# Earthquake Simulation with Rate-and-State Friction

Here we study a case of earthquake simulation on strike-slip fault.

The fraction on the fault is governed by rate-and-state friction with a aging law:

where $V$ is the fault slip velocity, $\Psi$ is a state variable controlling the fricional process, $D_c$ is the state evolution distance, $a$ and $b$ are dimensionless frictional parameters, $V_0$ is a reference steady sliding velocity, and $f_0$ is the reference friction coefficient for the reference velocity $V_0$.

: steady state velocity weakening.

: steady state velocity strengthening.

Using quasi-dynamic approximation, the inertial term is approximated by a radiation damping term to account for seismic waves, so force balance on the fault becomes:

where $\eta V$ is the radiation damping term, $\sigma_n$ is the effective normal stress. We set $\eta = \frac{\mu}{2 c_s}$, where $\mu$ is the shear modulus and $c_s$ is the shear wave speed.