Three-dimensional micromechanical expression for the average strain tensor of granular materials

Abstract

In investigations of the behaviour of granular materials, the conversion of discrete contact information, specifically the force and displacement data, into macroscopic quantities such as stress and strain is a fundamental approach. The expression for the average stress tensor, a well-established formulation, involves the summation over all interparticle contacts while considering both the contact force and geometric parameters such as the branch vector. However, for the three-dimensional case, a general micromechanical expression for the average strain tensor is still missing.

In this study, a three-dimensional micromechanical expression is derived for the average strain tensor of granular materials. The new expression for the strain tensor involves only particle positions and relative displacements between particles, and does not depend on the tessellation method applied to the space occupied by the particles and interparticle voids. To validate the accuracy of the strain tensor, displacement data of granular assemblies were generated through Discrete Element Method simulations, and the strains of granular assemblies calculated by the derived strain tensor were compared with those calculated from the macroscopic deformation of granular assemblies. The results demonstrate that the proposed strain tensor is consistent with the macroscopic strain tensor, either in triaxial compression or in simple shear tests. This research conclusively addresses the fundamental question for the three-dimensional micromechanical strain tensor of granular materials and contributes to the development of accurate micromechanics-based constitutive models for granular materials.