Stress Distribution in Granular Media and Nonlinear Wave Equation

Abstract

We propose phenomenological equations to describe how forces “propagate" within a granular medium. The linear part of these equations is a wave equation, where the vertical coordinate plays the role of time, and the horizontal coordinates the role of space. This means that (in two dimensions) the stress propagates along “light-cones"; the angle of these cones is related (but not equal to) the angle of repose. Dispersive corrections to the picture, and various types of nonlinearity are discussed. Inclusion of nonlinear terms may be able to describe the “arching" phenomenon, which has been proposed to explain the nonintuitive horizontal distribution of vertical pressure (with a local minimum or “dip" under the apex of the pile) observed experimentally. However, for physically motivated parameter choices, a “hump", rather than a dip, is predicted. This is also true of a perturbative solution of the continuum stress equations for nearly-flat piles. The nature of the force fluctuations is also briefly discussed.