Energy Spatial Distribution of Behind-Armor Debris Generated by Penetration of Explosively Formed Projectiles with Different Length–Diameter Ratio

Abstract

To understand the influence of the length–diameter ratio (L/D) of explosively formed projectiles (EFPs) on the energy spatial distribution of behind-armor debris (BAD), three EFPs with different L/Ds were designed in this study. The scattering characteristics of the BAD formed by the EFP penetrating a steel target were investigated. High-speed photography was used to observe the shape of the BAD cloud. Fiber and foam plates were sequentially stacked to recover the fragments. The three-dimensional damaged area by the BAD was obtained based on the spatial position information of a large amount of BAD. Finally, the energy spatial distribution characteristics of the EFP and target material fragments were analyzed. The results showed that a large EFP L/D increased the total energy of the BAD, and the proportion of the energy of projectile fragments increased. The difference in the energy spatial distribution between EFPs with varying L/Ds was mainly in the scattering angle range of 3–17°. The total energy of fragments within 17° of scattering angle accounted for 85% of the total energy of all fragments. The BAD energy of the EFP with a large L/D (L/D = 3.86) was concentrated in a small scattering angle range in which the residual projectile was located. The average projectile fragment energy of the EFP with a moderate L/D (L/D = 2.4) was evenly distributed in the scattering angle range of 5–20°. As a result, the energy distribution of the BAD from EFP (L/D = 2.4) shifted towards the large scattering angle, thus leading to a uniform radial distribution of the striking area within the range of 500–1100 mm behind the target. However, with the increase in the distance behind the target, the radial direction of the striking area of the other two EFPs was gradually reduced. The reason was explained according to the force analysis of the fragments resulting from the bulge fracture of target. The spatial energy distribution of BAD is closely related to the damage ability of EFP in relation to the armored target. Thus, it is necessary to design EFPs with appropriate L/Ds in order to maximize the damaging effect behind the armor.

1. Introduction

Explosively formed projectiles (EFP) are used in long-range anti-armor weaponry such as terminal-sensitive projectiles because of their large standoff, long distance, large penetration aperture, and significant behind-armor effects [1,2,3]. Early studies mainly focused on the design of an EFP damage element with high velocity, high solidity, good forming shape, and stable flight performance [4,5,6,7]. To improve the penetration ability of EFP against armored targets, common methods include the use of high-density liner material and increases in the length–diameter ratio (L/D) of EFP. However, there is an increasing demand for assessment of the damage effectiveness of weapon systems due to improved ammunition technology and target defense technology. The effectiveness of EFP cannot be simply evaluated based on the penetration performance such as the penetration depth, incident perforation diameter, and emergent perforation diameter; the damaging effect of the secondary behind-armor debris (BAD) should also be evaluated [8,9]. EFP is required to have penetration and perforation ability as well as sufficient behind-armor damage power. Existing studies on this topic mainly have focused on the influence of EFP impact velocity and target thickness [10,11] on the number and mass distribution of BAD fragments. EFPs with a large L/D can increase the penetration depth, but it is unclear whether they can enhance the damaging effect behind the target. Thus, it is necessary to study the effect of the L/D of EFPs on the distribution of BAD.

Researchers previously used EFPs with a specific shape or simulated EFPs to study the influence of shape and L/D of an EFP. For instance, Kim et al. [12] studied the radial expansion velocity of the BAD cloud of simulated EFPs with different solidities and L/Ds. Mostert [13] found that the shape of the semi-hollow EFP leads to the existence of jets, formed inside the BAD cloud in the reverse flight direction of the EFP. However, the EFP formation is affected by numerous factors such as the liner and the charge structure, and thus it is challenging to quantitatively control a single forming parameter. Therefore, BAD characteristics for real EFP L/D are more complex relative to simulated projectiles. In terms of the hypervelocity impact of space debris and penetration of kinetic energy projectile, promising achievements have been made in BAD cloud modeling [14]. Many studies have been performed on the influence of projectile shapes on the BAD cloud. Piekutowski et al. [15] compared the debris cloud of projectiles with three L/Ds, i.e., 0.058 (disk), 1.00 (right cylinder), and 3.00 (long rod). BAD cloud images of long-rod and cylindrical projectiles (L/D = 1), which were in the process of penetrating a steel target, were taken by Hohler et al. [16]: The radial scattering range of a short cylindrical projectile is large. In contrast to EFP, the projectiles in these studies were ither regular or the L/D could be precisely controlled.

In addition, the spatial distribution of BAD is closely related to the damaging effect of the internal components of the armored vehicle [17,18]. Many studies evaluated the relationship between BAD mass and velocity and the scattering angle for long-rod projectiles [19,20,21]. However, the ultimate goal of studying the scattering characteristics of secondary fragments behind the target is to analyze the damaging ability to the internal crew and to instruments within the armored target; the damaging elemental energy is an effective criterion with which to measure the damaging effect. Therefore, it is necessary to investigate the spatial energy distribution of BAD.

There are currently limited methods avilable to measure the kinetic energy distribution of the BAD cloud; only numerical simulations are available. Jia et al. [22] obtained the specific kinetic energy of the BAD cloud through numerical simulations and then obtained the function of the sum of the specific kinetic energies of particles per unit annular area. He et al. [23] obtained the energy distribution of dangerous fragments with respect to the spatial scattering angle. In an experimental study, Karpenko [24] weighed the BAD collected by the foam layer of a witness package and calculated the energy of the fragments using X-ray technology. However, the author did not present the results of the BAD energy in detail. Based on the number of perforations in witness targets with 3–5 layers, Pedersen et al. [25] found that the increase in the diameter of the rod-type projectile significantly increased the penetration ability/energy of the fragments. Zheng et al. [26] evaluated the kinetic energy and chemical energy behind the target according to the perforated area of the multi-layered spaced aluminum witness plate. Bless et al. [27] used rubber materials to recover fragments and employed X-ray tomography to obtain the energy distribution of the tungsten fragments. The results showed that the number of high-energy fragments decreased with increasing impact velocity.

These studies mainly focused on the energy of BAD based on space debris, shaped charge jets, or long-rod projectiles; the shape and velocity are much different than that of EFP. Therefore, it is necessary to analyze the energy distribution and the potential striking area of the BAD resulting from EFP penetration. Dawe et al. [28] obtained the relationship between the cumulative lethality of the BAD and the scattering angle. Wang et al. [17] found that the kinetic energy of the BAD decreased exponentially with increasing scattering angle, but they did not distinguish between EFP fragments and target fragments. Xing et al. [29] reported that the BAD with large kinetic energy was mainly generated by EFP and located far from the target. However, the smoothed-particle hydrodynamics (SPH) algorithm has some limitations, such as unstable tension and difficulties in applying physical boundaries; thus, the results from the simulation may be different from the actual results. These studies all used numerical simulation methods to study the energy distribution of the BAD from EFP penetrating target and did not involve spatial energy distributions, discrimination of EFP and target fragments in the experiment. Due to the high degree of randomness in BAD scattering, it is necessary to use the experimental method of Akahoshi et al. [30] to perform statistical analysis on the large number of fragments to determine the BAD spatial energy distribution. Moreover, the use of mathematical modeling is critical for describing the space dispersion characteristics of fragment energy [19,31,32], which requires in-depth analysis on the force of fragments generated by target fragmentation.

In view of the irregular shape of EFPs, this study focused on the effect of the L/D of EFPs on the spatial energy distribution of BAD. Three EFPs with different L/Ds were used to penetrate #45 steel target. Multi-layered witness plates were used to establish the correspondence between the information of the BAD and the three-dimensional spatial position and thus obtain the 3D spatial distribution of the striking area of the BAD. The spatial energy distribution of the BAD of the three EFPs was analyzed. We then evaluated the influence of L/D on behind-armor damaging effects.

2. Experimental Design

To prepare EFPs with different L/Ds, three shaped charge structures with the same diameter (70 mm) were designed by adjusting the charge height and the structural parameters of the liner, as shown in Figure 1. The JH-2 explosive used for detonation had a density of 1.69 g/cm³ and a detonation velocity of 8425 m/s. The detonation point was set at the center of the charge bottom. The liner material was copper. Three EFPs with similar velocities and masses but different L/Ds were formed, and the corresponding X-ray images are shown in Figure 2. According to the total length and head diameter of the EFP, the calculated L/Ds were 1.66, 2.4, and 3.86, respectively. Through high-speed photography, the EFP forming velocity was maintained at 1475 ± 25 m/s.

To obtain the spatial energy distribution of the BAD and the striking area behind the target, a medium with a reasonable penetration depth was required for use to recover the fragments [8]. Here, multi-layered witness plates were used to intercept the fragments, achieving a stepwise weakening of the energy of the BAD group. To some extent, the energy of BAD was quantitatively estimated, reflecting the damaging effect of the BAD to the components behind the target. Moreover, information on the scattering of the BAD group was analyzed to obtain the spatial distribution.

Similar to previous studies [18,33], the fragments were recovered using multi-layered witness plates consisting of a stacked polystyrene foam plate (EPS, density 20 kg/m³) and fiber plate. The test setup is shown in Figure 3. The 9 mm-thick fiber plates gradually weaken the energy of the fragments, and each plate was separated by a 50 mm-thick piece of EPS to provide space for fragment scattering. The remaining positions of the fragments were collected to analyze the spatial distribution. Only the fiber plates were placed at the end to capture the residual projectile because its energy was too large. An aluminum witness plate with a thickness of 2 mm was used as the first layer to obtain the contour of the perforation. The distance h₀ between the penetration target and the witness plate was 500 mm. The size of the recovery plates was 1.2 m × 1.2 m to ensure that the witness plate could capture as much information about BAD as possible. The three types of EFPs then penetrated vertically into 20 mm-thick #45 steel targets at the same standoff. Two repeatability tests were carried out for each EFP to consider the randomness of the formation and the scattering of the BAD.

3. Experimental Results and Analysis

3.1. Spatial Distribution of BAD in the Striking Area

After an EFP penetrates the target, the fragments behind the target scatter in different directions, forming a debris cloud. The shape of the BAD cloud reflects the spatial distribution of the fragment velocity, and the striking area in multi-layered witness plates after the fragment penetrates reflects the spatial distribution of the fragment energy. Figure 4 shows the high-speed photographic images of the BAD clouds for the three EFPs.

Figure 4 shows that the BAD cloud expanded continuously in the axial and radial directions and that the overall shape is ellipsoid, as shown by the red line in the figure. The major axis of the debris cloud increased significantly with increasing L/D of the EFP. This was due to the velocity of the residual EFP, while the increase in the minor axis was weak, thus indicating that an increase in the EFP L/D led to a larger increase in the axial scattering velocity of the fragments versus the radial scattering velocity, i.e., the fragment axial energy was mainly increased due to the large L/D. Because a reduction in EFP diameter leads to an increase in specific kinetic energy and an increase in input energy received by the target material per unit mass, the overall scattering velocity of BAD increases. The velocity gradient of the fragments causes the shape of the BAD cloud to be approximately ellipsoid, but there are differences between the spatial distribution of the BAD energy, the damaged area and the spatial distribution of BAD velocity. Therefore, in-depth analysis of the BAD information in the recovery plates behind the target is required.

One key component of damage assessment is the striking area of the BAD in relation to multiple targets. The spatial density of BAD was large, and the fragments scattered in different directions and struck different parts of the targets. They eventually stopped in the recovery plates with varying depths due to the exhaustion of energy. Since the fragment deeply embedded in the multi-layered recovery plates must penetrate the previous layers of the recovery plate, the spatial distribution of the final positions of the BAD after penetrating the multi-layered recovery plates reflects the damaging effect behind the target to a certain extent. Thus, it is necessary to first quantitatively describe the distribution field of the BAD in the three-dimensional striking area to obtain the energy spatial distribution of the BAD.

The coordinate system shown in Figure 5 was established to characterize the striking area behind the target. The x axis is the incident direction of the EFP, the y axis is the transverse direction, the Z axis is the longitudinal direction, and the exit on the back of the steel target was the origin O. The positions where the fragments were embedded on each layer of the recovery plate were collected, namely, the positions where the fragments impact the recovery plate of this layer. Additionally, we calculated the coordinates of each fragment relative to the residual EFP perforation center O’ in the local coordinate system Y’O’Z’ of the witness plate. In turn, the spatial coordinates (x_i, y_i, z_i) of each fragment were obtained.

The distance R_i ($R_i=\sqrt{y_i{}^2+z_i{}^2}$) between the fragment position in the recovery plate and the center perforation of residual EFP was taken as the radial scattering distance of the fragment, a factor which reflects the radial penetration ability. Figure 6 shows a schematic diagram of the method for measuring R on the recovery plate. The scattering angle β of the fragment is the angle between the line formed by the fragment position and the origin versus the x axis (Figure 5). The scattering angle of each fragment could be calculated based on R and the distance h between the recovery plate and the back of the steel target:

$$\beta =\arctan \left(R/h\right)=\arctan \left(\frac{\sqrt{y^2+z^2}}{h}\right)$$

(1)

The fragments collected in each recovery plate were numbered. For example, Figure 6 also shows the fragments embedded in this layer of witness plate. Additionally, the he numbers next to each fragment indicate the number of the fiber plate or foam plate and the position of the fragment, which correspond to the position on the recovery plate in Figure 6. Thus, the relation between the mass and spatial coordinates of the fragments could be established.

According to the model proposed by Schafer [34], the BAD cloud is described from three aspects. The residual EFP, large fragments of the target plug in the last few layers of recovery plates, and the fragments in the large hole formed by the residual EFP were excluded; statistical analysis of the fragments scattered around the residual EFP was then performed.

We accumulated the two tests data of EFP with the same L/D and obtained the spatial distribution map of the BAD damage area behind the target according to the relation between the mass and 3D coordinates of the fragments (Figure 7). The fragments are represented by spheres. The color of the spheres reflects the fragment mass. A color closer to red implies a larger fragment mass. Some fragments in certain holes were not retrievable. Thus, these fragments only have position information and do not have mass information; such fragments are represented by black spheres. Figure 7 shows the spatial striking area after the fragments penetrated the multi-layered recovery plate. The fragments were densely distributed immediately after the target, and the fragments then became sparsely distributed along the residual EFP trajectory. Hence, there was a densely distributed striking area. There were some small fragments in the first few layers of the recovery plate which did not reach the deeper layers. In general, the large-mass fragments had a large penetration depth, thus indicating large energy.

The 3D distribution field is projected on the XZ plane as shown in Figure 8 to analyze the distribution of the fragment striking area. The scales of each coordinate axis in the graphs are kept consistent in order to reflect the actual striking position of the fragment. As can be seen from Figure 8, the fragment striking area distribution of the EFPs with a large L/D (L/D = 3.86) and small L/D (L/D = 1.66) were trapezoidal. The lateral scattering of the fragments became smaller and smaller with increasing distance behind the target. They then gradually converged to the vicinity of the ballistic line. The increase in the L/D led to an increase in the depth of the striking area. The fragment striking areas of the EFP with a medium L/D (L/D = 2.4) were distributed in a rectangle. With increasing distance behind the target, the lateral scattering of the fragments did not converge but remained fairly even, thus indicating that the striking area distribution of the fragments of the EFP with a moderate L/D was uniform in the range of 500–1100 mm behind the target.

The EFP material (copper) fragments and target material (steel) fragments were then separated for further research. Figure 8 shows that there are many fragments with a mass less than 0.1 g. Considering their small lethality, only the positions of fragments greater than 0.1 g were included in the subsequent analysis.

Since all three EFPs can penetrate the target, the EFP diameter will have a significant influence on the radial dispersion of BAD. The radial scattering distance R reflects the radial distribution of the strike area. Based on the radial distance R of each type of fragment in the multi-layered recovery plate, a scatter diagram of the axial distance behind the target vs. the radial scattering distance was obtained (Figure 9). It should be noted that there are copper–steel bonded fragments, which are marked with a different symbol.

Figure 9 shows that there are some fragments with a small R in the area beyond 1200 mm behind the target, as indicated by the pink box. The fragment penetration depth increased with increasing L/D. The fragments of EFP with a large L/D (L/D = 3.86) were almost all copper in the final few layers, suggesting that the increase in the L/D mainly increased the energy of the EFP fragments near the ballistic line. Moreover, the scattering distance of the EFP fragments was the smallest versus EFPs of other L/Ds; that of the target fragments was comparable. When the diameter of EFP increased from 19.18 mm to 24.73 mm, the average value of the radial scattering distance of EFP fragments increased by 61%, i.e., increasing the diameter of EFP can enhance the radial damage ability of projectile fragments. Due to the randomness in the experiment, some fragments had a large scattering distance. Overall, the radial scattering distance of most fragments was within 400 mm, and the radial scattering distance of the target fragments was larger than that of the EFP fragments. The fragment with the largest radial scattering distance was from the steel target. Thus, the target fragments were distributed in the outer layer of the BAD cloud, which is consistent with Hohler et al. [21]. These results indicate that with increasing the EFP diameter, the difference in the radial scattering distance between the EFP fragments and the target fragments gradually decreased.

In addition, for the EFP with an L/D of 2.4, as the depth of the recovery plate increased, the overall maximum radial scattering distance of the fragments was almost unchanged in the range of 500–1100 mm behind the target. For the other two types of EFPs, although the maximum radial scattering distance was large at the beginning, the penetration depth of the fragments with a large radial scattering distance was very small. Hence, the maximum radial scattering distance gradually decreased with an increase in the depth of the recovery plate. This was caused by the gradual weakening of the energy of the fragment group by the multi-layer recovery plates. This reflects the energy attenuation process of the BAD and the radial reduction of the damaged area. Therefore, we concluded that the EFP with a medium L/D (L/D = 2.4) had a larger radial distribution range of striking area in the farther position behind the target. This was also verified by the fragment distribution on the 2D projection surface in Figure 8. An increased distance behind the target leads to greater maximum radial scattering distance of the EFP material fragments produced by the EFP with a medium L/D (L/D = 2.4), as indicated by the two red lines in the blue box. The difference in the spatial distribution of the striking area between different EFPs is caused by the varying BAD energy, and thus the uniform distribution of the striking area of the EFP with a medium L/D (L/D = 2.4) may be related to the distribution of EFP material fragment energy with respect to the scattering angle scattering angle. Therefore, it is necessary to quantitatively analyze the spatial energy distribution of the BAD.

3.2. Total Energy Distribution of BAD

The effective large-area damage to the internal instruments and crew of the armored target is mainly inflicted by the BAD group. The focus of the damaging effect assessment is on the energy of the damaged element (i.e., the kinetic energy of the fragments). Thus, the spatial energy distribution of the BAD is key to damage assessment.

The layer number of the witness plates penetrated by the BAD reflects the total penetration depth of the fragment to some extent [35]. Akahoshi et al. [36] assumed that all the kinetic energy of fragments was spent in moving against the resistance force in the recovery material. They then established the energy conservation formula based on the penetration depth of the fragments in the recovery plate. Thus, the penetration depth is closely related to the energy of the fragments. This is the case for all fragments recovered in the experiment penetrated the first layer of aluminum plate, but the number of layers of fiber plate penetrated is different. Therefore, the length of penetration path in the fiber plate was used as a measure of the fragment energy for simplification. The energy of each fragment is calculated as follows:

$$E=ne/\cos \beta$$

(2)

where β is the scattering angle of the fragments, n is the number of fiber layers penetrated by the fragments, and e is the energy consumed by penetrating one layer of fiber plate vertically. It should be noted that E is the fragment energy after penetrating the first aluminum layer.

To facilitate the subsequent analysis of fragment energy distribution, the following assumptions were made:

(1) The energy consumed by a fragment vertically penetrating one layer of fiber plate is constant and is denoted as e.

(2) If a fragment is embedded in the fiber plate, then the energy consumed when impacting the plate is 0.5e.

(3) The fragments embedded in the foam plate are considered to have no ability to penetrate the next layer of fiber plate. Because the density of foam plate is far less than that of fiber plate [30,33], the energy consumed by the fragments moving in the foam plate is ignored.

Based on the above assumptions, Equation (2) can calculate the energy of all fragments. Since the lethality of fragments with a mass below 0.1 g is minimal, the energy of these fragments is not calculated. By averaging the total energy of fragments in the two tests of each EFP, the total energy of BAD and that of the fragments of the two materials are obtained, as shown in Figure 10. Some fragments in the recovery plate were not retrievable. For those fragments, only the position information of the fragments was obtained, and it was impossible to determine the fragment material. Therefore, the total energy was greater than the sum of the energy of the steel (target) and copper (EFP) fragments. The energy of the copper–steel bonded fragments was calculated separately.

Figure 10 shows that when the L/D increased from 1.66 to 3.86, the total energy of the all fragments and EFP fragments increased by 54.8% and 167.7%, respectively, whereas the total energy of the target fragments decreased slightly. When the steel target remained the same, the total energy of the target fragments gradually approached a certain level, and the total energy of the EFP fragments increased significantly with increasing L/D. Hence, the proportion of the energy of the EFP fragments increased from 35.6% to 61.6%. According to the experimental results, the total mass of the BAD (>0.1 g) of the three types of EFPs was 388.37 g, 304.93 g, and 202.93 g, respectively, of which the total mass of the target fragments was 239.84 g, 135.14 g, and 88.10 g, respectively. The relationship between fragment total mass and EFP cross-sectional area is shown in Figure 11.

It can be seen that there was a certain linear relationship between the total mass of all fragments, target fragments and the cross-sectional area of EFP. This was because the perforation area on the steel target increased with the increasing cross-sectional area of EFP. Then, the perforation volume increased, resulting in the gradual increase in target fragment mass, while the difference of EFP fragment mass was relatively small and so the total mass of BAD gradually increased. Based on the BAD cloud in the high-speed images, the scattering velocity of the residual EFP and secondary fragments after EFP penetrated the target of the same thickness increased as the L/D increased, as did the number of fragments. Although the total mass of fragments decreased, the total energy increased since the target fragments were mainly formed by the erosion of the target material. This was caused by the penetration behavior of EFP and the continuous action of the reflected tensile wave at the end of penetration. The scattering velocity of this was relatively low, and the energy was mainly affected by the mass. Therefore, the total energy of target fragments produced by EFP with a large diameter (D = 24.73 mm) is large.

3.3. Energy Distribution with Scattering Angle of BAD

Theoretically, the distribution of the BAD cloud is rotationally symmetrical along the projectile incident axis (i.e., the x axis) [23], and the radial scattering probability along the circumferential direction is the same. Based on the rotational symmetry, the 3D spatial distribution of the BAD was simplified into a 2D distribution, and the scattering angle was used as a variable to quantitatively analyze the spatial energy distribution characteristics of the BAD.

Based on the calculated fragment energy and the spatial position of the fragment, the relationships between the energy of the EFP fragments and target fragments and the scattering angle were obtained, as shown in Figure 12. Additionally, two sets of test data of EFP with the same L/D were accumulated. The green dots mean that no fragments were collected in the corresponding hole of recover plate and that the material of the fragments could not be determined.

Figure 12 shows that the BAD energy decreased in an approximately exponential relationship with increasing scattering angle. The energy of the fragments from 0–8° was affected by the residual EFP, decreasing significantly with increasing scattering angle. The energy distribution was concentrated in the range of 0–10e from 8–20°. When the L/D was 2.4, the maximum energy was almost unchanged as the scattering angle increased, as shown in the red box. The energy distribution was more uniform versus the other two types of EFPs. There were mainly target fragments in the scattering angle range of over 25°, which was the rear section of the BAD cloud. With increasing L/D, the energy of the EFP fragments within the small scattering angle (0–5°) increased, as did the number of fragments, thus indicating that the energy of the EFP fragments concentrated along the EFP incident axis.

According to the scatter plot of energy distribution, the average energy of all fragments in each scattering angle range was calculated, and the results of two tests of each EFP type were averaged to obtain the average energy distribution of the fragments, as shown in Figure 13.

At scattering angles less than 30°, the average energy of the fragments gradually decreased with increasing scattering angle, thus indicating that the energy of the BAD cloud gradually weakened from the front to the middle section. This was because the fragment velocity in the front section of the BAD cloud was high, and therefore, the energy was higher than in the middle and rear sections. The average energy of the fragments increased when the scattering angle was greater than 30°, and there was a peak in the average energy of the fragments. To analyze the causes, the energy of the EFP fragments and target fragments was separated, and the average energy distribution of the projectile and target fragments is shown in Figure 14.

Figure 14 shows that there was only target fragment with energy beyond 25°. The fragments with a large scattering angle were from the ring fragments, i.e., the spall ring formed by the stretching wave due to the reflection of the shock wave on the back surface of the target. Although the velocity of the fragments was low, their mass was quite large [11,19]; therefore, the average energy of the fragments in this section increased instead. With an increasing L/D of EFP, the average energy of the target fragments with a large scattering angle (35–40°) increased because the spall area reflected the energy absorbed by the target from the EFP [37]. The specific kinetic energy increased when the EFP diameter decreased, and the input energy on the target per unit mass increased, resulting in a small fragment mass and a high fragment velocity. Thus, the average energy of fragments in this region increased.

When the L/D was large (L/D = 3.86) or small (L/D = 1.66), the average energy of the target fragment and EFP fragments decreased exponentially with the increase in scattering angle from 0° to 25°. Conversely, the average energy of the EFP fragments did not decrease but basically remained at a certain level within the range of 5–20° when the L/D was moderate (L/D = 2.4). Versus the other two types of EFPs (L/D = 3.86 and 1.66), the energy of the fragments was more uniform over different scattering angle ranges, which is consistent with the results in Figure 12b. This suggests that it is not the case that a higher L/D is better; rather, there is an appropriate L/D that can lead to a uniform spatial distribution of fragment energy.

The average fragment energy of the EFP with a large L/D (L/D = 3.86) was small within a scattering angle range of 0–5° because of the large number of fragments in the range (Figure 12c). Additionally, the scattering of these fragments was greatly affected by the residual EFP and the fragments basically followed the residual EFP in the small scattering angle range. The diameter of the residual EFP with a large diameter (small L/D) was large (Figure 3), thus the perforation on the recover plate was large, allowing more small fragments to pass through. Therefore, the perforation information of such fragments is missing, and it is difficult to accurately calculate the number of perforations in the small scattering angle range in the experiment.

3.4. Spatial Distribution of Relative Cumulative Energy of BAD

The energy of fragments in each scattering angle range was accumulated and normalized to obtain the relative accumulated energy distribution, as shown in Figure 15.

Figure 15 shows that the overall rate of increase in the BAD energy gradually decreased with the increasing scattering angle. The differences in the spatial distribution of the BAD energy between the three EFPs were mainly concentrated in the scattering angle range of 3–17°, the distribution of other scattering angle ranges is basically the same. The total energy of the fragments within a scattering angle of 17° accounted for about 85% of the total energy of all fragments, indicating that the energy of the fragments with a large scattering angle was small. In the range of 3–17°, the curve of the EFP with an L/D of 2.4 was the lowest, which reflects the shift of energy distribution towards the large scattering angle. In contrast, the fragment energy of the EFP with an L/D of 3.86 was concentrated in the small scattering angle region. The data in the scattering angle range of the same distribution were fitted in Figure 15, and the spatial distribution of the relative cumulative energy followed an exponential function.

$$\frac{E(<\beta )}{E_0}=-90.97\exp (-\beta /9.81)+103.72$$

(3)

where E(<β) is the sum of the energy of the fragments with a scattering angle of less than β, and E₀ is the total energy of BAD. The energy of the residual EFP and the target plug fragments (i.e., the sum of the energy of the fragments with a scattering angle of 0°) accounted for 20–25% of the total energy of all fragments, thus indicating that a large part of the BAD energy was dispersed in scattering space around the projectile incident axis in a way that could threaten or even damage the internal components of an armored target. The relative cumulative energies of the EFP and target fragments were separated, as shown in Figure 16.

According to Figure 16, the difference in the BAD energy distribution between different EFPs was mainly expressed via the energy of the EFP fragments. When the L/D was 3.86, the energy of the EFP fragments was mainly distributed at 0–5°, and the EFP fragment energy then increased slowly as the scattering angle increased. In comparison, the energy of the EFP fragments of the other two types of EFPs increased approximately linearly over the entire scattering range. These results suggest that a very large L/D will cause the energy of the EFP fragments to be concentrated closely to the projectile axis. Additionally, the cumulative energy of EFP fragments hardly increased when the scattering angle was greater than 20°, indicating that the energy of EFP fragments was distributed in the range of 0–20°. However, the energy distribution of the target fragments was not different beyond 25°. Moreover, the energy distribution curves of the EFP with an L/D of 2.4 was the lowest from 7–13° for the EFP fragments and 7–16° for target fragments, thus indicating that the energy distribution of EFP fragments and target fragments in the corresponding range was concentrated at the large scattering angle and within the difference region of the energy distribution of all fragments, as shown in Figure 15.

According to previously conducted analysis, the energy of EFP fragments was mainly concentrated in the scattering angle range of 0–20°. When the steel target was the same, the energy distribution of the fragments within the large scattering angle range (>25°) caused by the spallation of the target was close. Thus, the difference in the energy distribution of the BAD in the scattering angle range of 0–20° was large due to the effects of the L/D and the shape of the EFP head. Because the three EFPs could penetrate the target, the difference in the radial distribution of fragments was mainly related to the diameter of the EFP rather than the length. The BAD was mainly formed by the rupture of the bulge on the back of the target [21]. During the penetration process, the head of the EFP was thickened to form a mushroom-shaped head, which resulted in a bulge on the back of the target. The bulge gradually expanded, cracks were generated under the action of tensile stress, and the cracks connected each other and finally penetrated the bulge wall. Then, the bulge broken, and the residual EFP together with a large number of projectile erosion fragments and target spall fragments are ejected from the rear of the target, forming a BAD.

Assuming that the shape of EFP mushroom head and bulge is spherical, then the bulge is completely broken, EFP erosion is complete, and the material in mushroom is no longer in a fluid state. At this time, the residual EFP velocity is usually greater than the target fragment velocity. Because of the velocity difference between the two, friction is generated along the contact surface. At the time of complete rupture, the force of a fragment generated by a fracture on the bulge, as is shown in Figure 17. The forces are expressed as:

$$F_p=m_f{a}_p$$

(4)

$$F_N=F_p\cos \theta$$

(5)

$$F_f={\mu }_s{F}_p\cos \theta$$

(6)

where F_p is the force acting on the contact surface between the projectile and the fragment, a_p is the corresponding acceleration, and m_f is the fragment mass. F_N is the normal force, F_f is the friction force, μ_s is the friction coefficient, and θ is the included angle between the projectile axis and the line which connects the fragment with the center of the mushroom. The friction force is decomposed in axial and radial directions:

$$F_{fx}=-{\mu }_s{F}_p{\cos }^2\theta$$

(7)

$$F_{fr}={\mu }_s{F}_p\sin \theta \cos \theta$$

(8)

where F_fx and F_fr are the force components in the axial and radial directions, respectively. In addition, the corresponding acceleration components are expressed as follows:

$$a_{fx}=-{\mu }_s{a}_p{\cos }^2\theta$$

(9)

$$a_{fr}={\mu }_s{a}_p\sin \theta \cos \theta$$

(10)

where a_fx and a_fr are the acceleration components in the axial and radial directions, respectively. The fragment velocity component under the residual EFP motion is:

$$v_{fx}=v_r-{\mu }_s{a}_p{\cos }^2\theta \cdot t_p$$

(11)

$$v_{fr}={\mu }_s{a}_p\sin \theta \cos \theta \cdot t_p$$

(12)

In the equation, v_r is the residual EFP velocity, t_p represents the time interval between the complete rupture of the bulge and the formation of the initial BAD cloud, after which the BAD cloud maintains a stable shape and expands proportionally. The fragment scattering angle β is calculated using the following:

$$\tan \beta =\frac{v_{fr}}{v_{fx}}=\frac{{\mu }_s{a}_p\sin \theta \cos \theta \cdot t_p}{v_r-{\mu }_s{a}_p{\cos }^2\theta \cdot t_p}=\frac{{\mu }_s\sin \theta \cos \theta \cdot t_p}{\frac{v_r}{a_p}-{\mu }_s{\cos }^2\theta \cdot t_p}$$

(13)

In the process of EFP penetrating the target, according to the axial stress balance equation of the projectile–target interface in the A-T model [38], the axial stress of the bulge element squeezed by EFP is:

$$P=\frac{1}{2}{\rho }_p{\left(v-u\right)}^2+Y_p$$

(14)

where Y_p and ρ_p is EFP strength and density.

When the bulge is completely broken, the residual EFP can freely rush out of the target bulge, and the erosion of the mushroom deformation zone stops because it depends on the target material which is completely separated from the main body of the target [39]. At this time, the velocity v_c of the undeformed part of the EFP is equal to the penetration velocity u_c [40], and the force of the fragment generated by the bulge fracture is:

$$F_p=A_f{Y}_p$$

(15)

where, A_f refers to the contact area between the fragment and projectile. Considering that the crack propagates along the bulge thickness direction and breaks through the bulge wall to produce fragments, taking the bulge wall thickness δ_b at the fracture time as the fragment thickness (Figure 17), the fragment mass can be expressed as:

$$m_f={\rho }_t{A}_f{\delta }_b$$

(16)

where ρ_t is the material density of the target. Substitute Equations (4), (15) and (16) into Equation (13) to obtain the following:

$$\beta =\arctan \left(\frac{{\mu }_s\sin \theta \cos \theta \cdot t_p}{\frac{v_r{\rho }_t{\delta }_b}{Y_p}-{\mu }_s{\cos }^2\theta \cdot t_p}\right)$$

(17)

It can be seen that, when the target material is determined, the friction coefficient μ_s is a constant, the fragment scattering angle is affected by the residual EFP velocity v_r and the bulge fracture thickness δ_b. When the L/D of EFP is large, i.e., the residual velocity of EFP is high, the scattering angle is primarily affected by it. Conversely, when the residual velocity of EFP is low, the bulge fracture thickness δ_b is the main factor affecting the scattering angle.

Since the diameters of three EFPs differ greatly, the L/D actually affected the fragment dispersion characteristics through the EFP diameter. The specific kinetic energy of EFP decreases with increasing EFP diameter. Additionally, the residual EFP velocity decreases through the high-speed images, which further led to a certain degree of increase in the scattering angle of the fragment at the same position θ (according to Equation (17)). The energy distribution of the fragments then shifted towards the large scattering angle region. When the resultant energy of a fragment is constant, the radial energy component increases and the radial scattering distance increases. As a result, the radial distribution of the striking area in Figure 8 and Figure 9 was uniform. However, the bulge diameter become large when the EFP diameter was too large. Meanwhile, the bulge thickness becomes larger at the fracture time, then the dispersion angle of fragment at the same position gradually decreases. The energy distribution of the fragments then no longer moved towards the large scattering angle region. In addition, the fragment resultant velocity and resultant energy decreased due to a small axial penetration ability of EFP, i.e., the radial scattered energy components of the fragments were weakened. Hence, in Figure 8 and Figure 9, the maximum radial scattering distance gradually decreased with the increase in the depth of the recovery plate. Therefore, there are obvious differences in the radial distribution of the striking area behind the target and the spatial energy distribution of BAD with different EFP L/Ds.

In summary, a larger L/D is not always better from the perspective of the damaging effect of the BAD. This is because, when EFP can penetrate the target, EFP diameter will have the most significant impact on the radial distribution of BAD. When the L/D is too small, the EFP penetration ability decreases and the energy of the BAD is small. Although an increase in the L/D can increase the penetration depth into armor and the BAD energy, a very large L/D will cause the spatial energy distribution of the fragments to become concentrated close to the residual projectile axis, thus reducing the degree of scattering and the radial distribution of the striking area. Therefore, in the damage assessment of EFPs and warhead structural design, a reasonable and moderate L/D can cause the spatial energy distribution of the BAD to shift towards the large scattering angle and a uniform radial distribution of the striking area while ensuring damage to the components behind the armor.

4. Conclusions

In this study, a multi-layered recovery plate was used to collect BAD from EFPs vertically penetrating steel targets. Based on the results, the spatial distribution of the striking area of EFPs with different L/Ds was obtained, and the spatial energy distribution of the BAD was analyzed. The main conclusions are as follows:

(1) In the range of 500–1100 mm behind the target, the maximum radial scattering distance of the fragments decreased with increasing distance from the target when the L/D was too small (L/D = 1.66) or too large (L/D = 3.86), thus indicating a reduction in the strike area in the radial direction. In comparison, the radial distribution of the striking area of the EFP with a moderate L/D (L/D = 2.4) was uniform, thus resulting in a larger radial distribution range of the strike area at a large distance from the target. Moreover, the radial scattering distances of the target fragments were larger than those of the EFP fragments.

(2) With increasing L/D, the total energy of the BAD and the energy of the EFP fragments both increased, and the energy of the EFP fragments account for a large portion of the total energy of the BAD. With increasing scattering angle, the average energy of the BAD did not simply decrease—it increased after 30° due to the effect of the spalled fragments on the back side of the target.

(3) The differences in the spatial energy distribution of BAD formed by EFP with different L/Ds was mainly concentrated in the scattering angle range of 3–17°. The spatial distribution of the relative cumulative energy in other scattering ranges was exponential. When the L/D was too large (L/D = 3.86), the energy of the EFP fragments became concentrated in the small scattering angle range, i.e., close to the residual EFP. The energy distribution of the target fragments was indistinguishable. For the EFP with a moderate L/D (L/D = 2.4), the average energy of the fragments was evenly distributed in the range of 5–20°, and the energy distribution of the BAD shifted towards the large scattering angle region. This led to a uniform radial distribution of the striking area within 500–1100 mm behind the target, with uniform damage to components outside the ballistic line. It should be noted that the L/D of EFP actually affected the radial dispersion of BAD through the EFP diameter. The damage assessment of armored vehicle targets and EFP warhead structural design makes it necessary to select a reasonable and moderate L/D (diameter D) to achieve the best damaging effect of the BAD behind the armor.

The results of the present study can be used in future work to determine the zone of damage inflicted by BAD on the internal components of an armored target. Furthermore, the results lay the foundations for evaluating the damage effectiveness behind armor due to EFP penetration and provide technical references for the structural protection of the internal instruments and crew of an armored target and for EFP warhead design.