Mathematical Models of Granular Matter

by Capriz, Gianfranco; Giovine, Pasquale; Mariano, Paolo Maria

• ISBN13: 9783540782766
• ISBN10: 3540782761
• Format: Paperback
• Copyright: 2008-07-01
• Publisher: Springer Verlag

Summary

Granular matter displays a variety of peculiarities that distinguish it from other appearances studied in condensed matter physics and renders its overall mathematical modelling somewhat arduous. Prominent directions in the modelling granular flows are analyzed from various points of view. Foundational issues, numerical schemes and experimental results are discussed. The volume furnishes a rather complete overview of the current research trends in the mechanics of granular matter. Various chapters introduce the reader to different points of view and related techniques. New models describing granular bodies as complex bodies are presented. Results on the analysis of the inelastic Boltzmann equations are collected in different chapters. Gallavotti-Cohen symmetry is also discussed.

Table of Contents

From Granular Matter to Generalized Continuum	p. 1
Introduction	p. 1
Mathematical Preliminaries	p. 3
Balances	p. 4
Micromechanics	p. 5
Granular Microstructure and Rotation	p. 5
Graph Theory for Extrinsic Modes	p. 7
Extrinsic Power	p. 12
Energy-Based Homogenization	p. 13
Intrinsic Moments and Continuum Fields	p. 15
Conclusions	p. 17
Appendix: Simplex and Edge-Complex Gradients	p. 17
References	p. 20
Generalized Kinetic Maxwell Type Models of Granular Gases	p. 23
Introduction	p. 23
Maxwell Models of the Boltzmann Equation	p. 26
Isotropic Maxwell Model in the Fourier Representation	p. 28
Models with Multiple Interactions	p. 30
Statement of the General Problem	p. 31
The General Problem in Fourier Representation	p. 33
Existence and Uniqueness of Solutions	p. 33
Large Time Asymptotics	p. 34
Existence of Self-Similar Solutions	p. 41
Properties of Self-Similar Solutions	p. 42
Main Results for Maxwell Models with Multiple Interactions	p. 45
Self-Similar Asymptotics	p. 45
Distribution Functions, Moments and Power-Like Tails	p. 47
Applications to the Conservative or Dissipative Boltzmann Equation	p. 51
References	p. 56
Hydrodynamics from the Dissipative Boltzmann Equation	p. 59
Introduction	p. 59
Modeling Dissipative Boltzmann Equation	p. 62
Hydrodynamic Limit and the Euler Equations	p. 65
Hydrodynamics from Homogeneous Cooling States	p. 67
Conclusions	p. 71
References	p. 72
Bodies with Kinetic Substructure	p. 77
Kinetics	p. 77
A Shadow Speck of Matter	p. 79
Straining and Allied Notions	p. 81
Balance Laws	p. 84
Balance of Kinetic Energy	p. 86
The First Principle	p. 89
References	p. 90
From Extended Thermodynamics to Granular Materials	p. 91
Introduction	p. 91
Boltzmann Equationand Moments	p. 92
The Closure of Extended Thermodynamics	p. 93
Macroscopic Approach of ET in the 13 Fields	p. 93
Extended Thermodynamics of Moments	p. 94
Maximization of Entropy	p. 97
Maximum Characteristic Velocity in Classical Theory	p. 98
Nesting Theories and Principal Subsystems	p. 99
Example of 13-Moments Principal Subsystems	p. 99
Lower Bound Estimate and Characteristic Velocities for Large n	p. 100
Qualitative Analysis	p. 102
Shizuta-Kawashima Condition	p. 103
Global Existence of Smooth Solutions	p. 103
Comparison with Experiments: Sound Waves and Light Scattering	p. 104
References	p. 105
Influence of Contact Modelling on the Macroscopic Plastic Response of Granular Soils Under Cyclic Loading	p. 109
Introduction	p. 109
Discrete Element Methods	p. 111
Boundary Conditions: Biaxial Test	p. 112
Molecular Dynamics	p. 113
The Normal-Dashpot Model	p. 114
Contact Dynamics	p. 115
Results	p. 116
Comparing MD and CD	p. 117
Comparing Different Visco-Elastic Laws	p. 118
Conclusions	p. 123
References	p. 123
Fluctuations in Granular Gases	p. 125
Introduction	p. 125
A Brief Introduction to Granular Gases	p. 127
Boundary Driven Gases	p. 128
Randomly Driven Gases	p. 129
Total Energy Fluctuations in Vibrated and Driven Granular Gases	p. 131
The Inhomogeneous Boundary Driven Gas	p. 131
The Homogeneously Driven Case	p. 135
A Large Deviation Theory for the Injected Power Fluctuations in the Homogeneous Driven Granular Gas	p. 138
The Cumulants	p. 141
The Solvable Infinite Dimension Limit	p. 145
Fluctuations of Injected Power at Finite Times: Two Examples	p. 146
The Homogeneous Driven Gas of Inelastic Hard Disks	p. 146
The Boundary Driven Gas of Inelastic Hard Disks	p. 153
The Dynamics of a Tracer Particle as a Non-Equilibrium Markov Process	p. 157
Detailed Balance	p. 158
Action Functionals	p. 160
Conclusions	p. 161
References	p. 162
An Extended Continuum Theory for Granular Media	p. 167
Introduction	p. 167
A First Model	p. 169
Rotations	p. 171
Balance of Interactions for Material Bodies with Affine Microstructure	p. 173
Observers	p. 175
Dilatant Granular Materials with Rotating Grains	p. 177
Inertia Forces and Balance of Granular Energy	p. 179
Constitutive Restrictions in the Thermoelastic Case	p. 182
Suspension of Rigid Granules in a Fluid Matrix	p. 185
Appendix: Kinetic Energy Coefficients	p. 187
References	p. 190
Slow Motion in Granular Matter	p. 193
Introduction	p. 193
Representation of the Granularity	p. 194
Balance of Interactions: <$>{\op R}^3 \times SO(3)<$> Invariance	p. 200
Evolution of the Local Numerosity of Granules	p. 204
A Single Granule Coinciding with the Generic Material Element	p. 207
References	p. 209
Index	p. 211
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