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Gears pp 439-537 | Cite as

Micropitting Load Capacity of Spur and Helical Gears

  • Vincenzo VulloEmail author
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Part of the Springer Series in Solid and Structural Mechanics book series (SSSSM, volume 11)

Abstract

In this chapter, a general survey is first done on the micropitting damage of spur and helical gears, which manifests itself to the roughness scale. The mechanism that trigger this type of damage as well as the characteristics that distinguish it from those typical of macropitting (the classical pitting) are described. The problem to be solved for a reliable calculation procedure of micropitting load carrying capacity of gears are then analyzed and, to this end, the ideal characteristics of a general micropitting model are described. An interesting tribological-dynamic analytical model for cylindrical spur gears is then described, which also consists of a three-dimensional analytical-numerical contact sub-model and a multiaxial fatigue sub-model. The procedure for calculating the surface durability of spur and helical gears in accordance with the ISO standards is described, highlighting when deemed necessary how the formulae used by the same ISO are anchored to the theoretical bases previously discussed. Finally, for a better understanding of micropitting mechanisms, attention is drawn to the need to introduce, instead of traditional profile parameters, the areal field parameters that best describe the topography and texture of surfaces.

References

  1. AGMA 925-A03:2003 Effect of lubrication on gear surface distressGoogle Scholar
  2. Abbott EJ, Firestone FA (1933) Specifying surface quality: a method based on accurate measurement and comparison. Mech Eng 55:569–572Google Scholar
  3. Al-Tubi IS, Long H (2013) Prediction of wind turbine gear micropitting under variable load and speed conditions using ISO/TR 15144-1:2010. Proc Inst Mech Eng, Part C: J Mech Eng Sci 227(9):1898–1914CrossRefGoogle Scholar
  4. Al-Tubi IS, Long H, Zhang J, Shaw B (2015) Experimental and analytical study of gear micropitting initiation and propagation under varying load conditions. Wear 328–329:8–16CrossRefGoogle Scholar
  5. Aver’yanova IO, Bogomolov DY, Porishin VV (2017) ISO 25178 standard for three-dimensional parametric assessment of surface texture. Russ Eng Res 37(6):513–516CrossRefGoogle Scholar
  6. Barber JR (1992) Elasticity. Kluwer Academic Publishers, DordrechtzbMATHCrossRefGoogle Scholar
  7. Bargis E, Garro A, Vullo V (1980a) Crankshaft design and evaluation. Part 1—Critical analysis and experimental evaluation of current methods. In: ASME, reliability, stress analysis and failure prevention methods in mechanical design. New York, pp 181–201Google Scholar
  8. Bargis E, Garro A, Vullo V (1980b) Crankshaft design and evaluation. Part 2—a modern design method: modal analysis. In: ASME, reliability, stress analysis and failure prevention methods in mechanical design. New York, pp 203–211Google Scholar
  9. Bargis E, Garro A, Vullo V (1980c) Crankshaft design and evaluation. Part 3—a modern design method: direct integration. In: ASME, reliability, stress analysis and failure prevention methods in mechanical design. New York, pp 213–218Google Scholar
  10. Bathgate J, Kendall RB, Moorhouse P (1970) Thermal aspects of gear lubrication. Wear 15(2):117–129CrossRefGoogle Scholar
  11. Becker AA (1992) The boundary element method in engineering: a complete course. McGraw-Hill Book Company, New YorkGoogle Scholar
  12. Benson R, Sroka GJ, Bell M (2013) The effect of the roughness profile on micropitting. GearSolutions, March pp 47–53Google Scholar
  13. Bernardini C, Ragnisco O, Santini PM (1993) Metodi matematici della fisica. Carocci Editore SpA, RomaGoogle Scholar
  14. Bernasconi A, Papadopulos IV (2005) Efficiency of algorithms for shear stress amplitude calculation in critical plane class fatigue criteria. Comput Mater Sci 34:355–368CrossRefGoogle Scholar
  15. Berthe D, Flamand L, Foucher D, Godet M (1980) Micro-pitting in Hertzian contacts, Transactions of the ASME. J Lubr Technol 102:478–489CrossRefGoogle Scholar
  16. BGA-DU P602 (2008) Gear micropitting procedure. Test procedure for the evaluation of micropitting performance of spur and helical gearsGoogle Scholar
  17. Blateyron F (2013) The areal field parameters. In: Leach Richard (ed) Characterization of areal surface texture. Springer-Verlag, Berlin HeidelbergGoogle Scholar
  18. Blok H (1937a) Theoretical study of temperature rise at surfaces of actual contact under oiliness lubricating conditions. Proc Inst Mech Eng (General Discussion on Lubrication) 2:222–235Google Scholar
  19. Blok H (1937b) Les températures de surface dans des conditions de graissage sous pressions extrêmes. In: Proceedings of the 2nd world petroleum congress, Paris, Section IV, vol. III, pp 151–182Google Scholar
  20. Blok H (1937c) Measurements of temperature flashes on gear teeth under extreme pressure conditions. Inst Mech Eng (Proceedings of the general discussion on lubrication) 2:18–22Google Scholar
  21. Blok H (1937d) Surface temperature measurements on gear teeth under extreme pressure lubricating conditions. Power Transm, 653–656Google Scholar
  22. Blok H (1940) Fundamental mechanical aspects of boundary lubrication. SAE Trans 35:54–68Google Scholar
  23. Blok H (1969) The thermal-network method for predicting bulk temperatures in gear transmissions. In: Proceedings of the 7th round-table discussion on marine reduction gears, stat-laval, Finspong, Sweden, pp 3–25 and 26–32Google Scholar
  24. Boussinesq J (1885) Application des Potentiels à l’étude de l’équilibre et du mouvement des solides élastiques avec des notes étendues sur divers points de physique mathématique et d’analyse. Gauthier-Villars, PariszbMATHGoogle Scholar
  25. Brandão JA, Scabra JHO, Castro MJD (2010) Gear micropitting: model and validation. WIT Trans Eng Sci 66:25–36CrossRefGoogle Scholar
  26. Bronshtein IN, Semendyayev KA (1997) Handbook of mathematics, 3rd edn. Springer-Verlag, New YorkzbMATHGoogle Scholar
  27. Buckingham E (1949) Analytical mechanics of gears. McGraw-Hill Book Company, New YorkGoogle Scholar
  28. Buzdygon KJ, Cardis AB (2004) A short procedure to evaluate micropitting using the new AGMA designed gears. AGMA Fall Tech MeetGoogle Scholar
  29. Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Oxford University Press, OxfordzbMATHGoogle Scholar
  30. Cauchy A-L (1828) Sur les équations qui expriment les conditions d’équilibre ou les lois du mouvement intérieur d’un corps solide élastique ou non élastique. Exerc. de Mathématiques 3:160–187Google Scholar
  31. Cerruti V (1881–82) Ricerche intorno all’equilibrio dei corpi elastici isotropi, Atti della Reale Accademia dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, Serie 3, Annata 279, vol. 13, pp 81–122; reprint with the same title, Roma, Salviucci, 1882Google Scholar
  32. Chang WR, Etsion I, Bogy DB (1987) An elastic-plastic model for the contact of rough surfaces. J Tribol 109(2):257–263CrossRefGoogle Scholar
  33. Clark A, Evans HP, Snidle RW (2015) Understanding micropitting in gears. In: Part C: J Mech Eng Sci (Proceedings of the institution of mechanical engineers)Google Scholar
  34. Clark A, Weets IJJ, Snidle RW, Evans HP (2016) Running-in and micropitting behavior of steel surfaces under mixed lubrication conditions. Tribol. Int, 101:59–68Google Scholar
  35. Conry TF, Seireg A (1973) A mathematical programming technique for the evaluation of load distribution and optimal modification for gear systems. J Eng Ind 95:1115–1122CrossRefGoogle Scholar
  36. Cook RD (1981) Concepts and applications of finite element analysis, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  37. Démidovitch B, Maron I (1973) Éléments de Calcule Numérique. Éditions MIR-Moscou, MoscouzbMATHGoogle Scholar
  38. Den Hartog JP (1985) Mechanical vibrations, 4th edn. Dover Publications Inc, New YorkzbMATHGoogle Scholar
  39. Dowson D, Higginson GR (1977) Elastohydrodynamic lubrication, 2nd edn. Pergamon, LondonGoogle Scholar
  40. Ehret P, Dowson D, Taylor CM (1998) On the lubricant transport conditions in elastohydrodynamic conjunctions. Proc R Soc Lond, Series A, 454Google Scholar
  41. Evans HP, Snidle RW, Sharif KJ (2011) Analysis of micro-elastohydrodynamic lubrication and surface fatigue in gear micropitting tests. In: ASME proceedings of 11th international power transmission and gearing conference, and 13th international conference on advanced vehicle and tire technologies, Washington, DC, USA, August 28–31, Vol. 8, Paper No. DETC2011-47714, pp 585–591Google Scholar
  42. Evans HP, Snidle RW, Sharif KJ, Shaw BA, Zhang J (2013) Analysis of micro-elastohydrodynamic lubrication and prediction of surface fatigue damage in micropitting tests on helical gears. J Tribol 135(1)Google Scholar
  43. Fatemi A, Shamsaei N (2011) Multiaxial fatigue: an overview and some approximation models for life estimation. Int J Fatigue 33(8):948–958CrossRefGoogle Scholar
  44. Fatemi A, Socie DF (1988) A critical plane to multiaxial fatigue damage including out-of-phase loading. Fatigue Fracture Eng Mater Struct 11(3):149–165CrossRefGoogle Scholar
  45. Favata A (2012) On the Kelvin problem. J Elast 109(2):189–204MathSciNetzbMATHCrossRefGoogle Scholar
  46. Fenner RT (1986) Engineering elasticity: application of numerical and analytical techniques. Ellis Horwood Limited Publishers, ChichesterGoogle Scholar
  47. Ferrari C, Romiti A (1966) Meccanica applicata alle macchine. Torino: Unione Tipografica–Editrice Torinese (UTET)Google Scholar
  48. Flamant A-A (1892) Sur la répartition des pressions dans un solide rectangulaire chargé trnsversalement. Comptes Rendus des Séances de l’Academie des Sciences, Paris 114:1465–1468zbMATHGoogle Scholar
  49. FVA-Information Sheet 54/7 (1993) Test procedure for the investigation of the micropitting capacity gear lubricantsGoogle Scholar
  50. Garro A, Vullo V (1979) Acoustic problems of vehicle transmission. Nauka I Motorna Vozila ’79, Bled, Slovenija, Jugoslavija, June 4–7Google Scholar
  51. Gauss CF (1815) Methodus nova integralium valores per approximationem inveniendi: auctore Carolo Friderico Gauss. H. Dicterich, GottingaeGoogle Scholar
  52. Gladwell GML (1980) Contact problems in the classical theory elasticity, Sijthoff & Noordhoff International Publishers B.V., Alphen aan den Rijn, the Netherlands Germantown, Maryland, USAGoogle Scholar
  53. Goglia PR, Cusano C, Conry TF (1984a) The effects of surface irregularities on the elastohydrodynamic lubrication of sliding line contacts. Part I-Single IrregulIties, ASME J Tribol 106(1):104–112CrossRefGoogle Scholar
  54. Goglia PR, Cusano C, Conry TF (1984b) The effects of surface irregularities on the elastohydrodynamic lubrication of sliding line contacts. Part II-Wavy SurfS, ASME J Tribol 106(1):113–119CrossRefGoogle Scholar
  55. Greco A, Sheng S, Keller J, Erdemir A (2013) Material wear and fatigue in wind turbine systems. Wear 302:1583–1591CrossRefGoogle Scholar
  56. Greenwood JA, Tripp JH (1970–71) The contact of two nominally flat rough surfaces. Proc Inst Mech Eng 185(48/71): 625–634CrossRefGoogle Scholar
  57. Greenwood JA, Williamson, J-BP (1966) Contact of nominally flat surfaces. In: Proc R Soc Lond, Ser A, Math Phys Sci 295(1442):300–319Google Scholar
  58. Gupta PK (1984) Advanced dynamics of rolling elements. Springer-Verlag, Berlin HeidelbergCrossRefGoogle Scholar
  59. Hein M, Stahl K, Tobie T (2017) Practical use of micropitting test results according to FVA 54/7 for calculation of micropitting load capacity according to ISO/TR 15144-1. In: International conference on gears Sept. 13-15, 2017, Technische Universität München (TUM), Garching/Munich, GermanyGoogle Scholar
  60. Henrici P (1988) Applied and computational complex analysis. Power series, integration, conformed mapping, vol 1. John Wiley & Sons, Inc, Location of Zeros, New YorkGoogle Scholar
  61. Höhn B-R, Oster P, Emmert S (1996) Micropitting in case-carburized gears - FZG micropitting test. In: International Conference on Gears, Dresden, Germany, VDI Berichte Nr. 1230, pp 331–334Google Scholar
  62. Incropera FP, DeWitt DP, Bergmann TL, Lavine AS (2006) fundamentals of heat and mass transfer, 6th edn. John Wiley & Sons Inc, New YorkGoogle Scholar
  63. ISO 12085:1996 Geometrical product specifications (GPS)—surface texture: Profile method—motif parametersGoogle Scholar
  64. ISO 1302:2002 Geometrical product specifications (GPS)—indication of surface texture in technical product documentationGoogle Scholar
  65. ISO 1328-1:2013, Cylindrical gears—ISO system of flank tolerance classification—part 1: definitions and allowable values of deviations relevant to flanks of gear teethGoogle Scholar
  66. ISO 13565-1:1996, Geometrical product specifications (GPS), surface texture: profile method; surfaces having stratified functional properties—part. 1: filtering and general measurement conditionsGoogle Scholar
  67. ISO 13565-2:1996 Geometrical product specifications (GPS)—surface texture: profile method; surfaces having stratified functional properties—part 2: height characterization using the linear material ratio curveGoogle Scholar
  68. ISO 13565-3:1998 Geometrical product specifications (GPS)—surface texture: profile method; surfaces having stratified functional properties—part 3: height characterization using the material probability curveGoogle Scholar
  69. ISO 16610-1:2015, Geometrical product specifications (GPS)—filtration—part 1: overview and basic conceptsGoogle Scholar
  70. ISO 4288:1996 Geometrical product specifications (GPS)—surface texture: profile method—rules and procedures for the assessment of surface textureGoogle Scholar
  71. ISO 4287:1997 Surface roughness testing: surface texture: profile method—terms, definitions and surface texture parametersGoogle Scholar
  72. ISO 53:1998 Cylindrical gears for general and heavy engineering—standard basic rack tooth profileGoogle Scholar
  73. ISO 25178-2:2012 Geometrical product specifications (GPS)—surface texture: areal-part 2: terms, definitions and surface texture parametersGoogle Scholar
  74. ISO 25178-3:2011 Geometrical product specifications (GPS)—surface texture: areal—part 3: specifications operatorsGoogle Scholar
  75. ISO/TS 6336-22:2018 Calculation of load capacity of spur and helical gears—part 22: calculation of micropitting load capacityGoogle Scholar
  76. ISO/TS 6336-31:2018(E) Calculation of load capacity of spur and helical gears—part 31: calculation examples of micropitting load capacityGoogle Scholar
  77. Jackson RL, Green I (2011) On the modeling of elastic contact between rough surfaces. Tribol Trans 54:300–314CrossRefGoogle Scholar
  78. Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge, United KingdomzbMATHCrossRefGoogle Scholar
  79. Karas F. (1941) Elastische formänderung und lastverteilung beim doppeleingriff gerader stirnradzähne, VDI—forschungheft 406, B, Bd. 12Google Scholar
  80. Kissling U. (2012) Application of the first international calculation method for micropitting. Gear Technol 54–60Google Scholar
  81. Leach RK (2009) Fundamental principles of engineering nanometrology. Elsevier, AmsterdamGoogle Scholar
  82. Li S, Kahraman A (2010a) A transient mixed elastohydrodynamic lubrication model for spur gear pair. ASME J Tribol 132(1), 011501-1-9Google Scholar
  83. Li S, Kahraman A (2010b) Prediction of spur gear mechanical power losses using a transient elastohydrodynamic lubrication model. Tribol Trans 53(4):554–563CrossRefGoogle Scholar
  84. Li S, Kahraman A (2011a) A fatigue model for contacts under mixed elastohydrodynamic lubrication condition. Int J Fatigue 33(3):427–436CrossRefGoogle Scholar
  85. Li S, Kahraman A (2011b) Influence of dynamic behavior on elastohydrodynamic lubrication of spur gears. Proc Inst Mech Eng, Part J, J Eng Tribol 225:740–753CrossRefGoogle Scholar
  86. Li S, Kahraman A (2013a) A physics-based model to predict micro-pitting lives of lubricated point contacts. Int J Fatigue 47:205–215CrossRefGoogle Scholar
  87. Li S, Kahraman A (2013b) Micro-pitting fatigue lives of lubricated point contacts: Experiments and model validation. Int J Fatigue 48:9–18CrossRefGoogle Scholar
  88. Li S, Kahraman A (2013c) A tribo-dynamic model for a spur gear pair. J Sound Vib 332:4963–4978CrossRefGoogle Scholar
  89. Li S, Kahraman A (2014) A micro-pitting model for spur gear contacts. Int J Fatigue 59:224–233CrossRefGoogle Scholar
  90. Li S, Kahraman A, Klein M (2012) A Fatigue model for spur gear contacts operating under mixed elastohydrodynamic lubrication conditions. ASME J Mech Des 134(4):041007-1-11Google Scholar
  91. Liu Y, Mahadevan S (2007) A unified multiaxial fatigue damage model for isotropic and anisotropic materials. Int J Fatigue 29:347–359zbMATHCrossRefGoogle Scholar
  92. Liu H, Lohner T, Jurkschat T, Stahl K (2018) Detailed investigation on the oil flow on dip-lubricant gearboxes by the finite volume GFD method. Lubricants 6:47CrossRefGoogle Scholar
  93. Liu H, Liu H, Zhu C, Zhou Y (2019) A review on micropitting studies of steel gears. Coatings 9(1):42CrossRefGoogle Scholar
  94. Lo CC (1969) Elastic contact of rough cylinders. Int J Mech Sci 11(1), 105–106, IN7-IN8, 107–115CrossRefGoogle Scholar
  95. Long H, Al-Tubi IS, Martinze MTM (2015) Analytical and experimental study of gear surface micropitting due to variable loading. Appl Mech Mater 750:96–103CrossRefGoogle Scholar
  96. Love AEH (1944) A treatise on the mathematical theory of elasticity, 4th edn. Dover Publications, New YorkzbMATHGoogle Scholar
  97. Manin L, Play D (1999) Thermal behavior of power gearing transmission, numerical prediction, and influence of design parameters, ASME. J Tribol 121:693–702CrossRefGoogle Scholar
  98. McGrew JM, Gu A, Cheng HS, Murray SF (1970) Elastohydrodynamic lubrication—preliminary design manual, U.S. Air force technical report # AFAPL-TR-70-27, air force aero-propulsion laboratory, air force systems command, Wright-Patterson Air Force Base, OhioGoogle Scholar
  99. Milovanović GV (2016) Generalized gaussian quadratures for integrals with logarithmic singularity. Filomat 30(4):1111–1126MathSciNetzbMATHCrossRefGoogle Scholar
  100. Moorthy B, Shaw BA (2013) An observation on the initiation of micro-pitting damage in as-ground and coated gears during contact fatigue. Wear 297(1):878–884CrossRefGoogle Scholar
  101. Morales-Èspejel GE, Rycerz P, Kadiric A (2018) Prediction of micropitting damage in gear teeth contacts considering the concurrent effects of surface fatigue and mild wear. Wear 398–399:99–115CrossRefGoogle Scholar
  102. Moser W, Duenser C, Beer G (2004) Mapped infinite elements for three-dimensional multi-region boundary element analysis. Int J Numer Meth Eng 61(3):317–328zbMATHCrossRefGoogle Scholar
  103. Naveros I, Ghiaus C, Ordoñez J, Ruiz DP (2016) Thermal networks considering graph theory and thermodynamics. In: Proceedings of the 12th international conference on heat transfer, fluid mechanics and thermodynamics, Costa del Sol, Spain, 11–13 July, pp 1568–1573Google Scholar
  104. Olver AV, Spikes HA, MacPherson PB (1986) Wear in rolling contacts. Wear 112(2):121–144CrossRefGoogle Scholar
  105. Oster P (1982) Beanspruchung der Zahnflanken unter Bedingungen der Elastohydrodynamic, Dissertation Technische Universität München, Forschungsheft/Forschungsvereinigung Antriebstechnik, vol. 131, Frankfurt, M, Frankfurt A.MGoogle Scholar
  106. Ozguven HN, Houser DR (1988) Mathematical models used in gear dynamics—a review. J Sound Vib 121:383–411CrossRefGoogle Scholar
  107. Ramdan RD, Setiawan R, Sasmita F, Suratman R, Taufiqulloh T (2018) Determination on damage mechanism of the planet gear of heavy vehicle final drive. IOP Conf Ser, Mater Sci Eng 307:1–7Google Scholar
  108. Ree T, Eyring H (1955a) Theory of non-newtonian flow. I. solid plastic system. J Appl Phys 26(7):793–800zbMATHCrossRefGoogle Scholar
  109. Ree T, Eyring H (1955b) Theory of non-newtonian flow. II? solution system of high polymers. J Appl Phys 26(7):800–809zbMATHCrossRefGoogle Scholar
  110. Reynolds O (1876) On rolling friction. Philos Trans R Soc Lond 166:155–171Google Scholar
  111. Rycerz P, Kadiric A (2019) The influence of slide-roll ratio on the extend of micropitting damage in rolling-sliding contacts pertinent to gear applications. Tribol Lett 67(2):1CrossRefGoogle Scholar
  112. Sadeghi F, Jalalahmadi B, Slak TS, Raje N, Arakere NK (2009) A review of rolling contact fatigue. J Tribol 141:041403-1–04140304140315Google Scholar
  113. Sheng S Ed (2010) Wind turbine micropitting workshop: a recap. National Renewable Energy Laboratory, Technical Report NREL/TP-500–46572, FebrGoogle Scholar
  114. Shotter BA (1981) Micropitting: its characteristics and implications on the test requirements of gear oils. In: Performance testing of gear oils and transmission fluids, Institute of Petroleum, pp 53–59 and 320–323Google Scholar
  115. Smirnov V (1975) Cours de Mathématiques Supérieurs. Tome IV, Première partie, MIR Éditions de Moscou, MoscouGoogle Scholar
  116. Sneddon IN (1974) The use of integral transforms, TMH edn. Tata McGraw-Hill Publishing Company Ltd, New DelhizbMATHGoogle Scholar
  117. Snidle RW, Evans HP, Alanou MP, Holmes MJA (2003) The control and reduction of wear in military platforms. Williamsburg, USA, 7–9 June also published in RTO-AVT-190Google Scholar
  118. Socie DF, Marquis G (1999) Multiaxial fatigue, R-234. SAE International Publisher, New YorkCrossRefGoogle Scholar
  119. Tanaka F, Edwards SF (1992) Viscoelastic properties of physically crosslinked networks. 1. Transient network theory, Macromolecules, 25(5):1516–1523CrossRefGoogle Scholar
  120. Theißen J (1998) Eignungsnachweise von Schmierölen für Industriegetriebe, 11th International Colloquium, 13–15.1.1998, Technische Akademie EsslingenGoogle Scholar
  121. Thomson W, Lord Kelvin - (1848) A note on the integration of the equation of the equilibrium of an elastic solid. Camb Dublin Math J 3:87–89Google Scholar
  122. Timoshenko S, Goodier JN (1951) Theory of elasticity. McGraw-Hill Book Company Inc, New YorkzbMATHGoogle Scholar
  123. Wang S, Cusano C, Conry TF (1991) Thermal analysis of elastohydrodynamic lubrication of line contacts using the ree-eyring fluid model. Trans ASME, J Tribol 113:232–244CrossRefGoogle Scholar
  124. Wang J, Li R, Peng X (2003) Survey of nonlinear vibration of gear transmission systems. ASME, Applied Mechanics Reviews 56(3):309–329CrossRefGoogle Scholar
  125. Warburton GB (1976) The dynamical behaviour of structures, 2nd edn. Pergamon Press, OxfordGoogle Scholar
  126. Whittaker ET, Watson GN (1990) A course in modern analysis, 4th edn. Cambridge University Press, Cambridge, EnglandzbMATHGoogle Scholar
  127. Yastrebov VA (2013) Numerical methods in contact mechanics. John Wiley&Sons, New YorkzbMATHCrossRefGoogle Scholar
  128. Yastrebov VA, Anciaux G, Molinari J-F (2014) From infinitesimal to full contact between rough surfaces: evolution of the contact area, arXiv: 1401.3800v1, [physics.class-ph] 16 JanGoogle Scholar
  129. Yastrebov VA, Anciaux G, Molinari J-F (2016) On the accurate computation of the true contact-area in mechanical contact of random rough surfaces. Tribology International, vol. 114Google Scholar
  130. Yu Z-Y, Zhou S-P, Liu Q, Liu Y (2017) Multiaxial fatigue damage parameter and life prediction without any additional material constants, Materials, MDPI Basel Switzerland, 10(8)CrossRefGoogle Scholar
  131. Zhou Y, Zhu C, Liu H (2019) A micropitting study considering rough sliding and mild wear. Coatings 9:639–653CrossRefGoogle Scholar
  132. Zhu D (2007) On some aspects of numerical solutions of thin-film and mixed elastohydrodynamic lubrication. Proc Inst Mech Eng, Part J, J Eng Tribol 221(5):561–579CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of Rome “Tor Vergata”RomeItaly

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