Stress intensity factors and weight functions for high aspect ratio semi-elliptical surface cracks in finite-thickness plates

Pergamon

Engineering Fracture Mechanics Vol. 57, No. I, pp. 13-24, 1997 :~ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0013-7944(97)00018-0 0013-7944/97 $17.00 + 0.00

STRESS INTENSITY FOR HIGH

FACTORS

ASPECT

CRACKS

AND

WEIGHT

RATIO SEMI-ELLIPTICAL

IN FINITE-THICKNESS

FUNCTIONS SURFACE

PLATES

XIN W A N G and S. B. LAMBERT Department of Mechanical Engineering, University of Waterloo, Ontario, Canada N2L 3Gl Abstract--Three-dimensional finite element analyses have been conducted to calculate the stress intensity factors for high aspect ratio semi-elliptical cracks. The stress intensity factors are presented for the deepest and surface points on semi-elliptic cracks with aspect ratios of 1.5 and 2, and a/t values of 0.2, 0.4, 0.6 and 0.8. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were combined with corresponding results for lower aspect ratio surface cracks to derive weight functions over the range 0.6<_ale< 2.0 and 0.0 <_a/t <_0.8, The weight functions were then verified against finite element data for parabolic or cubic stress distributions. Differences were less than 3% for the surface point and 6% for the deepest point. The present results complement weight functions for low aspect ratio cracks, 0.0_< ale<_ 1.0, developed previously by the authors. (c:2, 1997 Elsevier Science Ltd.

1. INTRODUCTION SURFACE CRACKS frequently initiate and grow at notches, holes or welded joints in structural components. A large percentage of the useful life of these components is consumed while the cracks are small compared to the component thickness. Hence, understanding and quantifying the severity of surface cracks are important parts of the development of life prediction methodologies. Current methodologies use the stress intensity factor to quantify the severity of cracks. Development of stress intensity factor solutions for surface cracks using analytical, numerical and semi-analytical methods has continued for the last two decades. The most accepted stress intensity factor solutions for surface cracks in finite thickness plates were obtained using the finite element method[I-3]; other methods have been compared with FEM solutions to confirm their accuracy and convergence. Reference[4] clearly demonstrates that even in FEM, the experience and insight of the analyst are needed to accurately model complex crack configurations and to extract stress intensity factor values. Stress intensity factor solutions for semi-elliptical surface cracks in finite thickness plates were obtained using the finite element method by Raju and Newman[l], Shiratori et al.[2], and Wang and Lambert [5]. Newman and Raju[1] obtained results for remote tension and bending, and presented their results using an empirical equation. Shiratori et al. [2], and Wang and Lambert [5] obtained results for constant, linear, parabolic or cubic stress distributions on the crack face. The stress intensity factor for an application can be obtained by superposition of these solutions provided the stress distribution has been expressed as a polynomial of order three. Wang and Lambert [5, 6] then used the finite element results to generate weight functions for semi-elliptic surface cracks in finite thickness plates. The weight function method removes the limitation on the stress distribution used. The weight function derived by Wang and Lambert [5] is applicable to cracks with aspect ratio (a/c--depth over the surface half length) between 0 and 1. In practice, surface cracks with ale larger than 1 may emerge during certain stages of the propagation phase of fatigue. It is necessary to calculate stress intensity factors for surface cracks with a/c larger than 1 using finite element methods and to extend the weight functions to higher aspect ratios in order to accurately account for nonlinear stress distributions which can occur as a consequence of stress concentrations and/or residual stresses. In this paper, three-dimensional finite element analyses have been used to calculate the stress intensity factors for high aspect ratio semi-elliptic surface cracks. The stress intensity factor results are presented for the deepest and surface points on semi-elliptic cracks with aspect 13

14

XIN WANG and S. B. LAMBERT

m

Fig. 1. Geometry and coordinate system. ratios, a/c, of 1.5 and 2, and a/t (depth over plate thickness) values of 0.2, 0.4, 0.6 and 0.8. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were combined with corresponding results for lower aspect ratios from Shiratori et al. [2] to derive the weight functions over the range 0.6 < a/c < 2.0 and 0.0 < a/t < 0.8. The weight functions are then verified using the results for parabolic or cubic stress distributions.

2. FINITE ELEMENT ANALYSIS FOR HIGH ASPECT RATIO SURFACE CRACKS 2.1. Three-dimensional finite element model Three-dimensional finite elements were used to model the symmetric quarter of a plate containing a semi-elliptic surface crack. Figure 1 shows the geometry and co-ordinate system used. The finite element analyses were made using ABAQUS, version 5.4 [7], with 20-noded isoparametric three-dimensional solid elements and reduced integration. In order to model the square root singularity at the crack tip, three-dimensional prism elements with four mid-side nodes at the quarter points (a degenerate cube with one face collapsed) were used and the separate crack tip nodal points were constrained to have the same displacement[8]. The stress intensity factor, K, was calculated from the J-integral which was calculated using the domain integral method [7]. The analyses were made with a linear elastic material model using a Young's modulus, E, of 207 GPa and Poisson's ratio, v, of 0.3. The relationship for plane strain between J and K was used to calculate K

K=

(l-v2)

'

(l)

except at the surface point of the crack where the relationship for plane stress was used K = 4~-E.

(2)

Loads were applied directly to the crack surface. Four types of loading were applied to each crack geometry, with the following stress distributions a ( x ) = or0

(:) 1 -

,

with n = 0, 1, 2 or 3; ao is the nominal stress and a is the crack depth.

(3)

Semi-elliptical surface cracks in finite-thickness plates

15

Fig. 2. Typical finite element mesh (part), a/c= 2.0, a/t = 0.2.

A mesh generator has been developed to generate all required input files for the analysis. An elliptical transformation was used to form the crack tip mesh. Therefore, the lines of elements around the crack tip were elliptic or hyperbolic, so that intersecting lines were orthogonal as required for the evaluation of the stress intensity factors [1,9]. A typical model for the present analysis used about 1000 elements and had 15000 degrees-of-freedom. A detail of a typical mesh is plotted in Fig. 2. The stress intensity factor results have been normalised as follows F -

K

(4)

~0 v/Yh-7~'

where F is the boundary correction factor, Q is the shape factor for an ellipse and is given by the square of the complete elliptic integral of the second kind. The following empirical equation for Q was used[2]: for a/c <_ 1.0 / a - X 1-65

Q = 1 . 0 + 1 . 4 6 4 1 7] \ /

;

(5a)

for a/c > 1.0

Note that the r -~/2 singularity vanishes at the intersection of three free surfaces[10] such as the surface point of the crack. That is, the r -~/2 singularity occurs only near crack front points embedded entirely in the material. However, as shown in ref. [10], for engineering materials with a Poisson ratio, v -- 0.3, the dominant singularity near the surface point of a surface crack is r -°4523, which in practical terms does not represent a dramatic departure from the r -~/2 singularity. Also, in the present calculation, the domain integral method was used to evaluate stress intensity factors. Therefore, the stress intensity factor calculated for the surface point was in fact an average value over the element size. The stress intensity factor for the surface point of the surface crack should be considered a reasonable physical approximation of the state of affairs at the surface.

2.2. Verification of the finite element model 2.2.1. Verification with exact solutions. In order to verify the finite element model, stress intensity factors for embedded elliptical cracks in an infinite body were calculated. In the finite element models, w and t were large enough (w/c and t/a > 10) so that the free boundary had a negligible effect on the stress intensity factors[l]. The number of elements used was similar to that used in the surface crack models. Comparisons were made between the stress-intensity factors calculated from the finite-element analysis and from the exact solutions for an embedded circular crack

16

X1N W A N G and S. B. L A M B E R T

Table 1. Comparison of boundary correction factor F from present F E M calculation and exact solution Fe for an embedded elliptical crack in an infinite body, constant load on crack surface, a/c = 2 2q~/rt

Exact solution Fe

Present calculation F

Difference % 100 x I F - Fel/F~

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1

1.4142 1.4040 1.3736 1.3241 1.2574 1.1780 1.0953 1.0273 1

1.3903 1.3804 1.3501 1.2997 1.2315 1.1507 1.0671 0.9991 0.9722

1.691 1.677 1.713 1.841 2.058 2.319 2.579 2.748 2.778

(a/c = 1) and an embedded high aspect ratio elliptical crack (a/c = 2). Constant loads were applied to the surface of the crack. For an embedded circular crack, the calculated K along the crack front was 0.33% above the exact solution. For an embedded elliptic crack of a/c = 2, the present solution for K along the crack front was within 2.78% of the exact solution. The stress intensity factor results of these calculations for a/c = 2 have been presented in Table 1 as a function of ~b, which is the parametric angle used to describe the position along the crack front, and are plotted graphically in Fig. 3. These results indicated that the present finite element models are suitable for the analyses of high aspect ratio elliptical cracks. 2.2.2. Verification with approximate solutions. In order to verify the finite element model's ability to handle nonlinear loading applied to the face of a surface crack, constant, linear, quadratic and cubic loads were applied to the crack surface of cracks with a/c = 1.0 and a/t = 0.2, 0.4, 0.6 or 0.8 to calculate the corresponding stress intensity factors. Comparisons for the deepest and surface points were made with results from Shiratori et al. [2]. The maximum difference was within 8.91% and most were within 5%. The results are presented in Table 2. Based on these results, the present finite element model was considered suitable for the analysis of high aspect ratio surface cracks loaded on the crack face.

2.00

1.60 u,.

2e

1.20

¢g

I

_

0.80

"0

¢ o m

Embedded crack a/c = 2.0 Constant load on crack face 0.40

~

FEM calculation Exact Solution

0.00

,,,,,,,,~l,,,,,~,,,i,,,,~,,,,I,,~,,,~,,I,,,,,,,, 0.00

0.20

0.40

0.60

0.80

1.00

Fig. 3. Verification for an embedded high aspect ratio elliptical crack in infinite body, a/c = 2.

Semi-elliptical surface cracks in finite-thickness plates

17

Table 2. Comparison of boundary correction factors F from present FEM calculation and Shiratori et al.'s [2] calculations Fs, for a/c= 1. (a) F from present finite element calculation. (b) Percentage of difference between F and Fs, 100 x

IF- #~oll&

(a) a(x)

Position

a/t = 0.2

a/t = 0.4

a/t = 0.6

a/t = 0.8

Constant

Surface Deepest Surface Deepest Surface Deepest Surface Deepest

1.0950 1.0427 0.9206 0.3282 0.8120 0.1812 0.7350 0.1274

1.1478 1.0667 0.9628 0.3441 0.8519 0.1950 0.7749 0.1376

1.2460 1.0860 1.0270 0.3477 0.8943 0.2014 0.8020 0.1440

1.3341 1.0837 1.097 0.3162 0.9582 0.1772 0.8503 0.1223

a(x)

Position

a/t = 0.2

a/t = 0.4

a/t = 0.6

a/t = 0.8

Constant

Surface Deepest Surface Deepest Surface Deepest Surface Deepest

4.36 0.41 4.09 7.97 3.55 6.61 2.79 8.91

5.92 1.50 4.67 4.28 3.08 3.20 1.27 2.75

5.42 1.80 4.92 1.68 4.25 0.73 3.60 0.75

7.41 2.10 6.52 3.28 5.21 3.14 3.69 3.69

Linear Parabolic Cubic (b)

Linear Parabolic Cubic

2.3. Finite element results f o r high aspect ratio surface cracks T h e stress intensity factors for high aspect ratio semi-elliptical surface cracks (a/c = 1.5 or 2.0) in a finite thickness plate with relative crack depths, aft, o f 0.2, 0.4, 0.6 o r 0.8, subjected to c o n s t a n t , linear, q u a d r a t i c o r cubic stress d i s t r i b u t i o n s as expressed in eq. (3) have been determined. The results are s u m m a r i s e d in Tables 3 a n d 4.

3. WEIGHT FUNCTIONS FOR

a[c FROM

0.6 TO 2

T h r e e - d i m e n s i o n a l finite element calculations o f stress intensity factors for high aspect ratio surface cracks with f o u r k i n d s o f stress d i s t r i b u t i o n have been presented in Section 2. M o r e c o m p l e x stress d i s t r i b u t i o n s m a y be e n c o u n t e r e d in practice due to t h e r m a l stresses, residual stress fields o r notches. O n e o f the m o s t efficient m e t h o d s to derive stress intensity factors for c o m p l e x stress d i s t r i b u t i o n s is to use weight functions. T h e weight f u n c t i o n m e t h o d was conceived by B u e c k n e r [ l l ] a n d Rice[12] a n d has been used b y several a u t h o r s to generalise the stress intensity factor solutions for cracks subjected to a r b i t r a r y l o a d i n g [13]. F o r a o n e - d i m e n s i o n a l v a r i a t i o n o f stresses acting across the p o t e n t i a l c r a c k plane, the basic relation between the stress intensity factor a n d stress d i s t r i b u t i o n is given by

K =

tr(x)m(x, a) dx,

(6)

where tr(x) is the u n c r a c k e d stress d i s t r i b u t i o n a p p l i e d to the crack face a n d m(x,a) is the weight function, which varies with the p o s i t i o n c o - o r d i n a t e x a n d the crack length a. Once the weight Table 3. Boundary correction factors F for high aspect ratio semi-elliptical surface cracks a/c = 1.5, F = K/csox/(na/Q) a(x)

Position

a/t = 0.2

a/t = 0.4

a/t = 0.6

a/t = 0.8

Constant

Surface Deepest Surface Deepest Surface Deepest Surface Deepest

1.3008 I. 0106 1. 1105 0.2407 0.9875 0.1281 0.8980 0.0876

1.3532 I. 0316 1.1473 0.2519 1.0157 0.1358 0.9209 0.0934

1.4091 1.0341 I. 1875 0.2527 1.0471 0.1361 0.9466 0.0936

1.4822 1.0422 1.2466 0.2418 1.0960 0.1227 0.9882 0.0803

Linear Parabolic Cubic

18

XIN WANG and S. B. LAMBERT

Table 4. Boundarycorrection factors F for high aspect ratio semi-ellipticalsurfacecracks a/c = 2, F = K/oox/(na/Q)

a(x)

Position

a/t = 0.2

a/t = 0.4

a/t = 0.6

a/t = 0.8

Constant

Surface Deepest Surface Deepest Surface Deepest Surface Deepest

1.4608 0.9833 1.2704 0.1871 1.1423 0.0921 1.0472 0.0610

1.5177 1.0082 1.3116 0.1969 1.1750 0.0977 1.0741 0.0645

1.5456 1.0004 1.3321 0.1920 1.1909 0.0940 1.0870 0.0616

1.6115 1.0156 1.3853 0.1862 1.2346 0.0845 1.1239 0.0518

Linear Parabolic Cubic

function is known for a particular situation, the stress intensity factor can be obtained for any given stress field through integration of eq. (6). There are several ways to obtain the weight function, m(x,a). Glinka and Shen [13] suggested a general form which could approximate weight functions for a variety of one- and two-dimensional cracks, and a method to derive the weight function from two reference stress intensity factor solutions. Using this method, Wang and Lambert [5] derived the weight functions for surface cracks with 0 < a/c ~ 1.0. Using the same approach and the finite element results for high aspect ratio cracks presented in Section 2, together with the finite element results for a/c = 0.6 and 1.0 by Shiratori et al. [2] for constant and linear stress fields, weight functions over the range 0.6 < a/c < 2.0 were obtained for the deepest and surface points. These results were then validated using the results for parabolic and cubic stress distributions.

3.1. General weight function forms Glinka and Shen[13] found that weight functions for a variety of one- and two-dimensional cracks could be approximated using the following expression:

2 m(x,a)_ /2rr(a_x ) [ I + M , ( I _ o ) x '/2 + M 2 (1 _ x ) +

M3 (1

x ) 3/2] ,

(7)

where the crack tip is at x = a, and Ml, M2 and M3 are constants. When applied to a surface crack, the weight function for the deepest point is[14] 2

mA(x,a)_x/2rr(a_x)[1.q._M1A( 1 x )

t/2

q-M2A ( 1 - x ) q- M3A(1 -- x)3/2],

(8)

where Mla, M2A and M3a can be decided from two reference solutions (constant and linear stress distributions) and a third condition. The weight function for the surface point is[14]

mB(X, a) = ~

2

[1

+ Mlu

( x ) '/2

+ M2u

( x ) ( x )

+ M3•

3/2]

,

(9)

where MIB, M2B and M3B can similarly be decided from two reference solutions and a third condition. As explained in ref. [14], the third condition for the deepest point is that the second derivative of the weight function be zero at x = 0 leading to MZA = 3.

(10)

The third condition for the surface point is that the weight function equals zero at x = a, which gives 1 + MIB + M2B + M3B = 0.

(11)

3.2. Weight function .for the deepest point of a semi-elliptical surface crack Two reference solutions are used to decide Mla, M2A and M3A in eq. (8): uniform tension and linear decreasing stress corresponding to n = 0 and n = 1 in eq. (3), respectively. The

Semi-elliptical surface cracks in finite-thickness plates

19

weight function for the deepest point of a surface crack is derived from these two reference stress intensity factor solutions and the condition given by eq. (I0). 3.2.1. Reference stress intensity factors. For the deepest point of surface crack, the numerical solutions for a/c = 1.5, 2.0 presented in Section 2, and the numerical solution for a/c = 0.6, 1.0 obtained by Shiratori et al. [2] were approximated by empirical formulas fitted with an accuracy of 2% or better. The range of applicability for these equations is 0.6
(12)

are: KAr =o'0

Yo,

(13)

where

Yo=Bo+B1

(t) (:)' +B2

B 0 = 1 . 1 1 2 - 0 . 0 9 9 2 3 ( a ) + 0.02,954(a) 2

B1 = 1.138- 1.134

(a)

+0.3073

(o; c

B2 = - 0 . 9 5 0 2 + 0 . 8 8 3 2 ( a ) - 0 . 2 2 5 9 ( a ) 2. The results for a linearly decreasing stress field

(14) are: KAr = tr0

Y1,

where:

YI = Ao + AI

+A2

(15)

(a)4 7

Ao=0.4735-0.2053(~)+0.03662(~) 2

A 2 = - 0 . 2 0 0 6 - 0.9829

+ 1.237

- 0.3554

.

20

XIN W A N G and S. B. L A M B E R T

3.2.2. Weight function. By substituting eqs (12)-(15) into eq. (6) and applying the condition represented by eq. (10) the constants in the weight function can be determined: MIA =

zr 24 ~/z(2~(4Y° - 6Y1) - ~-

(16) (17)

M2A = 3

M3A =

~ Y 0

--MIA

-4

.

(18)

The weight function for the deepest point of a semi-elliptic surface crack can then be determined directly from eq. (8).

3.3. The weight function for the surface point of a semi-elliptical surface crack The two reference stress intensity factor solutions used to determine the weight function for the surface point of a semi-elliptic crack, eq. (9), were for uniform tension and linear decreasing stress fields. 3.3.1. Reference stress intensity factors. For the surface point of surface cracks, the finite element results presented in Section 2 for high aspect ratio surface cracks (a/c = 1.5, 2.0), and the finite element results for a/c= 0.6 and 1.0 obtained by Shiratori et al. [2] were approximated as follows, with an accuracy of 2% or better. The range of applicability for these equations is 0.6 < a/c < 2.0 and 0 < a/t < 0.8. The results for uniform tension or(x) = ~r0

(19)

are: 0.50

Deepest point, parabolic Present, FEM 0.40 -

•

Shiratori, FEM

--

Weight Function

o. o .~

a/c = 0.6 -

. o

^ 0.10

o.00

o

~

a/c = L O

0

cr ._.__...........

a/c=l.5

o

o

a/c = 2.0

~

,,,,,,,,,1,,,,,,,,,1,,,,,,,,,i,,,,,,,,,i,,,,,,,,, .00

0.20

0.40

0.60

0.80

1.00

a/t

Fig. 4. Comparison of the weight function based stress intensity factor and F E M data for parabolic stress distribution (deepest point).

Semi-elliptical surface cracks in finite-thickness plates

21

1.60

~

r~ 1.20

a/c=2.0 ~

a/c= 1.5

~ •

a/c= 1.0

o c~

C..)

a/c = 0.6 0.80

Surfacepoint,lxeabolic

o

0.40

PresentFEM •

Shhatori,FEb{ WeightFunction

0.00

'''''''"l'''''""l'''"""l"'"'"'l"''""' 0.20

0.00

0.40

0.60

0.80

1.00

a/t Fig. 5. Comparison of the weight function based stress intensity factor and FEM data for parabolic stress distribution (surface point).

K,"r : O 0

F0,

(20)

where Fo

ICo

C1(7) 2+

Co = 1.340-0.2872

a

4

a

+0.06611

C2=-0.1493 + 0.01208 ( a ) + 0.02215 ( a ) 2.

The results for the linear decreasing field

o,x,:o0(,x)

(21)

are: K~r =
F1,

(22)

22

XIN WANG and S. B. LAMBERT 0,50

Deepest Point, Cubic

0,40

Present, FEM •

Shiratori,FF_/vl

0.30

WeightFunction

0.20

~

~

'

~

-

~

0.10 9

0.00

a/c =0.6

~

°

o

o

o

9

~

~,=1.o We= 1.5

~

a/c = 2.0

JiliilillllillillillillltiJtlllltlliilllliitlllt~

0.00

0.20

0.40

0.60

0.80

1.00

a/t Fig. 6. Comparison of the weight function based stress intensity factors and FEM data for cubic stress distribution (deepest point).

FI = [Do

a 2 (a)

Do = 1.120 - 0.2442

a 4

a

(a) 2 + 0.06708 c

(a) (a) 2 D1 = 1.251 - 1.173 +0.2973 c

D2 = 0 . 0 4 7 0 6 - 0 . 1 2 1 4 ( a ) + 0 . 0 4 4 0 6 ( a ) 2.

3.3.2, Weightfunction. By substituting eqs (19)-(22) into eq. (6), and applying the condition represented by eq. (11), the weight function constants for the surface point can be determined: MIB = ~ ( 3 0 F , 7~

M2B = ~ ( 6 0 F 0

- 18Fo) - 8

(23)

- 90F,) + 15

(24)

M3B = --(1 + M1B + M2B).

(25)

The weight function for the surface point can then be determined directly from eq. (9). 3.4. Validation of derived weight functions The weight functions for the deepest and surface points derived in Sections 3.2 and 3.3 were validated using finite element results for two non-linear stress fields. Using eq. (6), stress

Semi-elliptical surface cracks in finite-thickness plates

23

1.60

;z.,

1.20

g

-

~ .3 ~.

a/c = 2.0

~

a/c = 1.5 a/c- 1.0

0.80

~

a/c 06

Surface point, cubic

•

0.40

shoo., Present, ~ v l

WeightFunction

0.00

,I,,,,,,,I,,,,,,,,,I,,,,,,,,,l'',,,'',,I,'',l,''' 0.00

0.20

0.40

0.60

0.80

1.00

a/t Fig. 7. Comparison of the weight function based stress intensity factors and FEM data for cubic stress distribution (surface point).

intensity factors were calculated for the following stress fields: o'(x) = c r o ( l - x ) 2

(26)

and ( X )

o'(x)=~0 1 -

3

.

(27)

The stress intensity factors for the deepest and surface points calculated from weight functions for quadratic and cubic stress distributions are shown in Figs 4-7. For a/c from 0.6 to 2 and 0 < a/t < 0.8, the difference between the stress intensity factors calculated from the weight function and all the finite elements results were better than 3% for the surface point and 6% for the deepest point.

4. CONCLUSIONS Three-dimensional finite element analyses have been conducted to calculate the stress intensity factors of high aspect ratio semi-elliptical surface cracks. Constant, linear, quadratic and cubic stress distributions were applied to the crack surface. The analysis procedures and results were verified using both exact and approximate solutions. Based on the present finite element calculations for stress intensity factors for high aspect ratio surface cracks, and finite element data for lower aspect ratios from Shiratori et al. [2], weight functions for the deepest and surface points were derived. They are valid for aspect ratios in the whole region 0.6 < a/c < 2. These weight functions were verified using stress intensity factors for two different non-linear stress fields. The results were compared with finite element solutions and the difference was found to be less than 6% for the deepest point and 3% for the surface point.

24

XIN WANG and S. B. LAMBERT

The closed form weight functions derived here make them suitable for two-dimensional crack fatigue growth prediction and fracture analysis. The weight function can be used to calculate stress intensity factors for any given stress field a(x). When used together with the weight function developed for 0 < a/c < 1 [5], the stress intensity factors for semi-elliptical surface cracks with aspect ratio in the range 0 _< a/c < 2 and relative depths 0 _< a/t < 0.8 can be calculated. Acknowledgements--The authors gratefully acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada. They also would like to thank Prof. G. Glinka for helpful discussions. ABAQUS was made available from Hibbit Karlsson and Sorensen, Inc. under academic license to the University of Waterloo.

REFERENCES I. Newman, J. C. Jr and Raju, I. S., An empirical stress intensity factor equation for the surface crack. Engng Fracture Mech., 1981, 15, 185-192. 2. Shiratori, M., Niyoshi, T. and Tanikawa, K., Analysis of stress intensity factors surface cracks subjected to arbitrarily distributed surface stresses. In Stress Intensity Factors Handbook, vol. 2, ed. Y. Murakami et al., 1987, pp. 698 705. 3. Raju, I. S., Mettu, S. R. and Shivakumar, V., Stress intensity factor solutions for surface cracks in flat plates subjected to nonuniform stresses. Twenty Fourth National Symposium on Fracture Mechanics, American Society for Testing and Materials, Gatlinburg, Tennessee, U.S.A., 1992. 4. Tan. P. W., Raju, I. S., Shivakumar, V. and Newman, J. C., An evaluation of finite element models and stress intensity factors for surface cracks emanating from stress concentrations. NASA Technical Memorandum 101527, 1988. 5. Wang, X. and Lambert, S. B., Stress intensity factors for low aspect ratio semi-elliptical surface cracks in finite thickness plates subjected to nonuniform stresses. Engng Fracture Mech., 1995, 51, 517-532. 6. Wang, X. and Lambert, S. B., Local weight functions for semi-elliptical surface cracks in finite thickness plates. Theor. appl. Fracture Mech., 1995, 23, 199-208. 7. ABAQUS User's Manual, Version 5.4, Hibbit, Karlsson and Sorensen, Pawtucket, R.I., U.S.A., 1994. 8. Barsoum, R. S., Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements. Int. J. numer. Meth. Engng, 1977, II, 85-98. 9. Banks-Sills, L., Application of the finite element method to linear elastic fracture mechanics. Appl. Mech. Rev., 1991, 44, 447-461. 10. Sih, G. C. and Lee, Y. D., Review of triaxial crack border stress and energy behaviour. Theor. appl. Fracture Mech., 1989, 12, 1-17. I 1. Bueckner, H. F., A novel principle for the computation of stress intensity factors. Z. angew. Math. Mech., 1970, 50, 129-146. 12. Rice, J., Some remarks on elastic crack tip field. Int. J. Solids Structures, 1972, 8, 751 758. 13. Glinka, G~ and Shen, G., Universal features of weight functions for cracks in mode I. Engng Fracture Mech., 1991, 40, 1135-1146. 14. Shen, G. and Glinka, G., Weight function for a surface semi-elliptical crack in a finite thickness plate. Theor. appl. Fracture Mech., 1991, 15, 247 255. (Received 1 November 1996)