Newton-Like Solver for Elastoplastic Problems with Hardening and its Local Super-Linear Convergence

We discuss a new solution algorithm for quasi-static elastoplastic problems with hardening. Such problems are described by a time dependent variational inequality, where the displacement and the plastic strain fields serve as primal variables. After discretization in time, one variational inequality of the second kind is obtained per time step and can be reformulated as each one minimization problem with a convex energy functional, which depends smoothly on the displacement and non-smoothly on the plastic strain. There exists an explicit formula how to minimize the energy functional with respect to the plastic strain for a given displacement. Thus, by its substitution, an energy functional depending only on the displacement can be obtained. Our technique based on the well known theorem of Moreau from convex analysis shows that the energy functional is differentiable with an explicitely computable first derivative. The second derivative of the energy functional exists everywhere in the domain apart from the elastoplastic interface, which separates the deformed continuum in elastic and plastic parts. A Newton-like method exploiting slanting functions of the energy functional’s first derivative is proposed and implemented numerically. The local super-linear convergence of the Newton-like method in the discrete case is shown and sufficient regularity assumptions are formulated to guarantee local super-linear convergence also in the continuous case.