Contents 7
Preface 13
List of Contributors 17
I Static Properties 21
1 Stress in Dense Granular Materials 23
1.1 Introduction 23
1.2 Continuum Mechanics: A Brief Review 24
1.3 Constitutive Relations for Dense Granular Materials 25
1.3.1 Engineering Approaches 25
1.3.2 Recent Approaches 26
1.3.3 Experiments and Possible Reconciliation 26
1.4 A Microscopic Approach 27
1.4.1 Displacement and Strain 28
1.4.2 Microscopic Derivation of Elasticity 29
1.5 Forces, Stress and Response Functions 30
1.5.1 Force Models 30
1.5.2 Force Chains, Stress, Elasticity and Friction 31
1.5.3 Force Statistics 39
1.6 Concluding Remarks 39
References 40
2 Response Functions in Isostatic Packings 43
2.1 Introduction 43
2.2 Rigidity Considerations for Contact Networks 44
2.2.1 Formulation 44
2.2.2 Network Rigidity 45
2.2.3 Isostaticity in the Limit of Large Stiffness to Load Ratio 47
2.3 Consequences of Isostaticity 48
2.3.1 Green Functions and the Virtual Work Principle 48
2.3.2 Anomalous Fluctuations: Multiplicative Noise in Isostatic Networks 49
2.4 Specific Examples 52
2.4.1 Topologically and Positionally Regular Isostatic Networks 53
2.4.2 Topologically Regular Isostatic Networks with Positional Disorder 53
2.4.3 Topologically Disordered Positionally Regular Isostatic Networks 55
2.4.4 Topologically and Positionally Disordered Isostatic Networks 56
2.4.5 Non-sequential Isostatic Networks 57
2.5 Discussion 59
References 61
3 Statistical Mechanics of Jammed Matter 65
3.1 Introduction to the Concept of Jamming 65
3.1.1 Jamming in Glassy Systems 66
3.1.2 Jamming in Particulate Systems 68
3.1.3 Unifying Concepts in Granular Matter and Glasses 71
3.2 New Statistical Mechanics for Granular Matter 73
3.2.1 Classical Statistical Mechanics 73
3.2.2 Statistical Mechanics for Jammed Matter 74
3.2.3 The Classical Boltzmann Equation 79
3.2.4 \u201cBoltzmann Approach\u201d to Granular Matter 81
3.3 Jamming with the Confocal 84
3.3.1 From Micromechanics to Thermodynamics 84
3.3.2 Model System 85
3.4 Jamming in a Periodic Box 92
3.4.1 Simulating Jamming 93
3.4.2 Testing the Thermodynamics 97
References 103
II Granular Gas 107
4 The Inelastic Maxwell Model 109
4.1 Introduction 109
4.2 Uniform Gases: One Dimension 110
4.2.1 The Freely Cooling Case 110
4.2.2 The Forced Case 114
4.3 Uniform Gases: Arbitrary Dimension 116
4.3.1 The Freely Cooling Case 116
4.3.2 The Forced Case 120
4.3.3 Velocity Correlations 121
4.4 Impurities 122
4.4.1 Model A 123
4.4.2 Model B 126
4.4.3 Velocity Autocorrelations 128
4.5 Mixtures 128
4.6 Lattice Gases 129
4.7 Conclusions 131
References 133
5 Cluster Formation in Compartmentalized Granular Gases 137
5.1 Introduction 137
5.2 The Vertically Vibrated Experiment 139
5.3 Eggers\u2019 Flux Model 141
5.4 Extension to More than two Compartments 144
5.5 Urn Model 147
5.6 Horizontally Vibrated System 152
5.7 Double Well Model 154
5.8 Further Directions 155
References 156
III Dense Granular Flow 161
6 Continuum Modeling of Granular Flow and Structure Formation 163
6.1 Introduction 163
6.2 Order Parameter Description of Partially Fluidized Granular Flows 164
6.3 Avalanches on an Inclined Plane 167
6.3.1 Stability of Simple Solution 168
6.3.2 Avalanches in a Single-mode Approximation 169
6.3.3 Comparison with Experiment 170
6.4 Fitting the Theory with Molecular Dynamics Simulations 172
6.4.1 Order Parameter for Granular Fluidization: Static Contacts vs. Fluid Contacts 172
6.4.2 Stress Tensor 173
6.4.3 Couette Flow in a Thin Granular Layer 173
6.4.4 Fitting the Constitutive Relation 174
6.5 Surface-driven Shear Granular Flow Under Gravity 175
6.6 Stick-Slips and Granular Friction 178
6.7 Conclusions 182
References 183
7 Contact Dynamics Study of 2D Granular Media: Critical States and Relevant Internal Variables 185
7.1 A Geometry\u2013Mechanics Dialogue 185
7.2 A Granular Model 185
7.2.1 Contact Dynamics 186
7.2.2 Driving Modes 187
7.3 Macroscopic Continuum Description 188
7.3.1 Constitutive Framework 188
7.3.2 Relation Between Micro- and Macro-descriptors 189
7.3.3 Internal Variables 190
7.4 Numerical Results 191
7.4.1 Critical States 191
7.4.2 Stress\u2013Strain Relation 194
7.4.3 Dilatancy 196
7.4.4 Internal Variables 199
7.4.5 Evolution of Internal Variables 201
7.4.6 Frictional/Collisional Dissipation 204
7.5 Conclusion 205
References 206
8 Collision of Adhesive Viscoelastic Particles 209
8.1 Introduction 209
8.2 Forces Between Granular Particles 210
8.2.1 Elastic Forces 210
8.2.2 Viscous Forces 213
8.2.3 Adhesion of Contacting Particles 216
8.3 Collision of Granular Particles 219
8.3.1 Coefficient of Restitution 219
8.3.2 Dimensional Analysis 220
8.3.3 Coefficient of Restitution for Spheres 222
8.3.4 Coefficient of Restitution for Adhesive Collisions 225
8.4 Conclusion 227
References 228
IV Hydrodynamic Interactions 231
9 Fluidized Beds: From Waves to Bubbles 233
9.1 Introduction 233
9.2 Flow Regimes and Instabilities 234
9.3 Instability Mechanism 236
9.4 Governing Equations 238
9.5 Primary Instability 239
9.6 Rheology of the Particle Phase 242
9.7 Secondary Instability and the Formation of Bubbles 243
9.8 Conclusions 248
References 249
10 Wind-blown Sand 253
10.1 Introduction 253
10.2 The Wind Field 254
10.3 Aeolian Sand Transport 259
10.4 Dunes 266
10.5 Conclusion 269
References 270
V Charged and Magnetic Granular Matter 273
11 Electrostatically Charged Granular Matter 275
11.1 Introduction 275
11.2 Charged Granular Matter in Vacuum 276
11.3 Charged Granular Matter in Suspension 280
11.4 Agglomeration of Monopolar Charged Suspensions 282
11.4.1 Mean Field Rate Equation 283
11.4.2 Self-focussing Size Distribution 285
11.4.3 Brownian Dynamics Simulations 287
11.5 Coating Particles in Bipolarly Charged Suspensions 291
11.5.1 Coulomb Interaction vs. Translational Brownian Motion 293
11.5.2 Coulomb Interaction vs. Rotational Brownian Motion 296
11.6 Summary 297
References 298
12 Magnetized Granular Materials 301
12.1 Introduction 301
12.2 Background: Dipolar Hard Spheres 302
12.3 Experimental Technique 303
12.4 The Phase Diagram 305
12.5 The Non-equipartition of Energy 308
12.6 Cluster Growth Rates 310
12.7 Compactness of the Cluster 312
12.8 Migration of Clusters 313
12.9 Summary 313
References 315
VI Computational Aspects 317
13 Molecular Dynamics Simulations of Granular Materials 319
13.1 Introduction 319
13.2 The Soft-particle Molecular Dynamics Method 320
13.2.1 Discrete-particle Model 320
13.2.2 Equations of Motion 320
13.2.3 Contact Force Laws 321
13.3 Hard-sphere Molecular Dynamics 325
13.3.1 Smooth Hard-sphere Collision Model 325
13.3.2 Event-driven Algorithm 326
13.4 The Link between ED and MD via the TC Model 327
13.5 The Stress in Particle Simulations 329
13.5.1 Dynamic Stress 329
13.5.2 Static Stress from Virtual Displacements 330
13.5.3 Stress for Soft and Hard Spheres 330
13.6 2D Simulation Results 331
13.6.1 The Equation of State from ED 331
13.6.2 Quasi-static MD Simulations 332
13.7 Large-scale Computational Examples 336
13.7.1 Cluster Growth (ED) 336
13.7.2 3D Ring-shear Cell Simulation 338
13.8 Conclusion 341
References 342
14 Contact Dynamics for Beginners 345
14.1 Introduction 345
14.2 Discrete Dynamical Equations 346
14.3 Volume Exclusion in a One-dimensional Example 347
14.4 The Three-dimensional Single Contact Case Without Cohesion 349
14.5 Iterative Determination of Constraint Forces in a Multi-contact System 353
14.6 Computational Effort: Comparison Between CD and MD 356
14.7 Rolling and Torsion Friction 357
14.8 Attractive Contact Forces 359
14.9 Conclusion 360
References 361
Index 365