## 2 Prediction of Penetration Depth Using the Finite Cavity Pressure Method
### Free Surface Effect  
The cavity pressure was calculated from the assumption that the target material is infinite. When the distance between the projectile and target is short, the pressure applied to the element is low compared to the normal. The free surface effect means that when the position of the projectile is near the ground, no pressure is generated because the cavity pressure is not perfectly formed at that part of the projectile. In order to consider the reduced cavity pressure on the surface near the target, [PERSON] and [PERSON] [13] proposed the following method.  
In the calculation of the factor for the free surface effect, the aluminum material was considered as perfectly plastic and incompressible. In the calculation of the cavity pressure, a spherical cavity expansion problem with the infinite boundary and the finite boundary were employed. A schematic figure describing the free surface effect is presented in Figure 4. [PERSON] and [PERSON] [17] derived the decay function, which is a function of d, a, and v and demonstrates the free surface effect.  
Figure 3: Schematic figure of the finite cavity pressure method.  
First, calculate the pressure acting at an infinite distance from the particle. Then calculate the pressure at a certain distance. The pressure ratio at a certain distance can be obtained through the air ratio. As shown in Figure 2, the interface between the elastic region and the plastic region should be defined. For an incompressible material, the elastic-plastic interface (b) is defined as follows:  
\[\frac{\mathrm{a}}{\mathrm{b}}=(\frac{3\mathrm{Y}}{2\mathrm{E}})^{1/3} \tag{3}\]  
where a is the radius of the cavity. For metal materials, the pressure distribution depends on d. For the plastic region (d \(<\) b), the decaying function is defined as follows:  
\[\mathrm{f(d,a,v)=}\frac{2\mathrm{Y}\mathrm{In(\frac{d}{a})+\frac{1}{2}\rho v^ {2}[3+(\frac{a}{d})^{4}-4(\frac{a}{d})]}}{\frac{2\mathrm{Y}}{3}[1+\mathrm{ln }\left(\frac{2\mathrm{E}}{3\mathrm{Y}}\right)]+\frac{3}{2}\rho v^{2}},\ \ d<\mathrm{b} \tag{4}\]  
For the elastic region (d \(\geq\) b), the decaying function is defined as follows:  
\[\mathrm{f(d,a,v)=}\frac{\frac{2\mathrm{Y}}{3}[\mathrm{ln}(\frac{2\mathrm{E}}{ 3\mathrm{Y}})+1-\frac{2\mathrm{E}}{3\mathrm{Y}}(\frac{a}{d})^{3}]+\frac{1}{2} \rho\mathrm{a}\cdot^{2}[3+(\frac{a}{d})^{4}-4(\frac{a}{d})]}{\frac{2\mathrm{Y }}{3}[1+\mathrm{ln}\left(\frac{2\mathrm{E}}{3\mathrm{Y}}\right)]+\frac{3}{2} \rho v^{2}},\ \ d<\mathrm{b} \tag{5}\]  
Finally, the general solution according to d, a, and v is defined as follows:  
\[\frac{\sigma_{r}(\mathrm{d,a,v})}{\mathrm{Y}}=[\mathrm{A}+\mathrm{B}(\sqrt{ \frac{\rho_{0}}{\mathrm{Y}}}\mathrm{v})+\mathrm{C}(\sqrt{\frac{\rho_{0}}{ \mathrm{Y}}}\mathrm{v})^{2}]\mathrm{f(d,a,v)} \tag{6}\]  
The aforementioned equation presents a cavity pressure distribution with respect to d\({}_{\mathrm{I}}\) and v\({}_{\mathrm{I}}\). The first equation presents the compressibility, strain hardening, and strain rate dependent cavity pressure of the aluminum materials. The following equation is the decaying function and presents the effects of the free surface effects.