Energetic formulation of multiplicative elasto-plasticity using dissipation distances

Abstract.

We introduce a new energetic formulation for the inelastic rate-independent behavior of standard generalized materials. This formulation is solely based on the classical elastic energy-storage potential $\hat{\psi}$ and a dissipation potential $\hat{\Delta}$, and it replaces the classical variational inequalities which describe the flow rules for the inelastic variables like the plastic deformation and the hardening parameters.

The energetic formulation has the major advantage that it is defined for a larger class of processes since it does not involve any derivatives of the strains or the internal variables, thus allowing for an analysis of processes involving sharp interfaces, localization or microstructure. Two new quantities are derived from $\hat{\psi}$ and $\hat{\Delta}$. First, this is the global dissipation distance $\hat{D}$ on the manifold of internal states. Second, the reduced stored-energy density $\Psi^{\rm red}$ contains the comprised information of the elastic and plastic material properties via minimization of $\hat{\psi}{+}\hat{D}$ over the new internal variable. Several stability concept are derived and used to analyze failure mechanism. Finally, a natural incremental method is proposed which reduces to a minimization problem and can be solved efficiently using $\Psi^{red}$.