Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case

We study the spatially uniform case of the quasistatic evolution in Cam-Clay plasticity, a relevant example of small strain nonassociative elastoplasticity. Introducing a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity; the slow dynamics, when the stress evolves smoothly on the yield surface and plastic flow is produced; the fast dynamics, which may happen only in the softening regime, when viscous solutions exhibit a jump determined by the heteroclinic orbit of an auxiliary system. We give an iterative procedure to construct a viscous solution.