Nearly all solids are compised of grains. However most studies treat materials as a continious solid. The book applies analysis used on loose granular materials to dense grainular materials. This titles main focus is devoted to static or dynamic loadings applied to dense materials, although rapid flows and widely dispersed media are also mentioned briefly. Three essential areas are covered: Local variable analysis: Contact forces, displacements and rotations, orientation of contacting particles and fabric tensors are all examples of local variables. Their statistical distributions, such as spatial distribution and possible localization, are analyzed, taking into account experimental results or numerical simulations. Change of scales procedures: Also known as homogenization techniques, these procedures make it possible to construct continuum laws to be used in a continuum mechanics approach or performing smaller scale analyses. Numerical modeling: Several methods designed to calculate approximate solutions of dynamical equations together with unilateral contact and frictional laws are presented, including molecular dynamics, the distinct element method and non-smooth contact dynamics. Numerical examples are given and the quality of numerical approximations is discussed.
Introduction xiii
Chapter 1. Experimental and Numerical Analysis of Local Variables in Granular Materials 1
Farhang RADJAÏ and Jack LANIER
1.1. Introduction 1
1.2. Description of granular texture 3
1.3. Granular kinematics 20
1.4. Force transmission 31
1.5. Conclusion 44
1.6. Bibliography 45
Chapter 2. The Stress Tensor in Granular Media and in other Mechanical Collections 51
Jean-Jacques MOREAU
2.1. Introduction 51
2.2. Efforts and virtual power 59
2.3. Equilibrium 65
2.4. Comparison with the pair-by-pair approach 76
2.5. Directions of cut 83
2.6. Coarse graining the equation of Statics 90
2.7. One step into Dynamics 91
2.8. Bibliography 97
Chapter 3. Multiscale Techniques for Granular Materials 101
Bernard CAMBOU, Alexandre DANESCU
3.1. Introduction 101
3.2. Scale change andfabric tensors 102
3.3. Change of scale for static variables . 112
3.4. Change of scale for kinematic variables in granular materials 115
3.5. Statistical homogenization in granular materials 131
3.6. Bibliography 145
Chapter 4. Numerical Simulation of Granular Materials 149
Michel JEAN
4.1. Introduction 149
4.2. The actors of a contact problem 152
4.3. Kinematic relations 167
4.4. The dynamical equation 174
4.5. Frictional contact laws 179
4.6. The equations governing a collection of contacting bodies 196
4.7. Preparing numerical samples 198
4.8. Smooth DEM numerical methods 206
4.9. Non-smooth DEM numerical methods 216
4.10. Some illustrating examples 227
4.11. Quasi-static evolutions, equilibrium dedicated methods 234
4.12. Accuracy criteria 241
4.13. Indetermination in granular materials 246
4.14. Stability 265
4.15. Numerical integration schemes 275
4.16. More non-smooth DEM methods 284
4.17. Signorini -Coulomb derived laws 292
4.18. Conclusion 301
4.19. Appendix: basic convex analysis 303
4.20. Bibliography 307
Chapter 5. Frictionless Unilateral Multibody Dynamics 317
Patrick BALLARD
5.1. Introduction 317
5.2. The dynamics of rigid body systems 318
5.3. The dynamics of rigid body systems with perfect bilateral constraints 322
5.4. The dynamics of rigid body systems with perfect unilateral constraints 326
5.5. Bibliography 340
List of Authors 343
Index 345
