Variational formulations for discontinuous displacement fields in problems of the deformation theory of plasticity without hardening

Abstract

The problem of the construction and use of extended variational formulations which enable an explicit analysis to be made of discontinuous displacement fields for a wide class of problems of the deformation theory of plasticity is discussed. Three-dimensional, as well as plane problems with the Mises and Schleicher-Moreau criteria are investigated. In the case of a piecewise-continuous discontinuity line it is shown that the existence of a saddle point of an extended Lagrangian results in an integral inequality, which imposes certain conditions on the trace of the stress tensor on the line of discontinuity. Different arguments were used in [1–3] to obtain different versions of this condition for a number of problems of the theory of plasticity. When sufficient regularity of the stresses is assumed, then from the condition in question a simple algebraic relation follows connecting, at the line of discontinuity, the value of the stress tensor with the parameters determining the magnitude and direction of the discontinuity. Examples are given, which show that, generally speaking, only some of the stress states lying on the yield surface correspond to discontinuous solutions.