Determination of thickness parameters of detonation shell and safety performance analysis

On the premise of security, it is necessary to ensure good operational performance and technical indicators for nonlethal weapons such as flash detonation bombs. To make the weight of a flash detonation grenade more convenient for hand throwing and material costs save, the outer dimensions of the grenade body are fixed, and the optimal parameters of outer shell thickness of the vertigo grenade are explored. Based on the maximum deformation energy theorem, the strength of the shell with the obtained maximum thickness of 8 mm is checked, and then the optimal shell thickness under the ideal air explosion condition is obtained by reducing the thickness; Then check the thread strength to obtain the inner casing fracture, shock wave, and sound pressure level. The fragmentation performance test, detonation test, and sound pressure level test were carried out for the projectiles involving the shell thickness of 4 and 5 mm, and the simulation results were verified from three aspects of sound intensity, shell strength, and fragments. The results show that the maximum equivalent stress is 232.6 MPa and the maximum deformation is 1.07 × 10−4 mm when the outer shell thickness of the vertigo grenade is 5 mm, which is less than the material failure strength standard and no plastic deformation could occur. The structural connection strength of the missile body meets the standard. Most of the fragments are trapped in the cavity. The average sound pressure level at 1.5 m can reach 159.7 dB. The relative error between the sound pressure level at 3 m and the live ammunition test results is 4.8%, which meets the expected operational technical indicators and has good safety performance. The results of the numerical simulation can provide a reference for the design optimization and performance evaluation of similar vertigo grenades.


INTRODUCTION
With an increasingly prominent role of nonlethal weapons in antiterrorism, antiriot, and other social security incidents, the development of nonlethal weapons has become an important choice to solve local wars. 1 Baines and Defence 2 defined flash bombs in nonlethal weapons as combined effective devices, sensory effects, and combination effects from chemical-mechanical blunt shocks, and all underlying intended and unintended effects should be assessed. For a projectile of bombs near people without safety evaluation and design optimization, the explosion pressure wave, the fall, and the secondary projectile propelled by the explosion may cause serious damage to human tissue. 3 Scolan et al. 4 studied the people injured by a rubber stun grenade commonly used in France, which confirmed that the sound pressure of the weapon is 165 dB and the rubber projectile is easy to cause serious damage to the human head and other parts within 5 m. For this reason, the safety of nonlethal bombs has already been one of the focuses of domestic and foreign scholars. Ma and Zhao 5 conducted an explosion simulation analysis of a small high-light stun grenade with a single-layer oval-shaped projectile structure made of ABS material, the results showed that the largest fragment mass is 2.15 g with the maximum initial velocity V 0 = 97.6 m/s, and a large weight of low-speed fragmentation is generated in the middle with the killing radius of 2.33 m. Therefore, the stun grenades of the traditional single outer shell structure, which through the optimized design still have the risk of destroying the target or killing the personnel. In nonlethal weapons with strong light, the safety stun grenades involving a double-layer cavity structure combined with an open-hole outer shell and an inner shell have the characteristics of nonlethal fragments when exploding compared with traditional nonlethal weapons, and the sound pressure level of the explosion can meet the requirements of combat technical indicators through structural performance optimization, and the materials are easy to obtain as well as low cost. 6 As a temporary replacement for the MK-141 flash bomb, the American NICO BTV-1 hand-throwing flash bomb uses a metal fuselage with openings at the bottom to reduce the risk of explosion damage. 7 The research has shown that the bomb structure is one of the important factors affecting the safety of the ammunition and the parameters include the material of the bomb, the thickness of the bomb wall, and the strength of the connection between bomb components. 8 The outer shell is not only the carrier of the inner shell, but also the main component to prevent the inner shell fragments from scattering and hurting people. Therefore, when designing the outer shell, the outer shell must have good capabilities of impact attenuation and protection, as well as considering the economic role, the usage amount of material, and the reuse of the outer shell with good strength. 9 She et al. 10 used the finite element method to analyze the three-dimensional stress concentration of the finite-thick central elliptical orifice plate, and obtained the approximate relationship between the three-dimensional stress concentration factor and the plane solution, indicating that the thickness has a significant effect on the stress concentration and fatigue strength. Only Ma and Liu 6 chose the oblique opening double-layer cavity structure, and used the LS-DYNA software to analyze the influence of the thickness of the inner shell material, the number and the diameter of the nozzles on the sound pressure effect as well as optimizing the design scheme. However, no analysis, check or optimization of the force of the shell and thread was conducted as well as the analysis of its safety, thus the current research on this aspect is very insufficient. Therefore, this article reduces the wall thickness of the side opening detonation bomb based on the maximum shell thickness of the maximum design distance of the outer diameter of the inner and outer shells. Based on the maximum distortion energy theorem, LS-DYNA software is used to evaluate the mechanical properties of structures with shells involving baseline thickness and optimum thickness under the effect of an explosion. With the allowable stress value without plastic deformation as the standard, the minimum wall thickness is determined, and the structural strength of the body threads as well as upper and lower covers are checked. The strength calculation and verification method of the pressured vessel are used to check the selection of wall thicknesses. Finally, the fragmentation, shell strength and sound pressure level are tested and analyzed through live ammunition tests. Compared with the simulation results, the safety index and tactical index are checked, and the optimal shell thickness of this kind of detonation bomb in the air explosion state is obtained. It is expected to provide a new method for effectively promoting the operational throwing performance and the economy of the shock detonation bomb, as well as the design and development of a new type of shock detonation bomb, which has important practical significance.

2.1
Numerical simulation method

Physical model
Based on Reference 11, this article takes the double-cavity stun grenades as the object. It adopts three rows of parallel side openings and does not produce lethal fragments. The bomb structure mainly includes an insurance pin and a firing device, involving an inner and outer shell surrounded by an upper and lower connection seat. The outer shell and the connection seat are connected by a threaded connection, while the insurance pin and the firing device adopt the active standard components, as shown in Figure 1. 11 This article mainly optimizes and checks the inner and outer shells as well as the connection seat.
1. Inner shell. The inner shell is cylindrical and connected with the upper connection seat by threads. A plastic material with moderate deformation resistance is used, and the ABS material is selected as the inner shell with good mechanical and thermal properties considering the factors of shock resistance and damage adaptability. 2. Outer shell. The outer shell is made of 45# steel with good mechanical property to ensure good shock resistance during the explosion, and it can also ensure structural integrity, economical and universal role, as well as suitable weight of the bomb. 3. Upper/lower connection seat. To facilitate multiple assemblies and ensure the connection strength, the shape of the connection seat is a hexagonal prism, and six annular explosion vents are evenly distributed on the plane to better release the sound and light. The connection seat and the firing seat are connected by the tread of M20×1.5, and the outer cartridge is connected by the thread of M38×1. Both the connection seat and the thread are made of 45# steel to ensure consistent strength.
The simulation study of the stun grenades in this article focuses on the optimization of the shell thickness of the double-layer cavity. Due to a large number of parts in the stun grenades, the factors such as the shell strength and the influence of the combat technical index are ignored. The explosion process makes the following assumptions to reduce computational resources without compromising computational accuracy.
1. In the process of finite element analysis and modeling, the firing ignition and safety devices, which have little influence on the simulation results, are omitted. 2. To simulate the process of the firing tube to ignite the flash agent, the detonation point is set at the center of the circle on the grid model of the charge column, and the flash agent detonates linearly.
F I G U R E 1 Schematic diagram of the structure of a safe stun grenade.

F I G U R E 2
The overall geometric model of a stun grenade.

3.
A nonreflection boundary is set around the air to simulate the shock wave transmission effect of a free explosion in the air, which could avoid the influence of the shock wave reflection superposition on the simulation results. The reflection and superposition of the stress wave are ignored inside the shell cavity. 4. To simplify the model and improve the quality of the mesh, the thread of the connection seat, the inner and outer shell is simplified to a plane. The connection contact form is replaced by the surface-to-surface fixation failure keyword to minimize the error. 5. A failure criterion is added to the material of the inner shell to simulate the process of impact fragmentation, and the relevant collision and erosion can be simulated through the erosion contact keyword between the inner and outer shell.
Since the research object focuses on the force of the outer shell, the geometric model adopts the full model as well as adding the air and explosives. The inner diameter of the outer shell is 36 mm, the length is 120 mm, and the thickness is 8 mm. 15 vent holes are distributed evenly around the circumference, the thickness of the inner shell is 2 mm and its outer diameter is 28 mm with a length of 110 mm. The thickness of the connection seat is 3 mm, six vent holes are evenly distributed along the inner circle, and the overall length of the bomb is 125 mm. The geometric model is shown in Figure 2.

2.1.2
Finite element modeling and grid convergence test The explosive model mesh adopts the mapped hexahedron element of Solid164 eight-node solid element. According to the principle of finite element analysis, the finer the mesh, the higher the accuracy of the solution. This is an effective method to improve the calculation accuracy of the structural model, but the balance follows among the calculation efficiency, the accuracy and calculation time. Most computers have limited software and hardware capabilities. Thus it is necessary to select an appropriate grid generation method and grid number to obtain the best result with low computing cost. In addition, when the number of grids reaches a certain number, the improvement in calculation accuracy is not obvious. Because of the fluid-structure coupling of the explosive and air grid, the common node connection is adopted, and the grid size is consistent. This article attempts to change the grid size of explosive and air components first. Then the errors are compared among the peak value of explosive shock wave, overpressure rise time, and the positive pressure duration of explosive components at different proportional distances based on empirical formulas. The shell model with holes is simulated and verified based on verified air explosive model. The calculated values of radial, circumferential, and axial stresses under different grid divisions, on the edge of the 8 mm thick shell hole and away from the hole, are compared with the simulated stress distribution. And this manipulation can determine the grid convergence size of the shell element, and can evaluate the grid independence avoiding changing with the reduction of the grid.
To verify the correctness of the numerical simulation value of explosion overpressure, the existing research results of the explosion shock wave are compared and analyzed. Regarding literature 12,13 and literature, 14 it is found that when the error of peak value of explosion overpressure is small, the grid size R is between 1/8 of the side length of the explosive body and 3/80 of the side length of the explosive body. Therefore, R in this simulation is 1/4, 1/8, 1/16, and 1/32 of the explosive body diameter (d), and the explosive body diameter is 20 mm.
There are many empirical formulas for shock wave parameters, such as Ye Xiaohua formula, Ginny Graham formula, Sadovsky empirical formula, and so forth. Concerning literature, 12,13 the classic Henrich J empirical formula is selected for comparison. According to the experiment, Henrych proposed the expression of peak overpressure (MPa) of the shock wave as follows: Xu 17 gave the expression of the duration of the overpressure rise section of the shock wave (from atmospheric pressure rise to peak) as follows: Sadovsky suggested that the positive pressure action time of an explosion shock wave is: where r is the proportional distance, m/kg 1/3 ; S is distance from the explosion center, m; W is the TNT equivalent of an agent, kg; is the efficiency factor of aluminized explosives, indicating that the fraction involved in the explosion is regarded as 40%; W f is the total mass of agent, kg; Q f is the combustion heat of the agent is generally 25,166 J/g; Q TNT is explosion heat of TNT, generally taken as 4.61 × 10 3 kJ/kg; C is sound velocity, taking 340 m/s. As shown in Figures 3-5, the Y -axis on the left is the variation of overpressure peak value, overpressure rise time, and positive pressure duration divided by grids with 1/8, 1/16, 1/20, and 1/32 of the side length d of the explosive body at different proportional distances. The Y -axis is the change of average relative error, involving the absolute value of grid division F I G U R E 3 Peak value and average error of overpressure for different grid sizes. size at the same proportional distance, to find out the proportional distance when the average error of overpressure peak value, overpressure rise time, and positive pressure duration is the maximum. Furthermore, the error convergence diagram under different grid divisions is obtained when the error in Figure 6 is maximum. It can be seen from the left Y -axis in Figures 3-5 that the relative error between the simulation result of the overpressure peak value and the calculation result is smaller with the increase of the proportional distance; The overpressure peak value, positive pressure time and overpressure rise time of the explosion shock wave are close to the value of the empirical formula with the reduction of the element size, and the shock wave changes from a smooth curve to a triangle curve with an obvious rise and fall. It can be seen from the Y -axis of Figures 3-5 that with the largest proportional distance, the average error of the overpressure peak value and the duration of positive pressure increases firstly and then decreases. At about 25%, the average error value of the overpressure rise time decreases firstly and then increases reaching a maximum of about 1400%. It can also be seen that the proportional distance r, when the average error of overpressure peak value, overpressure rise time and positive pressure duration is maximum, is 0.3, 1, 0.6, respectively.
The impact of grid division on shock wave results is explored, the errors of different grid divisions are compared and analyzed concerning the shock wave simulation and empirical formulas to determine the accurate grid division method. Figure 6 shows the overpressure peak value of different grid divisions and the error of empirical formulas. It can be seen that when the grid size is larger, the overpressure peak value, overpressure rise time, and positive pressure duration are F I G U R E 6 Error of different meshing and empirical formulas at the same scale distance. less than the true value, indeed they are close to the true value with the continuous reduction of the grid. After the grid size is less than 1/20 of the explosive body diameter, the error range has dropped to less than 5%. Therefore, the geometric model in this article uses 1/20 of the explosive body diameter, that is, 1 mm grid size, to divide the grid. At this point, the influence of the grid size of 1 mm on the calculation accuracy is acceptable while minimizing the calculation cost.
To verify the grid independence of the outer shell model in the simulation in this article, the flash detonation bomb is approximately regarded as a pressure vessel, and its stress is checked by the strength calculation and verification method of the pressure vessel. In general, the ratio k of the outer diameter to the inner diameter of the container is used, determining whether the container is a thin-walled container or a thick-walled container. It is a thick-walled container when k > 1.2, and it is a thin-walled container k ≤ 1.2. The shell casing of the outer shell is assumed to be the thin-walled container first. For a thin-walled cylinder with internal pressure, the average axial stress is 15 : The hoop means stress: The thin-walled cylinder can be based on the no-torque theory, assuming that the axial stress is zero, the hoop stress is uniformly distributed along the wall thickness, and the stress of the thin-walled cylinder is set as b . The stress criterion can be obtained according to the balance equation: where where D i is the inner diameter of the cylinder, mm; D n is the diameter of the middle surface of the cylinder, mm; P C is the calculated pressure on the inner cylinder wall, MPa; is the calculated thickness of the cylinder, mm; [ ] T is the design temperature T, the cylinder allowable stress of bulk material, MPa. Substituting Equations (9) and (10) into Equation (8), after simplification, we obtain: If the inequality takes the equal sign, then Therefore, the formula for calculating the wall thickness of a thin-walled cylinder: From The formula of wall thickness is designed by a thin-walled cylinder: where d is the design thickness of the cylinder, mm; D o is the outer diameter of the cylinder, mm; is the welding joint coefficient, ≤ 1.0, = 0.9 is taken due to the high welding accuracy of the projectile; C 2 is the corrosion allowance, mm, ignoring the corrosion effect of the outer shell material, C 2 = 0. The formula for calculating the explosion pressure of the stun grenades is as follows.
where P C is the explosion pressure, MPa; x is taken as 2; is the charge density, kg/m 3 ; D is the explosion velocity, m/s; it can be seen from Section 2.1 that the charge mass is 50 g, the inner cylindrical shell is charged, the inner diameter is 24 mm, the length is 110 mm, and the available charge density is 603.17 kg/m 3 . Substitute into Equation (16), the stun grenade explosive calculates the explosion pressure P C = 59.828 MPa ≤ 0.6[ ] T = 127.98 MPa, and substitute P C into Equation (14), the calculated thickness of the cylinder is 5.87 mm, which is the design thickness of the cylinder d , at this time, the outer diameter and the inner diameter ratio is 1.326 (a = 1.326 > 1.2), which does not meet the requirement (a ≤ 1.2) for thin-walled containers, so the assumption of thin-walled containers does not hold.
Since the inner and outer sides of the thick-walled cylinder are constrained and restricted by deformation, the hoop stress is not uniformly distributed along the wall thickness, and the radial stress is generated that is not uniformly distributed along the wall thickness. The calculation formula of the cylinder wall thickness under the thick-walled theory can be derived from the strength criterion.
As the inner and outer sides of the thick-walled cylinder are subject to deformation constraints and restrictions, the circumferential stress is no longer uniformly distributed along the wall thickness direction, and the radial stress is nonuniformly distributed along the wall thickness. According to the geometric characteristics of the force equilibrium of the thick-walled cylinder, the stress formulas of the inner wall and outer wall of the thick-walled cylinder under internal pressure can be obtained. In the stress calculation of thick-walled cylinder, the equivalent stress of the average stress of the thick wall is used as the strength control stress in many projects at present (the average stress represents the average stress of the principal stresses in all directions obtained by dividing the integral along the wall thickness by the wall thickness based on the stress formula of the thick-walled cylinder). Through calculation, we can obtain the circumferential mean stress of a thick wall, p k is the same as that of a thin wall: F I G U R E 7 Stress value change curve of shell under different mesh sizes.
The average radial stress is: .
The average axial stress is: where t is the calculated wall thickness of the thick-walled cylinder, mm; D i is the average inner diameter of the projectile, mm; P is the pressure on the inner wall, MPa. According to the above stress calculation, the calculated values of the circumferential, radial and axial mean stresses of the 8 mm thick outer projectile body are 135, −24.55, and 55.23 MPa, respectively. By deriving the isotropic stresses of all elements of the shell except the hole edge, the average values are counted, and then the circumferential and axial stress values of all hole edge elements are derived. For the elasticity, there is a Giles solution to the problem of small hole stress, 16,17 because the solution formula is complex, this article will not repeat it here. When the orifice plate is subjected to a uniform load force q in the direction of 90 • , the maximum circumferential normal stress ( ) max is obtained at the hole edge, ( ) max = 3q. In other words, the circumferential or axial stress value at the hole edge is three times greater of the load on the cylinder. The air explosive element verified by the grid and the shell combination, involving the grid sizes of 1/8d, 1/16d, 1/32d, and 1/64d, are calculated separately to derive the circumferential and axial stress values at the hole edge as well as the circumferential, radial, and axial average stress values of the shell. The comparison between the simulated calculation value and stress formula calculation value is shown in Figure 7.
It can be seen from Figure 7 that after the shell grid size of 1/32d, the error between each stress value and the calculated value decreases, and the error variation is stable below 5%. It can be considered that when the grid size is reduced to 1/32d, that is 0.6 mm, the stress values of the shell have converged.
It can be seen from the above proof process that the size of the air explosive mesh is controlled as 0.2-1 mm, and the bomb body is controlled at about 0.6 mm. The component meshes are based on the overall model and the number of meshes is 1,200,000. Euler mesh is used for air and explosives, Lagrange mesh is used for bomb shell parts, and fluid-structure coupling algorithm is used for air, explosives and shells. The mesh division of the bomb body and the overall finite element 1/2 section mesh are shown in Figure 8.

Solution parameter settings
In the explosion simulation process, the bomb body adopts *MAT_PLASTIC_KINEMATIC, the plastic follow-up material model; the explosive composition is KClO 4 powder (60%) and Al powder (40%), and the charge is 50 g. agent adopts the material model of *MAT_HIGH_EXPLOSIVE_BURN using the state equation of * EOS_JWL. The JWL equation of state in LS-DYNA software is usually used for detonation products of high explosives, which defines the pressure as follows: where and are material constants, GPa; E 0 is the initial value of detonation energy per unit volume; V is relative volume, V 0 is just the initial relative volume, R 1 , R 2 are dimensionless. The solution time is set to 0.3 ms and the output is performed at intervals of 3 × 10 −6 s. According to the description of the mechanical properties of 45# steel in the literature, 18 the material mechanical properties are shown in Table 1, and the material mechanical properties of the plastic inner shell are determined in the literature, 19 as shown in Table 2. The flash agent material mechanical properties are from the JWL parameters of aluminized explosives in the AUTO-DYN Version 6.1 constitutive material library in the ANSYS simulation platform, as shown in Table 3. Many constitutive models in LS-DYNA do not allow failure and erosion. *MAT_ADD_EROSION keyword option provides a way to include fault removal in these models. This option can also be applied to constitutive models containing other failure/erosion criteria. Each fault criterion defined here is applied independently. Once a sufficient number of these criteria meet according to NCS (the number of fault conditions to be met before a fault occurs), the element fails and is automatically deleted from the subsequent finite element calculation. This option in the software applies to three-dimensional solid elements. Rodríguez-Martínez et al. 20 proposed that the use of failure criteria based on material deformation behavior in the Lagrange grid frame is a widely accepted approach when analyzing the metal structures under dynamic load. Johnson and Cook, 21 Bao and Wierzbicki, 22 and others also reached a similar conclusion. To show the fracture of the plastic inner shell and steel shell under explosion impact, the failure criterion based on equivalent plastic strain is adopted in this article. This criterion can be proved to be simple and efficient, and can be used to predict the failure of stainless steel sheets under projectile load. It is well known that the equivalent plastic strain failure is a function of the stress state. However, according to the judgment formula (21) of Johnson-Cook failure strain model equation, D 3 is usually negative when the stress triaxiality * is greater than a certain value, the input of D 2 will become irrelevant, and the failure strain f will tend to be a constant value.
where D 1 -D 5 fracture damage constant; * is stress triaxiality, * = m ∕ e , m is hydrostatic pressure, e is Von Mises equivalent stress; * is the normalized effective plastic strain rate; T * is homologous temperature, K; EFMIN is the lower bound for calculated strain at fracture. Dey et al. 23 also conducted target penetration experiments on three steel alloys and concluded that for ductile metals, when the stress triaxiality value is greater than a threshold, the failure strain is largely independent of triaxiality.
Both projectile load and near-field air blasting load will produce local deformation, and the main reason for shell deformation is tearing failure. Therefore, in this study, the material failure can be defined by the constant value of equivalent plastic strain. Teng and Wierzbicki 24 and Zhang and Ravi-Chandar 25 also proved its rationality. The failure strain value of the steel shell is 0.42, which is estimated according to the natural logarithm of the ratio of the cross-sectional area involving the initial and fractured specimen. Zhang et al. 26 checked the consistency between the numerical results and the experimental results involving the simplified estimation of failure strain. The piecewise plasticity model improved by Wang et al. in LSDYNA is in good agreement with the experimental data when describing the failure strain behavior of polypropylene plastic materials. It is determined that the failure strain in the thickness direction is f ,t = 0.072. 27 Therefore, in the same way, the plastic inner shell and steel shell are placed in *MAT_ADD_EROSION. The maximum effective strain EFFEPS defined in the EROSION keyword is 0.072 and 0.42.

Determination of thickness parameters and strength check of the outer shell
From a large number of mechanical experiments, it can be known that for plastic materials, when the stress reaches the yield stress s , the material part will produce plastic yield, 28 and the allowable stress of the material is: where s is the material yield strength, MPa; n is the safety factor. Then the strength criterion of the outer shell is the case that the equivalent stress should be less than or equal to the allowable stress, that is, ≤ [ ], 29 the tensile strength b and the yield stress s are listed in Table 1. After calculation, its yield ratio s ∕ b is 0.59. It can be seen that the recommended safety factor is 1.5-2.2 when affected by the shock load. Considering the economic role of the material and the rationality of the weight, the safety factor is 1.5, thus the stress value [ ] of the outer shell body is 237 MPa. For linear elastic isotropic materials, the maximum distortion energy theorem attributes the plastic yielding of the material to reach a critical value of the maximum shape-change specific energy. The equivalent stress of the materials takes into account three principal stresses and affects the strength equally, and the theory has a high degree of agreement with the experimental data of most plastic materials. 30 In the finite element analysis, the Von Mises equivalent stress criterion follows the maximum distortion energy theorem. As the explosion shock wave belongs to the impact load of dynamic load, the acceleration changes sharply in a short time. As the shell is very close to the explosion shock wave, the impact time and natural vibration period of the shell are very short. The maximum displacement amplitude mainly depends on the magnitude of the action impulse. This article is based on the peak value of the maximum Von Mises stress time of the element to judge the plastic yield. Therefore, the peak moment load of the explosion load is simplified as a transient quasi-static load. Von Mises stress of plastic materials is common in explosion impact analysis during postprocessing. In this article, in order not to affect the secondary use, the frequency of use is not high, which is different from the fracture phenomenon caused by the cumulative damage of the structure under the long-term action of variable load and strain. The stress level of fatigue failure is lower than the strength limit under static load. Therefore, it is safe to use static load. So when the maximum equivalent stress value of the Von Mises stress nephogram is greater than its allowable stress value during the simulation time, and it is considered to be plastic deformation.
Due to different thicknesses, the time for the structure to reach the maximum equivalent stress value is also different. Take the stress of the baseline model involving the outer shell thickness of 8 mm as an example. When the detonation wave develops to the bottom for the first time, the maximum equivalent stress element appears longitudinally at the inner side of the third row of circular holes for the first time. When the detonation wave acts on the shell for the second time, the maximum equivalent stress element appears longitudinally at the inner side of the third row of circular holes for the first time, as shown in Figure 9A, the change in the stress value is shown in Figure 9B. The equivalent stress value of the largest equivalent stress element in and element is taken as the maximum equivalent stress value of the shell.
To further analyze the deformation of the projectile, the resulting model is hidden along the axial 1/2 to facilitate the observation of the internal stress. When the thickness of the outer shell is 8 mm, the nephogram of the equivalent stress changes every 60 μs as shown in Figure 10. It can be seen from Figures 9 to 10 that the sharp change of the shell section due to the opening will generate local high stress, the round hole is affected by the stress concentration, and the stress concentration at the corner of the lobe is highly concentrated. From this phenomenon, it can be seen that the stress concentration at the lobe is one of the factors affecting the corner crack at the orifice of the shell with holes. 10 Due to the yielding of plastic materials, when the maximum stress at the stress concentration reaches the yield strength of the material, if the load continues to increase, the stress will not increase while the strain will increase, and the load will be borne by the rest of the materials that have not reached the yielding. The material loses its bearing capacity due to yielding until the stress at each point of the entire section tends to reach the yield strength, so the stress decays rapidly at a little distance from the local area.
It can be seen from Figure 9 that the outer shell passes through two equivalent stress peaks during the explosion process, and the shock wave and detonation products interact with the shell, respectively: the first time is the expansion and expansion stage of the shock wave, and the initial shock wave generated by the detonating compressed air is over pressured downward. When the propagation reaches the middle section of the shell due to the fact that the two ends of the outer shell are constrained by threads, the middle section structure is deformed more obviously than the two ends when affected by the detonation wave. It is more fragile and mostly concentrated in the longitudinal area inside the orifice with the form of subtrees. The maximum equivalent stress peak value of the element is 272.2 MPa, which appears at about 134 μs; the second time is the reflection and superposition stage of detonation products. After the shock wave is reflected, the subsequent detonation products are reflected and superimposed, and a secondary overpressure zone will be formed below the bomb. 31 The reverse rarefaction wave will gradually attenuate during the propagation process when encountering the outer shell wall, and the compression wave will release the pressure again through the orifice, 32 which is concentrated on the inside of the orifice in the longitudinal area with the form of tensile stress. The maximum equivalent stress peak value of the element is 208.1 MPa appearing around 296 μs.
The maximum deformation of the shell with a thickness of 8 mm is shown in Figure 11. It can be seen that due to the stress concentration effect of the circular hole, the deformation is mostly concentrated in the inner side of the hole and is distributed in a crossoblique strip. The maximum deformation is 6.5 × 10 −5 mm, and the elongation is far less than that of 16% when it breaks, so the bomb shell will not break and damage, and it can meet the requirements of repeated use.

F I G U R E 10
The maximum equivalent stress nephogram at the wall thickness of 8 mm.

F I G U R E 11
The maximum deformation variable cloud map of the outer shell under the thickness of 8 mm.

F I G U R E 12 Equivalent stress cloud map and deformation displacement cloud map of the upper connection seat.
Under the shell thickness of 8 mm, the Von Mises stress cloud diagram and deformation displacement cloud diagram of the upper and lower connection seats are shown in Figures 12 and 13. It can be seen that the maximum equivalent stress value of the upper connection seat is 233.7 MPa, the maximum deformation amount is 1.275 × 10 −4 mm, the maximum equivalent stress value of the lower connection seat is 945.8 MPa, and the maximum deformation amount is 3.654 × 10 −4 mm. Since the ammunition is detonated at the center of the upper surface, the initial detonation wave diffuses downward in the chemical medium, and the shock wave is reflected and superimposed at the bottom. Then the pressure will be released through the orifice of the lower connection seat, and the stress will be concentrated around and in the center of the round hole, thus the upper connection seat will not undergo plastic deformation, and the lower connection seat will exceed the yield limit of the material and undergo plastic deformation, which will affect its secondary use but will not break and damage. It is safe for the bomb to explode for the first time as a whole, so the thickness of the upper and lower connection seats can be 3 mm.

F I G U R E 13
Equivalent stress cloud map and deformation displacement cloud map of the lower connection seat.

F I G U R E 14 Determination process of outer shell thickness.
It can be seen from the above analysis that the concentrated stress value near the shell orifice is greater than the allowable stress value [ ] under the baseline involving a thickness 8 mm, and the shell is prone to failure and fracture under the action of explosion load while the maximum deformation does not exceed its allowable elongation. Because the external dimension of the shell body is considered as a fixed value in this article, thus the shell thickness is reduced from the inside to the outside. In this way, the stress concentration effect of the Angle between the transverse holes can be reduced and the maximum stress concentration value of the shell can be reduced. Change the maximum thickness of the bomb shell from 8 to 1 mm, the interval is 0.5 mm, and simulate the cases separately.
In this article, the design rationality of the size and weight of the bullet body is fully considered. The outer size of the bullet body and the inner shell size are fixed to reduce the volume facilitating the hand-held and fixed variable parameters. The distance between the outer radius D e of the shell and the outer radius D 0 of the inner shell is considered as the maximum wall thickness of the outer shell. Therefore, the initial thickness is D e − D 0 = 8 mm. Then, the inner radius D i of the shell is increased by 0.5 mm until the shell thickness is changed to 1 mm. A total of 15 parameters of the outer wall thickness are set, simulations are conducted separately. The parameter determination process is shown in Figure 14. The outer radius D e of the outer bullet body and the outer radius D 0 of the inner bullet body remain unchanged. The inner radius D i of the outer bullet body is enlarged as shown by the dotted line to reduce the wall thickness: The calculation results of the maximum equivalent stress and the deformation changed as a function of the shell thickness are summarized in Figure 15. It can be seen from Figure 15 that with the decrease of the shell thickness, the stress value initially decreases and then increases, and the deformation increases gradually. This is because in the thickness range of 5-8 mm, the stress concentration mainly occurs at the angle between the holes, and decreases with the thickness. When the included angle is reduced, the stress concentration effect is weakened; however, in the thickness range of 1-5 mm, the corner point of the convex angle of the round hole is more obviously affected by the stress concentration, and the maximum equivalent stress value reaches the lowest at 5 mm, which is 232.6 MPa < [ ] = 237 MPa. The maximum deformation is 1.07 × 10 −4 mm, and the elongation is far less than that of 16% when it breaks, so the bomb shell will not break and damage, and it can meet the requirements of repeated use. Therefore, it can be judged that the outer shell does not undergo plastic deformation when the shell thickness is selected as 5 mm.

Theoretical calculation verification of wall thickness selection
In this article, the strength requirements of the outer cartridge of the stun grenades are: under the action of detonation pressure, it will not undergo plastic deformation and strength damage, so it will not affect the next usage of the bomb. Because the strength check and wall thickness calculations have not appeared in the structural design specification and research of the stun grenades, it is difficult to measure the microscopic deformation and force of the shell in the test of detonation. To verify the rationality of the design of the wall thickness of the outer shell in this article, the stun grenades can be approximately regarded as a pressure vessel, and the strength calculation and checking method of the pressure vessel are used to check. Based on the network independence verification in Section 2.1.2, it is proved that the shell of the missile body can be regarded as a thick wall pressure vessel, the simulation calculation in this study is based on the Von Mises stress nephogram, that is, the maximum distortion energy theorem. The strength is judged by the fourth strength criterion and the strength judgment criterion formula are: The calculation formula of the wall thickness of the cylindrical thick-walled cartridge can be deduced under the maximum distortion energy theorem criterion: where [ ] is the allowable stress of the cylinder material, MPa.
It should be noted that the pressure on the inner wall P is mainly the radial internal pressure applied to the cylinder, which is not the maximum stress value in the circumferential or axial direction of the shell. In this article, P is considered as the calculated value of the instantaneous explosion load, P c = P; The existence of small holes must affect the stress distribution in the plate. According to the finite element simulation, when the uniform explosive load is transferred from the inner wall of the cylinder to the small hole, there is a stress concentration at the edge of the opening corner and the horizontal axis of the inner wall. Here, the author believes that the structural characteristics affect the stress distribution and transmission process of the projectile itself, rather than the process of applying the external explosive load to the cylinder. According to the Saint Venant principles influence is only limited to the area near the hole, and is significantly reduced in the distance from the hole edge; Therefore, this article put forward a reasonable assumption. When using the thick wall cylinder wall thickness checking formula, parameter P takes into account the external explosion load pressure on the whole body, and the internal pressure on the cylinder wall with holes is not affected by the stress concentration at the hole mouth. This article use this checking formula to check and select the thickness of the shell with holes.
From formula (25), the calculated wall thickness t of the thick-walled cylinder is 4.496 mm. At this time, the ratio of the outer radius to the inner radius a = 1.249 > 1.2, which meets the requirement of a > 1.2 for the thick-walled container, so the outer shell can be approximated as a thick-walled pressure vessel, and its calculated wall thickness is 4.496 mm. Considering the influence of the residual stress and plastic residual strain of the outer barrel on the secondary safe use of the outer shell, the wall thickness can be taken as 5 mm, which follows the thickness-stress curve in Figure 15.
The calculated thickness of the elliptical head is calculated according to the formula: where is the shape coefficient of the elliptical head: In the formula, h i is the length of the short axis b of the elliptical section. Since the upper and lower connection seats have no axial protrusions, the section can be approximately regarded as a rectangle, and h i = 0. The thickness of the upper and lower connection seats can be obtained from Equation (27) which is 2.11 mm. Considering the threaded connection, as well as the influence of stress and residual stress, the thickness of the connection seat can be taken as 3 mm.

Thread strength check
Since the thread between the upper cover and the inner shell is a broken thread, the strength of the threaded connection between the connection seat and the outer shell is mainly checked. The connection seat and the outer cartridge are connected by M38×1 thread, take out the maximum force bearing elements , of the upper and lower connecting threads. The location of the element is shown in Figure 16A. The simulated force is shown in Figure 16B. As shown in Figure 16, the maximum force values are 90,700 and 102,000 N, both of which are the same threaded connection. According to the international standard, 33 the guaranteed stress of the 45# steel M38×1 thread is 720 N/mm 2 , and the stress cross-sectional area of the thread is 720 N/mm 2 . The calculation formula is: where A s is the stress cross-sectional area of the thread, mm 2 ; d is the nominal radius of the thread, mm; P is the distance between threads, mm. The calculated stress cross-sectional area of the thread A s is 1324 mm 2 , and the guaranteed load is 720 × 1324 = 953,280 N, which can fully satisfy the connection strength demand. Therefore, the connection seat and the outer cartridge can be connected by M38×1 thread.

Analysis of the inner shell cartridge breakage and fragment dispersion
To facilitate observation and analysis, the air explosive model is hidden, and the bomb model is hidden by 1/2 to take a picture. Within 0-500 μs, the fragmentation distribution of the inner shell cartridge is shown in Figure 17. It can be seen from Figure 17 that in the period of 0-80 μs, since the shock wave enters the homogeneous explosive after a delay in the shock wave front, it is in the initial induction period of detonation, the detonation wave is not transmitted to the inner shell wall, and the inner shell cartridge is slightly deformed. At about 90 μs, the shocked explosive forms "overpressure detonation," the upper end of the inner cartridge fails and ruptures, the failed element reaches the stress limit and is deleted, and the stress is transmitted downward with the detonation wave. At about 480 μs, the inner cartridge was completely broken, most of the fragments were rectangular, the bottom was large fragments and stayed inside the shell, and a very limited number of small fragments flew out and the speed reached about 150 m/s and then quickly decayed to approximately 50 m/s, which is not lethal.

F I G U R E 17
Fragmentation process diagram of the inner barrel.

Sound pressure analysis
Under the condition of no boundary, when the influence of the outer shell is not considered, the relationship between the peak sound pressure generated by the explosion and the peak overpressure can be expressed by the following formula 34 : In the formula: L p is the peak sound pressure level, Pa; P peak is the explosion peak overpressure, Pa; in air, P 0 takes 20 μPa.
When the shock wave is released through the outer shell with openings, each orifice is equivalent to an independent sound source, then the total sound pressure level of the stun grenades is equivalent to the superposition of the sound pressure levels of all openings, and the formula for total sound pressure level is: where L ps is the total sound pressure level, Pa; L p0 is the sound pressure level of a single orifice, Pa; n is the total number of orifices on the outer shell.
Taking the reference position where the sound pressure starts to attenuate at 1 mm from the opening plane, and L ps is the total peak sound pressure level, the peak sound pressure level L p after r m attenuation can be calculated from the following formula (30): Take the three openings in the axial row as the research object as well as one opening, respectively, for the upper and lower connection seats, and take the air cells A, B, C, D, and E, at a distance of 1 mm from the center of the circular hole plane as shown in Figure 18, and the overpressure curve is shown in Figure 19. From the overpressure curve, it can be seen that the maximum overpressure air element is B, and the maximum overpressure peak is 2.42 MPa. Then change into the peak sound pressure level based on the formulation (30), the value is 221.65 dB.
There are 27 orifices on the up-and-down bomb. The superimposed sound pressure level is 235.96 dB calculated from formula (31), and the sound pressure level at 1.5 m can be obtained from formula (32) to be 159.7 dB. According to the expected combat technical index of the safe stun grenades, the sound pressure level acting on the human ear at a distance of 1.5 m from the explosion center should be 140-160 dB, 20 and the sound pressure level of this type of acousto-optic bomb reaches the expected combat technology index range.

F I G U R E 19
Overpressure curve of air element at the orifice of 1 mm.

EXPERIMENTAL VERIFICATION METHOD AND RESULT ANALYSIS
To check whether the design scheme of the stun grenades meets the safety design requirements, including the fragmentation and scattering of the inner shell and the strength of the outer shell, and whether the simulation optimization results correspond with the experiment, a real bomb test is required to validate. According to Figure 1, the sample bomb was manufactured. In this article, after crushing and shell strength testing of detonators with a shell thickness of 4 and 5 mm, in order to make the fragment shape more natural and close to the air explosion state, the chair leg is selected for the fixed detonation of the inner shell crushing and shell strength analysis; To fix the missile body and reduce the error influence on the air overpressure after the missile body moves, the acoustic pressure analysis uses the fixed steel pipe perpendicular to the ground to detonate, increasing the fixed constraint. In this article, through two detonation experiments, the sound intensity, shell case strength, and fragment power are analyzed.

F I G U R E 20
The field conduction of the stun grenades with a thickness of 5 mm.

Experimental method of shell breaking and shell strength analysis
To check whether the simulation of the outer shell thickness of the simulated stun grenades and the selection formula of the wall thickness are accurate, the bombshells with a thickness of 4 and 5 mm were tested with real bullets. For other bomb shells with other thicknesses, the difference between the simulation and experimental results is quite large, so the test is not considered here. Because the quantitative analysis of the shell strength in the explosion experiment is very difficult, this article uses the allowable stress and strength theory to predict and judge the deformation threshold of shell fracture and conducts a simulation to quantitatively analyze the shell stress strength. Therefore, after the experiment with the predicted fracture of the stress concentration in the simulation model, the shell strength analysis can only be compared by observing the deformation and fracture of the outer bullet. Whether the damage degree of the bomb can be used for secondary use is observed with the thickness of 4 and 5 mm, comparing it with the previous finite element results to test the simulation results and detect the power of the explosion fragments. The test equipment includes safe stun grenades with a thickness of 4 and 5 mm as well as a camera. First, fix the safe stun grenades of 5 mm, fasten one end of the rope with the safety ring, and pull the other end to the bunker, as shown in Figure 20. The experimenter pulled a rope to detonate the stun grenades at the bunker, and a camera was installed at a safe distance from the bunker to record the explosion process. After the test, the data was recorded and analyzed. The same process is conducted for the safe stun grenades of 4 mm. After all the tests were completed, the explosion power is analyzed by observing the fragmentation shape, fragmentation quality and shell cracking deformation degree.

Bomb sound intensity test method
The sound pressure intensity test mainly checks whether the sound intensity of the safe stun grenades meets the technical warfare index requirements, as well as the accuracy of the calculation of air overpressure and sound intensity in the simulation. In the free sound field, when the sound pressure level L p1 (dB) is measured at the distance L 1 (m) from the sound source, the sound pressure level L p2 (dB) at any distance L 2 (m) from the sound source can be calculated according to formula (33).
The test equipment is a flash and sound pressure test system, as well as a camera, fixing materials (stone or cement brick), iron wire, rope, sign pole, tape measure, and so forth. Before the test, an obstacle safety wall should be set up at a distance of 15 m away from the explosion point. The weather conditions are good, the temperature should be 20-35 • C, the humidity should not exceed 60%, and the wind speed should not exceed 6 m/s. However, when the wind speed exceeds 2.5 m/s, it is necessary to attach a spherical windshield to the acoustic sensor. Background noise should not exceed 60 dB.
The test site is divided into two parts: the test area and the recording area. Taking the placement point of the ammunition as the center of the equilateral triangle, three sound pressure testers were placed at the three vertices, and the F I G U R E 21 Schematic diagram of the layout of the test site.

F I G U R E 22
Test site diagram and detonator fixed connection. distance from the ammunition was 3 m. The microphone was aligned with the direction of the ammunition to record the peak sound pressure level of the explosion. The acousto-optic bullet characteristics tester is set at a bunker at a distance of 10 m away from the fixed point of the ammunition. The recording area is 30 m away from the test area, and a camera is placed facing the ammunition in the recording area to record the whole process of the explosion. The camera used in this experiment is Vision Research-LAB 310 model, with a pixel size of 20 μm. The resolution is 3200 fps, the throughput is 3.2 Gpx/s, and the camera is placed about 38 m away from the target. The layout of the test site is shown in Figure 21. The five-pointed star represents the ammunition placement point, and the circle represents the sound pressure tester. The fixed connection between the test site diagram and the detonator is shown in Figure 22

Analysis and verification of inner shell crushing
Through two tests, we found that the fragments were all controlled in the outer body. Figure 23 shows that after the test with a wall thickness of 5 mm, the outer shell and some fragments in the shell can be seen from the figure that the fragments are effectively controlled in the outer shell, and the shape of the fragments tends to be irregular strip, the upper and lower parts tend to be arcs, and it is not easy to fly out of the outer shell. By measuring the fragment quality after the test, the comparison of the fragment quality distribution after the test and simulation is shown in Table 4. It can be seen from the table that the value of the fragment quality in the simulation is less than the experimental fragment, and the number is slightly more than the experiment. This is due to the mass reduction caused by the failure deletion of a small number of crack elements set in LS-DYNA. The error analysis of the fragment quality is inevitable and acceptable. In the experiment, there are 15 fragments, the maximum fragment mass is 3.46 g, and the average fragment mass is 0.68 g. The

F I G U R E 23
Comparison of tested and simulated fragments of outer shell after explosion with a thickness of 5 mm. shape, quantity and quality are not enough to be lethal, and the error between the fragment shape and quality calculated by simulation is relatively low, so the simulation has certain reliability.

Analysis and verification of the strength of outer projectile
Under the charge condition of 50 g, the flash detonation bomb of safety thickness of 5 mm is intact, and the outer cartridge is still intact under the effect of explosion pressure, and can be reused, as shown in Figure 24. Figure 25 shows the photos and Von Mises stress nephogram calculated by simulation after the test of outer cartridge of a thickness of 4 mm. It can be seen that part of the base and the outer cartridge is damaged and deformed after the test, which is consistent with the plastic deformation stress concentration area of the cartridge body in the simulation at 300 μs. And it is concentrated in the axial area between the lower connecting seat and the lower hole. The stress concentration zone is the part where the projectile body is further deformed and damaged under the action of the shock wave. It can be seen that this test is consistent with the simulation results, thus verifying the accuracy of its strength check and material selection. Therefore, the flash detonation bomb of a safety thickness of 4 mm cannot be used again. The outer shell with a thickness of 5 mm can be reused, indicating that the thickness is safe. The threaded structure is well connected without failure and large-scale deformation. In practice, whether the shell is impacted by multiple explosion loads and whether its strength can meet the requirements of repeated use are still the focus of future safety optimization research.

Bomb sound intensity analysis and verification
When the ammunition is exploded, it produced a deafening sound and a dazzling flash, and the sound pressure tester recorded the sound intensity of the ammunition. The sound pressure measurement results of each measuring point are shown in Table 5.

F I G U R E 24
Photo after explosion of 4 mm wall thickness of outer cartridge.

F I G U R E 25
Comparison of photos and simulated cloud images after the explosion of the outer barrel with a wall thickness of 4 mm. From Table 5, it can be seen that the sound pressure intensity of the safe stun grenades is less than 160 dB, which means that it meets the nonlethality requirements. According to the simulation calculation of the air overpressure at 1 mm, the sound pressure at 3 m can be obtained by formula (33) to be 152.52 dB. It can be seen that the simulation calculation overpressure value and the sound pressure conversion formula are larger than the actual test value, and the relative error is 4.8%, but it is still between 140 and 160 dB. Under the premise of meeting the requirements of technical and tactical indicators, the sound pressure intensity can ensure the absolute safety of personnel.

CONCLUSION
In this article, LS-DYNA software and the maximum distortion energy theorem are used to simulate and analyze the strength of shells with different thicknesses when they are first impacted by explosion; Check its structural performance and threaded connection as well as the fragment scattering and sound pressure level; The safety analysis is carried out, and the fragment test analysis and sound pressure level test are carried out for the real bomb according to the design parameters. The following conclusions are drawn: 1. It can be seen from the simulation results that the stress concentration of the shell with a thickness of 5-8 mm occurs at the transverse included angle between the holes; The shell with a hole of a thickness of 1-5 mm is easy to produce corner cracks in the convex part of the hole opening. When the shock wave and detonation products interact with the shell, respectively, the shell pressure changes through two peaks. The maximum equivalent stress and deformation of the upper and lower connecting seats are mainly concentrated around and in the center of the circular hole. 2. As the outer shell wall thickness decreases, the maximum stress concentration value decreases first and then increases.
The allowable wall thickness calculation formula of the pressure vessel derived according to the maximum discharge energy theory and the bomb with a wall thickness of 5 mm in the live ammunition test have good structural integrity, which will not lead to failure and fall off of the threaded structure. Therefore, the outer shell with a thickness of 5 mm in the simulation meets the strength requirements, and has good safety to theoretically meet the requirements for repeated use. The maximum equivalent stress and maximum deformation of the upper connection seat meet the strength requirements. The first use of the bomb is safe, but the second use is limited. 3. In the simulation and real bomb tests, after the inner shell is broken by the detonation wave, most of the fragments remain in the bomb body without producing lethal fragments, so it has good security; The shape and quality of the fragments are very similar to those of real bombs, which verifies the reliability of the simulation; The sound pressure level acting on the human ear at 1.5 m from the explosion center meets the operational technical indicators, and the human body is safe outside this radius; The relative error between the sound pressure level at 3 m and the average sound pressure level obtained from the on-site sound pressure test is 4.8%, which is within the allowable range of operational technical indicators.

ACKNOWLEDGMENTS
The author is deeply grateful for the guidance given by Prof. Guo Sanxue from the School of Equipment Management and Security of the Armed Police Engineering University.

CONFLICT OF INTEREST
Authors have no conflict of interest relevant to this article.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.