Ill-posedness of the initial and boundary value problems in non-associative plasticity

Abstract

Associative plasticity theories do not predict correctly the volumetric plastic strain, in the course of plastic deformation, in the case of materials where the position and conformation of the yield surface are functions of the prevailing hydrostatic stress. Non-associative theories have been proposed and used to correct this deficiency. Such theories, however, lead to other serious difficulties. In this paper we establish clear criteria for the well-posedness of the initial, boundary/initial and boundary value problems when the plasticity theory is associative as well as non-associative. We further show cases where non-associativity leads to ill-posedness of these problems even when the material is not at failure. Specifically we demonstrate that the initial/boundary and boundary value problems either have no solution, or if they do, the solution is not unique. We also show by specific examples that the banding condition, i.e., the vanishing of the determinant of the acoustic tensor, is tantamount (a) to loss of hyperbolicity of the equation of motion and (b) lack of existence or loss of uniqueness of the solution of the boundary value problem, in certain situations. Finally, we show the existence of a fundamental criterion that governs the stability of infinitesimal as well as finite elastoplastic domains.