Numerical Simulation of Multiple Explosively Formed Projectile Warhead Forming Characteristics considering Various Materials

To study the influence of different liner structures and materials (copper, steel, and tungsten) on the forming characteristics of multiple explosively formed projectile (MEFP) with integrated liner and shell designs, three types of liners with different structures were designed. LS-DYNA was used for numerical simulation, and the results show that the thickness change at the center of the liner has no obvious influence on the shape of the explosively formed projectile (EFP). However, the curvature radius of the liner has a significant influence on the shape of the EFP.When the liner material is copper and the curvature radius of the liner is greater than 8mm, the EFP shape approximates an ellipsoidal or hemispherical shape and the EFP forming speed is between 1900m/s and 2400m/s. When the material of the liner is steel or tungsten and the curvature radius of the liner is thicker than 8mm, the liner is not able to form projectiles in the shape of a sphere, ellipsoid, or long rod. By comparing the forming speed from 1#EFP to 4#EFP, it can be said that MEFP with integrated liner and shell design displays a certain pressurization effect. Research results show that, for small-caliber MEFP warheads, subject to the size of the warhead, when the liner is steel or tungsten, the detonation energy generated by the limited charge does not result in the liner forming an effective EFP. However, when the liner material is selected as copper, the EFP forming shape and speed are more appropriate.


Introduction
During the Spanish Civil War from 1936 to 1939, the High-Explosive Antitank (HEAT) with a shaped charge effect began to be used. As a type of shaped charge projectile, explosively formed projectiles (EFP) began to appear in the 1970s. Multiple explosively formed projectile (MEFP) began to appear in the 1980s to improve the hit rate and damage probability of the projectile [1]. MEFP, a highly effective damage warhead, was developed based on a single EFP warhead [2]. MEFP warheads can be divided into three types according to their charge structure: integral type, combined type, and cutting type. Among these, the integral-type MEFP has a simple structure that enables the formed projectile to obtain a higher penetration performance, which has become the research focus [3][4][5]. In terms of MEFP penetration into the target, Zhao et al. [6] studied the penetration of MEFP into a target plate after cross-mesh cutting EFPs using static explosion tests and numerical simulations. Xiang et al. [7,8] carried out numerical simulations and experimental verification of grooved MEFP penetrating steel target plates. Fong et al. [9] studied the elimination of threat posed by mines and improvised explosive devices using the MEFP warhead technology. In terms of the initiation mode of the MEFP, Zhao et al. [10] studied the forming characteristics of MEFP under three different initiation modes through numerical simulation and detonation wave action theory. Fan et al. [11] studied the influence of central initiation and three-, four-, and eight-point simultaneous initiation on the molding characteristics of MEFP using numerical simulation. Li et al. [12] and Song et al. [13] analyzed the impact of the detonation mode on the performance of the MEFP warhead by combining a static detonation test with numerical simulation. In terms of the liner parameters of MEFP, Zhang et al. [14] studied the influence of charge spacing, charge, and curvature radius on the forming characteristics of MEFP using numerical simulation. Liang et al. [15,16] studied the influence of the structural parameters of the liner and warhead on the forming characteristics of MEFP by simplifying the model and conducting numerical simulations. Yin et al. [17] analyzed the influence of liner parameters (radius of curvature, caliber, and thickness) on the shaping of MEFP using numerical simulation. In terms of the influence of liner material on MEFP forming characteristics, Zhao et al. [18] conducted a numerical simulation of the MEFP forming process with different materials such as copper, aluminum, and iron used as liners and concluded that the projectile speed and radial dispersion angle were reduced by 58% and 56%, respectively, with an increase in the material density of the liner. Yuan et al. [19] studied the forming characteristics of circumferential MEFPs with different materials and shell thicknesses using numerical simulation and concluded that when the thickness of the liner was larger than that of the shell, the MEFP speed increased rapidly, whereas when the thickness of the liner was smaller than that of the shell, the MEFP speed increased slowly. e research on the forming performance of the MEFP, according to the flying direction of the formed projectile, can be divided into two categories. In the first category, the liner is along the axial direction of the charge arrangement, and the formed projectile is scattered along the axial direction [6-11, 14, 18], that is, an axial MEFP. In the second type, the liners are along the charge circumferential arrangement, forming the projectile along the charging circumferential direction of flying [12,13,[15][16][17]19], that is, circumferential MEFP. In the literature [6][7][8][9][10][11][12][13][14][15][16][17][18][19], whether referring to axial or circumferential MEFP, there are three characteristics of structure: first, the separation design of the liner and shell is adopted; second, the liner is designed with equal wall thickness; third, the charge caliber of the MEFP warhead is between 40 mm and 205 mm. With the exception of the charge caliber of the MEFP warhead in [13] and [19], which are 48 mm and 40 mm, respectively, the MEFP warheads have medium to large calibers. To study the forming performance of the circumferential MEFP with the integrated design of the smallcaliber shell and liner, first, three different structural types of MEFP warheads were designed, wherein the thickness of the liner was smaller than, equal to, and larger than that of the shell, respectively. Second, by using the fluid-solid interaction method, a numerical simulation of the MEFP warhead with three different structure types was carried out, and the forming characteristics of the MEFP were analyzed when the material of the liner and shell was copper, steel, and tungsten, and the simulation results were compared and analyzed.
In the numerical simulation, firstly, the structured mesh of the designed circumferential MEFP warhead was generated using the ANSYS/ICEM (Integrated Computer Engineering and Manufacturing) meshing tool, and then the unstructured mesh generated by ANSYS/ICEM was generated and imported into HyperMesh for LS-DYNA preprocessing. Finally, it was submitted to the LS-DYNA solver for calculation.

Structural Design
To investigate the forming performance of the circumferential MEFP integrated design of the small-caliber shell and liner, the structural parameters of the three different types of MEFP warheads designed are listed in Table 1. e three-dimensional (3D) model designed by the structural parameters of the MEFP warhead is shown in Table 1, and the structural sketch of the liner is shown in Figure 1.
In the structure (Figure 1), each column is composed of four liners in the axial direction of the charge; each row is composed of 12 liners in the circumferential direction of the charge. erefore, the circumferential MEFP warhead structure designed in this study had a total of 48 liners, and the liner was centrosymmetric.

Finite Element Model.
At present, there are many commonly used finite element meshing software, such as HyperMesh, Finite Element Model Builder (FEMB), True Grid, and ANSYS/ICEM. In this study, first, ICEM was used for finite element structured mesh division, then the divided mesh file was imported into HyperMesh for preprocessing of finite element analysis, and finally, the numerical calculation was carried out using LS-DYNA. To reduce the amount of calculation in the numerical simulation, according to the symmetry of the MEFP warhead structure, a quarter finite element model was established, as shown in Figure 2.

Material Model.
e material model of the shell, explosive, and air is the same as that in Section 3.2 of [20], and the material parameters of the liner are shown in Table 2.
ρ is the density of the materials, G is the young's modulus of the materials, A is the yield stress parameters of the materials, B is strain hardening coefficient of the materials, n is the hardening exponent of the materials, C is the strain rate constant of the materials, m is the thermal softening exponent, T m is the melting point temperature of the materials, T room is the reference temperature (it is usually room temperature), Γ is the Gruneisen coefficient, c 0 is the volume velocities of the materials, and S is the impact adiabatic parameters. Single-point detonation was adopted in the numerical simulation, and the position of the detonation point is shown in Figure 2. e results are shown in Figure 3, where the material of the integral circumferential MEFP warhead shell and liner is copper.

Forming Characteristics of Liner and Shell Materials Are
Because the structure of the monolithic circumferential MEFP warhead designed in this study is center-symmetrical and the detonation point is on the symmetrical centerline, each row of the formed projectiles of the circumferential MEFP warhead has similar characteristics. erefore, a column of formed projectiles was selected as the research object and labeled as 1#EFP, 2#EFP, 3#EFP, and 4#EFP, respectively. Figure 3 shows that when the liner and shell material is copper, the slender forming shape of the projectile gradually flattened with an increase in the liner curvature radius. Due to the single-point initiation and the selection of initiation position, the pressure of detonation waves front increased gradually with the detonation process, the different positions of detonation wave front acting on the liner lead to the shape difference in 1#EFP, 2#EFP, 3#EFP, and 4#EFP, and the direction of the 2#EFP, 3#EFP, and 4#EFP projectile heads deflects to the direction of detonation wave front propagation. When the radius of curvature of the liner is 8 mm, the shape approximates a rod, and when the radius of curvature of the liner is between 9 mm and 11 mm, the shape is similar to an ellipsoid.     e forming speed is shown in Table 3. From Table 3, it can be observed that, with the increase in the liner curvature radius, the forming speed contrast diagram of the MEFP warhead at t � Δt, t > Δt, and t < Δt is shown in Figure 4.
As shown in Figure 4, when t > Δt and t � Δt, the velocity of 1#EFP decreases from 2077 m/s and 2097 m/s to 2042 m/s and 2072 m/s, respectively, and then increases to 2085 m/s and 2135 m/s, respectively, resulting in an increase of 1.68% and 1.68% and decrease of 2.1% and 1.68%, respectively. e velocity of 2#EFP decreased from 2205 m/s and 2197 m/s to 2134 m/s and 2131 m/s, respectively, and then increases to 2186 m/s and 2141 m/s, resulting in an increase of 3.21% and 3% and decrease of 2.44% and 0.47%, respectively. e velocity of 3#EFP decreases from 2358 m/s and 2334 m/s to 2358 m/s and 2164 m/s, respectively, and then increases to 2189 m/s and 2175 m/s, respectively, resulting in an increase of 7.38% and 7.28% and decrease of 0.23% and 0.51%, respectively. e velocity of 4#EFP decreases from 2273 m/s and 2257 m/s to 2138 m/s and 2130 m/ s, respectively, and then increases to 2138 m/s and 2130 m/s, respectively, resulting in an increase of 5.94% and 5.63% and decrease of 1.82% and 0.05%, respectively. When t < Δt, the velocity of 1#EFP decreases from 2067 m/s to 1997 m/s and then increases to 2034 m/s, resulting in an increase of 3.39% and decrease of 1.85%, respectively, and the velocity of 1#EFP-4#EFP decreases from 2165 m/s, 2329 m/s, and 2241 m/s to 2165 m/s, 2329 m/s, and 2241 m/s, respectively, resulting in decreases of 3.14%, 8.93%, and 7.1%, respectively. It can be seen from the above data that when t > Δt and t � Δt, the forming speed first decreases and then increases with the increase in the liner curvature radius, the speed reduction range is between 1.5% and 8%, and the speed increase reduction range is between 0.05% and 2%. When t < Δt, the forming speed gradually decreases with the increase in the liner curvature radius, and the reduction is between 3% and 9%.
From Table 3, the speed change curve of 1#EFP to 4#EFP can be obtained, as shown in Figure 5.
As shown in Figure 5, when the liner curvature radius with three different structures is less than 11 mm, with the detonation process of the main charge and the pressurization effect of the integrated design of the liner and the shell on the detonation energy, the speed of 1#EFP to 3#EFP gradually ascends. With the detonation, the material of the shell breaks, and the pressurization effect of the integrated design of the liner and shell on the detonation energy of the explosive disappears; therefore, the speed from 3#EFP to 4#EFP reduces. In Figure 5   1#EFP to 3#EFP is between 4% and 12.5%, the increase gradually decreases with the curvature radius of the liner increases, the speed reduction range of 3#EFP to 4#EFP is between 0.5% and 4%, and the reduction rate gradually decreases.
When the shell thickness of the integral MEFP warhead remains unchanged and the curvature radius of the liner remains unchanged, only the thickness at the center changes for three different types of MEFP warheads. erefore, the influence of thickness change at the center on the change of forming speed can be obtained from the forming speed list shown in Table 3, as shown in Figure 6.
As shown in Figure 6, when the thickness at the center of the liner increases from 1.1 mm to 1.3 mm, the speed of the projectile gradually decreases. In Figure 6, when the curvature radius is different, the velocity of m/s, respectively, resulting in a decrease of 1.4%, 1.7%, 2.2%, and 4.3%, respectively. It can be seen from the above data that the speed of the formed projectile decreases by 1%-5% with the increase of the thickness at the center of the liner. e resulting forming result is shown in Figure 7.

Forming Characteristics of Liner and Shell Materials Are
In Figure 7, when the material of the liner and the shell is steel, the shape of the EFP starts to deteriorate with the increase in the liner curvature radius, and even the shape of a sphere, an ellipsoid, and a long rod cannot be formed. When the liner curvature radius is 8 mm, 1#EFP, 2#EFP, 3#EFP, and 4#EFP can form an approximate ellipsoidal projectile. When the liner curvature radius is 9 mm, 3#EFP and 4#EFP   will be able to form an ellipsoid projectile; when the liner curvature radius is greater than 9 mm, 1#EFP, 2#EFP, 3#EFP, and 4#EFP will be unable to form projectiles in the shape of spheres, ellipsoids, or long rods. When the shell thickness of the MEFP warhead and the liner curvature radius is constant, the shape does not change significantly with an increase in the thickness at the center. In addition, the forming results of the three structure types are similar.

Forming Speed.
e forming speed is shown in Table 4.
From Table 4, it can be obtained that, with the increase of the liner curvature radius, the forming velocity contrast diagram of the MEFP warhead at t � Δt, t > Δt, and t < Δt is shown in Figure 8.
In Figure 8, 2399 m/s, respectively, increasing by 6%, 5.5%, and 5.8%, respectively. It can be seen from the above data that when the material of the liner and shell is steel, the speed increase range of the formed projectile is between 2% and 6.5% with the increases of the liner curvature radius.
From Table 4, the speed change curve from 1#EFP to 4#EFP can be obtained, as shown in Figure 9.
When the shell thickness of the integral MEFP warhead remains unchanged and the liner curvature radius remains unchanged, only the thickness at the center changes for three different types of MEFP warheads. erefore, the influence of thickness change at the center on the change of forming speed can be obtained from the forming speed list shown in Table 4, as shown in Figure 10.
As shown in Figure 10

Forming
Results. e material in the finite element model of the MEFP warhead established in the third part was chosen as tungsten, and the remaining boundary conditions of the numerical simulation were set to be the same as in Section 4.1. e forming result is shown in Figure 11.
As shown in Figure 11, when the liner and shell material is tungsten and the liner curvature radius is greater than 8 mm, the projectile with a spherical shape, ellipsoid shape, and long rod shape cannot be formed. It is shown in Figures 11(a)-11(c) that the forming results of the three types of MEFP warheads are similar.

Forming Speed.
e forming speed is shown in Table 5.
From Table 5, it can be observed that, with the increase in the liner curvature radius, the formed speed contrast diagram of the MEFP warhead at t � Δt, t > Δt, and t < Δt is shown in Figure 12.
As shown in Figure 12, the velocity of 1#EFP increases from 1569 m/s, 1541 m/s, and 1511 m/s to 1622 m/s, 1597 m/ s, and 1571 m/s, respectively, resulting in an increase of 3.4%, 2.5%, and 4%, respectively. e velocity of 2#EFP increases from 1733 m/s, 1694 m/s, and 1658 m/s to 1833 m/s, 1806 m/ s, and 1768 m/s, respectively, resulting in an increase of 5.8%, 6.6%, and 6.6%, respectively. e velocity of 3#EFP increases from 1729 m/s, 1682 m/s, and 1660 m/s to 1850 m/ s, 1813 m/s, and 1780 m/s, respectively, resulting in an increase of 7%, 7.8%, and 7.2% respectively. e velocity of 4#EFP increases from 1612 m/s, 1585 m/s, and 1559 m/s to 1690 m/s, 1669 m/s, and 1644 m/s, respectively, resulting in an increase of 4.8%, 5.3%, and 5.5%, respectively. It can be seen from the above data that when the material of the liner and shell is steel, the speed increase range of the formed projectile is between 3% and 8% with the increases of the liner curvature radius.
From Table 5, the speed change curve from 1#EFP to 4#EFP can be obtained, as shown in Figure 13.
When the shell thickness of the integral MEFP warhead remains unchanged and the liner curvature radius remains unchanged, only the thickness at the center changes for three different types of MEFP warheads. erefore, the influence of thickness change at the center on the change of forming speed can be obtained from the forming speed list shown in Table 5, as shown in Figure 14.
As shown in Figure 14, when the thickness at the center of the liner increases from 1.1 mm to 1.3 mm, the velocity of the projectile gradually decreases. In Figure 14 m/s, respectively, resulting in a decrease of 3.3%, 3.1%, 2.9%, and 2.7%, respectively. It can be seen from the above data that the speed of the formed projectile decreases by 2%-5% with the ascent of the thickness at the center of the liner. Tables 3-5, it can be observed that, with the increase in the liner radius of curvature, the contrast diagram of the forming speed of the MEFP warhead with different materials of the liner is shown in Figure 15 when the liner curvature radius increases.

e Speed Comparison of Different Materials. From
In Figure 15, when the material is copper, steel, or tungsten in the liner, the forming speed of EFP is v steel > v copper > v tungsten . When the liner material is copper, the forming projectile speed is between 1900 m/s and 2400 m/s, and the formed projectile speed reduces with an increase in the radius of curvature; when the liner material is steel, the forming projectile speed is between 2250 m/s and 2550 m/s, and the speed of formed projectile increases with an increase in the radius of curvature; when the liner material is tungsten, the speed of the formed projectile is between 1500 m/s and 1850 m/s, and the speed of formed projectiles increases with an increase in the radius of curvature.
From Tables 3-5, we can obtain the speed curve of 1#EFP to 4#EFP when the liner is a different material, as shown in Figure 16.
In Figure 16, the speed variation law of 1#EFP to 4#EFP first increases and then decreases. Moreover, when the liner is of a different material, the speed of 4#EFP is greater than Shock and Vibration that of 1#EFP but less than that of 2#EFP and 3#EFP. is indicates that, in addition to the complete detonation of the main charge, the structure of the integrated design of the liner and shell has a certain pressurization effect. Tables 3-5, we can determine the influence of the thickness change at the center on the forming speed change when the liner is a different material, as shown in Figure 17.          1550  1600  1650  1700  1750  1800  1850  1900  1950  2000  2050  2100  2150  2200  2250  2300  2350    18 Shock and Vibration

From the forming speed list shown in
In Figure 17, when the material of the liner is copper, steel, or tungsten, the EFP forming speed gradually decreases with an increase in the liner center thickness.

Conclusion
To study the influence of different materials on the forming characteristics of different structural types of circumferential MEFP warheads, three structural types of circumferential MEFP warheads were designed: the warheads in which the thickness of the liner was smaller than that of the shell, those in which the thickness of the liner and the shell was equal, and those in which thickness of the liner was greater than that of the shell. ree materials, copper, steel, and tungsten, were used to simulate the designed circumferential MEFP warhead. e results show that when the material of the liner and shell is copper, the forming effect of EFP worsens with an increase in the curvature radius of the liners. e forming results of the three types of circumferential MEFP warheads are similar, and the forming effect is appropriate when the radius of curvature of the liner is between 9 and 10 mm; when the material of the liner and shell is steel and tungsten, the liner will not be able to form projectiles with spherical, ellipsoidal, or long rod shapes with an increase in the curvature radius (when the diameter is greater than or equal to 8 mm), and the forming results of the three types of circumferential MEFP warheads are similar. For the small-caliber MEFP warhead, if the material of the liner and the shell is selected as copper, the liner curvature radius is between 9 mm and 10 mm; if the material of the liner and the shell is selected as steel and the liner curvature radius is greater than 8 mm, the forming speed of the projectile is large, but the liner cannot form a better-shaped projectile. If the liner and the shell material are selected as tungsten, and the liner curvature radius is greater than 8 mm, the forming speed of the projectiles is slow, and the liners cannot form a better-shaped projectile.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.