deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 572d169..320fb74 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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The local increase of temperature can lead to two main effects creating elastic waves: (1) thermoelastic expansion and (2) ablation of medium.  In the thermoelastic regime, a local dilatation of the medium occurs. Adopting We suppose later on that the medium is homogenous and isotropic, and we adopt  a unidimensional analysis along $z$ (as 1D model, as  the depth of absorption is small compared to the beam diameter), the diameter. The  stress $\sigma_{zz}$ can be written is the sum between the axial strain component and the thermal expansion component  \cite{scruby1990laser}: \begin{equation}  \sigma_{zz} = (\lambda + 2  \mu) \frac{\partial u_z}{\partial z} - (3\lambda 3(\lambda  + 2\mu) \frac{2}{3}\mu)  \frac{\alpha E}{\rho C S \zeta} \label{eq:stressUnidim}  \end{equation}  where $\lambda + 2 \mu$ is the P-wave modulus and $\lambda + \frac{2}{3}\mu$ the bulk modulus, with  $\lambda$ and $\mu$are  respectively the first and second Lamé's coefficient, $\alpha$ is the thermal dilatation coefficient and $\zeta$ 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, 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:deplUnidim} \end{equation}  As in a biological soft tissues, $\mu \gg \lambda$, the displacement $u_z$ can be approximated as $3 \alpha \frac{E}{\rho C S}$. 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 slightly smaller than the typical ultrasound resolution of displacement, typically of a few micrometers. In a tridimensional model, displacements along X and Y axis would also occurs, as the local expansion acts as dipolar forces parallel to the surface. 
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