Work on this problem has proceeded on several fronts over the last few decades. The statistical physics community has focused on so-called random fuse type models which consist of networks of springs with randomly assigned breaking thresholds. Unfortunately, these models are geared for studying fractures which occur in simple tension and are ill suited for studying situations such as simple shear where one might expect frictional sliding along incipient failure surfaces to become important. Others have studied the dynamics of crack propagation or the dynamics of ruptures which run along geometrically simple frictional interfaces. In such models, one does not observe the evolution of the geometrically complex failure surfaces seen in laboratory experiments or at the geological scale in the field.
Mark Robbins and I are currently performing numerical simulations using a novel technique to introduce damage directly into particle based models. The virtue of such an approach is that it allows one to model both the evolution of the complex geometry of damaged regions along with the avalanche-like dynamics (earthquake ruptures) which leads to the evolution from within a single framework. The image here shows one such damaged configuration of particles, with connected clusters of damaged particles given the same color. This movie shows the evolution of the local deviatoric strain, while this movie shows the evolution of local rotations (allowing one to determine the sense of simple shear and observe the conjugate sets of so-called "left" and "right" lateral shears) as the top edge of the sample is moved to the right while the bottom is held fixed. Issues which we are currently investigating include patterns of correlated damage, statistics of avalanches of incurred damage, dynamics of individual rupture/damage pulses, finite-size effects, etc.