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Pol Grasland-Mongrain edited Simu disp maps.tex
over 8 years ago
Commit id: f3cc55fa32c862256a80832fcc7b282a8d7dff3b
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diff --git a/Simu disp maps.tex b/Simu disp maps.tex
index 97e4122..e36ccfb 100644
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This physical phenomenon was then modeled numerically. 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_y
= (x,y,z)= \frac{\cos \beta \sin \beta}{4\pi \rho c_p^2 r} \delta_P + \frac{-\sin \beta \cos \beta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos \beta \sin \beta}{4\pi \rho r^3} \int_{r/\alpha}^{r_\beta}{\tau \delta_{NF}}\\
G_z
= (x,y,z)= \frac{\cos^2 \beta}{4\pi \rho c_p^2 r} \delta_P + \frac{\sin^2 \beta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos^2 \beta-1}{4\pi \rho r^3} \int_{r/\alpha}^{r_\beta}{\tau \delta_{NF}}
\label{eq:akirichards}
\end{equation}
where $\beta$ is the angle between the applied force and the
axis y or z, direction of the considered point (x,y,z), $\rho$ the medium density, $c_p$ 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 then be computed by convoluting $G_y$ and $G_z$ with time and spatial extent of the force:
\begin{equation}