A revision of results for standard models in elasto-perfect-plasticity theory

Abstract

We consider two most studied standard models in the theory of elasto-plasticity in arbitrary dimension $$d\ge 2$$ , namely, the Hencky model and the Prandtl–Reuss model subjected to the von Mises condition. There are many available results for these models—from the existence and the regularity theory up to the relatively sharp identification of the plastic strain in the natural function/measure space setting. In this paper we shall proceed further and improve some of known estimates in order to identify sharply the plastic strain. More specifically, we rigorously improve the integrability of the displacement and the velocity (which was known only under a nonnatural assumption that the Cauchy stress is bounded), show the BMO estimates for the stress and finally also the Morrey-like estimates for the plastic strain. In addition, we shall provide the whole theory up to the boundary. As an immediate consequence of such improved estimates, we provide a sharper identification of the plastic strain than that known up to date. In particular, in two dimensional setting, we show that the plastic strain can be point-wisely characterized in terms of the stresses everywhere although the stress is possibly discontinuous and thus the natural duality pairing in the space of measures could be violated.