Weight function estimation of sif for mode I part-elliptical crack under arbitrary load

Engineering Fracrure Mechanics Vol. 41, No. 5, pp. 659-684. Printed in Great Britain.

WEIGHT MODE

0013-7944/92 s5.00 + 0.00 e 1992 Rrgamon PIUS pk.

1992

FUNCTION ESTIMATION OF SIF FOR I PART-ELLIPTICAL CRACK UNDER ARBITRARY LOAD G. S. WANG

Department of Structures, The Aeronautical Research Institute of Sweden, 161 1I Bromma, Sweden Abatroct-An approximate solution for stress intensity factors (SIFs) of a Mode I partclliptical crack under complex load was developed. The solution is based on a weight method to compute the stress intensity factor for a complex load by a reference SIF for a simple load case. A previously developed crack surface displacement solution (G. S. Wang, Approximate weight functions for RMS-averaged stress intensity factor, submitted to Inr. J. Frucrure] is used in the solution. Both self-consistency examinations and comparisons with available solutions for various crack geometries and loading cases show that errors of the SIFs computed by the solution are within several percent.

1. INTRODUCTION FOR A Mode I crack in a three-dimensional geometry subjected to load system (1) and load system (2) investigations[l-31 show that there exists the relation

where Ku’, II(‘),t”’ and f(l) are stress intensity factor (SIF), displacement, boundary load and body force for load system (1) and Kc2’,I$‘), t”’ and fc2)are SIF, displacement, boundary load and body force for load system (2). H is a general elastic modulus, which equals E under a plane stress condition and E/(1 - v’) under a plane strain condition, where E is Young’s modulus and v is Poisson’s ratio. 61 is a virtual crack growth at the crack front, L is the crack front, ST is the load prescribed boundary, S, is the displacement prescribed boundary and V is the volume of the corresponding geometry. Equation (1) can be simplified by taking consideration of load systems for which the body force is omitted, the prescribed boundary displacement on S, is equal to zero, and the prescribed boundary force on ST is equal to zero on all boundaries except on the crack surface. So far as the SIF is concerned, this load system can be used to compute the SIF under general load conditions by superposition principles[4]. From the above simplifications, eq. (1) can be rewritten as

s s ~ ;

[K”‘K”’

6f] dL =

ti” . &I” dS. s ST

Considering d(AS) = 61 dL, S, = S, H = E/l, and crack surface pressure n(2)= ti2’,eq. (2) can be written as [Kwp’]

AS

d(AS) = E

____ D

sS

ot2) 6~“’ dS ,

(3)

where AS is the virtual crack growth area, S is the crack surface, o is the crack surface load and /I is a function to account for the variation of the stress state along the crack front. It is clear from eq. (3) that once SIF and crack surface displacement for a load system (1) are known, the SIF for any other crack surface load crc2)can be determined. Consider a function of ratio K”’ and Ku’ K’2’ fi

=

da/c, al4 4). 659

G. S. WANG

660

Substitution

of relation (4) to eq. (3) gives &a/c, f

c+ 6~“’ dS.

a/t, &)[K”‘]’ dAS = E

AS B

s s

(9

By solving the ratio function g with eq, (5), the SIF for load system (2) will be determined by eq. (4). Let g(a/c, a/t, 6) be represented by a series of the form

WI where ($,(4)} is a given set of functions, Substituting (6) into (5) gives

then g can be specified by the coefficients {A,).

s

AS$~~~A.i.[KI’)]2d(AS)=

E

ss

.(2)&J’)dS.

(7)

The virtual crack growth AS can be arbitrarily assumed for given a/c and a/t. Therefore, a system of equations can be derived from eq. (7) for a set of independent virtual crack growths AS, as of’) 6uj” dS

ntO A, iAsi $ tj[K~i)]2d(ASJ = E

(i = 1,2 . . . m),

ss

(8)

where m is the number of independent virtual crack growths. Equation (8) can be written concisely as

$

(9)

(i=l,2...m)

A,Zi,=Dj tt=O

with Zi, =

$ $ [KC’)]’d(AS,) i AS,

When the g function, eq. (6), is approximated of independent virtual crack growths m as g(a/c, a/t, dr) = t

by a finite series which is equal to the number

MA/c,

a/t)$n(+)1

(12)

n=l

the linear system of functions of eq. (9) becomes (13) or (14) {A,} can be solved from eq. (14) as

t4m,1 = m&P1,,’ * Hence, the SIF for load system (2) can be approximately K”’ = K”’ i A&, , n-1 with (A,) solved by eq. (15).

(15) solved by relation (4) as (16)

661

Weight function estimation of stress intensity factors

2. CRACK

SURFACE

DISPLACEMENT

The knowledge of crack surface displacement for load system (1) is needed for {Di> in order to solve linear eq. (15). Although many useful SIF solutions for various part-elliptical Mode I crack problems are available in the literature, the corresponding crack surface displa~ent solutions are seldom provided. Various efforts have been made to derive approximate crack surface displacements from limited known condition@-91. Early approximation[6] gives very poor results, as shown in [5]. Other solutions have very much improved the accuracy for weight function applications. For the present investigation, the previously developed crack surface displacement solution[5], which is derived from only the SIF solution for uniform crack surface load, is used. The solution gives U(P, 4) = F(a/c, a/t, a, 4)(1 - p/r)‘/‘+

G(&,

a/t, a, (6)(1-

~/r)~”

(17)

with

W/c, 46

6

#I = *$j

G(a/c, a/t, a, t$) =

-t/4

(

Y(a/c, a/t, #)a g cos2f# + sin24

$

c2

Y2a2

-d+da B

-i

>

Fd+ -F. s d, >

(18) (19)

Geometry parameter definitions are shown in Fig. 1, and Y(a/c, a/t, 4) is the dimensionless SIF under uniform crack surface load Q defined by KC’)= Y(a/c, a/t, q5)J(na)ao.

(20)

4 is elliptic parameter angle, and rP is integration angle. @ depends on the crack problem type. di = n/2 for a two-axis symetrical crack problem, 9 = a for a one-axis symetrical problem, t9 = 2x for a general embedded crack, and Q, = any value ~2% for a general pan-ellipti~l crack problem. The crack surface displacement solution, eq. (17), meets the demands of exact near crack tip displacement field, the potential balance and crack surface displacement continuous. The investigation in [5] shows that the error of eq. (17) is less than 3%, compared to exact solution at the most unfavourable conditions for eq. (17).

3. ANALYSIS

PROCEDURE

With given reference SIF solutions for a uniform crack surface load, the corresponding crack surface displacement can be approximately solved by eq. (17). The SIF for any other crack surface load can be derived approximately by eq. (16) with (A, ) solved from eq. (15). Apparently, the order (m) of eq. (15) depends on the number of independent virtual crack growths AS,, For any part-elliptical crack, two independent virtual crack growths ASi can always be assumed with a virtual crack growth Aa in “a” direction, and a virtual crack growth AC in “c” direction while the crack front is still kept elliptically, as shown in Fig. 2. For general application,

Fig. 1. Geometric quantities of an elliptic crack.

642

G. S, WANG X

a+Aa a

a

(8)

(b)

Fig. 2. Two degrees of crack growth assumption.

m can be assumed to be m = 2 and corresponding respectively. The geometrical relations give

virtual crack growths are AS, and AS,,

AS,=n/4-c-Aa AS,==n/4*a*

AC,

(21)

and considering d(AS) = 61 dL d(AS,) = Aa . c +sin’ # dd, d(AS,) = AC - a . cos2 (b d#. g function in eq. (12) is approximated

(22)

as

g(aic, 46 (6) = A,$, + 4th.

(23)

It is clear that a good solution for g function depends strongly on the choice of $t and & as the terms in eq. (23) are rather limited. For the general crack surface load of*), a polynomial series can be used to represent it approximately as (24) with desired accuracy by the proper terms of nx and ny. By linear superposition principle, the SIF by load ~6’) in eq. (24) can be calculated by superposing the SIF with the individual terms of the load

i 00

6=;x

)’

a

-

c

i

(Cj 2 0).

(25)

The SIF results for load eq. (25) are enough to determine the SIFs for the general load eq. (24). Therefore, discussions will now be concentrated on the specific crack surface load o of eq. (25). The “weight” of a point force acting on a crack surface to the SIF depends on its location on the crack surface. As investigations in [5] show, the point force has larger “weight” near the centre of the crack surface than the rest of the crack surface except near the crack front. The “weight” of the point force approaches infinity when it approaches the crack front,, For the load form, eq. (25), the value of CIis very small near the crack centre, and the resultant SIF will be expected to be strongly determined by the load values near the crack front. g function is a “modification” of SIF under load a to the results for the uniform crack surface load. Jt is closely related to the crack surface load. It will be reasonable to assume that g is a function of the crack surface load, especially the load near the crack front. In the present investigation, g has been approximated as a function of the crack surface load at the crack front.

Weight function estimation of stress intensity factors

Using the coordinate

transformation

relation

the value of load CT,eq. (2S), at the crack front (p = r) is equal to sin’ C$cosi 4. The g function can be expressed as g(alc, a/t, 4) = A, (a/c, a/t) + A,(a/c, a/t)sin’+

cosj 4.

(27)

From the above knowledge, eq. (10) can be solved as I,, = noiac Aa iI2 = miac

Aa

r s *P y’ s

sin2 + d+

sin’+2

i$

cod

4

d4

(28)

.B

and 12,= naiac Aa

I22 = naiac Aa

Equation (11) can also be solved as

sy’ s *B

Y2

*B

cos’ t#~dd

sin’ C$COS~+~ C$d$.

(29)

(30) with u derived from eq. (17) and partial derivatives of u given as ;

=&,(I -p/r)-‘:2+f,,(l

-p/r)‘:‘+f,,(l

-p/‘)‘!2

g =_&(I - P/r)-“2 +f,2(l - p/r)‘/2 +h3(l - pfr)3J2,

(31) (32)

where

(33) and

(34)

G. S. WANG

664

Using coordinate D, = E Au(uc)

transformation

ds = acp dp d#,V, eq. (30) can be further derived as 3

I

2’“‘cz(i+j+ @

lYsin”b, cos’(bnL, 3

D, = E AC@)

‘“‘+‘(i +j + l)! sin’4 cos’#“T, i 02’

I+/+1

,!1, 2(k +‘,)_

lf,d6

I+/“!

knO 2(k +‘n, _ I f,d&. _

(35)

a in eqs (28)-(35) is assumed to be

n represents the number of intersections between the crack front and geometry surfaces for the crack. n = 1 for a surface crack, n = 2 for a corner crack and cos nd, 3 0 for an embedded crack. Hence, A, and A, in ey. (27) are solved (37)

(38) The SIF for load eq. (25) is obtained K,, =

gijK(‘),

and the SIF for the general load case, eq. (24), is obtained by superposition

(39) of eq. (39) as

for various values of i and j. When the crack surface load is fitted with a more general form (41) with L,, L, as scalars, the corresponding

SIF can also be derived from eq. (40) as (42)

The results computed from the above procedure will be discussed in the following sections for various part-elliptical crack problems in order to establish its accuracy.

4. EMBEDDED

CRACK

The elliptical embedded crack in an infinite body subjected to arbitrary polynomial load perpendicular to the crack plane is investigated. There exist analytical solutions of SIF for this problem[lO, 111. The accuracy of the present computations for SIF can be established by the comparisons of weight function computed results with analyti~l exact solutions. The exact dimensionless SIF for a uniform crack surface load is availabl~l21 as

Weight function ovation

665

of stress intensity factors

where for a/c d 1 k E(k) =

d2 J

(1 -k2sin2#)lRdd,

W

0

and for a/c > 1

E(k) = z E(k,).

(45)

Y in eq. (43) is defined by eq, (20). The dimensionless SIFs were computed for a crack surface load QZ

x A -

(46)

0a

and c=

0 Y” -

(47)

c

for various a/c ratios up to n =4 with the present weight function procedure (WF). The results are shown in Tables 1 and 2 for crack face loads of eqs (46) and (47), respectively. Self-consistency examination was performed by comparing the results for n = 0 to the results of exact solution, eq. 143). The error of WF results is less than 1.3%, and the maximum error occurs at a/c = 1. As inv~ti~tions in [q show, the approximate crack surface displacement representation, eq. (I?), has maximum error at this ~nfi~~tion. Table 1. SIFs for an elliptical embedded crack in an indite body subjected to crack face load $:(x/a)**n a/c

?I

##*2/z = 0

0.2

0.6

0.8

1.0

0.2

0

O&S7

OS734

0.7432

0.8606

0.9292

0.9518

:

0.0106 0.0235

0.0405 0.1352

0.1418 0.2962

0.2921 0.4542

0.4270 0.5675

0.6085 0.4811

:

0.0038 0.0059

0.0070 0.0150

0.0397 0.0733

0.1454 0.2022

0.2924 0.3466

0.3643 0.4106

0

0.5SOt

0.6089

0.7125

0.7983

0.8518

0.8698

: 3 4

0.027 0.05661 0.0158 0.0103

0.1657 0.0574 0.0250 0.0135

0.1513 0.3026

0.2860 0.4358

o,oai3

0.2006

0.0459

0.1458

0.53I2 0.40056 0.3325 0.2825

0.5657 0.4534 0.3906 0.3490

0

0.6069

0.63 11

0.6841

0.7360

0.7712

0.7835

1: 3 4

0.0744 0.0372 0.0223 0.0149

0.1839 0.0678 0.0314 0.0178

0.303s 0.1562 0.0859 0.0496

0.2780 0.4155 0.1976 0.1449

0.3845 0.4953 0.3186 0.2727

O-5241 0.4268 0.3715 0.3343

0

0.6303 0.0774 OSMO6 0.0251 0.0170

0.6386 0.1910 0.0722 0.034 1 0.0197

0.6589 0.3018 0.1582 0.0882 0.051s

0.6817 0.3983 0.2708 0.1945 0.1437

0.6986 0.4647 0.3663 0.3064 0.2640

0.7047 0.4885 0.4037 0.3547 0.3213

0.637 1 0.0713 0.0397 0.02s2 0.0174

0.6371 0.1910 0.0726 0.0345 0.0201

0.6371 0.2990 0.1586 0.089 1 0.0523

0.6371 0.3848 0.2650 0.1920 0.1427

0.6371 0.4398 0.3511 0.2961 0.2567

0.6371 0.4588 0.3840 0.3402 0.3099

0.4

0.6

0.8

1 : 4 1.0

0 :

3 4

K&&h*a)

666

G. S. WANG Table 2. SIFs for elliptical

embedded

crack

in infinite

body subjected

to crack

face load S:@/c)**n

0.2

K,l4l,/C~+a) 0.4

0.6

0.8

1.0

0.4257 0.3915 0.3745 0.3622 0.3521

0.5734 0.5027 0.4577 0.4211 0.3895

0.7432 0.5591 0.4346 0.3411 0.2690

0.8606 0.4805 0.2762 0.1610 0.0951

0.9292 0.2918 0.0998 0.0395 0.0193

0.9518 0.0411 0.0247 0.0170 0.0127

0.4

0.5501 0.4595 0.4176 0.3898 0.3687

0.6089 0.4862 0.4209 0.3739 0.3364

0.7125 0.4927 0.3658 0.278 1 0.2140

0.7983 0.4190 0.2345 0.1353 0.0801

0.8518 0.2682 0.0987 0.0443 0.0248

0.8698 0.0714 0.0418 0.0279 0.020 1

0.6

0.6069 0.4722 0.4136 0.3770 0.3505

0.6311 0.4703 0.3927 0.3407 0.3013

0.6841 0.4443 0.3192 0.2375 0.1799

0.7360 0.3685 0.2020 0.1155 0.0684

0.7712 0.2415 0.0917 0.043 1 0.0249

0.7835 0.0824 0.0466 0.0303 0.0214

0.8

0.6303 0.4646 0.3975 0.3568 0.3281

0.6386 0.4515 0.3683 0.3147 0.2753

0.6589 0.4083 0.2870 0.2105 0.1578

0.6817 0.3297 0.1780 0.101 I 0.0598

0.6986 0.2181 0.0838 0.0400 0.0234

0.7047 0.0861 0.0466 0.0295 0.0204

1.0

0.6371 0.4494 0.3787 0.3366 0.3073

0.637 1 0.43 17 0.3468 0.2934 0.2549

0.6371 0.3803 0.2634 0.1913 0.1425

0.637 1 0.3003 0.1603 0.0905 0.0533

0.637 1 0.1995 0.0769 0.0369 0.0216

0.6371 0.0877 0.0450 0.0278 0.0189

ale

n

0.2

0

4

4*2/x

= 0

For a polynomial crack face load, the exact solutions for SIF are available[lO]. The solution in [lo] can be solved analytically for n = 1 as Y=i

{~--fSin3”&[J(l

+sinb)-Jsinm]j.

(48)

Comparison of WF results and exact results is shown in Fig. 3. There are good agreements between WF results and exact results. The maximum error of WF results for a uniform crack face load is less than 1.3%. For a linear crack surface load, n = 1, the error of WF results at x = a, y = 0 (or Q, = n/2) is less than 1.5%. The agreement for the whole crack front is well within 1.5% compared to the value at x = a, except for the value at $J = 0 where the WF result is apparently lower than the exact value, with an absolute value 0.05 lower. The error at 4 = 0 is the cumulation of numerical error involving 0 calculation and approximation made in eq. (23). The error will be reduced when the polynomial coefficient n becomes larger since the load on the centre part of the crack face is reduced and eq. (23) can better describe the effect of stress field on SIF. 5. SURFACE

CRACK

For a semi-elliptical surface crack (Fig. 4a), the empirical SIF equations by [13], which are fitted with finite element (FE) computations, are used as the reference SIFs for uniform crack surface load to compute the SIF under polynomial crack face load. The empirical equations in [ 131 give Y = (M, + M*a2/t2 + M3a4/t4)gY’. (49) For a/c < 1 M, = 1.13 - o.o9a/c M2 = -0.54 + 0.89(0.2 + u/c)-’ M3 = 0.5 - (0.65 + u/c)-’ + 14(1 - a/~)~~ g = 1.0 + (0.1 + 0.35u2/t2)(l

- sin 4)’

Weight function estimation of stress intensity factors

361

1 a X E ;

-w .l

OExrct

.Ol 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 3. Comparison of WF results with exact results for an embedded crack in an infinite

L

Fig. 4. Crack located on different geometries.

667

668

G. S. WANG

and for a/e > 1 M, = [l.O + O.O4(a/c)-‘](a/c)-“2 h& = O.Z(a/c)-4 M, = -0.1 l(a/c)-4 g = I .O+ [O.1 + 0.35a2/r2(a/c>-‘I(1 - sin f$)*.

(51) eq. (49) is equal to the SIF for an embedded crack, eq. (43). The dimensionless SIFs for surface cracks subjected to polynomial load, eqs (46) and (47, were computed with WF method and are given in Tables 3 and 4, respectively, for various a/c and a/t Y’

in

Table 3. SIFs for semi~ejlipticai surface crack in a plate subjected to crack face load .$:@/a)**~ a/c

ait

n

9*2/n =o

0.2

0.2

0.2

0 1 2 3 4

0.5481 0.0153 0.01 IO 0.0072 0.0050

0.6988 0.1503 0.0451 0.0173 0.0085

0.4

0

0.6

0.8

1.0

0.8757 0.3362 0.1585 0.0814 0.0442

0.9989 0.5174 0.3246 0.2212 0.1575

1.0744 0.6490 0.4742 0.3778 0.3146

1.1002 0.6912 0.5346 0.4474 0.3916

0.6570 0.0285 0.020 t 0.0131 0.0090

0.8226 0.1789 0.0582 0.0249 0.0136

1.0186 0.3815 0.1794 0.093 1 0.0515

1.1556 0.5768 0.3541 0.2385 0.1686

1.2411 0.7189 0.5111 0.4007 0.3303

1.2708 0.7711 0.5744 0.4728 0.4095

2 3 4

0.8378 0.0659 0.0389 0.0241 0.0162

1.0198 0.2344 0.0828 0.0385 0.0222

1.2390 0.4538 0.2131 0.1117 0.0630

1.3931 0.6610 0.3960 0.2630 0.1844

1.4926 0.8120 0.5599 0.4310 0.3510

1.5281 0.8679 0.6262 OS058 0.4327

0 :

1.0800 0.0742 0.1524

1.2687 0.3278 0.1233

1so25 0.2582 0.5472

1.6686 0.7478 0.4407

1.7820 0.8947 0.6037

0.6699 0.9496

0.0601 0.0354

0.1371 0.0787

0.2899 0.2021

0.4586 0.3702

0.5339 0.4526

0.7185 0.1901 0.0665 0.0296 0.0164

0.8128 0.3385 0.1675 0.0897 0.0507

0.897 1 0.4817 0.3 108 0.2157 0.1555

0.9535 0.5858 0.4385 0.3548 0.2987

0.9734 0.6239 0.4899 0.4164 0.3684

I 2 3 4 0.6

0

1

0.8

3 4 0.4

1.8239

0.2

0 1 2 3 4

0.4

0 1 2 3 4

0.7764 0.0900 0.0458 0.0274 0.0181

0.7988 0.2164 0.0790 0.0368 0.0211

0.8929 0.3667 0.1816 0.0980 0.0561

0.9803 0.5114 0.3260 0.2249 0.1617

1MO4 0.6168 0.4547 0.3648 0.3056

1.0620 0.6557 0.5066 0.4268 0.3757

0.6

0

0.9194 0,1301 0.0646 0.0382 0.0251

0.9197 0.2569 0.0975 0‘047 1 0.0276

1.0086 0.4072 0.2016 0.1094 0.0634

1.0973 0.5513 0.3465 0.2373 0.1699

1.1618 0.6573 0.4759 0.3780 0.3 146

1.1857 0.6967 0.5282 0.4405 0.3853

1.0957 0.1920 0.0905 0.0523 0.0339

1.0578 0.3097 0.1210 0.0597 0.0353

1.1308 0.451 I 0.2236 0.1221 0.0713

1.2152 0.5871 0.3655 0.2490 0.1777

1.2824 0.6889 0.4926 0.3887 0.3221

1.3085 0.7273 0.5443 0.4508 0.3927

0.7396 0.0977 0.0487 0.0291 0.0194

0.7280 0.2138 0.0799 0.0376 0.0216

0.7629 0.3351 0.1716 0.0943 0.0546

0.8086 0.4486 0.2970 0.2095 0.1528

0.8439 0.5309 0.4072 0.3345 0.2845

0.8571 0.5611 0.4513 0.3894 0.348 1 con~in~ed

1 2 3 4 0.8

0

1 2 3 4 0.6

0.043 1 0.0219 _.__. ._l_____ll 0.6858 0.0660 0.0338 0.0202 0.0134

y?/J(“‘“)

0.2

.-..-

669

Weight function estimation of stress intensity factors Table I--continued. a/c

alt

n

4*2/n = 0

0.2

0.4

0 2 3 4

0.8109 0.1207 0.0596 0.0356 0.0236

0.7838 0.2346 0.0900 0.0434 0.0254

0.8116 0.353 1 0.1811 0.1001 0.0584

0.6

0 1 2 3 4

0.9224 0.1560 0.0759 0.0449 0.0297

0.8669 0.2652 0.1044 0.0515 0.0305

0.8

0 1 2 3 4

1.0581 0.2027 0.0960 0.0561 0.0367

0.2

0 1 2 3 4

0.4

0.6

0.8

I.0

0.8556 0.4643 0.3053 0.2147 0.1564

0.8917 0.5456 0.4147 0.3391 0.2876

0.9056 0.5757 0.4586 0.3938 0.3512

0.8806 0.3781 0.1940 0.1078 0.0634

0.9200 0.4852 0.3163 0.2215 0.1610

0.9566 0.5646 0.4245 0.345 1 0.2918

0.9713 0.5944 0.4680 0.3996 0.3552

0.9597 0.3014 0.1209 0.060l 0.0360

0.9504 0.4036 0.2074 0.1156 0.0684

0.9807 0.5025 0.3258 0.2276 0.1652

1.0164 0.5780 0.4313 0.3495 0.2949

1.0318 0.6068 0.4742 0.4035 0.3580

0.7522 0.1135 0.056 I 0.0338 0.0226

0.7213 0.2249 0.0867 0.0417 0.0244

0.7195 0.3289 0.1726 0.0965 0.0566

0.7333 0.4196 0.2844 0.2035 0.1500

0.7485 0.4838 0.3799 0.3166 0.2719

0.7549 0.5071 0.4177 0.3656 0.3301

0 1 2 3 4

0.8079 0.1339 0.0655 0.0393 0.0262

0.7608 0.2414 0.0949 0.0465 0.0274

0.7500 0.3405 0.1791 0.1006 0.0594

0.7601 0.4276 0.2888 0.2064 0.1520

0.7748 0.4899 0.3828 0.3183 0.273 1

0.7814 0.5127 0.4201 0.3669 0.3309

0.6

0 1 2 3 4

0.8956 0.1627 0.0788 0.0470 0.0312

0.8200 0.2645 0.1059 0.0527 0.0314

0.7930 0.3566 0.1878 0.1059 0.0629

0.7966 0.4385 0.2947 0.2102 0.1547

0.8100 0.4984 0.3869 0.3208 0.2748

0.8167 0.5207 0.4237 0.3690 0.3323

0.8

0 : 3 4

1.0044 0.1945 0.0933 0.0551 0.0364

0.8875 0.2893 0.1172 0.0587 0.0351

0.8367 0.3726 0.1963 0.1109 0.0660

0.8301 0.4484 0.3001 0.2137 0.1572

0.8414 0.5056 0.3905 0.323 1 0.2764

0.8481 0.5274 0.4269 0.3710 0.3338

0 :

0.7439 0.0588 0.1199

0.7041 0.0894 0.2285

0.6807 0.1719 0.3213

0.6706 0.2734 0.3950

0.6680 0.3562 0.4436

0.6678 0.3882 0.4608

3 4

0.0356 0.0240

0.0434 0.0255

0.0973 0.0575

0.1982 0.1474

0.3008 0.2608

0.3444 0.3139

0.4

0 1 2 3 4

0.7875 0.1382 0.0671 0.0404 0.0271

0.7320 0.2418 0.0961 0.0473 0.0280

0.6993 0.3287 0.1764 0.1002 0.0596

0.6851 0.3980 0.2753 0.1996 0.1484

0.6815 0.4444 0.3563 0.3ca7 0.2607

0.6813 0.4610 0.3876 0.3438 0.3133

0.6

0 1 2 3 4

0.8574 0.1620 0.0780 0.0467 0.0312

0.7748 0.2596 0.1048 0.0522 0.031 I

0.7262 0.3391 0.1824 0.1040 0.0621

0.7052 0.403 1 0.2782 0.2016 0.1499

0.6998 0.4470 0.3572 0.3012 0.2610

0.6994 0.4630 0.3880 0.3438 0.3132

0.8

0 1 2 3 4

0.9484 0.1821 0.0880 0.0525 0.0350

0.8271 0.2768 0.1124 0.0562 0.0336

0.7557 0.3501 0.1882 0.1074 0.0642

0.7248 0.4098 0.2818 0.2039 0.1516

0.7169 0.4523 0.3598 0.3029 0.2622

0.7163 0.4682 0.3905 0.3454 0.3144

1

0.8

1.0

0.2

l

G. S. WANG

670

Table 4. SIFs for semi-elliptical surface crack in a plate subjected to crack face load S:b/c)**n ale

all

n

9*2/n =o

0.2

0.2

0.2

0

0.6988

I 2 3 4

0.5481 0.4520 0.4222 0.4026 0.3875

0.4

0 1 2 3 4

0.6

0.6

0.8

1.0

0.5515 0.4902 0.4446 0.4070

0.8757 0.6014 0.4567 0.3535 0.2761

0.9989 0.5266 0.2988 0.1739 0.1035

I .0744 0.3503 0.1280 0.0560 0.0301

1.1002 0.1134 0.0555 0.0345 0.0240

0.6570 0.4945 0.4503 0.4238 0.4048

0.8226 0.5947 0.5154 0.4613 0.4189

1.0186 0.6487 0.4815 0.3683 0.2858

1.1556 0.5809 0.3266 0.1906 0.1146

1.241 I 0.4141 0. I598 0.0750 0.0428

I .2708 0.1859 0.0890 0.0542 0.0370

0 1 2 3 4

0.8378 0.5762 0.5044 0.4647 0.4379

1.0198 0.6756 0.5634 0.4937 0.4423

1.2390 0.7299 0.5234 0.3928 0.3012

1.3931 0.6615 0.3647 0.2120 0.1281

I .4926 0.4939 0.1967 0.0963 0.0568

1.5281 0.2617 0.1256 0.0758 0.0513

0

1.0800 0.7092 0.5920 0.5303 0.4905 _-__ 0.6858 0.5278 0.4672 0.4295 0.402 I

1.5025 0.8433 0.580 1 0.4248 0.3204

1.6686 0.7521 0.4023 0.2305 0.1384

1.7820

0.8021 0.6386 0.5445 0.4789

0.5550 0.2207 0.1091 0.0650

1.8239 0.2865 0.1442 0.0879 0.0595

0.7185 0.5301 0.4471 0.3912 0.3484

0.8128 0.5244 0.3802 0.2850 0.2173

0.8971 0.4495 0.2482 0.1430 0.0852

0.9535 0.3046 0.1 165 0.0550 0.03 19

0.9734 0.1150 0.0614 0.0393 0.0276

0.7764 0.5676 0.4923 0.4417 0.4163

0.7988 0.5608 0.4638 0.4015 0.3552

0.8929 0.5518 0.3933 0.2923 0.2218

0.9803 0.4774 0.2621 0.1514 0.0909

I .0404 0.3350 0.1325 0.0650 0.0387

I.0620 0.1477 0.0784 0.0497 0.0347

2 3 4

0.9194 0.6341 0.5345 0.4785 0.4405

0.9197 0.6102 0.4909 0.4185 0.3666

1.0086 0.5927 0.4 126 0.3027 0.2279

1.0973 0.5152 0.2797 0.1615 0.0977

I.1618 0.3725 0.1513 0.0766 0.0461

1.1857 0.1845 0.0980 0.0618 0.0429

0 1 2 3 4

1.0957 0.7280 0.5946 0.5225 0.475 I

1.1308 0.6404 0.4339 0.3134 0.2335

1.2152 0.5501 0.2935 0.1683 0.1017

I .2824 0.3969 0.1622 0.083 I 0.05 12

1.3085 0.1973 0.1080 0.0684 0.0475

0.2

0 1 2 3 4

0.7396 0.5421 0.4618 0.4142 0.3809

1.0578 0.6762 0.5276 0.4417 0.3823 -_-0.7280 0.5119 0.4159 0.355 1 0.3107

0.7629 0.4700 0.3295 0.2417 0.1815

0.8086 0.3888 0.2101 0.1198 0.0712

0.8436 0.2619 0.1018 0.0493 0 0292

0.857 I 0.1040 0.0578 0.0370 0.0259

0.4

0 1 2 3 4

0.8109 0.5752 0.4823 0.4289 0.3923

0.7838 0.5339 0.4273 0.3619 0.3150

0.8116 0.4865 0.3369 0.2455 0.1837

0.8556 0.4039 0.2176 0.1245 0.0745

0.8917 0.2779 0.1 108 0.0552 0.0334

0.905h 0.1212 0.0675 0.0432 0.0302

0.6

0 1 2 3 4

0.9224 0.6282 0.5152 0.4525 0.4106

0.8669 0.5677 0.4448 0.3722 0.3215

0.8806 0.5097 0.3468 0.2504 0.1863

0.9200 0.4236 0.2268 0.1300 0.0785

0.9566 0.2978 0.1218 0.0624 0.0385

0.97 13 0.1417 0.0794 0.0508 0.035i

0.8

0

1.0581 0.6985 0.5594 0.4843 0.4354

0.9597 0.6094 0.4665 0.3851 0.3296

0.9504 0.5341 0.3563 0.2544 0.1880

0.9807 0.4395 0.2328 0.1332 0.0807

I.0164 0.3098 0.1282 0.0668 0.0417

1.0318 0.1506 0.0863 0.0555 0.0388

0.8

1 2 3 4 0.4

0.2

0

1 2 3 4 0

0.4

I 2 3 4 0

0.6

I

0.8

-0.6

K,/.%/&*Q) 0.4

1 2 3 4

1.2687

continued

671

Weight function estimation of stress intensity factors Table 4-continued. 0.8

1.0

0.7195 0.4302 0.2947 0.2130 0.1582

0.7333 0.3417 0.1815 0.1025 0.0606

0.7485 0.2245 0.0872 0.0425 0.0252

0.7549 0.0875 0.0502 0.0322 0.0224

0.7608 0.5086 0.3980 0.3323 0.2862

0.7500 0.4402 0.2987 0.2149 0.1592

0.7601 0.3498 0.1855 0.1051 0.0626

0.7748 0.2328 0.0923 0.0461 0.0279

0.7814 0.0963 0.0559 0.0360 0.025 1

0.8956 0.6050 0.4873 0.4227 0.3801

0.8200 0.5321 0.4096 0.3387 0.2900

0.7930 0.4536 0.3038 0.2170 0.1602

0.7966 0.3602 0.1905 0.1084 0.0651

0.8100 0.2440 0.0994 0.0510 0.0315

0.8167 0.1093 0.0639 0.0413 0.0288

0 1 2 3 4

1.0044 0.6573 0.5197 0.4459 0.3982

0.8875 0.5586 0.4225 0.3457 0.2940

0.8367 0.4658 0.3076 0.2181 0.1603

0.8301 0.3682 0.1940 0.1108 0.0671

0.8414 0.2533 0.1054 0.0554 0.0347

0.8481 0.1213 0.0712 0.0461 0.0322

0.2

0 1 2 3 4

0.7439 0.5206 0.4244 0.3703 0.3341

0.7041 0.4722 0.3676 0.3055 0.2623

0.6807 0.3990 0.2689 0.1923 0.1418

0.6706 0.3049 0.1596 0.0893 0.0524

0.6680 0.1930 0.0741 0.0359 0.0212

0.6678 0.0702 0.0417 0.0268 0.0186

0.4

0 1 2 3 4

0.7875 0.5439 0.4389 0.3807 0.3421

0.7320 0.4847 0.3736 0.3087 0.2641

0.6993 0.4052 0.2710 0.1931 0.1421

0.685 1 0.3087 0.1614 0.0906 0.0535

0.6815 0.1963 0.0768 0.0380 0.0229

0.6813 0.0736 0.0448 0.0291 0.0203

0.6

0 1 2 3 4

0.8574 0.5785 0.4601 0.3958 0.3538

0.7748 0.5016 0.3814 0.3128 0.2663

0.7262 0.4127 0.2732 0.1937 0.1421

0.7052 0.3138 0.1641 0.0926 0.0553

0.6998 0.2027 0.0816 0.0416 0.0256

0.6994 0.0822 0.0505 0.0330 0.0232

0.8

0 1 2 3 4

0.9484 0.6178 0.4842 0.4130 0.3672

0.827 1 0.5180 0.3885 0.3161 0.2676

0.7557 0.4181 0.2741 0.1934 0.1416

0.7248 0.3186 0.1670 0.095 1 0.0575

0.7169 0.2124 0.0885 0.0466 0.0293

0.7163 0.0984 0.0591 0.0385 0.0270

4

n

f$*2/x =o

0.2

0.8

0.2

0 1 2 3 4

0.7522 0.5353 0.4443 0.3920 0.3564

0.7213 0.4923 0.3899 0.3277 0.2835

0.4

0 1 2 3 4

0.8079 0.5627 0.4612 0.4041 0.3657

0.6

0 1 2 3 4

0.8

1.0

$pI&*a)

0.6

a/c

ratios

for polynomial powers up to n = 4. The self-consistency check for WF results shows that the errors of WF results are less than 2% for a uniform crack face load. The WF results in Tables 3 and 4 were used to calculate SIFs for crack surface loads of

0 = cr,(l -x/a)

(52)

e = G(l --Y/C),

(53)

and

for which some FE results are available[l4]. The comparisons between WF results and FE results are shown in Figs 5 and 6 for load eqs (52) and (53), respectively, for several a/c and a/t ratios, shown in the insert. As shown in Figs 5b and 6b, the best agreements between WF results and FE results are at a/c = 0.5. Rather coarse meshes were used in [14] to compute SIFs. The mesh size in [14] was determined by calibrating SIF results with the “best estimate magnification factor” for a/c = 0.5 with a confidence band of rt 3%. Calibrations in [14] show that, at a/c = 0.5, most of the FE results

672

G. S. WANG

0.9 0.8 -

0

•I

l

.

0.7 ” 3

0.6 -

I z rn =. Y

0.5 -

sr L

04.

Y

m

n

e

,

0.3 -

a/c=0.2

0.2 0.1 -

0.0-j 0.0

6

a.75

m

an=.5

e

and5

-w

.

.

, 0.2

.

.

, 0.4

.

.

, 0.6

.

.

,

.

0.8

.

, 1.0

phl*!Upl

Fig. 5(a)

1.2

q

al19.75

I

n

Mm.5

\

a

aAm.

0.8

0.4

0.6

phlWpl Fig. 5(b)

0.8

1.0

.

. 1.2

673

Weight function estimation of stress intensity factors 1.47

l.O-

0.8 -

0.2: 0.0

0.2

0.6

0.4

q

all-75

.

ab.5

.

afb.25

0.8

1.0

2

Fig. 5(c) Fig. 5. Comparisons of WF results with FE nsults[l4] for S, = S(1 - x/a) crack surface load.

fall within the band except for the result for a/t = 0.75, when the FE results are 2-3% lower. The same disagreements are observed in the comparison with WF results as shown in Figs 5b and 6b. FE results for linear crack surface load are also lower at a large a/c ratio (a/c = 0.75). For a small a/c ratio and a large a/t ratio (a/c = 0.2, a/t = 0.75), the FE results in [14] are much lower than the results from eq. (49) (15 - 23%). Therefore, the SIFs from [ 141for linear crack surface load are also expected to be lower than WF results in the same order as can be found in the comparison of Figs 5a and 6a. The SIF results from (141 are much lower than WF results, especially for larger a/c ratios. The inaccuracy of results from [14] is considered to result from the coarse FE mesh used in the analysis, though it is calibrated at the configuration a/c = 0.5. Comparisons of SIF results for configuration a/c = 0.8 between FE and WF computations show a moderate agreement for linear crack surface load, as shown in Figs 5c and 6c. For this configuration (a/c = 0.8), which does not differ much from the calibrated configuration, the FE results from [14] are relatively reliable, as discussed above. For an internal surface crack in a cylinder (Fig. 4b), with R,/t = 10, which does not differ too much from a crack in a plate, computations were performed by different authors, e.g. [15, 161,with different methods for the polynomial crack surface load eq. (46). The results from these sources differ within 10%. Comparisons are made between WF results and FE results from [15] for which the error is estimated at less than 8% for a configuration a/c = l/3. Figure 7 shows the comparisons. The symbols represent FE results and solid curves represent WF results for different a/t ratios for n up to 3. The SIF is represented by a nondimensional magnification factor W(k)

H, = J(tra)(a’/c’

c0.s’4 + sin2 4)“’ *

(54)

The SIF for a uniform surface load is little higher at 4 = n/2 for the crack in a plate than that for the crack on a cylinder, 4 = 0, as shown in Fig. 7a-c for a/c = l/3 and a/t = 0.25-0.8. SIFs

G. S. WANG

674

phP2/pl

Fig. 6(a)

*

0.0

0.2

0.4

0.6

phl'Vpl

Fig. 6(b)

0.8

ab.25

1.0

1.2

675

Weight function estimation of stress intensity factors

0.0

0.2

0.4

0.6

0.6

1.0

1.2

Fig. 6(c) Fig. 6. Comparisons of WF results with FE resufts[lrl] for S, = S(1 -y/c)

crack surface load.

for polynomial load up to n = 3 also show higher values for the crack in a plate than the crack in a cylinder. For the crack front near the surface (r#~= 0), both the results for the crack in a plate and those for the crack in a cylinder are in good agreement for uniform crack surface load, as shown in Fig. 7. The corresponding SIFs for polynomial load also show rather good agreement, except for the case n = 1. The previous discussion shows that, for this loading case, WF calculation results are low because of the approximations made in its derivation. However, the error is rather limited and localized. In agreement with previous analysis, the accuracy of WF results near a crack surface improves with the increase of polynomial power, n, as shown in Fig. 7 (4 = 0). A better estimation of the SIF at 4 = 0 can be possibly made by increasing SIF values according to comparisons between WF results and exact or FE results for a crack in a simple case. WF computations show a rather good description of the SIF along the crack front for polynomial crack surface loadings which can eventually approximate a rather complex crack surface load. Comparisons in Fig. 7 show that the SIF ratios for polynomial load to uniform load, or the function g in eq. (12), seem to be approximately equal in spite of the location of the surfae crack in different geometries. Further investigations were performed for a surface crack on a plate and on an internal face of a cylinder for various configurations[l5-21]. Comparisons of results for K-/K,, (or g) at the deepest point of a surface crack subjected to linear loading in the x direction for cracks in both plates and cylinders are shown in Table 5 for various a/c and a/t ratios. In Table 5, present WF results are also compared to WF results from 1221, which is based on a different crack surface displacement representation. The maximum difference of present WF results compared to numerical results is within 5%, and present WF results show better agreement with numerical computations than those of [22]. Table 6 shows comparisons of g functions from present WF computation, the WF from [22], FE methodEl5, 19,211 and boundary element method (BEM)[16] for n = 1,2,3, 0.2 G a/c $ 1, and

676

G. S. WANG IO, ah-O.25

0.0

0.2

0.4

0.6

phlWpl

Fig. 7(a)

phlwpl

Fig. 7(b)

0.8

1.0

1.2

Weight function estimation of stress intensity factors

677

0.8

Fig. 7(c) Fig. 7. Comparisons of “‘magnification” factor H, from WF calculations and FE computations[l5] for surface crack on a place and on an internat face of a cylinder.

0.2 < a/t c 0.8. III this case, the results from [15,t6,21] correspond to the ~o~~orni~ loading of a crack on the inner surface of the pressure vessel (Fig. 4b). The present WF results are within 6% of the numerical results. The comparisons of the results shown in Tables 5 and 6 indicate that there is an insignificant difference:between the g function for the crack on a plate and that on the curvilinear surface of a cylinder. In addition, the comparisons also show that the present WF results are better than the WF result by [22]for a surface crack problem. The crack surface displacement Table 5. The values of K,/K, at the deepest point of a semi~ellipticalsurface crack subjected to linear loading in the x-direction o/c

air

WF

@I

UPI

0.2

0.2 0.4 0.6 0.8

0.621 0.593 0.566 OS4

0.630 O&O5 0.571 0.533

OS94 0.566 0.539 0.511

0.603 0.566 0.532 0.520

0.33

0.2 0.4 0.6 0.8

0.629 0.604 0.574 OS43

0.639 0.616 0.588 0.560

0.612 0.594 0.576 0.558

0617 0.591 0.567 0.558

0.5

ii?: 0:6

0.643 0.622 0.596

0.649 0.630 0.606

0.631 0.612 0.622

0.636 0.620 0.605

0.670 0.628 0.652

0.629 0.597 0.613

0.8

0.570

0.583

0.603

0.599

0.587

0.604

iti

0.690 0.676 0.661 0.653

0.667 0.654 0,640 0.634

0.673 0.676 0.679 0.682

0.699 0.691 0.684 0.686

1.0

0.6 0.8

Vl

u71

I201 0.578 a.555 OS17 0.517

0.710 0.695 0.691 0.705

678

G. S. WANG

Table 6. The values of K./K,, at the deepest point of a semi-elliptical surface crack subjected to polynomial loading in the x-direction a/c = 0.2

n ~-~ I

0.629 0.624 0.589 0.543

cI!C = l/3 [I91 [I51 P21 -.~I,__ 0.639 0.621 0.634 0.608 0.615 0.602 0.585 0.577 OSS9 0.558 0.561

[16] _.__._0.618 0.546 0.546

0.461 0.400 0.368

0.490 0.494 0.445 0.399

0.499 0.493 0.457 0.416

-

0.471 0.431 0.417

0.407 0.399

0.393 -

0.41s 0.417

0.422 0.416

-’ -

0.355 0.305

0.324 0.283

0.370 0.327

0.380 0.342

-

WF

[22]

[21]

WF

0.2 0.25 0.5 0.8

0.621 0.615 0.575 0.514

0.630 0.625 0.589 0.533

0.604 0.549 0.513

2

0.2 0.25 0.5 0.8

0.476 0.469 0.421 0.363

0.485 0.478 0.434 0.381

3

0.2 0.25

0.399 0.391

0.5 0.8

0.343 0.289

a/t

a/c = 0.4

0.634 0.630 OS97 0.553

[22] [21] . --._-.__~~___ 0.643 0.627 0.639 0.609 0.594 0.568 0.579

a/c = 1.0 [22] [21] ----__.0.690 0.667 0.704 0.687 0.654 -0.668 0.694 0.653 0.634 0.497

0.464 0.408 0.395

0.498 0.493 0.457 0.414

0.506 0.501 0.468 0.430

0.488 0.451 0.431

0.581 0.579 0.561 0.544

0.570 0.553 0,538 0.539

0.579 --0.568 0.567

0.404

0.382

0.423 0.418

0.431 0.426

0.411 -

0.516 0.513

0.510 0.502

0.504 .._

0.350 0.331

0.346 0.329

0.383 0.343

0.393 0.357

0.375 0.354

0.497 0.481

0.487 0.482

0.491 0.488

WF

WF

in [22] is derived from an embedded crack; it will be less effective when being used for surface crack problems. Comparisons in Table 6 suggest that a simple method can be used to estimate SIFs for a surface crack on a complex curvilinear surface of a structure under complex load from a known SIF solution for a simple load case (uniform crack surface load, for example) and the knowledge of {gj for a surface crack as shown in Tables 5 and 6, with errors within several percent. 6, CORNER

CRACK

The empirical SIF equations in [13] for a quarter-elliptical corner crack located at the edge of a finite plate (Fig. 4c), under uniform remote load, were used as reference SIFs for uniform crack face load for this investigation. The equations give Y = (M, + f&a2:tz + MJ/t”)g,

g, Y’.

(55)

For a/c 6 1 M, = 1.08 - O.O3a/c Mz = -0.44 -I- 1.06(0.3 + a/c)-’ Mj = -0.5 +

0.25a/c

+

14.8(1 -a/c)”

g, = I f (0.08 + 0.4a2/t2)(1 - sin 4)3 g, = 1 f (0.08 + 0.15a2/t2)(1 - cos 4)’

(56)

and for a/c > 1 M, = ~(~~~)(1.08 Mz

= 0.375c’/a2

154~ = g,

0.03cla)

=

- 0.25c2/a”

1 -f- (0.08 f 0.4c2/t2)(I - sin (6)”

g, = 1 - (0.08 + 0.1 5c2/t2)(l - cos @3.

(57)

Y’ in eq. (55) is equal to Y in eq. (43) for an embedded crack in an infinite body. Tables 7 and 8 show the WF SIF results for polynomial load equations (46) and (47), respectively, for powers up to n = 4 for various a/c and a/t ratios, as shown in the tables. Self-consistency examinations were performed for the reference loading case (n = 0) and the results show that the errors of WF computations are less than 2.2%. When the values in Tables 7 and 8 are used to estimate SIFs for nonunifo~ crack face load cases, several percent of extra error are expected to be added to errors of the reference SIF solutions [eq. (SS)]. According to analysis

679

Weight function estimation of stress intensity factors Table 7. SIFs for quarter-elliptical comer crack in a plate subjected to crack face load S:(x/a)**n 0.8

0.8557 0.3371 0.1632 0.0865 0.0487

0.9903 0.4970 0.3124 0.2136 0.1526

06240 0.4549 0.3625 0.3023

i.1815 0.7032 0.5377 0.4502 0.3946

0.8047

1.0064

1.1626

1.2876

1.4078

0.0398 0.0778 0.0237 0.0157

0.2072 0.0765 0.0366 0.0213

0.3847 0.1857 0.0994 0.0570

0.5595 0.3430 0.2313 0.1639

0.7015 0.4950 0.3870 0.3187

0.5892 0.7978 0.4835 0.4183

0.8818 0.1611 0.0767 0.0446 0.0290

1.0421 0.2956 0.1179 0.0600 0.0363

1.2787 0.4792 0.2323 0.1266 0.0746

1.4725 0.6583 0.3923 0.2605 0.1831

1.6422 0.8104 0.5501 0.4202 0.3408

1.8268 0.9307 0.6597 0.5283 0.4500

a/r

n

+*2/n =o

0.2

0.2

0.2

0 1 2 3 4

0.5333 0.057 1 0.0273 0.0159 0.0103

0.6761 0.1744 0.0606 0.0271 0.0150

0

0.6524

: 3 4 0

0.4

0.6

i 3 4

0.4

1.0923

0.8

0

I .2668 0.372 1 0.1588 0.0882 0.0559

1.4188 0.4771 0.2008 0.1055 0.0645

1.6965 0.6356 0.3120 0.1737 0.1051

1.9452 0.7881 0.4595 0.3015 0.2105

2.1902 0.9302 0.6100 0.4562 0.3647

2.4934 I .0758 0.7348 0.5755 0.483 1

0.2

0 :

0.6718 0.1047 0.0496

0.7000 0.2063 0.0783

0.7997 0.3359 0.1693

0.8955 0.2999 0.4652

0.9761 0.4243 0.5694

1.0525 0.4976 0.6371

3 4

0.0289 0.0189

0.0375 0.0217

0.0926 0.0537

0.2084 0.1506

0.3429 0.2886

0.4224 0.3739

0 :

0.7757 0.0630 0.1302

0.7862 0.0911 0.2327

0.8877 0.1836 0.3653

0.9922 0.4993 0.3167

1.0860 0.6104 0.4454

1.1836 0.5265 0.6893

3 4

0.0370 0.0242

0.0450 0.0267

0.1010 0.059 I

0.2182 0.1570

0.3557 0.2972

0.4415 0.3877

0.6

0 1 2 3 4

0.9483 0.1827 0.0877 0.0513 0.0335

0.9210 0.2787 0.1129 0.0575 0.0347

1.0201 0.4099 0.2055 0.1137 0.0674

1.1367 0.5462 0.3398 0.2316 0.1656

1.2527 0.6653 0.4732 0.3724 0.3083

1.3891 0.7619 0.5666 0.4679 0.4070

0.8

0 : 3 4

I.1818 0.2791 0.1284 0.0737 0.0476

1.0877 0.3477 0.1452 0.0755 0.0460

1.1741 0.4637 0.2329 0.1299 0.0779

1.3026 0.5924 0.3629 0.2454 0.1746

I.4493 0.7147 0.4978 0.3870 0.3178

1.6448 0.8328 0.6057 0.4940 0.4262

0.2

0 1 2 3 4

0.7285 0.1285 0.0619 0.0366 0.0242

0.7132 0.224 1 0.0887 0.9438 0.0259

0.7547 0.3310 0.1715 0.0957 0.0564

0.8116 0.4360 0.2876 0.2028 0.1481

0.8687 0.5217 0.3973 0.3254 0.2765

0.9319 0.5801 0.4628 0.3982 0.3557

0.4

0 1 2 3 4

0.8162 0.1555 0.0753 0.0446 0.0294

0.7773 0.2468 0.0999 0.0504 0.0303

0.8129 0.3512 0.1818 0.1019 0.0606

0.8725 0.4562 0.2976 0.2087 0.1519

0.9378 0.5451 0.4091 0.3325 0.2811

1.0169 0.6115 0.4805 0.4101 0.3645

0.6

0 : 3 4

0.9573 0.2067 0.0988 0.058 1 0.0382

0.8735 0.2842 0.1182 0.0610 0.037 1

0.8962 0.3801 0.1967 0.1110 0.0666

0.9589 0.4818 0.3103 0.2162 0.1568

1.0379 0.5736 0.4232 0.3408 0.2865

1.1450 0.6527 0.5037 0.4258 0.3762

0 1 2 3 4

1.1368 0.2939 0.1352 0.0782 0.0509

0.983 1 0.3366 0.1437 0.0755 0.0463

0.9829 0.4109 0.2138 0.1216 0.0738

1.0472 0.5007 0.3200 0.2221 0.1609

1.1442 0.5912 0.4312 0.3452 0.2891

0.4

0.6

~%‘&a)

I.0

0.6

a/c

0.8

1.2919 0.6846 0.5220 0.4386 0.3861 continued overleaf

680

G. S. WANG Table 7-continued.

1.0

1 2 3 4

0.1459 0.1343 0.0661 0.0397 0.0265

0.7114 0.2302 0.0926 0.0463 0.0277

0.7167 0.3247 0.1714 0.0969 0.0576

0.7410 0.4118 0.2770 0.1977 0.1456

0.7757 0.4823 0.3745 0.3105 0.2660

0.8264 0.5326 0.4334 0.3774 0.3399

0.4

0 I 2 3 4

0.8190 0.1594 0.0782 0.0469 0.0312

0.7598 0.2490 0.1022 0.0520 0.0314

0.7565 0.3387 0.1789 0.1015 0.0607

0.7807 0.4239 0.2830 0.2013 0.1479

0.8207 0.4958 0.3812 0.3144 0.2685

0.8837 0.5525 0.4448 0.3853 0.3459

0.6

0 1 2 3 4

0.9337 0.2055 0.099 I 0.0589 0.0390

0.8300 0.2792 0.1173 0.0608 0.037 I

0.8107 0.3578 0.1893 0.1080 0.0652

0.8340 0.4375 0.2896 0.2052 0.1506

0.8828 0.5102 0.3879 0.3181 0.2708

0.9671 0.5768 0.4590 0.3952 0.3536

0.8

0 1 2 3 4

I .0721 0.2812 0.1306 0.0763 0.0500

0.9037 0.3197 0.1377 0.0725 0.0446

0.8602 0.3754 0.2000 0.1152 0.0702

0.8811 0.4423 0.2922 0.2069 0.1518

0.9416 0.5123 0.3877 0.3174 0.2699

1.0557 0.5897 0.4670 0.4015 0.3589

0.2

0 1 2 3 4

0.7444 0.1298 0.0653 0.0398 0.0268

0.7007 0.2292 0.0927 0.0464 0.0279

0.6842 0.3181 0.1702 0.0969 0.0578

0.6838 0.3927 0.2685 0.1936 0.1436

0.6985 0.4502 0.3556 0.2980 0.2572

0.7376 0.4933 0.4084 0.3594 0.3261

0.4

0 1 2 3 4

0.8055 0.1524 0.076 1 0.0462 0.0310

0.7376 0.2449 0.1007 0.0512 0.0310

0.7118 0.3280 0.1756 0.1004 0.0603

0.7100 0.3997 0.2719 0.1956 0.1449

0.7283 0.4578 0.3592 0.2999 0.2584

0.7774 0.5063 0.4161 0.3649 0.3304

0.6

0 1 2 3 4

0.9001 0.1920 0.0940 0.0565 0.0378

0.7897 0.2690 0.1131 0.0585 0.0357

0.7476 0.3406 0.1829 0.1051 0.0636

0.7434 0.4064 0.2750 0.1975 0.1461

0.7679 0.4648 0.3620 0.3013 0.2590

0.8339 0.5215 0.4254 0.3718 0.3360

0.8

0 1 2 3 4

1.0109 0.2535 0.1197 0.0707 0.0468

0.8406 0.2996 0.1288 0.0676 0.0415

0.7755 0.3502 0.1895 0.1097 0.0670

0.7678 0.4048 0.2741 0.1970 0.1459

0.8007 0.4610 0.3588 0.2987 0.2568

0.8899 0.5272 0.4297 0.3758 0.3397

[13], eq. (55) has - 7 . . .4% differences compared to FE numerical results for 0.2 < a/c < 2 and 0 < a/t f 0.8 for 0 < b, < w/2. Estimated maximum error possibly introduced by values from Tables 7 and 8 for compiex crack surface load will be within 8-9% for a corner crack at the edge of the finite plate. Present WF computation results for a crack surface load, eq. (46), are compared to the numerical results [20], as well as to the WF results from [22]. The comparison results are shown in Table 9 as a function of g. There is a good agreement between present WF results and FE results from [20]. Present WF results also show a better agreement with numerical results than the WF results from [22]. The maximum difference between present WF results and FE results is less than 5% in general, except for a few points for which the FE results are lower than the WF results. WF computations for crack surface load, eq. (47), are compared to numerical results from [20]. In the surface crack case, comparisons show that g function is rather weakly dependent on the geometries on which the crack is located. A comparison is also made for WF results and numerical results for a corner crack located at the edge of a finite plate and a corner crack located at the edge of a hole[23] in a plate (Fig. 4d). These comparisons are shown in Table 10.

681

Weight function estimation of stress intensity factors Table 8. SIFs for qua~r~llipti~

comer crack in a plate subjeoted to crack face load S:ty/c)**u

olc

olt

?J

ip*z/?r =o

0.2

0.2

0.2

0

0.5333

0.6761

0.8557

:

0.4257 0.4587

0.4911 0.5557

:

0.3893 0.4048

0

0.4

0.6

0.8

0.4

0.2

1.0

0.9903

1.0923

1.1815

0.4604 0.6082

0.5325 0.3023

0.1277 0.3492

0.0532 0.0921

0.4444 0.4064

0.3559 0.2779

0.1047 0.1760

0.0301 0.0559

0.0250 0.0350

0.6524

0.8047

1.0064

1.1626

1.2876

1.4078

:

0.4603 0.5096

0.5186 0.6027

0.4878 0.6601

0.3338 0.5939

0.1638 0.4224

0.0932 0.1766

:

0.4115 0.4317

0.4195 0.4627

0.2886 0.3724

0.1172 0.1948

0.0443 0.0773

0.0407 0.0587

0

0.8818

1.0421

1.2787

1.4725

1.6422

1.8268

: 3 4

0.5397 0.6295 0.4917 0.4602

0.7147 0.5845 0.5068 0.4512

0.5458 0.7729 0.4062 0.3098

0.3845 0.7034 0.2225 0.1341

0.5223 0.2087 0.1028 0.0610

0.2576 0.1386 0.0868 0.0597

0

1.2668

0.6965 0.9726

0.8649 0.6898

1.9452 0.8686 0.453 1 0.2561 0.1528

2.1902 0.6282 0.2494 0.1245 0.0751

2.4934

:

1.4188 0.9289 0.7081 0.5886 0.5093

0.7997

0.8955 0.4559 0.2520 0.1452 0.0865

0.9761 0.3075 0.1182 0.0560 0.0326

1*OS25 0.1052 0.0630 0.0417 1.1836 0.1526 0.0866 0.0560

:

0.5472 0.6019 0.6718

0.6448 0.4620 0.3436

0.2737 0.1705 0.1101 0.0764

: 3 4

0.5348 0.4710 0.4320 0.4040

0.7600 0.5334 0.4479 0.3911 0.3480

0 1 2 3 4

0.7757 0.5808 0.5010 0.4545 0.4221

0.7862 0.5648 0.4649 0.4015 0.3548

0.8877 0.5596 0.3977 0.2952 0.2238

0.9922 0.4894 0.1555 0.0935

1.0860 0.3480 0.1389 0.0687 0.0412

0 1 : 4

0.9483 0.6646 0.5557 0.4952 0.4547

0.9210 0.6206 0.4955 0.4206 0.3674

1.0201 0.6075 0.4205 0.3076 0.2311

1.1367 0.5382 0.2920 0.1686 0.1020

1.2527 0.4005 0.1643 0.0838 0.0513

1.3891 0.2073 0.1154 0.0737 0.0513

0.8

0 1 2 3 4

1.1818 0.7954 0.6403 0.5577 0.5041

1.0877 0.7039 0.5416 0.4496 0.3869

1.1741 0.6714 0.4499 0.3226 0.2393

I .3026 0.5923 0.3143 0.1798 0.1086

1.4493 0.4450 0.1833 0.~47 0.0586

1.6448 0.2358 0.1360 0.0872 0.9606

0.2

0

0.7285 0.5470 0.4644 0.4159 0.3821

0.7132 0.5132 0.4158 0.3546 0.3101

0.7541 0.4746 0.3324 0.2438 0.1830

0.8116 0.3957 0.2144 0.1225 0.0730

0.8687 0.2690 0.1057 0.0517 0.0309

0.9319 0.1047 0.0626 0.9410 0.0289

: 3 4

0.8162 0.5876 0.4904 0.4349 0.3973

0.7773 0.5372 0.4280 0.3617 0.3144

0.8129 0.4938 0.3411 0.2483 0.1856

0.8725 0.4162 0.2247 0.1288 0.0774

0.9378 0.2934 0.1185 0.0598 0.0364

1.0169 0.1330 0.0776 0.0504 0.0353

0 :

0.9573 0.659 1 0.5360 0.4683 0.4236

0.8962 OS234 0.3540 0.2547 0.1891

0.9589 0.4440 0.2374 0.1361 0.0822

1.0379 0.3223 0.1331

3 4

0.8735 0.5780 0.4492 0.3741 0.3222

1.1450 0.1621 0.0947 0.0612 0.0428

0

1.1368 0.7666 0.6043 0.5182 0.4627

0.983 1 0.6366 0.4801 0.3926 0.3339

0.9829 0.5601 0.3688 0.2612 0.1918

1.0472 0.4699 0.2465 0.1402 0.0845

1.1442 0.3388 0.1398 0.0729 0.0455

0.4

0.6

0.6

0.8

0

K~~~~~~~~~a)O 0.4 6

: 3 4 0.4

0.6

0.8

0

1 2 3 4

0.5303 0.3835 0.2872 0.2189

0.2690

0.0689

0.0427

0.0297

0.0394

1.2919 0.1640 0.1020 0.0669 0.0471 continuedoverleaf

682

G. S. WANG Table &-continued.

ale

ait

n

+*2/n = 0

0.2

0.8

0.2

0 1 2 3 4

0.7459 0.5366 0.4449 0.3923 0.3566

0.4

0 1 2 3 4 0

0.4

0.6

0.8

1.0

0.7114 0.4909 0.3886 0.3264 0.2824

0.7167 0.4336 0.2973 0.2150 0.1598

0.7410 0.3500 0.1869 0.1061 0.063 I

0.1757 0.2368 0.0937 0.0464 0.0279

0.8264 0.0992 0.0584 0.0377 0.0263

0.8190 0.5717 0.4671 0.4086 0.3695

0.7598 0.5094 0.3976 0.3314 0.2852

0.7565 0.4464 0.3026 0.2 176 0.1612

0.7807 0.3629

0.1934 0.1102 0.0660

0.8207 0.252 I 0.1021 0.0518 0.0317

0.8837 0.1169 0.0684 0.0441 0.0308

0.9337 0.6322 0.5054 0.4364 0.3914

0.8300 0.5400 0.4128 0.3399 0.2902

0.8107 0.4653 0.3100 0.2208 0.1627

0.8340 0.3790 0.2005 0.1143 0.0689

0.8828 0.2683 0.1107 0.0575 0.0357

0.967 1 0.1326 0.0788 0.0511 0.0357

0.9037 0.5827 0.4344 0.3522 0.2976 -.0.7007 0.4682 0.3650 0.3036 0.2608

0.8602 0.4867 0.3172 0.2229 0.1628 _0.6842 0.4018 0.2715 d. 1946 0.1437

0.8811 0.3902 0.2029 0.1148 0.0689

0.9416 0.2711 0.1114 0.0581 0.0364

2 3 4

1.0727 0.7209 0.5616 0.4773 0.4233 -_~ 0.7444 0.5182 0.4230 0.3693 0.3334

0.6838 0.3153 0.1666 0.0940 0.0557

0.6985 0.2111 0.0836 0.0415 0.0249

1.0557 0.1226 0.0797 0.0529 0.0373 ___~. 0.7376 0.0930 0.0534 0.0340 0.0235

0.4

0 1 2 3 4

0.8055 0.5485 0.4422 0.3833 0.3444

0.7376 0.4825 0.3717 0.307 I 0.2626

0.7118 0.4104 0.2748 0.1960 0.1444

0.7100 0.3235 0.1707 0.0967 0.0577

0.7283 0.2209 0.0893 0.0453 0.0276

0.7774 0.1044 0.0603 0.0386 0.0267

0.6

0 2 3 4

0.9001 0.5994 0.4743 0.4068 0.3629

0.7897 0.5054 0.3825 0.3128 0.2657

0.7476 0.4222 0.2787 0.1972 0.1446

0.7434 0.3328 0.1747 0.0991 0.0596

0.7679 0.2304 0.0947 0.0491 0.0305

0.8339 0.1137 0.0674 0.0435 0.0303

0 1 2 3 4

1.0109 0.6715 0.5201 0.4400 0.3889

0.8406 0.5358 0.3972 0.3206 0.2699

0.7755 0.4338 0.2812 0.1968 0.1433

0.7678 0.3364 0.1742 0.0983 0.0589

0.8007 0.2284 0.0937 0.0489 0.0306

0.8899 0.1030 0.0668 0.0442 0.0312

0.6

I 2 3 4

1.0

4 IS, iJ(n*a)

0.8

0 1 2 3 4

0.2

0

1

I

0.8

Table 9. The values of K,,/& at 4 = x/2 for a quarter-elliptical corner crack subjected to crack face loading S*(x/a)**n n=3

n=2

n=l

P21

[20]

WF

f221

[20]

WF

[221

[201

0.596 0.564 0.511 0.441

0.618 0.588 0.536 0.468

0.584 0.558 0.515 0.489

0.456 0.417 0.362 0.301

0.474 0.437 0.384 0.325

0.427 0.403 0.358 0.330

0.381 0.342 0.290 0.236

0.387 0.359 0.308 0.256

0.343 0,322 0.280 0.252

0.2 0.4 0.6 0.8

0.608 0.584 0.550 0.511

0.632 0.609 0.577 0.540

0.618 0.602 0.564 0.537

0.466 0.446 0.409 0.372

0.495 0.467 0.432 0.397

0.461 0.450 0.412 0.384

0.395 0.374 0.338 0.303

0.419 0.391 0.357 0.324

0.374 0.366 0.333 0.306

1.0

0.2 0.4 0.6 0.8

0.695 0.665 0.639 0.602

0.691 0.662 0.620 0.588

0.699 0.678 0.659 0.636

0.569 0.547 0.519 0.490

0.573 0.548 0.515 0.478

0.563 0.539 0.521 0.494

0.497 0.477 0.452 0.427

0.504 0.482 0.453 0.424

0.483 0.456 0.441 0.427

2.0

0.2 0.4 0.6 0.8

0.788 0.779 0.766 0.749

0.787 0.779 0.766 0.749

0.766 0.763 0.745 0.730

0.684 0.676 0.664 0.650

0.692 0.684 0.673 0.660

0.651 0.652 0.629 0.610

0.621 0.613 0.602 0.590

0.632 0.624 0.613 0.603

0.575 0.580 0.555 0.535

ale

ait

WF

0.2

0.2 0.4 0.6 0.8

0.4

683

Weight function estimation of stress intensity factors

Table 10. The values of K./K, at 6 = 1 for a quarter-elliptical comer crack subjected to crack face loading S*(y/c)**n a/t

WF

0.2

0.2 0.4 0.6 0.8

0.861 0.778 0.715 0.698

0.864 0.785 0.728 0.708

0.5

0.2 0.4 0.6 0.8

0.775 0.736 0.698 0.680

0.773 0.738 0.704 0.691

1.0

0.2 0.4 0.6 0.8

0.673 0.667 0.662 0.667

0.700 0.700 0.706 0.708

1.5

0.2 0.4 0.6 0.8

0.638 0.634 0.632 0.637

0.657 0.654 0.653 0.659

2.0

0.2 0.4 0.6 0.8

0.623 0.621 0.620 0.624

0.646 0.644 0.641 0.638

n=3

n=2

n=l

P21

ale

PI

WF

PA

0.799 0.703 0.613 0.557

0.813 0.719 0.634 0.573

0.785 0.746 0.712 0.701

0.669 0.624 0.575 0.541

0.677 0.634 0.588 0.559

0.702 0.695 0.681 0.666

0.556 0.543 0.528 0.520

0.578 0.568 0.559 0.551

0.671 0.673 0.667 0.658

0.514 0.507 0.500 0.498

0.531 0.525 0.518 0.517

0.495 0.491 0.487 0.486

0.514 0.510 0.506 0.503

[23]

0.873 0.800 0.739 0.632

0.699 0.673 0.644 0.611

0.642 0.642 0.623 0.610

PO1

WF

WI

WI

0.776 0.659 0.559 0.486

0.760 0.677 0.581 0.502

0.745 0.656 0.581 0.439

0.665 0.618 0.596 0.578

0.605 0.559 0.507 0.467

0.615 0.570 0.521 0.485

0.561 0.559 0.545 0.532

0.488 0.473 0.454 0.441

0.507 0.493 0.478 0.461

0.534 0.535 0.528 0.517

0.444 0.437 0.428 0.423

[23]

0.800 0.712 0.640 0.506

0.563 0.534 0.504 0.468

0.492 0.497 0.473 0.460

0.424 0.420 0.414 0.412

0.439 0.436 0.431 0.427

[231

0.580 0.532 0.512 0.499 0.483 0.452 0.424 0.389

0.484 0.475 0.461 0.443

0.459 0.452 0.444 0.439

0.452 0.455 0.446 0.436

0.406 0.414 0.388 0.378

Table 10 shows that there is a close agreement between present WF results and FE results for an edge crack in a plate. The difference is less than 5% for all cases, except for deep crack configurations where WF results are generally higher than FE results. Present WF results are more close to FE results than the results from [22]. The results from [22] are higher than the present results. For this crack configuration, outer boundaries have strong influence on the crack surface displacement. Hence, the crack surface displacement assumption derived from an embedded crack in [22] is expected to be less than the present assumption, thus present WF results will be more accurate than those of [22]. Comparisons of g function (K,,/&) for an edge crack in a finite plate with that for an edge crack at a hole[23] show rather close agreement, as can be seen in Table 10. The maximum difference between WF results for a crack in a plate and numerical results for a crack at a hole is less than 5%, except for a few points for which the difference approaches 7%. The difference is well within the derivation band of WF results and FE results for the corner crack in a plate. This investigation shows that the whole corner crack problem can be rather simply solved by the knowledge of the corner crack in a plate with satisfactory accuracy by a superposition method described by eq. (40) [or eq. (42)] with {g} obtained from a corner crack on the edge of a plate for a polynomial crack face load. An example is given below. The example is to determine the SIF for an open hole under remote load. Consider a corner crack emanating from an open hole in a large plate loaded with a remote uniaxial stress 0 perpendicular to the crack plane. The corresponding SIFs for a uniform crack surface load were computed with the finite element-alternating method[23] for a hole t/R = 2. For a configuration a/c = 1.5 and a/t = 0.2, the SIF can be computed by the following steps. The first step is to determine the unflawed hoop stress CT~caused by the remote load. From the exact solution[24], gH is given. Then, least squares procedures are employed to express the results in polynomial expansion as 2 = 2.9410 - 5.3863(x/R) + 6.3767(x/R)* - 2.7347(x/R)’ for 0 < x/R < 1, with errors less than 2%. R in eq. (58) represents the hole radius. Now, by g, given in Table 10 and the relation eq. (42) (L, = R), we have g,, = 1.000

S;(c/R)O = 2.9410

g, = 0.638

S;(c/R)’ = - 1.4363

684

G. S. WANG g, =

0.514

g, = 0.444

s;(c/R)2 = 0.4535 &(c/R)3 = -0.0519.

Substituting the above results into eq. (42) with & = 0.6296s,,/(xa) K = 0.6296~~~~~)(2.9410

obtained from [23] yields

- 1.4363 x 0.638 + 0.4535 x 0.516 - 0.0519 x 0.444)

= 1.407fl&ra).

09) The corresponding result obtained by empirical equation [13] is K = 1.464aJ(xa) (a 3.9% difference). The difference is partly due to the error in fitting the exact hoop stress with the polynomial series (error 2%). In the same way, the SIF for residual stress resulting from a cold working process can also be caiculated[23]. 7. CONCLUSION A weight function procedure has been developed to analyse SIFs under complex crack surface load for part-elliptical Mode I crack problems. Its effectiveness and efficiency have been investigated. The investigation shows that a reference SIF solution can provide enough information to solve the same crack problem under a general load case with satisfactory accuracy. ~C~~o~Ze~ge~~f-Financial

support by the Swedish Board for Technical Development is gratefully acknowl~g~.

REFERENCES Me& 50(g), 529-546 (1970). J. R. Rice, Some remarks on elastic crack-tip stress fields. Int. J. Solids Struct. 8, 751-758 (1972). 0. L. Bowie and C. E. Freese, Cracked-m&angular sheet with linearly varying end displa~ents, &gq Fracture Me& 14, 519-526 (1981). X. R. Wu and J. Carlsson, The generalised weight function method for crack problems with mixed boundary conditions. J. Mech. Phys. Solids 31, 485497 (1983). G. S. Wang, Approximate weight functions for RMS-averaged stress intensity factor. Submitted for publication in Int. J. Fracture. C. Mattheck, P. Morawietz and D. Munz, Stress intensity factor at the surface and at the deepest point of a semi-elliptical surface crack in plates under stress gradients. Znt. J. Fracture 23, x1-212 (1983). V. A. Vainshtok and I. V. Varfofomeyev, Application of the weight function for determining stress intensity factors of semi-elliptical cracks. ht. J. Fracture 35, 175-186 (1987). T. Fett, The crack opening displacement field of semi-ellipti~l surface cracks in tension for weight functions applications. Int. J. Fracfure 36, 55-69 (1988). T. Fett, An approximat~ve crack opening displacem~t field for embedded cracks. Engng Fracture Mech. 32,731-737 (1989). H. Tada, The stress analysis of cracks. Handbook, Del Research Corporation (1986). M. K. Kassir and G. C. Sih, Three Di~ens~onol Crock Problems. Noordhoff, Leyden (1975). Y. Murakami et al., Stress Intensify Factors Handbook. Pergamon, Oxford (1987). J. C. Newman, Jr. and 1. S. Raju, Stress intensity factor equations for cracks in three dimensional finite bodies. NASA technical memorandum 83200, pp. l-49 (1981). X. R. Wu, Stress intensity factors for half-elliptical surface cracks subjected to complex crack face loadings. Engng

H. 0. Bueckner, A novel principle for the computation of stressintensity factors. 2. ungew. Math.

Fracture Mech. 19,387-405 (1984). 3.J. McGowan and M. Raymund, Stress intensity factor solutions for internal longitudinal semi-elliptical surface flaws _ in a cylinder under arbitrary loadings. ASTM STP 677, 365-380 (1979). 3. Heliot, R. C. Labbens and A. Pellissier-Tanon, Semi-elliptical cracks in a cylinder subjected to stress gradients. ASTM STP 677, 341-364 (1979). 1. J. McGowan, _Enp. Me& 20, 253-264 (1980). M. Isida. H. Noauchi and T. Yoshida. Znt. J. Fracrure 25. 157-188 (1984). J. C. Newman, J<. and 1. S. Raju, &z&g Fracture Mech. i5, 185-19i (1981). M. Shiratori, Bulletin 35 of the Faculty of Engineering, Yokohama National University, pp. 1-25 (1986). 1. S. Raju and J. C. Newman, Jr., J. Press. Vess. Technol. 104, 293-298 (1982). V. A. Vainshtok and I. V. Varfoiomeyev, Stress intensity factor analysis for part-elliptical cracks in structures. Int. J. Fracture 46, 1-24 (1990).

A. F. Grandt, Jr. and T. E. Kullgren, Stress intensity factors for corner cracked holes under general loading conditions. J. Engng Mater. Technol. 103, 171-176 (1981). [24] S. Timoshenko and J. N. Goodier, Theory of Elasticity, 2nd Edn. McGraw-Hill, New York. (Received 8 February 1991)