An exact renormalization model for earthquakes and material failure statics and dynamics

Earthquake events are well-known to posses a variety of empirical scaling laws. Accordingly, renormalization methods offer some hope for understanding why earthquake statistics behave in a similar way over orders of magnitude of energy. We review the progress made in the use of renormalization methods in approaching the earthquake problem. In particular, earthquake events have been modeled by previous investigators as hierarchically organized bundles of fibers with equal load sharing. We consider by computational and analytic means the failure properties of such bundles of fibers, a problem that may be treated exactly by renormalization methods. We show, independent of the specific properties of an individual fiber, that the stress and time thresholds for failure of fiber bundles obey universal, albeit different, scaling laws with respect to the size of the bundles. The application of these results to fracture processes in earthquake events and in engineering materials helps to provide insight into some of the observed patterns and scaling - in particular, the apparent weakening of earthquake faults and composite materials with respect to size, and the apparent emergence of relatively well-defined stresses and times when failure is seemingly assured.