Subsurface crack propagation under rolling contact fatigue in bearing ring

SCIENCE CHINA Technological Sciences

© Science China Press and Springer-Verlag Berlin Heidelberg 2013 tech.scichina.com www.springerlink.com

*Corresponding author (email: [email protected])

• RESEARCH PAPER • January 2013 Vol.56 No.1: 1–11

doi: 10.1007/s11431-013-5291-5

Subsurface crack propagation under rolling contact fatigue in bearing ring

DENG Song, HAN XingHui, QIN XunPeng* & HUANG Song

School of Automotive Engineering, Hubei Key Laboratory of Advanced Technology of Automotive Parts, Wuhan University of Technology, Wuhan 430070, China

Received April 22, 2013; accepted June 20, 2013

The influences of subsurface cracks, distributing along the axial direction, on the rolling contact fatigue (RCF) faliure in a bearing ring are investigated. A realistic three-dimensional model of the bearing ring containing three subsurface cracks is used to evaluate the fatigue crack propagation based on stress intensity factor (SIF) calculations. Moreover, the distributions of the subsurface cracks along the axial direction are varied to study their effects on RCF. The results provide valuable guidelines for enhanced understanding of RCF in bearings.

bearing ring, subsurface cracks, rolling contact fatigue, stress intensity factors

Citation: Deng S, Han X H, Qin X P, et al. Subsurface crack propagation under rolling contact fatigue in bearing ring. Sci China Tech Sci, doi: 10.1007/s11431-013-5291-5

1 Introduction

The failure of machine components due to fatigue is of ma-jor concern in most engineering applications. In machinery (rolling element bearings, gears, rail-wheel contacts, etc.), RCF is the key aspect of component failure. When two non- conformal bodies roll against each other, alternating stress is induced throughout a very small volume. Under the al-ternating stress, cracks first initiate in the subsurface, and then propagate toward the surface. Next, a chunk of materi-al lifts from the surface, leading to spalling, and RCF failure occurs. Moreover, the distribution of subsurface cracks in the raceway plays an important role on RCF failure. This failure can result in significant economic costs, particularly in the area of personal injuries. Thus, it is of great im-portance to study the subsurface crack propagation under RCF. For the bearings, bearing rings are made by the ring

rolling technology [1–6]. However, this technology usually leads to material defects such as dislocations, voids and microcracks in the raceway subsurface. Therefore, it is nec-essary to investigate fatigue crack growth in the raceway subsurface of bearings

Previous work has been carried out on RCF failure mecha-nism. Some studies [7–10] mainly investigated how material defects cause a very high local increase in stress and strain and studied the influences of load magnitude, contact geometry, and defect location and size on fatigue failure. Moreover, the RCF crack propagation mechanism in fluid stress field has also been studied [11–14]. Additionally, RCF life evaluation is important for predictions of RCF failure. Some criterions, such as the Jiang-Sehitoglu criterion [15], the Brown-Miller strain-life equation [16] and Pair’s law [17], have been used to calculate fatigue life of crack initiation and propagation. However, it should be noted that on the one hand, a majority of the studies above were primarily concentrated on RCF crack growth in railway wheel steel, while little attention was

doi: 10.1007/s11431-013-5291-5

2 Deng S, et al. Sci China Tech Sci January (2013) Vol.56 No.1

paid to the effect of the bearings contact condition on the RCF property. However, the contact condition of the bearings is more complex than that of the rail/wheel. Particularly, the contact between the ball and the bearing raceway is elliptical, and the aspect ratio between the two axes of the ellipse is suf-ficiently high. On the other hand, the effects of multiple cracks below the raceway surface on RCF remain unclear. Hence, it is necessary to study the multiple subsurface cracks propaga-tion mechanism. For example, the stress intensity factor (SIF) calculation and crack growth rate can be used to assess the crack propagation mode and the degree of fatigue damage. To date, fracture mechanics also have been used to predict RCF. Based on SIF calculations, the typical fatigue failure mode of rails was revealed during the passage of rail/wheel contact [18, 19]. The subsurface crack propagation when a pore exists be-low the Hertzian contact center of the rail was analyzed under rolling contact loading [20]. Correlation of the crack growth rates with various parameters, such as the surface Coulomb friction, the crack inclination, the crack depth and the crack face friction, were investigated to determine the significant conditions that influence RCF [21, 22]. However, the studies given above neglected the effects of multiple subsurface cracks on SIF calculations. Particularly, the distribution of subsurface cracks along the axial direction obviously influ-ences the crack propagation mechanism due to the elliptical contact. Therefore, it is of great importance to study the influ-ences of multiple subsurface cracks on RCF by developing an accurate 3D model of the bearing ring.

In this paper, a realistic 3D model of a bearing ring con-taining three subsurface cracks is established using the finite element (FE) software ABAQUS. First, this model is vali-dated according to the Hertzian contact theory and SIF cal-culations. Second, based on this reliable model, the distribu-tions of subsurface cracks along the axial direction are var-ied to investigate their influences on the RCF crack propa-gation mechanism in the bearing ring. The results provide valuable guidelines for enhanced understanding of RCF failure in bearings.

2 Subsurface crack observation

A failed bearing ring was found in an automobile repair shop, as shown in Figure 1(a). The bearing material is GCr15 bearing steel which corresponds to AISI52100 bear-ing steel. From Figure 1(a), it can be seen that a lot of spalls removed from the raceway surface of the inner ring. The failed inner ring was cut from the spalling zone along the axial direction, and the micro-section below the failed raceway surface was presented in Figure 1(b), where it is clear that two subsurface cracks are approximately parallel to the spalling surface along the axial direction. Thus, the distributions of multiple subsurface cracks are used to guide

the study on their effect on RCF.

3 Numerical model development and stress in-tensity factor calculations

3.1 Numerical model development

In this paper, the inner ring of a 6208 (bearing designation, SKF) deep groove ball bearing is considered as the RCF research object. The main geometric data are: bearing bore diameter d=40 mm, bearing width b=18 mm, steel ball di-ameter Dw=12.7 mm, diameter of ball’s centre in the bear-ing Dpw=60 mm, inner raceway groove curvature radius ri =6.54 mm, as shown in Figure 2. Subsurface cracks are modeled as three elliptical pores and located at an approxi-mately depth H (0.2 mm) below the raceway surface. The major size 2a, the minor size 2b and the thickness 2c of these cracks are 0.5, 0.25 and 0.25 mm, respectively, as shown in Figure 3. Symbol θ indicates the rotation angle of

Figure 1 Illustration of (a) appearance of raceway surface of the failed inner ring, (b) photo of the axial micro-section below the failed raceway surface by optical microscope.

Figure 2 Plane sketch of bearing inner ring.

Deng S, et al. Sci China Tech Sci January (2013) Vol.56 No.1 3

Figure 3 Elliptical crack size.

the inner ring. The rotation angle is defined to be positive when the contact position is to the right of the cracks. The moment M is exerted on the inner ring in the anticlockwise direction.

The numerical model is developed as shown in Figure 4(a). The material used in this study is AISI 52100 steel and the main parameters are: Young’s modulus E=2.10×105 MPa, Poisson’s ratio =0.3, yield stress y=1747 MPa, and failure stress r=2106 MPa. The inner ring was meshed with 20-noded quadratic hexahedron elements. To study the crack propagation, it is necessary to define the crack front, crack tip and crack extension direction [23], as shown in Figure 4(b). The elliptical surface is defined as the crack front, the edge is defined as the crack tip, the direction par-allel to the major size is assumed as the crack extension direction. Using this method, three subsurface cracks, dis-tributed along the “z” direction, are defined, as shown in Figure 5, and their interval s is varied to investigate RCF in the bearing ring. The three crack tips are called tips 1, 2 and 3. To get accurate stress and strain, the refined mesh is cre-ated around the subsurface cracks. The refined elements around the tip are C3D20 elements and the element size is an approximately 0.025-mm cube. The element size near the subsurface crack is an approximately 0.075-mm cube. Moreover, the refined meshes above the subsurface crack are 10 layers. The other part of the inner ring is roughly meshed for reducing the computer calculation time. To pre-vent the crack surface penetration, the crack face friction coefficient is assumed to be 0.35. The raceway surface and the ball’s surface are defined as a contact fair and the fric-tion coefficient is assumed to be 0.15. The ball is defined as a rigid body and its displacement degrees of freedom are constrained. A radial force Fr (3 kN) in the radial direction and a moment M (0.5 kN mm) in the circumferential direc-tion are exerted on the inner ring.

In a real contact between a steel ball and an inner ring, the pressure distribution is often complex and highly de-pendent on the exact profiles of the steel ball and inner ring as well as their relative positions. However, a significant amount of previous work on RCF (e.g., [7, 10, 15]) has em-ployed either a simple representative Hertzian line or ellip-tical contact. Using the boundary element modeling ap-proach, it is possible to directly simulate a ball contacting the inner ring. Hence, it is convenient to obtain a real pres-

Figure 4 Numerical model of inner ring containing elliptical cracks: (a) Whole numerical model; (b) subsurface crack definition.

Figure 5 Subsurface crack distribution along the “z” direction.

sure distribution on the raceway surface by defining the contact surfaces within the model.

3.2 Stress intensity factors calculation

The formulation of enriched crack tip elements for two- dimensional fracture problems dates back to the work by Benzley [24]. These elements contain closed-form asymp-totic crack tip displacement fields in addition to the usual polynomial interpolation function. The enriched element displacements u, v and w for the three-dimensional fracture problems are given by [25, 26]

1

1

0

1 1

, , , ,

, ,, , ,

, , , ,

r

j jj

u N u

K FZ

K G K H

(1)


1

2

0

2 2

, , , ,

, ,, , ,

, , , ,

r

j jj

v N v

K FZ

K G K H

(2)

1

3

0

3 3

, , , ,

, ,, , ,

, , , ,

r

j jj

w N w

K FZ

K G K H

(3)

where uj, vj and wj are the r unknown nodal displacements and Nj (, , ) are the conventional element shape func-tions in terms of the element’s local coordinates. KI(), KII() and KIII() represent modes I, II and III SIFs. The SIFs vary along the crack front as

1

,s

ii

i

K N K

1

,s

ii

i

K N K

(4)

1

.s

ii

i

K N K

where IiK , II

iK and IIIiK are the unknown SIFs at the ith

crack tip node in the enriched element shown in Figure 6. In most cases, Ni () is the element shape function along the element edge that coincides with the crack front. The crack front is defined by the crack tip nodal coordinates xi, yi, zi and given by

1

,s

i ii

x N x

1

y ,s

i ii

N y

(5)

1

,s

i ii

z N z

depending on which edge of the element touches the crack front, =, = or =. For example, for a 20-noded quadratic hexahedron that has a crack front located on the edge defined by =1, =1, with (1⫹⫹1), the mode I SIF variation along the crack front is given by

2 1

2 2 2 3

1

21

1 ,2

K

K K

K K

(6)

where 1IK , 2

IK , 3IK are the mode I SIFs at the nodes

located at =1, =0, =1, respectively. Mode II and III

Figure 6 An arbitrarily oriented 20-node enriched crack tip element.

SIFs are also defined in a similar manner. Thus, SIFs at crack tips can be calculated using the Python language (ABAQUS/CAE programming language) based on these matrices.

4 Crucial node determination

To investigate the important nodes at the crack tips, the SIFs distributions along the “z” direction at the crack tips when the rotation angle is 0° are shown in Figure 7 where it can be seen that the curves of SIFs show the typical para-bolic shapes. This means that the important nodes are lo-cated at the ends of the crack tips. Therefore, node 11 at tip 1, node 1 at tip 2 and node 1 at tip 3 are considered as the crucial nodes for studying the subsurface crack propagation.

5 Validation of numerical model

To validate the numerical model, the contact stress distribu-tion based on the numerical analysis is compared with that calculated from the Hertzian theory. From Figure 8, it can be seen that the contact stress distribution is in good agree-ment with the ideal distribution of the Hertzian theory. The contact half-width of the numerical analysis is 2.265 mm, and the ideal half-width is 2.481 mm calculated from the Hertzian theory. The half-width is different because it is not a point contact but a contact arc between the steel ball and the raceway surface. The maximum contact stress calculated from the numerical analysis is bigger than the ideal value from the Hertzian theory because there is an exterior tan-gency between the steel ball and the raceway surface, which results in the increase of the contact pressure with respect to

the point contact. The relative error ( 100%s h

s

) be-

tween the simulated result (s) and the ideal result (h) is approximately 9.8%.

The SIF calculations also are used to validate the numer-ical model. Liu et al. [20] presented a model in which an elliptical crack was located 6 mm below the tread face of the wheel and the SIFs history obtained from this model is shown in Figure 9(a). Figure 9(b) is given by this study. The elliptical crack is located 0.2 mm below the raceway surface. From Figure 9, it can be seen that modes II and III SIFs


presented by ref. [20] have the same variation rule as those derived from the current study, although the SIFs values are not equal. The difference can be attributed to the depth of the crack, which is significantly less here than that in ref. [20]. Therefore, the above analysis suggests that the numer-ical model in this paper is reliable.

6 Results and discussion

Based on this validated model, the SIFs are calculated using

Figure 7 SIFs distributions at crack tip along the “z” direction when the rotation angle is 0°: (a) Model I; (b) model II; (c) model III.

the matrixes in the previous section. The inner ring rotates within the range θ=±10° in the anticlockwise direction. The distributions of three subsurface cracks along the axial di-rection are varied to study their effects on RCF. Three dis-tinct conditions are detailed below.

Condition 1: Three subsurface cracks distribute along the “z” direction and one of them is located below the center of the Hertzian contact area, as shown in Figure 10, and their interval s is 0.3 mm. That is, the distribution of these cracks

Figure 8 Comparison of numerical calculation with Hertzian theory analysis.

Figure 9 SIFs history under rolling contact: (a) Derived from ref. [20]; (b) given by this study.


is dense below the Hertzian contact area. The other remain-ing parameters stated in subsection 3.1 remain unchanged.

Condition 2: To study the effect of the distribution den-sity of subsurface cracks on RCF, the interval is selected to be 0.85 mm, namely, cracks 1 and 2 are located below the contact area and crack 3 is near the contact area, as shown in Figure 10. The other remaining parameters in condition 1 remain unchanged.

Condition 3: To study the effect of the interval between subsurface cracks on RCF, the interval is defined to be 1.10 mm. That is, the number of cracks remains unchanged be-low the contact area with respect to condition 2 when the interval increases, as shown in Figure 10. The other re-maining parameters in condition 1 remain unchanged.

6.1 Influence of subsurface cracks distribution on SIF calculation under condition 1

Figure 11(a) presents the variation in mode I SIF at three crack tips under the dense subsurface crack distribution. It can be seen that mode I SIFs first decrease from 0 to the minimum values, and then increase to 0. The minimum values appear near the rotation angle of 0°. This is because that the downward load causes crack closures at these tips, i.e., the faces of these cracks trend to close when the ball rolls over the cracks, as shown in Figure 12(a). This means that mode I crack mechanism does not occur at these tips. Moreover, the decays of the values indicate that the down-ward load contributes less to the closing displacement at these tips as the Hertzian contact moves away from the subsurface cracks. Additionally, it is clearly noted that these minimum values are close but not equal to one another. Thus, the ranges of mode I SIF are close to one another at these tips, as shown in Figure 13(a). Therefore, mode I crack propagation mechanism does not occur at these tips and the cracks bear approximately equal downward loads when these subsurface cracks densely distribute along the axial direction.

The variation in mode II SIF under condition 1 is shown in Figure 11(b), where it can be seen that mode II SIFs ini-tially increase from 0 to the maximum values, and then de-crease to the minimum values when the ball rolls over the cracks. Further, mode II SIFs decay from the minimum values to 0 as the Hertzian contact goes away from the

Figure 10 Distribution of three subsurface cracks in the profile of bear-ing ring.

cracks. The maximum value means that the sliding displacement occurs between the faces of cracks, that is, mode II crack propagation occurs at these tips. The reason for this is that the tangential component of the downward load (TCDL) largely contributes to the sliding displacements at these tips when the Hertzian contact is to the left of the cracks, as shown in Figure 12(b). The minimum value indi-cates that an opposite sliding displacement occurs at the tips, that is, the sliding direction changes between the faces of cracks. This is because that TCDL contributes to the oppo-site sliding displacement when the Hertzian contact is to the right of the cracks, as shown in Figure 12(c). Thus, a re-peated sliding exists between the faces of cracks. Near 0°, the values of mode II SIFs are close to zero at these tips. This means that almost no sliding displacement occurs at these tips. The reason for this is that the downward load contributes little to the sliding displacement at the tips, i.e., the downward load located over the cracks, shown in Figure 12(a), only contributes to the closing displacement at the tips, and has little effect on the sliding displacement. More-over, it is clearly noted that the absolute values of the peaks are smaller than those of the valleys. The reason for this is that the downward load can significantly act on the crack tips when is about 1.0°, as shown in Figure 12(c), alt-hough the surface traction weakens the sliding displacement. Additionally, the maximum values are almost close to one another and the minimum values are nearly equal to one another. This suggests that the degrees of mode II crack damage are approximately equal at these tips because the degree of fatigue damage is governed by the range of SIF [27]. Therefore, mode II crack mechanism occurs at these tips and there is a repeated sliding between the faces of cracks during rolling of the ball. Moreover, the degrees of mode II crack damage are almost equal at these tips when these cracks densely distribute below the Hertzian contact zone.

The variation in mode III SIF is presented under condi-tion 1 in Figure 11(c), where mode III SIFs first increase from 0 to the maximum values and then decay to 0 during rolling of the ball. The maximum values appear near the rotation angle of 0°, which is because that the downward load induces a tear stress zone around the crack tips ac-cording to the elasticity theory. This means that mode III crack mechanism occurs at these tips. When the Hertzian contact moves away from the cracks, these tips are gone away from the tear stress zone, thus mode III SIFs are very close to 0. Moreover, it can be found that the maximum value at tip 1 is the largest and the one at tip 3 is the small-est. This indicates that the ranges of mode III SIFs are not equal at these tips, as shown in Figure 13(c), Therefore, mode III crack mechanism occurs at these tips and the dan-gerous mode III crack damage occurs below the center of the Hertzian contact zone when the distribution of the cracks is dense.


Figure 11 Variation in SIFs under condition 1: (a) Model I; (b) model II; (c) model III.

Figure 12 Schematic illustration of crack tip displacement: (a) =0°; (b) =1.0°; (c) =1.0°.

Figure 13 Ranges of SIFs under three different conditions: (a) KI; (b) KII; (c) KIII.

6.2 Influence of subsurface crack distribution on SIF calculation under condition 2

Figure 14(a) shows the variation in mode I SIFs under the sparse cracks distribution. From Figure 14(a), it can be seen that mode I SIFs first decrease from 0 to the minimum val-ues and then increase to 0 at tips 1 and 2 during rolling of the ball. However, mode I SIF has no obvious change at tip 3 relatively to that at tips 1 and 2. This is because that crack 3 is not below the Hertzian contact zone so that the down-ward load has little effect on mode I SIF calculation at tip 3. Moreover, it is clearly noted that the valley at tip 1 is larger than those at tips 2 and 3. This means that crack 1 under-goes a large downward load relatively to cracks 2 and 3 when the distribution of the subsurface cracks becomes sparse. Thus, the range of mode I SIF at tip 1 is larger than those at tips 2 and 3, as shown in Figure 13(a). By compar-

ing these ranges of mode I SIF under condition 2 with those under condition 1, it can be mentioned that the range in-creases at tip 1, and it reduces at tip 2 when the subsurface cracks sparsely distribute along the axial direction. Yet at tip 3, the decay of the range is attributed to the location of crack 3 outside of the Hertzian contact zone, not to the dis-tribution density of the cracks. Therefore, the crack below the center of the Hertzian contact zone undergoes larger downward loads than others when the distribution of the cracks is sparse.

Figure 14(b) shows the variation in mode II SIF under condition 2. From Figure 14(b), it can be seen that mode II SIF has an approximately sinusoidal variation rule at tips 1 and 2 during the rolling of the ball. But at tip 3, the peak and the valley are very small. Moreover, it is observed that the peak at tip 1 is larger than that at tip 2, and the valley at tip 1 is smaller than that at tip 2. This indicates that there is



a repeatedly sliding displacement at these tips during the rolling of the ball. But this couldn’t explain the degree of mode II crack damage at these tips. Thus, the ranges of mode II SIFs at the tips are shown in Figure 13(b). Under this condition, the range of mode II SIFs at tip 1 is almost equal to that at tip 2. This means that the degree of mode II crack damage at tip 1 is close to that at tip 2. At tip 3, the mode II crack damage is slight relatively to those at tips 1 and 2. Additionally, when comparing the ranges of mode II SIFs under condition 2 with those under condition 1, it is clearly noted that the range has no obvious change at tip 1, but it decreases at tips 2 and 3. This suggests that the mode II crack damage becomes slightly at these tips when the subsurface cracks sparsely distribute along the axial direc-tion. Therefore, the distribution density of the cracks im-portantly influences the mode II crack damage, namely, the denser the cracks distribute along the axial direction, the more serious the mode II crack damage at these tips, and vice versa.

The variation in mode III SIF under condition 2 is shown in Figure 14(c), whereIt mode III SIF has an approximately parabolic variation rule at these tips. Moreover, it is easily observed that the peak at tip 2 is the largest, and the one at tip 1 is approximately equal to that at tip 3. This means that the downward load induces a large tear stress zone around tip 2, and this tear stress zone covers tip 3. Thus, the tear damage occurs at tip 3 although crack 3 is outside of the Hertzian contact zone. The ranges of mode III SIFs are pre-sented in Figure 13(c) under this condition. By comparing the ranges of mode III SIF under condition 2 with those under condition 1, it is clearly indicated that the range de-creases at tip 1, but increases at tips 2 and 3. This illustrates that mode III crack damage becomes slightly at tip 1, and increases at tips 2 and 3, when the subsurface cracks sparsely distribute along the axial direction. Therefore, the distribution density of the cracks has a significant effect on the mode III crack damage.

6.3 Influence of subsurface cracks distribution on SIF calculation under condition 3

Figure 15(a) provides the variation in mode I SIFs under condition 3, where it can be seen that mode I SIF has the

same variation rule at tips 1 and 2 with that under condition 2. At tip 3, the curve of mode III SIF is approximately hor-izontal and the value is about 0 MPa (m)0.5. Moreover, it can be observed that the valley at tip 1 is larger than that at tip 2. This means that crack 1 undergoes a large downward load relatively to crack 2 when the interval is big between these cracks. Thus, the range of mode I SIF at tip 1 is larger than that at tip 2, as shown in Figure 13(a). By comparing the ranges of mode I SIF under condition 3 with those under condition 2, it can be found that the range of mode I SIF has no obvious change at tip 1 and it reduces at tip 2. Yet at tip 3, the decay of the range is attributed to that the influence of the normal stress zone on tip 3 decreases when the interval increases. Therefore, the interval significantly influences the ranges of mode I SIFs at these tips, namely, the bigger the interval between the subsurface cracks, the smaller the range of mode I SIF at these tips, and vice versa.

Figure 15(b) shows the variation in mode II SIFs under condition 3. From Figure 15(b), it can be seen that mode Ⅱ SIF has the same variation rule at tips 1 and 2 with that un-der condition 2. But at tip 3, the value of mode II SIF has no obvious change and it is close to zero. Moreover, it is clear-ly noted that the value of mode II SIF at tip 1 is larger than that at tip 2 during the rolling of the ball. This means that the sliding displacement occurs at these tips. But this couldn’t explain the degree of mode II crack damage at these tips. Thus, the ranges of mode II SIF are shown in Figure 13(b) under this condition, where it can be found that the range of mode II SIF at tip 1 is approximately equal to that at tip 2. Moreover, at tip 3, the mode II crack damage is small because crack 3 is outside of the Hertzian contact zone. Additionally, by comparing the ranges of mode II SIF under condition 3 with those under condition 2, it is clearly noted that the ranges of mode II SIF decrease with the in-creasing interval. Therefore, the interval plays an important role in mode II crack damage, that is, the bigger the interval, the smaller the mode II crack damage at these tips.

The variation in mode III SIF under condition 3 is shown in Figure 15(c). It can be seen from Figure 15(c) that mode III SIF has the same variation rule at tips 1 and 2 as that under condition 2. At tip 3, the value of mode III SIF is very small and close to 0. Moreover, by the comparison of these peaks, it is easily observed that the peak is the largest at tip



2, and the one at tip 3 is smaller than that at tip 1. This is because that the tear stress has little effect on mode III SIF calculation at tip 3 when the interval becomes big. The ranges of mode III SIF are presented in Figure 13(c) under this condition. By comparing the ranges of mode III SIF under condition 3 with those under condition 2, it is easily seen that the range has no obvious change at tip 1, but it decreases at tips 2 and 3. This means that the interval has little influence on mode III crack damage at tip 1 when it is big, but it plays an important role in mode III crack damage at tips 2 and 3. Therefore, it can be concluded that mode III crack damage decreases with the increasing interval at these tips, that is, the smaller the interval between the subsurface cracks, the more serious the mode III crack damage at these tips, and vice versa.

6.4 Influence of subsurface crack distribution on crack growth rate

According to the analysis above, it is obvious that a mixed SIF mode takes place at these tips during the rolling of the ball. That is, the crack propagation experiences a non-proportional cycle of mode I (closing), mode II (sliding) and mode III (tearing) displacement as the Hertzian contact passes, as shown in Figure 16. Therefore, it is necessary to use the effective stress intensity factor (ESIF) Keff to evalu-ate the RCF crack propagation. The equation for the ESIF is expressed as follows [28]:

0.5

2 2 .1

2III

eff I II

KK K K

(7)

According to eq. (7), the variation in ESIF at the tips can be obtained as shown in Figure 17, where the maximum ESIFs appear near the rotation angle of 0°. This means that mode I SIF has an indispensable effect on the crack propa-gation. The reason for this is that under the cyclic crack closure, the elastic strain energy is stored in a local metal zone around the crack tip. Once the crack propagates, the energy is released to accelerate the crack propagation [29]. Moreover, the range of ESIF keff can be calculated based on

n ,eff eff,max eff,mi∆ K K K (8)

where Keff, max is the maximum value of ESIF and Keff, min is the minimum value.

In the previous sections, the crack propagation modes were investigated at the tips under these conditions. How-ever, the crack growth rate remains unclear at the tips. Hence, it is necessary to study the crack growth under these

Figure 16 Variations in modes I, II and III SIFs at tip 1 under condition 2.

Figure 17 Variations in ESIF at crack tips under condition 2.


conditions. Paris et al. [27] postulated that the crack growth rate is governed by the SIF range. The equation for crack growth rate is presented as follows:

,eff

d

d

nLC K

N (9)

where L is the half-length of the crack, N is the number of loading cycles, C is the material constant, Keff is the ESIF range and n is the slope index. C and n are taken as 5.8×10-9 and 3, respectively for bearing steels [30].

According to eqs. (7)–(9), the crack growth rates are ob-tained under the three conditions, as shown in Table 1. From Table 1, it can be seen that under condition 1, the crack growth rate is large at tips 1, 2 and 3, and it is the largest at tip 2. This means that the crack can propagate easily when the cracks densely distribute below the raceway surface. Under condition 2, the crack growth rate at tip 2 is the largest. This suggests that the RCF crack can easily propagate at tip 2 so that the RCF damage is obviously ac-celerated. At tip 3, it is observed that the crack growth rate is very small because crack 3 is outside of the Hertzian contact region. By comparing the crack growth rates under condition 2 with those under condition 1, it is easily noted that the crack growth rates decrease at the tips when the subsurface cracks sparsely distribute. Therefore, it can be concluded that the sparser the subsurface cracks distribute, the slower the fatigue cracks propagate at these tips, and vice versa.

Under condition 3, the largest crack growth rate appears at tip 1, which is different from that under conditions 1 and 2. This indicates that the serious crack damage transfers to crack 1 when the interval becomes big. Moreover, by com-paring the crack growth rates between conditions 3 and 2, it is clear that the crack growth rates at these tips decrease with the increasing interval, that is, crack propagation slows down when the interval increases between the cracks. Therefore, it can be concluded that the interval between cracks also influences the RCF crack propagation, namely, the larger the interval is, the more difficult the crack propa-gates at these tips, and vice versa.

7 Conclusion

(1) The mode I crack mechanism does not occur at crack tips, but modes II and III mechanisms occur at the tips. That is, the shear stress mainly drives crack propagation in the subsurface of bearing raceway.

(2) The distribution density of the subsurface cracks has

Table 1 Crack growth rates under various conditions

Conditions Crack growth rate×10-4 ( mm/cycle )

Tip 1 Tip 2 Tip 3

Condition 1 298.8 376.1 298.2

Condition 2 253.3 305.2 4.2

Condition 3 241.3 176.8 0.09

an important influence on modes II and III crack damages. The denser the subsurface cracks distribute along the axial direction, the more serious the mode II crack damage is at these tips, and vice versa. Moreover, mode III crack damage slows down at tip 1, and increases at tips 2 and 3, when the subsurface cracks sparsely distribute along the axial direc-tion, and vice versa.

(3) The interval between cracks plays an important role in modes II and III fatigue damages. The bigger the interval is between the cracks, the smaller the modes II and III crack damages are at these tips, and vice versa.

(4) The crack growth rates are influenced significantly by the distribution density and the interval. The sparser the subsurface cracks distribute along the axial direction, the slower the fatigue cracks propagate at the tips, and vice versa. The larger the interval is, the more difficult the crack propagates at these tips, and vice versa.

This work was supported by the National Basic Research Program of Chi-na (Grant No. 2011CB706605), State Key Program of National Natural Science Foundation of China (Grant No. 51135007), Innovative Research Groups of the National Natural Science Foundation of Hubei Province (Grant No. 2011CDA12)and the Fundamental Research Funds for the Central Universities (Grant Nos. 2012-Ia-017, 2013-IV-014) for the sup-port given to this research.

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Subsurface crack propagation under rolling contact fatigue in bearing ring