deletions | additions       diff --git a/Simu disp maps.tex b/Simu disp maps.tex  index c7359b6..d5cbf2e 100644  --- a/Simu disp maps.tex  +++ b/Simu disp maps.tex 
...
% 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 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 opto\cite{Alimpiev_1995}: \cite{Alimpiev_1995}:  the energy can in reality be sufficient to get an ablative a "vaporization"  regime. 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 $G_y$ and $G_z$ as calculated by Aki Richards $G_zz$  \cite{aki1980quantitative}: \begin{equation}  G_z 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:akirichards3}  \end{equation}  where $\theta$ is the angle between the applied force and the considered point (r,$\theta$,z), $\rho$ the medium density, $c_p$ and $c_s$ the compression and shear wave speed respectively, $\delta_S$ and $\delta_P$ Dirac distribution indicating the position of the compression and shear waves along space and time, $\tau$ the time and $\delta_{NF}$ representing near-field effects. The three terms correspond respectively to the far-field compression wave, the far-field shear wave and the near-field component.  Displacement can again be computed by convoluting $G_z$ $G_zz$  with the 4-D matrix H(x,y,z,t) $H_z(x,y,z,t)$  of the applied force. We modeled here the vaporization as a point force directed along Z direction (so angle $\beta$ = 0) during 100 $\mu$s 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 approximate a Gaussian shape). The medium density $\rho$ was taken equal to 1000 kg.m$^{-3}$, the compression wave speed to 1500 m.s$^{-1}$ and the shear wave speed to 5.75 m.s$^{-1}$. Results are shown on Figure \ref{figGreenAbla} which represents displacement maps between each frame along 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}.    Ablation regime is confirmed visually by the observation of a disk of paler color of the same size as the beam diameter at the impact location of the laser on the phantom at high power, which is coherent to a vaporization of a fraction of the material.