Description
Book SynopsisA material's various proprieties is based on its microscopic and nanoscale structures. This book provides an overview of recent advances in computational methods for linking phenomena in systems that span large ranges of time and spatial scales. Particular attention is given to predicting macroscopic properties based on subscale behaviors. Given the book’s extensive coverage of multi-scale methods for modeling both metallic and geologic materials, it will be an invaluable reading for graduate students, scientists, and practitioners alike.
Table of ContentsForeword 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
