Transition regimes for growing crack populations.

Abstract

Numerous observational papers on crack populations in the material and geological sciences suggest that cracks evolve in such a way as to organize in specific patterns. However, very little is known about how and why the self-organization comes about. We use a model of tensile-like cracks with friction in order to study the time and space evolution of normal faults. The premise of this spring-block analog is that one could model crustal deformation for long time scales assuming a brittle layer coupled to a ductile substrate. The long time-scale physics incorporated into the model are slip-weakening friction, strain-hardening rheology for coupling the two layers, and randomly distributed yield strength of the brittle layer. We investigate how the evolution of populations of cracks depends on these three effects, using linear stability analysis to calculate the stable regimes for the friction as well as numerical simulations to model the nonlinear interactions of the cracks. We find that we can scale the problem to reduce the relevant parameters to a single one, the slip weakening. We show that the distribution of lengths of active cracks makes a transition from an exponential at very low strains, where crack nucleation prevails, to a power law at low to intermediate strains, where crack growth prevails, to an exponential distribution of the largest cracks at higher strains, where coalescence dominates. There is evidence of these different length distributions in continental and oceanic normal faults. For continental deformation the strain is low, and the faults have power-law frequency-size distributions. For mid-ocean ridge flanks the strain is greater, up to an order of magnitude higher than the continental strain, and faults have exponential-like frequency-size distributions. No theory has been offered to explain this difference in the distributions of continental and mid-ocean faults. In this paper we argue that they are indicative of different stages of evolution. The former faults are at an early stage of relatively small deformation, while the latter are at a later stage of the evolution. For high strain the faults reach a saturation regime with system size cracks evenly spaced in proportion to the brittle layer thickness. We asymptotically approximate the time space evolution of faults as a long time-scale phenomenon, thereby avoiding modeling the short time-scale earthquakes. We show that this assumption is valid, which implies that the faults that creep and faults with earthquakes display the same time and space evolutions.