Boundary-Value Problems for Hyperbolic Equations Related to Steady Granular Flow

Abstract

Boundary value problems for steady-state flow in elastoplasticity are a topic of mathematical and physical interest. In particular, the underlying PDE may be hyperbolic, and uncertainties surround the choice of physically appropriate stress and velocity boundary conditions. The analysis and numerical simulations of this paper address this issue for a model problem, a system of equations describing antiplane shearing of an elastoplastic material. This system retains the relevant mathematical structure of elastoplastic planar flow. Even if the flow rule is associative, two significant phenomena appear: (i) For boundary conditions suggestive of granular flow in a hopper, in which it seems physically natural to specify the velocity everywhere along a portion of the boundary, no such solutions of the equations exist; rather, we construct a solution with a shear band (velocity jump) along part of the boundary, and an appropriate relaxed boundary condition is satisfied there. (ii) Rigid zones appear inside deforming regions of the flow, and the stress field in such a zone is not uniquely determined. For a nonassociative flow rule, an extreme form of nonuniqueness—both velocity and stress—is encountered.