## 3 Numerical Modeling
For most of the materials, shock velocity \(U_{s}\) varies linearly, in relation to the particle velocity, \(U_{p}\), as \(U_{s}=C_{0}+S_{1}U_{p}\), where \(S_{1}\) is the unitless coefficient of the slope of the \(U_{s}\)-\(U_{p}\) curve. For expanded materials the following holds:  
\[P=\rho_{0}C_{0}^{2}\mu_{v}+(\gamma_{0}+\alpha\mu_{v})E \tag{9}\]  
A proper material-constitutive model for the metal was adopted, the Johnson-Cook (J-C) material model, which considers the effect of plastic strain, strain rate, and temperature. The flow stress is expressed as [22]:  
\[\sigma_{y}=(A+B\epsilon^{n})(1+Cln\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}})( 1-\frac{T-T_{r}}{T_{m}-T_{r}})^{m} \tag{10}\]  
where \(A\) is the yield stress, \(B\) is the hardening constant, \(\epsilon\) is the equivalent plastic strain, \(n\) is the hardening exponent, \(C\) is the strain rate sensitivity, \(\dot{\epsilon}/\dot{\epsilon}_{0}\) is the dimensionless plastic strain rate, \(\dot{\epsilon}_{0}=1\) s\({}^{-1}\) is a reference strain rate used to normalize the strain rate, \(T_{m}\) is the melting temperature of the workpiece, \(T_{r}\) is the room temperature, and \(m\) is the thermal softening exponent. The J-C material model also adopts a fracture model that considers the nucleation, growth, and coalescence of voids in a ductile material at high-strain rates. The equivalent plastic strain at the onset of damage is defined as:  
\[\epsilon_{f}=\left(\bigg{[}D_{1}+D_{2}expD_{3}\frac{p}{\sigma_{VM}}\bigg{]} \bigg{[}1+D_{4}In\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\bigg{]}\bigg{[}1+D _{5}\frac{T-T_{r}}{T_{m}-T_{r}}\bigg{]}\right)\,,\,\,\,D=\sum\frac{\Delta \epsilon}{\epsilon_{f}} \tag{11}\]  
where \(D_{1}\) is the initial failure strain, \(D_{2}\) is the exponential factor, \(D_{3}\) is the triaxility factor, \(D_{4}\) is the strain rate factor, \(D_{5}\) is the temperature factor, \(\sigma_{VM}\) is the Von Mises stress, and \(\epsilon_{f}\) is the plastic strain at fracture. The \(D_{1}\)-\(D_{5}\) constitute unitless failure parameters. Material fracture occurs when the damage parameter \(D\) reaches the value of 1 and the concerned elements are removed from the computation. For the study of the generation and propagation of SAW's under pulsed laser irradiation, we developed and demonstrated a series of FEM simulations that focus on monitoring the matter dynamics on the macroscopic level of the solid target. The developed models are capable to simulate the transient thermo-structural matter's response from solid to plasma state under ns and fs pulsed laser irradiation and to study the generation and propagation of SAWs [6; 7; 8; 9]. The common characteristic of these models is the generation of a locally adaptive mesh, around the irradiated area of the spot, capable to handle the phase changes of matter due to the high-temperature gradients. This sophisticated finite element grid allows for the smooth transition of the SAWs from their source in the epicenter of the target [7] and provides the appropriate nodal distribution to the entire computational domain. Thus, the computing of the new nodal positions and temperature values in the Lagrangian mesh were accurately performed without exaggerating the computational resources and the run time. Figure 2 shows a typical FEM locally adaptive model, that precisely simulates the pulsed laser-Al target interaction. The dimensions of the solid target were 1800 \(\upmu\)m \(\times\) 1800 \(\upmu\)m \(\times\) 20 \(\upmu\)m and the generated mesh consisted of approximately 2 million elements, as presented in Figure 2a. Moreover, the beam spot had an elliptical shape. The major and minor axes of the elliptical laser spot were 160 \(\upmu\)m and 75 \(\upmu\)m long, as indicated by the experiment. The rotation angle was 29\({}^{\circ}\).This model considered the phase changes of matter and provided identical monitoring of the ultrasounds' generation and propagation throughout the computational domain, as demonstrated in the vertical Z-displacement results in Figure 2b.