## 3 Fragment Penetration Calculation Based on Forest Environment
The average penetration path of fragments is:  
\[L_{\mathrm{p}}=S_{\mathrm{p}}/R=\rho\cdot d\cdot D^{2}/4 \tag{3}\]  
In the above formula, the following table \({}_{\mathrm{p}}\) represents penetration and \({}_{\mathrm{w}}\) represents wood. In order to verify the accuracy of the calculation method, the above method is used to verify different parameters in the following three cases:  
Figure 6: A model for the uniform dispersion of fragments in forests. (1/4 model) (**a**) is schematic diagram of evenly scattered fragments; (**b**) is penetration trajectory through trees. 1. Set the wood diameter and stand density as fixed values, change the horizontal dispersion angle between two adjacent fragments, and compare the average penetration trajectory length of fragments with formula (3). (1\({}^{\circ}\), 3\({}^{\circ}\), 5\({}^{\circ}\))
2. Set different wood diameters without changing the stand density and scattering angle, and compare the average penetration trajectory length of fragments with formula (3). (200 mm, 300 mm, 400 mm)
3. The average penetration trajectory length of fragments is compared with formula (3) by setting different stand densities without changing the tree diameter and scattering angle. (1/m\({}^{2}\), 0.5/m\({}^{2}\), 0.25/m\({}^{2}\))  
The comparison results are shown in Table 4.  
It can be seen from the error between the actual value and the calculated value corresponding to the three variables of different scatter angles, tree diameters, and stand densities that the calculated value was generally larger than the actual value, and the error rate increased with the increase in the scatter angle. It decreased with the increase in tree diameter, and with the increase in stand density. It can be seen that the accuracy of the simple calculation formula was closely related to the three variables. Therefore, on the one hand, the appropriate range of use of each variable could be obtained to improve the accuracy. On the other hand, the correction coefficient \(w\) for the three variables could be added to Equation (3). The dimensionless expression of \(w\) is:  
\[w=\frac{\alpha}{D\cdot\rho} \tag{4}\]  
Using the penetration area of the fragment, the length of the fragment penetrating the wood in the horizontal direction can be obtained, that is, the thickness of the target plate of the fragment penetrating the wood. The overall penetration path \(L_{xyz}\) can be obtained by using the penetration length \(L_{xy}\) of the horizontal plane and the angle \(\beta\) between the fragment velocity vector and the horizontal plane, as shown in formula (5). \[L_{xyz}=L_{\text{p}}\cdot w/\cos\beta \tag{5}\]  
According to the experimental results of 11 mm tungsten alloy fragments and 6 mm fragments, the relationship between the residual velocity and the final velocity is an exponential function when the thickness \(H\) of the target plate is constant, which is similar to the velocity attenuation law of the projectile in the air. Due to the low density and low quality of wood, the velocity attenuation formula of fragments in air is considered first [13]. When fragments fly in the air, they are affected by gravity and air resistance, and gravity makes the flight trajectory of fragments bend. Air resistance causes the attenuation of fragment velocity. Since the distance from the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & **Stand Density** & **Tree Diameter** & **Scattering** & **Model** & **Formula** & \\  & \(\rho\)/(n\(\cdot\)m\({}^{-2}\)) & **D/mm** & **Angle \(\alpha\)/\({}^{\circ}\)** & **Calculation** & **Calculation** & **Error Rate/\%** \\  & & & & & & \\  & & & & &  
fragment to the target is not too long and the time is very short, the influence of gravity can be ignored, and the fragment trajectory can be approximated as a straight line.