## 4 Verification on the Mesoscale Equivalent Model
### Model Verification with Penetration Tests
#### 4.2.2 Results and Discussion  
Figure 15 shows the numerical results of the penetration process for U-1, where the striking velocity is 410 m/s and the target thickness is 90 cm. The \"History Variable #4\" refers to the damage parameter \(D\) in the RHT model, where \(D=0\) means no damage and \(D=1\) means fracture of the material. It can be seen that severe damage mainly occurs around the ballistic trajectory, and the region of damage distribution expands as penetration depth increases. Figure 14: FE model for projectile penetration. \begin{table}
\begin{tabular}{c c c c} \hline
**Material** & **Material Model** & **Input Parameter** & **Value** \\ \hline \multirow{8}{*}{Outer shell of projectile} & & Shear modulus & 0.84 \\  & & Mass density & 7.83 \\  & & Poisson’s ratio & 0.25 \\  & & \(a/b/n/c/m\) & 0.01350/0.00477/0.18/0.012/1.0 \\  & *MAT\_015 & Failure stress & \(-2\) \\  & & \(D_{1}\) & 0.15 \\  & & \(D_{2}\) & 0.72 \\  & & \(D_{3}\) & 1.66 \\  & & \(C/S_{1}/\gamma/A\) & 0.4596/1.357/1.71/0.43 \\ \hline \multirow{8}{*}{Inner filling of projectile} & & Mass density \(RO\) & 1.63 \\  & & Young’s modulus \(E\) & 0.1 \\ \cline{1-1}  & *MAT\_003 & Poisson’s ratio \(PR\) & 0.45 \\ \cline{1-1}  & & Yield strength \(SIGNY\) & 0.0006 \\ \cline{1-1}  & & Tangent modulus & 0.001 \\ \cline{1-1}  & & _ETAN_ & Failure strain \(FS\) & 3.0 \\ \hline \end{tabular}
\end{table} 
Table 5: Parameters of material models for projectile [3] (units: cm-g-\(\upmu\)s). Frontal damage contours of targets U-1 and U-2 are shown in Figure 16. As the striking velocity of the projectile increases, the impact surface of the target tends to be subjected to more severe local damage. The numerical results of the crater diameter on targets U-1 and U-2 are 53.8 cm and 77.3 cm, respectively. Comparing the numerical and experimental results, the maximum relative error in the crater diameter is just 5.9%, showing perfect consistency. In addition to the crater, the projectile impact also forms cracks on the impact surface. The numerical results of cracks show that there are four main cracks on the impact face of target U-1 and eight main cracks on the impact face of target U-2, and these cracks develop radially and orthogonally. The width, number, and distribution range of cracks expand with increased striking velocity. For target U-1, the number and distribution of the simulated cracks are in good agreement with the experimental results. While for target U-2, the difference between the numerical results of the cracks and the experimental results is a little large, which may be caused by the slight yaw of the projectile penetrating into U-2 in the penetration tests. The time-displacement curves and time-velocity curves of the projectiles are shown in Figure 17. The numerical values of DOPs at striking velocities of 410 m/s and 664 m/s are 40.4 cm and 87.1 cm, respectively. The numerical results and experimental results of DOPs are compared in Figure 18, where the relative errors are \(-\)3.8% and 11.7%, respectively. The reason for the slightly larger error in the case of 664 m/s may be that the projectile has a slight yaw, as shown in Figure 7b. Although the yaw is not obvious, it could increase the resistance of the projectile penetration into the target and then reduce the DOP. The maximum relative error of the DOP is lower than 15%, which indicates that the numerical method could reasonably predict the DOP when projectiles penetrate UHPC targets. Figure 17 shows that the velocities of the three projectiles gradually decrease from positive values to negative values with the increase of the DOP. The negative values of the velocities mean that the projectiles bounce in the opposite direction, which is consistent with the experimental phenomenon. The rebound velocities of projectiles at striking velocities of 410 m/s and 664 m/s are 16.7 m/s and 22.6 m/s, respectively. Figure 15: Numerical results of the penetration process for U-1, where the striking velocity is 410 m/s, and the target thickness is 90 cm.