CHEN Chuang(陳闖)，WANG Xiao-ming(王曉鳴)，LI Wen-bin(李文彬)，LI Wei-bing(李偉兵)，CHEN Kui(陳奎)

(ZNDY of Ministerial Key Laboratory，Nanjing University of Science and Technology，Nanjing 210094，Jiangsu，China)

Introduction

The liner structure has an important effect on the performance of jet.Compared with traditional singleangle liner，the double-angle liner can better adjust the jet’s velocity and mass distribution.The jet with higher tip velocity and velocity gradient in the middle and rear can improve the penetration efficiency of the liner［1］.The double-angle liner is widely used in missile warhead，such as Hellfire and TOW antitank missiles.Our Red Arrow 9 antitank missile warhead adopts tandem charge structure，and the main charge uses a copper double-angle liner［2］.

The influence of liner structure on the jet formation has been studied extensively.LI，et al［3］studied the effect of liner structure parameter on the formation of multimode penetrator，and optimized a shaped charge structure.GU，et al［4］studied the jet formation of a cylinder-cone liner experimentally and obtained the variation laws of jet tip velocity and length.But，for the double-angle liner，the influence of its structure parameter on the jet formation was not studied at home and abroad.The stand-off distance influences the jet penetration greatly.The increase of stand-off distance will make the jet more fully stretched，and it is beneficial to the penetration.On the other hand，too large stand-off distance will lead to the jet break and affect the penetration seriously.This paper studies the formation of double-angle liner in large stand-off distance numerically.

By using analysis software LS-DYNA and changing some structure parameters of double-cone liner，such as top cone angle，wall thickness，liner height and height ratio of top cone，their influences on the jet formation are analyzed.

1 Simulation Model

1.1 Liner Structure

The structure of the double-angle liner analyzed in this paper is shown in Fig.1.It consists of a top cone and a bottom cone.The angle of top cone is less than that of bottom.There is a smooth transition between the top and bottom，and the top of liner is designed as the shape of arc-cone combination.The structure parameters of liner include the top cone anglea，wall thicknesst，liner heighthand height ratio of top coneh1.

1.2 Material Model and Algorithm

To solve the problem of large deformation，ALE algorithm to calculate the formation process［5］can be used.In the calculation，the high explosive material model and JWL equation of state are used.JWL equation precisely describes the pressure，volume，energy characteristics of detonation gas product of explosive.The expression is

whereA，B，R1，R2，ωare all constants，Eis the internal energy of explosive per unit volume.

The liner is made of red copper with density of 8.96 g/cm3.Johnson-Cook material model and Gruneisen equation are used as constitution and state equations respectively.The expression of Gruneisen equation is［6］

for compression state，and

for expansion state，whereμ=ρ0/ρ－ 1，Cis the velocity of material static body，S1，S2，S3are the parameters relevant to material Hugoniot，γ0is the Gruneisen coefficient，ais the first order volume correction toγ0.The air uses hollow material fluid model，the state equation is linear polynomial.The simulation model is shown in Fig.2.

Fig.2 Simulation model

2 Simulation Results and Discussions

2.1 Influence of Top Cone Angle

According to the theory of shaped charge，the smaller top cone angle can increase the jet tip velocity and penetration depth.In order to study the jet formation effected by top cone angle only，the charge height is taken as 180 mm，the liner height 140 mm，the wall thickness 2.6 mm，and the height ratio of top cone 50%.The jet formation parameters are calculated when the top cone angle varies from 24°to 32°.Fig.3 shows the parameters at 120 μs after ignition.

Fig.3 Formation parameters vs.top cone angle at 120 μs

It can be seen from Fig.3 that，with increase of top cone angle，both the jet tip velocity and tail velocity decrease，but the tail velocity declines slower than the tip velocity，and the jet length decreases gradually.The small top cone angle can increase the tip velocity and the velocity gradient，so that the jet can not extend for a long distance，thus the jet will break earlier.Seeing from the jet formation effect，the jet will break easily when the top cone angle is from 24 °to 26°，so it is not conducive to penetration.Therefore，if the top cone angle is selected as 28°，the jet has a higher tip velocity and ensures that the tip will not break.

2.2 Influence of Wall Thickness

In the simulation program，the top cone and bottom cone adopt the same wall thickness.Keeping other liner structure parameters unchanged，the influence of wall thickness on the jet formation can be studied.The charge height is taken as 180 mm，the liner height 140 mm，the top cone angle 28°，and the height ratio of top cone 50%.The wall thickness is selected as 1.8 mm，2.2 mm，2.6 mm，3 mm and 3.4 mm in the simulation.Fig.4 shows the jet formation parameters at 120 μs.

Fig.4 Formation parameters vs.wall thickness

It can be seen from Fig.4 that the jet tip velocity，tail velocity and length all decrease with increase of wallthickness.The reason is that the detonation energy is constant in the same charge volume and initiation mode.With the increase of wall thickness，the energy obtained by unit mass of liner decreases gradually，thus the collapsing velocity of liner element declines gradually，and it leads to the decrease of jet tip velocity.But，if the wall is too thin，the tip velocity and the velocity gradient are all also high，so it breaks easily.Therefore the wall thickness can be selected as 2.6 mm.

2.3 Influence of Liner Height

Keeping the charge height as 180 mm，the top cone angle 28°，the wall thickness 2.6 mm and the height ratio of top cone 50%，the jet formation is simulated when the liner height varies from 130 mm to 150 mm.The simulation results are shown in Fig.5.

Fig.5 Formation parameters vs.liner height

Seeing from the simulation results，with the increase of liner height，the tip velocity and jet length gradually rise;but the jet tail velocity tend to decline.The formation parameters change a little.The reason is that the jet can not stretch fully for the small liner height，and it leads to decrease of jet velocity.On the other hand，for the smaller liner height，the more effective charge acts on the liner element，and its motion velocity increases.In the common effect of two as-pects，the liner height influences the jet formation a little.

2.4 Influence of Height Ration of Top Cone

Selecting the height ratio of top cone from 30%to 70%，its influence on the jet formation is studied.The top cone angle，wall thickness and liner height are taken as 28°，2.6 mm and 140 mm，respectively.The simulation results are shown in Fig.6.

Fig.6 Formation parameters vs.height ratio of top cone

Figure 6 shows that the tip velocity and length of jet firstly increase and then decrease with the increase of the height ration of top cone.When the ration is 60%，the tip velocity is the maximum.When the ratio varies from 30%to 40%，the gain of tip velocity is larger.The reason is that the height of 40% corresponds to the top particles of jet，so the height ratio should not be too small.The tail velocity of jet presents the same trend to the tip velocity，but its change scope is less than that of tip velocity.Thus，the height ration of top cone can be taken as 60%.

Based on the above discussions，the final optimized liner structure parameters are the top cone angle of 28°，the wall thickness of 2.6 mm，the liner height of 140 mm and the top cone’s height ratio of 60%，as shown in Tab.1.The jet tip velocity and tail velocity are 9 030 m/s and 2 044 m/s，and its length is 761 mm.The formation effect figure shows that the jet does not break，and the tip velocity is higher.

Tab.1 Formation effect and parameters of optimized shaped charge

3 Penetration Experiments

Taken the stand-off distance as 8 times of the charge diameter，penetration experiments of doubleangle liner to 45#steel target are carried out，and the experiment layout is shown in Fig.7.The sizes of targets areφ160 mm ×500 mm andφ160 mm ×150 mm.The experiment and simulation results of penetration are shown in Fig.8 and Fig.9.Seeing from the picture，target 1－1 and 1－2 are all penetrated，and it shows that the double-angle liner has stronger penetration ability in large stand-off distance.

Fig.7 Layout of penetration experiment

Fig.8 Picture of steel specimen

The experiment and simulation results are shown in Tab.2.It shows that the inlet diameter is bigger，the target 1－1 fractures，and the experiment and simulation results are in good agreement.

Fig.9 Simulation result of penetration

Tab.2 Comparison of penetration experiment and simulation results

4 Conclusions

The influence of double-angle liner’s structure parameters on the jet formation is studied numerically，and the specific influences of the top cone angle，the wall thickness，the liner height and the height ratio of top cone on the jet formation are obtained.The jet tip velocity gradually decreases with the increase of top cone angle and wall thickness.With the increase of liner height，the tip velocity presents an improving trend.However，the tip velocity presents increasing first and then decreasing with the increase of the height ratio of top cone.The structure parameters are optimized.The simulation results show that the jet has better formation effect in the large stand-off distance.The penetration experiment shows that the penetration depth of double-angle liner can reach to 6.27 times of the charge diameter.This paper provides a reference for further study.

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［6］LI W B，WANG X M，LI W B.The effect of annular multi-point initiation on the formation and penetration of an explosively formed penetrator［J］.International Journal of Impact Engineering，2010，37(4):414 －424.