Mixed analytical–numerical relaxation in finite single-slip crystal plasticity

The modeling of the finite elastoplastic behaviour of single crystals with one active slip system leads to a nonconvex variational problem, whose minimization produces fine structures. The computation of the quasiconvex envelope of the energy density involves the solution of a nonconvex optimization problem and faces severe numerical difficulties from the presence of many local minima. In this paper, we consider a standard model problem in two dimensions and, by exploiting analytical relaxation results for limiting cases and the special structure of the problem at hand, we obtain a fast and efficient numerical relaxation algorithm. The effectiveness of our algorithm is demonstrated with numerical examples. The precision of the results is assessed by lower bounds from polyconvexity.