Synergy of stochastics and inelasticity at multiple scales: novel Bayesian applications in stochastic upscaling and fracture size and scale effects

The main goal of this review is to provide a thorough scientific understanding of the interplay between stochastics and mechanics, by classifying what can be achieved by representing mechanical system parameters in terms of deterministic values (homogenization) versus random variables or random fields (stochastic upscaling). The latter is of special interest for novel Bayesian applications capable of successfully handling the phenomena of fracture in both the quasi-static and the dynamic evolution of heterogeneous solids where no scale separation is present, which we refer to as stochastic upscaling. We seek to quantify the sensitivity of these phenomena with respect to the size-effect (changes in characteristic system dimension) and to the scale-effect (changes in characteristic time evolution). The challenge is to provide an answer as to why a system that is big does not break under quasi-static loads in the same way as a small system, even when both are built of the same material, and further extend this to inelasticity and fracture under dynamic loads. We plan to illustrate the crucial role of fine-scale heterogeneities and to develop the ground-breaking concept of stochastic upscaling that can capture their influence on instability and dynamic fracture at the system macro-scale. The stochastic upscaling is the key to size and scale laws in the proposed multi-scale approach, which can reach beyond homogenization to properly account for epistemic uncertainties of system parameters and the stochastic nature of dynamical fracture.