## 3 Fragment Penetration Calculation Based on Forest Environment  
When a warhead is detonated in the forest, various parameters of the warhead and the forest will cause different initial variables of fragment scattering, such as the height \(H\) of warhead initiation, the size and direction \(v\) of fragment velocity vector, the main wood species caused by forest environment, tree diameter \(D\), and forest density [11]\(\rho\), etc. Among them, due to the difference of tree species, the strength and density of bark and inner core of trees are different, and the thickness of bark is also different [12], so the difference between bark and inner core is ignored in this study. The initial velocity vector of the fragment is decomposed into a vector \(v_{xy}\) parallel to the ground and a vector \(v_{z}\) perpendicular to the ground, where \(v_{xy}\) mainly controls the flight direction of the fragment in the horizontal direction; \(v_{z}\) mainly controls the flight time of fragments in the vertical direction. It can be seen that \(v_{z}\) and \(H\) determine the flight time of fragments in the air, and then affect the flight distance of fragments. See Figure 5 for the fragment penetrating trees and the relationship between the variable parameters of fragments. In the \(x\)-\(y\) plane, the fragments may penetrate the wood in the forest in all directions, based on the forest density \(\rho\) and tree diameter \(D\). assuming that the trees are evenly distributed in the forest, the number of trees per unit area can be determined according to the stand density (n/m\({}^{2}\)), so as to determine the land area of each tree. If each tree is  
Figure 4: Comparison of experiment and simulation on the back of target plate. (**a**) is experiment; (**b**) is numerical simulation. Figure 5: Model of fragment penetrating trees in forest. (**a**) This is a schematic diagram of fragments penetrating trees; (**b**) This is an assumption of average distribution of trees (1/m\({}^{2}\)). placed in the center of its occupied area, the following forest average distribution diagram can be obtained. Taking the 1/4 model as an example, the lower left corner is the explosion center, the stand density is 1/m\({}^{2}\), and the tree diameter is 300 mm. From the top view, the trajectory of the fragments will fly around without considering the vertical movement time until \(v_{xy}\) is 0. By numbering the fragment trajectories at 3\({}^{\circ}\) intervals, it can be seen that the lengths of different fragments penetrating the wood in the same radius area are different, some through four trees, and some through only one tree. Although this difference can reflect the maximum damage radius and the minimum damage radius of fragments in the forest environment, individual differences cannot reflect the comprehensive damage degree of all fragments. The trajectory of fragments penetrating trees is shown in Figure 6.  
In this paper, a method of average distribution of wood area is adopted. The penetration trajectory area of fragments in the \(x\)-\(y\) plane is equivalent to the product of fragment diameter \(d\) and trajectory length \(L\). The wood area \(S\) penetrated in the penetration area can be calculated through the trajectory area it passes through. Ignoring the irregular part of the tree penetration surface, the path length \(L\) of the wood medium can be obtained by dividing the penetration area \(S\) by fragment diameter \(d\), as shown in formulas (1) to (3). The calculation formula of the average area of trees is:  
\[S_{\mathrm{w}}=\rho\cdot D^{2}/4 \tag{1}\]  
Among \(\rho\) is the density of trees per unit area (n/m\({}^{2}\)), and \(D\) is the average diameter of trees. The penetration area of the fragment with diameter \(d\) on the horizontal plane is:  
\[S_{\mathrm{p}}=S_{\mathrm{w}}\cdot d\cdot R \tag{2}\]  
\(R\) is the radius length of fragment flight.