YIN Jian-ping(尹建平)，WANG Zhi-jun(王志軍)，ZHANG Yun-yi(張?jiān)埔?，ZHOU Dong(周棟)

(1.College of Mechatronic Engineering，North University of China，Taiyuan 030051，Shanxi，China;2.Unit No.63936，Beijing 102202，China)

Introduction

To meet the technical requirements of air and missile defense warhead，a novel structure of semi-preformed fragment warhead against air targets such as armed helicopter is analyzed in the paper.The work principle of warhead is that a scoring array is machined on the end face inside the scored liner，so that the stress concentration at the notch tip causes the warhead to be fractured into multiple equal-mass fragments in a predefined manner.Compared with liner without notches，the scored liner can not form jets or be exploded into projectiles under the explosive pressure，instead，it forms a focused jet of metal fragments with certain speed and mass that travels at very high speed in the cone-angle direction.A tremendous amount of practices have proved that［1－3］，based on theoretical analysis，the fracture mechanism of the scored liner can be obtained by numerically simulating the fragment formation from the fracture of scored liner under explosive loading，and can be verified through experiment［4］.It is of significant importance to the warhead fragment control researches and the warhead power improvement.

1 Model of Scored Liner

The scored liner is formed by machining V-shaped notches at the outer surface of a conventional scored liner，as shown in Fig.1.Steel 45#with the density of 7 830 kg/m3is adopted as the raw material of scored liner.The scored liner has the included cone angle of 140°，the end opening diameter of 50 mm，and thet=4 mm thickness along the axis.The notch is V-shaped，and has the depthKtof 0.2tand the sharp angle 2θof 60°.There are six notches，and the diameter of charge is 50 mm.The local stress concentration at notch tips can force the scored liner to be fractured into six pieces of equal-mass fragments under explosive loading and then fly to the target in the cone-angle direction.During the process of machining the notches on the scored liner，three important factors:material properties of scored liner，basic layout of notches and section shape of notches，need to be taken into account.The fracture process of scored liner can be effectively controlled by using the broaching technique in machining notches，and the depth and interval of the machined notches determines the amount and the shape of fragments.When the notches are too deep，the travelling speed of formed fragments will be relative low due to the rapid power leakage before the explosives supply the scored liner of warhead with optimal energy;while when the notches are insufficient deep，the stress concentration will not be achieved，which will cause the incomplete fracture and the multiple connected fragments.In addition，too shallow notches may produce the explosion-formed pills with empennages due to the distribution of periodical loading on the scored liner［5］.

Fig.1 Structural scheme of scored liner

2 FractureMechanism ofScored Liner

Fracture of scored liner under explosive loading and subsequent formation of fragments are within the research scope of elastic-plastic fracture mechanics［6］.The fracture modes of elastic-plastic structure can be classified into brittle fracture and ductile fracture.In an object，when stress near the top region of a crack is weaker than yield stress，the fracture caused by crack propagation is known as an ideal brittle fracture.Ductile fracture is one of the damage modes of rheological materials such as metal and glassy polymers.A infinitesimal ring which is perpendicular to the generator on the scored liner is chosen as the research subject.The infinitesimal ring is sheared and spread to a plane.Its approximate treatment image is shown in Fig.2.

Fig.2 Approximate treatment image of infinitesimal section of scored liner

For a scored liner under explosive loading，the stress concentration emerges near its notch tip，which causes lattice slippage and the plastic flow of its metal material near the notch tip.From a macroscopic point of view，the tensile stress can not cause the material to be fractured quickly because the metal material is in plastic yield state，while the shear stress can cause the material to slip along the maximum shear stress tracing line(slip line);from a microscopic point of view，the dislocation and movement occur among metallic lattices in plastic flow so that the cavities are formed in the discontinuous regions of the metal material.The cavities continuously grow and assemble，and finally，form the continuous cracks which lead to the fracture of scored liner.Therefore，the dominant fracture mode of scored liner in this study is a ductile shear fracture，and the fracture track is along the line of maximum shear stress(fracture track or slip line).For the plane problem，the maximum shear stress is on a pair of mutually perpendicular planes which make the angle of 45°with the main stress plane according to the basic principles of material mechanics and elastic mechanics.The scored liner bears a single-way force with less complex situation in its main stress，so it can be treated using the method similar to the plane strain treatment method.

2.1 Dislocation Model of Notch Tip Crack Propagation

The threshold value of crack propagation depends on the elastic modulus and microstructure of material，especially the particle size，environment and stress ratio inside steel.Since some simplified models of threshold conditions take into account the critical fracture stress and the minimum critical shear stress of slippage，the two types of stresses can be used for judging the crack propagation.The cracks at the tip of the scored liner are generated due to the joint function of opening and slipping forces，which is similar to the trend of threshold-fatigue crack propagation.Therefore，the dislocation model proposed by Sadananda and Shahinian is adopted as the fracture criterion in the analysis of fracture mechanism of scored liner notches［7－9］.Fig.3 shows the scheme of this model.

Fig.3 Dislocation model

As the crack propagation generally results from the plastic deformation due to slippage，the threshold condition is presumed as the minimum critical shear stress required for slippage from crack tips.Sadananda and Shahinian deduce a general formula for this critical shear stress

whereτis the shear stress of a sharp crack，μis the shear modulus，vis the Poisson's ratio，ris the length of crack，bis the depth left by the notch，γeis the surface energy andσysis the effective yield stress.

For a sharp crack，according to the following formula

whereais the notch depth，xyis the sliding surface of dislocation，thexaxis is the crack extension direction，theyaxis is vertical to the crack extension direction on the sliding surface of dislocation.

It can be written as

As for a dislocation formed at the crack tip，τxyshould equal toτat least，andrshould equal tobat least.Therefore，the threshold condition is

whenθequals to 45°，the equation can be simplified as

The adopted scored liner is made of high-quality 45#low-carbon steel with the critical shear stress of 128 MPa.The threshold condition is

According to the empirical formula of single-side notched-slab model:

whereais the notch depth，Cis the coefficient related to the shape of crack，load mode and geometrical parameter of the liner，σis the normal stress of the crack tip.

In order to meet the threshold condition，the required stress should be

When the stress value at the notch tip of the scored liner exceedsσ1，the scored liner suffers from the shear stress and is fractured into six fragments.

2.2 Maximum Shear Stress Line and Its Direction

Given the condition of balance between the forces in horizontal and vertical directions，the location of the principal stress plane can be determined based on the above approximate treatment of the infinitesimal section of the scored liner and the relationship between the stress on arbitrary sections and section azimuth variation.According to the elastic mechanics，the principal stress plane is the plane on which the maximum and the minimum tensile stresses are，and the shear stress on the principal plane is zero;the shear stresses on a pair of planes which are perpendicular to the two principal planes should be maximal and minimal，respectively.Therefore，the planes，on which the maximum shear stress is，can be determined as the horizontal and vertical planes in Fig.4 according to the presumption of plane strain，and should pass through the two planes set by rotating the principal planes for 45°clockwise and anticlockwise，respectively，as shown in Fig.4.

Fig.4 Action scheme of explosive pressure

3 Numerical Simulation of Fracture Process of Scored Liner

Based on the established notch-tip crack fracture criterion，the non-linear finite element software is used to simulate the fracture process of the scored liner under explosive loading.The material model of MAT_ELASTIC_PLASTIC_HYDRO and Gruneisen state equation are used for the scored liner，and TNT detonator material model and JWL state equation are used for explosive charge.Given the special properties of the structure and the complete model adopted to simulate the prototyping process of scored liner，a suitable simplification is made in the modeling process，and the effect of the shell is ignored.The model includes a detonator and a scored liner.Grids are divided using the Lagrange method，the divided grids are in form of solid-164 hexahedrons，and a few of them are degenerated into tetrahedrons.The grid refinement is conducted in the concerned notch regions，and afterwards，the grid nodes are combined.The numerical simulation of the fracture process of scored liner under explosive loading is performed based on the above parameters and the fracture principle，as shown in Fig.5.

4 Experimental Verification of Scored Liner Fracture

The experimental device is machined based on the size of the finite element model.The experimental study is conducted to verify the performance of fragment formation of scored liner under explosive loading，the formation speed and the penetration performance of fragments against the targets.The experimental results are shown in Fig.6，from which it can be seen that the scored liner has been fractured into multiple fragments in the predefined manner owing to the existence of notches.The focusing capability of the fragments formed by the novel scored liner is also excellent.A fast focused fragment distribution is formed in form of shape similar to plum flower，which is quite in line with the results of the numerical simulation，and which indicates that the fracture principle of notch-tip cracks is basically tenable.

Fig.5 Simulation of fracture process of scored liner

Fig.6 Experimental results of fragment formation

5 Conclusions

1)The explosive loading causes stress concentration in the region near the notch tip of scored liner，which leads to the slippage of lattices and the plastic flow of the metal material of the scored liner in this region.Therefore，the dominant fracture mode of scored liner is a ductile shear fracture，and the fracture track is along the maximum shear stress tracing line.

2)According to the numerical simulation and the experimental verification，the dislocation model basically conforms to the actual practices in judging the crack fracture in the analysis of the fracture mechanism of the notches of scored liner，and can also be applied to the research on the fragment control and power design for air-defense and antimissile warheads.

3)Numerical simulation and experimental results show that the novel scored liner can form fast focused fragments with certain mass，quantity and directivity，and can be used to attack the light armored targets，such as armed helicopter.

4)The relative depth，shape and type of notches all have certain effect on the formation and damage effectiveness of fragments，which requires further research.

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