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

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 & Müller, 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.