# Study on Bulge State of PCnNi3 MoV Target Plates Penetrated by Square Fragments
## 3 Theoretical Analysis
\begin{array}{ll}\pi_{4}&=(\mathrm{L})^{\lambda_{1}}(\mathrm{ML}^{-3})^{ \lambda_{2}}(\mathrm{ML}^{-1}\mathrm{T}^{-2})^{\lambda_{3}}(\mathrm{ML}^{2} \mathrm{T}^{-2})^{\lambda_{4}}\\ &=\mathrm{L}^{\lambda_{1}-3\lambda_{2}-\lambda_{3}+2\lambda_{4}}\mathrm{M}^{ \lambda_{2}+\lambda_{3}+\lambda_{4}}\mathrm{T}^{\lambda_{3}+\lambda_{4}}\end{array}  \tag{6}\]  
The resulting system of algebraic equations is  
\[\left\{ \begin{array}{l}\lambda_{1}-3\lambda_{2}-\lambda_{3}+2\lambda_{4}=0 \\ \lambda_{2}+\lambda_{3}+\lambda_{4}=0\\ \lambda_{3}+\lambda_{4}=0\end{array} \right. \tag{7}\]  
and  
\[\left\{ \begin{array}{l}\lambda_{4}=-\lambda_{3}\\ \lambda_{2}=0\\ \lambda_{1}=3\lambda_{3}\end{array} \right. \tag{8}\]  
In this study, let \(\lambda_{4}=1\), and then \(\lambda_{3}=-1\) and \(\lambda_{1}=-3\). Thus,  
\[\pi_{4}=h^{3}E/\sigma_{t} \tag{9}\]  
Therefore, Equation (1) can be re-written as  
\[1=\left(\frac{l}{h}\right)^{a}\cdot\left(\frac{\rho_{f}}{\rho_{t}}\right)^{b} \left(\frac{h_{g}}{h}\right)^{c}\left(\frac{h^{3}E}{\sigma_{t}}\right)^{d} \tag{10}\]  
Through the above test, it is known that there is a linear relationship between the bulge height increment \(\Delta h_{g}\) and the fragment energy increment \(\Delta E\), and thus, c = -d = 1. Hence,  
\[\Delta h_{g}=\left(\frac{l}{h}\right)^{a}\cdot\left(\frac{\rho_{f}}{\rho_{t}} \right)^{b}\frac{1}{h^{2}\sigma_{t}}\cdot\Delta E \tag{11}\]  
where \(a\) and \(b\) are undetermined constants. According to the test results, the curve of the bulge height of the target plate with the fragment energy must pass through point (\(E_{0}\), 0), where \(E_{0}\) is the minimum energy for the fragment to cause plastic deformation on the back of the target plate. The fragment energy equation can be written as follows:  
\[h_{g}=\left(\frac{l}{h}\right)^{a}\cdot\left(\frac{\rho_{t}}{\rho_{f}}\right)^ {b}\frac{1}{h^{2}\sigma_{t}}\cdot\left(E-E_{0}\right) \tag{12}\]  
Since \(E_{0}\) is affected by the material properties of the target plate, the following equation can be established:  
\[E_{0}=\left(\frac{l}{h}\right)^{x}\cdot\left(\frac{\rho_{f}}{\rho_{t}}\right)^ {y}h^{3}\sigma_{t} \tag{13}\]  
By integrating the equations, the following equations are obtained:  
\[\left\{ \begin{array}{l}h_{g}=A\cdot\left(E-E_{0}\right)\\ A=\left(\frac{l}{h}\right)^{a}\cdot\left(\frac{\rho_{f}}{\rho_{t}}\right)^{b} \frac{1}{h^{2}\sigma_{t}}\\ E_{0}=\left(\frac{l}{h}\right)^{x}\cdot\left(\frac{\rho_{f}}{\rho_{t}}\right)^ {y}h^{3}\sigma_{t}\end{array} \right. \tag{14}\]As shown by Equation (14), under certain target conditions, \(A\) and \(E_{0}\) are constant values, and the penetration energy required increases with the increase in the fragment velocity. By fitting the data in Table 2 with Equation (14), the relationship between the fragment energy and the bulge height of the target plate was obtained, as shown in Figure 6. The fitting coefficient and fitting accuracy for different target plates are shown in Table 5.  
Figure 6 shows that the bulge height of the target plate increased linearly with the increase in the fragment energy. By fitting the data, the parameters were obtained (Table 5). As the ratio of the side length of the fragment to the thickness of the target plate increased, the coefficient A increased, whereas the fitting coefficient \(E_{0}\) decreased. In other words, the bulge height increased with the increase in the ratio of the side length of the fragment to the thickness of the target plate when the fragment energy remained the same, and the minimum energy for plastic deformation of the back of the plate was reduced accordingly. In addition, the fitting accuracies for the different target and projectile systems were 95% or better, indicating that Equation (14) is suitable for predicting the bulge height, and thus, it can be used in practical situations. By obtaining the values of \(A\) and \(E_{0}\) under different conditions, the four parameters \(a\), \(b\), \(x\), and \(y\) were calculated, as shown in Table 6.