Approximation of dynamic and quasi-static evolution problems in elasto-plasticity by cap models

This work is devoted to the analysis of elasto-plasticity models arising in soil mechanics. Contrary to the typical models mainly used for metals, it is required here to take into account plastic dilatancy due to the sensitivity of granular materials to hydrostatic pressure. The yield criterion thus depends on the mean stress and the elasticity domain is unbounded and not invariant in the direction of hydrostatic matrices. In the mechanical literature, so-called cap models have been introduced, where the elasticity domain is cut in the direction of hydrostatic stresses by means of a strain-hardening yield surface, called a cap. The purpose of this article is to study the well-posedness of such models in dynamical and quasi-static regimes. An asymptotic analysis as the cap is moved to infinity is also performed, which enables one to recover solutions to the uncapped model of perfect elasto-plasticity.