Derivation of Elasticity Equation
For completeness we start our discussion of the governing equations with a derivation of the elasticity equation.
Consider domain \(\Omega\) bounded by boundary \(\Gamma\).
Applying a Lagrangian description of the conservation of momentum gives
(1)\[\frac{\partial}{\partial t}\int_{\Omega}\rho(\vec{x})\frac{\partial\vec{u}}{\partial t}\, d\Omega=\int_{\Omega}\vec{f}(\vec{x},t)\, d\ + \int_{\Gamma}\vec{\tau}(\vec{x},t)\, d\Gamma.\]
The traction vector field is related to the stress tensor through
(2)\[\begin{equation}
\vec{\tau}(\vec{x},t) = \boldsymbol{\sigma}(\vec{u}) \cdot \vec{n},
\end{equation}\]
where \(\vec{n}\) is the outward normal vector to \(\Gamma\).
Substituting into equation (1) yields
(3)\[\begin{equation}
\frac{\partial}{\partial t}\int_{\Omega}\rho(\vec{x})\frac{\partial\vec{u}}{\partial t}\, d\Omega = \int_{\Omega}\vec{f}(\vec{x},t)\, d\Omega+\int_{\Gamma}\boldsymbol{\sigma}(\vec{u})\cdot\vec{n}\, d\Gamma.
\end{equation}\]
Applying the divergence theorem,
(4)\[\begin{equation}
\int_{\Omega}\boldsymbol{\nabla}\cdot\vec{a}\: d\Omega=\int_{\Gamma}\vec{a}\cdot\vec{n}\: d\Gamma,
\end{equation}\]
to the boundary integral results in
(5)\[\begin{equation}
\frac{\partial}{\partial t}\int_{\Omega}\rho(\vec{x})\frac{\partial\vec{u}}{\partial t}\, d\Omega=\int_{\Omega}\vec{f}(\vec{x},t)\, d\Omega+\int_{\Omega}\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}(\vec{u})\, d\Omega,
\end{equation}\]
which we can rewrite as
(6)\[\begin{equation}
\int_{\Omega}\left(\rho(\vec{x})\frac{\partial^{2}\vec{u}}{\partial t^{2}}-\vec{f}(\vec{x},t)-\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}(\vec{u})\right)\, d\Omega=\vec{0}.
\end{equation}\]
Because the domain \(\Omega\) is arbitrary, the integrand must be the zero vector at every location in the domain, so that we end up with
(7)\[\begin{gather}
\rho(\vec{x})\frac{\partial^{2}\vec{u}}{\partial t^{2}}-\vec{f}(\vec{x},t)-\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}=\vec{0}\text{ in }\Omega,\\
\boldsymbol{\sigma}(\vec{u})\cdot\vec{n}=\vec{\tau}(\vec{x},t)\text{ on }\Gamma_{\tau}\text{,}\\
\vec{u}=\vec{u}_0(\vec{x},t)\text{ on }\Gamma_{u},\text{ and}\\
\vec{u}^{+}-\vec{u}^{-}=\vec{d}\text{ on }\Gamma_{f}.
\end{gather}\]
We specify tractions, \(\vec{\tau}\), on boundary \(\Gamma_{f}\), displacements, \(\vec{u^{o}}\), on boundary \(\Gamma_{u}\), and slip, \(\vec{d}\), on fault interface \(\Gamma_{f}\).
