M A C Koenders

The Physics of the Deformation of Densely Packed Granular Materials


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target="_blank" rel="nofollow" href="#ulink_065cc985-b4c1-506f-bb41-138df1605dd6">1.4Contact laws

       1.5The frictional interaction

       1.6Contact laws in terms of material parameters

       1.7Interaction for small particles in a fluid environment

       Chapter 2Continuum Mechanics and Cartesian Tensor Calculus

       2.1Initial considerations

       2.2Rotations

       2.3The strain tensor

       2.4The stress tensor

       2.5Tensors

       2.6Material response

       2.7Isotropic materials

       2.8Elastic behaviour

       2.9Anisotropic materials

       2.10Coaxiality

       2.11Objectivity and pre-stressed materials, the Jaumann derivative

       2.12Frictional materials

       Chapter 3The Bounds of Static Equilibrium

       3.1Criteria for rupture

       3.2The context of second order, partial differential equations

       3.3Wave speeds and strong ellipticity

       Chapter 4Heterogeneity

       4.1General considerations

       4.2Isotropic materials

       4.3Effective stiffness moduli

       4.4Transverse anisotropic material

       Chapter 5Fabric Description

       5.1Voronoi tiling

       5.2Contact point distribution

       5.3Correlation

       5.4Strain in a granular medium

       5.5Stress in a granular medium

       5.6Calculating averages in a contacting granular material

       Chapter 6Stress-Strain Relations of Granular Assemblies: A Frictionless Assembly

       6.1General considerations

       6.2Kinematics

       6.3Mean-field approximation

       6.4Perturbations to the mean-field theory

       6.5Solution in two dimensions

       6.6Connectivity in a granular medium

       6.7Estimates of the correction to the moduli due to heterogeneity

       Chapter 7Stress-Strain Relations of Granular Assemblies: Normal and Tangential Interactions

       7.1Particle spin

       7.2The interaction and the quasi-static equilibrium equations

       7.3Mean-field stiffness estimate

       7.4Heterogeneity with tangential interactions

       7.5Solution of the strain fluctuations for a two-dimensional isotropic medium

       7.6Considerations for an isotropic contact distribution in two special cases

       7.7Anisotropic calculation

       7.8A few remarks on the theory

       Chapter 8Frictional Granular Materials

       8.1The frictional interaction

       8.2Mean-field estimate

       8.3Mean-field estimate with randomly distributed slip angles

       8.4Mean-field estimate with concentrated slip angles

       8.5Strain fluctuations

       8.6Fabric heterogeneity and other refinements

       8.7The evolution of an assembly in a biaxial cell test

       Appendix AMathematical Appendix

       A.1Isotropic tensors

       A.2Integrals of strings of unit vectors

       A.3Elastic constants

       A.4Fourier transforms and harmonic density

       A.5Bessel functions

       A.6Various