seas.markdown

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SEAS models capture the entire earthquake cycle, i.e. tectonic loading, nucleation, rupture, and afterslip, within one physical model {% cite erickson2020 %}. A fault is idealized as an infinitesimally thin fault surface embedded in linear elastic media, and on-fault behaviour is described via laboratory-derived rate and state friction laws. Friction couples the slip $$S$$, normal stress $$\sigma_n$$, and shear traction $$\tau$$ on the fault surface $$\Gamma_F$$, cf. the following conceptual illustration of a normal fault.

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The rate and state friction relations are given by

$$ \begin{aligned} -\tau_i &= \sigma_nf(|V|,\psi) V_i / |V| \\ \frac{dS_i}{dt} &= V_i\\ \frac{d\psi}{dt} &= g(|V|,\psi) \end{aligned} $$

The first equation states that shear traction is proportional to normal stress times a coefficient of friction $$f$$, where $$f$$ depends on the slip-rate and a single state variable. Moreover, slip-rate $$V$$ and shear stress $$\tau$$ are anti-parallel. The evolution of state is controlled by $$g$$.

The friction equations are coupled through the equations of linear elasticity in the domain $$\Omega$$, i.e.

$$ -\frac{\partial\sigma_{ij}(\bm{u})}{\partial x_j} = 0. $$

(Sums over indices appearing twice are implied.) A slip boundary is imposed in the linear elasticity problem, that is,

$$ \llbracket u_i\rrbracket = T_{ij}S_j \text{ on } \Gamma_F, $$

where $$T_{ij}(\bm{n})$$ is a $$D \times (D-1)$$ matrix, $$D$$ being the space dimension, which contains a tangential basis of a fault segment with normal n.

The linear elasticity problem omits modelling of seismic waves, which are relevant during an earthquake but can be neglected otherwise. In order to get a stable formulation, the outflow of energy due to seismic waves is approximated with the damping term $$\eta V_i$$ in the frictional relation. {% cite rice1993 %}

$$ -\tau_i = \sigma_nf(|V|,\psi) V_i / |V| \color{red} + \eta V_i $$

Adding the constitutive relation, Dirichlet and Neumann boundary conditions, and the damping term we get the following system of equations:

**Linear elasticity with slip BC**

$$ \begin{aligned} -\frac{\partial\sigma_{ij}(\bm{u})}{\partial x_j} &= F_i & \text{ in } & \Omega\\ \sigma_{ij}(\bm{u}) &= c_{ijkl}\epsilon_{kl}(\bm{u}) & \text{ in } & \Omega\\ u_i &= g_i& \text{ on } & \Gamma_D \\ \sigma_{ij}(\bm{u})n_j &= 0 & \text{ on } & \Gamma_N \\ \llbracket u_i\rrbracket &= T_{ij}S_j & \text{ on } & \Gamma_F \end{aligned} $$

($$F$$: body force, $$c$$: stiffness tensor, $$n$$: unit normal)

**Rate and state friction on $$\Gamma_F$$**

$$ \begin{aligned} -\tau_i &= \sigma_nf(|V|,\psi) V_i / |V| + \eta V_i \\ \frac{dS_i}{dt} &= V_i\\ \frac{d\psi}{dt} &= g(|V|,\psi)\\ \tau_i &= T_{ji}\sigma_{jk}(\bm{u})n_k \\ \sigma_n &= n_i\sigma_{ij}(\bm{u})n_j \end{aligned} $$

($$\delta$$: Kronecker symbol, $$n$$: unit normal)

Although seismic waves are neglected, tectonic loading, nucleation, rupture, and afterslip can be observed in a SEAS model:

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The above shows a 2D simulation of a normal fault (vertical axis) over 1500 years. Slip profiles are plotted along the horizontal axis, and displaced by time in the in-screen direction. An earthquake (in red) occurs about every hundred years.

Time-steps vary strongly: In the interseismic phase time-steps of days to months are possible, whereas in the coseismic phase time-steps in the order of milliseconds are required.

References

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