Shear flows in a plane viscoplastic layer with the plasticity criterion weakly depending on pressure

Abstract

A possible formulation of a spatial boundary value problem of viscoplastic flow is presented for the general case in which the criterion of plasticity depends on all invariants of the stress tensor. In the plane strain condition, this criterion involves pressure and stress intensity. If the behavior of the yield surface as a constitutive function of the medium weakly depends on the pressure, then one can seek solutions of the problems in the form of perturbations near the solutions obtained in using the classical von Mises-Hencky criterion. Linearized problems in perturbations simulating steady motion of a heavy viscoplastic layer on an inclined plane and a steady flow under the action of a pressure drop in a flat layer are investigated. In the first problem, the presence of weak dependence of the plasticity criterion on the pressure in the model keeps the one-dimensional nature of the problem, while, in the other problem, this makes the task essentially two-dimensional.