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Pol Grasland-Mongrain edited The_absorption_of_the_laser__.tex
over 8 years ago
Commit id: 143d0e593050aa77134bbc2bdaed33e4f2ed9f11
deletions | additions
diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex
index 22319c4..692e967 100644
--- a/The_absorption_of_the_laser__.tex
+++ b/The_absorption_of_the_laser__.tex
...
\sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - 3(\lambda + \frac{2}{3}\mu) \frac{\alpha E}{\rho C S \zeta}
\label{eq:stressThermo}
\end{equation}
where
$\lambda + 2 \mu$ is the P-wave modulus, $\lambda + \frac{2}{3}\mu$ the S-wave modulus with $\lambda$ and $\mu$ respectively the first and second Lamé's coefficient, $\alpha$ the thermal dilatation coefficient and $\zeta$ the depth over which there is a temperature rise. In the absence of external constraints normal to the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$, so that equation \ref{eq:stressThermo} can be integrated, giving a displacement $u_z$ from the surface:
\begin{equation}
u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho C S \zeta}
\label{eq:deplThermo}