Fracture mechanism of a cylindrical shell cut by circumferential detonation collision

He-yng Xu ,Wei-ing Li ,* ,Wen-in Li ,Qing Zhng ,Y-jun Wng ,Xio-wen Hong ,Ying Yng

a Nanjing University of Science and Technology,Nanjing,210094,China

b Jiangsu Yongfeng Mechanics Co.,Ltd,Nanjing,210094,China

Keywords: Detonation wave Circumferential detonation collision Fracture mechanism Fragmentation control

ABSTRACT The failure mechanism of a cylindrical shell cut into fragments by circumferential detonation collision was experimentally and numerically investigated.A self-designed detonation wave regulator was used to control the detonation and cut the shell.It was found that the self-designed regulator controlled the fragment shape.The macrostructure and micro-characteristics of fragments revealed that shear fracture was a prior mechanism,the shell fractured not only at the position of detonation collision,but the crack also penetrated the shell at the first contact position of the Chapmen-Jouguet (C-J) wave.The effects of groove number and outer layer thickness on the fracture behavior were tested by simulations.When the thickness of the outer layer was 5-18 mm,it has little effect on fragmentation of the shell,and shells all fractured at similar positions.The increase of the groove number reduced the fracture possibility of the first contact position of the C-J wave.When the groove number reached 7 with a 10 mm outer layer(1/4 model),the fracture only occurred at the position of detonation collision and the fragment width rebounded.

1.Introduction

The propagation of detonation waves is an important research topic in the field of explosive mechanics.With the increasing diversity of initiation methods,the collision of detonation waves has become common.The energy and the pressure converge and jump in a narrow area when two detonation collision.Due to the obvious flexibility and directionality,this accumulation effect of detonation collision is widely studied in basic research and military applications.

Dunne et al.[1] first theoretically deduced the calculation method of the Mach rod pressure when two waves collided in a condensed explosive.Ivanov [2] conducted relevant experiments on the deformation of inert materials by detonation collision to verify the accumulation phenomenon.Müller F [3].studied the detonation interaction process in coaxial charges and captured the shape of the Mach disk by X-ray photography.However,Ivanov and Müller F.both did not measure the definite gain,although the enhancement effect of collision was experimentally proved.Later,with more advanced technology,researchers could accurately measure and capture the process of detonation interaction and energy changes.Tang Mingjun et al.[4]measured the pressure and velocity of detonation in each section with GSJ high-speed photography and found that the velocity of the Mach wave was 1.9 times the Chapman-Jouguet (C-J) detonation velocity and the pressure was 3 times the C-J detonation pressure.WEN Shang-gang et al.[5,6]and A V Shutov et al.[7,8]also examined the interactions between multiple detonations and confirmed that the pressure and velocity in the collision area were greater than those of the C-J wavefront.

Detonation collision is widely studied in basic research and military applications due to the obvious flexibility and directionality.It can be applied to explosive shaped projectiles to achieve better flight stability [9-11] or can be applied to fragment warheads in a multi-point eccentric initiation method to achieve higher dispersion speed and better ejection angle [12-14].In addition,detonation collision can also be used to directly cut low ductility metals.W.Arnold et al.[15,16] proposed a method called high explosive pellet and applied the pressure enhancement effect at the collision area to cut shells or liners to generate multiple fragments;however,they mainly focused on the use of some devices to complete the exchange between different damage elements.Zipu Geng [17] simulated the fracture failure behavior of a liner resulting from the axial detonation collision by multi-point initiation and noticed that the detonation wave became superimposed on the liner surface and formed concentration zones with a strong stress,leading to fracture along these zones;however,no corresponding test was conducted to verify the result.Zhang K.F[18,19].conducted a similar test to further study the fracture mechanism of a steel plate cut by multi-detonation superposition.The fragment and simulation results were discussed and it was found that the steel plate shear fractured at local high-stress zones produced by the detonation collision.

However,most of these studies have mainly focused on cutting metal plates by axial detonations and rarely studied the fracture mechanism of cylindrical shells under circumferential detonation collision.When the charge is detonated,a cylindrical shell becomes exposed to a large strain in the circumferential direction.The cylindrical shell then breaks under the combined effects of this large tensile strain and detonation collision,and this phenomenon does not exist in axial metal cutting.In the present paper,the fracture mechanism of a cylindrical shell under circumferential detonation collision is discussed.

The soft recovery test was carried out for a cylindrical shell(with a self-designed detonation wave regulator) filled with explosives.The shape,quantity,and mass of fragments generated from the warhead with or without a regulator were compared.The fracture surfaces and shapes of fragments generated from the warhead with a regulator were analyzed by microscopic and macroscopic methods.The fracture mechanism of the cylindrical shell under circumferential detonation collision was subsequently studied.Further,the formation process of local high-pressure regions was analyzed by a finite element (FE) model.Finally,the influences of groove number and outer layer thickness on the fracture behavior of the cylindrical shell were investigated by the simulation model.

2.Experimental procedure

2.1.Self-designed regulator and action principle

In order to achieve a circumferential detonation collision,it is necessary to generate independent circumferential multi-waves.There is a high demand for initiation synchronization to generate circumferential multi-waves;however,it has a big impact on the obtained results.Similarly,the multi-point initiation technology cannot produce independent circumferential multi-waves reliably.Therefore,a detonation wave regulator was designed to realize the collision and superposition of circumferential multi-waves by a simple,single detonator initiation at the end face.The structure of the regulator is displayed in Fig.1.The regulator was a ring-shaped cylinder with an outer diameter of 60 mm and a thickness of 10 mm.It was made of phenolic resin,which has a small density and good explosion-proof performance.The sidewall was processed with evenly distributed grooves that ran through the entire thickness.When the charge was detonated,the detonation propagated outward,then became blocked,and finally was divided by the regulator.A part of the wave propagated through the grooves to the outer layer,and new wavelets were formed.Adjacent wavelets collided with each other and generated localized high-pressure regions on the shell.As a result of that,the shell would experience different distribution of pressure and thus produced different effects.

2.2.Experimental setup

In the test,the warhead structure shown in Fig.2(a) was used.The shell was a 6 mm-thick AISI 1045 steel cylindrical tube (outer diameter=92 mm and height=80 mm),and its two ends were connected with a ring (height=10 mm) made of phenolic resin with the same diameter(the ring was used to reduce the impact of axial rarefaction waves on the shell).Different from the common structure,a self-designed regulator was placed coaxially in the shell to control the detonation wave,divided the explosive into inner charge and outer layer.The charge was made of Comp B with the shapes of the shell and the regulator,and the grooves were filled evenly to connect the explosives on both sides.A cover was used to fixing the detonator and the booster explosive at the top,the charge was point-initiated with a booster explosive from the top center.

In order to retain more comprehensive fragment information and reduce the secondary fragmentation as much as possible,water was used as a soft medium to recover all fragments.Fig.2(b) presents the experimental setup that includes the specimen,an air chamber,and a recovery device.During the test,the specimen was placed in the middle of the air chamber and the water well,and a fine net was used to recover fragments at the bottom.Three warheads were used (Table 1):the treatment group had two repeated specimens with the structure shown in Fig.2(a),and the control group had a specimen without a regulator and filled with Comp B.

Table 1 Experimental conditions.

3.Experimental results and analysis

3.1.Mass distribution of fragments

All recovered fragments were cleaned,dried,and sorted,and as a result,Fig.3 displays partial typical fragments of the treatment group and the control group.It is evident that obvious differences in the fragment shape and size existed between these two groups.The fragment shape of the control group was normal and common,and the size was smaller than that of the treatment group(Fig.3(a)).In the treatment group,the fragment shape was more regular (Fig.3(b));however,two typical fragment shapes were observed:(i) large fragments with a regular shape (the overall shape tended to be rectangular),and fracture surfaces were oriented at～45°to the inner surface and (ii) long and elongated fragments;one side of these fragments was smooth,whereas the other side had a ridged surface.

In order to understand the impact of the regulator comprehensively and also to prove that the above-mentioned large shape difference was not caused by accidental factors or artificial selection,all recovered fragments were weighed and counted (fragments of mass>1 g were considered).The mass distributions of the three cases are displayed in Fig.4.It is evident that when a regulator was in charge,it had a great impact on shell fragmentation,regardless of quantity or mass.

The fragment mass distribution of case 3 (without a regulator,Fig.4(d)) is consistent with the known results of the shell:The fragmentation was relatively uniform,small fragments make up the majority,fragments were mainly concentrated in the range 0-4 g(accounting for～90% of the total number of fragments),the maximum fragment mass was less than 8 g,and the number of fragments decreased exponentially with the increase of the mass.In comparison to case 3,cases 1 and 2(with a regulator)manifested an obvious difference in mass distribution.The number of small mass fragments was much smaller than that without a regulator.Fragments of mass 1-2 g generated in the detonation cutting process or secondary breaking were not the main part.Moreover,an abnormal upward trend in the partition statistics was observed rather than a significant decrease in the fragment number with the increase of the fragment mass(Fig.4(a)).It is evident from Fig.4(b and c) that fragments were evenly distributed in the range of 5-20 g.These fragments were the main part from the shell cut and several individual fragments with a mass close to 30 g appeared as connected fragments.The regulator only processed grooves in the axial direction.Therefore,the shell became stretched under rarefaction waves (although two phenolic resin rings were connected with the shell to reduce rarefaction,it was not enough to completely offset the effect),and the fracture at different horizontal positions resulted in the fragment mass difference.

Fig.1.Structure of the regulator:(a) 3D partial section view,(b) and (c) are the side view and the A-A section view,respectively.

Fig.2.(a)Side view and the A-A section view of the specimen with a detonation wave regulator,(b)Experimental setup of water well(φ8 m×8 m)recovery,the centers of the well,the air chamber,and the specimen coincide and (c) Photos of the specimen with a regulator.

Fig.3.Partial experimental results(a large grid is 10 mm):(a)fragments of the control group with a normal and common shape and(b)fragments of the treatment group with two typical shapes.

Fig.4.Statistical results of fragment mass.

By comparing the results of mass distribution and typical fragment shape,it is confirmed that the addition of the regulator had a significant impact on the fracture behavior of the cylindrical shell.Although the charge mass of the control group was larger than that of the treatment group,it could not impact the distribution trend and resulted in these two extremely different mass distributions and fragment morphologies.This happened because of the special detonation propagation path caused by the regulator rather than a larger charge mass.Therefore,it can be inferred that the shell could be cut into large and regular fragments by detonation after adding the regulator.

The addition of the regulator completely changed the detonation propagation path,shown as Fig.5(a).After the detonation spread through regulator,formed wavelets,and collided at point I.With time,the high-pressure region moved to the shell at region M (M refers to the superposition position on the shell).Due to such high pressure,region M experienced a large deformation.In the subsequent expansion,the cylindrical shell underwent shear instability under a high strain.The shear band initiated at region M due to the large deformation and then grew into cracks through the shell,forming long strip fragments(shown as the red area in Fig.5(b)).The shell may also be fractured where the detonation first contacted position (region D) and formed regular large fragments (shown as the blue area in Fig.5(b));this will be discussed in the 3.2 section.

3.2.Mechanism of waves cut the shell

The macrostructure and microfracture characteristics of typical fragments in the treatment group were analyzed to understand the fracture mechanism of the cylindrical shell under circumferential detonation collision.Fig.6 presents the Scanning Electron Microscope (SEM) images of the fractured surface of a long-ridged fragment,and three different micro-characteristics were observed.Region I is the ridge part,which is the inner surface of the shell,and Fig.6(A)displays the enlarged microscopic feature of this part.This area presented a water ripple characteristic with obvious directionality.The direction of the microstructure was perpendicular to the direction of the fragment fracture,and similar characteristics were not observed on the inner surface of the shell with a normal charge.This happened because the pressure of the detonation collision was more than double the C-J pressure.Under such high pressure,the inner surface metal softened and flowed along the direction of detonation propagation.When the detonation collision area passed through the shell,the large pressure difference produced great shear stress and strain on both sides of region M.Severe plastic deformation and heat accumulation occurred in the metal,which reduced the shear strength of the material,slips occurred along a certain crystallographic plane and direction in the structure.Then micro-cracks appeared at region M and expanded rapidly to both sides,formed the elongated fragments with a ridged surface.During cracks propagated,fracture surfaces became squeezed and rubbed against each other,resulting in flat surfaces with obvious textures,consistent with Fig.6(B),which is indicated by a series of small bright surfaces macroscopically.However,the high pressure on region M only existed in a short period,and the detonation pressure gradually tended to be uniform during the subsequent expansion process.Therefore,region II did not cover the whole fracture surface,rather the uneven surface with elliptical dimples appeared in region III (Fig.6(C)).It is a typical shear fracture characteristic of the thin cylindrical shell without a regulator fractured at a high strain rate.Through the micro-characteristics,it can be known that a large compressive stress acted on ridged fragments,and embryos appeared after the passage of detonation collision.The initiation of embryos and the crack propagation occurred earlier than normal.These characteristics confirm that ridged fragments were formed in the position of detonation collision (Region M).

The fracture in region D was not expected.The test results showed that the width of regular fragments(upper row in Fig.3(b))in the treatment group was smaller than expected.The width of all regular fragments ranged between 1.3 cm and 1.6 cm.Assuming that the initial radius and thickness of the shell before expansion area0and δ0,respectively,the radius and thickness of the shell after fracture areafand δf,respectively;in addition,εsatisfiesaf=εa0due to the volume of the shell being constant,and then [20].

where,ν is shape factor,for cylindrical shell ν=1,ignoring highorder small quantities of δ0and δf,the thickness of the cylindrical shell at the moment of fracture is δf=δ0/ε.Ten fragments were selected randomly,and their thicknesses were measured.The average fragment thickness was calculated as 4.22 mm;thus ε=1.42.Further,the shell radius at fracture was calculated asaf=1.42× 4=5.68 cm.If the fracture only occurs at region M,the shell should be evenly broken into 12 parts and the width of each fragment is about 2.97 cm (without considering the loss of the shell).This value has a great deviation from the actual measured value (～2 times the actual value).

Further,four connected fragments in the treatment group were measured and observed.All four pieces were located at the end of the shell because of the low pressure caused by the axial rarefaction wave.The inner surface and the section are displayed in Fig.7.It is clear that the section had a trapezoidal shape (the width of the outer surface was smaller than that of the inner surface),and the widths of these four fragments were 2.72 cm,2.81 cm,2.50 cm,and 2.73 cm.Those results are consistent with the calculated values and the fragment shape between adjacent detonation collision positions shown in Fig.5(b).What’s more,obvious indentations and incomplete cracks were noticed in the middle of the inner surface of these four connected fragments.Therefore,it is proved that the fracture also occurred at the midpoint between adjacent detonation collision positions (region D),We guess the reason of the indentation is due to the presence of an adjusted wavefront with a large curvature,and the indentation directly leads to the generation of crack embryos.This is a unique phenomenon that cannot be observed in the process of axial detonation wave cutting.

Fig.5.(a) Detonation wave propagation and interaction process after the addition of the regulator and (b) Formation of two types of fragments in the treatment group.

Fig.6.SEM images of the long-ridged fragment.

Fig.7.Characters for four connected pieces.

Fig.8(a) displays the fracture micro-characters of region D;it presents a serpentine sliding feature and featureless flat surfaces.Macroscopically,the fracture was relatively bright with obvious plastic deformation traces,and the direction was about 45°to the inner surface,thus revealing a typical shear fracture feature.This proves that region D was fractured after the shell had enough expansion,similar to the fragmentation of normal cylindrical shell under explosively loads,and later than region M.Fig.8(b)displays a metallograph of the thickness section of regular fragments.Cavities and micro-cracks can be observed,but these cavities were not connected to a through crack.This is due to the earlier initiation and propagation of cracks on both sides,namely regions M and D,and the released waves from both sides shield the crack growth.Hence,the width of regular fragments depends not only on the positions of region D and M,but also on the released waves speed from those positions.

Therefore,the fracture process of the cylindrical shell with a regulator can be summarized as follows:the shell first deformed at regions M and D after the charge was detonated and generated a large number of important embryos.Embryos at region M developed into cracks earliest due to the high pressure.As the shell moved outward rapidly and deformed at a high strain rate,embryos at region D were gradually developed into cracks through the shell.Shear fracture is a prior mechanism for regions M and D.Microcracks in other locations were shielded by released waves and eventually could not penetrate the entire thickness.

4.Numerical simulation

4.1.Analysis of influencing factors

It would be time-consuming to discuss different influencing factors for circumferential detonation shell cutting through individual experiments.Therefore,a finite element software was used to reconstruct the above fracture process and simulated the influences of different factors on the fracture behavior of the shell.

When two waves propagate at an angle,the wave oblique collision occurs,and the pressure at the collision point of detonation increases.The entire process can be simplified as the detonation collides with the rigid wall at a certain angle,resulting in oblique shock waves after the collision.According to the conventional oblique collision theory and the unconventional oblique collision (Mach collision) theory,the pressure at the collision position [21] can be calculated as

Fig.8.(a) SEM image of fracture surface at region D and (b) Metallograph of the section of a regular fragment.

whereP2andPHare the collision position pressure and the explosion pressure,respectively,φis the incident angle,θis a parameter related to φ,φcis the critical angle that determines whether the collision belongs to the normal oblique collision or Mach collision,andkis the adiabatic index.

When the explosive type is determined and the change ofkis ignored,P2/PHat the collision point is a univariate function of φ,P2/PHonly changes with φ.According to Fig.5(a),the incident angle φ depends on the distance between adjacent grooves and the thickness of the outer layer between the adjuster and the shell.Therefore,the groove number and the outer layer thickness can change the collision position pressure and affect the cutting effect of detonation.Hence,in the following sections,the influences of groove number and outer layer thickness on shell cutting by circumferential detonation collision are discussed.

4.2.Finite element model

The AUTODYN software was used to simulate the cylindrical shell with the regulator based on the multi-material fluid-solid coupling algorithm.Considering the large deformation of explosive and regulator elements,the 3D Multi-material Euler solver was used,and the Lagrange solver was used to the shell.The unit was gcm-us.The statistics of the regular fragment length revealed that the length distribution obeyed a step-like nature distribution,which signifies that the length could be quantized,and the average length was 4.11 cm.It could be assumed that no axial fracture occurred within 4 cm and the model had a symmetrical threedimensional geometry and load.Therefore,a 1/4 model with a length of 4 cm was established to reduce unnecessary calculation time,and the other sizes are same as the specimen of treatment group.The proposed finite element model,air not shown,is displayed in Fig.9.Mesh sizes play a significant role in computation efficiency and accuracy,the average mesh sizes for the specimen and the shell were 0.5 mm and 0.3 mm,respectively[22].

The Johnson-Cook model [23] was applied to describe the deformation behavior of the specimen.This material model was sometimes used when the strain rate varied over a large range and the adiabatic temperature increased due to plastic heating,causing material softening.The equation of the material model can be expressed as

where σyis the yield stress,is the effective plastic strain,ε*is the normalized effective plastic strain,T*=（T-Tr）（TM-Tr）is the homologous temperature,TMandTrare the melting temperature and the room temperature,respectively,andA0,B0,C,n,andmare constants.The Johnson-Cook failure model was used for the shell and the strain at failure can be expressed as

where εfis the strain at failure,σ*is the ratio of pressure divided by effective stress,D1-D5are failure parameters.

Fig.9.Finite element model.

For the high explosive model,the Jones-Wilkins-Lee(JWL)[22]method was used to describe the detonation of Comp B.The pressure field was defined as a function of relative volume and internal energy per initial volume.

wherePis the detonation pressure,A,B,R1,R2,and ω are constants,andE’is specific KE.

The Von Mises model and the shock equation were applied to describe phenolic resin,and air used ideal gas equation with a density 0.001225 g/cm3,and an adiabatic exponent γ=1.4.The parameters used in simulations are listed in Table 2.

4.3.Model validation

The obtained simulation results are presented in Fig.10.Fig.10(a)displays the gauges and pressure distribution on the inner surface of the shell at 6.45 μs.The red and yellow areas indicate detonation collision areas,and they propagated along the entire length in the vertical direction.Five gauges were set on the shell at the same separation distance to further observe the pressure distribution,and Fig.10(b) displays the pressure history of these gauges.The purple line represents the interaction point of dual detonations.The pressure started to rise at 6.3 μs and then rapidly jumped to 65.79 GPa (about 2.5 times the C-J pressure) at 6.5 μs.Consequently,region M experienced a large deformation due to a huge pressure difference of～40 GPa.

The simulation result of shell cutting at 60 μs is displayed in Fig.11.It is clear that numerical and test results were the same.The cylindrical shell fractured at regions M and D and was cut into six large fragments and three long-ridged fragments (1/4 model).To reduce measurement error,the inner width and outer width of fragments at three different sections were measured.Widths of six large fragments were 1.66 cm,1.44 cm,1.32 cm,1.34 cm,1.40 cm and 1.61 cm,the maximum error with the actual width(～1.4 cm)is 15.6%,and the average error is less than 10%.Therefore,it can be inferred that the model accurately reflected the fragmentation of the shell with a regulator.Subsequent simulations will be carried out base on this model.

4.4.The effects of groove number

The number of grooves on the regulator seriously affected the fragment number and fracture location of the cylindrical shell.The fracture mechanism of the shell was simulated as the groove number increased from 3 to 7 (1/4 model).Table 3 displays the fragmentation of the cylindrical shell with different groove numbers,red arrows indicate the positions of detonation collision and black arrows indicate the action positions of the C-J wave.

Table 2 Material parameters for the simulation [24-27].

Table 3 Simulation of the fragmentation of the cylindrical shell for different groove numbers.

It is clear that the detonation superposition effectively cut the shell;however,the fracture at region D was affected by the groove number.When the number of grooves increases from 3 to 5,both regions M and D fractured and the fragment width decreased with the increase of the fracture position.When the groove number was 6,the partial region D generated connected fragments.When the groove number increased to 7,region D was not cut and the fragment width increased.The groove number did not affect the cutting of the shell at the detonation collision position;however,it weakened the effect of the C-J wave.With the increase of the groove number,the distance between adjacent detonation collision became smaller,and released waves spread to the entire distance in a shorter time,thus shielding the crack growth.Simultaneously,the smaller the curvature radius of the C-J wave also reduces the deformation at these regions.

4.5.The effects of outer layer thickness

The outer layer thickness affected the propagation of detonation and generated different pressures at the detonation collision position.When metal is cut axially,the distance between the initiation point and the metal plane seriously affects the cutting effect[19,28].In order to study the effect of outer layer thickness on the fracture behavior of the cylindrical shell,the fragmentation of the shell with the outer layer thickness of 5-18 mm was simulated.

Fig.12 displays the peak pressures of detonation collision at the same position on the inner surface and the fracture of the shell with the increasing outer layer thickness from 5 mm to 18 mm.It is evident from Fig.12 that although the peak pressure of detonation collision changed greatly with the increase of the outer layerthickness,the fracture position of the shell did not change,even when the thickness was only 5 mm and the peak pressure was only 45.17 GPa(1.7 times the C-J pressure),the shell still fractured at the similar position.It is because that the shell bore a large circumferential strain when the explosive expanded violently,which is much larger than the axial strain.The detonation did not need to cut the shell completely.As long as defects are produced,they will develop into cracks under stress.We also simulated the cutting process of the 10-mm-thick outer layer with horizontally distributed grooves,and it was found that the shell could not be cut along high-pressure regions without a huge strain.

Fig.10.Numerical results:(a) Distributions of five gauges and pressure at 6.45 μs and (b) Pressure history of gauges.

Fig.11.Simulation result of cutting shell at 60 μs.

Fig.12.Peak pressures of interaction on the shell and simulation results for outer layer thicknesses from 5 mm to 18 mm.

5.Conclusion

1.An annular detonation wave regulator with grooves was designed to realize circumferential detonation collision.Experiments and simulations were carried out to investigate the effect of the self-designed regulator on a cylindrical shell.The detonation wave interaction generated local high-pressure regions on the shell,and the shell fractured into large and regular fragments.

2.Due to the circumferential detonation collision,two types of fragments were formed and the macrostructure and micro characteristics of typical fragments obtained.It is found that the interaction of dual waves and the first contact position of the C-J wave played a key role in the fracture mechanism of the shell.In the process of shell expansion,these two positions preferentially initiate embryos and evolve into cracks,eventually leading to shell fracture at these positions.

3.The effects of outer layer thickness and groove number on the fracture behavior of the shell were analyzed by numerical simulations.When the thickness of the outer layer was 5-18 mm,the fracture intensity of the shell was very small,which was compensated by a large circumferential strain.When the outer layer thickness was 10 mm,the number of grooves on the regulator increased in the range of 3-6 and the fragment width became shorter with the increase of the fracture position.However,the groove number was not the main reason for the change of the fragment width.The increase of the groove number reduced the possibility of fracture at the first contact position of the C-J wave.When the groove number reached 7,the fracture only occurred at the position of detonation collision and the fragment width rebounded.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The work presented in this paper has been funded by the National Natural Science Foundation of China No.11972018,and the Defense Pre-Research Joint Foundation of Chinese Ordnance Industry No.6141B012858.