Solution of Elastoplastic Problems based on Moreau-Yosida Theorem

The quasi-static problem of elastoplasticity with isotropic hardening is considered, with particular emphasis on its numerical solution by means of a Newton-like method. The smoothness properties necessary for such method can be ensured by Moreau-Yosida’s theorem. After time discretization, the classical formulation of the elastoplastic problem is transformed to a minimization problem. The convex minimization functional depends on the displacement smoothly and on the plastic part of the strain non-smoothly. The minimization problem implicitly defines the plastic part of the strain as a function depending on the total strain tensor. Furthermore, this function may be calculated analytically. That way, the minimization problem depends on the displacement only. The Moreau-Yosida theorem, well-known in convex analysis, states that the minimization functional is smooth and also provides the Fréchet derivative of the functional. Hence, a method of the field of smooth optimization may be applied. After discretization in space, a Newton-like method is used to solve the discrete minimization problem. This method requires the Hesse matrix of the discrete minimization functional, which may be calculated in all material points apart from the elastoplastic interface. Numerical experiments in two dimensions show super-linear convergence of the Newton-like method. From the moment that elastic and plastic zones freeze, even quadratic convergence can be observed.