## 4 Validation and Application
### Application: High-Velocity Impact of Projectile on Ceramic Plate
The projectile velocity (absolute value of the velocity in \(z\)-direction) decreases monotonically with time and becomes stable at the time of 30 \(\upmu\)s.  
The residual velocity of the projectile predicted by the simulation is about 683 m/s. It is in good agreement with the experimentally measured one (682 m/s) [29]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Parameter & Value & Parameter & Value \\ \hline \(\rho_{0}\)/(kg/m\({}^{3}\)) & 17,162 & & \(S_{0}\)/(m/s) & 4029 \\ \hline Shear modulus/GPa & 124 & & \(S_{1}\) & 1.23 \\ \hline \multirow{6}{*}{Parameters in Johnson-Cook strength model} & \(a_{0}\)/MPa & 1200 & Parameters in Grüneisen EOS & \(S_{2}\) & 0 \\ \cline{2-5}  & \(b_{0}\)/MPa & 141 & & \(S_{3}\) & 0 \\ \cline{2-5}  & \(n_{0}\) & 0.12 & & \(S_{0}\) & 1.54 \\ \cline{2-5}  & \(c_{0}\) & 0.016 & & \(S_{4}\) & 0 \\ \cline{2-5}  & \(m_{0}\) & 1.0 & & \(S_{4}\) & 0.33 \\ \cline{2-5}  & \(\varepsilon_{0}\)/s\({}^{-1}\) & 1.0 & Parameters in damage model & \(S_{3}\) & \(-1.5\) \\ \cline{2-5}  & \(T_{\rm melt}\)/K & 1723 & & \(S_{4}\) & 0 \\ \cline{2-5}  & \(T_{\rm room}\)/K & 300 & & \(S_{4}\) & 0 \\ \cline{2-5}  & \(C_{0}\)/(J/(kg\(\cdot\)K)) & 134 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters for projectile material. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Parameter & Value & Parameter & Value \\ \hline \(\rho_{0}\)/(kg/m\({}^{3}\)) & 3921 & & \(p_{\rm T}\)/GPa & 0.2 \\ \hline Shear modulus/GPa & 90.16 & Parameters in strength model & \(\varepsilon_{0}\)/s\({}^{-1}\) & 1.0 \\ \hline \multirow{6}{*}{Parameters in strength model} & \(a_{1}\) & 0.93 & Conversion fraction of elastic energy loss & 1.0 \\ \cline{2-5}  & \(b_{1}\) & 0.31 & & \(K_{1}\)/GPa & 130.95 \\ \cline{2-5}  & \(n_{1}\) & 0.6 & & \(K_{2}\)/GPa & 0 \\ \cline{2-5}  & \(c_{1}\) & 0 & & \(K_{3}\)/GPa & 0 \\ \cline{2-5}  & \(m_{1}\) & 0.6 & & \(K_{3}\)/GPa & 0 \\ \cline{2-5}  & HEL/GPa & 2.79 & Parameters in damage model & \(D_{1}\) & 0.005 \\ \cline{2-5}  & \(P_{\rm HEL}\)/GPa & 1.46 & Parameters in damage model & \(D_{2}\) & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters for ceramic material. Figure 15: _Cont._  
Figure 17 shows the particle distributions and reconstructed surfaces of the ceramic plate and the projectile at different time instants with a section view. The particles of surfaces of the projectile and the ceramic plate are in different colors. Both the ceramic plate and the projectile experience very large deformation. Meanwhile, it can be clearly seen from this figure that the projectile forms a mushroom-shaped head under the resistance of the ceramic plate. Part of failed projectile material becomes fragments separating from the projectile and moves toward the opposite direction of the projectile motion. This indicates the mass loss of the projectile. Note that the mushrooming deformation and erosion of the projectile is also reported in the experimental results [29; 30]. Figure 18 shows a comparison between the simulation result of the mushrooming deformation of the projectile at 8 us and the corresponding experimental result. A good correlation between the simulated and experimental results is observed. Additionally, as seen from this figure, the interface between ceramic material and projectile material is kept clear under such a complex deformation condition. This proves that the present contact algorithm has the strong capability and good stability in dealing with large deformation. The present SPH simulation result is also compared with the simulation results of [PERSON] et al. [1], as shown in Figure 19. Two different approaches were employed by [PERSON] et al. to model the present problem with quarter models. One is the classical FEM with erosion technique, which deletes elements when the material fails. The other approach they proposed is to use a fragmentation model along with the conversion of FE elements to SPH particles (named \"fragmentation approach\" for short).