New stress and velocity fields for highly frictional granular materials

Abstract

The idealized theory for the quasi-static flow of granular materials which satisfy the Coulomb–Mohr hypothesis is considered. This theory arises in the limit as the angle of internal friction approaches π/2, and accordingly these materials may be referred to as being ‘highly frictional’. In this limit, the stress field for both two-dimensional and axially symmetric flows may be formulated in terms of a single nonlinear second-order partial differential equation for the stress angle. To obtain an accompanying velocity field, a flow rule must be employed. Assuming the non-dilatant double-shearing flow rule, a further partial differential equation may be derived in each case, this time for the streamfunction. Using Lie symmetry methods, a complete set of group-invariant solutions is derived for both systems, and through this process new exact solutions are constructed. Only a limited number of exact solutions for gravity-driven granular flows are known, so these results are potentially important in many practical applications. The problem of mass flow through a two-dimensional wedge hopper is examined as an illustration.