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

Abstract

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 H2-regularity of the displacement and H1 -regularity for the symmetric plastic strain εp up to the boundary, provided that the plastic strain of the previous time step is in H1 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 finite element convergence. As a result we may obtain estimates linear in the mesh-width h in error estimates. © 2008 Society for Industrial and Applied Mathematics.