deletions | additions       diff --git a/Simu disp maps.tex b/Simu disp maps.tex  index 0ce90a3..caae3ef 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}.    This physical phenomenon was then modeled numerically. If enough energy is deposited, the local increase of temperature could also vaporize a part of the surface of the medium \cite{scruby1990laser}. Substituting the experimental parameters to equation \ref{eq:eqChaleurApprox} ($\gamma^{-1} \approx$ 50 $\mu$m$^{-1}$, $S$ = 20 mm$^{2}$, $E$ = 0.2 J, $\rho \approx$ 1000 kg.m$^{-3}$, $C \approx$ 4180 J.kg$^{-1}$.K$^{-1}$) lead to 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). However, 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: the calculated increase of temperature can be enough at high power to have an ablative 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 \cite{aki1980quantitative}: \begin{equation}  G_z (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:akirichards}