deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 5ef02f7..5dc08a4 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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In the thermoelastic regime, a local dilatation of the medium occurs. In a unidimensional analysis along $z$ (as the depth of absorption is small compared to the beam diameter), the stress $\sigma_{zz}$ can be written \cite{scruby1990laser}:  \begin{equation}  \sigma_{zz} = (\lambda + \mu) \frac{\partial u_z}{\partial z} - 3 (\lambda (3\lambda  + \frac{2}{3}\mu) 2\mu)  \frac{\alpha E}{\rho C S \delta} \label{eq:stressUnidim} \end{equation}  where $\lambda$ and $\mu$ are respectively the first and second Lamé's coefficient, $\alpha$ is the thermal dilatation coefficient and $\delta$ the average depth of laser beam absorption. 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:stressUnidim} can be integrated:  \begin{equation}  u_z = \frac{3(\lambda \frac{(3\lambda  + \frac{2}{3} \mu)}{(\lambda 2\mu)}{(\lambda  + 2\mu)} \frac{\alpha E \delta}{\rho C S \delta} \approx 3 \alpha E}{\rho C S} \label{eq:deplUnidim} \end{equation}  as in a biological soft tissues, $\mu \gg \lambda$. Taking as an order of magnitude $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), $E$ = 200 mJ, $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 kg.m$^{-3}$ (water calorific capacity) and $S$ = 20 mm$^2$, we obtain a displacement $u_z$= 0.5 $\mu$m. This value is still smaller than the typical displacement resolution with ultrasound, of a few micrometers. This unidimensional analysis cannot explain the induction of shear waves, as displacement is curl-free. However, in a tridimensional model, the local expansion acts as dipolar forces parallel to the surface, so displacements along X and Y axis can be higher. 
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