| | Contents | |
| | | |
| |
| | | |
| | Preface | XI |
| | List of Contributors | XV |
| | | |
| I | Static Properties | 1 |
| | | |
| 1 | Stress in Dense Granular Materials
(I. Goldhirsch and C. Goldenberg) | 3 |
| 1.1 | Introduction | 3 |
| 1.2 | Continuum Mechanics: A Brief Review | 4 |
| 1.3 | Constitutive Relations for Dense Granular Materials | 5 |
| 1.3.1 | Engineering Approaches | 5 |
| 1.3.2 | Recent Approaches | 6 |
| 1.3.3 | Experiments and Possible Reconciliation | 6 |
| 1.4 | A Microscopic Approach | 7 |
| 1.4.1 | Displacement and Strain | 8 |
| 1.4.2 | Microscopic Derivation of Elasticity | 9 |
| 1.5 | Forces, Stress and Response Functions | 10 |
| 1.5.1 | Force Models | 10 |
| 1.5.2 | Force Chains, Stress, Elasticity and Friction | 11 |
| 1.5.3 | Force Statistics | 19 |
| 1.6 | Concluding Remarks | 19 |
| | References | 20 |
| 2 | Response Functions in Isostatic Packings
(C. F. Moukarzel) | 23 |
| 2.1 | Introduction | 23 |
| 2.2 | Rigidity Considerations for Contact Networks | 24 |
| 2.2.1 | Formulation | 24 |
| 2.2.2 | Network Rigidity | 25 |
| 2.2.3 | Isostaticity in the Limit of Large Stiffness to Load Ratio | 27 |
| 2.3 | Consequences of Isostaticity | 28 |
| 2.3.1 | Green Functions and the Virtual Work Principle | 28 |
| 2.3.2 | Anomalous Fluctuations: Multiplicative Noise in Isostatic Networks | 29 |
| 2.4 | Specific Examples | 32 |
| 2.4.1 | Topologically and Positionally Regular Isostatic Networks | 33 |
| 2.4.2 | Topologically Regular Isostatic Networks with Positional Disorder | 33 |
| 2.4.3 | Topologically Disordered Positionally Regular Isostatic Networks | 35 |
| 2.4.4 | Topologically and Positionally Disordered Isostatic Networks | 36 |
| 2.4.5 | Non-sequential Isostatic Networks | 37 |
| 2.5 | Discussion | 39 |
| | References | 41 |
| 3 | Statistical Mechanics of Jammed Matter
(H. A. Makse, J. Bruji , and S. F. Edwards) | 45 |
| 3.1 | Introduction to the Concept of Jamming | 45 |
| 3.1.1 | Jamming in Glassy Systems | 46 |
| 3.1.2 | Jamming in Particulate Systems | 48 |
| 3.1.3 | Unifying Concepts in Granular Matter and Glasses | 51 |
| 3.2 | New Statistical Mechanics for Granular Matter | 53 |
| 3.2.1 | Classical Statistical Mechanics | 53 |
| 3.2.2 | Statistical Mechanics for Jammed Matter | 54 |
| 3.2.3 | The Classical Boltzmann Equation | 59 |
| 3.2.4 | “Boltzmann Approach” to Granular Matter | 61 |
| 3.3 | Jamming with the Confocal | 64 |
| 3.3.1 | From Micromechanics to Thermodynamics | 64 |
| 3.3.2 | Model System | 65 |
| 3.4 | Jamming in a Periodic Box | 72 |
| 3.4.1 | Simulating Jamming | 73 |
| 3.4.2 | Testing the Thermodynamics | 77 |
| | References | 83 |
| | | |
| II | Granular Gas | 87 |
| | | |
| 4 | The Inelastic Maxwell Model
(E. Ben-Naim and P. Krapivsky) | 89 |
| 4.1 | Introduction | 89 |
| 4.2 | Uniform Gases: One Dimension | 90 |
| 4.2.1 | The Freely Cooling Case | 90 |
| 4.2.2 | The Forced Case | 94 |
| 4.3 | Uniform Gases: Arbitrary Dimension | 96 |
| 4.3.1 | The Freely Cooling Case | 96 |
| 4.3.2 | The Forced Case | 100 |
| 4.3.3 | Velocity Correlations | 101 |
| 4.4 | Impurities | 102 |
| 4.4.1 | Model A | 103 |
| 4.4.2 | Model B | 106 |
| 4.4.3 | Velocity Autocorrelations | 108 |
| 4.5 | Mixtures | 108 |
| 4.6 | LatticeGases | 109 |
| 4.7 | Conclusions | 111 |
| | References | 113 |
| 5 | Cluster Formation in Compartmentalized Granular Gases
(K. van der Weele, R. Mikkelsen, D. van der Meer, and D. Lohse) | 117 |
| 5.1 | Introduction | 117 |
| 5.2 | The Vertically Vibrated Experiment | 119 |
| 5.3 | Eggers’ Flux Model | 121 |
| 5.4 | Extension to More than two Compartments | 124 |
| 5.5 | Urn Model | 127 |
| 5.6 | Horizontally Vibrated System | 132 |
| 5.7 | Double Well Model | 134 |
| 5.8 | Further Directions | 135 |
| | References | 136 |
| | | |
| III | Dense Granular Flow | 141 |
| | | |
| 6 | Continuum Modeling of Granular Flow and Structure Formation
(I. S. Aranson and L. S. Tsimring) | 143 |
| 6.1 | Introduction | 143 |
| 6.2 | Order Parameter Description of Partially Fluidized Granular Flows | 144 |
| 6.3 | Avalanches on an Inclined Plane | 147 |
| 6.3.1 | Stability of Simple Solution | 148 |
| 6.3.2 | Avalanches in a Single-mode Approximation | 149 |
| 6.3.3 | Comparison with Experiment | 150 |
| 6.4 | Fitting the Theory with Molecular Dynamics Simulations | 152 |
| 6.4.1 | Order Parameter for Granular Fluidization: Static Contacts vs. Fluid Contacts | 152 |
| 6.4.2 | Stress Tensor | 153 |
| 6.4.3 | Couette Flow in a Thin Granular Layer | 153 |
| 6.4.4 | Fitting the Constitutive Relation | 154 |
| 6.5 | Surface-driven Shear Granular Flow Under Gravity | 155 |
| 6.6 | Stick-Slips and Granular Friction | 158 |
| 6.7 | Conclusions | 162 |
| | References | 163 |
| 7 | Contact Dynamics Study of 2D Granular Media:Critical States and Relevant Internal Variables
(F. Radjaï and S. Roux) | 165 |
| 7.1 | A Geometry–Mechanics Dialogue | 165 |
| 7.2 | A Granular Model | 165 |
| 7.2.1 | Contact Dynamics | 166 |
| 7.2.2 | Driving Modes | 167 |
| 7.3 | Macroscopic Continuum Description | 168 |
| 7.3.1 | Constitutive Framework | 168 |
| 7.3.2 | Relation Between Micro- and Macro-descriptors | 169 |
| 7.3.3 | Internal Variables | 170 |
| 7.4 | Numerical Results | 171 |
| 7.4.1 | Critical States | 171 |
| 7.4.2 | Stress–Strain Relation | 174 |
| 7.4.3 | Dilatancy | 176 |
| 7.4.4 | Internal Variables | 179 |
| 7.4.5 | Evolution of Internal Variables | 181 |
| 7.4.6 | Frictional/Collisional Dissipation | 184 |
| 7.5 | Conclusion | 185 |
| | References | 186 |
| 8 | Collision of Adhesive Viscoelastic Particles
(N. V. Brilliantov and T. Pöschel) | 189 |
| 8.1 | Introduction | 189 |
| 8.2 | Forces Between Granular Particles | 190 |
| 8.2.1 | Elastic Forces | 190 |
| 8.2.2 | Viscous Forces | 193 |
| 8.2.3 | Adhesion of Contacting Particles | 196 |
| 8.3 | Collision of Granular Particles | 199 |
| 8.3.1 | Coefficient of Restitution | 199 |
| 8.3.2 | Dimensional Analysis | 200 |
| 8.3.3 | Coefficient of Restitution for Spheres | 202 |
| 8.3.4 | Coefficient of Restitution for Adhesive Collisions | 205 |
| 8.4 | Conclusion | 207 |
| | References | 208 |
| | | |
| IV | Hydrodynamic Interactions | 211 |
| | | |
| 9 | Fluidized Beds:From Waves to Bubbles
(É. Guazzelli) | 213 |
| 9.1 | Introduction | 213 |
| 9.2 | Flow Regimes and Instabilities | 214 |
| 9.3 | Instability Mechanism | 216 |
| 9.4 | Governing Equations | 218 |
| 9.5 | Primary Instability | 219 |
| 9.6 | Rheology of the Particle Phase | 222 |
| 9.7 | Secondary Instability and the Formation of Bubbles | 223 |
| 9.8 | Conclusions | 228 |
| | References | 229 |
| 10 | Wind-blown Sand
(H. J. Herrmann) | 233 |
| 10.1 | Introduction | 233 |
| 10.2 | The Wind Field | 234 |
| 10.3 | Aeolian Sand Transport | 239 |
| 10.4 | Dunes | 246 |
| 10.5 | Conclusion | 249 |
| | References | 250 |
| | | |
| V | Charged and Magnetic Granular Matter | 253 |
| | | |
| 11 | Electrostatically Charged Granular Matter
(S. M. Dammer, J. Werth, and H. Hinrichsen) | 255 |
| 11.1 | Introduction | 255 |
| 11.2 | Charged Granular Matter in Vacuum | 256 |
| 11.3 | Charged Granular Matter in Suspension | 260 |
| 11.4 | Agglomeration of Monopolar Charged Suspensions | 262 |
| 11.4.1 | Mean Field Rate Equation | 263 |
| 11.4.2 | Self-focussing Size Distribution | 265 |
| 11.4.3 | Brownian Dynamics Simulations | 267 |
| 11.5 | Coating Particles in Bipolarly Charged Suspensions | 271 |
| 11.5.1 | Coulomb Interactionvs. Translational Brownian Motion | 273 |
| 11.5.2 | Coulomb Interactionvs. Rotational Brownian Motion | 276 |
| 11.6 | Summary | 277 |
| | References | 278 |
| 12 | Magnetized Granular Materials
(D. L. Blair and A. Kudrolli) | 281 |
| 12.1 | Introduction | 281 |
| 12.2 | Background: Dipolar Hard Spheres | 282 |
| 12.3 | Experimental Technique | 283 |
| 12.4 | The Phase Diagram | 285 |
| 12.5 | The Non-equipartition of Energy | 288 |
| 12.6 | Cluster Growth Rates | 290 |
| 12.7 | Compactness of the Cluster | 292 |
| 12.8 | Migration of Clusters | 293 |
| 12.9 | Summary | 293 |
| | References | 295 |
| VI | Computational Aspects | 297 |
| 13 | Molecular Dynamics Simulations of Granular Materials
(S. Luding) | 299 |
| 13.1 | Introduction | 299 |
| 13.2 | The Soft-particle Molecular Dynamics Method | 300 |
| 13.2.1 | Discrete-particle Model | 300 |
| 13.2.2 | Equations of Motion | 300 |
| 13.2.3 | Contact Force Laws | 301 |
| 13.3 | Hard-sphere Molecular Dynamics | 305 |
| 13.3.1 | Smooth Hard-sphere Collision Model | 305 |
| 13.3.2 | Event-driven Algorithm | 306 |
| 13.4 | The Link between ED and MD via the TC Model | 307 |
| 13.5 | The Stress in Particle Simulations | 309 |
| 13.5.1 | Dynamic Stress | 309 |
| 13.5.2 | Static Stress from Virtual Displacements | 310 |
| 13.5.3 | Stress for Soft and Hard Spheres | 310 |
| 13.6 | 2D Simulation Results | 311 |
| 13.6.1 | The Equation of State from ED | 311 |
| 13.6.2 | Quasi-static MD Simulations | 312 |
| 13.7 | Large-scale Computational Examples | 316 |
| 13.7.1 | Cluster Growth(ED) | 316 |
| 13.7.2 | 3D Ring-shearCellSimulation | 318 |
| 13.8 | Conclusion | 321 |
| | References | 322 |
| 14 | Contact Dynamics for Beginners
(L. Brendel, T. Unger, and D. E. Wolf) | 325 |
| 14.1 | Introduction | 325 |
| 14.2 | Discrete Dynamical Equations | 326 |
| 14.3 | Volume Exclusion in a One-dimensional Example | 327 |
| 14.4 | The Three-dimensional Single Contact Case Without Cohesion | 329 |
| 14.5 | Iterative Determination of Constraint Forces in a Multi-contact System | 333 |
| 14.6 | Computational Effort: Comparison Between CD and MD | 336 |
| 14.7 | Rolling and Torsion Friction | 337 |
| 14.8 | Attractive Contact Forces | 339 |
| 14.9 | Conclusion | 340 |
| | References | 341 |
| | Index | 345 |
