## 4 Results and Discussions
### Effect of Target Diameter  
Figure 6 shows the comparison of normalized target resistance and penetration depth with \(r_{\rm t0}/R_{\rm P}\) at initial impact velocities of 1000 m/s, 1500 m/s, and 2000 m/s. Figure 7 shows the comparison of penetration velocity and tail velocity with penetration depth at different \(r_{\rm t0}/R_{\rm P}\) values of 5, 10, and 20. From Figures 6 and 7, the following can be seen:  
* When \(r_{\rm t0}/R_{\rm P}<25\), \(R_{\rm t}/R_{\rm t}\)* decreases rapidly as \(r_{\rm t0}/R_{\rm P}\) decreases, while \(R_{\rm t}/R_{\rm t}\)* decreases slowly as \(r_{\rm t0}/R_{\rm P}\) decreases when \(r_{\rm t0}/R_{\rm P}>25\). This suggests that \(R_{\rm t}/R_{\rm t}\)* rapidly decreases as the elastic-plastic interface approaches the target boundary (\(r_{\rm t0}/R_{\rm P}=16.63\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Statistical Error**} & **Experimental Results**[23] & **Numerical Results**[23] & **Experimental Results**[24] \\ \hline \multirow{2}{*}{The present model} & MPE * & 2.29\% & 1.75\% & 9.47\% \\  & RMSE & 3.56 & 4.12 & 11.78 \\  & MAE & 0.06 & 0.06 & 0.22 \\ \hline \multirow{2}{*}{[PERSON] [25]} & MPE & 19.53\% & 19.01\% & 94.73\% \\  & RMSE & 15.03 & 14.39 & 71.60 \\  & MAE & 0.20 & 0.20 & 1.07 \\ \hline \multirow{2}{*}{[PERSON] [26]} & MPE & 15.48\% & 15.30\% & 93.12\% \\  & RMSE & 11.93 & 11.59 & 70.28 \\  & MAE & 0.16 & 0.16 & 1.06 \\ \hline \multirow{2}{*}{[PERSON] [23]} & MPE & 20.30\% & 19.44\% & 95.13\% \\  & RMSE & 15.59 & 14.65 & 71.78 \\ \cline{1-1}  & MAE & 0.21 & 0.20 & 1.08 \\ \hline \hline \end{tabular}  
* MPE, RMSE, and MAE are mean percentage error, root mean square error, and mean absolute error, respectively.  
\end{table} 
Table 4: Summary of statistical errors. Figure 5: Comparison of calculation results of penetration depths using the model in [23; 25; 26] and in this article with existing experimental results in [24]. * When \(r_{\rm 10}/R_{\rm P}<25\), the penetration depth increases exponentially with decreasing \(r_{\rm 10}/R_{\rm P}\). However, when \(r_{\rm 10}/R_{\rm P}>25\), the penetration depth slowly increases with decreasing \(r_{\rm 10}/R_{\rm P}\). This is because of the decrease in \(R_{\rm t}/R_{\rm t}\)*. When \(r_{\rm 10}/R_{\rm P}=25\), the deviation of the penetration depth from the semi-infinite target is less than 10%, which means that the target can be treated as a semi-infinite target when \(r_{\rm 10}/R_{\rm P}>25\). * As the circumferential constraint effect decreases, the initial penetration velocity and penetration depth gradually increase. Figure 8 compares the decay functions of the target resistance in this article with those in [17]. As shown in Figure 8, the lateral boundary effect in the elastic-plastic stage is relatively small. When the elastic-plastic interface approaches the lateral boundary, the target resistance decreases by about 13%. Different from the decay functions in [17], the following can be seen:  
Figure 6: Comparison of normalized target resistance and penetration depth with \(r_{\rm 10}/R_{\rm P}\). Figure 7: Comparison of penetration velocity and tail velocity with penetration depth of different \(r_{\rm 10}/R_{\rm P}\). * Because the proposed decay function is based on the average cavity pressure rather than the instantaneous cavity pressure, the proposed decay function decreases more slowly than that in [17];
* The diameter of the target is greater as the elastic-plastic interface intersects the target boundary in the proposed decay function because of the use of the finite cylindrical cavity expansion model;
* The proposed decay function will not decrease to 0 before the initial target radius decreases to 0. The reason for this is that the target deformation is taken into account in this article;
* Due to the fact that the target resistance of long-rod penetration is independent of velocity, the decay function in this article is independent of velocity, whereas the decay function in [17] decreases significantly with increasing velocity.