PSI - Issue 7

Sho Hashimoto et al. / Procedia Structural Integrity 7 (2017) 453–459 Sho Hashimoto / Structural Integrity Procedia 00 (2017) 000–000

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2. Mode II stress intensity factor of a crack emanating from the edge of a drilled hole under rolling contact 2.1. Method of Analysis Figure 1 shows the deep-groove ball bearing used for the RCF tests, as produced from JIS-SUJ2. The stress field generated under Hertzian contact between the inner ring and the steel ball was analyzed via FEM, using the commercial software, MSC Marc 2013r1. A half-symmetric model was used for the analysis. Some external force in the z direction was applied to the rolling component, so that the two elements entered into contact with each other. Node displacement was constrained on the x-z symmetric plane in a y direction. Young’s modulus and Poisson’s ratio of the inner ring were 208 GPa and 0.3, respectively. The inner ring was modelled as an elastic-flat plane so as to simplify the modelling and analysis processes. In accordance with such simplification, the rolling element was modelled as a rigid barrel with an equivalent curvature radius, calculated from the curves of the inner ring and ball so that the contact ellipse resembled a realistic scenario - a combination of elastic inner ring and elastic ball. As indicated in Fig. 2, a small drilled hole was introduced in the middle zone of the inner ring raceway. Figure 3 displays a micrograph of the cross-section of the drilled hole, after the RCF test of the rolling bearing at the maximum contact pressure of q max = 2.5 GPa, in which the diameter and depth of the hole were d = 0.1 mm and h ′ = 0.064 mm, respectively. No flaking occurred in this bearing after the stress cycles of N = 2.6× 10 8 . As illustrated in Fig. 3, small cracks were observed at the edge of the hole. Similar cracks were also found in other un-flaked bearings at various diameters and depths. The length of the crack, a ′ , as measured from the edge of the drilled hole, was a maximum of 0.01 mm. Therefore, these cracks were assumed to be either non-propagating, or such that propagated at an extremely slow growth rate. Similar cracks were previously discovered in un-flaked specimens after RCF tests of JIS-SUJ2 (Komata et al., 2012, 2013). The SIF of a ring-shaped crack with a length, a ′ , of 0.010 mm around the hole-edge was analyzed by FEM, so as to quantify the flaking limit of the rolling bearing as a non-propagation crack limit. Figure 4 displays the flat-plate, FEM model, with a drilled hole of d = 0.100 mm and h ′ = 0.100 mm which was in contact with the rolling element. In this study, the hole diameter, d, was varied at four levels ( i.e. , 0.050 mm, 0.075 mm, 0.100 mm and 0.200 mm), with the hole-edge depth, h ′ , also varied at four levels ( i.e. , 0.050 mm, 0.100 mm, 0.220 mm and 0.345 mm). To assess the change in Mode II SIF K II at Point A in Fig. 5 under the movement of the rolling component, the rigid-barrel element was placed in contact with the flat plate at a position sufficiently distant from the hole, then moved through the hole along the contact surface. Maximum contact pressure, q max , was altered at three levels, 3.0 GPa, 2.5 GPa and 2.0 GPa. During the analysis, the value of K II was calculated at each position in increments of 0.020 mm. The stress extrapolation method was used to calculate K II (Ishida, 1976). Based on the shear-stress distribution ahead of the crack-tip, the provisional SIF, K II *, was calculated at each node using Eq. (1):

Outer ring

Inner ring

Rolling element (Ball)

Contact ellipse

Major axis 2 s a

Minor axis 2 s b

y

x z

Rolling element (Ball)

Inner ring

(a) Radial-type rolling bearing.

(b) Ellipse-type contact area on rolling bearing.

Fig.1: Target of FEM analysis. Contact ellipse is generated between ball and ring.