On the through-thickness crack with a curve front in center-cracked tension specimens

Engineering Fracture Mechanics 73 (2006) 2600–2613 www.elsevier.com/locate/engfracmech

On the through-thickness crack with a curve front in center-cracked tension specimens Zhixue Wu

*

Mechanical Engineering College, Yangzhou University, Yangzhou 225009, PR China Received 10 February 2006; received in revised form 22 April 2006; accepted 24 April 2006 Available online 19 June 2006

Abstract Three-dimensional finite element analyses are performed on through-thickness cracks with slightly wavy front in centercracked plates. Considering there is an inherent relationship between the crack shape and the corresponding stress intensity factor (SIF) distribution of a crack, the curved configuration of the crack is determined using a heuristically derived iterative procedure if the SIF distribution function is known. Several simple SIF distribution functions, for instance the constant SIF distribution along the crack front, are assumed to determine the crack shape. Under the assumption that the rate of fatigue crack growth depends on the SIF range or the effective SIF range, possible effects of plate thickness, crack length and crack closure level gradient on the behaviour of crack tunneling are investigated. The stability of the curved shape of a through-thickness crack in fatigue is also discussed, i.e. whether a crack can maintain its shape satisfying the conditions of constant SIF distribution or other distribution along the crack front during fatigue growth. This study will be useful for a better understanding of the behaviour of crack tunneling and help to evaluate the validity of the twodimensional linear elastic fracture mechanics in cracked plates. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Stress intensity factor; Through-thickness crack; Fatigue crack growth; Thickness effect; 3-D finite element analysis

1. Introduction It is well known that the state of deformation near the crack tip is always three-dimensional (3-D). The 3-D nature of this kind of cracks results in a SIF that is not only varying along the crack front but is also highly sensitive to the crack shape. For a through-thickness crack with straight front in a center-cracked plate of finite thickness, attempts have been made by many authors [1–5] to obtain the stress distribution close to the crack front. From their results, several conclusions can be reached: (i) the state of deformation at the crack front is plane strain except for the surface point; (ii) the SIF varies over the thickness and drops to zero at the plate surface; (iii) a boundary layer exists near the plate free surface. The complexity of the stress state at the crack front causes the question about the validity of two-dimensional (2-D) linear elastic fracture mechanics. *

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For a through-thickness crack under fatigue load, the case may become more complicated. It can be observed in experiments that an initially straight-fronted crack in specimens such as a center-cracked specimen usually grows to a slightly curved shape under a fatigue load. This phenomenon is called crack-front tunneling, i.e. the interior, plane-strain portion of a fatigue crack front often tunnels ahead of the surface, plane-stress portion of the crack front. Leblond et al. [6] investigated the possible bifurcation from the straight configuration of a crack front in an infinite elastic solid by applying the three-dimensional weight-function theory proposed by Rice [7]. They found that there is a bifurcation mode that corresponds to a sinusoidal shape of the crack front under the condition that the SIF be equal to a constant along the crack front. Bakker [8] predicted a relative tunneling depth of 2.5–3% based on constant SIF distribution for self-similar crack propagation using finite element method. However, experimentally observed tunneling depths on fatigue cracks are generally larger, close to 5% [9]. This means that the SIF distribution may not be, generally, constant for a fatigue crack. For a center-cracked plate of finite width and thickness, the SIF is generally evaluated by pffiffiffiffiffiffi K ¼ r0 paF ; ð1Þ where r0 is the applied tension stress remote from the crack, a is the half-crack length, and F is the boundarycorrection factor. For a through-thickness crack with a slightly wavy front, as shown in Fig. 1, the SIF varies, in general, over the thickness. The SIF is generally estimated according to the crack length measured at the surface because surface crack extension can be easily measured. The surface measured value may be the shortest crack length if the specimen is experiencing crack-front tunneling. Then, it is necessary to elucidate the meaning of the SIF solution evaluated using Eq. (1) and its relation to the 3-D solution. It is of some importance to estimate the possible error of the SIF due to the influence of crack-front tunneling. Assuming we want to estimate the SIF at mid-plane point based on the crack length measured at the surface, an error is inevitable due to the curved shape a crack. According to Eq. (1), the relative error of SIF caused by Da can be expressed approximately as DK 1 Da ¼  : K 2 a

ð2Þ

Our objective here, then, is try to understand the behaviour of crack tunneling in order to evaluate the validity of the 2-D linear elastic fracture mechanics in cracked plates. In the present paper, Three-dimensional finite element analyses are performed on center-cracked plates with through-thickness cracks with a slightly wavy front. Assuming the SIF distribution for a crack satisfying the given functions, the curved crack shape is determined using a heuristically derived successive iterative procedure. Under the assumption that the fatigue crack growth law is that of Paris or that of Elber [10], the stability of the curved shape of a crack in fatigue and the possible effects of plate thickness, crack length and crack closure level gradient on the behaviour of crack tunneling are investigated. y Fatigue surface x Crack front

K

a Δa

The SIF along the crack front z

B

W 2

The mid-plane point

The surface point

Fig. 1. Schematic of a through-thickness cracks with a slightly wavy front.

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2. The gradientless iterative method This section briefly introduces the heuristically derived iterative method, by which the crack shape can be determined if the SIF distribution function for the crack is given. The method has been used to achieve the shape of a surface crack in a plate by the author [11]. Fig. 2(a) shows sketchily a through-thickness planar crack with arbitrary shape in a plate subjected to remote Mode-I load. The SIF at a point on the crack front is denoted as K(v), where v is a general variable representing the position of the point. Here, v = 2z/B (B is the thickness of the plate) is selected as the variable. Generally, K(v) depends on the loading, the geometry of the crack and the position on the crack front. For simplicity, K(v) is normalized by the value at a reference point, for instance at mid-plane point A, which determines the location of the crack front. Then, the SIF distribution along a crack front can be described by kðvÞ ¼

KðvÞ : KðAÞ

ð3Þ

k(v) is called the SIF distribution function, which depends only on the crack geometry. k(v)  1 means a constant SIF distribution (Iso-KI) along the crack front. The crack front is defined by using cubic splines, which are determined by a set of control-points. Moving control-points can alter the crack front shape, and correspondingly the SIF distribution along it. For an arbitrary crack front shape, the SIF distribution is, generally, random along the crack front as shown schematically in Fig. 2(b). However, the final SIF distribution should satisfy the given function defined by Eq. (3). The objective is to achieve the shape of the crack, along which the final SIF distribution satisfies k(v), by iteratively adjusting the crack front shape. Here, a control-point is moved in the normal direction n, apart from the surface point, which must move in the surface direction. The SIF value at the control-point i is K ji at the jth iteration. However, the fictitious-correct value at this point should be kðvi Þ  K jA according to Eq. (3). Comparing K ji with k(vi) Æ Kj(A), the control-point i should be moved at the (j + 1)th iteration in order to make K jþ1 ¼ kðvi Þ  K jþ1 ðAÞ. A heuristically derived logic is used i to adjust iteratively the crack front shape such that the SIF distribution around the crack front satisfies Eq. (3). A control-point should be moved inwards where SIFs are considered to be high and outwards where SIFs are considered to be low for a Mode-I crack. About the amount by which to move the control-point i at the (j + 1)th iteration, it is proposed that the amount d jþ1 is, to a first approximation, directly proportional to i the relative difference between K ji and k(vi) Æ Kj(A), which is expressed as ¼ d jþ1 i

K ji  kðvi Þ  K jA  s; maxfK ji ; kðvi Þ  K jA g

i ¼ 1; 2; . . . ; k;

ð4Þ

where k is the number of control-points; s is an arbitrary step length, which should be determined by trial. After moving all control-points on the basis of Eq. (4), the SIF distribution along the new crack front will become more consistent with the known k(v), as shown schematically in Fig. 2(b). This procedure continues K(χ)

x

Given SIF distribution Initial crack front

Control-point

New SIF distribution

n d ij A

1

i-1 2 i i+1 New crack front Final crack front

O Mid-plane

Surface (B/2)

(a)

Initial SIF distribution

C k 1(A) z

Corner node position

Mid-plane

k(C)

Surface (B/2)

(b)

Fig. 2. Explanation of the iterative method: (a) geometry and notation with crack shape defined by cubic spline and (b) typical SIF distribution around the crack front.

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in an iterative manner until the SIF distribution along the crack front satisfies k(v) within a prescribed tolerance. It is particularly worth indicating that one should be able to effectively determine the curved-front of 3-D crack satisfying a given SIF distribution from an initial crack front, using the derivative of SIF with respect to virtual crack shape extension presented by Hwang et al. [12,13]. If the derivatives of SIF at crack front i at the jth iteration, oK ji =oai , are computed, the amount of crack extension at the (j + 1)th iteration can be approximated using the linear relation d jþ1 ¼ i

K ji  K ji-virtual ; oK ji =oai

i ¼ 1; 2; . . . ; k:

ð5Þ

K ji-virtual is virtual correct value at point i at the jth iteration, which is assumed to be kðvi Þ  K jA in Eq. (4). Comparing Eq. (4) with Eq. (5), no arbitrary step length is needed in the 3-D virtual crack extension method. Thus, it is anticipated that the curved-front of 3-D planar crack can be determined more effectively and robustly based on the derivatives of SIF with respect to virtual crack shape extension. To monitor solution convergence, a performance index PI is defined as below   jK i  kðvi Þ  K A j P I ¼ max ; i ¼ 1; 2; . . . ; k: ð6Þ maxfK i ; kðvi Þ  K A g As PI approaches zero, the SIF distribution along the crack front becomes consistent with k(v). The user can specify a tolerance s for PI to control the error of the SIF around the crack front. For example, s = 0.01 indicates that the relative error of the estimated SIF at a point along the crack front is less than 1% of KA in the case of an Iso-KI crack. Then, the termination criterion can be set as PI 6 s. 3. Finite element analysis method Fig. 3 shows a through-thickness cracked plate subjected to uniform stress r0. Value of H/W = 2 is used so that the influence of the specimen ‘height’ on the SIF is at a negligible level [14]. The material is assumed to be homogeneous and isotropic with Young’s modulus of 210 GPa and Poisson’s ratio of 0.3. Considering the computational efficiency, the 1/4-point displacement method is used to estimate the SIFs along a crack front. This method is proposed by Henshell and Shaw [15] and Barsoum [16], and has been widely used in evaluating SIF for various planar crack. The SIF can be estimated from the crack-surface displacement at the 1/4-point,

σ0

B

2W y Crack plane

z C

2H

A

x

2a Crack

σ0 Fig. 3. Geometric parameters of through-thickness cracked plate subjected to uniform stress r1.

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Crack front

A

Fig. 4. Finite element model (one-eighth of the plate).

sffiffiffiffiffiffiffiffiffiffi E0 2p uzð1=4Þ ; K¼ 4 rð1=4Þ

ð7Þ

where E 0 = E in plane stress state and E 0 = E/(1  m2) in plane strain state; r(1/4) is the distance of the 1/4-point away from the crack tip; and E and m are Young’s modulus and Poisson’s ratio of the material. The presence of a boundary layer effect leads to the difficulty to obtain a correct estimate of SIF in the region close to the free surface. In this study, when the SIF is calculated based on Eq. (7), only the surface point is considered to be in plane stress, while all the interior points along the crack front are considered to be in plane strain since the thickness of free surface layer is very small [1,2]. The 1/4-point displacement method theoretically requires an orthogonal finite element mesh along the crack front, otherwise errors may occur. In present study, non-orthogonal mesh correction technique, as suggested by Lin and Smith [17], is used to calculate the SIFs along a curved crack front. Due to the symmetry in the specimen geometry and loading, only one-eighth of the specimen is modeled in the finite element analysis. All finite element models are developed by the ANSYS 8.0 package using 20-node hexahedron elements and wedge elements with mid-side nodes relocated to their 1/4-point. The point at the mid-plate A, as shown in Fig. 3, is selected as the reference point, which represents the crack half-length and is fixed during the iterative process. Fig. 4 shows a typical three-dimensional finite element model. To verify the accuracy of the above method for SIF evaluation, the profile of stress intensity K(2z/B) along the thickness direction are obtained for various plate thicknesses, as shown in Fig. 5. K should drop to zero at 1.4 1.3

K σ0√ πa

1.2 B increase

B increase B = a/10 B = a/5 1 B = a/2.5 B=a 0.9 B = 2a B = 3a 0.8 B = 4a

a/W = 0.5 W = 100mm

1.1

0.7 0

0.2

2-D plane strain

0.4

0.6

0.8

2z/B Fig. 5. SIF for various plate thicknesses.

1

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the plate free surface due to the weaker singularity than square root [4], but this is difficult to obtain by the finite element [1] or boundary element [2] analysis. The present result is in pretty good agreement with that in [1], which is obtained using global-local finite element technique with sub-modeling. This demonstrates that the 1/4-point displacement method can produce a good SIF accuracy for through-thickness cracked plate of intermediate thickness. The result in Fig. 5 clearly indicates that the 2-D plane strain solution of SIF is a good approximation of the value at the mid-plane for only very thick plates. 4. The crack tunneling satisfying Iso-KI distribution Assuming that a through-thickness crack has a constant distribution of SIF along the front, it can be anticipated that the crack will tend to a particular configuration to satisfy this assumption. The shapes and the corresponding SIFs of the through-thickness cracks for various plate thicknesses are shown in Fig. 6, where the crack length measured at the mid-plate is also a/W = 0.5. These results (and the following results) are achieved setting the tolerance s for the performance index PI to be s = 0.002–0.005. Examining Fig. 6, a number of features can be recognized. (i) The intersection angles of the crack front and plate surface for all the Iso-KI cracks is in the range of 96– 102°, which is close to the critical value—101°—estimated by Bazˇant and Estenssoro [18] for Poisson’s ratio of 0.3. This demonstrates that the present method can produce a reasonable estimation of SIF at the surface point. (ii) The thickness of plate has a great influence on the configuration of the Iso-KI cracks. In thinner plates (B 6 a), the crack-front tunneling is evident. However, such tunneling disappears gradually in thicker plates, and anti-tunneling is observed in very thick plates (B > 2a). (iii) It is noted that, except for the plate with very large thickness (B = 4a), the normalized K solutions seem to converge to the value of 1.25 within 1% for various plate thicknesses, which is about 5% higher than the value of 2-D plane strain solution. Contrary to the result in Fig. 5, the plane strain solution cannot be used for the mid-plane even for very thick plates (say, B = 4a) since the 3-D solution at the mid-plane for the Iso-KI crack is almost 8% higher than the value of 2-D plane strain solution. The above results strengthen the view that the SIF distribution is sensitive to shape variations, and vice versa. Define Da/B as relative tunneling depth, where Da is the difference between the crack length measured at the surface and that measured at the mid-plane, see Fig. 1. The variation of relative tunneling depth with thickness

0.58

0.56 0.55

x/W

0.54

2-D plane strain

1.35 1.30 1.25

K σ0√ πa

0.57

1.40

a/W = 0.5 W = 100mm

B = a/10 B = a/5 B = a/2.5 B=a B = 2a B = 3a B = 4a

1.20

0.53 0.52

B = a/10 B = a/5 B = a/2.5 B=a B = 2a B = 3a B = 4a

1.15

0.51

1.10

0.5 1.05

0.49 0.48 0

(a)

0.2

0.4

0.6

2z/B

0.8

1.00

1

(b)

0

0.2

0.4

a/W = 0.5 W = 100mm

0.6

0.8

1

2z/B

Fig. 6. The shapes and the SIFs of the through-thickness cracks satisfying Iso-KI for various plate thicknesses: (a) crack configuration and (b) SIF distribution.

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5

a/W = 0.5 W = 100mm

4

Δa/B (×100)

3 2 1 0 -1 -2 -3 0

1

2

3

4

B/a Fig. 7. The variation of relative tunneling depth with thickness of plate for Iso-KI crack.

of plate is shown in Fig. 7, where negative value of Da/B indicates anti-tunneling. Fig. 7 shows that the relative tunneling depth increases with the thickness reduction. The Paris fatigue crack growth relation can be expressed as da m m ¼ CðDKÞ ¼ Cð1  RÞ K mmax ; dN

ð8Þ

where a is the crack length; DK is the SIF range; Kmax is the maximum applied SIF; R is stress ratio of minimum to maximum applied stress; C and m are, generally, considered as the material constants. Assuming that a through-thickness crack has always a constant distribution of SIF, the Da is also a constant along the crack front according Eq. (8), i.e. the crack will maintain the same shape during the fatigue process. This is the case of equal spacing of striations, which can be observed on a fracture surface using the scanning electron microscope or the transmission electron microscope. Examining the variation of the crack shape can reveal the stability of a crack in fatigue, i.e. whether the above assumption of Iso-KI crack is correct. The variation of the shape of the Iso-KI crack in the plate of B = 20 mm and W = 100 mm is shown in Fig. 8, where a is the crack length measured at the mid-plane. It can be seen that the crack shape varies markedly for the crack with smaller length (a < 0.3W) or larger length (a > 0.8W), i.e. a crack can approximately maintain the same shape only within the intermediate length (say, 0.3W 6 a 6 0.8W) satisfying the conditions 0.2 0.1 0

a = 0.1W a = 0.15W a = 0.2W a = 0.3W a = 0.4W a = 0.5W a = 0.6W a = 0.7W a = 0.8W a = 0.9W

-0.1

x-a

-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8

0

0.2

W = 100mm B = 20mm

0.4

0.6

0.8

1

2z/B Fig. 8. The variation of the shape of the Iso-KI crack in the plate of B = 20 mm and W = 100 mm.

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of constant SIF distribution. This result suggests that it is possible for an Iso-KI crack to propagate steadily in fatigue within the intermediate length. The variation of the crack shape can also be characterized by the evolution of the relative tunneling depth, as shown in Fig. 9, where the variation of Da/a with the normalized crack length is also plotted to give some information about the possible relative error of SIF caused by Da. The relative tunneling depth is 2.5–3%, which agrees well with Bakker’s prediction [8], and varies little within the intermediate crack length. The maximum relative error of SIF caused by Da, based on Eq. (2), occurs at the crack length of about a/W = 0.15, and decreases as the crack grows. The SIF error due to the influence of crack-front tunneling can be ignored since the maximum relative error of SIF is less than 1% in the intermediate crack length for the Iso-KI crack. Eq. (1) has been used in many standards, for instance the ASTM standard, for center-cracked specimens to estimate the SIF. The boundary-correction factor, F, is generally given by F ¼ ½secðpa=W Þ1=2 ;

ð9Þ

which is developed by Feddersen [19]. To evaluate the validity of the 2-D linear elastic fracture mechanics, the results pffiffiffiffiffiffi of SIFs for various crack lengths are plotted in Fig. 10, where all the SIFs is non-dimensionalized by r0 paF . For comparison, the results of the straight-front crack are also plotted (measured at the mid-plane). 5

W = 100mm B = 20mm

Δa/B or Δa/a (×100)

4

Δa/B 3

Δa/a

2

1

0

0

0.2

0.4

0.6

0.8

1

a/W Fig. 9. The variations of Da/B and Da/a for Iso-KI crack in the plate of B = 20 mm and W = 100 mm.

1.1

The straight-front crack

W = 100mm B = 20mm

1.08

Eq. (15) 1.06

1.04

The Iso-KI crack

1.02

1

The crack satisfying the given SIF distribution

0

0.2

0.4

0.6

0. 8

1

a/W Fig. 10. The SIFs of cracks satisfying different SIF distribution for various crack length in the plate of B = 20 mm and W = 100 mm.

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As can be seen in Fig. 10, the 3-D SIF for the Iso-KI crack is larger than the 2-D KI and varies with crack length. However, the variation of SIF is very small within the intermediate crack length (say, 0.2W 6 a 6 0.7W). It should be noted that the 3-D SIF for the Iso-KI crack is smaller than that for the straight-front crack when 0.15 6 a/W. The average of the 3-D SIFs for the Iso-KI crack is 1.055 (1.07 for the straight-front crack), which means that the 3-D SIF is averagely 5.5% higher than the value of the 2-D KI for the Iso-KI crack in the range of 0.1 6 a/W 6 0.9. 5. The crack tunneling satisfying other SIF distributions 5.1. The SIF distribution function The phenomenon of crack closure led to new concepts in fatigue crack growth. Closure is important because it tends to alter the relation between the applied SIF range (DK) and that actually experienced by the crack tip. Under constant amplitude loading, crack closure tends to decrease the applied DK, resulting in a corresponding reduction in crack growth rate. Elber [10] contributed the phenomenon of closure to the mechanism of plasticity-induced crack closure. On account of crack closure behaviour, the DK factor in Eq. (8) was replaced by the effective SIF range DKeff, and thereby gives Eq. (8) as da m m m m m ¼ CðDK eff Þ ¼ CðK max  K op Þ ¼ CðU DKÞ ¼ Cð1  RÞ ðUK max Þ ; dN

ð10Þ

where Kop is the SIF at which the crack starts to become fully open, U = DKeff/DK is the effective SIF ratio. Several assumptions are made to analyze the stability of the curved crack shape in fatigue and the possible effects of plate thickness, crack length and crack closure level gradient on the behaviour of crack tunneling. It is assumed that the Da is always a constant along the crack front (the case of equal spacing of striations). This assumption is reasonable for a through-thickness crack since it has only a slightly curved shape in fatigue. On the other hand, the effect of the shape error on the SIFs along the crack front becomes less significant because of the 1/m (m > 1) power relation between da and Kmax in Eqs. (8) and (10). The second assumption is that Eq. (10) is always valid at any point along the crack front. Under these assumptions, the relation between the SIF distribution function k(v) and that of the effective SIF ratio function u(v) can be easily derived as kðvÞ ¼

1 ; uðvÞ

ð11Þ

where k(v) = K(v)/K(A); u(v) = U(v)/U(A). u(v) is a function describing the variation of the effective SIF ratio along a crack front. A is the mid-plane point. Eq. (11) shows that only the gradient of the effective SIF ratio has an influence on the crack shape, although a constant U along a crack front has a great effect on the rate of fatigue crack growth. Fatigue crack closure is inherently a three-dimensional problem. The levels of crack closure at the surface and at interior point are different due to the effects of the transition from plane stress conditions at the surface to plane strain elsewhere [20–24]. Newman et al. [23,24] proposed that the ratio of a crack opening, Kop/Kmax, depends on R and also on the ratio between the maximum stress rmax and the material flow strength rfl, and on a plane stress/strain constraint factor a, ranging from a = 1 for pure plane-stress to a = 1/(1  2m) for pure plane-strain, where m is Poisson’s ratio: ( maxðR; A0 þ A1 R þ A2 R2 þ A3 R3 Þ; R P 0 K op ¼ ð12Þ K max A0 þ A1 R; 2 6 R < 0 where the polynomial coefficients are given by 8 1=a A0 ¼ ð0:825  0:34a þ 0:05a2 Þ  ½cosðprmax =2rfl Þ ; > > > < A1 ¼ ð0:415  0:071aÞ  rmax =rfl ; > A2 ¼ 1  A0  A1  A3 ; > > : A3 ¼ 2A0 þ A1  1:

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1.1 1

σmax/σfl =0.7

Uσ /Uε

0.9

0.5 0.8

0.3

0.7

0.1 0.6 0.5 -1

-0.8 -0.6 -0.4 -0.2

0

0.2 0.4

0.6 0.8

1

R Fig. 11. Effect of rmax and R on the ratio between Newman’s effective SIF ratios under plane-stress and plane-strain, Ur/Ue.

From the definition of Newman’s crack opening ratio, the effective SIF ratio U can be written as U¼

DK eff 1  K op =K max ¼ : DK 1R

ð13Þ

Fig. 11 shows that Eq. (13) can predict very different effective SIF ratios for plane-stress, Ur, and for planestrain, Ue, especially when rmax/rfl is low, assuming m = 0.3. Obviously, the function k(v) (or u(v)) should be known while applying the present method to determine the crack shape. Unfortunately, the function u(v) is unavailable to date due to the difficulty in experimental measure. In present study, it is assumed that the state of deformation at mid-plane point is always plane strain, but at the surface point is always plane stress for a plate of intermediate thickness while accounting for the variation of the effective SIF ratio along a crack front. Thus, an arbitrary function k(v) has to be designed to consider the variation of u(v) along a crack front. Define kðvÞ ¼

KðvÞ ¼ c 1 þ c 2 vn KðAÞ

ðn P 1Þ;

ð14Þ

where v = 2z/B, n P 1. Considering the end values of k(v), k(0) = 1 and k(1) = K(C)/K(A) = Ue/Ur, k(v) can be rewritten as   Ue kðvÞ ¼ 1 þ ð15Þ vn ðn P 1Þ: Ur  1 k(v) is a simple increasing function in the interval 0 6 v 6 1 if Ue/Ur > 1. This means that the maximum SIF occurs at the surface point. The parameter n controls the style of SIF distribution function. The larger n is, the greater the variations of the SIF distribution as well as the effective SIF ratio are near the surface point. 5.2. The effects of SIF-distributions on the crack tunneling The effects of various SIF distributions on the crack tunneling are investigated using the plate of B = 20 mm, W = 100 mm and a = 50 mm. Assume that the SIF distribution satisfies Eq. (15) and Ur/Ue = 0.9. Oh and Song [21] reported that, for surface cracks, the ratio of crack opening in the crack length direction to that in the crack depth direction is close to 0.9 regardless of the applied loading types and the stress ratios. Fig. 12 shows the shapes and the corresponding SIFs of through-thickness crack with various values of n, where the results of the Iso-KI crack are also presented for comparison. It can be seen that the style of SIF distribution function has marked effect on both the crack shape and the SIF. The relative tunneling depth of the cracks is larger than that of the Iso-KI crack, and increases as n decreases while considering the gradient

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1.3

0.494

Iso-KI

0.492

n = 100

K σ0√ πa

x/W

Uσ /Uε =0.9

1.35

Uσ /Uε =0.9

0.496

1.25 Iso-KI

1.2

n = 10

0.49 0.488

n = 10 1.1

n=2

0.486

1.05

0.484

1

0

0.2

(a)

0.4

n = 100

1.15

n=5

0.6

0.8

1

n=5 n=2 0

0.2

0.4

(b)

2z/B

0.6 2z/B

0.8

1

Fig. 12. The shapes and the SIFs of the through-thickness crack satisfying Eq. (14) with various values of n: (a) crack configuration and (b) SIF distribution.

of the effective SIF ratio. However, the SIF at mid-plane point is smaller than that of the Iso-KI crack and decreases with n. It is suggested [22,25] that crack closure occurs predominantly near the material surface, under plane-stress conditions. In the following, n = 10 is assumed to account for the large variation of the effective SIF ratio near the surface. As shown in Fig. 11, the value of Ur/Ue might be in a large range depending on R and rmax/rfl. Fig. 13 shows the shapes and the corresponding SIFs of the curved crack with various values of Ur/Ue. Ur/Ue = 1 corresponds to the case of the Iso-KI crack. As expected, the value of Ur/Ue has significant effect on both the crack shape and the SIF. The larger the value of Ur/Ue is, the greater the relative tunneling depth is. Similarly, the SIF at mid-plane point decreases as the value of Ur/Ue increases. The above results demonstrate that, to evaluate SIF with high accuracy, a reasonable description for the closure level gradient along a crack front is necessary even if the values of Ur and Ue can be estimated precisely. Whether the above assumptions of equal spacing of striations and the given SIF distribution function are possible correct can be validated by examining the stability of a crack in fatigue. The variation of the shape for a fatigue crack in the plate of B = 20 mm and W = 100 mm is shown in Fig. 14 (a is the crack length measured at the mid-plane), assuming that the SIF distribution satisfies Eq. (15), n = 10 and Ur/Ue = 0.9. The evolution

1.5 0.5

1.45

0.498 0.496

1.4 n = 10

1.3

K σ0√ πa

x/W

0.494

Uσ /Uε = 1

0.492 0.49 0.488

0.484 0

0.2

1.25

Uσ /Uε = 0.944

1.2

Uσ /Uε = 0.900

1.15

Uσ /Uε = 0.944

Uσ /Uε = 0.833

1.1

Uσ /Uε = 0.900

1.05

Uσ /Uε = 0.833

0.486

(a)

n = 10

1.35

0.4 0.6 2z/B

0.8

1

1

(b)

Uσ /Uε = 1

0

0.2

0.4

0.6

0.8

1

2z/B

Fig. 13. The shapes and the SIFs of the through-thickness crack satisfying Eq. (14) with various values of Ur/Ue: (a) crack configuration and (b) SIF distribution.

x-a

Z. Wu / Engineering Fracture Mechanics 73 (2006) 2600–2613

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.10

a = 0.1W a = 0.15W a = 0.2W a = 0.3W a = 0.4W a = 0.5W a = 0.6W a = 0.7W a = 0.8W a = 0.9W 0.2

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W = 100mm B = 20mm

0.4

0.6

0.8

1

2z/B Fig. 14. The variation of the shape for a fatigue crack in the plate of B = 20 mm and W = 100 mm, assuming that the SIF distribution satisfies Eq. (14), n = 10 and Ur/Ue = 0.9.

of relative tunneling depth is shown in Fig. 15, where the variation of Da/a with the normalized crack length is also plotted to give some information about the possible relative error of SIF caused by Da. It can be seen that, similar to the Iso-KI crack, the crack can almost maintain the same shape within the intermediate length (0.3W 6 a 6 0.8W). This result suggests that it is also possible for a crack to propagate in fatigue within the intermediate length satisfying the given SIF distribution. It is interesting to note that the relative tunneling depth is about 5% in this case, which agrees well with experiment observation [9], and varies little within the intermediate crack length. The relative error of SIF caused by Da is larger than that of the Iso-KI crack and decreases as the crack grows. The maximum relative error of SIF due to the influence of crack-front tunneling is about 3%, occurring at the crack length of about a/W = 0.1. However, the relative error of SIF is less than 1.5% in the intermediate crack length. The SIFs measured at the mid-plane for various crack lengths are plotted in Fig. 10 to evaluate the validity of the two-dimensional linear elastic fracture mechanics. It can be seen that the 3-D SIF is smaller than that of the Iso-KI crack, but is larger than the 2-D KI. The average of the non-dimensionalized 3-D SIFs is 1.04, which means that the 3-D SIF is averagely 4% higher than the value of the 2-D KI for the crack satisfying the given SIF distribution in the range of 0.1 6 a/W 6 0.9. 7

W = 100mm B = 20mm

Δa/B or Δ a/a (×100)

6 5 4

Δ a/B 3 2

Δa/a

1 00

0.2

0.4

0.6

0. 8

1

a/W Fig. 15. The variations of Da/B and Da/a for a fatigue crack in the plate of B = 20 mm and W = 100 mm, assuming that the SIF distribution satisfies Eq. (14), n = 10 and Ur/Ue = 0.9.

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Z. Wu / Engineering Fracture Mechanics 73 (2006) 2600–2613

Kwon and Sun [1] proposed a simple technique without 3-D calculations for evaluating approximate 3-D K of the straight-front crack at the mid-plane of the plate, based on the results of the crack of a/W = 0.5. The 3-D SIF, K3-D, at the mid-plane was obtained from the 2-D SIF, K2-D, with the relation rffiffiffiffiffiffiffiffiffiffiffiffiffi K 3D 1 : ð16Þ ¼ 1  m2 K 2D Examining Fig. 10, it is interesting to note that Eq. (16) is more suitable for evaluating approximately 3-D K at the mid-plane of the plate in the range of 0.1 6 a/W 6 0.9 for both the Iso-KI crack and the crack satisfying the given SIF distribution. 6. Concluding remarks Using a heuristically derived iterative procedure, the behaviour of crack tunneling in center-cracked plates is numerically investigated. Several simple SIF distribution functions are assumed to determine the crack shape. The possible effects of plate thickness, crack length and crack closure level gradient on the crack tunneling are examined. Assuming that the crack advance is governed by the SIF through Paris’s law or Elber’s law, the stability of the curved shape of a through-thickness crack in fatigue is also discussed. The following conclusions can be made based on the numerical results: (1) There are important differences in the shape and the SIF between the Iso-KI crack and the straightfronted crack. For the Iso-KI crack (the crack length measured at the mid-plate is a/W = 0.5), the thickness of plate has a great influence on the configuration of the crack. In thinner plates (B 6 a), the crack-front tunneling is evident. However, such tunneling disappears gradually in thicker plates, and anti-tunneling might be observed in very thick plates (B > 2a). Contrary to the straight-fronted crack, the plane strain solution cannot be used for the mid-plane for very thick plates (say, B = 4a) since the 3-D solution at the mid-plane for the Iso-KI crack in very thick plate is almost 8% higher than the value of 2-D plane strain solution. (2) It is possible for an Iso-KI crack in the plate of B = 20 mm and W = 100 mm to propagate steadily within the intermediate length (say, 0.3W 6 a 6 0.8W) under the assumption that the fatigue crack growth law is that of Paris. (3) The gradient of the effective SIF ratio has a significant influence on the SIF distribution along the crack front, and correspondingly on the crack shape. Thus, it is necessary to describe rationally the closure level gradient along a crack front in order to evaluate SIF with high accuracy. (4) It is also possible for a crack in the plate of B = 20 mm and W = 100 mm, satisfying a given SIF distribution, to propagate steadily within the intermediate length under the assumption that the fatigue crack growth law is that of Elber. (5) The relative error of SIF due to the influence of crack-front tunneling is less than 1.5% and decreases as the crack grows in the intermediate crack length for the crack satisfying the conditions of constant SIF distribution or the given SIF distribution. Therefore, the SIF error can be ignored while estimating the SIF according to the crack length measured at the surface in the intermediate crack length. (6) Without 3-D calculations, the 3-D SIF of a curved-front crack at the mid-plane of the intermediatethickness plate can be obtained approximately from the 2-D SIF using Eq. (16). Acknowledgement This work is supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars. References [1] Kwon SW, Sun CT. Characteristics of three-dimensional stress fields in plates with a through-the-thickness crack. Int J Fract 2000;104:291–315. [2] Agrawal AK, Kishore NN. A study of free surface effects on through cracks using BEM. Engng Fract Mech 2001;68:1297–316.

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