced base functions so that Eq. (1) can be written as:

m

U = Na + "'f-irdNbj, L ,J

j=l

(7)

302 J.J. C. Remmers, R. de Borst and A. Needleman

where a denotes the regular degrees of freedom and hj the additional degrees of freedom corresponding to the discontinuity r d,j' The matrix N contains the finite element shape functions. The strains, displacement jumps and the displacement variations, Eqs. (2), (3) and (5), are directly computed from (7). Also, the accelerations are obtained by time differentiation of (7). Inserting these expressions into Eq. (6) leads to:

The various terms in the mass matrix are:

fext a

(8)

Mab j = / pNTN dO;

n+ J

Mbjbk= / pNTNdO. (9)

ntnnt

The internal forces are given by:

fint = / B T udO . a , fi;t = / BTudO + / NTtjdr, (10)

n nt rd,j

where the matrix B contains the derivatives of the shape functions. The corre­sponding expression for the external forces is:

fext = / NTt"dr . a , fgxt = /1-lr d .NTt"dr. J ,J

(11)

rt rt

A variant of the Newmark-jJ explicit time integration scheme is used to discretise Eq. (8) in the time domain. Numerical experiments show that using a conventionally lumped mass matrix leads to inaccurate results. With the lumped mass matrix, essential information on the coupling of the regular and additional degrees of freedom is lost and this gives rise to the spurious transmission of stress waves through cohesive surfaces. In the results presented here, the consistent mass matrix is used, which increases the computational cost. It remains to be seen whether or not an accurate mass lumping procedure can be developed.

5. Implementation

The procedure has been implemented using a four-node quadrilateral con­tinuum element. A new cohesive segment is added when the stress state at

A Cohesive Segments Approach/or Dynamic Crack Growth 303

J.

1 ~: -- -

.!,; -0- - 0 -... -~

)"--1. ,r--<. ,r--<. r (a) (b)

Figure 2. (a) A single cohesive segment in a quadrilateral mesh. The segment passes through an integration point I8i where the cohesive strength is attained. The solid nodes contain additional degrees of freedom bj that determine the magnitude of the displacement jump. The gray shade denotes the elements that are influenced by the cohesive segment. (b) Interaction of two cohesive segments. The hatched elements have two sets of additional degrees of freedom. The solid circles denote the nodes that contain the additional degrees of freedom for the segment on the left and the solid squares denote those for the segment on the right.

an integration point reaches the cohesive strength of the material. The seg­ment passes through the integration point into the neighbouring elements, see Fig. 2 (a). In general, the orientation of the segment depends on the cohesivere­lation employed. However, in the examples presented here we confine attention to mode-I crack growth along a specified path.

The segment is straight so that the normal vector nd,j is constant along the patch of elements. The magnitude of the displacement jump of the segment is governed by a set of additional degrees of freedom, which are added to the nodes of the central element. The nodes that support the edges of the patch do not contain additional degrees of freedom in order to simulate a zero opening at the tip of the cohesive segment [9]. Since the additional degrees of freedom cannot be condensed at the element level, they influence all the surrounding elements.

A key feature of the method is the possibility of having overlapping segments. In the situation sketched in Fig. 2 (b) a new segment is added next to an existing segment. The added segment is independent of the existing segment.

6. Numerical examples In the first example, the cohesive segments method is used to model mode­

I dynamic crack growth. A square block with dimension 2L = 3 mm and an edge crack of initial length a = 0.25 mm is analysed, see Fig. 3. The block is loaded in tension by a prescribed normal velocity Vo = 3.0 mls on the boundaries X2 = ±L with a rise time fixed at 0.01 J-ls. The normal velocity on the boundaries Xl = ±L is prescribed zero. The block consists of an isotropic linear elastic material with Young's modulus E = 3.24 GPa and Poisson's ratio 0.25. The density, p, is 1.19 . 10-6 g/m3 .

304

1.0

a [krnI.

0.6

0.2

J.J.c. Remmers, R. de Borst and A. Needleman

L 2L

Figure 3. Geometry of the square block with initial edge crack.

v.v t lJ'SJ (a) (b)

- [MPa] 110

-35

Figure 4. (a) Crack speed versus time. The dashed line shows the Rayleigh wave speed, eR = 938m1s. (b) Contours of the opening stress, 0"22, throughout the block at t = 7.5 jlS

for the calculation with an initial edge crack. The bold line shows the position of the cohesive segments.

The crack is constrained to propagate in the Xl -direction so that the fracture mode is mode-I. A linear cohesive relation is used:

for 0 < Vn < vsep , (12)

Here, tn is the normal traction across the cohesive segment, Vn is the corre­sponding normal displacement jump, it is the cohesive strength and vsep is the magnitude of the displacement jump at which the load carrying capacity van­ishes, which is specified by vsep = 2gc/ ft, where gc is the work of separation. Inthecalculations,ft = 324.0 MPaand gc = 352.3Jm-2. A cohesive segment is added when the local stress normal to the crack plane, 0"22, exceeds ft. The mesh consists of 240 x 239 quadrilateral elements and the time step is 10-9 s. The initial crack consists of a number of traction free cohesive segments.

The crack speed history is shown in Figure 4 (a). The crack accelerates to the Rayleigh wave speed, the theoretically expected maximum crack speed [to]. Contours of the opening stress, 0"22, together with the initial crack and the added cohesive segments are shown in Figure 4 (b).

In order to illustrate the capability of the cohesive segments method to model crack nucleation, the previous example is slightly modified. There is no initial

A Cohesive Segments Approach for Dynamic Crack Growth 305

- ! [MPa] 200

-100

Figure 5. Contours of 0"22 at t = 9.0 J.ts for the calculation where the block has an initial central cavity. The bold lines show the positions of the cohesive segments.

crack, but there is an initial cavity of size 0.25 x 0.125 mm, positioned at the centre of the block. The same boundary conditions are used as in the first example except that in this calculation the maximum applied velocity is Vo = 6.0 mls. Decohesion is only permitted along a central weak plane. Two cracks nucleate at the cavity sides at t ~ 8.2 J.Ls. Contours of a22 in the block and the position of the cohesive segments are shown in Figure 5 at t = 9.0 J.Ls.

7. Concluding remarks

The cohesive segments method has been used to model dynamic crack growth. We focused on the derivation of the governing equations and the numerical implementation. It was found that use of a consistent mass matrix was required to obtain accurate results. Results were presented for two example problems; one illustrating the ability to model crack nucleation. In principle, the cohesive segments method provides a framework for modelling the complex crack patterns typically seen in fast fracture of brittle and quasi-brittle solids. However, this capability remains to be demonstrated.

Acknowledgement

AN is grateful for support from the Office of Naval Research through grant NOO014-97-1-0179.

References

[1] Barenblatt GL The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics 1962; 7: 55-129.

[2] Dugdale DS. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 1960; 8: 100-108.

[3] Hillerborg A, Modeer M, Petersson, PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 1976; 6: 773-782.

306 J.J.c. Remmers, R. de Borst and A. Needleman

[4] Needleman A, A continuum model for void nucleation by inclusion debonding. Journal of Applied Mechanics 1987; 54: 525-531.

[5] Xu XP, Needleman A. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 1994; 42: 1397-1434.

[6] Remmers JJC, de Borst R, Needleman A. A cohesive segment method for the simulation of crack growth. Computational Mechanics in press.

[7] Babuska T, Melenk, 1M. The partition of unity method. International Journal for Numer­ical Methods in Engineering 1997; 40: 727-758.

[8] Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journalfor Numerical Methods in Engineering 1999; 46: 131-150.

[9] Wells GN, Sluys LJ. A new method for modeling cohesive cracks using finite elements. International Journalfor Numerical Methods in Engineering 2001; 50: 2667-2682.

[10] Freund, L.B., 1998. Dynamic Fracture Mechanics, Cambridge University Press, Cam­bridge.

A LINEAR MODEL OF PROCESSING PATH IN CUBIC-ORTHOTROPIC SYSTEM

D.S. Li, H. Garmestani

School of Materials Science and Engineering, Georgia Institute of Technology Atlanta, GA, 30332, USA

Abstract: Modeling the texture evolution in polycrystalline materials during thennomechanical processing is an important aspect of the newly developed design methodology, microstructure sensitive design (MSD). The texture coefficients are used in this study as a representation of microstructure and their evolution describes the processing path. A new methodology based on a conservation principle in orientation space is utilized in this work to model the processing path in cubic-orthotropic system. Least squares error method was used to improve the accuracy of simulation results from the processing path function developed in this study. The processing path function developed in this study empowers the MSD users to find out how to optimize and tailor microstructures with desired properties.

Key words: microstructure, texture, thennomechanical processes, polycrystalline material, processing path

1 INTRODUCTION

Recently Adams et al. (2001;2002) proposed a novel methodology, microstructure sensitive design (MSD), which introduced a new paradigm in materials by design. MSD is an approach to meet the requirements of several different properties by adjusting the microstructure. Traditional design methodologies tried to link the microstructures to properties but dealt with properties as individual entities in the absence of a correct microstructure representation. MSD in its present form is applicable to most engineering materials, which are polycrystalline. The primary motivation to cut cost prompts material scientists to use anisotropy as a parameter to optimise microstructure by thermomechanical processing.

To achieve this goal, a suitable texture representation should be used. It should describe the texture completely and concisely. Texture is

307

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization o/Elastic-Inelastic Behavior o/Engineering Materials, 307-314. 1D2004 Kluwer Academic Publishers.

308 D.S. Li, H. Garmestani

mathematically represented as a set of spherical harmonics weighted with texture coefficients. Pole figures and orientation distribution functions (ODF) can easily be quantified in this representation. Using texture coefficients, microstructures can be represented as points in a multidimensional space with coordinates as texture coefficients (Adams et al. 2001; 2002, Lyon et al. 2003). That is to say, each point in this Fourier space stands for a unique microstructure, associated with corresponding properties. To utilize MSD, it is very important to understand the texture evolution during the thermomechanical processing.

Modelling the evolution of texture is always a challenge and can benefit from micromechanics to crystal plasticity. To achieve the final microstructure with desired properties can be facilitated using a mathematical representation of the evolution of texture. A conservation principle proposed by Clement and Coulomb (1979; 1982) was used in this work to model the microstructure evolution. Using the continuity function proposed by them, Bunge and Esling (1984) studied the flow field of single crystals of face centered cubic (fcc) materials. They also pointed out the possibility of describing the texture evolution of polycrystalline materials using this method.

Based on these works, Li et al. (2003a; 2003b) proposed an alternate approach using polycrystalline materials description rather than single crystal orientation description. It established a linear relationship between the rate of change of the texture coefficients and the deformation descriptor. Further progress is made in this study to find out direct relationship between texture coefficients and deformation descriptors. This work established a processing path function to describe the evolution of texture coefficients. To examine the accuracy and applicability range of this linear approach, a modified Taylor model proposed by Kalidindi et al. (1992a; 1992b) was used for comparison. This model provides a fairly accurate approximate solution for the texture evolution of single phase, especially for the highly symmetric lattice structures (Garmestani et al. 2002).

2 PROCESSING PATH FUNCTION IN CUBIC­ORTHOTROPIC SYSTEM

To establish a functional form for the processing path which represents the evolution of the texture coefficients during thermomechanical processing, two parameters will be introduced, one of which is for processing step or ". In the case of uniaxial tension, " represents the drawing strain and for compression, it represents the compression ratio. The

other parameter is for the texture representation, which is a set of F/mn (,,)

coefficients. Here f(g,'ll) is used to represent texture as a function of the

processing step ". Texture at any "can be expressed as a series of

A linear model of processing path in cubic-orthotropic System 309

generalized spherical harmonic functions (Bunge 1965) where ~mn are the

weighting factors, as shown in Eq. (1) 00 M(l)N(/) •

f(g, 1]) = L L L~mn (1])~mn (g) (1) 1=0 m=O n=O

Texture evolution is regarded by Clement (1982) as a fluid flow in orientation space, which is composed of three Euler angles. At any point in the orientation space (g), the orientation density is defined asf(g) and flow rate is R(g). According to the conservation principle, the continuity equation is:

af~~1]) +div[f(g,1])R(g)] = 0 (2)

The first term in Eq. (2) describes the increase of the quantity of matter per unit time in an infinitesimal volume element in the orientation space. The second term refers to the quantity of matter in the orientation space moving out of that infinitesimal volume element. With Eq. (1), Eq. (2) is expanded in a series of spherical harmonics:

LF:P (1])div(f:p (g)R(g»)+ L d~mn(1]) ~mn(g) = 0 (3) ).up Imn d1] Using similar Fourier expansion method, the divergence function in the

first term is further exranded into:

div(f:p(g)R(g»)= LA/~nC1P~mn(g) (4) Imn

A;rn, called texture evolution coefficients, are weights of the spherical

harmonics. Substituting back to Eq. (3), the relationship between the texture coefficients and their rate of change is derived:

d~ mn (1]) = L A/~nC1p F;.C1P (1]) (5) d1] ;'C1p

This linear relationship was used by Bunge (1984) to predict the texture evolution for single crystal orientations. After the integration of this equation, the processing path function describing the evolution of the texture coefficients with the deformation parameter is obtained:

F(1]) = e A 1/ F(1]o = 0) (6)

The elements of A, texture evolution matrix, represent the texture evolution coefficients. In this study, Eq. (6) was used to simulate the texture evolution of fcc materials with random texture to generate process path. If the number of interesting texture coefficients ~mn is limited to N, then

texture data at N+ 1 different strains will be needed to obtain a solution for

the texture evolution coefficients A :mn .

310 D.S. Li, H Garmestani

3 SIMULATION RESULT OF PROCESSING PATH IN CUBIC-ORTHORTROPIC SYSTEM

In the present study, raw data from Taylor Model, not the experimental

data is used to provide the evolution of ~mn at corresponding strains

because Taylor model gives numerous estimates with acceptable accuracy at different strains which can be used as input. The goal of our study is to propose a functional form for the processing path and check its validity, limitation and range of applicability.

The initial texture in this study is arbitrarily assumed to be an aggregate of 400 crystals somehow evenly distributed in the orientation space. The accuracy of Taylor model in this case is assumed to be a correct measure of

deformation path. Fd1, F11, F12 and F413 for this texture are 1.00,0.04, -

0.05, and 0.002 respectively. All the other texture coefficients are zero. They

are very close to the ideal random texture where only F d I is 1 and all the

other ~mn are zeros.

Here cubic crystal system is assumed along with orthotropic sample symmetry. In this system, the high order texture coefficients F[mn with />4

will not affect elastic properties (Bunge 1982). As a result the number of nonzero F[mn decreases to 4. Furthermore,Fd 1 is a constant. This reduces

interesting texture coefficients to F411 , F12 and F13. That is to say, N=3.

From Eq. (5), we have:

[dF11/dTJ] [A!!11 A!!12 dF412 / dTJ = A!;11 A!; 12

dF413 /dTJ A!!11 A!!12

(7)

Here A[~n<TP are components of a sixth order tensor. Most components of

this tensor are useless because most of F[ mn are either 0 or not of our

interest. After the texture evolution matrix A is obtained, texture coefficients along the deformation history can be calculated by this processing path function:

F(1J) = eA(1/-1/o) F(1Jo) (8)

In the first set, a strain step dry of 5% was used. ~mn at strains of 25%,

30%, 35% and 40% were calculated from Taylor model. Here these strains compose an initial strain set. From these strains, A;",n were obtained.

Further the texture coefficients at strains from 20% to 50% were simulated.

A linear model of processing path in cubic-orthotropic System 311

The evolution of these three texture coefficients with the strain is shown in

Fig!. In the strains between 25% and 40%, the recalculated F;' from A;rn are very close to those obtained from Taylor model according to Figure 2a.

When the strain is lower than 25% or higher than 40%, the recalculated F4" begin to deviate from raw data. The same trend is observed in the evolution

curves of F4'2 and F413 from Figure 1 b and 1 c. In the range of initial strain set, from 25% to 40%, the results from the linear model have a first order agreement with the raw data. In a typical case when the strain is 30%, the difference between recalculated values and raw data is 0.0. It deviates from the raw data of Taylor model when the strain is far from the initial strain set.

(a)

(c)

(b)

Fig. 1. Simulated processing path

function of (a) F;' ,(b)F;2 and

( )F13 • I . . C 4 usmg texture evo utlOn matrIx

A obtained by strain step as 1 %, 2% and 5%, respectively. Evolution of

F " F12 F13 4' 4 and 4 of raw data obtained from Taylor model and simulation curve by least squares error method is also illustrated

For the comparison a strain step of 2% was used. The initial strain set includes 28%, 30%, 32% and 34%. Figure 2 shows that the processing path function using A from this strain set works well in the strains from 20% to 40%. If the strain is larger than 40%, the deviances of the texture coefficients become larger with the increasing of the strain.

In the third set, a strain step of I % was used. The initial strain set includes 30%, 31%, 32% and 33%. Texture evolution matrix A obtained from this procedure works well in the strain range close to initial strain set.

312 D.S. Li, H Garmestani

When the strain is smaller than 20%, the simulated texture coefficients evolution curve deviates from the curve from the raw data. With the decrease

of strain, Fit and F12 are overpredicted sharply. It is clear from the results above that the procedure introduced in this

paper provides the best results when used for interpolation. This means that if the texture coefficients are predicted at a strain which is within the range of initial strain set, the error is negligible. If this prediction is extended to strains out of the range of the initial strain set, the farther the strain is from the range of the initial strain set, the worse the prediction of the texture evolution is. How to choose the initial strain set is critical in improving the accuracy of modelling processing path. In the real world, experimental data of texture at different strains are limited. Fully utilizing the strain range of these experimental data will increase the accuracy of this model to predict the texture evolution behaviour out of the experimental strain range.

Another way to improve the accuracy is to take account of more experimental data in the range of initial stain set using least squares error method. From M texture data (M> N) at different strains, A was calculated from the over-determined system below:

[dF,mn (1])/ d1]] = [A HF,mn (1])] (9)

Let us try the initial strain set from 30% to 41 %, and the strain step d 1] as I %. Least squares error method is used to recalculate the evolution of texture coefficients during the deformation from 20% to 50%. The results are also illustrated in Figure 1. It shows clearly that using the least squares

error method describe the behaviour better than using the AT" obtained

from strain step of 5%. The agreement with raw data of the simulated results using least squares method is almost the same as that using strain step of 5% in the strain range of 25% to 40%, which is included in the initial strain set. When extrapolated out of the initial strain range, the recalculated texture coefficients evolution curves obtained by least squares error method are closer to the curves from raw data than any other simulated curves do. The texture evolution coefficients obtained using least squares method give a more accurate description of the texture evolution behaviour during mechanical deformation.

A linear model oj processing path in cubic-orthotropic System 313

F412

d1J=5% d1J=2% .. ..-

.. 1/ /

":~:.r~J~~.':--'Jo:-:_/_:'-""""--"""'---raw data least squares \.

F411

error method

F413

Fig2. Processing path from the raw data and simulated result presented in the microstructure space whose coordinates are three texture coefficients.

The processing paths of cubic-orthotropic system from raw data and simulated results are illustrated in Fig2. These processing paths describe the texture evolution from a strain of 20% to a strain of 50% during uniaxial tension. The black solid processing path from raw data is indistinguishable from the red dash processing paths simulated using least squares error method. The green dashed line processing path simulated by strain step as 5% deviates a little from the processing path of raw data. The blue dashed line processing path simulated by strain step as 2% deviates more. This model described a simple but effective methodology to connect the evolution of microstructure and processing.

4 CONCLUSION

A linear model of processing path in cubic-orthotropic system is proposed in this paper to describe microstructure evolution during plastic deformation. In the processing path function developed by this model, texture evolution matrix A is a critical parameter. It is calculated from textures at different strains which compose initial strain set. The simulated processing path works well if the strain is in the range of the initial strain set. Increasing the range of the initial strain set will improve the performance of simulated result in a large strain range. If more data are available to calculate

314 D.S. Li. H. Garmestani

the texture evolution matrix, using least squares error method will increase the accuracy of simulated processing path.

REFERENCES

Adams B.L., Henrie A., Henrie B., Lyon M., Kalidindi S.R., Garmestani H., 200l. Microstructure-sensitive design ofa compliant beam. 1. Mech. Phys. Solids 49,1639-1663.

Admas B.L. Lyon M., Henrie 8., 2002. Microstructure by design: linear problems in elastic­plastic design. Int. 1. Plast. submitted

Bunge H.I., 1982. Texture analysis in materials science: mathematical methods. London: Butterworth & Co; p.340.

Bunge H.J., 1965. Zur Darstellung allgemeiner Texturen. Z. Metallkde 56,872-874.

Bunge H.J., Esling C., 1984. Texture Development by plastic deformation. Scripta Met. 18,191-195.

Clement A, Coulomb P., 1979. Eulerian Simulation of deformation textures. Scripta Met. 13, 899-901.

Clement A, 1982. Prediction of Deformation Texture Using a Physical principle of Conservation. Mater. Sci. Eng. 55, 203-210.

Garmestani H., Kalidindi S.R., Williams L., Bacaltchuka C.M., Fountain c., Lee E.M., Se­Said O.S., 2002. Modeling the evolution of anisotropy in Al-Li alloys: application to Al-Li 2090-T8E41. Int. 1. Plast. 18, 1373-1393.

Kalidindi S.R., Bronkhorst C.A, Anand L., 1992a. Crystallographic texture evolution in bulk deformation processing offcc metals. J. Mech. Phys. 40, 537-569.

Kalidindi S.R., Anand L., I 992b. An approximate procedure for predicting the evolution of crystallographic texture in bulk deformation processing of fcc metals. Int. 1. Mech. Sci. 34, 309-329.

Li D.S., Garmestani H., 2003a. Evolution of crystal orientation distribution coefficients during plastic deformation. Scripta Materialia, accepted

Li D.S., Garmestani H., 2003b. Processing path of texture coefficients during plastic deformation. Plasticity '03 Symposium, Quebec City, Quebec, July 2003.

Lyon, M., Adams B.L., 2003. Gradient based non-linear microstructure design. Int. J. Plast. Submitted

TAYLOR THEORY WITH MICROSCOPIC SLIP TRANSFER CONDITIONS

B. L. Adams!, B. S. EI-Dasher, R. Menill l , J. Basingerl and D.S. Li

/ Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA lMaterials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Abstract

Keywords

The main focus of the paper is an extension of the classical Taylor theory of plasticity to include the microscopic conditions for slip transfer at grain boundaries. It is demonstrated that such leads to consideration of the grain boundary character distribution function, in concert with the usual local state distribution function. The primary result is an expression for the generalized Taylor Factor that includes an inverse grain size dependence (Hall-Petch).

crystal plasticity, grain bmmdaries, microscopic conditions

1. INTRODUCTION

Most studies in crystal plasticity are based upon Taylor's original 1938 work. I Within Taylor's framework the dependence of yield strength on microstructure, beyond lattice orientation, is carried within the critical resolved shear stress for slip. Thus, as the grain size decreases, the critical resolved shear stress is required to increase. This increase in critical resolved shear stress is applied, uniformly across the entire interior of the slipping grains according to the basic assumption of the model (uniform plastic strain or strain rate). It is well known that slip patterns are not uniform over the grain interior. It is known, from the evidence of transmission electron microscopy, that certain microscopic conditions must exist near grain boundaries and triple junctions within polycrystalline materials, leading to differences in the patterns of dislocation slip near the boundaries, as compared with the grain interior.2-4

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316 B. L. Adams, B. S. EI-Dasher, R. Merrill, J. Basinger and D.S. Li

The purpose of this paper is to introduce a framework in which these microscopic conditions can be incorporated within the classical Taylor model. It will be shown how these considerations lead to a grain-size and grain-boundary­character dependence in the initial yield stress. The results are expressed in the Fourier space of microstructures.

2. BRIEF REVIEW OF TAYLOR PLASTICITY

Let F denote the macroscopic plastic deformation gradient tensor. Our focus shall be on initial plastic yielding, and therefore F can be separated into 1 + & + (jj , where 1 is the second-order identity tensor, & the infinitesimal strain and (jj the infinitesimal rotation. Let F, 6 and 0) denote the local (crystallite) plastic deformation gradient, infmitesimal strain and infinitesimal rotation tensors, respectively. Taylor's theory assumes that all grains within the material will undergo the same shape change imposed upon a representative macroscopic sample of the material. Thus,

8=&. (1) Crystal plastic strain can be accommodated by slip on any compatible set of slip systems, chosen from the set S. The deformation gradient associated with slip system s E S is defined by it associated deformation gradient, strain and rotation tensors: F(s), is), and O)(S) , respectively. If b(s)represents the unit slip

direction vector, and ,;(s) the unit slip plane normal direction vector associated

with slip system s, and if y<s) is the scalar slip strength, then

F(s) F(sl F(s) F(sl F(s) = 1 +y(s)b(s) ® ,;(s) 8(s) = + 1 O)(s) = - (2)

, 2 ' 2 where superscript T denotes the transpose and ® denotes the dyadic product.

Whenever the slip strengths are sufficiently small (r(s) «1 for all s E S) then to a good approximation

F=1 + "L(F(s) -1), 8="L8(s), 0) = "LO)(s) , (3) s s s

where S' c S denotes the set of slip systems that have r(s) > o. If T£8) is taken to be the operative critical resolved shear stress associated with slip system s, then the plastic work done is W:

W = "LT£s)r(s). (4) seS'

The question naturally arises as to what are the possible sets, S', that satisfy the basic compatibility relation embodied in (1). Given that the plastic strain tensor 8 is volume conservative, tr(8)= 0, it follows that there will generally be five independent components of 8. Therefore, five slip systems,

Taylor theory with microscopic slip transfer conditions 317

selected from among the set S, will be required to satisfy relation (1). Define S to be the set of all possible combinations of slip systems such that the compatibility relation is satisfied; thus,

S = {S' IS' c S; L &(s) = &}. (5) SE S'

Taylor then postulates that the particular operative set of slip systems is the set S" that minimizes the plastic work. In mathematical terms

S" = S' c S 3 W = min. (6) In cases where more than one set S" satisfies the same minimum plastic work criterion, then the solution is ambiguous. The so-called Taylor Factor (TF),

M(eO), is defmed to be a function of the unit strain, eO, according to the relation

a:e=1]a:eo = LT~s)r(s)=Tg 'L,a(s)y<s)=1JM(eO )Tg. (7) SES' SES'

Here the parameter 1] scales the imposed strain to the unit strain: e = 1]eo ; the

coefficients a(s) scale the critical resolved shear stress to the reference critical

resolved shear stress, Tg: a(s) = T~s) / Tg; and a denotes the Cauchy stress tensor. It is evident that the TF is dependent upon the reference critical resolved

shear stress, Tg, the scaling coefficients, a(s), and the slip strain tensors, e(s).

The latter depend upon the orientation of the crystal lattice in which b(s) ® Ti(s)

is fixed; the former (scalar parameters) are dependent upon crystal phase, composition, and other local state parameters. Let h = h(t/J,g,c, ... ) denote the complete set of local state parameters, including lattice phase t/J and orientation g, chemical composition c, and any other pertinent parameters. Then

M =M(h 1&0). (8)

The macroscopic TF, M(&O), is obtained by averaging the local Taylor

factors, M(h leO), over the representative volume of the sample. If the volume fraction distribution of the local state is defined by

dV/V=f(h)dh, (9) where dh is the appropriate invariant measure on local state spaces, then

M(eo) = J f(h)M(h leo)dh, (10) H

where H denotes the complete local state space.

Let ¢1[~v (h) represent the complete set of orthogonal basis functions for

the set of real-valued, square-integrable functions of the form F: H --)0 !R. Also, defme the Fourier representation ofthe local state function be

318 B. L. Adams, B. S. El-Dasher, R. Merrill, J Basinger and D.S. Li

f(h) = I rPF/:vrPT/:V(h), (11) all ¢,I,r,,u, v

and the local TF to be M(hleO)= I ¢mfrV(eO)¢r/:V(h). (12)

all ¢,l,r,,u, v

It foIlows that relation (10) can be expressed, in terms of the Fourier coefficients

that define the local state distribution function, ¢ Fi~v , and the coefficients that

define the local Taylor Factor, rP mfrv (eo), by the expression

M(eo) = I ¢mfrV(eO)¢F/:v , (13) all ¢,I,r,,u, v

where ¢mfrV(eO) = !1/mfrV(eO) . Further details on the Fourier description

are given in the works of Bunge.s It is known that relatively small numbers of the coefficients ¢mfrV(eO) are required for convergence in

relation (12), and hence the number of coefficients of the microstructure that are important in (13) is also similarly limited.6

3. INTRODUCING MICROSCOPIC CONDITIONS

3.1 Experimental Evidence

30.00 ~m Figure I. Typical orientation boundary layer in 30%

plastically-deformed <00\> aluminum.

Taylor theory with microscopic slip transfer conditions 319

Figure 1 illustrates the experimental evidence for the modifications to Taylor's fully constrained theory. The sample consists of <001> directionally solidified aluminum, deformed plastically in compression to -0.30 height reduction. The starting grain size of the material is - 5 millimeters. Using fine­scale orientation imaging microscopy7, the orientation field of material adjacent grain boundaries is observed. The step size of the scan is 0.5 microns. Lattice

orientation is determined at each step point, to within ±O.SO. In Figure 1 all

scan points that lie with 2.5 0 of the average orientation in the near boundary zone are shaded black. It is evident that a boundary layer develops adjacent the grain boundaries; and this boundary layer has a distinct orientation compared to that found in the grain interior. The two regions are separated by a system of geometrically necessary dislocations.

3.2 Taylor Theory with Microscopic Conditions

Transmission electron microscopy suggests that certain microscopic conditions favor dislocation slip transmission across grain boundaries.2-4 These include: a) a minimum of residual net burgers vector is left behind in the grain boundary, b) the shear stress ahead of a slip system pileup, resolved in the adjacent grain, is maximized, and c) the angle of intersection between two adjacent slip systems in the grain boundary plane is minimized.

Here we maintain the basic compatibility requirement that pertains to Taylor's theory, embodied in relation (1). We shall relax, in the vicinity of the grain boundary, the minimum plastic work hypothesis of Taylor, embodied in relation (6). In its place we postulate the existence of a microscopic condition that governs the slip patterns near the grain boundary.

Let a local element of the grain boundary have unit normal vector ;, _ Let it separate grain A and grain B. Let hA and hB denote the local state of grain A and B. We suppose that the nominal thickness of the grain boundary layer is a in grain A and b in grain B. We shall assume that it is only in these regions that Taylor's minimum plastic work criterion must be altered_ Let SA denote the set of all possible slip system combinations that satisfy relation (l) in grain A. Similarly, let SB denote the set of all possible slip system combinations that

satisfy relation (1) in grain B. Let SA • and SB" represent the patterns of slip in grains A and B that satisfy the minimum plastic work criterion of Taylor. The

associated TFs shall be M(hA leo)and M(hB leD).

320 B. L. Adams, B. S. EI-Dasher, R. Merrill, J. Basinger and D.S. Li

Next, consider the boundary layers associated with grains A and B. It is evident from the experimental observations that the slip pattern in the boundary layer in grain A is dependent upon the slip pattern in the boundary layer in grain

B, and vice-versa. Let SA"' denote the correct slip pattern in grain A in the boundary layer of thickness a, after due consideration for the pertinent

microscopic condition. Similarly, let SB lit be the associated slip pattern in B

within the boundary layer of thickness b. Also, let M(hA I ii,hB,co) be the TF

in the boundary layer of grain A, associated with slip pattern SA"'; and let

M(hB I ii,hA ,CO) represent the TF in the boundary layer of grain B, associated

with slip pattern SB"'. Also, since it is known that

M(hA I ii,hB,cO) 2M(hA leO), M(hB I ii,hA,cO) 2M(hB leo), it is useful to defme the Excess TF, llM, in the following way:

llM(hA lii,hB,co)=M(hA lii,hB,cO)-M(hA leO),

The new macroscopic TF, M(cO) now contains two terms:

(14)

(15)

M(cO)=M(cO)+M(c°)' (16)

where M(cO) is the original TF, defined via the local state distribution function

through relation (10), and M(cO) is defined by

- ° 1 f " {allM(hA I ii,hB,CO)} " M(c )="2 Sv(hA,n,hB hAdndhB, (17) HxS2 xH' +bllM(hB I ii,hA ,co)

where Sv(hA,ii,hB) is the grain boundary character distribution function. 8 In relation (17) the integration is over the space of possible grain boundary

characters, H x S2 X H', where character is defined by local state hA E H on

side A of the grain boundary, unit normal ii E S2 defming the inclination of the boundary plane when passing from side A to side B, and local state hB E H' on side B. In homophase materials H = H'. (For details about the product space see Adams and Olson.'1 Note that Sv(hA,ii,hB)dhAdizdhB is equal to the surface area per unit volume of grain boundary that has local state lying in the range dh A of h A on side A of the boundary, normal direction lying in the range

dn of direction n, and local state lying in the range dhB of hB on side B. When multiplied by the Excess TFs, weighted by their thicknesses, contained in the term [ ... ] in (17), we estimate the additional plastic work done due to the

Taylor theory with microscopic slip transfer conditions 321

local microscopic criterion. The approximation is valid if a,b« d, where d is the grain size of the material.

Note that the dimension of the grain boundary character function is d-1,

and therefore the grain boundary term in the Excess TF, M(eO ) , is also inversely proportional to the grain size of the microstructure. The remainder of its functional dependence is found in the distribution of grain boundary character types within the microstructure.

Constructing a complete orthogonal system of basis functions on

H x S2 x H' is achieved with products of the eigenfunctions defined on each separate space.9 Thus, the Excess TF accepts the form

M(hA I n,hB ,co) = I ifJifJ~mfzt:r;v' (eO)ifJTI~v (hA )kz(n)ifJ'1i~:v' (hB) (18) all ifJ,ifJ',I,I',r, r',Jl,p.', v, v'

where kz(n) are the surface spherical harmonic functions. A similar Fourier

representation exists for the grain boundary character distribution function, with

coefficients ifJifJ~Gfz~;;v'. Thus, an expression like (13) for M(eO) is

M(eO) = I ifJifJ'q-,nf.JJ.l'vv'(eO)#'qGf.Jf.J'VV' (19) P ll'rr' P ll'rr' , ap "~',l,r,,r,

r ,f.J,f.J ,v,v ifJifJ'q ~ f.Jf.J'vv' _ ifJifJ'q f.Jf.J'VV'

where pmU'rr' - fill' pmU'rr' .

3.3 Exemplary Microscopic Conditions

We shall consider only criterion a), minimum net Burger's vector in the grain boundary. Let n denote the normal to the grain boundary. The net density of Burger's vector left behind in the grain boundary is just

bOB = ~y<S) ~(S) . n )(S) _ ~y<S) ~(S) . n )(S) . (20)

In accordance with microscopic condition a) we shall select SA, So so that

IlboB II:::: min. These will be considered to be the operative sets.

Consider the three independent ~ 0 To X 112 0) slip systems in hexagonal

crystals. Restrict consideration to (0001) columnar polycrystals. For this case

the orientation distribution function is simply defined over the angular interval [0,K/3) of rotations about (0001). Figure 2 shows the Excess TF calculated

from the minimum residual Burger's vector criterion, where ()A, Os describe the orientations of grains A, B relative to a common reference frame. For this

322 B. L. Adams. B. S. El-Dasher. R. Merrill. J Basinger and D.S. Li

calculation, efl = 1 is the only non-zero component of plastic strain, and lies in

the (0001) plane. (The symmetry about BA = BB is a consequence of the

homophase nature of the boundary, and setting a=b. The value of the Excess TF is precisely zero on this line.) Evidently, the Excess TF is a complex function of macroscopic grain boundary parameters.

1.04~-----:::-----------'

0.52 9 s

1.04

Figure 2. Excess Taylor Factor as a function of orientation parameters BA, BB (radians).

4. Discussion and Conclusions

.' .0~' . '

. "' .05

.0.95· '

. 0 9-0.95

. 0 85.Q 9

. 0 8-0.85

.0 7~O.8

.0 7·0 75

.0.6S-O.7

. 0.8·0.65

.0.55.Q.6

.0 5·0.55

. 04S-O.5

. 0.4·0.45

. 0 35-0 4

. 0.3-0.35 025-03 02-0.25 o IS-02

01-015 005-0.1 0.0 05

We conclude that the incorporation of any of the observed microscopic conditions for slip transfer at grain boundaries, within the classical Taylor theory of plasticity, gives rise to an inverse grain size dependence of the Excess TF. All eight 'macroscopic parameters' of grain boundary character associated with the grain boundary character distribution function are also predicted to affect the Excess TF.

It is evident that the uniform strain criterion of Taylor enforces rather restrictive requirements on plastic deformation, and these will often violate local conditions of stress equilibrium. However, within the Taylor framework it will

Taylor theory with microscopic slip transfer conditions 323

be important to examine the details of the geometrically-necessary dislocations that are observed to form at the transition region between the boundary layers shown in Figure 1, and the grain interior region. Such considerations may provide sensitive insight into the most appropriate microscopic conditions to apply in conjunction with the Taylor theory.

Acknowledgements

This work was supported primarily by the MRSEC Program of the National Science Foundation under DMR-0079996.

References

1. Taylor, G. I. Plastic strains in metals. Journal of the Institute of Metals 1938; 62:307-24. 2. Shen, Z., Wagoner, R. H. and Clark, W.A.T. Dislocation and Grain Boundary Interactions

in Metals. Acta Metallurgica 1988; 36:3231-42. 3. Jacques, A., Michaud, H.-M., Baillin, X. and George, A. New results on dislocation

transmission by grain boundaries in elemental semiconductors. Journal de Physique 1990; 51:Colloq. CI, supple. au nOI, CI-531-6.

4. Lee, T. C., Robertson, I.M. and Birnbaum, H. K. An in-situ transmission electron microscope deformation study of the slip transfer mechanisms in metals. Metallurgical Transactions 1990; 2IA:2437-47.

5. Bunge, H.-J. Texture Analysis in Materials Science. London, Butterworths, 1982. 6. Park, N. J., Klein, H. and Dahlem-Klein, E. Physical Properties of Textured Materials.

Gottingen, Cuvillier Verlag, 1993. 7. Sun, S., Adams, B. 1. and King, W. E. Observations oflattice curvature near the interface

of a deformed aluminum bicrystal. Philosophical Magazine A 2000; 80:9-25. 8. Adams, B. 1. and Field, D. P. Measurement and representation of grain boundary texture.

Metallurgical Transactions 1992; 23A:2501-2513. 9. Adams, B. 1. and Olson, T. The mesostructure-properties linkage in polycrystals. Journal

of Progress in Materials Science 1998; 43:1-88.

DYNAMICS OF NANOSTRUCTURE FORMA­TION DURING THIN FILM DEPOSITION

Daniel Walgraef

Center for Nonlinear Phenomena and Complex Systems, Free University of Brussels, CP 231, Bd du Triomphe, B-J050 Brussels, Belgium. E-mail:dwaelgr@ulb.ac.be

Abstract: Coverage evolution, during atomic deposition on a substrate may be described, on mesoscopic scales, by dynamical models of the reaction-diffusion type, which combine reaction terms representing chemical processes such as adsorption­desorption and nonlinear diffusion terms. Below a critical temperature, uniform deposited layers are unstable, which leads to the formation ofnanostructures cor­responding to regular spatial variations of substrate coverage. For increasing mean coverage close to one-half, the dynamics is of the Cahn-Hilliard type, and one should observe a succession of structures going from hexagonal arrays of high coverage dots, to stripes and finally to hexagonal arrays of low coverage dots. For mean coverage close to one, the nanostructures are highly nonlinear, and have to be obtained numerically. Structures obtained in the case of Allayers deposited on TiN substrates are presented. Comparisons between weakly and highly nonlinear structures are performed.

Key words: thin film deposition, nanostructure, reaction-diffusion dynamics, instability

1. INTRODUCTION

Modeling thin film growth actually one of the most challenging activity in materials science. Since growth mechanisms usually determine film proper­ties and textures, it is of capital technological and scientific interest to under­stand and master this phenomenon. Thanks to the numerous computer simu­lation methods which have been developed since the 1970's, detailed infor­mation on growing films, such as island shapes, step formation and surface

325

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 325-332. ©2004 Kluwer Academic Publishers.

326 Daniel Walgraef

roughening are now available [1,2]. Unfortunately, the computational time required to simulate thin film growth under realistic deposition rates remains excessive, or requires unrealistically high deposition rates, although small polycrystalline films with linear dimension under the J.1m may now be satis­factorily simulated [3].

On the other hand, continuous models, based on rate equations of the POE type, remain good candidates to describe film evolution on mesoscopic scales (between the J.1ffi and the mm), i.e. at scales inaccessible by both equipment (macroscopic) and feature scales (microscopic) models [4]. In this framework, mesoscopic models have been recently proposed to describe spontaneous ordering of nanostructures or self-assembled quantum dots in multi-component epilayers on a substrate [5-8]. They are based on an under­lying instability of the alloy or solid solution which fonn the film, and which is stabilized by concentration-<iependent surface stresses or atomic surface deposition. As a result, the phases may order into periodic patterns, such as alternating stripes or disks lattices, with nanometric spacings.

Up to now, these models had limited predictive capability, because of a rough description of kinetic processes, such as atomic diffusion or deposi­tion. Nevertheless, such capability could be greatly enhanced, if the results of atomistic simulations could be properly fed into mesoscopic descriptions, in the framework of Multiscale Materials Modelling [9], where mesoscopic models should link micro- and macroscales, and provide not only qualitative, but also quantitative descriptions of thin film growth. In this spirit, it has recently been shown that the evolution of deposited layers may be described by dynamical models of the reaction-diffusion type, where the competition between atomic deposition and reactions, and nonlinear diffusion may stabi­lize nanoscale spatial patterns, even for monoatomic layers [10]. Close to the spinodal decomposition point of the adsorbed layer, the dynamics is of the Cahn-Hilliard type, and nanostructure fonnation may be studied through a weakly nonlinear analysis. However, if one tries to rely these results to ex­periments or simulations on deposited AI, Ti or TiN films, for example, one has to consider non critical adsorbed layers, since desorption is usually much lower than adsorption, and the mean coverage is far from its critical value. In this case, nonlinearities are strong and weakly nonlinear analysis breaks down. The aim of this paper is then to address this problem, and to present numerical results for nanostructure formation for deposition conditions where the mean atomic coverage is far from critical.

The paper is organized as follows. The reaction-diffusion model, which describes the evolution of a deposited atomic layer on a substrate, is pre­sented in section 2. The instability ofunifonn layers is discussed in section 3 and pattern selection in the weakly nonlinear regime is reviewed in section 4. Numerical analysis of nanostructure formation, in the conditions of Al layer deposition, is presented in section 5. Finally, conclusions are drawn in sec­tion 6.

Dynamics ofNanostructure Formation during Thin Film Deposition 327

2. THE DYNAMICS OF A DEPOSITED LAYER ON A SUBSTRATE

As discussed in previous publications, the evolution of a monoatomic layer, deposited on a substrate, may be described by a continuous dynamical model of the reaction-diffusion type [10,11]. Relevant examples of such sys­tems are Al or Cu layers deposited on Si substrates, or Si02 and TiN layers deposited on Ti or Al substrates. In such cases, the dynamics is governed by atomic adsorption and thermal desorption on the substrate, but also by diffu­sion. In conditions of homoepitaxy, elasticity and stress effects may be ne­glected, and the film evolution may be described by atomic coverage dynam­ics only. On mesoscopic scales, its kinetic equation has the following struc­ture [10]:

(I)

where c = c(r,t) is the local atomic coverage, which is defined as the average occupancy number, or average atom number per lattice site. R(c) represents reaction terms and J is the diffusion current in the deposited layer. For direct absorption, without precursor molecule dissociation, and in the absence of chemical reaction with the substrate, R(c) = ex. (I-c) -13 c, where ex. and 13 are the adsorption and desorption rates. In a frrst approximation, they will be considered as constant with respect to coverage. In order to build an reliable dynamical model for films deposited on a sub­

strate, the main problem is to adequately model atomic motion. Linear non­equilibrium thermodynamics provides the necessary tools to describe this situation. Since the chemical potential is the functional derivative of the free energy, one has [10]:

J = - LV. (0 10 c)

where is the free energy of the adsorbed layer. It has been evaluated in [11], and writes, in the mean field approximation:

(3)

where c (r) is the local coverage, fer) = (l-c (r» In (I-c (r» + c (r) In c(r). For nearest neighbor attractive interactions between deposited atom, &0 = 1&

and Ei = yea2, where 1 is the lattice coordination number, & is the pair inter­action energy and a is the lattice constant. The local chemical potential may then be easily obtained from the resulting free energy as:

328 Daniel Walgraej

J.1 (r) = (0 /0 c) = Es - EoC (r) + kaT In [c(r)/I-c (r)] + ~02V2C (r) (4)

This is the equation of state of the system which defmes the coverage as a function of temperature, interaction energies, etc. Note, for T < Tc = Eo/4ks

single homogeneous phases are unstable, and the system separates into two distinct phases, one with low coverage (c < 0.5), the other one with high coverage (c > 0.5).

3. STABILITY OF UNIFORM DEPOSITED LAYERS

Taking into account the results of the preceding section, equation (I) may be written as:

to t C = Co - C + r V2 [ J.11n (I +2C / 1-2C) - C - V2C] (5)

where C = c - 0.5 and Co = a. / (a. + J3), t -1 = a. + J3, r = (4tDoTJa2T), J.1 = (T/4Tc), y = aVand D is the surface diffusion coefficient. The stability of the uniform steady state of eq. (5), Co, is given through linear evolution of small perturbations, cr = C - Co, which may be written, in Fourier transform, as:

(6)

with J.1* = (TIT*c) and T*c = Tc(l - 4C02). The corresponding marginal sta­bility curve is given by J.1* = 1 - q2 - (1/ rq2 ), and bifurcation or instability point is given by

(7)

with K = 4tDoIa2(1 - 4C02). Beyond instability, the maximum growth rate corresponds to spatial modulations of scaled wavelength ~ = 27t/qm with q2 m = (T*c - T)/ T*c. Hence, for temperatures T higher than Ti, uniform steady state coverage is stable, while it is linearly unstable versus spatial inho­mogeneities for T < Ti. For T close to Ti, these inhomogeneities are ex­pected to have critical wavenumber qj. It is also interesting to note that Ti is maximum for critical coverage, where instability is favored. On the other hand, critical wavenumber is maximum at critical coverage and decreases for decreasing adsorption rate.

Dynamics ofNanostructure Formation during Thin Film Deposition 329

4. THE WEAKLY NONLINEAR REGIME

Close to instability and criticality, the modified Cahn-Hilliard dynamics (5) may be written as:

(8)

where E = (Ti-T)/q;2Ti, ~2 = T*clTi q?, v = 4Co /q?(1-4C02), U = 4 (1+ 12 C02)/3q?(1-4 c02i and 1<-1) = D* q? cr may be considered as an order pa­rameter, and this order parameterlike dynamics corresponds to a modified Swift-Hohenberg equation. Such equations have been extensively studied in the framework of pattern formation theory [12]. It has to be noted here, that the quadratic kinetic coefficient, v, is proportional to Co - 0.5 and is positive for steady uniform coverage below critical, and negative for steady uniform coverage above critical. Hence, below critical, the so called 0 hexagons should be selected, where the maxima of the pattern amplitude are located at the center of the hexagons [12], which corresponds to an equilateral triangu­lar lattice of high coverage spots. When coverage increases towards its criti­cal, hexagonal patterns become unstable versus stripes [12], and one should obtain regular arrays of high coverage bands. Finally, on increasing coverage further beyond critical, on gets 1t hexagons, where the minima of the pattern amplitude are located at the center of the hexagons [12], which correspond to low coverage dots (see Figure 1 [13]).

These predictions have also been numerically tested on the complete dy­namics (5), for parameter's values corresponding to typical Al deposition experiments [14]. Al crystallizes in FCC lattices, with a lattice constant aIAI}= 4.05.A, and an attractive pair interaction potential of 0.0553 eV [15]. If Al atoms are deposited on (100) TiN surfaces, the lattice coordination num­ber is 4 and Tc ~ 641 0 K. The lattice mismatch is very small (about 4%) and interfacial stress effects may be neglected. Surface diffusion coefficient on (100) Al have been obtained by molecular dynamics simulations [3].

(0) (b) (c)

Figure 1. Numerical solutions of a dynamical model of the type (5), obtained by Verdasca et al., close to threshold (& = 0.005: (a) below critical coverage, (b) at critical coverage, and (c) above critical coverage (dark regions correspond to low coverage.

330 Daniel Walgraej

Equation (5) has been solved numerically, in a one-dimensional geometry, with periodic boundary conditions, for T = 0.4 Tc -2560 K, and Co = 0.1, which corresponds to r = 200, J.L = 0.1. In these conditions, the uniform steady state is unstable, since T j = 573~ and bm = 11.63, while .k , the preferred wavelength, is equal to l3.96. A random initial condition is ob­served to evolve asymptotically into a sinusoidal pattern with A = 14.15 (5.8 nm), close to.k, in agreement with the predictions of the weakly nonlinear analysis [14].

5. THE STRONGLY NONLINEAR REGIME

For Co far from critical, weakly nonlinear analysis breaks down, and nanostructures become highly nonlinear, as it may be seen through the nu­merical analysis of equation (5). As in the preceding section, equation (5) has been solved for r = 200, J.L = 0.1, in a one-dimensional domain, with pe­riodic boundary conditions. In this case, uniform coverage is unstable for ICol < 0.3654. Strongly nonlinear patterns corresponding to regularly spaced low coverage holes are obtained for Co = 0.365 (cf. Figure 2). The observed wavelength of this structure is A ~ 32 nm, while the low coverage holes ex­tend over about 0.25 of a wavelength.

If one considers now desorption rates about 0.1 of the adsorption rates, the mean coverage should be of the order of 0.9, and Ti - 212~. Hence, in the conditions of our numerical analysis (T - 256°K > Ti - 2120K), the uniform steady state is linearly stable. However, stable localized structures, corre­sponding to isolated low coverage holes, have been obtained from random initial conditions (cf. Figure 3). These local structures are a consequence of co-existence between a uniform state and periodical spatial state [14].

( t) 'c ~ ~x,

0 ,..,

OMS x

O~5 ,

50

Figure 2. Numerical solution of equation (5) in one-dimensional geometry for r=200. IF 0.1. c(x.t) = u(x.t) + Co. Co = 0.865 and obtainedfrom a random initial condition. The wavelength

of the pattern is about 78 (- 32 nm for Al films). far from any value obtained by weakly nonlinear analysis.

Dynamics of Nanostructure Formation during Thin Film Deposition 331

~~c(x,t) -- Co 0.9

o .~ --"'"

X

I

I ~ 100 l~

Figure 3. Numerical solution of equation (5) in one-dimensional geometry for T=200, f.J= 0.1, c(x,t) = (j (x,t) + Co, Co = 0.9 and obtained from a random initial condition. Although Co is stable, a localized solution, corresponding to a drop in coverage, develops.

6. CONCLUSIONS AND OUTLOOK

A dynamical model for nanostructure formation in adsorbed atomic layers has been proposed in this paper. It is based on the combined action of ad­sorption-desorption processes and nonlinear diffusion, and has been studied by analytical and numerical methods. It has been shown that for Al or Cu layers, deposited on substrates like TiN, Si2, Ta, etc., uniform layers are un­stable below instability temperatures which are well in the experimentally accessible range, while critical wavelengths are in the nanometer range. For realistic parameters, corresponding to Al layers, regularly spaced, or local­ized low coverage holes are found.

Elasticity effects have been neglected, but should be considered when uni­form coverage breaks down, since they may generate internal stresses, able to affect nanostructure evolution, and generate textures of grains with differ­ent orientations with respect to the substrate. This problem will be addressed in forthcoming publications.

ACKNOWLEDGEMENTS

Fruitful discussions with Prof. H. Hanchen and the participation of Prof. M.G. Clerc in the numerical analysis are gratefully acknowledged.

REFERENCES

1) Director of Research at the Belgian National Fund for Scientific Research (F.N.R.S.). [1] F. Abraham and G. White, J. Appl. Phys. 41, 1841 (1970). [2] G.H. Gilmer and P. Bennema, 1. Appl. Phys. 43,1347 (1972).

332 Daniel Walgraef

[3] H. Huang, G.H. Gilmer and T. Diaz de la Rubia, "An Atomistic Simulator for Thin Film Deposition in Three Dimensions", J. Appl. Phys. 84, 3636 (1998).

[4] Gao, H. and Nix, W.D., "Surface Roughening of Heteroepitaxial Thin Films", Annual Review of Materials Science 29, 173 (1999).

[5] Leonard, F., Laradji, M. and Desai, R.C., "Molecular beam epitaxy in the presence of phase separation", Phys.Rev. B 55, 1887 (1997).

[6] Leonard, F. and Desai, R.C., "Alloy decomposition and surface instabilities in thin films''Phys.Rev. B 57,4805 (1998).

[7] Suo, Z. and Lu, W., "Self-organizing nanophases on a solid surface", in Multiscale De­formation and Fracture in Materials and Structures, The James R. Rice 60th Anniversary volume, edited by T.J. Chuang and J.W. Rudnicki, Kluwer Academic Publishers, Dor­drecht, 107-122 (2000)

[8] Suo, Z. and Lu, W., "Forces that drive nanoscale self-assembly on solid surfaces", Journal ofNanoparticle Research 2, 333-344 (2000).

[9] Ghoniem, N. M., Heinisch, H., Huang, H., Yip, S. and Yu, J., eds., Journal of Computer Aided Materials Design, Special Issue for Multiscale Materials Modeling Symposium of the 5th IUMRS, Kluwer, Dordrecht (1999).

[10] Walgraef, D., "Nanostructure Initiation during the Early Stages of Thin Film Growth", Physica E 15, 33 (2002).

[11] Walgraef, D., "Reaction-Diffusion approach to nanostructure formation during thin film deposition ", to appear, Phil. Mag. A.

[12] Walgraef, D., "Spatio-Temporal Pattern Formation (with examples in physics, chemistry and materials science", Springer Verlag, New York, 1996. [13] Verdasca, J., Borckmans, P. and Dewel, G., "Chemically frozen phase separation in an

adsorbed layer ", Phys. Rev. E. 52, R4616 (1995). [14] Walgraef, D. and Clerc, M.G., to appear. [15] Mishin, Y., Farkas, D., Mehl, M.J. and Papaconstantopoulos, D.A., "Interatomic poten­

tials for monoatomic metals from experimental data and ab initio calculations", Phys.Rev. B 59, 3393-3407 (1999).

PREDICTION OF DAMAGE IN RANDOMLY ORIENTED SHORT-FIBRE COMPOSITES BY MEANS OF A MECHANISTIC APPROACH!

Ba Nghiep Nguyen and Mohammad A Khaleel

Pacific Northwest National Laboratory

P.O. Box 999 Richland, WA 99352, USA

333

Abstract: A micro-macro mechanistic approach to damage in short-fibre composites is developed in this paper. At the micro-scale, the damage mechanisms such as matrix cracking and fibre/matrix debonding are analysed to define the associated damage variables. The stiffness reduction law dependent on these variables is then established using micromechanical models and average orientation distributions of fibres and microcracks. The macroscopic response is obtained by means of thermodynamics of continuous media, continuum damage mechanics and a finite element formulation.

Key words: Short-fibre composites, matrix cracking, fibre/matrix debonding, mechanistic approach

1. INTRODUCTION

When a short-fibre composite structure is submitted to increasing loading, two main damage mechanisms are observed: matrix cracking and fibre matrix debonding, which may lead to fibre pullout at a later loading stage. In this paper, a micro-macro mechanistic approach to damage in short-

!This manuscript has been authored by Battelle Memorial Institute, Pacific Northwest division, under Contract No. DE-AC06-76RLO 1830 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid­up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

333

S. Ahzi et al. (eds.). Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 333-341. ©2004 Kluwer Academic Publishers.

334 B.N Nguyen and M.A. Khaleel

fibre composites is developed based on micromechanical and continuum damage mechanics descriptions to determine the macroscopic response of these materials suffering from matrix cracking and fibre/matrix debonding. We consider two stages of the damage development. The first stage is characterised by progressive degradation of the composite due to these mechanisms. The second stage corresponds to final failure for which the composite cannot carry loads any more. For modelling matrix cracking in randomly oriented short-fibre composites, we explore our previously developed model (Nguyen et al., 2002, Nguyen & Khaleel, 2003) in which a reference aligned fibre composite containing parallel matrix microcracks was used to compute the stiffuess of the random fibre composite having random matrix microcracks. The matrix cracking model is extended here to include the fibre/matrix debonding mechanism, which often occurs simultaneously with matrix cracking. During the first stage, fibre/matrix debonding is the result of progressive fibre/matrix sliding and deterioration of the bonding between fibre and matrix. This mechanism is modelled using Qu's concept of a spring-layer of vanishing thickness surrounding the fibre (Qu, 1993). In addition, the homogenisation procedure based on the self­consistent model is modified by replacing the Eshelby tensor by the modified Eshelby tensor by Qu. Finally, the macroscopic response is obtained by means of a continuum damage mechanics (COM) formulation, which extends Renard et al. 's model (1993) for continuous fibre composites to randomly oriented short-fibre composites. Failure as a result of excessive matrix cracking, fibre/matrix debonding, fibre pullout and rupture leading to initiation and propagation of a macroscopic crack is modelled using a vanishing element technique (Nguyen & Khaleel, 2003).

2. PROBLEM FORMULATION

2.1 Stiffness reduction due to matrix cracking

In this section, we first give a summary of the micro-macro mechanistic approach to matrix cracking developed by Nguyen and Khalee1 (2003). After that the model is extended to account for fibre/matrix debonding coupled with matrix cracking. We start from a short fibre composite in which the fibres and fibre-shape matrix microcracks are unidirectional. This composite is served as the reference composite to compute the crack density. During the damage development before final failure, if the linking-up process of matrix microcracks does not occur, the crack density in a misoriented fibre composite can then be computed from the solution for the aligned fibre composite system containing the same number of microcracks, which are parallel to each other.

Prediction of damage in randomly oriented short-fibre composites by 335 means of a mechanistic approach

(a)

Laws et aJ.'s model (1983) §] Distribution of

-" I ~ cracks and fibers / I '\ over all orientations

• Jr~'~~~~';f (c) (b)

Figure J. Schematic of the method to compute the stiffuess of a random fibre composite subject to matrix cracking.

Figure 1 gives a schematic description of the method used in this paper to determine the reduced stiffness for a misoriented short-fibre composite subject to matrix cracking. Figure la corresponds to an aligned fibre composite system containing parallel slit cracks. If its volume is sufficiently large, to a first approximation, the properties of this cracked medium are identical to those of a composite layer containing the same crack density according to Laws et aI.'s model (1983) (Figure Ib). The situation described in Figure 1 b leads to defme the crack spacing L, which gives rise to a crack density -9 =1/L. Consequenly, the damage variable is defined as I = elL where e is the layer thickness. If the fibre diameter is much smaller than the crack length, the stiffness (C) and compliance (S) matrices of a cracked layer are expressed in terms of the damage variable as (Laws et aI., 1983):

(1)

where the superscript "0" denotes the initial state, and 4 is a tensor dependent on the compliance coefficients. Next, the stiffness of the random fibre composite containing random microcracks is obtained from that of the reference composite averaged over all possible orientations and weighted by an orientation distribution function:

336 B.N. Nguyen and M.A. Khaleel

o !t12 J R(a,O)k-A,(-fJ)dO+ JR(a,O)k-A.8dO

C(a) = -,,12 0 ,,12

2 Jk-A,8 dO o

(2)

where R is the global stiffness matrix of the reference composite, which is obtained by transforming CCl) into the global coordinate system. This coordinate system is defined such that the fibres are assumed in the layer plane 1-2 with the orientation angle n measured relative to the 1-axis. In (2), Karcir et al.' s orientation distribution function dependent on parameter L'is used (Karcir et al., 1975). Our analysis considers random orientations of fibres and microcracks; hence L' is rather small and tends to zero when the fibres and microcracks are completely random.

2.2 Stiffness reduction due to matrix cracking coupled with fibre/matrix debonding

The computation of the effective stiffness of the composite from the properties of the constituents is carried out using the self-consistent model (Chou et al., 1980). Accordingly, the composite stiffness reads:

2

CO =Cm + L/;(Cfl -Cm)A; ;=1 (3)

in which up to two types of fibres with volume fraction Ii (i = 1,2) are considered. C, S and A are the stiffness, compliance and concentration matrices, respectively; and E is the Eshelby tensor. The subscript «f' and "m" designate the fibre and matrix constituents. Relation (3) provides the self-consistent approximation of the stiffness matrix for a virgin composite with perfect fibre/matrix interfaces, and containing no matrix microcracks.

When degradation occurs at the interface, the fibre/matrix bonding becomes imperfect. We assume that this interfacial deterioration has not led to fibre pullout yet so that the concept of a spring layer of vanishing stiffness proposed by Qu (1993) can still be used to characterise the imperfect bonding. This assumption is reasonable during the first stage of damage prior to final failure. In such a case, the interfacial tractions are still continuous, but a displacement discontinuity may happen at the interface. These conditions are expressed as (Qu, 1993):

Prediction of damage in randomly oriented short-fibre composites by means of a mechanistic approach

/1a ij n j = [a ij (x) I T+ - a ij (x) I T- ] n j = 0

/1u i = ui (x) IT+ -ui (x) Ir = 1]ija jknk

337

(4)

where T+ and T- indicate approaching the interface (with its unit outward normal ni ) from outside or inside of the inclusion (fibre), respectively. 1]ij denotes the compliance of the spring layer: 1] ij = 0 corresponds to a perfect interface while 1] ij -+ 00 represents complete debond. In this analysis, a special form of 1]ij proposed by Qu is used:

(5)

In particular, we consider the case where r == 0 and f3 is small, which allows relative sliding between the two surfaces without separation, and for which complete debonding causing fibre pUllout does not occur. By doing so, and using Qu's development, this leads to the modified Eshelby tensor (Qu, 1993):

E*=E + (1 -E)HCf (I-E), (6)

where the tensor H has the dimension of the compliance matrix, and depends on the interface property and the geometry of the inclusion. H is of the form: H = f3 (P - Q) in which the inclusions considered are fibres whose lengths are much greater than their radii. Consequently, simplified expressions for cylinder-shape fibres can be used. In these expressions, the nonzero terms of P and Q are proportional to 1/a, where a is the fibre radius. Furthermore, between two limiting cases for 1]ij discussed above, the compliance of the spring layer should be sufficiently smaller than the compliance of the matrix material to represent the slightly weakened interface. Therefore, one possible way to define!) is:

(7)

where Em is the matrix Young modulus, and f3 *is a variable. f3 * is employed here as a damage variable to characterise the interfacial degradation. Since matrix cracking and fibre/matrix debonding are not independent mechanisms, a relationship between them needs to be established. Figure 2 provides a schematic description of the possible evolutions of the damage variables, which are used to assume a relation between these variables. Curve 1 represents the situation where matrix cracking occurs first; and fibre/matrix debonding is initially negligible and

338 B.N. Nguyen and MA. Khaleel

will develop after the damage variable I associated with matrix cracking has attained a certain value. Curve 2 depicts a linear evolution of P * versus I until the ultimate value while curve 3 represents a non-linear evolution where fibre/matrix debonding happens first.

Figure 2. Schematic of the evolutions of the damage variables

Hence, we assume the following relationship:

(8)

in which co' C1 and c2 are the material coefficients dependent on the properties of the fibre/matrix interface and of the matrix. Next, by introducing (8), (7) in (6) we determine the modified Eshelby tensor in terms of the interfacial weakening for a given type of fibre. Finally, replacing the Eshelby tensor by the modified Eshelby tensor in (3), and with the use of (3) in (1) and (2), we compute the stiffness of the random fibre composite affected by matrix cracking coupled with fibre/matrix debonding.

2.3 Damage evolution relation

The damage evolution law in terms of the local strains &j can be obtained using thermodynamics of continuous media and a continuum damage mechanics formulation similar to that used by Renard et al (1993) for continuous fibre composites. Only independent state variables are considered. These are I and &j in this analysis:

Prediction of damage in randomly oriented short-fibre composites by 339 means of a mechanistic approach

dCij --eide;

da=- da (9) 1 d 2C.. dF ---Ye.e.--c 2 da 2 I J da

where Fc (a) is the damage threshold function which can be determined using the experimental data for applied stress versus crack density. The method to compute this function for short-fibre composites containing random matrix microcracks is given in Nguyen et al. (2002) and Nguyen & Khaleel (2003). Finally, the macroscopic response is determined by introducing this damage model into a small strain kinematically admissible finite element formulation. Total failure occurs when the damage variable has attained the maximum value, a L , and can be modeled using a vanishing element technique that consists of reducing the local stiffness and stresses to zero in a certain number of steps in order to avoid numerical instabilities (Nguyen & Khaleel, 2003). It is assumed that at total failure the stiffness matrix governed by the stiffness reduction parameters kij tends to zero as:

(10)

where m =1,2,3, ... with m=1 when a=aL (no summation on i and}).

3. NUMERICAL APPLICATION

The damage model implemented in the ABAQUS code is used for the simulation of the tensile stress/strain response of a short glass fibre (600tex)/epoxy composite. The material data employed in the analysis were taken from Meraghni & Benzeggagh (1995). Figure 3 illustrates the tensile stress/strain responses predicted using the damage model accounting for i) only matrix cracking, and ii) matrix cracking coupled with fibre/matrix debonding. The later case was studied assuming Co = 0,c1 = C2 = 1 in (8). On the same figure are also presented the experimental results by Meraghni & Benzeggagh (1995). In each case, the highest value of the crack density, which was still measured experimentally, provided an estimate for the ultimate value at which total failure occurs. The simulated applied stress versus crack density curves are presented in Figure 4 for this material. The experimental results by Meraghni & Benzeggagh (1995) are also shown for comparison. A good correlation with the experimental results has been found with the use of the damage model accounting for both damage mechanisms.

340 B.N Nguyen and M.A. Khaleel

200 E>perimeots. Meraghoi & Beozeogagh o E.periments. Meraghni & Benzeogagh ¢ E.perimenlS. Mer8ghoi & Beozeggagh

--- Matri> Crac~i/lg Model -- Matri> Crackiog & loterfacial Deboodiog Model

00

0.016

Figure 3. Tensile stress/strain responses of the 600tex glass/epoxy specimens.

4. CONCLUSION

A micro-macro mechanistic approach to damage in short-fibre composites has been developed in this paper. At the micro-scale, the use of Laws et al.'s model and the spring layer model by Qu has enabled the numerical characterisation of damage due to matrix cracking and fibre/matrix debonding. A relationship between the associated damage variables were proposed to express the interaction between these damage mechanisms. Finally, the macroscopic response was determined using a continuum damage mechanics formulation associated with a vanishing element technique to capture final failure.

Prediction of damage in randomly oriented short-fibre composites by 341 means of a mechanistic approach

ISO

o Eloperimeots CMel8gMj, Be~iegg8\lN -- Mal,l. C,acNnglil«fel

'30 -- Mal,i. C,actcing,& Intertac:i81 Deb«lding Model

o o.os 0.1 O.IS

CraCK DePI3ity

Figure 4. Applied stress versus crack density the 600tex glass/epoxy specimens.

REFERENCES

Chou, T.-W., Nomura, S., Taya, M., 1980. A self-consistent approach to the elastic stiffness of short-fiber composites. 1. Compo Mater. ,14, 178-188.

Karcir, L., Narkis, M., Ishai, 0., 1975. Oriented short glass-fiber composites: I Preparation and statistical analysis of aligned fiber materials. Polym. Engn. Sci., 15, 525-531.

Laws, N., Dvorak, G.1., Hejazi, M., 1983. Stiffness changes in unidirectional composites caused by crack systems. Mech. Mater., 2, 123-137.

Meraghni, F., Benzeggagh, M.L., 1995. Micromechanical modeling of matrix degradation in randomly oriented discontinuous-fibre composites. Compo Sci. & Tech. , 55, 171-186.

Nguyen, B.N., Ahn, B.K., Khaleel, M.A., 2002. Continuum damage modeling of short-fiber composites subject to matrix cracking. In: Chang F.K., editor. Proceedings of the 10th US­Japan Conference on Composite Materials, Destech Publications, 531-540.

Nguyen, B.N., Khaleel, M.A., 2003. A mechanistic approach to damage in short-fiber composites based on micromechanical and continuum damage mechanics descriptions (Submitted).

Qu, 1., 1993. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech. Mater., 14,269-281.

Renard 1., Favre 1.-P., Jeggy T., 1993. Influence of transverse cracking on ply behavior: Introduction of a characteristic damage variable. Compo Sci. & Tech., 46, 29-37.

NONSTEADY PLAIN-STRAIN IDEAL PLASTIC FLOW CONSIDERING ELASTIC DEAD ZONE

Wonoh Lee, Kwansoo Chung, Tae Jin Kang and Jae Ryoun Youn School of Materials Science and Engineering, Seoul National University, 56-1, Shinlim-tiong, Kwanak-/cu, Seoul 151-742, Korea

Abstract: Ever since the ideal forming theory has been developed for process design purposes, application has been limited to sheet forming and, for bulk forming, to two-dimensional steady flow. Here, application for the nonsteady case was made under the plane-strain condition. In the ideal flow, material elements deform following the minimum plastic work path (or mostly proportional true strain path) so that the ideal plane-strain flow can be effectively described using the two-dimensional orthogonal convective coordinate system. Besides kinematics, schemes to optimize preform shapes for a prescribed final part shape and also to define the evolution of shapes and frictionless boundary tractions were developed. For demonstration purposes, numerical calculations were made for an automotive part under forging.

Key words: Nonsteady ideal bulk forming, Rigid perfect plasticity, Method of characteristics, Orthogonal convective coordinate system

1. INTRODUCTION

In order to improve trial-and-error based conventional practices for optimizing forming processes, a direct design theory, called the ideal forming theory, has been previously developed [1-9]. Here, the ideal forming theory is applied for the non-steady plane-strain case, considering isotropic rigid-perfect plasticity. For such cases, to account for the minimum plastic work condition in the ideal flow, principal stretch lines are materially embedded and its kinematics can be effectively described using the two­dimensional orthogonal convective coordinate system. In this coordinate system, the orthogonal base vectors represent principal directions of deformation, which are materially fixed during ideal flow. A numerical code

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344 Wonoh Lee, Kwansoo Chung, Tae Jin Kang and Jae Ryoun Youn

to generate the orthogonal convective coordinate system was developed based on the characteristic method in this work. Besides kinematics, the equilibrium condition and schemes to optimize preform shapes for a prescribed fmal part shape and also to define the evolution of shapes and frictionless boundary tractions were established. For demonstration purposes, numerical calculations were made for an automotive part under forging.

2. KINEMATICS AND FORCE EQUILLIBRIUM CONDITION

In the ideal flow, material elements deform following the minimum plastic work path (or mostly proportional true strain path) so that the ideal plane-strain flow can be effectively described using the two-dimensional orthogonal convective coordinate system with uniform local area. When ~ and TJ are the variables of the orthogonal convective coordinate system, representing principal lines fixed on material lines, the following quasi­linear system of equations is obtained:

(1)

where h is the scale factor of ~ -base vector and rp is the orientation of ~ -line with respect to the x -direction of the Cartesian coordinate system.

The method of characteristics applied to Equation 1 provides

dlnh-drp = 0 for which TJ,~ = _h 2 (a-line)

d In h + drp = 0 for which TJ,~ = h2 (p -line) (2)

and a numerical code to generate the orthogonal convective coordinate system was developed based on Equation 2. The plane-strain force equilibrium condition for isotropic rigid plastic materials leads to

L+lnh = C+ 2k

L-lnh = C-2k

for 1,>0

for I, < 0

(3)

Here, p is the mean stress (the hydrostatic pressure) and k is the shear yield stress, while C+ and C- are arbitrary constants.

Nonsteady plain-strain ideal plastic flow considering elastic dead zone 345

3. APPLICATION: PRESCRIBED PART SHAPE

y D G

c

A B Yo E

x Xo

Figure 1. Schematic view of a final part shape prescribed for ideal forming design

The final part geometry to be formed is shown in Figure 1, which is symmetric with respect to the y -axis. Here, the boundary contours are supposed to match with the materially embedded principal lines in order to have frictionless boundary tractions. For this particular part, the analytic solution of Equation 1 was available [10] and the solution provides the following parametric expressions for the boundary contours,

2 tan(O) x(O)=x +x--

o 1 y1d(O)

yeO) = Yo + y{l- d;O»)

x(O) = xo + x~ tan(O) + /-ro Fa (cos 0 + sinO) y1d(O)

yeO) = Yo + Yl (l __ l_)_eo-ro Fa (cosO -sinO) d(O)

for Be (4)

for FG (5)

346 Wonoh Lee, Kwansoo Chung, Tae Jin Kang and Jae Ryoun Youn

where 0 ~ () ~ 7r / 2 and d(O) == (1 + (XI tan () / YI)2)0.5 . Here, Xo, XI , yo , yl and ro{ = -(In yo) /2) are constants describing the part geometry shown in Figure 1: xo=6.0, XI = 2.4117, yo =1.0, YI=4.8373. The rest of the contours are straight lines. The numerically obtained final part shape agrees well with the one obtained from the parametric expressions in Figure 2, confirming the validity of the numerical method.

y

14 Numerical result

12

E o

• • • Characteristic lines o Parametric expression

Final shape

2 4 6 F 8 10 12 14

Figure 2. The principal line coordinate system numerically and analytically obtained to describe the final part geometry along with characteristic lines

4. OPTIMUM INITIAL SHAPE, INTERMEDIATE SHAPES AND BOUNDARY TRACTIONS

In order to defme the initial shape, a set of rectangular shapes with a uniform local scaling factor was considered here for simplicity. Also, as an

Nonsteady plain-strain ideal plastic flow considering elastic dead zone 347

optimization criterion for the initial shape, the condition of the minimum average absolute strain was introduced. The average absolute strain K IS

defined as

flel dA K=-­

A (6)

where e = In(hf / hi). Here, the subscripts i and f refer to the initial and final shapes, respectively and A is the area of the part shape. Therefore, the following condition provides the optimum hi:

dK =0. dh,

1.2 ,-----------,---------, ,

1.0

I< = .; 0.8 .c fIJ Q,j

]1 0.6 = fIJ .c ~ Q,j

~0.4

'" ~ 0.2

(e f> 0): :

~ I no elastic I.---t- (e f< 0)

i 1 dead zoner i .-___ --:-,--,------,i

Formation of elastic dead zone

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 initial scaling factor (h~)

(7)

Figure 3. The average absolute strain (K ) with respect to the initial scaling factor ( hi )

The average absolute strain K as a function of the uniform initial scaling factor hi is plotted in Figure 3. The figure shows that the minimum K is obtained for hi = 0.2575 . Note that the characteristic line, MN in Figure 2, has the constant h value, 0.2575, so that there is no deformation along this line: the elastic dead line.

348 Wonoh Lee, Kwansoo Chung, Tae Jin Kang and Jae Ryoun Youn

As for the evolution of intermediate shapes and boundary tractions, the following boundary conditions are imposed for the geometric parameters, h and cp, along the ABeD line (1] = 0):

h(ah) = ahhf + (1-ah)h;

cp(all') = all'CPf + (1- all')cp; (8)

Here, a. and a~ vary from 0.0 to 1.0, which represents the initial and the final shapes, respectively.

When a. and a., are constant along the boundary line during the forming process, discontinuous boundary traction conditions with non-vanishing internal tractions are obtained [11]. In order to have continuous boundary traction conditions without internal tractions, a. and a., are prescribed as variables of ~ along the boundary line ABeD as shown in Figure 4 for this particular application. The resulting evolutions of the intermediate shapes and the boundary traction along the line ABeD are shown in Figures 5 and 6, respectively.

1.1,.-------------,

1.0 ---,

-- stagel

~O.8 --- stage2 ---- stage 3

i c:! 0.6

------ .tage 4 --_._.- stage 5

i ~ 0.4

0.1

0.0 +----,---.,....----,---.,....----'

o 10 10~ 30 40

I.S

I f f

/..-

/

----------------- stagel --- stage 2 ---- stagel ------ stage 4 ------- stage 5

0.0 +---'------------1

o 10 10~ 30 40

Figure 4. Distribution ofthe geometric parameters, h and rp ,at each stage of forming

Nonsteady plain-strain ideal plastic flow considering elastic dead zone 349

16 Y

14

12

10

8

6

4

2

0

stage 1 stage 2 stage 3 stage 4 stage S

-----" /

/

-2 +---r---.-~r--.---,--~--~---.---i.X

-2 0 2 4 6 8 10 12 14 16

Figure 5. Evolution of the part shape

0.5 ...------------------,

0.0 +--.--------,,:""'7-...--------I /.~;.-;;;. ,.....

~ -0.5 b'"

" ... -1.0 Ib '-'

i -1.5

i al -2.0

~ EI -2.5 .. g

-3.0

\ I /

b " / i' Ii I' 'i I )~i II!\ l' "-

stage 1 stage 2 stage 3 stage 4 stage 5

I~ ...... " ,---------------h , f \ I '\ I ', __________________ _

----1 ._._.i

-3.5 +----r---"T"""--...---...----I o 10 20

~ 30 40 so

Figure 6. Evolution of the nonnalized stress (Uq) distribution

350 Wonoh Lee, Kwansoo Chung, Tae Jin Kang and Jae Ryoun Youn

5. SUMMARY

The ideal forming theory, previously developed for process design purposes, was successfully applied for the non-steady plane-strain case. The minimum plastic work path, which is the kinematical constraint for ideal flow, was well accounted for using the orthogonal convective coordinate system having uniform local area. Besides kinematics, the equilibrium condition, schemes to optimize preform shapes for a prescribed final part shape and also to define the evolution of shapes and frictionless boundary tractions were developed. Numerical solutions were obtained for a part under forging for demonstration purposes.

ACKNOWLEDGEMENT

The authors would like to thank Drs. Sergei Alexandrov and Paul Wang for sharing the example data and also for their discussions. This work was supported by the Ministry of Science and Technology in Korea through the National Research Laboratory for which the authors feel so thankful.

REFERENCES

1. Richmond, O. and Devenpeck, M.L., 1962, A die profile for maximum efficiency in strip drawing, Proc. 4th U.S. Natn. Congo Appl. Mech., 1053-1057.

2. Hill, R., 1967, Ideal forming operations for perfectly plastic solids, J. Mech. Phys. Solids, 15,223-227.

3. Richmond, O. and Morrison, H.L., 1967, Streamlined wire drawing dies of minimum length, J. Mech. Phys. Solids, 15, 195-203.

4. Chung, K. and Richmond, 0., 1993, A deformation theory of plasticity based on minimum work paths, Int. J. Plasticity, 9, 907-920.

5. Chung, K. and Richmond, 0., 1994, The mechanics of ideal forming, J. Appl. Mech., 61, 176-181.

6. Chung, K., Yoon, J.W. and Richmond, 0., 2000, Ideal sheet forming with frictional constraints, Int. J. Plasticity, 16, 595-610.

7. Richmond O. and Chung, K., 2000, Ideal stretch forming processes for minimum weight axisymmetric shell structures, Int. J. Mech. Sci., 42, 2455-2468.

8. Richmond, O. and Alexandrov, S., 2000, Nonsteady planar ideal plastic flow: general and special analytic solution, J. Mech. Phys. Solids, 48, 1735-1759

9. Richmond O. and Alexandrov, S., 2002, The theory of general and ideal plastic deformation ofTresca solids, Acta Mechanica XXX, 1-10.

10.Alexandrov S., 2001, private communication. II.Chung, K., Lee, W. and Richmond, 0., 2002, Nonsteady plane-strain ideal plastic flow, Int.

J. Plasticity (submitted)

MULTISCALE MODELLING OF NON-LINEAR BEHAVIOUR OF HETEROGENEOUS MATERIALS: COMPARISON OF RECENT HOMOGENEISATION METHODS

Pascale Kanoute and Jean-Louis Chaboche

ONERA 29 avenue de La Division Leclerc 92322 Chatillon Cedex, France E-mail: pascale.kanoute@onera.fr

Abstract: This paper aims at evaluating recent homogenisation methods for scale changes in the micro-to-macro elastoplastic analysis of composites as well as polycrystalline aggregate. The corresponding localisation rules are recalled including, the T.F.A scheme, the incremental tangent approach of Hill, and the more recent affine method. These different schemes are finally applied to predict the overall behaviour of metal-matrix composites. With the help of simulations performed by the Finite Element method, we will discuss the limitations and the advantages of these procedures.

Key words: Multiscale analysis, Inhomogeneous material, Non-Linear behaviour,

Homogenisation

1. INTRODUCTION

Over the past few years, phenomenological approaches, which consist of a macroscopic description of experimental observations, have been progressively supplanted by micro-macro approaches, namely because of their inadequacy to describe the evolution of localised phenomena. Multi­scale modelling has the particular advantage to provide both the local and the overall mechanical behaviour of heterogeneous structures. This article

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S. Ahzi et al. (eds), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 351-358. ©2004 Kluwer Academic Publishers.

352 Pascale Kanoute and Jean-Louis Chaboche

lies within tbis context by presenting the application of several micro-macro procedures commonly used or recently developed to predict the overall behaviour of non-linear composites and polycrystals.

Non-linear estimates for the effective properties of heterogeneous materials can be traced back to the classical work of Taylor [1] on elastoplasticity of polycristals, followed by the work of Kri)ner [2] through a self-consistent procedure. An "incremental" formulation of the self­consistent procedure is proposed later by Hill [3] in order to incorporate additional micro-structural information. Tbis work has spawned different models, namely the formulations of Berveiller and Zaoui [4] and Molinari and al. [5]. The analysis presented here are based on recent formulations: the Transformation Field Analysis developed by Dvorak [6,7] and the "Affine" method of Masson and Zaoui [8,9].

The frrst approach analysed here is based on the idea of a purely elastic redistribution of the macroscopic stress and strain, and of the local eigenstresses or eigenstrains. First are presented the main lines of the transformation field analysis namely some of its properties for two-phase and multiphase exploitations. Then, we will focus on a second class of approaches based on a linearisation of the local constitutive laws: the Hill's incremental formulation and the "Affine" one. Finally, all these different schemes are applied to predict the overall behaviour of metal-matrix composites. We will discuss the limitations and the advantages of these procedures together with comparisons to simulations performed by a finite element based on periodic homogenisation.

2. THE TRANSFORMATION FIELD ANALYSIS

The Transformation Field analysis is based on the idea of a purely elastic redistribution of the macroscopic stress and strain, and of the local eigenstresses or eigenstrains. In tbis method, initially proposed by Dvorak, the plastic strain and the thermal expansion are considered as given eigenstrains.

Consider a representative volume V of an heterogeneous material with small inhomogeneities compared to that of V. The volume V may be subdivided into several local volumes V,.r = 1,2, ... N, such that each contains one phase material. The local constitutive relations in each sub­volume are written in the following form:

CTr(x)=Lr : a, (x) + Ar(X) a, (x) = ae r (x) + f.lr (x) (1)

Multiscale modelling of non-linear behaviour of heterogeneous materials:353

comparaison of recent homogeneisation methods

The J-lr (x) denotes a prescribed distribution of local eigenstrains and

Ar (x) = -LrJ-lr (x) is the corresponding eigenstress field. The eigenstrain and eigenstress fields may consist of contributions of distinct physical origin like thermal strains, plastic strains and transformation strains.

The macroscopic stress L and the macroscopic strain E are defined as the volume average of the local stress and strain fields while the local and the overall eigenstrain and eigenstress fields are connected by the generalised Levin formula. The relation between the local and overall fields is then given by the following localisation rule:

8 r = Ar : E + I Drs : J-ls (3)

The tensors Ar and B r are the mechanical strain and stress concentration

tensor factors, Dsr and Fsr the "transformation influence tensors" (fourth

rank tensors). All the tensors Ar, Br, Dsr ' Fsr depend on the local and

overall elastic moduli, and on the shape and the volume fraction of the phases, and are then derived once, independently of the inelastic process. These tensors are determined by solving a set of linear problems (6 for the concentration tensors and 6*N for the influence tensors) by a finite element method.

In the case of periodic microstructure, the transformation fields analysis corresponds to a simplification of the periodic homogenisation provided each phase is subdivided into a large number of sub-domains. On the contrary, in the particular case of a two-phase medium not subdivided into sub-domains, it reduces to a mean fields method. It has long been recognized that the TF A method, which is based on a purely elastic redistribution of the macroscopic and of the local eingenstresses or eigenstrains, delivers in this case "too stiff' response. It can also be noticed that in the case of polycrystals and the self-consistent estimate, the TF A method is a generalisation of the Kroner localisation rule.

A correction method that takes advantage of the "tangent formulation" has then been proposed by Chaboche et al. [11] and T. Pottier [12], in order to reduce the stiffness of this localisation rule when applied to two-phase materials not subdivided into sub-domains. The correction consists of writing the total elastic localisation rule with corrected values for the eigenstrains using the asymptotic tangent stiffness of the local constitutive equation. The application of the method to two-phase medium [11, 12] has shown that this correction has the advantage to give an

354 Pascale Kanoute and Jean-Louis Chaboche

improved estimate of the overall strain-stress response of the elasto-plastic behaviour of composites. Alternative formulations, which are expected to yield more realistic predictions of the effective properties of non-linear materials, have also been considered: the incremental tangent and the affine formulations.

3. TANGENT FORMALISM: THE INCREMENTAL TANGENT AND AFFINE METHODS

The Incremental tangent and affine methods have been proposed in order to introduce an elastoplastic redistribution of the macroscopic and of the local fields, in order to correct the too stiff estimates of approaches based on a purely elastic description of the internal interactions between the phases in heterogeneous materials. We will not present here the Hill's incremental formulation, this method being already well known. Only the main lines of the affine formulation, published recently by Zaoui and co-workers [8,9] are recalled in this paper.

This method calculates directly, as the secant formulation [4], the response of material to a given request by using a linearisation of the constitutive laws. As Molinari et al., these authors have resorted to a tangent linearisation of the constitutive laws:

- 0 CY,(x)=L, :G,(x)+cy ,(x) (4)

- 0 0 - 0 G,(x)=M,:CY,+G ,(x) CY ,(x)=-L,G ,(x)

- -where the tensors L, and M, are the local tangent modulus and

compliance. The tensor G,o represents the prestrain and will express the stress dependence of the inelastic strain in a linearised way.

This formulation is neither an incremental nor a secant one; it uses the tangent modulus or compliance, associated with some given initial prestress or prestrain. Moreover, in contrary to the approach of Molinari et al., the affine formulation is based on a linear thermoelastic comparison medium. The overall constitutive equations of the corresponding thermoelastic medium then read:

Considering the Self-consistent or Mori-Tanaka estimates, the corresponding localisation scheme is then given by:

Multiscale modelling of non-linear behaviour of heterogeneous materials :355

comparaison of recent homogeneisation methods

Note that in these equations, as for the TF A procedure, the mechanical fields are supposed to be uniform inside each phase. In this model, the localisation tensors, the tangent tensors (anisotropic) and the strain green tensor P have to be evaluated at each time step at a given loading.

4. APPLICATION TO THE MICROMECHANICAL ANALYSIS OF ISOTROPIC TWO-PHASE COMPOSITE

Three examples have served to compare the capabilities of the methods with the exact reference solution obtained by a finite element based on periodic homogenisation. The applications correspond to a composite with a high contrast between the inclusions (E f = 400GPa, v f = 0.2) and the matrix (Em = 75GPa, v m = 0.3). The inclusions are elastic and distributed isotropically in an elastoplastic matrix. The matrix is governed by either a J2-flow theory with a power law or a linear isotropic hardening.

The application of the incremental and affine methods are made following two ways which differ from the determination of the tangent polarisation tensor: • The anisotropic formulation which takes into account the anisotropy of the

tangent elastoplastic operator. The polarisation tensor is then defined as follows and has to be evaluated numerically:

P= 4~--l fH(x~lq-1 ~r3 dS,.x) "1':>11x=~ (8)

H = [X®K-1 ®x](S) K=xi.x

where q characterises the geometry of the ellipsoid (here a sphere).

• The isotropic simplified formulation which neglect the anisotropic term in the expression of the elastoplasti,£ tangent operator. The polarisation tensor P reduces then to P = S: L-1 where the Eshelby tensor S is obtained analytically.

The results of the prediction are shown in Figures 1, 2 and 3. A few observations can be drawn from the comparison between these schemes:

356 Pascale Kanoute and Jean-Louis Chaboche

• The predictions of the model based on the TF A seem to be very stiff when used with no subdivision of the phases into sub-domains. This feature of the TF A procedure has already been noted and has led Chaboche and al and T. Pottier to propose a correction method that takes advantage of the ''tangent formulation" in order to reduce the stiffness of this localisation rule. It can be noticed in these figures the fairly good predictions of this approach by comparison with F.E.M simulations.

• The predictions of the incremental method appear also very stiff when used with the correct anisotropic localisation tensor. As already reported, the affme method delivers a softer response than the incremental one. However, the predictions largely overestimate the overall strength and hardening. It can also be noticed that in the case of a matrix governed by a linear isotropic hardening there is a discontinuity in the response for this approach.

• In contrary to the correct anisotropic formulation of the incremental and affine methods, "the isotropic formulation" gives better response with a small underestimation of the overall strength and hardening for the incremental approach except for an elastic-perfectly plastic matrix for which the incremental method and the corrected T.F.A formulation give the F.E.M simulation.

5. CONCLUDING REMARKS

This paper provides a review of different methods that have been developed for some of them recently to estimate the effective behaviour of non-linear heterogeneous materials. It has been shown by the means of different examples on isotropic two-phase composites that the TF A, the incremental tangent and the affme methods significantly underestimate the overall strength and hardening of the composite. On the other hand, better predictions are obtained when using an "isotropic formulation" of the incremental and affme methods. Indeed, by neglecting the anisotropic part of the elastoplastic tangent operator, we obtain slight underestimation of the reference solution. Therefore, it can be enticing to use the "isotropic" version of the incremental and affine methods, regarding the fact that it recovers sufficiently well the reference overall response. But it leads to a controversial question concerning the validity of such a formulation. The "corrected" formulation of the T.F.A method, which delivers fairly good predictions appears in this sense more adapted.

Multiscale modelling oJnon-linear behaviour oj heterogeneous materials:357

comparaison oj recent homogeneisation methods

350

300

250

tOO

50

o 0.005 0.01

_ F.Emethod

--- T.F.A method .... _ .•.• Affine ~ Inc;::remental --- T.Pottler formulation .-. Isotropic Affine ---- Isotropic Incremental ......... Matrix .

0.015 0.02 0.025 0.03 Strain

0.035 0.04

Figure 1: Predictions of the overall response of the composite. The matrix governed by a J2·flow theory with a power law isotropic hardening (H=416, a=O.3895). Example proposed by Suquet [13] in 1997.

160

140

o 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Strain

Figure 2: Predictions of the overall response of the composite. The matrix is governed by a J2-flow theory with a linear isotropic hardening (H=1000, a=l).

358

180

160

140

Pascale Kanoute and Jean-Louis Chaboche

" " " , ,

,/,---,--=-=--.-,------, ~ ., . .,. -eo- F.E melhod ~' --- T.F.A method

, . " .... . Affine - Incrementa' --- T.Pottl.t formulaUon

:.:.: ::!~~~~ ~~~~-:nent.1

,i· 'V • _ ...... _ . ....... .. '" ........ _ ....... -. ... _ ............. .ol .... , .. _ ..

~ . _ . I . ,..,., - ~

"~sotropic fotriiiilatioii" . . , .

o 0-'101 0 .002 0 ,003 0 ,004 0 ,005 0 ,006 0 .007 0 ,008 0 .009 0 .01 Strain

Figure 3: Predictions of the overall response of the composite by several schemes (H =1, a=I).

REFERENCES

[I] GJ.Taylor, Plastic strains in metals, J. Inst. Metals, 62 307-324, 1938. [2] E.Kroner, Zurplastichen Verformung des Vielkristalls,Acta Metalla. 9,155-161. [3] R.HiII, Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids., 13, 89-10 1. [4] M. Berveiller and A. Zaoui, An extension of the self-consistent scheme to plastically­flowing polycrystals, J. Mech. Phys. Solids., 26, 325-344, 1979. [5] A. Molinari, G.R. Canova, and S. Ahzi" A self-consistent approach for the large deformation polycrystal viscoplasticity, Acta Metall.,35:2983-2994, 1987. [6] G.Dvorak, Transformation fields analysis of inelastic composite materials. Proceedings of the Royal Society of London, A.437, 311-327, 1992. [7] G.Dvorak, Y.Bahei-EI-Din, A.Wafa, Implementation of the transformation field analysis for inelastic composite materials, Comput. Mech., 14,201-228, 1994. [8] R.Masson and A.Zaoui, Self-consistent estimates for the rate-dependent elastoplastic behaviour of poly crystalline materials, 1. Mech. Phys. Solids, 47,1543-1568,1999. [9] R.Masson, M.Bomert, P. Suquet and A. Zaoui, An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J. Mech. Phys. Solids, 48: 1203-1227,2000. [10] J. L. Chaboche, S. Kruch, J. F. Maire, T. Pottier, Towards a micromechanics based inelastic and damage modeling of composites, International Journal of Plasticity, 17, 411-439,2001. [11] T.Pottier, Modelisation multiechelle du comportement et I'endommagement de composites Ii matrice metallique. Doctorat d'Universite, Ecole nationale des Ponts et Chaussees, 1998. [12] P. Suquet, Effective properties of nonlinear composites, Continuum Micromechanics, P.Suquet Ed., CISM Lectures No. 377, Springer-Verlag, 197·264, 1997.

THERMOMECHANICAL BEHAVIOUR OF SHAPE MEMORY ALLOY TAYLOR'S MODEL

M.O. Bensalah1), L. Boulmane1) , A. Hihe) l)University Moulay Ismail, GPSEM,Faculty of Sciences Meknes Morocc

2) University Mohamed V, L.MM, Faculty of Sciences Rabat Morocco.

E-mail: M-bensal(iiJ.(sr.ac.ma

Abstract In this work, we study the shape memory alloys (S.M.A) micromechanical behaviour of single and polycrystal in the perfect transformation plasticity using Taylor's approach. This study is based on the kinematical description of the physical mechanisms at the origin of the phenomenon, and on the thermodynamical analysis of the irreversible processes presented by these alloys in such a transformation. The pseudoelastical behaviour of S.M.A is described by defining a pseudoelastical potential through the Gibbs theory, and by application of the second principle of the thermodynamic. We also put out the contribution of the variant's reorientation mechanism to the micromechanical behaviour.

1. INTRODUCTION

The shape memory alloys (S.M.A) undergoing a thermoelastic martensitic transformation, which could be induced by variation of temperature or by application of the macroscopic sufficient stresses [1,2], have received a considerable attention in the last three decades because of their particular shape memory effect (S.M.E) and pseudoelastical behaviours. The intensive investigations have allowed the understanding of the physical mechanisms of these alloys, during such a transformation, in order to exploit them in a large field of industrial applications. Hence, the interest of establishing some modellings which are able to describe the principal characteristics of the thermomechanical behaviour of S.M.A and satisfy the engineers' requirements. The S.M.A thermomechanical behaviour modellings, undertaken since the eighties, are registered either in the

359

S. Ahzi et aJ. (eds.). Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 359-366. ©2004 Khmer Academic Publishers.

360 MOo Bensalah, L. Boulmane, A. Hihi

macroscopic thermodynamic framework [3,5] or in a micromechanical description framework which is based on a scale transition method (micro­macro) [6,8]. Our contribution is situated in the latter.

In this study, we limit ourselves to the pure transformation plasticity and to the formation and development of the martensite from the austenite, in the case of a pseudoelastical behaviour. For this, we present firstly a kinematic study associated to such a transformation in small deformations. The pseudoelastical behaviour is described by defining a pseudoelastical potential, through the Gibbs theory, and by applying the second thermodynamic principle. 2-Kinematic of the transformation

The choice of the used description is an important stage of the modelling. In order to consider the structure of a single crystal and the existence of twenty-four martensite variants cristallographically equivalents [10,11], one adopts, as a description scale, the martensite variant. One considers, therefore, as element of basis for the material global behaviour modelling, the transformation strain associated to the formation of a martensite variant h in the single crystal. This transformation strain, noted eh , is a function of only the strain amplitude g and the orientation R h of the considered variant[7];

(1)

(2)

where jih is the normal of the variant h and mh is the transformation direction.

We consider an homogeneous and elastic material of volume V undergoing a phase transformation, the macroscopic transformation strain ETr is the volume average of the local transformation strain eTr (r),

E~r =.!. I e~r (r) dv Vv

(3)

Tr J't" h h)d ·th £:\ "" I h

{OSi r d vh

Eij =- Leij0 (r v; WI ~ (r)= Vv h lsirevh

(4)

Thermomechanical behaviour o/shape memory alloy taylor's model 361

Equation (4) becomes:

.. (5)

where fh (= Vh IV) is the volume fraction of the variant h.

In the case of the polycrystal the macroscopic transformation strain is taken as the middle strain on all grains that form the polycrystal. The expression of this strain is given then by:

(6)

where F, is the volume fraction occupied by the grain I in the polycrystal.

2. PSEUDOELASTICAL POTENTIAL OF AUSTENITE-MARTENSITE TRANSFORMATION

In order to describe the S.M.A pseudoelastical behaviour, we adopt a thermodynamic potential through the Helmoltz's free energy <I> associated to the thermoelastic martensitic transformation. This energy is defined, in a unit volume, as the sum of the variation of chemical free energy ilG ch' of the

interfacial free energy Wsur associated to the creation of the interfaces

during the transformation and of the elastic deformation energy (mechanical energy) W [12]. In the stage of propagation, the observed elongation shape

of the martensite plates shows that Wsur is negligible compared to the

mechanical energy; this contribution will be neglected in the following of this study. <I> is thus written:

(7)

This energy depends on the control variables (the total strain E and the temperature T) and on the internal variables y k' The kinematic study

showed that the volume fractions of the different martensite variants could correctly describe the microstructural state of the material. We choose, therefore, these parameters as internal variables.

362 MD. Bensalah, L. Boulmane, A. Hihi

The Gibbs's free energy \f is best adapted to the description of the S.M.A thermomechanical behaviour because the martensitic transformation can be induced by a drop in temperature or by applying a stress :E:

(8)

The variation of chemical free energy, ~G ch' per unit of volume,

associated to the formation of martensite from an austenite state is well represented by a linear function of the difference between the current

temperature and the equilibrium transformation temperature To. ~G ch is

proportional of the total amount f of martensite formed in the unit volume[l2]:

(9)

where B is a coefficient of proportionality.

The mechanical energy W is made up of two terms which have different physical origins. On the one hand the elastic energy of deformation We caused by the applied macroscopic stress field :E :

(10)

On the other hand, the stored energy Wb associated to the internal

stresses field cr(r) generated by the incompatibilities of the transformation

strain:

1 f TT Wb = --2 crij(r)e jj (r)dv vv

(11)

Considering the martensite variant h as ellipsoidal inclusion characterised by the Eshelby's tensor Sh , the expression of the internal stress in the variant h (crh ) is given by [13]:

(12)

where C is the elasticity tensor and I is the identity fourth order tensor.

Thermomechanical behaviour of shape memory alloy taylor's model 363

It is important to note that the relationship (12) is, as reported by many authors [14,16] ,an approximate way to determine the internal stresses.

The stored energy Wb becomes:

(13)

From equations 9,10 and (13) the pseudoelastical potential \{' is written as:

(14)

The pseudoelastical potential \{' describes the austenite-martensite system state in terms of the applied stress L, the temperature T, the active martensite variants volume fractions fh (internal variables), the morphologies of the latter as well as their orientations.

The extension of this potential to the case of the polycrystal gets himself comfortably under the following shape:

1 ()h Tr ( )b ( )h h - -2 I I C ijkl (Iklmn - Sklmn r )(E mn - Emn I) Emn r fr FJ J h

(15)

The equation (15) is optimised while considering that among the twenty four variants of martensite, only six variants are active by grain. The game of six variants solution is the one for which the work of strain is minimal.

3. NEW APPROACH

In this new approach the calculation of the deformation of transformation is appreciably different. Indeed ,one applies the theory twice inclusions.

364 M 0. Bensalah. L. Boulmane • A. Hihi

Once on the level of the grain by considerate the alternatives like inclusions in the grain. It the same like the old approach.

The second time one applies this theory to the level of the polycrystal and in this case inclusions are the grains in the polycrystal. The defonnation of transfonnation of the grain in the polycrystal is obtained by:

(16)

Finally the defonnation of transfonnation in the polycrystal is taken as the average of the defonnations of transfonnations on all the grains.

(17)

4. RESULTS AND DISCUSSIONS

We apply the proposed modelling in order to determine the Cu 18.4% Zn 7.3% Al SMA thermomechanical behaviour. In this class of alloys the martensite variants are of the family ii(0.182, 0.669, 0.721) and m(0.165,0.737,0.655) [24]. The choice of this alloy contains multiple advantages[25] :

• These materials don't present any coupling between the classical plasticity and the transformation plasticity,

• Interaction between the moving interface and dislocations of the crystal in the transfonnation.

We suppose that the elastic behaviour of single crystal is isotropic and homogeneous with an elastic shear modulus ~ equal to 40000 MPa and a Poisson's ratio v of 0.333. The martensite variants shape are ellipsoidal flattened « penny shape» of half axes a) = a2 = IOOa3 •

First, we consider forward transformation using Taylor's approach and we compare to the experimental result. Nevertheless, comparison between our results and experiments (Figure 1) present a difference .In fact, our model does not take into account the nucleation mechanism involved in the earlier stages oftransfonnation.

Figure 2 deals with cooling at imposed constant stress. We obtain no dependence of the maximal transformation strain with the level of applied stress. This discrepancy with the experimental results stems from an under­evolution of the influence of the internal stress. More realistic description should take into account the contribution of the stored energy. Nevertheless, this simplify scheme put in evidence some interesting features. A

Thermomechanical behaviour of shape memory alloy taylor's model 365

fundamental result, obtained in that way, is the non intrinsic character of the end transformation stress. Figure 3 present a polycystal behaviour in uniaxial tension using a new approach and comparison with Taylor's approach and experiment one. The above results allow us to conclude that this work constitutes an attempt modelling of SMA behaviour.

l !.

!

600

500 I==::'J 400

300

200

100

o 0.01 0.02 0.03 a.a. 0.05 0.08 0.07 0.08

Strain

Figure 1. Polycrystal behaviour in uniaxial

tension at T=20 C, g=O.23, b=O.23 MPa K,

Ms=-30C

200

.s;

I

0.1

0.08

0.08

0.04

0.02

0 .. 0 10 20 30

Temparaltn rC)

Figure 2. Cooling-heating CUNes at

constant imposed stress for the polycrystal.

100 .f.l!::....~::::::..----

o 0 0.01 0.02 0.03 0.04 0.05 0.08 0.01 OM .....

Figure 3. Polycrystal behaviour in uniaxial tension at T=20 C using a new approch comparaison with Taylor's approch and experiment.

5. CONCLUSION

We have present the element of micromechanical behaviour of shape memory alloy using the Taylor's model and a new approch based to the

366 M.D. Bensalah, L. Boulmane, A. Hihi

autocoherent model.The results show that the autocoherent model give a good approch to Taylor's model.But, this approch us insufficient to descript the real behaviour of the SMA.This phenomenon can be obtained using an affined approch study which can show the real behaviour of SMA(the influence of the cristallographic texture and the interaction between variants}.

6. REFERENCES

[1] L. Delaey, R. V. Krishnan, H. Tass and H. Warlimont, Mater Sci 9, p 1536 (1974)

[2] N. Nakarishi Archs. Mech 35, 1 (1983)

[3] M. Fremond, C. R. Acad. Sci. Paris, 304, serie II, p. 239-245, (1987)

[4] I. Muller, H. Xu, Acta Metall. Mater. 39, p. 263, (1991)

[5] B. Raniecki, C. Lexcellent, K. Tanaka, Arch. Mech, 44, 3, p. 261-284, (1992)

[6] F. Falk, Int. J. Engng. Sci. Vol. 27, W3, p. 277-284, (1988)

[7] E. Patoor, A. Eberhardt, M. Berveiller, Arch. Mech., 40, p. 775, (1988)

[8] E. Patoor, M. O. Bensalah, A. Eberhardt, M. Berveiller, La revue de Met. p. 1587-

1592, (1993)

[9] Q. P. Sun and K. C. Hwang, J. Mech Phys. Solids Vol. 41, No.1, p. 1-17 (1993)

[10] H. Warlimont, L. Delaey, R. V. Krishnan, H. Tass, J. Mater Sci 9, p. 1536, (1974)

[II] T. Saburi, C. M. Wayman, K. Tanaka, S. Nenno, Acta Metall., 28, p. 15, (1980)

[12] J. Ortin, A. Planes, Acta Mettall., 36, p.1873, (1988)

[13] T. Mura, Micromechanics of defects in solids, Martinus Nijhoff, the Hague, (1987)

[14] H. Sabar these de doctorat d'ingenieur, Universite de Metz, (1990)

[15] E. Patoor, A. Eberhardt, M. Berveiller, J. Phy. IV, CI-277-293, (1996)

[16] S. Agouram, L. Abdous, M. O. Bensalah, Eur. Phys. J. AP 1,341-346, (1998)

[17] C. Lexcellent, C. Licht, J. de Phys. Appl. IV p. 35-39, Novembre (1991)

[18] J. Van Humbeek, L. Delay, J. Phys., C5, N° 10,42, p. 1007-1011, (1981)

[19] J. Ortin, A. Planes, J. de Phys. III, C4, Vol. 1, p. 13-23, Novembre, (1991)

[20] D. Fran~ois, A. Pineau, A. Zaoui, Comportement mecanique des materiaux, Hermes,

Paris, (1991)

[21] Z. S. Basinski et J. W. Christian, Acta. Met., 2, (1954)

[22] M. Berveiller, S. Dominiak A. Eberhardt et E. Patoor. Plasticite des materiaux solides,

A.T.P., Mat. 4 Metz, (1985)

[23] T. Katata et H. Saka, Electron-microscopic in situ observation of the motion of

interfaces in Cu-Zn-Al thermoelastic martensite, Phil. Mag. A, 59, N° 3, 677-686,

(1989)

[24] J. Devos, L. Delaey, E. Aemond, Academic Press, p. 438, (1978).

[25] E. Patoor, these de docteur ingenieur presentee a l'Universite de Metz (1986)

MULTISCALE ANALYSIS OF DYNAMIC DEFORMATION IN MONOCRYSTALS

M. A. Shehadeha, H. M. Zbib"·, T. Diaz de la Rubiab and V. Bulatovb

• School of mechanical and Materials Engineering, Washington State University, Pullman, WA 99163-2920, USA b Material Science and technology Division, Chemistry and Material Science Directorate, Mail Stop L-353, Lawrence Livermore National Laboratory, 7000 East Avenue Livermore, CA 9450, USA

Abstract: The dynamic deformation in FCC single crystals is investigated using a multiscale dislocation dynamic plasticity model. We examine the effect of strain rate, pulse duration, nonlinear elastic properties and crystal anisotropy on wave profiles and dislocation microstructures. The morphologies of the relaxed configurations of dislocations microstructures show formation deformation bands.

Key words: Dislocation Dynamics, High Strain Rate, Dynamic Plasticity, Multiscale

1. INTRODUCTION

Advancements in experimental capabilities over the years have improved our understanding of the dynamic response of materials. The Hopkinson bar and plate impact techniques can now be used to study the dynamic inelastic response of materials over strain rates ranging from 102 S-1 to 5x 106 s-1 (Clifton, 2000, Naser, 1992). Recently, Laser based experiments have been used to study the plastic deformation in metals single crystals. Short pulse duration (in the order on nanoseconds) was used to generate extremely strong pressure waves that propagate through the tested samples (Loveridge-Smith, 2001, Kalanter et al., 2001, Kanel et al., 2001, Meyers, et at, 2001), In all of these experimental techniques, simple loading conditions (uniaxial stress or uniaxial strain) are applied on the specimens to facilitate the interpretation of the results. During shock or impact loading, plastic deformation (occurs as a result of fast moving dislocations) is often localized in narrow band like structure. These bands generally lie on the slip planes and contain a very large number of dislocations (Coffey, 1992) that

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S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 367-378. ©2004 Kluwer Academic Publishers.

368 M Shehadeh, HM, Zbib, T Diaz de la Rubia and V Bulatov

arrange themselves in certain structures depending on the loading conditions. The dislocation structure forms almost instantaneously up to 106 S-1 strain rate (Kuhlmann-Wilsdorf, 2001). The observed microstructures consist of dislocation cells, deformation micro bands and deformation twins (Rivas et aI., 1995, Mogilevskii and Bushnev, 1990). The microstructures generated by the stress wave depend on material properties such as stacking fault energy and shock wave parameters, namely strain rate and pulse duration (Meyers, 1994).

Deformation physics at high strain rate is a complex multi scale dynamic problem. The level of pressure and temperature involved under extreme conditions may make it difficult to address the deformation process using physical experiments. In fact, current experimental capabilities cannot address material response at pressures larger than 100 GPa. In addition the costs of full scale testing in this area of research is high and escalating (Mayer, 1992). Therefore, computer simulations methodologies are used to study the dynamic deformation phenomena by attempting to bridge the length scales from atomistic to macroscopic scales. In the atomistic scale, molecular dynamic simulations are used to investigate the response of single crystals to high strain rate loading (kadau et aI., 2002, Horstemeyer et aI., 2001, 2002). Smimov, et al. (1999) introduced a combined molecular dynamics and finite element approach to simulate the propagation of laser induced pressure in a solid. In spite of all computational advances we have, atomistic simulations can model no more than billions of atoms (Clifton and Bathe, 1999). In the micorscale, discrete dislocation dynamics provides an efficient approach to investigate the collective behavior of many interacting dislocations.

Recently, multi scale dislocation dynamics plasticity has emerged as an excellent numerical tool to simulate the collective behavior of dislocations in a bulk material. Dislocation dynamics can simulate sizes much larger than the current atomistic simulation capabilities. In our attempt to understand the response of FCC single crystal to high strain rates, we used a multiscale dislocation dynamics plasticity (MDDP) model to study the interaction between stress waves and dislocations. In this study, the effects of strain rate, shock pulse duration, crystal anisotropy, and the dependence of elastic properties on pressure were studied. Therefore, computer simulations of dislocation motion under impact loading hold great promise for investigating deformation processes in regimes that cannot be probed by current experiments.

Multiscale analysis of dynamic deformation in monocrystals 369

2. DISLOCATION DYNAMIC PLASTICITY MODEL

The MDDP model is based on fundamental physical laws that govern dislocations motion and their interactions with various defects and interfaces. The model merges two scales, the nano-microscale where plasticity is determined by explicit three dimensional dislocation dynamics analysis providing the material length scale, and the continuum scale where energy transport is based on basic continuum mechanics laws. The detail of the framework and basic equations can be found in numerous articles provided by Zbib and co-workers. The model is based on the basic laws of continuum mechanics, i.e. linear momentum balance and energy balance:

divS=pv

(1) P cj =KV2T + S.i P

(2)

where v = it is the particle velocity, p, Cv and K are the displacement vector field, mass density, specific heat at constant volume and thermal conductivity. Then, the strain rate tensor i is decomposed into an elastic

part i e and plastic part i P which when combined with the classical Hooke's law yields:

(3)

where Ce is, in general, the anisotropic elastic stiffness tensor, OJ is the spin of the microstructure and it is given as the difference between the material

spin Wand plastic spin W p. The evaluation of the plastic strain increment is performed in the discrete dislocation dynamics component of the model, involving massive computations of dislocation-dislocation interaction, motion, multiplication, annihilation, etc. The reader is referred to various papers dedicated to the development of this model (Hirth et aI., 1998, Zbib et aI, 1998-2002, Rhee et aI, 1998). The resulting system of equation is solved numerically using that standard finite element method.

370 M. Shehadeh, H.M., Zbib, T. Diaz de la Rubia and V. Bulatov

3. MDDP SIMULATIONS Velocily controlled B.C

Symmetric S.C

2SjJITI

Figure 1. Setup of the simulation cell and the finite element mesh.

The simulations MDDP are designed to mimic uniaxial strain loading at extreme conditions of high strain rate ranging between 105 /s to 107 Is, and short pulse durations of few nanoseconds. As illustrated in Fig. 1, a computational block, with dimensions 2.5 !-tm x2.5 !-tm x25 !-tID is used. Velocity-controlled boundary condition was applied on the upper surface over a short period of time t* to generate the stress waves. In this case, the applied velocity corresponds to the average strain rate over the entire domain, whereas t* corresponds to the required pulse duration. The upper surface is then released and the simulation continues for the elastic wave to interact with the existing dislocations. In order to achieve a uniaxial strain involved in shock loading, the four sides of the block are confined so that they can move in the loading direction only. The bottom surface was rigidly fixed. In order to isolate the effect of the reflected wave, the length of the cell (25 !-tID) is chosen such that once the wave front reaches the bottom surface, the value of the stresses in the position where the dislocations are located is very small so that dislocation relaxation process can take place.

At high-pressure loading, the elastic constants become pressure dependent, we adjust for this effect by fitting the experimental results of the

Multiscale analysis of dynamic deformation in monocrystals 371

shear modulus (G) and Poisson's ration (v) obtained by Hayes et al (1999) such that

(4)

(5)

{Go +0.89P

G-Go +53.4+0.40P

v = Vo + 1.70 x P X 10-12

0<P<60

60<P<100

where, P is the mean stress in GPa, Go and Vo are the shear modulus and Poisson's ratio under normal static loading conditions. For copper, Go and Vo are 46.6 GPa and 0.33 respectively. The implementation of the above nonlinear elastic model in our FE framework required the adjustment of the stiffness matrix at every time step. Furthermore, due to their geometry, metal single crystal exhibits cubic symmetry that leads to the anisotropy in their behavior. This effect is also implemented and investigated in this study.

One of our main objectives in this study is to investigate the plasticity induced by transmitted waves. Frank Read loops randomly distributed on four different slip planes are used as agents for dislocation generation as the interaction with the stress wave takes place. The length of each source ranges from 1200 to 3000 b (0.30-0.80 /lm) where b is the magnitude of the Burgers vector. It is worthy mentioning that while in FEA, the fours sides of the cell were constrained to move in the x and y direction so that confined state of stress is achieved, periodic or symmetric boundary condition are used in the DD part in order to take into account the periodicity of single crystals in infinite media.

4. RESULTS AND DISCUSSION

4.1 Wave Propagation Characteristics

The application of a uniaxial strain on the crystal results in propagating a three-dimensional state stress, which can be decomposed into mean stress and deviatoric stresses. Figure 2 shows snap shots of a wave propagating in defect free copper crystal shocked to 7x 105 /s for 1.3 nanoseconds. As the wave propagates, additional peaks appear as a result of the FE mesh effect. In addition, the values of the lateral stresses (ax, oy) are equal as predicted by the elasticity theory:

372 M Shehadeh, HM, Zbib, T Diaz de la Rubia and V Bulatov

(6)

va a =a = __ z x Y I-v

where oz is the stress in the loading direction. From the above equation the mean stress can be calculated in terms of oz. Upon doing that, the expression for the mean stress (P) can be written as:

P = - (1 + 2v )a Z (7) 3(1-v)

Figure 2. Stress wave snap shots in copper crystal shocked to strain rate 7x 105 /s for 1.3 nanoseconds

- 1.10ns.

- 221n,

- - - 3.31 nto

~4.41ns

4.0

\! .... ;\ ~~ . .. \:., . .

".:

position (11m)

Dislocation activities cause stresses to have different values than those predicted by elasticity theory (Weertman, 1981). Figure 3 shows that the dislocation motion and multiplication result in reducing the values of the effective shear stress. The decrease in peak stress reveals that plastic deformation produced by dislocation motion and multiplication will cause material softening. The stress field is calculated by subtracting the plastic component of the strain field from the total strain. The energy used to generate plastic strains is converted into heat, which results in local temperature increase.

The effect of elastic properties dependence on pressure is examined in Figure 4 by plotting the wave profiles for both the linear elastic and nonlinear elastic constitutive relations. Clearly, the qualitative features of the two profiles are similar. However, as a result of the increase in the elastic properties, nonlinear elastic model leads to faster wave propagation and higher values of peak stresses.

Multiscale analysis of dynamic deformation in monocrystals 373

Figure 3. The effect of dislocations activities on the effective shear stress for copper shocked at 7xl05/s for 1.3 nanoseconds

4.2 Dislocation History

positiop(JimJ

Figure 4. The effect of pressure dependence of the mean stress profile.

Figure 5 shows a representative dislocation density history for a simulation carried out on copper at strain rate 10% using reflective boundary condition in DD. This curve suggests the existence of three distinct regimes of interaction between dislocations and the propagating elastic waves. These regimes are: A) no interaction regime, where the wave has not yet impacted the sources, B) the interaction regime, where the wave impacts the dislocation sources leading to avalanche in the dislocation density, and C) the relaxation regime, in which, wave surpasses the region where the dislocations are located leading to saturation in the dislocation density.

Figure 5. The effect of shock pulse duration on dislocation density histories

J.SOE+13

l.OOEH3

2.SOE+13

"'i 2.00E+13

~ UOE+13

1.00E+13

5.00E+12

~ ~

r-.. ···· .. ,-..... '--11JOn5

f I

I j-~

O.OOBHlO.\-o_-_~~-_-~_-_----<

tanr:(nanosecondl)

Elastic waves launched at high strain rates carry higher stress levels and propagate at velocities faster than that of lower strain rates waves. Under shock loading conditions, wave speed is proportional to the shock strength

374 M Shehadeh, H.M, Zbib, T. Diaz de fa Rubia and V Bulatov

because the shear modulus is pressure dependent and so the interaction between the dislocations and wave starts at higher strain rate before lower strain rates. As the wave passes the location where the dislocations are situated, the dislocations start to relax and organize themselves in certain morphologies that depend on the stress level carried by the wave and the pulse duration of the wave.

4. 3 Dislocation Microstructure

The dislocation microstructures generated by shocks depend strongly on the shock pulse duration and the stress level of the wave. It is worthwhile to

mention that for relatively low strain rates ([;' ~ 5 x 105 ), the combination of the stress level and pulse durations (1-4 nanoseconds) is not sufficient for the dislocations to organize themselves in a regular microstructure. However, as the strain rate increases, the state of stress renders so high that it allows the dislocations to form long deformation band of submicron dimension coincident with the {Ill} planes. Typical results of the morphologies of dislocations at different strain rates is illustrated in Figure 6 that show the microstructure in slices within the computational cell. The lengths of these bands do not appear to be dependent on the strain rate. However, the thickness of the bands appears to correlate inversely with the applied strain rate. The shock pulse duration is one of the most important parameters in controlling the microstructure of the dislocations. Pulse duration is related to the time required for the dislocations to reorganize (Meyers 1994). The effect of pulse duration on the dislocation microstructure is mainly to give more time for dislocations to reorganize. The microstructure at strain rates less than 5 xl 05/ S consists of irregular dislocation entanglements. As the pulse duration increases, these entanglements become more distinguishable. However, at strain rates (~ 5xl05 Is), the microstructure consists of micro bands. These bands become more defined as the shock pulse duration increases. Within these bands, areas of high dislocation densities, surrounded with relatively lower dislocation density areas.

.1. Figure 6. The effect of pulse

duration on dislocation microstructures of copper single crystal. The applied strain rate for of this simulations is 1 x 106 Is. The microstructures of each pulse duration are shown in slices within that domain. (a) pulse duration = 1.3 nanoseconds (b) pulse duration = 3.6 nanoseconds

Multiscale analysis of dynamic deformation in monocrystals 375

4.4 Crystal orientation and anisotropy

Crystal orientation is among the most important factors that affect the deformation process in single crystals. This effect was studied by performing simulations for three different orientations such that the number of slips activated in each orientation are different. For the first, second and third orientation, the computational cell was oriented with respect to the reference frame i.e. (I 00, ° 1 0, ° ° 1) in the (1 00, ° 1 0, ° 01), (1 -1 0, 1 1 -2, -1 1 1) and (1 2 1, -1 ° 1, 1 -1 1) directions respectively and loaded in the [0 ° -1] direction. There are 12 possible slip systems in FCC crystals, which can contribute in the deformation process. Table 1 below lists the 12 slip systems as combinations of the slip planes and slip directions. Figure 7 shows that the dislocation density histories are very sensitive to crystal orientation. This may be attributed to the fact that the type, number and interaction of slip systems differ from one orientation to another. The influence of crystal anisotropy appears to have a minor effect when compared to that of orientation. The dislocation activity on the slip systems before and after the load is applied is shown in Figure 8a and 8b.

l 0

__ Third(arisolropic)

--- • Third(isolropic) 1.62E+13

-Second(arisolropic)

••••••• Second(isolropic)

1.22E+13

8.20E+12

4.20E+12

2.00E+ll 0 2

Figure 7. The effect of crystal orientations and

1~~!!~~!!!!!!!!!!!!!!~ anisotropy on the ~ dislocation history.

3 6

time(nanosecond)

T bl 1 81" t a e lp sys ems 10 FCC tal me s. Slip Burgers Slip Plane Slip Burgers Slip Plane

System Vector System Vector

I [-1 1 0] (-1 -1 1) 7 [1 0 -1] (1 -1 1)

2 [-1 1 0] (I 1 1) 8 [1 0 -1] (1 1 1)

3 [0 1 -1] (-1 11) 9 [0 -1-1] (I-I 1)

4 [0 1 -1] (1 1 1) 10 [0 -1-1] (-1-11)

5 [-10-1] (-1 1 1) 11 [-I -I 0] (1-11)

6 [-10-1] (-1 -1 1) 12 [-1 -10] (-I I 1)

376 M Shehadeh, HM, Zbib, T. Diaz de la Rubia and V. Bulatov

. ~ :f; . I: ~ If

0""" a Soc;ond .....

) , .... " ...

Figure 8. The dislocation density distribution on the slips systems at different orientations ( a) before loadinl! (b) after loadinl!

In conclusion, mutliscale simulations were carried out to study the stress wave propagation and interaction with dislocations in copper single crystals. These simulations were designed to mimic the loading conditions in recent laser based experiments, where the pulse duration is a few nanoseconds. It is shown that avalanche of dislocations is a natural consequence of the interaction between dislocations and stress waves. The results of our calculations show that dislocation density is proportional to strain rate and pulse duration; however, the dislocation microstructure is controlled mainly by strain rate. Dislocation micro bands coincident with the {Ill} planes were formed at strain rate larger than 106 Is, whereas, dislocation entanglements were formed at strain rates lower than 106 Is. Deformation process was found to be very sensitive to crystal orientation. However, the effect of crystal anisotropy was found to have a minor effect of the wave profiles at ultra-high rates of deformation. More information can be found in a forthcoming article by the authors.

ACKNOWLEDGEMENT

The support of Lawrence Livermore National Laboratory to WSU is greatly acknowledged. This work was performed, in part under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory (contract W-7405-Eng-48). Fruitful discussions with Dr. John Hirth are greatly appreciated.

REFERENCES

Clifton, R.O, 2000. Response of materials under dynamic loading. International Journal of Solids and Structures, 37: 105-113.

Multiscale analysis of dynamic deformation in monocrystals 377

Cilfton R. 1., Bathe, N, 1999. Bridging length scales in dynamic plasticity simulations. In: Furnish, M. D., Chhbildes, L. C., Hixson, R. S., (Eds.), shock Compression of Condensed Matter, Snowbird Utah.

Coffey, C. S, 1992. Dislocation microscopic approach to shear band formation in crystalline solids during shock or impact. Shock Waves and High Strain Rate Phenomena in Materials, Meyers, M. A., MUff, L. E., Staudhammer, K. P., (Eds.), Marcel Dekker.

Hayes, D., Hixson, R. S., McQueen, R.G, 1999. High Pressure Elastic Properties, Slid-Liquid Phase Boundary and Liquid Equation of State From Release Wave Measurements in Shock Loaded Copper. In: Furnish, M. D., Chhbildes, L. c., Hixson, R. S., (Eds.), shock Compression of Condensed Matter, 1999 June 27- July 2 in Snowbird Utah.

Hirth 1. P., Zbib, H. M., Lothe, J., 1998, Forces on high velocity dislocations. Modeling Simlu. Mater. Sci. Eng. 6, 165-169.

Horstemeyer, M. F., Baskes, M. I., Godfrey, A., Hughes, D. A., 2000. A large deformation atomistic study examining crystal orientation effects on the stress strain relationship. International Journal of Plasticity. 18,203-229.

Horstemeyer, M. F., Baskes, M. I., Plimpton, S. J., 2001. Length scale and time scale effects on the plastic flow offcc metals. Acta Mater. 49,4363-4374.

Kadau, K., Germann, T. C., Lomdhal, P. S. Holian, B., 2002. Microscopic view of structural phase transitions induced by shock waves. Science. 296, 1681-1684.

Kalantar, D. H. et a1.2001. Laser driven high pressure, high strain-rate materials experiments In: 12th bienneial International Conference of the APS Topical Group on Shock Compression of Condensed Matter, 24-29, in Atlanta, Georgia.

Kanel, G. I., Razorenov, S. V , Baumung, K., Singer, J., 200l.Dynamic yield and tensile strength of aluminum single crystals at temperature up to the melting point. J. App. Phys. 90, 136-143.

Kuhlmann-Wilsdorf, D., 2001. Q: Dislocations structure-how far from equilibrium? A: Very close indeed. Material Science and Engineering, A315: 211-216

Loveridge-Smith, A., 200l. Anomalous elastic response of silicon to uniaxial shock compression on nanosecond time scales. Phys. Rev.Lett, 86(11),2349-2352.

Mayer, G., 1992. New directions in research on dynamic deformation of materials. Shock Wave and High strain Rate Phenomena in Materials, M. Meyers, L. MUff, and K. Staudhammer, (Eds.) Marcel Dekker.

Meyers, M. A., 1994. Dynamic Behavior of Materials, John Wiley & Sons, INC.

Meyers, M. A.,et a!., 200l. Plastic Deformation in Laser-Induced Shock Compression of Monocrystalline Copper. In: Furnish, M. D., Thadhani, N. N., Horie, Y., (Eds.), shock Compression of Condensed Matter, Atlanta, Georgia.

Mogilevskii, M. A., Bushnev L. S., 1990. Deformation structure development in Al and Cu single crystals on shock-wave loading up to 50-100 GPa. Combustion, Explosion, and Shock Waves. 26, 215-220.

378 M. Shehadeh, HM., Zbib, T. Diaz de la Rubia and V. Bulatov

Naser, S., 1992. Dynamic defonnation and failure. Shock Wave and High strain Rate Phenomena in Materials, M. Meyers, L. Murr, and K. Staudhammer, (Eds.) Marcel Dekker.

Rivas, J. M., Quinones, S. A., Murr, L. E., 1995. Hyper velocity impact cratering: microstructural characterization. Scripta Metallurgica et Materialia, 33(1),101-107.

Smirnova J. A., Zhigilei L. V., Garrison B. 1., 1999, A combined molescular dynamics and finite element method technique applied to laser induced pressure wave propagation. Computer Science Communications, 118, 11-16.

Rhee, M., Zbib, H. M., Hirth, J.P., Huang, H. & de La Rubia, T.D., 1998. Models for long/short range interactions and cross slip in 3D dislocation simulation of BCC single crystals. Modeling and simulations in Maters.Sci. & Eng. 467-492.

Weertman, 1., 1981. Moving dislocations in a shock front. Shock-Wave and High strain Rate Phenomena in Metals, M. Meyers, and L. Murr, (Eds.) Plenum press, New York,

Wright, R. N., Mikkola, D. E., LaRouche, S., 1981. Short Duration Shock Pulses as a Tool to Study the Time Dependence of Plastic Defonnation." In: Meyers, M. A., Murr, L. E., (Eds.), Shock Waves and high Strain Rate Phenomena in Metals, Plenum, New York.

Zbib, H. M., Rhee,M.& Hirth, J.P., 1998. On plastic defonnation and dynamics of 3D dislocations. Int.1.Mech.Sci. 40,113-127.

Zbib, H. M., Diaz de la Rubia, T., 2002. A multiscale model of plasticity. International Journal of Plasticity, 18, 1133-1163.

Zbib, H. M., Shehadeh, M., Khan, S.M, and Karami, G., 2003. Multiscale Dislocation Dynamics Plasticity. International Journal for Multiscale Computational Engineering.

MICROIMESO-MODELING OF POLYMERIC COMPOSITES WITH DAMAGE EVOLUTION

F. Ellyin, Z. Xia and Y. Zhang

Department of Mechanical Engineering. University of Alberta Edmonton, Alberta, Canada T60 20

Abstract: In the analysis of fiber-reinforced composites a structure composed of micro­cells can be defined which could represent the periodic nature of material. In this paper a three-dimensional multi-cell volume element is analyzed by viscoelastic finite element method. The effect of viscoelastic matrix, time­dependent behavior of residual stresses arising from the manufacturing process, and damage initiation and propagation due to externally applied loads are presented.

Key words: micro/meso modeling, polymer composites, viscoelasticity, damage evolution

1. INTRODUCTION

A class of materials which are gaining increasing industrial applications are polymers and polymer composites. A sub-class is epoxy polymers and fiber-reinforced epoxy composites.

Epoxy resins are cross-linked polymers and partially crystalline. Because of complexity of the spatial arrangement of molecular chains of these polymers and disorders contained therein, the constitutive modeling of the epoxy resins has been mostly based on a rheological and/or phenomenological formulation [1,2].

In fiber reinforced polymeric composites, a structure composed of unit cells can be defmed which represents the periodic nature of material. The scale of cells in this structure is of the order of the fiber diameter, generally in f.Drl. We will defme this scale as the micro-structure and will analyze a

unidirectional reinforced laminate, using a three-dimensional micro-cell. A further higher scale can be constructed by the homogenization of fibers

through the lamina thickness into strands embeded in the matrix in a meso-

379

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 379-388. ©2004 Kluwer Academic Publishers.

380 F. EUyin, Z. Xia and Y Zhang

scale (of the order of 100 J.lI1l). This will be demonstrated by analyzing a

cross-ply laminate by a three-dimensional multi-cell volume element. The analysis of the micro/meso-scale volume elements is carried out by finite element method (FEM). It predicts the stress and strain at the fiber, matrix and their interface, between the layers, as well as the overall (macroscopic) mechanical behavior.

These micro/meso/macro-analyses are based on the known properties of the constituents (fiber and matrix) and the assumption that the composite material posses a periodic structure. The epoxy matrix is represented by a nonlinear viscoelastic model whereas the fiber is assumed to remain elastic.

Damage initiation and growth in cross-ply laminates is also studied. The effect of viscoelastic matrix and the resulting time-dependent behavior on the evolution of residual stresses arising from the manufacturing process, and damage initiation and propagation due to externally applied load are presented.

2. MATERIAL MODELS FOR THE FmER AND MATRIX

The glass fiber is assumed to remain linear elastic with Young's modulus E=72.4 GPa, Poisson ratio, ~0.22 and coefficient of thermal expansion (CTE) equal to 5xl0-6/°C. The epoxy matrix is modeled by a nonlinear viscoelastic constitutive relation recently developed by Xia and Ellyin [1]. For the sake of completeness, only a brief description will be given below.

The multiaxial constitutive equations are summarized below: {et}={ee}+{ee} (1)

{ee}= i:(ai[A]cr~-I{O"}-bi{eci}) (2) i=1

{o-}= E[A]-I {ee} (3)

In the above, {et }, {ee} , {ee}, {o-} are the total strain-rate, elastic strain-rate, creep strain-rate and stress-rate vectors (each having six components),

respectively. 0" eq = ~2sijsij /3 is the von Mises equivalent stress and

S ij = 0" ij - oijO" kk 13 is the deviatoric stress tensor. The matrix [A] is related to

the value of Poisson's ratio. Although any number of n can be chosen in eq. (2), it was shown in Ref.

[1] that n=2 will suffice. Based on test results, the material constants defined in eq. (1) to (3) for the epoxy matrix were determined as follows:

al =5xlO-8(MPa)-2, al =2, b l =0.01,

a2 =lxlO-8(MPa)-2, a2 =1, b2 =lxlO-6 ,

E=2600MPa, v=OA andCTE=63xl0-6 rC.

Micro/meso-modeling of polymeric composites with damage evolution 381

3. MICRO-MECHANICAL MODELING

The manufacturing process of polymer composites involves heating the material to a cure temperature, holding it at this temperature until it begin to solidify and then cooling it to ambient temperature. During this cooling process residual stresses are generated inside the composite because of the mismatch between the coefficients of thennal expansion of the fiber and matrix. However, due to the viscoelastic property of the epoxy matrix, these residual stresses decrease through a process of stress relaxation. In general, experimental methods to measure residual stresses are complicated and the results have been rather difficult to interpret [3]. In this section we will detennine the residual stresses and their evolution through a three­dimensional micro-mechanical analysis.

A number of investigators [4,5] have used a micro-mechanical model to study the generation of residual stresses in metal matrix composites. A few researchers, e.g. [6], have extended this methodology to polymer-based composites, but the evolution of residual stresses has not been addressed.

(a) (b)

Aa1IY, .. . * -1.1 . . ...

~ -l. .. . ~" .---. -, . 1.1$ 1--..1 - .. . . '1 B ."un c::J 5 . In: ,......., l.I. . l.t:, L.........II 1.1J . IU

~ !!::::

Fig. 1. Micro-mechanical modeling of a unidirectional laminate: (a) Micro-cell; (b) FEM mesh and longitudinal residual stress distribution

Figure l(a) represents a schematic of a unidirectional laminate in which a cell portrays the periodic structure of the laminate. Due to the symmetry in X­Y plane, only 114 of the cell needs to be analyzed. The material properties are independent of the Z-direction; thus, only one layer of finite elements is required when a generalized plane-strain boundary condition is applied to the two boundary surfaces in the Z-direction. The model length in Z-direction will depend on the FEM mesh in the X-Y plane, here it is taken to be 114 of dimensions in the X and Y directions. With the origin at the center of the fiber, the symmetric conditions at surfaces, X = 0, Y = 0 and Z = 0, are enforced by

u(O,Y,Z) = v(X,O,Z) = w(X,Y,O) = 0

(4) The boundary conditions on the other three surfaces, X = 1, Y = 1 , and

382 F. Ellyin. Z. Xia and Y. Zhang

Z = 0.25 , indicating periodicity in all directions, are given by:

u(l, Y,Z) = u(l,O,O), v(X,l,Z) = v(O,l,O), w(X,Y,0.25) = w(0,0,0.25) (5)

The above boundary conditions indicate that each plane remains plane during the deformation, but is free to displace in its own plane.

In the micro-mechanical analysis, it is assumed that the overall strain and stress are dermed by the average values over the volume of the cell [7, 8],

(6)

where V is the volume of the cell. By using Gauss's theorem, the average strain in the cell can be expressed as an integration around the boundary surfaces[7]. The micro-cell in Fig. 1 (b) was analyzed by the finite element method. The interface between fiber and matrix was meshed by rmer elements. The fiber volume fraction was taken to be 52.5% which is representative of industrial applications. The thermal residual stresses generated during cooling from T = 149°C to T = 23°C are determined. At the cure temperature the laminate begins to gel: thus, it is assumed to be stress free at this temperature.

The cooling process is simulated by applying incremental temperature decrease at a specified rate. The result of the finite element analysis at a cooling rate of 1.4°C / min indicated a variation for the longitudinal stresses, U zz, from a maximum tensile value of 26.41 MPa in the matrix near the

interface to a maximum compressive value of 19.34 MPa in the fiber, see Fig. 1 (b). In the case of transverse stress, u xx' the maximum tensile stress of

23.92 MPa occurred in the matrix near the interface, and the maximum compressive stress of 19.20 MPa was in the fiber near the interface. Following cooling to room temperature, creep and relaxation occur due to the viscoelastic properties of the epoxy matrix. The change of the stress/strain in the matrix causes the change of the stress/strain in the elastic fiber in order to reach a new equilibrium state for the laminate. The numerical analysis was continued for 2600 minutes (~3 hrs) while the temperature was held at 23°C. Figure 2 shows the relaxation of these maximum stresses with time. It is seen that after about 43 hrs, the stresses tend to an asymptotic value of approximately 2 MPa .

Micro/meso-modeling of polymeric composites with damage evolution 383

30

10

-;. -10 0"

-·-a xx

-A-a zz

-30 -r-.-,-..,......,,,....,.-.....-~,.....-,-..-,-.-,.......-,-..---,-..-,.......-,-~ o 200 400 600 800100012001400160018002000220024002600

t (min.)

Fig. 2. Evolution o/the maximum residual stresses in a unidirectional laminate.

4. MESO-MECHANICAL MODELING

Figure 3 shows a meso-mechanical representation of a thick cross-ply [0/90]n laminate with the FEM mesh. Each unidirectional layer is represented by a unit cube with a single fiber having the same fiber volume fraction as the ply. Two stacked unit cubes with fibers in perpendicular directions can represent the periodic structure of [0/90] lay-up. It is assumed that there are many [0/90] plies alternately arranged in the thickness direction. Therefore, periodical boundary conditions are applied in all three coordinate directions for the meso-mechanical model. Only 118 of the model needs to be analyzed due to the symmetry of geometry and load conditions. In this analysis, the fiber volume fraction was taken to be 52.5%, which is the same for each layer in the laminate.

In a unidirectional lamina there can be many fibers dispersed across its thickness. The use of a single unit cube to represent such a non-monofilament layer neglects fiber interaction within a lamina. Thus, in such a case the representation is strictly a 'meso' homogenization process for each lamina.

Similar to the unidirectional laminate, symmetric boundary conditions on three planes containing the origin and periodic conditions on the remaining boundary surfaces were applied. Also, a uniform temperature distribution was assumed in the analysis, and the same material constants as the previous example were used. The curing temperature of 149°C is much higher than that of ambient temperature, 23°C, therefore it may be more appropriate to incorporate temperature dependency for the material constants. To investigate the possible bounds for the residual stresses, two sets of material constants were employed in the present study. First, the values at room temperature were used for the entire temperature range similar to the previous analysis of a unidirectional laminate. Second, the temperature-dependent relations were

384 F. Ellyin, Z. Xia and Y. Zhang

used in the analysis. Due to lack of data in high temperatures, an exponential function was assumed:

E(T) = 2600e-k(T-23) ; a) (T) = 5xIO-8 ek(T-23) ; a2 (T) = Ix 10-8 ek(T-23) (7)

with k=-0.0183. The above relations indicate that at the curing temperature of 149°C, the elastic modulus, E is 1110 times, and the constantsa)oa2are 10

times of their values at ambient temperature.

The residual stress, (j zz' distribution induced by cooling from 149°C to 23°C

at a cooling rate of 1.4°C/min for temperature dependent material properties is shown in Fig. 3 (b). It is seen that the maximum tensile stress is 21.36 MPa in the matrix, and the maximum compressive stress is 40.71 MPa in the fiber. For the [0/90]n cross-ply laminate, the stress component u xx has the same

maximum values and distribution as u zz' When the material constants at

ambient temperature are used for the entire temperature range, much higher values of residual stresses are obtained. These are denoted as "1.4,T.I." in Fig.4. For the same cooling rate, the maximum tensile stress of 42A2MPa and compressive stress of 77.04 MPa were obtained. The results from the temperature-independent constants overestimated the residual stresses, because the modulus of an epoxy material increases with the temperature decrease.

Following cooling to room temperature, the numerical analysis was also continued for 2600 minutes (:::43 hrs). Figure 4 shows the evolution of the maximum tensile stress in the matrix and the maximum compressive stress in the fiber. It is seen that both tensile and compressive stresses decrease with time and tend to an asymptotically small value. For example, for the temperature dependent material constants, these values for the tensile and compressive stresses are 3.44MPa and .5.70MPa, respectively. The results of the temperature-independent constants are also depicted in FigA, which indicate a similar trend but with slightly higher asymptotic values. Note that if matrix is assumed to be linearly elastic, very high residual stresses will be predicted as shown in Fig. 4.

Micro/meso-modeling of polymeric composites with damage evolution 385

.. ... D' .. ' .. ... .. ...

a) ,/,/ L' L'L (b)

A:. I I I'

IG mi • 1m.

~f( ~ ~ • CD ele I •• • Ie •• •• •• j I"

I •• • I I' e l!! • • 01G •• Ot! • •• v

Figure 3. Meso-mechanical modeling o/the cross-ply laminate,

. :~ : ;~: o -1I . 'll o _!l . 11I o -U.UI o -1 . ln

DD . nllil

' . 111

D. 14•fll U .,U

(a) Schematic representation, (b) Finite element mesh and residual stress distribution

co c: :c .a '0, c:

60

~-100~------------------------------~ o 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600

Time (min.)

Cooling Rates

(,el min.) - . -1,4, T.I, - - 1.4,T.D. -o-Elastic

Fig. 4. Residual stress relaxation predicted by temperature-independent material constants (T.!.) and temperature-dependent ones (T.D.).

5. FREE EDGE EFFECT AND DAMAGE ANALYSIS

The micro/meso-mechanical modeling can also be effectively used in damage analysis of composites. The FEM was used to study the free edge effect and the damage initiation and propagation in a cross-ply laminate. Numerical tests on this model have shown that the free edge effect is localized near the free surface with a depth less than half of the fiber diameter [9]. Therefore, in the Z-direction, the FEM model now contains a full diameter of the fiber. A traction free boundary condition was applied to the surface Z=2 instead of the plane-remains-plane condition. A damage criterion and a post-damage constitutive model were also introduced in the damage analysis. The maximum principal strain was used as the damage criterion for

386 F. Ellyin, Z. Xia and Y. Zhang

matrix cracking, i.e. the matrix would crack when cl ~ c j = 0.03. Upon

reaching this damage state, the following post-failure constitutive relation governs the element response,

{a} = ~ [A ]-1 {Et }-17{O} (8)

where P is a small number (=10-4) and 17 is a constant (=0.01). The above

constitutive relation reduces the stress vector to a very small value in a short duration of time ( {O'} = 0 is the limit case) and thus, the matrix element is no

longer able to carry a load.

Damaged elements

Free edge surface

Interior surface

&0 =0.6%

t = 60sec.

&0 =0.6%

t = 124sec.

&0 =0.6%

t = 700sec.

Fig. 5. Damage mll,ratumjrom mechanical load and evolution with the strain held at 0.6%

A monotonically increasing displacement with a constant strain rate of 10.3

is applied along the X-direction. Figure 5 shows the damage initiation due to a mechanical loading and its evolution with time when the applied displacement is held constant. As the applied strain reaches 0.52%, damage initiates near the free edge at the interface and upon further loading to 0.6%, damage also starts in the matrix along 90° fiber direction but is still localized near the free edge. The other side of the model representing the interior surface is also shown in the same figure. It is seen that for an applied global strain, damage first initiates on the free edge surface of the cross-ply laminate. In addition, it is noted that the free edge effect is localized. When the applied

Micro/meso-modeling o/polymeric composites with damage evolution 387

global strain of 0.6% is held constant, one sees that although the initial damage was confined to the near free edge area, damage in the matrix propagates across the full width of the laminate after 124 seconds. At 700 seconds, the damage reaches a stable state. Note, however, that the interface damage is still localized at the free surface area.

6. CONCLUSIONS

A micro/meso-mechanical modeling methodology has been developed for fiber-reinforced laminated composites. For a unidirectional laminate, a micro­scale cell model can be used as a periodic element. For a cross-ply laminate, a meso-scale multi-unit cell model can be constructed which represent the periodic structure.

Viscoelastic fmite element analyses were carried out on these micro/meso models. The analyses predict the residual stresses/strains arising from the manufacturing process as well as their evolution. It was found that due to the time-dependent nature of the viscoelastic matrix, the residual stresses relax and asymptotically approach small values with time.

The free edge effect, damage initiation and propagation in the cross-ply laminates were also studied. The analysis indicated that damage initiates near fiber/matrix interface on the free edge. The transverse matrix crack also initiates from the free edge. However, the transverse matrix crack propagates faster to the interior area of the laminate either under a further increase of the applied load or when the global load is held constant.

REFERENCES

[I] Xia, Z. and Ellyin, F., "Time-dependent Behavior and Viscoelastic Constitutive Modeling of an Epoxy Polymer", Polymers & Polymer Composites, 6, 1998, 75-83.

[2] Hu, Y., Xia, Z. and Ellyin, F. "Mechanical Behaviour of an Epoxy Resin under Multiaxial Loadings, Part II: Comparison of Viscoelastic Constitutive Model Predictions", Polymers & Polymer Composites, 8, 2000, 157-166

[3] Majumdar, S., et aI., "Application of Neutron Diffiaction of Measure Residual Strains in Various Engineering Composite Materials," ASME Journal 0/ Engineering Materials and Technology, 113,1991,51-58.

[4] Chandra, N., Ananth, C. R. and Garmestani, H., "Micromechanical Modeling of Process­Induced Residual Stresses in Ti-24AINb/SCS Composites," Journal 0/ Composite Technology & Research, Hi, 1994,37-46.

[5] Bigelow, C. A., "Thermal Residual Stresses in Silicon-CarbidelTitanium [0/90] Laminate," Journal o/Composite Technology & Research, 15, 1993,304-310.

[6] Gardner, S. D., Pittman, C. U. and Hackett, R.M., ''Residual Thermal Stresses in Filamentary Polymer-matrix Composites Containing an Elastomeric Interphase," Journal o/Composite Materials, 27, 1993,830-860.

[7] Aboudi, J., Mechanics of Composite Materials, A Unified Micromechanical Approach. Amsterdam: Elsevier Science Publishers, 1991

[8] Sun, C. T. and Vaidya, R. S., Prediction of Composite Properties from a Representative Volume Element. Composites Science and Technology, 56, 1996, 171-179.

388 F. Ellyin, Z. Xia and Y. Zhang

[9] Ellyin, F., Xia, Z. and Chen, Y., "Viscoelastic micromechanical modeling of free edge and time effects in glass fiber/epoxy cross-ply laminates", Composites: Part A, 33, 2002, 399-409.

AN ALTERNATIVE APPROACH FOR HETEROGENEOUS MATERIAL BEHAVIOUR MODELLING

O. Bouaziz, P. Buessler

IRSID, ARCELOR Group R&D, Voie Romaine-BP303230, 57283 Maizieres-les-Metz Cedex, France

Abstract: Assuming an equi-incremental mechanical work in each constituent in stress-strain mixture law (i.e 0\. dEl = 02' dE2), this article focuses on the interest of this approach

trough comparison with miscellaneous experimental results and with others modelling methods as Taylor, Sachs, self-consistent or 3D finite element modelling.

Keywords: Heterogeneous material, Mechanical properties, Modelling, Mechanical-work

1. INTRODUCTION

1.1 General framework

Mechanical behaviour modelling of the heterogeneous materials is one of important field of physical metallurgy. The available solving methods can be classified in three main families. The first one is based on classical mixture rule widely used because very simple but it is not realistic or needs arbitrary fitting parameter to improve the prediction. The second one is based on homogeneization techniques like self-consistent method. The third one is based on finite element description of either a unit cell or a representative microstructure. It is able to be more accurate but much more difficult and heavy to perform, and therefore less used. To keep easy running mixture law modelling but to avoid any fitting parameter and improve prediction accuracy, an equi-incremental mechanical work has been assumed in each constituent in stress-strain mixture law.

389

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 389-396. ©2004 Kluwer Academic Publishers.

390 0. Bouaziz, P. Buessler

1.2 Easy running method

Despite the unrealistic iso-strain assumption (the strains in each constituent are taken equal so-called Voigt assumption in elasticity and Taylor assumption in plasticity) and a weak predictibility the simple stress mixture law is so far the most widely used way to quantify the heterogeneous material behaviour. In the case of two constituents, this law is expressed as : o(e) = (l- F).ol (e) + F.02(e) (1)

Where F is the volume fraction of the second phase, 8 is the strain and a, ah a2 are respectively the flow stress of the aggregate and of the two constituents.

The prediction accuracy becomes very degraded when the constituents exhibit very different behaviour laws (strength and ductility) [1,2,3]. To improve this previous approach, authors proposed to introduce a phenomenological non linear function of volume fraction in equation one. The first one was Gladman using a power law volume fraction function [3] : a(E) = (l- Fn).OI (E) + pn .02 (e) (2)

A second approach keeps the linear volume fraction mixture law for stress but applies the same mixture law to the strains. Such an intermediate mixture law was first proposed by Tomota&al. [4] : a(E) = (1- F).al (EI) + F.02 (E2) (3a) E = (1- F).EI + F.E2 (3b)

where 81 is the average strain in the first constituent and 82 the average strain in the second one.

This modelling seems more realistic than the previous one but needs always a fitting parameter. Indeed, it should be noted that Eq. 3 does not indicate anything about the absolute amount of the stress and strain transfer. That is the reason why several authors have proposed to fix the following quantity [2,5]

l02 - OIJ = ~ 82-E)

(4)

where /3 is an arbitrary fitting constant with 0 ::; ~ ::; 00 (figure I).

For Dual-Phase steels /3 can be fitted at values in the range between 3000 and 6000 MPa [2] and values in the range between 4500 and 11000 MPa [5]. The greater the value of /3, the more closely does the situation approach iso-strain condition expressed by Eq.l. The iso-stress condition is given for /3=0. The exact nature of the dependence of /3 with composite arrangement and microstructure is clearly not yet well known.

In order to avoid the need for such an arbitrary fitting parameter, a new assumption is proposed in this work. For a disordered microstructure, whatever material state mechanical work increment is taken equal in each constituent. This equality is expressed as following:

An alternative approach for heterogeneous material behaviour modelling 391

al' dEl = a2' dE2 (5)

In the following this assumption is called IsoW.

2. APPLICATION TO ELASTICITY

Assuming the constituents obey to Hook elastic law: a) = E).E) (6)

a2 = E2·E2 where El and E2 are respectively the Young modulus for the first and the second constituent. IsoW assumption written in Eq. 5 gives immediately:

~= [fu (7)

E2 VEt Combining Eq.3, 6 and 7 the equivalent Young modulus of the mixed material is given by :

(1- F).EI'.,JE;. + F.E2'.JE; (8) E=~~~~=---~~

(1- F) . .,JE;. + F . .JE;

In order to evaluate this approach by comparison to more classical theories, results of Eq.8 have been drawn on figure 2 showing a comparison with upper bound model (Voigt), lower bound model (Reuss), Self­consistent modelling and Hashin-Shtrikman bounds assuming isotropic materials and spherical inclusions. Iso W predictions appear to be always between theoretical Hashin-Shtrikman bounds and give very close results than self-consistent approach up to a 40% volume fraction of the more rigid material.

------,'i' ........ - C<>nI'lltutM2

".,.,' : ,.~'<., : " : ", :

,/ ! ...... j

Ccnttitulnt 1

Son

Figure 1. : Graphic illustration of stress­strain mixture law and ~ parameter.

1000 -,""W

· ·'SeIf~.nI . ,. . ""_ ...... . """""' ...... 700 - "-......

···-I.O¥wttboufld

-: 550 £

"' ' 00

2"

100 0.2 0.- 0.6 0.8

Figure 2. : Calculated Young modulus versus volume fraction with Iso W assumption, self­consistent, Hashin-Shtrikman bounds, upper and lower bound models (E1=lOO GPa and E2=lOOO GPa).

392 0. Bouaziz, P. Buessler

3 APPLICATION TO MONOTONIC PLASTICITY

3.1 Experimental validation

In order to validate Iso W approach through comparison with experimental results, literature studies including measurements of the stress-strain curve for each constituent and for an aggregate and of local strains have been selected. The first one is dedicated to a composite where 18% silver is in inclusion in an iron matrix [6] especially developed by powder metallurgy to improve the classical self-consistent approach in plasticity [7]. Behaviour of iron, silver and composite have been measured and local average strains along the tensile axis in the two phases have been experimentally investigated and computed using self-consistent scheme by authors. Experimental data previously determined with 50% silver [8] have been reported too on figure 3. Flow stresses and volume fraction of each constituent have been included in Iso W model. Figure 4 and 5 illustrate model predictions of the composite behaviour and the local strains. The modelling is always in well agreement with experimental results.

The second study is related to a Ti-Mn alloy exhibiting by by suitable thermal treatment either a fully Tia either a fully Tip or a duplex Tia -Tip microstructure [9,10]. As shown on figures 6 and 7, if single phase behaviour is known Iso W approach captures nicely the stress-strain curve of the composite structure and the ratio q between the local strain in soft phase and the local strain in the hard phase.

<0 0.. :::!:

300

-;; 200

'" ~ Ci5

100 ---... -;:;;-'.'-/

---'-1 ---- ·_·_ ·1

_._ .• ____ ._ ._ ._._. ________ ___ ._.1

.~-------~~. -,

.. ---_.­_._._.-- --------_ ._--

- iron silver

• iron+18%silver .. iron+50%%silver

-lsoW modelling - lsoW modellin

O ~------~------~--------~------~------~

0.00 0.05 0.10 0.15 0.20 0.25

Global strain

An alternative approachfor heterogeneous material behaviour modelling 393

Figure 3 : Comparison between experiment and IsoW modelling for the flow stress of iron­silver composites

... r---------------------~ --- Sih ... 05CI - Noses ........... ........

I

~~~D~--~~~O----~,~~--~U7.A~--~ ... OIobol_<'II»

.. ,

Figure 4: Experiment local strains and self- Figure 5 : IsoW modelling results for Fe-consistent modelling computation [11] Ag\8% composite

,,..

0100 · --------. - - - .-. ---

1 == 1 ~ ¥--_-_-_-__ - -__ - _-,...:-'====-=:" .:=;::;;:':-:='-="":!::::;-===:L. • oo 1)..(12 OJ)') .... .... --Figure 6 : Comparison between experiment

and Iso W modelling for the flow stress Ti~ + 1 7% Tia composite

3.2 Discussion

f# 1,.

o.+-~----__ ~ __ --__ ~ __ ~ Q,oQIXI 0.008 0.010 0015 0.020 o.OU a.hO .0.03$ 0040 O.Ot$

.- ..... Figure 7 : Comparison between experiment and Iso W modelling for the ratio q between Tia average strain and Tip average strain given by Iso W modelling.

In order to discuss futher the interest of the Iso W assumption in plasticity, comparison with finite element computations have been performed for a macroscopic uniaxial tensile strain state. Zebulon® code with a microstructure mesh tool have been used as illustrated on figure 8. A two constituent mechanical behaviour has been investigated assuming both material follow a Mises yield locus and obey to an Hollomon plastic law:

0"1 = KI.&tl (9)

0"2 = K2·&l2

In this case IsoW assumption written in Eq. 5, gives the necessary relation between average strain in the second constituent and the average strain in the

394 0. Bouaziz, P. Buessler

first one assuming the total plastic work (LO"ij.de .. ) is totally given by the •. 1J

plastic work in tensile direction: 1 (10)

_ (KI 1 +n2 l+nl)l+n2 E2- -·--·EI K2 l+n\

I,J

The material parameters are summarized in table 1. They have been chosen to have obvious different mechanical behaviour. Simulations have been performed for respectively 11, 30 and 50% volume fraction of hard phase using a microstructure meshing [11] as shown on figure 8. The output especially studied are : the flow stress of the mixture (figure 9.a), local average strain in tensile direction in each phase (Figure 9b,c,d), the ratio of plastic works in tensile direction between the hard phase and the soft one (figure 10) and the ratio between the total plastic work and the plastic work in tensile direction (figure 11). IfFEM computations are taken as reference it is clear that :

the results of Is oW are always close for stress-strain curves, Iso W modelling tends to overestimate the strain heterogeneity between the soft and the hard phase especially when hard phase volume fraction increases, Iso W assumption is realistic because figure 10 indicates the ratio of plastic works in tensile direction between the hard phase and the soft one is close to unity and because around 90% of the total plastic work is given by the plastic work in tensile direction.

Figure 8. : Microstructure meshing used (30% hard phase in black)

constituent I-soft phase

2-hard phase

E (GPa) 210 210

v 0.3 0.3

K(MPa) 750 1250

n 0.5 0.1

Table 1. : Material parameters used for investigation.

An alternative approach/or heterogeneous material behaviour modelling 395

... 0"' 0.' 0." • .2

. .. ' .1 ... " GIOboI ..... """"-

a. Flow stress b. Local strain for II % hard phase volume fraction

.. , ...

1 •.. !

....

0 .. .. """"' .....

.. ..

.. t 0\

!

c. Local strain for 30% hard phase volume d. Local strain for 50% hard phase volume fraction fraction

Figure 9. : Comparison between FEM and IsoW results

I." ,-' . - . - - - - - - - - - - - - - - - - -

i "" s 1

i"" . - . - . - . - . - -- --I :: r --0.5 ____ 0 _____ • ___ . ---O~ _._. - - - - - - - - - - - - - - - - -

o ~--~-----------------• 0015 0.' CI .1 15 o. ---

• .2>

1,2 ---- .-.-.-.-. -~---- ... g i t ,1S

l, ,~~'_ I~-----------------------o 0." 0.' _ .. aIn 01' ...

Figure 10 : Ratio between plastic work Figure I I : Ratio between total plastic work increment in each phase in tensile direction increment and plastic work increment in

tensile direction

4 CONCLUSION

We have proposed to assume an equi-incremental mechanical work in each constituent in intermediate mixture law. This approach has shown :

a more realistic description than simple mixture rule on stress, the non necessity of any arbitrary fitting parameter for intermediate mixture rule on stress and strain, the respect of theoretical classical bounds of homogeneization methods, an improvement of the ratio between accuracy and sophistication of heterogeneous material behaviour modelling method.

396 0. Bouaziz, P. Buessler

It would be interesting now to check if our assumption doesn't violate the more restrictive upper bound than the Taylor bound [12,13] and to compare Iso W to multi-axial FEM computations.

References

[I] Fishermeister, H., & Karlsson, B. (1977). Z. Metallkde 68, 311, [2] Karlson, B., & Linden, G. (1975). Mat. Sc. Eng. 17,209. [3] Gladman, T., Mc Ivor, 1.0., & Pickering, F.B. (1972). nSI 210, 916. [4] Tamura, I., Tomota, Y., & Ozawa, H. (1973). Proc. 3rrl Int. Conf. on Strength of Metal and Alloys, Cambridge, vol. I , 611. [5] Sangal, S., Goel, N.G., & Tangri, K. (1985). Met. Trans. A 16,2023. [6] Bomert, M., Herve, E., Stolz, C., & Zaoui, A. (1994). Appl. Mech. Rev. 47, 66. [7] Berveiller, M., & Zaoui, A. (1979). J. Mech. Phys .Sol. 26, 325. [8] Le Hazif, R. (1977). Acta Met. 26, 247. [9] Sreeramamurthy, A., Margolin, H. (1982). Met. Trans. A 13, 595. [10] Sreeramamurthy, A., Margolin, H. (1982). Met. Trans. A 13,603. [l1]Barbe, B., Forest, S., Cailletaud, G., Decker, L., Jeulin, (2001) Int. J. Plast. 17,513 [12] Ponte Castaneda, P. (1991). J. ofMech. Phys. Solids. 39,45 [13] Gilormini, P. (1995).C.R. Acad. Sci. Paris, serie II, liS.

ON ANISOTROPIC FORMULATIONS OF THE ELASTIC LAW WITHIN MULTIPLICATIVE INELASTICITY

Carlo Sansour

School of Petroleum Engineering, University of Adelaide, Adelaide SA 5005, Australia

carlo.sansour@adelaide.edu.au

Abstract The paper addresses an anisotropic fonnulation of finite strain viscoplasticity. We focus on the fonnulation of the anisotropic constitutive law for the stress tensor. The essential ingredients of the theory are: 1) the multiplicative setting of inelasticity, 2) the fonnulation of an anisotropic elastic law based on a mixed­variantly transfonned structural tensors. A numerical example is included as well.

1. Introduction

Anisotropic response is a common phenomena of material behaviour. It arises as a result of the micro structure of the material and constitutes itself in the different constitutive laws describing the material behaviour. In the general inelastic material description, the constitutive laws split into two classes, namely, that of the thermodynamical forces, which may be derivable from a thermodynamical potential, and that of the evolution equations for the internal variables. Within multiplicative theories of inelasticity, the constitutive law for the stress tensor as a thermodynamical force plays a dominant role as it dictates to a wide extent the solution schemes to be adopted in numerical analysis. In fact, the treatment of the elastic constitutive law as an isotropic function allows for a considerable simplifications in the resulting numerical schemes and updating procedures.

The anisotropic formulation of the constitutive law for the stress tensor is faced from the very beginning with certain difficulties. On the side of the

397

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 397-404. ©2004 Kluwer Academic Publishers.

398 Carlo Sansour

theory, it is not evident of how the anisotropic formulation of the thermody­namical stress is to be accomplished. On the side of computations, the resulting multiplicative structures are in general very much involved. Dealing with the exponential map necessitate special technique and must be paid special atten­tion.

In this paper, we address the formulation of an anisotropic law for the stress tensor. The anisotropic formulation is based on the introduction of specific structural tensors defined first for a reference configuration. Under the action of the inelastic part of the deformation gradient, the structural tensors are then transformed and the elastic laws is then assumed to depend on those transformed quantities.

2. Basic relations of multiplicative inelasticity

For the description of the inelastic deformation we assume the well estab­lished multiplicative decomposition of the deformation gradient in an elastic part Fe and an inelastic part F p according to F = F eF p. For metals, the inelastic part of the above decomposition is accompanied with the assumption of incompressibility: Fp E 8L+(3, R). Here, 8L+(3, R) is the special linear group with determinant equal one: det(F p) = l.

On the base of the above decomposition we define the following right Cauchy­Green-type deformation tensors

(1)

The time derivative of the deformation gradient can be formulated as

F=lF, F=FL, (2)

where 1 and L are the left and right rate, respectively. Finally, based on the fact thatF p E 8L+(3, R), two inelastic rates can be defined. We confine ourselves to the right one given according to

(3)

which proves sufficient for our purposes.

On Anisotropic Formulation of the Elastic law witin Multiplicative Inelasticity

3. The constitutive framework of elastic anisotropy

3.1 General aspects

399

Consider the definition of the internal power W = T : 1, where T is the Kirchhoff stress tensor and 1 is defined in (2). The double dot product of two second order tensors a, b is defined by a : b = trab T, where tr is the trace operation. Using the material rate, the internal power takes W = 8 : L. The combination of both expressions for W together with (2h provides the definition of the material stress tensor 8: 8 = FTTF-T . The stress tensor 8 is the mixed-variant pull-back of the Kirchhoff tensor. Up to a spherical part and a sign it coincides with Eshelby stress tensor (see e.g.[3]). Accordingly, we refer to it as Eshelby-like tensor. We mention also that the symmetry condition imposed on T can be recast in terms of S to give ST = C-1SC.

We denote a typical set of internal variables by Z. The free energy function 7/J can be stated as 7/J = 7/J (Ce, Z). The localized form of the dissipation inequality for an isothermal process then reads

'D = T : 1 - Pref-¢ = 8 : L - Pref-¢ 2 0 , (4)

where Pref is the density at the reference configuration.

3.2 A first formulation of elastic anisotropy

The description of elastic anisotropy is carried out using the method of struc­tural tensors as developed by Rivlin, Spencer, Smith, Wang, and Boehler (see [2]). To be specific the developments are carried out for the case of orthotropy. Essentially the same results hold for any kind of anisotropy described by struc­tural tensors. A first formulation is carried out which gives motivation for a modification to be addressed in the next subsection.

A structural tensor is defined as the tensor product

M = VQ9V, (5)

where v is a privileged direction of the material in the reference configuration. In case of more than one privileged direction corresponding more tensors are introduced. The case of orthotropy is described by means of three structural tensors:

Mi = Vi Q9 Vi, i = 1,2,3 (no summation), Vi' Vj = Oij, (6)

where Oij is Kronecker's delta.

In the case of orthotropy, a complete irreducible representation of 7/Je(Ce) in given when the function depends on the following set of invariants which

400 Carlo Sansour

define the integrity basis (see e.g. [2]):

h = tr(MI Ce) , 14 tr(MI C~), h = tr(C~).

12 = tr(M2Ce), Is = tr(M2C~),

13 = tr(M3Ce), 16 = tr(M3C~),

(7)

The elaboration of (4) and the use of classical thermodynamical arguments provides an expression for the material stress tensor, which reads as follows:

'=' -.... -

In fact, (7) is the most general possible constitutive law for the thermodynam­ical force :=: in the case of orthotropy. The explicit evaluation of the expression depends on the choice of 'lj;e. We observe first that the quantity CCp1 is a ma­terial tensor. The quantity enters the formulation in the isotropic case as well. On the other hand it is not the constant structural tensors Mi which are present in the formulation but modified ones which are mixed-variantly transformed structural tensors: Mi = F~MiF~-1 . In fact this observation will motivate a second approach to anisotropy to be addressed in the following section.

3.3 A second formulation of elastic anisotropy

Modified structural tensors. Motivated by the observations in the last subsection we consider from the very beginning modified structural tensors according to

(no summation). (9)

The transformation is motivated in cases when the privileged directions are material and can be imagined to be affected also by an inelastic deformation.

In analogy with (7) we consider the following set of invaraints

II = tr(M!Ce),

14 = tr(M!C~), h = tr(C~).

12 = tr(~Ce),

Is = tr(M;C~),

The invariants can alternatively be rewritten as

13 = tr(M3Ce),

16 = tr(M3C~) (10)

Ii = tr(MiCp1C), Ii+3 = tr(Mi(C~IC)2), i = 1,2,3. (11)

On Anisotropic Formulation of the Elastic law witin Multiplicative Inelasticity

401

The following remarks are pertinent. The transformation (9) does not change the invariants of the structural tensor itself. The resulting invariants (11), with all quantities defined with respect to the reference configuration, strongly suggest that the mixed-variant choice is correct. It is also stressed that an alternative transformation rule such as F pMF~, which would preserve symmetry, cannot be a correct choice. In fact, the subsequent multiplication with Ce practically excludes inelasticity from the linear terms. Inelasticity would then appear as an effect of higher order.

Evaluation of the dissipation inequality. The inclusion of modified struc­tural tensors changes the form of the reduced dissipation inequality. We start again with the dissipation inequality (4), which is a general thermodynamical statement and as such independent of the kind of anisotropy under consider­ation. The evaluation of the new form of the free energy function leads first to

'T'I _ (,=, _ FTa'lj;(Ce , az)) . L + = . L _ 'lj;(aCe, Z) . z > 0 L/ - .... Pref F . ..... p Pref az -'

From (12) we first deduce

- FTa'lj; .::. = Pref aF' y _ _ 'lj;(aCe , Z)

- Pref az .

The reduced dissipation inequality has then the form

TJp := 2: : Lp + Y . Z 2: O.

Rewritten in terms of C e instead of F, the expression reads:

(12)

(13)

(14)

'=' = CF-1 [a¢(Ce, Z) + (a¢(ce, Z))T] FT-1 (15) .... Pref p aCe aCe P

Explicitly for the anisotropy under consideration, and with (11), we may elab­orate

8 Pref t, [~i (CMiCp1 + CCplMi)

+~ (CM.C-1CC-1 + CC-1M'CC-1 (16) a1 I p P P z P i+3

+CCplCMiCpl + CCplCCplMi)] .

Having established the expression for the material stress tensor 8, we pay now our attention to the formulation of the reduced dissipation inequality. In fact due to the dependency of the structural tensors on the inelastic deformation,

402 Carlo Sansour

the reduced dissipation inequality is modified. It is the modified stress tensor 5 which acts as thermodynamically conjugate to the inelastic rate Lp. The modified stress tensor S is given by the expression

5 = E + Pref t [:~ (MiCCpl- CCp1Mi) + z=l z

8'1j;e (MoCC-1CC-1 _ CC-1CC-1Mo)] (17) 81 z p p p p z·

i+3

Consequently, flow rules should be formulated in terms of S. This provides us with a natural access to a formulation with a plastic spin or, depending on its definition, a contribution to it.

3.4 Inelastic constitutive model

The formulation of the constitutive law for the thermodynamical forces is to be complemented with the formulation of the evolution equations for the internal variables. To be specific we cqnsider the inelastic model suggested in [1] and modified to fit into the framework of multiplicative inelasticity by Sansour and Kollmann in [4]. Major modification refers to the flow rule which now reads:

·-T Lp = cpv , v = 3 devS 2 II

J3 - -II = 2devE : devE.

For details the reader is referred to the mentioned literature.

4. Example

(18)

To conclude the treatment we present a numerical example, the extension of a thin specimen, to illustrate the applicability of the formulation stated above. The details of the numerical treatment are extremely involved and are to be presented elsewhere.

The geometry of the specimen loaded at its free end by a longitudinal force is shown in the Figure. The boundary conditions on the second end of the bar allow for a free contraction of the specimen. Here, only the left comer point is fixed. The specimen is modelled using 10 x 26 finite elements developed in [4]. Three computations are provided for the privileged direction of orthotropy given by the angle 0°, 60°, and 90°. The prescribed deformation velocity of the left top comer is 0.035 mml sec. The load-displacement curves are given in the Figure. The following material data is considered (see [4])

Do = 10000 lisee, Zo = 1150 N/mm2 , Zl = 1540 Njmm2 ,

m = 100, N = 1,

On Anisotropic Formulation of the Elastic law witin

Multiplicative Inelasticity

403

As the to the elastic constitutive law we restrict ourselves to a linear one ac­cording to which the relation holds

where we have

H=

and

a = 48968. N /mm2 ,

d = 4669.2 N/mm2, g = 4804.3 N /mm2,

a f e 0 0 0

f b d 0 0 0 e d c 0 0 0 0 0 0 9 0 0 0 0 0 0 h 0 0 0 0 0 0 j

b = 13588. N /mm2 ,

e = 1874.7 N/mm2, h = 4804.3 N /mm2 ,

(19)

(20)

c = 13414. N/mm2, f = 3469.2 N/mm2, j = 4214.3 N/mm2 .

I~ _I 1=26 b=4 h=l

404 Carlo Sansour

400.---~-----r----.----.r----r----'-----r---~-----r----'

350

300

250

~ 200 :! Ii

150 Ii

50

/;:::':::-:::0-'" i ::

0--30 -------60 --------90 ------

0~ __ ~ ____ ~ ____ L-__ ~ ____ ~ ____ 4_ ____ ~ __ ~ ____ ~ __ ~

o 2 4 6 8 10 Displacement

12 14 16 18 20

Figure 1. Extension of a thin sheet. Problem definition and load-displacement curves - 10 x 26 elements.

References

[1] S. R Bodner and Y. Partom, "Constitutive equations for elastic-viscoplastic strain­hardening materials", ASME, J. Appl. Meek, 42, 385-389 (1975).

[2] J.P. Boehler, (eel.), Applications of Tensor Functions in Solid Mechanics, Springer (1987).

[3] Maugin, G.A., 1995. Material forces: Concepts and applications. Appl. Mech. Rev. 48, 213-245.

(4] C. Sansour and EG. Kollmann, " Large viscoplastic defonnations of shells_ Theory and finite element fonnulation", Computational Meek, 21, 512-525 (1998).

DEEP DRAWING PROCESS OF THE AISI 304 STAINLESS STEEL CUP: INTERACTION BETWEEN DESIGN TOOLS AND KINETIC OF PLASTIC STRAIN INDUCED MARTENSITE

Zoubeir TOURKI1) and Mohamed CHERKAouf)

I) Laboratoire de Mecanique Materiaux et procedes Ecole Superieure des Sciences et Techniques de Tunis5, Rue Taha Hussein 1008 Bab Menara Tunis TUNISIE E-mail: zoubeir.tourki@esstt.rnu.tn 2) Laboratoire de Physique et Mecanique des Materiaux CNRS, ISGMP, Universite de Metz 57045 Metz Cedex OI-France E-mail cherk@lpmm.univ-metzfr

Abstract: Mechanisms of transformation plasticity deformation are reviewed for martensitic transformation of stainless steel material. the transformation strains may be accommodated elastically, plastically or by self accommodation. The main of the present work consists in the analysing of the interaction between the kinetic of plastic strain induced martensite (pSIM) in the AISI 304 stainless steel and the design parameters in the deep drawing process.

Key words: kinetic of martensitic transformation, stainless steel, deep drawing, self accommodation, yield surface, hardening behavior

1. INTRODUCTION

The drawing sheet-forming process plays an important part in the automobile and mechanical product industries. The sheet metal named blank is placed onto a drawing die and an adequate amount of pressure is applied to the blank holder. Next, the blank is pushed into the die cavity by the punch to form a cylindrical cup after drawing. The thickness distribution fracture phenomena, the drawing force and the limiting drawing ratio (LDR) for the drawing process are the most important variables usually studied in this case. The factors which influence the formability of drawing process are

405

S. Ahzi et al. (eds.). Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 405-412. ©2004 Kluwer Academic Publishers.

406 Zoubeir Tourki and Mohamed Cherkaoui

the pressure of the blank-holder, the radius of the die-arc value, the lubricant conditions of the work-piece and the type of sheet material used for cup drawing process. These factors determine the maximum punch load in drawing, the sheet-thickness variation after drawing, and the maximum limit drawing ratio. Shinagawa et al. [1] used the thermo-rigid-plastic finite element method to simulate the axisymmetric drawing process for austenitic stainless steel, which could generate the deformation mode, the distribution of temperature and plastic deformation being investigated. Simandiri [2] discussed the high speed continuous deep drawing of aluminum cans, a detailed discussion being made regarding the friction force, lubrication and wear. Huang and Chen [3] analyzed the influence of the die-arc radius on formability in the axisymmetric deep drawing process. Li [4] Gonzalez B. H. et al. [5] and Bargui H. et al. [6] have adopted uniaxial tensile drawing to predict the values of the strain-hardening coefficient (n) and the anisotropy coefficient (r) for metal performance in the deep drawing. As the nand r values are different for various annealing structures, the optimal dimension of the material can be predicted. A larger r value will provide better deep­drawing formability. A discussion is undertaken in this present study regarding the influence of the material hardening behavior of the AISI 304 stainless steel metal sheet in connection with the die-arc radius. Additionally, a rigid-plastic differential finite-element computer code based on a new description of the kinetic of the plastic strain induced martensite (PSIM) volume fraction was adopted to simulate the process of drawing a cylindrical cup. Finally, the comprehensive numerical simulation was verified by making a comparison with the punch load and thickness distribution obtained by a second stable stainless material AISI 316 [6] took as a reference material.

2. EXPERIMENTAL WORK

The chemical composition of the two austenitic stainless steels used in this work are given in Table 1. Steel A is a commercial grade of AISI 304 and steel B is an experimental alloy with a higher percent of a Nickel, Chrome and Molybdenum fraction. The microstructure of the alloyed layer was analyzed by scanning electron microscopy (S.E.M) and optical microscopy. The composition along the depth of material layer was determined by energy-dispersive X-ray spectroscopy (E.D.S).

Table!: Chemical composition of the studied materials and instability parameter.

Steel € Sf .~ Ct Nt ·······Mo N:·'· Cd Fe M.I3o A 0.05 < 0.41 1.14 8.04 9 0.193 0.04 0.348 balance 38.8

B 0.06 0.055 1.85 16.8 12.3 2.59 0.03 0.057 balance -26.5

Deep drawing process of the AISI 304 stainless steel cup 407

The phases fonned in the surface layer was determined by X-ray diffractometry (XRD) using eu K as the radiation source at 40kV and 35 rnA, with Ni filter. Figures 1 a and 1 b display the results of X-ray analysis on the A and B material at both initial and defonned state. The quantitative exploitation of the spectrum obtained allows to detennine the quantity of a' -martensite characterized by (200)a' and (211 )a' peaks. They show clearly that material B does not contain any a' peaks. Figure lb highlights no tendency of structure change and the B type steel was than took as a material reference.

(11 ) ., (211) ,,' a)

Intensity (I)

r-~~------~~--~~t= 6 5

60 70

Inte~nsity (I) (2 0) Y

(1~1)y ( OOlY ,

,--'

80

(311 )y

4 3

2

90 28

b)

(A Y

o 50 60 70 80 90 100 2 9

Figure 1. XRD patterns of the stainless steel sample at different deformation level: (a) steel A; [1: reception, 2:+22°C, 3:0°C, 4: -45°C, 5: -78°C, 6:-196°C]; (b) steel B.

Figure 2 illustrates a metallographic analyse by S.E.M. of A steel specimens defonned by an uniaxial tension at a room temperature. This figure shows the distribution of a' -martensite that can be associated to an intennediary E phase. At this analyse scale under a room temperature process, some shear bands intersections appeared and seems to act as nucleation sites for a' embryos as is shown in bright field micrograph. These observations indicate that the growth process of a'-martensite involves the coalescence at a band shear intersections [6]. According to Olson and Cohen [7] and other authors [8,9,10], the higher rate of fonnation of stress-induced

408 Zoubeir Tourki and Mohamed Cherkaoui

E and a' -martensites observed in steel A is a direct consequence of its lower stacking fault energy Huang Y. M. [11]. This also implies that austenite is more stable in steel B, that is, it must have a lower Md30 temperature than steel A, as in fact was observed in our precedent work and shown on Table 1. Cherkaoui et al. [12] and [6].

Figure 2. Nucleation of an a'-particles from some plaques in the intermediary Ii

phase bv a twinninf,! mode at 2]oC temperature at fracture.

2.1 Kinetic behavior of P .S.I.M.

In order to evaluate experimentally the kinetic of volume fraction of plastic induced martensite on the A steel, a suitable testing on a specimen with trapezoidal shape is conducted. The dosage of volume fraction is represented on Figure 3 and the mechanical characteristics of two types of material are displayed on Table 2. According to the Abrassart F. [13] model, the kinetic behavior of the mixture components : austenite y and martensite a' can be described by the following equation:

(J'e(s) = Kysn + {Ka.S m - Kysn )/(s) (1)

where the kinetic of the martensite volume fraction is given by :

/(s) = 1- expl- p(l- exp(- as ))4.5 J (2)

the hardening coefficients K y' K a" m and n are given just for two

temperatures (+22 and -196°C).

Table 2. Mechanical and kinetic characteristics of the two types steel

Steel & tempe- Hardening coefficients (Mpa) Mechanical Kinetic characteristics {M~ characteristics

rature (0C) K., n K",. m Ruo.2 It" A(%) a a

A +22 730 0.14

4500 0.7 315 690 58 3.5 0.6

-196 980 0.1 13 2.02 B +22 1370 0.41 320 625 58

Deep drawing process o/the AfSf 304 stainless steel cup 409

The kinetic coefficients a and fJ are determined by a dosage of volume

fraction plastic strain induced martensite in a uniaxial test and they are given on Table 2. The model predicts a relationship between the martensite fraction transformed and the transformation strains, including volume change, that results from the formation of martensite. The fitted curves of this behavior are plotted on Figure 3. This figure shows clearly that the low is the experimental temperature level is, the more is the volume fraction transformation. This volume reaches 85% at -196°C of the initial austenite volume.

100 -.!,96:£

'; 0' 80 __ -7e '~_

•• b -...: ~ -,,~ - ,-

. ; E 60

j 'i 40 ~ ~ . . ~ -~ • • .; $ 20

> • ~ _r-:- +22 "C 0

0 10 20 30 40 50

True Strain (x100)

Figure 3. Volume fraction evolution of the PSIM at different temperature levels.

2.2 Plasticity Modeling

The plasticity model used in this work applies for orthotropic sheets under a state of plane stresses (0'1,0'2)' For the current axisymmetric deep

drawing problem, just a planer isotropic form of this model is considered Ferron G. et al. [14] and given by the following expression:

f (0' ij ) = <1>(0'1 , 0' 2 ) - (j e (e) = 0 (3)

where 0'1 and 0'2 are the principal stress components. iT e (e) is obtained

from Eq. 1 by an new identification Tourki Z. et al. [15] from uniaxial to the biaxial behavior using the yield surface polar radius g('I'). The hardening

expression becomes than: (je(e)=t;(O'e,g) (4)

where the polar radius is chosen as an extension of the isotropic Drucker criterion [16]. 'I' is the polar angle of M point situated on the yield surface.

3. CUP-DRAWING PROCESS: NUMERICAL DESCRIPTION

The analytical formulation of the cup-drawing process use a yield surface in the plane stress Eq. 3, coupled to the new hardening behavior Eq. 4, the

410 Zoubeir Tourki and Mohamed Cherkaoui

normality law, the plastic incompressibility and the plastic work. It becomes easy to obtain the strain rate ratio p = 82 /81 , the equivalent stress (j, the

plastic strain ratee P and the polar angle '1/. By using the kinematics

equations for each "zone" of the cup and by denoting u(r) the radial

displacement and h the current thickness, the plastic incompressibility condition can described by:

du u it -+-+-=0 (5) dr r h

Respecting the differential model integration and the process geometry given on Figure 4, it is possible to determine, using a numerical techniques [15] for each increment llrna , all physical components:

(!!.hi' hi,rj>Gri,Ga,Ciri,Cia,Cii,8j>".). All these parameters will be used for analyzing simulation of the cup-drawing process using the two types of stainless steel material A and B.

Fp : Punch loading Fh : Holder loading rd : Die-arc radius

Figure 4. Design discription of the axisymmetric deep drawing process

~ Punch I

4. RESULTS AND DISCUSSION

Holder

(i) The variation in the relative amount of a'-martensite with the true strain, determined by XRD in specimens of two steels deformed in tension at room temperature, is shown on Figure 1 -a and -b;

(ii) The change in the strain hardening exponent, n, with true strain from 0.14 to 0.1 at a higher strain given on Table 2 for the steel A and 0.41 for the steel B;

(iii) The change of the volume fraction of a' -martensite as a function of the deformation temperature for samples of steels A and B are shown on Figure 3; All these observed behavior of the two types of stainless steel A and B are

well discussed in our work (submitted to the UP [12]) and summarized in the present paper, indicate clearly that the steel B is more stable than steel A. The results presented therein are in a good agreement with experimentally observations. In the following, a numerical deep drawing simulation will be discussed by considering the so-called stable B steel as a reference material. We will change the design geometry of the tools (die-arc radius) and will

Deep drawing process of the AISI 304 stainless steel cup 411

compare the thickness strain distribution on both A and B steel. The relationship between the a'-martensite transformation and the die-arc radius will be exhibited.

4.1 The Die-arc effect on the thickness strain

Figure 5. illustrates two different results. The first one is the good agreement obtained between ABAQUS thickness strain C1 and our specific differential program C2 curve (A steel with rd=5.0mm). The second notice is the low thickness strain distribution represented by dash points C3 (A steel with rd=6.0mm). This result adjusts well the C4 curve, obtained using B steel with rd=5.0mm, from the edge of the rim until cylindrical part. At the bottom part of the punch, C3 points decrease in absolute value to reach again the C2

curve. The obtained behavior means that by modifying the die radius, it could be possible to drop the thickness strain which leads to a higher stretch formability and decreasing the wrinkling instability phenomena even if the unstable stainless steel material is used. Consequently, steel A reaches an as well as formability, than steel A as it is shown on the following Figure.

25.00%

20.00%

15.00%

c

~ 10.00%

I 5.000/.

0 .00%

-5.00%

-10.00%

-15.00%

I I

~------~==~~~--~ I i

0.07 0 .08

Figure 5. Thickness strain distribution as afunction ofa current radius using two types of materials and two values of die-arc radius.

5. CONCLUSION

The volume fraction of martensite increases rapidly with strain and low temperature, obeying an exponential relationship. The stress-induced martensite of AISI 304 stainless steel (sheet A) is much more easily produced than that of AISI 316 (sheet B) because of their different nucleation mechanisms of the martensite Bargui H. et al. [6]. the two material types behavior have been well assessed in our previous work. We

412 Zoubeir Tourki and Mohamed Cherkaoui

succeeded in finding a new relation who connects the geometry to this behavior by taking material B as a reference. The PSIM can be compensated by a simple modification in the die-arc radius. A good agreement was found between the stress and thickness strain distribution of a reference material B and those obtained with the metastable material A after a die-arc radius adjustment.

6. REFERENCES

(I) K. Shinagawa, K. Mori and K. Osakada, Finite element simulation of deep-drawing of stainless steel sheet with deformation-induced transformation, J. Mater. Process. Technol., 27(13) (1991) 301-310.

(2) S.S. Simandiri, Optimizing the aluminum deep-drawing process, SME technical paper, MF91-415 (1991).

(3) Y. M. Huang and J. W. Chen, Influence oflubricant on formability of cylindrical cup­drawing, J. Mater. Process. Technol. Volume 63, Issues 1-3, January (1997),77-82.

(4] M. Li, Prediction of optimum dimension of the crystalline grains for the deep-drawing of metals, J. Mater. Process. Technol., 26(3) (1991) 349-354.

(S) B. M. Gonzalez, C.S.B. Castro, V.T.L. Buono, 1M.C. Vilela, M.S. Andrade, J.M.D. Moraes and M.l. Mantel, The influence of copper addition on the formability of AlSI 304 stainless steel. Mat. Scien. And Engineering A 343 (2003) 51-56.

(6] H. Bargui, H. Sidhom and Z. Tourki (2000), Martensite induite et comportement en ecrouissage de l'acier AISI304, Materiaux et Techniques N° 11-12, pp. 31,41.

(7) G.B. Olson, and M. Cohen, (1975), Kinetics of Strain Induced Martensitic Nucleation, Metall. Trans. N°6A, pp.791-795.

(8) J. A. Venables, Philos, Magazine 7 (1962) 35. [9] W. O. Binder, Metal Progress 58 (1950) 201-207. (10] B. Cina, J. Iron Steel Inst. 177 (1954) 406-422. (11] Y. M. Huang, Jia-Wine Chen, Influence of the toll clearance in the cylindrical cup­

drawing process, Journal Materials Processing Technology 57 (1996) 4-13. (12] M. Cherkaoui, Z. Tourki, H. Bargui and H. Sidhom, Micro-mechanical

modeling coupled to the quantitative and microstructure analyses on the plastic induced martensite in the TRIP steels (AISI304IIAISI316), submitted to the Int. Journal of Plasticity.

(13] F. Abrassart (1973), Stress-induced 17(1' Martensitic Transformation in two Carbon Stainless Steels. application to Trip Steels Metallurgical Transactions, Vol. 4, Sep. 73, p.p 2205-2216.

[14] G. Ferron, R. Makkouk and 1. Morreale, (1994) A parametric description of orthotropic plasticity in metal sheets, Int. J. of Plasticity, vol. 10, n05, pp. 431-449.

(IS] Tourki Z., Makkouk R., Zeghloul A. and Ferron G. (1994), Orthotropic plasticity in metal sheets: a theoretical framework, J. of Materials Processing Technology. 45 pp. 453-458.

(16] D.C. Drucker, Journal Appl. Mech., (1949) 16349.

MODELING AND SIMULATION OF DYNAMIC FAILURE IN DUCTILE METALS

L. Campagne, L. Daridon and S. Ahzi

IMFS-UMR 7507 CNRS Universite Louis Pasteur 2 Rue Boussingault, 67000 Strasbourg, France

Abstract: In the present work, we propose a physically based model describing these processes for planar impact. This model combines the Mechanical Threshold Stress (MTS) model for the evolution of the flow stress and a void nucleation and growth model. This paper presents results from numerical simulation of planar impact test and their comparison with experimental results for OFHC copper.

Key words: Void Nucleation, Void Growth, Planar impact, Dynamic Plasticity, Dynamic Failure

1 INTRODUCTION

It is well established that high rate failure of structural materials takes place by rate processes occurring at the micro level and involving nucleation, growth, and coalescence of voids or cracks. These mechanisms of failure and plasticity in polycrystalline materials is often dislocation controlled. Most of existing failure models are based on phenomenological approaches rather than mechanistic approaches (Tuler-Butcher 68, Cagnoux 85, Zhurkov 65, Klepaczko 90).These have been reviewed by Hanim and Ahzi (2001). The well known mechanistic approaches for dynamic failure are the work of Curran et al (87) where a nucleation and growth model is proposed and the work of Addessio et al (93) where the Gurson model for nucleation and growth is used. The main thrust of the paper is the development of failure model based on microscopic assumptions. We propose a model in which the nucleation and growth processes are linked to the dislocation motion. The new model will be applied to dynamic failure that occurs by spallation during planar impact test. The material used is

413

s. Ahzi e/ al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 413-420. ©2004 Kluwer Academic Publishers.

414 L. Campagne, L. Daridon and S. Ahzi

OFHC copper. We used various impactor shapes (cylindrical and conical target). In these applications, the new failure model along with the mechanical threshold plasticity model were implemented in the finite element code ABAQUSlExplicit. Results will be compared with the existing experimental results.

2 MODELLING OF NUCLEATION AND GROWTH PROCESS

2.1 Nucleation process

The mechanism of failure in metallic materials is dislocation controlled (McClintock and Argon, 1966). The process of deformation and failure can therefore be described by plastic glide, which involves the mechanism of dislocation pile-ups. In the work of Stroh (1957), a probability function was suggested for brittle failure due to dislocation pile-up. Since the brittle and ductile nature of failure can be associated to the process of release or non­release of dislocations near a pile up, we assume the brittle and ductile failure are complementary events. So, the probability for ductile failure, P, can be given by the following equation(Campagne et al 02):

p = t-ex{ -rotex{ -k~U)) (1)

where t is the duration of the loading. ~U is the energy of activation that can be related to the shear strain rate by the well-known Arrhenius law. This probability of ductile failure can be expressed in terms of strain rate (eq.2):

p= I-ex{ -rot n (2)

where Y is the shear rate and Yo is a pre-exponential factor

We assume that this probability, P, describes the void nucleation process in the case of ductile failure. Thus, we express the rate of the nucleated sites per unit volume for ductile failure as (Hanim and Ahzi, 2000):

(3)

where No is the maximum rate of the nucleated sites per unit volume

(Curran and al , 1987). The number of nucleation sites per unit of volume,

Modeling and simulation of dynamic failure in ductile metals 415

I1N , which have nucleated during a time interval, M, and the associated

relative void volume, !1v;ew, are given by the following equations:

!1N = N!1t and (4)

Here,Ro is the radius of the nucleated voids which are assumed of spherical shape

2.2 Growth process based on microscopic assumption

At the sub microscopic level, the growth process can also be explained in terms of dislocation motion. Among the extensive contributions in this area can be found in the book of McClintock and Argon (McClintock and Argon ,1966), and in the recent work ofWeertman (1981, 1996). Following the work of Weertman (Weertman , 1981) on the modeling of ductile growth process, we obtain the following growth equation (champagne et al 02):

(5)

where R is the radius of spherical void at time t, 0" mean is the mean stress,

and (0" mean - 0" gO t is the positive part of (0" mean - 0" go)' To grow, the

void should have nucleated and the mean stress 0" mean should exceed growth • threshold stress 0" gO' t is time for one loading cycle and g is a coefficient

that depends upon the ratio of the theoretical tensile strength 0" T to

theoretical shear strength r T, V is the Poisson ratio and G the shear modulus

The loading cycle t' is evaluated by the shock theory (Meyers, 1994). The initial duration of the peak shock pulse in an impact is approximately equal to twice of travel time of the shock wave through the impactor. Thus,

to determine the loading cycle t', we use the following approximate expression:

• 2do t ~­

Us (6)

where do is the thickness of the impactor and Us is the velocity of the

shock wave. The expression for relative void volume by growth of the crown can be

expressed by:

416 L. Campagne, L. Daridon and S. Ahzi

where No is the initial number of pre-existing void per unit volume. This

constant can be expressed in term of initial porosity of the material. The total relative void volume, denoted Vr , is the sum of initial relative

void volume, Vo, and of the contributions associated respectively with

nucleation and growth processes.

(8)

From the total relative void volume, we determine the porosity of the material, f, and the associated damage parameter, d, obtained via a self consistent method:

and d = lS{l-u) . f 7-Su

(9)

To account for the deactivation of the damage in compressive loading and for damage initiation after a threshold porosity, the following expression for the damage law is proposed:

(10)

where £5 (0" meaJ = 1 if 0" mean> 0 and B( cr mean) = 0 if cr mean 5, 0, It is the

threshold porosity for damage initiation and!c the critical porosity for failure.

3 APPLICATION TO PLANAR IMPACT

A spall is a material failure produced by the action of tensile stresses developed in the inside of a solid body when two tensil waves collide(Meyer et al). In the following, we show results from our numerical simulations of the planar impact test of OFHC copper. These results are obtained by FEM analysis where our failure model, along with the mechanical threshold stress

Modeling and simulation of dynamic failure in ductile metals 417

model for dynamic plasticity (Kocks, 1987 ; Follansbee and Kocks, 1988), are implemented in Abaqus finite element code. The framework and details of the MTS model are given in the work of Follansbee and Kocks (Follansbee and Kocks, 1988). Determination of the failure model

parameters h, (0, No, Ie' (Y gO are estimated for the planar impact test

with cylindrical geometry and are given in Tablel.

3.1 FEM analysis.

The element type used in the FEM analysis is the 4-node, reduced­integration, first-order, with hourglass control, axisymetric solid element CAX4R. The kinematic predictor/corrector contact algorithm is used with balanced master-slave concept to minimize the penetration of the contacting bodies and we considered no rebound. We studied the planar impact of a cylindrical and conical target with a cylindrical impactor. Both the impactor and target materials are OFHC copper and their respective geometries are shown in figure 1.

-I. 2rnrn :

................................... ................................... ...................................

19mm

Fig. l.a. Geometries of the cylindrical target and impactor

Parameters

Nucleation coefficient No [no./cm3 -s]

Frequency (0 [Is]

Hydrostatic stress (Yo [Mpa]

Critical porosity Ie

Threshold porosity I,

25mm

25mm

!""""'"""'"'" : 2mm

:+~---... 19mm

Fig. 1.b. Geometries of the conical target and cylindrical impactor

Mechanistic approach

3.2 x 1024

10°

5

0.44

0.0

418 L. Campagne, L. Daridon and S. Ahzi

Table I: Parameters for the mechanistic failure model for OFHC Copper

3.1.1 Cylindrical geometry

For this test where the impactor velocity is 185 ms-I . According to

Meyers (1994), we use for this test the value for loading cycle t* = 0.98.10-6S. The predicted curve of the free surface velocity history for a node located on the free surface in the target is shown in Figure 2. in comparison with the experimental results of Rajendran et al. (Rajendran and ai, 1988).

';

~ .~~ 150 ~ e - '-" ~~ Ole:> 100 I "'..,. .:!M - x- experimental .. ::I ..

50 ",-g - mechanistic ~ I: ..

'"" 0 2 3 4 5 times in microseconds

Fig. 2. Comparison between numerical result and the experimental results of Rajendran and ai, 1988

The agreement of this numerical simulation with the experimental data shows that the choice of the model parameters is fair. Now, considering the failure map in the target. Our approach predict the spalling plane at a distance equal the height of the projectile from the free surface velocity,(see Fig 3). This is in good agreement with both the experimental observations and the elastic shock wave analysis. Note the figure 3 shows damage evolution at time 2.5 and3.0 Ils. Dark area is highly damaged and

corresponds to the value of the critical porosity Ie .

Damage map at t = 2.5 ~s Damage map at t = 3.0 ~s Fig3. Damage map for cylindrical target

Modeling and simulation of dynamic failure in ductile metals 419

3.1.2 Conical geometry

For this test where the impactor velocity is 183 ms· l . According to

Meyers (Meyers, 1994), we use for this test the value for loading cycle t" = 0.98 lO-6s, all of the other parameters of our model are the same as in the previous test. We present in figure 4 damage map at different times. These results are qualitatively in good agreement with the elastic shock wave analysis, but the lack of experimental data, for OFHC Copper with this geometry, does not allow us to evaluate the quantitative side of the results.

Damage map at t = 3.6 f.!S Damage map at t = 4.8 f.!S Fig4. Damage map for conical target

4 CONCLUSION

In this work we have proposed a new physically based dynamic failure model. The new Nucleation and Growth model (NAG) also takes into account the initial porosity of the material. This model is valid 3 dimensional analysis (conical target) The advantage of a NAG approach is that it predicts damage evolution in the target as well as in the impactor. For the cylindrical test, the numerical damage map is in good agreement with the experimental observations

REFERENCES

Addessio F.L., Johnson J.N., Maudlin PJ., 1993. The effect of void growth on taylor cylinder impact experiments. J. Appl. Phys., 73, 7288-7297.

Cagnoux J, 1985. Deformation et ruine d'un verre pyrex soumis a un choc intense: etude experimentale et modelisation du comportement, PhD dissertation, France.

Caracostas CA., Shodja H.M. and Weertman 1., 1996, The double slip plane model for the study of short cracks., Mechanics of Materials, 20 ,195-208

Curran, D. R., Seaman, L. and Shockey, D.A., 1987, Dynamic Failure of Solids. Physics reports. 5-6(147), 253-388.

420 L. Campagne, L. Daridon and S. Ahzi

Follansbee, P.S., Kocks, U. F., 1988, A Constitutive Description of the Defonnation of Copper Based on the Use of the Mechanical Threshold Stress as an Internal State Variable, Acta metall. 36(1), 81-93.

Hanim S. and Ahzi S., 2001, A unified approach for pressure and temperature effects in dynamic failure criteria. Int. J. Plast.,17, 1215-1244.

Klepaczko J.R, 1990,Dynamic crack initiation, some experimental methods and modeling for FCC and BCC metals. In: Amman et al (Eds), Impact Effect of Fast Transient Loading. Balkerna, Rotterdam, 3-35.

Kocks, U.F., 1987, Constitutive behaviour based on crystal plasticity. In Unified Constitutive Equations for Creep and Plasticity, ed. A.K.Miller, Elsevier.

McClintock, F. A. and A. S. Argon, 1966, Mechanical behaviour of materials. Addison­Wesley Pub. Company, Inc.

Meyers, M. A., 1994, Dynamic behaviour of materials, Ed. Wiley-Interscience, 179-180

Rajendran, A. M., Dietenberger, M. A. and Grove, D. J., 1988, A void nucleation and growth based failure model for spallation. J. Appl. Phys. 65,1521-1527.

Shockey, D. A., Curran, D. R., Seaman, L., Rosenberg, J. T., and Peterson, C. F., 1974, Int. J. Rock Mech. Sci. & Geome

Stroh N., 1957, A theory of the fracture of metals. Advances In Physics, 6, 418-465.

Tuler F.R., Butcher B.M., 1968, A criterion for the time dependence of dynamic fracture. Int. J. Fract. Mech. 4, 431

Weertman J., 1981, Fatigue crack growth theory for ductile material. In Three Dimensional Constitutive Relations and Ductile Fracture, ed. Nemat-Nasser S., pp. 111-122. North­Holland Publishing Company.

EFFECTS OF POLYMERIC ADDITIVES ON THE MORPHOLOGY AND THE STRUCTURE OF CALCIUM CARBONATE MATERIAL

AmaneJada

Institut de Chimie des Surfaces et Interfaces, 15 rue Jean Starcky, B.P. 2488, 68057 Mulhouse, France. Email: A.Jada@univ-mulhousejr

Abstract: In this work, we investigate the crystallization of calcium carbonate (CaC03)

from supersaturated solutions and in the presence of polymeric additives such as an anionic polyelectrolyte (polystyrene sulfonate) and a non-ionic copolymer (polystyrene-polyethylene oxide). Depending on the nature and concentration of the additive used, we obtain various calcium carbonate particles having different sizes, crystalline structures, and morphologies. Thus, in the presence of the non-ionic copolymer alone, the mineral crystallization gives birth to CaC03 agglomerates of primary particles. The size of the primary particles, which present mostly rhombohedral morphologies, is about 4 J..I.m and the aggregate size is the range 10-20 J..I.m. On the other hand, in the presence of the anionic polyelectrolyte alone, we obtain CaC03 crystals having smooth spherical shape and size about 2 J..I.m. However, in the presence of copolymer-polyelectrolyte mixture, the calcium carbonate crystallization gives birth to modified CaC03 crystals morphologies. Such morphologies are function of the ratio I, I = CcopolymerlCpolyelectrolyte' The Ccopol)'lllOl' and Cpolyelectrolyte, are respectively, the copolymer and the polyelectrolyte concentrations used. Hence, at value of 1=2.9, the calcium carbonate crystallization gives rough CaC03 spherical having size about 2 J..I.m, while at values ofI=3.1 and 1=18.6, we observe elongated CaC03 particles having size (maximum crystal length) in the range 3.4-4.3 J..I.m. The CaC03 crystal structures obtained by the X-ray analyses were pure calcite when the additive is the non-ionic copolymer and pure vaterite in the presence of the anionic polyelectrolyte. The data obtained indicate that the pure polyelectrolyte alone inhibits the crystal growth in all direction and lead to homogenous CaC03 spherical particles. However in the presence of the copolymer-polyelectrolyte mixture, the crystal growth seems to be inhibited in preferential directions. This study shows that the morphology and structure of the CaC03 crystals can be controlled by the use of either pure or mixture polymer additives. Finally, the effects of the polyelectrolyte and non-ionic copolymer on the precipitation of the CaC03 are explained in term

421

S. Ahzi et al. (eds.), Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, 421-428. ©2004 Kluwer Academic Publishers.

422 AmaneJada

of two main mechanisms involving ion exchange and/or ion complexation and preferential adsorption.

Key words: crystal growth; calcium carbonate; adsorption; polymer, vaterite; calcite.

1. INTRODUCTION

Calcium carbonate, in its various forms, is the most widely used of the mineral fillers and pigments. However large-scale production of CaC03

particles with the same morphology and uniform size is still a challenge. Compounds such as surface-active additives are widely used in the

industrial production of CaC03 in order to control the morphology and size of the mineral [I]. This mineral exists in various polymorph phases that are, in order of decreasing solubility: calcium carbonate hexahydrate, calcium carbonate monohydrate, vaterite, aragonite, and calcite. The two first polymorphs are unstable and are converted into the most thermodynamically stable form, calcite. The vaterite is also unstable and is formed first in aqueous solution under conditions of spontaneous precipitation. This phase transforms rapidly into calcite, but may be stabilized in the crystallization medium in the presence of polymeric additives.

Various approaches to the preparation of calcium carbonate have been described, which include crystal growth inhibition and morphology change in solution [2] or under a Langmuir monolayer [3]. Further, in the biogenic synthesis of CaC03, it has been shown that acidic protein [4] interacts in solution with the growing CaC03 crystals. Others studies on morphology changes have included growing crystals, using an organic template, such as [3-chitin or an ammonium surfactant to form aragonite or vaterite [5,6].

We have shown in previous studies [7] that the precipitation of calcium carbonate in the presence of anionic polyelectrolyte alone, such as polystyrene sulfonate, gave smooth spherical vaterite particles. On the other hand, the precipitation of calcium carbonate in the presence of non-ionic copolymer alone, such as polystyrene-polyethylene oxide, the size, and the morphology of the calcite mineral are not affected. However, in the presence of polyelectrolyte-copolymer mixture and at the weight ratio I, I = Ccopo1yme/Cpolyelectrolyte = 2.9, the CaC03 particles formed are rough spherical vaterite particles. The Ccopolymer and CpOlyeIectrolyte, are respectively, the copolymer and the polyelectrolyte concentrations expressed in g.rl.

In the present study we show that, under similar polyelectrolyte­copolymer mixture conditions and by varying the weight ratio 1= Ccopo1yme/Cpolyelectrolyte up to 18.6, various CaC03 morphologies and sizes are obtained. Thus at 1=2.9, rough CaC03 spherical particles having size about 2 1JlIl., are observed. However, at 1 values 1= 3.1 and 1= 18.6, elongated CaC03

particles having size in range 3.4-4.3 IJlIl. were observed.

Effects o/polymeric additives on the morphology and the structure o/the 423 calcium carbonate material

2. EXPERIMENTAL

2.1 Materials The anionic polyelectrolyte used as additive in the preparation of calcium

carbonate particles was the polystyrene sulfonate (PSS 1). This polyelectrolyte having molecular weight Mw = 7.104 glmol, was purchased from Aldrich. The non-ionic diblock copolymer is a polystyrene­polyethylene oxide (SE1O-30), having block molecular weights equal to, respectively, 1000 glmol, and 3000 glmol. All the samples are the same as used elsewhere [7]. The sodium carbonate anhydrous (Na2C03), and the calcium nitrate tetrahydrate Ca(N03h 4H20, were purchased, respectively, from Fluka Chemie AG, and from Prolabo. All the reagents were used as received without further purification.

2.2 Preparation of calcium carbonate particles Supersaturated solutions for crystal growth experiments were prepared, at

ambient temperature, by addition of equal volumes of sodium carbonate (Na2C03) to calcium nitrate (Ca(N03)2 4H20) solutions in a water-jacketed Pyrex glass vessel of 300-ml capacity. In each experiment the concentrations of calcium and carbonate were kept constants and equal to 0.01 M. In addition, the polymers aqueous solutions of PSSI and SE1O-30, were first added to calcium nitrate solution and the mixtures were stirred for about 15 minutes prior the addition of Na2C03 solutions. The final copolymer concentration varied from 1 to 5.2 gil while the polyelectrolyte concentration varied slightly in the range 0.28-0.36 gil. It should be emphasized that prior the CaC03 crystallization takes place, the polyelectrolyte-calcium­copolymer mixture at the equilibrium, forms homogenous solutions. At the end of the precipitation, CaC03 crystals were recovered by filtration and the dried particles were gold coated in vacuum and examined by Scanning Electron Microscope (SEM) at magnifications from x 2,000 to 30,000.

3. RESULTS

3.1 CaC03 shape modification in the presence of polymeric additives

The crystallization of the calcium carbonate in the presence of polystyrene-polyethylene oxide (SEIO-30) copolymer alone gives birth to agglomerate of primary particles, which present rhombohedral morphologies, as shown in Fig. 1.

424 AmaneJada

Figure 1. SEM micrograph of CaCO) particles obtained in the presence of the copolymer

SEW-30, 0.55 gil. [Ca2+]=[CO/]=0.01 M.

The X-ray analysis of such CaC03 particles indicated that the crystal

Figure 2. SEM micrograph of CaCO) particles obtained in the presence of the polyelectrolyte PSS1, 0.34 gil. [Ca2+]=[CO/]=0.01 M

structure is calcite. In the presence of polystyrene sulfonate (PSS 1) alone, Fig. 2, we obtain CaC03 crystal having smooth spherical shape.

Moreover, the CaC03 crystal structure obtained by the X-ray analysis is vaterite in the presence of the polyelectrolyte alone.

Figure 3. SEM micrograph of growing CaC03 particles obtained at earlier precipitation stages, in the presence of the copolymer SEW-30, 1 gil and the polyelectrolyte PSS1, 0.34 gil. The concentration ratio 1= 2.9. [Ca2+]=[CO/]=0.01 M

The growing CaC03 particles at earlier precipitation stages, is a regular polyhedron, Fig. 3, and it is obtained in the presence of the PSSl/SEIO-30 polymer-copolymer mixture at weight I ratio = 2.9. Under similar experimental conditions and at later precipitation stages, rough spherical particles, were obtained as shown in Fig. 4.

Effects of polymeric additives on the morphology and the structure of the 425 calcium carbonate material

Figure 4. SEM micrograph of CaCO) particles obtained at later precipitation stages, and grown under similar experimental conditions as in Figure 3.

Further, elongated CaC03 particles resembling somewhat to the ellipsoidal shape as shown in Figs 5 and 6, were obtained, respectively, at 1== 3.1 and 1==18.6. The ellipsoidal shape ofvaterite calcium carbonate was also observed in batch-precipitation of calcium carbonate at 45°C [8].

Figure 5. SEM micrograph of growing CaC03 particles obtained at earlier precipitation stages, in the presence of the copolymer SEIO-30, 1.12 gil and the polyelectrolyte PSSI, 0.36 gil. The concentration ratio 1= 3.1. [Ca2+]=[CO/,]=0.O 1 M.

Figure 6. SEM micrograph of CaCO) particles obtained at later precipitation stages, in the presence of the copolymer SEJO-30, 5.2 gil and the polyelectrolyte PSS1, 0.28 gil. The concentration ratio [= 18.6. [Ca2+]=[CO/]=0.01 M

3.2 CaC03 size modification in the presence of polymeric additives

As can be observed from Figures 4 to 6, the size of the crystal increases from 2 to about 4.3 J.Ul1 by increasing the ratio I from, 2.9 to 18.6. Further, by considering that the CaC03 particle is embedded in a rectangle having a

426 AmaneJada

long side L and short side 1, we can define the CaC03 particle radius R, as R = (L+I)/4 and the particle anisotropy A, as A = Lll. Thus, the Table 1 summarizes the values found for R and A, as function of various values of!.

Table 1: Values of A and R for CaC03 particles prepared under various copolymer and polyelectrolyte concentrations.

Ratio 1

1= Ccopolyme/Cpolyelectrolyte A=LlI R=(L+I)/4

(!-1m)

1=2.9 1 2

1=3.1 1.23 2.41

1=18.6 1.37 2.86

As can be seen in the Table 1, the particle radius R and the anisotropy A, increase with the increase of the ratio I, indicating that the shape and size of CaC03 particles are under influence of the copolymer (SE 10-30) and polyelectrolyte (PSS 1) concentrations.

4. Discussions

It should be emphasized that vaterite CaC03 particles can be formed without any additives and are converted into the most thermodynamically stable form, calcite. However, in the present study the precipitation of calcium carbonate in the presence of the polystyrene sulfonate stabilizes the vaterite CaC03 structure for prolonged times.

The data obtained in this work indicate clearly the effects of the copolymer and the polyelectrolyte on the shape and size of the CaC03

colloidal particle. The CaC03 size and shape modifications result from the structure, the electrical surface charge, and the surface affinity of the polymeric additives.

The anionic polyelectrolyte used in the present work bears sulfonate groups which are negatively charged, it is thus expected that such polymer when present in the crystallization medium will accumulates calcium and repels carbonate ions.

Further, in the crystallization medium and prior the addition of carbonate, the polyelectrolyte forms ion pairs with the calcium ions. Such ion pairs result from the exchange of the polyelectrolyte counterions, i.e. the sodium ions, with the calcium ions present in the aqueous solution. The ion exchange is function of the electrolyte type and leads to the modification of polyelectrolyte chain conformation. It results from this ion exchange mechanism that the anionic polyelectrolyte may block sites essential to the incorporation of new solute into the crystal lattice [9] leading to reduction in the CaC03 crystal size. On the other hand, the non-ionic copolymer SElO-

Effects of polymeric additives on the morphology and the structure of the 427 calcium carbonate material

30, containing hydrophilic (polyethylene oxide) and hydrophobic (polystyrene) blocks, may adsorb preferentially onto some growing crystal surfaces leading to changes in the relative free energies of the crystal faces. The faces on which the copolymer is adsorbed will stop growing while the non-modified faces will continue to growth leading to an anisotropic shape of the resulted crystal. Simulation studies of the vaterite structure indicated that the crystal faces have calcium, and carbonate terminated planes, and the carbonate plane {01O} is the dominant surface [10], i.e. the most stable crystal face. Thus, since the non-ionic copolymer is not expected to interact with the growing CaC03 charged crystal faces but rather may adsorb onto some less charged crystal planes, it is likely that the growing faces are the carbonate planes. Similar morphological modification on calcite was reported in studies dealing with crystallization of CaC03 in the presence of PEO-block-PMMA copolymer [11].

In summary, the observed growing CaC03 crystal having polyhedral morphology, formed at earlier stage of precipitation (Fig. 3) and obtained at value of 1= 2.9, is in agreement with the simulated morphology of the unhydrated vaterite crystal [10]. Thus, the smooth CaC03 particles (Fig. 2) prepared in the presence of the anionic polyelectrolyte alone may result from the aggregation or homogeneous settling of the vaterite nanocrystallites on the CaC03 polyhedral crystal. When the non-ionic copolymer is added to the anionic polyelectrolyte in the CaC03 precipitation medium, the preferential adsorption of the copolymer on the CaC03 growing crystal, will induce heterogeneous crystal surfaces, leading to the crystal growth in a preferential direction (Figs. 5 and 6). In addition, the roughness of the CaC03 particle surfaces may result from the heterogeneous settling of the nanocrystallites onto the growing polyhedral crystal surfaces having various energies, as shown in Figs 4 and 6.

5. CONCLUSIONS

The effects of the polyelectrolyte and non-ionic copolymer on the precipitation of the CaC03 involve calcium exchange and/or calcium complexation by the polyelectrolyte and preferential adsorption of the copolymer onto the growing crystal. Thus, the anionic polyelectrolyte affects the crystal nucleation and structure while the non-ionic copolymer controls the crystal growing morphology. The size of the crystal seems to be under influence of the both polymeric additives. This work shows that suitable additives can be used to modify crystal morphologies in the required way. Further the vaterite morphologies found in the present study depend on the experimental conditions and agree with the calculated morphologies found in the literature. Further work is in progress for a systematic study of the

428 AmaneJada

CaC03 crystal structure, morphology, and surface charge in the presence of non-ionic copolymer/anionic polyelectrolyte mixtures.

REFERENCES

[1] J. P. Canselier, "The effects ofsurfactants on crystallization phenomena", J. Dispersion Sci Technol, vol. 14, pp. 625-644,1993.

[2]. H. COIf en and M. Antonietti, "Crystal design of calcium carbonate microparticles using double-hydrophilic block copolymers", Langmuir, vol. 14, pp. 582-589, 1998.

[3] S. Mann, B. R. Heywood, S. Rajam and J. D. Birchall, "Controlled crystallization of CaC03 under stearic acid monolayers", Nature, vol. 334, pp. 692-695, 1988.

[4] G. Falini, S. Albeck, S. Weiner and L. Addadi, "Control of aragonite or calcite polymorphism by mollusk shell macromolecules", Science, vol. 271, pp. 67-69,1996.

[5] D. Walsh, S. Mann, "Fabrication of hollow porous shells of calcium carbonate from self­organizing media", Nature, vol. 377, pp. 320-323, 1995.

[6]. A. Litvin, L. A. Samuelson, D. H. Charych, W. Spevak, D. L. Kaplan, "Influence of supramolecular template organization on mineralization", J. Phys. Chern, vol. 99, pp. 12065-12068, 1995.

[7] A. Jada, E. Peiferkom, "Smooth and rough spherical calcium carbonate particles", Journal of Materials Science Letters, vol. 19, pp. 2077-2079, 2000.

[8] S. Kabasci, W. Althaus, P. -M. Weinspach, "Batch-precipitation of calcium carbonate from highly supersaturated solutions", Chemical-engineering-research-and-design, vol. 74, pp. 765-772, 1996.

[9] R. J. Davey, "The effect of impurity adsorption on the kinetics of crystal growth from solution", Journal of Crystal Growth, vol. 34, pp. 109-119, 1976.

[10] N. H. de Leeuw, S. C. Parker, "Surface structure and morphology of calcium carbonate polymorphs calcite, aragonite, and vaterite: An atomistic approach", J. Phys.Chem B, vol. 102,pp.2914-2922,1998.

[11] J. M. Marentette, J. Norwig, E. StOckelmann, W. H. Meyer, G. Wegner, "Crystallization ofCaC03 in the presence ofPE0-block-PMAA copolymers", Adv. Mater, vol. 9 pp. 647-651, 1997.

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61. LG. Goryacheva: Contact Mechanics in Tribology. 1998 ISBN 0-7923-5257-2 62. O.T. Bruhns and E. Stein (eds.): IUTAM Symposium on Micro- and Macrostructural Aspects

of Thermoplasticity. Proceedings of the IUTAM Symposium held in Bochum, Germany. 1999 ISBN 0-7923-5265-3

63. Ee. Moon: IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics. Proceedings of the IUTAM Symposium held in Ithaca, NY, USA. 1998

ISBN 0-7923-5276-9 64. R. Wang: IUTAM Symposium on Rheology of Bodies with Defects. Proceedings of the IUTAM

Symposium held in Beijing, China. 1999 ISBN 0-7923-5297-1 65. Yu.L Dimitrienko: Thermomechanics of Composites under High Temperatures. 1999

ISBN 0-7923-4899-0 66. P. Argoul, M. Fremond and Q.S. Nguyen (eds.): IUTAM Symposium on Variations of Domains

and Free-Boundary Problems in Solid Mechanics. Proceedings of the IUTAM Symposium held in Paris, France. 1999 ISBN 0-7923-5450-8

67. EJ. Fahy and w.G. Price (eds.): IUTAM Symposium on Statistical Energy Analysis. Proceedings of the IUTAM Symposium held in Southampton, U.K. 1999 ISBN 0-7923-5457-5

68. H.A. Mang and EG. Rammerstorfer (eds.): IUTAM Symposium on Discretization Methods in Structural Mechanics. Proceedings of the IUTAM Symposium held in Vienna, Austria. 1999

ISBN 0-7923-5591-1 69. P. Pedersen and M.P. Bendsl'le (eds.): IUTAM Symposium on Synthesis in Bio Solid Mechanics.

Proceedings of the IUTAM Symposium held in Copenhagen, Denmark. 1999 ISBN 0-7923-5615-2

Mechanics

SOUD MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell

70. S.K. Agrawal and B.C. Fabien: Optimization of Dynamic Systems. 1999 ISBN 0-7923-5681-0

71. A. Carpinteri: Nonlinear Crack Modelsfor Nonmetallic Materials. 1999 ISBN 0-7923-5750-7

72. F. Pfeifer (ed.): IUTAM Symposium on Unilateral Multibody Contacts. Proceedings of the IUTAM Symposium held in Munich, Germany. 1999 ISBN 0-7923-6030-3

73. E. Lavende1is and M. Zakrzhevsky (eds.): IUTAMIlFToMM Symposium on Synthesis ofNon­linear Dynamical Systems. Proceedings of the IUTAMlIFToMM Symposium held in Riga, Latvia. 2000 ISBN 0-7923-6106-7

74. J.-P. Merlet: Parallel Robots. 2000 ISBN 0-7923-6308-6 75. J.T. Pindera: Techniques of Tomographic Isodyne Stress Analysis. 2000 ISBN 0-7923-6388-4 76. G.A. Maugin, R. Drouot and F. Sidoroff (eds.): Continuum Thermomechanics. The Art and

Science of Modelling Material Behaviour. 2000 ISBN 0-7923-6407-4 77. N. Van Dao and EJ. Kreuzer (eds.): IUTAM Symposium on Recent Developments in Non-linear

Oscillations of Mechanical Systems. 2000 ISBN 0-7923-6470-8 78. S.D. Akbarov and A.N. Guz: Mechanics of Curved Composites. 2000 ISBN 0-7923-6477-5 79. M.B. Rubin: Cosserat Theories: Shells, Rods and Points. 2000 ISBN 0-7923-6489-9 80. S. Pellegrino and S.D. Guest (eds.): IUTAM-IASS Symposium on Deployable Structures: Theory

and Applications. Proceedings of the IUTAM-IASS Symposium held in Cambridge, U.K., 6-9 September 1998. 2000 ISBN 0-7923-6516-X

81. A.D. Rosato and D.L. Blackmore (eds.): IUTAM Symposium on Segregation in Granular Flows. Proceedings of the IUTAM Symposium held in Cape May, NJ, U.S.A., June 5-10, 1999.2000 ISBN 0-7923-6547-X

82. A. Lagarde (ed.): IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics. Proceedings of the IUTAM Symposium held in Futuroscope, Poitiers, France, August 31-September 4,1998.2000 ISBN 0-7923-6604-2

83. D. Weichert and G. Maier (eds.): Inelastic Analysis of Structures under Variable Loads. Theory and Engineering Applications. 2000 ISBN 0-7923-6645-X

84. T.-J. Chuang and J.W. Rudnicki (eds.): Multiscale Deformation and Fracture in Materials and Structures. The James R. Rice 60th Anniversary Volume. 2001 ISBN 0-7923-6718-9

85. S. Narayanan and R.N. Iyengar (eds.): IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. Proceedings of the IUTAM Symposium held in Madras, Chennai, India, 4-8 January 1999 ISBN 0-7923-6733-2

86. S. Murakami and N. 0hno (eds.): IUTAM Symposium on Creep in Structures. Proceedings of the IUTAM Symposium held in Nagoya, Japan, 3-7 April 2000. 2001 ISBN 0-7923-6737-5

87. W. Ehlers (ed.): IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials. Proceedings of the IUTAM Symposium held at the University of Stuttgart, Germany, September 5-10, 1999. 2001 ISBN 0-7923-6766-9

88. D. Durban, D. Givoli and J.G. Simmonds (eds.): Advances in the Mechanis of Plates and Shells The Avinoam Libai Anniversary Volume. 2001 ISBN 0-7923-6785-5

89. U. Gabbert and H.-S. Tzou (eds.): IUTAM Symposium on Smart Structures and Structonic Sys­tems. Proceedings of the IUTAM Symposium held in Magdeburg, Germany, 26-29 September 2000.2001 ISBN 0-7923-6968-8

90. Y. Ivanov, V. Cheshkov and M. Natova: Polymer Composite Materials -Interface Phenomena & Processes. 2001 ISBN 0-7923-7008-2

Mechanics SOLID MECHANICS AND ITS APPLICATIONS

Series Editor: G.M.L. Gladwell

91. RC. McPhedran, L.c. Botten and N.A. Nicorovici (eds.): IUTAM Symposium on Mechanical and Electromagnetic Waves in Structured Media. Proceedings of the IUTAM Symposium held in Sydney, NSW, Australia, 18-22 Januari 1999. 2001 ISBN 0-7923-7038-4

92. D.A. Sotiropoulos (ed.): IUTAM Symposium on Mechanical Waves for Composite Structures Characterization. Proceedings of the IUTAM Symposium held in Chania, Crete, Greece, June 14-17,2000.2001 ISBN 0-7923-7164-X

93. Y.M. Alexandrov and D.A. Pozharskii: Three-Dimensional Contact Problems. 2001 ISBN 0-7923-7165-8

94. J.P. Dempsey and H.H. Shen (eds.): IUTAM Symposium on Scaling Laws in Ice Mechanics and Ice Dynamics. Proceedings of the IUTAM Symposium held in Fairbanks, Alaska, U.S.A., 13-16 June 2000.2001 ISBN 1-4020-0171-1

95. U. Kirsch: Design-Oriented Analysis of Structures. A Unified Approach. 2002 ISBN 1-4020-0443-5

96. A. Preumont: Vibration Control of Active Structures. An Introduction (2nd Edition). 2002 ISBN 1-4020-0496-6

97. B.L. Karihaloo (ed.): IUTAM Symposium on Analytical and Computational Fracture Mechan­ics of Non-Homogeneous Materials. Proceedings of the IUTAM Symposium held in Cardiff, U.K., 18-22 June 2001. 2002 ISBN 1-4020-0510-5

98. S.M. Han and H. Benaroya: Nonlinear and Stochastic Dynamics of Compliant Offshore Struc-tures.2002 ISBN 1-4020-0573-3

99. A.M. Linkov: Boundary Integral Equations in Elasticity Theory. 2002 ISBN 1-4020-0574-1

100. L.P. Lebedev, 1.1. Vorovich and G.M.L. Gladwell: Functional Analysis. Applications in Me­chanics and Inverse Problems (2nd Edition). 2002

ISBN 1-4020-0667-5; Pb: 1-4020-0756-6 101. Q.P. Sun (ed.): IUTAM Symposium on Mechanics of Martensitic Phase Transformation in

Solids. Proceedings of the IUTAM Symposium held in Hong Kong, China, 11-15 June 2001. 2002 ISBN 1-4020-0741-8

102. M.L. Munjal (ed.): IUTAM Symposium on Designingfor Quietness. Proceedings of the IUTAM Symposium held in Bangkok, India, 12-14 December 2000.2002 ISBN 1-4020-0765-5

103. J.A.C. Martins and M.D.P. Monteiro Marques (eds.): Contact Mechanics. Proceedings of the 3rd Contact Mechanics International Symposium, Praia da Consola~ao, Peniche, Portugal, 17-21 June 2001. 2002 ISBN 1-4020-0811-2

104. H.R. Drew and S. Pellegrino (eds.): New Approaches to Structural Mechanics, Shells and Biological Structures. 2002 ISBN 1-4020-0862-7

105. J.R Vinson and RL. Sierakowski: The Behavior of Structures Composed of Composite Ma-terials. Second Edition. 2002 ISBN 1-4020-0904-6

106. Not yet published. 107. J.R. Barber: Elasticity. Second Edition. 2002 ISBN Hb 1-4020-0964-X; Pb 1-4020-0966-6 108. C. Miehe (ed.): IUTAM Symposium on Computational Mechanics of Solid Materials at Large

Strains. Proceedings of the IUTAM Symposium held in Stuttgart, Germany, 20-24 August 2001. 2003 ISBN 1-4020-1170-9

Mechanics SOLID MECHANICS AND ITS APPLICATIONS

Series Editor: G.M.L. Gladwell

109. P. Stillile and K.G. Sundin (eds.): IUTAM Symposium on Field Analyses for Determination of Material Parameters - Experimental and Numerical Aspects. Proceedings of the IUTAM Symposium held in Abisko National Park, Kiruna, Sweden, July 31 - August 4, 2000. 2003

ISBN 1-4020-1283-7 110. N. Sri Namachchivaya and Y.K. Lin (eds.): IUTAM Symposium on Nonlnear Stochastic Dynam­

ics. Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26 - 30 August, 2000.2003 ISBN 1-4020-1471-6

111. H. Sobieckzky (ed.): IUTAM Symposium Transsonicum IV. Proceedings of the IUTAM Sym-posium held in Gottingen, Germany, 2-6 September 2002,2003 ISBN 1-4020-1608-5

112. J.-c. Sarnin and P. Fisette: Symbolic Modeling of Multibody Systems. 2003 ISBN 1-4020-1629-8

113. A.B. Movchan (ed.): IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics. Proceedings of the IUTAM Symposium held in Liverpool, United Kingdom, 8-11 July 2002. 2003 ISBN 1-4020-1780-4

114. S. Ahzi, M. Cherkaoui, M.A. Khaleel, H.M. Zbib, M.A. Zikry and B. LaMatina (eds.): IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials. Proceedings of the IUTAM Symposium held in Marrakech, Morocco, 20-25 October 2002.2004 ISBN 1-4020-1861-4

115. H. Kitagawa and Y. Shibutani (eds.): IUTAM Symposium on Mesoscopic Dynamics of Fracture Process and Materials Strength. Proceedings of the IUTAM Symposium held in Osaka, Japan, 6-11 July 2003. Volume in celebration of Professor Kitagawa's retirement. 2004

ISBN 1-4020-2037-6

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