Stochastic multiscale modeling of crack propagation in random heterogeneous media

A stochastic approach to model crack propagation in random heterogeneous media, using mesoscopic representations of elastic and fracture properties, is presented. In order to obtain reference results, Monte-Carlo simulations are first conducted on microstructural samples in which a pre-existing crack is propagated by means of a phase-field approach. These computations are used to estimate the subscale-induced randomness on the macroscopic response of the domain. Mesoscopic descriptors are then introduced to investigate scale transition. Elasticity tensor random fields are specifically defined, at that stage, through a moving-window upscaling approach. The mesoscopic fracture toughness, which is assumed homogeneous and deterministic, is identified by solving an inverse problem involving the macroscopic peak force. A stochastic model is subsequently constructed in which the mesoscopic elasticity is described as a non-Gaussian random field. This model allows the multiscale-informed elastic counterpart in the phase-field formulation to be sampled without resorting to computational homogenization. The results obtained with the sample-based and model-based mesoscopic descriptions are finally compared with those corresponding to the full-scale microscopic model. It is shown, in particular, that the mesoscopic elasticity-phase-field formulation associated with statically uniform boundary conditions enables the accurate predictions of the mean elastic response and mean peak force.