00 JAD

oCONTRACT REPORT ARBRL-CR-00461

AN UNSTEADY TAYLOR ANGLE FORMULA

FOR LINER COLLAPSE

Prepared by :

Dyna East Corporation Ito227 Hemlock Road

Wynnewood, PA 19096

August 1981

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

Approved for public release; distribution unlimited.

81 928 157

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Additional copies of this report may be obtainedfrom the National Technical Information Service.U.S. Department of Commerce, Springfield, Virginia22161.

The findings in this report are not to be construed asS!Han official Department of the Army position, unless

so designated by other authorized docunents.

IrMlaIw rs nam NIt% 4$iiiomiemflC of my 0fpmr2dia '

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-UNCLASSIFIED-ýR IT Y CL A~ tý 1$ 1( A' CN OF TH I$ r- A GE ( r t')0:0 Ft- W 0d)

IEu OCUM TION PAGE NO BEFORE COMPLETING FORM

1. EPOT UM1ER2. GOVT ACCESSION N .RECIPIENT'S CATALSG NUMBER

CONTRACT REPORTýA ý ýR-00461 V O___________

An Unsteady Taylor Angle Formula for Liner Final -/.UOVRE

ui n

ARE T WOKUI NB

Dyna East Corporatio

227 Hemlock RoadWynnewood, PA 19096

It. CONTROLLING OFFICE NAME AND ADDRESS "If0

U.S. Army Armament Research & Development Command IV /U.S. Army Ballistic Research Lab. (flP1)ARBL) a

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Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (a! the abstract entered In Block 20, If different fraom Report)

It. SUPPLEMENTARY NOTES

IS. KEY WORDS (Continue an reveree oid* It noceeaaary and identify by block number)

Explosive-Metal Interaction Taylor AngleConical Shaped Charge Liner CollapseExploding CylinderCollapse Angle.

20. ABSTRACT (Continue an revere* aide If necessary and identify by black number)

An analytical formula for determining the direction of motion for an ex-plosively driven metal liner under unsteady conditions is presented. Thisdirection is defined by the angle iS between the velocity vector of the linerelement and the perpendicular to the initial liner surface.

'~--A formnula for determining the angle was first proposed by G.I. Taylor assin> V/2U, where V is the final liner element velocity And U is the velocity

DD i JOI1- 1473 EDITION OF I NOV 65 IS OBSOLETE UNCLASSIFIED

60- ( ~ SECURITY CLASSIFICATION OF THIS PAGE~ (We Dotes Xnte00

-...... UINCLASS.IFIEA.-141 CL. ALS ýI C0 "( LI ON F T MIS P A G C.( 14hem V'.' 4 nI~red) __________

"by which the detonation wave front sweeps past the liner. This formula is,however, accurate only under steady-state conditions where the detonationwave sweeps past identical cross sections of the explosive-liner geometry.For non-steady cases, the Taylor formula is not applicable since the exis-tence of a velocity gradient or a gradient of the typical accelerationduration along the liner may significantly affect the angle S.

The new formula is tested against both numerical calculations andexperimental data and predicts the angle 6 more accurately than the steadyTaylor formula.

The derivation of the formula along with a comparison of its predictionsfor the angle & with previous experimental work and two-dimensional codecalculations for both a conical-shaped charge and exploding cylinder arepresented in this report.

UNCLASSIFIEDZBCURITY CLAWIFICATION OP THII VAG9fftM DMIe. DnOrn.

ON. .. •

FOREWORD

This report represents Part C of a four-oart final report forContract No. DAAKll-79-C-0124. These four parts address the topics:(A) "Maximum Jet Velocity and Comparison of Jet Breakup Models"; (B)"The Virtual Origin Approximation in Hemi Charges and Shaped Charges";(C) "An Unsteady Taylor Angle Formula for Liner Collapse"; and (D) "Jet

Formation of an Implosively Loaded Hemispherical Liner." Each part isbound separately for convenience.

Accession ForNTIS C".,.

DTIC T.e0

D v fi.-- t,

Dirt 3.

I I,

1

Table of Contents

PageList of Illustrations ............. ................. 7

I. Introduction .................. ..................... 9

II. Derivation of Basic Equations ..... ............. ... 11

III. Integration of Basic Equations ...... ............ .. 15

IV. Application to Exponential Acceleration ........... ... 16

V. Constant Acceleration Equation ..... ............ .. 18

VI. Comparison with Previous Work and Two-Dimensional CodeCalculations .............. .................... .. 19

A. Conical Shaped Charge ....... ............... ... 19

B. Exploding Cylinder .......... ................ .. 24

VII. Summary and Conclusions. .... ................ 35

Acknowledgment .............. .................... .. 36

References ................ ...................... .. 37

Distribution List ............... ................... 39

JPN4ZMM .A U.AI-IIOT FIUM

List of TIIus trnt onn

Page

1. Liner acceleration diagram showing how the additionalvelocity dý contributes to the increase in magnitudeand change in the direction of the current velocity, V . 12

2. Liner position diagram showing the relation between its 13

angular velocity and its linear velocity gradient ....... .

3. Drawing of 81.3mm diameter 42* conical shaped charge. . .. 20

4. Fit of exponential velocity vs. time equation to the TEMPScomputer code data for Mass Point Number 10 (2=73.64mm) 21

5a. Final velocity V0 as a function of location Z along theliner's formation line. ........... .................. .. 22

5b. Plot showing r as a function of location k from TEMPScalculation of the 420 conical charge .... ........... ... 25

6. Comparison of the projection angle 6 from formulasand two-dimensional calculations ...... .............. .. 26

7. Exploding cylinder charge used for comparisons ......... ... 27

8. Comparison of TEMPS, HEMP and experimental data forV0 of an exploding cylinder ....... .................

9. Comparison of TEMPS, HEMP, and experimental data for theliner projection angle S of an exploding cylinder ..... ... 30

10. Exponential fit of TEMPS Mass Point velocity V vs. timeat Z=-64.38mm' ............... ....................... .. 31

11. Plot showing T as a function of location Z fittedfrom TEMPS calculation of exploding cylinder charge . ... 32

12. Comparison of projection angle 6 calculated fromformulas and TEMPS Code ............... .................. 33

7

PAM .IT a

I. Introduction

When a metal liner is driven to a velocity V by an explosive charge,it is often important to know not only the magnitude of its velocitybut also the direction in which each liner element moves. This direc-tion is conveniently defined by the angle 6 between the velocity vectorof the liner element and the perpendicular to the initial liner surface.We shall call 6 the liner projection angle. For a fragmentation charge(or exploding bomb), the projection angle will determine the final angu-lar distribution of the scattered fragments. For explosive cladding orshaped-charge liner collapse, it will determine the collapse angle Bof the given charge. The specified qualities of either the claddingbond or the jet, crucially depend on this collapse angle.

A formula for determining the projection angle 6 from the velocityof the liner V and the velocity U by which the detonation wave frontsweeps past the liner surface was first proposed by Taylor [1]:

sins = V/2U (1)

This formula, which has been the most extensively used one todate (see References 2-8), is, however, accurate only under steady stateconditions where the detonation wave sweeps past identical cross sec-tions of the explosive-liner geometry. For non-steady cases, the Taylorformula is not applicable since the existence of either a gradient,-• of the velocity V along the liner or a gradient, L, of the typical

acceleration duration r(Z denotes the length tangential. to the linersurface) may very significantly affect the angle 6, sometimes causingvery big deviations between the Taylor predictions and experimentalmeasurements.

A more accurate formula for the liner projection angle is there-fore needed. It will lead to a more accurate description and under-

standing of processes such as the collapse of the conventional shapedcharge or the hemispherical liners and the formation of the self-forg-ing fragment.

Recently, Randers-Pehrson [9] derived a non-steady liner projectionangle formula by empirically fitting a formula to the numerically cal-culated results.

In the present paper an analytically derived formula is obtained.It is based on the assumptions that (1) the detonation pressure actsnormally on the liner, and (2) the angle (9-6), which will be definedlater, is small.

We found that for the small angle assumption to be valid, the

initial radius of curvature must be small in comparison with the dis-tance traveled by the liner during acceleration. A third assumption isthat the internal forces in the liner metal can be neglected. The newformula was tested against both numerical calculations and experimentaldata and was found to predict the projection angle 6 more accurately

"than the steady Taylor formula.

9

pk M .... AAMK.NOT 11AM

In this report we address only the equation governing the angle 6and the velocity parameters V0 and T. No attempt is made here todetermine values of V0 and T as functions of explosive and liner geometryand properties. It is obvious that values of V0 and T are needed forthe eventual application of this equation. A simple method in deter-mining V0 and T will be the objective of future research. For thepresent purpose of ascertaining the accuracy of the unsteady Taylorangle equation, we shall use values of V0 and T determined either bytwo-dimensional computer code, or by expelimental measurement.

10

II. Derivation of Basic Equations

We shall first derive the differential equations describing theliner's motion. We shall limit ourselves to liners with very largeinitial radii of curvature. In other words, the liner has an almoststraight formation line. The liner could either be an axisynimetricshell, or a "plane strain" plate. The explosion wave front. is assumedto be cylindrically symmetric for the shell case with its axis of sym-metry coinciding with that of the liner's. We denote by k the lengthcoordinate along the liner and by U the velocity by which the detona-tion wave front sweeps past the liner surface along the direction ofthe liner surface.

The first differential equation deals with the increase of linervelocity with time. Let V and 6 be the magnitude and direction of theliner velocity at the point 2 on the liner at a specific time, t. Wemake the assumption that the gas pressure always acts perpendicularto the liner surface. When a liner segment elongates or shrinks as itis pushed or pulled by its neighboring segments, a force component inthe liner direction may also appear. We assume that this force can beneglected during the acceleration time. We denote by 9 the angle be-tween the original liner formation line direction and the current for-mation line direction, at Z. Then, we can see from Fig. 1 that theadditional velocity vector dl, at a given position Z, induced by theforce acting perpendicularly to the liner during the time dt, adds tothe current velocity vector ý(t) to form the new velocity vectorý(t+dt). As a result the velocity increases in magnitude by theamount dV = dIVI = IdtI cos(G-6) where Id•I is the magnitude of theabove mentioned additional velocity vector. At the same time, thetangential velocity component IdaI sin(@-6) causes the velocity direc-tion to change by the ani~je dU. We can relate the angle d6 to thiscomponent by the equation:

IdvI sin(0-6) = V'd6.

Substituting d@I = dV/cos(0-6) we get the scalar equation forthe velocity magnitude:

dV tan(@-6) - V.d6. (2)

Dividing by dt and denoting the differentiation with respect to thetime by a dot above the symbol we finally obtain

Vi6iII- •"tan(9-6). (3)

The next equation describes the influence of the existence ofa velocity gradient along the liner. We make the assumption that aliner will not elongate while being accelerated, or that its elonga-tion may be neglected.

We are interested in calculating the rate-of-change of liner

slope L. Referring to Fig. 2, we note that during the time dt,

at1

Initial Liner Contour U

Normal to InitialLiner Face

Current LinerContour (FormationI' Line)

Parallel tonormal at 2i

Normal to current Linert \- Contour att

S(t + dt)

dd / , /d'• cos (0 - 6)

(8-6- d6 )

Figure 1. Liner acceleration diagranL shov•ing how the additionalvelocity dV coutributes to the increase in magnitude_nd change in the direction of the current velocitv,

.V.

12

Cj

4-J4

4-j-

F-a

4-41

CQ u0

60 Qu

CC

C1

point A will move the distance V(t)dt to point C in a direction makingthe angle 0(t) - 8(t) with the normal to the current liner surfaceat point A. Point B will move at the same time the distance

V + Lj- d • dt to point D. The angle at A (see Fig. 2) between the

initial and current liner direction is equal by definition to 0(t).

The angle dO between CD and AB is determined by the componentsof V dt and (V + dV)dt normal to the current liner at 9 thus:

dO * tan(dG) M CF-DE - [V- (V+dV)] cos(9-_ dt. (4)d di dl

At the limit of small dt this yields:

6 - - V' cos(@-6) (5)

where the prime denotes differentiation with respect to t, and dotwith respect to time.

14

III. Integration of Basic Equations

Eqs. (3) and (5) represent two equations governing the threequantities V, 0, and 6. Assuming that the spatial and temporal dis-tribution of V for a given problem is known, we can solve for 9 and 6.Under the approximation tan(O-6) 0 9-6. Eq. (3) now becomes

+ .. (6)

Eq. (6) is a first order differential equation which for theinitial conditions 9=6=V-O at t < T has the general solution:

6 jI QX dt i- 0 f1 V6 dt (7)=V IT Vat= -V IT

where T is the instant when the liner begins to accelerate. FromEq. (5) we obtain under the above approximation:

9 I V'cos(O-6)dt I - V'dt. (8a)

When we substitute 9 from Eq. (5), the second term of Eq. (7) becomes

1 f Vedt = I V VV' cos(9-6)dt ..-1 VV'dt. (8b)V fT T VI

We, therefore, obtain the general solution for 6

6= - V'dt + -1 ft (V2 )'dt. (9)IT 2VT

15

" ... ....... .. ._ , • L ,= i " " -_ .-- i ...."

IV. Application to Exponential Acceleration

In order to apply Eq. (9) to a practical problem, we shall assumefor V the exponential form:

V(1,t) - Vo (t) 1i - exp (10)

when V0 ,T, and T are functions of X only. Though this form may notfit perfectly for all cases, it is reasonably accurate for cases wehave studied and easily integrable for the calculation of 6 given byEq. (9). The corresponding acceleration is

av V0 F/t-Ta -- Texp

The quantity T(k) is the arrival time of the detonation wave at thepoint k on the liner, T is the characteristic acceleration time ofthis point and VO(£) is the final asymptotic velocity reached by theliner. The time t-0 is taken when the detonation wave reaches a con-venient point on the metal say Z-0.

To facilitate the integration, we interchange the order of timeintegration and differentiation with respect to Z.

6- -- V dt +1- V'dt. (11)at 2 T 2V 31 T

(the differentiation with respect to T vanishes since V 0 at t - T).

For the V function given by Eq. (10),

V dt - V0 (t-T) - TV (12)T0

V2 dt - V0 [V0 (t-T) - TV] - 1 V213)

Substituting Eq. (12) and (13) into Eq. (11) yields:

i --|i--~~ ~~ [.1v(tT 1 T0v2°6 - 2[T V-V (t-T) -i-i _L IV0 'TV-V (t-T)] + 1V2 (14)

atieL 0 2

To compare this equation with experimental results, we shall beinterested in the value 6 will obtain as t - c. At this limitjV be-comes equal to VO, V' becomes equal to V0 ' and the expression for 6becomes:

16

--------'i-......--- ----

VQ2 - '2 T'V (15)

2 - 2 TV 4 0'15

Let U be the detonation wave sweep speed, then T' is equal toI/U for a detonation on the axis of symmetry and Eq. (15) becomes:

V0 1 1 I'

TO2U 2 tV'0 + Z VO" (16)

The first term is, for a small angle 6, identical to the Taylorformula, Eq. (l),and the other two are the correction terms.

17

V. Constant Acceleration Equation

In deriving Eq. (16) we assumed the exponential form for thevelocity time dependence. Even though this form very well simulatesthe experimental measurements it is not the only form one can chooseto fit the experimental data. When solving Eq. (9) with a constantacceleration assumption at the period of time T < t <(T + -c) i.e.,

V - VoI\t-T) (17)

we find

V0T' 1 1 V (18)

2 6 cV0 6 c 0(8

and for the form:

1/2

V =V 0 ( )t-T (19)

we obtain

I V0T' T V + T V. (20)2 6 SR 0 12 SR 0

It can easily be seen that the TSR parameter of Eq. (19) ex-actly corresponds to 3T, when T is the corresponding parameter inthe exponential acceleration (Eq. 10).

Using an acceleration form such as Eq. (17) or Eq. (19) may bemore convenient in obtaining a closed form solution in more compli-cated cases, e.g., when the effect of the liner curvature on theangle 6 cannot be neglected.

18

I

VI. Comparison with Previous Work and Two-Dimensional Code Calculations

Let us now compare Eq. (16) with the equation obtained empiri-cally by Randers-Pehrson [9] using Two-D code calculations andexperiment. The formula given in Ref. 9 is as follows:

V 1 , 1 t,)2=2t 2 - -TV)- " (21)

(Note that Eq. (21) agrees with Eq. (16) in the first two terms).

To perform this comparison we need to know V0 and V0 ',T and T'.There exists some experimental data for V0 and Vol from explodingcylinder tests. Experimental data for T (excepc in special steadystate cases) seems lacking, however. We, therefore, performed Two-Dcode simulations to calculate Vo,V0 ',t,T' and 6 for two specificgeometries. Our general approach is to use output from a Two-D codeof V0 and T at a number of £ locations to numerically calculate V0'and T'. Then substituting these values of V0 and VO', T and T'into Eq. (16), we obtain a value for 6 which we shall denote 6F(F - formula). The value of 6 F is then compared with the Two-D codevalue of 6 denoted 6 c(c-code). Wherever experimental measurements of6 are available for a specific charge we shall denote them 6ex. Weexpect to find a good agreement between 6c and 6 ex when using areliable Two-D code. Therefore, a check against experimental data is,in fact, a check of the Two-D code validating its use for checkingEq. (16). For completeness, these values are also substituted intoEq. (21) and the result denoted 6RP

The two dimensional code we found most convenient to apply tothis purpose is a Lagrangian code named TEMPS in which the explosiveis treated by a two-dimensional finite-difference grid similar toHEMP, TOODY and other Two-D Lagrangian codes. The liner in TEMPS,however, is described as an array of mass points having tensile andbending forces between each two neighboring points. The liner isthus being treated as one-dimensional. This not only saves compu-tation time but also facilitates the data reduction from the code out-put.

A. Conical Shaped Charge

The first configuration studied is the 81.3 mm diameter, 420,conical shaped charge depicted in Fig. 3. In this example we choose£-0 at the cone liner apex and £ is the distance from this point along

the formation line. The charge is initiated by plane detonation atthe rear of the explosive.

From the TEMPS code calculations, the velocity history for eachliner mass-point position Z is recorded and plotted. A typical plotis shown in Fig. 4. The asymptotic final velocity reached is definedas Vo(Z) for each mass point. These are tabulated in Table 1 forthis case. This final velocity is then plotted as a function of Zas shown in Fig. 5a. This curve is then numerically differentiated

19

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using the simple difference formula AV0 /Ak to obtain V0 '(Z) which isalso contained in Table 1.

Next we must estimate the value of T for each mass point. Thiswas obtained by fitting Eq. (10) to the TEMPS velocity data. Thearrival time T used in Eq. (10) is the theoretical arrival time at theexplosive metal interface and obtained by dividing the distance be-tween the plane of initiation and the point X by the detonationvelocity UD. As is indicated in Fig. 4, the wave arrival time pre-dicted by the TEMPS code does not coincide with the chlculated theo-retical arrival time. This inaccuracy in the code simulction Iscaused by the code's smearing of the detonation front in the finite-difference calculation scheme.. Therefore, in performing the fit for T,we did not choose T as the value of t when V - Vo(l-e-1) = 0.632 VO,as is indicated by Eq. (10). This value of T would be inaccuratesince it is close to and heavily influenced by the code's calculationof wave arrival time. We found it more reliable to fit the curvesclosely in the region V > .86 V0 . The values of T obtained from thisprocedure are given in Table 1. These values of T are also plottedas a function of Z in Fig. 5b. A line is fitted through the pointsand T' calculated by simple numerical differentiation. The valuesare shown in Table 1. We see that for this charge T is approximatelyconstant and therefore T' is a very small quantity.

These values are then substituted into Eq. (16) and Eq. (21) aswell as Taylor's relation and compared to the code calculations. Thequantity 6 c is the angle of the velocity vector as given by TEMPSwhen the liner reaches a final velocity V0 . The detailed data aregiven in Table 1 and plotted graphically in Fig. 6. We see that theEq. (16) gives a significant improvement over the Taylor relationand agrees well with the TEMPS code. Note that since the last term inboth Eqs. (16) and (21) is small, the difference between Eq. (16j andEq. (21) is small for this example.

B. Explcding Cylinder

The second configuration studied is an exploding cylinder whichis identical co the example used by Randers-Pehrson [9,10]. It con-sists of a steel pipe segment 101.6 mm in length and a diameter of50.8 mm filled with OCTOL as shown in Fig. 7. For this example,direct experimental measurements of 6 and V0 are presented in Ref. 10.In these experiments, the expanding cylindrical pipe breaks into frag-ments. The speed and direction of motion of the fragments are measuredby means of x-ray shadowgraphs taken at several predefined times.Similar experiments were also conducted in References 11 and 12.

To provide a complete verification of Eq. (16), however, thequantity T is also needed. Since no acceleration data was measuredin these experiments, we cannot provide an experimental value for Tand hence a full experimental verification of Eq. (16) at this pointin time is not possible. In fact, as mentioned in the previous sec-tion, measurements of metal motion during acceleration have bean re-ported for only a very special steady state case (see [13,14) in

24

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25

10.0-

TEMPS (2-D Simulation)

9.0

Present Theory

8.0

Randers-Pehrson

4 7.0 Taylor (6 = V0/2U)

d00

22

z

5.0Present Theory

V 0 v~ 8T "V 02U 2 4

4.0 Randers--Pehrson

V Vo VT (V20T) Liner Mess

2U 2 5 Point Number

10 11 12 13 14 15 16 17 18. i I I I I I I II

60 70 80

POSITION ALONG LINER t(mm)

Figure 6. Comparison of the projection angle 8 from formulasand two-dimensional calculations.

26

I-

WRT

_._ 1o1.6mm -

3. 25mm

Explosives - OCTOL

p= 1.70q gm/cm3 50.8mmi (mm)

100 -.50 0

p = 7.8 gm/cm3

Figure 7. Exploding cylinder charge used for comparisons.

27

which Vol and T' are identically zero. Therefore, in our presentcomparison, TEMPS code calculations will again be used to supply thequantities T and T' (as well as V0 and Vol). The experimental datawill thus be used to verify the final values for V0 and 6 given bythe code calculations as well as the 6 given by Eqs. (16) and (21).

The results of the TEMPS calculations for V0 and 6 are comparedto the experimental data in Figs. 8 and 9. Also shown ir, these fig-ures are the HEMP simulation results presented by Randers-Pehrson in[9]. Both TEMPS and HEMP results are within the spread of the expezi-mental data.

The procedure to determine T, T', V and V ' from the TEMPS datawas described in the previous section. R velocity vs. time plot fora typical mass point is shown in Fig. 10. A plot of T vs. £ is givenin Fig. 11. The values of all the necessary quantities are listed inTable 2. These values were then substituted into Eqs. k16)and (21)to compute 6. and 6 . Fig. 12 is a comparison of 6F, 6Rp' theTaylor formula and &e TEMPS result 6 c. We see that considerableimprovement in the estimation of the angle 6 is achieved. Again, wenote that the third term in both Eqs. (16) and (21) is small, there-fore, 6 F and 6Rp do not differ significantly.

28

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VII. Summary and Conclusions

In this report a formula for the explosive-metal Taylor angle 6in the unsteady case was derived from basic physical principles underthe assumptions

(a) the explosive T ressure always acts normalto the current liner surface.

(b) total angular motion of the metal is small(i.e., 9-6 is small)

(c) forces in the liner are small during theacceleration time.

Also, the present case is restricted to liners whose meridionalcurvature is small. Under the assumption of an exponential decayingacceleration in time of the metal, this formula is given by

V O 0 1 , 1 + I I'

2U 2 0 4 0

where V0 is the final velocity, T the characteristic acceleration time,and U the detonation sweep speed; the prime indicates spatial dif-ferentiation along the liner. We point out that the first term repre-sents the steady-state Taylor result. The remaining two terms repre-sent the unsteady effects. Note that the first two terms areidentical to the semi-empirical formula given in [9].

for the collapse of a conical shaped charge and the explosion ofa metal cylinder we have found this formula to yield accurate resultswhen compared to Two-D hydrocode calculations and to provide a signifi-cant improvement over the Taylor relation. The experimental dataavailable also shows that the formula is accurate.

More recently, liners with large amounts of curvature in themeridional direction (formation line) such as hemispheres have shownpromise as candidates for certain warhead applications. For suchcases, the present formula is not applicable, It is therefore suggestedthat the current analysis be extended to include these more complexgeometries.

35

-l- - - - - - - - --i '

Acknowledgement

The authors would like to thank Dr. William Walters and Mr. John

Kineke of BRL for their valuable advice and cotmments and Mr. Richard

Foedinger and Mr. Charles Dabundo of Dyna East Corporation for their

assistance in the reduction of data from the simulations and for their

assistance in the preparation of this document.

36

References

1. Taylor, G.I., "Analysis of the Explosion of a Long Cylindrical BombDetonated at One End," (1941), Scientific Papers of G.E. Taylor,Vol. III, Cambridge University Press, 1963, pp. 277-286.

2. Birkhoff, G., MacDougall, D.P., Pugh, E.M., and Taylor, G.,"Explosives with Lined Cavities," J. Appl. Phys., Vol. 19,June 1948, pp. 563-582.

3. Pugh, E.M., Eichelbeiger, R.J. and Rostoker, N., "Theory of JetFormation by Charges with Lined Conical Cavities," J. Appl. Phys.,Vol. 23, No. 5, May 1952, pp. 532-536.

4. Eichelberger, R.J. and Pugh, E.M., "Experimental Verification ofthe Theory of Jet Formation by Charges with Lined Conical Cavities,"J. Appl. Phys., Vol. 23, No. 5, May 1952, pp. 537-542.

5. Eichelberger, R.J., "Re-Examination of the Nonsteady Theory ofFormation by Lined Cavity Charges," J. Appl. Phys., Vol. 26,No. 4, April 1955, pp. 398-402.

6. Hirsch, E., "A Simp-e Representation of the Pugh, Eichelberger andRostoker Solution to the Shaped Charge Jet Formation Problem,"J. Appl. Phys., Vol. 50, No. 7, July 1979, pp. 4667-4670.

7. Carleone, J. and Chou, P.C., "A One Dimensional Theory to Predictthe Strain and Radius of Shaped Charge Jets." Proceedings of theFirst International Symposium on Ballistics. Orlando, Florida,November 13-15 (1976).

8. Defourneaux, M., "The Push Plate Test for Explosives," Proceedingsof the First International Symposium on Ballistics. Orlando,Florida, November 13-15 (1976).

9. Randers-Pehrson, Glenn, "An Improved Equation for CalculatingFragment Projection Angle," Proceedings of 2nd InternationalSymposium on Ballistics, Daytona Hilton, Daytona Beach, Florida,March 9-11 (1977).

10. Karpp, R.R., Kronman, S., Dietrich, A.M., and Vitali, R.,"Influence of Explosive Parameters on Fragmentation," BRLMemorandum Report No. 2330, USA Ballistic Research Laboratory,Aberdeen Proving Ground, Maryland, October 1973. (AD #917248L)

11. Allison, F.E. and Watson, R.W., "Explosively Loaded MetallicCylinders I," J. Appl. Phys., Vol. 31, No. 5, May 1960, pp. 842-845.

12. Allison, F.E. and Schriemf, J.T., "Explosively Loaded MetallicCylinders II," J. Appl. Phys., Vol. 31, No. 5, May 1960, pp. 846-851.

37

13. Kury, J.W., Horning, H.C., Lee, E.L., McDonnell, J.L., Ornellos,D.L., Finger, M., Strange, F.M., Wilkins, M.L., "Metal Accelera-tion by Chemical Explosives," Fourth International Symposium onDetonation ONR ACR-126, 1965.

14. BJarenholt, G.,OEffect of Aluminum and Lithium Flouride Admixtureson Metal Acceleration Ability of Couip. B," Proceedings of the 6thInternational Symposium on Detonation, Coronado, California,August 1976.

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