Local and nonlocal material models, spatial randomness, and impact loading

In many material systems, both man-made and natural, we have an incomplete knowledge of geometric or material properties, which leads to uncertainty in predicting their performance under dynamic loading. Given the uncertainty and a high degree of spatial variability in properties of geological formations subjected to impact, a stochastic theory of continuum mechanics would be useful for modeling dynamic response of such systems. In this paper, we examine spatial randomness in local and nonlocal material-mechanics models. We begin with classical linear elasticity. Then, we consider nonlocal elasticity and, finally, peridynamic theory. We discuss a formulation of stochastic peridynamic theory and illustrate this formulation with examples of impact loading of geological materials with uncorrelated versus correlated properties, sampled in a Monte Carlo sense. We examine wave propagation and damage to the material. The most salient feature is the absence of spallation, referred to as disorder toughness, which, in fact, generalizes similar results from earlier quasi-static damage mechanics.