On the Problem of the Determination of Force Distributions in Granular Heaps Using Continuum Theory

The determination of the horizontal and vertical force distributions at the base of a sand‐pile is by now a famous problem in granular theory. In 1981 it was suggested from experimental work that the peak vertical force at the base does not occur directly beneath the vertex of the pile, but at some intermediate point so that there is a ring of maximum vertical pressure. Practising engineers have some reservation on this result and since that time numerous discrete theoretical and computational models of granular sand‐piles have been proposed to explain this phenomenon, with varying degrees of success. For two‐dimensional sand‐piles, following other authors, we attempt to estimate the horizontal and vertical force distributions using the proper continuum mechanical theory of granular materials. For an infinite sand‐pile we determine the force distributions at a certain height, and we argue that these forces should approximate those for a sand‐pile of finite height resting on a horizontal surface. We exploit an adaptation of the classical Jenike radial flow solutions for granular flow in a converging wedge. This solution involves a stress function ψ which may be positive or negative. For both cases, numerical results indicate there is no solution satisfying all of the conditions of the stated problem, except in the special case when the angle of internal friction is ninety degrees. Except in this special case, the difficulty in determining a numerical solution might well be due to the fact that the zero surface stress conditions are generally incompatible with the yield condition. For ψ negative a formal exact parametric solution is presented for the special case of the angle of internal friction equal to ninety degrees and which coincides with an independent numerical solution. This special exact solution which is a bona fide solution of the stated problem may well be the ‘only’ solution of the problem, but does not predict the dip in the vertical force which is suggested from experimental work.