RateandStateReibungsgesetze

Rate-and-State Friction Laws describe how the frictional resistance on sliding surfaces evolves with both the instantaneous slip rate and the evolving contact state of the surfaces. They are foundational in rock mechanics and seismology for modeling fault strength, earthquake nucleation, afterslip, and slow earthquakes. The central idea is that the friction coefficient mu depends on the current sliding velocity V and a state variable theta that encodes the history and maturity of contact asperities.

A common phenomenological form is mu = mu0 + a log(V/V0) + b log(theta/theta0), where mu0, V0, and theta0 are reference values, and a and b are empirical parameters. The term a log(V/V0) captures instantaneous rate dependence, while b log(theta/theta0) captures evolving state of contact interfaces. The state variable theta is not directly measurable in most experiments but represents, for example, the real area of contact or contact maturity.

Theta evolves according to one of two typical evolution laws. The aging law (often attributed to Dieterich) is d theta/dt = 1 - (V theta)/Dc, where Dc is a characteristic slip distance. The slip law (often attributed to Ruina) is d theta/dt = - (V theta)/Dc ln(V theta / Dc). Each law leads to different transient and stability behavior, though both reproduce many laboratory observations.

If the velocity-weakening regime (b > a) dominates, the system tends toward stick-slip and earthquakes; if velocity-strengthening (a > b) dominates, sliding tends to be stable. Calibrating parameters from experiments is challenging, and natural faults may exhibit substantial heterogeneity and scale effects.