Stochastic elastic–plastic finite elements

A computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic–plastic materials. Uncertain elastic–plastic material properties are modeled as random fields, which appear as the coefficient term in the governing partial differential equation of mechanics. A spectral stochastic elastic–plastic finite element method with Fokker–Planck–Kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non-linear (elastic–plastic) partial differential equation with stochastic coefficient. To this end, the random field material properties are discretized, in both spatial and stochastic dimension, into finite numbers of independent basic random variables, using Karhunen–Loève expansion. Those random variables are then propagated through the elastic–plastic constitutive rate equation using Fokker–Planck–Kolmogorov equation approach, to obtain the evolutionary material properties, as the material plastifies. The unknown displacement (solution) random field is then assembled - using polynomial chaos - as a function of known basic random variables and unknown deterministic coefficients, which are obtained by minimizing the error of finite representation, by Galerkin technique.