Geometrical derivation of frictional forces for granular media under slow shearing

We present an alternative way to determine the frictional forces at the contact between two particles. This alternative approach has its motivation in a detailed analysis of the bounds on the time integration step in the discrete element method for simulating collisions and shearing of granular assemblies. We show that, in standard numerical schemes, the upper limit for the time integration step, usually taken from the average time t _c of one contact, is in fact not sufficiently small to guarantee numerical convergence of the system during relaxation. In particular, we study in detail how the kinetic energy decays during the relaxation stage and compute the correct upper limits for the time integration step, which are significantly smaller than the ones commonly used. In addition, we introduce an alternative approach based on simple relations to compute the frictional forces that converges even for time integration steps above the upper limit.