deletions | additions       diff --git a/Let_s_describe_the_phenomenon__.tex b/Let_s_describe_the_phenomenon__.tex  index 3b39826..52dbb5c 100644  --- a/Let_s_describe_the_phenomenon__.tex  +++ b/Let_s_describe_the_phenomenon__.tex 
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I(z)=(1-R) I_0 \exp(- \gamma z)  \label{eq:expontentialDecay}  \end{equation}  where $R$ is the reflection coefficient of the material (supposed negligible on a black mat material as the one used here) and $\gamma$ the absorption coefficient of the medium. The absorption coefficient $\gamma$ can be estimated in two ways: by measuring the fraction of light going through different thickness of the medium or by calculating the skin depth $\delta = \frac{2}{\gamma}$ (the factor 2 is due to the fact that $\delta$ is related to magnitude of the electrical field while $\gamma$ is related to the magnitude of the optical energy, which is the square of the electrical field magnitude). Skin depth $\delta$ is equal to $(\pi \sigma \mu_r \mu_0 \nu)^{-\frac{1}{2}}$, where $\sigma$ is the electrical conductivity of the medium, $\mu_r \mu_0$ its permeability and $\nu$ the frequency of the radiation. Substituting $\sigma \approx$ 0.1 S.m$^{-1}$, $\mu_r \mu_0$ = 4 $\pi \times 10^{-7} H.m^{-1}$ and $\nu$ = 3 10$^8$ / 532 10$^{-9}$ = 5.6 10$^{14}$ Hz, the skin depth for our medium is about 70 $\mu$m: it means that about 63\% of the radiation energy  is absorbed in the first 70 micrometers of the sample. We have validated experimentally this value by measuring the fraction of light which go through different thicknesses of the medium (respectively 0, 30, 50 and 100 $\mu$m) with a laser beam power measurement device (QE50LP-S-MB-D0 energy detector, Gentec, Qu\'ebec, QC, Canada). We found respective transmitted powers of 100\%, 88\%, 71\% and 57\%. 57\%, as illustrated in the Figure \ref{figAbsorpExp}.  An exponential fit indicated that $\gamma^{-1} \approx$ 50 $\mu m$ in our sample, as illustrated in the little graph in Figure 1. sample.  The absorption of the laser beam by the medium gives then rise to an absorbed optical energy $\gamma I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T$ in absence of convection and of phase transition can be computed using the heat equation:  \begin{equation} 
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Substituting the experimental parameters ($\gamma^{-1} \approx$ 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.  This local increase of temperaturecan  lead to  a local dilatation of the medium occurs. We suppose that the medium is homogeneous and isotropic, and as the depth of absorption (about 40 $\mu$m) is small compared to the beam diameter (5 mm), we adopt 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}  \end{equation}  where $\lambda + 2 \mu$ is the P-wave modulus and modulus,  $\lambda + \frac{2}{3}\mu$ the bulk modulus with $\lambda$ and $\mu$ respectively the first and second Lamé's coefficient, $\alpha$is  the thermal dilatation coefficient and $\zeta$ the depth over which there is a temperature rise. In the absence of external constraints normal to the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$, so that equation \ref{eq:stressThermo} can be integrated, giving a displacement $u_z$ from the surface: \begin{equation}  u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho C S \zeta}  \label{eq:deplThermo}