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The Physics of Granular Media

The Physics of Granular Media

Haye Hinrichsen (Editor), Dietrich E. Wolf (Editor)

ISBN: 978-3-527-60448-7

Mar 2006

364 pages

$164.99

Description

Despite extensive empirical experience, there is both a scientific challenge and a technological need to develop an understanding of the mechanisms underlying the flow of grains. This new reference provides quick access to the current level of knowledge, containing review articles covering recent developments in the field of granular media from the viewpoints of applied, experimental, and theoretical physics.
In short, a must-have for advanced researchers and specialists as well as a useful starting point for anyone entering this field.

The authors represent different directions of research in the field, with their contributions covering:

- Static properties
- Granular gases
- Dense granular flow
- Hydrodynamic interactions
- Charged and magnetic granular matter
- Computational aspects



Preface XI

List of Contributors XV

I Static Properties 1

1 Stress in Dense Granular Materials 3
(I. Goldhirsch and C. Goldenberg)

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 AMicroscopic 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

2 Response Functions in Isostatic Packings 23
(C. F. Moukarzel)

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

3 Statistical Mechanics of Jammed Matter 45
(H. A. Makse, J. Brujic, and S. F. Edwards)

3.1 Introduction to the Concept of Jamming 45

3.1.1 Jammingin Glassy Systems 46

3.1.2 Jammingin Particulate Systems 48

3.1.3 Unifying Concepts in Granular Matter and Glasses 51

3.2 NewStatistical 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 Jammingwith theConfocal 64

3.3.1 From Micromechanics to Thermodynamics 64

3.3.2 Model System 65

3.4 Jamming in aPeriodicBox 72

3.4.1 Simulating Jamming 73

3.4.2 Testing the Thermodynamics 77

II Granular Gas 87

4 The Inelastic Maxwell Model 89
(E. Ben-Naim and P. Krapivsky)

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 Lattice Gases 109

4.7 Conclusions 111

5 Cluster Formation in Compartmentalized Granular Gases 117
(K. van der Weele, R. Mikkelsen, D. van der Meer, and D. Lohse)

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

III Dense Granular Flow 141

6 Continuum Modeling of Granular Flow and Structure Formation 143
(I. S. Aranson and L. S. Tsimring)

6.1 Introduction 143

6.2 Order Parameter Description of Partially Fluidized Granular Flows 144

6.3 Avalanchesonan 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

7 Contact Dynamics Study of 2D Granular Media: Critical States and Relevant Internal Variables 165
(F. Radjaï and S. Roux)

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–StrainRelation 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

8 Collision of Adhesive Viscoelastic Particles 189
(N. V. Brilliantov and T. Pöschel)

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

IV Hydrodynamic Interactions 211

9 Fluidized Beds: From Waves to Bubbles 213
(E. Guazzelli)

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 theParticle Phase 222

9.7 Secondary Instability and the Formation of Bubbles 223

9.8 Conclusions 228

10 Wind-blown Sand 233
(H. J. Herrmann)

10.1 Introduction 233

10.2 The Wind Field 234

10.3 Aeolian Sand Transport 239

10.4 Dunes 246

10.5 Conclusion 249

V Charged and Magnetic Granular Matter 253

11 Electrostatically Charged Granular Matter 255
(S. M. Dammer, J. Werth, and H. Hinrichsen)

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 Interaction vs. Translational Brownian Motion 273

11.5.2 Coulomb Interaction vs. Rotational Brownian Motion 276

11.6 Summary 277

12 Magnetized Granular Materials 281
(D. L. Blair and A. Kudrolli)

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

VI Computational Aspects 297

13 Molecular Dynamics Simulations of Granular Materials 299
(S. Luding)

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-shear Cell Simulation 318

13.8 Conclusion 321

14 Contact Dynamics for Beginners 325
(L. Brendel, T. Unger, and D. E. Wolf )

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

CD-ROM

The enclosed CD-ROM contains the figures of the articles, many of them colored, as well as related movies.

"An excellent, but very specialised book - one that I will place on my bookshelf and refer to regularly." (Chromatographia, Vol. 61, No. 5/6, March 2005)