AFML-TR-76-!2

SSTRESS ANALYS1.1 OF PLASTIC ROTATNG EANDS

CALI)FORNIA RrSEAR Cl & 7ECHINOLOGk'. INC.6289 VAJIIE AVTZNVD. SJITE f20QWOODLAND HILLS. CA i,13.4

FEBRUARY 1976

TECHNI6CAL ?.EPOK1 AiML-TR.70L12 .FrN4AL RE1VOE F3R PERIOD APRIL 1915 - S'ETMMEFR 1075

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AIR Vt ACE MATERIAIZLk 14 BO~RYAIr F ORCE WRIGHT iE *OTUAUTiCAL LAFIOMOAMLAIR'ORCE SiSTEMS COMAND ~ 44.

-t0TICE

When Govsrnment drawings, specifications, or other data are used -forany purpose other Chan in connection with a definitely related Govern-wsent procurement operation, the United States Government therebyincurs no responsibility nor any obligation whatsoever; and the factthat the Government may have formulated, furnished, or in any waysupplied the said drawings, specifications, or other data, is not tobe regarded by implicatiov or otherwise as in any manrer licensingthe holder or any other person or corporation, or conveylng any rightsor permission to manufacture, use, or sell any patented Inventionthat may be related in any way thereto.

This report has been reviewed by the Information Office (01) oi• isreleasable to the National Technical Information Service (NTIS). AtNTIS, it will be available to the general public, includiPg foreignnat.ons.

Thi3 technical report has been reviewed and is approved for publication.

Project Monitor

THE m DIRECTOR

S. W. Tast, ChawfHechanics & Surface lnteracticns BranchNoamatallic Materials Division

C.-pies of tals report should net be retarned unless return In requiredby security coesi-eratioda, contractual oblipatiots, or notice on aspecific dec=Kut.

-. ". 0, - ,.I

EPRTDOCMENTATIO*4 PAGE REA INSTRLJCTO"S

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CRT361 -1--

California Research & technology, Inc.6269 Varell Avenue. Suite 200 ILIKO073Woodland Hills,-•A-91364

Air Force HeterialA Laborator- (AFML/W) Fe *76Air Force Wright Aeronautical Laboratories

Wni t-PFt~er&,x, A_ _ _ Ohio 45433 93

Uniclassified

-- "n.roved foz Ptubli rel e ritmo, ua limited.

Size

Interlor ballls'.liarotatlg bandsdynamic strese analysia

finite-difference codes

-~Dynxamic stress analyst* of plastic rotating :,ands, used on poectiles to iepart spin, have been obtained by meann of a tvo-.isnsic-alU ffnite-diffter.nc computer code. The firs-: 5base of the interaction withinthe barrel. involy;ng the axisymsetrIc convergence of the plastic band mnzothe forcing come. hau been investigated in thia study. N-wnerical solutionsfor three rotatlog band prablems were ohtaioed. The results of those solutionsshowed that (a) *bearl•g foxces rcting %long the b4nd/barel InterC:ct &uo to

I '-

I ie•l•li 'l - .-".•--•,,..._-7lllir [ 'II 'i l ll l ii a d

i •x. J. NClAbI~FIFD

!riction can have a ma,#or effect on rotating band perforstance, and (b) thatvith a current band deteign. the sharply-beveled rear section of thir handbecomes grossly distorted and develops a lar-.e an-nular lip protruding frorthe rear of the band (I., , feathering). The third case considered a designvzoditication vade to alleviate the extrusion problea. These initiai analysesshow that finite-difference code solutions of the dynamics nf rotatic,; bandsare practical and can provice a usefsl tool to investignte a number of keyfactors bearing on plastic band perfor"nce. su-h as a.iterial properties,vtress-strain conditions experienced, and band geomercies.

W I.

I -|

PREFACE

This report describes a research program to analyze

the dynamic stresses and deformations in plastic rotating

bands (such as used on projectiles) as they are forced into

a converging forcing cone at high speed. A two-dimensional

finite-difference computer code was used far the analyses.

The program was conducted during the period April through

September 1975, at California Research & Technology, Inc.

6269 Variel Ave., Woodland Hills. CA 91364, under Contract

F33615-7S-C-5206, initiated under Task No. ILIR9073. The

research was performed by M, H. Wagner and K. N. Kreyej;'agen.

W. S. Goerke, C. K. Wilson, and C.C. Fulton provided assis-

tance in cod- development and programming. The Project En-

gineer for the Air Force Materials Laboratory was S. W. Tsai,

Chief, Mechanics & Suiface Interactions Rranch, Nonmetallic

Materials Division (AFML/MBM). The authors submitted this

report in October 1975.

iii!

____ ____ ____ ____ ___

TABLE OF CONTENTS

Page

SECTION I. INTRODUCTION, SUMMARY AND RECOMMENDATIONS . 1

A BACKGROUND .. . . . .. . .. . .. . . . 1

B OBJECTIVES ............ ................. 3

C NUMERICAL APPROACH FOR DYNAMIC STRESSANALYSIS .............. .................. 3

[ SUMMARY OF RESULTS ..... ............... 5

E CONCLUSIONS AND RECOMIMENDATIONS .... ...... 7

I Utility of Numerical Analysis of DynamicStresses and Distortions in Rotating Bands 7

2 Recommendations .......... .............. 9

SECTION II. NUMERICAL SOLUTIONS ......... ............ 12

A PROBLEM DEFINITION ..... ............. .. 12

B MATERIAL PROPERTIES ..... ............ .. 13

C NUMERICAL METHOD ................. "... 14

0 NUMERICAL SOLUTION OF CASE 1 - PLASTIC BANDINTERACTION WITH FRICTION ..... ......... 15

E NUMERICAL SOLUTION OF CASE 2 - PLASTIC BANDINTERACTION WITHOUT FRICTION ........... .. 16

F NUMERICAL SOLUTION OF CASE 3 - MODIFIEDPLASTIC BAND DESIGN ......... ............ 18

APPENDIX A: PRINCIPAL STRESS FIELD PLOTS .... ....... 57

APPENDIX B: TIME HISTORIES 0 STRESS AND VElOCITY. .. 66

REFERENCES .................. ...................... .. 86

iv

LIST OF ILLUSTRATIONS

Figure Pg

1. Field of Analysis for Computations of a Proj-

ectile/Rotating Band Tunneling into a ForcingCone ............... ....................... ... 21

2. Problem Conditions for Numerical Solutions . . . . 22

3. Deformation and Flow in Plastic Band at 11.4 usec,Case 2 (Baseline design, no friction) ......... ... 23

4. Deformation and Flow in Plastic Band at 7.7 psec,Case 3 (Modified design) ..... ............. ... 24

S. Deformation of Plasts Bazd at 20.7 and 27.6 usec,Case 3 (Modified designr ........... ........... 25

6. Axial Force at Band/Barrel •n~erface vs Time forBaseline and Modified Plastic Band Designs . . . . 26

7. Suggested Method for Plane Strain Analysis ofRotating Band-Rifling Interaction ............ ... 27

8. Baseline Plastic Band Geometry and ComputationalGrid (Cases 1 and 2 - Baseline Design) ........ .. 28

9. Grid Configuration at 1.5 ;isec, Case withFriction (Case 1 - Baseline Design) ........... .. 29

10. Ver.,7i.y Field at 1.S usec. Case with Friction. 1 - Baseline Design) ..... ............ .. 30

ii. C:iul Configuration at 3.6 usec, Case with Friction(Cise 1 - Baseline Design) ..... ............ .. 31

12. Velocity Field at 3.6 psec, Case with Friction(Case 1 - Baseline Design) ..... ............ .. 32

13. Velocity Field at 1.5 psec, Case without Friction(Case 2 - Baseline Design) ..... ............ .. 33

14. Grid Configuration at 3.1 usec, Case withoutFriction (Case 2 - Baselino Design) ........... .. 34

v

i, !

LIST OF ILLUSTRATIONS (Cont'd.)

D.9g1 re Page

15. Velocity Field at 3.1 usec, Case without Friction(Case 2 - Baseline Design) .... ............ ... 35

16. Grid Configuration at 7.9 usec, Case withoutFriction (Case 2 - Baseline Design) ........... ... 36

17. Velocity Field at 7.9 psec, Case without Friction(Case 2 - Baseline Design) .... ............ ... 37

18. Grid Configuration at 11.4 usec, Case withoutFriction (Case 2 - Baseline Design) .... ......... 38

19. Velocity Field at il.4 usec, Case withoutFriction (Case 2 - Baseline Design) ........... ... 39

20. Grid Configuration at Rear Section of PlasticBand at 11.4 usec, Case without Friction (Case 2 -Baseline Design) . ........ ................. .. 40

21. Comparison of Axial Forces on Barrel with andwithout Surface Friction (Cases 1 and 2 -Baseline Design) ........ ................. ... 41

22. Comparison of Radial Forces on Barrel with andwithout Surface Friction (Cases 1 and 2 -Baseline Design) ...... ................. ... 42

23. Axial Force at Band/Barrel interface vs Time,Case without Friction (Case 2 - Baseline Design) . 43

24. Radial Force at Band/Barrel Interface vs Time,Case uithout Friction (Case 2 - Baseline Design) . 44

25. Normal Stress (on) and Tangential Stress (T)Distributions on Barrel, Case with Friction(Case I - Baseline Design) ......... ............ 4S

26. Normal Stress (on) Distributions on Barrel,Case without Friction (Case 2 - Baseline Design) . 46

27. Modified Plastic Band Geometry and ComputationalGrid (Case 3 - voaified Design) .... .......... ... 47

vi

li ir I i • l , ... ...

LIST OF ILLUSTRATIONS (Cont'd.)

FigurePage

28. Grid Configuration at 7.7 usec, ModifiedPlastic Band Design (Case 3) ..... ............ ... 48

29. Velocity Field at 7.7 psec, Modified PlasticBand Design (Case 3) ........ ................ ... 49

30. Grid Configuration at 15.S usec, ModifiedPlastic Band Design (Case 3) .... ............ ... SO

31. Grid Configuration at 20.7 usec, ModifiedPlastic Band Design (Case 3) ..... ............ . .. S1

32. Grid Configuration at 27.6 Usec, ModifiedPlastic Band Design (Case 3) ........ ............ 52

33. Velocity Field at 27.6 psec, Modified PlasticBand Design (Case 3) ........ .................. 53

34. Radial Force at Band/Barrel InterZace vs Timefor Baseline and Modified Plastic Band Designs . . . 54

35. Distributions of Normal Stress along Band/BarrelInterface, Modified Plastic Band Design .... ....... 55

36. Peak Compressive Stresses within Band, Baselineand Modified Plastic Band Designs (Cases 2 and 3). , 56

vii

"33

SECTTON I

INTRODUCTION, SUMMARY, AND RECOMMENDATIONS

A. BACKGROUND

Rotating banus are used on projectiles to impart spin

(by transmitting torque from rifling to the projectile), to

improve the propellant gas seal, and to reduce wear of the

barrel (by minimizing direct contact between the hard pro-

jectile and the walls). Rotating bands must generally be

of pliable and/or ductile material, since they must with-

stand substantial plastic flow in accommodating to the bar-

rel and to the rifling. The material must also have suffi-

cient strength to resist the propellant gas pressure, to

transmit torque to the projectile, and to renain attached

to the rotating projectile as it leavez; the muzzle.

Forcin Cone Rifling

Projectile)

Rotating Side View End ViewBand

This sketch (not to scale) shows a typical rotating

band. The projectile is somewhat smaller than the minor

diameter of the tore, but the OD of the rotating band is

larger, so that there is dimensional interference. When

the projectile is fired, it "tumtels" into a forcing conewherein the rotating band is squeezed down until it fits

into the barrel. To accomplish this squeezing, the rotat-

ing band material:

1B

(1) eku eiecufc-entially into the rifling grooves

(2) , com,[reveed radially under the high confining

pressure of the barrel

(3) Flo,)e (aially to relieve the volumetric compres-

sion (i.e., the rotating band becomes longer).

Within the barrel, the rifling can have either a fixed

or a variable twist (pitch) alon5 the barrel length. This

twist, acting through the iotating band, imparts angular

acceleration as the projectile itself undergoes linear ac-

ccleration down the barrel. As the projectile leaves the

muzzle, the radial compression of the band is relaxed. The

band remains attached to the p-ojectile until it reaches

the target.

Some nylon-type plastics are desirable materials for

rotating bands, amd they are successfully used in this ap-

plication. However, failures of nylon rotating bands have

been experienced, especially in projectiles wr.ich are de-

signed to reach higher muzzle velocities. Information

about the timing of such iailures (i.e., the location with-

in the barrel at which failure occurs) ard the cause there-

of is not available. nor is it readily attainable through

ex eriments. Such information is needed in order to design

better rotating band-rifling systems, and in order to selP

ect er develop better materials for rotating bands.

Numerical hydro-elastic-plastic code analyses of the

dynamics of rotating band distortion as tbe projectile fun-

nels into the forcing cone provide an approach for obtaýn-

ing information about the dynamic stresses and deformations

in rotating bands. This report presents the results of

three such analyses.

i i

F. OBJECTIVES

The objectives of this preliminary program were:.

(1) To adapt a two-dimensional numerical code for

analysis of the axisy'metric stresses and dis-

tortions of rotating bands entering the forcing

cone,

(2) Through analyses of a representative, or "base-

line" rotating band design, to show the utility

of numerical dynamic stress and deformation ana-

lyses as a tool for designing, evaluating, and

improving rotating bands, and

(3) To identify the stress and deformation histories

experienced in critical regions in the represen-

tative rotating band design.

C. NUMERICAL APPROACH rOR DYNAMIC STRESS ANALYSIS

When the projectile enters the barrel, the significant

interference fit requires that the rotating band be severe-

ly compressed and that it be deformed in a very short in-

terval of time. To accommodate the nonlinear, dynamic

processes involved, a finite-difference code was used,

specifically, the CRT WAVE-L code. This is a Lagrangian

code based on Wilkins' HEMP method.' The WAVE-L code %asbeen applied to a wide range of impulsive loading problems,

includig hvpersonic impact, 2' 3 projectile penetration,4''and dynamic :tress analysis.'

WAVE-L is a "first principle" code which obtains "time-

marching" solutions of the conservation equations and the

3

,.o:stjtutive equations describing the properties (elastic,

jiastic, and iractu'e) of the materials involved. The codeIs .to-dirensional, i.c., problems must be described in two-,j e dimensions. ThLs includes 2-D plane strain problems,

.i- well as 3-0 probl,.ms which are axisymmetric.

Tne geometry of the projectile/rotating band tunneling

into a smooth-walled forcing cone and traveling through aimooth bore is axisymmetric, aid caa thus be describee us-ing two space dimensions. Wbvu rifling grooves in tre bar-

rel are encoantered, however, :he problem is so longer

strictly axi!ymmetric, in tha, 1-D non-symmetrical effectsoccur (i.e., circumferentiAl flow). In the present stud-,

-- he have decoupled the pronle., treating only the axisyiame-

tric aspects of the initial tunineling processes.

kA separate but couplid -lane strain analyfis can be

made of the nnn-syimetyic interactions between the rotating

bans and the rifling 4n the barrel. Recommendations for

thi s ýype aiialysis, togetLhr •ith a suggested approach, a'e

give- in Seztion I El (,f this report.)

Fiyoire I is a schomatic defining the computationalfield for analysis of a projectile/rotating band tunneling

into a forciiig cone. It is assumed that neither the projec-tile nor the barrel deform significantly, so they are treat-ed as rigid iiounda±e,,. which confine and deform the nylon

rotating hani. (This is a reasonable assumption; the dis-

tortions in the barrel and projectile due to stress levels

imposed as the nylon compresses are very small compared tothe ro*at.ng band Jietortions.) The a:tual field of analy-

sis can thus be cinfined to the rotating band itself,

For the code analysis, the geometry of the rotating

band siot (i~e, the slot in the projectile containing tho

4

-.

rotating band) and the geometry of the forcing cone are

specified, as well as the velocity at which the pro)ectile

is moving. For the present analysis, thi. was specified to

be 500 fps. (For convenience in possible future compari-

sons with experiments, the projectile was aqsumed to bestationary, with the bairel moving over it at 500 fps.This transformation has no effect on the stresses in the

rotating band.)

The rotating band itself is described by a Lagrangian

network of computational cells, and by a set of equations

(based on measured properties) which characterize the

elastic and plastic stress-strain behavior of the rotating

band material (in the cases treated here, Nylon 6/12.) io

failure criteria were used in these analyses, alt'ough the

capability to treat certain types of failure exists in the

WAVE-L code.

D. SUMMARY OF RESULTS

Num,.rical solutions were performed for three rotating

band problems as defined in Figure 2. The baseline design

(Cases 1 and 2) approximately corresponds with the rotating

band for a 20mm M-S6 projectile. Representative major andminor diameters for a 20mm barrel are .81? in. anA .786 in.,

respectively. The nominal barrel diameter (Db - .8G9 in.)chosen for these axisymmetric analyses gives thu same over-all volumetric compression of the rotating banu entcring

the forcing cone as occurs in a rifled barrel.

The presence of friction (coefficient 1' - 0.1) in Casi:

I produced almost iFmediate locking between the nylon andtie forcing cone. This was because the normal stresses,

an, across the converging interftcc build up very quicklyto the level where 4c, exceeds the shear strength of the

nylon. Nylon near the interface thereafter distorts at

54

essentially the full trojectile-barrel relative -elocity.

This clearly shows that friction can be very important in

rotating band design. However, we consider the results of

Case 1 to be an exaggerated indication of the effects of

friction, at least in a plastic material like nylon. Ini-

tial friction and plastic distortion will produce signifi-cant heating of the surface, thereby softening or melting

the material such that the surrace cannot thereafter sup-

port large frict-onal stresses.

we did not ha.e properties to quantitatively define

this softening Laid weakening of the surface; it appeared

reasonable for the purposes of these analyses to assume

that the friction lecomes so small as to be negligible.

In Case 2, we therefore repeated the analysis of the base-

line rotalirg band design, with no friction. Figure 3 in-

dicates the way in which the frictionless basic band die-

torts as it tunnels into the forcing cone. The nylon isinitially squeezed inward. The stress on the surface of

the nyic:ý builds up to about 40,000 psi. This compression

causes the nyi.n to begin to flow rearward, producing gross

distortion of the sharply-beveled trailing surface, as seen

in Figvre 3. As a result, nylon extrudes out of the rotat-

ing band slot and into the converging region between the

projectile and forcing cone walls. Eventually, this will

leae to a very thin, probably irregular annulus of nylon

between the aft end of the projectile and the barrel (i.e.,

feathering). Portious of this badly extruded annulus may

break off, either in the barrel or after the projectile

leaves the %uzzle. Presumably thiv irregular extrusion

will have undesirable effects on the in-flight aerodynamics

of the prcjectile.

6I

MATo avoid, or at least lessen, the rearward extrusion

of nylon out of the rotating band slot, a modified design

was considered (Case 3) As seen in Figure 2, this denign

substituted a more giadual slope for the trailing surface.

Figures 4 and S show the pattern of distortion whicn deve-

lops in the modified design. The gradually-sloping trail-

ing -urface bulges out towards the forcing cone. Some

rearwa-d extrusion of nylon cut of the rotatinE band slot

eventually occurs, but it is greatly reduced, as compared

to the basic design.

Figure 6 compares the axial force history applied to

the projectile by the basic and the modifed rotating band

desrgns. The force is higher in t0- modified case because

the more gradual trailing 3urface closes against the fcrc-

ing cone, thereby increasing the surface area of the rotat-

ing band which acts oa the forcing cone. The increased

force does not appreciably affect the projectile dynamics,

howuver; the impulse transmitted to the projectile with

the modified rotating band as it tunnels into the forcing

cone will affect its velocity by only about 1 fps.

Detailed information about the stresses and distor-

tions in the basic design (Case 2) and the modified design

(Caae 3) is given in the text and in the appendices. This

includes time histories of the stresses at several points

within the rotating bands.

E. CONCLUSIONS AND RECOM.BENDATIONS

1. Utility of Numerical Analysis of Dynamic Stresses.nd Distor1ions in Rotating ands

the three analyses which were performed show that

finite-•ifferenct solttions of rotating bands tunneling

jA

7

| |_ _ . .... . .. . ...• •ni mHi• |[]I l~ll n~ - il--------- -

into forcing cones arc practical, ind are specifically use-

ful for these purposes:

"o To identify material properties which have major

effects on design and performance

"o To identify stress-strain conditions which are

exper;enced by rotating band materials

o To identify design weaknesses

o To suggest mcthods for correcting such weaknesses

o To evaluate design modifications.

Examples of these uses are seen in the three cases

which were analyzed:

In Case 1, a code solution was used to identify a

criticaZ material propertu (friction), showing that it can

have a major effect on rotating band design and performance.

(The lack of data on frictional properties for candidate

materials under pertinent conditions of stress, temperature,

and sliding velocity is a deficiency which needs to be rem-edied. The pertinent conditions for these and other pro-perty measurements can be determined froQ numerical solu-tions.)

In Case 2, a code solution was used to identify a de-

cign weaknese in the baseline rotating band geometry, andto augget a mearla for correcting the problem. (The

sharply-beveled trailing surface of the baseline geometry

leads to gross distortion and eventually to extrusion ofmaterial out of the rotating band slot and into the narrow-ing region between the projectile and the barrel. A more

gradual slope for the trailing surface may alleviate thisproblem.)

8

In Case 3, a code solution was used to evaluate a

possible design imiprov.-•ený in the baseline georietry.

(This improvement involved the gradually sloping trailing

surface. It greatly reduced, but did not wholly eliminate,

the extrusion problem.)

In all the cases, st'.ea and strain levels, and strain

rate, were determined at several stations. (Maximuam com-

pressive stresses were of the order of 2.5 kb, or about

38,000 psi. Maximum strain rates in these 500-fps cases

were of the order of 104 _ 105/sec.)

2. Recommendations

Applications. The above uses of the code can be of

substantial value in design and evaluation, and

in material selection or development. These ap-

plications should be pursued.

"o Candidate new designs, as well as existing

designs which are not performing well,

should be analyzed to identify critical

regions where failures may occu'. Possible

design modifications should then 6e numer-

ically evaluated.

"o A systematic study should be made to deter-

mine the desirable mechanical property at-

tributes of rotating band materials.

Code Validation. Although the basic WAV!-L nz-

merical method has been validated in a number of

prior experimental comparisons for other applica-

tions, it would be desirable to verify the abil-

ity of the code (and material models) to predict

-~ WE-

rotating hand processes. A possible experiment

sould involve reverse ballistic firing of a

"barrel" over a fixed "projectile" which is sup-

ported on a rod.* Force-time measuremcnts could

he made in the support rod as the barrel pushes

over the proJectilz.

Interactions with Rifling. In addition to the

axisymmetric processes occurring as the rotating

hand is squeezed into the barrel (which are an-

alyzed in the three cases treated so far), non-

symmetric processes are involved as the rifling

lands force deep grooves into tile rotating band,

and as the projectile travels down the barrel.

1hese processes involve circumferential flow of

material as the rifling lands impinge into the

band, and the application of tangential forces

to the rotating band by the twist of the rifling.

Analyses of the stresses and deformations in r-o-

tating bands due to these asymmetric processes

should be made. A possible approach would be to

treat the problem in plane strain, as indicated

in Figure 7. In this appioach, the projectile

would be represented by the rigid-body central

core, and the barrel would also be represented

as a rigid body (concentric to the core). The

rotating band (between the rigid projectile and

barrel) would be described with the elastic-

plastic rotating band materipi properties.

This experxnnnt has been suggested by It. Swift of UOR1

lI lIlI

There would be a sliding interface between the

band and the barrel. IFriction could be speci-

fied in the z-direction (U.erpendiýalar to the

plane of solution), thereby allowing the effects

of sliding friction in the third dimension (z-

direction) to be incorporeted in the plane

strain solution. The band would be locked to

the piojectile body.

Initiallf, the rifling lands, as represented by

the periodic protrusions around the rigid

barrel, would be nonexistent (i.e., the barrel

surface would be smooth). The flat surfaceswould thereafter be gradually forced into the

rotating baird by prcscribing their displacement,

r(t) a function of time. The angular accelera-

tion produced by the rifling would bc represented

by prescribinc rotating of the barrelby w(t),

as determined by the projectile velocity and bythe pitch of the rifling.

The same types of information could be obtained

from the plane strain solutions -s from the .,xi-

svmmetric solutions which are described in the

report; i.e., field plots of distortions, parti-

cle velocitie*, and principal stresses, and para-

meter vs time plots at selected stations.

}S

illl

SECTION 11

NUMERICAL SOLUTIONS

A. PROBLEDI DEFINITION

A representative "baseline" rotating band of interest

was chosen as an initial vehicle to test and demonstrate

the finite-difference cede technique of analysis. This

is illustrated in Figure 1 (page 21). The specific base-

line dimensions, and the computational grid used to repre-

sent the band, arc shown in Figure 8. The band design

shown is one which has been considered for a 20mm M-56 pro-

jectile.

The analyses which were perfoimed were confined to

the axisymmetric convergence of the band as it engaged with

the forcing cone, ani therefore did not explicitly treat

the rifling. Instead, a uniform barrel diameter (.809 in.),

between he major and minor diameters of the barrel (.817

In. and .786 in., respectively), was chosen which provides

the same overall volumetric compression of the rotating

band as occurs in the rifled barrel. The initial .829-4in.

01) of the rotating band is thus subjected to a maximum

.adial convergence of .010 in. in the forcing cone.

A nominal constant S0C fps was chosen for the velo-

city at which the projectile enters the forcing cone. in the

"actual gun fizing, the projectile is accelerated from an in-

itial rent position during the tunneling process. This mode

of engagewent can be handled by the code, hut we did not have

information defining a typical acceleration-time history in the

forcing cone. This problem is not felt to be particularly

-I

sensitive to velocity, so long as the velocity is anywhere

in the low subsonic range (which cover; the range of velo-

cities of interest here), and so long as the material pro-

perties are not rate sensitive (as was the case witn the

material model used in these analyses).

In the code solution, the forcing cone and barrel

were set up to move at the prescribed velocity, and the

projectile was fixed. This was done for convenience and

efficiency, since it immediately gives the flow in the band

due to the interaction, rather than centering the flow

about the projectile velocity. This arrangement also cor-

responds to a reverse ballistic laboratory test (in which

the band is mounted on rigid rod, and a heavy ring simula-

ting the barrel is projected against the band), which

could be used to confirm the validity of the code solut:ons.

Mi -ATERIAL PROPERTIES

A material of current interest, Nylon 6/12, was

chosen for the plastic hand. The following initial value

properties were employed in the code solution for thi-

material: 7

Density, P0o = 1.08 gm/cm3

Young's modulus, E = 300 ksi - 20.7 kb

Poisson's ratio, v - .4

These values imply the following other properties:

Bulk modulus, K - 500 ksi - 34.5 kb

Shear modulus, G = 107 ksi - 7.4 kb

Dilatational wave speed, c. = b640 ft/sec

2026 m/sec.

13

- -n

ihe equation of state was assumned to be linear elas-

tic. An elastic-plastic constitutive model was -ised, en-

ploying the von klises yield criterion and a non-as~c;iated

flow rule. The value of yield strength used was 16.4 ksi

(1.13 kb). Tensile stress was limited by imposing a mini-

mum va!ue for 1ýydrostatic tension, Pmin = -2.93 ksi

(-.202 kb). Under uniaxial stress, this gives a maximum

permizsible tensile stress of 8.8 ksi (.61 kb), correspond-

ing to the tensile strength of the material.

The stcel projectile and barrel were treated as rigid

bodies. The plastic band was assumed to be locked to the

proecticle all along the mating interface.

"* In the Case I analysis, shearing forces due to fric-

tion %,ere generated at the band/barrel interface, in addi-

tion to the normal forces developed as the band is deformed.

A nominal constant value for the coefficient of friction

of 0.1 was used. Cases 2 and 3 were run with no friction,

both for comparison and to provide a lower bound on the

shear deformation of the band surface. (The solution re-

sults from Case I indicated that the frictional effects re-

sulting from a friction coefficient of 0.1 were excessive.)

C . NUMERICAL MIETHIOD

The WAVE-L code was employed for these calculations.

WAVE-L is a two-dimensional code which solves the equations

of motion for elastic-plastic bodies by means of a finite-

difference Lagrangian-cell technique. The mathematical

formulation is basically the same as that described by

hiikins.1 An important feature of the code is its provi-

sion for sliding interfaces. (In a normal Lagrangian grid,

the cells are locked together, and no sliding can occur.)

14

ii 4i __

, Ii ding interface ;,as used for the hand/barrel interfare.

Another important capability of WAVE-L is it ability to

treat moving rigid bodies (in this ca.se, the barrel).

code computes the loading forces on the rigid body and

.hus the deceleration can be included in the calculations,

given the mass of the rigid body. This rig;d-body formula-

tion was previously developed to treat the penetration

dynamics of projectiles.'.' Some additional development of

the code wa.s required to handle the piastic band interac-

tion geomet r).

- . NUMERICAL SOLUTION OF CASE I - PLASTIC BAND INTERAC-

TION WITH FRICTION

The initial Lagrangian grid configuration set up for

the problem described above is shown in Figure 8. Tie

plastic band was resolved with 8 cells across its width

and 53 cells along its length. One hundred six lattice

points were used along the barrel to compute the distribu-

tion of stresses and force components acting on the barrel.

The code results of this first case, which included

frictinal effects, are depi ted in Figures 9 to 12,

which are plots of the g-id configuration and velocity

fielI in the band for times of l.S and 3.6 tisec.

The plots of the grid also show which material is ýur-

rently undergoing plastic deformation; i.e., is on the

yield surfaze. This is denoted by cells containing an x

or + , sith x indicating a compressive prea:sare, P > ',

and * indicating hydrostatic ten.ion, P < 0

The velocity %ector field plots show the direction and

nagnitude of the particle velocity at each lattice point in

the computing grid.

Sis

fhe.e rc-lt. now that a strong upward axial flow

i,. induced :n the band next to the contact surfacv. Note

that the band ;urfac, alipears to be dragged along oy the

a barrel, cauu ng severe distottion of the materiaI next tothe ban(. lhe frictional forces built up across the in-

terface we~e sufficient to lock a portion of the nylon

surf'ace -,;itc the barrel, so that the svrface was actually

aoving up.ard at the full barni, 'elocity. This clearly

shw•ws that friction is im'portant, and incicates that fric-

tional propertie5 under high stress, high velocity condi-

t.ons need to be defined. However, we consider the resultsobtained in Case I to br- an exaggeration of the actual

fe'ictional effects. Sioce fheire is a sign~ificant hcating

of the suirface due to frictiun and plastic deformation, it

1s likely that the serface material softens s:ich that it

could not support frictional stre.-ses as large as that

compuited with a coefficient of fr:ction of 0.1. it aos

tOus decided to discontinue the running of Case I and toinstead run .. second case, %ith no friction.

Additional re~ult: of Case I are given in the next

secticn in comparative plots.

E. NUiMEmICAL SOLUTION OF CASE 2 - PLASTIC FND IXTERAC-TiO. wiTnourf EiICTION

The problem conditions for Case 2 were oept exactly

the same as for Case 1, except that the bhind/barrel inter-face uas assumed to be frictionless. Thus no -hearing

forces are generated at the intcrface.

16

____________E___________

VQ-4

The results of the code solution for th:ý ci.e are dc-

pscted in Figurcs 13 to l4, which show plots of the gi i,

configuration and/or velocity field for tiaes of 1.5, 3.i,

7.9, and 11.4 Lisec. Note that the early tine flow i,. noi,

radially inward, as opposed to the strong axial flo,, seen

in the case with friction. At later times, however, a

strong axial flow occurs. as the upper end of the bmnd i.,

extrudeo into the open cone.

By 7.9 usec, the extrusion action at the rear of the

band had produced such severe grid distortion that the so-

lution ccu)d not be continued. To allow continued inte-

gration, a rezone of the grid was performed. Rezones are

used to reposition the comput3tional grid in a distorted

region so ab to givw more regular cell shapes; a compre-

hensive rezone processor correctly redistributes the cell

variables among the new cells. Cede modifications re-

quired to treat rezones of the plastic band problem were

developed and chezked out for this first rezone and will

be available in the future as needed.

Following the rezone, the solution was contnued out

to a time of 11.4 usec. The extrusion action noted earl,-

2r in the solution continued in this later phase of the

solution, causing formation of a pronounced annular lip

protruding out from the rear of the band. The presence

of such a lip would presumably degiade the aerodynamic sta-

bility of the bullet. The solution was terminated at this

point, since the behavioral pattern of this band design

was clear (for this material) and because the axial forces

were significantly diminished. An enlarged view of the

extruded rear section of the band at the end of the solu-

tion is shown in Figure 20. As the projectile moves

17

S.... 'n . . . -- ,,_ i • IVIN I |l l • _NN INN|I! • l llml_ mm__

further into the converging forcing cone, this extruded

annult's wculd becom,a thinner and longer.

Comparatve time histories of the axial and radial

forces acting at the band/barrel interface for Cases 1 and

2 (with and withotut friction), are shown in Figures 21 and

12. Iigure 21 shows the increase in axial forcc at early

tlOrcs caused by friction. However, by 3.5 tsec, the aximll

fo-Ce in the case without friction surpasses that with

friction. Significantly higher normal stresses build up tn

the case without friction, since the )..nd ba-ierial next to

the barrel does not flow ul- ,oxially) as much, and is thus

subjected to greater radial compression by the forcing cone.

Coeplete time hibtories of the j~al nd radza, forces for

Case 2 are shown in Figuios 23 and 24. The peak axial

forcc was 840 lb, occurring at 5.6 psec. and the peak ra-

dial force was 1670 ib/rad, occurring at 6.7 usec. At

11.4 iisec, at the end of the solution, the axial force had

dropped off to 15% of its peak value.

Stress distributions on *ne forcing cone are shown inFigures 2S and 26 for the two cases.

F. NUIMERICAL WOLUTIO'N OF LASE 3 - MODIFIEZ PLASTIc: HAND

DESIGN

The solution results of Case 2 indicated that the band

distortion was concentrated at the junLtion of the 450

beveled rear section and the projectile. For the purpose of

alleviating the severe distortion occurring at the rear ofthe hand, it was decided to evaluate the behavior of a

modified design, having a longer, Aore giadual sloping

rear section. The modi;ied design is shown in Figure 27.

It has a long, 100 sloping rear section replacing the 45•0

18

..- -...... . .... !-.' ,=!r. mai r -• -. l~.. .-.

beveled so'tion of the original design. The flat section

and the 60 sloping forward section making initial contact

with the forcing cone were kept the same as in the baseline

design so as to isolate the effect of the Poditied rear

section. The !ront of the band ahead of the start of the

forcing cone was shortened, since it has little effect

on the problem dynamics. The overall length of the -odi-

fied band was .32 in., compared with .28 in. for the base-

line design.

The initial Lagrangiar. grid for the code solution of

Case 3 is aio indicated in rigure 27. The zoning uas

-" kept the same as in ýhe previous problem; i.e., 8 cells

across thie width of tne band. Fifty-nine cells were used

along its length.

The results of tht Case 3 solution %jere very interest-

ing: with the long, shallow-sloping rear section, the re-

spnse of the band was much more regular, smoothly bowing

out as it was forced back. The solution has carried out

to 27 6 psec, by wlhich time the axial relative displace-

,ent of thb band/berrel was .166 in. As an indication of

the uniform deformation oehavior, no rezones of the griJ

-,ere required duritg thli period. (In the previous problem,

severe distortion necessitated a rezone at 7.9 usec.) A

sequence of the hand/barrel configurations and/or velocity

fields for times of 7.7, 15.5, 20.7. and 27.6 usec are

shown in Fi-ure5 28 to 33. 3y 27.6 Usec, sc-e distortion

has built up, at the rear of the band, but it is less than

seen with the previous design.

Time histories of the axial and radial forces at the

band/barrel interface in the modified design (Case 3) are

iv iuiv, (, and 34, plus those for the b'aseline

- , , "1 for compai ison. At ecrly times, the re-

,,-e ee:n to be the same, coufjrming that the geometry

,-i,,t, n i, l cont.ict surface wa- the same betueen the

;-i.l *:I. The miodified design produces a significantly

., ioi,, e pulse, corresponding to the increased length

S,,itid I)chind the forcing tip (.27 in. compared with .17

It.. Al-o, the band bows out until it teaches the forc-

i, 1 ,0e. He resulting high pressure between the band and

0i, one produces a large axial force. Assuming a projec-

ztl, weight of -. 4 lb, the peak deceleration imparted to

t,ic pro.ectile due to convergence in the modified design

nirv- g2-W g's. rhis deceleration for 20 jisec would reduce

tc pioectile velocity by 1.3 fps.

01-tributions of the normal stress along the band/

t-r� �interface for several times are shown in Figure 35.

ith- peak stres- levels experienced within the band

. Ine' interaction are indicated in Figure 36. The

, ,,rpre-sstie stresses along the inner and outer edge

A:u)t PAid:ay across the band vs axial distance are

t~ed. Ilso sh.own for comparison are the corresponding

f(r thz baselir.i band design.

1Uditional results from the n'imerical solutions are

I, ,,; in the Ap-cndices.

20

Projectile and barrel are 1 -considered to be rigid materials

Fil of Anelyl

IR -I1014 j I

I I '

i4 0

~4) 0 0.0

A- 10 0

ii,,

Q .;

0Z==7777.7771111111111/1 'A 0 .

,'IIIIIIIIIIIII 0

U~ 0 u .-

_/ ______,_,_,_....,• 0

00

01.

e, .4 a 0

WOM 0.

-- '• ,, .,',',-,,,,4, .,,,4".i •

a °I

,..0.

Velocity Scl.le

500 fps

-: :

I ~I-

*1 i

[ 4 '5Tl Velocity

~:S (500 fps)

In ltdicatesraterial isyielding 1:::

Grid Distortion Particle Velocity Field

41$

Fi.ure 3. Deformetion amd Flow In Plastic -&-A at 11.4 psec, Case 2(Baseline design, no friction) j

23

i . .._ _I _ _II _l _ Ili_ _ _ . ..._ _ ..._. . .._ .. . .. . ... .

f Velocity Sc le

20{0 fps

* .5

Indicates Barrel Velo itymaterial , (50 fps)

Grid Distortion Particle Velocity Field

Figure 4. Deformation gad Flowic ?jtic lamd at 7.7 psec, Case 3(Modified design)

24

- 4

2 Indicates materialis yilldift

c 20.7 gac t 27.6 psec

Figure S. Delfoatioe of Plastic IUa at 20.7 and 27.6 vasc, Case 3

2S

KGLNm Wmm w nmmin. ag.mu. 3"1-L~ #msc "wmo

Baeline Design (Case 2)

200

0 0 20 30TIME NiU~cW

Figure 6 *Axial Forcel at Band/Barrel- Iterface vs Usme for Baseline and

26

WWI.

Potation by rifling reprqached inner surfact is smooth at t . 0by rotating outer rigidbody (birrel) accardi I] bands Impingeto specified (t .... rotating band asape e y r(t)

S/ Rota itin and is

aragiz-d ri:dr nandby ela y stic a

plasticil irprte-

of material •Figure 7 Su g iet d Metody f rPa e Sri nlss o

Rotat Is Bagid-Rotating band isBarre is igidbodydescribed byS~Lsgranglan• gridS~and by elastic-

Plastic Propertiesof material

I"Flgare 7- SuSPated Method for P'lans Str&tn Aa2ysis of

]Rotating IBond-R•ifling lute raetion y

27-1 °-

W AL 75 mjimMw KO M 11sa off cogso l. Sh.2 r. U ,•CJ I9oInrf g eSa

| .829" Dia•

SU " .745" Dia----O

.28'

0o S O0 ft/slec

0L

RADIUS IN. 0. UMC

Figure 8. Baseline Plastic Band Geometry and Computational Grid(Caez I and 2 Baseline Design)

tO-~

iaL. s ISNin m m MM inisIm L M16&rafl M K - OM

a" -

I

L|In

(Case I

W29RADIUS IN. 1.509 UJSC

Figure 9. Gil~d Configuraiton at 1.5 usec. Case vith Friction

(Case 1 - Baseline Design)

ii I

21J ati Q"inU WWM 90 iW L WKnCm. Wa. U ST LWO AMfl~

CMI@. IN . 1.509 LU * .

Figre10 VloctyFild at 15 se, Cs hFito

(Case~~~ I 01~eeDein

ERN~ ~ 9Sw

U AnL w o m nvmu m m 1us WW4 CIK

U)

iP4

-4]

.30 .35 .40 .'S .9)RDIUIS IN. 3.640 USEC

flgure 11. Grid Zonfiguration at 3.6 pstC, Case vitb Friction(Case 1- Baseline Design)

31

X.

A W- ALMZ 7- y.' onfl MMMWW

LIs&.re Vet3IL

is's

.3D 3 W~1 A~SRADIUS IN. S. V1DLr. USC

Yigurp 12. vt1otwty Fi eld at S-6 ptmwcl Caiic witb Friction(cas~e 1-USCcLIe D"11a)

32

-Or

O. O Nano ONUT "aa OWW UOm L Use-t tLwu omm )m

CM I" wt Lo .10

RI

Barre

(500 fl

A3 S

RRIU M aM-UE

f tgo1.vlciyFeda um. aewihu ."0

LIsan

go-Nore

a as Mi C - o "mo"a

RRIL INI08,UE-iueI rdcdluýla t31-Oe,ý-Wau

Vviciou Cats 2-Osolle Deigi

* IMSA Raum oma .mamu SMIU

*7_1

BarrelVelocity(500 Eps)

* 6*.?

-,RAOIW- ~ IN *-78*tE

a, -ý1 6cty~rfod-ti. 3*1-sc " lb t rcI

4,~S

14.

a.IS0 U.S .M .SBLK MIc

MAIUS IN. 7.908 -.USEC

Figure 16. arid coofigurAtion at 7.9 goW Coas v4tbat Vrictoa a (Cast 2 - Baealiae fealgs)

36

"Smw *-M

48 OL 10. MAIM g Umb nqMg - M~ Lumm .7M-

I.I-:

.30 .35 -. 405% 5

R- U IN 17A USE

71ue 7 eoiYPeda .9 --Ic CatwtotFito11420r1f * I I

I I

U~-i OnR 11" ~

mms w~~s aa. um c~~*m

Fl~uv 1. Grd Cuf~patln a ,!!c. az-c winou

(case

-I

30 16 ftv6 UNIV i 0sm U N" ~

- I-

! :-! s:-,:

Bare

I. _

I

•- ...I Barrel__. j Velocity

(5O0 fps+

Figure 19. Voincity Field lat 11 c Cs sib.tFicih (Case 2"Dsavlir Dsign)

1o.

3t

a

3 333 it

3 it ~

RMI3 IN3 1-MUE

3 40

- 3 - 4R l

06 ...m ..& 8 m anII

TotaI axial forcevit-a friction

y,""KAxlal forcef/ - without friction

I Fr-ctional force component

Cog

Normal force corn

- Case with friction (1)

Case without friction (2)

4. .. 0 2.0 LaTIM HI(ROSEC

W rinsure 21. Comparioon of Axi•l Force. on bo-rel with &ad v•-hout Surifterrictioe (Cases 1 sai 2 - Neseollas N0esig)

41

mii

am ~ ~ ~ ~ ~ ~ ~ s UmM. A n ~ O O0 e

Radial force without friction/(Case 2)

Cn

I Total radial force with friction

FrictionAl force compone 2

I -

2.6 .

rigure 22. Conprison, of Rad- I. forces on Warel wiith arid without SurrfaceIrriction Cdoi'.amIad 2 - Uiteline Deutini)

42

W&-M =wa m~ MMM. mmiu ww

Fw-

1200

Fisgue 23. Axial Ferce &z.Bauifarval Interface -vs T7u, qas witboutaFriction (Case 2-. Saaeltaa Design)

43"

4x

Im mCWYI~ua 2. WII iorc -atft;1Ijr~j Titrfee vsTie, -4gstwitou 1fiF to 1Cs "ltbIa

am W. MIDsL fLW= inmm UWHO"

mzzlorcizag ccue

.5 Time (vaec)

Axial Distance ftom TIpu ocn~ oa g

FItjur* 25. Normal Stress( )an Tanjantiml Stress (Distribution oLaIrol Cese with Fricti~on

45

~ -R

A.A

3.

I114

0.

Axial Disatnce f rom Tip of Forcing Cone, i1.

F~gure 26. Normal Stress to D fistributions on Barrel, Case witimutFriction (Cate t avellse D#Sign)_

46l

cog 2

orciflg Cone

.7*'1

.89 i

32"v .

rRADISctil.OOeOUEFipire~~ 27)o~ e lsiand as" t~ n opu~li~~ .1

(Cas 30 Idft a1U

47 1rre

InI

S! -

RRDIUS IN. 7.747 USEC

Figure 28. Grid Coufigursttoa at 7.7 usec, Nodlited Plastic 3and Design(Case 3)

48

..iI. Mm or IS 01U I IM n SlOn 10 ur*IC

n o ----00.--u ýf

Li: Barrela I Velocity

Ow 0fp"

So.30 .35o .4 o OSRMOlt• IN. 7.747 MEC

Fiupre 29. Velocity Fleld at 7.7 usec, Hodified Plastic- Bnd Design(Case 3)

-49

lmaW aII&rma un lm n IIMMIl-.rlmi Ut l0, M II/E m I~mmom "-L C

_. UL -NMA fwS

sqso

0N I. IC°

'. Figure 30. Grid Coo~iguretiou at 15.5 isec. kiodified Plastic flaud Design

I (Case 3)

SO

RMa SI. -- 0.1 u s mm M E

l r-'I -

" 3)

- I* I

i 'I

* v•'•-_ 31. Grid Configuration at 20.7 •s~ec. .4olftd flaetic 3ki~d Desipti(ca.. 3)

ai 173 OmwmM M m .uWK-L USK

PMI. S94 tIN. -,-7.0 USEl

rigr_-32 CrA onfswtiU a *27. oe ~ild ls ADsg(cse3

I ---- i

Wk

MLI =0S Mm alaQ uvocwmu =I4 tam we MUMt

Wi lo * ty

Figure 33. Vel~ity Y1014 4*..6 " . It*f~ w~c" e~

(Cse3

53 ere

G M a& Imgo. A&M 41 2bI um %o

20W

Il 1Baseline Design (Lase 2)

IlWI

Figure 34. RsdadIj, FoCe At SabiS/Uerre Aftutefecii va, Tim- -for "Ilaes ieendModified Plastic ti:4 Deele i

Forcing

01I

-.41 0IL276a uis0.c fro 12. p of. ForcingCoe. in.

EllI I l

5S 43I

* . IR

I777-7

>1 I 'Flots Shnj peak

Muzzle s 8t-fess prof£1iBcrre, urfalc~s

F. r iForcing cone -

St Deslgn

444

Basel-'.e oified

.DesigrI.

*1 Inner EdgeI ~~~of Band ----

Midway across --. " -B]and

Outer Edge of .. - .e_

___-.___ •I ,I!

Axial Distance from Tip of Forcing Cone (Initial Position), in.

Figure 36. Peak Compressive Stresses within Bond; Baseline avi MediftedPlastic Band Design.- (Cases 2 sad 3)

56

i -I . i ll i isi ,•6

F -

A•PENDI K A

PRINCIPAL STRESS PIFLD PLOTS

In eddition to the plots of the grid configuration

and the velocity field presented in Section II of the re-

port, plots of toe principal stress field cccurring at

several times during each of the solutions were obtained.

A representative selection of these for Cases 1, 2, and 3

are contained in this Appendix.

In the stress field plots, the principal comportents

of the stress tensor for each cell cre shown as follows:The magnitude of the two principal stresses in the r-z

plane are plotted in their corresponding principal direc-

tions. The third principal stress (in the azimuthal di-

rection) is plotted along the line bisecting the other two

principal directions. Vectors pointed to the rigtt are

compressive, to the left, tensile. An examrle of hew a

stress tensor is plotted is sketched below:

Tensile .- j.--.------ %(N3roasive

tPrincipal Stressesin r-z Plane

Principal Streas

The scale of zhe stress vectors is given rt the top of each

plot. The plots listed below 2re given in the followingset of figures:

SCase L. 1. 5, 3.6 -Vseco -

Cose* 2 2.1, 7,9, 1. usecCase 3: 7.7, IS.S, 27.6 4sec

57

-. , . n n iiiii0iNunl W,

EI

s a K am an mmap as"'at INL-- A

RAO~~~~ It LI IN.150 U

Figure t~. Priwips Srsfilat 1t. 5 sc ewt

li it

Frcto •d cI-lsl- einSS

7 a-

- cton (Cs....... i"t esgn

58

II!

3 .. .1ninU n mn iW t P00" . Ira" cogas 0. MOS4. RfIC Wt AM 1IS lf to

3 US.C

S(ne ign)

. .

a 75 GLMS .oo n TMW 41-WML. UN4. NUT IC wvm Mi¶II' 9

!'

a

ini

RADIUS IN. 3.078 USEC

Fig.ue A-3. Principal Stress Field at 3.1 usec, Case vithout Friction(Case 2 - Baseline Design) -

60

-7-77

J-° !

I

a"~ ms4 f ITC m imP 0-4

.35 .. 5..fRADIUS IN. 7.908 LJSEC

Figure A-4. Principal Stress Field at 7.9 vsec, Case without Friction(Case 2 BaUseline. Design)

61

U' ii ~mW= .la

- -U ei• Il'mlm Il Piaslm .IIL ml_--" mm OS4. Ialf %lb .s

" "

_-=No I I

S-.3o .5s .o . .5oRlOIUS IN. 11.383 USEC

Figure A-S. Principal Stress Field it li.4 usec. Case without Friction(Case 2 - Baseline Design)

62

~'~1 -

-!§'-

W O&NNL 3RU46 RO ONKMm~ soCPnU M m tm" Amnv .

m _ - ,RW a

S.Iture A,-6. Principal Stress field at 7.7 useci Modified Plastic Band Deig

(Case 3)

1K

CP arn wim' ianAM

I 64

WeV

it Us 71 inmiMw SSWR W VIIK -4. Cm,.__NG- MiC WWX INOI 3

mA

.30 .35 .40 .45 .50RMAIUS IN. 27.606 USEC

Figure A-8. Principal Stress Field at 27.6 Usec. Modified Plastic BandDesign (Case 3)

' 65

I-•

APPENDIX B

TIME HISTORIES OF STRESS AND VELOCITY

Parameter-time histories were recorded at several lo-

cations in the plastic band. The locations of the stations

are shown in Figure B-1 for Cases I and 2 and in Figure B-2

for Case 3. The stations for Case 3 were placed at analo-

gous positions to those used in the first design so that

the response of the two bands could be compared. The fol-

lowing parameters are plotted:

a. Radial stress or

Axi stress oz

Hoop stress a0

Shear stress Orz

b. Radial particle velocity

Axial particle velocity

PFsitive stresses are compressive. Positive r is

radially outward. Positive ; is axially upward (in the

direct:on of the barrel velocity).

Tie first set of figures (B-3 to B-9) are comparative

plots of the response for Cases 1 and 2 (the basic plastic

band geometry with and without friction) at selected sta-

tions. The previously noted "locking-on" of the surface

of the band to the barrel due to friction during the Case

I solution is clearly shown in Figure B-", the velocity

plot at Station 8. The second set of figures (3-10 to B-19)

are comparative plots from Cases 2 and 3 (the basic and

modified plastic band geometry). Note that the extrusion

action that occurred at Station 9 in the original design

$ did not occur with thi revised design (Figure B-19).

66

0 JA 73 COUFMOM UZS( *6 XOPOLOR WWL Urn4w O. IMo6. nmirc ML 'me Mu1OUCOU a

ini

InI

04

+ Stress Station

* Velocity Station

RADIUS IN. 0. USECFigure B-1. Locatlonsa of Primster-Time Stations (Cases I and 2-

Baseline Design)

- 67

Its ____W1K

11 IT- IS CMn.!rrlln r.L:Jf1 M ITr.*'Xt SIE CCCO 113. 3MO64. eUs5TIC V C IU #!Z;tICM

Cn

.30 M11 .11RADIUS IN. 0.000 USEC

Figure 5-2. Loc~ation~s of Paranater-Tine Stetions. Nodifl*d M~astic CacdDesign (Case 3)

m 2

ft CLM04 " InsinsXgmm qwn.um ano

-- -With friction (Case 1)

-Without friction (Case 2)

&ae

TIE ICROSC

Figure B-3. Stres& CoLponents at Station 2UeleDsg

69

9 . - ~s',

- isi

.. -- -- Wit fric'tion (Case 1)

.. .........

TIME NICRMSC

Fi-ure B-4. Stress Components at Station r Baieline Design

C70

i i

SM S71 1 n

Sm MUaKn. ag..3~ aMa 31094. KMIC M" 3I~ctfm fUflenm) 40 7B

A -- -With friction (Case 1)

Without friction (Case 2)

/ .'

A o- I

' TIME ICROSC

S • iFigure 3,-5. Velocity Components at Station 5 -Ba~e~lna

• Design

I7

urn 7

GLmrn 038 m rTausw. m.Mm sML~ 2004 AMI am wMS iSM 10 M It""

S- -With friction (Case 1)

-Without friction (Case 2;

Itil

Figur B-.Vlct 5pe tSain7BslnDeigC 72

(-3ý

a"11110 ... . IM- t-

M W& low ftom asf"N Mno swMIM as IS

-- Vith friction (Case 2)

--- Without ftiction (Case 2)

• . .. /

U)

TIE HICROSEC

Figure 3-7. Stress Components at Station 8 - laseline Des1ign

73 I

• ,

No & w*L MCInan AINM NCTILMam"a

I--- With friction (Case 1)-Without frietion (ease 2)

ImI

TIMEMIROSEC

Figure Tý-8. Velw.ity Cooponents at sttetion 8 - flseji~e90ais1n

74

QSNm 4m sM6'.4 . UL

f - with friction (Case 12)C'41 -. Wichout friction

(Case 2)

IN

TIME MICRMSC

Firjre B-9. Vvlcc-ity Componente at Station 9- uia

Desig7 -

M IL WID-I, F ?T . mM

22

Case 2 CaseA

itv -. baseline lesagn (Case 2)"-- H fodlied Design (Cas* 3)

0 s 20 S

TJME INCIcUE

figure 3-10. Stress Components at Strtiou 2, lmnelitu and ModifledPlastic Bond Designs

76

|i

swim SI I

2

B iaseline Design (Case 21

Modified Design (Case 3)

1I0 20o

All

T:: uIFAriguro s-.1, Stres Copoosets at Station 3, lasellne stid Iodifled

rlastic Sai Desia"s

77

I - -

1.5

-.- a'.Baseline Design (Case 2)

- odified Design (Case 3)

*Case PF e

10 20 I30~e

Figure B-12. Tl IfCStveia C.Po~aots 4t Ststine 4, fslhsele and Noif ledPlastic land Designs

78

ro ip-ý -

OIM MmU -,

WMLm m90 m wmwnm SIff.

2.0

-- Baseline Design (Came 2)

Modiffed Design tCase 3)

1z1Figure B1,-trs2Cm'~tsa SainS:jslL ~_Adi s

Plati Bn- D slg _

W Halsmoa mommam

V. ll eseleDsg C"2

I I

1--

S,-

S•f • Modi•fied De•sig (¢z•" 3)

800

Ii~ ~~~~~~~~~1 .OF... *:;.•...- *..-..

- -M Wmm. ULW UL US4. RO wU " wmm

Basellive Design (ease 2)150 Modified Design (Case)ISO

100Csej

10 1

FiV~AM-3-15. Veioclt7 Caýapooete a Station 6. liseliue aa4 Nodifietplactic idDis

Wkft 0

W LW&L RAMt -%M

2r

tlii

iriareF,1. tre copomto4r-racumF; se~rt~dlHaifeig ease

Pla"K, I=&mln

m--.- gm. s

!1-koeine Design (C-se 2)

M- Modified Design (Case 3)

-- I

, rr

'-4

itoo--

Case 2 'Case 3

""-0

"?Odified Plastic Sarnd Dsl~um

83i

PRO ~ WIX

mmuinmm .

"W NL 3" 11

Case 2 Case 3\

l i -- '-- celtne'Deelta (Case 2)

SHModified Desip (Case 3)

Am

A A - maE

to 3TIMa Pli~Oc"=

Figure B5-18. Stresm Components at Stationý 9, 31so1i" and ModifiedPlastic Band Deigns

S 84

- gg r.w

. . . .-.

.mm .. ,.

-0-IrI

- -

' ' -- Baseline Design (Case 2)

M- odified Design (Case 3)I )!-/

I

iCase 2 1)Case 3

100

Iv

10 2D 30

Figure 3-19. Velocity Cooponents at Station 9. Baseline and ModifiedPlastic Band Designs

852 - _ _ _ _ _ _ _ _ _.°+

I.? ii , ii a i i l I I I I I I II I

REFERENCES

1. M. L. Wilkins, Calculation of Blastic-Plastic Flow,UCRJ.-7322, Rev. 1, Lawrence Radiation Laboratory,Livermore, CA, Januu.:y 1965)

2. K. N. Kreyenhagen, M. H. Wagner, and W. S. GoerkeDirect Impact Effects in Hypersonic Erosion, SAMSO-TR-71339', Space 4 Missile Systems Organization, LosAngeles, CA August 1973

3. K. N. Kreyenhagen, M. Rosenblatt, and T. R. Isbelle(California Research & Technology) Cratering, Mass Loss,and Residual Damage to Hypersonic Particle Imps onAblat-.ve Materials, AFWL-TR-75-184, Air Force WeaponsLaboratory, Kirtland AFB, NM, June 1975

"4. M. H. Wagner, K. N. Kreyenhagen, and W. W. Goerke

(California Research & Technology), Numerical Analysisof DNA Earth Penetrator Experiment at , DNA 537F,Defense Nuclear Agency, washington, D.C., October 1974

5. M. H. Wagner, K. N. Kreyenhagen, and W. S. Goerke(California Research & Technology), Numeric Analysisof Projectile Impact and Deep Penetrat into EartMedia, WES 5-75-4. Waterways Experiment Station,Vic-sourg, Miss., August 1975

6. Y. M. Ito, K. N. Kreyenhagen, G. E. Egguw, and W. S.Goerke, (California Research & Technology), Anal sof Dynamic Stresses within a Terminal Delivery VehicleDuring Penetration of a Hard Earth Target, WES S-75-1,Waterways Experiment Station, Vicksburg, miss.,February 1975

7. S. W. Tsai, Air Force Materials Laboratory (PersonalCommunication)

iAi

86 J•i~ ~ iS i- tf•l flb flilb IIIII lb - , illlli llll~i