Nucleation of rupture under slip dependent friction law: Simple models of fault zone

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Abstract
The initiation of frictional instability is investigated for simple models of fault zone using a linearized perturbation analysis. The fault interface is assumed to obey a linear slip‐weakening law. The fault is initially prestressed uniformly at the sliding threshold. In the case of antiplane shear between two homogeneous linearly elastic media, space‐time and spectral solutions are obtained and shown to be consistent. The nucleation is characterized by (1) a long‐wavelength unstable spectrum bounded by a critical wave number; (2) an exponential growth of the unstable modes; and (3) an induced off‐fault deformation that remains trapped within a bounded zone in the vicinity of the fault. These phenomena are characterized in terms of the elastic parameters of the surrounding medium and a nucleation length that results from the coupling between the frictional interface and the bulk elasticity. These results are extended to other geometries within the same formalism and implications for three‐dimensional rupture are discussed. Finally, internal fault structures are investigated in terms of a fault‐parallel damaged zone. Spectral solutions are obtained for both a smooth and a layered distribution of damage. For natural faults the nucleation is shown to depend strongly on the existence of a internal damaged layer. This nucleation can be described in terms of an effective homogeneous model. In all cases, frictional trapping of the deformation out of the fault can lead to the property that arbitrarily long wavelengths remain sensitive to the existence of a fault zone.