On the existence of minimisers for strain-gradient single-crystal plasticity

TL;DR

This paper proves the existence of minimisers in a strain-gradient single-crystal plasticity model with single-slip constraints, ensuring lower-semicontinuity of the energy through advanced mathematical techniques.

Contribution

It introduces a novel existence proof for minimisers in relaxed single-crystal plasticity models with single-plane conditions, using exclusion lemmas and div-curl methods.

Findings

Existence of energy minimisers under single-plane slip constraints.

Lower-semicontinuity of dislocation and elastic energies established.

Applicability of mathematical techniques to complex elastoplastic models.

Abstract

We prove the existence of minimisers for a family of models related to the single-slip-to-single-plane relaxation of single-crystal, strain-gradient elastoplasticity with $L^p$-hardening penalty. In these relaxed models, where only one slip-plane normal can be activated at each material point, the main challenge is to show that the energy of geometrically necessary dislocations is lower-semicontinuous along bounded-energy sequences which satisfy the single-plane condition, meaning precisely that this side condition should be preserved in the weak $L^p$-limit. This is done with the aid of an 'exclusion' lemma of Conti \& Ortiz, which essentially allows one to put a lower bound on the dislocation energy at interfaces of (single-plane) slip patches, thus precluding fine phase-mixing in the limit. Furthermore, using div-curl techniques in the spirit of Mielke \& M\"uller, we are able to show that the usual multiplicative decomposition of the deformation gradient into plastic and elastic parts interacts with weak convergence and the single-plane constraint in such a way as to guarantee lower-semicontinuity of the (polyconvex) elastic energy, and hence the total elasto-plastic energy, given sufficient ($p>2$) hardening, thus delivering the desired result.