Engineering Fracture Mechanics 128 (2014) 1–7

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Engineering Fracture Mechanics

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Technical Note

Stress intensity factors of a corner crack emanating from apinhole of a solid cylinder

http://dx.doi.org/10.1016/j.engfracmech.2014.07.0210013-7944/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +60 3 79675341.E-mail addresses: judha@um.edu.my, judha.purbolaksono@gmail.com (J. Purbolaksono).

R.P. Joseph, J. Purbolaksono ⇑, H.L. Liew, S. Ramesh, M. HamdiCentre of Advanced Manufacturing and Materials Processing, Department of Mechanical Engineering, Faculty of Engineering, University of Malaya,Kuala Lumpur 50603, Malaysia

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 January 2014Received in revised form 9 May 2014Accepted 18 July 2014Available online 26 July 2014

Keywords:Fatigue crack growthCorner crackBoundary element methodStress intensity factor

This paper presents stress intensity factors (SIFs) and fatigue growth analyses of a cornercrack emanating from a pinhole of a cylinder under cyclic tension. No such work has beenreported in the literature. Stress intensity factors for different crack aspect ratios andpinhole sizes in different diameters of solid cylinder are presented. A larger SIF was foundat the smaller crack size of a or c in a larger diameter of a pinhole of the smaller-diametercylinder. Crack growths at the pinhole edge were found to be faster than those at thecylinder edge.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Many cylindrical parts with geometrical discontinuities in mechanical components are subjected to cyclic loading. Solidcylinders may contain sharp edge, pinhole, notch, slot or keyway that can lead to stress concentration and potentiallypromote a defect. In practices, there are often some inherent micro cracks in structures due to metallurgical defects, man-ufacturing processes or service handling. These cracks subjected to cyclic loading will propagate when the stress intensityfactor is greater than the threshold SIF of the material. In the past several years special considerations have been taken toevaluate stress intensity factor for a corner crack emanating from a hole in a prismatic specimen under cyclic Mode I loading[1,2]. Lazzarin et al. [3] presented rapid calculations of notch stress intensity factors based on strain energy density usingcoarse meshes of finite element models. Next, Afshar et al. [4] proposed some simple analytical expressions generated fromthe strain energy density approach through a combination of finite element models and multi-scaling technique for calcu-lating the notch stress intensity factors of periodic sharp notches. Berto and Lazzarin [5] reported recent developments onfatigue and fracture assessments in brittle and quasi-brittle of engineering materials.

With respect to practical applications, the SIFs of cracks in cylindrical components with geometrical discontinuities areactively sought. Some relevant works have been reported in literature. Raju and Newman [6] estimated SIFs for circumfer-ential surface cracks in pipes and rods with different aspect ratios. Carpinteri [7] presented the SIFs of surface cracks in roundbars. Thompson and Sheppard [8] conducted experimental and numerical works on fatigue crack growths in notched andplain shafts under axial and torsion loading. Carpinteri and Brighenti [9] reported fatigue growths of a part-through crackin a round bar based on the Paris’s law for different ellipse aspect ratio, crack depth ratio and ellipse shifting ratio. Linand Smith [10,11] presented crack shape evolutions and variation of the aspect ratio of a surface crack in a semi-circularly

Nomenclature

a crack lengthc’ crack depthC crack growth rate coefficientda/dN crack growth rateD diameter of cylinderE Young’s modulusK stress intensity factorKI Mode I stress intensity factorKc fracture toughnessKeff effective stress intensity factorLEFM linear elastic fracture mechanicsn NASGRO exponentP NASGRO coefficientq NASGRO coefficientR stress ratiox arc length along the crack frontY normalized distance along crack front.z total arc length along the crack front

Special symbolsa plane stress/strain constraint factort Poisson ratioKo stress intensity factor in infinite domainDK stress intensity factor rangeDKth threshold stress intensity factor rangeR applied stress£ pinhole diameter

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notched round bar during crack growth. A linear elastic 3D-finite element analysis was used to estimate SIFs along the crackfront during fatigue crack growths based on the Paris’s law. Carpinteri et al. [12] and Guo et al. [13] evaluated the effect ofstress concentration of notch on the SIFs of a surface crack in a round bar. Fonte and Freitas [14] reported the behavior ofsurface flaws in round bar under torsion (Mode III) loading. Citarella and Cricrì [15] investigated the growth path of a surfacecrack in a round bar by the FEM and BEM. Carpinteri et al. [16] reported their researches in the last two decades onevaluations of stress intensity factors and fatigue growth of surface cracks in notched shells and round bars.

Although many efforts have been devoted to estimate fatigue life of surface and corner cracks under tension and bending[17], understanding the behaviors of the fatigue crack growth in cylindrical components with geometrical discontinuitiesunder Mode I loading remains open for updates. Due to complexities of the experimental setup, a numerical analysis isalways sought to reduce the experimental work. In this paper, the SIFs and fatigue growth analyses of a crack emanatingfrom pin hole of a cylinder under axial loading using the boundary element software package of BEASY [18] are presented.No such SIFs and fatigue crack growth analysis have been reported in the literature. The generalized NASGRO equation andthe mechanical and fatigue properties of 6061-T6 aluminum alloy are used in this work. A quarter-elliptical corner crack isintroduced at the edge of the pinhole under. Effect of the cylinder and pinhole diameters and the initial crack aspect ratio onthe SIFs and fatigue crack growths are studied.

2. Materials and method

The schematics of the cracked models are illustrated in Fig. 1. The validation model for a semi-elliptical crack in smoothsolid cylinder is shown in Fig. 1a. Fig. 1b shows a schematic of a quarter-elliptical corner crack emanating from a pinhole of asolid cylinder. A tensile load of 100 MPa was used in evaluating the SIFs and fatigue growths of a corner crack (crack lengtha = 0.5 mm) emanating from the pinhole. The cylinder diameters D of 30 mm, 40 mm, and 50 mm and the pinhole diametersØ of 4 mm, 5.89 mm and 8 mm were considered in this work. Fig. 1c-d show the details of the notations used in semi-elliptical and corner cracks. Table 1 lists the mechanical and fracture properties of 6061-T6 aluminum alloy that are usedin the NASGRO equation. All the analyses are carried out by using the BEASY software [18] that utilizes the dual boundaryelement method.

Fig. 1. Schematic diagrams: (a) a semi-elliptical crack in solid cylinder; (b) a corner crack emanating from a pinhole in a solid cylinder; (c) details of thenotations used in a semi-elliptical crack; (d) details of the notations used in a corner crack.

Table 1Mechanical and fracture properties of 6061-T6 aluminum alloy [18].

Poisson’s ratio 0.33

Yield stress (0.2%) (MPa) 282.69Ultimate tensile strength (MPa) 310.26Plane stress fracture toughness, KC (MPa

pmm) 1250.95

Plane strain fracture toughness, KIC (MPap

mm) 903.46Crack growth rate coefficient (C) 6.53E�10NASGRO N exponent (N) 2.3NASGRO P coefficient (P) 0.5NASGRO Q coefficient (Q) 0.5Threshold SIF range at R = 0 (DKth) (MPa

pmm) 121.62

Plane stress/strain constraint factor, a 2.0Smax/S0 0.3

R.P. Joseph et al. / Engineering Fracture Mechanics 128 (2014) 1–7 3

3. Numerical results and discussion

To demonstrate the accuracy of the results obtained by using BEASY [18], several comparisons associated with asemi-elliptical surface crack in smooth round bar (Fig. 1a) with the published solutions/results [6,8,10,12,19] are madeand they are plotted in Fig. 2. The diameter of the smooth round bar D = 30 mm and a surface crack with a crack lengtha = 6 mm are used in evaluating the SIFs and the results are presented in Fig. 2a. The stress intensity factor Ko shown inFig. 2a is defined as

K0 ¼ rffiffiffiffiffiffiffiffip ap

ð1Þ

where r is the applied remote tension stress and a is the crack length. It can be seen from Fig. 2a that the results of bothsolutions are in good agreement. The largest difference is found at the mid-point of the crack front and is within 3%.

To perform fatigue crack growth analyses, the generalized NASGRO equation that is widely used in many practicalsimulations is used in this study. The NASGRO equation incorporates the fatigue crack closure feature for considering theeffect of the stress ratio on crack growth rate and is written as [20]:

dadN¼

Cð1� f Þn DKn 1� DKthDK

� �p

ð1� RÞn 1� DKð1�RÞ KC

� �q ð2Þ

Fig. 2. Comparisons of the results associated with a semi-elliptical surface crack in a smooth solid cylinder: (a) the Mode I SIFs obtained by using BEASY [18]and Raju–Newman’s solution [6]; (b) variations of crack aspect ratio c/a during crack growth by Lin and Smith [10] (numerical), BEASY [18] and Casperset al. [19] (experiment); (c) fatigue crack growths under cyclic tension by Thompson and Sheppard [8] (experiment) and BEASY [18] with NASGRO equation;(d) Fatigue crack growths based on the Paris’s law under cyclic tension numerically done by Carpinteri et al. [12] and BEASY [18].

Fig. 3. Normalized Mode I stress intensity factors of a corner crack from a pinhole for different crack aspect ratios (a = 0.5 mm and K0 ¼ rffiffiffiffiffiffiffiffip ap

): (a) pinholediameter of 4 mm; (b) pinhole diameter of 5.89 mm; (c) pinhole diameter of 8 mm; (d) effect of the pinhole diameter on the SIFs.

4 R.P. Joseph et al. / Engineering Fracture Mechanics 128 (2014) 1–7

Table 2Normalized stress intensity factors (KI/Ko) with a = 0.5 mm and K0 ¼ r

ffiffiffiffiffiffiffiffip ap

.

a/c

D (mm) Ø (mm) Location (see Fig. 1) 0.5 0.75 1.0 1.25 1.5 2.0

30 4 A 2.323 2.138 1.987 1.755 1.642 1.410C 0.982 1.170 1.342 1.436 1.530 1.563

5.89 A 2.609 2.394 2.229 1.974 1.783 1.555C 1.129 1.392 1.590 1.697 1.794 1.827

8 A 2.792 2.593 2.458 2.072 1.930 1.624C 1.317 1.683 1.810 1.944 2.063 2.056

40 4 A 2.261 2.064 1.930 1.707 1.625 1.383C 0.970 1.159 1.339 1.455 1.517 1.527

5.89 A 2.508 2.285 2.109 1.856 1.721 1.412C 1.105 1.340 1.514 1.611 1.706 1.672

8 A 2.744 2.500 2.284 2.010 1.758 1.486C 1.265 1.530 1.714 1.827 1.894 1.859

50 4 A 2.234 2.070 1.904 1.671 1.593 1.341C 0.915 1.169 1.339 1.429 1.483 1.527

5.89 A 2.472 2.233 2.075 1.806 1.688 1.443C 1.103 1.326 1.464 1.577 1.669 1.658

8 A 2.602 2.389 2.205 1.944 1.811 1.529C 1.233 1.474 1.646 1.736 1.846 1.838

R.P. Joseph et al. / Engineering Fracture Mechanics 128 (2014) 1–7 5

where N is the fatigue cycles, a is the crack length (crack depth), R is the stress ratio, DKth is the threshold stress intensityfactor range, DK is the range of stress intensity factors, KC is the plane stress fracture toughness, and the constants C, n, p, andq are the NASGRO coefficients, and f is the crack opening function that is defined as [20]:

f ¼ MaxðR;A0 þ A1Rþ A2R2 þ A3R3Þ ð3Þ

for R P 0, and

f ¼ A0 þ A1R ð4Þ

for �2 6 R < 0. The polynomial coefficients are defined as [20]:

A0 ¼ ð0:825� 0:34 aþ 0:05 a2Þ ðcos½p=2 � ðSmax=S0Þ�Þ1=a ð5Þ

A1 ¼ ð0:415� 0:071 aÞ � Smax=S0 ð6Þ

A2 ¼ 1� A0 � A1 � A3 ð7Þ

A3 ¼ 2A0 þ A1 � 1 ð8Þ

where a is the plane stress/strain constraint factor, and Smax/S0 is the ratio of the peak stress to the material flow stress in astress cycle. The constants a and Smax/S0 are available as the fitting constants [18].

To show the accuracy of the results for fatigue crack growth obtained by BEASY [18], variations of aspect ratios c/a of asemi-elliptical surface crack during crack growth, in which the Paris’s law with a stress ratio R = 0.5 is used, are comparedwith those reported by Lin and Smith [10] and Caspers et al. [19] and presented in Fig. 2b. Lin and Smith [10] simulated thecrack shape evolutions using a three-dimensional finite element method. Caspers et al. [19] quantified the crack extensionsbased on the beach marks on the fracture surface. Next, based on the NASGRO equation, the results for fatigue growth of Al2024-T351 by BEASY [18] are compared with the experimental results reported by Thompson and Sheppard [8] (Fig. 2c).The fatigue crack growths fairly show similar patterns, but the estimations for Al 2024-T351 and Al 2024-T3 obtained byBEASY [18] are shown to have shorter fatigue lives. However, conservative estimations demonstrated by the numerical sim-ulations may be desirable in engineering design. Further, a numerical comparison on fatigue crack growth based on the Par-is’s law with the results reported by Carpinteri et al. [12] is also made and plotted in Fig. 2d. It is shown that the estimationmade by BEASY [18] has slightly a shorter fatigue life than that by Carpinteri et al. [12]. However, they are shown to be in agood agreement.

It is essential to show the effect of the cylinder and pinhole diameters and the crack aspect ratio a/c on the SIFs. The stressintensity factors at the initial increment for different pinhole diameters and crack aspect ratio a/c of the 30 mm-diametercylinder are presented in Fig. 3. It can be observed from Fig. 3a–c that the Mode I SIF at the cylinder edge increases andthe SIF at the pinhole edge decreases as the crack aspect ratio a/c increases. A smaller a/c tends to lead to the ascending SIFsalong the crack front whereas a larger a/c shows the descending SIFs. Fig. 3d shows that a larger pinhole diameter results inlarger SIFs along the crack front that may lead to a shorter fatigue life. It may be explained that a larger ratio of the pinholediameter to the crack size causes a higher stress concentration factor, in turn leading to a larger SIF. More SIF results fordifferent cylinder and pinhole diameters and crack aspect ratios a/c are presented in Table 2. It can be generally justified from

Fig. 4. (a) Effect of the crack aspect ratio a/c on the fatigue life at R = 0; (b) crack shape evolutions during the fatigue crack growth.

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Table 2 that a larger SIF was found at the smaller crack size of a or c in a larger diameter of a pinhole of the smaller-diametercylinder.

Fatigue growths of a corner crack for different crack aspect ratios a/c emanating from the pinhole of 8 mm in 30 mm-diameter solid cylinder at the stress ratio R = 0 are presented in Fig. 4a. In all cases, the crack growths at the pinhole edgeare observed to be faster than those at the cylinder edge. For a given a small crack size, a shorter fatigue life was observedat the model which has initial a/c – 1 (larger a/c or smaller a/c). In other word, the initial crack aspect ratio of a/c � 1 willresult in a longer fatigue life. Fig. 4b shows the crack shape evolutions of a corner crack for different crack aspect ratios. Thecrack shape evolutions also indicate the faster crack growths at the pinhole edge. For the first few increments, the initialcrack aspect ratio a/c = 0.5 leads to larger crack extensions at the pinhole edge, whereas the initial a/c = 2 has a larger crackgrowth rate at the cylinder edge. Prior to having unstable crack growth, larger crack extensions are generally observed at thecylinder edge. The final evolving crack aspect ratios a/c are shown to be greater than 1.

4. Conclusion

Evaluations of the stress intensity factors and behaviors of the fatigue growth of a part-through crack in a hole/notch of asolid cylinder are of interest in researches on mechanical engineering design. Numerical modeling is always sought in astructural design process to minimize the use of experimentations due to the complexities of experimental setup. Here,stress intensity factors and fatigue growths of a corner crack emanating from pinhole of a cylinder were presented.A few statements may be fairly drawn as follows:

– A smaller crack size of a or c in a larger diameter of a pinhole of the smaller-diameter cylinder would have a largerSIF.

R.P. Joseph et al. / Engineering Fracture Mechanics 128 (2014) 1–7 7

– Initial crack aspect ratio a/c – 1 (larger a/c or smaller a/c) led to a shorter fatigue life.– Crack growths at the pinhole edge were found to be faster than those at the cylinder edge– The final evolving crack aspect ratio a/c was shown to be greater than 1.

Acknowledgements

The authors wish to thank the Ministry of Higher Education, Malaysia, through the High Impact Research Grant (UM.C/625/1/HIR/MOHE/ENG/33). Special thanks to Dr. Sharon Mellings, Senior Fracture Simulation Engineer of C.M Beasy Ltd.Southampton UK for the suggestions and advices during the completion of this study.

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