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 title s 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.

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 title's 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 xiiiChapter 1. Experimental and Numerical Analysis of LocalVariables in Granular Materials 1Farhang RADJAÏ and Jack LANIER1.1. Introduction 11.2. Description of granular texture 31.3. Granular kinematics 201.4. Force transmission 311.5. Conclusion 441.6. Bibliography 45Chapter 2. The Stress Tensor in Granular Media and in otherMechanical Collections 51Jean-Jacques MOREAU2.1. Introduction 512.2. Efforts and virtual power 592.3. Equilibrium 652.4. Comparison with the pair-by-pair approach 762.5. Directions of cut 832.6. Coarse graining the equation of Statics 902.7. One step into Dynamics 912.8. Bibliography 97Chapter 3. Multiscale Techniques for Granular Materials101Bernard CAMBOU, Alexandre DANESCU3.1. Introduction 1013.2. Scale change andfabric tensors 1023.3. Change of scale for static variables . 1123.4. Change of scale for kinematic variables in granularmaterials 1153.5. Statistical homogenization in granular materials 1313.6. Bibliography 145Chapter 4. Numerical Simulation of Granular Materials149Michel JEAN4.1. Introduction 1494.2. The actors of a contact problem 1524.3. Kinematic relations 1674.4. The dynamical equation 1744.5. Frictional contact laws 1794.6. The equations governing a collection of contacting bodies1964.7. Preparing numerical samples 1984.8. Smooth DEM numerical methods 2064.9. Non-smooth DEM numerical methods 2164.10. Some illustrating examples 2274.11. Quasi-static evolutions, equilibrium dedicated methods2344.12. Accuracy criteria 2414.13. Indetermination in granular materials 2464.14. Stability 2654.15. Numerical integration schemes 2754.16. More non-smooth DEM methods 2844.17. Signorini mu-Coulomb derived laws 2924.18. Conclusion 3014.19. Appendix: basic convex analysis 3034.20. Bibliography 307Chapter 5. Frictionless Unilateral Multibody Dynamics317Patrick BALLARD5.1. Introduction 3175.2. The dynamics of rigid body systems 3185.3. The dynamics of rigid body systems with perfect bilateralconstraints 3225.4. The dynamics of rigid body systems with perfect unilateralconstraints 3265.5. Bibliography 340List of Authors 343Index 345