GONG Chao-an， CHEN Zhi-gang， YIN Li-kui

(National Defense Key Laboratory of Underground Damage Technology, North University of China, Taiyuan 030051, China)

龔超安， 陳智剛， 印立魁

(中北大學(xué) 地下目標(biāo)毀傷技術(shù)國防重點學(xué)科實驗室， 山西 太原 030051)

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Analysis of movement laws of fragment and shock wave from a blast fragmentation warhead

GONG Chao-an， CHEN Zhi-gang， YIN Li-kui

(NationalDefenseKeyLaboratoryofUndergroundDamageTechnology,NorthUniversityofChina,Taiyuan030051,China)

By studying shock wave propagation and fragment movement after the explosion of a blast fragmentation warhead, a whole calculation process about the place where fragments meet shock wave was proposed. A computing system for movement laws of fragments and shock wave was developed based on VC++. Numerical segment integration method is used for the calculation of shock wave velocity and displacement, which makes the calculation be more convenient. The movement of preformed fragments and shock wave was simulated by ANSYS/LS-DYNA. The results show that the simulation is nearly equal to calculation.

fragment; shock wave; LS-DYNA; VC++

0 Introduction

Blast fragmentation warhead mainly as anti-missile warhead has been researched a long time. Blast warhead mainly has three target damage effects: direct effect of products, damage effect of shock wave and kill effect of fragments. As two target damage ways, shock wave and fragment has different action principles therefore action order affects the effectiveness of damage[1-3]. The Third Research Institute of General Staff Corps of Engineers has fitted the distances with different ratios, load coefficient with the time during which fragment meets shock wave according to experiment data[4-7]. Therefore, study on the position where fragment meets shock wave is extremely important.

1 Calculation of TNT equivalence

1.1 Transformation from shelled charge to uncovered charge

When shelled explosive explodes, according to energy conservation law, the whole energy is transformed to the kinetic energy (KE) and internal energy (IE) of explosion products and the kinetic energy of the fragments, that is[8]

(1)

whereE1is the IE of explosion products;E2is the KE of explosion products;E3is the KE of fragments; andQvis explosion heat. The IE of explosion products is expresed as

wherer0is initial radius of the shell;ris expansion radius;γis polytropic exponent and equal to 3; andbis shape factor and equal to 2 for cylinder shell.

The KE of explosion products is expressed as

wherev0is the expansion velocity of shell;ais shape factor and equal to 2 for cylinder shell.

Thewholeenergyofexplosionproductsis

E1+E2=mQv-E3=mbeQv.

Theshelledchargecanbetransformedtouncoveredchargeby

(2)

whereβis load coefficient,β=m/M.

1.2 Transformation from other charges to TNT sphere charge

To transform other charges to spherical TNT charge, supposing that the explosion heat of explosive isQviand the mass ismbe, the transformation is done by

(3)

whereQvTis the explosion heat of TNT. Almost all of blast warheads are cylinders, therefore it can be calculated approximately as spherical charge when the propagation distance of shock wave is farther than the length of charge.

2 Calculation for velocity of shock wave

2.1 Calculation for overpressure of TNT

For the uncovered spherical TNT charge explosion in the infinite air, according to the specifications for design of national defense engineering in China[8], positive overpressure peak is calculated by

(4)

2.2 Relationship between overpressure and velocity of shock wave

The Rankine-Hugoniot formula shows the relationship between overpressure and velocity of shock wave[9], that is

(5)

(6)

(7)

Calculating the integral of reciprocal of Eq.(6), the relationship betweentandris got by

(8)

Due to the complexity of analytic solution for Eq.(8), Eq.(8) is converted into numerical integration in experimental calculation. After getting the step, a series of integral points are got and then they are drawn in the axis. The density of points shows the accuracy directly.

3 Calculation of fragment velocity and its decline

3.1 Calculation of fragment velocity

The Gurney formula is deduced according to energy conservation law[10], that is

(9)

(10)

3.2 Description for velocity decline of fragment

The formula for fragment velocity decline is[11]

(11)

Table1Modificationfactorandattenuationcoefficientofinitialvelocity

KindsoffragmentsKH(m-1kg12)Naturalfragment1320PreformedfragmentsSphere0.7548Cylinder0.8416Cube0.9399Sector0.9363Half-preformedfragment0.95320

Calculating the integral of reciprocal of Eq.(11), the relationship betweentandLcan be got by

(12)

4 Determination of the place where fragment meets shock wave

For computing the displacementR, lett=T, the expression of the place and time of fragment can be got first according to initial velocity and velocity dedine. But there is not an analytic solution for Eq.(8). What is worse, the integral calculation is complex. In view of the theories and convenience, the author developed the computing systems for fragments and shock wave movement by using Visual C++. For Eq.(12), the system drew the curve in the coordinate system. For Eq.(8), the system used trapz function to compute numerical integration. After that, the point set was got and drawn in the coordinate system. MFC controls the data processing while TeeChart controls the drawing. The system shows the movement of fragment and shock wave directly. What’s more, it shows the distance between fragments and shock wave at any time. The framework of the system is shown in Fig.1.

Fig.1 Framework of computing system of fragments and shock wave’s movement

5 Comparison of calculation and numerical simulation

For the preformed steel rectangle fragment warhead, the radius of TNT cylinder charge is 4.8 cm, and the height is 50 cm. There are 42 fragments one layer in radial direction, and 22 layers in axial direction. There are 924 preformed fragments in all. Each fragment’s mass is 0.04 kg. The whole mass of fragments is 37 kg.

5.1 Calculation

These parameters are input to the computing system, as shown in Fig.2.The curves are drawn, as shown in Fig.3. After explosion, there are two stages of movement[12]. The first stage: the velocity of shock wave is faster than that of fragment, and shock wave is further than fragment. The second stage: the velocity decline is more than shock wave decline, and fragment meets shock wave. After that, fragment is further than shock wave. The system shows that fragment meets shock wave at 7.1 ms while the displacement is 4.73 m.

Fig.2 Interface of inputting parameters

Fig.3 Curves of time-displacement for fragment and shock wave

5.2 Numerical simulation

The numerical simulation is performed by using ANSYS/LS-DYNA. Because of structural symmetry, 1/4 model is created to make calculation easy. Both charge and air are arbitrary Lagrange-Eular elements (ALE) while fragments are Lagrange elements. Liquid-solid coupling is used between ALE elements and Lagrange elements. The radius of air is 7 m. The pressure of air shows shock wave. The density of TNT charge is 1.59 g/cm2, and its constitution relationship is described by MAT_HIGH_EXPLOSIVE_BURN. Its state equation is JWL. The detonation way is point detonation. The density of fragment is 7.83 g/cm2, and its constitution relationship is Johnson-Cook. Its state equation is Gruneisen. The finite element model of warhead is shown in Fig.4. Fig.5 is the nephogram of air pressure and the small cubic blocks are preformed fragments. Fragments meet shock wave at 7 ms while the displacement is 4.69 m.

Fig.4 Finite element model of warhead

Fig.5 Nephogram of air pressure

It can be seem from Table 2 that the simulation is nearly equal to calculation.

Table 2 Comparison of displacement where fragment meets shock wave

FragmentkindDisplacement(m)CalculationSimulationSteelcube4.734.69

6 Conclusion

The whole calculation process is based on theories, therefore it is helpful to applicable for all structures. The system shows the distance between fragment and shock wave at any time. The characteristics of the two stages are reflected in the picture. It is more accurately control the order and the distance between shock wave and fragment.

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某殺爆戰(zhàn)斗部破片與沖擊波運(yùn)動規(guī)律研究

通過對某殺爆戰(zhàn)斗部爆炸后破片運(yùn)動及沖擊波傳播的研究， 理論推導(dǎo)了破片與沖擊波相遇位置的求解全過程。 采用Visual C++開發(fā)破片與沖擊波運(yùn)動規(guī)律計算系統(tǒng)， 并用數(shù)值積分方法對沖擊波速度與位移進(jìn)行分段積分。 通過該系統(tǒng)計算了某預(yù)制破片戰(zhàn)斗部破片與沖擊波的相遇位置， 并運(yùn)用ANSYS/LS-DYN數(shù)值模擬了該預(yù)制破片戰(zhàn)斗部爆炸后破片與沖擊波的運(yùn)動， 數(shù)值模擬結(jié)果與計算結(jié)果基本相符。

破片； 沖擊波； LS-DYNA； VC++

GONG Chao-an， CHEN Zhi-gang， YIN Li-kui. Analysis of movement laws of fragment and shock wave from a blast fragmentation warhead. Journal of Measurement Science and Instrumentation, 2015, 6(3)： 218-222. [

龔超安， 陳智剛， 印立魁

(中北大學(xué) 地下目標(biāo)毀傷技術(shù)國防重點學(xué)科實驗室， 山西 太原 030051)

10.3969/j.issn.1674-8042.2015.03.003]

GONG Chao-an (gca_bm@sina.com)

1674-8042(2015)03-0218-05 doi： 10.3969/j.issn.1674-8042.2015.03.003

Received date： 2015-05-05

CLD number： TJ410.3+3 Document code： A