Stochastic Plasticity - A Variational Inequality Formulation and Functional Approximation Approach. I: The Linear Case

In this paper we formulate and study the existence and uniqueness of the solution for a class of stochastic mixed variational inequalities arising in problems of infinitesimal elastoplasticity described by uncertain parameters. As a particular example we consider the quasi-static von Mises elastoplastic rate-independent evolution problem with linear elastic behaviour and hardening. For such a problem under the neccessary assumptions we show the equivalency between the variational inequality and a quadratic minimization problem described by a strictly convex, continuous, G^{a}teaux differentiable, and coercive functional on a Hilbert space. In order to find the unique minimiser we propose the stochastic closest point projection method, obtained by extension of the well known classical return alogorithms to the more general stochastic case. The method is, similarly to its deterministic counterpart, described by non-dissipative and dissipative operators.