A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

Abstract

This article presents an efficient nonlinear stochastic finite element method to solve stochastic elastoplastic problems. Similar to deterministic elastoplastic problems, we describe history‐dependent stochastic elastoplastic behavior utilizing a series of (pseudo) time steps and go further to solve the corresponding stochastic solutions. For each time step, the original stochastic elastoplastic problem is considered as a time‐independent nonlinear stochastic problem with initial values given by stochastic displacements, stochastic strains, and internal variables of the previous time step. To solve the stochastic solution at each time step, the corresponding nonlinear stochastic problem is transformed into a set of linearized stochastic finite element equations by means of finite element discretization and a stochastic Newton linearization, while the stochastic solution at each time step is approximated by a sum of the products of random variables and deterministic vectors. Each couple of the random variable and the deterministic vector is also used to approximate the stochastic solution of the corresponding linearized stochastic finite element equation that can be solved via a weakly intrusive method. In this method, the deterministic vector is computed by solving deterministic linear finite element equations, and corresponding random variables are solved by a non‐intrusive method. Further, the proposed method avoids the curse of dimensionality successfully since its computational effort does not increase dramatically as the stochastic dimensionality increases. Four numerical cases are used to demonstrate the good performance of the proposed method.