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 ...
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 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
