doc/user/slip.tex

\section{Slip on a Fault}
\label{Slip CHAP}

In this next example we want to calculate the displacement field $u\hackscore{i}$ for any time $t>0$ by solving the wave equation:
\index{wave equation}
\begin{eqnarray}\label{WAVE general problem}
\rho u\hackscore{i,tt} - \sigma\hackscore{ij,j}=0
\end{eqnarray}
in a three dimensional block of length $L$ in $x\hackscore{0}$
and $x\hackscore{1}$ direction and height $H$
in $x\hackscore{2}$ direction. $\rho$ is the known density which may be a function of its location.
$\sigma\hackscore{ij}$ is the stress field \index{stress} which in case of an isotropic, linear elastic material is given by
\begin{eqnarray} \label{WAVE stress}
\sigma\hackscore{ij} & = & \lambda u\hackscore{k,k} \delta\hackscore{ij} + \mu ( u\hackscore{i,j} + u\hackscore{j,i})
\end{eqnarray}
where $\lambda$ and $\mu$ are the Lame coefficients 
\index{Lame coefficients} and $\delta\hackscore{ij}$ denotes the Kronecker symbol\index{Kronecker symbol}.
On the boundary the normal stress is given by
\begin{eqnarray} \label{WAVE natural}
\sigma\hackscore{ij}n\hackscore{j}=0
\end{eqnarray}
for all time $t>0$.
1	gross	2647	\section{Slip on a Fault}
2			\label{Slip CHAP}
3
4			In this next example we want to calculate the displacement field $u\hackscore{i}$ for any time $t>0$ by solving the wave equation:
5			\index{wave equation}
6			\begin{eqnarray}\label{WAVE general problem}
7			\rho u\hackscore{i,tt} - \sigma\hackscore{ij,j}=0
8			\end{eqnarray}
9			in a three dimensional block of length $L$ in $x\hackscore{0}$
10			and $x\hackscore{1}$ direction and height $H$
11			in $x\hackscore{2}$ direction. $\rho$ is the known density which may be a function of its location.
12			$\sigma\hackscore{ij}$ is the stress field \index{stress} which in case of an isotropic, linear elastic material is given by
13			\begin{eqnarray} \label{WAVE stress}
14			\sigma\hackscore{ij} & = & \lambda u\hackscore{k,k} \delta\hackscore{ij} + \mu ( u\hackscore{i,j} + u\hackscore{j,i})
15			\end{eqnarray}
16			where $\lambda$ and $\mu$ are the Lame coefficients
17			\index{Lame coefficients} and $\delta\hackscore{ij}$ denotes the Kronecker symbol\index{Kronecker symbol}.
18			On the boundary the normal stress is given by
19			\begin{eqnarray} \label{WAVE natural}
20			\sigma\hackscore{ij}n\hackscore{j}=0
21			\end{eqnarray}
22			for all time $t>0$.