Formulation, analysis and solution algorithms for a model of gradient plasticity within a discontinuous Galerkin framework

Doctoral Thesis


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University of Cape Town

An investigation of a model of gradient plasticity in which the classical von Mises yield function is augmented by a term involving the Laplacian of the equivalent plastic strain is presented. The theory is developed within the framework of non-smooth convex analysis by exploiting the equivalence between the primal and dual expressions of the plastic deformation evolution relations. The nonlocal plastic evolution relations for the case of gradient plasticity are approximated using a discontinuous Galerkin finite element formulation. Both the small- and finite-strain theories are investigated. Considerable attention is focused on developing a firm mathematical foundation for the model of gradient plasticity restricted to the infinitesimal-strain regime. The key contributions arising from the analysis of the classical plasticity problem and the model of gradient plasticity include demonstrating the consistency of the variational formulation, and analyses of both the continuous-in-time and fully-discrete approximations; the error estimates obtained correspond to those for the conventional Galerkin approximations of the classical problem. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution for both hardening and softening problems. It is well known that classical finite element method simulations of softening problems are pathologically dependent on the discretisation.

Includes bibliographical references (p. [221]-239).