Stochastic Multiscale Coupling of Inelastic Processes in Solid Mechanics

Abstract

Here we consider the inelastic nonlinear response of heterogeneous materials, possibly undergoing localised failure. We regard the material to be heterogeneous on many scales, but for simplicity we only look at one scale transition. The micro-scale is regarded as incompletely known and hence uncertain, therefore modelled probabilistically. Two alternative approaches are discussed: one for localised regions where a rather detailed micro-description is necessary to capture relevant effects, and the other in domains where it is accurate enough to define phenomenological models of ‘generalised standard materials’ on the macro-scale, which have to be identified via micro-scale computations. Apart from a proper transfer of mechanical quantities across scales, the same has to be achieved for the stochastic part of the model. Several main ingredients of the proposed approaches are discussed in detail, including micro-structure approximation with a structured mesh, random field representation, and Bayesian updating.