IMA Journal of Applied Mathematics Advance Access originally published online on December 17, 2004
IMA Journal of Applied Mathematics 2005 70(1):92-118; doi:10.1093/imamat/hxh054
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New stress and velocity fields for highly frictional granular materials*
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia
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 
/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.
Keywords: granular materials; exact solutions; Lie symmetries; double-shearing theory; highly frictional materials.
* Dedicated to Ray W. Ogden on the occasion of his 60th birthday