Macroscopic Constitutive Model Of Granular Materials Based On Microscopic Mechanics

Granular material is one of the materials widely existing in nature and used in engineering, such as soil, sands, concrete. On the other hand, the geologic disasters endangering people's lives and properties, such as earthquake, collapse of coal mine are related to the properties of granular materials. Hence, the study of mechanical properties and responses of granular materials is valuable to engineering practice.It is of great significance to derive the micromechanically informed constitutive model of the continuum equivalent to a discrete particle assembly to explore macro-mechanical behaviors in granular materials. Computational homogenization method is a direct multi-scale micro-macro constitutive modelling based on the conception of representative volume element (RVE) in the framework of average-field theory. RVE is a volume element with finite size centered by a classical sampling point containing enough information of microscopic structures for granular material in order to achieve the purpose of representivity. To derive the micromechanically informed constitutive model of the equivalent continuum, the relation between local strain defined by relative contact displacements of particles and the average strain of RVE should be given. The usual approach is using the Voigt hypothesis assuming that local strain in RVE is uniform. However, local strain, in fact, is not uniform in RVE.Firstly, in light of the average field theory a single Voronoi cell of effective continuum composed of a particle and its neighboring interspaces able to describe different local micro-structures and characterize different contact topologies of a typical particle with its neighboring particles is defined. Assuming the local strain is uniform in the scale of Voronoi cell, however, Voigt hypothesis is not adopted. The constitutive model for the Voronoi cell is formulated by means of the relations of inter-contacting forces and relative displacements between two particles in contact. The constitutive relations and the associated elastic modular tensors for different types of Voronoi cells with regular polygon shapes representing different local micro-structures of granular materials are given. The coordinate invariance of the formulated constitutive relations for the Voronoi cells is demonstrated. Finally the perturbation analysis of contacts to the Voronoi cell is fulfilled. Secondly, overall constitutive relationship for effective continuum of granular material is derived based on the assumption that displacements obtained from uniform assumption of Voronoi cell best fitting the real displacements of particles. After that average strain of equivalent continnum is defined. Uniform local strain hypothesis and best fit hypothesis for Voronoi cell with regular polygon shapes are compared and verified by a numeric example.At last overall constitutive relationship of equivalent Cosserat continuum is derived from expressing the continuum displacement field of granular material by polynomial function in RVE. The relations between the material parameters of equivalent continuum and micro parameters of particles are formed in isotropic contact situation.