deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 886801c..8f917fb 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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\frac{\partial T}{\partial t} = \kappa \nabla ^2 T + \frac{q}{\rho C}  \label{eq:eqChaleur}  \end{equation}  where $\rho$ is the density, $C$ the heat capacity and $\kappa$ the thermal diffusivity. If melting temperature is reached, a part of the absorbed heat willbe used to  melt the solid without increase of temperature. The thermal diffusion path, equal to $\sqrt{4\kappa t}$, \tau}$,  with $t$ $\tau$ = 20 ns  the laser emission duration(20 ns)  and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is approximately  equal here  to 0.1 $\mu$m. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, so and  term $k \nabla ^2 T$ can be neglected in equation \ref{eq:eqChaleur}. 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 expansion, a local dilatation of the medium occurs. In an unbounded solid, this would lead to a curl-free displacement, so no shear wave would occur. However, in the case presented, the solid is semi-infinite (the laser beam is absorbed on one side of the medium), and the local expansion acts as dipolar forces parallel to the surface.  In metals, transition from first to second regime occurs approximately about 10$^7$ W.cm$^{-2}$. This is equal to the energy of the laser we used, so the predominant regime in our experiment cannot be determined yet.  In the thermoelastic expansion, regime,  a local dilatation of the medium occurs. In an unbounded solid, this would lead to a curl-free displacement, so no shear wave would occur. However, in the case presented, the solid is semi-infinite (the laser beam is absorbed on one side of the medium), and the local expansion acts as dipolar forces parallel to the surface. In the ablative regime, the local increase of temperature is so high that the surface of the medium melts and creates a point-force in the medium. In this case, the net stress $\sigma$ is given by Newton's second law of motion and is equal A unidimension analysis lead  to \cite{scruby1990laser}: a local displacement $u_z$ along $z$ #:  \begin{equation}  \sigma u_z  = \frac{1}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2} \frac{3 \lambda + 2 \mu}{\lambda + 2\mu} \frac{\alpha \E}{\rho C S}  \label{eq:deplUnidim}  \end{equation}  where $\lambda$, $\mu$ are respectively the first and second Lamé's coefficient and $\alpha$ is the thermal dilatation coefficient. In a biological soft tissues, $\mu \gg \lambda$, so that \ref{eq:deplUnidim} becomes:  \begin{equation}  u_z = \frac{3\alpha E}{\rho C S}  \end{equation}  Taking as an order of magnitude $\alpha = ???$, $E$ = 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 $u_z$= ??? $\mu$m. This value is still smaller than the typical displacement resolution with ultrasound, of a few micrometers. However, displacements along X and Y axis are   In the ablative regime, the local increase of temperature is so high that the surface of the medium melts. This phenomenon creates a point-like force $f$ in the medium, given by \cite{scruby1990laser}:  \begin{equation}  f = \frac{S}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}  \label{eq:ablation}  \end{equation}  where $L$ is the lastent heat required to vaporize the solid, $T_V$ and $T_0$ the initial and vaporization temperatures. 
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