Isostaticity in Cosserat continuum

Abstract

Under conditions of isostaticity in granular media, the contact forces for all particles are statically determinate and forces can be computed without recourse to deformation equations or constitutive relationships. Given that stresses represent spatial averages of inter-particle forces, the stress-equilibrium equations for the isostatic state form a hyperbolic system of partial differential equations that describe the internal stress state using only boundary tractions. In this paper, we consider a Cosserat medium and propose closure relationships in terms of stresses and couple stresses from observations of stress variations in the critical state regime from discrete element simulations and experiments on sand, even though the isostatic condition is only satisfied in an average sense. It is shown that the governing equations are hyperbolic, which can be solved using the method of characteristics. Examples of both analytic and numerical solutions are presented. These examples clearly demonstrate that stress chains (characteristic lines) form oblique angles with the assumed direction of the force chains.