Page 11 - The Combined Finite-Discrete Element Method

viii CONTENTS

3.5 Direct Mapping Algorithm for Discrete Elements of Similar Size 87
3.6 Screening Contact Detection Algorithm for Discrete Elements of Similar Size 89
3.7 Sorting Contact Detection Algorithm for Discrete Elements of a Similar Size 94
3.8 Munjiza-NBS Contact Detection Algorithm in 2D 102
3.8.1 Space decomposition 103
3.8.2 Mapping of discrete elements onto cells 104
3.8.3 Mapping of discrete elements onto rows and columns of cells 104
3.8.4 Representation of mapping 104
3.9 Selection of Contact Detection Algorithm 117
3.10 Generalisation of Contact Detection Algorithms to 3D Space 118
3.10.1 Direct checking contact detection algorithm 118
3.10.2 Binary tree search 118
3.10.3 Screening contact detection algorithm 118
3.10.4 Direct mapping contact detection algorithm 120
3.11 Generalisation of Munjiza-NBS Contact Detection Algorithm to
Multidimensional Space 120
3.12 Shape and Size Generalisation–Williams C-GRID Algorithm 128

4 Deformability of Discrete Elements 131
4.1 Deformation 131
4.2 Deformation Gradient 132
4.2.1 Frames of reference 132
4.2.2 Transformation matrices 139
4.3 Homogeneous Deformation 141
4.4 Strain 142
4.5 Stress 143
4.5.1 Cauchy stress tensor 143
4.5.2 First Piola-Kirchhoff stress tensor 145
4.5.3 Second Piola-Kirchhoff stress tensor 149
4.6 Constitutive Law 150
4.7 Constant Strain Triangle Finite Element 156
4.8 Constant Strain Tetrahedron Finite Element 166
4.9 Numerical Demonstration of Finite Rotation Elasticity in the Combined
Finite-Discrete Element Method 174

5 Temporal Discretisation 179
5.1 The Central Difference Time Integration Scheme 179
5.1.1 Stability of the central difference time integration scheme 182
5.2 Dynamics of Irregular Discrete Elements Subject to Finite Rotations in 3D 185
5.2.1 Frames of reference 185
5.2.2 Kinematics of the discrete element in general motion 186
5.2.3 Spatial orientation of the discrete element 186
5.2.4 Transformation matrices 187
5.2.5 The inertia of the discrete element 188
5.2.6 Governing equation of motion 188
5.2.7 Change in spatial orientation during a single time step 190
5.6.8 Change in angular momentum due to external loads 191
5.6.9 Change in angular velocity during a single time step 193
5.6.10 Munjiza direct time integration scheme 194