Existence of minimisers for nonlinear strain‐gradient elastoplasticity with finite or infinite cross‐hardening

Consideration is given to the existence of minimisers for a family of variational models of finite‐strain single‐crystal elastoplasticity with infinite cross‐hardening. The non‐convex cross‐hardening condition on the plastic slip necessitates the use of special analytical tools, in particular the combination of the div‐curl Lemma with a slip‐exclusion Lemma of Conti & Ortiz [1], if one wishes to prove existence for physically reasonable parameters. A regularised model with a cross‐hardening matrix is also briefly discussed ‐ existence of minimisers for this model also follows by a div‐curl argument, at least if one goes over to the case of linearised elasticity. Moreover, in this case one can also prove that the regularised model Γ‐converges to the infinite‐cross‐hardening model as the hardening matrix becomes unboundedly large. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)