# Impact of Soil on the Penetration of Prefabricated Spherical Fragments and Its Protective Effect
## 2 Experiments on Fragment Penetration into Soil
### Experimental Results
In Table 2, \(R_{0}\) is the material density; \(G\) is the shear modulus; \(E\) is the modulus of elasticity; \(PR\) is [PERSON]'s ratio; \(A\), \(B\), \(N\), \(C\), and \(M\) are parameters of the relevant equation; \(TM\) is the melting temperature; \(TR\) is the room temperature; and \(EPSO\) is the effective plastic strain rate. In Tables 3 and 4, \(R_{0}\) represents the material density; \(G\) is the shear modulus; \(BULK\) is the bulk modulus; \(A_{0}\) is the yield constant of the plastic yield function; \(VCR\) is the volume crushing option; \(REF\) is the initialization pressure using the reference geometry; \(EPS_{\text{n}}\) is the volumetric strain value of the soil; and \(P_{\text{n}}\) is the pressure value corresponding to the volumetric strain value. The \"Soil_and_Foam\" (keyword name) model was proposed by [PERSON] [21] in 1972, which can be used for soil and flattening foam, and has been well matched in the soil penetration test of high-speed projectiles by many scholars. The author used the same initial conditions as the experiment, with a fragment size of 11 mm and a soil medium size of 300 \(\times\) 300 \(\times\) 300 mm. All fragments were vertically shot into the soil medium. The comparison of the residual velocity results between the numerical simulation and the experiment is shown in Table 5.  
From Table 5, it can be seen that the numerical simulation results are more stable than the experimental results. This is mainly because the experimental conditions have a certain  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(\text{EPS}_{\text{1}}  
degree of uncertainty, but the average error of the remaining velocity of fragments is 25.94%, and the numerical simulation results of fragments penetrating the soil have credibility. Based on the above experimental and numerical simulation data, combined with the Poncelet formula, the residual velocity of fragments is determined. Equation (1) is a general expression of the Poncelet formula, which is characterized by a proportional relationship between the square of the initial velocity and the square of the residual velocity. Due to the limited available experimental data for fragments with a diameter of 6 mm, the author performed a formula fitting on fragments with a diameter of 11 mm, and the resulting fitting formula is shown in Equation (3). \[v^{2}={v_{0}}^{2}\exp(-\frac{2c_{3}Ah}{m})+\frac{c_{1}}{c_{3}}\exp(-\frac{2c_{ 3}Ah}{m})+1 \tag{1}\]  
\[v^{2}={v_{0}}^{2}\exp(-\frac{c_{3}h}{r\rho})+\frac{c_{1}}{c_{3}}\exp(-\frac{c_ {3}h}{r\rho})+1 \tag{2}\]  
\[v^{2}={v_{0}}^{2}\exp(-\frac{3051.1h}{r\rho})+\frac{1165239166.7}{3051.1}\exp( -\frac{3051.1h}{r\rho})+1 \tag{3}\]  
\[v^{2}={v_{0}}^{2}\exp(-\frac{3240.5h}{r\rho})+\frac{636696645.8}{3240.5}\exp( -\frac{3240.5h}{r\rho})+1 \tag{4}\]  
Based on the numerical simulation calculation model, the following research mainly focuses on two aspects: the angle of attack of fragment penetration and the boundary effect of finite-thickness soil.