Micromechanical definition of the strain tensor for granular materials

In order to develop constitutive relations for granular materials from the micromechanical viewpoint, general expressions relating macroscopic stress and strain to contact forces and particle displacements are required. Such an expression for the stress tensor under quasi-static conditions is well established in the literature, but a corresponding expression for the strain tensor has been lacking so far. This paper presents such an expression for two-dimensional assemblies. This expression is verified by computer simulations of biaxial and shear tests. As a demonstration of the use of the developed expression, a study is made of the elastic moduli of two-dimensional, isotropic assemblies of bonded, nonrotating disks. Theoretical expressions are given for the elastic moduli in terms of micromechanical parameters, such as coordination number and contact stiffnesses. Comparison with the results from computer simulations show that the agreement is fairly good over a wide range of coordination numbers and contact stiffness ratios.