deletions | additions       diff --git a/Simu disp maps.tex b/Simu disp maps.tex  index 2c0e019..49b0a55 100644  --- a/Simu disp maps.tex  +++ b/Simu disp maps.tex 
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% This physical phenomenon was then modeled numerically. The thermal dilatation was simulated by calculating the displacement created by two opposite forces decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm), with a depth of 100 $\mu$m. Propagation as a shear wave was calculated using Green operator \cite{aki1980quantitative}, using a medium density $\rho$ of 1000 kg.m$^{-3}$, a compression wave speed of 1500 m.s$^{-1}$ and a shear wave speed of 4 m.s$^{-1}$. Results are shown on Figure \ref{Figure3} which represents displacement maps along Y and Z axis 0.8, 1.6, 2.4, 3.2 and 4.0 ms after force application. The displacement maps present many similarities with the experimental results of the Figure \ref{Figure2}.    If enough energy is deposited, the local increase of temperature could can  also vaporize a part of the surface of the medium \cite{scruby1990laser}. Using an energy Ejection  of 200 mJ in particles leads to a displacement inside the medium which can create shear waves: this constitutes the \textit{ablative regime}.  Solving  equation \ref{eq:eqChaleurApprox}, \ref{eq:eqChaleurApprox}  with same experimental parameters as before, before but a laser energy of 200 mJ,  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 about  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 laser  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}  
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u_z = \frac{\zeta}{\rho \lambda}\frac{I^2}{(L+C(T_V-T_0))^2}  \label{eq:deplAblaApprox}  \end{equation}  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), $C$ = 4180 J.kg$^{-1}$.K$^{-1}$,  $\rho$ = 1000 kg.m$^{-3}$ (water density), kg.m$^{-3}$,  $E$ = 200 mJ (high laser energy), 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 This displacement along Z can create shear waves. We  modeled the ablation  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}