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.

Initiation propagation and termination of elastodynamic ruptures associated with segmentation of faults and shaking hazard

“…Nevertheless, the complex step over geometry displays many interesting features which are quite suggestive of real faults. For example, the distribution of segment lengths [Spyropoulos et al, 2002] evolves in ways similar to laboratory analogue systems [Spyropoulos et al, 1999], and normal faults on Venus where the best data set on fault population lengths has been found [Scholz, 1997]. Other aspects of the fault behavior, such as slip distributions along faults and at step overs, look very realistic when compared with field observations [Dawers and Anders, 1995;Manighetti et al, 2005].…”

“…The numerical scheme proceeds by first evolving the fault system quasistatically, taking advantage of the dependence of the fault system evolution on the total slip, rather than slip increments, on the faults. Previous work has shown the fault system which evolves is insensitive to the details of the slip increments, since the interactions occur through the total slip on the faults [Spyropoulos et al, 2002]. This allows us to evolve rapidly through geological time.…”

“…It begins from some initial unbroken strength having some overall value plus a spatially uncorrelated random component. A long term geological slip weakening localizes the slip onto faults and leads to a slow geological evolution of the fault system [Spyropoulos et al, 2002]. Dynamic weakening during slip events leads to sudden stick-slip events.…”

“…For large F 0 , we can operate in a regime where the saturating term a is small and irrelevant. The brittle strain excess e (nt À F 0 )/x 0 gives the relevant strain [Spyropoulos et al, 2002], so the amplitude x 0 of the uncorrelated random strength heterogeneity scales out of the problem.…”

“…With friction at high slip rates being an open question, we use a friction which has a minimum of parameters, is computationally efficient, and spans a range of frictional instabilities, including slip, time, and velocity weakening [Shaw, 1995;Shaw and Rice, 2000]. Specifically, we use a F which combines long-term geological strength F S which weakens with accumulated geological slip [Spyropoulos et al, 2002] and a dynamic strength F Q which weakens during events [Shaw, 1997] …”