Abstract
Predicting rolling bearing fatigue life requires knowledge of the three-dimensional (3D) stress fields in the roller and raceway near the lubricated contact. Owing to the increasingly severe operating conditions, the effect of localized features such as surface roughness, subsurface inclusions, and even the crystallographic structure of the material becomes important. Achieving such detail requires (locally) extremely dense gridding in simulations, which in 3D is a major challenge. Multigrid techniques have been demonstrated to be capable of solving such problems. In this study, multigrid techniques are shown to further increase the efficiency of the solution by exploiting local grid refinement while maintaining the simplicity of a uniform discretization. This is achieved by employing increasingly finer grids only locally, where the highest resolution is required. Results are presented for dry contact and elastohydrodynamically lubricated contact cases, circular as well as elliptic, with varying crystallographic structure, and with surface roughness. The results show that the developed algorithm is very well suited for detailed analysis, with also excellent prospects for computational diagnostics involving actual material crystallographic structure from electron backscatter diffraction measurements.
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Abbreviations
- a :
-
Half width of Hertzian contact in x direction (m)
- b :
-
Half width of Hertzian contact in y direction (m)
- c 11, c 12, c 44 :
-
Elastic constants for cubic material (Pa)
- C i,j,k,l :
-
The elastic stiffness matrix (Pa), i, j, k, l are the indexes of the fouth-rank tensor
- d :
-
The dimension of the problem
- E :
-
Young’s modulus (Pa)
- F :
-
Applied load (N)
- h x, y, z :
-
Grid mesh
- h :
-
Film thickness (m)
- h f :
-
Fine grid
- H c :
-
Coarse grid
- H :
-
Dimensionless film thickness, h/(a2/Rx)
- H af :
-
Anisotropy factor (Pa)
- L, M :
-
Moes dimensionless parameters for point \(L = \alpha E{\left({{{{\eta _0}{u_{\rm{m}}}} \over {E{R_x}}}} \right)^{1/4}},\,\,M = {F \over {E{R^2}}}{\left({{{{\eta _0}{u_{\rm{m}}}} \over {E{R_x}}}} \right)^{- 3/4}}\)
- p :
-
Pressure (Pa)
- p H :
-
Maximum Hertzian pressure (Pa)
- P :
-
Dimensionless pressure, p/pH
- R x, R y :
-
The radius of a sphere in x and y directions (m)
- S 11, S 12, S 44 :
-
Elastic compliance constants for cubic material (Pa−1)
- u m :
-
Entrainment velocity (m/s)
- u, v, w :
-
Displacements in x, y, and z directions (m)
- W :
-
Displacement in z direction of the top surface, w/a
- U EHL, W EHL :
-
Hamrock and Dowson dimensionless parameters for point contacts
- GEHL :
-
\({U_{{\rm{EHL}}}} = {{{\eta _0}{u_{\rm{m}}}} \over {{E^\prime}{R^\prime}}},\,\,{W_{{\rm{EHL}}}} = {F \over {{E^\prime}{R^2}}},\,\,{G_{{\rm{EHL}}}} = {\alpha ^\prime}{E^\prime}\)
- z 0 :
-
Pressure viscosity index
- X, Y, Z :
-
Dimensionless coordinate system, x/a, y/a, z/a
- α :
-
Pressure viscosity coefficient (Pa−1)
- α 2 ,β, γ :
-
Euler rotation angle (rad)
- \(\overline \lambda \) :
-
Dimensionless parameter in EHL for point contacts, \(\overline \lambda = {\left({{{128{\pi ^3}} \over {3{M^4}}}} \right)^{1/3}}\)
- λ v ,μ v :
-
Lame’s constants (Pa)
- η 0 :
-
Atmospheric lubricant viscosity (Pa·s)
- σ ij :
-
Stiffness tensor (Pa)
- \(II_{{H_{\rm{c}}}}^{{h_{\rm{f}}}}\) :
-
High order interpolation operator
References
Echeverri Restrepo S, Ooi S W, Yan P, Andric P, Vegter R H, Lai J. Dark etching regions under rolling contact fatigue: A review. Mater Sci Technol 37(4): 347–376 (2021)
Lai J B, Lund T, Rydén K, Gabelli A, Strandell I. The fatigue limit of bearing steels—Part I: A pragmatic approach to predict very high cycle fatigue strength. Int J Fatigue 38: 155–168 (2012)
Bhadeshia H K D H. Steels for bearings. Prog Mater Sci 57(2): 268–435 (2012)
Johnson K L. Contact Mechanics. Cambridge (UK): Cambridge University Press, 1987.
Bakolas V, Mihailidis A. Analysis of rough line contacts operating under mixed elastohydrodynamic lubrication conditions. Lubr Sci 16(2): 153–168 (2004)
Kuo C H. Contact stress analysis of an elastic half-plane containing multiple inclusions. Int J Solids Struct 45(16): 4562–4573 (2008)
Warhadpande A, Sadeghi F. Effects of surface defects on rolling contact fatigue of heavily loaded lubricated contacts. Proc Inst Mech Eng J: J Eng Tribol 224(10): 1061–1077 (2010)
Bomidi J A R, Sadeghi F. Three-dimensional finite element elastic—plastic model for subsurface initiated spalling in rolling contacts. J Tribol 136(1): 011402 (2014)
Moghaddam S M, Sadeghi F, Paulson K, Weinzapfel N, Correns M, Bakolas V, Dinkel M. Effect of non-metallic inclusions on butterfly wing initiation, crack formation, and spall geometry in bearing steels. Int J Fatigue 80: 203–215 (2015)
Ting T C T. Anisotropic Elasticity: Theory and Applications. Oxford (UK): Oxford University Press, 1996.
Stroh A N. Dislocations and cracks in anisotropic elasticity. Philos Mag 3(30): 625–646 (1958)
Paulson N R, Bomidi J A R, Sadeghi F, Evans R D. Effects of crystal elasticity on rolling contact fatigue. Int J Fatigue 61: 67–75 (2014)
Paulson N R, Sadeghi F. EHL modeling of nonhomogeneous materials: The effects of polycrystalline anisotropy on RCF. Tribol Int 112: 137–146 (2017)
Wang Z J, Wang W Z, Wang H, Hu Y Z. Stress analysis on layered materials in point elastohydrodynamic-lubricated contacts. Tribol Lett 35: 229–244 (2009)
Wang Z J, Yu C J, Wang Q. Model for elastohydrodynamic lubrication of multilayered materials. J Tribol 137(1): 011501 (2015)
Wang Z J, Yu H, Wang Q. Layer-substrate system with an imperfectly bonded interface: Coupled dislocation-like and force-like conditions. Int J Solids Struct 122–123: 91–109 (2017)
Liu S B, Wang Q, Liu G. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear 243(1–2): 101–111 (2000)
Zhang S G, Wang W Z, Zhao Z Q. Elastohydrodynamic lubrication analysis of point contacts with consideration of material inhomogeneity. J Tribol 136(4): 041501 (2014)
Eshelby J D. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 241(1226): 376–396 (1957)
Golmohammadi Z, Sadeghi F. A 3D finite element model for investigating effects of refurbishing on rolling contact fatigue. Tribol Trans 63(2): 251–264 (2020)
Boffy H, Baietto M C, Sainsot P, Lubrecht A A. An efficient 3D model of heterogeneous materials for elastic contact applications using multigrid methods. J Tribol 134(2): 021401 (2012)
Boffy H, Venner C H. Multigrid solution of the 3D stress field in strongly heterogeneous materials. Tribol Int 74: 121–129 (2014)
Boffy H, Venner C H. Multigrid numerical simulation of contact mechanics of elastic materials with 3D heterogeneous subsurface topology. Tribol Int 92: 233–245 (2015)
Zhang B B, Boffy H, Venner C H. Multigrid solution of 2D and 3D stress fields in contact mechanics of anisotropic inhomogeneous materials. Tribol Int 149: 105636 (2020)
Zhang B B, Liu H C, Quiñonez A F, Venner C H. Effects of 3D anisotropic heterogeneous subsurface topology on film thickness, pressure and subsurface stresses in an elastohydrodynamically lubricated point contact. Tribol Int 151: 106471 (2020)
Zhang B B, Quiñonez A F, Venner C H. Effect of material anisotropy on rolling contact fatigue life under dry and lubricated point contact conditions: A numerical study. Tribol Int 152: 106584 (2020)
Gu H F, Réthore J, Baietto M C, Sainsot P, Lecomte-Grosbras P, Venner C H, Lubrecht A A. An efficient multigrid solver for the 3D simulation of composite materials. Comput Mater Sci 112: 230–237 (2016)
Liu X D, Réthoré J, Baietto M C, Sainsot P, Lubrecht A A. An efficient strategy for large scale 3D simulation of heterogeneous materials to predict effective thermal conductivity. Comput Mater Sci 166: 265–275 (2019)
Wang Z J, Jin X Q, Zhou Q H, Ai X L, Keer L M, Wang Q. An efficient numerical method with a parallel computational strategy for solving arbitrarily shaped inclusions in elastoplastic contact problems. J Tribol 135(3): 031401 (2013)
Fretigny C, Chateauminois A. Solution for the elastic field in a layered medium under axisymmetric contact loading. J Phys D: Appl Phys 40(18): 5418–5426 (2007)
Lubrecht A A, Ioannides E. A fast solution of the dry contact problem and the associated sub-surface stress field, using multilevel techniques. J Tribol 113(1): 128–133 (1991)
Zhang X, Wang Q J, He T. Transient and steady-state viscoelastic contact responses of layer—substrate systems with interfacial imperfections. J Mech Phys Solids 145: 104170 (2020)
Lubrecht A A, Venner C H, ten Napel W E, Bosma R. Film thickness calculations in elastohydrodynamically lubricated circular contacts, using a multigrid method. J Tribol 110(3): 503–507 (1988)
Brandt A, Venner C H. Multilevel evaluation of integral transforms on adaptive grids. In: Multigrid Methods V. Lecture Notes in Computational Science and Engineering. Hackbusch W, Wittum G. Eds. Berlin: Springer, 1998.
Bai D, Brandt A. Local mesh refinement multilevel techniques. SIAM J Sci Stat Comput 8(2): 109–134 (1987)
Briggs W L, Henson V E, McCormick S F. A Multigrid Tutorial. Philadelphia (USA): Society for Industrial and Applied Mathematics, 2000.
Boffy H, Venner C H. Multigrid numerical simulation of contact mechanics of elastic materials with 3D heterogeneous subsurface topology. Tribol Int 92: 233–245 (2015)
Venner C H, Lubrecht A A. Multi-level Methods in Lubrication. Amsterdam (the Netherlands): Elsevier Science, 2000.
Zhang B B. Multigrid methods for anisotropic heterogeneous material simulation in dry and elasto-hydrodynamically lubricated contacts. Ph.D. Thesis. Enschede (the Netherlands): University of Twente, 2020.
Zhang M Q, Wang Q, Wang Z J, Zhao N, Peng Y J. Multiscale computational scheme for semi-analytical modeling of the point contact of inhomogeneous materials. Int J Solids Struct 168: 90–108 (2019)
Brandt A, Livne O E. Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, Revised Edition. Philadelphia (USA): Society for Industrial and Applied Mathematics, 2011.
Trottenberg U, Oosterlee C W, Schuller A. Multigrid. Washington, DC (USA): Academic Press, 2000.
Noyel J P, Ville F, Jacquet P, Gravouil A, Changenet C. Development of a granular cohesive model for rolling contact fatigue analysis: Crystal anisotropy modeling. Tribol Trans 59(3): 469–479 (2016)
Anderson P M, Hirth J P, Lothe J. Theory of Dislocations. Cambridge (UK): Cambridge University Press, 2017.
Hill R. The elastic behaviour of a crystalline aggregate. Proc Phys Soc A 65(5): 349–354 (1952)
Milton G W. The Theory of Composites. Cambridge (UK): Cambridge University Press, 2002.
Morito S, Tanaka H, Konishi R, Furuhara T, Maki T. The morphology and crystallography of lath martensite in Fe-C alloys. Acta Mater 51(6): 1789–1799 (2003)
Swarr T, Krauss G. The effect of structure on the deformation of as-quenched and tempered martensite in an Fe-0.2 pct C alloy. Metall Trans A 7(1): 41–48 (1976)
Zhang J M, Zhang Y, Xu K W, Ji V. Young’s modulus surface and Poisson’s ratio curve for cubic metals. J Phys Chem Solids 68(4): 503–510 (2007)
Paulin C, Ville F, Sainsot P, Coulon S, Lubrecht T. Effect of rough surfaces on rolling contact fatigue theoretical and experimental analysis. Tribol Ser 43: 611–617 (2003)
Goulden J, Pinard P, Gholinia A, Kocun M, Proksch R. Characterization of materials properties by EBSD, EDS and AFM. Microsc Microanal 24(S1): 594–595 (2018)
Brandt A, Livne O E. Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics. Philadelphia (USA): Society of Industrial and Applied Mathematics, 1984.
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The authors would like to thank Mr. Bernie van Leeuwen, SKF Research and Technology Development (RTD) Director, for his kind permission to publish this article, and to thank Dr. Armando Felix-Quinonez for his helpful suggestions to improve the manuscript.
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Binbin ZHANG. He obtained his M.S. degree in mechanical engineering from Qingdao University of Technology, China, in 2016. Since October 2016, he started his Ph.D. study under supervision of Prof. C. H. VENNER in the Faculty of Engineering Technology, University of Twente, the Netherlands, and obtained his Ph.D. degree in August 2020. Then, he worked as a post-doctoral researcher until September 2020. His Ph.D. and postdoc research focused on developing novel, efficient and practical computational methods to optimize the 3D topology and anisotropic heterogeneity of materials for the maximum life in contact mechanics and lubrication. Since December 2021, he has been working as a senior engineer at Schaeffler (China) Co., Ltd. in Shanghai, China. His work focuses on the system simulation of advanced rolling bearings and related software development.
Tristan G. VLOGMAN. He received his M.S. degree in mechanical engineering from the University of Twente, the Netherlands, in 2021. His thesis work focused on developing a Multigrid method with local grid refinement to solve contact problems involving heterogeneous anisotropic materials. Since April 2021 he is working as a Ph.D. researcher at the University of Twente under supervision of Dr. K. JAIN and Prof. dr. ir. R. HAGMEIJER. His research concerns the computational modelling of fluid and particle transport in the liver to improve radioembolization therapy. The aim of his Ph.D. work is to couple a particle transport model with the Lattice-Boltzmann method to enable massively-parallel simulations of fluid and particles in complex geometries such as the human liver.
Predrag ANDRIC. He received M.S. degree in aerospace engineering from University of Belgrade, Serbia. In April 2015, he joined the Laboratory for Multiscale Mechanics Modeling in École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, to carry out Ph.D. studies under supervision of Prof. Bill CURTIN. His Ph.D. work was mostly focused on understanding the fundamental crack—tip processes using atomistic simulations. Since June 2019, he has been working as a researcher at SKF’s RTD center in the Netherlands. His research focuses on understanding the effect of steel microstructure on the surface and subsurface initiated fatigue in rolling contact.
Ton C. BOR. He received his M.S. (1994) and Ph.D. (2000) degrees in materials science from the Technical University of Delft, the Netherlands. Currently, he is an associate professor within the Department of Mechanics of Solids, Surfaces & Systems in the research chair of Production Technology at the University of Twente, the Netherlands. His research interest covers various areas including fiber reinforced composite materials, X-ray and mechanical analysis of crystalline materials, and solid-state additive manufacturing of high strength aluminium and magnesium alloys.
Cornelis H. VENNER. He obtained his M.S. (1987) and Ph.D. (1991) degrees in mechanical engineering (tribology) from the University of Twente, the Netherlands, on the development of optimally efficient and robust Multigrid/Multilevel methods for elastohydrodynamic lubrication. He was a post-doctoral researcher at the Weizmann Institute of Science, Rehovot, Israel, with Prof. A. BRANDT. He co-authored a book on Multilevel methods for lubrication with Prof. A. A. (Ton) LUBRECHT. Since 2014, he chairs the Engineering Fluid Dynamics group in the faculty of Engineering Technology of the University of Twente with research teams in experimental and theoretical aerodynamics/aeroacoustics for aeronautical applications, fluid mechanics of functional materials, and biomedical fluid mechanics. He is an expert in research and teaching of Multigrid/Multlievel methods for engineering problems, in thin layer flow, contact mechanics, and in (visco)elastohydrodynamic lubrication including the solution of subsurface stress fields and computational diagnostics. He is also active in the development of novel concepts for teaching/learning environments in STEM education.
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Zhang, B., Vlogman, T.G., Andric, P. et al. Local grid refinement in multigrid method for point contact problems of polycrystalline anisotropic material under dry and lubricated conditions. Friction 10, 2086–2110 (2022). https://doi.org/10.1007/s40544-021-0582-5
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DOI: https://doi.org/10.1007/s40544-021-0582-5