deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index 59256af..73d7239 100644  --- a/The_absorption_of_the_laser__.tex  +++ b/The_absorption_of_the_laser__.tex 
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Substituting low-energy experimental parameters ($\gamma^{-1} \approx$ 50 $\mu$m$^{-1}$, $S$ = 20 mm$^{2}$, $E$ = 10 mJ, $\rho$ = 1000 kg.m$^{-3}$, $C$ = 4180 J.kg$^{-1}$.K$^{-1}$) lead to a maximum increase of temperature of 3 K. This local increase of temperature gives rise to a local dilatation of the medium. The induced displacements can then lead to shear waves: this constitutes the \textit{thermoelastic regime}.  To describephysically  this thermoelastic regime, dilatation,  we take a homogeneous and isotropic medium and as the depth of absorption (about 50 $\mu$m) is hundred times smaller than the beam diameter (5 mm), we have adopted a 1D model. The stress $\sigma_{zz}$ is the sum between the axial strain component and the thermal expansion component \cite{scruby1990laser}: \begin{equation}  \sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - 3(\lambda + \frac{2}{3}\mu) \frac{\alpha E}{\rho C S \zeta}  \label{eq:stressThermo} 
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Dilatation along X and Y axis also occurs and lead to stronger displacements than along Z. We modeled thus the thermoelastic regime as two opposite forces during 100 $\mu$s directed along Y axis with a depth of 100 $\mu$m and decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm), to simulate an approximate Gaussian shape \cite{Davies_1993}. Propagation as a shear wave along Z axis was calculated using Green operators $G_{yz}$ as calculated by Aki Richards \cite{aki1980quantitative}:  \begin{equation}  G_{yz} (r,\theta,z)= (r,\theta,z,t)=  \frac{\cos \beta \sin \theta}{4\pi \rho c_p^2 r} \delta_P f(t-\frac{r}{c_p})  + \frac{-\sin \theta \cos \theta}{4\pi \rho c_s^2 r} \delta_S f(t-\frac{r}{c_s})  + \frac{3\cos \theta \sin \theta}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau \delta_{NF}} f_{NF}}  \label{eq:Gyz}  \end{equation}  where $\theta$ is the angle between the applied force and the considered point (r,$\theta$,z), $c_p$ 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 $\delta_{NF}$ $f_{NF}$  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 along space and time Displacements  are then computed by convoluting $G_{yz}$ with time and spatial extent of the force:  \begin{equation}  u_z(x,y,z,t) = H_y * G_{yz}  \label{eq:uz}  \end{equation}  where force  $H_y$ is a 4-D matrix of the applied forces (directed along Y) along space and time. : $u_z=G_{yz}*H_y$.  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 5.5 m.s$^{-1}$. Results are shown on Figure \ref{figGreenThermo} which represents displacement maps between each frame along Z axis 1.0, 1.5, 2.0, 2.5 and 3.0 ms after force application. The normalized displacement maps present many similarities with the low-energy experimental results of the Figure \ref{Figure2}.