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Fracture analysis of a corner crack in a pinhole of a solid cylinder under torsion loading

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Introduction
The cylindrical bars are widely used in engineering applications for machine components and structures. Surface cracks or flaws are frequently initiated in these components due to cyclic applied loads, material defects and improper manufacturing processes. During service loading, the crack or flaw grows into a critical stage, which can then result in an undesirable failure. To ensure the safety of the component to be fit in service, scheduled inspections shall be performed. Thus, understanding of the fatigue crack behavior in components is essential. Machinery components such as turbines, electric motors, power transmissions, etc. commonly have cylindrical parts. In services, those components are often subjected to cyclic torsion loading. Sharp edge, pinhole, notch, slot or keyway in solid cylinders can potentially lead to stress concentration and promote a defect. Some inherent micro cracks in structures are unavoidable as a result of metallurgical defects, manufacturing processes or service handling. When a component has a micro crack and its stress intensity factor is greater than the threshold value of the material fracture toughness, the crack will propagate to become a macro crack.
Various works on fracture analyses using different methods, materials and geometries have been reported by researchers (Alshoaibi, 2020;Abd-Elhady 2013, Abd-Elhady 2020, Aliha & Ayatollahi 2008, 2010Aliha & Ayatollahi, 2012;Aliha & Ayatollahi 2013;Citarella et al., 2014;Predan et al., 2013;Zakavi et al., 2019). In particular, fatigue crack growths of a surface crack in a smooth round bar have been reported in literature (Carpinteri & Brighenti, 1996;Carpinteri & Vantadori, 2009;Couroneau & Royer, 1998;Lin & Smith, 1997;Lin & Smith, 1998;Thompson & Sheppard, 1992;Toribio et al., 2009;Toribio et al., 2011;Toribio et al., 2014). Although many works for notched and smooth shafts have been done, only a few fatigue crack growth analyses on a round solid bar with a transverse hole are available in literature. Comprehensive work on life estimations of crack initiation in solid cylinders under multiaxial in-phase and out-of-phase loadings was reported by Jen and Wang (2005). Next, Yip and Jen (1996) reported fatigue life of crack initiation in transverse circular holes of a solid cylinder under combined axial and torsion loading. Joseph et al. (2014) presented the fatigue crack growths of a corner in a pinhole subjected to cyclic tension. It simply indicates that numerical methods were often used for evaluating the local stress and strain at components with a geometrical discontinuity.
Considering many mechanical components in services being subjected to torsion loading, the updates on the phenomena of fracture behaviors are always desired. In this work, the stress intensity factors (SIFs) and fatigue crack growths of a corner crack emanating from a pinhole of a solid cylinder under cyclic torsion loading are presented. A numerical analysis is often sought to reduce the experimental works that usually have the complex setup. Here, all the analyses are carried out by using the dual-boundary element software package of BEASY (2014). The BEASY software uses the DBEM which was developed by Mi and Aliabadi (1992) and Cisilino and Aliabadi (1999) for treating the crack boundaries and extensions during the crack growth. The generalized NASGRO equation and the mechanical and fatigue properties of 6061-T6 aluminum alloy (BEASY 2014) are used in this study. A quarter-elliptical corner crack for different crack orientations is introduced at the edge of the pinhole. Effects of the initial crack aspect ratio on the SIFs are studied.

Materials and Method
The schematic illustrations of the cracked models are presented in Fig. 1. The schematic model of a semi-elliptical crack in a smooth solid cylinder is shown in Fig. 1a. Figure 1b shows a schematic model of a quarter-elliptical corner crack emanating from a pinhole of a solid cylinder. In modeling, all the surfaces of cylinder region are defined to be in outward normal direction. Two-dimensional quadratic elements were used to discretize all the non-cracked surfaces. A maximum shear stress τs of 100 MPa (Mt = 530 x 10 3 N.mm) was used in all cases for calculating the SIFs. The J-integral method is used to evaluate the stress intensity factors. The pinhole diameter Ø of 6 mm in 30 mm-diameter cylinder was considered for crack growth analyses. Figures 1c-d show the details of the notations used in a semielliptical crack and a corner crack. Table 1 lists the mechanical and fracture properties of 6061-T6 aluminum alloy that are used in the NASGRO equation. The minimum strain energy density criterion for crack extension is adopted in BEASY (2014). The modeling strategy of three-dimensional crack problems may be summarized as follows (Mi & Aliabadi, 1992;Cisilino & Aliabadi, 1999): -Crack surfaces are modeled with discontinuous elements.
-Surfaces intersecting a crack surface are modeled with edge discontinuous quadrilateral or trian gular elements. -The displacement integral equation is applied for collocation on one of the crack surfaces, i.e. th e upper surface. -The traction integral equation is applied for collocation on the opposite crack surface, i.e. the lo wer surface.

Results and Discussion
Prior to introducing an initial crack, the model of a solid cylinder with a pinhole shall be first established. The principal stresses around the pinhole of the twisted shaft are shown in Fig. 2(a). The maximum and minimum principal stresses are shown, respectively, at 45 o (225 o ) and 135 o (315 o ) of the pinhole perimeter. The wireframe initial meshing of a corner crack model is shown in Fig. 2 where τs is the applied maximum shear stress and c is the crack length. Both the solutions, except at Y = 0.8, are shown to be in good agreements. It is widely known that the FEA has an inaccuracy of calculating the SIFs at the crack front near the surface intersecting the crack surface (Y is approaching 1) of having corner point singularities. Pook (1994) raised the implication of singularities in the vicinity of a corner point where a crack front intersects a free surface. The consideration of corner point singularities may be necessary for the situations that promote mixed mode loadings, particularly for analyzing mixed mode fatigue crack growths. It is reported that stresses in the vicinity of a corner point are known to be the combined stresses due to two different singularities of different orders: stress intensity factors and corner point singularities (Pook et al., 2015;Pook et al., 2017;Berto et al., 2017). The phenomenon of corner point singularities at the cracked bodies was also reported in later works by Aliha et al. (2015Aliha et al. ( , 2017Aliha et al. ( , 2018Aliha et al. ( , 2019, Pour et al. (2018), Aliha and Bahmani (2017) and Bahmani et al. (2017Bahmani et al. ( , 2020. Meanwhile, the DBEM uses the edge discontinuous quadrilateral or triangular elements to treat the singularities at the intersecting surface, resulting in more accurate results (Mi & Aliabadi, 1992;Cisilino & Aliabadi, 1999). At the initial increment, the KI, KII, KIII, and Keff values of a corner crack (α = 45 o ) in a 6 mmpinhole diameter of a 30 mm-diameter cylinder for different crack aspect ratio a/c are presented in Fig.  4. It can be seen from Fig. 4a that the KI values at the cylinder edge are shown to increase as the crack aspect ratio a/c decreases. It indicates that, in respect to the crack geometry on KI, the shorter section of a or c would lead to a larger local normal stress. The KI values along the crack front for a/c = 0.67 are shown to be nearly constant. The KI value for a/c around 0.67 may be the turning point to be the lowest SIF at the pinhole edge.
At the crack orientation α = 45 o and for a given aspect ratio a/c, the absolute KII values at the pinhole edge (Fig. 4b) are observed to be larger than those at the cylinder edge. In respect to the crack geometry on KII, the shorter section of a or c would lead to a larger local shear stress. In contrast to the KII values, the KIII values at the pinhole edge (Fig. 4c) are generally shown to be smaller than those at the cylinder edge. A crack aspect ratio smaller than 0.67 seems to trigger a larger KIII values. However, the location of the largest SIF at the crack front varies and depends on the crack aspect ratio a/c. The Keff values along the crack fronts are shown in Fig. 4d. The Keff is defined as Keff = [(KI + |KIII|) 2 + 2KII 2 ] 0.5 . The patterns of the Keff values along the crack front (Fig. 4d) are shown to be similar to those of KI values (Fig. 4a). The SIFs for Mode I are observed to contribute dominantly to the Keff values. It can also be evidenced by the magnitudes of the KI in comparison to those of KII and KIII.
It is necessary to evaluate the effect of the crack orientation α (see Fig. 1b) on the SIFs. To demonstrate this purpose, a corner crack (a/c = 1 = 0.5 mm/0.5 mm) in a 6 mm-pinhole diameter of a 30 mm-diameter cylinder is used, and the corresponding SIF results are presented in Fig. 5. Figure 5a confirms that the largest KI values are found at α = 45 o where the maximum principal stress is known to be at this angle. Meanwhile, the SIFs along the crack front at α = 0 o and 90 o are found to be nearly zero. The KII and KIII values are presented in Figs. 5b and 5c, respectively. At α ≠ 45 o , the absolute KII values at the pinhole edge (Fig. 5b) are generally observed to be smaller than those at the cylinder edge. At α = 22.5 o , the KII values toward the pinhole edge are found to be asymptotically descending to zero. The maximum absolute KIII values (Fig. 5c) are found at the crack front away from the cylinder and pinhole edges. Again, the patterns of the Keff values (Fig. 5d) are shown to be similar to those of the SIFs for Mode I (Fig. 5a). It means that at α ≠ 0 o and 90 o , the KI values are the dominant magnitudes in determining the effective stress intensity factors.   More Mode I SIFs for different pinhole diameters, crack orientation α, and crack aspect ratios a/c are presented in Table 2. Table 2 generally shows that for a given crack aspect ratio a/c, a larger KI value are observed at the larger pinhole diameter. It is also confirmed that the larger Mode I SIFs are found at α = 45 o .
To perform fatigue crack growth analyses, the generalized NASGRO equation that is widely used in many practical analyses is used in this study. The NASGRO equation is written as (Forman et al., 1994): where N is the fatigue cycles, a is the crack length (crack depth), R is the stress ratio, ΔKth is the threshold stress intensity factor range, ΔK is the range of stress intensity factors, KC is the plane stress fracture toughness, and the constants C, n, p, and q are the NASGRO coefficients, and f is the crack opening function that is defined as (Forman et al., 1994): for R ≥ 0, and for -2 ≤ R < 0. The polynomial coefficients are defined as (Forman et al., 1994): where α is the plane stress/strain constraint factor, and Smax/S0 is the ratio of the peak stress to the material flow stress in a stress cycle. The constants α and Smax/S0 are available as the fitting constants (BEASY 2014).
To validate the estimations obtained by using BEASY (2014), the fatigue growth with a constant stress ratio R = 0.1 of a surface crack (a = 0.5 mm and c = 0.5 mm) in a 15.9 mm-diameter smooth cylinder is compared with the experimental result reported by Thompson and Sheppard (1992) as presented in Fig.  6. The cylinder that is made of Al 2024-T351 is subjected to a maximum cyclic torsion loading of 134.28 MPa. At the stable stage, even though having a similar slope of the curve, the estimation done by using BEASY (2014) shows more conservative than that by Thompson and Sheppard (1992). Very good agreement is shown for the estimation at the unstable stage. However, this is desirable in the preliminary engineering design process.
Fatigue growths of a corner crack at α = 45 o for different crack aspect ratios a/c emanating from a 6 mm-diameter pinhole of a 30 mm-diameter solid cylinder under a constant amplitude load of R = 0 (a maximum shear stress of 265 MPa or Mt = 1.4 x 10 6 N.mm) are presented in Fig. 7. The crack lengths c (= c0 + Δc) and a' (= a'0 + Δa') are, respectively, measured at the pinhole and cylinder edges (see Fig.  1), where c0 and a'0 are the initial crack length and depth, and Δc and Δa' are the crack increments. In all cases, initially, the faster crack growth is found at the pinhole. Next, after the evolving crack aspect ratio a/c be around 1, the crack growths at the cylinder edge are observed to be faster than those at the pinhole edge. It leads to the final evolving crack aspect ratio a/c to be larger than 1. It can also be seen from Fig.  7, for the same given initial crack length a, a smaller crack depth c was found to lead to a shorter fatigue life. It may be due to the fact that the shorter crack length at the pinhole edge has a larger KI value, causing a larger opening mode. Meanwhile, for the same initial crack depth c, different crack lengths comparatively result in similar fatigue lives.  Thompson and Sheppard (1992) (experimental work) and present work with NASGRO equation  8 shows the crack path and crack shape evolutions of a corner crack for different crack aspect ratios. The crack planes are shown to be waved and twisted during the crack growths. This is as a result of the changes of the principal stress distribution around the crack front. It can also be from Fig. 8 that a shorter fatigue life tends to yield a larger α of the crack path.

Conclusions
Stress intensity factors and fatigue crack growths of a corner crack emanating from pinhole of a cylinder under torsion loading were presented. Some concluding remarks may be fairly drawn as follows: -For given crack aspect ratio a/c, larger Mode I SIFs were observed at a larger pinhole diameter. -At α = 45 o and for a given aspect ratio a/c, the absolute KII values at the pinhole were observed t o be larger than those at the cylinder edge. Meanwhile, the KIII values at the pinhole edge were g enerally found to be smaller than those at the cylinder edge. -For given crack aspect ratios, the crack growth at the pinhole edge was initially found to be faste r than that at the cylinder edge. Then, after the crack aspect ratio a/c approached a unity, the cra ck growths at the cylinder edge were observed to be faster than those at the pinhole edge, leadin g to the final evolving crack aspect ratio a/c to be larger than 1. -For the same given crack length a, a smaller crack depth c resulted in a shorter fatigue life. -A shorter fatigue life produced a larger evolving angle α of the crack path.