New stress and velocity fields for highly frictional granular materials

The idealized theory for the quasi-static flow of granular materialswhich satisfy the Coulomb–Mohr hypothesis is considered.This theory arises in the limit as the angle of internal frictionapproaches /2, and accordingly these materials may be referredto as being ‘highly frictional’. In this limit,the stress field for both two-dimensional and axially symmetricflows may be formulated in terms of a single nonlinear second-orderpartial differential equation for the stress angle. To obtainan accompanying velocity field, a flow rule must be employed.Assuming the non-dilatant double-shearing flow rule, a furtherpartial differential equation may be derived in each case, thistime for the streamfunction. Using Lie symmetry methods, a completeset 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-drivengranular flows are known, so these results are potentially importantin many practical applications. The problem of mass flow througha two-dimensional wedge hopper is examined as an illustration.