An explanation for the central stress minimum in sand piles

Abstract

A KEY component of a continuum-mechanical description of a sand pile (viewed as an assembly of hard particles in frictional contact) is the requirement of stress continuity—the forces acting on a small element of material must balance. But an additional physical postulate is required to close the equations. Standard continuum approaches postulate that the material is everywhere just at the point of slip failure^1,2. But these approaches have been unable to explain a startling experimental observation³—that the weight exerted by a conical sand pile on a surface has a minimum, not a maximum, below the apex. Here we propose a new closure, which embodies an intuitive model of arching^4,5 within a fully consistent continuum theory. Our assumption is that the principal stress axes have a fixed angle of inclination to the vertical. In two dimensions, this is sufficient to close the equations. In three dimensions, a second closure relation is required, but our results are relatively insensitive to the choice made. Our model, which contains no adjustable parameters, can account for the vertical stress distribution in real sand piles. This supports the idea that stresses propagate within a granular medium according to local rules that depend on its construction history.