Constitutive relations for fault slip and earthquake instabilities

Constitutive relations for fault slip are described and adopted as a basis for analyzing slip motion and its instability in the form of earthquakes on crustal faults. The constitutive relations discussed include simple rate-independent slip-weakening models, in which shear strength degrades with ongoing slip to a residual frictional strength, and also more realistic but as yet less extensively applied slip-rate and surface-state-dependent relations. For the latter the state of the surface is characterized by one or more variables that evolve with ongoing slip, seeking values consistent with the current slip rate. Models of crustal faults range from simple, single-degree-of-freedom spring-slider systems to more complex continuous systems that incorporate nonuniform slip and locked patches on faults of depth-dependent constitutive properties within elastic lithospheric plates that may be coupled to a viscoelastic asthenosphere. Most progress for the rate and state-dependent constitutive relations is at present limited to single-degree-of-freedom systems. Results for stable and unstable slip with the various constitutive models are summarized. Instability conditions are compared for spatially uniform versus nonuniform slip, including the elastic — brittle crack limit of the nonuniform mode. Inferences of constitutive and fracture parameters are discussed, based on earthquake data for large ruptures that begin with slip at depth, concentrating stress on locked regions within a brittle upper crust. Results of nonlinear stability theory, including regimes of complex sustained stress and slip rate oscillations, are outlined for rate and state-dependent constitutive relations, and the manner in which these allow phenomena like time-dependent failure, restrengthening in nearly stationary contact, and weakening in rapidly accelerated slip, is discussed.