Dynamic Multiprojectile Attack and Killing Effects of Detonation Warheads

Dynamic spatial distribution and killing characteristics of warhead fragments are important topics in the ﬁeld of weapon ef-fectiveness and protection. However, there is little research on the fragment distribution formed by continuous dynamic attacks of multiple projectiles that explode above the ground. This study analyzes spatial distributions of warhead fragments using witness boards in a rectangular target test. The results show that the fragment distribution of multiple projectiles in continuous dynamic attacks demonstrates a spatial superposition characteristic. The superimposed distribution is the sum of the distributions of two independent fragment distributions. The distribution characteristics are consistent with fragment scattering behavior. Therefore, they can be used to analyze the killing eﬀects of multiple projectiles conveniently. The eﬀects of falling speed, falling angle, and explosion height on the damage range of fragments were explored by a fragment spatial distribution model obtained from experiments. Analysis indicates that the prefabricated fragment distribution conformed to a spatial superposition relationship under dynamic multiprojectile continuous attacks, and the superposition obeyed fragment scattering characteristics. As the projectile falling angle increased at the explosion center, or as the falling height decreased, the positive pressure duration increased gradually. The falling speed had the greatest impact on the speciﬁc impulse of overpressure. The falling angle had the greatest impact on the peak value of overpressure. Both the falling angle and the explosion height had the greatest impact on the positive pressure acting time.


Introduction
Detonation warheads are the most common type of warheads. When a warhead explodes, it disperses many highspeed fragments radially, causing damage to living targets and the environment, including structures and machinery. Warhead parameters and the combined damage capability of multiple projectiles determine damage efficiency. Many researchers have been interested in this topic for decades and have developed many models to study the maximum warhead fragment speed [1][2][3]. e Taylor formula [4] and the Shapiro formula are two commonly used models to solve axial fragment speed and fragment spatial dispersion.
An et al. [5] and Felix et al. [6] studied fragment axialspeed distribution characteristics for various explosions. König [7] and Huang et al. [8] studied a formula for the fragment dispersion angle of cylindrical warheads with end effects by experimental testing and numerical simulation. Jiang et al. [9] studied the dispersion distribution of fragments under the charge of cylindrical shell during central initiation at one end. Wang et al. [10] studied a calculation method of full-scale fragment distribution based on the CDEM method. Fu et al. [11] determined a fragment flying trajectory with a simulation model of shooting tracks based on LS-DYNA calculation results. Guo et al. [12] used the LS-DYNA software to simulate warhead fragment distribution and investigate the effects of falling angle, explosion height, and falling speed on fragment distribution. Wang and Li [13] investigated an engineering calculation method for estimating the damage probability of fragments against ground targets. Li et al. [14] from Nanjing University of Science and Technology analyzed the damage power of forward-killing grenades against personnel. Jiang et al. [15] used simulation to analyze the distribution of shock wave overpressure during an explosion.
However, few studies exist on the dynamic killing effects and fragment spatial distributions of multiple projectiles under continuous attacks. erefore, it is necessary to investigate killing effects under continuous dynamic attacks of multiple projectiles.
Based on the theory of rectangular target test, this paper studies the dynamic spatial distribution relationship of warhead.
rough the fan-shaped target test under the dynamic attack of multiple bombs, the dynamic distribution and spatial superposition relationship under the flying of multiple bombs are analyzed. e test is verified with the theory, and the verification results are integrated and programmed. e influence of different falling velocity, angle of fall, and blast height on the ground damage area was studied by self-programming method, and the relationship between the overpressure distribution and falling velocity angle and blast height of the killing projectile under dynamic condition was analyzed, which provided data support and theoretical basis for the establishment of the corresponding macro damage analysis model.

Analysis of Dynamic Fragment Distribution.
Before the spatial distribution of prefabricated fragments formed by multiple projectiles is calculated, the spatial fragment distribution formed by a single projectile can be first analyzed via a probability estimate on the number of fragments. When the prefabricated-fragment warhead explodes, each fragment is projected normal to the surface of the warhead. Taylor presented the basic idea to predict static fragment distribution characteristics, and Shapiro applied it. Shapiro assumed that the warhead consisted of a series of rings with their centers located on the symmetric axis of the projectile. e detonation wave starts from the explosion center and propagates outwards in the form of a spherical wavefront. e fragment dispersion angle is the main research focus in such an approach. e angle ϕ 1 is formed between the normal line of the prefabricated fragment and the symmetric axis of the projectile body and ϕ 1 � π/2 for the spherical prefabricated fragment with cylindrical charge. e angle ϕ 2 is formed between the normal line of the detonation wave and the symmetric axis of the projectile body. e deviation angle θ of the fragment speed vector relative to the normal line of the shell is given by the Shapiro formula as follows: After the warhead explodes, the fragment dispersion angle and flying speed are affected by the projectile body's tractive speed. Assuming the tractive speed along the projectile axis direction as V c , the fragment speed after the static explosion as V 0 , and the dispersion angle as φ 0 , the following equations can calculate the dynamic fragment dispersion angle φ and the initial dynamic speed V with the consideration of tractive speed V c : Assuming the number of fragments in the spherical zone from the spatial angle θs to θs + dθs as dN and the number of fragments generated by the entire projectile as N0, then the quantity probability density of fragments in the space is given by e unit spherical solid angle is given by dΩ � sin θ s · dθ s · dΨ, where dΨ is the unit angle in the circumferential direction of the projectile. e quantity density of the unit spherical solid angle of the fragment is given by Moreover, Previous experimental results show that the fragment spatial distribution curve is similar to the normal distribution curve given by where θ s is the fragment space angle at the radius R and θ 0 is the mathematical expectation of the space angle θ s with a value usually close to π/2. e parameter s is the mean square deviation of the space angle θ s , and (5) is a function related to θ s . erefore, the number of fragments in a flying region (θ i ∼ θ i+1 ) can be determined by the following formula: Assuming that the relative coordinates of the centroid of multiple projectiles are (Δx, Δy, Δz), and the projectile axis direction is y, the Cartesian coordinate system can be transformed into a polar coordinate system by using When the relative polar coordinate distance of the mass center of multiple projectiles along the projectile axis direction is greater than the covering range of the dispersion angle of a single projectile, the multiple-projectile fragment distribution cannot be obtained by superposition, and the spatial density distribution of multiple-projectile fragments is given by 2 Shock and Vibration When the relative polar coordinate distance of the mass center of multiple projectiles along the projectile axis direction is less than the covering range of the dispersion angle of a single projectile, the multiple-projectile fragment distribution can be superimposed within a certain range of the dispersion angle. e polar coordinate system of the first projectile is assumed as a reference system. e spatial density distribution of multiple-projectile fragments can be obtained as follows:

Analysis of Explosion Shock Wave Overpressure.
Assuming that a spherical or similar shaped charge of TNT explodes in an infinite air medium, and the resulting spherical shock wave is not influenced by other interfaces, the peak value of overpressure ΔP m at any distance R from the explosion center can be calculated by the following formulae: where R is the relative distance, determined by where R is the distance to the explosion center (m) and m ω is TNT explosive mass (kg). e explosive mass can be converted to a TNT equivalent in the above formula for other explosives. e above calculation assumes an explosion condition in an infinite air medium. It is generally believed that when the explosion height H satisfies the following relationship: Because the warhead is simplified as a cylindrical symmetric shell charge, the equivalent explosive charge mass can be expressed as where m is charge mass (kg), m ω is the equivalent explosive of detonation products (kg), α is the loading coefficient, r 0 is the charge radius (m), and r m the radius where the fragment reaches its maximum speed (m). r m of the steel shell is usually 1.5r0 to 2.1r0, according to test data. According to the similarity law for an explosion, when a warhead explodes in air, e magnitude of a specific impulse directly determines the degree of shock wave damage. eoretically, the specific impulse is determined by a time integral of the air shock wavefront overpressure:

Experiment Design
In order to study the fragment dynamic spatial distribution under multiple-projectile continuous attacks and verify the spatial distribution formulae in Section 2, an experimental test using rectangular targets under continuous dynamic attacks were designed. As shown in Figures 1 and 2, two rectangular targets were arranged on both sides of the explosion center, respectively, and one rectangular target was arranged in front of the explosion center with the target board thickness 3 mm made of Q235 steel. e radial distance between the rectangular side target and the explosion center was 10 m, and each side target was located along the 45°direction from the ballistic trajectory line. e left-side rectangular target had a size of 6 × 3 m 2 . e width of a single component of the target is 1 m, with a height of 3 m and a recovery angle Δφ � 65°. e right-side rectangular target had a size of 5 × 3 m 2 and a recovery angle Δφ � 49°. e front rectangular target was 13 × 3 m 2 and 30 m away from the explosion center. A Model 105 cannon launched the warhead specimen with flat firing. e warhead was detonated with a center fuse installed at one end. e firing Shock and Vibration 3 distance from the cannon to the explosion center was 200 m. e explosion height was 1.5 m above the ground. Four measurement sensors for shock wave overpressure signals were arranged along the line of the firing direction, and four more were arranged perpendicular to the firing direction its perpendicular line. e sensors were placed at 2 m, 3 m, 4 m, and 5 m away from the explosion center along both axes. e angle between the two columns of the sensors was 90°. e sensor sampling frequency was 1 MHz, the pressure sensing range was 1.5 MPa, and the triggering method was internally triggered. Figure 3 shows the layout of the sensors. e distances of the sensors are shown in Table 1. Moreover, a high-speed camera was placed 50 m away from the explosion center. e high-speed camera system was a Fastcamnltima APX high-speed camera produced by the Photron Company. e photo shooting rate was set at 24,000 frames per second during the test to capture the instantaneous fragment speed when the fragments penetrated the target. e fragment trajectory, the initial speed, and the flight attitude of the warhead at the explosion center were also traced. e overpressure data, the initial warhead speed and attitude, and the target perforation status under multiple-projectile continuous attacks were obtained during the test.
In order to identify the number of holes on the rectangular targets during the process of prefabricated fragment penetration and find out the corresponding fragment spatial distribution, the Q235 steel target was painted before the test. e holes on the targets made by prefabricated fragments and natural fragments could be easily distinguished. e warhead used in the test was a 105 mm grenade, as shown in Figure 4. e charge diameter was 105 mm, and the charge length was 280 mm. e charge was a polyblack aluminum-2 press-fit charge with a density of 1.71 g/cm3, detonation pressure of 29.5 GPa, and detonation velocity of 8425 m/s. e charge mass was 1.672 kg. e shell material was 58SiMnVB with 12.18 kg mass. e material properties of the warhead are shown in Table 2. e shell wall thickness was 20 mm. A tungsten ball cap with a diameter of 3.3 mm and mass of 1.56 kg, made by injection molding, was installed at one end of the shell. A thread connection fixed the shell of the tungsten ball cap and the hood. A dynamic explosion test was repeated three times.

Analysis of Warhead Attitude at
Explosion. e warhead flying attitude at the explosion and the evolution of the shell fragment flight trajectories recorded by high-speed photography are shown in Figure 5. e test results show that there was obvious fire after the charge exploded. In time, the fire grew stronger (larger and brighter) and then faded away. e initial moment of the fire was set as 0 ms. e duration of the fire was 131 ms. Each high-speed photograph in Figure 5 was adjusted with the same scale. e size of the Q235 target in the background was used as a reference to measure the fragment speed and the flying direction at different moments. e initial warhead speed, fragment speed, and flight trajectory were calculated using the time markers on the photos after the explosion and the target penetration events filmed by the high-speed camera. e warhead flight direction at the explosion was parallel to the ground and was 45°away from the target board. At the explosion center, the initial warhead speeds were 540 m/s, 563 m/s, and 523 m/s for the three projectiles. e fireball and the fragments began to separate at 1.394 ms after the explosion, and the fragments moved faster than the fireball. e fragment data within the angle is collected by the witness plate. It can be found that after the explosion drive, the bright spot caused by fragment breakdown appeared in the witness plate at 30.8 ms, and then gradually reached 38.4 ms. Fractures spread and increased from the middle to both sides of the witness plate. It can be seen that the flying velocity of fragments at the warhead first increased and then decreased from the initiation end to the detonation center to the tail end. To calculate the fragment motion, it was assumed that the trajectory was a straight line. e influences of air lift and fragment gravitational force was ignored. e aerodynamic resistance effect was considered in the calculation. e initial fragment speed was calculated using (2) and (3) based on the measured speeds at the measurement locations. In the initial speed calculation, the camera shooting errors were removed to prevent using excessively low fragment speeds. e fragment speeds measured by the experiments of three specimens were compared. e spatial distributions of the fragment speed and dispersion angle under dynamic explosion conditions were obtained. e distribution of the fragment axial perforation speed within the angular range covered by the target boards was analyzed, as shown in Figure 6.

Analysis of Rectangular Target Perforation.
After the experiments of continuous attacks, the fragment perforation conditions of the left Q235 steel target were analyzed. e images were segmented to extract hole shape information based on the abrupt change of the pixel grayscale value at the edge of the hole area on the witness board. e Image-Pro Plus and ImageJ image analysis software packages were used to distinguish the perforation hole contours on the witness boards. e holes made by natural fragment perforation and prefabricated fragment perforation were classified and counted. Figure 7 shows the fragment distribution on the left witness board after the tests of three projectiles. e grayscale bitmaps of fragment perforation were obtained by Image-Pro processing of the target boards. e holes made by prefabricated fragment perforation were counted according to the pre-divided angular regions on the witness boards for target-shooting conditions under continuous dynamic attacks with three projectiles.
Due to the differences in the explosion position and initial launch velocity of continuous dynamic attacks and the fact that the witness boards were placed at only 10 m away from the explosion center, there were not many fragments hitting the target boards. e test results of the fragments hitting the witness boards with three projectiles are shown in Figure 8. e spatial distribution of fragment quantity on the witness board was consistent with the normal distribution curve of the static fragment dispersion. e fragments are concentrated in regions 9-12 on the witness board. Many fragments hit the witness board in the overlapping area within regions 6-14 under multiple-projectile explosion conditions. is area is the superimposed spatial area created by continuous dynamic attacks of multiple projectiles.

Analysis on Damage Distribution under Multiple-Projectile Continuous Attacks.
e fragment spatial distribution characteristics in Figure 8 were analyzed using (7) in Section 1 to calculate the dynamic multiple-projectile fragment distribution. e fragments were mainly concentrated in the angular range of 23°-31°on the Q235 target located 10 m away from the explosion center. e distribution illustrates the fragment dispersion characteristics of the prefabricated fragment warheads. e explosion centers of multiple projectiles had an interval of 0.91 m. e detonation direction was the same for all the projectiles. e fragments created by the multiple projectiles were mainly concentrated in the angular range of 12°-51°at 10 m away from the explosion center with a superimposed spatial area located at 19°-36°. ese results are consistent with the experimental data of the rectangular targets shown in Figure 8. erefore, it can be concluded that the spatial distribution of multiple dynamic projectiles is the superposed result of the spatial distributions of every single projectile, and the superposition characteristics satisfy the normal distribution, meeting the requirement of (11). ese data help analyze the effects of continuous attacks of multiple projectiles.   Table 3 presents the results of three projectiles obtained by the high-speed camera, as shown in Figure 5, including the warhead attitude at the explosion and initial falling angle. e dispersion angle and speed data of the natural fragments and the prefabricated fragments were integrated by computer programming to obtain the ground fragment distribution maps of three projectiles, as shown in Figure 9. en, considering different locations, falling speeds, and falling attitudes of multiple warheads, the falling locations were calculated to obtain the combined fragment distribution of the multiple projectiles, as shown in Figure 10.

Analysis of Dynamic Shock Wave Field Created by
Explosion. In the explosion-in-air test, the ambient pressure of the pressure sensor is the initial air pressure p0 before the air shock wave generated by the explosion reaches the freefield pressure sensor. e overpressure of the air shock wave is defined as Δp � p − p0. When the air shock wave reaches the sensor, the air pressure rapidly increases to p, and then the overpressure slowly decays to the initial air pressure. e pressure measured by the sensor in this test is the overpressure of the air shock wave, Δp. When the pressure is measured at the distance R, the time-dependent Δp(t) curve of the air shock wave overpressure can be obtained. e shock wave overpressure curves measured in the warhead explosion test are shown in Figure 11. It is observed that the pressure curves have the same trend with respect to time. As the shock wave propagates, the parameters, including air pressure, decrease rapidly since the energy per unit area on the shock wave front decreases rapidly as the shock wavefront expands with an increase in the propagation distance. Figure 11(b) shows that the initial oscillating shock wave is caused by the expanded and broken shell. e shock wave overpressure curve obtained in the test rapidly decays in the initial stage and then slowly decays. Shock waves with 8 Shock and Vibration smaller amplitudes are found immediately after the positive pressure zone or the negative pressure zone. ese smaller waves are believed to be secondary shock waves. Peak and valley pressures and propagation speed can be used to characterize the air shock wave caused by warhead explosion. In addition, wave arrival time, peak pressure, and shock wave positive pressure acting time can quantify the intensity of the instantaneous energy release of the explosive. e positive pressure acting time t+ is a characteristic parameter of the explosion air shock wave, and it is an important parameter indicating the target damage effect. When the air shock wave reaches the pressure sensor, the air pressure suddenly rises to a certain peak value, called the overpressure peak of the air shock wave. en, the air pressure slowly decays to the ambient pressure with time t+. erefore, the time duration when the air pressure is greater than the ambient pressure is the positive pressure acting time t+. e overpressure peak Δp, the positive pressure acting time t+, and the specific impulse i at different locations were measured in the warhead explosion test, as shown in Table 4. Figure 12 also shows the overpressure peak, the positive pressure acting time, and the specific impulse measured at different locations. In general, these parameters at different explosion heights and falling angles demonstrate certain trends. For the overpressure signals measured at distances less than 4 m away from the explosion center, the overpressure peak of the second projectile decayed faster than those of the other two projectiles. For the overpressure signals measured beyond 4 m, the overpressure peaks of the three projectiles decayed at similar rates. At a distance of 2 m away from the explosion center, the secondary shock wave appeared behind the negative pressure zone. At a distance of 5 m away from the explosion center, the secondary shock wave appeared behind the positive pressure zone (Y1, Y2), indicating that the arrival time of the secondary shock wave was related to the overpressure amplitude and the distance from the explosion center. At distances less than 3.5 m away     from the explosion center, the specific impulse of the shock wave of the second projectile decayed faster than those of the other two projectiles. Beyond 3.5 m, the specific impulses of the three projectiles decayed at similar rates. ese phenomena are mainly due to different explosion locations of different projectiles. As the falling angle at the explosion center increased, or as the falling height decreased, the positive pressure acting time gradually increased from 2 m to 5 m, with an increased amplitude of 57.2%.
At the distances of 2 m to 4 m (i.e., in the near field), the air shock wave's specific impulse and overpressure peak decreased significantly due to the influence of falling speed.
e test result showed that the falling speed had the greatest impact on the specific impulse of overpressure; the falling angle had the greatest impact on the peak value of overpressure; both the falling angle and the explosion height had the greatest impact on the positive pressure acting time.
us, properly increasing the falling angle, falling height, and falling speed of the projectile may lead to an optimal damage effect in terms of increased overpressure peak and specific impulse.
is topic is worth exploring in future research on dynamic damage effects.

Results and Discussion
is paper explores the performance and damage effects of continuous dynamic attacks of multiple grenade warheads, including dynamic warhead speed, fragment dispersion distribution, and dynamic overpressure distribution. Conclusions are drawn as follows: (1) e warhead fragments mainly concentrated in the angular range of 23°-31°, demonstrating a dispersion characteristic of the prefabricated fragment warhead. When the explosion centers were 0.91 m apart from each other, and the detonation directions of the multiple projectiles were the same, the fragments mainly concentrated in the angular range of 12°-51°, with a superimposed spatial area located in the range of 19°-36°. ese findings were consistent with the test data on the rectangular target boards. It is concluded that the dynamic fragment spatial distribution of multiple projectiles is the superposition of the spatial distributions of every single projectile, and the superimposition obeys the normal distribution. us, the experimental data conforms to the theoretical model of fragment distribution of multiple projectiles under continuous attacks. (2) e secondary shock wave appeared behind the negative pressure zone for the distances in the near field away from the explosion center. When the distance increased to the far-field, the secondary shock wave appeared behind the positive pressure zone. e arrival time of the secondary shock wave was related to the overpressure amplitude and the distance. As the falling angle increased at the explosion center, or as the falling height decreased, the positive pressure acting time gradually increased. (3) e falling speed had the greatest impact on the specific impulse of overpressure. e falling angle had the greatest impact on the peak value of overpressure. Both the falling angle and explosion height had the greatest impact on the positive pressure acting time. erefore, properly increasing the warhead falling angle, falling height, and falling speed can result in an optimal damage effect due to increased overpressure peak and specific impulse.

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