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Non-planar mixed-mode growth of initially straight-fronted surfacecracks, in cylindrical bars under tension, torsion and bending, using thesymmetric Galerkin boundary element method-finite element methodalternating method

L. -G. TIAN1,2, L. -T. DONG2, N. PHAN3 and S. N. ATLURI2

1Department of Geotechnical Engineering, College of Civil Engineering, Tong ji University, Shanghai 200092, China, 2Center for Aerospace Research andEducation, University of California, Irvine USA, 3Structures Division, Naval Air Systems Command, Patuxent River MD, USA

Received Date: 1 August 2014; Accepted Date: 19 January 2015; Published Online: 2015

ABSTRACT In this paper, the stress intensity factor (SIF) variations along an arbitrarily developingcrack front, the non-planar fatigue-crack growth patterns, and the fatigue life of a roundbar with an initially straight-fronted surface crack, are studied by employing the 3Dsymmetric Galerkin boundary element method-finite element method (SGBEM-FEM)alternating method. Different loading cases, involving tension, bending and torsion ofthe bar, with different initial crack depths and different stress ratios in fatigue, areconsidered. By using the SGBEM-FEM alternating method, the SIF variations alongthe evolving crack front are computed; the fatigue growth rates and directions of thenon-planar growths of the crack surface are predicted; the evolving fatigue-crack growthpatterns are simulated, and thus, the fatigue life estimations of the cracked round bar aremade. The accuracy and reliability of the SGBEM-FEM alternating method are verifiedby comparing the presently computed results to the empirical solutions of SIFs, as well asexperimental data of fatigue crack growth, available in the open literature. It is shownthat the current approach gives very accurate solutions of SIFs and simulations of fatiguecrack growth during the entire crack propagation, with very little computationalburden and human–labour cost. The characteristics of fatigue growth patterns of initiallysimple-shaped cracks in the cylindrical bar under different Modes I, III and mixed-modetypes of loads are also discussed in detail.

Keywords fatigue crack growth; round bar; SGBEM-FEM alternating method; stressintensity factor.

NOMENCLATURE a = crack deptha0 = initial crack depthA = area of the cross section of the round bar

BIE =boundary integral equationc = crack surface arc length of the round barC = parameter of Paris Law

CT = loading type, cyclic tensionCB = loading type, cyclic bending

CTR = loading type, cyclic torsionCTCA = loading type, cyclic torsion and cyclic axial loadingda/dN = fatigue crack growth rate

D = diameter of the round barDa = surface tangential operatorE =Young’s modulus

FEM =finite element methodFEAM =finite element alternating method

Correspondence: L.-T. Dong. E-mail: [email protected]

© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13 1

ORIGINAL CONTRIBUTION doi: 10.1111/ffe.12292

I = the moment of initial of the cross section of the round barKI =mode I stress intensity factorKII =mode II stress intensity factorKIII =mode III stress intensity factorKIC = fracture toughness of the materialL = length of the specimenM = the bending moment applied to the specimenn = parameter of Paris LawN =number of load cyclesP = the axial loading applied to the specimenR = stress ratio

SGBEM = symmetric Galerkin boundary element methodSIF = stress intensity factorΔK = range of stress intensity factorΔx = crack growth length in x-directionΔy = crack growth length in y-directionν =Poisson’s ratioθ = the angle between the crack growth orientation and the plane of the cross

section of the round barσ = the applied nominal stressσ0 =monotonic yield strength of the materialσb = the applied bending nominal stressσm = ultimate tensile strength of the materialσt = the applied tension nominal stress

I NTRODUCT ION

Round bars are widely used in various mechanicalsystems to connect structural components. They maybe subjected to different types of loads, such as tension,compression, bending and torsion. Fatigue failure oftenoccurs when such round bars are subjected to cyclicloads, and such failures mostly originate from smallsurface flaws. Several studies have been carried out tocompute the stress intensity factors (SIFs) along thecrack front of defective round bars and to estimatethe shape of the crack surface during fatiguegrowth.1–5 The crack front is most usually treated inprior literature as an equivalent elliptical arc duringfatigue growth, but the initial surface flaw is oftenapproximated as a straight-fronted crack. Also,straight-edged surface flaws are often manually createdin experiments, to investigate the fatigue growth ofsmall surface flaws in round bars.

The SIFs, and fatigue growth of cracks in a roundbar, have been studied using various numericalmethods6–10 and by conducting experiments.11 Linand Smith12 used the 3D finite element analyses tosimulate Mode I fatigue crack propagation in a roundbar. It was shown that the crack shape after fatiguegrowth is not sensitive to the aspect ratio of the initialsmall surface flaw. Tschegg13 studied the fatiguegrowth of Mode III cracks in a round bar and showed

that a significant reduction of crack growth rate wasobserved. This is because of the ‘crack closure’ effectdue to the interaction between the fatigue crack sur-faces. By employing the R-curve method, Yu et al.14 es-timated the crack extension rate and discussed theinfluence of crack surface contact on the crack growthrate. Meanwhile, several experiments were conductedto study the fatigue behaviour of cracked round bars.Branco et al.15 determined the Paris Law constants ina round bar made of S45 steel. They used a three-stepreverse engineering technique, from the analysis offinal fracture surfaces. Yang and Kuang16,17 conductedexperiments to study the fatigue crack growth ofsurface cracks in round bars under various types ofloads. The fatigue life, surface crack extension directionand the crack growth rate were investigated in theirexperiments. It was shown that a compression loadsuperimposed on the cyclic Mode III (torsion) loadleads to a significant reduction on the crack growthrates.

In spite of its widespread popularity, the traditionalfinite element method (FEM), with simple polynomialinterpolations, is unsuitable for modelling cracks andtheir propagations. This is partially because of the high-inefficiency of approximating stress and strain singulari-ties using polynomial FEM shape functions. In order toovercome this difficulty, embedded-singularity elementsby Tong, Pian and Lasry,18 Atluri, Kobayashi and

2 L.-G. TIAN et al.

© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–13

Nakagaki,19 and singular quarter-point elements byHenshell and Shaw20 and Barsoum,21 were developed inorder to capture the crack tip/crack front singular field.Many such related developments were summarised inthe monograph by Atluri,22 and they are now widelyavailable in many commercial FEM software. However,the need for constant re-meshing makes the automaticfatigue crack growth analyses, using traditional FEM, tobe very difficult.

In a fundamentally different mathematical way,after the derivation of analytical solutions for embed-ded elliptical cracks in an infinite body whose facesare subjected to arbitrary tractions,23 the first paperon a highly accurate Finite Element (Schwartz–Neumann)Alternating Method was published by Nishioka andAtluri.24 The finite element (Schwartz–Neumann)alternating method uses the Schwartz–Neumann alter-nation between a crude and simple finite elementsolution for the uncracked structure and the analyticalsolution for the crack embedded in an infinite body.Subsequent 2D and 3D variants of the finite elementalternating methods were successfully developed andapplied to perform structural integrity and damagetolerance analyses of many practical engineeringstructures.25 Recently, the symmetric Galerkin bound-ary element method-finite element method (SGBEM-FEM) alternating method, which involves the alterna-tion between the very crude FEM solution of theuncracked structure and an SGBEM solution for a smallregion enveloping the arbitrary non-planar 3D crack,was developed for arbitrary 3D non-planar growth ofembedded and surface cracks by Nikishkov, Park andAtluri26 and Han and Atluri.27 An SGBEM ‘superelement’ was also developed in Dong and Aluri28–31 forthe direct coupling of SGBEM and FEM in fractureand fatigue analysis of complex 2D and 3D solidstructures and materials. The motivation for this seriesof works, by Atluri and many of his collaborators sincethe 1980s, is to explore the advantageous features ofeach computational method: model the complicateduncracked structures with simple FEMs and model thecrack singularities by mathematical methods such ascomplex variables, special functions, boundary integralequations (BIEs) and by SGBEMs.

In the present paper, by employing the SGBEM-FEM alternating method, the SIFs at the evolving frontof a non-planarly growing surface crack in round barsare computed and compared with the available empiri-cal solutions. The process of fatigue growth of theinitial surface crack is also simulated, consideringvarious types of load cases. The SIFs along the crackfront are computed during each step of crack incre-ment. An additional layer of boundary elements isadded to the crack front, in the direction determined

by the Eshelby force vector,32 with the crack frontadvance size determined locally by Paris’ Law. Thefatigue crack growth patterns and fatigue lives arethereafter predicted for a damage tolerance evaluation.The validity of the SGBEM-FEM alternating methodis illustrated, through the comparison of the presentSGBEM-FEM alternating results to the availableempirical and other numerical solutions and experimen-tal observations in the open literature.

SGBEM-FEM ALTERNAT ING METHOD : THEORYAND FORMULAT ION

The SGBEM has several advantages over traditional anddual BEMs by Rizzo,33 Portela, Aliabadi and Rooke,34

such as resulting in a symmetrical coefficient matrix ofthe system of equations and the avoidance of the needto treat sharp corners specially. Early derivations ofSGBEMs involve regularisation of hypersingularintegrals; see the work by Frangi and Novati35; Bonnet,Maier and Polizzotto36; Li, Mear and Xiao37; and Frangi,Novati, Springhetti and Rovizzi.38 A systematic proce-dure to develop weakly singular symmetric Galerkinboundary integral equations was presented by Han andAtluri.39,40 The derivation of this simple formulationinvolves only the non-hypersingular integral equationsfor tractions, based on the original work reported inOkada, Rajiyah and Atluri.41,42 It was used to analysecracked 3D solids with surface flaws by Nikishkov, Parkand Atluri26 and Han and Atluri.27

For a domain of interest as shown in Fig. 1, withsource point x and target point ξ, 3D weakly singularsymmetric Galerkin BIEs for displacements and tractionswere developed by Han and Atluri.39

Fig. 1 A solution domain with source point x and target point ξ.

MIXED -MODE FAT IGUE ANALYSES OF CYL INDR ICAL BARS US ING THE SGBEM-FEM ALTERNAT ING METHOD 3


The displacement BIE is as follows:

12∫∂Ωvp xð Þup xð ÞdSx

¼ ∫∂Ωvp xð ÞdSx∫∂Ωtj ξð Þu*pj x; ξð ÞdSξ

þ∫∂Ωvp xð ÞdSx∫∂ΩDi ξð Þuj ξð ÞG*pij x; ξð ÞdSξ

þ∫∂Ωvp xð ÞdSx∫∂Ωni ξð Þuj ξð Þφ*pij x; ξð ÞdSξ:

(1)

And the corresponding traction BIE is as follows:

�12∫∂Ωwb xð Þtb xð ÞdSx

¼ ∫∂ΩDa xð Þwb xð ÞdSx∫∂Ωtq ξð ÞG*qab x; ξð ÞdSξ

�∫∂Ωwb xð ÞdSx∫∂Ωna xð Þtq ξð Þφ*qab x; ξð ÞdSξ

þ∫∂ΩDa xð Þwb xð ÞdSx∫∂ΩDp ξð Þuq ξð ÞH�abpq x; ξð ÞdSξ:

(2)

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

Da ξð Þ ¼ nr ξð Þersa ∂∂ξ s

Da xð Þ ¼ nr xð Þersa ∂∂xs

:

(3)

Kernels functions u*pj ;G*qab ; φ

*qab ;H

�abpq can be found in

the paper by Han and Atluri,39 which are all weaklysingular, making the implementation of the current BIEsvery simple.

As shown in Fig. 2, by applying Eq. (1) to Su, wheredisplacements are prescribed, and applying Eq. (2) to Stwhere tractions are prescribed, a symmetric system ofequations can be obtained as follows:

App Apq Apr

Aqp Aqq Aqr

Arp Arq Arr

264

375

pqr

8><>:

9>=>;¼

f pf qf r

8><>:

9>=>;; (4)

where p, q, r denotes the unknown tractions at Su,unknown displacements at St ’, and unknown displace-ment discontinuities at Sc, respectively.

With the displacements and tractions being firstdetermined at the boundary and crack surface, thedisplacements, strains and stress at any point in thedomain can be computed using the non-hyper singularBIEs in the paper by Okada, Rajiyah and Atluri.41,42

Therefore, the path-independent or domain-independentintegrals can be used to compute the SIFs. Alternatively,with the singular quarter-point boundary elements at thecrack face, SIFs can also be directly computed using the

displacement discontinuity at the crack front elements;see the paper by Nikishkov, Park and Atluri26 for detaileddiscussion. For fatigue growth of cracks, there is no needto use any other special techniques to describe the cracksurface, such as the level sets. The crack surface is alreadyefficiently described by boundary elements. In each fatiguestep, a minimal effort is needed: One can simply extend thecrack by adding a layer of additional elements at the cracktip/crack front. This greatly saves the computational timefor fatigue crack growth analyses.

However, SGBEM is unsuitable for carrying outlarge-scale simulations of complex structures. This isbecause the coefficient matrix for SGBEM is fullypopulated. To further explore the advantages of bothFEM and SGBEM, Han and Atluri27 coupled FEM andSGBEM indirectly, using the Schwartz–Neumannalternating method. As shown in Fig. 3, simple FEM isused to model the uncracked global structure, andSGBEM is used to model the local cracked subdomain.By imposing residual loads at the global and localboundaries in an alternating procedure, the solution ofthe original problem is obtained by superposing thesolution of each individual sub-problem. The detailedprocedures are described as follows.

(1) As shown in Fig. 3(a), firstly solve the problem ofthe entire domain ΩFEM using FEM, with all theexternally prescribed displacements and tractionsbut without considering the cracks. The resultingtractions on crack surfaces SSGBEM

c can be obtainedas pSGBEM

c ¼ �pFEMc .

(2) As shown in Fig. 3(b), model a local subdomainΩSGBEMwith all the cracks by SGBEM, only consider-ing the tractions pSGBEM

c on the crack surface SSGBEMc .

The prescribed tractions tSGBEM on SSGBEMt are set

to be 0. Then the resulting tractions on theintersection surface SI are obtained, denoted aspSGBEMu on SSGBEM

u .(3) Applying the tractions on the intersection surface

as residual loads to ΩFEM, denoted as pFEM ¼�pSGBEM

u on SI in Fig. 3(c). Resolve the problem byFEM, and obtain again the residual crack surfacetractions pSGBEM

c on SSGBEMc .

(4) Repeat steps 2–3 until pFEM are very small. Thesolution of the original problem is obtained bythe superposition of the FEM solution and theSGBEM solution, as shown in Fig. 3(d).

The great flexibility of this SGBEM-FEM alternatingmethod is obvious. The SGBEM mesh of the crackedsub-domain is totally independent of the crude FEMmesh of the uncracked global structure. Because theSGBEM is used to capture the stress singularity at crack

4 L.-G. TIAN et al.


tips/crack fronts, a very coarse mesh can be used for theFEM model of the uncracked global structure. BecauseFEM is used to model the uncracked global structure,large-scale structures can be efficiently modelled.

In this study, the SGBEM-FEM alternating method isused to simulate the fatigue growth of both Mode I andmixed-mode cracks in round bars. The detailed discus-sions of numerical simulations and their comparisons tothe available empirical solutions and experimental resultsare given in the next section.

GEOMETRY OF SPEC IMENS AND EXPER IMENTALCONF IGURAT IONS

The SGBEM-FEM models in the present study arecreated on the basis of the geometries given in the studiesby Yang and Kuang,16,17 in which many experiments werecarried out on the fatigue behaviour of round bars with aninitially straight-fronted surface crack. The geometricalconfiguration of the specimen is shown in Fig. 4. Thediameter D is 12mm, and the length L is 90mm. Straight-fronted surface cracks were cut in the mid-plane usinglinear cutting machines, with different initial depths a0.Fig. 2 A defective solid with arbitrary cavities and cracks.

(a)

(c) (d)

(b)

Fig. 3 Superposition principle for SGBEM-FEM alternating method: (a) the uncracked body modelled by simple FEM, (b) the local domaincontaining cracks modelled by SGBEM, (c) FEM model subjected to residual loads and (d) alternating solution for the original problem.



The material is carbon steel S45, and the mechanicalproperties of the test material at room temperature arelisted in Table 1. According to Yang and Kuang,17 fatiguegrowth of cracks for this material under cyclic loadingcan be characterised by Paris Law43

dadN

¼ C ΔKð Þn

C ¼ 2:4866�10�14; n ¼ 3:2560:(5)

In Eq. (5), da/dN is inmm/cycle, andΔK is inMPaffiffiffiffiffiffiffiffimm

p.

Various far field sinusoidal cycling loads are applied tothe specimen. As listed in Table 2, four types of loadcases are considered in this study.

Type I Cyclic tension (CT) represents a simple CT loadwith stress ratio 0.1, to study the Mode I SIFs and theMode I fatigue behaviour of the straight-fronted crackin the round bar with pure tension load.Type II Cyclic bending represents a simple cyclicbending load with stress ratio 0 to study the Mode I SIFs

and the Mode I fatigue behaviour of the straight-frontedcrack in the round bar with pure bending load.Type III Cyclic torsion represents a simple cyclic torsionload with stress ratio 0 to study the mixed-mode fatiguebehaviour of the straight-fronted crack in the round barwith pure torsion load.Type IV Cyclic torsion and cyclic axial loading (CTCA)represents a combination of cyclic torsion with CT/compression to investigate the effect of cyclic axial stress.The cyclic torsion and cyclic axial loads are synchronoussinusoidal with the same frequency. The stress ratios fortension/compression and torsion loads are both 0.

These load cases are modelled by the SGBEM-FEM al-ternating method. As shown in Fig. 5, the meshes for theSGBEM-FEM alternatingmethod consist of only 515 finiteelements and 14 boundary elements. On the other hand, forpure finite element-based methods, it is typical to use morethan a million degrees of freedom to model a cracked 3Dcomplex structure; see Levén and Rickert44 for examples.Thus, the current SGBEM-FEM alternating methodgreatly reduces the burden of computation and the humanlabour for preprocessing, compared with pure FEM-basednumerical methods, by several orders of magnitude.

NUMER ICAL RESULTS

In this section, the computational results of the SIFs andfatigue growths of cracks in round bars are presented.

Fig. 4 Geometry of the specimen, all dimensions are in millimetre.

Table 1 Mechanical properties of S45 steel at room temperature

Ultimate tensile strength, σm 775MPa

Monotonic yield strength, σ0 635MPaYoung’s modulus, E 206GPaPoisson’s ratio, ν 0.3Fracture toughness, KIC 104 MPa

ffiffiffiffim

p

Table 2 Loading conditions for the specimens

ModelID

Initial crackdepth a0 (mm)

Cyclic axialloading (N)

Cyclic bendingmoment (N m)

Cyclic torsionmoment (N m)

Pmax Pmin Mmax Mmin Tmax Tmin

CT1 0.918 25 000 2500 0 0 0 0CT2 1.2 25 000 2500 0 0 0 0CB 0.918 0 0 45 0 0 0CTR 0.918 0 0 0 0 100 0CTCA1 1.0 15 000 0 0 0 100 0CTCA2 1.0 0 �15 000 0 0 100 0

CT, cyclic tension; CB, cyclic bending; CTR, cyclic torsion; CTCA, cyclic torsion and cyclic axial loading.

6 L.-G. TIAN et al.


Comparisons are made between the results obtained bySGBEM-FEM alternating method and other empirical/numerical solutions and experimental data reported inthe literature. All the numerical simulations using theSGBEM-FEM alternating method are carried out on aPC with an Intel Core i7 CPU.

Stress intensity analyses

In order to verify the accuracy of the SGBEM-FEMalternating method, the computed SIFs are comparedwith the available empirical solutions. James and Mills45

obtained an expression for the SIF at the midpoint of astraight-fronted surface crack in a round bar subjectedto tension, by fitting a polynomial through several otheranalytical and empirical results. The equation theyobtained is

Kσ

ffiffiffiffiffiπa

p ¼ 0:926� 1:771aD

� �þ 26:421

aD

� �2

�78:481aD

� �3þ 87:911

aD

� �4;

(6)

where K is the Mode I SIF, D is the diameter of theround bar, a is the depth at the midpoint of the crackfront and σ is the applied nominal stress.

Daoud and Cartwright46 also derived an equation forthe SIF of a straight-fronted surface crack in a roundbar subjected to bending load, by a 2D finite elementanalyses with variable thickness

Kσ

ffiffiffiffiffiπa

p ¼ 1:04� 3:64aD

� �þ 16:86

aD

� �2� 32:59

aD

� �3

þ28:41aD

� �4:

(7)

It should be noted that, in their derivation, the thicknesswas changed across the round bar diameter to represent thecircular contour of the bar, thus, Eq. (7) only predicts an‘average’ value of SIF along the straight crack front.

On the basis of Eq. (6), the empirical normalised SIF atthe midpoint of the crack front for the cracked roundbar subjected to axial tension load is 0.9130. Thecorresponding computed SIF by SGBEM-FEM alternat-ing method is 0.9555, with 515 finite elements and 14boundary elements as shown in Fig. 5. On the basis ofEq. (7), the empirical normalised SIF for the cracked roundbar subjected to a bendingmoment along the straight crackfront is 0.8466 (an ‘average’ value), and the correspondingcomputed SIF by SGBEM-FEM alternating method is0.8617. The computed SIFs along the initial crack frontfor the round bar subjected to a bending moment arecompared with the empirical solution in Fig. 6. Asexpected, the computed SIFs are higher in the midpointand lower at the free-boundary, in contrast to the constantvalue solution of Daoud and Cartwright.46

It is also worth to mention that the computation timefor the presently reported SGBEM-FEM alternatingmethod for this problem is about 6 s in a regular PC withIntel i7 CPU. In addition, the human–labour cost is alsovery minimal, to generate the independent coarse meshesof FEM and SGBEM.

Evolving non-planar crack surface under fatigue

The process of fatigue crack growth for each specimen isdivided into nine analysis steps. In each fatigue analysis

(a)

(b)

Fig. 5Meshes for the problem of a round bar with a straight-frontedsurface crack: (a) the uncracked global structure is moulded with 515finite elements (10-noded tetrahedron elements); (b) the initial cracksurface is moulded independently with only 14 boundary elements(eight-noded quadrilateral elements).

Fig. 6 Normalised SIFs at the initial crack front for the round barsubjected to a bending moment.



step, the SIFs along the crack front are firstly computedby the SGBEM-FEM alternating method. The size ofcrack advance at each SGBEM node along the crackfront is then determined by the Paris’ Law, based onthe computed SIFs at each node. The direction of crackpropagation at each SGBEM node is further consideredto be the same as the Eshelby force vector (the vectorJ-integral), which can be obtained by their relations withthe computed SIFs.32 And finally, the crack growth in anarbitrary 3D surface direction is modelled by adding alayer of additional elements at the crack front, with thedetermined crack advance sizes and directions at eachSGBEM node.

First, pure cyclic tensile load is considered. Themaximum magnitude of the axial cyclic tensile load is25KN, and the stress ratio is 0.1. The numerical modelis denoted as ‘CT1’ in Table 2. The relevant materialconstants are listed in Eq. (5). The non-planar evolution

of the crack surface of model ‘CT1’ is shown in Fig. 7.The photograph of the final-cracked specimen and thesimulated crack surface by SGBEM-FEM alternatingmethod is shown in Fig. 8.

The load case presented earlier is relatively simple,because pure Mode I SIFs result in an in-plane crackgrowth. However, if cyclic torsion load is considered,the initial crack will grow in a non-planar way. The valueof crack growth angle θ is a function of the load, materialproperties, stress ratio and geometry of the specimen. Inthe studies by Makabe and Socie47 and Liu and Wang,48

it was found that when the shear stress ratio R is negative,the crack often branches into two directions on both sidesof the crack, as shown in Fig. 9(a). But when the shearstress ratio R is non-negative, the crack propagates onlyalong one direction at each side of the crack, as shownin Fig. 9(b). In this case, the non-planar evolution ofthe crack surface can be depicted by Δx (the projection

Fig. 7 The propagation of a straight-fronted surface crack in a round bar subjected to cyclic tension (Model ID: CT1): 2D view.

(a) (b)

Fig. 8 Final fatigue crack shape of model CT1 for the cracked round bar: (a) photograph of the cross section of the specimen, (taken from Yangand Kuang17); (b) simulated by SGBEM-FEM alternating method.

8 L.-G. TIAN et al.


of the actual crack growth length on the horizontalx-direction) and θ (the angle between the crack growthdirection and the cross section of the round bar), whichare shown in Fig. 10.

In the following case, we consider that the crackedround bar is subjected to a combination of cyclic torsionand CT load. The cyclic torsion and tension loads aresynchronous sinusoidal functions with the same stressratio. The numerical model used here is labelled as‘CTCA1’, as detailed in Table 2. The mixed-modefatigue growth of the crack is shown in Fig. 11 in a 2Dview. The photograph of the final crack shape by experi-ment and the simulated crack shape by SGBEM-FEMalternating method is compared in Fig. 12. The finalfatigue crack shape is also plotted in different perspec-tives in Fig. 13.

The computational results show that, when the roundbar is subjected to both cyclic torsion and tension, theinitially straight-fronted planar crack grows into a non-planar surface. In fact, both the cyclic torsion and CTloads contribute to the crack propagation. When thecrack depth is rather small, it is the torsion that predom-inates, and the crack growth angle is very large. As thecrack depth gradually increases and the uncracked areaof the round bar decreases, the bending effect at the cracksurface is magnified, and the crack extension angle θ isgradually decreased.

Fatigue life estimation

In this section, the fatigue life of the cracked round bar isestimated by using the SGBEM-FEM alternatingmethod. Paris Law is used to characterise the fatiguecrack growth in the round bar. According to Paris Law,the fatigue life of the cracked round bar is as follows:

N ¼ ∫af

a0

1C ΔKð Þn da: (8)

Thus, at each fatigue growth step, the maximum valueof ΔK along crack front is computed by the SGBEM-FEM alternating method. And Eq. (8) is numericallyevaluated using the Trapezoidal rule. For the crackedround bar under CT (numerical model labelled as‘CT1’), the predicated fatigue life is plotted in Fig. 14,and the final fatigue life of the specimen is listed andcompared with the experimental result in Table 3. Excel-lent agreements are found between the simulation bySGBEM-FEM alternating method and experimentalresults.

For the round bar under cyclic tensile loading withinitial crack depth of 1.2mm (model labelled as ‘CT2’),the computed fatigue life by SGBEM-FEM alternatingmethod is compared with the experimental results byYang and Kuang17 in Fig. 15, which also agree well withthe experimental results.

Fig. 9 Surface crack extension behaviour of the round bar under cy-clic torsion loading.

Fig. 10 Surface crack extension paths for the cracked round bar.



Fig. 11 The propagation of a straight-fronted surface crack in a round bar subjected to cyclic torsion and cyclic tension (Model ID: CTCA1):2D view.

(a) (b)

Fig. 12 Final fatigue crack shape of model CTCA1 for the cracked round bar: (a) photograph of the crack path on the surface of the specimen,(taken from Yang and Kuang16); (b) simulated by SGBEM-FEM alternating method.

Fig. 13 The final crack shape of model CTCA1 after fatigue growth under combined cyclic torsion and cyclic tension from differentperspectives.

10 L.-G. TIAN et al.


The fatigue lives of the cracked round bar under cyclictorsion and cyclic axial loadings are also studied using thecurrent method. The crack arc length on the surface of

the round bar is plotted against the number of load cyclesfor models ‘CTCA1’ and ‘CTCA2’ in Fig. 16. It is clearlyshown that the axial CT superimposed on the cyclictorsion will significantly reduce the fatigue life of thecracked round bar. On the contrary, for the model of‘CTCA2’, the axial compression reduces the SIFs duringthe process of fatigue crack growth and significantlyincreases the fatigue life of the round bar.

It should be noted that, for modes II and III crackpropagations, plasticity-induced or roughness-inducedcrack closure may significantly affect lifetime estimationof the specimen. In this paper, simple Paris Law is usedto predict the fatigue growth rate. However, othermodels that account for the crack closure effects can alsobe incorporated in the framework of the currentSGBEM-FEM alternating method, which will beimplemented in our future studies.

CONCLUS IONS

In the present paper, an elastic round bar with aninitially straight-fronted surface crack is studied byemploying the 3D SGBEM-FEM alternating method.Several numerical models of the cracked round barare built, considering the different loading cases, differ-ent initial crack depths and different stress ratios. SIFvariations, the evolving non-planar surface crack pat-terns and fatigue lives of the cracked round bars arethoroughly investigated using the 3D SGBEM-FEMalternating method. By carefully comparing the numer-ical results with empirical solutions and experimentalobservations available in the open literature, thefollowing conclusions can be drawn:

Fig. 14 Surface crack arc length versus the number of load cycles forthe round bar subjected to cyclic tension (model labelled as CT1).

Table 3 The predicted fatigue life of model CT1 by SGBEM-FEM alternating method and the corresponding experimentalresult by Yang and Kuang17

MethodExperiment(cycles)

SGBEM-FEM alternatingmethod (cycles)

Fatigue lifeof specimen

323 637 333 137

CT, cyclic tension; SGBEM-FEM, symmetric Galerkin boundaryelement method-finite element method.

Fig. 15 Fatigue crack surface arc length versus the number of loadcycles for the round bar subjected to cyclic tensile loading (modellabelled as CT2).

Fig. 16 Fatigue crack surface arc length versus the number of loadcycles for the round bar subjected to cyclic torsion and cyclic axialloading (model labelled as CTCA1 and CTCA2).



(a) The SGBEM-FEM alternating method requires verysimple independent and very coarse meshes for boththe uncracked global structure and the crack surface;it only requires minimal computational burden andvery minimal human–labour efforts for modellingthe fatigue growth of cracks in 3D structures.

(b) The computed SIF variations for the cracks using theSGBEM-FEM alternating method are in goodagreement with the empirical solutions.

(c) The predicted non-planar fatigue crack growthpatterns and fatigue lives of the cracked round barsby using the SGBEM-FEM alternating methodagree well with the experimental observations,showing the reliability and high accuracy of thecurrent method.

(d) For the cyclic pure tension of the round bar, thecrack propagates in the same plane, but the finalcrack front does not strictly stay in a straight-line;for combined cyclic torsion and CT loads, the crackgrowth is along a curved path, and the crackextension angle decreases during the propagation.

(e) The axial cyclic loading plays an important role inthe fatigue life of the cracked round bar. A significantreduction of crack growth rate can be achieved, bysuperimposing axial compression load to the torsionload.

Acknowledgements

The first author gratefully acknowledges the financialsupport of China Scholarship Council (Grant No.201306260034); National Basic Research Program ofChina (973 Program: 2011CB013800); and New CenturyExcellent Talents Project in China (NCET-12-0415).This work was supported at UCI by a collaborativeresearch agreement with ARL. The encouragement ofMessrs. Dy Le and Jarret Riddick is thankfullyacknowledged.

REFERENCES

1 Carpinteri, A. and Brighenti, R. (1996) Part-through cracks inround bars under cyclic combined axial and bending loading.Int. J. Fatigue, 18, 33–9.

2 Carpinteri A. (1993) Shape change of surface cracks in roundbars under cyclic axial loading. Int. J. Fatigue, 15, 21–6.

3 Daoud, O. E. K. (1984) Strain energy release rates for a straight-fronted edge crack in a circular bar subject to bending. Eng.Fract. Mech., 19, 701–7.

4 McFadyen, N. B., Bell, R. and Vosikovsky, O. (1990) Fatiguecrack growth of semi-elliptical surface cracks. Int. J. Fatigue,12, 43–50.

5 Thompson, K. D. and Sheppard, S. D. (1992) Stress intensityfactors in shafts subjected to torsion and axial loading. Eng.Fract. Mech., 42, 1019–34.

6 Newman, J. C. and Raju, I. S. (1981) An empirical stress-intensity factor equation for the surface crack. Eng. Fract. Mech.,15, 185–192.

7 NG, C. K. and Fenner D. N. (1988) Stress intensity factors foran edge cracked circular bar in tension and bending. Int. J.Fract., 36, 291–303.

8 Lin, X. B., Smith, R. A. (1997) Shape growth simulation of thesurface cracks in tension fatigue round bars. Int. J. Fatigue,19, 461–469.

9 Lin, X. B. and Smith, R. A. (1999) Shape evolution of surfacecracks in fatigue round bars with a semicircular circumferentialnotch. Int. J. Fatigue, 21, 965–973.

10 Caspers, M. and Mattheck, C. (1987) Weighted averaged stressintensity factors of circular-fronted cracks in cylindrical bars.Fatigue Fract. Eng. Mater. Struct., 9, 329–341.

11 Sonsino, C. M. (2001) Influence of loading and deformation-controlled multiaxial tests on fatigue life to crack initiation. Int.J. Fract., 23, 159–167.

12 Lin, X. B. and Smith, R. A. (1997) Shape growth simulation ofsurface cracks in tension fatigued round bars. Int J Fatigue,19, 461–9.

13 Tschegg, E. K. (1982) A contribution to mode III fatigue crackpropagation. Mater Sci. Eng., 54, 127–36.

14 Yu, H. C., Tanaka, K. and Akiniwa, Y. (1998) Estimation oftorsional fatigue strength of medium carbon steel bars with acircumferential crack by the cyclic resistance-curve method.Fatigue Fract. Eng. Mater. Struct., 21, 1067–1076.

15 Branco, R., Antunes, F. V., Costa, J. D., Yang, F. P. and Kuang,Z. B. (2012) Determination of the Paris law constants in roundbars from beach marks on fracture surfaces. Eng. Fract. Mech.,96, 96–106.

16 Yang, F. P. and Kuang, Z. B. (2005) Fatigue crack growth for asurface crack in a round bar under multi-axial loading condition.Fatigue Fract. Eng. Mater. Struct., 28, 963–970.

17 Yang, F. P., Kuang, Z. B. and Shlyannikov, V. N. (2006) Fatiguecrack growth for straight-fronted edge crack in a round bar. Int.J. Fatigue, 28, 431–437.

18 Tong, P., Pian, T. H. H. and Lasry, S. J. (1973) A hybrid-element approach to crack problems in plane elasticity. Int. J.Numer. Methods Eng., 7, 297–308.

19 Atluri, S. N., Kobayashi, A. S. and Nakagaki, M. (1975) Anassumed displacement hybrid finite element model for linearfracture mechanics. Int. J. Fract., 11, 257–271.

20 Henshell, R. D. and Shaw, K. G. (1975) Crack tip finiteelements are unnecessary. Int. J. Numer. Methods Eng., 9,495–507.

21 Barsoum, R. S. (1976) Application of triangular quarter-point el-ements as crack tip elements of power law hardening material.Int. J. Fatigue, 12, 463–446.

22 Atluri, S. N. 1986 Computational Methods in the Mechanics ofFracture. North Holland, Amsterdam (also translated intoRussian, Mir Publishers, Moscow).

23 Vijayakumar, K., and Atluri, S. N. (1981) An embedded ellipticalcrack, in an infinite solid, subject to arbitrary crack-facetractions. ASME J. Appl. Mech., 103, 88–96.

24 Nishioka, T. and Atluri, S. N. (1983) Path-independent inte-grals, energy release rates, and general solutions of near-tipfields in mixed-mode dynamic fracture mechanics. Eng. Fract.Mech., 18, 1–22.

25 Atluri, S. N. (1997) Structural Integrity and Durability. TechScience Press, Forsyth.

12 L.-G. TIAN et al.


26 Nikishkov, G. P., Park, J. H. and Atluri, S. N. (2001) SGBEM-FEM alternating method for analyzing 3D non-planar cracksand their growth in structural components. CMES: Comput.Model. Eng. Sci., 2, 401–422.

27 Han, Z. D. and Atluri, S. N. (2002) SGBEM (for cracked localsubdomain) – FEM (for uncracked global structure) alternatingmethod for analyzing 3D surface cracks and their fatigue-growth. CMES: Comput. Model. Eng. Sci., 3, 699–716.

28 Dong, L. and Atluri, S. N. (2012) SGBEM (using non-hyper-singular traction BIE), and super elements, for non-collinear fatigue-growth analyses of cracks in stiffened panelswith composite-patch repairs. CMES: Comput. Model. Eng. Sci.,89, 417–458.

29 Dong, L. and Atluri, S. N. (2013) SGBEM Voronoi cells(SVCs), with embedded arbitrary-shaped inclusions, voids,and/or cracks, for micromechanical modeling of heterogeneousmaterials. CMC: Comput. Mater. Continua, 33, 111–154.

30 Dong, L. and Atluri, S. N. (2013) Fracture & fatigue analyses:SGBEM-FEM or XFEM? Part 1: 2D structures. CMES:Comput. Model. Eng. Sci., 90, 91–146.

31 Dong, L. and Atluri, S. N. (2013) Fracture & fatigue analyses:SGBEM-FEM or XFEM? Part 2: 3D solids. CMES: Comput.Model. Eng. Sci., 90, 379–413.

32 Eshelby, J. D. (1951). The force on an elastic singularity. Philos.Trans. R. Soc. A: Math., Phys. Eng. Sci., 244, 87–112.

33 Rizzo, F. J. (1967) An integral equation approach to boundaryvalue problems of classical elastostatics. Quart. Appl. Math,25, 83–95.

34 Portela, A., Aliabadi, M. H. and Rooke, D. P. (1992) The dualboundary element method: effective implementation for crackproblems. Int. J. Numer. Meth. Eng., 33, 1269–1287.

35 Frangi, A. and Novati, G. (1996) Symmetric BE method in two-dimensional elasticity: evaluation of double integrals for curvedelements. Comput. Mech., 19, 58–68.

36 Bonnet, M., Maier, G. and Polizzotto, C. (1998) SymmetricGalerkin boundary element methods. Appl. Mech. Rev.,51, 669–704.

37 Li, S., Mear, M. E. and Xiao, L. (1998) Symmetric weak formintegral equation method for three-dimensional fracture analy-sis. Comput. Methods Appl. Mech. Eng., 151, 435–459.

38 Frangi, A., Novati, G., Springhetti, R. and Rovizzi, M. (2002)3D fracture analysis by the symmetric Galerkin BEM. Comput.Mech., 28, 220–232.

39 Han, Z. D. and Atluri, S. N. (2003) On simple formulations ofweakly-singular traction & displacement BIE, and theirsolutions through Petrov–Galerkin approaches. CMES: Comput.Model. Eng. Sci., 4, 5–20.

40 Han, Z. D. and Atluri, S. N. (2007) A systematic approach forthe development of weakly-singular BIEs. CMES: Comput.Model. Eng. Sci., 21, 41–52.

41 Okada, H., Rajiyah, H. and Atluri, S. N. (1988) A novel displace-ment gradient boundary element method for elastic stressanalysis with high accuracy. J. Appl. Mech., 55, 786–794.

42 Okada, H., Rajiyah, H. and Atluri, S. N. (1989) Non-hyper-singular integral representations for velocity (displacement)gradients in elastic/plastic solids (small or finite deformations).Comput. Mech., 4, 165–175.

43 Paris, P. C. and Erdogan, F. A. (1963) A critical analysis of crackpropagation laws. J. Basic Eng. Trans. SME, Series D, 55, 528–534.

44 Levén, M. and Rickert, D. (2012) Stationary 3D crack analysiswith Abaqus XFEM for integrity assessment of subsea equip-ment. Master’s thesis, Chalmers University of Technology.

45 James, L. A. and Mills, W. J. (1988) Review and synthesis ofstress intensity factor solutions applicable to cracks in bolts.Eng. Fract. Mech., 30, 641–654.

46 Daoud, O. E. K. and Cartwright, D. J. (1984) Strain-energyrelease rates for a straight-fronted edge crack in a circular barsubject to bending. Eng. Fract. Mech., 19, 701–707.

47 Makabe, C. and Socie, D. F. (2001) Crack growth mechanism inprecracked torsional fatigue specimens. Fatigue Fract. Eng.Mater. Struct., 24, 607–615.

48 Liu, K. C. and Wang, J. A. (2001) An energy method forpredicting fatigue life, crack orientation, and crack growth undermultiaxial loading conditions. Int. J. Fatigue, 23, S129–S134.



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