Sands, Powders, and Grains

1. Introduction.- 1.1 Some Orders of Magnitude Defining the Problem.- 1.2 Economic Implications and Industrial Problems.- 1.2.1 Industrial Processing of Granulars.- Construction Materials.- Processing Industries.- An Example: Casting by Sacrificial Polystyrene.- The Agriculture Industry.- 1.2.2 Flow Problems.- 1.2.3 Problems of Segregation.- 1.3 Granular Materials and Geophysics.- 1.4 A Brief Historical Review.- 1.5 Prerequisites and Selected Bibliography.- 2. Interactions in Granular Media.- 2.1 A Single Particle and Its Environment.- Laminar Drag.- Turbulent Drag.- Granular Dendrites.- Humidity, Electrostatic Interactions, and Other Perturbations.- Classification of Granular Materials and Definitions.- 2.2 Interactions between Two Particles.- 2.2.1 The Laws of Friction between Solids.- The Three Fundamental Laws of Solid Friction.- A Microscopic Explanation.- Gliding and Rotations: Frustrated Rotations.- Rolling without Gliding.- Gliding without Rolling.- Transition from One Regime to the Other.- Stick—Slip Motion.- 2.2.2 Collisions and Deformations of Elastic Spheres.- Frontal Elastic Collision.- Nonfrontal Elastic Collision and Rotation of Particles.- A Ball Thrown Against a Wall.- Nonfrontal Collision Between Two Elastic Spheres with Friction.- The Tangential Restitution Coefficient.- Penetration During Frontal Collision:Hertz’s Problem.- Inhomogeneous Spheres: The Soft Crust Model.- 2.3 A Single Particle on Top of a Granular Medium.- 2.4 Interactions Between Several Particles.- 2.4.1 The Laws of Friction in a Granular Medium.- 2.4.2 Bagnold’s Number.- 3. Fluidization, Decompaction, and Fragmentation.- 3.1 The Static Properties of a Granular Pile.- 3.1.1 First Principle: The Role of Friction.- The Stacking of Cannon Balls.- Indetermination of Solid Friction: Hysteresis.- Distribution of Stresses in a Granular Medium.- Arch in Equilibrium Under its Own Weight.- Arch Supporting an Evenly Distributed Load.- 3.1.2 Stress—Strain Relations.- Identical Spherical Granules.- Granules of Different Sizes: Power Law and Electrical Analogy.- 3.1.3 Second Principle: Reynolds’s Dilatancy.- Deformation of a Simple Parallelogram.- Deformation of a Row of Parallelograms Placed Between Two Walls.- 3.1.4 Cylindrical Container: Janssen’s Model.- Generic Model: The Silo Problem.- Specific Applications.- Cylindrical Container of Diameter D.- Two-Dimensional Container.- 3.2 Dynamic Properties of a Granular Pile.- 3.2.1 A Column of Spheres Subjected to a Vertical Vibration.- Some Orders of Magnitude.- Mathematical Analysis of the Problem.- Results: Fluidized Phase and Condensed Phase.- Fluidization and Condensation as Functions of Acceleration.- Fluidization and Condensation as Functions of Height.- 3.2.2 Two-Dimensional Stack of Frictionless Spheres.- Some Comments on Scaling.- 3.2.3 Two-Dimensional Stack of Spheres with Friction.- Generic Model.- Levitation of Cylindrical Stacks.- Levitation of Two-Dimensional Stacks.- Experimental Observation of Decompaction and Convection in a Two-Dimensional Granular Structure.- Experimental Technique: Image Processing.- Measurements of the Velocity of Moving Particles.- Measurements of the Relative Motion of Particles.- Convection and Pile Formation.- Threshold of Pile Formation and Decompaction.- Dynamics of Pile Formation in Two Dimensions.- Experimental Verifications of the Decompaction Model.- Short-Term Decompaction: Fragmentation.- 3.2.4 Fragmentation of a Stack in Guided Fall.- Two-Dimensional Experiment.- Theoretical Modeling.- Fall Without Fragmentation.- Where Do Fractures Initially Appear’?.- Numerical Simulation of a Stack in Guided Fall.- Distribution of Pressure in a Stack: Arch Effects.- Self-Organization of Rotations.- 3.2.5 Surface Instabilities in an Extended Granular Medium.- Extended Three-Dimensional Stacks.- Wavelength Dependence on Vibration Frequency.- Extended Two-Dimensional Stacks.- Summary.- 4. Granular Media in a State of Flow.- 4.1 A Sand Pile in Equilibrium: The Angle of Repose.- The Embankment Angle of a Pile Made of a Small Number of Particles.- Transition from Intermittent to Continuous Regime—Power Laws.- Power Law for a Newtonian Fluid.- Power Law for a Granular Surface.- 4.2 Avalanche Models.- 4.2.1 Cellular Automaton Model (CAM).- The Principle.- Implementations of the Cellular Automaton Model (CAM).- Lifetime and 1/f Noise.- The Statistics of Avalanches.- Piles Involving Large Numbers of Particles.- Piles Involving Small Numbers of Particles.- Relaxation of the Critical Angle—Granular Temperature.- 4.2.2 Stick–Slip Model of Avalanches.- Different Friction Models.- Burridge–Knopoff (BK) Pads.- Other Friction Laws F(v).- A Few Remarks Concerning the Stick–Slip Model of Avalanches.- 4.2.3 Avalanche Models Based on Coupled Variables.- Upward Propagation of Perturbations.- Simulation of Avalanches.- De Gennes’s Modified Model.- Rotating Drum Experiment.- 5. Mixing and Segregation.- 5.1 Introduction.- 5.1.1 Oyama’s Cylindrical Drum.- 5.1.2 Potential Energy of a Heterogeneous Pile.- Superposition of Stacks—Two Compact Stacks.- Where Are the Defects Concentrated?.- 5.2 Segregation by Vibration.- 5.2.1 Simulation of Segregation by Size.- Two-Dimensional Model.- Three-Dimensional Model.- 5.2.2 Experiments on Segregation by Vibration.- Experiments on Continuous and Intermittent Ascent.- Convection or Arch Effect?.- Convection and Segregation in Three Dimensions.- Convection and Segregation in Two Dimensions.- 5.3 Segregation by Shearing.- 5.3.1 A Single Particle in a Uniform Medium.- 5.3.2 Segregation of Two Populations of Particles of Different Size.- Segregation Speed and Particle Size.- Segregation Speed and Rotation Velocity.- Fractal Growth of the Central Cluster.- 5.4 Segregation in Oyama’s Three-Dimensional Drum.- 5.4.1 Experimental Observations.- 5.4.2 Savage’s Model.- 6. Numerical Simulations.- 6.1 Introduction.- 6.1.1 The Challenges of Numerical Simulation.- 6.1.2 The Different Simulation Methods.- Hard Spheres and Soft Spheres.- Duration of Collisions and Chronology Problems.- 6.1.3 The Transition from a Discrete to a Continuous Description.- 6.2 Simulations of Collisions.- 6.2.1 Introduction.- 6.2.2 One-Dimensional LRV Procedure.- 6.3 Molecular Dynamics (MD) Simulations.- 6.3.1 Elastic and Friction Forces.- Linear and Nonlinear Equations.- Mechanical Analogies.- 6.3.2 MD Collision Model.- Linear Model of a Binary Collision.- Nonlinear Model of a Binary Collision.- The Detachment Effect.- 6.4 Simulation of the Dynamics of Contacts.- 6.5 Monte Carlo (MC) Simulations.- Monte Carlo Technique for Stacking and Relaxation.- 6.6 Sequential Model of a Pile.