# A Multiscale Overview of Modelling Rolling Cyclic Fatigue in Bearing Elements

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Background

## 2. Bearing Steel

## 3. Material Modelling

^{6}cycles). However, for the later life of RCF (i.e., 10

^{8}≤ N ≤ 10

^{10}cycles), the simulation results underpredicted the formation of residual stress, in which the bearing material experienced significant metallurgical changes under RCF [43]. It was stated that the later formation of residual stresses, accompanied by metallurgical change, depends upon the relative volume fraction of the decayed microstructure. However, no direct relationship of compressive residual stresses with the subsurface microstructural alterations was considered in detail. The underprediction of calculated residual stresses from finite element simulation leads to the argument that a more complex material model is required to compensate for the mechanical-metallurgical response associated with the RCF. Later, it was also reported [63] that the bearing material subjected to rolling cycles exhibits nonlinear kinematic hardening, which can be described by the Ramberg–Osgood model. A later study [10] presented J2 plasticity-based finite element simulations incorporating linear kinematic hardening and nonlinear kinematic hardening, consecutively. Nevertheless, the cyclic behaviour of a hardened bearing under RCF is complex and challenging to define with simplistic isotropic hardening [64] and kinematic hardening [65,66,67].

_{y}’ is the instantaneous yield strength. The depth distribution of the change in hardness of the evolved microstructure of the case-hardened bearing steel was integrated into the NIKH material model. Figure 7 shows the ratcheting response of bearing steel subjected to uniaxial loading in a stress-controlled environment. It can be seen that equivalent or von Mises stresses indicate substantial work hardening, with continuous accumulation of PEEQ. Simulation results showed that the cyclic hardening via ratcheting is promoted at the scale of the carbide microstructure. The hard spheroidised carbide particles act as local stress increasers and are the primary driver for the localised subsurface hardening. A similar approach was extended in a more recent research work [75] in order to simulate the deep zone residual stresses in an evolved bearing steel microstructure. The highly localised subsurface hardness change was incorporated into the Tabor rule [76] to convert the nanoindentation hardness to equivalent flow stresses. The evolved flow stresses were then fed into the newly developed 3D finite model in order to mimic the cyclic hardening response of the evolved bearing steel microstructure.

## 4. Microstructural Modelling

_{3}C, whereas the initial carbides remain unchanged. APT results have revealed that three forms of carbides coexist in the DER zone, e.g., θ-Fe

_{3}C, η-Fe

_{2}C, and ε-Fe

_{2.4}C [102]. Moreover, the lenticular carbides formed during WEBs have relatively less carbon content, as compared to θ-Fe

_{3}C [105]. Following extensive experimental investigations, a semi-empirical model for predicting WEBs [106] was presented based on the saturation density of LABs and HABs under a range of contact pressures and stress cycles. Based on the growth pattern of WEBs, it was suggested that the WEBs formation is mainly driven by the diffusion process. In a recent mechanistic study [107], it was reported that both LABs and HABs arise due to recrystallisation from energy build-up in the initial microstructure, which later transforms to the elongated ferrite grains via the grain rotation/coalescence recovery mechanism owing to subsurface plasticity. Mustafa et al. [17] suggested that HABs are formed in the densely formed areas of LABs. However, at elevated temperatures and loads, HABs prior to LABs have also been reported [88,94]. The early formation of HABs suggests that the reversal of the sequence of WEBs (e.g., HABs before LABs) does not depend upon applied load, but rather the temperature-load combination.

## 5. Overview and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kang, J.-H.; Hosseinkhani, B.; Rivera-Díaz-del-Castillo, P.E. Rolling contact fatigue in bearings: Multiscale overview. Mater. Sci. Technol.
**2012**, 28, 44–49. [Google Scholar] [CrossRef] - El Laithy, M.; Wang, L.; Harvey, T.J.; Vierneusel, B.; Correns, M.; Blass, T.J.T.I. Further understanding of rolling contact fatigue in rolling element bearings—A review. Tribol. Int.
**2019**, 140, 105849. [Google Scholar] [CrossRef] - Voskamp, A.; Österlund, R.; Becker, P.; Vingsbo, O. Gradual changes in residual stress and microstructure during contact fatigue in ball bearings. Met. Technol.
**1980**, 7, 14–21. [Google Scholar] [CrossRef] - Lundberg, G.; Palmgren, A. Dynamic Capacity of Rolling Bearings. J. Appl. Mech.
**1949**, 16, 165–172. [Google Scholar] [CrossRef] - Khurram, M.; Mufti, R.A.; Bhutta, M.U.; Afzal, N.; Abdullah, M.U.; Rahman, S.U.; Rehman, S.U.; Zahid, R.; Mahmood, K.; Ashfaq, M.; et al. Roller sliding in engine valve train: Effect of oil film thickness considering lubricant composition. Tribol. Int.
**2020**, 149, 105829. [Google Scholar] [CrossRef] - Abdullah, M.U.; Shah, S.R.; Bhutta, M.U.; Mufti, R.A.; Khurram, M.; Najeeb, M.H.; Arshad, W.; Ogawa, K. Benefits of wonder process craft on engine valve train performance. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2018**, 233, 1125–1135. [Google Scholar] [CrossRef] - Klecka, M.A.; Subhash, G.; Arakere, N.K. Microstructure–property relationships in M50-NiL and P675 case-hardened bearing steels. Tribol. Trans.
**2013**, 56, 1046–1059. [Google Scholar] [CrossRef] - Merwin, J.; Johnson, K. An analysis of plastic deformation in rolling contact. Proc. Inst. Mech. Eng.
**1963**, 177, 676–690. [Google Scholar] - Ruud, C. A review of selected non-destructive methods for residual stress measurement. NDT Int.
**1982**, 15, 15–23. [Google Scholar] [CrossRef] - Warhadpande, A.; Sadeghi, F.; Evans, R.D.; Kotzalas, M.N. Influence of plasticity-induced residual stresses on rolling contact fatigue. Tribol. Trans.
**2012**, 55, 422–437. [Google Scholar] [CrossRef] - Khan, Z.A.; Hadfield, M.; Tobe, S.; Wang, Y. Residual stress variations during rolling contact fatigue of refrigerant lubricated silicon nitride bearing elements. Ceram. Int.
**2006**, 32, 751–754. [Google Scholar] [CrossRef] - Shen, Y.; Moghadam, S.M.; Sadeghi, F.; Paulson, K.; Trice, R.W. Effect of retained austenite—Compressive residual stresses on rolling contact fatigue life of carburized AISI 8620 steel. Int. J. Fatigue
**2015**, 75, 135–144. [Google Scholar] [CrossRef] - Nazir, M.H.; Khan, Z.A.; Saeed, A. Experimental analysis and modelling of c-crack propagation in silicon nitride ball bearing element under rolling contact fatigue. Tribol. Int.
**2018**, 126, 386–401. [Google Scholar] [CrossRef] - Bush, J.; Grube, W.; Robinson, G. Microstructural and residual stress changes in hardened steel due to rolling contact. Trans. ASM
**1961**, 54, 390–412. [Google Scholar] - Polonsky, I.; Keer, L. On white etching band formation in rolling bearings. J. Mech. Phys. Solids
**1995**, 43, 637–669. [Google Scholar] [CrossRef] - Šmeļova, V.; Schwedt, A.; Wang, L.; Holweger, W.; Mayer, J. Electron microscopy investigations of microstructural alterations due to classical Rolling Contact Fatigue (RCF) in martensitic AISI 52100 bearing steel. Int. J. Fatigue
**2017**, 98, 142–154. [Google Scholar] [CrossRef] - El Laithy, M.; Wang, L.; Harvey, T.J.; Vierneusel, B. Re-investigation of dark etching regions and white etching bands in SAE 52100 bearing steel due to rolling contact fatigue. Int. J. Fatigue
**2020**, 136, 105591. [Google Scholar] [CrossRef] - Kanetani, K.; Ushioda, K. Mechanism of White Band (WB) Formation due to Rolling Contact Fatigue in Carburized SAE4320 Steel. Mater. Transations
**2020**, 61, 1750–1759. [Google Scholar] [CrossRef] - Su, Y.-S.; Li, S.-X.; Yu, F.; Lu, S.-Y.; Wang, Y.-G.J.I.J.o.F. Revealing the shear band origin of white etching area in rolling contact fatigue of bearing steel. Int. J. Fatigue
**2021**, 142, 105929. [Google Scholar] [CrossRef] - Arakere, N.K. Gigacycle rolling contact fatigue of bearing steels: A review. Int. J. Fatigue
**2016**, 93, 238–249. [Google Scholar] [CrossRef] - Yin, H.; Wu, Y.; Liu, D.; Zhang, P.; Zhang, G.; Fu, H.J.M. Rolling Contact Fatigue-Related Microstructural Alterations in Bearing Steels: A Brief Review. Metals
**2022**, 12, 910. [Google Scholar] [CrossRef] - Voskamp, A. Material response to rolling contact loading. ASME Trans. J. Tribol.
**1985**, 107, 359–364. [Google Scholar] [CrossRef] - Morris, D.; Sadeghi, F.; Chen, Y.-C.; Wang, C.; Wang, B. Effect of Residual Stresses on Microstructural Evolution Due to Rolling Contact Fatigue. J. Tribol.
**2018**, 140, 061402. [Google Scholar] [CrossRef] - Cherolis, N.E.; Benac, D.J.; Pridemore, W.D. Fatigue in Rotating Equipment: Is it HCF or LCF? J. Fail. Anal. Prev.
**2016**, 16, 828–841. [Google Scholar] [CrossRef] - Beswick, J.M. Bearing Steel Technology: Advances and State of the Art in Bearing Steel Quality Assurance: 7th Volume; ASTM International: West Conshohocken, PA, USA, 2007. [Google Scholar]
- Bhadeshia, H.; Honeycombe, R. Steels: Microstructure and Properties; Butterworth-Heinemann: Oxford, UK, 2017. [Google Scholar]
- Barrow, A.; Rivera-Díaz-del-Castillo, P. Nanoprecipitation in bearing steels. Acta Mater.
**2011**, 59, 7155–7167. [Google Scholar] [CrossRef] - Song, W.; Choi, P.-P.; Inden, G.; Prahl, U.; Raabe, D.; Bleck, W. On the spheroidized carbide dissolution and elemental partitioning in high carbon bearing steel 100Cr6. Metall. Mater. Trans. A
**2014**, 45, 595–606. [Google Scholar] [CrossRef] - Abdullah, M.U. Finite Element Modelling of Deep Zone Residual Stresses in Rolling Contact Bearing Elements; Bournemouth University: Poole, UK, 2022. [Google Scholar]
- Radhakrishnan, V.; Ramanathan, S. Plastic deformation in rolling contact. Wear
**1975**, 32, 211–221. [Google Scholar] [CrossRef] - Lorösch, H.-K. Influence of Load on the Magnitude of the Life Exponent for Rolling Bearings, in Rolling Contact Fatigue Testing of Bearing Steels; ASTM International: West Conshohocken, PA, USA, 1982. [Google Scholar]
- Bhattacharyya, A.; Pandkar, A.; Subhash, G.; Arakere, N. Cyclic constitutive response and effective S–N diagram of M50 NiL case-hardened bearing steel subjected to Rolling Contact Fatigue. J. Tribol.
**2015**, 137, 041102. [Google Scholar] [CrossRef] - Turteltaub, S.; Suiker, A. Transformation-induced plasticity in ferrous alloys. J. Mech. Phys. Solids
**2005**, 53, 1747–1788. [Google Scholar] [CrossRef] - Johnson, K. The strength of surfaces in rolling contact. Proc. Inst. Mech. Eng. Part C Mech. Eng. Sci.
**1989**, 203, 151–163. [Google Scholar] [CrossRef] - Bhattacharyya, A.; Subhash, G.; Arakere, N. Evolution of subsurface plastic zone due to rolling contact fatigue of M-50 NiL case hardened bearing steel. Int. J. Fatigue
**2014**, 59, 102–113. [Google Scholar] [CrossRef] - Zaretsky, E.V. Rolling bearing steels—A technical and historical perspective. Mater. Sci. Technol.
**2012**, 28, 58–69. [Google Scholar] [CrossRef][Green Version] - Chaboche, J.L. A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast.
**2008**, 24, 1642–1693. [Google Scholar] [CrossRef] - Jhansale, H.; Topper, T. Engineering Analysis of the Inelastic stress Response of a Structural Metal under Variable Cyclic Strains, in Cyclic Stress-Strain Behavior—Analysis, Experimentation, and Failure Prediction; ASTM International: West Conshohocken, PA, USA, 1971. [Google Scholar]
- Österlund, R.; Vingsbo, O. Phase changes in fatigued ball bearings. Metall. Mater. Trans. A
**1980**, 11, 701–707. [Google Scholar] [CrossRef] - Sadeghi, F.; Jalalahmadi, B.; Slack, T.S.; Raje, N.; Arakere, N.K. A review of rolling contact fatigue. J. Tribol.
**2009**, 131, 041403. [Google Scholar] [CrossRef] - Arakere, N.; Subhash, G. Work hardening response of M50-NiL case hardened bearing steel during shakedown in rolling contact fatigue. Mater. Sci. Technol.
**2012**, 28, 34–38. [Google Scholar] [CrossRef] - Jhansale, H. A new parameter for the hysteretic stress-strain behavior of metals. ASME J. Eng. Mater. Technol.
**1975**, 97, 33–38. [Google Scholar] [CrossRef] - Hahn, G.; Bhargava, V.; Rubin, C.; Chen, Q.; Kim, K. Analysis of the rolling contact residual stresses and cyclic plastic deformation of SAE 52100 steel ball bearings. J. Tribol.
**1987**, 109, 618–626. [Google Scholar] [CrossRef] - Swahn, H.; Becker, P.; Vingsbo, O. Martensite decay during rolling contact fatigue in ball bearings. Metall. Mater. Trans. A
**1976**, 7, 1099–1110. [Google Scholar] [CrossRef] - Hahn, G.T.; Bhargava, V.; Chen, Q. The cyclic stress-strain properties, hysteresis loop shape, and kinematic hardening of two high-strength bearing steels. Metall. Trans. A
**1990**, 21, 653. [Google Scholar] [CrossRef] - Christ, H.J.; Sommer, C.; Mughrabi, H.; Voskamp, A.; Beswick, J.; Hengerer, F. Fatigue behaviour of three variants of the roller bearing steel SAE 52100. Fatigue Fract. Eng. Mater. Struct.
**1992**, 15, 855–870. [Google Scholar] [CrossRef] - Kang, J.-H.; Hosseinkhani, B.; Vegter, R.H.; Rivera-Díaz-del-Castillo, P.E. Modelling dislocation assisted tempering during rolling contact fatigue in bearing steels. Int. J. Fatigue
**2015**, 75, 115–125. [Google Scholar] [CrossRef] - Fu, H.; Galindo-Nava, E.; Rivera-Díaz-del-Castillo, P. Modelling and characterisation of stress-induced carbide precipitation in bearing steels under rolling contact fatigue. Acta Mater.
**2017**, 128, 176–187. [Google Scholar] [CrossRef] - Warhadpande, A.; Sadeghi, F.; Evans, R.D. Microstructural alterations in bearing steels under rolling contact fatigue part 1—Historical overview. Tribol. Trans.
**2013**, 56, 349–358. [Google Scholar] [CrossRef] - Hamilton, G. Plastic flow in rollers loaded above the yield point. Proc. Inst. Mech. Eng.
**1963**, 177, 667–675. [Google Scholar] - Bhargava, V.; Hahn, G.; Rubin, C. An elastic-plastic finite element model of rolling contact, Part 1: Analysis of single contacts. J. Appl. Mech.
**1985**, 52, 67–74. [Google Scholar] [CrossRef] - Crook, A. Simulated gear-tooth contacts: Some experiments upon their lubrication and subsurface deformations. Proc. Inst. Mech. Eng.
**1957**, 171, 187–214. [Google Scholar] [CrossRef] - Nazir, M.H.; Khan, Z.A.; Saeed, A.; Siddaiah, A.; Menezes, P.L. Synergistic wear-corrosion analysis and modelling of nanocomposite coatings. Tribol. Int.
**2018**, 121, 30–44. [Google Scholar] [CrossRef] - Howell, M.; Hahn, G.; Rubin, C.; McDowell, D. Finite element analysis of rolling contact for nonlinear kinematic hardening bearing steel. ASME J. Tribol.
**1995**, 117, 729–736. [Google Scholar] [CrossRef] - Bower, A.; Johnson, K. The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact. J. Mech. Phys. Solids
**1989**, 37, 471–493. [Google Scholar] [CrossRef] - Jacq, C.; Nélias, D.; Lormand, G.; Girodin, D. Development of a three-dimensional semi-analytical elastic-plastic contact code. J. Trib.
**2002**, 124, 653–667. [Google Scholar] [CrossRef] - Chaise, T.; Nélias, D. Contact pressure and residual strain in 3D elasto-plastic rolling contact for a circular or elliptical point contact. J. Tribol.
**2011**, 133, 041402. [Google Scholar] [CrossRef] - Chen, W.W.; Wang, Q.J.; Wang, F.; Keer, L.M.; Cao, J. Three-dimensional repeated elasto-plastic point contacts, rolling, and sliding. J. Appl. Mech.
**2008**, 75, 021021. [Google Scholar] [CrossRef] - Walvekar, A.A.; Sadeghi, F. Rolling contact fatigue of case carburized steels. Int. J. Fatigue
**2017**, 95, 264–281. [Google Scholar] [CrossRef] - Golmohammadi, Z.; Walvekar, A.; Sadeghi, F. A 3D efficient finite element model to simulate rolling contact fatigue under high loading conditions. Tribol. Int.
**2018**, 126, 258–269. [Google Scholar] [CrossRef] - Bomidi, J.A.R.; Sadeghi, F. Three-Dimensional Finite Element Elastic–Plastic Model for Subsurface Initiated Spalling in Rolling Contacts. J. Tribol.
**2013**, 136, 011402. [Google Scholar] [CrossRef] - Warhadpande, A.; Sadeghi, F.; Kotzalas, M.N.; Doll, G. Effects of plasticity on subsurface initiated spalling in rolling contact fatigue. Int. J. Fatigue
**2012**, 36, 80–95. [Google Scholar] [CrossRef] - Popescu, G.; Morales-Espejel, G.E.; Wemekamp, B.; Gabelli, A. An engineering model for three-dimensional elastic-plastic rolling contact analyses. Tribol. Trans.
**2006**, 49, 387–399. [Google Scholar] [CrossRef] - Hill, R. The Mathematical Theory of Plasticity; Oxford University Press: Oxford, UK, 1998; Volume 11. [Google Scholar]
- Prager, W. A new methods of analyzing stresses and strains in work hardening plastic solids. J. Appl. Mech. (ASME)
**1956**, 23, 493–496. [Google Scholar] [CrossRef] - McDowell, D. A two surface model for transient nonproportional cyclic plasticity, Part 1: Development of appropriate equations. J. Appl. Mech. (ASME)
**1985**, 52, 298–302. [Google Scholar] [CrossRef] - Mroz, Z. On the Theory of Steady Plastic Cycles in Structures; Institute of Fundamental Problems of Technology: Warsaw, Poland, 1971. [Google Scholar]
- Pandkar, A.S.; Arakere, N.; Subhash, G. Microstructure-sensitive accumulation of plastic strain due to ratcheting in bearing steels subject to Rolling Contact Fatigue. Int. J. Fatigue
**2014**, 63, 191–202. [Google Scholar] [CrossRef] - Chaboche, J.-L. Time-independent constitutive theories for cyclic plasticity. Int. J. Plast.
**1986**, 2, 149–188. [Google Scholar] [CrossRef] - Kabo, E.; Ekberg, A. Fatigue initiation in railway wheels—A numerical study of the influence of defects. Wear
**2002**, 253, 26–34. [Google Scholar] [CrossRef] - Taraf, M.; Zahaf, E.; Oussouaddi, O.; Zeghloul, A. Numerical analysis for predicting the rolling contact fatigue crack initiation in a railway wheel steel. Tribol. Int.
**2010**, 43, 585–593. [Google Scholar] [CrossRef] - Portier, L.; Calloch, S.; Marquis, D.; Geyer, P. Ratchetting under tension–torsion loadings: Experiments and modelling. Int. J. Plast.
**2000**, 16, 303–335. [Google Scholar] [CrossRef] - Pandkar, A.S.; Arakere, N.; Subhash, G. Ratcheting-based microstructure-sensitive modeling of the cyclic hardening response of case-hardened bearing steels subject to Rolling Contact Fatigue. Int. J. Fatigue
**2015**, 73, 119–131. [Google Scholar] [CrossRef] - Bomidi, J.A.R.; Weinzapfel, N.; Slack, T.; Mobasher Moghaddam, S.; Sadeghi, F.; Liebel, A.; Weber, J.; Kreis, T. Experimental and Numerical Investigation of Torsion Fatigue of Bearing Steel. J. Tribol.
**2013**, 135, 031103. [Google Scholar] [CrossRef] - Abdullah, M.U.; Khan, Z.A.; Kruhoeffer, W.; Blass, T. A 3D Finite Element Model of Rolling Contact Fatigue for Evolved Material Response and Residual Stress Estimation. Tribol. Lett.
**2020**, 68, 122. [Google Scholar] [CrossRef] - Tabor, D. Indentation Hardness and Its Measurement: Some Cautionary Comments, in Microindentation Techniques in Materials Science and Engineering; ASTM International: West Conshohocken, PA, USA, 1985. [Google Scholar]
- Kanetani, K.; Mikami, T.; Ushioda, K. Effect of Retained Austenite on Sub-surface Initiated Spalling during Rolling Contact Fatigue in Carburized SAE4320 Steel. ISIJ Int.
**2020**, 60, 1774–1783. [Google Scholar] [CrossRef] - Jiang, Y.; Sehitoglu, H. Modeling of cyclic ratchetting plasticity, part I: Development of constitutive relations. J. Appl. Mech.
**1996**, 63, 720–725. [Google Scholar] [CrossRef] - Walvekar, A.A.; Morris, D.; Golmohammadi, Z.; Sadeghi, F.; Correns, M. A Novel Modeling Approach to Simulate Rolling Contact Fatigue and Three-Dimensional Spalls. J. Tribol.
**2018**, 140, 031101. [Google Scholar] [CrossRef] - Preparata, F.P.; Shamos, M.I. Computational Geometry: An Introduction; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Jalalahmadi, B.; Sadeghi, F. A Voronoi Finite Element Study of Fatigue Life Scatter in Rolling Contacts. J. Tribol.
**2009**, 131, 022203. [Google Scholar] [CrossRef] - Raje, N.; Sadeghi, F.; Rateick, J.R.G.; Hoeprich, M.R. A Numerical Model for Life Scatter in Rolling Element Bearings. J. Tribol.
**2007**, 130, 011011. [Google Scholar] [CrossRef] - Weinzapfel, N.; Sadeghi, F.; Bakolas, V. An approach for modeling material grain structure in investigations of Hertzian subsurface stresses and rolling contact fatigue. J. Tribol.
**2010**, 132, 041404. [Google Scholar] [CrossRef] - Weinzapfel, N.; Sadeghi, F.; Bakolas, V.; Liebel, A. A 3D Finite Element Study of Fatigue Life Dispersion in Rolling Line Contacts. J. Tribol.
**2011**, 133, 042202. [Google Scholar] [CrossRef] - Shen, F.; Zhou, K. An elasto-plastic-damage model for initiation and propagation of spalling in rolling bearings. Int. J. Mech. Sci.
**2019**, 161, 105058. [Google Scholar] [CrossRef] - Martin, J.; Borgese, S.; Eberhardt, A. Microstructural alterations of rolling—Bearing steel undergoing cyclic stressing. ASME J. Basic Eng.
**1966**, 88, 555–565. [Google Scholar] [CrossRef] - Lund, T. Structural alterations in fatigue-tested ball- bearing steel. Jernkontorets Ann.
**1969**, 153, 337–343. [Google Scholar] - Mitamura, N.; Hidaka, H.; Takaki, S. Microstructural development in bearing steel during rolling contact fatigue. Mater. Sci. Forum
**2007**, 539–543, 4255–4260. [Google Scholar] [CrossRef] - Warhadpande, A.; Sadeghi, F.; Evans, R.D. Microstructural Alterations in Bearing Steels under Rolling Contact Fatigue: Part 2—Diffusion-Based Modeling Approach. Tribol. Trans.
**2014**, 57, 66–76. [Google Scholar] [CrossRef] - Johnson, K. Formation of Shear Bands in Ball-Bearing Races; University of Cambridge, Department of Engineering: Cambridge, UK, 1988. [Google Scholar]
- Bhargava, V.; Hahn, G.; Rubin, C. Rolling contact deformation, etching effects, and failure of high-strength bearing steel. Met. Mater Trans A
**1990**, 21, 1921–1931. [Google Scholar] [CrossRef] - Zwirlein, O.; Schlicht, H. Werkstoffanstrengung bei Wälzbeanspruchung–Einfluß von Reibung und Eigenspannungen. Mater. Und Werkst.
**1980**, 11, 1–14. [Google Scholar] [CrossRef] - Zwirlein, O.; Schlicht, H. Rolling Contact Fatigue Mechanisms—Accelerated Testing versus Field Performance, in Rolling Contact Fatigue Testing of Bearing Steels; ASTM International: West Conshohocken, PA, USA, 1982. [Google Scholar]
- Abdullah, M.U.; Khan, Z.A.; Kruhoeffer, W.; Blass, T.; Vierneusel, B. Development of white etching bands under accelerated rolling contact fatigue. Tribol. Int.
**2021**, 164, 107240. [Google Scholar] [CrossRef] - Buchwald, J.; Heckel, R. An analysis of microstructural changes in 52100 steel bearings during cyclic stressing(Microstructural changes in 52100 steel bearing inner rings during cyclic stressing, obtaining thickening rate data on white-etching regions and lenticular carbides). ASM Trans. Q.
**1968**, 61, 750–756. [Google Scholar] - Mirza, M.; Sellars, C.; Karhausen, K.; Evans, P. Multipass rolling of aluminium alloys: Finite element simulations and microstructural evolution. Mater. Sci. Technol.
**2001**, 17, 874–879. [Google Scholar] [CrossRef] - Lindahl, E.; Österlund, R. 212 transmission electron microscopy applied to phase transformations in ball bearings. Ultramicroscopy
**1982**, 9, 355–364. [Google Scholar] [CrossRef] - Pernach, M.; Pietrzyk, M. Numerical solution of the diffusion equation with moving boundary applied to modelling of the austenite–ferrite phase transformation. Comput. Mater. Sci.
**2008**, 44, 783–791. [Google Scholar] [CrossRef] - Zhao, L. Modeling of oxygen diffusion along grain boundaries in a nickel-based superalloy. J. Eng. Mater. Technol.
**2011**, 133, 031002. [Google Scholar] [CrossRef] - Fu, H.; Rivera-Díaz-del-Castillo, P. Evolution of White Etching Bands in 100Cr6 Bearing Steel under Rolling Contact-Fatigue. Metals
**2019**, 9, 491. [Google Scholar] [CrossRef] - Cottrell, A.H.; Bilby, B. Dislocation theory of yielding and strain ageing of iron. Proc. Phys. Society. Sect. A
**1949**, 62, 49. [Google Scholar] [CrossRef] - Fu, H.; Song, W.; Galindo-Nava, E.I.; Rivera-Díaz-del-Castillo, P.E. Strain-induced martensite decay in bearing steels under rolling contact fatigue: Modelling and atomic-scale characterisation. Acta Mater.
**2017**, 139, 163–173. [Google Scholar] [CrossRef] - Fu, H.; Rivera-Díaz-del-Castillo, P.E. A unified theory for microstructural alterations in bearing steels under rolling contact fatigue. Acta Mater.
**2018**, 155, 43–55. [Google Scholar] [CrossRef] - Abdullah, M.U.; Khan, Z.A.; Kruhoeffer, W. Evaluation of Dark Etching Regions for Standard Bearing Steel under Accelerated Rolling Contact Fatigue. Tribol. Int.
**2020**, 152, 106579. [Google Scholar] [CrossRef] - Fu, H. Microstructural Alterations in Bearing Steels under Rolling Contact Fatigue; University of Cambridge: Cambridge, UK, 2017. [Google Scholar]
- El Laithy, M.; Wang, L.; Harvey, T.J.; Vierneusel, B. Semi-empirical model for predicting LAB and HAB formation in bearing steels. Int. J. Fatigue
**2021**, 148, 106230. [Google Scholar] [CrossRef] - El Laithy, M.; Wang, L.; Harvey, T.J.; Schwedt, A.; Vierneusel, B.; Mayer, J. White etching bands formation mechanisms due to rolling contact fatigue. Acta Mater.
**2022**, 232, 117932. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematic of a deep groove ball bearing subjected to axial and radial load; (

**b**) axial and circumferential cross-section of the inner ring is demonstrated [2].

**Figure 2.**(

**a**) Microstructure of martensitic bearing steel; (

**b**) the needle-like structure containing primary θ carbides, adapted from [29].

**Figure 3.**Schematic of the cyclic hardening response of bearing steel subjected to RCF [35].

**Figure 4.**The centreline hardness is shown as a function of depth. Hardening takes place at a depth of nearly 450 µm. A softened region can also be observed at 500–700 µm depth. Maximum hardening can be observed after 246 million RCF cycles [35].

**Figure 5.**(

**a**) Zone of plastic action in the steady state. (

**b**) The steady cycle of plastic shear strain, indicating forwarding and backward shearing [8].

**Figure 6.**(

**a**) The elastic linear kinematic plastic (ELKP) stress–strain curve acquired from the torsional testing of bearing steel; (

**b**) the shear cyclic stress–strain hysteresis loop generated by multiple rolling cycles [62].

**Figure 7.**(

**a**) Stress-controlled loading for ratcheting simulation employing the NIKH material model; (

**b**) cyclic hardening with continuous accumulation of plastic strains [73].

**Figure 9.**Activation energies of rolling contact fatigue life originated from microstructural alterations at different applied loads (adapted from [88]).

**Figure 10.**(

**a**–

**d**) Maximisation of relative normal stress and relative shear stress across various orientations (angle θ) of WEBs; (

**e**) represents schematic of WEBs orientation with rolling direction (RD) (adapted from [29]).

**Figure 11.**(

**a**) Contour plot for dissipated plastic energy per unit volume; (

**b**) carbon centration plot as the load moves from left to right in a 2D domain [89].

**Figure 12.**(

**a**,

**c**) Schematic of carbide thickening within DERs and WEBs; (

**b**) experimental evidence for carbon redistribution and precipitation; (

**d**) formation of lenticular carbides [103].

**Figure 13.**Experimental and numerical prediction of DERs formation at (

**a**) 4 GPa and (

**b**) 5 GPa contact stress. Adapted from [29].

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abdullah, M.U.; Khan, Z.A. A Multiscale Overview of Modelling Rolling Cyclic Fatigue in Bearing Elements. *Materials* **2022**, *15*, 5885.
https://doi.org/10.3390/ma15175885

**AMA Style**

Abdullah MU, Khan ZA. A Multiscale Overview of Modelling Rolling Cyclic Fatigue in Bearing Elements. *Materials*. 2022; 15(17):5885.
https://doi.org/10.3390/ma15175885

**Chicago/Turabian Style**

Abdullah, Muhammad U., and Zulfiqar A. Khan. 2022. "A Multiscale Overview of Modelling Rolling Cyclic Fatigue in Bearing Elements" *Materials* 15, no. 17: 5885.
https://doi.org/10.3390/ma15175885