Multiscale Modeling of Heterogenous Materials

Oana Cazacu is Professor in the Department of Mechanical and Aerospace Engineering at the University of Florida, REEF, Shalimar, USA.

Foreword xiii

Chapter 1. Accounting for Plastic Strain Heterogenities in Modeling Polycrystalline Plasticity: Microstructure-based Multi-laminate Approaches 1
Patrick FRANCIOSI

1.1. Introduction 1

1.2. Polycrystal morphology in terms of grain and sub-grain boundaries 2

1.2.1. Some evidence of piece-wise regularity for grain boundaries 2

1.2.2. Characteristics of plastic-strain due to sub-grain boundaries 3

1.3. Sub-boundaries and multi-laminate structure for heterogenous plasticity 5

1.3.1. Effective moduli tensor and Green operator of multi-laminate structures 6

1.3.2. Multi-laminate structures and piece-wise homogenous plasticity 10

1.4. Application to polycrystal plasticity within the affine approximation 10

1.4.1. Constitutive relations 10

1.4.2. Fundamental properties for multi-laminate modeling of plasticity 14

1.5. Conclusion 15

1.6. Bibliography 15

Chapter 2. Discrete Dislocation Dynamics: Principles and Recent Applications 17
Marc FIVEL

2.1. Discrete Dislocation Dynamics as a link in multiscale modeling 17

2.2. Principle of Discrete Dislocation Dynamics 19

2.3. Example of scale transition: from DD to Continuum Mechanics 21

2.3.1. Introduction to a dislocation density model 21

2.3.1.1. Constitutive equations of a dislocation based model of crystal plasticity 22

2.3.1.2. Parameter identification 24

2.3.1.3. Application to copper simulations 25

2.3.1.4. Taking into account kinematic hardening 26

2.4. Example of DD analysis: simulations of crack initiation in fatigue 29

2.4.1. Case of single phase AISI 31GL 29

2.5. Conclusions 32

2.6. Bibliography 33

Chapter 3. Multiscale Modeling of Large Strain Phenomenain Polycrystalline Metals 37
Kaan INAL and Raj. K. MISHRA

3.1. Implementation of polycrystal plasticity in finite element analysis 39

3.2. Kinematics and constitutive framework 41

3.3. Forward Euler algorithm 44

3.4. Validation of the forward Euler algorithm 46

3.5. Time step issues in the forward Euler scheme 49

3.6. Comparisons of CPU times: the rate tangent versus the forward Euler methods 51

3.7. Conclusions 52

3.8. Acknowledgements 52

3.9. Bibliography 52

Chapter 4. Earth Mantle Rheology Inferred from Homogenization Theories 55
Olivier CASTELNAU, Ricardo LEBENSOHN, Pedro Ponte CASTAÑEDA and Donna BLACKMAN

4.1. Introduction 55

4.2. Grain local behavior 57

4.3. Full-field reference solutions 59

4.4. Mean-field estimates 62

4.4.1. Basic features of mean-field theories 62

4.4.2. Results 64

4.5. Concluding observations 66

4.6. Bibliography 68

Chapter 5. Modeling Plastic Anistropy and Strength Differential Effects in Metallic Materials 71
Oana CAZACU and Frédéric BARLAT

5.1. Introduction 71

5.2. Isotropic yield criteria 72

5.2.1. Pressure insensitive materials deforming by slip 72

5.2.2. Pressure insensitive materials deforming by twinning 73

5.2.3. Pressure insensitive materials with non-Schmid effects 76

5.2.4. Pressure sensitive materials 78

5.2.5. SD effect and plastic flow 80

5.3. Anisotropic yield criteria with SD effects 80

5.3.1. Cazacu and Barlat [CAZ 04] orthotropic yield criterion 80

5.3.2. Cazacu Plunkett Barlat yield criterion [CAZ 06] 82

5.4. Modeling anisotropic hardening due to texture evolution 83

5.5. Conclusions and future perspectives 86

5.6. Bibliography 87

Chapter 6. Shear Bands in Steel Plates under Impact Loading 91
George Z. VOYIADJIS and Amin H. ALMASRI

6.1. Introduction 91

6.2. Viscoplasticity and constitutive modeling 92

6.3. Higher order gradient theory 97

6.4. Two-dimensional plate subjected to velocity boundary conditions 102

6.5. Shear band in steel plate punch 105

6.6. Conclusions 108

6.7. Bibliography 109

Chapter 7. Viscoplastic Modeling of Anisotropic Textured Metals 111
Brian PLUNKETT and Oana CAZACU

7.1. Introduction 111

7.2. Anisotropic elastoviscoplastic model 113

7.3. Application to zirconium. 115

7.3.1. Quasi-static deformation of zirconium 115

7.3.2. High strain-rate deformation of zirconium 120

7.4. High strain-rate deformation of tantalum 124

7.5. Conclusions125

7.6. Bibliography 126

Chapter 8. Non-linear Elastic Inhomogenous Materials: Uniform Strain Fields and Exact Relations 129
Qi-Chang HE, B. BARY and Hung LE QUANG

8.1. Introduction 129

8.2. Locally uniform strain fields 130

8.3. Exact relations for the effective elastic tangent moduli 136

8.4. Cubic polycrystals 139

8.5. Power-law fibrous composites 144

8.6. Conclusion 149

8.7. Bibliography 149

Chapter 9. 3D Continuous and Discrete Modeling of Bifurcations in Geomaterials 153
Florent PRUNIER, Félix DARVE, Luc SIBILLE and François NICOT

9.1. Introduction 153

9.2. 3D bifurcations exhibited by an incrementally non-linear constitutive relation 155

9.2.1. Incrementally non-linear and piecewise linear relations 155

9.2.2. 3D analysis of the second-order work with phenomenological constitutive models 157

9.3. Discrete modeling of the failure mode related to second-order work criterion 165

9.4. Conclusions 173

9.5. Acknowledgements 174

9.6. Bibliography 174

Chapter 10. Non-linear Micro-cracked Geomaterials: Anisotropic Damage and Coupling with Plasticity 177
Djimédo KONDO, Qizhi ZHU, Vincent MONCHIET and Jian-Fu SHAO

10.1. Introduction 177

10.2. Anisotropic elastic damage model with unilateral effects 179

10.2.1. Homogenization of elastic micro-cracked media 179

10.2.1.1. Micromechanics of media with random microstructure 179

10.2.1.2. Application to micro-cracked media 180

10.2.2. Micro-crack closure condition and damage evolution 181

10.2.2.1. Micro-crack closure effects and unilateral damage 181

10.2.2.2. Damage criterion and evolution law 182

10.2.3. Non-local micromechanics-based damage model 183

10.2.4. Application of the model 184

10.2.4.1. Uniaxial tensile tests 184

10.2.4.2. Predictions of the anisotropic damage model for William’s test 185

10.2.4.3. Numerical analysis of Hassanzadeh’s direct tension test 188

10.3. A new model for ductile micro-cracked materials 188

10.3.1. Introductory observations 188

10.3.2. Basic concepts and methodology of the limit analysis approach 190

10.3.2.1. Representative volume element with oblate voids 190

10.3.2.2. The Eshelby-like velocity field 191

10.3.3. Determination of the macroscopic yield surface 192

10.3.3.1. The question of the boundary conditions 192

10.3.3.2. Principle of the determination of the yield function 193

10.3.3.3. Closed form expression of the macroscopic yield function 193

10.3.4. The particular case of penny-shaped cracks 195

10.4. Conclusions 197

10.5. Acknowledgement 198

10.6. Appendix 198

10.7. Bibliography 198

Chapter 11. Bifurcation in Granular Materials: A Multiscale Approach 203
François NICOT, Luc SIBILLE and Félix DARVE

11.1. Introduction 203

11.2. Microstructural origin of the vanishing of the second-order work 205

11.2.1. The micro-directional model 205

11.2.2. Microstructural expression of the macroscopic second-order work 206

11.2.3. From micro to macro second-order work 208

11.2.4. Micromechanical analysis of the vanishing of the second-order work 210

11.3. Some remarks on the basic micro-macro relation for the second-order work 212

11.4. Conclusion 213

11.5. Bibliography 214

Chapter 12. Direct Scale Transition Approach for Highly-filled Viscohyperelastic Particulate Composites: Computational Study 215
Carole NADOT-MARTIN, Marion TOUBOUL, André DRAGON and Alain FANGET

12.1. Morphological approach in the finite strain framework 216

12.1.1. Geometric schematization 216

12.1.2. Localization-homogenization problem 217

12.1.2.1. Principal tools and stages 217

12.1.2.2. Solving procedure 219

12.2. Evaluation involving FEM/MA confrontations 221

12.2.1. Material geometry, relative representations 221

12.2.2. Loading paths, methodology of analysis 223

12.2.3. MA estimates compared to FEM results for hyperelastic constituents 225

12.2.4. Evaluation involving viscohyperelastic behavior of the matrix 229

12.3. Conclusions and prospects 232

12.4. Bibliography 234

Chapter 13. A Modified Incremental Homogenization Approach for Non-linear Behaviors of Heterogenous Cohesive Geomaterials 237
Ariane ABOU-CHAKRA GUÉRY, Fabrice CORMERY, Jian-Fu SHAO and Djimédo KONDO

13.1. Introduction 237

13.2. Experimental observations on the Callovo-Oxfordian argillite behavior 238

13.2.1. Microstructure and mineralogical composition of the material 238

13.2.2. Brief summary of the macroscopic behavior of the material 239

13.3. Incremental formulation of the homogenized constitutive relation 240

13.3.1. Introduction 240

13.3.2. Limitations of Hill’s incremental method 242

13.3.3. Modified Hill’s incremental method 243

13.4. Modifying of the local constituents’ behaviors 244

13.4.1. Elastoplastic behavior of the clay phase 244

13.4.2. Elastic unilateral damage behavior of the calcite phase 245

13.5. Implementation and numerical validation of the model 247

13.5.1. Local integration of the micromechanical model 247

13.5.2. Comparison with unit cell (finite element) calculation 248

13.6. Calibration and experimental validations of the modified incremental micromechanical model 248

13.7. Conclusions 249

13.8. Acknowledgement 251

13.9. Bibliography 251

Chapter 14. Meso- to Macro-scale Probability Aspects for Size Effects and Heterogenous Materials Failure 253
Jean-Baptiste COLLIAT, Martin HAUTEFEUILLE and Adnan IBRAHIMBEGOVIC

14.1. Introduction 253

14.2. Meso-scale deterministic model 254

14.2.1. Structured meshes and kinematic enhancements 255

14.2.2. Operator split solution for interface failure 257

14.2.3. Comparison between structured and unstructured mesh approach 258

14.3. Probability aspects of inelastic localized failure for heterogenous materials 259

14.3.1. Meso-scale geometry description 260

14.3.2. Stochastic integration 261

14.4. Results of the probabilistic characterization of the two phase material 263

14.4.1. Determination of SRVE size 263

14.4.2. Numerical results and discussion 264

14.5. Size effect modeling 266

14.5.1. Random fields and the Karhunen-Loeve expansion 267

14.5.2. Size effect and correlation length 269

14.6. Conclusion 271

14.7. Acknowledgments 272

14.8. Bibliography 272

Chapter 15. Damage and Permeability in Quasi-brittle Materials: from Diffuse to Localized Properties 277
Gilles PIJAUDIER-CABOT, Frédéric DUFOUR and Marta CHOINSKA

15.1. Introduction 277

15.2. Mechanical problem – continuum damage modeling 279

15.3. Permeability matching law 281

15.3.1. Diffuse damage 281

15.3.2. Localized damage – crack opening versus permeability 281

15.3.3. Matching law 283

15.4. Calculation of a crack opening in continuum damage calculations 283

15.5. Structural simulations 286

15.5.1. Mechanical problem – Brazilian splitting test 287

15.5.2. Evolution of apparent permeability 289

15.6. Conclusions 291

15.7. Acknowledgement 291

15.8. Bibliography 291

Chapter 16. A Multiscale Modeling of Granular Materials with Surface Energy Forces 293
Pierre-Yves HICHER and Ching S. CHANG

16.1. Introduction 293

16.2. Stress-strain model 294

16.2.1. Inter-particle behavior 296

16.2.1.1. Elastic part 296

16.2.1.2. Plastic part 296

16.2.1.3. Interlocking influence 297

16.2.1.4. Elastoplastic force-displacement relationship 298

16.2.2. Stress-strain relationship 298

16.2.2.1. Micro-macro relationship 298

16.2.2.2. Calculation scheme 300

16.2.3. Summary of parameters 301

16.3. Results of numerical simulation without surface energy forces consideration 302

16.4. Granular material with surface energy forces: the example of lunar soil 306

16.4.1. Van der Waals forces 308

16.4.2. Triaxial tests with consideration of surface energy forces 311

16.5. Summary and conclusion 314

16.6. Bibliography 315

Chapter 17. Length Scales in Mechanics of Granular Solids 319
Farhang RADJAI

17.1. Introduction 319

17.2. Model description 320

17.3. Force chains 321

17.3.1. Probability density functions 321

17.3.2. Bimodal character of stress transmission 322

17.3.3. Spatial correlations 324

17.4. Fluctuating particle displacements 325

17.4.1. Uniform strain and fluctuations 325

17.4.2. Scale-dependent pdfs 326

17.4.3. Spatial correlations 328

17.4.4. Granulence 329

17.5. Friction mobilization 330

17.5.1. Critical contacts 330

17.5.2. Evolution of critical contacts 330

17.5.3. Spatial correlations 331

17.6. Conclusion 332

17.7. Acknowledgements 333

17.8. Bibliography 333

List of Authors 337

Index 341