Original French edition published by Eyrolles, France, 1997
2000, XIV, 214 p.
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This is an introduction to the physical principles underlying the behavior of materials consisting of grains. These can be found in an enormous variety of places, ranging from the powders used to make vitamin pills to the rings of Saturn, from beaches to grain elevators, and from pottery clay to interstellar dust. Granular materials have interested physicists from Coulomb to Faraday to Reynolds and Rayleigh, but only recently have mathematical and experimental methods been developed to analyze their properties in detail. This introductory text develops the fundamental physics of the behavior of granular materials. It covers the basic properties of flow, friction, and fluidization of uniform granular materials; discusses mixing and segregation of heterogeneous materials (the famous "brazil-nut problem"); and concludes with an introduction to numerical models. The presentation begins with simple experiments and uses their results to build concepts and theorems about materials whose behavior is often quite counter-intuitive; presenting in a unified way the background needed to understand current work in the field. Developed for students at the University of Paris, the text will be suitable for advanced undergraduates and beginning graduates; while also being of interest to researchers and engineers just entering the field.
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.