Weight functions and stress intensity factors for internal surface semi-elliptical crack in thick-walled cylinder

Pergamon

EngineeringFractureMechanicsVol. 58, No. 3, pp. 207-221, 1997 PII: S0013-794,1(97)00083-0

© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0013-7944/97 $17.00 + 0.00

WEIGHT FUNCTIONS A N D STRESS INTENSITY FACTORS FOR INTERNAL SURFACE SEMI-ELLIPTICAL CRACK IN THICK-WALLED CYLINDER X. J. ZHENG, A. KICIAK and G. GLINKA University of Waterloo, Department of Mechanical Engineering, Waterloo, Ont. N2L 3G1, Canada Abstract--Calculation of stress intensity factors for a crack subjected to a complex stress distribution can be highly facilitated by using the weight function method. The method separates influences of a stress field and the geometry of a cracked body on a stress intensity factor. In this paper, mode I weight functions were derived for the deepest and surface points of an internal, radial-longitudinal, surface, semi-elliptical crack in an open-ended, thick-walled cylinder with internal radius to wall thickness ratio Rift = 2.0. Generalized weight function expressions for deepest and surface points of the crack were utilized. A method of two reference stress intensity factors was applied to determine coefficients of the weight functions. The weight functions were validated for several crack face stress fields against finite element data. Closed-form relations for calculation of stress intensity factors were obtained for a variety of one-dimensional stress distributions applied to crack faces. The paper complements a set of previously published weight function solutions for cracks in cylinders with other radius to thickness ratios. © 1997 Elsevier Science Ltd Keywords---surface crack, stress intensity factor, weight function, thick-walled cylinder.

NOMENCLATURE a

Ai ¢

Ci Di Fo, Ft K~, KB K~,K~

Kg, K~ mA(x,a), mB(x,a) AliA, M~B n

P Q Ri

t X

Yo, Y~ a(x) ¢70

depth of surface semi-elliptical crack coefficients of fit to geometry correction factor, Y0 (i = 0,1,2,3) coefficients of fit to geometry correction factor, Yl (i = 0,1,2,3) half-length of surface semi-elliptical crack coefficients of fit to geometry correction factor, F0 (i = 0,1,2,3) coefficients of fit to geometry correction factor, FI (i = 0,1,2,3) geometry correction factor for surface point, B, of semi-elliptical crack subjected to uniform or linearly increasing stress field applied to crack faces, respectively stress intensity factor at deepest, .4, or surface, B, point of semi-elliptical crack stress intensity factor at deepest point, A, of semi-elliptical crack for uniform or linearly increasing stress field applied to crack faces, respectively stress intensity factor at surface point, B, of semi-elliptical crack for uniform or linearly increasing stress field applied to crack faces, respectively weight function for deepest, A, or surface, B, point of semi-elliptical crack, respectively coefficients of weight function for deepest, .4, or surface, B, point of semi-elliptical crack, respectively (i = 1,2,3) exponent of power-law stress distribution applied to crack faces along crack depth direction pressure in cylinder elliptical crack front shape factor (square of complete elliptic integral of second kind) internal radius of cylinder wall thickness of cylinder coordinate along crack depth geometry correction factor for deepest point, A, of semi-elliptical crack subjected to uniform or linearly increasing stress field applied to crack faces, respectively stress distribution applied along crack depth reference stress or maximum value of tr(x).

1. INTRODUCTION INTERNAL, SURFACE, semi-elliptical cracks are occasionally found in pressure vessels and pipes

during service or manufacturing. Subsequent fatigue and fracture analyses of such cracks require determination of stress intensity factors for a wide range of encountered crack shapes and sizes. Although several stress intensity factor handbooks were published, e.g. refs [1,2], the available solutions are not always adequate for particular engineering applications. This is especially true 207

208

X.J. ZHENG et al.

for cracks subjected to non-uniform, e.g. thermal or residual, stress fields. In such cases, the weight function approach [3, 4] is particularly useful. A unique feature of the weight function method is that it separates influences of a stress field and the geometry of a cracked body on a stress intensity factor. Once a weight function for a particular cracked body has been determined, a stress intensity factor for any loading applied to that body can be obtained by integrating the product of the loading and the weight function. The superposition principle allows us to simplify calculations, since the stress intensity factor can be computed from a stress distribution resulting from the loading applied to the uncracked body, normal to the plane of a prospective crack and acting on crack faces. Although weight functions for many geometries were determined[l, 5], only a few solutions for surface cracks exist. This paper presents mode I weight functions and a method of calculating stress intensity factors for an internal, radial-longitudinal, surface, semi-elliptical crack in an open-ended, thick-walled cylinder with an internal radius to wall thickness ratio Rift = 2.0, subjected to one-dimensional stress distributions. Both deepest and surface points of the crack were considered. A method of two reference stress intensity factor solutions [6] was used to determine coefficients of the weight functions. The derived weight functions were validated using independent stress intensity factor data from the literature. Present results complement a set of previously obtained weight functions for internal and external surface cracks in thick-walled cylinders with various diameter ratios [7-11]. More general solutions for a wide range of radius to thickness ratios can also be obtained using the present method, provided that a more complete set of reference data becomes available.

2. DERIVATION OF WEIGHT FUNCTIONS Shen and Glinka [6, 12] found generalized forms of mode I weight functions for the deepest point, A, eq. (1), and the surface point, B, eq. (2), of a surface, semi-elliptical crack in a flat plate. These expressions were subsequently used to determine weight functions and stress intensity factors for semi-elliptical cracks in plates, thin- and thick-walled cylinders[7-11, 13-15].

2 ..\1/2 mA(x,a)--/2n(a_x)[l+M,n(1--a) +M2A(I X)+M3n(l_X)3'2],

m

(x,a) =

1/2

,-

(1)

) j.

/" X "~3/2"]

In this paper, eqs (1) and (2) were used to derive the weight functions for the case of an internal surface crack in a thick-walled cylinder, see Fig. 1. It should be noted that the weight functions, eqs (1) and (2), were derived for a unit stress, go = 1, distributed uniformly along a line perpendicular to the minor semi-axis of the ellipse and intersecting the crack front, as shown in Fig. 1. In the case of a continuously distributed stress field, the stress intensity factor at the deepest point, A, or at the surface point, B, can be determined by integrating the product of the weight function, mA(x,a) or mB(x,a), respectively, and the stress distribution, a(x), varying in the crack depth direction only, and normal to the plane of a prospective crack.

KA =

J2a(x)mA(x,a) dx,

(3a)

Ks= I i a (x)ms(x,a ) d x.

(3b)

In order to calculate stress intensity factors by using eq. (3a) or eq. (3b), the parameters,

Alia or MiB (i = 1,2,3), of the weight functions, mA(x,a) or ms(x,a), respectively, have to be determined, each from three independent equations. They can be derived [6] from two reference stress intensity factor solutions and from the properties of weight functions.

Weight functions and stress intensity factors

\

209

\

I

~A,..~ I / / / J

Fig. 1. Internal, radial-longitudinal, surface, semi-elliptical crack in a cylinder.

In the present case of a crack in a thick-walled cylinder, finite element results[16] for uniform and linearly increasing stress distributions applied to crack faces, Fig. 2, were used as the reference data. Stress intensity factors for these distributions were given in the form of geometry correction factors, Y0 and Yl or F0 and F1, for the deepest or the surface point, respectively. (a) For the deepest point, A:

t

~(x)

°°/

I

(a)

I:I

x

(b)

Fig. 2. Reference stress distributions for derivation of weight functions: (a) uniform stress field; (b) linear stress field.

X.J. ZHENGetaL

210

• uniform stress field, tr(x) = a0 (0 < x < a) K~ = a0

(4a)

Yo;

• linearly increasing stress field, a(x) = ao(x/a) (0 < x < a) K A __ ao

Yl.

(4b)

K ~ = tro~/--~ 0;

(5a)

(b) For the surface point, B: • uniform stress field, a(x) = ao (0 < x < a)

• linearly increasing stress field, a(x) = ao(x/a) (0 < x < a) K ~ = ao

(5b)

F1.

Herein, Q is the elliptical crack shape factor.

(a)165 ( Q=1+1.464

) 0_
c

(6)

The geometry correction factors, Y0, Y1, F0 and Fl, were given[16] for relative crack depths, 0 < a/t < 1.0, and crack aspect ratios, 0.2 < a/c < 1.0. It should be noted that the results for a/t = 0 and 1.0 were obtained in ref.[16] by using the Hermite extrapolation, see also ref.[13].

2.1. Weight function for the deepest point, A Substitution of eqs (1) and(4a)-(b) into eq. (3a) results in two linearly independent relations, eqs (7a)--(b), containing the parameters MiA (i = 1,2,3).

KA=tro~oYo=JotTO/27z~a_x)ana

2

[I_k_M1A(I_~)I/2wM2A(1 x).k_M3A(1 x)3/2]dx, (7a)

aax

2

)1'2+ 2A(1

)3'2]dX (7b)

The third equation necessary to find the parameters Mia was derived[12, 17] by requiring that the curvature of a crack opening profile vanishes at the mouth of a surface crack (x = 0). As a result, also a second derivative of the weight function vanishes there. 02mA(x,a)

OZx

x=0 ~---0.

(8)

By solving eqs (7a)-(b) and (8), the following set of relations between the parameters MiA and the geometry correction factors, Y0 and Y1, was obtained.

Weight functions and stress intensity factors

211

2n 24 MIA = ~/z~d~(-Y° + 3Yl) - -~-,

(9a)

M2A = 3,

(9b)

M3A = ~

6n

8 (Y0 - 2 Y1) + ~.

(9c)

Numerical values of the correction factors, Y0 and Y1, given by Mettu et al. [16] for crack aspect ratios, 0.2 < a/c _< 1.0, were supplemented with edge crack data, a/c = 0, by Andrasic and Parker [18]. In order to enable calculation of the parameters M;A for a variety of crack depths and aspect ratios, all the reference data was fitted with closed-form expressions. For a uniform stress distribution, eq. (4a), Fig. 2(a): Y0 = A0 + Al ( t )

+ A2 ( t ) 2 + A3 ( at) 4

(10)

where Ao = O.07e[-5°Sl(a/c)] + 1.044,

(10a)

A1 = 0.665e [-3"393(a/c)1 - 0.433,

(lOb)

A2 = 1.161e [-3386(a/c)] + 0.711,

(10c)

A3 = 1.46e [-4A65(a/c)] - O. 179.

(lOd)

(/) (~)2 (/)4

For a linear stress distribution, eq. (4b), Fig. 2(b):

q-- B3

(11)

Bo = -2.16e [-°'°35(a/c)] + 2.825,

(lla)

B1 = 0.265e [-5"574(a/c)] - 0.225,

(llb)

B2 = 0.753e [-4"°25(a/c)1 + 0.307,

(llc)

B3 = -1.284e [°'°79(a/c)] -¢- 1.398.

(I ld)

Y1 = Bo + BI

--I--B2

where

Equations (10) and (10a)-(d) and eqs (11) and (1 la)-(d) approximate the numerical data[16] with accuracy better than 1.5% for crack aspect ratios, 0 < a/c < 1.0, and relative crack depths, 0 < a/t < 0.8. However, the accuracy of the foregoing relations could not be verified for aspect ratios, 0 < a/c < 0.2, due to lack of reference data. Therefore, it can only be assured for aspect ratios, 0.2 < a/c < 1.0 and a/c = 0, at present. 2.2. Weight function for the surface point, B A weight function, mn(x,a), for the surface point B can also be determined from two reference stress intensity factors and one additional condition [12]. Substitution of eqs (2) and(5a)(b) into eq. (3b) results in a set of two equations, eqs (12a)-(b), containing the parameters of the weight function MiB (i = 1,2,3). Kff=a0

=

a 0 -f f- - ~

1 + M1B~a )

+ M2s

+ M3o~a )

J dx,

(12a)

X. J. ZHENGet al.

212 If:G0

F1 :

0"0

a ~

J dx.

--{-m3B~a )

I+

(12b)

The third equation necessary to determine the parameters M m was derived[12] by assuming that the weight function m s ( x , a ) vanishes at the deepest point of the crack (x = a). (13)

0 = 1 -t- M1B + M2B + M3B.

By solving eqs (12a)-(b) and(13), the parameters M m were found as functions of the reference geometry correction factors, F0 and F1.

37T

M1, = ~ ( 2 F o 4~

- 5F1) - 8,

(14a)

15n

MZB = ~ ( - F o q- 3El) -k-15,

(14b)

M3B = 3n,-~(3Fo - 10F1) - 8. 4~d

(14c)

Since in the limiting case of an edge crack, the stress intensity factor vanishes at the surface point B, geometry correction factors, F0 = F~ = 0 for a/c = 0, were added to the reference data [16]. The numerical values of the geometry correction factors, F0 and F~, were then fitted with closed-form expressions, eqs (15) and (15a)-(d) and eqs (16) and (16a)-(d), for 0 < a/c < 1.0. For a uniform stress distribution, eq. (5a), Fig. 2(a):

=

c3( -]41( a]

+

(15)

where Co = 5.163e [-5061(a/c)+l'568(a/c)2] + 0.972,

(15a)

C1 = - 10.239e [-46"053(a/c)-4"°°9(a/c)~] - O. 199,

(15b)

C2 ---- 8.784e [-4"081(a/c)+l'°92(a/c)~] 9-0.119,

(15c)

C3 = 28.13 3e[-9'959(a/c)-9"817(a/e)2l __O. 104.

(15d)

For a linear stress distribution, eq. (5b), Fig. 2(b):

= [oo+o (:)

a)

(16)

where D o = 1.033 - 4 . 8 4 2 ( a ) + 9 . 7 0 8 ( a ) 2 - 8 . 3 9 7 ( a ) 3 + 2 . 6 9 0 ( a ) 4 ,

D1 -------3.448 -4- 24.231

(_~)

(~)2 -- 50.221

(a)3 + 42.498 C

(16a)

(0)4 -- 13.099 C

'

(16b)

Weightfunctionsand stressintensityfactors D2 = 6 . 5 3 5 - 3 0 . 6 2 2 ( a )

+45.644(a)2-25.05(a)3+3.636(a)4,

(a) D3 = 2.243 - 21.677

213

(a) 2 + 65.546 c

(a) 3 - 76.555

(16c)

(a) 4. + 30.433

(16d)

The accuracy of the foregoing approximations is better than 1.3% within the ranges of aspect ratios, 0_< a/c<_ 1.0, and relative crack depths, 0 <_a/t <_0.8. However, the accuracy could not be checked for the aspect ratios, 0 < a/c < 0.2, due to a shortage of reference data.

3. VALIDATION In order to verify the accuracy of the derived weight functions, mA(x,a) or mB(x,a), stress intensity factors, KA and Ks, were calculated at points A and B of a semi-elliptical crack, Fig. 1, and compared to available reference data.

3. I. Stress intensity factors for power-law crack face loading Stress intensity factors for several power-law stress distributions applied to crack surfaces, eq. (17), were calculated at the deepest or the surface crack point by using eq. (3a) or eq. (3b).

The exponent of the stress distribution, eq. (17), was set to n = 0; 1/2; 1; 3/2; 2; 3, so that the results could be compared to the numerical data from ref. [16]. 3.1.1. Stress intensity factors for the deepest point, A. Stress intensity factors for the deepest point, A, were calculated by integrating the product of the weight function, ma(x,a), and the stress distribution, a(x). Closed forms of expressions (1) and (17) enabled us to derive closedform relations for the stress intensity factors: • uniform stress distribution, a(x) = a0

KA

v/'2--Q 2+MlA +-~M2A +~M3A

• square root stress distribution, a(x) = ffo(x/a) 1/2 a O ~ / - ~ -~ -

+

MIA + ~ MzA +

= Yo,

4)

M3A ,

(18)

(19)

• linear stress distribution, a(x) = ao(x/a)

X,~

0 . 0 ~

•

--

1 ~2-Q 4 1 A + ~---~M2A+-~M3A) -£ (5+~M~ =Y1,

power-law stress distribution, a(x)

=

¢70(x/a)3/2

4)

KA V"2-0{ 3~ 2 ~04~-h7-0- ~ ~--ff-+gMIA q- "~ M2A + -~ M3A ,

(20)

(21)

214

•

X.J. ZHENG

quadratic stress distribution,

a(x)

=

ao(x/a) 2

x/"~Q(16

KA

et al.

1

)

(22)

• cubic stress distribution, ~r(x) = ~o(x/a) 3 KA ~70~

--

Vc2--Q(32 1 3~5 1 ) 7~ "~'~+-~ MIA + M2A + M3A •

(23)

Stress intensity factors for the uniform and linear stress distributions were used in the derivation of the weight function, mA(x,a), as reference cases. Therefore, both sides of eqs (18) and (20) should be equal to I10 and Yl, respectively. 3.1.2. Stress intensity factors for the surface point, B. Stress intensity factors for the surface point, B, were calculated by integrating the product of the weight function, ms(x,a), eq. (2), and the stress distribution, a(x), eq. (17). The stress intensity factors were written as closed-form expressions of the parameters Mm: • uniform stress distribution, a(x) = Oo Ks

170~/-~-"~

4 + 2Mla + -~M2B + M3B = Fo,

--

• square root stress distribution, o(x) o'0~/-x-~

(24)

crO(x/a)1/2

=

--

(4

zt

,

(25)

=F1,

(26)

- ~ M l s + M2B

• linear stress distribution, a(x) = ao(x/a) Ks ~/-Q(4 4 2 ) ao~/-~-dT-~ - x "~+ M l s + -~M2s + "~M3s

•

power-law stress distribution, a(x) = So(x/a) 3/2 Ks

o'0~- ~

-

~

117

( 1 + ~4M I B W - ~2M 2 B + - ~ M43 s , )

(27)

• quadratic stress distribution, a(x) = ao(x/a) 2 Ks

~0,/-¢~-

Vq-Q(4

2

4

1

)

,

(28)

1 ~.] =...~_ ( ~4 + ~Mis + 4~M2s + ~M3s" 2

(29)

~-

~+sM~s+~M28+2M3B

• cubic stress distribution, a(x) = ~ro(x/a) 3 a oKs~

As before, since the stress intensity factors for the uniform and linear stress distributions were used as reference cases in the derivation of the weight function, ms(x,a), both sides of eqs (24) and (26) should be equal to F0 and F1, respectively.

Weight functions and stress intensity factors

215

3.2. Comparison with the finite element data A comparison of stress intensity factors at the deepest point, A, for the six stress distributions, eq. (17), calculated using eqs (18)-(23), to the finite element data[16] is shown in Fig. 3. (a) 3.0--

Rilt = 2,0

/n=O

ale :0'2weisht function 2.5

~ Ifiniteeleraentdata

/

2.0

1.5 <

1.0

0.5

0.0 0.1

0.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Crack depth a/t

(b) 1.8__

Rilt = 2.0 alc = i.O

1.6

weisht function

1.4

8

1.2

finite element data ref. [16] n=0

/ X

.............~ n= 1/2

|.0

<

O.S

X'-'------.__

b. ,,

0.6

~

~ n = 2 .

0

0

~

n

=

3

[3 0.4

0.2 0.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

O.S

0.9

1,0

Crack depth a/t Fig. 3. Comparison of stress intensity factors for the deepest crack point, A, calculated by using the weight function, mA(x,a), eq. (1), with the finite element data[16]: (a) crack aspect ratio a/c = 0.2; (b) crack aspect ratio a / c = 1.0.

x.J. ZHENG et al.

216

Results are presented for two crack aspect ratios only, a/e = 0.2 and 1.0. Similar agreement was obtained for other aspect ratios within the range, 0.2
3.3. Stress intensity factors for the Lamd stress distribution The weight functions, mA(x,a) and mn(x,a), were used to calculate stress intensity factors for a crack in a pressurized cylinder, subjected to the Lam6 tangential stress distribution with pressure acting on crack faces, see Fig. 5. In this case, the stress distribution applied to the crack surfaces was given by

{

R2[

(Ri+t')2] ]

a(x) = p 1 + (2Ri + t)t 1 + \ ~ ]

J •

(30)

By integrating the product of the weight function, eq. (1) or eq. (2), and the stress distribution, eq. (30), closed-form solutions for the stress intensity factor at point A or B, respectively, were derived. For the deepest point, A

KA -- VP2-~(AM1A + BMEA + CM3A + D) p ,/YdT-Q

(31)

where

F ~-E, (Ri/t)u

A- - -

Fw F ln(Ri/t) F 2 B = - 2(a/t)3/2ul/2 q- 2(a/t)3/2ul/2 d- (a/t)(Ri/t) + ~E,

(33)

FIn u

CD -

F F In(Rift) 1 ------~ (a/t) + (a/t)(Ri/t) -t- (a/t)2 I- ~ E,

(32)

Fw F ~ 2u3/2(a/t) 1/2 (Ri/t)u E = 1+

F ln( Ri/ t) + 2E, 2(a/t)l/2u3/2

(34)

(35)

(Ri/t)2 2(Ri/t) + 1'

(36)

F - (Ri/t)2(Ri/t + 1)2 2(Ri/t) + 1

(37)

w = ln[u + (a/t) + 2ul/2(a/t)l/2],

(38)

u=(Ri/t)+(a/t).

(39)

Weight functions and stress intensity factors

217

(a) 2.00

gilt = 2.0 edc= 0.2

1,75

weight function

]

1.50

t-q

1.25

n=O

8

-

finite element data ref, [16]

1.00 m

n=i/2

0.75

0.50

-

n=l r.=312

0.25

n.-~.2 n--3

0.00 0.0

0,1

0.3

0.2

0.4

0,5

0.7

0.6

0.8

0.9

!.0

Crackdcptha/t

(b) 2.0--

gilt = 2.0 ale = 1.0

1.8-

--

1.6-

weight

function

I finite element data ref, [161

1.4

y

O

n=O

1.2' v

1.0 ~ m

0.8 0.6 n=l12

0.4 ~

/t

:r

0.2 2[

~

.... .

nzl

:t "

0.0 ~

I

0,0

""I O.t

0.2

'

i: 0,3

~

t.,,.it

i ' } ' 0.4 0,5

I 0.6

'

I 0,7

'

I

0.6

'

I 0.9

"

I

1.0

Crack depth a/t Fig. 4. Comparison of stress intensity factors for the surface point, B, calculated by using the weight function, ms(x,a), eq. (2), with the finite element data[16]: (a) crack aspect ratio a/c -- 0.2; (b) crack aspect ratio a/c ~ 1.0.

E F M 5S/3--C

X.J. ZHENG et al.

218

t

aL(x)

~,(x) X

(

p

(a)

X

(b)

Fig. 5. Stress distribution in a thick-walledcylinder due to internal pressure p: (a) Lam6 tangential stress distribution; (b) crack face loadingin a pressurizedcylinder. For the surface point, B

KB -- 2Wr-Q(AMlB + pC-YffUQ

GM2B + HM3B + I )

(40)

where

G=

F arctan[(a/Ri) 1/2] (a/t)3/2(Ri/t)l/2

F H -

I =

(a/t)-----u+

F 2 (a/t)u I-~E,

Fin(1 +a/Ri)

E

(a/t) z

+ -~,

F arctan[(a/Ri) 1/2] F (a/t)l/2(Ri/t)3/2 + ~ +

2E.

(41)

(42)

(43)

Comparison of stress intensity factors calculated by using eqs (31)-(43) with available data from the literature [19-23] is shown in Fig. 6. For the deepest point, A, the agreement of the weight function results with the reference data is good for relative crack depths, a/t < 0.6. However, differences are larger for deeper cracks, especially for lower crack aspect ratios, a/c. For the surface point, B, stress intensity factors calculated by using the weight function, mB(x,a), are within the range of the reference data. The discrepancies between the weight function predictions and the literature data may be partly attributed to differences between models used to derive the reference data. The accuracy of stress intensity factors calculated by using e.g. the finite element method is influenced by imposed boundary conditions, the size of a finite element model, mesh refinement near the crack tip, etc. For example, Atluri and Kathiresan [19] modeled a quarter of the cylinder with relatively few elements using a half-cylinder length to external radius ratio close to 3, while Mettu et al. [16] used many more elements, and generally higher half-cylinder length to radius ratios resulting from convergence studies, see also ref. [24]. It should also be noted that the stress intensity factor at the surface point, B, represents only an average quantity associated with a near-

Weight functions and stress intensity factors

219

(a) 10 q

Rilt = 2.0

9 weight

" 7 --

function

r e f [19] " Q a/c = 0.6 ref. [201 Oa/c=0.2 }

• a/c=025 I " i • a/c = 0.5 I . . . . . O a / c = 0 . 7 5 1 ret'tzal

0 a/c=0.6

0 a/c-- 1.0 )

•t ' a / c : 0 2

8--

•

]rer.[21]

6--

a/c = o.8 I & a/c=0,2 I @a/c=0,6 }ref.[22[

5 --

[] a/c=0.981

.

alc = 0 2 /

/

"=/

o

J

•

/

_

/

el.

4--

0.8

3

1.0

2 n

1

0

' 0.0

I

'

I

'

0.2

0. !

I

'

0.3

I

'

0.4

I

'

0.5

I

'

0.6

I

'

0.7

I

'

I

0.8

'

0.9

I i .0

a/t

(b) 8 --

Rilt=2.0

7

~

weight f u n c t i o n a/c = 0.2 [19] a l e = 0.6 1201

• a/c = 0.25 I

A a/c=0.2 i

. . Oa/c=0.75 / reLtz'l

0 ale = 0.6 / [221

• a/c = 1.0 /

•

6 5

el

[] a/c

e~

I

•

a/c = 0.5

.

I

.

.

. /

/

/

= 0.98

i

/

I~1

/

/

,

"

/

0,8

06 ° ~_o.4 " . _ . _ ~ _ ~ ~

3

~c = 0.2 ~

'

0.0

I

0. I

'

:

•

~

I

0.2

'

I

0.3

'

I

0.4

'

I

0,5

'

I

0.6

i

I

0.7

'

I

0.8

'

I

0.9

'

I

1.0

a/t

Fig. 6. Comparison of stress intensity factors calculated using weight functions with literature data for a surface crack in a pressurized thick-walled cylinder: (a) for the deepest point, A; (b) for the surface point, B.

surface region o f a c r a c k front[12]. M o d e l l i n g a crack-tip singularity there requires special attention a n d the results are p r o n e to large variations. In cases o f m o r e c o m p l e x stress distributions, c l o s e d - f o r m stress intensity factor solutions m a y be unattainable. Hence, a simple numerical technique for integration o f eqs (3a)-(b) was d e v e l o p e d [25]. T h e m e t h o d can be used with a n y stress distribution applied to the c r a c k faces.

220

X.J. ZHENG et al.

4. CONCLUSIONS Stress intensity factors for an internal, radial-longitudinal, surface, semi-elliptical crack in an open-ended, thick-walled cylinder, with an internal radius to wall thickness ratio Ri/t = 2.0, were derived by using the weight function approach. Mode I weight functions for the deepest and surface points of the crack were obtained by using generalized weight function expressions [12], and a method of two reference stress intensity factors [6]. Parameters of the weight functions were determined from reference stress intensity factors [16] for uniform and linear stress distributions applied to crack faces. Stress intensity factors were then calculated for several power-law stress distributions applied to the crack faces. Good agreement with the reference data [16] was observed over the entire range of crack aspect ratios, 0.2 < a/c < 1.0, and crack depths, 0 _< a/t <_ 0.8. In the case of an internally pressurized cylinder, a comparison of the weight function predictions to several sets of stress intensity factor data showed good agreement for the deepest point, A, of the crack. For the surface point, B, the weight functions predicted values of stress intensity factors within the range of the available data. Differences between results calculated by using the weight functions and the data from the literature were ascribed to the differences between numerical models used to obtain the reference stress intensity factors. The derived weight functions enable rapid estimation of stress intensity factors for cracks subjected to complex stress distributions. This makes them particularly suitable for fatigue crack growth analysis. Acknowledgements--The authors are grateful to Dr. S. R. Mettu and Dr. V. Shivakumar from Lockheed ESC, to Dr. I. S. Raju from Analytical Services and Materials, Inc., and to Dr. R. G. Forman from NASA/JSC for making available their finite element data. Financial support for this research was provided by a grant from Natural Sciences and Engineering Research Council of Canada.

REFERENCES 1. Tada, H., Paris, P. C. and Irwin, (3. R., The Stress Analysis of Cracks Handbook, 2nd edn. Del Research Corp., St. Louis, MO, 1985. 2. Murakami, Y. et al., Stress Intensity Factor Handbook, The Society of Materials Science Japan. Pergamon Press, Oxford, 1989. 3. Bueckner, H. F., A novel principle for the computation of stress intensity factors. Zeit. Angew. Math. Mech., 1970, 50,529-546. 4. Rice, J. R., Some remarks on elastic crack-tip stress field. International Journal of Solids and Structures, 1972, 8,751758. 5. Wu, X. R. and Carlsson, A. J., Weight Functions and Stress Intensity Factor Solutions. Pergamon Press, Oxford, 1991. 6. Shen, G. and Glinka, G., Determination of weight functions from reference stress intensity factors. Theoretical and Applied Fracture Mechanics, 1991, 15,237-245. 7. Shen, G. and Glinka, G., Stress intensity factors for internal edge and semi-elliptical cracks in hollow cylinders. In High Pressure-Codes, Analysis and Applications, ASME PVP, Vol. 263, ed. A. Khare. ASME, 1993, pp. 73-80. 8. Zheng, X. J., Glinka, G. and Dubey, R. N., Calculation of stress intensity factors for semi-elliptical cracks in thickwall cylinders. International Journal of Pressure Vessels and Piping, 1994, 62,249 258. 9. Zheng, X. J. and Glinka, G., Weight functions and stress intensity factors for longitudinal semi-elliptical cracks in thick-wall cylinders. A S M E Journal of Pressure Vessel Technology, 1995, 117,383-389. 10. Wang, X. J. and Lambert, S. B., Stress intensity factors and weight functions for longitudinal semi-elliptical surface cracks in thin pipes. International Journal of Pressure Vessels and Piping, 1996, 65,75-87. 11. Kiciak, A., Glinka, G. and Burns, D. J., Weight functions for an external longitudinal semi-elliptical surface crack in a thick-walled cylinder. A S M E Journal of Pressure Vessel Technology, 1997, 119,74-82. 12. Shen, G. and Glinka, G., Weight functions for a surface semi-elliptical crack in a finite thickness plate. Theoretical and Applied Fracture Mechanics, 1991, 15,247-255. 13. Forman, R. G., Mettu, S. R. and Shivakumar, V., Fracture mechanics evaluation of pressure vessels and pipes in aerospace applications. In Fatigue, Fracture and Risk, ASME PVP, Vol. 241, eds W. H. Bamford et al. ASME, 1992, pp. 25-36. 14. Shen, G., Liebster, T. D. and Glinka, G., Calculation of stress intensity factors for cracks in pipes. In Proceedings of the 12th International Conference on Offshore Mechanics and Arctic Engineering, OMAE-93, Vol. 3B, Materials Engineering, eds M. H. Salama et al. ASME, 1993, pp. 847-854. 15. Raju, I. S., Mettu, S. R. and Shivakumar, V., Stress intensity factor solution for surface cracks in fiat plates subjected to nonuniform stresses. In Fracture Mechanics: Twenty-Fourth Volume, ASTM STP 1207, eds J. D. Landes et al. ASTM, 1994, pp. 560-580. 16. Mettu, S. R., Raju, I. S. and Forman, R. G., Stress intensity factors for part-through surface cracks in hollow cylinders. NASA Technical Report No. JSC 25685, LESC 30124, 1992. 17. Fett, T., Mattheck, C. and Munz, D., On the calculation of crack opening displacement from the stress intensity factor. Engineering Fracture Mechanics, 1987, 27,697-715.

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18. Andrasic, C. P. and Parker, A. P., Dimensionless stress intensity factors for cracked thick cylinder under polynomial crack face loading. Engineering Fracture Mechanics, 1984, 19,187-193. 19. Atluri, S. N. and Kathiresan, K., 3D analyses of surface flaws in thick-walled reactor pressure vessels using displacement-hybrid finite element method. Nuclear Engineering and Design, 1979, 51,163-176. 20. Blackburn, W. A. and Hellen, T. K., Calculation of stress intensity factors for elliptical and semi-elliptical cracks in blocks and cylinders. CEGB Report No. RD/B/13103, U.K., 1974. 21. Kendall, D. P. and Perez, E. H., Comparison of stress intensity factor solutions for thick walled pressure vessels. In High Pressure-Codes, Analysis, and Applications, ASME PVP, Vol. 263, ed. A. Khare. ASME, 1993, pp. 115-119. 22. Kobayashi, A. S., Emery, A. F., Love, W. J. and Jain, A., Further studies on stress intensity factors of semi-elliptical cracks in pressurized cylinders. In Structural Mechanics in Reactor Technology, SMiRT "79, Transactions of the 5th International Conference, eds A. Jaeger and B. A. Boley, Paper G4/1. 23. Guozhong, C., Kangda, Z. and Wu, D., Stress intensity factors for internal semi-elliptical surface cracks in pressurized thick-walled cylinders using the hybrid boundary element method. Engineering Fracture Mechanics, 1995, 52, 1055-1064. 24. Raju, I. S. and Newman, J. C. Jr., Stress intensity factors for internal and external surface cracks in cylindrical vessels. ASME Journal of Pressure Vessel Technology, 1982, 104,293-298. 25. Moftakhar, A. and Glinka, G., Calculation of stress intensity factors by efficient integration of weight functions. Engineering Fracture Mechanics, 1992, 43,749-756.

(Received 8 January 1997, in final form 11 June 1997, accepted 15 June 1997)