## 4 Validation and Application
### Application: High-Velocity Impact of Projectile on Ceramic Plate
and the pressure of material under tension (\(\mu<0\)) is calculated with \(P=K_{1}\mu\). In Equations (35) and (36), \(K_{1}\), \(K_{2}\) and \(K_{3}\) are material constants, \(\mu=\rho/\rho_{0}-1\) where \(\rho\) and \(\rho_{0}\) are, respectively, the current density and the initial density of the material, and \(\Delta P\) is introduced for the consideration of conversion of elastic energy loss to potential hydrostatic energy. The Johnson-Cook (JC) constitutive model along with the Gruneisen equation of state (EOS) is implemented to model the dynamic behavior of the projectile material. With the JC model, the yield strength of material is expressed as  
\[\sigma_{y}=\Big{[}a_{0}+b_{0}(\varepsilon_{\rm P})^{n_{0}}\Big{]}\Big{(}1+c_{ 0}\ln\hat{\varepsilon}^{*}\Big{)}(1-T^{*m_{0}}), \tag{37}\]  
where \(a_{0}\), \(b_{0}\), \(n_{0}\), \(c_{0}\) and \(m_{0}\) are constants, \(\varepsilon_{\rm P}\) is the plastic strain, and \(T^{*}=(T-T_{\rm room})/(T_{\rm melt}-T_{\rm room})\) where \(T\) is the current temperature, \(T_{\rm room}\) is the room temperature and \(T_{\rm melt}\) is the melt temperature. In this work, the temperature of material is determined based on the assumption that the internal energy increment converted into heat energy. It is calculated as  
\[T=T_{0}+\Delta u/C_{\rm v}, \tag{38}\]  
where \(T_{0}\) is the initial temperature which is assumed to be equal to the room temperature \(T_{\rm melt}\), \(\Delta u=u-u_{0}\) where \(u\) and \(u_{0}\) are the current and initial internal energy per unit mass, respectively, and \(C_{\rm v}\) is the specific heat. Since \(u_{0}\) for the projectile material is zero, \(\Delta u\) can be simplified as \(u\). The Gruneisen EOS using cubic shock velocity (\(v_{\rm s}\))-particle velocity (\(v_{\rm p}\)) relation [27] is used to calculate the pressure of the projectile material. When material is compressed (\(\mu\geq 0\)), the pressure is calculated as  
\[P=\frac{\rho_{0}S_{0}^{2}\mu\Big{[}1+\Big{(}1-\frac{\gamma_{0}}{2}\Big{)}\mu- \frac{\sigma}{2}\mu^{2}\Big{]}}{\Big{[}1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu +1}-S_{3}\frac{\mu^{3}}{(\mu+1)^{2}}\Big{]}^{2}}+(\gamma_{0}+a\mu)E. \tag{39}\]  
When material is expanding (\(\mu<0\)), the pressure is calculated as  
\[P=\rho_{0}S_{0}^{2}\mu+(\gamma_{0}+a\mu)E. \tag{40}\]  
In Equations (39) and (40), \(S_{0}\) is the intercept for the cubic \(v_{\rm s}\)-\(v_{\rm p}\) curve, \(S_{1}\), \(S_{2}\) and \(S_{3}\) are the coefficients for the slope of the curve, \(\gamma_{0}\) is the Gruneisen parameter, \(a\) is the parameter for the correction to \(\gamma_{0}\), and \(E\) is the internal energy per initial volume. In our SPH calculation, \(E\) is solved with \(E=\rho_{0}u\). The damage model for the projectile material has the same form in Equation (33). However, the failure strain \(\varepsilon_{\rm f}\) is defined as  
\[\varepsilon_{\rm f}=(d_{1}+d_{2}\exp(d_{3}\sigma*))\Big{(}1+d_{4}\ln\hat{ \varepsilon}^{*}\Big{)}(1+d_{5}T*) \tag{41}\]  
where \(d_{1}\)-\(d_{5}\) are damage parameters, and \(\sigma^{*}=\sigma_{\rm m}/\sigma_{\rm e}\) is the stress triaxiality where \(\sigma_{\rm m}=\big{(}\sigma^{11}+\sigma^{22}+\sigma^{33}\big{)}/3\) is the mean stress and \(\sigma_{\rm e}\) is the von Mises equivalent stress. The material parameters for the projectile material and the ceramic material are the same as those in reference [9] and summarized in Tables 1 and 2, respectively. Figure 15a shows the impact process simulated by the SPH method with the present contact algorithm. It can be seen that the ceramic material is ejected from the front face. The back-face of the ceramic plate begins to move at about 10 \(\upmu\)s, and a slight smooth bulge is formed at this time. The bulge expands as the impact process progresses. The back-face motion of the ceramic plate in the experiment was clearly captured by [PERSON] et al. [30] with a high-speed camera. The back-face deformation motion reproduced by the simulation agrees well with the experimental observation (see Figure 15b). Figure 16 shows the projectile velocity history.