## 3 Numerical Modeling  
A 3D coupled thermal-mechanical FEM analysis was performed in LS-DYNA [21]. The temperature distribution was found by solving the heat conduction equation:  
\[\rho(T)C_{p}(T)\frac{\partial T(x,y,z,t)}{\partial t}-\
abla[k(T)\
abla T(x,y, z,t)]=Q(x,y,z,t)-L_{i} \tag{1}\]  
where \(x,y,\) and \(z\) are the space coordinates while \(\rho,\)\(C_{p},\)\(k\) are the mass density, specific heat at constant pressure, and the thermal conductivity of the target material, respectively, and \(L_{i}\) is the latent heat of melting, which is equal to zero in the thermoelastic regime. The source term, \(Q(x,\)\(y,\)\(z,\)\(t)\), has an elliptic spatial form and represents the absorbed laser energy per unit volume per unit time by the sample:  
\[ \begin{array}{l}Q(x,y,z,t)=I_{0}(1-R)e^{-4\up i\up i\up i\up i\up j\up e^{-[(X/t_{ \alpha})^{2}+(Y/t_{\beta})^{2}]}}a_{b}e^{-a_{b}z}\\ X=(x-x_{\alpha})cos\theta\ +\ (y-y_{\alpha})sin\theta\\ Y=(x-x_{\alpha})sin\theta\ -\ (y-y_{\alpha})cos\theta\end{array}  \tag{2}\]  
where, \(I_{0}\) is the incident laser intensity on target, \(R\) is the optical reflectivity of the sample, \(a_{b}\) is the optical absorption coefficient, \(t_{0}\) is the laser pulse duration at full width at half maximum (FWHM), \(r_{\alpha}\) is the semi-major axis and \(r_{b}\) is the semi-minor axis of the laser spot, \(x_{\alpha}\) and \(y_{\alpha}\) are the center coordinates, and \(\theta\) is the rotation angle. The conservative equations of mass, momentum, and energy are also solved:  
\[ \begin{array}{l}\frac{\partial\rho}{\partial t}+\rho\
abla v=0\\ \rho\frac{\partial^{2}U_{i}}{\partial^{2}t}=\mu\frac{\partial^{2}U_{i}}{ \partial^{2}k}+(\lambda+\mu)\frac{\partial}{\partial i}(\frac{\partial u_{k} }{\partial k})-(3\lambda+2\mu)\alpha_{T}\frac{\partial T}{\partial i}\\ \rho\frac{\partial E}{\partial t}=\sigma_{ij}\
abla v\end{array}  \tag{3}\]  
where \(\rho\) is the density, \(v\) is the velocity, \(U\) is the displacement, \(\lambda\) and \(\mu\) are the _Lame_ coefficients, \(\alpha_{T}\) is the thermal expansion coefficient, \(E\) is the energy, and \(\sigma_{ij}\) is the stress tensor. The strain tensor is given by:  
\[\varepsilon_{ij}=\frac{1}{2}\left(\frac{\partial U_{i}}{\partial U_{j}}+ \frac{\partial U_{j}}{\partial U_{i}}\right) \tag{4}\]while for the stress tensor it holds:  
\[\sigma_{ij}=2\mu\epsilon_{ij}+\lambda\epsilon_{kk}\delta_{ij}-(3\lambda+2\mu)a_{T} \;(T-T_{0})\delta_{ij} \tag{5}\]  
In continuum mechanics theory, the complete stress tensor describes the material condition state. It is divided into two components, the deviatoric and the hydrostatic tensors:  
\[\sigma_{ij}=s_{ij}-P\delta_{ij} \tag{6}\]  
where \(s_{ij}\) is the deviatoric stress component. The hydrostatic component of stress is associated to the pressure in the material, which is equal to the trace of the complete stress tensor  
\[P=-\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}=-\frac{\sigma_{\rm{rx}}}{3} \tag{7}\]  
The hydrodynamic and deviatoric behavior of the metallic target is considered by using simultaneously an analytical equation of state (EOS) coupled with a strength material model. The analytical Gruneisen EOS is used and allows the extension of the application range to high-pressure values and to the liquid phase. The analytical Gruneisen EOS provides the pressure for compressed materials [21]:  
\[P=\frac{\rho_{0}C_{0}^{2}\mu_{v}\big{[}1+\big{(}1-\frac{\gamma_{0}}{2}\big{)} \mu_{v}-\frac{\delta}{2}\mu_{v}{}^{2}\big{]}}{\big{[}1-(S_{1}-1)\mu_{v}\big{]} ^{2}}+(\tau_{0}+\alpha\mu_{v})E \tag{8}\]  
where \(C_{0}\) is the sound speed, \(\gamma_{0}\) the unitless, Gruneisen parameter that defines the effect on the atom's vibration due to the change in energy and \(a\) is the unitless, first-order volume correction to \(\gamma_{0}\); while \(\mu_{v}\) is a volumetric parameter that holds \(\mu_{v}=\rho/\rho_{0}-1\).