deletions | additions       diff --git a/Simu disp maps.tex b/Simu disp maps.tex  index 4f9850e..03e5093 100644  --- a/Simu disp maps.tex  +++ b/Simu disp maps.tex 
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If enough energy is deposited, the local increase of temperature could also vaporize a part of the surface of the medium \cite{scruby1990laser}. Using an energy of 200 mJ in equation \ref{eq:eqChaleurApprox}, with same experimental parameters as before, we find a maximum increase of temperature of 60 K, which is close to the vaporization point of our medium, that we could approximate in a first hypothesis as 373K (water vaporization temperature). Besides the uncertainty of some values, it has been demonstrated that the presence of small particles like the graphite particles in our medium acts as nucleation sites for vaporization, which facilitate the vaporization of the medium at lower temperature \cite{Alimpiev_1995}: the energy can in reality be sufficient to get a "vaporization" regime.  This phenomenon can be modeled as a stress $\sigma_{zz}$ in the medium, as the sum of the P-wave modulus and a term given by the second law of motion \cite{scruby1990laser}:  \begin{equation}   \sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - \frac{1}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}  \label{eq:stressAbla}  \end{equation} where $L$ is the latent heat required to vaporize the solid, $T_0$ and $T_V$ the initial and vaporization temperatures.  Similarly to the thermoelastic regime, this leads to a displacement $u_z$:  \begin{equation}  u_z = \frac{\zeta}{\rho (\lambda + 2 \mu)}\frac{I^2}{(L+C(T_V-T_0))^2}\approx \frac{\zeta}{\rho \lambda}\frac{E^2/S^2 \tau^2}{(L+C(T_V-T_0))^2}  \label{eq:deplAbla}  \end{equation}  As in a biological soft tissues, $\mu \ll \lambda$, the displacement $u_z$ can be approximated as $\frac{\zeta}{\rho \lambda}\frac{E^2/S^2 \tau^2}{(L+C(T_V-T_0))^2}$. Estimating $\zeta$ equal to 50 $\mu$m (average depth of absorption), $\lambda$ = 2 GPa (first Lamé's coefficient of water), $L$ = 2.2 MJ.kg$^{-1}$ (vaporization latent heat of water), $C$ = 4180 J.kg$^{-1}$.K$^{-1}$ (water heat capacity), $T_V-T_0$ = 373-293 = 80 K (water vaporization temperature minus laboratory temperature), $\rho$ = 1000 kg.m$^{-3}$ (water density), $E$ = 200 mJ, $S$ = 20 mm$^2$ and $\tau$ = 10 ns, we obtain a displacement $u_z$ approximately equal to 3.9 $\mu$m. This is in the order of magnitude of the experimental measured displacement (2 $\mu$m).  A numerical calculation was then performed.  The vaporization was modeled as a point force directed along Z direction with a depth of 50 $\mu$m and increasing linearly from -2.5 to 0 mm and decreasing symmetrically from 0 to 2.5 mm, to simulate an approximate Gaussian shape. Propagation as a shear wave was calculated using Green operators $G_{zz}$ \cite{aki1980quantitative}: \begin{equation}  G_{zz}(r,\theta,z) = \frac{\cos^2 \theta}{4\pi \rho c_p^2 r} \delta_P + \frac{\sin^2 \theta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos^2 \theta-1}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau \delta_{NF}}  \label{eq:Gzz}