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for VARIATIONAL AND COMPUTATIONAL ASPECTS OF PROBLEMS IN CLASSICAL AND GRADIENT PLASTICITY

VARIATIONAL AND COMPUTATIONAL ASPECTS OF PROBLEMS IN CLASSICAL AND GRADIENT PLASTICITY

Daya Reddy
University of Cape Town

Abstract

The initial boundary value problem of elastoplasticity takes the form of a set of equations and inequalities which describe equilibrium behaviour, the elastic relation, and the plastic flow rule. The problem may be formulated as a variational inequality in two alternative ways, depending on the form of the flow rule used. This talk emphasises the formulation in which plastic flow is written in terms of a dissipation function, and examines features of the resulting variational inequality for classical plasticity and some models of gradient plasticity. The latter theories have been developed in response to the absence of an inherent length scale in classical theories of plasticity, and their consequential inability to capture size-dependent effects particularly at the mesoscale.

Conforming and discontinuous Galerkin finite element approximations are constructed, for the classical and gradient problems respectively. The convergence of these approximations and of the associated algorithms are established, and some numerical examples presented. 
 
A further focus of the presentation is on single-crystal gradient plasticity, for which a set of questions similar to those explored for the polycrystal case are addressed. A unified variational framework is adopted, in which gradient effects in the form either of gradients of slip rates or of the Burgers tensor, are accounted for. The development of convergent algorithms is explored for rate-independent systems, for both small and large strains, guided by the theoretical results, and with a view to elucidating the mathematical, physical and computational influence of the gradient terms.

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