Indian Journal of Engineering & Materials Sciences Vol. 10, October 2003 , pp. 38 1-389

Effect of the variation of conical liner apex angle and explosive ignition point on shaped charge jet formation

I Gokhan Aksoy a & Sadri Senb*

a Department of Mechanical Engineering, Engineering Faculty , Inonu University, Malatya. Turkey

b Department of Mechanical Engineering, Engineering Faculty, Ataturk Uni versity. Erzurum, T urkey

Received 6 March 2003; accepted 10 July 2003

Shaped charge technology has an important ro le in the military and commercial areas, as bullets and warheads made by the shaped charge princ iple are especially used for penetration to target and demolition purposes. In early studi es, the so luti on of shaped charge problems has been obtained by using anal ytical methods. However, over the past decade, numerical methods based on fi nite difference and finite e lement methods are used fo r simulating this complex process. In this study, the effects of varying parameters (conical liner apex angle and explosive ignition place) are investigated by a numerical modelling of the shaped charge perfo rmed using the Dyna2D hydrodynamic code. The 8 1 mm precision shaped charge warhead which is designed by Ball istic Research Laboratory (BRL) is taken as a reference charge. The simulations are carried out from 13° to 46° of the apex angle because both the jets do not occur completely for the apex angles equal or smaller than 13° and the velocity decreases fo r the apex angles larger than 46°. The velocity of the jet ti p increases very slow ly with the decrease of the apex angle from 38° to 13°. When the apex angle in this range decreases the jet tip becomes shorter while the jet slug gets longer and thicker. Additionally, it is realized that it needs longer time for the jet formation, and moreover, the jet does not occur complete ly. For those reasons, the parametric evaluations are reali zed by changing the coni cal liner apex angle (38°, 40°, 42°, 44°, 46°) and explosive ignition place (central point, plane and ring). As a result of the vari ation of the parameters, some important vari ati ons are determined in pressure, velocity gradient and jet formation of the shaped charge through simulati on. The present results are found in good agreement with the reported results.

The high-speed jet resulting from the detonation of shaped charge is used to penetrate and demolish hardened targets in military and commerci al areas. Particularly , the shaped charge is mainly used to penetrate armored vehicles such as tanks and to demolish bunkers, fuel tanks and bridge constructions. In addition, warheads designed by using the shaped charge principle are also used against airplanes and submarines. In the industry, shaped charges are generally used in geophysical areas, e.g., petroleum research, mining, steel industry and well bore penetration, underwater trenching and demolition t. Two- and three-dimensional numerical simulation of shaped charges and targets play an important role in testing and designing of shaped charges. In recent studies, computer programs based on Eulerian and Lagrangian calculations such as "hydrocode" are utilized to design shaped charges. Generally, the programs provide a good tool facility to analyze problems under very large material deformations.

*For correspondence (E-mail : sasen @atauni .edu .tr)

The purpose of this study is to perform an optimum modelling in order to benefit the shaped charge design in an effective manner. Especially, the effects of varying parameters are investigated by numerical modelling of the shaped charge using the Dyna-2D hydrodynamic code, which based on the Lagrangian formulation, is developed for analyzing large transient dynamic deformations and the hydrodynamic behaviour of inelastic solids. In the present shape charge simulations Dyna-2D is used for solutions of the high explosive detonation and the jet formation after the collapse of the conical liner. In addition, the characteristics of the jet created are calculated by "2DJET" subrouline based on the analytic "PER" theory, which is executed in conjunction with Dyna-20.

Description of Shaped Charges The shaped-charge prinCIples were first used in

developing weapons by the Germans during World War 1. During the Second World War years by making some modifications, it was used to demolish of concrete walls and hardened armors with a small

382 INDIAN J. ENG. MATER. SCI., OCTOBER 2003

quantity of explosive . The best-known application of these principles was in American and German bazookas against tanks in an effective manner' -3. At present, the geometry of the modern shaped charges consists of a cylinder of explosive with a hollow cavity in one end and a detonator at the other end. The hollow cavity is usually lined with a thin layer of metal, glass and ceramjcs . The cross-section of a typical shaped charge is given in Fig. l.

The ignition of detonator causes a shock wave, which detonates the high explosive. The detonation waves which propagate spherically into the explosive moves at a very large velocity, around 5-10 kmls. When the detonation wave reaches the conical liner surface, a conical liner is accelerated from the apex to the base on the cylinder axis under the high detonation pressure, collapsing the liner. Pressure on the stagnation point is approximately 0.5 MPa. As the conical liner material is loaded with high energy, it collapses on the axis, which causes a hypervelocity jet. It is commonly known that the inner liner material on the cavity side forms the jet tip which has an extremely high velocity, while the outer of the liner material which is in contact with high explosive forms a jet tail called "slug" which is massive but has a low velocity. The front of the jet Get tip) has a velocity range 2-10 kmls, while the back of the jet Get tail) has a velocity range 0.5-1 kmls ~ The conical liner collapse and jet formation process in a shaped charge after initiation of detonation is depicted in Fig. 2.

A highly energetic jet causes a deep crater when it penetrates on a metal biock or armor. The effect of penetration of the shaped charge on a target is as schematically shown in Fig. 3. When a jet strikes the metal target it induces a high pressure of around 100-200 GPa and then the pressure decreases to around 10-20 GPa. The penetration occurs at a rate of 106

_

107 lis. The crater on the metal plate with respect to jet-target interaction causes very large pressure effect with lateral displacements. The penetration capability increases when the shaped charge is detonated at a

DelonalOr

Booster

Ca..c

Fig. 1- Cross-section of a typical shaped charge

finite distance from the base to the metal plate or target, which is called "stand-off' distance. Generally, the stand-off distance is taken up to 4 or 8 times of the charge diameter2

. When the stand-off distance is not exactly adjusted, the penetration capability of jet decreases. For instance, if a stand-off distance is very short, a penetration occurs before a jet formation is not completed. On the contrary if it is very long, a jet elongates breaks into small particles (break-up) before it reaches the target due to velocity gradient along jet3

.

Then, a precision adjustment of a shaped charge is made for a whole effect of jet into target before it breaks-up into small particles.

Several numerical simulation and experimental studies of the jet formation and stretching processes have been carried out4

- ". Mayselees et al.4 have focused on the effect of the explosive energy on the shaped charge jet formation characteristics . They found that the jet formation flow cannot be considered as a steady-state process . Titov5 dealt with the brief discussion of experimental data on ultimate elongation of metals in a jet before its break-up, and defined the optimal velocity distribution for the

Detonation Wave

High Pressure Detonation

Siu

opper Liner Cone

Jet Movement

\ Jet

High Explosive

Fig. 2- Liner collapse and jet formation process

Stand­Off

Fig. 3- Penetration of a shaped charge jet into a target

AKSOY & SEN: SHAPED CHA RGE JET FORMATION 383

penetration into the target. Wlodarczyk et aL.6 studied on the analysis of the jet virtual origin concept. In their study, the theoretically determined virtual origin location for a test charge was used for the estimation of break-up times from flash radiographs of particulated jets and prediction of penetration depth . Rice el aL. 7 carried out a study on comparison of the fragmentation characteristics of tungsten, tantalum and steel. The material ranking generated during the process shows a correlation between the case expansion and the initial fragment velocity. Marriott et aL.s presented a computer modelling of small fragmenting warheads in 3D and evaluated several computer models by comparing predicted fragment velocities and distributions with those from firing three small, pre-formed fragment warheads. The jet formation and its penetration processes into targets, formed by the conical shaped charge with inhibitor, were simulated9 by a hydrocode, and the numerical analysis includes (i) phase change of the liner material, (ii) the decrease of jet mass during the flight, and (iii) the hollow and low-density effect of jet. Fukuoka el aL. 10 described the focusing methods utilizing the shaped charge and presented the results of the converging run. Lee' s II purpose was to obtain a good shaped charge design using a 20 Eulerian code such that the high speed jet of the liner creates a large hole and small hole in the gun pipe, and showed that the geometry and thickness of the liner are the most pertinent parameters for a good shaped charge design .

Analysis of Problem and Model Generation Maze-2D, Dyna-2D and Orion-2D computer

programs have been used in the numerical modellingI2.17. Firstly, Maze-2D as a pre-processor and mesh generator is used to prepare an input file for Dyna-2D, in which the geometry of the model , the finite element meshes, the material types, the initial

and the boundary conditions are defined. Then, Dyna-2D analysis is run which is an explicit, non­linear, hydrodynamics finite element code used for analyzing transient dynamic behaviour of materials under dynamic loading. Finally, Orion-2D as a post­processor is used to display and produce graphics output from the analysis Dyna-2D. The numeric formulation of the shaped charge problem is based on continuum mechanics. Dyna-2D uses mass, momentum and energy conservation equations to compute a continuum solution of the explosive detonation, shaped charge liner collapse and jet formation processes . Governing equations related to the theory are given briefly in Appendix A. In this study, BRL 81 mm precision shaped charge geometry is taken as a reference shaped charge geometry, and the effect of the variation of apex angle is analyzed.

o B

656

N ., ,,'

N ..

.., N

Fig. 4- The basic geometry and dimensions of the BRL 81 mm preci sion shaped charge (dimensions in mm).

Table I-The parameters used in the analysis

Apex angle ( 2a)

Explosive ignition place

Liner wall thickness (mm)

High explosive material

Central Point Plane Peripheral (Ring)

Central Point: Point 0 in Fig. 4

Plane: A circu lar surface with diameter AB in Fig. 4

Peripheral (Ring): The ci rcumference of the circul ar surface with diameter AB in Fig. 4

1.91

Comp-B (which is a mixture of 60% RDX (C3H6N60 6, cyclotrimethylene-trinitramine) and 40% TNT (C7HsN30 6, trinitrotoluene) by weight

384 INDIAN J. ENG. MATER. SCI., ocrOBER 2003

The BRL 81 conical liner is made of copper with a constant wall thickness of 1.91 mm and apex angle of 42° (2a) and a base diameter of 81.3 mm as shown in Fig. 4. BRL 81 mm precision shaped charge are formed with a cylinder of high explosive COMP-B surrounded by an aluminum case and a conical liner is placed at one end of the high explosive. In addition, the total head length and the diameter are 179.8 (::114.8+42+23) mm and 90 mm, respectively. In the s mulations, conical liner apex angle and explosive ignition point are changed while the head length and cone base dimensions being held fixed. The apex angle and the explosive ignition place analyzed in the present numerical model are given in Table 1. The ex plosi ve ignition place are selected as three types, which are as follows: (i) central point (point type burning); the explosive ignition place is selected as a point 0 placed at the center of line AB in Fig. 4, (ii) plane (plane wave burning); the explosive ignition place is selected as a circular surface (because shape charge geometry is axisymmetric) whose diameter is AS line; (iii) ring (peripheral type burning); the explosive ignition place is selected as a line which is the circumference of the circular surface defined in (ii ). The geometry and dimension of the shaped charge are given in Fig. 4. The whole mesh in the model is an arrangement of 347 elements, and the total number of nodes is 478 as shown in Fig. Sa. The number of elements for the conical copper liner, high explosive and case material is 100, 171 , and 76, respectively. In the simulation, copper and aluminum material types are selected for conical liner and case materials. The material behaviour of the liner and case materials are simulated using Steinberg-Guinan high elastic-plastic ' 3 material model which is suitable in high deformation rates (> 105 s·'). According to Steinberg-Guinan model, shear stress is a function of temperature and pressure. As an equation of state, Gruneisen equation-of-state used to define pressure in compressed materials is selected.

Comp-B which is a secondary high explosive type is selected as a high explosive material in the simulations. It is a mixture of 60% RDX (C3H6N60 6,

cyclotrimethylene-trinitramine) and 40% TNT (C7H5N30 6, trinitrotoluene) by weight. The density, pressure, and detonation velocity of Comp-B is 0.72 g/cm3, 0.2950 Mbar and 0.7980 cm/J..lS, respectively'8. "Programmed Burn Model" is used for the detonation of high explosive'9, which assumes that the detonation wave travels at a constant velocity

equal to the Chapman-Jouguet (eJ) detonation velocity. The times at which the wave arrives and leaves a particular cell are calculated and the high explosive is deposited linearly in the cell during the time interval that detonation wave is within the cell, and the total chemical energy has been added to the cell. JWL equation of state is used for defining the pressure for the reaction products. The values of the materials of the liner (copper), case (aluminum) and high explosive (Comp-B) are selected from the values of DYNA-2D material database' 8.

Results and Discussion

BRL 81 mm precision shaped charge is taken as a reference shaped charge in the numerical analysis. In the simulations, the parameters (apex angle and explosive ignition point analyzed in this study) that affect the jet characteristics are changed (Table 1). The simulations are carried out from 13° to 46° of the apex angle because both the jet does not completely occur for the apex angles equal and smaller than 13° and the velocity decreases for the apex angles larger than 46°. The velocity of the jet tip increases very slowly with the decrease of the apex angle from 38° to 13° and when the apex angle in this range decreases, the jet tip becomes shorter while the jet slug gets longer and thicker. Additional ly, it is realized that it needs longer time for the jet formation, and moreover, the jet does not occur completely. It means that the penetration depth and its diameter at the target become smaller than the completed jet's. These results can be explained that as the apex angle decreases, the mass of the liner increases while the mass of the high explosive decreases because of fixed diameter and the length of the shaped charge penetrator. These variations affect the gradient of the velocity of the jet and the jet forming process. For these reasons, the results are given and criticized for the shaped charge cone of the apex angle of 38°, 40°, 42°, 44°, 46°, and three different explosive initiation positions, namely, central point (poi nt), plane (plane wave) and ring (peripheral) initiated at the rear of the explosive.

In Fig. 5a-h, the deformation processes and the mesh positions are given at different time instances for the BRL 81 mm precision standard shaped charge (the apex angle of 2a=42°, the explosive ignition place is point 0 which is the center of AB line in Fig. 4, the liner thickness is 1.9 mm and high explosive material is Comp-B) from the initiation of high

AKSOY & SEN: SHAPED CHARGE JET FORMATION 385

explosive detonation to the jet formation. The initial finite element mesh is shown in Fig. Sa. As seen in the figure, the copper liner cone starts collapsing from the apex portion in a very short time interval (-16 JlS)

after initiation of detonation (t=0 JlS). With the collapsing of all liner elements, a hypervelocity jet is formed with the jet tip and jet tail (slug) portions. The collapse of all liner elements and jet formation takes about 41 Ils. In addition, as a result of the expansion of detonation products causes the case material to laterally expand.

~.BII

~ • • • ~ • • • • T T i ,. ,; • .. . a) t=o !lS

"1 .•

U . ..

14 . •

11 ••

11 ••

.... '.11

<4."

.... • 11

• • • • ~ • • • • ,. or i 'i oW • .. ,;

b) t=12 !ls

In the simulations performed for different apex angles (38°, 40°, 42°, 44°, 46°), it is observed that as an increase of apex angle, the jet tip length increases and slug part decreases. In addition, as an increase of apex angle, the forward jet (tip) diameter increases.

The velocity of the inner liner elements on the cavity side at the time (-41 Ils for our study in Fig. 5h) at which the collapse of all liner elements and jet formation taken is plotted versus the normalized initial liner axial position given as percentage in Figs 6 and 7 for three different apex angles (38°, 42°, 46°)

• • • , T T

c) t=16 !lS

111."

lA ••

12._

11."

..-I." 1._

....

- R • •

• • • ~ .; T

d) t=20 ~s

• • f or

• I t 'i

~ • ..

I .; • •

.;

• ..;

• .. • • ,; ::

• I

386 IN DI AN J. ENG. MATER. SC I. , OCTOBER 2003

l l .!J1I 16. 8111

II.U 16. M

17.58 1"'1. !tI!I

15 . l1li lZ. R

1~ . 58

liL Iili

18 . 1111

I .•

' . 51 ... 5 . 1 11 ..... ! . Sg) ..... . .. ..

- 1 . 511

-Ol._ -S . DIII

-" . l1li -1 . 511

- (i •• -11 . "

i I I I Ii ~ I I I i

~ . oj T i 7 N ,; ,; .. N R I II " I II I R I II • ~ . t '/ .. ,; ~ .. .. .

e) 1=26 ~s j'

g) 1=38 ~s

! 1iI . al

n.58 1B.U

11 . 1.

U .BIII

l ' . SII

H .D

1' .• ' 12 .11

11.:1'

li1 . QIi

111.811

1 .10

? 5&

"_ .. S.11iI .... r. sl

l .. 81

. 11 ... - Z.:I.

-r •• - :::I.el

- C. ft

- f.58

-I . • ' -1 • . ••

- 8." -ll' . SI I

I I I ,. • I I ~ I I I I I I • I I I • I • ~ IJ I I I • I • ~ ~ ~ 'i .; i T oj .. ,; .. ~ ~ ! ~ ~ . ~ ,: ., T .; .,; .: ~ ~ ~ ~ , ,

f) 1=32 ~S h) 1=41 ~S

Fig. 5- Deformation processes and mesh positions: (a) t=O /-ls, (b) t=12 /-lS, (c) t=16 /-lS, (d) t=20 /-lS, (e) 1=26 /-lS , (f) t=32 /-lS, (g) t=38 /-lS, and (h) t=41 /-ls

Table 2-Jet tip velocity (Poi nt- initiated shaped charge)

Apex angle

Jet tip velocity (Iun/s)

38°

8.628

40°

8.339

42°

7.962 7.733

46°

7.558

AKSOY & SEN: SHAPED CHARGE JET FORMATION 387

and for three different explosive initiation positIOns defined in Table 1. The comparison of the velocities is made between the jet tip velocities (the front of the jet) in Tables 2 and 3. According to Fig. 7a-b for the apex angle 38°, 42° and 46°, the velocities increase rapidly up to 10% of the liner length from point C, then the velocity remains nearly same between 10-40%, and then between 40-100% of the liner length the velocity of collapsing liner elements decreases. Similar characteristics in the velocity of the elements are observed for the other conditions (2a=40°, 44°) .

Case

y

r.

1 , "1 .t <; Ii 7 Jt Q 0 Initial Liner Position (%)

• h------~

Fig. 6--Normalized initi al liner position of the BRL 81 mm precision shaped charge.

liner Thickness : 1 .9 0 5 mm 0 .9

0 .8

0 .7

i 0 .6

2:- 0 .5

~ o .~ ~

0 .3

,./ Comp·8 ---...,

~ ""- 38' _

/;~ ~ I--~ ~ 'u.... 42°

III "--~~ ~ .......... 46°

I ~ J ~ 1 ~ ~

0 .2 ~ ~

0 .1 o 10 20 30 40 50 60 70 80 90 100

Ini ti al Liner Position (y/h %)

0 .9

0 .8 . 0 .7 "-5 0 .6 ~ 8 0 .5

~ 0.4

0 .3

.Apex Angle: 3 8°

~ ~ ~ I Comp-B -

-w;~ "0..... Point

'b.... Peripheral

t '" .......... Plane wsw

I ""'I ~

L "-I f'..

0.2 .... ~

0.1 o 10 20 30 ~O 50 60 70 80 90 100

Initial Liner Pos ition (y/h %)

b) 2a::3So, CompoS, ignition type : point, peripheral and plane wave.

Because of forming the front of the jet formed by the fast elements of the liner mentioned as earlier. In the study, it is focused the high velocity region of the liner which is between the 10-40% of liner position.

As seen in Fig. 7a, the velocities decrease as the liner apex angle increases. The decreasing of the jet velocities increase between about the 10-60% liner position and decrease in the region of 60-100%. For instance, the jet velocities of the liner elements at 30% for 38°, 42°, and 46° cones are 8.618, 7.911 , 7.448 kmls, respectively. The jet tip velocities for the five different copper liner apex angles are given in

Table 3- A comparison of jet tip velocities for the reference shaped charge (BRL 81 mm)

Researcher

Murphy

Bolstad and Mandell

Thiel and Levatin

In thi s study

0.9

"'*' 0 .8

0 .7 rr-. "- 0.6

~ 0 .5

8 ~

0.4

0 .3

1/ r I /

0.2

0 .1

Jet tip velocity (krn/s)

8.04

8.00

7.00

7.962

~ ,., I&.

"'"

References

17

20 21

.Apex .Allgle: 42° ' . -I Comp-B

'u.... Point

u.... Peripheral

.......... Plane wsw

"'\... i'.. ~ ~-

-o 10 20 30 ~O 50 60 70 80 90 100

Initial Liner Pos ition (y/h %)

c) 2a- n o, Comp""B, ign ition type: point, peripheral and plane wave.

Apex Angle: 46° 0 .9 lcomp.,e -0.8 "0..... Point

0.7

i 0 .8

2:- 0.5

:B 0 .4 ~

0.3

0 .2

~ -... 'D...... Peripheral

j, ....,~ ............. Plane wa~

# ~ i ~

/' ~ t ~

0 .1 o 10 20 30 40 50 60 70 80 90 100

InitialUner Position .(ylh %)

d) 2a::.46°, Comp-S, igni t~n type: point. peripheral and plane wave.

Fig. 7- The velocities of the elements of normalized initial liner position (liner thickness= 1.905 mm): (a) 2a=38°-42°-46°, Comp-B, the point ignition, (b) 20.=38°, Comp-B, the ignition of point, peripheral and plane wave, (c) 2a=42°, Comp-B, the ignition of point, peripheral and plane wave, and (d) 2a=46°, Comp-B, the ignition of point, peripheral and plane wave

388 INDIAN 1. ENG. MATER. SCI., OCTOBER 2003

Table 2. As seen in this table, the jet tip velocity decreases when the apex angle increases.

In Fig. 7b-d, the velocity distribution curves are given at different apex angles (38°, 42°, 46°) with a constant liner thickness (l.905 mm) for each different explosive initiation place. As seen in the figures, the liner elements velocities in a position 2-8% of the liner length does not change, and the velocity increases in the direction of the case of central point, peripheral and plane wave initiation of explosive in a 8-40% of the liner length. In addition, between the 40-100% of the liner length, the jet velocity nearly remains same. For instance, for the central point, the peripheral and the plane wave initiated at 38° of the apex angle, the jet velocities are 0.8258, 0.8623, 0.8775 crn/I-ls at a point of 16%, respectively. Similar results are observed for 42° and 46° of the cone angle.

Table 3 shows a comparison of the jet tip velocity with respect to the other studies given in the litera­ture l

? 20, 21 for a BRL 81 mm precision shaped charge. The obtained results show a good agreement with the first two studies 1?20.

Conclusions The purpose of this study is to perform an optimum

modelling in order to benefit the shaped charge in an effective manner. It was realized that the velocities for the point ignition decrease especially in between 10-40% of liner position with increasing apex angle when the other parameters are the same. The highest velocity and the lowest velocities between all ignition pl.aces were obtained at the apex angle of 38° and 46° respectively. For each apex angles, the highest velocity was obtained in the point ignition while the lowest velocity was at the plane wave ignition types. As the apex angle decreases, the velocity of the jet tip increases slowly, but the shape and the velocity gradient of the liner become different at each angle. These differences give different effects to any target such as thin and deep penetration or large and thick crater. A decision for the selection of dimensions of a shaped charge penetrator should be performed according to the aim of what type of effect is needed.

References Walters W P & Zukas J A, Fundamentals of shaped charges (John Wiley & Sons, New York), 1989.

2 Held M, J Explos Propellants, 7 (1991) 1-7. 3 Wa lters W P & Summers R L, An allalytical expression for

the velocity difference between jet particles frol1/ a shaped charge (A merican Institute of Physics), 1994, 1861 - 1864.

4 Mayseles M , Hirsch E, Lindenfeld A & Schwartz A, Effect of explosive )n the shaped-charge jet form ation characteristics, 16th Int Symp Ballistics, San-Francisco. USA, Sept 23-28, 1996.

5 Titow M & Vladimir M, Ultimme elollgation of metallic shaped-charge j ets , 16th [nt Symp Balli stics, San-Francisco, USA, Sept 23-28, 1996.

6 W lodarczyk E, Jack K, Trebinski R, Mroczkowski R, Swierczynski R & Cudzilo S, Allalaysis of the jet virtual origill cOllcept, 16th [nt Symp Ball istics, San-Francisco, USA , Sept 23-28, 1996.

7 Donna J R, Kreider W, Garnett C & Wilson T, Leonard, Comparillg fragmentation characteristics of tungsten, tantalum and steel, 16th Int Symp Balli stics, San-Francisco, USA, Sept 23-28, 1996.

8 Marriot C 0, LR MacMahon D J, Fairlie G E & Ran son H J, Computer modellillg of small fragment warheads ill 3D, 16th

Int Symp Ba llistics, San-Francisco, USA, Sept 23-28. 1996. 9 Katayama M , Takeba A, Toda S & Ki be S, Int J Impact Eng,

23 (1999) 443-454. 10 Fukuoka H, Fujiwara K, Matsuo H & Hiroe T, J Mater

Process Teclmol, 85 (1999) 60-63. 11 Lee W H, Int J Impact Eng (In press). 12 Engelmann B E & Hallquist J 0 , A nOIl -lill ear, Implicit, 1W0-

dimellsional finite element code for solid mechanics, NlKE2D User Manual , April 199 1

13 Whirley R G, Engelmann B E & Hallquist J 0 , A non-linear, explicit, two-dimensional finite eleml!lll code for solid mechallics, DYNA-2D User Manual , Jan 1995.

14 Murphy M 1, Users Guide for 2DJET Analysis of Shaped Charges, Lawrence Livermore National Laboratory, UCID-21451,1988,1-21.

15 Hallqui st J 0, MAZE- An Input Gellerator for DYNA-2D alld NIKE-2D, Lawrence Livermore National Laboratory , UCID-19029, 1983, Rev. 2, 1- 114.

[6 Hallquist J 0& Levatin J L, ORION- An interactive colour post-processor for two-dimensional finite element codes, Lawrence Livermore National Labora tory, UCID 19310, 1985, Rev. 2, I-53 .

17 Murphy M J, Description of a PC-Based COl/pled "PER Theory/Finite Elemellt Method for Calcularing Shaped Charge Jet Characteristics", 12th Int Symp Balli stics, San Antonio, Texas, 30 Oct.- I Nov., 1990,2, 456-465.

18 DYNAMATS, Hydrosoft International Material Model Database, Lawrence Livermore National Laboratory, Jan 1995.

19 Mader C L, Numerical Modelling of Detollation, (University of California Press, California), 1979, 1-4 .

20 Bolstad J & Mandell D, Calcularioll of a shaped charge jet using MESA-2D and MESA-3D hydrodynamic computer codes, Los Alamos National Laboratory, Technical Report, 1992, LA-12274, 1-33.

21 Thiel M V & Levatin 1, J Appl Phys, 51 (1 2), 6107-6114. 22 ANSYS/LS-DYNA-3D Theoretical Manual, Houston, USA,

June 1996.

Appendix A

Formulation 12, 13,22

Consider the body shown in Fig. A I. The a im is time­

dependent deformation in which a point in b initially at Xa (cx= I,2,3) in a fixed rectangular Cartesian coordinate system

AKSOY & SEN: SHAPED CHARGE JET FORMATION 389

aB,

Fig. AI-Notation

moves to a point Xi (i= I, 2, 3) in the same coordinate system. Since a Langrangian formulation is considered, the deformation can be expressed in terms of the convicted coordinates Xu and time t:

... (AI)

At time 1=0 the initi al conditions are written:

(A2)

X; (X u ,0) = V; (X u) (A3)

where Vi defines the initial velocity.

Governing equations

Equation of motion Eq. (4) as follows:

... (A4)

Eq. (AI) should satisfy the traction boundary conditions:

... (AS)

on boundary obi ' the displacement boundary condition:

... (A6)

on boundary CJb2

the contact discontinuity:

. . . (A7)

along an interior boundary ob3 when xt = x~ . Here cr ij is the Cauchy stress, p is the current density,1, is the

body force density , X; is acceleration, the comma denotes

covariant differentiation, and n j is a unit outward normal to a boundary element of a b.

Mass conservation is trivially stated:

pv =po ... (A8)

where V is the relative volume, i.e., the determinant of deformation gradient matrix, Fij:

Ox F.. =--'

'j ax . j

And Po reference density. The energy equation:

. . . (A9)

... (A 10)

is integrated in time and is used for equation of state evaluations and global energy balance In Eq. (10), Sij and p represent the deviatoric stresses and pressure:

'" (All)

... (AI2)

respectively, q is the bulk viscosity, Oij is the Kronecker delta

(Oij=1 if i=j; otherwise Oij=O) and £ij is the strain tensor.

It can be written as:

f (px; - a;j.j - PJ; )8x;dv + f (a ijnj - I; )8x;ds ab,

+ f (a: - aij )nj8x;ds = 0

a"" . .. (AI3)

where OXi satisfies all boundary conditions on ob2 , and the

integrations are over the current geometry application of the divergence theorem, gives:

f (aij8x) j dv = f a;jnj8x;ds + f (a: - aij )nj8x;ds ab, a""

... (AI4)

and noting that

. . . (AIS)

leads to the form of the equilibrium equations:

8n = f px;8x;dv + f aij 8x;. jdv - f pJ;8x;dv - f 1;8x;ds = 0 ab,

... (A16)

a statement of the principle virtual work.