Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity

In this paper we investigate the question of higher regularity up to the boundary for quasi-linear elliptic systems which originate from the time discretization of models from infinitesimal elasto-plasticity. Our main focus lies on an elasto-plastic Cosserat model. More specifically we show that the time discretization renders H-2-regularity of the displacement and H-1-regularity for the symmetric plastic strain epsilon(p) up to the boundary, provided that the plastic strain of the previous time step is in H-1 as well. This result contrasts with classical Hencky and Prandtl-Reuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity such as viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elasto-plasticity problem, provide the update functional for one time step, and show various preliminary results for the update functional (Legendre-Hadamard/monotonicity). Using nonstandard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of infinite element convergence. As a result we may obtain estimates linear in the mesh-width h in error estimates.