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Mon Sep 7 03:39:45 2009 UTC (10 years ago) by jfenwick
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1 \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 fault}
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 fault}
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 fault}
20 \sigma\hackscore{ij}n\hackscore{j}=0
21 \end{eqnarray}
22 for all time $t>0$.

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