An exact solution for a model of pressure-dependent plasticity in an un-steady plane strain process

Abstract

Abstract An incompressible material obeying a pressure-dependent yield condition is confined between two planar plates which are inclined at an angle 2 α . The plates intersect in a hinged line and the angle α slowly decreases from an initial value. An initial/boundary value problem for the flow of the material is formulated and solved for the stress and the velocity fields, the solution being in closed form. The material is assumed to obey a special case of the double slip and rotation model, which generalises the classical plastic potential model and is also a variant of the double shearing model. The solution for the velocity field may exhibit sliding or sticking at the plates. Solutions which exhibit sticking may have a rigidly rotating zone in the region adjacent to the plates. It is shown that sliding occurs when the value of α is less than a certain critical value α c ; that sticking occurs without a rigid zone if α exceeds or equals α c but is less than a second critical value α 0 ; and that sticking with a rigid zone adjacent to the plates occurs if α exceeds α 0 . The values of α c and α 0 coincide for a certain range of model parameters. Solutions which exhibit sliding are singular. Qualitative features of the solution found are compared with those of the solution for the classical plastic potential model.