Effective elastic moduli of heterogeneous granular solids

Abstract

A micro-mechanical model is employed to study the elastic stress-strain behavior of heterogeneous granular solids. The granular material is idealized as a collection of spherical particles interacting through inter-particle contacts. Based on this idealized model an equivalent continuum description of the granular solid is envisaged and the overall stiffness tensor of the granular solid is determined in terms of the stiffness of the inter-particle deformation. To facilitate the derivation of overall stiffness tensor, the granular solid is considered to be composed of continuum cells made of a single particle and the associated void space. A local stiffness tensor is defined for each cell. The local stiffness tensor is obtained in terms of the inter-particle stiffness, the number of contacts and the relative position of the neighboring particles. The local stiffness tensor is utilized to obtain the overall behavior of a representative volume of granular solid through the “self consistent” averaging technique. The overall stress and strain for the representative volume are determined as a volume average of the corresponding local quantities. To account for the heterogeneity of deformation in the granular medium, a “concentration” factor is defined for each cell. Based on the concept of volume averaging and the “concentration” tensor an overall stiffness tensor is derived for the granular solid. The applicability of the derived micro-mechanical model is evaluated by comparing its results with those obtained from the computer simulation method.