Mechanics of shear banding in a regularized two-dimensional model of a granular medium

A regularized two-dimensional model for the buckling of force chains is presented, comprising identical rigid discs sitting initially in a conventional close-packed arrangement. As linear elastic constitutive laws are used throughout, the only nonlinearity in the system comes from large rotations as the resulting force chains are obliged to buckle under imposed end-shortening. The evolving deflected shapes are seen to develop and interact in a highly complex bifurcation structure. Analysis by the nonlinear continuation code Auto exposes at realistic load levels an energy landscape rich in local minima. A number of such states are identified, amongst them families of solutions with the familiar appearance of shear bands over a finite number of discs. A well-known “snakes and ladders” pattern is identified as the mechanism for the addition of extra discs to increase the width of the band.