Stress intensity factor approximate formulae for uniform crack arrays in pressurized or autofrettaged cylinders

Engineering Fracture Mechanics

Vol. 43,

0013-7944/92

No. 5, pp. 725-132, 1992

$5.00 + 0.00

Pergamon Press Ltd.

Printed in Great Britain.

STRESS INTENSITY FACTOR APPROXIMATE FORMULAE FOR UNIFORM CRACK ARRAYS IN PRESSURIZED OR AUTOFRETTAGED CYLINDERS M. PERLt Department of Mechanical Engineering, Florida International Miami, FL 33199, U.S.A.

University,

Abstract-The

presently available large number of stress intensity factor (SIF) values for uniform arrays of radial cracks, emanating from the bore of partially or fully autofrettaged cylindrical pressure vessels, results from extensive research into the fatigue and fracture processes of these vessels. The data consist of more than 400 values for K,p, the SIF due to internal pressure, for different crack arrays and for a wide range of crack lengths, as well as more than 200 values for K,“, the negative SIF due to the compressive residual stress field induced by the autofrettage process, for different configurations of crack arrays and various levels of autofrettage. A detailed analysis of all the data herein reveals that any of the above results pertains to only one of two possible categories designated “sparse crack arrays”, or “dense crack arrays”. While for sparse arrays both K,p and K,” are directly proportional to the crack length, I, in the dense case, they depend primarily on the inter-crack spacing, d, and are practically crack length independent. Furthermore, the inter-crack aspect ratio, Z/d, is found to be the sole parameter determining to which of the two categories each case belongs. Based on these results, and using a least-squares approach, SIF approximate formulae for K,p and K,A for sparse and dense crack arrays are developed, yielding simple expressions of very good engineering accuracy which are applicable to the whole gamut of existing results.

INTRODUCTION THE GROWING interest on the part of the nuclear, chemical, and armament industries in the fatigue and fracture processes in pressurized, most likely autofrettaged, cylindrical pressure vessels during the past two decades has resulted in considerable research into this problem. As an initial step towards its solution, it was necessary to evaluate the prevailing stress intensity factors for various possible configurations of cracks in such vessels. As arrays of radial cracks are sometimes observed at the bore of pressurized cylinders, one of the first approaches to the problem consisted of a two-dimensional model of the cracked cylinder, bearing a uniform array of n equal, radial cracks emanating from its inner surface as schematically described in Fig. 1. Two fundamental cases were considered:

(4 The

(W

evaluation of&---the

stress intensity factor due to the internal pressure, assuming that this pressure acts fully on the crack faces. More than 400 values of KIp are presently available. These results were obtained by different analytical and numerical techniques, though mainly by the Finite Element (FE) method, and by various researchers, such as Bowie and Freese [l], Tweed and Rooke [2], Grandt [3], Goldthorpe [4], Barrata [5], Pu and Hussain [6, 71, Kendall [8], Per1 and Aront [9, lo], and Per1 et al. [ll, 121. Most available KIp data are for a cylinder with w = b/a = 2, including values for numerous arrays with n = l-1024 cracks, and for a wide range of crack lengths, I/a = 0.001-0.65. A few values of Krp for cylinders with w = 1.5-3.0 are also available, e.g. refs [5,6]. The evaluation of &-the negativez stress intensity factor due to the compressive residual stress field induced at the inner layer of the cylinder during the autofrettage process. More than 200 values of KrA are presently available. Most results were obtained by the FE method, simulating the residual stress field by an active thermal load. However, some results were obtained by other techniques, such as the modified mapping-collocation method. K,” values

tvisiting Research Professor, on sabbatical leave from the Technion-Israel Institute of Technology, Haifa 32000, Israel. $Physically, a negative SIF cannot exist as it would mean an overlapping of the crack faces. However, as an intermediate result, K,,, (due to the residual stress) can conveniently be considered negative when it is superimposed on the positive K,p (due to pressurization) to yield a net SIF, which is either positive for K,p > IK,” ( or nil in the case of lKIA 1> K,p. It is only in this context of superposition of loads that an SIF can be considered negative. 125

M. PERL

126

were evaluated by several researchers, such as Pu and Hussain [7], Pu and Chen [13], Parker and Farrow [ 141,Parker et al. [ 151,and Per1 and AronC [9, 10, 161.As in the previous case, most results were obtained for a cylinder with M:= b/a = 2 bearing crack arrays of iz = l-1024 cracks, of lengths l/a = 0.001-0.25, and for autofrettage levels of t = 30-100%. Only recently, a first attempt to develop general expressions of engineering accuracy for K,, for regular crack arrays in pressurized cylinders was successfully made by Pook [17]. On the one hand, by closely examining the analytical expressions for the stress intensity factors of three relevant multiple crack cases: the Benthem and Koiter [18] solution for an infinite array of edge cracks in a semi-infinite sheet subjected to uniaxial tension, Wu’s [19] results regarding a hypocycloid of n cusps in an infinite sheet under internal pressure, and the Westman [20] expression for a pressurized star crack, and on the other hand, by reviewing the existing & values [7-lo], Pook [17] was able to develop approximate expressions for K,p. Two distinct expressions were derived: one for closely spaced arrays with a spacing ratio, Z/d > 0.16 (see Fig. l), and the other for smaller spacing ratios, l/d < 0.16. Though Pook’s [17] two approximate formulae are of very good engineering accuracy (about 5%), these expressions are limited to certain values of the governing parameters. His expression for closely spaced arrays is only valid for arrays of n > 9, and a relative crack length shorter than l/a < 0.25. His expression for a smaller spacing ratio is only applicable to arrays bearing less than seven cracks (n < 7), and shorter than l/a < 0.25. Due to these restrictions, Pook’s expression [17] can merely be used to approximate K,p in only about half the cases for which numerical values are available. Furthermore, Pook’s suggestion [17] for the limiting value of the spacing ratio Z/d = 0.16, which distinguishes between the two cases, was empirically deduced from the numerical results without any physical interpretation. Consequently, the present analysis has three major objectives: (a) To provide, at least partially, a physical rationale as to the role that the inter-crack ratio, I/d, plays in classifying crack arrays as dense or sparse; as well as to define the critical numerical value of Z/d for the entire range of the available data. (b) To develop approximate formulae of engineering accuracy for K,,, applicable to the whole range of relevant parameters, and covering all existing data for both dense and sparse crack arrays. (c) To develop, for the first time, approximate formulae of engineering accuracy for KIA, applicable to all relevant crack arrays, crack lengths, and levels of autofrettage, which are valid for all existing data.

Fig. 1. Schematic representation

of a cross-section of a multicracked autofrettaged cylindrical pressure vessel and the relevant notation.

Stress intensity factor approximate formulae

121

THE CLASSIFYING PARAMETER l/d Per1 and Arone [lo], upon obtaining, for the first time, & values for large arrays of up to 1024 radial cracks, and for a wide range of crack lengths, pointed out the fact that while in some cases Kip is highly dependent on the crack length, 1, it is virtually crack length independent for the following range of crack arrays and crack lengths: n > 256 and I/a = 0.005-o. 1. Furthermore, they have also shown [9] that the SIFs for large arrays of short radial cracks (n > 128; l/a < 0.025), emanating from the bore of a pressurized cylinder, coincide in value with those obtained by Benthem and Koiter [18] for an infinite array of edge cracks in a semi-infinite sheet under tension. Pook [17], as previously mentioned, was the first to demonstrate that in several cases of multi-crack configurations, the SIF characteristics depend on the density of the cracks in the array, which is in turn governed by their spacing ratio. In these cases, K, for low density arrays is crack length dependent, while for closely spaced crack arrays, K, is proportional to the crack spacing. The inter-crack stress field (ICSF), prevailing in between any two adjacent cracks in a uniform array of radial cracks emanating from the inner bore of a pressurized cylinder, was studied by Per1 et al. [ll, 121. In this study, special attention was given to the hoop component of the ICSF as it directly affects &. Per1 et al. [l 1, 121 have found that:

(a) Under certain geometrical conditions, the hoop stress in the inter-crack region (ICR), usually @I

w

a tensile stress, becomes compressive, thus affecting the KLpcharacteristics considerably. The sole parameter controlling this phenomenon is the spacing ratio, l/d, therein [l l] referred to as /It---the inter-crack aspect ratio (ICAR). There exists a threshold value for the inter-crack aspect ratio (ICAR), /I < 0.2,$ beneath which no compressive circumferential stress develops in the ICAR. This value was found to be universal for all the cases studied in refs [l 1, 121, i.e. crack arrays bearing n = 20-1024 cracks of lengths l/a = 0.05-0.25.

Based on all the above findings, it can be concluded that all possible geometrical configurations of uniform arrays of inner-radial cracks in a pressurized cylinder belong to one of two possible types: Dense crack arrays:

crack arrays of the crack Sparse crack arrays: crack arrays of the crack

for which the SIF is directly proportional spacing, i.e. Jd. for which the SIF is directly proportional length, i.e. 4.

to the square root to the square root

Furthermore, it could be expected that the classifying parameter distinguishing between these two types will be the inter-crack aspect ratio, l/d, whose critical value for arrays bearing n 2 20 is anticipated to be (l/d), = 0.2. As no information is available for smaller crack arrays (n < 20), it will be necessary to establish empirically an l/d critical value for this range. By sorting about 450 available values of Kip for n = l-1024, and l/a = 0.001-0.65, it is found that for arrays with n > 8 cracks, the expected (l/d),, = 0.2 is valid. Dense arrays are, therefore, defined as those with an ICAR of: l/d > 0.2

for 1024 > n > 8.

(1)

As for arrays with less than eight cracks, it is found that the critical value of the ICAR is n-dependent. Based on the available data, a simple empirical expression for this value is derived using linear regression. Thus, for n < 8, dense arrays are defined as those bearing an ICAR of: l/d < 0.2[0.156 + O.l055n]

for 8 > n 2 1.

(2)

tThe definition of the inter-crack aspect ratio, /I, in ref. [l l] differs slightly from Pook’s definition of l/d [17]. While Pook determines the inter-crack spacing along the circumscribing circle (see Fig. l), i.e. d = 2nr,/n, Per1 et al. [l I] determine it along the inscribing circle, i.e. d = Zna/n. In the present analysis, Pook’s [17] definition is adopted as it is found to be more convenient. $The threshold value, in terms of Pook’s spacing ratio, happens to be identical in these cases, namely I/d < 0.2.

728

M. PERL

APPROXIMATE

FORMULAE

FOR &

Once the critical value of the classifying parameter, I/d, for the entire range of available data was determined, approximate formulae for & for both dense and sparse crack arrays could be readily derived. KIP for dense crack arrays The available data for dense crack arrays, as well as Pook’s [ 171analysis, suggest the following expression for KIp: KLp= F, (n )a,Jz.

(3)

where o, is the effective crack tip hoop stress defined by Pook [17] (see Appendix A). Numerical values for F, (n) are calculated from the 351 available values of K,p, and presented as a function of n in Fig. 2. F,(n) numerical values can be approximated by the following function of n: F,(n)=C,+-

ci

~

C3

C4

____

n+1+(n+1)2+(n+1)3+(n+l)“’

~

C*

The constants C,-Cs are determined by a least-squares procedure that is subjected to an additional constraint of minimizing the maximum absolute error. This process yields the following expression for F,(n): F,(n)=--

1.5416

0.05595

---_+-------,li-n [ 1-l*+l (n + l)-

12.681 (n + 1)3

22.715

I

(n -I- 1)4

.

(5)

F,(n) for large n tends to (270)‘,“, as could have been anticipated [17-201. Equation (3), using expression (5), reproduces all the available Klp values with a maximum absolute error of less than 5.5%, and an average error of 0.10% with a standard deviation of 2.0%. Furthermore, for dense arrays with n >, 40 cracks, as F,(n) is practically (211)~~“‘,eq. (3) can, without any loss of accuracy, be further simplified to: K,P = 01

d

J ‘z

for II 2 40

The applicability of eq. (3) to cylinders of different geometries was also verified. The only relevant data regarding other geometries are presented by Pu and Hussain [6]. Their entire data, presented in graphical form, include some values of KIp for dense crack arrays in cylinders with diameter ratios of w = 1.5 and 2.5 in addition to the common configuration of w = 2. Equation (3) is found to be fully applicable to these cases as it reproduces all the relevant values of KIp, obtained from the above graphs, with the same accuracy as for a cylinder with w = 2.

10

100

n - number of orecks in the array Fig. 2. The function F,(n); the solid line represents the approximating

function of eq. (5).

129

intensityfactor approximateformulae

Stress

K,, for sparse crack arrays

A natural choice to approximate following form:

&

for sparse crack arrays would be an expression of the (7)

Kp = &(lld)a,&

where aa is the effective hoop stress at the cylinder’s bore (see Appendix A). Numerical values for F,(l/d) are evaluated from the 100 available values of Ku. and are presented as a function of I/d in Fig. 3. F2(I/d) can be approximated by a second order polynomial of l/d, whose constants are determined by the previously mentioned least-squares procedure, yielding the following expression: F,(l/d)

= 1.12[1-0.1465(1/d) - 5.128(,/d)‘].

(8)

Equation (7), using expression (8), generates Krp for sparse crack arrays with a maximum absolute error of 5%, and an average error of -0.88% with a standard deviation of 1.6%. As the crack array becomes sparser (I/d -+ 0), F,(I/d) approaches the value of 1.12, the well-known solution for a single edge crack, as could have been anticipated. For geometrical configurations with l/d < 0.085, eq. (7) can, without any loss of accuracy, be further simplified, to: Kip = l.lZa,JZ

(9)

APPROXIMATE FORMULAE FOR K,” Most results for &, the negative SIF due to the compressive residual stress field induced by autofrettage, are for a cylinder with a diameter ratio of w = 2, which underwent an L = 100% overstrain. About 135 values of &, are available for this case, obtained by Pu and coworkers [7, 131, Parker et al. [14, IS], and Per1 and Aront [9, 10, 161. Per1 and Arone [lo, 161 have also evaluated & for a partially autofrettaged cylinder of the same geometry. About 40 values of & are available for a cylinder with an c = 60% overstrain, and 33 values for the case of c = 30%. These over 200 K,,_, values are for crack arrays of n = l-1024 cracks of length I/a = 0.001-0.25. Per1 and AronC [lo] have shown that KIA , for all levels of autofrettage (30-lOO%), exhibits a similar dependence on the crack length, I, as Kip; namely, that while in some cases K,,, is highly dependent on I, in other cases & is virtually crack length independent. Furthermore, in a limited study of the inter-crack stress field for a fully autofrettaged cylinder, Arone and Per1 [ 161have found that the ICSF behaves similarly as in the Z& case. Thus, based on these observations, it would be reasonable to try to sort all available %A values using I/d (the inter-crack aspect ratio) as the classifying parameter.

F*(l/d)= 1.12[l -0.1485(1/d).S.l28(1/d)']

0.6 4 0

I 0.02

0.04

0.66

0.06

0.1

0.12

0.14

0.16

0.16

0.2

I/d - inter-crack aspect ratio

Fig. 3. The function &Q/d); the solid line represents the approximating function of eq. (8).

730

M. PERL

Using eqs (1) and (2) to sort all &, values for all three levels of autofrettage proved to be successful. It also revealed the fact that all possible geometrical configurations of uniform arrays of inner-radial cracks in a cylinder loaded by the residual stress field due to autofrettage are either dense or sparse crack arrays in the same sense as in the case of K,P as discussed in the previous section. K,, for dense crack arrays The available K,A data for dense crack arrays include 113 values for 6 = 100% overstrain, 36 values for t = 60% and 29 values oft = 30%. Based on the similarity between the present and the previous case, the following expression is suggested for K,A: Ku = F3 (n )af m,

(10)

where o, is the effective crack tip residual hoop stress (see Appendix B). Numerical values for Fj (n) are calculated for the 178 values of K,A to find that E;(n) is practically constant and equals: (11) Substituting this value into eq. (10) yields: I2 K,A = a, : 0

(12)

Equation (12), which is valid for dense crack arrays for all three levels of autofrettage,

reproduces

K,A values with a maximum absolute error of 4.8% for t = lOO%, and 4.1% for c = 6.0%. As for the case of L = 30%, most values of K,A are reproduced with a maximum absolute value of 6% except for two particular cases, i.e. n = 20 and l/a = 0.1, 0.15, for which eq. (12) yields results with

9.9% and 14.6% error respectively. No explanation

has yet been found for this deviation.

K,, for sparse crack arrays An appropriate

expression to approximate

K,A for sparse crack arrays would be:

K,A = F4(l/d)af&.

(13)

Numerical values for F,(l/d) are evaluated for all relevant K,A including 22 values for t = 100%. four values for c = 60%, and four values for t = 30%. Using the previously discussed least-squares procedure, it is found that all these numerical values of F,(l/d) can be approximated by the same second-order polynomial of i/d as for the respective case of K,P, i.e. F4(l/d) = F,(Z/d) = 1.12[1 -0.1465(1/d)

- 5.128(1/d)2].

(14)

Equation (13) generates KIA values for sparse crack arrays with a maximum absolute error of 5.2% for L = lOO%, 2.4% for t = 60%, and 2.6% for t = 30%. It is worthwhile noting that these results are based on a very small sample of KjA for the cases of t = 30% and 60%. CONCLUDING

REMARKS

The present analysis has demonstrated that the particular geometrical configuration of the cracked cylinder, as expressed by the inter-crack aspect ratio, l/d, determines two distinct categories of crack arrays, designated as sparse and dense crack arrays. Both stress intensity factors, K,P and K,A, are found to be directly proportional to the crack length, 1, in the sparse case, and primarily dependent on the inter-crack spacing, d, in the dense case. Based on this classification, approximate formulae for K,P and KIA for both sparse and dense crack arrays were derived. These simple expressions, which are of very good engineering accuracy, can be conveniently applied to fracture and fatigue evaluations of thick-walled, autofrettaged, pressurized cylinders. In case more data concerning & and K,* for uniform radial crack arrays become available, the approximate formulae for the classifying parameter, l/d, as well as those for KIP and KIA, can be adjusted to accommodate these data.

Stress intensity factor approximate formulae Acknowledgement-The computational phase.

731

author wishes to express his deep gratitude to his student, Mr H. Hamerov, for his help in the

REFERENCES [l] 0. L. Bowie and C. E. Freese, Elastic analysis for a radial crack in a circular ring. Engng Fracture Mech. 4,315322 (1972). [2] J, Tweed and D. P. Rooke, The stress intensity factor for a crack in a symmetric array originating at a circular hole in an infinite elastic solid. Znf. J. Enmra Sci. 13, 653-661 (1975). [3] A. F. Grandt, Jr., Two dimensional &&s intensity factor solution for radially cracked rings. AMFL-TR-75-121, Air Force Material Laboratory (1975). [4] B. D. Goldthorpe, Fatigue and fracture of thick walled cylinders and gun barrels, in Case Studies in Fruerure technics (Edited by T. P. Rich and D. J. Carrot). Army Materials and Mechanics Research Center, TR MS-77-5 (1977). [S] F. I. Barrata, Stress intensity factors for internal multiple cracks in thick-walled cylinder stressed by internal pressure using load relief factors. Engng Fracture Mech. 10, 691-697 (1978). [6] S. L. Pu and M. A. Hussain, Stress intensity factor for a circular ring with uniform array of radial cracks using cubic isoparametric singular elements. Fracture Mechanics, ASTM STP 677, 685-699 (1979). (71 S. L. Pu and M. A. Hussain, Stress intensity factors for radial cracks in a partially autofrettaged thick-walled cylinder. Fracture Mechanics: Fourteenth Sym~s~um-V~i. 1. Theory and Anatysir, ASTM STP 791, 1-194-1-215 (1983). [8] D. P. Kendall, Stress intensity factors for pressurized tuck-wall cylinders. Inf. J. Fracture 30, R17-R19 (1986). (91 M. Per1 and R. Arone, Stress intensity factors for large arrays of radial cracks in thick-walled steel cylinders. Engng Fracture Mech. 25, 150-156 (1986). [lo] M. Per1 and R. AronC, Stress intensity factors for a radially multicracked partially-autofrettaged pressurized thick-walled cylinder. J. Press. Vess. Technol. 110, 200-207 (1988). [l I] M. Perl, K. H. Wu and R. Arone, Uniform arrays of unequal-depth cracks in thick-walled cylindrical pressure vessels, Part I-Stress intensity factors’ evaluation. J. Press. Vess. Technol. 114, 340-345 (1990). [12] K. H. Wu, M. Per1 and R. Arone, Uniform arrays of un~ual-depth cracks in thick-walled cylindrical pressure vessels, Part II-The inter-crack stress field. J. Press. Vess. Tech&. 114, 346-352 (1990). [13] S. L. Pu and P. C. T. Chen, Stress intensity factors for radial cracks in pre-stressed, thick-walled cylinder of strain-hardening materials. J. Press. Vess. Technol. 105, 117-123 (1983). [14] A. P. Parker and J. R. Farrow, Stress intensity factors for multiple radial cracks emanating from the bore of an autofrettaged or thermally stressed thick cylinder. Engng Fracture Mech. 14, 237-241 (1981). [IS] A. P. Parker, J. H. Underwood, J. F. Throop and C. P. Andrasic, Stress intensity and fatigue crack growth in pressurized autofrettaged thick cylinder. Fracture Mechanics: Fourteenth Symposiwp-Vol. 1: Theory and Analysis, ASTM STP 791, f-216-1-237 (1983). fl6] R. Ato& and M. Perl, Influence of autofrettage on the stress intensity factors for a thick-walled cylinder with radial cracks of unequal lengths. Znt. J. Fracture 39, R29-R34 (1989). [17] L. P. Pook, Stress intensity factor expressions for regular crack arrays in pressurized cylinders. Fatigue Fracture Engng Mater. Structures 13, 135-143 (1990). [18] J. P. Benthem and W. T. Koiter, Asymptotic approximations to crack problems, in Methods of Analysisand Solutions of Crack Problems (Edited by G. C. Sih), pp. 131-170. Noordhoff, The Netherlands (1972). [19] C. H. Wu, Unconventional internal crack, Part II-Methods of generating simple crack. J. appl Mech. 49, 383-388 (1982) 1201 R. A. Westman, Pressurized star crack. J. math. Phys. 43, 191-198 (1964).

APPENDIX

A: THE NORMALIZING

STRESS

FOR &,a

The hoop stress at the bore of a pressurized cylinder is @en by: a,(r=a)=p-

wZ+l wz-1

where p is the internal pressure and w is the cylinder diameter ratio, b/u. As we assume that the internal pressure fully penetrates the crack cavity and acts on its faces, it is convenient to define an effective hoop stress at the bore as:

where ga is used to normalize KLpfor sparse crack arrays [see eq. (7)]. For dense crack arrays, Pook [17] has suggested the use of the effective hoop stress at the crack tip, u,. rather than u,. a, is evaluated by eq. (A2) using an apparent diameter ratio w’ defined as: (A3) where I is the relevant crack length, and l/a is its normaliid u,‘,=

zp

w2

(w?2=2=

value. Substituting this value into eq. (AZ) yields:

w2

w* - (1 + //a)2

where u, is used to normalize KEpfor dense crack arrays [see eq. (3)].

132

M. PERL

APPENDIX B: THE NORMALIZING The residual

hoop

stress at the bore of a partially

STRESS FOR &

or fully autofrettaged

*II ) c7,jncr= a) = -;4w21n ,.:3(w- - 1) i

cylinder

(1 ;

is given by [IO]:

(Bl)

+ 2[(P)Z - I] I

where err,is the yield stress of the cylinder’s material and p is the normalized radius of the autofrettaged interface, ~/a (see Fig. 1 and ref. [lo]). Following the same rationale as in the previous case, a residual hoop stress at the crack tip, CT,,is suggested for the normalization of K,A. of is evaluated by eq. (Bl) using an apparent diameter ratio, w’, defined as: h M.‘= _. u-t1 as well as an apparent

normalized

radius

of the autofrettaged

= __“_ I +lla

(B2)

interface

defined

as:

‘“‘-=ilfi=-C--.

(B3)

1 -+1/a Substituting

eqs (B2) and (B3) into (Bl) yields: (B4)

where o, is used to normalize

K,A for both

sparse

and dense crack

(Received 5 December

arrays 1991)

[see eqs (10) and (13)].