Single-slip elastoplastic microstructures

We consider rate-independent crystal plasticity with constrained elasticity, and state the variational formulation of the incremental problem. For generic boundary data, even the first time increment does not admit a smooth solution, and fine structures are formed. By using the tools of quasiconvexity, we obtain an explicit relaxation of the first incremental problem for the case of a single slip system. Our construction shows that laminates between two different deformation gradients are formed. Plastic deformation concentrates in one of them, the other is a purely elastic strain. For the concrete case of a simple-shear test we also obtain a completely explicit solution