Some properties of the dissipative model of strain-gradient plasticity

Abstract

A theoretical and computational investigation is carried out of a dissipative model of rate-independent strain-gradient plasticity and its regularisation. It is shown that the flow relation, when expressed in terms of the Cauchy stress, is necessarily global. The most convenient approach to formulating the flow relation is through the use of a dissipation function. It is shown, however, that the task of obtaining the dual version, in the form of a normality relation, is a complex one. A numerical investigation of problems in two space dimensions casts further light on the response using the dissipative theory in situations of non-proportional loading. The elastic gap, a feature reported in recent investigations, is observed in situations in which passivation has been imposed. The computational study indicates that the gap may be regarded as an efficient path between a load-deformation response corresponding to micro-free boundary conditions, and that corresponding to micro-hard boundary conditions, in which plastic strains are set equal to zero on all or part of the boundary.

Keywords:

• Strain gradient plasticity
• dissipative model
• flow relation
• elastic gap

Acknowledgements

BDR acknowledges many helpful discussions on the topic of this paper with JW Hutchinson. The work reported in this paper was carried out with support through the South African Research Chair in Computational Mechanics to BDR and ATMcB. This support is gratefully acknowledged. PS acknowledges support through the Collaborative Research Center 814.

Notes

No potential conflict of interest was reported by the authors.

1 More generally, one considers a free energy that depends in addition on the plastic strain, the plastic strain gradient and, possibly, hardening internal variables. Details may be found, for example, in [7].

2 The subdifferential of a convex function F is defined by(20)

That is, is the set of tangents at the point . If F is smooth at then comprises a single member, viz. the tangent to F at , or equivalently the gradient or normal to the level set For this and other concepts from convex analysis, see for example [18,19].