RateandStateModellen

RateandStateModellen, or rate-and-state friction models, are a framework used to describe how the frictional resistance on a sliding contact evolves with slip rate and the evolving state of the contact. They are widely applied in geophysics to model fault friction, laboratory rock experiments, and processes such as earthquake nucleation and afterslip.

A central feature is that the friction coefficient μ depends on the instantaneous slip rate v and a state variable θ that captures the contact history and age. A common form is

μ = μ0 + a ln(v/v0) + b ln(θ/θ0),

where μ0 is a reference friction, v0 and θ0 are reference values, and a and b are empirical constants. The parameter Dc, the critical slip distance, sets the scale over which state evolves.

Two standard evolution laws describe how θ changes with time:

- Aging (or first-order) law: dθ/dt = 1 − (v θ)/Dc. This law represents gradual strengthening with contact aging and gradual weakening due to slip.

- Slip law: dθ/dt = − (v θ)/Dc ln(1 + Dc/(v θ)). This form emphasizes changes driven primarily by slip history.

Behavior depends on a − b: velocity-strengthening (a − b > 0) tends to produce stable sliding, while velocity-weakening (a − b < 0) can lead to stick-slip and earthquake-like instability.

History and use: developed independently by Dieterich and Ruina in the late 20th century, rate-and-state models remain a standard tool for interpreting frictional dynamics, though they are phenomenological and require careful calibration to specific materials and conditions.

---