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Mixed mode weight functions for an elliptical subsurface crack under shear loadings
Engineering Fracture Mechanics 136 (2015) 58–75
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Mixed mode weight functions for an elliptical subsurface crack under shear loadings Rahmatollah Ghajar a,⇑, Javad Alizadeh Kaklar b a Mechanical Properties Research Laboratory (MPRL), Department of Mech. Eng, K.N. Toosi University of Technology, Pardis St., Mollasdra Ave., Vanak Sq., P.O. Box 19395-1999, Tehran, Iran b Department of Mech. Eng., Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Received 17 June 2014 Received in revised form 13 January 2015 Accepted 15 January 2015 Available online 7 February 2015 Keywords: Elliptical subsurface cracks Mixed mode weight functions Coupling of fracture modes Two dimensional stress distributions
a b s t r a c t In this paper, mixed mode weight functions (MMWFs) for elliptical subsurface cracks under shear loadings are derived. Reference mixed mode stress intensity factors (MMSIFs), calculated by finite element analysis (FEA), under uniform shear loadings are used to derive MMWFs. The cracks have aspect ratios and crack depth to crack length ratios in the ranges of 0.2–1.0 and 0.05 to infinity, respectively. MMWFs are verified to calculate MMSIFs for any point of the crack front under linear and nonlinear two dimensional (2D) loadings. The derived MMWFs could be effectively used for evaluation of fatigue crack growth under complicated 2D stress distributions. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Calculation of stress intensity factor (SIF) is a prerequisite step in applied fatigue and fracture analysis. Embedded elliptical cracks are one of the most common types of cracks in engineering structures. Elliptical subsurface cracks are embedded cracks that often initiate and propagate under rolling contact fatigue (RCF). Due to the existence of large pressures under the rolling contact condition, the subsurface cracks propagate mainly in shear fracture modes [1,2]. Fractures of two railway wheels caused by the fatigue crack growth of two elliptical subsurface cracks are represented in Fig. 1. Many studies have been conducted so far on determination of the SIFs for the embedded elliptical cracks in an infinite space. Irwin [3] analytically obtained an exact relation for mode I SIF of the cracks under uniform normal loadings. Kassir and Sih [4,5] investigated the cracks under linear normal and constant shear loadings, and developed the closed-form solutions for SIFs. Shah and Kobayashi [6] studied such cracks under two dimensional (2D) third order polynomial normal loadings and obtained exact relation for SIF. Vijayakumar and Atluri [7] determined SIF for the cracks under any-order polynomial normal loadings. Nishioka and Atluri [8] solved the problem for any-order polynomial shear loadings. Bueckner [9] presented the weight function method (WF) for the computation of SIF under complex stress distributions. In this method, derived WF can be utilized to calculate SIF under any arbitrary loading. Rice [10] demonstrated that the WF can be obtained using crack opening displacement and SIF of a reference load. Roy and Saha [11,12] applied analytical mode I, II, III WFs and obtained the mode I, II, III SIFs for any point of the cracks fronts under any arbitrary normal and shear loadings. Also, Atroshchenko, Potapenko and Glinka [13,14] calculated mode I SIFs for the cracks under arbitrary normal loadings using analytical WF. Wang and Glinka [15] proposed approximate form of WF for the cracks, and calculated the SIF for arbitrary normal loadings. ⇑ Corresponding author. Tel.: +98 2184063240; fax: +98 2188677273. E-mail addresses: [email protected] (R. Ghajar), [email protected] (J.A. Kaklar). http://dx.doi.org/10.1016/j.engfracmech.2015.01.024 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.
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Nomenclature A crack face area Aim constant coefficients AEmax maximum absolute error a semi-major axis of ellipse, crack length Bnim constant coefficients c semi-minor axis of ellipse Cj constant coefficients c1 distance of ellipse from the major axis in normal direction c2 distance of ellipse center from the tangent of the ellipse Di WF coefficients DIn mode I WF coefficient for a shear stress acting normal to the crack front DIIn mode II WF coefficient for a shear stress acting normal to the crack front DIIIn mode III WF coefficient for a shear stress acting normal to the crack front DIt mode I WF coefficient for a shear stress acting tangent to the crack front DIIt mode II WF coefficient for a shear stress acting tangent to the crack front DIIIt mode III WF coefficient for a shear stress acting tangent to the crack front E elastic modulus E(a) second kind of elliptical integral E0 material constant F nim constant coefficients Gjn constant coefficients Gjt constant coefficients h depth of the crack K(a) first kind of elliptical integral K FEA SIFs calculated by FEA KI mode I SIF K II mode II SIF K III mode III SIF K Ix mode I SIF caused by unit uniform shear stress along x K IIx mode II SIF caused by unit uniform shear stress along x K IIIx mode III SIF caused by unit uniform shear stress along x K Iy mode I SIF caused by unit uniform shear stress along y K IIy mode II SIF caused by unit uniform shear stress along y K IIIy mode III SIF caused by unit uniform shear stress along y Kr reference SIF K WF SIFs calculated by MMWFs L point of loading MSE mean square error Q arbitrary point of the crack front R ellipse radius for the corresponding point of L R2 coefficient of determination r polar coordinate of point L s shortest distance of the crack front from L point ur ðx; aÞ reference displacement of crack face Wðx; aÞ one dimensional WF Wðx; y; u0 Þ two dimensional WF W In mode I WF for a shear stress acting normal to the crack front W IIn mode II WF for a shear stress acting normal to the crack front W IIIn mode III WF for a shear stress acting normal to the crack front W It mode I WF for a shear stress acting tangent to the crack front W IIt mode II WF for a shear stress acting tangent to the crack front W IIIt mode III WF for a shear stress acting tangent to the crack front a aspect ratio of the crack b crack depth to the crack length ratio u angle of point L u0 angle of crack front, point Q m Poisson’s ratio q distance between the point L and point Q r normal stress distribution of the unflawed structure, across the crack faces
59
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s0 sn ðx; yÞ st ðx; yÞ szx ðx; yÞ szy ðx; yÞ
uniform shear stress distribution shear stress distribution acting normal to the crack front shear stress distribution acting tangent to the crack front shear stress distribution acting on the crack faces along x shear stress distribution acting on the crack faces along y
Fig. 1. Fractured railway wheels due to the propagation of the elliptical subsurface cracks [2].
Fig. 2. An elliptical subsurface crack in a semi-infinite space under normal and tangent shear stresses. minimum dimensions of the space are evaluated in Ref. [16].
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If the embedded crack in an infinite space changes to a subsurface crack by approaching one of boundary surfaces of the space (Fig. 2), coupling of the fracture modes happens due to the nonsymmetrical geometry with respect to the crack face [16]. In such case, mixed mode WFs (MMWFs) with a matrix formulation is necessary to consider the coupling of the fracture modes and calculate mixed mode SIFs (MMSIFs). Beghini, Bertini and Fontanari [17] evaluated the WFs of a subsurface crack in a half-plane and investigated the coupling of modes I and II. Mazzu [18] studied a 2D subsurface micro-crack using WF approach, regardless of coupling of the modes. In three-dimensional space, coupling between all fracture modes exists, and a subsurface crack experiences all three fracture modes. Ghajar and Alizadeh [16] conducted 50 finite element analysis (FEA) and derived explicit expressions for mode I, II and III SIFs of elliptical subsurface cracks under uniform normal loadings. They concluded that under uniform normal loadings, the coupling is considerable between mode II and I and is negligible between mode III and I. The problem of an elliptical subsurface crack in a half-space under shear loadings plays a basic role in the investigation of RCF. Calculation of SIFs for elliptical subsurface cracks in a half-space under arbitrary shear loadings has not been studied yet. Also, it is known that the determination of SIFs under any arbitrary 2D stress distributions can be performed using 2D WF [19,20]. So, the main purpose of this paper is to develop MMWFs for any point of the elliptical subsurface crack fronts with different aspect ratios and ratios of crack depth to crack length under shear loadings. Having MMWFs makes it possible to calculate MMSIFs for any point along the subsurface elliptical crack front under any arbitrary 2D shear loading acting on the crack faces. 2. Weight function method Bueckner [9] showed that the SIF of a crack, K I , under any loading can be calculated from the integral of the stress distribution across the crack plane in the unflawed body, r(x), multiplied by WF of the crack geometry,Wðx; aÞ,:
KI ¼
Z
a
rðxÞ Wðx; aÞdx
ð1Þ
0
in which, a is the crack length. The WF of the crack geometry can be derived using crack face displacement, ur ða; xÞ, and mode I SIF for a reference loading, K r , by:
Wðx; aÞ ¼
E0 @ur ðx; aÞ @x Kr
ð2Þ
where, E0 is E for plane stress and E=ð1 m2 Þ for plane strain, and x is the axis along the crack. Rice [10] showed that the WF is a unique geometrical property of a cracked body and is independent of the loading system. The surface displacements have been rarely determined for the cracks and therefore, Eq. (2) is not suitable and applicable for deriving the WF. So, Petroski and Achenbach [21] offered an approximate relation for the surface displacements of cracks base on which, other researchers developed some general forms for WF [22–24]. By considering general form of WF, one can derive the WF by determining the coefficients of general form using reference SIFs under reference loadings. The 2D WF, Wðx; y; u0 Þ, represents the SIF at point Q along the crack front, caused by unit point loads imposed on the crack faces at point Lðx; yÞ, Fig. 3. So, to calculate the SIF at point Q under a 2D stress distribution, rðx; yÞ, the Eq. (1) is modified to the following integration:
Kðu0 Þ ¼
Z Z A
rðx; yÞ Wðx; y; u0 ÞdA
ð3Þ
where A is the crack surface area. Wang and Glinka [15] proposed an effective general form of WF for the embedded elliptical cracks as:
Wðx; y; u0 Þ ¼
pffiffiffiffiffi " 2s
p3=2 q2
i # n X rðuÞ 1þ Di ða; u0 Þ 1 RðuÞ i¼1
Fig. 3. An elliptical crack and representation of the geometrical parameters.
ð4Þ
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here, s and q are the distances of the load point Lðx; yÞ from the crack front and the point Q, respectively, r and u define the polar coordinates of point Lðx; yÞ, and R(u) is the polar radius of corresponding point on the crack front, as indicated in Fig. 3. The WF coefficients for any point of the crack front, Di , depend on the aspect ratio of the elliptical crack. In Eq. (4), the WF is the summation of two parts, namely, the singular term, and the term accounting for the effect of crack configurations [15]. Ghajar and Saeidi [25,26] showed that this general form of WF can also be used for surface cracks. 3. Mixed mode weight functions (MMWFs) In this study, subsurface cracks with different aspect ratios, a = c/a, and crack depth to crack length ratios, b = h/a, are considered, while the crack faces are parallel to the boundary surface of the half-space, as indicated in Fig. 2. Therefore, the WF coefficients, Di , in Eq. (4) for any point of the crack front, Q(u0 Þ, are functions of two geometrical parameters, a and b. In fracture mechanics, existence of any non-symmetry in material, geometry or loading system with respect to the crack faces will cause coupling of fracture modes. Thus, a subsurface crack experiences all fracture modes under shear loading. So, mixed mode WFs are necessary to be derived for elliptical subsurface cracks under shear loadings. MMWFs are shown in matrix form as:
2
K I ða; b; u0 Þ
3
6 7 4 K II ða; b; u0 Þ 5 ¼ K III ða; b; u0 Þ
Z Z
2
W In ðx; y; a; b; u0 Þ
W It ðx; y; a; b; u0 Þ
3
6 7 4 W IIn ðx; y; a; b; u0 Þ W IIt ðx; y; a; b; u0 Þ 5 A W IIIn ðx; y; a; b; u0 Þ W IIIt ðx; y; a; b; u0 Þ
sn ðx; yÞ dA st ðx; yÞ
ð5Þ
here, K I ; K II and K III are respectively the mode I, II and III SIFs of the crack front, W In ; W IIn and W IIIn are respectively the mode I, II and III WFs when the direction of the applied shear loading is normal to the crack front at point Q ; W It ; W IIt and W IIIt are respectively the mode I, II and III WFs for the shear loading which is tangent to the crack front at point Q, and sn and st are shear loadings applied to the crack faces along normal and tangential directions of the crack front at point Q, respectively (Fig. 2). It should be emphasized here that three fracture modes for a 3D planar crack are considered as normal to the crack face (mode I), in the crack face along normal direction of crack front (mode II or sliding mode) and in the crack face along tangent direction of the crack front (mode III or tearing mode). So, to evaluate the mixed fracture modes of an elliptical crack, stress distributions along these three directions, r; sn and st , should be considered. This study investigates the elliptical subsurface cracks under shear stresses (sn and st Þ. In the WF matrix of Eq. (5), W IIn and W IIIt are called direct elements, while the others are coupling elements. Direct elements return to the direct effect of the loading on the SIFs, whereas the coupling elements are developed by the nonsymmetrical geometry and show the coupling effect of fracture modes on the calculated SIFs. The direct elements of the WF matrix are derived utilizing the general form of Eq. (4). Present analyses indicate that one term of Eq. (4), n = 1, accurately approximates the WF of the subsurface cracks, i.e.:
i pffiffiffiffi h rðuÞ W IIn ðx; y; a; b; u0 Þ ¼ p3=22sq2 1 þ DIIn ða; b; u0 Þ 1 Rð uÞ i pffiffiffiffi h rðuÞ W IIIt ðx; y; a; b; u0 Þ ¼ p3=22sq2 1 þ DIIIt ða; b; u0 Þ 1 Rð uÞ
ð6Þ
where DIIn and DIIIt are the WF coefficients for W IIn and W IIIt , respectively. The general form of Eq. (6) has to be modified for the coupling elements of the WF matrix. It is obvious that with the b parameter approaching infinity, the coupling effect of the fracture modes is eliminated. So, the coupling elements of the WF matrix must approach zero by increasing the b. This condition will be satisfied if the singular term of Eq. (6) is removed and the WF coefficients approach zero by b approaching infinity. Therefore, general form of WF for the coupling elements of WF matrix should be as follows:
W j ðx; y; a; b; u0 Þ ¼
pffiffiffiffiffi 2s
rðuÞ ; D ð a ; b; u Þ 1 j 0 RðuÞ p3=2 q2
j ¼ In; IIIn; It; IIt
ð7Þ
where DIn ; DIIIn ; DIt and DIIt are the WF coefficients for W In ; W IIIn ; W It and W IIt , respectively. By determining the DIn ; DIIn ; DIIIn ; DIt ; DIIt and DIIIt , the calculation of the SIFs for any arbitrary 2D shear stress distribution can be performed using Eq. (8) as follows:
3 rðuÞ rðuÞ DIn ða; b; u0 Þ 1 Rð DIt ða; b; u0 Þ 1 Rð u Þ u Þ pffiffiffiffiffi 6 K I ða; b; u0 Þ 7 Z Z 7 sn ðx; yÞ 2s 6 6 7 rðuÞ 7 6 1 þ DIIn ða; b; u Þ 1 rðuÞ dA ð a ; b; u Þ 1 D 4 K II ða; b; u0 Þ 5 ¼ IIt 0 0 RðuÞ RðuÞ 7 3=2 q2 6 Ap 4 5 st ðx; yÞ K III ða; b; u0 Þ rðuÞ rðuÞ DIIIn ða; b; u0 Þ 1 Rð 1 þ DIIIt ða; b; u0 Þ 1 Rð uÞ uÞ 2
3
2
ð8Þ
4. Reference MMSIFs and calculation of WF coefficients Alizadeh and Ghajar [27] developed a parametric FEA code written in Python scripting language of ABAQUS 6.12 software to calculate MMSIFs for the subsurface cracks under uniform shear loadings imposed to the crack surfaces along x and y
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63
directions, show in Fig. 3. The FEA code was validated using analytical solutions presented by Saha and Roy [12] for embedded elliptical crack under uniform shear stress with mean relative error of 0.5%. The validated FEA code was used to calculate MMSIFs of the crack fronts Q(u0 Þ, for 0.2 6 a 6 1.0, 0.05 6 b 6 4ffi infinity and a = 1 mm. Then, six equations were fitted to the FEA solutions with mean relative error of 3% as follow [27]:
pffiffiffiffiffiffiffiffi
K Ix ða; b; u0 Þ ¼
4 s0 pc1 X F n ða; bÞ cosn u0 EðaÞ n¼1
# pffiffiffiffiffiffiffiffi " 4 X 4as0 pc2 cos u0 þ F n ða; bÞ cosn u0 I1 ða; mÞ n¼1 # pffiffiffiffiffiffiffiffi " 4 X 4ð1 mÞs0 pc2 n sin u0 þ F n ða; bÞ sin u0 K IIIx ða; b; u0 Þ ¼ I1 ða; mÞ n¼1 pffiffiffiffiffiffiffiffi 4 s0 pc1 X n K Iy ða; b; u0 Þ ¼ F n ða; bÞ sin u0 EðaÞ n¼1 # pffiffiffiffiffiffiffiffi " 4 X 4s0 p c 2 n K IIy ða; b; u0 Þ ¼ sin u0 þ F n ða; bÞ sin u0 I2 ða; mÞ n¼1 # pffiffiffiffiffiffiffiffi " 4 X 4ð1 mÞs0 pc2 n cos u0 þ F n ða; bÞ cos u0 K IIIy ða; b; u0 Þ ¼ I2 ða; mÞ n¼1 K IIx ða; b; u0 Þ ¼
ð9Þ
where K Ix ; K IIx and K IIIx , and K Iy ; K IIy and K IIIy are mode I, II and III SIFs when the unit uniform shear stress is separately applied along x and y directions, respectively, m is the Poisson’s ratio, s0 is the uniform shear stress. The c1 and c2 are the length of the normal of the ellipse between the crack front and the x axis, and the distance of ellipse center from the tangent of the ellipse, respectively and are equal to [16]: 2
0:5
c1 ¼ cða2 cos2 u0 þ sin u0 Þ c c2 ¼ 0:5 2 2 2 ða cos u0 þ sin u0 Þ
ð10Þ
Also, I1 ða; mÞ and I2 ða; mÞ are defined as [27]:
4 2 ma KðaÞ þ ð1 a2 mÞEðaÞ 1 a2
4 I2 ða; mÞ ¼ ma2 KðaÞ þ ð1 a2 þ ma2 ÞEðaÞ 1 a2
I1 ða; mÞ ¼
ð11Þ
where K(a) and E(a) are the first and second kind of elliptical integrals, respectively, and the functions F n (a; b) are as follows:
F n2 ðaÞ F n3 ðaÞ F n4 ðaÞ þ 0:5 þ 0:75 þ 1:5 b0:25 b b b 3 X F ni ðaÞ ¼ F nim am ; i ¼ 1; . . . ; 4 F n ða; bÞ ¼
F n1 ðaÞ
ð12Þ
m¼0
where, F nim are constant coefficients provided in [27]. The normal and tangential shear stresses on the crack front of u0 versus the
szx and szy can be written as:
a cos u0 szx þ sin u0 szy cos u0 sin u0 sn ðx; yÞ ¼ ¼ c2 szx þ szy 0:5 2 a c ða2 cos2 u0 þ sin u0 Þ sin u0 szx þ a cos u0 szy sin u0 cos u0 st ðx; yÞ ¼ ¼ c2 s s zx þ zy 0:5 2 c a ða2 cos2 u0 þ sin u0 Þ
ð13Þ
So, normal and tangential shear stresses caused by the unit uniform shear stress along x and y directions are obtained as:
szx ðx; yÞ ¼ 1; szy ðx; yÞ ¼ 0
)
szx ðx; yÞ ¼ 0; szy ðx; yÞ ¼ 1
)
cos u0 ; a sin u0 sn ðx; yÞ ¼ c2 ; c
sn ðx; yÞ ¼ c2
sin u0 c cos u0 st ðx; yÞ ¼ c2 a
st ðx; yÞ ¼ c2
ð14Þ
Substituting the reference MMSIFs and stress distributions of Eq. (14) in Eq. (8), results in a system of two equations with two unknowns for each fracture mode. By solving the systems of equations, the coefficients of weight functions are obtained as follows:
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8 ffi R R pffiffiffi c 2s > cosau0 K jx þ sincu0 K jy Gcjn Djn ða; b; u0 Þ ¼ R R pffiffiffi 2 > 3=2 q2 dA > A p rðuÞ > 2 2s < 1RðuÞ dA A p3=2 q2 j ¼ I; II; III ffi R R pffiffiffi c2 > 2s sincu0 K jx þ cosau0 K jy Gcjt > dA > A p3=2 q2 > Djt ða; b; u0 Þ ¼ R R pffiffi2sffi rðuÞ 2 : 1 dA A p3=2 q2
RðuÞ
where GIIn and GIIIt are unity while other Gs are equal to zero.
Fig. 4. Mode I, II and III WF coefficients of the elliptical subsurface cracks for shear loading normal to the crack front, DIn ; DIIn ; DIIIn .
ð15Þ
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65
Fig. 5. Mode I, II and III WF coefficients of the elliptical subsurface cracks for shear loading tangent to the crack front, DIt ; DIIt ; DIIIt .
5. Results and discussion 5.1. MMWFs for elliptical subsurface cracks under shear loadings A program is used to calculate the numerical parametric singular integration by Wolfram Mathematica 7 software. The integration algorithm is verified using the FEA results. By integration, the MMWFs coefficients, Dj (a,b,u0 Þ, are calculated for 0.2 6 a 6 1.0, 0.05 6 b 6 4ffi infinity and 0 6 u0 6 p=2. The mode I, II and III WFs coefficients for normal and tangent shear loadings are plotted in Figs. 4 and 5, respectively. It should be emphasized that due to the symmetry, the coefficients DIt ; DIIt and DIIIn are zero for circular cracks (a =1). Also, the coefficients DIn and DIt approach zero for b ¼ 1. To formulate the coefficients of MMWFs, Dj ða; b; u0 Þ, a nonlinear fitting process with regression analysis is conducted using Data-fit software. To
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fit an equation to a set of data, various general forms of equations can be utilized. Among all tried forms of equations, the best fitted one is as follows:
Dj ða; b; u0 Þ ¼
"
Cj 2
ða2 cos2 u0 þ sin u0 Þ
5 X n Aða; bÞ þ ð1 aÞ Bn ða; bÞ sin u0
0:25
# ð16Þ
n¼1
where, j = In, IIn, IIIn, It, IIt, IIIt and A(a, b), Bn (a, b) and C j are represented by:
Aða; bÞ ¼ A0 ðaÞ þ Bn ða; bÞ ¼
Bn1 ðaÞ 0:25
b
A1 ðaÞ 0:25
b
þ
þ
Bn2 ðaÞ 0:5
b
A2 ðaÞ 0:5
b þ
þ
A3 ðaÞ
Bn3 ðaÞ 0:75
b
þ
0:75
b
þ
A4 ðaÞ b
Bn4 ðaÞ 1:5
b
1:5
;
;
Ai ðaÞ ¼
3 X Aim am ; i ¼ 0; . . . ; 4 m¼0
2 X Bni ðaÞ ¼ Bnim am ; i ¼ 1; . . . ; 4
ð17Þ
m¼0
C In ¼ C IIn ¼ C IIIt ¼ 1; C IIIn ¼ C It ¼ C IIt ¼ cos u0 in which, Aim and Bnim are constant coefficients provided in appendix. The numbers of coefficients in presented equations are 80 for DIIn and DIIIt , 76 for DIn and 60 for DIIIn ; DIt and DIIt . To evaluate the conformity of the Eq. (16) with the data points used for curve fitting, the results of Eq. (16) are indicated in Figs. 4 and 5 by continues lines. Comparison between the values calculated by Eq. (16) and data points represents excellent accuracy for the fitted equations. Almost 754–1050 data points are utilized in fitting process for each Dj . Accurate fitting of a three-variable equation to this amount of data points indicates the appropriateness of the selected general form for the Eq. (16). 5.2. Validation To use the proposed MMWFs to determine MMSIFs for a loading applied to the boundary surface of the space, stress distribution has to be first calculated in a semi-infinite body without a crack and then imposed on the crack faces. It should be noted here that all elliptical (or circular) cracks studied with other researchers are in infinite space. Therefore, they can be used to validate the MMWFs of the subsurface cracks in the special case with b ¼ 1. Fig. 6 indicates accuracy of MMWFs solutions comparing with the analytical values provided in Ref. [12] for elliptical crack in an infinite space under linear 2D stress distribution. To validate the efficiency and accuracy of the obtained MMWFs as well as the fitted equations, Eq. (16), first, Dj for the cracks with a = 0.3, b = 0.12 and a = 0.7, b = 0.07 were calculated using Eq. (16). These values of a and b were not used in fitting process of Eq. (16). Then, MMSIFs of the subsurface cracks are calculated under five different linear and nonlinear stress distributions for a = 1 mm using FEA method and MMWFs. All FEA and MMWFs calculations are performed for stress distributions imposed on the crack faces whereas the surface of the semi-infinite body is free. The parametric FEA code which was developed and validated by Alizadeh and Ghajar [27] is used for calculation of MMSIFs. The FEA model is analyzed statically for five contour integrals including the singularity of the crack front. It should be emphasized here that more details of the FEA model were presented in our previous studies [16,27]. Comparison between the FEA and MMWFs results will demonstrate the efficiency and accuracy of obtained MMWFs for calculating MMSIFs of the elliptical subsurface cracks with any a and b under any arbitrary shear stress distributions. Due to the geometrical symmetry, the WF coefficients, Dj , were only calculated for the 0 6 2u0 =p 6 1. The magnitudes of the WF coefficients for other trigonometric quadrants are as follows:
Fig. 6. Comparison of analytical and MMWFs solutions for embedded elliptical crack in an infinite space under linear 2D stress distribution.
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8 Dj ðu0 Þ > > > < D ðp u Þ j 0 Dj ðu0 Þ ¼ > Dj ðu0 pÞ > > : Dj ð2p u0 Þ
67
0 6 2u 0 = p 6 1 1 6 2u0 =p 6 2 2 6 2u0 =p 6 3
ð18Þ
3 6 2u0 =p 6 4
In addition, the signs of the WF coefficients in trigonometric quadrants are important. Based on the MMSIFs calculated by full model of FEA under unit uniform shear loadings, the signs of the Dj are determined. Considering the sings of Dj and Eq. (18), the MMSIFs for all points of the crack front under arbitrary shear loadings are calculated by:
3 rðuÞ rðuÞ D D ð u Þ 1 ð u Þ sgnðsin 2 u Þ 1 It 0 0 7 6 In 0 2 3 RðuÞ RðuÞ pffiffiffiffiffi 6 KI 7 Z Z 6 2s 6 rðuÞ rðuÞ 7 7 sn ðx; yÞ 6 7 D dA ð u Þ 1 ð u Þ sgnðsin 2 u Þ 1 1 D 7 6 4 K II 5 ¼ IIn IIt 0 0 0 3=2 q2 6 RðuÞ RðuÞ 7 st ðx; yÞ Ap 7 6 K III 5 4 rðuÞ rðuÞ 1 þ DIIIt ðu0 Þ 1 DIIIn ðu0 Þ sgnðsin 2u0 Þ 1 RðuÞ RðuÞ 2
ð19Þ in which, the sgn(x) is the sign function that extracts the sign of x. 5.2.1. Linear stress distributions Two linear shear stress distributions are imposed on the subsurface crack faces and the MMSIFs are calculated from FEA and Eq. (19). The linear stress distributions are as follows:
Fig. 7. Comparison of SIFs calculated by FEA and MMWFs for a = 0.3, b = 0.12 and a = 0.7, b = 0.07 under linear 2D stress distributions.
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x a y szx ðx; yÞ ¼ szy ðx; yÞ ¼ 0:5 þ 1 c
szx ðx; yÞ ¼ szy ðx; yÞ ¼ 0:5 þ 1
ð20aÞ ð20bÞ
Comparison between FEA results and results of the MMWFs, as well as schematic representation of the stress distributions on the crack surface are illustrated in Fig. 7. It is clear from this figure that all fracture modes occur, although only the shear stresses are applied. The ratios of the maximum mode I SIF to maximum mode II and mode III SIFs, K I;max =K II;max and K I;max =K III;max , are 0.53 and 0.72 for stress distribution of Eq. (20a)a, and are 0.50 and 0.65 for stress distribution of Eq. (20b), respectively. Therefore, tension mode caused by the coupling between shear modes and mode I is important under linear shear stress distributions. Maximum absolute error (AEmax Þ, mean square error (MSE) and coefficient of determination ð0 6 R2 6 1Þ between SIFs calculated by FEA and MMWFs, are calculated as follows:
AEmax ¼ max jK WF K FEA j N X ðK WF i K FEAi Þ2 MSE ¼ N1
ð21Þ
i¼1
PN
2 i¼1 ðK WF i K FEAi Þ R2 ¼ 1 P ; 2 N i¼1 ðK FEAi KÞ
N X ¼1 K K FEAi N i¼1
where N is the total number of data points for each mode of SIFs, and K FEA and K WF are SIFs calculated by FEA and MMWFs, respectively. Based on the calculated errors (provided in Table 1), the obtained MMWFs are validated to be accurately used for linear loadings imposed on the subsurface elliptical cracks with any as and bs. Under uniform shear loadings, maximum SIFs of mode I, II and III appear for 2u0 =p = 0, 1, 2, 3 or 4 [27]. Linear loading in Eq. (20a), changes the mode I, II and III critical points of the crack to 2u0 =p = 0.57, 0.40 and 3.42 for a = 0.3, b = 0.12, and to 2u0 =p = 0.42, 0.42 and 3.51 for a = 0.7, b = 0.07, as shown in Fig. 7. So, it may be expressed that the stress distributions affect the location of critical points along the crack front. One of the most important benefits of WF method is time saving in calculation of the SIFs for different crack lengths, when the fatigue phenomenon is to be investigated. In this study, run time of the FEA for each model with individual a, b and stress distribution, was about 2 h, whereas the MMWFs took about 2 min to calculate the MMSIFs. So, WF method is 60 times faster than the FE method in calculation of MMSIFs for present problem. 5.2.2. Elliptic paraboloid stress distributions MMSIFs of the subsurface cracks are calculated from FEA and MMSIFs (Eq. (19)) under two elliptic paraboloid normal stress distributions as follows:
x2 y2 þ a2 c2 2 x y2 szx ðx; yÞ ¼ szy ðx; yÞ ¼ 2 2 2 a c
szx ðx; yÞ ¼ szy ðx; yÞ ¼ 1 þ
ð22aÞ ð22bÞ
Table 1 AEmax , MSE and coefficient of determination between SIFs calculated by FEA and MMWFs. Stress distribution
Linear function of x
Type of err.
Max. AE MSE 2
R Linear function of y
Max. AE MSE R2
Elliptic paraboloid, Eq. (22a)
Max. AE MSE R2
Elliptic paraboloid, Eq. (22b)
Max. AE MSE R2
Trigonometric functions
Max. AE MSE R2
KI
K II
K III
a = 0.3 b = 0.12
a = 0.7 b = 0.07
a = 0.3 b = 0.12
a = 0.7 b = 0.07
a = 0.3 b = 0.12
a = 0.7 b = 0.07
0.149 0.00196 0.982
0.248 0.02160 0.978
0.037 0.00050 0.994
0.095 0.00310 0.997
0.078 0.00032 0.997
0.096 0.00369 0.996
0.082 0.00254 0.966
0.219 0.03002 0.969
0.025 0.00019 0.998
0.136 0.00698 0.993
0.010 0.00005 0.999
0.155 0.00282 0.997
0.057 0.00108 0.991
0.167 0.00773 0.995
0.051 0.00070 0.994
0.191 0.01483 0.990
0.042 0.00029 0.997
0.084 0.00272 0.998
0.057 0.00094 0.995
0.159 0.00641 0.997
0.049 0.00064 0.997
0.183 0.01549 0.994
0.039 0.00031 0.998
0.085 0.00251 0.999
0.046 0.00057 0.996
0.138 0.00558 0.997
0.043 0.00075 0.995
0.104 0.00290 0.998
0.018 0.00010 0.999
0.134 0.00906 0.995
69
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Fig. 8. Comparison of SIFs calculated by FEA and MMWFs for a = 0.3, b = 0.12 and a = 0.7, b = 0.07 under elliptic paraboloid stress distributions.
The FEA and MMWFs results are compared in Fig. 8. The graphical illustrations of the stress distributions are also provided in Fig. 8. According to the values of K I;max =K II;max and K I;max =K III;max ratios, which are 0.48 and 0.64, and 0.78 and 1.05 respectively for stress distributions of Eqs. (22a) and (22b), it may be concluded that under shear loadings, the mode I SIF caused by the coupling effect can be more significant than mode III SIF. Maximum AE between FEA and MMWFs SIFs are 0.167, 0.191 and 0.085 for mode I, II and III, respectively, as included in Table 1. So, it can be concluded that derived MMWFs are capable of quickly and accurately calculating MMSIFs of subsurface elliptical cracks with any as and bs under nonlinear shear loadings. As can be seen in Fig. 8, even stress distributions with two symmetry planes affect the critical points of the crack front. Run time of the FE analysis or numerical integration of the MMWFs is almost independent of the stress distributions on the crack faces. So, the MMWF method is time efficient relative to the FE method for all stress distributions. 5.2.3. Trigonometric paraboloid stress distribution So far, the loadings used for validation have symmetry plane(s). To generalize the validation and indicate the efficiency of the MMWFs for all stress distributions, a trigonometric paraboloid stress distribution without any symmetry plane is considered. The stress distribution, applied to the crack faces, is shown in Fig. 9 and defined as follows:
szx ðx; yÞ ¼ szy ðx; yÞ ¼ 2 sin
p x 6 a
p y þ 2 cos þ1 6 c
ð22cÞ
70
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Fig. 9. Comparison of SIFs calculated by FEA and MMWFs for a = 0.3, b = 0.12 and a = 0.7, b = 0.07 under trigonometric 2D stress distributions.
MMSIFs are calculated and compared in Fig. 9 using FEA and MMWFs. Similar to the previous stress distributions, the values of K I;max are considerable compared to K II;max and K III;max values, K I;max =K II;max = 0.66 and K I;max =K III;max = 0.70, and therefore the coupling effect of shear loadings on the mode I is compelling. Also, MSE between K I ; K II and K III calculated by FEA and MMWFs are less than 0.0056, 0.0030 and 0.0091, respectively, as provided in Table 1. According to the calculated errors, it is deduced that the results of derived MMWFs and FEA are in excellent agreement and so the MMWFs are reliable for calculation of MMSIFs under any kind of stress distribution. Mode I SIFs are negative for some angels of the crack front in Figs. 7–9. These negative values indicate that the crack faces are compressed due to the coupling effect and the crack front is closed in that area. The use of negative mode I SIFs are applicable only under combined tension and shear loadings, when there is sufficient tension to make the total SIFs positive. It should be emphasized that the negative sing for the shear modes of SIFs indicates only the sliding and tearing directions of the crack faces. Such MMWFs which provide accurate MMSIFs are desirable, regarding to the time and endeavor required to determine MMSIFs utilizing numerical procedures such as FEA. Moreover, these MMWFs are very comfortable to apply in industrial applications, and very efficient for evaluating fatigue behavior of the cracks under complicated 2D loadings.
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71
6. Conclusions Reference MMSIFs under unit uniform shear loadings along x and y directions are utilized to derive MMWFs for elliptical subsurface cracks. To consider the coupling of the fracture modes for subsurface cracks, the WFs is derived in matrix form as MMWFs. The dimensionless parameters are aspect ratio, a = 0.2–1.0, and crack depth to crack length ratio, b = 0.05 to infinity (=4). Different approximate forms of WFs, with only one coefficient for each mode, are considered for direct elements (DIIn ; DIIIt Þ and coupling elements of WF matrix (DIn ; DIIIn ; DIt ; DIIt Þ. Explicit equations with about 80 coefficients are fitted to the MMWFs coefficients with high accuracy. These equations make it possible to calculate MMSIFs for any values of a, b, and the angle of crack front. Five linear and nonlinear 2D stress distributions are applied to the crack faces with as and bs, different from those used for fitting, to verify the accuracy of MMWFs including explicit equations of their coefficients. Maximum AE between FEA and MMWFs SIFs are 0.248, 0.119 and 0.155 for mode I, II and III, respectively. Hence, it can be deduced that MMSIFs of MMWFs method are in high accordance with FEA results, while MMWFs method is 60 times faster than FE method in calculation of MMSIFs for present problem. MMSIFs under different shear stress distributions indicate that, tension mode caused by the coupling effect is significant for an elliptical subsurface crack under shear loadings. Also, from the results it can be concluded that the stress distribution may affect the location of critical points along the crack front. The MMWFs are applicable for subsurface cracks under (complicated) 2D stress distributions, developed in industrial processes. The MMWFs are convenient to apply for calculation of MMSIFs of the crack front, and so for study of the fatigue crack growth phenomenon. Appendix A The coefficients DIn ; DIIn ; DIIIn ; DIt ; DIIt and DIIIt in Eq. (17):
ForDIn : A0 ðaÞ ¼ 0 3 2 1:58166 13:93044 32:70718 14:81751 7 6 6 2:99029 25:71956 53:46568 22:62742 7 7 Aim ¼ 6 7 6 8:00360 5 4 1:50322 12:46105 21:24514 0:02550
0:16134
0:17826 3 1:08087 5:37478 2:42958 7 6 0:12297 7 6 0:99970 4:62350 7 ¼6 7 6 4 0:36664 0:96027 1:22056 5 0:00220
0:09917
0:01472
2
B1im
2
18:88158
0:19380
118:14770
99:80799
3
7 6 6 37:59783 226:67235 200:32192 7 7 B2im ¼ 6 7 6 4 20:92382 115:08795 102:87613 5 2
0:90431
3:32284
2:37514
104:34370
334:86668
140:98134
3
7 6 6 191:75323 602:58060 289:30164 7 7 B3im ¼ 6 7 6 279:53472 142:78811 5 4 93:09366 2:79312
5:76304
1:20763
3 122:64739 203:90187 115:59134 7 6 6 219:46861 349:48549 166:25360 7 7 ¼6 7 6 72:22723 5 4 101:85484 147:78358 2
B4im
2:53231 2
41:74191
6 6 73:98725 B5im ¼ 6 6 4 33:54795 0:71505
0:66230
5:44052 3
3:76035 146:26313 3:84032 3:69127 1:52014
7 235:48160 7 7 7 103:65580 5 4:02565
72
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ForDIIn : 2 Aim
0:88378
11:61606
24:86375
12:30724
3
6 1:24037 16:60548 47:87198 26:30990 7 7 6 7 6 7 ¼6 0:22664 8:81657 37:42833 22:78568 7 6 7 6 4 0:22851 0:90828 10:86887 7:48242 5
B1im
0:00059 8:57843
0:07389 24:00878
0:28287 3 7:75538 6 5:17970 0:93101 26:48050 7 7 6 ¼6 7 4 0:55498 7:57167 20:55390 5 2
0:18802
0:06149 0:54520 0:92253 3 45:80331 131:59403 188:85024 6 63:21148 259:53782 419:93866 7 7 6 ¼6 7 4 26:18417 129:91757 220:38683 5 2
B2im
5:42200 8:57518 3 1018:08027 1124:69965 6 355:32515 1581:49995 1956:86574 7 7 6 ¼6 7 4 146:42928 708:74192 929:65748 5 2
B3im
21:68296 1534:19365
5:38222 157:88003
29:47513 692:34427
2:36307
12:97763
39:17136 3 765:40007 6 223:34218 1069:70969 1274:61658 7 6 7 ¼6 7 4 94:86533 476:44652 590:93739 5 2
B5im
3:88439 366:18861
29:96757 3 1678:17466 6 506:46739 2357:10961 2813:50650 7 6 7 ¼6 7 4 212:49352 1047:64850 1309:81518 5 2
B4im
0:87249 264:33234
17:08923
ForDIIIn : Aða; bÞ ¼ 0 3 2 30:27640 61:17297 39:34168 6 24:68252 27:96486 10:31798 7 7 6 B1im ¼ 6 7 4 6:40971 0:64290 4:46884 5
B2im
2
0:05193 139:11855
0:10892 0:10233 3 435:15787 308:93640 6 134:50260 370:09131 199:57364 7 7 6 ¼6 7 4 43:00209 100:54789 29:26046 5 2
0:46552 206:11330
6 195:75243 6 B3im ¼ 6 4 59:90716
0:07646 672:16893
1:68623 3 426:10573 476:05888 63:34884 7 7 7 82:75417 111:23319 5
0:15529 114:03271
4:90365 302:73847
6 85:45329 6 B4im ¼ 6 4 16:27209
23:36571
2
B5im
130:32902
10:63187 3 61:19341 486:34038 7 7 7 360:00592 5
0:92480 10:71258 17:91138 2 3 15:19507 8:38198 105:04459 6 3:40349 179:40496 378:42555 7 6 7 ¼6 7 4 8:52000 122:50314 223:76205 5 0:70873
5:89961
9:12622
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ForDIt : Aða; bÞ ¼ 0 3 2 0:23045 17:21569 8:10737 7 6 6 0:78543 22:05137 6:65161 7 7 B1im ¼ 6 7 6 3:74902 1:62828 5 4 0:65320 0:04988
0:06715
0:09856
3 74:31918 195:58061 82:70268 7 6 6 125:38163 311:23804 102:63547 7 7 ¼6 7 6 4 52:93653 119:75685 19:73653 5 2
B2im
2
0:78532
0:74829
184:72728
336:53259
1:78081 82:73465
3
7 6 325:41572 7 6 293:05927 436:62244 7 B3im ¼ 6 7 6 4 113:74918 108:11323 250:22801 5 0:30568
14:98046
6:64026
3 141:26500 85:02430 455:98801 7 6 72:28089 1048:43665 7 6 196:01158 7 ¼6 7 6 607:21023 5 4 59:59866 152:36591 2
B4im
2:10854 2
25:12465
16:80942 78:61129
27:18545 3
319:56135
7 6 6 17:53596 247:23216 681:72215 7 7 B5im ¼ 6 7 6 168:61712 370:72230 5 4 6:39574 1:80105
10:29909
14:95897
ForDIIt : Aða; bÞ ¼ 0 2 3 67:11344 183:89028 144:84641 6 7 6 59:99010 146:98985 116:34040 7 7 B1im ¼ 6 6 7 38:89261 31:96874 5 4 17:84391 0:21966
0:57556
0:61031
3 348:55352 1253:06876 1067:28537 7 6 6 333:86595 1182:23473 1006:14973 7 7 ¼6 7 6 379:92824 328:02472 5 4 107:67367 2
B2im
2
1:39093
5:25377
5:21579
673:18837
2669:27915
2389:11828
6 6 643:70897 2566:87311 B3im ¼ 6 6 853:90058 4 209:42177 2
2
7 2326:62799 7 7 7 798:07310 5
2:60615
12:00353
13:38122
584:76392
2423:39238
2223:03715
6 6 543:04630 B4im ¼ 6 6 4 173:89911
2297:87697 769:77111
2:03300
11:15474
192:93964
817:55412
6 6 172:87929 B5im ¼ 6 6 4 54:27985 0:61620
759:52515 255:53289 3:95171
3
3
7 2161:91529 7 7 7 760:07394 5 13:83491 3
758:53762
7 731:35244 7 7 7 262:66800 5 5:29287
73
74
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ForDIIIt : 2
Aim
3 3:24823 9:58283 6:05990 1:00364 7 6 6 0:26605 8:43388 7:75604 14:41867 7 7 6 7 6 ¼ 6 0:11388 6:03134 13:32155 17:34155 7 7 6 6 0:04888 1:02286 6:00136 6:59591 7 5 4 0:00611 2
0:02852
13:63671
0:17230
48:76125
0:13682 3
41:81282
7 6 6 12:49967 45:09159 38:08483 7 7 B1im ¼ 6 7 6 14:17798 11:65905 5 4 3:93807 2
0:06536
0:21669
0:15687
55:13826
60:06507
9:41365
3
7 6 6 44:43512 26:33747 100:44682 7 7 B2im ¼ 6 7 6 45:61408 68:99298 5 4 8:73708 0:45058 2 B3im
183:20630
6 6 142:95837 ¼6 6 4 27:76059 2
2:62891 479:80914 256:88799 4:14522
2:55356 292:11856
7 60:18886 7 7 7 65:15533 5
1:36787
5:91419
5:62024
190:13110
630:86827
467:04586
6 6 141:17300 B4im ¼ 6 6 4 24:39263 1:57077
426:59566 62:31095 5:14416
3
3
7 265:15284 7 7 7 17:88071 5 4:50146
3 67:10750 252:93670 202:89995 7 6 6 47:09674 185:53421 140:45279 7 7 ¼6 7 6 35:83472 25:84049 5 4 6:82296 2
B5im
0:62271
1:64843
1:30190
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Mixed mode weight functions for an elliptical subsurface crack under shear loadings Rahmatollah Ghajar a,⇑, Javad Alizadeh Kaklar b a Mechanical Properties Research Laboratory (MPRL), Department of Mech. Eng, K.N. Toosi University of Technology, Pardis St., Mollasdra Ave., Vanak Sq., P.O. Box 19395-1999, Tehran, Iran b Department of Mech. Eng., Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Received 17 June 2014 Received in revised form 13 January 2015 Accepted 15 January 2015 Available online 7 February 2015 Keywords: Elliptical subsurface cracks Mixed mode weight functions Coupling of fracture modes Two dimensional stress distributions
a b s t r a c t In this paper, mixed mode weight functions (MMWFs) for elliptical subsurface cracks under shear loadings are derived. Reference mixed mode stress intensity factors (MMSIFs), calculated by finite element analysis (FEA), under uniform shear loadings are used to derive MMWFs. The cracks have aspect ratios and crack depth to crack length ratios in the ranges of 0.2–1.0 and 0.05 to infinity, respectively. MMWFs are verified to calculate MMSIFs for any point of the crack front under linear and nonlinear two dimensional (2D) loadings. The derived MMWFs could be effectively used for evaluation of fatigue crack growth under complicated 2D stress distributions. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Calculation of stress intensity factor (SIF) is a prerequisite step in applied fatigue and fracture analysis. Embedded elliptical cracks are one of the most common types of cracks in engineering structures. Elliptical subsurface cracks are embedded cracks that often initiate and propagate under rolling contact fatigue (RCF). Due to the existence of large pressures under the rolling contact condition, the subsurface cracks propagate mainly in shear fracture modes [1,2]. Fractures of two railway wheels caused by the fatigue crack growth of two elliptical subsurface cracks are represented in Fig. 1. Many studies have been conducted so far on determination of the SIFs for the embedded elliptical cracks in an infinite space. Irwin [3] analytically obtained an exact relation for mode I SIF of the cracks under uniform normal loadings. Kassir and Sih [4,5] investigated the cracks under linear normal and constant shear loadings, and developed the closed-form solutions for SIFs. Shah and Kobayashi [6] studied such cracks under two dimensional (2D) third order polynomial normal loadings and obtained exact relation for SIF. Vijayakumar and Atluri [7] determined SIF for the cracks under any-order polynomial normal loadings. Nishioka and Atluri [8] solved the problem for any-order polynomial shear loadings. Bueckner [9] presented the weight function method (WF) for the computation of SIF under complex stress distributions. In this method, derived WF can be utilized to calculate SIF under any arbitrary loading. Rice [10] demonstrated that the WF can be obtained using crack opening displacement and SIF of a reference load. Roy and Saha [11,12] applied analytical mode I, II, III WFs and obtained the mode I, II, III SIFs for any point of the cracks fronts under any arbitrary normal and shear loadings. Also, Atroshchenko, Potapenko and Glinka [13,14] calculated mode I SIFs for the cracks under arbitrary normal loadings using analytical WF. Wang and Glinka [15] proposed approximate form of WF for the cracks, and calculated the SIF for arbitrary normal loadings. ⇑ Corresponding author. Tel.: +98 2184063240; fax: +98 2188677273. E-mail addresses: [email protected] (R. Ghajar), [email protected] (J.A. Kaklar). http://dx.doi.org/10.1016/j.engfracmech.2015.01.024 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.
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Nomenclature A crack face area Aim constant coefficients AEmax maximum absolute error a semi-major axis of ellipse, crack length Bnim constant coefficients c semi-minor axis of ellipse Cj constant coefficients c1 distance of ellipse from the major axis in normal direction c2 distance of ellipse center from the tangent of the ellipse Di WF coefficients DIn mode I WF coefficient for a shear stress acting normal to the crack front DIIn mode II WF coefficient for a shear stress acting normal to the crack front DIIIn mode III WF coefficient for a shear stress acting normal to the crack front DIt mode I WF coefficient for a shear stress acting tangent to the crack front DIIt mode II WF coefficient for a shear stress acting tangent to the crack front DIIIt mode III WF coefficient for a shear stress acting tangent to the crack front E elastic modulus E(a) second kind of elliptical integral E0 material constant F nim constant coefficients Gjn constant coefficients Gjt constant coefficients h depth of the crack K(a) first kind of elliptical integral K FEA SIFs calculated by FEA KI mode I SIF K II mode II SIF K III mode III SIF K Ix mode I SIF caused by unit uniform shear stress along x K IIx mode II SIF caused by unit uniform shear stress along x K IIIx mode III SIF caused by unit uniform shear stress along x K Iy mode I SIF caused by unit uniform shear stress along y K IIy mode II SIF caused by unit uniform shear stress along y K IIIy mode III SIF caused by unit uniform shear stress along y Kr reference SIF K WF SIFs calculated by MMWFs L point of loading MSE mean square error Q arbitrary point of the crack front R ellipse radius for the corresponding point of L R2 coefficient of determination r polar coordinate of point L s shortest distance of the crack front from L point ur ðx; aÞ reference displacement of crack face Wðx; aÞ one dimensional WF Wðx; y; u0 Þ two dimensional WF W In mode I WF for a shear stress acting normal to the crack front W IIn mode II WF for a shear stress acting normal to the crack front W IIIn mode III WF for a shear stress acting normal to the crack front W It mode I WF for a shear stress acting tangent to the crack front W IIt mode II WF for a shear stress acting tangent to the crack front W IIIt mode III WF for a shear stress acting tangent to the crack front a aspect ratio of the crack b crack depth to the crack length ratio u angle of point L u0 angle of crack front, point Q m Poisson’s ratio q distance between the point L and point Q r normal stress distribution of the unflawed structure, across the crack faces
59
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s0 sn ðx; yÞ st ðx; yÞ szx ðx; yÞ szy ðx; yÞ
uniform shear stress distribution shear stress distribution acting normal to the crack front shear stress distribution acting tangent to the crack front shear stress distribution acting on the crack faces along x shear stress distribution acting on the crack faces along y
Fig. 1. Fractured railway wheels due to the propagation of the elliptical subsurface cracks [2].
Fig. 2. An elliptical subsurface crack in a semi-infinite space under normal and tangent shear stresses. minimum dimensions of the space are evaluated in Ref. [16].
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If the embedded crack in an infinite space changes to a subsurface crack by approaching one of boundary surfaces of the space (Fig. 2), coupling of the fracture modes happens due to the nonsymmetrical geometry with respect to the crack face [16]. In such case, mixed mode WFs (MMWFs) with a matrix formulation is necessary to consider the coupling of the fracture modes and calculate mixed mode SIFs (MMSIFs). Beghini, Bertini and Fontanari [17] evaluated the WFs of a subsurface crack in a half-plane and investigated the coupling of modes I and II. Mazzu [18] studied a 2D subsurface micro-crack using WF approach, regardless of coupling of the modes. In three-dimensional space, coupling between all fracture modes exists, and a subsurface crack experiences all three fracture modes. Ghajar and Alizadeh [16] conducted 50 finite element analysis (FEA) and derived explicit expressions for mode I, II and III SIFs of elliptical subsurface cracks under uniform normal loadings. They concluded that under uniform normal loadings, the coupling is considerable between mode II and I and is negligible between mode III and I. The problem of an elliptical subsurface crack in a half-space under shear loadings plays a basic role in the investigation of RCF. Calculation of SIFs for elliptical subsurface cracks in a half-space under arbitrary shear loadings has not been studied yet. Also, it is known that the determination of SIFs under any arbitrary 2D stress distributions can be performed using 2D WF [19,20]. So, the main purpose of this paper is to develop MMWFs for any point of the elliptical subsurface crack fronts with different aspect ratios and ratios of crack depth to crack length under shear loadings. Having MMWFs makes it possible to calculate MMSIFs for any point along the subsurface elliptical crack front under any arbitrary 2D shear loading acting on the crack faces. 2. Weight function method Bueckner [9] showed that the SIF of a crack, K I , under any loading can be calculated from the integral of the stress distribution across the crack plane in the unflawed body, r(x), multiplied by WF of the crack geometry,Wðx; aÞ,:
KI ¼
Z
a
rðxÞ Wðx; aÞdx
ð1Þ
0
in which, a is the crack length. The WF of the crack geometry can be derived using crack face displacement, ur ða; xÞ, and mode I SIF for a reference loading, K r , by:
Wðx; aÞ ¼
E0 @ur ðx; aÞ @x Kr
ð2Þ
where, E0 is E for plane stress and E=ð1 m2 Þ for plane strain, and x is the axis along the crack. Rice [10] showed that the WF is a unique geometrical property of a cracked body and is independent of the loading system. The surface displacements have been rarely determined for the cracks and therefore, Eq. (2) is not suitable and applicable for deriving the WF. So, Petroski and Achenbach [21] offered an approximate relation for the surface displacements of cracks base on which, other researchers developed some general forms for WF [22–24]. By considering general form of WF, one can derive the WF by determining the coefficients of general form using reference SIFs under reference loadings. The 2D WF, Wðx; y; u0 Þ, represents the SIF at point Q along the crack front, caused by unit point loads imposed on the crack faces at point Lðx; yÞ, Fig. 3. So, to calculate the SIF at point Q under a 2D stress distribution, rðx; yÞ, the Eq. (1) is modified to the following integration:
Kðu0 Þ ¼
Z Z A
rðx; yÞ Wðx; y; u0 ÞdA
ð3Þ
where A is the crack surface area. Wang and Glinka [15] proposed an effective general form of WF for the embedded elliptical cracks as:
Wðx; y; u0 Þ ¼
pffiffiffiffiffi " 2s
p3=2 q2
i # n X rðuÞ 1þ Di ða; u0 Þ 1 RðuÞ i¼1
Fig. 3. An elliptical crack and representation of the geometrical parameters.
ð4Þ
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here, s and q are the distances of the load point Lðx; yÞ from the crack front and the point Q, respectively, r and u define the polar coordinates of point Lðx; yÞ, and R(u) is the polar radius of corresponding point on the crack front, as indicated in Fig. 3. The WF coefficients for any point of the crack front, Di , depend on the aspect ratio of the elliptical crack. In Eq. (4), the WF is the summation of two parts, namely, the singular term, and the term accounting for the effect of crack configurations [15]. Ghajar and Saeidi [25,26] showed that this general form of WF can also be used for surface cracks. 3. Mixed mode weight functions (MMWFs) In this study, subsurface cracks with different aspect ratios, a = c/a, and crack depth to crack length ratios, b = h/a, are considered, while the crack faces are parallel to the boundary surface of the half-space, as indicated in Fig. 2. Therefore, the WF coefficients, Di , in Eq. (4) for any point of the crack front, Q(u0 Þ, are functions of two geometrical parameters, a and b. In fracture mechanics, existence of any non-symmetry in material, geometry or loading system with respect to the crack faces will cause coupling of fracture modes. Thus, a subsurface crack experiences all fracture modes under shear loading. So, mixed mode WFs are necessary to be derived for elliptical subsurface cracks under shear loadings. MMWFs are shown in matrix form as:
2
K I ða; b; u0 Þ
3
6 7 4 K II ða; b; u0 Þ 5 ¼ K III ða; b; u0 Þ
Z Z
2
W In ðx; y; a; b; u0 Þ
W It ðx; y; a; b; u0 Þ
3
6 7 4 W IIn ðx; y; a; b; u0 Þ W IIt ðx; y; a; b; u0 Þ 5 A W IIIn ðx; y; a; b; u0 Þ W IIIt ðx; y; a; b; u0 Þ
sn ðx; yÞ dA st ðx; yÞ
ð5Þ
here, K I ; K II and K III are respectively the mode I, II and III SIFs of the crack front, W In ; W IIn and W IIIn are respectively the mode I, II and III WFs when the direction of the applied shear loading is normal to the crack front at point Q ; W It ; W IIt and W IIIt are respectively the mode I, II and III WFs for the shear loading which is tangent to the crack front at point Q, and sn and st are shear loadings applied to the crack faces along normal and tangential directions of the crack front at point Q, respectively (Fig. 2). It should be emphasized here that three fracture modes for a 3D planar crack are considered as normal to the crack face (mode I), in the crack face along normal direction of crack front (mode II or sliding mode) and in the crack face along tangent direction of the crack front (mode III or tearing mode). So, to evaluate the mixed fracture modes of an elliptical crack, stress distributions along these three directions, r; sn and st , should be considered. This study investigates the elliptical subsurface cracks under shear stresses (sn and st Þ. In the WF matrix of Eq. (5), W IIn and W IIIt are called direct elements, while the others are coupling elements. Direct elements return to the direct effect of the loading on the SIFs, whereas the coupling elements are developed by the nonsymmetrical geometry and show the coupling effect of fracture modes on the calculated SIFs. The direct elements of the WF matrix are derived utilizing the general form of Eq. (4). Present analyses indicate that one term of Eq. (4), n = 1, accurately approximates the WF of the subsurface cracks, i.e.:
i pffiffiffiffi h rðuÞ W IIn ðx; y; a; b; u0 Þ ¼ p3=22sq2 1 þ DIIn ða; b; u0 Þ 1 Rð uÞ i pffiffiffiffi h rðuÞ W IIIt ðx; y; a; b; u0 Þ ¼ p3=22sq2 1 þ DIIIt ða; b; u0 Þ 1 Rð uÞ
ð6Þ
where DIIn and DIIIt are the WF coefficients for W IIn and W IIIt , respectively. The general form of Eq. (6) has to be modified for the coupling elements of the WF matrix. It is obvious that with the b parameter approaching infinity, the coupling effect of the fracture modes is eliminated. So, the coupling elements of the WF matrix must approach zero by increasing the b. This condition will be satisfied if the singular term of Eq. (6) is removed and the WF coefficients approach zero by b approaching infinity. Therefore, general form of WF for the coupling elements of WF matrix should be as follows:
W j ðx; y; a; b; u0 Þ ¼
pffiffiffiffiffi 2s
rðuÞ ; D ð a ; b; u Þ 1 j 0 RðuÞ p3=2 q2
j ¼ In; IIIn; It; IIt
ð7Þ
where DIn ; DIIIn ; DIt and DIIt are the WF coefficients for W In ; W IIIn ; W It and W IIt , respectively. By determining the DIn ; DIIn ; DIIIn ; DIt ; DIIt and DIIIt , the calculation of the SIFs for any arbitrary 2D shear stress distribution can be performed using Eq. (8) as follows:
3 rðuÞ rðuÞ DIn ða; b; u0 Þ 1 Rð DIt ða; b; u0 Þ 1 Rð u Þ u Þ pffiffiffiffiffi 6 K I ða; b; u0 Þ 7 Z Z 7 sn ðx; yÞ 2s 6 6 7 rðuÞ 7 6 1 þ DIIn ða; b; u Þ 1 rðuÞ dA ð a ; b; u Þ 1 D 4 K II ða; b; u0 Þ 5 ¼ IIt 0 0 RðuÞ RðuÞ 7 3=2 q2 6 Ap 4 5 st ðx; yÞ K III ða; b; u0 Þ rðuÞ rðuÞ DIIIn ða; b; u0 Þ 1 Rð 1 þ DIIIt ða; b; u0 Þ 1 Rð uÞ uÞ 2
3
2
ð8Þ
4. Reference MMSIFs and calculation of WF coefficients Alizadeh and Ghajar [27] developed a parametric FEA code written in Python scripting language of ABAQUS 6.12 software to calculate MMSIFs for the subsurface cracks under uniform shear loadings imposed to the crack surfaces along x and y
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63
directions, show in Fig. 3. The FEA code was validated using analytical solutions presented by Saha and Roy [12] for embedded elliptical crack under uniform shear stress with mean relative error of 0.5%. The validated FEA code was used to calculate MMSIFs of the crack fronts Q(u0 Þ, for 0.2 6 a 6 1.0, 0.05 6 b 6 4ffi infinity and a = 1 mm. Then, six equations were fitted to the FEA solutions with mean relative error of 3% as follow [27]:
pffiffiffiffiffiffiffiffi
K Ix ða; b; u0 Þ ¼
4 s0 pc1 X F n ða; bÞ cosn u0 EðaÞ n¼1
# pffiffiffiffiffiffiffiffi " 4 X 4as0 pc2 cos u0 þ F n ða; bÞ cosn u0 I1 ða; mÞ n¼1 # pffiffiffiffiffiffiffiffi " 4 X 4ð1 mÞs0 pc2 n sin u0 þ F n ða; bÞ sin u0 K IIIx ða; b; u0 Þ ¼ I1 ða; mÞ n¼1 pffiffiffiffiffiffiffiffi 4 s0 pc1 X n K Iy ða; b; u0 Þ ¼ F n ða; bÞ sin u0 EðaÞ n¼1 # pffiffiffiffiffiffiffiffi " 4 X 4s0 p c 2 n K IIy ða; b; u0 Þ ¼ sin u0 þ F n ða; bÞ sin u0 I2 ða; mÞ n¼1 # pffiffiffiffiffiffiffiffi " 4 X 4ð1 mÞs0 pc2 n cos u0 þ F n ða; bÞ cos u0 K IIIy ða; b; u0 Þ ¼ I2 ða; mÞ n¼1 K IIx ða; b; u0 Þ ¼
ð9Þ
where K Ix ; K IIx and K IIIx , and K Iy ; K IIy and K IIIy are mode I, II and III SIFs when the unit uniform shear stress is separately applied along x and y directions, respectively, m is the Poisson’s ratio, s0 is the uniform shear stress. The c1 and c2 are the length of the normal of the ellipse between the crack front and the x axis, and the distance of ellipse center from the tangent of the ellipse, respectively and are equal to [16]: 2
0:5
c1 ¼ cða2 cos2 u0 þ sin u0 Þ c c2 ¼ 0:5 2 2 2 ða cos u0 þ sin u0 Þ
ð10Þ
Also, I1 ða; mÞ and I2 ða; mÞ are defined as [27]:
4 2 ma KðaÞ þ ð1 a2 mÞEðaÞ 1 a2
4 I2 ða; mÞ ¼ ma2 KðaÞ þ ð1 a2 þ ma2 ÞEðaÞ 1 a2
I1 ða; mÞ ¼
ð11Þ
where K(a) and E(a) are the first and second kind of elliptical integrals, respectively, and the functions F n (a; b) are as follows:
F n2 ðaÞ F n3 ðaÞ F n4 ðaÞ þ 0:5 þ 0:75 þ 1:5 b0:25 b b b 3 X F ni ðaÞ ¼ F nim am ; i ¼ 1; . . . ; 4 F n ða; bÞ ¼
F n1 ðaÞ
ð12Þ
m¼0
where, F nim are constant coefficients provided in [27]. The normal and tangential shear stresses on the crack front of u0 versus the
szx and szy can be written as:
a cos u0 szx þ sin u0 szy cos u0 sin u0 sn ðx; yÞ ¼ ¼ c2 szx þ szy 0:5 2 a c ða2 cos2 u0 þ sin u0 Þ sin u0 szx þ a cos u0 szy sin u0 cos u0 st ðx; yÞ ¼ ¼ c2 s s zx þ zy 0:5 2 c a ða2 cos2 u0 þ sin u0 Þ
ð13Þ
So, normal and tangential shear stresses caused by the unit uniform shear stress along x and y directions are obtained as:
szx ðx; yÞ ¼ 1; szy ðx; yÞ ¼ 0
)
szx ðx; yÞ ¼ 0; szy ðx; yÞ ¼ 1
)
cos u0 ; a sin u0 sn ðx; yÞ ¼ c2 ; c
sn ðx; yÞ ¼ c2
sin u0 c cos u0 st ðx; yÞ ¼ c2 a
st ðx; yÞ ¼ c2
ð14Þ
Substituting the reference MMSIFs and stress distributions of Eq. (14) in Eq. (8), results in a system of two equations with two unknowns for each fracture mode. By solving the systems of equations, the coefficients of weight functions are obtained as follows:
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8 ffi R R pffiffiffi c 2s > cosau0 K jx þ sincu0 K jy Gcjn Djn ða; b; u0 Þ ¼ R R pffiffiffi 2 > 3=2 q2 dA > A p rðuÞ > 2 2s < 1RðuÞ dA A p3=2 q2 j ¼ I; II; III ffi R R pffiffiffi c2 > 2s sincu0 K jx þ cosau0 K jy Gcjt > dA > A p3=2 q2 > Djt ða; b; u0 Þ ¼ R R pffiffi2sffi rðuÞ 2 : 1 dA A p3=2 q2
RðuÞ
where GIIn and GIIIt are unity while other Gs are equal to zero.
Fig. 4. Mode I, II and III WF coefficients of the elliptical subsurface cracks for shear loading normal to the crack front, DIn ; DIIn ; DIIIn .
ð15Þ
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65
Fig. 5. Mode I, II and III WF coefficients of the elliptical subsurface cracks for shear loading tangent to the crack front, DIt ; DIIt ; DIIIt .
5. Results and discussion 5.1. MMWFs for elliptical subsurface cracks under shear loadings A program is used to calculate the numerical parametric singular integration by Wolfram Mathematica 7 software. The integration algorithm is verified using the FEA results. By integration, the MMWFs coefficients, Dj (a,b,u0 Þ, are calculated for 0.2 6 a 6 1.0, 0.05 6 b 6 4ffi infinity and 0 6 u0 6 p=2. The mode I, II and III WFs coefficients for normal and tangent shear loadings are plotted in Figs. 4 and 5, respectively. It should be emphasized that due to the symmetry, the coefficients DIt ; DIIt and DIIIn are zero for circular cracks (a =1). Also, the coefficients DIn and DIt approach zero for b ¼ 1. To formulate the coefficients of MMWFs, Dj ða; b; u0 Þ, a nonlinear fitting process with regression analysis is conducted using Data-fit software. To
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fit an equation to a set of data, various general forms of equations can be utilized. Among all tried forms of equations, the best fitted one is as follows:
Dj ða; b; u0 Þ ¼
"
Cj 2
ða2 cos2 u0 þ sin u0 Þ
5 X n Aða; bÞ þ ð1 aÞ Bn ða; bÞ sin u0
0:25
# ð16Þ
n¼1
where, j = In, IIn, IIIn, It, IIt, IIIt and A(a, b), Bn (a, b) and C j are represented by:
Aða; bÞ ¼ A0 ðaÞ þ Bn ða; bÞ ¼
Bn1 ðaÞ 0:25
b
A1 ðaÞ 0:25
b
þ
þ
Bn2 ðaÞ 0:5
b
A2 ðaÞ 0:5
b þ
þ
A3 ðaÞ
Bn3 ðaÞ 0:75
b
þ
0:75
b
þ
A4 ðaÞ b
Bn4 ðaÞ 1:5
b
1:5
;
;
Ai ðaÞ ¼
3 X Aim am ; i ¼ 0; . . . ; 4 m¼0
2 X Bni ðaÞ ¼ Bnim am ; i ¼ 1; . . . ; 4
ð17Þ
m¼0
C In ¼ C IIn ¼ C IIIt ¼ 1; C IIIn ¼ C It ¼ C IIt ¼ cos u0 in which, Aim and Bnim are constant coefficients provided in appendix. The numbers of coefficients in presented equations are 80 for DIIn and DIIIt , 76 for DIn and 60 for DIIIn ; DIt and DIIt . To evaluate the conformity of the Eq. (16) with the data points used for curve fitting, the results of Eq. (16) are indicated in Figs. 4 and 5 by continues lines. Comparison between the values calculated by Eq. (16) and data points represents excellent accuracy for the fitted equations. Almost 754–1050 data points are utilized in fitting process for each Dj . Accurate fitting of a three-variable equation to this amount of data points indicates the appropriateness of the selected general form for the Eq. (16). 5.2. Validation To use the proposed MMWFs to determine MMSIFs for a loading applied to the boundary surface of the space, stress distribution has to be first calculated in a semi-infinite body without a crack and then imposed on the crack faces. It should be noted here that all elliptical (or circular) cracks studied with other researchers are in infinite space. Therefore, they can be used to validate the MMWFs of the subsurface cracks in the special case with b ¼ 1. Fig. 6 indicates accuracy of MMWFs solutions comparing with the analytical values provided in Ref. [12] for elliptical crack in an infinite space under linear 2D stress distribution. To validate the efficiency and accuracy of the obtained MMWFs as well as the fitted equations, Eq. (16), first, Dj for the cracks with a = 0.3, b = 0.12 and a = 0.7, b = 0.07 were calculated using Eq. (16). These values of a and b were not used in fitting process of Eq. (16). Then, MMSIFs of the subsurface cracks are calculated under five different linear and nonlinear stress distributions for a = 1 mm using FEA method and MMWFs. All FEA and MMWFs calculations are performed for stress distributions imposed on the crack faces whereas the surface of the semi-infinite body is free. The parametric FEA code which was developed and validated by Alizadeh and Ghajar [27] is used for calculation of MMSIFs. The FEA model is analyzed statically for five contour integrals including the singularity of the crack front. It should be emphasized here that more details of the FEA model were presented in our previous studies [16,27]. Comparison between the FEA and MMWFs results will demonstrate the efficiency and accuracy of obtained MMWFs for calculating MMSIFs of the elliptical subsurface cracks with any a and b under any arbitrary shear stress distributions. Due to the geometrical symmetry, the WF coefficients, Dj , were only calculated for the 0 6 2u0 =p 6 1. The magnitudes of the WF coefficients for other trigonometric quadrants are as follows:
Fig. 6. Comparison of analytical and MMWFs solutions for embedded elliptical crack in an infinite space under linear 2D stress distribution.
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8 Dj ðu0 Þ > > > < D ðp u Þ j 0 Dj ðu0 Þ ¼ > Dj ðu0 pÞ > > : Dj ð2p u0 Þ
67
0 6 2u 0 = p 6 1 1 6 2u0 =p 6 2 2 6 2u0 =p 6 3
ð18Þ
3 6 2u0 =p 6 4
In addition, the signs of the WF coefficients in trigonometric quadrants are important. Based on the MMSIFs calculated by full model of FEA under unit uniform shear loadings, the signs of the Dj are determined. Considering the sings of Dj and Eq. (18), the MMSIFs for all points of the crack front under arbitrary shear loadings are calculated by:
3 rðuÞ rðuÞ D D ð u Þ 1 ð u Þ sgnðsin 2 u Þ 1 It 0 0 7 6 In 0 2 3 RðuÞ RðuÞ pffiffiffiffiffi 6 KI 7 Z Z 6 2s 6 rðuÞ rðuÞ 7 7 sn ðx; yÞ 6 7 D dA ð u Þ 1 ð u Þ sgnðsin 2 u Þ 1 1 D 7 6 4 K II 5 ¼ IIn IIt 0 0 0 3=2 q2 6 RðuÞ RðuÞ 7 st ðx; yÞ Ap 7 6 K III 5 4 rðuÞ rðuÞ 1 þ DIIIt ðu0 Þ 1 DIIIn ðu0 Þ sgnðsin 2u0 Þ 1 RðuÞ RðuÞ 2
ð19Þ in which, the sgn(x) is the sign function that extracts the sign of x. 5.2.1. Linear stress distributions Two linear shear stress distributions are imposed on the subsurface crack faces and the MMSIFs are calculated from FEA and Eq. (19). The linear stress distributions are as follows:
Fig. 7. Comparison of SIFs calculated by FEA and MMWFs for a = 0.3, b = 0.12 and a = 0.7, b = 0.07 under linear 2D stress distributions.
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x a y szx ðx; yÞ ¼ szy ðx; yÞ ¼ 0:5 þ 1 c
szx ðx; yÞ ¼ szy ðx; yÞ ¼ 0:5 þ 1
ð20aÞ ð20bÞ
Comparison between FEA results and results of the MMWFs, as well as schematic representation of the stress distributions on the crack surface are illustrated in Fig. 7. It is clear from this figure that all fracture modes occur, although only the shear stresses are applied. The ratios of the maximum mode I SIF to maximum mode II and mode III SIFs, K I;max =K II;max and K I;max =K III;max , are 0.53 and 0.72 for stress distribution of Eq. (20a)a, and are 0.50 and 0.65 for stress distribution of Eq. (20b), respectively. Therefore, tension mode caused by the coupling between shear modes and mode I is important under linear shear stress distributions. Maximum absolute error (AEmax Þ, mean square error (MSE) and coefficient of determination ð0 6 R2 6 1Þ between SIFs calculated by FEA and MMWFs, are calculated as follows:
AEmax ¼ max jK WF K FEA j N X ðK WF i K FEAi Þ2 MSE ¼ N1
ð21Þ
i¼1
PN
2 i¼1 ðK WF i K FEAi Þ R2 ¼ 1 P ; 2 N i¼1 ðK FEAi KÞ
N X ¼1 K K FEAi N i¼1
where N is the total number of data points for each mode of SIFs, and K FEA and K WF are SIFs calculated by FEA and MMWFs, respectively. Based on the calculated errors (provided in Table 1), the obtained MMWFs are validated to be accurately used for linear loadings imposed on the subsurface elliptical cracks with any as and bs. Under uniform shear loadings, maximum SIFs of mode I, II and III appear for 2u0 =p = 0, 1, 2, 3 or 4 [27]. Linear loading in Eq. (20a), changes the mode I, II and III critical points of the crack to 2u0 =p = 0.57, 0.40 and 3.42 for a = 0.3, b = 0.12, and to 2u0 =p = 0.42, 0.42 and 3.51 for a = 0.7, b = 0.07, as shown in Fig. 7. So, it may be expressed that the stress distributions affect the location of critical points along the crack front. One of the most important benefits of WF method is time saving in calculation of the SIFs for different crack lengths, when the fatigue phenomenon is to be investigated. In this study, run time of the FEA for each model with individual a, b and stress distribution, was about 2 h, whereas the MMWFs took about 2 min to calculate the MMSIFs. So, WF method is 60 times faster than the FE method in calculation of MMSIFs for present problem. 5.2.2. Elliptic paraboloid stress distributions MMSIFs of the subsurface cracks are calculated from FEA and MMSIFs (Eq. (19)) under two elliptic paraboloid normal stress distributions as follows:
x2 y2 þ a2 c2 2 x y2 szx ðx; yÞ ¼ szy ðx; yÞ ¼ 2 2 2 a c
szx ðx; yÞ ¼ szy ðx; yÞ ¼ 1 þ
ð22aÞ ð22bÞ
Table 1 AEmax , MSE and coefficient of determination between SIFs calculated by FEA and MMWFs. Stress distribution
Linear function of x
Type of err.
Max. AE MSE 2
R Linear function of y
Max. AE MSE R2
Elliptic paraboloid, Eq. (22a)
Max. AE MSE R2
Elliptic paraboloid, Eq. (22b)
Max. AE MSE R2
Trigonometric functions
Max. AE MSE R2
KI
K II
K III
a = 0.3 b = 0.12
a = 0.7 b = 0.07
a = 0.3 b = 0.12
a = 0.7 b = 0.07
a = 0.3 b = 0.12
a = 0.7 b = 0.07
0.149 0.00196 0.982
0.248 0.02160 0.978
0.037 0.00050 0.994
0.095 0.00310 0.997
0.078 0.00032 0.997
0.096 0.00369 0.996
0.082 0.00254 0.966
0.219 0.03002 0.969
0.025 0.00019 0.998
0.136 0.00698 0.993
0.010 0.00005 0.999
0.155 0.00282 0.997
0.057 0.00108 0.991
0.167 0.00773 0.995
0.051 0.00070 0.994
0.191 0.01483 0.990
0.042 0.00029 0.997
0.084 0.00272 0.998
0.057 0.00094 0.995
0.159 0.00641 0.997
0.049 0.00064 0.997
0.183 0.01549 0.994
0.039 0.00031 0.998
0.085 0.00251 0.999
0.046 0.00057 0.996
0.138 0.00558 0.997
0.043 0.00075 0.995
0.104 0.00290 0.998
0.018 0.00010 0.999
0.134 0.00906 0.995
69
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Fig. 8. Comparison of SIFs calculated by FEA and MMWFs for a = 0.3, b = 0.12 and a = 0.7, b = 0.07 under elliptic paraboloid stress distributions.
The FEA and MMWFs results are compared in Fig. 8. The graphical illustrations of the stress distributions are also provided in Fig. 8. According to the values of K I;max =K II;max and K I;max =K III;max ratios, which are 0.48 and 0.64, and 0.78 and 1.05 respectively for stress distributions of Eqs. (22a) and (22b), it may be concluded that under shear loadings, the mode I SIF caused by the coupling effect can be more significant than mode III SIF. Maximum AE between FEA and MMWFs SIFs are 0.167, 0.191 and 0.085 for mode I, II and III, respectively, as included in Table 1. So, it can be concluded that derived MMWFs are capable of quickly and accurately calculating MMSIFs of subsurface elliptical cracks with any as and bs under nonlinear shear loadings. As can be seen in Fig. 8, even stress distributions with two symmetry planes affect the critical points of the crack front. Run time of the FE analysis or numerical integration of the MMWFs is almost independent of the stress distributions on the crack faces. So, the MMWF method is time efficient relative to the FE method for all stress distributions. 5.2.3. Trigonometric paraboloid stress distribution So far, the loadings used for validation have symmetry plane(s). To generalize the validation and indicate the efficiency of the MMWFs for all stress distributions, a trigonometric paraboloid stress distribution without any symmetry plane is considered. The stress distribution, applied to the crack faces, is shown in Fig. 9 and defined as follows:
szx ðx; yÞ ¼ szy ðx; yÞ ¼ 2 sin
p x 6 a
p y þ 2 cos þ1 6 c
ð22cÞ
70
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Fig. 9. Comparison of SIFs calculated by FEA and MMWFs for a = 0.3, b = 0.12 and a = 0.7, b = 0.07 under trigonometric 2D stress distributions.
MMSIFs are calculated and compared in Fig. 9 using FEA and MMWFs. Similar to the previous stress distributions, the values of K I;max are considerable compared to K II;max and K III;max values, K I;max =K II;max = 0.66 and K I;max =K III;max = 0.70, and therefore the coupling effect of shear loadings on the mode I is compelling. Also, MSE between K I ; K II and K III calculated by FEA and MMWFs are less than 0.0056, 0.0030 and 0.0091, respectively, as provided in Table 1. According to the calculated errors, it is deduced that the results of derived MMWFs and FEA are in excellent agreement and so the MMWFs are reliable for calculation of MMSIFs under any kind of stress distribution. Mode I SIFs are negative for some angels of the crack front in Figs. 7–9. These negative values indicate that the crack faces are compressed due to the coupling effect and the crack front is closed in that area. The use of negative mode I SIFs are applicable only under combined tension and shear loadings, when there is sufficient tension to make the total SIFs positive. It should be emphasized that the negative sing for the shear modes of SIFs indicates only the sliding and tearing directions of the crack faces. Such MMWFs which provide accurate MMSIFs are desirable, regarding to the time and endeavor required to determine MMSIFs utilizing numerical procedures such as FEA. Moreover, these MMWFs are very comfortable to apply in industrial applications, and very efficient for evaluating fatigue behavior of the cracks under complicated 2D loadings.
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6. Conclusions Reference MMSIFs under unit uniform shear loadings along x and y directions are utilized to derive MMWFs for elliptical subsurface cracks. To consider the coupling of the fracture modes for subsurface cracks, the WFs is derived in matrix form as MMWFs. The dimensionless parameters are aspect ratio, a = 0.2–1.0, and crack depth to crack length ratio, b = 0.05 to infinity (=4). Different approximate forms of WFs, with only one coefficient for each mode, are considered for direct elements (DIIn ; DIIIt Þ and coupling elements of WF matrix (DIn ; DIIIn ; DIt ; DIIt Þ. Explicit equations with about 80 coefficients are fitted to the MMWFs coefficients with high accuracy. These equations make it possible to calculate MMSIFs for any values of a, b, and the angle of crack front. Five linear and nonlinear 2D stress distributions are applied to the crack faces with as and bs, different from those used for fitting, to verify the accuracy of MMWFs including explicit equations of their coefficients. Maximum AE between FEA and MMWFs SIFs are 0.248, 0.119 and 0.155 for mode I, II and III, respectively. Hence, it can be deduced that MMSIFs of MMWFs method are in high accordance with FEA results, while MMWFs method is 60 times faster than FE method in calculation of MMSIFs for present problem. MMSIFs under different shear stress distributions indicate that, tension mode caused by the coupling effect is significant for an elliptical subsurface crack under shear loadings. Also, from the results it can be concluded that the stress distribution may affect the location of critical points along the crack front. The MMWFs are applicable for subsurface cracks under (complicated) 2D stress distributions, developed in industrial processes. The MMWFs are convenient to apply for calculation of MMSIFs of the crack front, and so for study of the fatigue crack growth phenomenon. Appendix A The coefficients DIn ; DIIn ; DIIIn ; DIt ; DIIt and DIIIt in Eq. (17):
ForDIn : A0 ðaÞ ¼ 0 3 2 1:58166 13:93044 32:70718 14:81751 7 6 6 2:99029 25:71956 53:46568 22:62742 7 7 Aim ¼ 6 7 6 8:00360 5 4 1:50322 12:46105 21:24514 0:02550
0:16134
0:17826 3 1:08087 5:37478 2:42958 7 6 0:12297 7 6 0:99970 4:62350 7 ¼6 7 6 4 0:36664 0:96027 1:22056 5 0:00220
0:09917
0:01472
2
B1im
2
18:88158
0:19380
118:14770
99:80799
3
7 6 6 37:59783 226:67235 200:32192 7 7 B2im ¼ 6 7 6 4 20:92382 115:08795 102:87613 5 2
0:90431
3:32284
2:37514
104:34370
334:86668
140:98134
3
7 6 6 191:75323 602:58060 289:30164 7 7 B3im ¼ 6 7 6 279:53472 142:78811 5 4 93:09366 2:79312
5:76304
1:20763
3 122:64739 203:90187 115:59134 7 6 6 219:46861 349:48549 166:25360 7 7 ¼6 7 6 72:22723 5 4 101:85484 147:78358 2
B4im
2:53231 2
41:74191
6 6 73:98725 B5im ¼ 6 6 4 33:54795 0:71505
0:66230
5:44052 3
3:76035 146:26313 3:84032 3:69127 1:52014
7 235:48160 7 7 7 103:65580 5 4:02565
72
R. Ghajar, J.A. Kaklar / Engineering Fracture Mechanics 136 (2015) 58–75
ForDIIn : 2 Aim
0:88378
11:61606
24:86375
12:30724
3
6 1:24037 16:60548 47:87198 26:30990 7 7 6 7 6 7 ¼6 0:22664 8:81657 37:42833 22:78568 7 6 7 6 4 0:22851 0:90828 10:86887 7:48242 5
B1im
0:00059 8:57843
0:07389 24:00878
0:28287 3 7:75538 6 5:17970 0:93101 26:48050 7 7 6 ¼6 7 4 0:55498 7:57167 20:55390 5 2
0:18802
0:06149 0:54520 0:92253 3 45:80331 131:59403 188:85024 6 63:21148 259:53782 419:93866 7 7 6 ¼6 7 4 26:18417 129:91757 220:38683 5 2
B2im
5:42200 8:57518 3 1018:08027 1124:69965 6 355:32515 1581:49995 1956:86574 7 7 6 ¼6 7 4 146:42928 708:74192 929:65748 5 2
B3im
21:68296 1534:19365
5:38222 157:88003
29:47513 692:34427
2:36307
12:97763
39:17136 3 765:40007 6 223:34218 1069:70969 1274:61658 7 6 7 ¼6 7 4 94:86533 476:44652 590:93739 5 2
B5im
3:88439 366:18861
29:96757 3 1678:17466 6 506:46739 2357:10961 2813:50650 7 6 7 ¼6 7 4 212:49352 1047:64850 1309:81518 5 2
B4im
0:87249 264:33234
17:08923
ForDIIIn : Aða; bÞ ¼ 0 3 2 30:27640 61:17297 39:34168 6 24:68252 27:96486 10:31798 7 7 6 B1im ¼ 6 7 4 6:40971 0:64290 4:46884 5
B2im
2
0:05193 139:11855
0:10892 0:10233 3 435:15787 308:93640 6 134:50260 370:09131 199:57364 7 7 6 ¼6 7 4 43:00209 100:54789 29:26046 5 2
0:46552 206:11330
6 195:75243 6 B3im ¼ 6 4 59:90716
0:07646 672:16893
1:68623 3 426:10573 476:05888 63:34884 7 7 7 82:75417 111:23319 5
0:15529 114:03271
4:90365 302:73847
6 85:45329 6 B4im ¼ 6 4 16:27209
23:36571
2
B5im
130:32902
10:63187 3 61:19341 486:34038 7 7 7 360:00592 5
0:92480 10:71258 17:91138 2 3 15:19507 8:38198 105:04459 6 3:40349 179:40496 378:42555 7 6 7 ¼6 7 4 8:52000 122:50314 223:76205 5 0:70873
5:89961
9:12622
R. Ghajar, J.A. Kaklar / Engineering Fracture Mechanics 136 (2015) 58–75
ForDIt : Aða; bÞ ¼ 0 3 2 0:23045 17:21569 8:10737 7 6 6 0:78543 22:05137 6:65161 7 7 B1im ¼ 6 7 6 3:74902 1:62828 5 4 0:65320 0:04988
0:06715
0:09856
3 74:31918 195:58061 82:70268 7 6 6 125:38163 311:23804 102:63547 7 7 ¼6 7 6 4 52:93653 119:75685 19:73653 5 2
B2im
2
0:78532
0:74829
184:72728
336:53259
1:78081 82:73465
3
7 6 325:41572 7 6 293:05927 436:62244 7 B3im ¼ 6 7 6 4 113:74918 108:11323 250:22801 5 0:30568
14:98046
6:64026
3 141:26500 85:02430 455:98801 7 6 72:28089 1048:43665 7 6 196:01158 7 ¼6 7 6 607:21023 5 4 59:59866 152:36591 2
B4im
2:10854 2
25:12465
16:80942 78:61129
27:18545 3
319:56135
7 6 6 17:53596 247:23216 681:72215 7 7 B5im ¼ 6 7 6 168:61712 370:72230 5 4 6:39574 1:80105
10:29909
14:95897
ForDIIt : Aða; bÞ ¼ 0 2 3 67:11344 183:89028 144:84641 6 7 6 59:99010 146:98985 116:34040 7 7 B1im ¼ 6 6 7 38:89261 31:96874 5 4 17:84391 0:21966
0:57556
0:61031
3 348:55352 1253:06876 1067:28537 7 6 6 333:86595 1182:23473 1006:14973 7 7 ¼6 7 6 379:92824 328:02472 5 4 107:67367 2
B2im
2
1:39093
5:25377
5:21579
673:18837
2669:27915
2389:11828
6 6 643:70897 2566:87311 B3im ¼ 6 6 853:90058 4 209:42177 2
2
7 2326:62799 7 7 7 798:07310 5
2:60615
12:00353
13:38122
584:76392
2423:39238
2223:03715
6 6 543:04630 B4im ¼ 6 6 4 173:89911
2297:87697 769:77111
2:03300
11:15474
192:93964
817:55412
6 6 172:87929 B5im ¼ 6 6 4 54:27985 0:61620
759:52515 255:53289 3:95171
3
3
7 2161:91529 7 7 7 760:07394 5 13:83491 3
758:53762
7 731:35244 7 7 7 262:66800 5 5:29287
73
74
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ForDIIIt : 2
Aim
3 3:24823 9:58283 6:05990 1:00364 7 6 6 0:26605 8:43388 7:75604 14:41867 7 7 6 7 6 ¼ 6 0:11388 6:03134 13:32155 17:34155 7 7 6 6 0:04888 1:02286 6:00136 6:59591 7 5 4 0:00611 2
0:02852
13:63671
0:17230
48:76125
0:13682 3
41:81282
7 6 6 12:49967 45:09159 38:08483 7 7 B1im ¼ 6 7 6 14:17798 11:65905 5 4 3:93807 2
0:06536
0:21669
0:15687
55:13826
60:06507
9:41365
3
7 6 6 44:43512 26:33747 100:44682 7 7 B2im ¼ 6 7 6 45:61408 68:99298 5 4 8:73708 0:45058 2 B3im
183:20630
6 6 142:95837 ¼6 6 4 27:76059 2
2:62891 479:80914 256:88799 4:14522
2:55356 292:11856
7 60:18886 7 7 7 65:15533 5
1:36787
5:91419
5:62024
190:13110
630:86827
467:04586
6 6 141:17300 B4im ¼ 6 6 4 24:39263 1:57077
426:59566 62:31095 5:14416
3
3
7 265:15284 7 7 7 17:88071 5 4:50146
3 67:10750 252:93670 202:89995 7 6 6 47:09674 185:53421 140:45279 7 7 ¼6 7 6 35:83472 25:84049 5 4 6:82296 2
B5im
0:62271
1:64843
1:30190
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