Solution of One-Time-Step Problems in Elastoplasticity by a Slant Newton Method

We discuss a solution algorithm for quasi-static elastoplastic problems with hardening. Such problems can be 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. By substitution, the energy functional can be written as a functional depending only on the displacement. The theorem of Moreau from convex analysis states that the energy functional is differentiable with an explicitly computable first derivative. The second derivative of the energy functional does not exist, hence the plastic strain minimizer is not differentiable on the elastoplastic interface, which separates the continuum in elastically and plastically deformed parts. A Newton-like method exploiting slanting functions of the energy functional’s first derivative instead of the nonexistent second derivative is applied. Such method is called a slant Newton method for short. The local super-linear convergence of the slant Newton method in the discrete case is shown and sufficient regularity assumptions are formulated, which would guarantee the local super-linear convergence also in the continuous case.