Formulation of a heterogeneous stochastic elasto-plastic material

Heterogeneities at the micro-structural level are usually subjected to a number of uncertainties. There it is assumed that the heterogeneous material behaves according to an elasto-plastic model, but with uncertain parameters, which are modeled as random fields like Young's modulus, yield stress etc. Actually, we try to model the simplest variant which is isotropic elasto-perfectly plastic behavior. The evolution law for the internal variables now becomes a stochastic law. The internal variables are also modeled as random fields and we develop evolution equations for these. This uses the polynomial chaos expansion to project the evaluation laws of the internal variables. The internal variables are also modeled as random fields, and we develop evolution equations for these. This uses the polynomial chaos expansion to project the evaluation laws of the internal variables, giving evolution equations for each component in the expansion. By truncating the expansion, a numerical procedure is derived to actually compute the evolution of the internal parameters as random variables. This in turn then enables us to compute various quantities of interest; which are functional of the internal parameters, like their mean and variance, or for example the exceedence probability of a certain level.