## 3 Analyses and Discussions  
### Scaling Theory for the Impact Crater  
\(\pi\)-group scaling is a useful tool for analyzing the factors affecting crater size. Holsapple 3 provided a generalized equation for the crater volume of spherical projectiles in strength-dominated impacts. Considering the effect of the length-to-diameter ratio (\(k_{p}\)) of the long-rod projectile, the crater volume \(V_{c}\) is determined by the listed variables:  
\[V_{c}=f(D_{p},k_{p},m_{p},\rho_{p},\rho_{t},\sigma_{t},v_{0}), \tag{2}\]  
where \(D_{p}\), \(m_{p}\), \(\rho_{p}\), and \(v_{0}\) are the long-rod projectile diameter, mass, density, and velocity. \(\rho_{t}\) is the target density and \(\sigma_{t}\) is a measure of target strength. In the case of Equation (2), there are seven parameters, with the three independent dimensions of mass, length, and time. Therefore, the crater volume can be expressed non-dimensionally and Equation (2) can be written as:  
\[\frac{V_{c}}{D_{p}^{3}}=f\bigg{[}k_{p},\frac{m_{p}}{D_{p}^{3}\rho_{t}},\frac{ \rho_{t}}{\rho_{p}},\frac{\rho_{t}v_{0}^{2}}{\sigma_{t}}\bigg{]}, \tag{3}\]  
as  
\[\frac{\rho_{t}V_{c}}{m_{p}}\propto k_{p}^{\mu_{1}}\cdot\bigg{(}\frac{m_{p}}{D _{p}^{3}\rho_{t}}\bigg{)}^{\mu_{2}}\cdot\bigg{(}\frac{\rho_{t}}{\rho_{p}}\bigg{)} ^{\mu_{3}}\cdot\bigg{(}\frac{\rho_{t}v_{0}^{2}}{\sigma_{t}}\bigg{)}^{\mu_{4}}, \tag{4}\]  
or as  
\[\frac{V_{c}}{V_{p}}\propto k_{p}^{\
u_{1}}\cdot\bigg{(}\frac{\rho_{t}}{\rho_{p }}\bigg{)}^{\
u_{2}}\cdot\bigg{(}\frac{\rho_{t}v_{0}^{2}}{\sigma_{t}}\bigg{)} ^{\
u_{3}}, \tag{5}\]  
where \(V_{p}\) is the projectile volume. Here, \(\
u_{3}\) is defined as \(\
u_{3}=1\), which means that \(V_{c}\) increases at the rate of energy scaling. The scaling exponents \(v_{1}\) and \(v_{2}\) should be determined for certain types of projectiles and target materials. [PERSON] et al., 1 conducted steel projectile penetration tests on mortar targets with the compressive stress of 50 MPa. The flat-nose cylindrical projectiles with a length-to-diameter ratio of 5 and a mass of 6.58 g were used in the test. The impact velocity ranged from 510 to 1850 m/s. [PERSON] et al., 2 conducted tungsten alloy projectile penetration tests on concrete targets with a compressive stress of 42.7 MPa. Flat-nose cylindrical projectiles with a length-to-diameter ratio of 3 and a mass of 1.72 g were used in the test. The impact velocity ranged from 1970 to 3660 m/s. As shown in Figure 14, the experiment results of [PERSON] et al., 1 and [PERSON] et al., 2 were used to fit Equation (5), and the fitted function shown is:  
\[\frac{V_{c}}{V_{p}}=9.6k_{p}^{-0.2}\cdot\bigg{(}\frac{\rho_{t}}{\rho_{p}} \bigg{)}^{3/2}\cdot\frac{\rho_{p}v_{0}^{2}}{\sigma_{t}}. \tag{6}\]  
This can be used as an empirical formula for predicting the crater volume of a rod-shaped projectile impacting a concrete-like target. Figure 15 shows the present experimental crater volumes as well as the calculating result of Equation (6), from which good match can be observed. From the present experimental result, the crater shape is similar to a spherical cap. The same phenomenon can be observed from the experimental results of [PERSON] et al. 2. The ratio of crater depth to diameter (\(\gamma\)) at different impact velocities of the present experiment as well as the test data of [PERSON] et al. 1 is shown in Figure 16. The value of \(\gamma\) is randomly distributed between 0.1 and 0.2, which is not related to the impact velocity. Here, the empirical value of \(\gamma\) is defined as 0.15. Then, the empirical formulas for crater diameter (\(D_{c}\)) and crater depth (\(H_{c}\)) can be obtained as  
\[\frac{D_{c}}{D_{p}}=9.11k_{p}^{0.267}\Bigg{(}\frac{\rho_{t}}{\rho_{p}}\Bigg{)} ^{1/2}\Bigg{(}\frac{\rho_{p}v_{0}^{2}}{\sigma_{t}}\Bigg{)}^{1/3}, \tag{7}\]  
\[\frac{H_{c}}{D_{p}}=1.37k_{p}^{0.267}\Bigg{(}\frac{\rho_{t}}{\rho_{p}}\Bigg{)} ^{1/2}\Bigg{(}\frac{\rho_{p}v_{0}^{2}}{\sigma_{t}}\Bigg{)}^{1/3}. \tag{8}\]  
Figure 17 shows the crater diameters and the predicted result of the present experiment and [PERSON] et al. [2].