deletions | additions       diff --git a/Absorption_of_the_laser_beam__.tex b/Absorption_of_the_laser_beam__.tex  index c658279..b972942 100644  --- a/Absorption_of_the_laser_beam__.tex  +++ b/Absorption_of_the_laser_beam__.tex 
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\sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - 3(\lambda + \frac{2}{3}\mu) \alpha \frac{ E}{\rho C S \zeta}  \label{eq:stressThermo}  \end{equation}  where $\lambda$ and $\mu$ are respectively the first and second Lamé's coefficients, $\alpha$ is the thermal dilatation coefficient, and $\zeta$ is the average depth of absorption. This equation can be simplified by the fact that in most soft media, including biological tissues, $\mu \ll \lambda$. Moreover, in the absence of external constraints normal to the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$. This allows the equation \ref{eq:stressThermo} to be integrated, giving at the surface a displacement, displacement  $u_z = (3 \alpha E)/(\rho C S)$ S)$.  Substituting the same experimental parameters used previously along with $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), we obtain a displacement $u_z$ of 0.025 $\mu$m. This value is very close to the measured experimental displacement (about 0.02 $\mu$m). Note that both the experimental and theoretical central displacements are directed towards the outside of the medium (see white circle arrow in Figures \ref{figElastoPVA}-(A) and \ref{figGreen}-(A)). To calculate the propagation of the displacements as shear waves, we must first consider the transverse dilatation, which leads to stronger displacements than those occurring along the Z axis. We thus modeled the thermoelastic regime in 2D as two opposite forces directed along the Y axis with a depth of 40 $\mu$m and with an amplitude decreasing linearly respectively from 2.5 to 0 mm and from -2.5 to 0 mm. The magnitude of the force along space and time is stored in a matrix, $H_y^{thermo}(y,z,t)$. Displacements along the Z axis are then equal to the convolution between $H_y^{thermo} (y,z,t)$ and $G_{yz}$ \cite{aki1980quantitative}:  \begin{equation}