Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution

Abstract

Cam-Clay plasticity is a well-established model for the description of the mechanics of fine grained soils. As solutions can develop discontinuities in time, a weak notion of solution, in terms of a rescaled time s, has been proposed in Dal Maso, DeSimone and Solombrino (Calc Var Partial Equ 40:125–181, 2011) to give a meaning to this discontinuous evolution. In this paper we first prove that this rescaled evolution satisfies the flow-rule for the rate of plastic strain, in a suitable measure-theoretical sense. In the second part of the paper we consider the behavior of the evolution in terms of the original time variable t. We prove that the unrescaled solution satisfies an energy-dissipation balance and an evolution law for the internal variable, which can be expressed in terms of integrals depending only on the original time. Both these integral identities contain terms concentrated on the jump times, whose size can only be determined by looking at the rescaled formulation.