Transmission of stress in granular materials as a problem of statistical mechanics

Abstract

We consider the problem of stress transmission in granular materials. We formulate the simplest statically determinate problem of stress transmission through a static granular material. This is the case when grains are rigid and have an average coordination number of z ̄ =d+1 . Under this condition the system of Newton's equations of interparticle force and torque balance is complete. This means that there exists a complete set of equations for the macroscopic stress tensor σ ij( r ) i.e., the d (where d is the dimension of the problem) equations of force balance ∇ jσ ij( r )=g i( r ) have to be supported by d( d−1)/2 equations. These equations have their origin in Newton's laws of interparticle force and torque balance and incorporate tensorial geometrical characteristics of the packing. We conjecture that in order to have a coherent and self-consistent continuum theory of stress transmission in static granular media it is necessary to link the averaging procedure to the concept of compactivity. We emphasize that although real granular materials have many features ignored within the proposed framework it is essential for making progress to derive equations of stress transmission for the simplest model, as opposed to guessing and postulating.