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Pol Grasland-Mongrain edited Simu disp maps.tex
over 8 years ago
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index 89a3673..0ce90a3 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 (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 \beta}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau \delta_{NF}}\\
G_z (r,\theta,z)= \frac{\cos^2 \theta}{4\pi \rho c_p^2 r} \delta_P + \frac{\sin^2 \theta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos^2 \theta-1}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau \delta_{NF}}
\label{eq:akirichards}
\end{equation}
where $\theta$ is the angle between the applied force and the considered point (r,$\theta$,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 again be computed by convoluting
$G_y$ and $G_z$ with
time and spatial extent of the
force:
\begin{equation}
u_y = G_y * H(x,y,z,t)\\
u_z = G_z * H(x,y,z,t)
\label{eq:akirichards2}
\end{equation}
where H is a 4-D matrix
H(x,y,z,t) of the applied
force along space and time. force. We modeled here the vaporization as a point force directed along Z direction (so angle $\beta$ = 0) during 100 $\mu$s 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 approximate a Gaussian shape). 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 5.75 m.s$^{-1}$.
Results are shown on Figure \ref{Figure3} which represents displacement maps between each frame along
Y and 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}.