A Model for the Flow of Granular Materials and its Application to Initial/Boundary Value Problems
A plasticity type model is here considered for the flow of a dry granular material such as grain or sand. The physical and kinematic basis for the model is briefly summarised and the equations governing the model are presented in terms of the components of the deformationrate, spin and stress tensors. The equations comprise a set of six first order partial differential equations of hyperbolic type for which there are five distinct characteristic directions. An idealised application to a hopper is considered for the flow in the vicinity of the upper free surface. A simple analytic solution is given in which (a) the velocity field is linear in the space coordinates and represents a dilatant or contractant shear, (b) two possible stress fields are proposed, one linear and one exponential in space, which satisfy the stress equilibrium equations, the yield condition and the traction-free condition at the free surface, (c) the density is homogeneous in space and exponential in time. Finally, a method is proposed for defining an intrinsic time-scale for the deformation, which enables a physically realistic density field to be obtained via a sequence of dilatant and contractant shearing motions. Full advantage is taken of the hyperbolic nature of the governing equations to allow the solution to have discontinuities in the field variables, or their derivatives, in crossing characteristic lines.