## 3 Theoretical Penetration Model and Model Verification
### Modified Alekseevskii-Tate Model and Calculation of Penetration Depth  
According to [PERSON] [5; 6], it is assumed that the interface of the high-velocity long-rod and the target is semi-fluid and the initial shape of the long-rod's nose has negligible effects on the penetration depth. As shown in Figure 2, \(R_{\rm P}\) is the radius of the rod; \(v_{0}\) and \(L_{0}\) are the initial impact velocity and the initial length of the projectile, respectively; and \(u\), \(v\), \(P\), and \(L\) are the penetration velocity, tail velocity, penetration depth, and remaining length of the rod, respectively. The following basic equations can be drawn according to the Alekseevskii-Tate model [5]:  
\[\frac{1}{2}\rho_{P}(v-u)^{2}+Y_{P}=\frac{1}{2}\rho_{T}u^{2}+R_{T}, \tag{26}\]  
\[\frac{dl}{dt}=-(v-u), \tag{27}\]  
\[\frac{dP}{dt}=u, \tag{28}\]  
\[\frac{dv}{dt}=-\frac{Y_{P}}{\rho_{P}L}. \tag{29}\]  
Figure 2: Schematic diagram of long-rod and cylindrical metal thick target with finite diameter: (**a**) before impact; (**b**) shortly after impact.  
The initial conditions are as follows:  
\[P(0)=0,L(0)=L_{0},v(0)=v_{0}, \tag{30}\]  
where \(\rho_{P}\) and \(\rho_{T}\) are the density of the rod and target, respectively; \(Y_{P}\) is the dynamic flow stress of the rod; and \(R_{T}\) is the target resistance.  
\(Y_{P}\) can be taken as [29]  
\[Y_{P}=(1+\lambda)\sigma_{\rm{yp}}. \tag{31}\]  
By using (25), \(R_{T}\) can be modified as  
\[R_{T}=f\Big{(}r_{cf},r_{t0}\Big{)}R_{T}^{*}, \tag{32}\]  
\[R_{T}^{*}=Y_{t}\Bigg{(}\frac{2}{3}+\ln\frac{2E}{\big{(}4-e^{-\lambda}\big{)}Y_ {t}}\Bigg{)}, \tag{33}\]  
where \(\sigma_{\rm{yp}}\) is the yield stress of the rod; \(\lambda\) is usually taken as 0.7; \(R_{\rm{t}}\)* is the target resistance of a semi-infinite target [29]; and \(r_{\rm{cf}}\) can be taken as the channel radius [8]:  
\[r_{cf}=R_{P}\Big{(}1+0.287v_{0}+0.148v_{0}^{2}\Big{)}, \tag{34}\]  
where \(v_{0}\) is measured in km/s.  
Using the above equations, the depth of penetration can be calculated in two situations determined by the relationship between \(Y_{P}\) and \(R_{t}\).  
* \(Y_{P}<R_{t}\)  
In this situation, the long-rod only undergoes eroding penetration. When \(v=v_{\rm{c}}\) and \(u=0\) are reached, the penetration ends, and the remaining long-rod continues to erode until the residual velocity reaches 0. The penetration depth can be obtained as follows:  
\[P=\frac{\rho_{P}L_{0}}{Y_{P}}\int_{v_{\rm{c}}}^{v_{0}}\frac{1}{1-\lambda^{2}} \Big{(}v-\lambda\sqrt{v^{2}+A}\Big{)}\bigg{(}\frac{v(v)}{v(v_{0})}\bigg{)}^{k _{1}}\exp\{k_{2}[\theta(v)-\theta(v_{0})]\}\mathrm{d}v, \tag{35}\]  
\[v_{c}=\sqrt{\frac{2(R_{T}-Y_{P})}{\rho_{P}}},\lambda=\sqrt{\frac{\rho_{T}}{ \rho_{P}}},A=\frac{2(R_{T}-Y_{P})}{\rho_{T}}\Big{(}1-\lambda^{2}\Big{)}, \tag{36}\]  
\[v(x)=x+\sqrt{x^{2}+A},\theta(x)=x\sqrt{x^{2}+A}-\lambda x^{2}, \tag{37}\]  
\[k_{1}=\frac{R_{T}-Y_{P}}{\lambda Y_{P}},k_{2}=\frac{\lambda\rho_{P}}{2(1- \lambda^{2})Y_{P}}. \tag{38}\]  
* \(Y_{P}>R_{t}\)  
In this situation, the penetration depth can be divided into two parts: eroding penetration and rigid body penetration. First, the high-speed long-rod undergoes eroding penetration until the penetration speed is equal to the tail velocity (\(u=v=v_{r}\)). Then, the remaining long-rod (with remaining length \(L_{\rm{r}}\)) penetrates as a rigid body at the remaining velocity \(v_{t}\). The depth of penetration can therefore be obtained as  
\[P_{1}=\frac{\rho_{P}L_{0}}{Y_{P}}\int_{v_{\rm{c}}}^{v_{0}}\frac{1}{1-\mu^{2}} \Big{(}v-\mu\sqrt{v^{2}+A}\Big{)}\bigg{(}\frac{v(v)}{v(v_{0})}\bigg{)}^{k_{1} }\exp\{k_{2}[\theta(v)-\theta(v_{0})]\}\mathrm{d}v, \tag{39}\]\[v_{r}=\sqrt{\frac{2(Y_{P}-R_{T})}{\rho_{T}}}, \tag{40}\]  
\[P_{2}=\frac{\rho_{P}L_{r}R_{P}^{2}v_{r}^{2}}{2r_{cf}^{2}R_{T}}, \tag{41}\]  
\[r_{cf}=R_{P}\Big{(}1+0.287v_{0}+0.148v_{0}^{2}\Big{)}, \tag{42}\]  
\[L_{r}=L_{0}\bigg{(}\frac{v(v_{r})}{v(v_{0})}\bigg{)}^{k_{1}}\exp[k_{2}(\vartheta (v_{r})-\vartheta(v_{0}))], \tag{43}\]  
\[P=P_{1}+P_{2}. \tag{44}\]