## 3 Theoretical Penetration Model and Model Verification
### Validation of the Theoretical Model  
In Section 3.2, \(b\) = 0.367, i.e., the Mises yield criterion, is used in the proposed model. Experiments on long rods with different initial impact velocities penetrating metal targets with different diameters were carried out in [23; 24]. The mechanical parameters of the rod and target materials are summarized in Table 1. As the flow stress is not always given, it can be estimated from the Brinell hardness [30]:  
\[\sigma_{y}=3.92\times\text{BHN}, \tag{45}\]  
where \(\sigma_{y}\) is the flow stress, in MPa; BHN is the Brinell hardness. It is clear from the figure in [23] that the target undergoes significant radial expansion after penetration, which is consistent with the model in this article. In addition, numerical simulation calculations show that the elastic-plastic interface just reaches the lateral boundary of the target when \(r_{\text{t0}}/R_{\text{P}}\) is approximately 15 [23], while the theoretical model in this paper and other references predicts that the elastic-plastic interface reaches the lateral boundary of the target when \(r_{\text{t0}}/R_{\text{P}}\) = 16.63, 19.92 [23], 19.3 [25], and 17.4 [26]. The proposed model is closest to the numerical simulation results. Figure 3 shows the comparison of experimental and numerical results [23] with the calculation results of penetration depths for different \(r_{\text{t0}}/R_{\text{P}}\). Tables 2 and 3 summarize the initial penetration conditions, penetration depth results, and predicted penetration depths of different theoretical models in [23; 24], respectively. From Figure 3 and Table 2, the following conclusions can be drawn:  
* The proposed model is in good agreement with the experimental and numerical results, while all the other theoretical models overestimate the penetration depth;
* Although [PERSON]'s [26] model has better prediction results than [PERSON]'s [25] and [PERSON]'s [23] models, there is a sudden change in the penetration depth when \(r_{\text{t0}}/R_{\text{P}}\) is around 18.5, which will result in multiple results within a certain range of \(r_{\text{t0}}/R_{\text{P}}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Reference** & \(\sigma_{\text{\bf{yp}}}\) **(MPa)** & \(\boldsymbol{Y_{\text{t}}}\) **(MPa)** & \(\boldsymbol{\rho_{\text{P}}}\) **(kg/m\({}^{3}\))** & \(\boldsymbol{\rho_{\text{T}}}\) **(kg/m\({}^{3}\))** & \(\boldsymbol{L_{0}}\) **(mm)** & \(\boldsymbol{R_{\text{P}}}\) **(mm)** \\ \hline [PERSON] et al. [23] & 1300 & 1055 1 & 17,730 & 7850 & 77.9 & 3.895 \\ [PERSON] et al. [24] & 1140 [13] & 400 [13] & 7830 & 2710 & 74.655 & 3.555 \\ \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of long-rod and target material in references. Let the ratio of the resistance of a finite-diameter target to the resistance of a semi-infinite target \(R_{\rm T}/R_{\rm T}\)* be the normalized target resistance. Figure 4 shows the comparison of the calculated normalized target resistance with the numerical results [24]. From Figure 4, the following can be concluded:  
* The proposed model is in good agreement with the numerical results. The calculated results are smaller than the numerical results when \(r_{0}/R_{\rm P}>6.5\). The reason is that the model does not consider the propagation time of elastic waves during the elastic-plastic stage;
* Due to the assumption that the target material is incompressible during the plastic stage, the target resistance is overestimated when \(r_{0}/R_{\rm P}<6.5\) compared to the numerical results.