Micromechanics of Granular Materials

Micromechanics of Granular Materials......Page 5
Contents......Page 7
Introduction......Page 15
1.1. Introduction......Page 19
1.2.1. Particle connectivity......Page 21
1.2.2.1. General case......Page 23
1.2.2.2. Case of 2D data......Page 25
1.2.2.3. Case of 3D data......Page 29
1.2.3. Branch vectors......Page 31
1.2.4. Evolution of granular texture......Page 32
1.2.5.2. Dirichlet cells......Page 35
1.2.5.4. Neighborhoods and local void ratios......Page 36
1.3.1. Particle displacements and rotations......Page 38
1.3.2. Rolling versus sliding......Page 39
1.3.3.1. Uniform strain and fluctuations......Page 41
1.3.3.2. Probability densities......Page 43
1.3.3.3. Spatial correlations......Page 44
1.3.4. Local and global strains......Page 45
1.3.4.1. Particle-scale strain......Page 46
1.3.4.2. Strain localization......Page 47
1.4. Force transmission......Page 49
1.4.1. Probability density functions......Page 50
1.4.2. Bimodal character of stress transmission......Page 57
1.4.3. Force anisotropy......Page 60
1.5. Conclusion......Page 62
1.6. Bibliography......Page 63
2.1. Introduction......Page 69
2.1.1. Motivation......Page 70
2.1.2. The theoretical background......Page 71
2.1.3. Dynamics......Page 73
2.1.4. Pertinence......Page 74
2.2.1. Resultant and moment of an effort......Page 77
2.2.2. Internal efforts......Page 78
2.2.4. Efforts of order greater than zero......Page 80
2.2.5. Contact actions......Page 82
2.3.1.1. Case of a continuous body......Page 83
2.3.1.2. Case of a granular material......Page 84
2.3.2. Classical continuous body......Page 85
2.3.3. Piece of string......Page 86
2.3.4. Finite collection of points......Page 89
2.3.5. Interaction bridges......Page 91
2.3.6. Saturated soil......Page 93
2.4.1. The classical definition......Page 94
2.4.1.1. The pair-by-pair calculation......Page 95
2.4.2. Numerical discussion of a tri-axial test......Page 97
2.5.1. Force transmitted across a cut......Page 101
2.5.2. Proof of the cutting direction law......Page 102
2.5.3.1. Free surface law......Page 105
2.5.4. Conical pile......Page 106
2.6.1. The divergence operator......Page 108
2.7.1. Introducing the acceleration field......Page 109
2.7.2.1. Introducing the mass center......Page 110
2.7.2.2. Introducing principal axes......Page 111
2.7.2.3. Invoking rigid body dynamics......Page 112
2.7.2.5. 2D models......Page 113
2.7.3. Percussions......Page 114
2.8. Bibliography......Page 115
3.1. Introduction......Page 119
3.2.1.2. Shape of particles......Page 120
3.2.2.1. Coordination number and compactness......Page 124
3.2.2.2. Definition of the overall anisotropy of a sample......Page 126
3.2.3. Voids description......Page 129
3.3. Change of scale for static variables......Page 130
3.4.1. Definition of local kinematic variables......Page 133
3.4.2. Method based on an energetic approach......Page 135
3.4.3.1. Strain proposed by Kruyt and Rothenburg in 2D......Page 136
3.4.3.2. Strain proposed by Cambou et al. in 2D......Page 139
3.4.4.1. Strain proposed by Cundal......Page 141
3.4.4.2. Strain proposed by Liao et al......Page 142
3.4.5. Analysis of the different microstructural definitions of strain and comparison with the macro strain defined at the considered sample scale......Page 144
3.5. Statistical homogenization in granular materials......Page 149
3.5.1.1. Model based on kinematic localization......Page 150
3.5.1.2. Model based on static localization......Page 151
3.5.2.1. Voigt-type hypothesis for kinematic localization......Page 153
3.5.2.2. Static localization hypothesis......Page 154
3.5.3. Extension to nonlinear elasticity......Page 157
3.5.4. Definition of a yield surface from a local criterion......Page 158
3.5.5. Difficulties and limitations for statistical homogenization in granular materials......Page 161
3.6. Bibliography......Page 163
4.1. Introduction......Page 167
4.2. The actors of a contact problem......Page 170
4.2.1. Bodies, contactors and candidates to contact......Page 171
4.2.2. Some bodies and contactors used in numerical simulation......Page 174
4.2.3. Sorting......Page 182
4.3.1. Usual rigid body kinematics......Page 185
4.3.2. Local variables......Page 186
4.3.3. The distance function......Page 187
4.3.4. Relations between generalized and local variables......Page 188
4.3.4.2. Non-uniqueness......Page 190
4.3.5. Boundary conditions, driven or locked degrees of freedom......Page 191
4.4. The dynamical equation......Page 192
4.4.1. 2D or 3D bodies......Page 193
4.4.3. Shocks, momentum, impulses and percussions......Page 194
4.4.4. Energy formulae......Page 196
4.5. Frictional contact laws......Page 197
4.5.1.1. Signorini conditions......Page 198
4.5.1.2. Complementary relation and convex analysis......Page 199
4.5.1.3. Flexibility models......Page 200
4.5.1.4. Shock laws......Page 202
4.5.2.1. Coulomb’s low......Page 204
4.5.2.2. Coulomb’s law and convex analysis......Page 206
4.5.2.4. Dynamical friction, static friction coefficients......Page 207
4.5.3. Choosing a frictional contact law......Page 209
4.5.4. Cohesive behavior......Page 210
4.5.4.2. Rolling grains, welded grains......Page 213
4.6. The equations governing a collection of contacting bodies......Page 214
4.7. Preparing numerical samples......Page 216
4.7.1. Boundary conditions......Page 217
4.7.2. Initial state......Page 221
4.7.3. Size of samples......Page 223
4.8.1. Molecular dynamics methods......Page 224
4.8.2.1. Discretizing the dynamical equation......Page 225
4.8.3. PCDEM methods......Page 226
4.8.3.1. Discrete form of the frictional law Reac......Page 229
4.8.3.3. Numerical scheme......Page 230
4.8.3.4. Remarks......Page 231
4.8.4. Choosing the time-step......Page 232
4.9.1.1. Computing a collision......Page 234
4.9.2.1. Discretization of the dynamical equation......Page 238
4.9.2.2. Discrete form of kinematic relations......Page 239
4.9.2.3. Discrete forms of frictional contact relations......Page 240
4.9.2.4. Restriction of the dynamical equation to candidates to contact......Page 241
4.9.2.7. Solving the Signorini µ -Coulomb standard problem......Page 242
4.9.2.9. Solving the frictional contact problem, Gauss Seidel nesting......Page 244
4.10.1. The bouncing ball problem......Page 245
4.10.2. Frictional contact examples by explicit or implicit methods......Page 248
4.10.2.2. Example 2......Page 249
4.10.2.4. Example 4......Page 251
4.11. Quasi-static evolutions, equilibrium dedicated methods......Page 252
4.11.1. A strongly viscous contact law......Page 253
4.11.2. Flexibility models......Page 254
4.11.3. Rigid bodies and Signorini, µ-Coulomb law......Page 255
4.11.4. Quasi-static evolutions versus dynamics......Page 256
4.12. Accuracy criteria......Page 259
4.12.1. Implicit methods......Page 260
4.12.3.1. Mean and quadratic violations......Page 261
4.12.3.2. Bipotential violation......Page 263
4.13. Indetermination in granular materials......Page 264
4.13.1.1. A rigid wedged disk example......Page 266
4.13.1.2. Analyzing the kinematic indetermination......Page 267
4.13.1.3. A classical example of deformable model......Page 268
4.13.1.5. The single wedged disk......Page 271
4.13.1.6. Loading experiment, domains of attraction......Page 273
4.13.1.7. Another view of domains of attraction: rigid model......Page 275
4.13.1.9. Indetermination in the deformable model......Page 278
4.13.2.1. Three wedged disks......Page 279
4.13.2.2. 200 disks sample......Page 281
4.14. Stability......Page 283
4.14.1. Perturbations......Page 284
4.14.2. Coulomb stable sample......Page 285
4.14.3. Left reactions perturbations of the single wedged disk......Page 286
4.14.4. Left reactions perturbations of a 2,400 polygon sample......Page 287
4.14.5. Further comments......Page 292
4.15. Numerical integration schemes......Page 293
4.15.1. θmethod......Page 294
4.15.2. Consistency of the discrete approximations......Page 296
4.15.3. Newmark method......Page 298
4.15.4. Deformable grains......Page 299
4.15.5. Further comments......Page 301
4.16.1. The NSCD method, Gauss–Seidel nesting......Page 302
4.16.2. The NSCD method, Jacobi nesting......Page 303
4.16.3. The bi-potential method......Page 304
4.16.4. The generalized Newton method......Page 305
4.16.5. Gradient-type methods......Page 307
4.16.7. Multigrid computation......Page 308
4.16.8. Parallel computation......Page 309
4.17. Signorini µ-Coulomb derived laws......Page 310
4.17.1.1. Status SIGNORINI_CONTACT......Page 311
4.17.2. Change of variables......Page 312
4.17.3. Algorithm NSCD and derived laws......Page 314
4.17.5. Inelastic quasi-choc law and Coulomb’s law......Page 315
4.17.8. Velocity Signorini condition and Coulomb’s law with static or dynamic friction coefficient......Page 316
4.17.10. Mohr Coulomb cohesive law......Page 317
4.17.11. A simple cohesive example......Page 318
4.18. Conclusion......Page 319
4.19.2. Convex functions, conjugates, subdifferential......Page 321
4.19.4. Standard Coulomb’s law......Page 323
4.19.5. Bipotentials......Page 324
4.20. Bibliography......Page 325
5.1. Introduction......Page 335
5.2.1.The geometric description......Page 336
5.2.2. Formulation of the dynamics......Page 337
5.2.3. Well-posedness of the dynamics......Page 339
5.3.1. The geometric description......Page 340
5.3.2. Formulation of the dynamics......Page 341
5.3.3. Well-posedness of the dynamics......Page 343
5.4.1. The geometric description......Page 344
5.4.2.1.Equation of motion......Page 345
5.4.2.2. The impact constitutive equation......Page 347
5.4.2.3. The evolution problem......Page 351
5.4.3. Well-posedness of the dynamics......Page 352
5.4.3.1. Comments......Page 357
5.5. Bibliography......Page 358
List of Authors......Page 361
Index......Page 363