deletions | additions
diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex
index fefd2e0..ef59c99 100644
--- a/The_absorption_of_the_laser__.tex
+++ b/The_absorption_of_the_laser__.tex
...
\label{eq:eqChaleurApprox}
\end{equation}
Substituting low-energy experimental parameters ($\gamma^{-1} \approx$ 50 $\mu$m$^{-1}$, $S$ = 20 mm$^{2}$, $E$ =
0.01 J, $\rho \approx$ 10 mJ, $\rho$ = 1000 kg.m$^{-3}$, $C \approx$ 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}.
This local increase of temperature lead to a local dilatation of the medium occurs. We suppose that the medium is homogeneous and
isotropic, isotropic and as the depth of absorption (about
40 50 $\mu$m) is
small compared to 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}
...
u_z = \frac{3 \alpha E}{\rho C S}
\label{eq:deplThermoApprox}
\end{equation}
Substituting
same experimental parameters as previously and $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient),
$E$ = 10 mJ (low laser energy), $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 kg.m$^{-3}$ (water calorific capacity) and $S$ = 20 mm$^2$, we obtain a displacement $u_z$= 0.025 $\mu$m. This value is slightly higher than the experimental displacement (about 0.02 $\mu$m).
This local displacement can Dilatation along X and Y axis also occurs and lead to
shear waves because of the limited size of the source.
To verify this physical model, a numerical calculation was performed. The thermal dilatation was stronger displacements than along Z. We modeled
them as two opposite forces directed along Y direction with a depth of 50 $\mu$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)= \frac{\cos \beta \sin \theta}{4\pi \rho c_p^2 r} \delta_P + \frac{-\sin \theta \cos \theta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos \theta \sin \theta}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau \delta_{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 $\delta_{NF}$ 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 along space and time are then
be computed by convoluting $G_yz$ with time and spatial extent of the force:
\begin{equation}
u_z u_z(x,y,z,t) =
H_y(x,y,z,t) H_y * G_{yz}
\label{eq:uz}
\end{equation}
where $H_y$ is a 4-D matrix of the applied
force forces (directed along Y) along space and time.
Results are shown on Figure \ref{figGreenThermo} 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}.
