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Let's describe now Next, we sought to characterize the other
regime. regim observed in our experiments. Solving
the equation \ref{eq:eqChaleurApprox} with
the same experimental parameters
as before used previously, but
with a laser energy of 200 mJ, we
find found a maximum increase
of in temperature of 60 K, i.e., a maximum medium temperature of about
298+60 = 360
K assuming a room temperature of about 298 K. While slightly below
to the vaporization point of our medium, supposed close to 373K (water vaporization temperature),
it can its proximity to the water vaporization temperature (373 K) may be sufficient to vaporize the
medium, as medium. Indeed, it has been demonstrated that
the presence of small particles like the graphite
and other small particles
acts can act as nucleation sites
and to facilitate
thus the vaporization of the medium at
lower temperature
lower than the vaporization point \cite{Alimpiev_1995}.
Vaporization A series of
the particles reactions then leads
by reaction to displacements inside the medium, which generate
then shear
waves: waves; this constitutes the \textit{ablative regime}.
To estimate the initial displacement amplitude in this regime, we
supposed again
assumed that the medium
as was homogeneous and
isotropic isotropic, and we discarded any boundary effect. The
stress $\sigma_{zz}$ stress, $\sigma_{zz}$, is now
defined as the sum of the axial strain component and a term given by the second law of motion
due to caused by the reaction
to of the particles ejected outside the medium
when they reach upon reaching the vaporization point \cite{ready1971effects}:
\begin{equation}
\sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - \frac{1}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}
\label{eq:stressAbla}
\end{equation} where $L$ is the latent heat required to vaporize the solid,
and $T_0$ and $T_V$
are the initial and vaporization
temperatures. temperatures, respectively.
By assuming $\mu \ll \lambda$ and a zero stress
state at the medium surface, equation \ref{eq:stressAbla} leads to a
displacement $u_z$: displacement, $u_z$, of:
\begin{equation}
u_z = \frac{\zeta}{\rho \lambda}\frac{I^2}{(L+C(T_V-T_0))^2}
\label{eq:deplAblaApprox}
\end{equation}
Using high-energy experimental
parameter, parameters, $\zeta \approx \gamma^{-1} = 40 \mu$m (average depth of absorption), $\lambda$ = 2 GPa (first Lamé's coefficient of water), $L$ = 2.2 MJ.kg$^{-1}$ (vaporization latent heat of
water) water), and $T_V-T_0$ = 373-298 = 75 K, we obtain a displacement $u_z$ of 2.9 $\mu$m.
This Again, this value is
again in
good high agreement with
experimental the experimentally obtained displacement (2.5 $\mu$m). Both theoretical and experimental displacements are directed
towards the inside
of the medium (see
white circle arrow in the
white circle in Figure \ref{figElastoPVA}-(B)).
To calculate the propagation of the displacement
along as a function of space and time, we modeled the
ablation ablative regime as a point force directed along
the Z
direction axis with a depth of 40 $\mu$m and
of a constant value from -2.5 to 2.5 mm. The magnitude of the force is stored in a
matrix matrix, $H_z^{abla}(y,z,t)$. Displacements along
the Z
axis are again equal to the convolution between $H_z^{abla}$ and $G_{zz}$ \cite{aki1980quantitative}:
\begin{equation}
G_{zz}(r,\theta,t) = \frac{\cos^2 \theta}{4\pi \rho c_p^2 r} \delta(t-\frac{r}{c_p}) + \frac{\sin^2 \theta}{4\pi \rho c_s^2 r} \delta(t-\frac{r}{c_s})
\label{eq:Gzz}
...
\begin{equation}
+ \frac{3\cos^2 \theta-1}{4\pi \rho r^3} \int\limits_{r/c_p}^{r/c_s}{\tau \delta(t-\tau)d\tau}
\end{equation}
with
the same notations as
presented in equation \ref{eq:Gyz}.
Using
same values for the physical quantities
as previously, results are illustrated in Figure \ref{figGreen}-(B) which represents values previously defined, displacement maps along
the Z
axis, axis were calculated 1.0, 1.5, 2.0,
2.5 2.5, and 3.0 ms after force
application. application, as illustrated in Figure \ref{figGreen}-(B). The displacement maps present many similarities
with to the experimental results of
the Figure \ref{figElastoPVA}-(B), with initial displacement directed
towards the inside
of the
medium medium, and propagation of only two half cycles.
Ablation The ablative regime was also confirmed
visually: at visually. At high power, a
disk of paler
color colored disk of the same size as the beam diameter
appears appeared at the
impact location site of
the laser
impact on the phantom, which is consistent with the theory of
a partial vaporization of the medium. This was not observed in the low-energy experiments.
