LEV EI<AMMRC TR 81-37

STRESS INTENSITY AND FATIGUE CRACK GROWTH

IN MULTIPLY-CRACKED, PRESSURIZED,

PARTIALLY AUTOFRETrAGED THICK CYLINDERS

DT !CANTHONY P. PARKER DE ',MECHANICS OF MATERIALS DIVISION DEC 3 0 19J

August 1981

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ARMY MATERIALS AND MECHANICS RESEARCH CENTER9 Watertown, Massachusetts 02172

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"" ARC--TR-81-37 p e , IAMF___/_44. TITLd (and Subtitle) S. TYPE OF REPORT I PERIOD COVERED

STRESS INTENSITY AND FATIGUE CRACK GROWTH INMULTIPLY-CRACKED, PRESSURIZED, PARTIALLY FIN G ROrtAUTOFRETTAGED THICK CYLINDERS . PERPORMNG ORG. REPORT NuWeR

, 7. AUTHOR(@) G. CONTRACT OR GRANT NUMlER(s)

Anthony P. Parker*

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Army Materials and Mechanics Research Center I/A Project: 1L162105Af184Watertown, Massachusetts 02172 kCMS Code: 612105.H846011DRXMR-TM i

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*Professor at the Royal Military College of Science, England.

IS. KEY WORDS (Continue un Powerssaide #I necessary and Identify by block nuymber)

Autofrettage Fracture (mechanics) RatesCrack propagation Gun barrels Residual stressCylinders Life expectancy Stress intensityFatigue (mechanics)

20. ABSTRACT (Continue an •everso side If necessary, and Identify by block number)

(SEE REVERSE SIDE)

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Block No. 20

ABSTRACT

Stress intensity factors are determined for i.nternally and externallycracked, pressurized thick cylinders with partial autofrettage (less than100% overstrain). The solutions are based on a superposition of existingsolutions which does not involve any loss of accuracy.6 'p

Implications of the stress intensity factor results for the safe-lifedesign of gun tubes are discussed. Various suggestions for future work 23are presented.

/4

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!-i

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CONTENTS

Page

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . 1

,ASIC EQUATIONS FOR PRESSURIZED, AUTOFRETTAGED TUBES... 2

SUPERPOSITION METHOD. . . . . . . . . . . . . . .. .. . 3

A. Internal Cracking, Pressure in Bore and Cracks,Autofrettage Stresses .............. 3

B. External Cracking, Pressure in Bore, ResidualAutofrettage Stresses. . . . . . . . . 4

EXTERNALLY CRACKED, PARTIALLY AUTOFRETTAGED TUBE ..... 4

INTERNALLY CRACKED, PARTIALLY AUTOFRETTAGED TUBE. . . . . 6

IMPLICATIONS FOR FATIGUE LIFE PREDICTION OF GUN TUBES . . 8

A. Externally Cracked Tubes ............ 9

B. Internally Cracked Tubes . . . . . . ...... 9

C. Stability of Crack Growth Patterns . . . . . . 10

D. Characterization of Autofrettage Stresses .... 10

E. Effects of Crack Shape .... . . . . . . .. 10

SUMMARY OF A 'AILABLE EXPERIMENTAL WORK. . . . . . . . . . 10

".c.' I 1.1

i . . , 3 . . • : , • , •. _

L . .u- '-

INTRODUCTION

Fatigue crack growth arising from the cyclic pressurization of thick cylinders canproduce a regular array of up to 50 equal-length radial cracks emanating from thebore.1 A knowledge of the crack tip stress intensity factor K is necessary to predictthe fatigue growth rate and critical length of such cracks.

Several solutions for the case of a cracked, pressurized-thick cylinder are avail-Sable.-6 It is likely that the most accurate of these solutions are those derived by

use of the modified mapping collocation (MMC) method. These include the solution inReference 5 for up to four internal or external radial cracks, and that in Reference 6for up to forty internal radial cracks. The errors associated with the MMC techniqueare generally estimated as being less than 1%.

To inhibit fatigue growth of internal cracks it is common practice to produce amore advantageous stress distribution involving residual compressive hoop stresses nearthe bore, by autofrettage treatment of the cylinder prior to use. 7

K solutions exist for a multiply-cracked, fully autofrettaged (100% overstrain)*tube. 6 , 8 Reference 6 is an MMC solution.

Note: It has been demonstrated that the stress distributionarising from full autofrettage is essentially equivalent to

that arising from steady-state thermal loading of the tube, 9

and that K values for one of these loadings may be obtainedfrom the other by a simple scaling operation. 8

However, the optimum autofrettage condition may not be 100% overstrain7 sincefatigue cracks may develop at the outside radius as a result of the relatively hightensile residual stress. Clearly, the choice of the optimum overstrain condition willinvolve a consideration of the rates at which external cracks will grow radially inward,and the rates at which internal cracks will grow outward. In each case, prediction ofcrack growth rate, critical crack length, and residual strength will depend on a know-ledge of the crack-tip stress intensity factor.

"Overstrain is the proportion of the cylinder wall thickness that is subjected to plastic strain during the autofrettage process.1. GOLDTHROPE, B. D, Fatigue and Fracture of Thick- Walled Cylinders and Gun Barrels in Case Studies in Fracture Mechanics, Army

Materials and Mechanics Research Center, AMMRC MS 77-5, June 1977. p. 3.8.1-3.8.15.2. PU, S. L., and HUSSAIN, M. A. Stress Intensity Factors for a Circular Ring with Uniform Array of Radial Cracks Using Cubic

Isoparametric Singular Elements. Trans. 24th Conference of Army Mathematicians, Army Research Office, Report 79-1, 1979.3. TWEED, J., and ROOKE, D. P. The Stress Intensity Factor for a Crack in a Symmetric Array Originating from a Circular Hole in an

Infinite Solid. International Journal of Engineering Science, v. 13, 1975, p. 653-662.4. BARATTA, F. 1. Stress Intensity Factors for Internal Multiple Cracks In Thick- Walled Cy0inders Stressed by Internal Pressure Using Load

Relief Factors. Engineering Fracture Mechanics, v. 10, 1978, p. 691-697; also Army Materials and Mechanics Research Center,AMMRC TN 77-3, July 1977.

5. TRACY, P. G. Elastic Analysis of Radial Cracks Emanating from the Outer and Inner Surfaces of a Circular Ring, Engineering FractureMechanics, v. 11, 1979, p. 291-300.

6. PARKER, A. P., and ANDRASIC, C. P. Stress Intensity Prediction for a Multiply-Cracked, Presawrised Gun Tube with Residual andThermal Stresses in Army Symposium on Solid Mechanics, 1980 - Designing for Extremes: Envirnonment, Loading, and Structural Behavior,Army Materials and Mechanics Research Center, AMMRC MS 80-5, 1980, p. 35-39.

7. KAPP, J. A., and EISENSTADT, R. Crack Crowth in Extcrnalh blawed, Autofrettaged Thick-Walled Cylinders and Rings in FractureMechanics - A Symposium, ASTM STP 677, C. W. Smith, ed., 1979, p. 746-756.

8. PARKER, A. P., and FARROW, J. R. Stress Intensity Factors for Multiple Radial Cracks Emanating from the Bore of an Autofrettagedor Thermally Stressed Thick Cnlinder. Engineering Fracture Mechanics, v. 14, 1981, p. 237-.41.

9. PARKER, A. P., and FARROW, J. R. On the Equivalence of Axi-Svmmetric Bending, Thermal and Autofrettage Residual Stress Fields.Journal of Strain Analysis, v. 15, no. 1, 1980, p. 51-52.

[ A.lm........-

The designer requires accurate stress intensity factors for both internally cracked* and externally cracked tubes with internal pressure, and any amount of autofrettage

from zero to I00% overstrain (full autofrettage). In this report it is demonstratedthat many solutions for the apparently more complicated problems of internally andexternally cracked, partially autofrettaged tubes may be obtained, without any loss ofaccuracy, by a straightforward superposition of existing solutions.

BASIC EQUATIONS FOR PRESSURIZED, AUTOFRETTAGED TUBESF

Consider a tube, internal radius a, external radius b, which is subjected to aninternal pressure p, Figure I. The distribution of hoop (ao) stress in this case isgiven by Lame's equation as:

B a2

e b2_a2 1r- (0)

where r is the radius at which the stress is defined, and the superscript B indicatespressure in the bore.

Ia r Figure 1. Pressurized, elastic thick cylinder.

pressureP

b

Assuming elastic-perfectly plastic material properties and plane strain conditions,and we employ Tresca's yield criterion, but omit the analysis, the pressure p* to causeyielding of the tube out to a radius r = c (Figure 2) is given by:

*Y 2 2p =Y n(c/a) + - (b -c) (2)2h

where Y is the uniaxial yield stress for the material. This will give directly thepressure for initial yielding at the bore:I* Y 2 2Pi =--(b -a) (3)

and the pressure for complete yielding of the tube:

py Y Xn(b/a) (4)

2

1

a s Figure 2. Pressurized, partially plastic thick cylinder.

If the cylinder is subjected to a pressure p*, [pi*<p*<Py*], there will be partialyielding of the tube out to a radius c, Figure 2. The hoop streEses produced by thispressurization are:

-p + YD + nr/a)] a<r<c (5)

= 1 + 2 c<r<b (6)

If the pressure p* is subsequently removed completely, assuming that the unloadingis entirely elastic, with no reversed yielding (valid provided b/4<2.22), the residualhoop stress distribution OR is given by: 1 0

0 R= -p + Y[l + Zn(r/a)] - p*a 2/(b 2 - a2)[W + (b 2/r 2)] a<r<c (7)

a = R [(Y 2 /2b 2 ) (p* a2/(b 2 a2))][1 + (b 2 /r 2) ] , c<r<b (8)

Clearly, a repressurization of the tube to a pressure p<p* will produce a stressdistribution which may be calculated by the addition of (7) and (1). for r.c, and (8)and (1) for r>c.

SUPERPOSITION METHOD

The superposition principle applies to any linearly elastic body subjected to astatically determinate loading system. By use of this principle it is a straight-forward procedure to demonstrate that the stress intensity factor for a crack in a bodysubjected to external stress is identical to that caused by stresses acting on the sur-face of the crack equal but opposite to those which wculd be present in the uncrackedbody under external load. The implications of this result for the particular cases tobe considered are presented.

A. Internal Cracking, Pressure in Bore and Cracks, Autofrettage Stresses (Figure 3)

The appropriate crack-line loading `11 comprise:

(i) A normal loading on the crack surface, equal and opposite to thatpredicted by Lame's Equation I

(ii) A constant pressure p which has infiltrated the cracks from the bore.10. HILL, R. The Mathematical Theor.y* of Plasticity. Oxford University Press, Oxford, England, 1967.

3

7

fPFigure 3. Internally cracked, autofrettaged thick cylinder •

, pressure in bore and cracks.

(The total effect of (i) and (ii) above is an equal and opposite stress given by:

212

where the superscript BC indicates pressure in bore and cracks.]

(iii) A residual distribution equal and opposite to that predicted by (7) and (8).

B. External Cracking, Pressure in Bore, Residual Autofrettage Stresses (Figure 4)In this case the necessary crack-line loading will comprise:

(i) A normal loading equal and opposite to that predictad by (1).

(ii) A residual distribe~tion equal and opposite to that predicted by (7) and (8).Ir I

,Figure 4. Extornally cracked, autofrettaged thick cylinder •S nroun• )pressure in bore.

EX>%:RNALLY CRACKED, PARTIALLY AUTOFRETTAGED TUBE

If a partially autofrettaged, externally cracked tube is repressurized to a pres-sure p<p*, then the total stress distribution a4 is given by the addition of (1) and(7), and (1) and (8), thus:

T B RS+ a (10)

or:

T 2F2 + - 2 I + 2 c2<r<bb a 2b b2 a2 r2 '(

i' 4

I

Apart from a multiplying constant, the above expression is identical to Equation 1.Thus, if stress intensity factors have been derived for a cylinder with radial edgecrack(s) of length I, lt<(b-c)J, it is only necessary to scale these results by S,where,:

SbK~ 2bc bY2I2 (12)

in order to predict K for the externally cracked, pressurized, partially autofrettagedtube,

'Example I

A thick cylinder has an external radius twice that of its internal radius. It issubjected to 50% overstrain in the autofrettage process. The cylinder is subsequentlyrepressurized to p - Y/N (<p*), where N is any suitable number, What is the scalingfactor S to be applied to K results for the externally cracked, nonautofrettaged, pres-surized 'ube in order to solve for the partially autofrettaged case? [The solution isto be limited to t<(b-c).]

Solution

From (2), the autofrettage pressure p* is given by

Y 2p = Y 2zn(c/a) + 2 (b2-c2 (13)

in the example c - 1.Sa, b - 2a; hence:

p* = 0.624 Y (14)

Substitute from (14) into (12), noting that Y m Np, to obtain:

S - I + 0.2196 N. (15)

Typical value would be N 3, giving S 1.659, hence:

K50% overstrain + p,7essure = 1.659 Kzero overstrain + pressure-

Note: The solution is valid only provided t<(b-c).

A set of results for external cracks with internal pressure and 50% overstrain, basedon the results of Reference 5 and the superposition outlined in this section, is pre-sented graphically in Figure 5.

5 .------.-

W3.0 N.

9M ftntrsai KO26 N~3 u Ov~raln3 KO N:1.0 3 b41N'to

WIA

U2 aCracks

I Crack*

0 0,1 0.2 0'3 ;A . 5 .4. 0.1 0.2 0.3 0.A 03 0.6 0.1it

I N-3.0

~.-jjj N.2. MW Owstrain 44 N3O'

b 2'0 N: 5 5 OwsrtrainN1 a N- 0

0z N1. - 2

'a'1

eCeralckrialsclndr

INTERNALLY ~CrackEDs ATAL UTFETGDTB

-' Fiu~e 7714~0.S - a~ed. u§Ej

Fiur S.winitest atosfrethrfullycakdprtly autofrettage d n thely cy16)er

Now, returning to the case of partial autofrettage, the required crack-line stresssystem is 4iven by the superposition of (9) and (7), thus:

T R + BCa e 0 (17)

YD -p + Y[l * Ln(r/a)] - 1 +

bu_ a r(18)

( b le)a jSubstituting into (18) from (2) we obtain:

C p - Y In(c/a) - 2 (b-c2) + Y Zn(b/a)] 1 + a 1 +

2; (b 2[ 212

- Y Zn(b/a) I a 2 1 + 2J)+ Y[l + Zn(r/a)]

On inspecting the above equation we note that the first term is merely a scalingof the stress field for pressure in bore and cracks, Equation 9, while the second andthird terms represent the stress field arising from full autofrettage, Equation 16.Making the substitution Y - Np and writing in terms of superposition of K solutions, weobtain the expression:

K 11+ N Ln(b/c) N (b-c2 )]K

partial 22b pressure (20)

autofret tage+ pressure

+ Kfull autofrettage

Example II

The tube described in Example I has internal cracking. What is the stress inten-sity factor in terms of that for full autofrettage without internal pressure, and thatfor internal pressure without autofrettage? [The solution is to be limited to t<(c-a).]

Solution

Substituting c - 1.5a, b = 2a into (19) gives:

K partial autofrettage ( C1 + 0.0689 N) Kpressure + Kfull+ pressure autofrettage (21)

Thus if N 3: Kpartial autofrettage + pressure (1.2067) Kpressure + Kfull autofrettage.

7

-Ak j

Note: The solution is valid only provided t<(c-a).

A set of results for intern~al cracks with internal pressure and 50% over .:rain,based on the results of Reference 6 and the superposition described in this section, ispresented graphically in Figure 6.

Nofl-Autlfrottqf (Rof.t,

1.0 overstrain 1.0 Non-Autofroent!ae (Rel.

0.8-5So Overstrain

01-0.6-N1.

4 Cracks10 Cracks

0 0*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.? 0.3 0.4 0.5 0.6 0.7

L.4 -b-a -

12- 12"m

1.0 .0 2.0

0.a Non-Autoflrottaga (Rot, 6) 0.8... .Non-Autatofradagd (Rol, 6)

0. ovaitrmn 0.

0.4 N 25K pr.- 0.4. %oesri

000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0. 0.6 0.7

b-a b-a

Figure 6. Stress intensity factors for internally cracked, partially autofrettaged thick cylinder.

IMPLICATIONS FOR FATIGUE LIFE PREDICTION OF GUN TUBES

The fatigue growth rate of cracks subjected to cyclic loadiiug may be expressed interms of Paris' Law,ll namely:

dR.(22)

11. PARIS, P. C., and ERVOGAN, F. A CW tial Analysis of Crack Prop~gation Laws. Trans. ASME, Journal of Basic Engineering, v. 85,1963, p. 528-534.

8

where dL.dN is the fatigue crack growth per loading cycle, C and m are empirical con-stants, and AK is the range of stress intensity defined by:

AK - Kmax - Kmin (Kmin O)

-AK - max (Kmin 0)

and Kmax and Kmin are the maximum and minumum values of stress intensity during theloading cycle. (Note, the possibility of "overlapping" of the crack surfaces at somepoint on the crack line remote from the crack tip is not considered in this report.)

During the lifetime of a particular cracked tube the crack will propagate fromsome iritial length ti to some final, critical length 4c at which Kmax approaches thefracture toughness of the material. In order to predict the fatigue life, (21) isrearranged to give:

i9:

c

N N. (23)

C(AK)m c1

where (Nc - Ni) is Lhe number of cycles to propagate from initial to critical cracklength.

The implications of the results presented herein for the safe-life design of gun

tubes may be summarized as follows.

A. Externally Cracked Tubes

All loading contributions in the case of external cracking tend to produce posi-tive contributions to K, thus crack closure is not a possibility, and the only contri-

bution to the stress intensity range (AK) is the cyclii: pressurization term. The stressintensity arising from residual stressing will simply serve to increase Kmax, while notaffecting AK. In the case of steels this generall' causes a relatively small increasein crack growth rate12. In addition, the increase in Kmax will cause the criticalcrack length 4c to be reduced in comparison with the nonautofrettaged case, hence totallifetime may also be reduced somewhat. This reduction in lifetime may not be signifi-cant, since relatively little of the component's lifetime is expended at longer cracklengths.

1 3

B. Internally Cracked TubesIn this case autofrettage will produce a negative K contribution. Positive K

values, and the possibility of crack growth, cannot occur until the bore pressure has

been raised to produce a positive K contribution which exceeds the autofrettage effect.Hence autofrettage tends to reduce the value of AK for internal cracks, and thus alsoreduces the crack growth rate in comparison with the nonautofrettaged tube. Further-more, since Kmax is also reduced, the critical crack length will increase and thus tendto increase compor. I, ifetime.

.2. POOK, L. P. Analysis and Application of Fatigue Crack Growth Data. Journal of Strain Analysis, v. 10, 1975, p. 242-250.13. DAVIDSON, T. E., and THROOP, J. F. Practical Fracture Mechanics Applications to the Design of High Pressure Vessels in Application

of Fracture Mechanics to Design, J. J. Burke and V. Weiss, ed., Plenum Press, New York, 1979.

9

- - ' -,-------------- .. .

C. Stability of Crack Growth Patterns

Stable arrays of 40 to 50 near-equal length, internal radiil cracks have beenobserved in rifled, nonautofrettaged gun tubes, Life estimates based on this number ofcracks appear to be borne out in practice. Goldthorpel has supgested that the case ofmultiple (say 40) crack growth is the stable configuration for the nonautofrettagedcylinder, This stability derives from the strong negative slope of the K versus . Jcurve at short crack lengths. However, this feature is abscnt in the case of auto-frettaged tubes. Initial results from other work 1 4 indicate that life estimates forinternally-cracked, fully autofrettaged tubes should be based on the assumption of sixcracks propagating.

In view of the significant effect on K arising from the assumption of a particularnumber of cracks, it is clearly important that future work should fully investigate thet stability of crack growth patterns in nonautofrettaged, partially autofrettaged, andfully autofrettaged tubes. This work should also address the question of the effect ofrifling on crack pattern stability. Only then will it be possible to establish a properfracture mechanics design procedure. ID. Characterization of Autofrettage Stresses

There is some evidence to suggest that the residual stress field arising from theautofrettage process may not attain the magnitudes predicted by an elastic/perfectlyplastic analysis which ignores strain hardening and Bauschinger effects. O,13 Theeffect of this reduction in residual stress has been included as a simple multiplyingfactor in the determination of stress intensity factors.1 5 Nevertheless, in order tohave confidence in the fracture mechanics design procedure, it will be necessary toinvestigate the true nature of the autofrettage stress field, and the effect of thisfield on stress intensity calculations.

E. Effects of Crack Shape

Another important feature of the crack growth pattern and rates in gun tubes iscrack shape, and the change in shape (and hence growth rate and relative interaction)of thumbnail or semielliptical cracks during the fatigue life.

An understanding of crack-pattern stability, residual stresses, three-dimensionaleffects and (possibly) crack closure will also be of importance in the fracture me-chanics design of other types of structural elements in general use, such as weldedcomponents.

SUMMARY OF AVAILABLE EXPERIMENTAL WORK

Experimental work relating to the fracture mechanics of autofrettaged tubes is

scarce. Work has commenced in the United Kingdom* and in Australiat on the determination

*AUSTIN, B. A., Private discussion, 1979.t DMORTON, M. E., Private discussion, 1979.14. TABONE, M. V., BURNS, I. W., and GIBSON, A. F. A Review of Fatigue Life Pediction for In.Service Ordnance. Army Staff Course,

Division II Project Study, Royal Military College of Science, Shrivenham, Swindon, England, 1980.15. UNDERWOOD, J. H., and THROOP, J. F. Residual Stress Effects on Fatigue Cracking of Pressurized Cylinders and Notched Bending

Specimens. Presented at SESA Spring Meeting, Boston, Massachusetts, May 1980.

10

Z t 4t. ...

of K calibrations. Other work is not currently accessible.* However, some open litera-ture crack-growth rate measurement is available. 7 , 1 5 In Reference 7 the fatigue growthrate of a single external crack was determined, using the test specimen configurationillustrated in Figure 7. The ring specimens tested had zero or 50% or 100% overstrain.Crack growth rates for 50% and 100% overstrain are generally within 25% of one another,this difference being easily explained on the basis of scatter and the relative values

of Kmax. However, the nonautofrettaged ring produced suprisingly low crack growth rates,bearing in mind that it was subjected to the same stress intensity range.

AF

.- //Figure 7. Configuration used in crack growth rate/ dtermination for utofrettaged ring (Ref. 7).

, crak

Reference 15 contains experimentally determined fatigue crack growth rate data for

na single, internal, elliptical crack propagating in an autofrettaged barrel with vary-ing degrees of autofrettage from 0% to 60%. The cyclic loading in this case was pro-duced by internal pressure. The crack growth rates show general agreement with thosepredicted from a simple, two-dimensional, linearly-varying stress analysis modified by

* appropriate correction factors.

A. 4

*JOHANSSON, S., Private discussion, 1980.

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Air Force Flight Dynamics Laboratory, Wright-Patterson Brown University, Providence, RI 02912Air Force Base, OH 45433 1 ATTN: Prof. J. R. RiceATTN: AFFOI (FBS), C. Wallace 1 Prof. W. N. Findley, Division of Engineering, Box 0

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I Prof. D. K. Fulbeck, Department of Mechanical Engineering, 1 Dr. 0. J. Ayres, 9487-421, Combustion Engineering Inc.,

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20th and Parkway, Philadelphia, PA 19103 1 Author

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