A formal exact mathematical solution for a sloping rat-hole in a highly frictional granular solid

Abstract

This paper provides a formal exact analytical solution to a rat-hole with a sloping base in two and three dimensions for a highly frictional granular material. A rat-hole is the general term used to describe those stable cavities, which frequently occur in storage hoppers and stock piles, whose formation prevents further material falling through the outlet. Figure 1a depicts the typical geometric configuration, comprising upper and lower sloping surfaces that form a channel or cylindrical cavity. In granular industries this is a commonly occurring situation, for example, where the flow of material from a hopper ceases due to the formation of a stable almost cylindrical vertical cavity. Despite their practical importance, the only analytical solution applies to the perfectly cylindrical cavity, assumed infinite in length with no upper sloping surface. In order to determine analytical solutions to more realistic situations, it is necessary to make compromises with regard to both geometric and constitutive considerations. Here, for both two and three-dimensional rat-holes, we present analytical parametric solutions for the special case of a highly frictional granular material, where the angle of internal friction is equal to ninety degrees. In addition, we assume that the highly frictional granular material is at the point of yield on a sloping rigid base, and with an infinitesimal central outlet as shown in Fig. 1b. The solutions given here are bona fide exact solutions of the governing equations for a Coulomb-Mohr granular solid, and satisfy exactly the free surface conditions on the sloping upper surface and a frictional condition along the sloping rigid base. We emphasize that while all zero-stress boundary conditions are correctly satisfied, and the solutions constitute the only known exact analytical solutions for a realistic rat-hole geometry, the solutions for both geometries exhibit infinite values of the other stress component on the free surface. This feature arises as a consequence of assuming an angle of internal friction equal to ninety degrees, and throws doubt on the physical applicability of the formal exact solution.