## 3 Numerical-Simulation Study
### Fragmentation Characteristics of Projectiles with Different Wave-Impedance Distributions  
The breaking characteristics of projectiles vary under different wave-impedance distributions compared with aluminum alloy. The wave-impedance gradient material changes the shock-wave characteristics of the projectile target. This not only increases the conversion rate of the initial kinetic energy to the internal energy, but also changes the breaking characteristics of the projectile and enhances the space-diffusion ability of the debris. The spatial diffusion of kinetic energy is mainly determined by the fragmentation degree of the projectile during the action of the projectile target. In this part, the numerical simulation method is mainly used to study the impact of the projectile on different wave-impedance gradient materials.  
Three kinds of wave-impedance gradient materials, Ti-Al-Mg-Ny, Ti-Al-Ny, and Ti-Ny, which have the same surface density as 1.5-millimeter thick aluminum alloy, were selected for the numerical calculation. In order to prevent the influence of the lateral sparse wave on the impact pulse in the central region of the projectile, a one-dimensional strain-collision model was established to observe the impact of the impedance gradient on the spallation breaking in one dimension. The projectile material was aluminum alloy, the projectile's thickness was 6 mm, the maximum tensile-stress failure criterion was adopted, the maximum tensile stress that could be withstood was 2.0 GPa, and the impact speed was selected as 3.0 km/s-6.0 km/s. The calculation-model parameters are shown in Table 6.  
In order to better analyze the fragmentation of the projectile under different impact speeds, the SPH particle size of the projectile was taken as 0.1 mm, as was the SPH particle size of the protective screen. In order to accurately obtain the maximum spallation breaking area of the projectile, observation points were set at the center of the model with intervals of 0.1 mm. The typical calculation-model parameters are shown in Figure 8.  
Figure 7: Internal-energy transformation under different impact conditions.  
The judgment of spallation breaking can be performed by analyzing the pressure-time-history curve of the typical observation points. A positive value in the time-history curve indicates that the material bears the compressive stress, and a negative value indicates that the material bears the tensile stress. When the tensile stress is greater than the set spallation-breaking strength, the material undergoes spallation breaking, as shown in Figure 9.  
The typical numerical calculation results of the projectile's impact on three kinds of wave-impedance gradient materials are shown in Figure 10. The statistics of the spallation fracture areas of the projectile's impact on three kinds of wave-impedance gradient materials under the speed conditions of 3 km/s, 4 km/s, 5 km/s, and 6 km/s are shown in Table 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Working Condition** & **Projectile Thickness (mm)** & **Projectile Material** &
\begin{tabular}{c} **Wave-Impedance** \\ **Gradient Material** \\ \end{tabular} & **Impact Velocity (km/s)** \\ \hline
1 & 6 & AL 2024 & Ti-Al-Mg-Ny & 3, 4, 5, 6 \\
2 & 6 & AL 2024 & Ti-Al-Ny & 3, 4, 5, 6 \\
3 & 6 & AL 2024 & Ti-Ny & 3, 4, 5, 6 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Calculation-model parameters.  
Figure 8: One-dimensional strain-collision model.  
Figure 9: Pressure\(-\)time\(-\)history curve of observation point at spalling position.