Statics and dynamics of sandpiles: Some phenomenological ideas

Abstract

We propose phenomenological equations to describe how the contraints ‘propagate’ within a granular media. 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 the stress propagates along ‘light’ cones. The opening angle of these cones is simply the maximum angle of stability of heaps. Inclusion of non linear terms is argued to describe the ‘arching’ phenomenon, which has been proposed to explain the non intuitive vertical distribution pressure observed experimentally. The analogue of Peclet and Reynolds numbers can be defined, suggesting possible classifications of sandpiles. In a second part, we propose a new continuum description of the dynamics of sandpile surfaces, which involves two population of grains: immobile and rolling. We introduce a simple bilinear form for the interconversion processes of “sticking” (below the angle of repose, and “dislodgement” (for greater slopes. We find a “spinodal” angle, larger than the angle of repose, at which the surface of a tilted immobile sandpile first becomes unstable to an infinitesimal perturbation.