Three-Dimensional SGBEM-FEM Alternating Methodfor Analyzing Fatigue-Crack Growth in and the

Life of Attachment LugsLonggang Tian1; Leiting Dong2; Nam Phan3; and Satya N. Atluri4

Abstract: In the present paper, stress intensity factor (SIF) analyses and fatigue-crack-growth simulations of corner cracks emanating fromloaded pinholes of attachment lugs in structural assemblies are carried out for different load cases. A three-dimensional (3D) symmetric Galer-kin boundary-element method (SGBEM) and FEM alternating method is developed to analyze the nonplanar growth of these surface cracksunder general fatigue. The 3D SGBEM-FEM alternating method involves two very simple and coarse meshes that are independent of eachother: (1) a very coarse FEMmesh to analyze the uncracked lug, and (2) a very coarse SGBEMmesh to model only the growing crack surface.By using the SGBEM-FEM alternating method, the nonplanar growth of cracks in 3D (surfaces of discontinuity) up to the failure of structuresare efficiently simulated, and accurate estimations of fatigue lives are made. The accuracy and reliability of the SGBEM-FEM alternatingmethod are verified by comparing them to other FEM solutions, as well as experimental data for fatigue-crack growth available in the openliterature. The SIF calculations, crack surface evolutions, and fatigue-life estimations are all in good agreement with other detailed FEM sol-utions and experimental observations. It is noted that for fracture and fatigue analyses of complex 3D structures such as attachment lugs, a pureFEM requires several hundreds of thousands or even millions of elements, whereas the present 3D SGBEM-FEM alternating method requiresonly up to1,000FEMelements and∼100 SGBEM elements. It thus demonstrates that the present SGBEM-FEM alternating method, among themany Schwartz-Neumann-type alternating methods developed in the past 20–30 years are suitable for analyzing fracture and fatigue-crackpropagation in complex 3D structures in a very computationally efficient manner, as well as with very low human labor costs.DOI: 10.1061/(ASCE)EM.1943-7889.0000870. © 2014 American Society of Civil Engineers.

Introduction

In various mechanical systems, attachment lugs are employed toconnect different components. They are mostly subjected to cyclicloading and are transferred through a single bolt or pin withoutclamping. This leads to a combination of high stress concentration andfretting and can potentially result in crack initiation and fatigue-crackgrowth near the pinhole. To properly ensure the operational safety ofthesemechanical systems, it is of great importance to perform fatigue-crack growth analyses and damage control of attachment lugs.

Over the past several decades, several numerical and experimentalstudies have been conducted to understand the fatigue behavior ofattachment lugs, with either constant or variable amplitude loads(Smith et al. 1977; Schijve and Hoeymakers 1979; Moon 1980).Meanwhile, several fatigue lawswith appropriate criteria for either theinitiation phase or the crack growth phase were also developed. As iswell known, Paris and Erdogan (1963) introduced an empirical re-lationship between the crack growth rate and the range of effective

stress intensity factor (SIF), which laid a solid foundation for futureresearch. Elber (1971) proposed the crack closure concept by mod-ifying Paris’ law. Erdogan and Roberts (1965), Walker (1970), andKujawski (2001) examined the effect of mean stress and introducedthe two-parameter driving force model, in which both the maximumSIF and the rangeofSIF areused to determine the crack growth rate. Inall these fatigue laws, the evaluation of SIFs, which characterize thesingular stress field near the crack front, are crucial for the fatigue-crack growth analyses. Thus, various methods were developed toobtain the approximate solutions of SIFs of different cracked struc-tural components (Carpinteri 1993, 1994; Qian and Fatemi 1996;Tada et al. 2000;Carpinteri et al. 2006;Rozumek 2009;Carpinteri andVantadori 2009a, b; Boljanovi�c and Maksimovi�c 2011; Maksimovicet al. 2011; Boljanovi�c 2012).

The initial crack shape in the attachment lug can often be approx-imated as either a quarter-elliptical corner crack or a through-the-thickness crack; and the former can grow and become the latter. Rajuand Newman (1979) obtained the SIFs for different corner cracks byusinga three-dimensional (3D)FEM.Schijve (1985)developeda simpleinterpolationmethod tocompute theSIFs for corner cracks in lugs.AtluriandKathiresan (1978) developed the hybrid 3Dcrack elements tomodelvarious embedded and surface cracks. Heliot et al. (1980) obtained theSIFsof cracksbyapplying theboundary integral equation (BIE)method.Shah and Kobayashi (1972) studied the same problems employing thealternating method. Sih and Li (1990) suggested that the strain energydensity function can be used for the computation of SIFs.

Despite its widespread popularity, the traditional FEM, withsimple polynomial interpolations, is unsuitable for modeling cracksand their fatigue growth. This is partially due to the inefficiency ofapproximating stress and strain singularities using polynomial FEMshape functions. To overcome this difficulty, embedded singularityelements by Tong et al. (1973) and Atluri et al. (1975), and singular

1Ph.D. Candidate, Dept. of Geotechnical Engineering, College of CivilEngineering, Tongji Univ., Shanghai 200092, China.

2Associate Professor, Dept. of Engineering Mechanics, Hohai Univ., Nan-jing, Jiangsu 210098, China (corresponding author). E-mail: dong.leiting@gmail.com

3BranchHead, StructuresDivision, Naval Air SystemsCommand, PatuxentRiver, MD 20670.

4Distinguished Professor, Center for Aerospace Research and Education,Univ. of California, Irvine, CA 92697.

Note.Thismanuscriptwas submittedonJune3,2014; approved onAugust18, 2014; published online on September 10, 2014. Discussion period openuntil February 10, 2015; separate discussions must be submitted for individ-ual papers. This paper is part of the Journal of Engineering Mechanics,© ASCE, ISSN 0733-9399/04014142(12)/$25.00.

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quarter-point elements by Henshell and Shaw (1975) and Barsoum(1976), among others, were developed in the 1970s to capture thecrack-tip/crack-front singular field. Many related developmentswere summarized in the monograph by Atluri (1986), and they arenow widely available in commercial FEM software. However, theneed for constant remeshing makes the automatic fatigue-crackpropagation analyses with FEM extremely difficult. In a funda-mentally different way, the first paper on a highly accurate finite-element (Schwartz-Neumann) alternating method (FEAM) waspublished by Nishioka and Atluri (1983). The FEAM uses theSchwartz-Neumann alternation between a crude and simple finite-element solution for an uncracked structure, and the analyticalsolution for an infinite body containing the crack. The success of thismethod is largely attributed to the work of Vijayakumar and Atluri(1981), in which analytical solutions were derived for an ellipticalcrack in an infinite domain subjected to arbitrary crack surfacetractions. Subsequent 3D and two-dimensional (2D) variants of theFEAMs were successfully developed and applied to performstructural integrity and damage tolerance analysis of many practicalengineering structures (Atluri 1997). Recently, the SGBEM-FEMalternating method, which involves the alternation between the verycrude FEM solution of the uncracked structure and a SGBEM so-lution for a small region enveloping the arbitrary nonplanar 3Dcrack, was developed for arbitrary 3D nonplanar growth of em-bedded as well as surface cracks (Nikishkov et al. 2001; Han andAtluri 2002). Dong and Atluri (2012, 2013a, c) also developeda SGBEM superelement for direct coupling of SGBEM and FEM,for fracture and fatigue analysis of complex 2D solid structures andmaterials. The motivation for this series of works, by Atluri andmany of his collaborators since the 1980s, is to explore the ad-vantageous features of each computational method: modelingcomplicated and large-scale uncracked structures with simpleFEMs, and modeling crack singularities by mathematical methodssuch as complex variables, special functions, BIEs, and SGBEMs.

In the present paper, the fatigue growth of corner cracks ema-nating from the pinholes of attachment lugs is studied in detail byemploying the 3D SGBEM-FEM alternating method (Han andAtluri 2002; Dong andAtluri 2013b). The SIFs of the crack front arecomputed during each step of crack increment. The crack growthrates are determined by Paris’ law. The crack paths and number ofloading cycles are predicted for damage tolerance of the attachmentlugs. The validations of the 3D SGBEM-FEM alternating methodare illustrated by the comparison of numerical results with availableempirical solutions and experimental observations.

SGBEM-FEM Alternating Method: Theoryand Formulation

The SGBEM has several advantages over collocation/direct anddual BEMs (Rizzo 1967; Hong and Chen 1988). It leads to a sym-metrical coefficient matrix and avoids the need to have a specialtreatment of sharp corners. Early derivations of SGBEMs involveregularization of hypersingular integrals (Frangi and Novati 1996;Bonnet et al. 1998; Li et al. 1998; Frangi et al. 2002). A systematicprocedure to develop weakly singular symmetric Galerkin BIEs waspresented by Han and Atluri (2003, 2007). The derivation of thissimple formulation involves only the non-hypersingular integralequations for tractions, based on the original works by Okada et al.(1988, 1989). It was used to analyze 3D solids with embedded andsurface flaws by Nikishkov et al. (2001) and Han and Atluri (2002).

For a domain of interest in Fig. 1, with the source point denotedby x and the target point denoted by j, 3D weakly singular sym-metric Galerkin BIEs for displacements and tractions were de-veloped by Han and Atluri (2003).

The displacement BIE is

12

ð∂V

vpðxÞupðxÞdSx ¼ð∂V

vpðxÞdSxð∂V

tjðjÞuppj ðx, jÞdSj

þð∂V

vpðxÞdSxð∂V

DiðjÞujðjÞGppij ðx, jÞdSj

þð∂V

vpðxÞdSxð∂V

niðjÞujðjÞwppij ðx, jÞdSj

(1)And the corresponding traction BIE is

212

ð∂V

wbðxÞtbðxÞdSx

¼ð∂V

DaðxÞwbðxÞdSxð∂V

tqðjÞGpqabðx, jÞdSj

2

ð∂V

wbðxÞdSxð∂V

naðxÞtqðjÞwpqabðx, jÞdSj

þð∂V

DaðxÞwbðxÞdSxð∂V

DpðjÞuqðjÞHpabpqðx, jÞdSj

(2)

In Eqs. (1) and (2), Da is a surface tangential operator

DaðjÞ ¼ nrðjÞersa ∂∂js

DaðxÞ ¼ nrðxÞersa ∂∂xs

(3)

Kernels functions uppj , Gpqab, w

pqab, and H

pabpq can be found in the paper

by Han and Atluri (2003), which are all weakly singular, making theimplementation of the current BIEs very simple.

As shown in Fig. 2, by applying Eq. (1) to Su, where displace-ments are prescribed, and applying Eq. (2) to St where tractions areprescribed, a symmetric system of equations can be obtained

App Apq Apr

Aqp Aqq Aqr

Arp Arq Arr

264

3758<:

pqr

9=; ¼

8<:

fpfqfr

9=; (4)

where p, q, and r 5 unknown tractions at Su, unknown displace-ments at St9, and unknown displacement discontinuities at Sc,respectively.

Fig. 1. Solution domain with source point x and target point j

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With the displacements and tractions being first determined atthe boundary, the displacements, strains, and stress at any point in thedomain can be computed using the non-hypersingular BIEs in thepapers by Okada et al. (1988, 1989). Therefore, the path-independentor domain-independent integrals can be used to compute the SIFs.

However, with the singular quarter-point boundary elements at thecrack face, SIFs can also be directly computed using the displacementdiscontinuity at the crack-front elements, as shown byNikishkov et al.(2001). For fatigue-crack growth, there is no need to use any otherspecial technique to describe the crack surface, such as the level sets.The crack surface is already efficiently described by boundary ele-ments. In each fatigue step, aminimal effort is needed: one can simplyextend the crack by adding a layer of additional elements at the cracktip/crack front. This greatly saves computational time for fatigue-crack growth analyses.

However, a SGBEM is unsuitable for carrying out large-scalesimulations of complex structures. This is because the coefficientmatrix for a SGBEM is fully populated. To further explore theadvantages of both the FEM and SGBEM, Han and Atluri (2002)coupled the FEM and SGBEM indirectly using the Schwartz-Neumann alternating method. As shown in Fig. 3, a simple FEMis used tomodel the global uncracked structure and a SGBEMis usedtomodel the local cracked subdomain. By imposing residual stressesat the global and local boundaries in an alternating procedure, thesolution of the original problem is obtained by superposing thesolution of each individual subproblem.

The great flexibility of this SGBEM-FEM alternating method isobvious. The SGBEM mesh of the cracked subdomain is totallyindependent of the crude FEM mesh of the uncracked globalstructure. Because a SGBEM is used to capture the stress singularity

Fig. 2. Defective solid with arbitrary cavities and cracks

Fig. 3. Superposition principle by Han and Atluri (2002) for SGBEM-FEM alternating method: (a) the uncracked body modeled by simple FEM;(b) the local domain containing cracksmodeled by SGBEM; (c) FEMmodel subjected to residual loads; (d) alternating solution for the original problem

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at crack tips/fronts, a very coarse mesh can be used for a FEMmodelof the uncracked global structure. Because a FEM is used to modelthe uncracked global structure, large-scale structures can be effi-ciently modeled.

In this study, the SGBEM-FEM alternating method is used tomodel the fatigue growth of corner cracks emanating from thepinholes of lugs. The detailed configurations of numerical experi-ments are given in the next section.

Geometry of Specimens and Configurationof Experiments

The FEM-SGBEM models in the current study are created basedon the geometries given in the comprehensive report by Kathiresanand Brussat (1984), in which a large number of experiments werecarried out. Four models are considered with different maximumgross stresses (represented by s0max) and different stress ratios(represented by R). The corresponding specimens’model numbersand load cases are given in Table 1. All four models have the samegeometrical parameters, as shown in Fig. 4. Note that the initialcrack surface is considerably small compared with the size of theattachment lug. In fact, the thickness of the attachment lug is20 times the size of the radius of the initial crack surface (T=Rc

5 0:5=0:0255 20), so the initial crack surface is magnified in Fig. 4.The applied external load is an axial cyclic load with a constant

amplitudeofs0max, as shown in Fig. 5. The contact pressure betweenthe pin and lug is assumed to have the following form, based on Rajuand Newman (1979):

sðuÞ ¼ A sin2 u (5)

From the condition of equilibrium

ðp

0

sðuÞRi sin udu ¼ s0maxH (6)

And thus the contact pressure is determined by

sðuÞ ¼ 3s0maxH4Ri

sin2 u (7)

An attachment lug with a considerably small initial quarter-circularsurface crack emanating from one corner of the pinhole is modeledby the SGBEM-FEM alternating method. As shown in Fig. 6, themeshes for the SGBEM-FEMalternatingmethod consist of only 268finite elements and 15 boundary elements. However, for pure FEMs,including the popular extended FEM (XFEM), it is typical to usemore than 1 million degrees of freedom to model a cracked 3Dcomplex structure [see Levén and Rickert (2012) for examples].Thus, the current SGBEM-FEM alternating method greatly reducesthe burden of computation as well as the human labor of pre-processing, compared with FEM-based numerical methods, in-cluding XFEM, by several orders of magnitude.

The attachment lugs are manufactured with 7075-T651 aluminumalloy. The Young’s modulus of this material is 71,705,475.8491 kPa(10,400 ksi), and Poisson’s ratio is 0.33. Paris’ law (Paris and Erdogan1963) is used to analyze the fatigue-crack growth. Based on Paris’ law,the crack growth rate is determined by

dadN

¼ CðDKÞn (8)

Based on the report by Kathiresan and Brussat (1984), the param-eters of Paris’ law for this material are C5 1:18973 1029 andn5 3:8585 for stress ratio R5 0:1, and C5 6:2963 1029 andn5 3:6999 for stress ratio R5 0:5. The units for DK and da=dNare 34,748:4823 kPa

ffiffiffiffiffiffiffiffimm

p(1 ksi

ffiffiffiffiffiffiin:

p) and 25:4 mm=cycle (1 in:

=cycle), respectively. Refer to Tables 1-4 and 1-5 in the report byKathiresan and Brussat (1984) for more details.

Table 1. Different Models of Attachment Lugs

Material R s0max [kPa (ksi)] Model number

7075-T651 aluminum alloy 0.1 41,368.5438 (6) ABPLC6,56103,421.3594 (15) ABPLC8,57

0.5 41,368.5438 (6) ABPLC5,54103,421.3594 (15) ABPLC55,38

Fig. 4. Geometry of the attachment lug (unit: inches) (1 in:5 25:4 mm)

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Numerical Results

SIF Analyses of the Initial Crack

To verify the accuracy of the SGBEM-FEM alternating method, thecomputed SIFs are compared with the empirical solution of Newman(1973)

KI ¼ SffiffiffiffiffiffiffipaQ

rMef1G1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

cos�pD2H

�vuuut gf (9)

where KI 5 empirical mode I SIF; S 5 applied gross stress;f1 5 Bowie correction factor (Bowie 1956) for a single crack ema-nating from a circular hole;G1 5 stress correction factor for a loadedlug with one crack emanating from the pinhole;D5 diameter of thepinhole of the lug; H 5 width of the lug; gf 5 curve fitting angularfunction; Q 5 elastic shape factor; and Me 5 boundary correctionfactor. See Boljanovi�c and Maksimovi�c (2014) for further details ofthe calculation for each correction factor.

For thegeometryand loadcases shown inTable 1 and Figs. 4 and 5,the SIFs of the initial quarter-circular corner crack is computed usinga SGBEM-FEM alternating method with 268 finite elements and 15boundary elements, as shown in Fig. 6. Note that the initial cracksurface is quarter-circular, so the initial semimajor and semiminor axesare the same, i.e., a0 5 c0 5 0:635 mm ð0:025 in:Þ Comparisons be-tween the computed SIFs by the alternating method and by the em-pirical solution are given in Tables 2 and 3, which demonstrate goodagreement. It is alsoworthmentioning that the computation time for theSGBEM-FEM alternating method presented here is less than 2 s ina regular PC with an Intel i7 CPU (Core i7-3820 CPU at 3.60 GHz,Intel Corporation, Santa Clara, California). In addition, the cost ofhuman labor to generate the independent coarse meshes of the FEMand SGBEM is also very minimal.

Tables 2 and3 show that the empirical results ofNewman (1973) areapproximately 5–7% in error compared with the present solutions.However, it should be emphasized that the Newman solutions areempirical formulas based on brutal-force finite-element analyses. Theyare only valid for small quarter-elliptical cracks as compared with thesize of the hole. The crack shape during the fatigue growth does notalways stay strictly quarter-elliptical, and there is no analytical orempirical solution as the crack starts to break through the thickness.Because the computation burden for the SGBEM-FEM alternatingmethod is very small and the independent meshing of the structure andthe crack takes only minimal human labor, it is very convenient to

Fig. 5. Loads applied on the attachment lug

Fig. 6. Meshes for the problem of a quarter-circular surface crackemanating from the pinhole in an attachment lug: (a) the uncrackedglobal structure is modeled with 268 finite elements; (b) the initial cracksurface is modeled independently with 15 boundary elements

Table 2. Computed SIFs for Quarter-Circular Crack of Lug UsingSGBEM-FEM Alternating Method [s0max 5 41,368:5438 kPa ð6:0 ksiÞ]

f

(degrees)

Kempirical

[34,748:4823 kPaffiffiffiffiffiffiffiffimm

p(1 ksi

ffiffiffiffiffiin:

p)]

KSGBEM-FEM

[34,748:4823 kPaffiffiffiffiffiffiffiffimm

p(1 ksi

ffiffiffiffiffiin:

p)]

Difference(%)

0 232,064.2642 (6.6784) 242,249.0444 (6.9715) 4.399.02 226,073.6258 (6.506) 226,869.3661 (6.5289) 0.3518.04 221,542.4238 (6.3756) 211,458.4142 (6.0854) 24:5525.5 218,894.5894 (6.2994) 206,378.1861 (5.9392) 25:7140.04 216,371.8496 (6.2268) 202,309.1388 (5.8221) 26:5047.12 216,243.2802 (6.2231) 203,275.1466 (5.8499) 26:0054.37 216,715.8596 (6.2367) 203,963.1666 (5.8697) 25:8861.62 217,681.8674 (6.2645) 204,668.5607 (5.89) 25:9868.84 219,019.6839 (6.303) 208,647.262 (6.0045) 24:7376.06 220,625.0638 (6.3492) 212,671.1362 (6.1203) 23:6083.03 222,348.5885 (6.3988) 215,086.1557 (6.1898) 23:2790 224,186.7833 (6.4517) 217,462.9519 (6.2582) 23:00

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simulate the entire fatigue-crack growth process for the damage tol-erance analyses of attachment lugs.

Simulations of Crack Growth Path

The process of fatigue-crack growth for each specimen is divided into40 analysis steps. In each fatigue analysis step, the SIFs along the

crack front are computed by the SGBEM-FEM alternating method.Then, the crack growth is modeled by adding a layer of additionalelements at the crack front, in the direction determined by the Eshelbyforce vector (Eshelby 1951), with the size determined by Paris’ law.

The development of fatigue cracks of Model No. ABPLC6,56 isshown in Fig. 7. Crack shapes of other models are similar to those ofModel No. ABPLC6,56, which are given in Figs. 8 and 9.

As shown in Fig. 7, the process of fatigue-crack growth in the at-tachment lugs canbebroadlydivided into fourphases: the initial quarter-circular crack; a quasi-quarter-elliptical crack in growth; transition froma quasi-quarter-elliptical to a through-the-thickness crack; and, finally,a through-the-thickness edge crack. Initially, the crack grows faster inthe depth direction than in the surface direction; once the crack cutsthrough the thicknessof the lug, it growsvery fast in the surfacedirectionand quickly becomes a through-the-thickness edge crack.

The simulated crack growth paths for Model No. ABPLC6,56(stress ratio5 0:1) are comparedwith available experimental resultsby Kathiresan and Brussat (1984) in Figs. 8 and 9. It should be notedthat only the crack sizes in the surface and depth directionwere givenin Kathiresan and Brussat (1984), whereas the exact crack shapeswere not given. Therefore, it was assumed that the crack shapesobserved in the experiments were quarter-elliptical, as plotted inFig. 8. In addition, simulated crack growth paths for Model No.ABPLC5,54 (stress ratio5 0:5) are also compared with experi-mental results in Fig. 9. Acceptable agreements can be observedbetween simulations and experimental results.

Table 3. Computed SIFs for Quarter-Circular Crack of Lug UsingSGBEM-FEMAlternating Method [s0max 5 103,421:3594 kPa ð15:0 ksiÞ]

f

(degrees)

Kempirical

[34,748:4823 kPaffiffiffiffiffiffiffiffimm

p(1 ksi

ffiffiffiffiffiin:

p)]

KSGBEM-FEM

[34,748:4823 kPaffiffiffiffiffiffiffiffimm

p(1 ksi

ffiffiffiffiffiin:

p)]

Difference(%)

0 580,160.6605 (16.696) 605,624.3483 (17.4288) 4.399.02 565,187.5395 (16.2651) 567,175.1526 (16.3223) 0.3518.04 553,852.5845 (15.9389) 528,646.0355 (15.2135) 24:5525.5 547,232.9987 (15.7484) 515,948.94 (14.8481) 25:7240.04 540,929.624 (15.567) 505,771.1096 (14.5552) 26:5047.12 540,609.9379 (15.5578) 508,193.0788 (14.6249) 26:0054.37 541,791.3863 (15.5918) 509,906.179 (14.6742) 25:8961.62 544,206.4058 (15.6613) 511,671.4019 (14.725) 25:9868.84 547,545.735 (15.7574) 521,616.4175 (15.0112) 24:7376.06 551,562.6595 (15.873) 531,676.1031 (15.3007) 23:6083.03 555,871.4714 (15.997) 537,715.3894 (15.4745) 23:2790 560,468.6956 (16.1293) 543,657.3798 (15.6455) 23:00

Fig. 7. Growth of a quarter-circular corner crack of the attachment lug (Model No. ABPLC6,56): 2D view

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Fig. 8.Crack shapes for Model No. ABPLC6,56 [initial crack a0 5 c0 5 0:635 mm ð0:025 in:Þ]: (a) a5 2:2352 mm ð0:088 in:Þ; (b) a5 3:5052 mmð0:138 in:Þ; (c) a5 4:6736 mm ð0:184 in:Þ; (d) a5 6:9342 mm ð0:273 in:Þ; (e) a5 9:1186 mm ð0:359 in:Þ; (f) a5 11:3284 mm ð0:446 in:Þ

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Fig. 9. Crack shapes for model number ABPLC5,54 [initial crack a0 5 c0 5 0:635 mm ð0:025 in:Þ]: (a) a5 3:6068 mm ð0:067 in:Þ; (b) a5 3:6068 mm ð0:142 in:Þ; (c) a5 5:6388 mm ð0:222 in:Þ; (d) a5 7:9756 mm ð0:314 in:Þ; (e) a5 10:2616 mm ð0:404 in:Þ; (f) a5 12:2682 mmð0:483 in:Þ

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Estimation of Fatigue Lives

In this section, the fatigue lives of the attachment lug are estimatedby employing the SGBEM-FEM alternating method together withParis’ crack growth law. Paris’ law is used to characterize thefatigue-crack growth in the attachment lug. According to Paris’ law,it is easy to estimate the fatigue life of the cracked attachment lug by

N ¼ðaf

a0

1CðDKÞn da (10)

At each fatigue growth step, the maximum value of DK along thecrack front is computed by the SGBEM-FEM alternating method,and Eq. (10) is numerically evaluated using the trapezoidal rule.

The crack extensions in both the depth direction and surfacedirection of the lug are investigated, and the results are shown inFigs. 10–13, where it is clear that the stress ratio plays an importantrole for the fatigue behavior of the attachment lugs. Generallyspeaking, for the same geometry and the same amplitude of cyclicloading, the smaller the stress ratio, the faster the fatigue crackpropagates and the shorter the life of the attachment lug.

Fig. 10. Crack length in the depth direction versus number of load cycles up to failure for the attachment lug (stress ratio R5 0:1) (1 in:5 25:4 mm)

Fig. 11. Crack length in the surface direction versus number of load cycles for the attachment lug (stress ratio R5 0:1) (1 in:5 25:4 mm)

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Conclusions

In this paper, several numerical models of attachment lugs withan initial quarter-circular surface crack emanating from the pinholeare considered. The SIF analyses and fatigue-crack growth simu-lations with different load cases are carried out by employing the3D SGBEM-FEM alternating method. By carefully comparing thenumerical results with empirical solutions and experimental obser-vations available in the literature, the following conclusions can bedrawn:1. The SGBEM-FEM alternating method requires independent

and very coarse meshes for both the uncracked structure andthe crack; it requires only minimal computational burden and

human labor efforts for modeling the fatigue growth of 3Dcracks.

2. The computed SIFs for the initial crack using the SGBEM-FEM alternating method are in good agreement with empiricalsolutions.

3. By employing the SGBEM-FEM alternating method, thewhole crack growth path up to failure of the attachment lugcan be easily simulated.

4. The predicted fatigue life of lug specimens by the SGBEM-FEM alternating method agrees well with experimentalobservations.

It is thus concluded that the 3D SGBEM-FEM alternatingmethod, among the many alternating methods developed in the past

Fig. 12. Crack length in the depth direction versus number of load cycles up to failure for the attachment lug (stress ratio R5 0:5) (1 in:5 25:4 mm)

Fig. 13. Crack length in the surface direction versus number of load cycles for the attachment lug (stress ratio R5 0:5) (1 in:5 25:4 mm)

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20–30 years by Atluri and his many collaborators, are considerablyefficient, accurate, and reliable for analyzing 3D fracture and fatigue-crack propagations, which is crucial for the damage tolerance ofattachment lugs as well as other 3D complex civil and mechanicalstructures. The implementation of the 3D SGBEM-FEM alternatingmethod in general-purpose off-the-shelf commercial software isbeing pursued by the authors.

Acknowledgments

The first author gratefully acknowledges the financial support of theChina Scholarship Council (grant 201306260034), theNational BasicResearch Program of China (973 Program, grant 2011CB013800),and the New Century Excellent Talents Project in China (grantNCET-12-0415).

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