Weight functions and stress intensity factors for corner quarter-elliptical crack in finite thickness plate subjected to in-plane loading

Weight functions and stress intensity factors for corner quarter-elliptical crack in finite thickness plate subjected to in-plane loading

PII: Engineering Fracture Mechanics Vol. 60, No. 2, pp. 221±238, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0013-...
Download PDF
548KB Sizes 0 Downloads 46 Views
Report

PII:

Engineering Fracture Mechanics Vol. 60, No. 2, pp. 221±238, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0013-7944/98 $19.00 + 0.00 S0013-7944(98)00006-X

WEIGHT FUNCTIONS AND STRESS INTENSITY FACTORS FOR CORNER QUARTER-ELLIPTICAL CRACK IN FINITE THICKNESS PLATE SUBJECTED TO IN-PLANE LOADING A. KICIAK G. GLINKA and M. EMAN Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 M. SHIRATORI Department of Mechanical Engineering and Materials Science, Yokohama National University, 156 Tokiwadai, Hodogaya-ku, Yokohama, 240, Japan AbstractÐApproximate weight functions for the pro®le and frontal plane crack front points of a corner quarter-elliptical crack in ®nite thickness plate, subjected to mode I, in-plane loading, were derived by using the method of universal weight functions by Shen and Glinka (Engineering Fracture Mechanics, 1991, 40, 1135±1146; Theoretical and Applied Fracture Mechanics, 1991, 15, 247±255.) Closed form expressions were obtained for the coecients of the weight functions. The coecients were derived from the reference stress intensity factor solutions obtained by Shiratori and Miyoshi (Stress Intensity Factors Handbook, Vol. 3, ed. Murakami et al. Pergamon, New York, 1992, pp. 591±597.) using the ®nite element method. A comparison of the stress intensity factors calculated using the weight functions with the ®nite element data for various applied stress distributions shows good accuracy of the present results. # 1998 Elsevier Science Ltd. All rights reserved KeywordsÐCorner quarter-elliptical crack, Stress intensity factor, Weight function..

1. NOMENCLATURE a Ai, A*i Bi, B*i c Ci, C*i Di, D*i Ei, E*i Fi G Gi, G*i K KA , KB KA i KBi B KA b , Kb

m mA, mB MiA MiB n Q r t W x y Yi G  s s(y) s0

pro®le plane length of quarter-elliptical corner crack coecients of ®t to geometry correction factor Y0 coecients of ®t to geometry correction factor Y1 frontal plane length of quarter-elliptical corner crack coecients of ®t to geometry correction factor Y2 coecients of ®t to geometry correction factor F0 coecients of ®t to geometry correction factor F1 geometry correction factor for frontal plane crack point, B, subjected to power law stress distribution strain energy release rate coecients of ®t to geometry correction factor F2 stress intensity factor stress intensity factor for point A or B, respectively stress intensity factor for pro®le plane crack point, A, for crack subjected to power law stress distribution stress intensity factor for frontal plane crack point, B, for crack subjected to power law stress distribution stress intensity factor for point A or B, respectively, for crack subjected to bending stress distribution weight function weight function for point A or B, respectively coecients of weight function for pro®le plane crack point, A (i = 1, 2, 3) coecients of weight function for frontal plane crack point, B (i = 1, 2, 3) exponent of power law stress distribution applied to crack surfaces semi-elliptical crack shape factor distance from crack tip plate thickness plate width coordinate along pro®le plane crack length coordinate along frontal plane crack length geometry correction factor for pro®le plane crack point, A, subjected to power law stress distribution contour around crack tip Poisson's ratio surface tractions stress distribution applied to crack surfaces reference stress. 221

222

A. KICIAK et al.

2. INTRODUCTION Corner cracks occur naturally in many structural components and may result in a premature failure [1±3]. Accurate stress intensity factor solutions for such cracks are needed for reliable prediction of crack growth. These available in handbooks [4, 5], are often insucient for particular engineering applications, especially for cracks subjected to non-uniform stress ®elds near notches or to thermal, or residual stresses. In such cases, stress intensity factors can be calculated by using numerical methods such as the ®nite element method. However, this usually involves a substantial computational e€ort. Instead, the weight function method, proposed by Bueckner [6] and Rice [7], can be used. With this method, in the absence of body forces, a stress intensity factor, K, can be calculated by integrating the product of surface tractions, s , and a weight function, m, along a perimeter, G. Z …1† K ˆ s m dG: G

Therefore, once the weight function, m, for a particular crack and body geometry has been determined, a stress intensity factor for any loading system applied to the elastic body can be calculated. Many approaches to the determination of weight functions were proposed [5, 8±10]. In particular, Glinka and Shen, [11], found universal expressions for weight functions for one-dimensional cracks. Subsequently, they developed weight functions for two-dimensional cracks [12, 13]. In this paper, weight functions and stress intensity factors were derived for a corner quarter-elliptical crack in a ®nite thickness plate, loaded by a stress distribution varying along the plate width (in-plane loading). They complement the previous solution by Zheng et al. [13], for a corner quarter-elliptical crack in a ®nite thickness plate loaded by a stress ®eld varying through the plate thickness. 3. WEIGHT FUNCTIONS FOR A CORNER CRACK SUBJECTED TO IN-PLANE LOADING Universal forms of mode I weight functions for a surface semi-elliptical crack in a plate shown in Fig. 1 were given [12], as:

Fig. 1. Surface semi-elliptical crack in a ®nite thickness plate subjected to loading varying through the thickness.

Stress intensity factors for corner quarter-elliptical crack

223

. for the deepest point, A

     2 x1=2 x x3=2 ‡M2A 1 ÿ ‡ M3A 1 ÿ ; mA …x; a† ˆ p 1 ‡ M1A 1 ÿ a a a 2p…a ÿ x†

…2†

. for the surface point, B

 x1=2 x x3=2  2 ‡M2B ‡ M3B : mB …x; a† ˆ p 1 ‡ M1B a a a px

…3†

Because of the limited number of terms used, the weight functions, (2) and (3), are only approximations. However, it was found [11, 12] that they enable reasonably accurate estimation of stress intensity factors. It should also be noted that both weight functions were derived for a uniformly distributed line load, s(x) = 1, being only a function of x-coordinate. Therefore, they only enable calculation of stress intensity factors for cracks subjected to stress distributions varying along the x-direction. The weight functions were successfully used earlier, [12±16], to determine stress intensity factors for semi-elliptical cracks in plates and thin-walled cylinders. The same general forms of weight functions were assumed to be valid for a corner quarterelliptical crack in a ®nite thickness plate, Fig. 2, loaded by a stress distribution, s(y), varying only along the plate width (in-plane loading). In this case, the location of the characteristic crack points, A and B, and the coordinate system are unchanged from the case of a surface crack in a plate (Fig. 1), with the x-axis directed through the thickness and the y-axis extending along the plate width. Since the crack loading changes along the plate width, the weight functions, (2) and (3), were rewritten to correspond to the direction of the stress ®eld variation. For the pro®le plane crack point, A  y1=2 y y3=2  2 ‡M2A ‡ M3A : …4† mA …y; c† ˆ p 1 ‡ M1A c c c py For the frontal plane crack point, B      2 y1=2 y y3=2 ‡M2A 1 ÿ ‡ M3A 1 ÿ : mB …y; c† ˆ p 1 ‡ M1B 1 ÿ c c c 2p…c ÿ y†

Fig. 2. Corner quarter-elliptical crack in a ®nite thickness plate subjected to loading varying along the width.

…5†

224

A. KICIAK et al.

A stress intensity factor for the point A or B can be calculated by using a special case of eq. (1) and by integrating the product of a stress distribution, s(y), and the weight function, mA(y,c) or mB(y,c), respectively. Zc s…y†mA …y; c† dy …6† KA ˆ 0

KB ˆ

Z

c 0

s…y†mB …y; c† dy:

…7†

The weight functions, eq. (4) or (5), should be considered only as approximate quantities associated with a surface layer near the point A or B, because of a change of strength of a stress ®eld singularity at a perpendicular intersection of the crack front with a free surface. Benthem [17], and Bazant and Estenssoro[18], showed that this strength changes from rÿ1/2 to a weaker rÿ0.4523 (for v = 0.3), r being the distance from the crack tip. This implies [19, 20] that the classical stress intensity factor, K and strain energy release rate, G, vanish at such a point. Shivakumar and Raju [21] investigated a possibility of existence of two singular stress ®elds in the neighborhood of a surface point, one being the classical rÿ1/2 cylindrical singularity, and the other one being a spherical singularity of strength rl. They found that the rÿ1/2-singularity predominates over 96± 98% of the crack front in the interior of a body. Conversely, the spherical singular stress ®eld is almost zero in the interior and attains the strength, l, equal to that obtained by Benthem [17] at the surface point. Since the majority of the crack front is dominated by the rÿ1/2-singularity, therefore, it is anticipated that the behavior of the surface layer is also governed by such a singularity. This was shown [18, 22] to cause the crack to intersect the free surface at an angle di€ering from 908. For this research, by assuming that the crack has a quarter-elliptical shape, the crack front intersects the free surface at the right angle. The weight functions, (4) and (5), assume existence of the rÿ1/2-singularity at the points A and B. It is not possible to change their character to re¯ect the weaker singularity near the free surface by adding more terms to the expressions (4) and (5), or by improving accuracy of integration in eqs (6) and (7). This problem also exists in other techniques for deriving stress intensity factors (e.g. ®nite element method) and must be specially addressed. However, the di€erence between the actual and classical singularities does not seem to be very signi®cant from an engineering analysis point of view. Since the stress intensity factors are calculated using the weight functions conforming to the classical de®nition of a stress intensity factor, they can be applied with currently available fracture mechanics criteria for crack growth. If the varying strength of the crack tip singularity were accounted for by using a parameter di€erent than K, then also the crack growth criteria would have to be expressed in terms of such a parameter. One of the possibilities might be the strain energy density factor as postulated by Sih et al.[19, 20].

4. DERIVATION OF WEIGHT FUNCTIONS FOR CORNER QUARTER-ELLIPTICAL CRACK Derivation of weight functions for a particular geometry using relations (4) and (5) requires determination of their parameters, MiA or MiB. They can be obtained from three independent equations for each point A or B. Such relations can be derived from three independent reference B loading cases with known stress intensity factors KA i or Ki . Alternatively, fewer independent reference stress intensity factor solutions can be used, provided that certain properties of weight functions or a displacement ®eld associated with a loaded crack are utilized. In particular Shen and Glinka[12, 23] developed a method for deriving the parameters using only two crack loading reference cases. However, because of the approximate nature of the weight functions, mA(y,c) and mB(y,c), the method based on two reference cases may sometimes lead to inaccurate results for certain types of stress distributions. In such a situation an improvement of accuracy is achieved by using three reference solutions. For this research, the reference stress intensity factors for corner quarter-elliptical cracks in a ®nite thickness plate of a large width (c/WE 0.2) were taken from Shiratori and Miyoshi [24] who obtained them for several stress distributions applied to crack surfaces by using the ®nite

Stress intensity factors for corner quarter-elliptical crack

225

element method. The cases of uniform, linearly decreasing and quadratically decreasing stress distributions were used to derive the parameters of weight functions, MiA and MiB (Fig. 3). The stress intensity factors for these stress distributions were given [24] in the form of geometry correction factors Yi or Fi (i = 0, 1, 2) for the point A or B, respectively. For the pro®le plane crack point A . Uniform stress distribution, s(y) = s0, K0A

r pa Y0 ; ˆ s0 Q

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) r pa A Y1 ; K1 ˆ s0 Q . quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 r pa Y2 : K2A ˆ s0 Q For the frontal plane crack point, B . Uniform stress distribution, s(y) = s0 K0B ˆ s0

r pa F0 ; Q

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) r pa B F1 ; K1 ˆ s0 Q . quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 r pa B F2 : K2 ˆ s0 Q

…8†

…9†

…10†

…11†

…12†

…13†

Q is a semi-elliptical crack shape factor (square of a complete elliptic integral of a second kind), approximated with the formulae by Rawe [25]:

Fig. 3. Reference loading cases: (a) uniform stress distribution; (b) linearly decreasing stress distribution; (c) quadratically decreasing stress distribution.

226

A. KICIAK et al.

8 ÿ < 1 ‡ 1:464 a 1:65 c Qˆ : 1 ‡ 1:464ÿc1:65 a

a c

1

a c

>1

:

…14†

The geometry correction factors: Y0, Y1, Y2 and F0, F1, F2 were given [24] for a variety of relative pro®le crack lengths, a/t, and aspect ratios, a/c. 4.1. Weight functions for pro®le plane crack point A Substitution of eqs (8)±(10) and (4) into (6) results in three linearly independent eqs (15)± (17), containing the parameters MiA.  r Zc y1=2 y y3=2  pa 2s0 Y0 ˆ 1 ‡ M ‡ M dy; …15† ‡M K0A ˆ s0 p 1A 2A 3A Q c c c py 0  r Zc y1=2  y y3=2  pa 2s0  y Y1 ˆ 1 ‡ M1A ‡ M3A ÿM2A dy; p 1 ÿ Q c c c c py 0

…16†

 r Zc y1=2 y y3=2  pa 2s0  y 2 Y2 ˆ 1 ‡ M1A ‡M2A ‡ M3A dy: p 1 ÿ py Q c c c c 0

…17†

K1A ˆ s0 K2A ˆ s0

By solving eqs (15)±(17), the parameters MiA (i = 1, 2, 3) were found in terms of the three reference geometry correction factors, Y0, Y1 and Y2.   42p 1 13 48 …18† M1A ˆ p Y0 ÿ Y1 ‡ Y2 ÿ ; 14 5 Q 7 M2A

  315p 1 ˆ p ÿ Y0 ‡ Y1 ÿ Y2 ‡ 21; 6 2 Q

  132p 2 21 64 Y0 ÿ Y1 ‡ Y2 ÿ : M3A ˆ p 22 5 Q 11

…19†

…20†

For convenience, the geometry correction factors, Yi (i = 0, 1, 2), were approximated with closed form expressions ®tted to the ®nite element data from ref. [24]. 4.1.1. Crack aspect ratios 0.2 E a/c E 1. . Uniform stress distribution, s(y) = s0 Y0 ˆ A0 ‡ A1 where

‡5:959579

 a

…21†

a3=2 a2 ‡1:36777 ; …21a† c c c c  a a2 a3 a4 a5 ÿ1 ‡9:960679 ÿ6:452157 ‡1:488832 ; A1 ˆ 0:031838 ‡ 3:97961 ÿ 7:220462 c c c c c A0 ˆ 1:454238 ÿ 2:793695

a1=2

aA2 ; t

ÿ 4:85518

…21b† A2 ˆ 4:911137 ÿ 20:190738

a1=2 c

 a

a3=2 a2 ‡43:871524 ‡10:202967 ; ÿ 36:55704 c c c

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) aB2 ; Y1 ˆ B0 ‡ B1 t

…21c†

…22†

Stress intensity factors for corner quarter-elliptical crack

227

where  1=2    2  3=2 a a a a ‡ 5:101376 ‡ 1:115533 ; …22a† ÿ 3:999018 c c c c     3  2 a a a ‡ 8:316849 ÿ 7:404074 B1 ˆ 0:031838 ‡ 5:157572 c c c  3  5 ÿ1 a a ÿ 4:427894 ‡ 0:879987 ; …22b† c c  1=2    2  3=2 a a a a ‡ 48:743199 ‡ 11:88452 ; …22c† ÿ 41:984596 B2 ˆ 4:910979 ÿ 21:1374 c c c c B0 ˆ 1:454223 ÿ 2:71323

. quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 aC2 Y2 ˆ C0 ‡ C1 ; t

…23†

where  3=2   1=2    2 ÿ1 a a a a ‡ 2:67 ÿ 3:55 ÿ 0:733 ; …23a† C0 ˆ 0:6876 ‡ 2:111 c c c c     3  2 a a a ‡ 5:084952 ÿ 6:808536 C1 ˆ 0:031838 ‡ 6:11831 c c c  4  5 ÿ1 a a ÿ 1:08445 ÿ 0:082662 ; …23b† c c  3=2  1=2    2 a a a a ÿ 51:008636 ‡ 56:855612 ‡ 14:803463 : C2 ˆ 4:910628 ÿ 23:048094 c c c c …23c† Expressions (21)±(23) were ®tted to the numerical data, [24], with accuracy better than 3% for relative pro®le plane crack lengths 0.1 Ea/tE 0.8. 4.1.2. Crack aspect ratios 1 < a/c E 2. . Uniform stress distribution, s(y) = s0 Y0 ˆ A0 ‡ A1

a a2 a4 ‡A3 ; ‡ A2 t t t

…24†

where   a a2 a3 ÿ1 ‡ 0:358198 ÿ0:100964 ; A0 ˆ 0:161688 ‡ 0:476333 c c c a a2 a3 ÿ 0:244615 ÿ0:0215 ; A1 ˆ ÿ0:531835 ‡ 0:95675 c c c aÿ1 a a2 ÿ0:07925 ÿ 1:489081 ; ‡ 0:566342 A2 ˆ 1:136289 c c c a a2 a3 ÿ0:091557 ; A3 ˆ 0:146955 ‡ 0:176902 ‡ 0:1155 c c c

…24a† …24b† …24c† …24d†

228

A. KICIAK et al.

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) a a2 a4 ‡B3 ; ‡ B2 Y1 ˆ B0 ‡ B1 t t t where

 a a2 a3 ÿ1 ˆ 0:103991 ‡ 0:772729 ÿ0:072192 ; ‡ 0:251772 c c c  a a2 a3 ‡ 0:182299 ÿ0:073218 ; B1 ˆ ÿ0:0755098 ‡ 0:074228 c c c aÿ1 a a2 ÿ0:831951 ‡ 0:133087 ; ÿ 0:016881 B2 ˆ ÿ0:772606 c c c  a a2 a3 ÿ 0:016218 ‡0:070209 ; B3 ˆ 0:76262 ÿ 0:522912 c c c B0

. quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 a a2 a4 ‡ C2 Y2 ˆ C0 ‡ C1 ‡C3 ; t t t where

a1=2 aÿ1 ˆ ÿ0:02351 ‡ 0:05337 ‡1:17 ; c c  a a2 ÿ 0:01505 ; C1 ˆ ÿ0:06741 ‡ 0:1752 c c aÿ1  a ÿ0:232 ÿ 0:113 ; C2 ˆ 0:3544 c c aÿ1  a ÿ0:3632 ‡ 0:1777 : C3 ˆ 0:4473 c c

C0

…24†

…24a† …24b† …24c† …24d†

…25†



…25a† …25b† …25c† …25d†

Equations (24) and (25) approximate the numerical data, [24], with accuracy better than 3% for relative pro®le plane crack lengths 0.1 Ea/tE 0.8.

4.2. Weight functions for frontal plane crack point B Analogously to the previous case, the coecients, MiB (i = 1, 2, 3), of the weight function, mB(x,a), for the frontal plane crack point, B, (Fig. 2) were determined from three reference stress intensity factors KB0 , KB1 , KB2 , for the uniform, linearly decreasing and quadratically decreasing stress ®elds, respectively. Substitution of relations (11)±(13) and (5) into (7) results in three independent equations, (26)±(28), containing the parameters, MiB.   r Zc    pa 2s0 y1=2 y y3=2 p 1 ‡ M1B 1 ÿ F0 ˆ ‡M2B 1 ÿ ‡ M3B 1 ÿ dy; …26† K0B ˆ s0 Q c c c 2p…c ÿ y† 0 K1B ˆ s0

  r Zc     pa 2s0 y y1=2 y y3=2 p 1 ÿ F1 ˆ 1 ‡ M1B 1 ÿ ‡M2B 1 ÿ ‡ M3B 1 ÿ dy; Q c c c c 2p…c ÿ y† 0 …27†

K2B ˆ s0

  r Zc     pa 2s0 y 2 y1=2 y y3=2 p 1 ÿ F2 ˆ 1 ‡ M1B 1 ÿ ‡M2B 1 ÿ ‡ M3B 1 ÿ dy: Q c c c c 2p…c ÿ y† 0 …28†

Stress intensity factors for corner quarter-elliptical crack

229

By solving eqs (26)±(28), the parameters, MiB, are determined from the geometry correction factors, F0, F1 and F2   90p 1 14 48 p  F0 ÿ F1 ‡ F2 ÿ ; …29† M1B ˆ 15 5 Q 5 M2B

  315p 1 ˆ p ÿ F0 ‡ F1 ÿ F2 ‡ 21; 6 Q

…30†

M3B

  252p 1 20 64 F0 ÿ F1 ‡ F2 ÿ : ˆ p 21 5 Q 7

…31†

The geometry correction factors, Fi (i = 1, 2, 3) for the relative pro®le plane crack lengths 0.1 E a/tE 0.8 and aspect ratios 0.2 E a/c E 2, were approximated with the following closed form expressions. 4.2.1. Crack aspect ratios 0.2 E a/c E 1. . Uniform stress distributions, s(y) = s0 F0 ˆ D0 ‡ D1

aD2 t

;

…32†

where a a2 a3 a4 a5 ‡3:482843 ÿ0:816993 ÿ0:005562 ; ÿ 5:038149 c c c c c a1=2  a1=2  a a3=2 a2  5:385 ÿ 4:042 ÿ9:44 ÿ4:515 D1 ˆ ‡ 13:29 ; c c c c c a  a a2 a3 a4 ÿ1 0:1132 ÿ 0:4537 ‡ 2:051 ÿ1:634 ‡0:4411 ; D2 ˆ c c c c c D0 ˆ 3:50086

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) aE2 ; F1 ˆ E0 ‡ E1 t

…32a† …32b† …32c†

…33†

where a2  a a2 a3 a4 ÿ1 ‡ 4:72 0:549 ÿ 0:3666 ÿ2:411 ‡0:5391 ; c c c c c a1=2   a a2 a3 a4 ÿ1 0:4568 ‡ 0:03256 ÿ3:794 ‡1:222 ; ‡ 4:43 E1 ˆ c c c c c a1=2   a a2 a3 a4 ÿ1 0:08081 ‡ 0:3305 ÿ0:8905 ‡0:3131 ; ‡ 0:6883 E2 ˆ c c c c c E0 ˆ

. quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 aG2 F2 ˆ G0 ‡ G1 ; t where

…33a† …33b† …33c†

…34†

230

A. KICIAK et al.

a2  a1=2 a a3=2 a2 ÿ1 3:29 ÿ 6:759 ‡6:695 ÿ0:283 ; …34a† ‡ 2:684 c c c c c a1=2  a a2 a3 a4 ÿ1 ‡ 3:743 0:6158 ‡ 0:6989 ÿ2:603 ‡0:6941 ; …34b† G1 ˆ c c c c c a a1=2  a a3=2 a2 a5=2 ÿ1 ‡0:9581 ÿ2:184 ‡0:8117 : …34c† 0:1685 ÿ 0:7799 ‡ 1:549 G2 ˆ c c c c c c G0 ˆ

The above expressions were ®tted to the numerical data, [24], with accuracy better than 3% for aspect ratios a/ce 0.4. For aspect ratios a/c < 0.4 the same numerical accuracy was achieved as for the higher aspect ratios. However, due to low values of the geometry correction factors at low relative pro®le crack lengths, the relative di€erences were larger, especially for the quadratically decreasing stress distribution.

4.2.2. Crack aspect ratios 1 < a/c E 2. . Uniform stress distribution, s(y) = s0 F0 ˆ D0 ‡ D1 where

aD2 ; t

…35†

a1=2 a a2 ‡1:864 ; ÿ 0:063 c c c  a a2 a3 ÿ1 ‡ 1:131 ÿ0:1117 ; D1 ˆ 0:07077 ‡ 0:3945 c c c aÿ1=2  a a2 ‡5:136 ÿ 2:177 ; ‡ 0:419 D2 ˆ ÿ1:445 c c c D0 ˆ 4:271 ÿ 4:949

…35a† …35b† …35c†

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) F1 ˆ E0 ‡ E1 where E0 ˆ 0:6103 ÿ 0:3874

a1=2 c

aE2 ; t

‡0:1172

 a c

…36†

ÿ 0:01029

a2 ; c

a1=2  a a2 ÿ 0:02394 ‡2:188 ; c c c  a a2 a3 ‡0:8971 ; ÿ 3:605 E2 ˆ 0:9961 ‡ 3:627 c c c

E1 ˆ 4:16 ÿ 5:898

…36a† …36b† …36c†

. quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 F2 ˆ G0 ‡ G1 where

aG2 t

;

…37†

Stress intensity factors for corner quarter-elliptical crack

a ; c c  a a2 ÿ1 ‡ 1:978 ; G1 ˆ 0:2167 ‡ 0:9536 c c a1=2 a2 ÿ0:05278 : G2 ˆ 2:231 ÿ 0:2672 c c G0 ˆ 0:1599 ‡ 0:07636

a1=2

ÿ0:05853

231

…37a† …37b† …37c†

Expressions (35)±(37) approximate the numerical data, [24], with accuracy better than 4%.

5. VALIDATION 5.1. Stress intensity factors for power law crack face loading To check accuracy of the derived weight functions, eqs (4) and (5), stress intensity factors at the points A and B of a corner quarter-elliptical crack loaded by power law stress distributions, eq. (38), were calculated by using eqs (6) and (7).  y n n ˆ 0; 1; 2; 3: …38† s…y† ˆ s0 1 ÿ c Since the uniform, linearly decreasing and quadratically decreasing stress distributions were used to derive the weight function coecients, a comparison of the calculated and reference stress intensity factors allows for evaluation of the self-consistency and accuracy of the method. Equations (4), (5) and (38) enabled derivation of closed form expressions for the stress intensity factors. 5.1.1. Stress intensity factors for point A. . Uniform stress distribution, s(y) = s0 p   K0A 4 Q q ˆ 4 ‡ 2M1A ‡ M2A ‡ M3A ˆ Y0 ; 3 p s0 pa

…39†

Q

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) p   Q 8 K1A 8 1 q ˆ ‡ M1A ‡ M2A ‡ M3A ˆ Y1 ; p 3 15 3 s0 pa Q . quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 p   Q 32 2 K2A 32 1 q ˆ ‡ M1A ‡ M2A ‡ M3A ˆ Y2 ; p 15 3 105 6 s0 pa Q . cubically decreasing stress distribution, s(y) = s0(1 ÿ y/c)3 p   Q 64 1 K3A 64 1 q ˆ ‡ M1A ‡ M2A ‡ M3A : p 35 2 315 10 s0 pa Q

…40†

…41†

…42†

5.1.2. Stress intensity factors for point B. . Uniform stress distribution, s(y) = s0 p   K0B 2 1 2Q q ˆ 2 ‡ M1B ‡ M2B ‡ M3B ˆ F0 ; 3 2 p s0 pa Q

…43†

232

A. KICIAK et al.

. linearly decreasing stress distribution, s(y) = s0(1 ÿ y/c) K1B q ˆ

s0

pa Q

p   2 1 2Q 2 1 ‡ M1B ‡ M2B ‡ M3B ˆ F1 ; 3 2 5 3 p

…44†

. quadratically decreasing stress distribution, s(y) = s0(1 ÿ y/c)2 K2B q ˆ

s0

pa Q

p   2 1 2Q 2 1 ‡ M1B ‡ M2B ‡ M3B ˆ F2 ; 5 3 7 4 p

…45†

. cubically decreasing stress distribution, s(y) = s0(1 ÿ y/c)3 K3B q ˆ

s0

pa Q

p   2 1 2Q 2 1 ‡ M1B ‡ M2B ‡ M3B : 7 4 9 5 p

…46†

5.2. Comparison of stress intensity factors for power law stress distributions Stress intensity factors calculated using the weight functions are compared to the ®nite element data from ref. [24] for cracks loaded by the power law stress distributions, eq. (38), in Figs 4, 5 and 6. Only the results for the aspect ratios, a/c = 0.2, 1, 2, are shown, but similar or better agreement was achieved in other cases. Relative di€erences are less than 4% for the pro®le plane crack point, A, within the whole range of crack aspect ratios, a/c, and relative pro®le crack lengths, a/t. For the frontal plane crack point, B, the di€erences are less than 6% for aspect ratios 0.4 E a/c E 2 within the whole range of relative pro®le crack lengths. For lower crack aspect ratios similar numerical accuracy was achieved as that for the higher aspect ratios. However, because of the low magnitudes of the reference ®nite element data, the relative di€erences between the solutions at the point B are high. Accuracy of the stress intensity factors calculated using the current weight functions increases with the increasing crack aspect ratio. Additionally, stress intensity factors calculated using the present weight functions are compared in Fig. 7 to the ®nite element data by Raju and Newman [26] for a crack in a plate under uniform tensile loading. The solutions di€er by less than 8% for the point A, and less than 10% for the point B. 5.3. Comparison of stress intensity factors calculated using weight functions to ®nite element data for bending stress distribution For the in-plane bending of a plate with the ratio of the frontal crack length to plate width, c/W, (the stress distribution being given as s(y) = s0(2y/W)), the stress intensity factor can be calculated as . for the pro®le plane crack point A

    KbA 2c 2c q ˆ Y0 1 ÿ ‡ Y1 ; pa W W s0

…47†

Q

. for the frontal plane crack point B

    KbB 2c 2c q ˆ F0 1 ÿ ‡ F1 : pa W W s0 Q

…48†

Stress intensity factors for corner quarter-elliptical crack

233

Fig. 4. Comparison of stress intensity factors calculated using the present weight functions with the ®nite element data, [24], for a crack loaded by power law stress distributions, eq. (38). Crack aspect ratio, a/c = 0.2:Ðweight function, W n = 0, . n = 1, Q n = 2, r n = 3; (a) pro®le plane crack point A; (b) frontal plane crack point B.

Stress intensity factors for c/W = 0.2 calculated using the weight functions are compared to the ®nite element data by Raju and Newman [26] in Fig. 8. Relative di€erences between these results are less than 4% for the pro®le crack point A and less than 10% for the frontal plane crack point B. In cases of relatively simple stress distributions, e.g. eq. (38), it is possible to obtain closed form stress intensity factor solutions. In more complex situations, the integration in eqs (6) and (7) must be performed numerically. An ecient method for calculating stress intensity factors using the weight functions was developed in ref. [27].

234

A. KICIAK et al.

Fig. 5. Comparison of stress intensity factors calculated using the present weight functions with the ®nite element data, [24], for a crack loaded by power law stress distributions, eq. (38). Crack aspect ratio, a/c = 1:Ðweight function, W n = 0, . n = 1, Q n = 2, r n = 3; (a) pro®le plane crack point A; (b) frontal plane crack point B.

6. CONCLUSIONS Weight functions and stress intensity factors for the pro®le and frontal plane points of a corner quarter-elliptical crack in a ®nite thickness plate subjected to a stress ®eld, s(y), varying along the plate width were derived using the universal weight functions by Shen and Glinka [11, 12]. Closed form expressions for the coecients of the weight functions were derived from three reference stress intensity factor solutions by Shiratori and Miyoshi [24] in the

Stress intensity factors for corner quarter-elliptical crack

235

Fig. 6. Comparison of stress intensity factors calculated using the present weight functions with the ®nite element data, [24], for a crack loaded by power law stress distributions, eq. (38). Crack aspect ratio, a/c = 2:Ðweight function, W n = 0, . n = 1, Q n = 2, r n = 3; (a) pro®le plane crack point A; (b) frontal plane crack point B.

range of relative pro®le crack lengths, 0.1 E a/t E0.8, and aspect ratios, 0.2 Ea/c E2. Stress intensity factors calculated using the present weight functions were compared to the ®nite element data for several stress distributions. Good numerical accuracy was found for cracks subjected to the power law stress distributions, eq. (38), applied to the crack surfaces. Comparisons with the ®nite element data by Raju and Newman [26] for plate under tension and in-plane bending showed di€erences not exceeding 10% within the whole range of relative pro®le crack lengths.

236

A. KICIAK et al.

Fig. 7. Comparison of stress intensity factors calculated using the present weight functions with the ®nite element data, [26], for a crack in a plate subjected to uniform tension:Ðweight function, W a/c = 0.2, . a/c = 0.4, Q a/c = 1, R a/c = 2; (a) pro®le plane crack point A; (b) frontal plane crack point B.

The present weight functions complement a set of solutions published earlier, for the surface and corner cracks subjected to one-dimensional stress distributions. They are particularly useful for computation of stress intensity factors in the analysis of fatigue crack growth. AcknowledgementsÐThe authors would like to thank Mr T. Matsumoto from the Department of Mechanical Engineering and Materials Science of Yokohama National University for his help with the ®nite element data for corner cracks.

Stress intensity factors for corner quarter-elliptical crack

237

Fig. 8. Comparison of stress intensity factors calculated using the present weight functions with the ®nite element data [26], for a crack in a plate subjected to in-plane bending:Ðweight function, W a/c = 0.2, . a/c = 0.4, Q a/c = 1, R a/c = 2; (a) pro®le plane crack point A; (b) frontal plane crack point B.

REFERENCES 1. Gran, R. J., Orazio, F. D., Paris, P. C., Irwin, G. R. and Hertzberg, R. H., Investigation and analysis development of early life structural failures. AFFL±TR-70-149, Air Force Laboratory, Wright-Paterson Air Force Base, Ohio, 1971. 2. McDermott, M. E. and Stephens, R. I., Experimental determination of K1 for hollow rectangular tubes containing corner cracks. In Fracture Mechanics, ASTM STP 677, ed. Smith, C. W., 1979, pp. 719±733. 3. Sahli, A. and Albrecht, P., Fatigue life of welded sti€eners with known initial cracks. In Fracture Mechanics: Fifteenth Symposium, ASTM STP 833, ed. Sanford, R. J.. American Society for Testing and Materials, 1984, pp. 193±217. 4. Tada, H., Paris, P. C. and Irwin, G. R., The Stress Analysis of Cracks Handbook, 2nd edn. Del Research Corp, St. Louis, Ms, 1985.

238

A. KICIAK et al.

5. Wu, X. R. and Clarsson, A. J., Weight Functions and Stress Intensity Factor Solutions. Pergamon Press, Oxford, 1991. 6. Buckner, H. F., A novel principle for the computation of stress intensity factors. Zeitschrift fuÈr Angewandte Mathematik und Mechanik, 1970, 50, 529±546. 7. Rice, J. R., Some remarks on elastic crack-tip stress ®eld. International Journal of Solids and Structures, 1972, 8, 751±758. 8. Besuner, P. M., The in¯uence function method for fracture mechanics and residual fatigue life analysis of cracked components under complex stress ®elds. Nuclear Engineering and Design, 1977, 43, 115±154. 9. Petroski, H. J. and Achenbach, J. D., Computation of the weight functions from a stress intensity factor. Engineering Fracture Mechanics, 1978, 10, 257±266. 10. Oore, M. and Burns, D. J., Estimation of stress intensity factors for embedded irregular cracks subjected to arbitrary normal stress ®elds. Journal of Pressure Vessels Technology, Transactions of the ASME, 1980, 102, 202±211. 11. Glinka, G. and Shen, G., Universal features of weight functions for cracks in mode I. Engineering Fracture Mechanics, 1991, 40, 1135±1146. 12. Shen, G. and Glinka, G., Weight functions for a surface semi-elliptical crack in a ®nite thickness plate. Theoretical and Applied Fracture Mechanics, 1991, 15, 247±255. 13. Zheng, X. J., Glinka, G. and Dubey, R. N., Stress intensity factors and weight functions for a corner crack in a ®nite thickness plate. Engineering Fracture Mechanics, 1996, 51, 49±61. 14. Shen, G. and Glinka, G., Stress intensity factors for internal edge and semi-elliptical cracks in hollow cylinders. In High Pressure±Codes, Analysis and Applications, ASME PVP-Vol. 263, ed. Khare, A., 1993, pp. 73±80. 15. Shen, G., Liebster, T. D. and Glinka, G., Calculation of stress intensity factors for cracks in pipes. In Proceedings of the 12th International Conference on O€shore Mechanics and Arctic Engineering, OMAE-93, ASME, 3B, 1993, pp. 847±854. 16. Forman, R. G., Mettu, S. R. and Shivakumar, V., Fracture mechanics evaluation of pressure vessels and pipes in aerospace applications. In Fatigue, Fracture and Risk, ASME PVP-Vol. 241, 1992, pp. 25±36. 17. Benthem, J. P., State of stress at the vertex of a quarter-in®nite crack in a half-space. International Journal of Solids and Structures, 1977, 13, 479±492. 18. Bazant, Z. P. and Estenssoro, L. F., Surface singularity and crack propagation. International Journal of Solids and Structures, 1979, 15, 405±426. 19. Sih, G. C. and Lee, Y. D., Review of triaxial crack border stress and energy behavior. Theoretical and Applied Fracture Mechanics, 1989, 12, 1±17. 20. Sih, G. C. and Li, C. T., Initiation and growth characterization of corner cracks near circular hole. Theoretical and Applied Fracture Mechanics, 1990, 13, 69±80. 21. Shivakumar, K. N. and Raju, I. S., Treatment of singularities in cracked bodies. International Journal of Fracture, 1990, 45, 159±178. 22. Pook, L. P., Some implications of corner point singularities. Engineering Fracture Mechanics, 1994, 48, 367±378. 23. Shen, G. and Glinka, G., Determination of weight functions from reference stress intensity factors. Theoretical and Applied Fracture Mechanics, 1991, 15, 237±245. 24. Shiratori, M. and Miyoshi, T., Weighting function for a quarter-elliptical crack in a plate under basic mode of stress distribution. In Stress Intensity Factors Handbook, Vol. 3, ed. Murakami, et al.. Pergamon Press, 1992, pp. 591±597. 25. Merkle, J. G., A review of some of the existing stress intensity factor solutions for part-through surface crack, ORNLTM-3983. Oak Ridge Laboratory, 1973. 26. Newman, J. C. and Raju, I. S., Stress intensity factor equations for cracks in three-dimensional ®nite bodies. In Fracture Mechanics: Fourteenth SymposiumÐVol. I: Theory and Analysis, ASTM STP 791, ed. Lewis, J. C. and Sines, G.. American Society for Testing and Materials, 1983, pp. I±238±I±265. 27. Moftakhar, A. and Glinka, G., Calculation of stress intensity factors by ecient integration of weight functions. Engineering Fracture Mechanics, 1992, 40, 749±756. (Received 1 August 1997, in final form 20 November 1997)