Contents
1 Introduction 1
1.1 Engineering Demands of Granular Mechanics 2
1.2 Basic Physical and Mechanical Properties of Granular Materials 8
1.2.1 Friction Law 8
1.2.2 Grain Silo Effect 9
1.2.3 Extrusion and Shear Expansion of Granular Materials 10
1.2.4 The Flow State of Granular Materials 12
1.3 Computational Analysis Softwares for Computational Granular Mechanics 15
References 17
Part I Fundamentals of Computational Granular Mechanics
2 Constructions of Irregular Shaped Particles in the DEM 23
2.1 Bonding and Clumping Models Based on Spherical Particles 24
2.1.1 Bonding Models Based on Spheres 24
2.1.2 Clumping Models Based on Spheres 26
2.2 Super-Quadric Particles 30
2.2.1 Super-Quadric Particles 31
2.2.2 Ellipsoidal Particles Based on Super-Quadric Equation 34
2.3 Polyhedral and Dilated Polyhedral Particles 36
2.3.1 Polyhedral Particles 36
2.3.2 Dilated Polyhedral Particles Based on Minkowski Sum 38
2.4 Advanced Constructions of Novel Irregular Shaped Particles 40
2.4.1 Random Star-Shaped Particles 40
2.4.2 B-Spline Function Models 42
2.4.3 Combined Geometric Element Method 43
2.4.4 Potential Particle Model 44
2.4.5 Poly-superellipsoid Model 46
2.5 Summary 47
References 47
3 Contact Force Models for Granular Materials 51
3.1 Visco-Elastic Contact Models of Spherical Particles 52
3.1.1 Linear Contact Model 53
3.1.2 Nonlinear Contact Model 55
3.2 Elastic-Plastic Contact Models of Spherical Particles 58
3.2.1 Normal Elastic-Plastic Contact Model 58
3.2.2 Tangential Elastic-Plastic Contact Model 60
3.3 Rolling Friction Models of Spherical Particles 62
3.3.1 Rolling Friction Law 63
3.3.2 Rolling Friction Model of Spherical Particles 64
3.4 Bonding-Breakage Models of Spherical Particles 66
3.5 Contact Models of Non-spherical Particles 71
3.5.1 Contact Model Between Super-Quadric Particles 71
3.5.2 Contact Model of Dilated Polyhedral Particles 73
3.6 Non-contact Physical Interactions Between Particles 76
3.6.1 Adhesion Force Between Spherical Particles 77
3.6.2 Liquid Bridge Force Between Wet Particles 80
3.6.3 Heat Conduction Between Particles 83
3.7 Summary 93
References 94
4 Macro-Meso Analysis of Stress and Strain Fields of Granular Materials 97
4.1 Computational Homogenization Method Based on Mean Field Theory 98
4.1.1 Variational Representation of Frictional Contact Problems 99
4.1.2 Macro-Meso Two Scale Boundary Value Problems 101
4.1.3 Macro-Meso Scale Solution Procedures Based on Mean Field Theory 104
4.2 Meso Analysis of Stress Field of Granular Materials 107
4.2.1 Average Stress Description of the Micro Topological Structure 108
4.2.2 Stress Characterization of Particle Aggregates 112
4.2.3 Description of Macro Stress Based on Virtual Work 114
4.2.4 The Average Stress of the RVE in a Cosserat Continuum 121
4.3 Meso Analysis of Strain Field of Granular Materials 123
4.3.1 Definition of Strain by Bagi 124
4.3.2 Definition of Strain by Kruyt-Rothenburg 125
4.3.3 Definition of Strain by Kuhn 127
4.3.4 Definition of Optimal Fitting Strain by Cundall 128
4.3.5 Definition of Optimal Fitting Strain by Liao et al 129
4.3.6 Definition of Optimal Fitting Strain by Cambou et al 130
4.3.7 Definition of Volumetric Strain by Li et al 131
4.4 Summary 134
References 135
5 Coupled DEM-FEM Analysis of Granular Materials 137
5.1 Combined DEM-FEM Method for the Transition from Continuum to Granular Materials 138
5.1.1 Contact Algorithm 139
5.1.2 Deformation of Element 142
5.1.3 Failure Model of Materials 144
5.2 Coupled DEM-FEM Model for the Continua-Discontinua Bridging Domain 150
5.2.1 Weak Form of Governing Equations for the Bridging Domain 151
5.2.2 Coupling Interface Force 154
5.2.3 Coupling Point Search 157
5.3 Coupled DEM-FEM Method for the Interaction Between Continua and Discontinua 159
5.3.1 Global Search Detection of Particle-Structure Contacts 159
5.3.2 Local Search Detection of Particle-Structure Contacts 165
5.3.3 Transfer of Contact Forces 168
5.4 Summary 171
References 172
6 Fluid-Solid Coupling Analysis of Granular Materials 175
6.1 DEM-CFD Coupling Method for Granular Materials and Fluid 175
6.1.1 Basic Governing Equations of Particles 176
6.1.2 DEM-CFD Coupling Solution Method 176
6.1.3 Governing Equations of Fluid Domain 177
6.1.4 Momentum Exchange Between Fluid and Solid Particles 177
6.1.5 Fluid Volume Fraction 179
6.1.6 Convection Heat Transfer Term 179
6.2 DEM-SPH Coupling Method for Granular Materials and Fluid 182
6.2.1 Integral Representation of Function and Particle Approximation in SPH 183
6.2.2 SPH Form for Navier-Stokes Equations 184
6.2.3 The EISPH Method for Incompressible Fluid 188
6.2.4 DEM-SPH Coupling Model 189
6.3 DEM-LBM Coupling Method for Granular Materials and Fluid 195
6.3.1 Lattice Bol
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