Regularity results for two standard models in elasto-perfect-plasticity theory with hardening

TL;DR

This paper establishes uniform interior and boundary regularity estimates for stress and hardening parameters in elasto-plasticity models with hardening, including Sobolev and fractional derivatives, independent of the dimension.

Contribution

It provides new boundary regularity estimates for elasto-plasticity models with hardening, including uniform control of derivatives near the boundary, extending known interior regularity results.

Findings

Interior Sobolev regularity for stress and hardening

Boundary estimates for tangential and normal derivatives

Fractional derivative control for time derivatives

Abstract

We consider two most studied standard models in the theory of elasto-plasticity with hardening in arbitrary dimension $d\ge 2$, namely, the kinematic hardening and the isotropic hardening problem. While the existence and uniqueness of the solution is very well known, the optimal regularity up to the boundary remains an open problem. Here, we show that in the interior we have Sobolev regularity for the stress and hardening while for their time derivatives we have the "half" derivative with the spatial and time variable. This was well known for the limiting problem but we show that these estimates are uniform and independent of the order of approximation. The main novelty consist of estimates near the boundary. We show that for the stress and the hardening parameter, we control tangential derivative in the Lebesgue space~$L^2$, and for time derivative of the stress and the hardening we control the "half" time derivative and also spatial tangential derivative. Last, for the normal derivative, we show that the stress and the hardening have the $3/5$ derivative with respect to the normal and for the time derivative of the stress and the hardening we show they have the $1/5$ derivative with respect to the normal direction, provided we consider the kinematic hardening or near the Dirichlet boundary. These estimates are independent of dimension. In case, we consider the isotropic hardening near the Neumann boundary we shall obtain $W^{\alpha,2}$ regularity for the stress and the hardening with some $\alpha>1/2$ depending on the dimension and $W^{\beta,2}$ with some $\beta > 1/6$ for the time derivative of the stress and the hardening. Finally, in case of kinematic hardening the same regularity estimate holds true also for the velocity gradient.