Solution of Elastoplastic Problems by a Newton-like Method Utilizing Slanting Functions
Dipl.-Ing. Peter GruberNov. 28, 2006, 3:30 p.m. T 1010
We discuss a new solution algorithm for quasi-static elastoplastic problems with hardening. Such problems are described by a time dependent variational inequality of the second kind, where the displacement and the plastic strain serve as primal variables. After discretization in time, the variational problem is reformulated as the minimization of each one convex energy functional per time step. Each functional 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 fixed displacement. Thus, by the substitution, a minimization functional depending on the displacement only can be obtained.
Due the our new approach which utilizes well known Moreau's theorem from convex analysis, we show that this functional in necessarily differentiable with the explicitely computable first derivative. The second derivative of the energy functional does not exist. However, a Newton-like method exploiting slanting functions of the energy functionals' first derivative can be applied.