Spatial regularity for general yield criteria in dynamic and quasi-static perfect plasticity

TL;DR

This paper proves spatial regularity of solutions in dynamic and quasi-static perfect plasticity models under specific convexity conditions, showing solutions are Sobolev inside but may have boundary singularities.

Contribution

It establishes new regularity results for the stress and velocity in perfect plasticity models with general yield criteria, including boundary behavior analysis.

Findings

Stress admits locally square integrable spatial derivatives.

Velocity regularity is established in the dynamic case.

Boundary singularities can occur, affecting boundary conditions.

Abstract

This work addresses the question of regularity of solutions to evolutionary (quasi-static and dynamic) perfect plasticity models. Under the assumption that the elasticity set is a compact convex subset of deviatoric matrices, with $C^2$ boundary and positive definite second fundamental form, it is proved that the Cauchy stress admits spatial partial derivatives that are locally square integrable. In the dynamic case, a similar regularity result is established for the velocity as well. In the latter case, one-dimensional counterexamples show that, although solutions are Sobolev in the interior of the domain, singularities may appear at the boundary and the Dirichlet condition may fail to be attained.