reverse yielding of a fully autofrettaged tube

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NOLuR 63-123

REVERSE YIELDING OF A FULLY

AUTOFRETTAGED TUBE OF

LARGE WALL RATIO

DOC

DE 1 3 1963

•.CTiSliAl B

t~~iiI~ 11 ugust96

UNITE STATES NAVALORDNANCELABORATORY,WHITEOAK,MARYLAND

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Without restr ict ions 7j For Release to Militatr and Covernzsezt

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UNCLASSIFIEDNOLTR 63-123

Bal l i s t i c s Research Report 106

REVERSE YIELDING OF A FULLY AUTOFRETTAGEDTUBE OF LARGE WALL RATIO

Prepared by:Victor C. D. Dawson and Arnold E. Seigel

ABSTRACTs The equations are developed for the case of areverse yielded thick-walled cylinder. It is assumed thata cylinder is subjected to an internal pressure which causesplas t ic flow throughout the walli the size of the cylinderis such tha t the r e s idua l s t r e s ses developed during pressurerelease cause the cylinder to reyield in compression. Th estress equations for the subsequent reapplication of pressureto the reyie lded cyl inder are a l so developed.

U. S. NAVAL ORDNANCE LABORATORYWHITE OAK, MARYLAND

iUN~CLA SSI FI ED



NOLTR 63-123 21 August 1963

REVERSE YIELDING OF A FULLY AUTOFRETTAGED TUBE OF LARGEWALL RATIO

This repor t is the r e s u l t of a need to provide high-s t reng thchambers for use in hyperveloci ty launchers. The ca lcu la t ionspresented provide understanding about the present limitation,and reverse yielding of autofrettaged cylinders.

This work was sponsored by the Re-Entry Body Section of theSpecial Projects Office, Bureau of Naval Weapons.

R. E. ODENINGCaptain, USNCommander

R. KENNETH LOBBBy di rec t ion

ii



NOLTR 63-123

CONTENTSPage

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . iv

INTRODUCTION . . . . . . . . . . . . .DERIVATION OF REYIELDING EQUATIONS ('>2.22) . . . 5PRESSURE APPLICATION TO THE REVERSE YIELDED

CYLINDER . . . . . . . . . . . . . . . . . . . . . 10CONCLUSIONS. . . . . . . . . . . . . . . . . . . . 16APPENDIX A . . . . . . . . . . . . . . . . . . .. . A-i

I LLUSTRATIONS

Figure Ti t l e

1 Limiting Curves for Internal PressureApplication to a Cylinder

2 Scale Drawing of Reyielded Cylinder with WallRatio of 5

3 Wall Ratio of Reyielded Plas t i c Core vs. WallRatio of Cylinder

4 Stress Dis t r ibu t ion in Roylelded Cylinder ofWall Ratio 5

5 Pressure Required to Cause Plas t i c Flow in aRoyielded Cylinder of Wall Ratio 5

REFERENCES

(1) Nadai, A., P las t i c i ty, McGraw-Hill Book Company,New York

(2) Hil l , R., Plas t ic i •y, Oxford Univers i ty Press,

London,1950

(3) Dawson, V. C. D., Elast ic and P l a s t i c StressEquations for Hollow Cyl inders . NavOrd Report 6786

iii



NOLTR 63-123

LIST OF SYMBOLS

a Inner rad ius o f cyl inder

b Outer rad ius o f c y l i n d e r

c Radius of i n t e r f a c e between t h r i c e y i e l d e d r eg ion(first in t e n s i o n , then in compress ion , finallyin t e n s i o n ) and the twice y i e l d e d r eg ion

d Radius of i n t e r f a c e between once y i e l d e d reg ion int e n s i o n and twice y i e l d e d reg ion (once in t en s ion ,then in compress ion)

D Diameter

m Diameter ratio i n s ide reg ion where tube is elastic(i.e., m is grea t e r than n)

n Diameter ratio to which plastic flow has occur red

p In t e rna l p r e s s u r e a p p l i e d to cy l i nde r after r e v e r s ey i e ld ing has occur red

P In t e rna l pressure appl ied to cy l i nde r before reversey i e ld ing has occur red

r Radius

w Diameter ratio i n s ide reg ion where plastic flow hasoccur red (i.e., w is l e s s than n)

Yo Yie ld s t r e n g t h

Yoe Yie ld s t r eng th in compress ion

Yot Yield s t r eng th in t en s ion

a Stress

W Wall ratio ( b / a )

Superscr ip t

* Res idua l (stress)

iv



NOLTR 63-123

Subscr ip t s

t Tangent ia l stress)

r Radia l (stress)

z Longi tud ina l (stress)

max Maximum stress)

i In t e r f ace

v



NOLTR 63-123

INTRODUCTION

The pressure capability of a closed-end cylindricalpressure vessel is limited for elastic operation. Based uponthe Distortion Energy Theory the pressure at which yieldingbegins at the bore is given iy

S- Y WI)1)NF W (1)

Thus, even for very large wall ratios the maximum pressure acylinder will hold elastically is given by P - Ya/5-

One of the methods of increasing the elastic pressurecapability of a cylinder in he use of autofrettage. Thisprocess consists of inducing plastic flow in the cylinderduring manufacture by pressurizing it with a pressure (the autofrettage pressure ) greater than that given by equation (1).The plastic flow of the metal begins at the bore and progressesthrough the wall as the pressure is increased. This non-uniformflow is such that when pressure is released, the wall is leftwith a residual stress distribution such that the bore has acompressive t angen t i a l s t ress . The cyl inder is then sa id tobe autofrettaged. Subsequent pressure application can be madeup to the autofrettage pressu.'e with the cylinder reactingelastically.

The equations for the au to f re t t age process (based on aper fec t ly plas t ic mater ia l ) have been derived by numerous

investigators(see for example, refs. (1) (1) and (3)).

The s t r e s s - s t r a i n curve for a per fec t ly p i a s t i c mater ia l issketched below.

Y

1.

O 63 122



NOLTR 63-122

According to reference (3) the pressure required to deform acylinder of wall diameter r a t i o u) , plas t ica l ly, to a diameterra t io ra is

a)2. - M 2

Upon re lease of the pressu re given by equation (2) th eres idua l s t ress dis t r ibu t ion is given by

tha t was p las t i ca l ly deformed. In the par t tha t was e las t icX r

where m is the position diameter ratio. It is assumed thatt he res idual s t resses a t the bore are not la rge enough to

cause thu bore, which had previous ly been yie lded in tension,to yield in compression, that is, to reyield or reverseyield .

As the pressure is increased during autofrettage a point

is reached wherethe tube is entirely plast ic , ioe., n - wU

This represents the maximum pressure which can be appl iedwithout ruptur ing the cyl inder for a perf~ectly p la s t i cmater ia l and is, according to equation (2),

2



NOLTR 63-123

The r e s idua l s t resses in th i s case are given by inse r t ing th evalue of PMAX for p in equations (3) and (4).

From equation (7) it is apparent that as W is increasedthe pressure required to cause plas t ic flow throughout th ewall increases. This, in turn, produces larger and largerres idual compressive s t resses at the bore upon pressure re lease .For the fu l ly plas t ic case, then, there is some par t i cu la r vai lra t io at which the res idual s t resses wi l l be la rge enough tojus t cause yie lding at the bore in compression upon re lease ofthe autofre t tage pressure .

To determine the wall ra t io at which the res idual s t resses

a t the bore of a fu l ly autofre t taged cyl inder are large enoughto jus t cause it to yie ld in compression the yield c r i t e r i onused in reference (3),

t

wil l be employed. Thus, the condition of yield, equation (8),becomes, when applied to the res idual s t resses at the bore,

T (9)

Here it has been assumed that the yield in compression is equalto the negative of the y ie ld in t ea s ion .

Subst i tu t ing the values of the res idual s t resses fromequation (5) and equation (6) in to equation (9) with kj- se tequal to 1 ( i . e . , at the bore), one obtains

*It was assumed t h a t 0 -' r(-0 ) so tha t the yield condi t ion

becomes C - See reference (2) for a discuss ionof the val idi ty of' thib assumption.

3

NOLTR 63 123



NOLTR 63-123

with = for the case of the ful ly autofre t tagedcylinders, tre equation above becomes

z YO z YO - .

or

U (10)

Solv ing equat ion (10) g ive s WA . 2.22. Thus, a cy l i nde rhav ing a wal l ratio o f 2.22, if a u t o f r e t t a g e d to the fullyplastic state, deve lops r e s i d u a l s t r e s s e s o f such magnitudet h a t t h e bore is an t h e verge o f r e y i e l d i n g ( r eve r s e y ie ld ing )in compression upon pre s su re r e l ea se . If 034 2 .22 th e

r e s i d u a l s t r e s s e s developed are l ess than those r e qu i r ed fo rr e y i e l d i n g for the f u l l y plastic case and if a) ) -2 .22 theses t r e s s e s will cause r e y i e l d i n g for the f u l l y plastic case .

It is fu r the r, found t h a t as t h e wal l ratio i n c r ea se sabove 2.22 the va lue o f n to j u s t l eave the bore at th ecompressive y i e l d l i m i t decreases ( r e f . (3)). This meanstthat, if the r ey i e ld ing cond i t i on is the l im i t ing des igncondition, there is a limit to the autofrettage pressure.This limit is calculated to be just twice the pressure tocause initial y i e l d i n g at t h e bore. Hence, according toequat ion (1) t h e l im i t ing au to f r e t t age p re s su re for i nc ip i en tr e y i e l d i n g in t h e case o f WO , 2.2 is

S(11)

For any cy l i nde r t he r e are, t he r e fo re , t h r ee l i m i t in gcurves as shown in figure 1. These are the followings

4

NOLTR 63-123



NOLTR 63 123

the pressure at which yielding in i t ia l ly occursy

S- ,(7)

the pressure necessary to make the cylinder fully plast ic ,which for CJ A 2.22, leaves the residual stresses low enoughto prevent reyieldingy

the pressure l imit for large wall rat ios to just leave th ebore at the yield point in compression after pressure release.

It is apparent that if a cylinder could be operated atpressures given by equation (7), a sizeable increase inpressure capability over that given in equation 11) wouldbe possible. However, as noted before, in th is circumstance,there would occur reyielding of the bore in compression whenthe pressure is released.

It is the purpose of this s tudy to i n v e s t i g a t e r eve r s eyie ld ing in th ick-wal led cyl inders tha t have been pressurizedto the fu l ly plas t i c s ta te during au to f re t t age .

DERIVATION OF REYIELDING EQUATIONS (u) >2.22)*

The assumptions made are the followings

1. The mater ia l in assumed per fec t ly p l a s t i c2. T* = 112. (dt +6)3. The yield cr i t e r ion is given by the Dis to r t ion

Energy Theory

Assumptions (2) and (3) r e s u l t in the fol lowing yield cr i t e r ion ,

S-r ± (12)

Let us consider the case of the fully autofrettaged cylinderof u) > 2.2 subjected to the pressure PMAX, where

See Appendix A for an alternate derivation of the reyieldingequations.

5

63 123



NOLTR 63-123

S- (7)Yo

At every point in the plast ical ly deformed cylinder

P. --31(13)

The equation of equilibrium is

T/L (14)

The radial stress at the bore is equal to -PMAX, that is

Equations (13), (14), and (15) may be combined to obta in th ep l a s t i c stresses due to the in ternal pressure PMjM tha t exis tin the cy l inder (as was done in reference (3)). Thesestresses a re

Zo 16),

L - = - -.- (k -b (17)

An the i n t e rna l p r e s s u r e is re leased , the cy l inderdeforms elast ical ly unti l the bore reaches the yield pointin compression. Thereafter, as the pressure is furtherreduced, plast ic flow progresses outward from the bore. Whenthe internal pressure reaches zero, the cylinder will consist

of two sones, an inner core which has reyielded in coqpression(reverse yielded) and an outer elast ic jacket that has beenpreviously yielded in tension during autofrettage.

6



NOLTR 63-123

The equations describing the reyielded inner core whenthe pressure has been released are the yield cri terionequation in compression, and the equilibrium equation (14),

via. r•0L - 18)

•-•Tt- 0;--Q o

o 14 )dtt.

These equations with the boundary condition that the residualradial stress a t the bore is zero lead to the followingequations for the inner reyielded core stresses after pressurer e l ea se t

-- 19)

=A (20)19

a6,L-dwhere d denotes the radius a t the in terface. These arethe res idual s t resses af te r pressure release in the reyie ldedinner core.

The stresses in the outer jacket before pressure releaseare expressed by equation (16) and equation (17). Sinceduring pressure release the outer jacket is only deformedelastically, the stresses may be obtained by superposition ofelastic s t r e s s e s . Thus,

T t after p re s su re - b e f o r e p re s su re + a due tore lease release change in

effective (21)

pressureat th einterface

The change in effective pressure AP at tha interface is given

by,

7



NOLTR 63-123



~~r 0-~?~Y4 (26)

b tLdEquations (25) and (26) are thus the residual stresses in th eouter once yie lded jacket af ter the pressure has been released.

Since the tangential stress at the interface r - d mustbe equal in each zone, one obtains by equating equation (26)to equation (20):

(dI.- a b (27) ,W

The extent of the reverse yielding can be calculated from thisequation by solving for the inner core radius d.

The residual stresses may be rewritten by insertingequation

(27) into equations (25) and (26) to gives

-l (;i- (28)

t Y /L 29)

Equations (19), (20), (28), and (29) are thus th eresidual stresses developed in a fully autofrettaged thick-walled cylinder ( i .e . , 0 **2.2) after pressure release. Theextent of the reversed yielded plast ic core of radius d isobtained from equations (27).

Figure 2 shows the plastic and elastic zone dimensionsof a cylinder of wall ratio W equal to 5. The value ofd/a is calculated from equation (27) to be 1.41.

It isseen that the plastic core is relatively small. Figure 3 isa plot of d/a for various wall ratios

9

NOLTR 63-123



Figure 4 shows the res idual s t ress d i s t r ibu t ion ( LT )in a reyielded cyl inder of 9A) - 5. Included in the plot , asdotted l ines , are the res idual CM* and 6* that would existif the cyl inder had no l imi t ing compressive yield strength. It

can be seen t ha t these res idual s t r e s ses are only s l i g h t l ymodified in the e las t i c zone.

The resu l t s indicate tha t a cyl inder with wall ra t io grea terthan 2.22, if autofre t taged to the fu l ly plas t i c condit ion, willhave a reyie lded core a f t e r pressure re lease . This plas t i c corehas re la t ive ly small dimensions compared to the or ig inaldimensions of the cyl inder.

PRESSURE APPLICATION TO THE REVERSE YIELDED CYLINDER

If pressure, p, is reapplied to the reverse yielded cyl in-der, then the core wil l initially deform elas t i ca l ly. However,if the pressure becomes s u f f i c i e n t l y high, the t ens i l e s t r e s sesin the ccore wil l cause it to begin to yie ld in tension at th ebore. Further pressur iza t ion wi l l cause the region of plas t i cdeformation to extend radia l ly from the bore to, say, a radius c . For t h i s plas t i c region the yie ld c r i t e r i o n

Yo

the equil ibrium equation (14), and the boundary condit ion thatthe radia l s t ress a t the bore is equal to minus the appliedpressure, r e su l t in the following:

T ._-. c 30)

G.. -1-)-~ ) Lj CL 31 )

These are the s t r e s ses in the thr ice yielded core of th e

cylinder. The cylinder at th i s time appears as sketched below.

10

NOLTR 63-123



NOLTR

Once yielded in tens ion

Twice yielded (first intension, then incompression)

Thrice yielded (first in

tension, then in com-pression, now yielded intension)

Since the deformations in the cyl inder o ther than In th ethr ice yielded core are e las t i c , the s t resses may be obtainedin these e lao t ica l ly deforming regions by the use of super-posi t ion. Thus, for the regions of radi i greater than r - e,

f- b e f o r e pressure + due to change in effec t iveappl ica t ion pressure a t the in te r face

The s t resses before pressure app l i ca t ion are the res iduals t resses ; the change in the effect ive pressure at the in te r faceis equal to the negative of the radia l s t ress change at th ein terface . Thus, the above expression becomes

S- * + ( due to effect ive pressure of value On, (rA.at in te r face

Hence, using equations (23) and (24) and denoting thein terface radius by /ZL

S= T& b L (32)

11

NOLTR 63-123



= . c41(33)

For the region

equations (32) and (33) become with /i=-c

/Y1) + L4 (35)C, +

Equating the tangent ia l s t r e s ses equation (35) and equation (31)

a t the in terface r - c, one obta ins an expression for th eapplied pressure, p, in terms of the radius, c, viz.,

-f5 I z C- 36 )

If a - c is subs t i tu t ed in to equation (36), there resu l t s

12

NOLTR 63 123



NOLTR 63-123

which is i d e n t i c a l to equa t ion (11). Thus, the reverse y i e l d e dc y l i n d e r beg ins to y i e l d a t h i r d t ime (at t h e bore s u r f a c e )when the r e a p p l i e d p r e s s u r e is t h a t given by e q u a t i o n (11); i.e.,the cy l i nde r, upon r e a p p l i c a t i o n o f pressure , w i t h s t a n d selastically the same p r e s s u r e t ha t it would have w i t h s t o o d if ithad been au to f r e t t aged in such a way as to leave the res idua lstresses a t the bore a t the compressive y i e l d s t r e n g t h .

The stress equa t ions (34) and (35) may be t r ans formed byuse o f equa t ion (36) to y ie ld

__P ItL fJ

0- 38)

For the reg ion

equa t ions (32) and (33) become with ri - d

13

NOLTR 63-123



These equations simplify to

b~'a /d

which for c - d becomes

S[(41 )

Equation (41) is independent of r and s ta tes that the outermostregion which was e las t i c i the reyielded cyl inder becomesplas t i c instantaneously when r reaches d during r eapp l i ca t ionof the pressure. Also, from equations (36) and (27) with r - d

14

NOLTR 63 123



NOLTR 63-123

which is simply equation (7), i .e., the pressure required tocause p las t i c flow throughout the en t i r e wall. Thus, it is

seen tha t upon reappl ica t ion of pressure, p, the inner borebegins to yie ld in tension for a th i rd time when the pressurereaches the value

zYO

and the yielding progresses to la rger radi i as the pressure

is increased. When the pressure reaches the value

zY0

the yie ld ing reaches the radius, d, at which time suddenlythe ent i re wall becomes p las t i c .

The s t ress - s t ra in his tory of elements in the tube wall issketched below. A 2A I

dor Tot

ý I-i

A plot of equation (36), giving the pressure required toextend the p las t i c zone when pressure is applied to areyielded cyl inder of wall r a t io 5, is shown in f igure 5,

15

NOLTR 63-123



CONCLUSIONS

Within the assumptions made, the equations for reyie lding

of a cyl inder of la rge wall ra t io ( ca 2.22), autofre t tagedto the fu l ly p las t i c condi t ion , have been derived. Theseequations indicate that the reyie lded plas t i c zone has re la t ive lysmall dimensions while the res idual s t resses in the outere las t ic par t of the tube are s l i g h t l y al tered from what theywould have been had the inner core had no l imi t ing compressiveyield s t r eng th .

It has also been shown that subsequent appl ica t ion ofpressure to the reyielded cyl inder causes the bore to s ta r ty ie ld ing at the pressure that is the l imi t pressure for th eautofre t taging of th ick-wal led cylinders. However, as pressureis bui l t up the plast ic zone grows but at a considorably slowerra te than it did during the or ig ina l autofre t tage process.When the e las t ic -p las t ic in terface reaches the outs ide radiusof the or ig ina l reyielded core, the en t i re cyl inder becomesplas t i c .

It thus appears that the repe t i t ive ly applied in ternal

pressure capabi l i tyof cyl inders

maybe the ful ly autofre t taged

pressure

even for thick-walled cyl inders where reverse yie ld ing occurs(i.e., where W>2.2). This conclusion requires experimentalconfirmation.

16

NOLTR 63-1231.6 -



1.5-

1.4 - FULLYPLASTIC

CYLINDER1.3 - P2 = Irw

Yo V~3

1.2-

1.1

1.0 -

"r -MAXIMUM AUTOFRETTAGE

I- 0.9- PRESSURE WITHOUTz REVERSE YIELDING

: 0 . 8 P 2(w 2 -1)SY'o r3 2

0 .7-w

S0.6

i 0.5w

0.3

YoV3 w 20.2

0.1.

0 I I I I1 2 3 4 5

WALL RATIO-cu

FIG. ILIMITING CURVES FOR INTERNAL PRESSURE

APPLICATION TO A CYLINDER

NOLTR 63-123



PLASTIC CORE

ad

b

b/0=5 d/o:I.41

FIG. 2SCALE DRAWING OF REYIELDEDCYLINDER WITH WALL RATIO OF 5.

NOLTR 63-123



wJ

z W U

z0 -D

00

w U) L L O C L L

M,. ,0

_ i i I i Io

/ O -0I0 D -j

-Ja

0z

.- JU

5N ( -

D/P-OI.V8 TIYM 38003 OliSV-ld

NOLTR 63-123



Lu zJ0 0

.- 0

L))

P cri

•./m_.- l'-n''-J

cq 00/

0-

N

od uj10

SrCLI-J

0i~

30

cp > .in 4D LLIL)

2-1-

N N LL

sniva~

NOLTR 63-123



1.9 LIMIT-ENTIRECYLINDER BECOMES PLASTIC

1.8

I. -

1.6 a

1.5P/ Yot

1.4

1.3

1.2

BORE BEGINS TO YIELD

1.0I.I -

I.I 1.2 1.3 1.4 1.5

r2 /a

FIG.5

PRESSURE REQUIRED TO CAUSEPLASTIC FLOW IN A REYIELDEDCYLINDER OF WALL RATIO 5.

NOLTR 63-123



APPENDIX A

ALTERNATE DERIVATION OF THE REYIELDING EQUATIONS

When the pressure is released in a ful ly plas t i c largewall ra t io tube, the cylinder wil l consist of a plast icreyielded center core and an outer elast ic jacket. Thus, thecore can be considered as a tube under external pressure, p,which has caused the core to be ful ly plas t i c . The outerjacket can be considered as an e las t i c tube with an in ternalpressure, p, which produces a f inal s t ress which is the sum ofthe res idual and Lame s t resses .

For a tube subjected to external pressure the followingequations apply

/~2

-(A-2)

It is assumed that

The yield cr i t e r ion is

T. (A-4)

Consider the external pressure, q, to increase on thecylinder unti l the bore begins to yield in compression. Theboundary condit ions for th is inner core are at r - a

-(A-5)

and

A-i

NOLTR 63-123



0= (A-6)

From (A-4) the re fo re

f-1 01.) oe A-6)

As the ex t e rna l pressure is increased the plastic zone spreads.

In the p l a s t i c zone

an d

IL d zY,dA.

Thus,

-=.Y, t/,, (A-7)

Q,-t- 2.Yv Lw-~It A-8)

For a fu l ly p l a s t i c tube, a t r - b, Th: :§ and a t r - a,C'ý O

- c, 2Y, ,C- 2 Yr A-9)

A-

NOLTR 63-123



Thus,

T -L f -,Yý4 A-10

- Y ~ ~ ~ A-li)

To cause plastic flow throughout t he tube

Yo, A-12)

Therefore , in the f u l l y plastic tube under e x t e r n a l p r e s s u r e

2- YA-Ox b(A-13)

V 4Aq (A

Consider now t he case where r eve r se y i e l d i n g occurs . Onthe o u t s i d e o f the plastic core t he re is a pressure , q, whichhas caused central core to be plast ic

To causeplastic flow throughout

the core, from (A-12)

A-15

In the core e stress distribution is from (A-13) and (A-14)

OIL1

A-3

A-3

NOLTR 63-123



C•L See equa t ion (19)

- iY~(+ (A- 7)See equa t ion (20)

In the elastic par t of the tube the r e s i d u a l elastic stresses

w i l l be the Lame-s t resses plus the r e s i d u a l s t r e s s e s crea ted by

the au to f r e t t age process . Thus,

t = z4 tJ A-18)G - / C .*4-+

C - A-19)

At r - d, the yie ld criterion holds so tha t

* + - (A-20)

From re fe rence (3) o r equa t ions (3) and (4) with =YO L O

VT•d-- dz&O -I •'-

A- 4

NOLTR 63-123



Hence, from (A-20) with Yoc - -Yo t

- •t(,-'- ))

A-21)

Subst i tu t ing C3 in (A-19), and noting that a t r - bj ' i= 1gives

•= o =c -C-

so that

A -T - -- w -- A-22)

I I

and Ct can be w r i t t e n as

V~b\- ~W) Jr A-23)

2 Yo( "

For the fu l ly plas t ic core, from (A-16)

This must equal • at r - d, so that

A-5



NOLTR 63-123



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