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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
deformation-rate, 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.
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