Regularity lost: the fundamental limitations and constraint qualifications in the problems of elastoplasticity

Abstract

We investigate the existence and non-existence of a function-valued strain solution in various models of elastoplasticity from the perspective of the constraint-based ``dual'' formulations. We describe abstract frameworks for linear elasticity, elasticity-perfect plasticity and elasticity-hardening plasticity in terms of adjoint linear operators and convert them to equivalent formulations in terms of differential inclusions (the sweeping process in particular). Within such frameworks we consider several manually solvable examples of discrete and continuous models. Despite their simplicity, the examples show how for discrete models with perfect plasticity it is possible to find the evolution of stress and strain (elongation), yet continuum models within the same framework may not possess a function-valued strain. Although some examples with such phenomenon are already known, we demonstrate that it may appear due to displacement loading. The central idea of the paper is to explain the loss of strain regularity in the dual formulation by the lack of additivity of the normal cones and the failure of Slater's constraint qualification. In contrast to perfect plasticity, models with hardening are known to be well-solvable for strains. We show that more advanced constraint qualifications can help to distinguish between those cases and, in the case of hardening, ensure the additivity of the normal cones, which means the existence of a function-valued strain rate.