Numerical simulation and experimental study on forming of pentaprism directional MEFP

Abstract

Explosively formed projectile (EFP) has the advantages of high flight speed, high burst height, insensitivity to initial velocity, etc. To study the forming performance of multiple explosively formed projectile (MEFP) and to analyze the factors influencing the forming results. A directional pentagonal prism MEFP with five liners is proposed in this paper. Experiments were conducted to validate the performance of this MEFP. The experimental results show that the generated EFPs have high flight speeds. However, the EFP hit position on the target plate was offset, causing multiple positional penetrations. To analyze the causes of the phenomenon, a numerical simulation model of directional MEFP is established and the forming process of MEFP is numerically simulated. The simulation results indicate that the EFP fracture occurs during the forming process due to the asymmetric detonation wave, and its tail cannot be fully closed. Combined with the numerical simulation and experimental results, the forming performance of the liners used in the MEFP has been further developed. Further analysis was carried out on the effect of liner thickness and height on forming performance. It can be seen that as the thickness of the liners increases, the length of the head of EFP increase, the incidence of EFP tail fracture decreased, the aspect ratio increases and the EFP velocity decrease. As the height of the MEFP increases, the velocity of the EFP increases, the head ratio increases, the shape of the detonation wave tends to be symmetrical, and the tendency of the EFP to appear as an unclosed tail decreases.

Introduction

Explosively formed projectile (EFP) has been widely used in civil blasting, oil well perforating and so on. The main research direction of domestic and foreign experts is to improve the forming performance of Explosively Formed Projectiles (EFP) so that the generated EFP has high muzzle velocity, high aspect ratio and high compactness. At present, it is mainly divided into two aspects: selecting the material of the high ductility liner¹ and the multi-point initiation design^2–4 or wave shaper to control the shape of the detonation wave^5–7. The Multiple Explosively Formed Projectile (MEFP) was initially designed to enhance the space coverage performance of the EFP device^8–10. Typically, MEFP incorporates multiple EFP units on a single device. According to the distribution mode of liner, MEFP can be divided into axial combined type, circumferential combined type and other forms. The circumferential composite type is generally a cylindrical structure, and the multiple liners are arranged along the outer surface of the cylinder in the circumferential direction, it can form a space cover of EFP around the cylinder during detonation, improving the hit rate of the warhead. However, this design also leads to the low energy utilization rate of the MEFP. In order to improve the forming performance of the circumferential MEFP, multiple detonation points are placed on the axis of the MEFP¹¹ and high ductility liner materials¹¹ are selected. The design of the detonation points on the axis ensures a uniform distribution of EFPs, but also limits the charge height of each EFP, thereby reducing the overall energy utilization.

The damage elements fly towards a certain direction owing to the rotation and deformation of the directional warhead, although its structure is similar to that of the circumferential MEFP. Furthermore, the charge height of the directional MEFP is higher than that of the circumferential one as a result of eccentric initiation. Zhang et al.¹² proposed a deformable warhead, and built the calculation formulas for shell parameters, explosive charge parameters, peak fragment velocity and distribution density of fragment cloud. Guo et al.^13,14 proposed a directional fragment warhead with D-section, the fragmentation velocity distribution of the warhead is analyzed. Dhote et al.¹⁵ proposed a directional fragment generator warhead, researched the spatial dispersion of the fragments in two configurations of FGWs having a circular shaped fragment generating surface by experiments¹⁶, analyzed the importance of knowing the standard deviation value for assessing damage in terms of fragment hit density and kill probability¹⁷. Huang et al.¹⁸ analyzed the velocity profile of fragments in a square warhead with eccentric initiation. Li et al.¹⁹ analyzed the velocity profile of a two-line asymmetrically initiated warhead, compared with one-line asymmetrically initiated warhead, the non-uniform velocity distribution enables an asymmetrically initiated aimable warhead to be constructed. Deng et al.²⁰ proposed the formula for predicting the radial velocity distribution of the fragments. About the MEFP warhead, Li et al.²¹ proposed a directional MEFP warhead with a cylindrical structure similar to the circumferential MEFP, but with liners only on one side and an eccentric detonation point on the opposite side. It was found that the MEFP formation and penetration performance achieved by the eccentric detonation was higher than that of the central detonation structure. Ji et al.²² proposed a directional MEFP with a D-shaped section for directional single-point detonation, and analyzed the forming of the generated EFP and the relative position of the liners and the detonation point. This structure provides a high energy utilization rate. Based on this, Wang et al.²³ changed the warhead design to a multi-point initiation system and analyzed the performance formation law and divergence angle of EFP affected by the initiation mode. It can be seen that the D-shaped warhead has detonation asymmetry on the edge liner. Gao et al.²⁴ proposed an eccentric liner for the edge of the warhead based on the shape of the detonation wave to improve the forming effect. The directional MEFP mentioned above use an eccentric initiation structure, which improves the energy efficiency of the explosive compared to the traditional central initiation structure. However, only one side is provided with liners.

To enhance the performance of directional MEFP, a warhead with five liners featuring a regular pentagonal section is proposed in this paper. Each of the five prisms has a liner on both sides and a bus bar with an initiation point opposite each liner. When an initiation point is detonated, the detonation wave successively shapes the other four liners, and finally forming the final liner opposite the initiation point. The purpose of this design is to enhance the formability of the main directional liner by utilizing the high energy efficiency of the directional structure and the reflection of the detonation wave when it passes through the other four liners. This paper investigates the formability of directional MEFP main direction through experiments and numerical simulations, and analyses the factors that affect the forming of the liner. The design aims to increase the energy utilization rate of the overall structure through eccentric initiation. Additionally, the detonation wave generated from the initiation point passes through the other four liners before reaching the opposite side liner, resulting in a pressure increase of twice through reflection. This enhances the power of the final formed opposite liner. This paper is structured as follows, and the first part conducts an experimental study of directional MEFP. In the second part, the numerical simulation model of directional MEFP is established, and the forming process is analyzed according to the experiment and simulation results. In the third part, the influencing factors of MEFP forming are analyzed by numerical simulation.

Methods

Experiment

A directional MEFP warhead consisting of a five-prism structure with a regular pentagon section is proposed in this study. A liner is arranged on each of the five sides of the five-prism bus, and an initiation point is placed at the midpoint of the opposite bus of each liner. During use, the warhead rotates along the middle line and activates one of the initiation points. The liner on the opposite side of the initiation point is formed accordingly. The design aims to increase the energy utilization rate of the overall structure through eccentric initiation. Additionally, the detonation wave generated from the initiation point passes through the other four liners before reaching the opposite side liner, resulting in a pressure increase of twice through reflection. This enhances the power of the final formed opposite liner. The warhead has five identical sides leading to low external power and high hit rate. The warhead of the study has a pentagonal outer circle diameter of 135 mm and height of the five prisms is 75 mm. The liner, made of copper, features a spherical cover of varying thickness with a diameter of 55 mm. The warhead is depicted in Fig. 1a and b. The MEFP is contained within a cylindrical cover made of aluminium.

Figure 1.

The directional MEFP used in the experiment for (a) diagram, (b) image.

The experiment layout are shown in Fig. 2a and b, in which a wooden support was placed in the center of the experimental field and the MEFP was laid on the support. A liner was taken as the effective direction, and an electric detonator was used to connect the detonation point on the opposite side of the direction. Two velocity measurement targets were placed 8.5 m and 9 m away from the MEFP along the effective direction. The support for the target plate is positioned 10 m from the device center. The target plate fixed onto the support is a homogeneous steel plate with the dimension of 800 mm* 500 mm*30 mm.

Figure 2.

Experimental setup for (a) image and (b) diagram of the experiment field.

Simulation

In this paper, a numerical simulation of directional MEFP device is carried out. In the calculation, the explosive and the liners are assumed to be continuous uniform medium, and the whole explosion process after detonation is isentropic adiabatic process. In the EFP forming process, the influence of gravity action is not considered, and the simulation model is shown in Fig. 3a. The dimensions of the liner and charge are consistent with the experiment. ALE multi-material group elements are employed in the explosives, liners and air domains. The mesh size is 1 mm. The model is assumed to be symmetric, in order to reduce the calculation load, a 1/4 model is used for simulation as shown in Fig. 3b.

Figure 3.

Simulation model of directional MEFP for (a) diagram of MEFP, (b) diagram of simulation model.

The liner material selected is copper, which is described using the Johnson–Cook model and Gruneisen equation of state. The material parameters are listed in the Table 1.

Table 1.

Constitutive model parameters of copper (liner)²⁶.

Material	Part	Material property
Cu	Liner	$G$/GPa	$A$/MPa	$B$/MPa	$\text{n}$
		46.5	90	292	0.31
		$C$	$\text{m}$	$\mathrm{\Gamma }$	$S$
		0.025	1.09	2.02	1.49

The explosive material composition B has a density of 1.67 g/cm³, which is described using the high explosive burn model and JWL equation of state. The material parameters are listed in the Table 2.

Table 2.

Constitutive model parameters of composition B (explosive)²⁷.

Material	Part	Material property
Composition B	Explosive	$D\text{/m}{\text{s}}^{-1}$	$P_{\mathit{CJ}}/GP\text{a}$	$A/GP\text{a}$	$B/GP\text{a}$
		7840	24	52.423	7.678
		$R_1$	$R_2$	$\omega$
		4.2	1.1	0.34

Results and discussion

Experiment results

The experiment was conducted three times in total. Figure 4 shows the pictures taken after each target plate was hit, and Table 3 is the recorded flight speeds for each trial. In the experiment 3, due to the impact of shell debris during the explosion, the velocity target did not measure the effective result.

Figure 4.

Results of experiment for (a) experiment 1, (b) experiment 2 and (c) experiment 3.

Table 3.

EFP velocity measured in experiments.

Case	Height/g	Interval of measurement targets/m	Flight time/μs	EFP velocity/m s−1
1	2505.8	0.43	175	2457.143
2	2516.3	0.42	164	2560.976
3	2509.8	0.41	–	–

Figure 5 illustrates the comparison of hit positions among the three experimental results. The target plate was hit by the first shot, resulting in four holes. The maximum diameter of the holes was 41 mm, and the penetration depth was 27 mm. The distance from the center of the target plate was 21 mm in the negative direction of the axis and 250 mm in the positive direction of the axis. The second shot resulted in two holes. The maximum diameter of hole is 40 mm, and they penetrated the target plate. The aperture on the back of the target plate measures 24 mm, and the distance from the center of the target plate is 193 mm in the negative direction of the axis and 253 mm in the positive direction of the axis. The target plate was hit by the third shot, resulting in four holes. These holes were located 132 mm away from the center of the target plate in the negative direction of the axis and 90 mm away in the positive direction of the axis.

Figure 5.

EFP hit position in experiments.

The three experiments resulted in divergent hit positions due to various reasons. Firstly, the EFP broke during the forming process, causing the head and the collapsed part to hit the target plate separately, resulting in multiple bullet holes. Secondly, the flight stability of the generated EFP was poor, leading to the deviation of its flight direction. Finally, in both experiment 2 and experiment 3, a bullet hole penetrated the target plate. It is evident that during the forming process, the EFP broke. The head structure remained intact and had a large amount of kinetic energy, which allowed it to penetrate the target plate. However, the tail did not form an effective penetrating body shape and had a small amount of kinetic energy, which resulted in its failure to penetrate the target plate upon impact. Furthermore, the intense fire and smoke produced by the explosion hindered the effectiveness of high-speed photography in capturing the EFP shape. As a result, additional research was conducted through numerical simulation.

Simulation verification

The numerical simulation is carried out according to the conditions adopted in the experiment, and the specific reasons affecting the EFP forming performance are analyzed.

The simulation result is shown in Fig. 6, the EFP velocity obtained by simulation is 3197.167 m/s, The experiment revealed that the error in measuring flight speed in experiment 3 was too large. Additionally, the average speed of EFP flying 9 m in experiments 1 and 2 was 2509.060 m/s. As per the literature²⁵, the formula for EFP velocity attenuation is as follows:

$$V=V_0{e}^{-kx}$$ 1

Figure 6.

Shape distribution for liner at specific moments.

In the formula, $k=\frac{C_x\rho S}{2m}$; $S=\pi {\left(\frac{D_m}{2}\right)}^2$, $C_x=C_{\mathit{xn}}+C_{\mathit{xt}}+C_{\mathit{xd}}+C_{\mathit{xf}}$;

$V$ is the impact velocity of EFP, $V_0$ is the initial velocity of EFP, $x$ is the flight distance, $k$ is the projectile velocity attenuation coefficient, $C_x$ is the projectile resistance coefficient, $C_{\mathit{xn}}$, $C_{\mathit{xt}}$, $C_{\mathit{xd}}$, $C_{\mathit{xf}}$ is the drag coefficient related to the dimensions and form of the EFP, and the values can be obtained from literature²⁵. $\rho$ is the air density, $S$ is the windward area of EFP, $D_m$ is the diameter of EFP, $m$ is the mass of EFP.

At a distance of 9 m, the EFP velocity decreases to 2907.777 m/s, resulting in a 15.9% deviation from the average velocity observed during experiment.

The origin of the coordinate system for the hit location of the EFP head was used in all three experiments. A hit location map of the EFP was obtained and compared with the numerical simulation results. The distribution of the hit location of the EFP tail debris in the three experiments was similar to the distribution location in the numerical simulation, as shown in the Fig. 7.

Figure 7.

Hit location in experiments compared with simulation result.

At the beginning of the reversal process, the liner is affected by the pressure of the detonation wave, as shown in Fig. 8. It can be seen from Fig. 8a that at 13 μs the detonation wave reaches the liner and the distribution of the detonation wave on the liner is not a circle but an ellipse. At 14 μs, the longitudinal pressure distribution of the liner is obviously greater than the transverse pressure distribution, which is shown in Fig. 8b. In addition, the central part of the liner eventually forms the cylindrical area of the EFP head, which is completely closed. The part in the dashed area is closed from both sides to the center due to the large gap between the velocity and the central part, as shown in Fig. 6 and Fig. 8. Finally, the unclosed area of the EFP tail is formed, and the velocity gradient of the final overturning deformation of the liner is too large, and the tail on both sides breaks at 120 μs. As a result, the EFPs do not form the cylindrical or spherical shape of conventional EFPs, and their flight stability is greatly reduced.

Figure 8.

Pressure distribution on liner at (a) t = 13 μs, (b) t = 14  μs.

To enhance the charge height, an eccentric initiation design was employed for the directional Multiple Explosively Formed Penetrator (MEFP) in this study. In this section, the formability outcomes of this configuration with those of a circumferential MEFP utilizing central initiation and an equivalent charge quantity to validate the enhanced forming performance achieved by this structure will be compared. The charges and liners utilized in both configurations are identical. The initiation point of the circumferential MEFP is positioned 37.5 mm above the axis of its circumscribed cylinder, which coincides with the geometric center of its pentagonal shape. Figure 9 illustrates a comparative analysis of these two structures' forming processes.

Figure 9.

Simulation results of circumferential MEFP and directional MEFP.

It can be seen from the Fig. 10 that after using the center initiation, there are still problems of asymmetric molding and unable to close the tail. The aspect ratio of EFP formed by circumferential MEFP is 0.878, which is smaller than the aspect ratio of directional MEFP (1.253).

Figure 10.

The curves of EFP velocity of circumferential MEFP and directional MEFP.

Compared with the speed curve of directional MEFP, as shown in the figure, it can be seen that due to the reduction of the charge distance of the circumferential MEFP, the time of EFP starting forming is 7 μs earlier than that of the directional MEFP, and the flight speed of the circumferential MEFP after molding is 2593.245 m/s, which is 18.9% lower than that of the directional MEFP.

In conclusion, the directional MEFP device studied in this paper demonstrated superior performance in comparison to the central initiation circumferential MEFP. The EFP formed in the main direction exhibited enhanced closure, higher flight speed and a larger aspect ratio. It can be demonstrated that the directional MEFP device utilizes energy in the main direction to a greater extent than the circumferential MEFP.

Analysis of influencing factors of forming

The numerical simulation of the experiment conditions indicates that the EFP penetration performance is primarily affected by the velocity gradient of the EFP when it is deformed, which is attributed to the detonation asymmetry caused by the charge structure. This results in a generated EFP shape with poor flight stability and partial fracture. In order to mitigate the impact of detonation asymmetry on forming, the improvement scheme proposed in this paper primarily addresses the influence of detonation asymmetry on EFP forming by modifying the charge structure of MEFP, specifically the height of MEFP and thickness of the liner, as illustrated in Fig. 11. The MEFP in the experiments, $l_1$ = 75 mm and $h_1$ = 4 mm. This chapter presents numerical simulations for the working conditions of $l_1$ = 75 mm, 85 mm, 95 mm, 105 mm and $h_1$ = 4 mm, 5 mm, 5.5 mm, 6 mm and 6.5 mm.

Figure 11.

Influence factor of EFP forming for (a) height of MEFP $l_1$, (b) thickness of liner $h_1$.

In the experiment condition, the head of the EFP is a solid structure that has been turned over in a normal manner, whereas the tail is not closed in a normal manner and is fractured. The experiment condition of $l_1$ = 75 mm was subjected to analysis. Figure 12 depicts the cross-sectional diagram of the EFP along the central plane perpendicular to the paper surface when $l_1$ = 75 mm was employed.

Figure 12.

Sectional view of EFP with different $h_1$.

As illustrated in Fig. 6, the incomplete tail is primarily comprised of the dashed portion of the liner depicted in Fig. 8b. During the overturning process, the detonation pressure experienced by the dashed section of the liner is less than that observed in other regions. Consequently, the velocity gradient between this section and the remaining parts of the liner is significantly elevated, leading to the formation of an unclosed tail illustrates the section of the liner. Despite the lack of improvement in the asymmetry of detonation following an increase in the thickness of the liner, the thickness of the liner does result in an increase in the thickness of the unclosed tail during the overturning process. This, in turn, enhances the strength of the tail and reduces the tendency for fracture.

The Fig. 13a demonstrates that the length of the EFP head increases with the $h_1$ increase. This solid area is formed by the displacement along the axial direction driven by the detonation wave at the center of the liner. The thickness of the solid area is determined by the axial velocity gradient of the liner. Consequently, the thickness of the top of the liner is greater. The solid region at the top of the generated EFP is also thicker. The Fig. 13b illustrates the variation in EFP aspect ratio with $h_1$. With an increase in $h_1$, the length-to-diameter ratio demonstrates an upward trend.

Figure 13.

The curves of (a) length of the EFP, (b) aspect ratio varying with $h_1$.

The Fig. 14 shows the change in velocity of the EFP with liner thickness $h_1$. As the thickness of the liner $h_1$ increases, the mass of the liner increases and the velocity of the EFP decreases.

Figure 14.

The curves of EFP velocity when (a) $l_1$ = 75 mm, (b) $l_1$ = 85 mm, (c) $l_1$ = 95 mm, (d) $l_1$ = 105 mm.

The Fig. 15 shows the change in velocity of the EFP with height of MEFP $l_1$. As the $l_1$ increases, the mass of the charge increases so that the velocity of the EFP increases.

Figure 15.

The curves of EFP velocity when (a) $h_1$ = 4 mm, (b) $h_1$ = 5 mm, (c) $h_1$ = 5.5 mm, (d) $h_1$ = 6 mm, (e) $h_1$ = 6.5 mm.

The total length of EFP decreases with the $l_1$ increase, as shown in Fig. 16a.

Figure 16.

The curves of (a) total length of EFP, (b) head ratio of EFP varying with $l_1$.

Let $n$ be:

$$n=\frac{l_{\mathit{head}}}{l_{\mathit{total}}}$$ 2

where $l_{\mathit{head}}$ is the length of the generated EFP head and $l_{\mathit{total}}$ is the total length of the EFP, which gives the proportion of the length of the unclosed tail of the generated EFP. This means that the higher the value of $n$, the better the flight stability of the EFP.

As shown in Fig. 16b, the EFP head ratio increases with the MEFP height $h_1$, resulting in a gradual decrease in the length of the unclosed tail and an improvement in EFP flight performance. The numerical simulation results indicate that as $h_1$ increases, the weight of the liner increases and the flight speed gradually decreases. Increasing $l_1$ from 75 to 105 mm results in an increase in the total charge of MEFP and a gradual increase in flight speed.

Further design

Four groups of working conditions with $h_1$ = 6 mm were analyzed. The pressure distribution on the back of the liner when the 14 μs detonation wave reaches the liner is shown in Fig. 17. As can be seen from Fig. 17, when $h_1$ = 6 mm, with the increase of $l_1$, the shape of detonation wave will gradually change from ellipse to circle when it reaches the liner. As the symmetry of the detonation wave shape increases, so does the symmetry of the generated EFP.

Figure 17.

Pressure distribution on the liner when $h_1$ = 6 mm with (a) $l_1$ = 75 mm, (b) $l_1$ = 85 mm, (c)$l_1$ = 95 mm and (d) $l_1$ = 105 mm.

Figure 18 shows the MEFP forming results with variable $l_1$ and $h_1$, When $h_1$ = 6 mm and $l_1$ = 105 mm, the EFP no longer has an open tail. The overturning deformation shape of the EFP is similar to the traditional EFP structure, and it still maintains a high flight speed of 2970 m/s, and aspect ratio of 2.509.

Figure 18.

Simulation results of further design.

Conclusions

This paper proposes a directional MEFP warhead with a five-prismatic structure. Experiment and simulation results has revealed that the EFP generated by this process exhibits asymmetric detonation characteristics, resulting in fractures. Additionally, the tail cannot be fully closed during the forming process, leading to poor flight stability. Numerical simulation was used to analyze the structure of the charge and liner of MEFP. The influence of the height of the MEFP and the thickness of liner on its forming performance was determined.

1. Through experimental research, the directional MEFP structure proposed in this study has been found to have a higher flight speed. However, in the test condition, the EFP generated has exhibited a broken phenomenon. However, the EFP generated during testing exhibited a phenomenon of fracture. Numerical simulation analysis of the test conditions determined that the main reason for EFP fracture was the uneven effect of detonation on EFP caused by the asymmetry of detonation.

2. Compared with circumferential MEFP with same charge and liner, the EFP formed by the directional MEFP has a higher velocity and a greater aspect ratio.

3. With the thickness of liner increase, the length of the head of EFP increase, the incidence of EFP tail fracture decreased, the aspect ratio increases and the EFP velocity decrease.

4. The velocity of EFP increases with the height of the MEFP, while the total length of the EFP head show an increasing trend. The head ratio increases with the height of the MEFP. The shape of the detonation wave gradually becomes more symmetrical, and the tendency of EFP with an unclosed tail decrease.

5. When $h_1$ = 6 mm and $l_1$ = 105 mm, the EFP no longer has an open tail. The overturning deformation shape of the EFP is similar to the traditional EFP structure, and it still maintains a high flight speed of 2970 m/s, and aspect ratio of 2.509.

Author contributions

Tao X.G. conceived the experiment. Song J.G. conducted the experiment and simulation, and Tao X.G., Song J.G. Yang Z.L. performed statistical analysis. All authors reviewed the manuscript.

Data availability

All data generated or analyzed during this study are included in this article.

Competing interests

The authors declare no competing interests.