A micromechanical model for a non-linear elastic granular material based on local equilibrium conditions

A micromechanical model is developed to study the non-linear elastic stress-strain relation of a homogenous isotropic granular solid consisting of a random package of identical elastic spheres with contacts equally distributed in all directions. In order to obtain the effective tangent elastic moduli of the random packing subject to an isotropic pressure, the equilibrium of each particle is imposed in an approximate way and the strain energy of a representative volume element containing a huge number of particles is determined. To this aim a variational approach is adopted by decomposing the displacement field into the sum of a mean field and a fluctuation field. It is shown that the elastic moduli so determined differ considerably from the values obtained by using classical homogenization procedures based on the uniform strain (Voigt) hypothesis and are in closer agreement with the experimental results.