Granular Physics

The field of granular physics has burgeoned since its development in the late 1980s, when physicists first began to use statistical mechanics to study granular media. They are prototypical of complex systems, manifesting metastability, hysteresis and bistability, and a range of other fascinating phenomena. This book is a wide-ranging account of developments in granular physics, and lays out the foundations of the statics and dynamics of granular physics. It covers a wide range of subfields, ranging from fluidization to jamming, and these are modeled through a range of computer simulation and theoretical approaches. Written with an eye to pedagogy and completeness, this book will be a valuable asset for any researcher in this field. It includes the most recent ideas and contains discussions at the end of each chapter. The book also contains contributions from Professor Sir Sam Edwards, with Dr Raphael Blumenfeld; Professor Isaac Goldhirsch; and Professor Philippe Claudin. Half-title......Page 3 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 Preface......Page 12 1 Introduction......Page 15 1.1 Statistical mechanics framework, packing and the role of friction......Page 16 1.2 Granular flow through wedges, channels and apertures......Page 18 1.3 Instabilities, convection and pattern formation in vibrated granular beds......Page 19 1.4 Size segregation in vibrated powders......Page 22 1.5 Self-organised criticality – theoretical sandpiles?......Page 25 1.6 Cellular automaton models of sandpiles......Page 27 1.7 Theoretical studies of sandpile surfaces......Page 29 2.1 Granular structures – Monte Carlo approaches......Page 32 2.2 Granular flow – molecular dynamics approaches......Page 36 2.3 Simulations of shaken sand – some general remarks......Page 38 3.1 Details of simulation algorithm......Page 41 3.2 The structure of shaken sand – some simulation results......Page 43 3.3 Vibrated powders: transient response......Page 54 3.4 Is there spontaneous crystallisation in granular media?......Page 58 3.5 Some results on shaking-induced size segregation......Page 60 4.2 On bridges in sandpiles – an overarching scenario......Page 66 4.3 Some technical details......Page 68 4.4 Bridge sizes and diameters: when does a bridge span a hole?......Page 69 4.5 Turning over at the top; how linear bridges form domes......Page 72 4.6 Discussion......Page 75 5.1 Coupled nonlinear equations: dilatancy vs the angle of repose......Page 80 5.2 Bistability within δθB: how dilatancy ‘fattens’ the angle of repose......Page 82 5.3 When sandpiles collapse: rare events, activated processes and the topology of rough landscapes......Page 84 5.5 Another take on bistability......Page 86 6 Compaction of disordered grains in the jamming limit: sand on random graphs......Page 96 6.1 The three-spin model: frustration, metastability and slow dynamics......Page 98 6.2 How to tap the spins? – dilation and quench phases......Page 99 6.3.1 Fast dynamics till SPRT: every grain for itself!......Page 101 6.3.2 Slow dynamics of granular clusters: logarithmic compaction......Page 103 6.3.3 Cascades at the dynamical transition......Page 104 6.4 Results II: realistic amplitude cycling – how granular media jam at densities lower than close-packed......Page 107 6.5 Discussion......Page 110 7.2 Definition of the model......Page 111 7.3 Results I: on the packing fraction......Page 113 7.4 Results II: on annealed cooling, and the onset of jamming......Page 114 7.5 Results III: when the sandbox is frozen......Page 117 7.6 Results IV: two nonequilibrium regimes......Page 119 7.7 Discussion......Page 120 8 Shaking a box of sand II – at the jamming limit, when shape matters!......Page 121 8.1 Definition of the model......Page 122 8.2 Zero-temperature dynamics: (ir)retrievability of ground states, density fluctuations and anticorrelations......Page 123 8.3 Rugged entropic landscapes: Edwards’ or not?......Page 125 8.4 Low-temperature dynamics along the column: intermittency......Page 130 8.5 Discussion......Page 131 9.1 Avalanches type I – SOC......Page 132 9.1.1 Review of sandpile cellular automata – Type I......Page 133 9.2 Avalanches type II – granular avalanches......Page 135 9.2.1 Dynamical scaling for sandpile cellular automata......Page 136 9.2.2 Qualitative effects of avalanching on surfaces......Page 137 9.2.3 The effect of avalanching on sandpile surfaces – some observations of material properties......Page 142 9.2.4 Spatial and temporal roughening of sandpile surfaces......Page 146 9.3 Discussion and conclusions......Page 148 10.1 Avalanches in a rotating cylinder......Page 149 10.2 The model......Page 150 10.3.1 Rotated sandpile......Page 152 10.3.2 Sandpile driven by random deposition......Page 158 10.4 Discussion......Page 163 11.1.1 Some general remarks......Page 165 11.1.2 Sand in rotating cylinders; a paradigm......Page 166 11.2 Review of scaling relations for interfacial roughening......Page 167 11.3.1 Analysis of the decoupled equation in h......Page 168 11.3.2 Some caveats......Page 171 11.4 Case B: when moving grains abound......Page 173 11.4.1 Numerical analysis......Page 174 11.4.2 Homing in on the physics: a discussion of smoothing in Case B......Page 178 11.5 Case C: tilt combined with flowing grains......Page 179 11.5.1 Results for the single Fourier transforms......Page 180 11.5.2 Results for the double Fourier transforms......Page 181 11.6 Discussion......Page 184 11.7 A more complicated example: the formation of ripples......Page 185 11.8 Conclusions......Page 191 12.1 Introduction......Page 193 12.2 Qualitative considerations......Page 194 12.2.1 Clustering......Page 195 12.2.2 Collapse......Page 198 12.2.3 Granular gases are mesoscopic......Page 199 12.3 Kinetic theory......Page 201 12.3.1 Some technical details and constitutive relations......Page 207 12.4 Boundary conditions......Page 213 12.5 Weakly frictional granular gases......Page 217 12.6 Conclusion......Page 223 13.1 Introduction......Page 226 13.2 Statistical mechanics......Page 228 13.3 Volume functions and forces in granular systems......Page 232 13.4 The stress field......Page 239 13.4.1 First approach......Page 240 13.4.2 Second approach – coarse-graining a microscopic theory......Page 243 13.5 Force distribution......Page 247 Equilibrium conditions......Page 250 Multiplicity of static solutions......Page 252 14.1.2 Force probability distribution......Page 254 14.1.3 Texture and force networks......Page 257 Presentation of the model......Page 259 Force distribution and the exponential tail......Page 261 14.2 Large-scale properties......Page 262 Silos......Page 263 Sandpiles......Page 265 Response functions......Page 267 From contact forces to stresses......Page 270 The effective medium theory......Page 276 14.2.3 Theoretical descriptions......Page 278 Elasto-plasticity......Page 279 Elasticity formalism......Page 281 Mohr–Coulomb yield criterion......Page 285 Janssen’s approach......Page 287 OSL model......Page 288 14.3 Conclusion......Page 290 References......Page 291 Index......Page 314 Half-title 3 Title 5 Copyright 6 Dedication 7 Contents 9 Preface 12 1 Introduction 15 1.1 Statistical mechanics framework, packing and the role of friction 16 1.2 Granular flow through wedges, channels and apertures 18 1.3 Instabilities, convection and pattern formation in vibrated granular beds 19 1.4 Size segregation in vibrated powders 22 1.5 Self-organised criticality – theoretical sandpiles? 25 1.6 Cellular automaton models of sandpiles 27 1.7 Theoretical studies of sandpile surfaces 29 2 Computer simulation approaches – an overview 32 2.1 Granular structures – Monte Carlo approaches 32 2.2 Granular flow – molecular dynamics approaches 36 2.3 Simulations of shaken sand – some general remarks 38 3 Structure of vibrated powders – numerical results 41 3.1 Details of simulation algorithm 41 3.2 The structure of shaken sand – some simulation results 43 3.3 Vibrated powders: transient response 54 3.4 Is there spontaneous crystallisation in granular media? 58 3.5 Some results on shaking-induced size segregation 60 4 Collective structures in sand – the phenomenon of bridging 66 4.1 Introduction 66 4.2 On bridges in sandpiles – an overarching scenario 66 4.3 Some technical details 68 4.4 Bridge sizes and diameters: when does a bridge span a hole? 69 4.5 Turning over at the top; how linear bridges form domes 72 4.6 Discussion 75 5 On angles of repose: bistability and collapse 80 5.1 Coupled nonlinear equations: dilatancy vs the angle of repose 80 5.2 Bistability within δθB: how dilatancy ‘fattens’ the angle of repose 82 5.3 When sandpiles collapse: rare events, activated processes and the topology of rough landscapes 84 5.4 Discussion 86 5.5 Another take on bistability 86 6 Compaction of disordered grains in the jamming limit: sand on random graphs 96 6.1 The three-spin model: frustration, metastability and slow dynamics 98 6.2 How to tap the spins? – dilation and quench phases 99 6.3 Results I: the compaction curve 101 6.3.1 Fast dynamics till SPRT: every grain for itself! 101 6.3.2 Slow dynamics of granular clusters: logarithmic compaction 103 6.3.3 Cascades at the dynamical transition 104 6.4 Results II: realistic amplitude cycling – how granular media jam at densities lower than close-packed 107 6.5 Discussion 110 7 Shaking a box of sand I – a simple lattice model 111 7.1 Introduction 111 7.2 Definition of the model 111 7.3 Results I: on the packing fraction 113 7.4 Results II: on annealed cooling, and the onset of jamming 114 7.5 Results III: when the sandbox is frozen 117 7.6 Results IV: two nonequilibrium regimes 119 7.7 Discussion 120 8 Shaking a box of sand II – at the jamming limit, when shape matters! 121 8.1 Definition of the model 122 8.2 Zero-temperature dynamics: (ir)retrievability of ground states, density fluctuations and anticorrelations 123 8.3 Rugged entropic landscapes: Edwards’ or not? 125 8.4 Low-temperature dynamics along the column: intermittency 130 8.5 Discussion 131 9 Avalanches with reorganising grains 132 9.1 Avalanches type I – SOC 132 9.1.1 Review of sandpile cellular automata – Type I 133 9.2 Avalanches type II – granular avalanches 135 9.2.1 Dynamical scaling for sandpile cellular automata 136 9.2.2 Qualitative effects of avalanching on surfaces 137 9.2.3 The effect of avalanching on sandpile surfaces – some observations of material properties 142 9.2.4 Spatial and temporal roughening of sandpile surfaces 146 9.3 Discussion and conclusions 148 10 From earthquakes to sandpiles – stick–slip motion 149 10.1 Avalanches in a rotating cylinder 149 10.2 The model 150 10.3 Results 152 10.3.1 Rotated sandpile 152 10.3.2 Sandpile driven by random deposition 158 10.4 Discussion 163 11 Coupled continuum equations: the dynamics of sandpile surfaces 165 11.1 Introduction 165 11.1.1 Some general remarks 165 11.1.2 Sand in rotating cylinders; a paradigm 166 11.2 Review of scaling relations for interfacial roughening 167 11.3 Case A: the Edwards–Wilkinson equation with flow 168 11.3.1 Analysis of the decoupled equation in h 168 11.3.2 Some caveats 171 11.4 Case B: when moving grains abound 173 11.4.1 Numerical analysis 174 11.4.2 Homing in on the physics: a discussion of smoothing in Case B 178 11.5 Case C: tilt combined with flowing grains 179 11.5.1 Results for the single Fourier transforms 180 11.5.2 Results for the double Fourier transforms 181 11.6 Discussion 184 11.7 A more complicated example: the formation of ripples 185 11.8 Conclusions 191 12 Theory of rapid granular flows 193 12.1 Introduction 193 12.2 Qualitative considerations 194 12.2.1 Clustering 195 12.2.2 Collapse 198 12.2.3 Granular gases are mesoscopic 199 12.3 Kinetic theory 201 12.3.1 Some technical details and constitutive relations 207 12.4 Boundary conditions 213 12.5 Weakly frictional granular gases 217 12.6 Conclusion 223 13 The thermodynamics of granular materials 226 13.1 Introduction 226 13.2 Statistical mechanics 228 13.3 Volume functions and forces in granular systems 232 13.4 The stress field 239 13.4.1 First approach 240 13.4.2 Second approach – coarse-graining a microscopic theory 243 13.5 Force distribution 247 14 Static properties of granular materials 250 14.1 Statics at the grain scale 250 14.1.1 Static solutions 250 Equilibrium conditions 250 Multiplicity of static solutions 252 14.1.2 Force probability distribution 254 14.1.3 Texture and force networks 257 14.1.4 The q-model 259 Presentation of the model 259 Force distribution and the exponential tail 261 14.2 Large-scale properties 262 14.2.1 Stress measurements in static pilings 263 Silos 263 Sandpiles 265 Response functions 267 14.2.2 From micro-to macroscopic scales 270 From contact forces to stresses 270 The effective medium theory 276 14.2.3 Theoretical descriptions 278 Elasto-plasticity 279 Elasticity formalism 281 Mohr–Coulomb yield criterion 285 Janssen’s approach 287 OSL model 288 14.3 Conclusion 290 References 291 Index 314 This Book Provides A Wide-ranging Account Of Developments In Granular Physics, And Lays Out The Foundations Of The Statics And Dynamics Of Granular Physics. It Covers A Wide Range Of Subfields, Ranging From Fluidisation To Jamming, And These Are Modelled Through A Range Of Computer Simulation And Theoretical Approaches. Written With An Eye To Pedagogy And Completeness, This Book Will Be A Valuable Asset For Any Researcher In This Field. In Addition To Professor Mehta's Detailed Exposition Of Granular Dynamics, The Book Contains Contributions From Professor Sir Sam Edwards, Jointly With Dr. Raphael Blumenfeld, On The Thermodynamics Of Granular Matter; From Professor Isaac Goldhirsch On Granular Matter In The Fluidised Stale; And Professor Philippe Claudin On Granular Statics.--book Jacket. Introduction -- Computer Simulation Approaches- An Overview -- Structure Of Vibrated Powders- Numerical Results -- Collective Structures In Sand- The Phenomenon Of Bridging -- On Angles Of Repose: Bistability And Collapse -- Compaction Of Disordered Grains In The Jamming Limit: Sand On Random Graphs -- Shaking A Box Of Sand I- A Simple Lattice Model -- Shaking A Box Of Sand Ii- At The Jamming Limit, When Shape Matters! -- Avalanches With Reorganizing Grains -- From Earthquakes To Sandpiles-stick-slip Motion -- Coupled Continuum Equations: The Dynamics Of Sandpile Surfaces -- Theory Of Rapid Granular Flows / Isaac Goldhirsch -- The Thermodynamics Of Granular Materials / Sir Sam Edwards And Raphael Blumenfeld -- Static Properties Of Granular Materials / Philiippe Claudin. Anita Mehta ; With Contributions From Sir Sam Edwards ... [et Al.]. Includes Bibliographical References (p. 274-296) And Index.

The field of granular physics has burgeoned since its development in the late 1980s, when physicists first began to use statistical mechanics to study granular media. They are prototypical of complex systems, manifesting metastability, hysteresis and bistability, and a range of other fascinating phenomena. This 2007 book is a wide-ranging account of developments in granular physics, and lays out the foundations of the statics and dynamics of granular physics. It covers a wide range of subfields, ranging from fluidisation to jamming, and these are modelled through a range of computer simulation and theoretical approaches. Written with an eye to pedagogy and completeness, this book will be valuable asset to any researcher in this field. The book also contains contributions from Professor Sir Sam Edwards, with Dr Raphael Blumenfeld, Professor Isaac Goldhirsch and Professor Philippe Claudin.