rateandstate

Rate-and-state friction is a constitutive model used to describe how friction on sliding interfaces, particularly tectonic faults, depends on both slip rate and the evolving state of contact. It integrates a memory of the contact surface through a state variable, typically denoted θ, which changes with slip and time. The framework accounts for frictional healing at low slip rates and velocity-weakening or strengthening at higher rates, enabling realistic representations of stick-slip earthquakes and fault behavior observed in experiments.

A widely used form expresses the friction coefficient μ as a function of slip velocity V and state θ: μ = μ0 + a ln(V/V0) + b ln( (θ V0) / Dc ), where μ0 is a reference friction, V0 is a reference velocity, Dc is a characteristic slip distance, and a and b are empirical parameters. The term a controls direct velocity effects, while b governs the evolution of friction through θ. The combination a − b determines the steady-state velocity dependence of friction.

The model employs two canonical state evolution laws. The aging (or Da) law is dθ/dt = 1 − (V θ)/Dc, describing gradual strengthening with time at rest and gradual weakening with slip. The slip (or Ruina) law is dθ/dt = − (V θ)/Dc ln( (V θ)/Dc ), emphasizing changes in θ driven by slip history. Both laws cause θ to adjust toward a state compatible with the current slip rate, producing corresponding changes in μ.

Steady-state friction for either law yields μss = μ0 + (a − b) ln(V/V0), illustrating rate-dependent behavior. Rate-and-state friction is applied in modeling fault slip, earthquake rupture, laboratory rock friction experiments, and numerical simulations of seismic processes. Limitations include parameter calibration challenges and potential inadequacy to capture all material and boundary-condition effects.