CONTRACT REPORT NO. 331

SHAPED CHARGE JET BREAKUP STUDIES USING

RADIOGRAPH MEASUREMENT AND SURFACE

INSTABILITY CALCULATIONS

Prepared by

Dyna East CorporationWynnewood, PA 19096

April 1917

Approved for public release; dlstrlbutlon unlimited.

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Ci II2

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REPOR DOCMENTTIONPAGERXAD INSTRUCTIONS"'REPOT DCUMNTATON AGEBEFORE COMPLETING FORM

I. REPORT NUMBER 2.GOVT ACCES$SION NO., 3._RltFI* AAQ ta•

BRL CONTRACT REPORT NO. 337

-SHAPED CHARGE JET BREAKUP)TUDTES USING /AD - Final/GRAPH XEASUREMENT AND SURFACE INSTABILITYCALCUL.EONS.RORMING OR. REPORT NUMBER

K DE-TR-7A-lr 11. AUT OR~. .... .. I O'WACTO• 'GRANT NUMBER(.)

, P. C.hou C. A. Tanzio 'DAADOS-75-C-07s3 .J. Carleone R. D. Cicarelli ii

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IS. SUPPLEMENTARY NOTES

This work was supported by the US Army Ballistic Research Laboratory underContract No. DAADOS-7S-C-O753 and this report comprises the final report ofthis contract.

19 KEY WOROS IConfirv. on -@ra.0c .,de it neo-V and Ideatify by block n.o.b.r)

Shaped ChargeJetWarheadExplosiveSurface Instability

0 ABSTRACT fCon.tnin -a ,e* olid If ns e e.oary ad Identify by block n-.bor)

A study of the shaped charge jet breakup phenomenon is undertaken by twoapproaches. In the first approach, jet velocity, jet breakup time, and jetradius distrlibutions are measured using timed flash radiographs of broken jetsfrom eleven different shaped charge designs. Experimental results are cor-related to theoretical predictions and a semi-empirical jet breakup time vs. letradius curve is presented for copper lined charges. The second approach appliedthe concepts of hydrodynamic instability to shaped charge jets. A.

(continued)nn FORM - 147 EDITIONMO F I NOV 5ISOBSOLETE l ISArq TPTFp

UNCLASSIFIEDSCCUmiTY CLAWFICATION OF T.IS PAGO 4e. Daa Siuus0

Item 20 (Cont'd)

. numerical study of the effects of yield strength, inertia forces, surfacedisturbance wavelength, and irregular or "r~andom`ý"-surface disturbances ispresented. Results indicato that high yield strength and low density jetmaterials will both cause earlier breakup. A critical range of disturbancewave-lengths exists in shaped charge jets; if a random surface disturbance isimposed, a critical wavelength will prevail and eventually cause the jet tobreakup into segments with lengths approximately equal to experimentallymeasurad values.

Table of Contents

Page

List of Illustrations .............................................. v

List of Tables ..................................................... ix

I. INTRODUCTION .................................................. I

II. JET RADIOGRAPH DATA

A. General Approach ...................................... 4

B. Results of Radiograph Measurements ...................... 7

C. Comparison of Data with Theory ........................... 8

III. STABILITY OF SHAPED CHARGE JETS AS A CAUSE FOR BREAKUP

A. Background ................................................ i5B. Preliminary Analytical Stability Study

of Shaped Charge Jets .................................... 17

C. Numerical Study of Shaped Charge Jet Stability ........... 19

IV. CONCLUSIONS ................................................ 34

V. REFERENCES ................................................. 36

Appendix A Dimensions of Charges Studied ....................... 39

Appendix B Plots of Jet Velocity, Breakup Time, andRadius Distributions Measured from Radiographs ...... 45

Appendix C Tables of Reduced Dati. from RadiographMeasurements .....................................

DISTRIBUTION LIST ................................................... 89

iN 14M $041

* ;1;tt It TII P

iii

List of Illustrations

Figure Title Page

1 Radiographs showing the jet breakup phenomenon(radiographs courtesy of R. Jameson, BRL). 2

2 Position-time plot of two typical jet segments show-ing the method of determining the breakup time fromjet radiograph measurements. 5

Theoretical jet stroin-rate, strain, radius andexperimental breakup time vs. relative cone positionfor the 38.1 mm, 60" Cu charge (charge No.3). 12

4 Breakup time vs. jet segment radius for variouscharges. Ten charges are included. Three datapoints selected from each charge. 14

5 The resemblance between the breakup of a liquid jet

and a shaped charge jet. 16

6 Initial and boundary conditions for the numericaljet stability computations. 20

7 Comparison of stretching jets with various yieldstrengths. 22

8 Schematic showing position-time plot of a jet elementwhich stretches to the typical breakup length. Dis-turbances are initiated separately at each of thetimes indicated for independent HEMP calculations ofthe element. 24

9 Relative growth vs. initial strain rate for fivesuccessive element configurations at 18 usec afterthe disturbance initiation in each configuration. 26

10 Relative growth vs. time for three different dis-turbance initiation times in a typical jat element. 27

11 Relative growth vs. time for surface disturbancesof four different wavelengths. In all cases the jetsinitially have the same radius and strain rate. 28

12 Results of HEMP calculation of a stretching jet withan irregular surface disturbance. 30

13 Results of HEMP calculations of a stretching jetwith a "random" surface disturbance. 31

14 Relative growth vs. time for three stretching jets ofdifferent densities as predicted by HEMP. 33

PT2L~Dr".7 ?TL'

Figure Title Page

Al Geometry of 38.1 mm copper and aluminum lined chargeswith various cone angles (charge nos. 1-4,9-11) 41

A2 Geometry of 50.8 mm, 42', copper lined charges withdifferent wall thicknesses (charge nos. 5-7). 42

A3 Overall dimensions of the 81.3 mm, 42" tapered wall,charge (charge no. 8). 43

A4 Details of the 81.3 mm, 42" tapered liner (charge no. 8). 44

Bl Theoretical and experimental jet velocity vs. jet particleposition for the series of 38.1 mm, 1.168 me wall, copper

lined charges with various cone angles at various timesafter the arrival of the detonation wave at the cone apex. ?

(charge nos. 1-4).

B2 Theoretical and experimental jet velocity vs. jet particleposition for the series of 38.1 mm, 1.626 mm wall, aluminumlined charges with various cone angles, at various timesafter the arrival of the detonation wave at the cone apex.(charge nos. 9-11). 48

B3 Theoretical and experimental jet velocity vs. jet particleposition for the 50.8 mm, 420, 0.762 mm wall, copper linedcharge at 130.6 wsec after detonation wave arrival at coneapex. (charge no. 5). 49

B4 Theoretical and experimental jet velocity vs. jet particleposition for the 50.8 mm, 42, 1.524 mm wall, copper linedcharge at 130.5 4sec after detonation wave arrival at coneapex. (charge no. 6). 50

B5 Theoretical and experimental jet velocity vs. jet particleposition for the 50.8 mm, 2.54 mm wall, copper lined chargeat 130.7 osec after detonation wave arrival at cone apex.(charge no. 7). 51

B6 Theoretical and experimental jet velocity vs. jet particleposition for the 81.3 mm, 42%. tapered wall, copper lined

charge at 130.5 isec after detonation wave arrival at coneapex. (charge no. 8). 52

B7 Jet breakup time vs. jet particle position at t-107.1 .secfor the 38.1 mm, 200, 1.168 mm wall, copper lined charge

(charge no. 1).

B8 Jet breakup time vs. Jet particle position at t-93.4 usecfor the 38.1 mm, 400, 1.168 mm wall, copper lined charge(charge no. 2).

B9 Jet breakup time vs. jet particle position at t=122.7 usecfor the 38.1 mm, 1.168 mm wall, copper lined charge(charge no. 3). 5

310 Jet breakup time vs. jet particle position at t=133.3 usecfor the 38.1 mm, 900, 1.168 rm wall, copper lined charge

(charge no. 4). 56

V I

Figure Title Page

ll Jet breakup time vs. jet particle position att-113.1 usec for the 38.1 mm. 40, 1.626 mm wall1aluminum lined charge (charge no. 9). 57

B12 Jet breakup time vs. jet particle position att-97.7 usec for the 38.1 mm, 60%, 1.626 mm wall,aluminum lined charge (charge no. 10). 58

B13 Jet breakup time vs. jet particle position att-132.8 wsec for the 38.1 a, 90%, 1.626 mm wall,aluminum lined charge (charge no. 11). 59

B14 Jet breakup time vs. jet particle position att-130.6 Psec for the 50.8 mm, 42, 0.762 mm wall,copper lined charge (charge no. 5). 60

B15 Jet breakup time vs. jet position at t-130.5 psecfor the 50.8 mm, 42, 1.524 mm wall, copper linedcharge (charge no. 6). 61

B16 Jet breakup time vs. jet position at t-130.7 Psecfor the 50.8 mm, 42, 2.54 mm wall, copper linedcharge (charge no. 7). 62

B17 Jet breakup time vs. jet particle position att-130.5 ueec for the 81.3 mm, 42%, tapered wall,copper lined charge (charge no. 8). 63

B18 Theoretical and experimental jet radius vs. jetparticle position at t-107.7 wsec for the 38.1 mm,20*, 1.168 mm wall, copper lined charge (charge no.1) 54

B19 Theoretical and experimental jet radius vs. jetparticle position at t-93.4 usec for the 38.1 mm,40%, 1.168 mm wall, copper lined charge (charge no.2) 65

B20 Theoretical and experimental jet radius vs. jetparticle position at t=122.7 wsec for the 38.1 mm,60% 1.168 mm wall, copper lined charge (charge no.3) 66

B21 Theoretical and experimental jet radius vs. jetparticle position at t-133.3 ;sec for the 38.1 mm,90%, 1.168 mm wall, copper lined charge (charge no.4) 67

B22 Theoretical and experimental jet radius vs. jetparticle position at t-113.1 ;isec for the 38.1 mm,40', 1.626 ,m= wall, aluminum lined charge 68(charge no. 9).

P23 Theoretical and experimental jet radius vs. jetparticle position at t-97.7 ;sec for the 38.1 mm,60%, 1.626 mm wall, alurainum lined charge 69(charge no. 10).

Figure Title Page

B24 Theoretical and experimental jet radius vs. jetparticle position at t-132.8 Psec for the 38.1 mm,90S, 1.626 m wall, aluminum lined charge 70(charge no. 11).

B25 Theoretical and experimental jet radius vs. jetparticle position at t=130.6 usec for the 50.8 mm,42, 0.762 -m wall, copper lined charge (charge no.5) 71

B26 Theoretical and experimental jet radius vs. jetparticle position at t-130.5 Psec for the 50.8 mm,42, 1.524 mm wall, copper lined charge (charge no.6) 72

B27 Theoretical and experimental jet radius vs. jetparticle position at t-130.7 Psec for the 50.8 mm,42,2.54 mm wall, copper lined charge (charge no. 7). 73

B28 Theoretical and experimental jet radius vs. jetparticle position at t-130.5 Psec for the 81.3 m,,42% tapered wall copper lined charge (charge no. 8). 74

viii

List of Tables

Table Page

I List of Charges Examined .. ..... ........... 6

II Average Aspect Ratio (L/d) of Jet Segments ..... 9

III Average Jet Velocity Difference BetweenNeighboring Segments (AV ............ .... ... 10

IV Analytical Approaches to Shaped ChargeJet Stability ..... .............. 18

V Dimensions of Jet Element Studied at VariousTimes After Its Formation .... ..... ............ 23

lx

4L

Acknowledgment

The authors would like to express appreciation to Robert Karpp,Robert Jameson, Julius Simon and john Majerus of BRL for supplying themany jet radiographs used in the present study, and also for the manyhelpful discussions and suggestions during the course of this work. Wewould also like to thank Michael Khalil for his help with the manyradiograph measurements, computer calculations and data reductioncontained here.

xi

1. INTRODUCTION

In recent years, tremendous advances have been made in the basicunderstanding of shaped charge mechanics. These include the processesof liner collapse, jet formation, and formulas for jet radius andstrain. Both two-dimensional computer codes and one-dimensional model-ing have been used successfully for the analysis of shaped charges.One important area, however, has not been well understood and cannotbe simulated easily by computer methods. This area is the breakup ofthe shaped charge jet. This breakup, or segmentation, occurs in alljets that have a large velocity gradient with higher speeds at the tipand lower speeds near the tail. This process can be seen in Fig. 1which shows flash radiographs of a typical shaped charge jet at threesuccessive times, displayed in the proper position-time coordinates.At the earliest time the jet segmentation has begun near the tip; mostof the jet is still continuous. The later times show the completelysegmented jet.

In (1[ and [2], we have presented initial studies on the jetbreakup phenomenon. Let us now briefly review the results of theseinitial studies. Formulas for the strain and radius of shaped chargejets based on a one-dimensional model are presented in [1]. Further,in (1] and [2], a method to detLrzine the jet breakup time distribu-tion from timed flash jet radiographs was developed. This method wasthen applied to jets from a series of identical BRL 81.3mm standard420 copper-lined charges and an unconfined 105mm 42* copper linedcharge in [1]. Despite the scatter in the results, the breakup timedoes show a definite "trend" as is indicated in [1]. In fact, for theparticular charge studied, the trend indicates that the jet breaksfirst near the tip with a progressively increasing breakup time towardsthe tail. Thus, shaped charge jets do not necessarily break simulta-neously along their length, as assumed previously by some investigators.The resulting jet breakup time distribution was then contrasted to theone-dimensional theoretical jet strain and radius distributions. Thisis presented in [11 and [2] where it is concluded that, for copperliners, the breakup tize distribution is related to the jet radius dis-trib'ition. No other correlation was indicatad in [1] or [2].

The present study addresses the problem of jet breakup through theuse of two approaches: (1) triple-flash radiographs of various jets tudetermine jet velocity, breakup time, etc., and (2) surface instabilityas a cause of breakup.

The first approach is actually an extension or continuation of thework we have presented in [1]. The method of [1] to measure breakuptime, jet velocity, etc. from timed flash radiographs is applied hereto many jets from various shaped charges having different cone angles,different liner wall thicknesses, liners with tapered walls, differentliner materials, and light or heavy confinement. This approach hasenabled us to obtain many characteristics of jets from a large cross-section of shaped charges. The data from all of these charges are

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conveniently tabulated and graphically displayed in this report.The geometry of the charges studied are given in Appendix A. Themeasured jet velocity, breakup .ime and jet radius are plottedgraphically in Appendix B. Appendix C contains a tabular list ofthis data and in addition includes the aspect ratio of the jet seg-ments and difference in velocity between neighboring particles. Allof this data has been analyzed to obtain trends in the breakup phe-nomenon. The breakup trends have been contrasted with the one-dimensional strain and radius predictions using the formulas of [1].Within an individual jet no general correlation has been observedbetween breakup time and one-dimensional parameters. The resultsof the breakup time measurements of all copper jets studied, however,do exhibit a definite trend. This trendyields a semi-empirical deaign curve for breakup time which can beused in conjunction with a one-dimensional shaped charge model topredict breakup time for copper jets. This first approach is pre-sented in Section II.

The second approach involves the study of the instabilitycaused by various disturbances in the jet with the goal of deter-mining whether or not this instability may cause the jet to breakup.We know from classical hydrodynamics that a continuous liquid jetwill break into small segments because of surface instability. Thebreakup of a shaped charge jet resembles this very much, and ourstudy shows that the shaped charge jet is indeed subject to surfaceinstability. In principle, there are three forces which may causethis instability: surface tension, aerodynamic force, and materialstrength (elastic-plastic force) in the jet. Both analyticalstudies and two-dimensional finite-difference numerical calculationswere made, and the results indicate that material strength is themain cause for shaped charge jet instability and breakup. In addition,the effect of strain-rate, time of disturbance initiation, and inertiawere considered. The results of this stability study are discussedin Section III. Finally, general conclusiois are stated in Section IV.

We would like to note that che results presented in this reportsummarize the work conducted during the entire contract period. Someof these results were presented previously in quarterly progress re-ports.

3

II. JET RADIOGRAPH DATA

The results which we presented in [1] indicate that the measure-ment of jet radiographs provides an efficient and useful procedure toobtain breakup data. Also breakup trends are indicated in [1] whichwarrant further investigation. In this study, we have thetefore con-tinued using this approach to study jats from a large number ofcharges having different geometries and liner materials.

A. General Approach

The detailed development and equations for the computation ofbreakup time, jet velocity, etc. from the radiograph measurements aregiven in [1]. Here, we will give only a brief description of thismethod. Figure 2 shews the position-time plot of two typical neigh-boring jet segments labeled as K and K+l. Suppose we have radiographsof these two segments at times t, and t 2 as indicated in Figure 2.(Note: in the actual case,, we have analyzed three timed flash radio-graphs, as supplied by BRL). We may then compute the velocity of eachsegment by measuring how far dhe segments have traveled during thetime t 2 -tI. In the two radiographs the segments are separated by agap as shown. Under the assumption that the segments remain at con-stant length and constant velocity after breakup (which is a goodapproximation after the jet is completely broken as evidenced by theradiographs), it is a simple matter to trace back the front of seg-ment (K+l) and the rear of segment K until they meet, i.e. the gap be-comes zero. The time of this meeting tb is the breakup time of seg-ment K from segment (K+1). This procedure may be applied successivelyto each pair of particles, i.e. K and (K+I), (K+l) and (K+2), etc.,until the complete breakup time distrib'ition is obtained. Therp is acertain degree of approximation to this procedure. Since the particlesdo not break successively from one end of the jet to the other, wesometimes have the situation where a larger segment breaks from themain jet, continues to stretch, and then finally breaks into smaller,constant velocity, constant length segments. However, the time betweenthe initial break and the succeeding breaks is small, therefore theprocedure may still be applied approximately to this breakup situation.The resulting breakup times have been found to be within the accuracyof the experimental procedures.

This method of determining the breakup time was then applied tojet radiographs from many various shaped charges. These radiograp:iswere obtained from BRL through the courtesy of R. Jameson, R. Karpp,J. Simon and J. Majerus. For convenience, the differeni charges arelisted in Table I; the detailed geometry and drawings ot each aregiven in Appendix A. It cain be seen from this table that we havestudied the effects of cone angle, wall thickness, wall taper, andliner material on the breakup mechanism.

SEGI'EN K+iIsllL

t2 RADIOGRAPH 2///

F- I :' :2F i t /___-_________ t tw)

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Table I

List of Charges ExaminedA

Copper Lined

Charge Cone Diameter Wall ThicknessNo. Angle Mm -

1 200 38.1 1.1682 400 38.1 1.1683 600 38.1 1.1684 90* 38.1 1.1685 42° 50.8 0.7626 42' 50.8 1.5247 42* 50.8 2.5408 420** 81.3 tapered wall

Aluminum Lined

Charge Cone Diameter Wall ThicknessNo. Angle mm mm

9 400 38.1 1.62610 60" 38.1 1.62611 906 38.1 1.626

* Note that we have also performed a breakup study of jets from the BRLStandard 81.3m- charge and the 105m unconfined charge in [1].

** Angle of inside wall

B. Results of Radiograph Measurements

From the measurements of the jet radiographs, we have determinedthe jet velocity, jet breakup time, and jet radius distributions forall of the charges listed in Table I. Plots of all of the data ob-tained are given in Appendix B. Figure B1 shows the jet velocitydistribution along the length of the jet at a particular time for theseries of 31.8mm copper lined charges having cone angles of 20, 40,60" and 90" (Charge Nos. 1-4). Test results of these charges werefirst reported in [3], where the average breakup time of the completejet was given. The time base used in all of the present studies isgiven as t - 0 when the detonation wave reaches the apex of the cone.*From Fig. B1 we see that, for a given liner material, the tip velocityof the jet decreases as the cone angle increases. It should be notedthat, for the 200 copper cone, the resulting jet possessed a bifur-cated region at the tip portion. We therefore made our radiographmeasurements starting from the first fully coherent particle of thejet, which has a velocity of 8. 4mm/vsec, much smaller than 9.9mm/usecreported in [3]. We should further note that it has been determinedthat this bifurcated region is caused by supersonic flow in the jetformation region (See (4] for details). None of the other chargesexamined under this study possessed this bifurcated region.

The jet velocity distributions for the remaining charges areshown in Figures B2-B6. It is interesting to note that Fig. B6 in-dicates that the 81.3mm tapered wall charge studied here has a higherjet velocity (8.3 km/sec at the tip) than the 81.3mm standard chargewith constant wall thickness studied in [1] (7.9 km/sec at the tip).This somewhat surprising result has a fairly simple explanation. Be-cause of the wall taper (thicker at the cone apex tapering linearlytoward the cone base), as we proceed from the apex toward the base theliner elements reach higher collapse velocity than corresponding ele-meats of the charge with a constant wall thickness. This effect pro-vides the tapered wall liner with an overall decrease in the collapseangle (i.e. the angle of the collapsing liner with the charge axis,denoted as a in [I]). From the basic one-dimensional jet formationtheory [5] a decrease in collapse angle, in general, produces ahigher jet velocity. Finally, from i'igs. Bl-B6 we also observe thatthe velocity distribution for each of the charges studied is aproxi-mately a straight line.

Plots of the breakup time of the jet particles versus theirposition in the jet at a particular time for the eleven charges ex-amined are given in Figs. B7-B17. The scatter in the results is tobe expected because of the nature of the breakup mechanism. In allcases a definite trend in breakup time exists. To demonstrate thistrend, a straight line was passed through the data using the methodof least squares. This line is also shown in Figs. B7-B17. All ofthe jets studied exhibit trends which indicate that the tip portionbreaks earlier than the rear portion except for the 38.1mm, 40'copper charge (Charge No.2) and the 81.3mm, 42* copper tapered wall

* Except for the data which is tabulated in Appendix C: The data thereis based on the initiation time of the main explosive charge. SeeAppendix C for details.

charge (Charge No.8). Breakup trends for these two charges show thatthese jets breakup almost simultaneously. Note that since only partof the jet for each charge appeared in the radiographs, these con-clusions are only valid for the portion of the jets measured.

Next, the measured jet radius distributions for the eleven chargesstudied are given in Figures B18-B28. We see from Figs. 818-B21 thatfor copper lined charges of a given diameter and wall thickness theaverage jet radius increases with increasing cone angle. This effectis not so pronounced in the case of the aluminum lined charges as canbe seen from Figs. B22-B24. Further, we note irom Figs. B25-B27 that,all else being equal, relatively large incrtases in wall thicknessgive only small increases in jet radius.

Finally, all of the data resulting from the radiograph measure-ments are tabulated in Appendix C. In addition to the jet velocity,breakup time, and jet radius, which were graphically displayed inAppendix B, we have also tab*Aated the aspect ratio (length to diameter,L/d) of each segment and the jet velocity difference between neighbor-ing segments A.V The average R/d and average AVi are summarized inTables II and III. in these tables, we have also summarized thisdata for the BRL Standard 81.3mm and 105mm unconfined charges whichwere studied in [l]. Since there was a certain amount of scatter inthese quantities, the coefficient of variation (standard deviation di-vided by tha mean) in each case was also computed. From Table II wesee that the average Z/d varies between 4 and 6 for copper linedconical charges of constant wall thickness. The tapered wall conehas an X/d ratio of 3.3 which may indicate that, even though thischarge has a higher tip velocity than the corresponding constant wallcharge, its penetration performance at longer standoffs may be poorerbecause of the small segment size and early breakup time. The alumi-num lined charges studied have average Z/d ratios of 3.89, 4.92, and8.39.

From Table III we observe that the overall average velocitydifference between neighboring segments for the copper lined chargesis approximately llOm/sec. Recently, Held [6) has published that theaverage AV for the copper lined German charges he studied was approx-imately l08m/sec. For the aluminum lined charges studied here, wefound an overall average AVj of 145m/sec. Finally, we obtained radio-graphs of a copper lined French charge, the ISL "S2T", from Perez [7] andfound the average L/d to be 5.62 and the average !Vj to be 137m/sec. Dataof this nature is useful in obtaining rough estimates of the number ofjet segments in newly designed charges.

C. Comparison of Data with Theory

In (1] and [8] an improved one-dimensional theoretical shapedcharge model was developed. This improved model is based on theclassical Pugh-Eizhelberger-Rostoker [5] theory but uses a semi-empirical collapse formula to describe the explosive-metal interactionprocess. Further, formulas for jet strain and jet radius were

/

Table II

Average Aspect Ratio (i/d) of Jet Segments

Copper LinedGeometry

Charge (Dia., Cone Angle, I/d Coeff. -fNo. Wall Thk.) (Average) Variacion

1 38.1 mm, 20%, 1.168 -m 4.69 .5202 38.1 -, 40%, 1.168 m 5.95 .4113 38.1 mm, 60%, 1.168 mm 5.91 .4474 38.1 mm, 90, 1.168 mm 4.49 .3015 50.8 mm, 42%, 0.762 mm 4.18 .4126 50.8 mm, 42", 1.524 m. 4.48 .3617 50.8 mm, 42%, 2.540 mm 4.48 .3368 81.3 mm, 42", tapered 3.31 .417* 81.3 mm, 42%, 1.905 mm 5.93 .375* 86.4 mm, 42, 2.921 mm 5.21 .324

Average Z/d for all copper jets 4.86

C.V. .180

Aluminum LinedGeometry

Charge (Dia., Cone Angle, Z/d Coeff. ofNo. Wall Thk.) Variation

9 38.1 mm, 40%, 1.626 mm 3.89 .45410 38.1 umm, 60%, 1.626 mm 4.92 .42011 38.1 mm, 90%, 1.626 mm 8.39 .426

Average Z/d for all Ak jets 5.73C.V. .411

* These charges were studied in [l] and this data is included here

for completeness

9

fI

Table III

Average Jet Velocity Difference Between Neighboring Segments (AV.)

Copper Lined

GeometryChar&* (Dia., Cone Angle, %Vj Coeff. of

No. Wall Thk.) (m/sec) Variation

1 38.1 =6, 20%, 1.168 m- 106 .4742 38.1 me, 40', 1.168 mm 111 .3673 38.1 ma, 60%, 1.168 ma 135 .3354 38.1 mm, 90%, 1.168 mm 109 .3445 50.8 ma, 42%, 0.762 mm 110 .4976 50.8 -m, 42%, 1.524 mm 115 .5227 50.8 zz, 42%, 2.540 - 119 .4498 81.3 ma, 42%, tapered 108 .519* 81.3 mm, 42, 1.905 m 96 .557* 86.4 mm, 42%, 2.921 mm 113 .445

Average AV for all copper jets 112J C.v. .090

Aluminum Lined

GeometryCharge (Dia., Cone Angle, 6Vj Coeff. of

No. Wall Thk.) (m/sec) Variation

9 38.1 me, 40, 1.626 mm 143 .36210 38.1 mm, 60, 1.626 mm 130 .52811 38.1 mm, 90%, 1.626 mm 162 .154

Average AV for all At jets 145C.V. .111

* These charges were studied in [1] and this data is included here for

completeness.

10

j.9

developed for this model. Since the publication of (1] and [8), wehave improved this model further by incorporating the effect of lineracceleration during collapse. This enables the prediction of the in-verse jet velocity gradient region during formation.

This improved one-dimensional model was applied to the elevencharges studied here. In Figs. Bl-B6 the theoretical jet velocitydistribution of each charge is shown in contrast to the experimentaldata. We see that excellent agreement is obtained between theory andexperiment. Note that in Fig. B4 and B5 there is some discrepancybetween the experimental tip velocity and the tip velocity predictedby the theory for Charge Nos. 6 and 7. This prediction could be im-proved by using a different value for the acceleration. At present,the acceleration values used in all the cases are given by a simpleempirical formula which depends only ou liner density and thicknesa.For these two particular charges the liner thickness to liner diameterratio is much higher than the other charge& studied and therefore thi.:simple acceleration formula may be inaccurate for these two charges.Theoretical jet radius distributions were also computed from the one-dimensional model at the breakup times tb. The tb lines in Figs. B7-

B17 were used. As shown in Figs. B18-B28, the theoretical radius dis-tributions compare reasonably well with experimental values.

We next compared the breakup time distribution with the theoret-ical distribution of other jet properties. The goal here was to findsome correlation among these distributions which would indicate con-trolling parameters in the breakup mechanism. We compared trends inthe following properties to the breakup trend: let strain n, jet radiusrj, and jet strain rate ý. The quantities n and rj are defined in[ ], and ý is simply the first time derivative of n defined there. Wealso contrasted these properties to the amount of time elapsed fromthe formation of an element until it breaks up. This quantity is theabsolute breakup time minus the absolute time when the element is firstformed, and is denoted tb-tf. The quantity tb is from the experimentalleast squares line and the quantity tt is computed from the one-dimensional model.

Since the quantities jet strain, jet radius, and jet strain ratebefore breakup are continuously changing with time, it is appropriateto compare the breakup time of each segment with these quantities atthe time when each segment breaks up. To do this, we first trace backeach segment in the radiograph to its original position in the cone,x, using the one-dimensional model. We are then able to compute -', rj,and j of thp segment at its own particular breakup time from the leastsquares experimental trend. Figure 3 shows a plot of i, rj, A, tb andtb-tf versus x/h for the 38.1mm, 60* copper charge (Charge No.3). Notethat x denotes the original liner position and h denotes the originalheight of the cone. Also plotted in Fig. 3 is r at t 55 -sec whichis a time before any breakup has occurred.

11

.50 28 lO1 r .19

.45 26 90 .17-

.40 24 wr

U

:t.35 22 70 .13-U

, 30 20 .".,0-.

.25 18 50tb-tri@ 55 ;Lsec

.20 16 40 - .07-

•-----"• r, @ tb

.15 R T L (650,6 0.7 0.8

RELATIVE CONE POSITION (X/H)

FIGURE 3. Theoretical jet strain-rate, strain,radius and experimental breakup time

vs. relative cone position for the

38. Imm, 60°' Cu charge. (Charge No. 3)

12

Figure 3 indicates that there is no reasonable correlation be-tween the breakup time distribution and n at tb or n at tb. Both therj at tb and rj at 55 usec appear to follow the general trena of thebreakup time. Plots similar to Figure 3 for the remaining 10 chargeswere also plotted but are not presented here in the interest of space*.However, the results indicate that within one jet even rj at tb and rjat a time before breakup may not follow the trend of breakup time.Results do indicate that for copper lined charges rj at a time beforebreakup seems to increase as the breakup time increases but no generalconclusion may be drawn.

We next looked at the breakup data from all the copper linedcharges together rather than just throughout a single jet. In Fig. 4we have made a plot of tb-tf vs- r, at tb for all of the copper linedcharges studied here (Charge Nos. 1-8). Also, in Fig. 4, we haveplotted the data for the 81.3mm BRL Standard Charge and the 105mm un-confined charge studied in El]. Three data points were selected fromeach jet. With the exception of one charge, the 38.1mm, 900 copper(Charge No.4), all of the data points for tb-tf fall within a narrowregion monotonically increasing with increasing rj. We may speculatethat the 90* charge does not fall in this region because the lowcollapse velocities in a charge with a large cone angle results inlower pressures and velocities at the stagnation point during forma-tion. Therefore, the properties of this jet may be different fromthose of smaller angle cones. Figure 4 indicates a correlation be-tween the two quantities for the other charges howevir. In fact, ifwe draw a line through these points, we obtain a useful breakup timevs. radius curve, which can be used for design purposes. For example,suppose we design a new copper lined charge and wish to obtain itsbreakup time distribution. We can compute the radius of the elementsof the jet as functions of time. If we then plot the radius vs. timecurve of a particular jet element on the breakup time vs. radius co-ordinates, we find that this curve will intersect the breakup curve.This inters'.rion gives the breakup time of that element. We thenrepeat this procedure for a series of jet elements. This then yieldsa breakup time distribution for the new charge. This breakup time in-formation may then be used together with other information on a pene-tration analysis to evaluate the final performance of the new design.

* These plots for the remaining 10 charges may be found in [9] and [10].

120 0

Lw 80

*0

80 .0 8 ' ,,3 t;tCAG O

I- 0

~60

a HARGE NO. s CHARGEau o .6CHARGE £ No.2 CHAGE No. 7

A ACHARGfk M.3 ElCHARGEND. 8CAcRGENo. 4 M4083

0 CHARGE No. 5 0 L90,86.LHmi

2-0 1 I I I

0.4 10.6 01,g 1.0 1.2 1.4JET RADlIUS AT BREAKUP Tim~ r.(i)

Figure 4. Breakup time vs. jet segment radius forvarious charges. Ten charges are included.Three data points selected from each charge.

I,.

III. STABILITY OF SHAPED CHARGE JETS AS A CAUSE FOR BREAKUP

A. Background

The concept of liquid jet stability In classical hydrodynamicshas been studied by many authors (e.g. [11]-[141). The idea of apply-ing a stability approach to shaped charge jets was motivated by theremarkable resemblance of the breakup of shaped charge jets to that ofliquid jets. This resemblance is shown in Fig. 5 where the breakup ofa glycerine-water jet [15) is contrasted with a typical shaped chargejet. Further, we note that the average 1/d ratio for all of the copperjet segments studied, 4.86 (see lable II), is very close to the valueof the critical X/2r 0 ratio of 4.5 predicted by classical stabilityanalyses of liquid jets, where X is the wavelength of the surface dis-turbance, and r0 is the radius of the undisturbed jet. These factssuggest that the shaped charge breakup phenomenon may be caused bysurface instability.

The first stability analysis of a liquid jet was published in1879 by Lord Rayleigh [11]. He considered an inviscid fluid jet mov-ing at a constant uniform axial velocity subjected to surface distur-bances about the equilibrium position. He obtained the most unstablewavelength and the perturbation growth rate using energy considera-tions. His values of wavelength and growth rate'have been verified byexperiments, and his work is still considered the foundation for thestudy of jet stability.

Weber [12] extended Rayleigh's work to include viscous Newtonianfluids. The wavelength of the most unstable surface disturbance for aviscous fluid does not deviate too much from the ideal fluid case of(0/2r0 ) - 4.5. He also found that Newtonian viscosity tends to dampenthe instability. Anno (16,17] gave a more general derivation of theanalyses by Rayleigh and Weber.

Goldin, et al [15] studied the breakup of non-Newtonian viscousfiuid jets. They used essentially the same approach as Levich j13]and obtained results on critical wavelength, growth rate, and breakuptime for fluids possessing general viscoelastic stress-strain re-lations. In [15], it is shown that the Newtonian fluid is the moststable one among viscous fluids. Experimental evidence for differenttypes of fluids seems to verify their results.

In our initial work, we have determined that three effects maycause jet breakup: surface tension, aerodynamic forces, and strengthof the jet material. We have applied all the essential results ofthe classical analyses to the conditions of a typical shaped rhargejet. We have also examined many experimental breakup records in thelight of stability considerations. These results have ruled out theimportance of the surface tension and aerodyiiamic effects in thestability of a shaped charge jet. This leaves the effect of jetstrength. Since strength effects are not readily amenable to analyt-ical treatment, we have turned to numerical approaches to study the

15

I,O

LIQUID JET SHAPED CHARGE JET

75 % GLYCERINE- 2 5 % WATER COPPER

REFERENCE 15 81.3ni BRL STANDARD(RADIOGRAPH COURTESY R. JNVESON, BRL)

FIGURE 5. The resemblance between the breakup of

a liquid jet and a shaped charge jet.

effects of jet strength.

In the next section of this report, we will first prebent resultsof a prelimJnary analytical study. 1hen more detailed results on thenumerical study will be presented, including the study of various sur-face disturbances, time of disturbance initiation, strain rate, andinertia effects in jets with strength.

B. Preliminary Analytical Stability Study of Shaped Charge Jets

The results and formulas developed in the classical analyses ofliquid jet stability were applied to the case of a typical shapedcharge copper jet. Analyses for non-stretching and stretching jetswere used. The results of this study are summarized in Table IV andwill be briefly discussed below.

1. Non-stretching Jets

(a) Surface Tension. In our initial analysis, we haveapplied the classical formulas of Rayleigh (11] and Weber [12] Lo theconditions of a typical copper shaped charge jet. Rayleigh's workgoverns the case of an ideal fluid under the effect of surface tensionand Weber's results are applicable to a viscous fluid under the effectof surface tension. The growth rate of disturbances on the surfaceof the jet as predicted from these formulas is quite small as comparedto that observed in radiographs of shaped charge jets. In fact, accord-ing to these formulas the amplitude of the initial disturbance onlygrows 8% in 100 wsec, whereas it is observed from experiments that theshaped charge jet used for this analysis breaks up at approximately100 Psec. The value of surface tension for copper used was 1.0 N/m.

(b) Aerodynamic Force. We have applied the formulas of0 Levich [13] to the problem of air passing over a shaped charge jet. A

very large disturbance growth rate was predicted by this analysis.We feel, however, that the analysis is not too realistic for the pres-ent problem since Levich only considers linear incompressible aero-dynamics and the present case is actually in the hypersonic regime.We would also like to indicate two experimental observations whichdemonstrate that aerodynamic effects are not important in the breakupof shaped charge jets. Vitali [18] has pointed out that superplastizjets do not breakup along their length in the typical manner of acopper jet, yet both jets are subject to the same aerodynamic force.Also, Frey (19] has studied photographs of copper jets in a vacuum,and found the typical surface disturbance growing within 100 .sec, sim-ilar to those found in jets traveling through air.

2. Stretching Jets

Mikami, et al [14] has developed an analytical approach to studythe growth of disturbances on the surface of a stretching viscousthread surrounded by another viscous medium. We have modified thisanalysis to make it applicable to the case of a shaped charge jet. Asshown in Table IV, appreciable growth rates and reasonable [/2r ratios

17

41'

aa

41 .k 4.0

*0

di 41 41 wa U 0

00r.0 c 0 -4 14 4 1

41

a ,

411

t0 rb u 0 0a.

I.' u 44 (a4U~ ~ 0 ~ 0.144

(n 41 0W A

-4 4. ca ora ca

41 41 0A &*'.A-4 u cu 3tu 3

.0cd>

41 00 a)

41 00 ~00 a0

041w 03 4

O '~U "40 ~ 1844

/

were predicted by this analysis at times in the regime of typicalbreakup times: (Note that in the stretching Jet case A/2r changeswith time thus making the comparison more complicated.) However, the

approach of (141 has neglected inertia forces to make the governingequations analytically tractable. Because of the high strain ratespresent in the shaped charge problem inertia effects will be important.This importance of inertia effects will be verified by numerical cal-culations in the next sub-section. Thus, we feel that the results ofthis analytical method of 114] as applied to the shaped charge problemare not conclusive.

C. Numerical Study of Shaped Charge Jet Stability

In the previous sub-section we have studied surface tension andaerodynamic effects on Jet stability by relatively simple analyticaltechniques. Since effects such as material strength are, at present,not readily amenable to analytical treatment, we have used numericaltechniques to study this and other effects. One advantage of numericalstudies over analytical studies is that in the numerical treatment allof the pertinent effects may be included at the outset, whereas ana-lytical treatments necessitate the use of certain simplifying assump-tions.

1. General Approach

The numerical study of jet stability was undertaken through theuse of the two-dimensional code HEMP [20,21]. The HEMP code is ageneral purpose code which solves the conservation equations of two-dimensional elastic-plastic flow in plane coordinates or in axisym-metric coordinates. The solution is by the method of finite differ-ences and uses the Lagrangian formulation. The code has the capabil-ity of handling many various boundary and initial conditions.

Karpp [22] first applied HEMP code calculations to the problem ofa stretching elastic-plastic jet with the surface slightly disturbed.After calculating various wavelengths of the surface disturbance, hefound a broad range of most unstable wavelengths with reasonablegrowth rates. We have applied the same basic method to study thebreakup problem in more detail. The boundary and initial conditionscf this method will now be briefly described.

A stretching shaped charge jet is modelled by a prismatic circu-lar bar fixed at one end, with the other end moving at a constantvelocity. A linear velocity distribution in the axial direction isimposed as the initial condition, and the surface of the bar is ini-tially perturbed in the shape of a cosine function, as shown in Fig. 6.The perturbed surfaces are free from any tractions and the end sur-faces are free of any shear stresses. Let the axial velocity beV(x,r,t), the radial velocity be u(x,r,t), the stress vector on thelateral surface be •(x,rs,t), and the shear stresses on the end sur-faces be -rx(O,r,t), Txg(0,r,t), Trx(L,r,t), 1x 0 (L,r,t), then theboundary conditions are

19

VO

V(:,rOL

LL

FREE

A0

rol _ _ _ _

L/Lf

FTGURF. 6. Tnitial and boundary conditions for thenumerical jet stability computations.

V(0,r,t) 0 0V(L,r,t) 0 V0 (1)

A(x, r.,t) -Trx(Or,t) = tx 9 (Or~t) Trx(Lrt) - tx,(L,r,t) -0

and the initial conditions are

V(x,rO) - (Vo/L)x (2)u(x,r,0) - 0

The calculation may often be limited to only one cycle of the surfacewave along the axial direction because of the symetry of the problem.*

2. The Effect of Yield Strength

In the analytical studies discussed previously all of the jetswere assumed to be fluid in nature and the driving forces for the in-stability were restricted to surface tensions, viscosity, and aero-dynamic forces. Now we will study the effect of yield strength on thestability of an elastic-plastic jet.

To study this, we computed four jet segments having differentyield strengths but being otherwise identical. In these calculationswe selected a copper jet segment with one cycle surface disturbancehaving an initial length of 7.5mm and an initial mean radius of 1.5mm.The initial amplitude of the wave was taken as 10% of the initialradius. The velocity difference between the two ends of the segmentwas 0.0625 mm/,sec. In a real jet this initial time correspondstypically to a time 50 usec after the segment was formed or approxi-mately 100 usec after the detonation of a BRL 81.3mm charge. Thisinitial configuration and calculation mesh is shown in Fig. 7. In allcases the copper equation of state a4 specified in HEMP was used.The value of density used was 8.9x103 kg/m 3 (8.9 g/cm3 ) and the elas-tic shear modulus was taken as 4.56xlol Pa (456 kbar). The first ofthe four segments has no strength, i.e. purely fluid; the next threehave yield strengths of 2x10 7 Pa (.2 kbar), 2x10 8 Pa (2 kbar), and2x10 9 Pa (20 kbar). The configuration after 32 ijsec for each of thesefour cases is also shown in Fig. 7. To compare results we have com-puted a quantity . , the relative growth of amplitude, which is definedas the difference between the amplitude of the disturbance afterstretching, A, and the initial amplitude, A0 , divided by the initialamplitude, i.e. (A-Ao)/AO. This quantity is also given for each casein Fig. 7. We observe from these calculations that the disturbancegrows faster for jets iith larger values of yield strength.

* We are studying a typical element in the middle of the jet. This element isbeing stretched by the force of the forward part of the jet on one side andthe rear part of the jet on the other. The details of how the momentum andenergy is transferred from one portion of the jet to another are currentlybeing studied.

** Note that Karpp and Simon [23] have recently studied the strength inshaped charge jets using the experimental results of rotating chargesand also by using two-dimensional numerical calculations.

21

ALL CASES

hi+~~If~I ~STRENGTH

ts32jA~sec

Yz 2XIOT Po

&=:+194 %

F-E-L Y=2 x IO1Pa

A=+247%

INITIAL LENGTH: 7.5MM

INITIAL M1EAN RADIUS: 1.5MM

VELOCITY AT MOVING END: .0,n525mm/piSEC

A- RELATIVE GROWTH OF DISTURBANCE = (A-A 0)/A0

Y =YIELD STRESS (ELASTIC-PERFECT PLASTIC)

F 1(;LR[E 7. Ci'ipair I n of 4t nutin-, !t s

withi varirhis vik Id L~t rtngths.

I

For static applications, we are used to the concept that, when

comparing two materials, the one with higher strength will sustainlarger stress, larger strain, and more stretching. Our present re-sults indicate, however, that under dynamic conditions, the oppositecase, which seems contrary to common sinse, prevails. That is, underdynamic conditions of stretching, the material with higher yieldstress is more unstable, will neck more, and break sooner.

3. The Effect of Disturbance Initiation Time

In this sub-section the effects of initiating the disturbanceat various times in a realistic shaped charge jet are presented. Con-sider an element of a jet at several stages from the time when it isfirst formed until the time of its breakup as shown schematically inFig. 8. This element of jet has an initial velocity difference of0.13 mm/visec between its two ends. The initial length and velocitygradient were selected such that, at the breakup time, the elementwill have a length equivalent to the average measured length of atypical jet segment from a 81.3mm BRL standard charge. We may there-fore consider only one wavelength disturbance over this selected ele-ment. When this jet element is first formed one-dimensional calcula-tions indicate its length whould be approximately 1.1mm and its radiusshould be approximately 3mm. The times chosen to introduce the dis-turbance, labelled t, through t 5 in Fig. 8, correspond to times 0 Psec,7.92 vsec, 17.62 lsec, 25.69 iisec, and 45.71 psec after the elementwas first formed, respectively. The overall dimensions of this ele-

T ment at these different times are summarized in Table V.

Table V

Dimensions of Jet Element Studied at Various Times After Its Formation

Run Time after Element Element A/2r Amplitude of Strain RateNo. formation Length radius Imposed

(psec) (mm) (mm) Disturbance (Psec-)r(mm)

l 0 1.10 2.98 0.18 0.149 0.1182 7.92 2.13 2.13 0.5 0.107 0.0613 17.62 3.39 1.69 1.0 0.085 0.0384 25.69 4.44 1.48 1.5 0,074 0.0295 45.77 7.05 1.17 3.0 0.059 0.018

23

LENGTH OF BRqOKEN

TRA.JECTORY S E GMIEDff E,.TR ORYbft OF THE BACK.,y/ OF THE FRONT/

tb OF THE OF THE ELEMENT

DISTURBANCE INITIATEDIN THE ELEMENT AT EACH

t 4 OF THESE TIMES

t,-

t,

t, *"-ELEMENT FIRST FORMED

POSITION IN THE JET (

FIGURE 8. Schematic showing a position-time plotof a jet element which stretches to thetypical breakup length. Disturbancesare initiated seperately at each of thetimes indicated for independent HEMPcalculations of the element.

24

The HEMP calculations were performed using each of these times as aseparate starting point. The disturbance was introduced independentlyat each of these times and each was treated as a separate problem.The equation of state and other constants for copper stated in theprevious problem were again used. The yield stress was 2xlO9 Pa (2 kbar).The amplitude of the imposed disturbance was taken at 5% of theradius in each instance and is also given in Table V.

The five HEMP calculations indicate that, during short timesafter the initiation of the disturbance, the amplitude of the surfacedisturbance in Run No.4 grows the largest amount. This can be seen inFig. 9, where the relative growth, A, at a time 18 usec after the dis-turbance initiation for each case is plotted versus the initial strainrate. This gives an indication of how soon each disturbance begins togrow. As time proceeds, however, and we continue the calculations,the earlier configurations (Run Nos. 1-3) eventually reach a con-figuration similar to Run No.4, and then the disturbance begins togrow quickly. This can be seen in Fig. 10 where the relative growthof Run Nos. 2,3, and 4 are plotted versus timc. Examining Run No.2,we see that the disturbance is stable until a time of approximately30-35 wsec after formation, then the growth rate drastically in-creases. Run Nos. 3 and 4, in which the disturbance is initiated attimes later than Run No.2, also begin to grow more rapidly in the re-gion 30-35 psec. Thus we see that, no matter how early we initiatethe disturbance, when a critical time is reached, the disturbance willbegin to grow and eventually the same final breakup configuration willbe obtained.

4. The Effect of Disturbance Wavelength

The effect of disturbance waveletgth was studied by consider-ing a jet of realistic radius and straii, rate at a reasonably earlytime after formation and introducing surface disturbances of differ-ent wavelengths. A jet element having an initial cadius of 2.13mmand an initial strain rate of 0.061 usec-I was used. The equation ofstate and material constants for copper stated in the previous problemwere again used. Four separate surface dis-urbances having initialwavelengths of 1.065mm, 2.13mm, 4.26mm, and 8.52mm were introduced andHEMP calculations made for each case. Note that we have chosen thisproblem such that the case A - 2.13mm will eventually grow to theobserved segment length after the experimentally determined breakuptime for a typical shaped charge jet. Let us denote t'is "correct"initial wavelength as X0 - 2.13mm. We can then denote the othercases as X - A0 /2, A = 2\0 and X = 4AO. The relative growth, ascomputed using the HEMP code for each of these cases, is plotted vs.time in Fig. 11. From this plot we see that the disturbances havinginitial wavelengths of - 0 /2 and A 4- 0 grow very slowly. Thewavelength X= A0 , which will gro4 into the proner segment size, growsvery quickly but not quite as fast as the case X 2XO.

25

RUN4 No. 4

160-

140

120 4RUNJ-% NO. 5 .- RUN NO.3

t- =181ASEC AFTERDISTURBANCE IS

S0INITIATED60-

40

20-RUN NO.I -2 RUN NO.2 - . M NU/

.0 .04 .6,1 1-20 ...

INITIAL STRAIN RATE (/LSE(' )

FIGURE 9. Relative growth vs. initial strainrate for five successive element

configurations at 18osec after

disturbance initiation in each

configuration.

26

IMI

C:CN4

..r

C)

CNC

cu.J CE

V RLM9 3A IV13

27S

600-500

, • 400Xo 0 2.13mm

• <200i

LLI.

100

0 , -- - - . , _ _ ,

0 10 20 30 40 50 60

TIME (pLSEC)

FIGURE 11. Relative growth vs. time for surface disturbances offour different wavelengths. In all cases, the jetsinitially have the same radius and strain rate.

28

J t

5. Irregular Surface Disturbance

To examine these wavelength effects further, we have numericallystudied jets having an irregular surface disturbance. In previouscases, we have considered jets having a surface of a single sinusoidalcurve. Next, we have combined the four wavelengths of the previousstudy to obtain an irregular surface disturbance which is describedby the following function:

4 2•rxr= r0 + A r cos k

where kl X0 /2, k2 - A0, k 3 - 2XO, k4 - 4 X0. We have taken numer-ical values to be ro - 2.13-m, A0 - 2.13umn, A0 = 0.026mm, and theinitial strain rate equal to 0.061 usec- 1 . We again used the materialconstants for copper described previously. The initial configurationwhich has a length of 2X0 is shown in Fig. 12, together with the con-figuration after 52 wsec. We observe that the jet appears to neck intwo places. This indicates that AO is the dominant or fastest grow-ing wavelength, which is in agreement with the experimentally ob-tained segment length. To demonstrate this more quantitatively, wehave fit a Fourier series to the outei surface cf the configurationafter 52 isec shown in Fig. 12 and have examined the coefficients ofthe terms of each wavelength. Denoting the length after 52 wsec as2A1 , we observe that the largest of all the Fourier cuefficients isthat of the term containing the length -= ". A comparison of thecoefficients of the four wavelengths of interest are also shown inFig. 12. Tlhus, this result indicates that the component with aninitial wavelength of A - 10, which stretches into A= l after52 ;sec is the most critical one.

In the case above, :-e initial disturbance consists of fourwaves of different wavelength, Which are all "in phase". We performedyet another irreeular surface disturbance calculation. In this calcu-lation, a "random" disturbance was used which was comprised of fivewaves of different wav-iength and phase angle.-, so that the waveswould not be "in phase" at the ends. Thus, this new disturbancerepresents more closely a random one. The function used for the sur-face was

r,, A- cos A-•= ~ ~ \ r ii ki

w e .= " ', k, k 3 = 4' k i,/-, = -- , aid -1 -,36 3 : 54, :5 = "3. Numerical values were

ta;sr as r 2.i3, , Ail = 0.053am, and the initial strainratý --ciai ••h.l _.sec-1 7he initial configuration and the con-fi.4-iri.tian iv-, r sec no stretching as t-.culated by the HiEMPcode are ;'-_;. :n z. 13. We ee the -et necking in two placeswhicnl agai1 .- .:ates that -- e zrawth of the wave predominatesand that tit -roper et segrent length will be attair.ed at oreakup.

I6A

1-17

LA--

.- I• A

0 9

01

Ar

tino a)

.. e' ,4.4 C

I 0 41

cli

,. C,

-A: I -

4.4 "a

4 ý4

c~%l04 UCD LA H

II II c~ w

E

)7-A

' -C '3 1

6. The Effect of Inertia

To study the importance of inertia effects on the growth ofsurface disturbances in stretching shaped charge jets, we have madeHEMP code calculations of three jets. Each jet was identical in allrespects except for the density of the material used in the calcu-lations. The configuration used was the standard single wave surfacedisturbance depicted in Fig. 6 with r 0 - 2.13mm and wavelength equalto 2.13-m. The initial amplitude used was AO - 0.107mm and theinitial strain rate equal to .061 usec- 1 . The equation of state andmaterial constants described previously for copper were used exceptnow three different values of density were used PO - lx103 kg/m 3,p0 - 8.9xl03 kg/m 3 , and PO - 16.5xl0 kg/m 3. The relative growth vs.time for each case is plotted in Fig. 14. We observe that increaseddensity, i.e. increased inertia force, has a retarding effect on dis-turbance growth.

32

F--

600

5W0 po~lXlO'kg/m 3

30 poz8.9xlOkg/ms

200

lODp =16.5xIO~kg/m 3

0 10 20 30 40 50 60TIME (ILSEC)

FI"ItURE 14. Relative growth vs. time for the stretching jets ofdifft-rent densities Ais predicted by HEMP.

13

IV. CONCLUSIONS

The major results and conclusions of the present research maybe summarized as follows:

A. Measurement of Jet Radiographs

1. Data from jet radiographs, including jet velocity,breakup time, and segment size, for jets from elevencharges were compiled and plotted for easy reference.

2. These measured jet velocity and jet radius dataagree closely with those calculated from a one-dimensionalshaped charge model.

3. Analysis of this data together with using one-dimensional calculations resulted in a semi-empiricalbreakup time vs. jet radius curve for copper jets. Thiscurve can be used to estimate an approximate breakuptime for new charge designs.

B. Stability of Shaped Charge Jets

1. Simple stability analyses and experimental resultsindicate that surface tension and aerodynamic forcesare not important in the breakup of shaped charge jets.

2. Numerical HEMP code calculations of stretchingelastic-plastic jets subjected to surface disturbanceswere conducted. The effects of material yield strength,time of disturbance initiation, wavelength of the distur-bance, irregular surface disturbances and jet densitywere examined. The following results were found:

a. Jets with higher yield strengths break sooner, allelse being equal.

b. Jets with lower densities will break sooner, allelse being equal.

c. For the shaped charge jet calculated, there is acritical time for the growth of the disturbanceamplitude. Disturbances introduced early do not growappreciably before this time, but grow rapidly afterthis critical time. Disturbances introduced after thistime also grow rapidly.

d. A critical wavelength (or a range of wavelengths)exists; disturbances having this wavelength grow fasterthan all others. The length of the broken jet segmentcaused by this critical wave is in the range of measuredjet segment lengths.

e. When irregular, or random, surface disturbances areintroduced, the growth is again domirated by the distur-bance component with the critical wavelength. The jetsurface grows into a shape similar to that obtained if

34

only the wave with the critical length were introduced.

3. Materials with high density and low yield strengthare promising liner materials which will likely retardthe breakup of the resulting jet.

35

V. REFERENCES

1. Chou, P.C. and Carleone, J., "Calculation of Shaped ChargeJet Strain, Radius, and Breakup Time," U.S. .. rmy BallisticResearch Laboratories (BRL) Contract Report No. 246,July 1975. (A[ 'BO07240L)

2. thou, P.C. and Carleone, J., "The Breakup of Shaped Charge

Jets," Proceedings of the 2nd International Symposium onBallistics, March 9-11, 1976, Daytona Beach, Florida, Sponsoredby the American Defense Preparedness Association.

3. DiPersio, R., Jones, W.H., Merendino, A.B., and Simon, J.,I "Characteristics of Jets from Small Caliber Shaped Chargesk with Copper and Aluminum Liners," U.S. Army Ballistic

Research Laboratories (BRL) Memorandum Rept. No. 1866,Sept. 1967. (AD #823839)

4. thou, P.C., Carleone, J., and Karpp, R.R., "Criteria forJet Formation from Impinging Shells and Plates," J. AppliedPhysics, Vol. 47, July 1976, pp. 2975-2981.

5. Pugh, E.M., Eichelberger, R.J., and Rostoker, N., "Theoryof Jet Formation by Charges with Lined Conical Cavities,"J. Appied Physics, Vol. 23, No. 5, May 1952, pp. 532-536.

6. Held, M4., "Air Target Warheads," lrternational DefenseReview, No. 5, 1975.

7. Perez, E., private communication, French-German Instituteof Saint Louis (ISL), 12 Rue de £'Industril, St. Louis58 France.

8. Carleone, J. and Chou, P.C., "A One-Dimensional Theory toPredict the Strain and Radius of Shaped Charge Jets,"Proceedings of the First International Symposium onBallistics, Nov. 13-35, 1974, Orlando, Florida, sponsoredby the American Defense Preparedness Association.

9. Chou, P.C. and Carieone, J., "Shaped Charge Jet BreakupResearch," Quarterly Tech. Rept. for Contract No. DAAD05-75-C-0753 for the period June 12-Sept. 11, 1975.

1 0. Chou, P.C. and Carleone, J., "Shaped Charge Jet BreakupResearch," Quarterly Tech. Rept. for Contract No. DAADO5-

75-C-0753 for the period Sept. 12-Dec. 1]., 1975.

11. Lord Rayleigh, The Theory of Sound, 2nd. ed., Vol. 2,Dover, New York, 1945, pp. 351-363.

12. Weber, C.: "Zum Zerfall eines Flussigkeitsstrahles"(Disentegration of a Liquid Jet), Zeitschrift fur

Angewandte Mathematik und Mechanilk, Vol.11, No. 2, April1931, pp. 136-153.

13. Levich, V.G., Physicochemical Hydrodynamics, Prentice-Hall1962.

-H

14. Mikami, T., Cox, R.G., and Mason, S.G., "Breakup of Ex-tending Liquid Threads," Post-Graduate Research LaboratoryReport, Pulp and Paper Research Institute of Canbda,PGRL/72, October 1974.

15. Goldin, M., Yerushalmi, J., Pfeffer, R. and Shinnar, R.,"Breakup of a Laminar Capillary Jet of a ViscoelasticFluid," J. Fluid Mech., Vol. 38, part 4, 1969, pp. 689-711.

16. Anno, J.N. and Walowit, J.A., "Integral Form of theDerivation of Rayleigh's Criterion for the Instability ofan Inviscid Cylindrical Jet," American Journal of Physics,Vol. 38, No. 10, Oct. 1970, pp. 1255-1256.

17. Anno, J.N., "Influence of Viscosity on the Stability of aCylindrical Jet," AIAA Journal, Vol. 12, No. 8, Aug. 1974,pp. 1137-1138.

18. Vitali, R., private communication, U.S. Army BallisticResearch Laboratories.

19, Frey, R., private communication, U.S. Army Ballistic1esearch Laboratories.

S20. Wilkins, M.L., "Calculation of Elastic-Plastic Flow,"University of California, Lawrence Livermore Laboratory,Rept. UCRL-7322, Rev. 1, Jan. 24, 1969.

21. Giroux, E.D., "HEMP User's Manual," University ofCalifornia, Lawrence Livermore Labora-ory, Rept. UCRL-51079, June 24, 1971.

22. Karpp, R.R., private communication, U.S. Army BallisticResearch Laboratories.

23. Karpp, R.R. and Simon. J.. "An Estimate of the Strength ofa Copper Shaped Charge Jet and the Effect of Strength on theBreakup of a Stretching Jet," U.S. Army Ballistic Research

Laboratories (BRL) Report No. 1893, June 1976. (AD #B012141)

24. Simon, J., "The Effect of Explosive Detonation Characteristicson Shaped Charge Performance," Army Science Conference,West Point, N.Y., June, 1974.

37

Appendix A

This appendix contains the specifications of the eleven chargesstudied in this report.

ZCZuDfIIfl P".ZsBLNT IL D

K 1.875 IN (47.6 mm)

1.5 IN - I (38.1 m)

0-- NSTEEL CONTAINER2.20 IN (HEAVY CONFINEMENT)

(55.88 mm~)0or A COMPSITIONB

EXPLOSIVE

(REF) A

CHARGE LINER E ANO. MAT'L a IN IN

(M) (m)

1 Cu 100 0.046 4.2531.168 108.0

0.046 2.0612 Cu 200 52.35

0 0.046 1.2993 Cu 300 1.168 33.00

40 0.046 0.7501.168 19.05

9 AL 200 0.064 2.0611.626 52 35

10 AL 300 0.064 1.29911.626 33.00

0.064 0.750II AL 4 1 1.626 19.05

FIGURE Al. Geometry of the 38.1 mm Copper and Aluminumlined charges with various cone angles.(Charge Nos. 1-4,9-11)(fror, Rhf.

)= ' 2.5 IN _l(63.5 w)2 IN (50.8 mm)

2 . INALLMINUM C(ONTAINER(503 I) (LIGHT CONFINEMENT)

COM SITIONB

2.6MIN

(66.17 mm) COPPER LINER

E

CHARGE INNO. (i

0.0305 0.762

0.0606 1.524

0.100

2.540

F!GURE A2. Geometry of the 50.8 rmm, 420 Copper linedcharges with different wall thicknesses.(Charge Nos. 5-7)(Courtesy of R. ,amneson)

12

1 6o

UI-

0c

LI5

EE

cr tr

000

LC40-

0 6

LL7

+1 IC'I 0

0O~- 4-1

__________I _______

loo* T

44lMl

Appendix B

This appendix contains a graphical display of jet velocity, jetbreakut time, and jet radius data computed from the jet radiographsof the eleven charges studied in this report.

I1

9

COPPER LINERS

"0

8 00(0 0

00 0 0-200

93.4 SEC o0 0

-00 000~ .. 6l

0L 0

000 0

300000 0 000000

j0 000 00oo

00

0 0O

3 0 0 EXPERIMENT

-THEORY

2 1a300 1400 :)ti 5 00 700 900 91

JET PARTICLE POSITIVM C (W)

FIG[URE BI. 'heoretical and experimental jet velocity vs. let

particle position for the series of 38.lImm, 1.168rmn

wal I , copper lined charg,,es with various cone angle5at various times after the. arrival of the detonaticwave at the cone apex. (charge nos. 1-4).

UJ

9 L00

ALUMINUM LINERS 00

0 00

600 00 0

0 0= 7 97,7 7,uSEC oO0 0 113,1p- SEC

0 0 000 o0

0 0 ~ Q0 %,-goo132.8 SEC

5 00 0 EXPERIM~ENT

- THEORY

- iI I I I i

400 500 600 700 800 900 1900

,JET PARTICLE POSITION C (tM)

FIGURE 82. Theoreticnl .and experinentla I j t velocity vs.jet part i- le position for thi' series of 38. 1mr,I.6?6mm wall, aluminum iined charges withvarious cone angles, at various times after thearrival of the detonation wave at the cone apex.(Charge Nos. 9-11)

00

zo

0: LA >. to4

'-4n

zC Co '-z

w ~ ~ 0 0 0 ~:

0 0

a C

-o

C:)t'01CD_rC-4

U-N~~~~ ~ ~ ~ ~ C)z)C) ul D LI D ul = -06 -ý -ý , t6 16 U"% 11%-TCo o * 1.

(:)3S-d~ ~ ~ ~ /W)rA l-01A

0

zU C: 1: 9

C W 0:041 '-U 0. w , - M

x =IoL I.-u

00 :c

- -S CL

00 -- N-N

Pgs~~ ~ ~ ~ ~ 7f/w A'01AA

50~

iI. - - - - - -

(/, 0

0 x z"- -

0 0 C-:) *00 00 0) N

0 -3

= o

0t . w -Co= ý t,C -.

Lrý~~ -3-r t

(AS~~ ~ ~ /W)r JJ-1AAAO~1-

"-I-

C -e

0 u

0

0~

00"

6. X

c -::

I~~~- o,!II• .

(33 W'•) rALI1 -74 J,.F

0001

000 0

00

Lni

(03S~~C- 7'W-Ak13-a 3

70 COPPER-20 00 0

0

300

O0 500 T00 700 1 300 900

JET PARTICLE POSITITIJ a (mm)

FIGURE B7. Jet breakup time vs. jet particle positionat t-107.7 'sec for the 38.1 mm, 20%, 1.168 mm wall,copper lined charge (charge no. 1).

5.1 ;

90"COPPER-400

0 00 0 0

~70 0 0 0 00 0 00 0 0

0 0 O 0 0 ~00 000

30300 400 500 600 700

JET PARTICLE POSITION C (mm)

FIGURE B8. Jet breakup time vs. jet particle

position at t-93.4 wsec for the38.1mm, 40 , 1.168 mm wall copper

lined charge.(Charge No.2)

i4

120COPPER-60°

100

0 0 0 0

S60LU

00800

W 0 0

0 0

'100

20 I I I 0 I

300 Lo0 500 600 700 800

JET PARTICLE POSITION .• (MM)

FIGURE B9. Jet breakup time vs. jet partic:le pos'itionat t=122.7 Uspc fo the 3.m,60,1.168 mm wall Copper lined charge.(Charge No. 3)

330

i0

120COPPER-90 0

0

100

68040

00~60-0

0

'40

201300 400 500 600 700 800

JET PARTICLE POSITION C (MM)

FIGURE B1O. Jet breakup time vs. jet particle positionat t-133.3 tsec for the 38.1mm, 900,1.!68 mm wall copper lined charge.(Charge No. 4)

S f

ALU1IINUM-40°_0I o

D 0 O 0 0

*60 1

W- 00,. 0 0

CL-20 0,• 40 0

LUU20 0

600 700 800 900 0OO0

JET PARTICLE POSITION • (MM)

FIGURE B11. Jet breakup time vs. jet particleposition at t113.1 isec for

the 38.1mm, 40°, 1.626mm wall aluminum

lined charge.(Charge No.9)

57

i

S~i

120ALUMINUM-60 0

0

100 0.

w 00GO

• -" 0

80 0 00 0

0 0

LU

0

00 0

40040-

20 I I i i i

400 500 600 700 800 900

JET PARTICLE POSITION • (MM)

FIGURE B12. Jet breakup time vs. jet particle positionat t=97.7 lisec for the 38.1rmw, 60°

1.626 nun wall aluminum lined charge.(Charge No. 10)

"58 i

160ALUM!INUM-90 0

140

.120

LU

S100 00

800 0

60 'I L - I - -

'400 500 600 700 800 900

JET PARTICLE POSITION (mm)

FIGURE B13. Jet breakup time vs. jet particle posic onat t=132.F iisec for the 38.1mm, 901.626 mm wall aluminum lined charge.(charge No. 11)

v5

A -o

8!

0 c

0Q

00

00

00 0.

0

0 0 L6-C

(33S

0 -_

oo0

0 .

/ o-o o L

I I L .. I I

(Y3sli) 3111 dl~V~fti liP

I

0

80

o -� �a.

oo -

44.D00

00 � r4

0 'a

0 00 -

-- .0

0 -

0

0

00I..

0 I-.- 'a

0 -�

oo (J�

0 0 � C- -� -

0 -

0�-, 0.Q-c�.- I�

0 � 'a

00

0

8 S

K

5.

ion 00

00

U0

-,AJ

0 020

S000• 0 0

500 00 0W9J

400

PARTICLE POSITIcri HI TIE JET (tv)

FIGURE B16. Jet breakup time vs. jet position att-130.7 tusec for the 50.8 umm, 42%, 2.54 rm wall,

copper lined charge (charge no. 7).

u w

000 r ..,

0 0

> -

00

00

00

0 00 =• •

0 -. •-

0 L4 - .

0 W)u

0 0. ' ." -,"

0

0 '

0 ••

0 _ • - -..

(!3SIi 1 I I ,. •_Ii

1.2-

80 000

, .4 o000ooo 00 0 0 00 00 0 0 0 00=00

L__ r, M th

0400 500 600 700 800 900

PARTICLE POSITION! IN JET C (MM)

FIGURE B18. Theoretical and experimental jet radius vs.jet particle position at t=107.7 Wsec for the 38.1 mm, 200,1.168 mm wall, copper lined charge (charge no. I).

1.2

S.8

0-4 0 0 0 0 0o000 CO 000: 4 0

_r. Ot 7t. 7

LU

0- ------

300 409 500 600 700

PARHCLE POSITION IN JET f (mM)

FIGURE B19. Theoretical and experimental jet radius vs.

jet particle position at t=93.4 11sec for the 38.1 mmr, 400,

1.168 rma wall, copper lined charge (charge nc. 2).

05

1.6

00

1.20

0 000 0"00 0 00

.8

0 0 0

0 70o0-

rj at

I b _ _

0300 400 500 600 700 800

PARTICLE POSITION IN JET • (MM)

FIGURE B20. Theoretical and experimental jet radius vs. jetparticle position at t-122.7 Psec for the 38.1 mm, 60',1.168 mm wall copper lined charge (chuaig fL.. 37.

661

II

2.4

2.0

1.6

0 0

1.20

C 0 0

00

• .8 00 0 0LAJ 00 0S~000

i b

0 L I a

300 400 500 600 700

PAPTICLE POSITIO'I III JET C(MM)

FIGURE B21. Theoretical and experimental jet radius vs. jet

particle position at t-133.3 *.sec for the 38.1 m, 900,

1.168 m wall copper lined charge (charge no. 4).

1,6

1.2

000 0500000 00 800 900 0

S00 0 0

.4 rA t

500 600 700 800 900 1000

PARTICLE POSITION IN JET C(MM)

FIGURE B22. Theoretical and experimental jet radius vs. jetparticle position at t-113.1 wsec for the 38.1 mm, 40Q,1.626 mmwall aluminum lined charge (charge no. 9).

1.2

3000000 0= 00 00 0 0000P .8 0

tr stt t

S .4LiJ

300 1400 500 600 700 3000

PARTICLE POSITION IM JET C(mm)

FIGURE B23. Theoretical and experimental jet radius vs. jet

particle position at t-97.7 wsec for the 38.1 mm, 600%

1.626 mm wall aluminum lined charge (charge no. 10).

()9

2.4 0

2.0

1.60

0

•1.2 0

-. 0 0 0 0 0 000 0

.8

r at t h

.4 I I

500 600 700 800 900

PARTICLE POSITION IlN JET C(MM)

FIGURE B24. Theoretical and experimental jet radius vs. jetparticle position at t=132.8 Psec for the 38.1 mm, 900,1.626 m wall aluminum lined charge (charge no. 11).

"-o

--. .. .. A

C4I

84L

00 cn -

o 0)

40

0 0C .ca.

0a0~H

0 E-44

o Cý

00Z

St 1 ICVd 131

07

S~-~-.----~ 41

CL C0r) f

4AP

00

0 Fn 14 '

08 '-4 1J '

0 - -4

4- .0 VA i 4

U .W 14

Qw 0. w

d -H

0 m

00 p

i-14

72

IU

01.6

12 00 0 00 000 00000

.8 0 0

0

0

t. " 00

0 0 0 0•

400 200 600 700 800 900

PART ICLI POSITION IN JET W(41

FIGURE B27. Theoretical and experimental jet radius vs. jetparticle position at t=130.7 usec for the 50.8 mm,420, 2.54 mm wall copper lined charge (charge no. 7).

73

2.0

10

1.6 00o10 0 0 0 0

000 00 000 0

0,8 r• alt°

0 0

h

0,LI . i I __500O700 3OO 900 0000

PARTICLE POSI]IlJ II lTHE JET C("i)

FIGURE B28. Theoretical and experimental jet radius vs.jet particle position at t=130.5 p'sec for the 81.3 mm420, tapered wall copper lined charge (charge no. 8).

(Note: The lead particle at 1003 mm in the jet has a

radius of 4.6 mm and is not plotted here for convenience.)

74

Appendix C

This appendix contains tabulated data for the jet velocity, break-up time, length, diameter, radius, aspect ratio and AV- of the jetsegments as computed from radiograph measurements for the elevencharges studied in this report.

The breakup times in these tables are referred to t=O when thecharge is first initiated. Throughout the rest of the report, includingthe plots of appendix B, all times are referred to t=O when thedetonation wave first reaches the apex of the liner cone. The timeit takes for the wave to travel from the point of initiation to theapex of the cone for charges 5,6 and 7 is 6.4lisec; for all othercharges this time is 7.Oisec.

75

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rr n ra e

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