Next, we examine the physical characteristics of the other regime observed in our experiments. Solving the equation \ref{eq:eqChaleur} with the same experimental parameters used previously, but with a laser energy of 200 mJ, we find a maximum increase in temperature of 60 K, i.e., a maximum medium temperature of about 360 K assuming a room temperature of about 298 K. While slightly below the vaporization point of our medium, supposed close to 373 K (water vaporization temperature), its proximity to the water vaporization temperature may be sufficient to vaporize the medium. Indeed, it has been demonstrated that graphite and other small particles can act as nucleation sites to facilitate the vaporization of the medium at temperature lower than the vaporization point \cite{Alimpiev_1995}. A series of reactions then leads to displacements inside the medium, which generate shear waves; this constitutes the ablative regime.
To estimate the initial displacement amplitude in this regime, we again assume that the medium was homogeneous and isotropic, and we discard any boundary effect. The stress, \(\sigma_{zz}\), is now defined as the sum of the axial strain component and a term given by the second law of motion caused by the reaction of the particles ejected outside the medium upon reaching the vaporization point \cite{ready1971effects}:
\begin{equation} \label{eq:stressAbla} \label{eq:stressAbla}\sigma_{zz}=(\lambda+2\mu)\frac{\partial u_{z}}{\partial z}-\frac{1}{\rho}\frac{I^{2}}{(L+C(T_{V}-T_{0}))^{2}}\\ \end{equation}where \(L\) is the latent heat required to vaporize the solid, and \(T_{V}\) and \(T_{0}\) are the vaporization and initial temperatures, respectively.
By assuming \(\mu\ll\lambda\) and a zero stress state at the medium surface, equation \ref{eq:stressAbla} leads to a displacement \(u_{z}=\zeta I^{2}/(\rho\lambda(L+C(T_{V}-T_{0}))^{2})\). Using high-energy experimental 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), and \(T_{V}-T_{0}\) = 373-298 = 75 K, we obtain a displacement \(u_{z}\) of 2.9 \(\mu\)m. Again, this value is in agreement with the experimentally obtained displacement (2.5 \(\mu\)m). Both theoretical and experimental displacements are directed towards the inside of the medium (see arrow in the white circle in Figures \ref{figElastoPVA}-(B) and \ref{figGreen}-(B)).
To calculate the propagation of the displacement as a function of space and time, we modeled the ablative regime as a point force directed along the Z axis with a depth of 40 \(\mu\)m and a constant value from -2.5 to 2.5 mm. The magnitude of the force was stored in a 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} \label{eq:Gzz} \label{eq:Gzz}G_{zz}=\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\limits_{r/c_{p}}^{r/c_{s}}{\tau\delta\tau d\tau}\\ \end{equation}with the same notations as presented in equation \ref{eq:Gyz}.
Using the physical quantity values previously defined, displacement maps along the Z axis were calculated 1.0, 1.5, 2.0, 2.5, and 3.0 ms after force application, as illustrated in Figure \ref{figGreen}-(B). The displacement maps present many similarities to the experimental results of Figure \ref{figElastoPVA}-(B), with initial displacement directed towards the inside of the medium, and propagation of only two half cycles.