Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling

Cam-Clay nonassociative plasticity exhibits both hardening and softening behaviour, depending on the loading. For many initial data the classical formulation of the quasistatic evolution problem has no smooth solution. We propose here a notion of general- ized solution, based on a viscoplastic approximation. To study the limit of the viscoplastic evolutions we rescale time, in such a way that the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized through an energy-dissipation balance, that can be written in a natural way using the rescaled time. As shown in Dal Maso and DeSimone (Math Models Methods Appl Sci 19:1–69, 2009) and Dal Maso and Solombrino (Netw Heterog Media 5:97–132, 2010), the proposed solution may be discontinuous with respect to the original time. Our formulation allows us to compute the amount of viscous dissipation occurring instantaneously at each discontinuity time.