Abstract
The current work presents a quantitative approach for the prediction of minimum film thickness in elastohydrodynamic-lubricated (EHL) circular contacts. In contrast to central film thickness, minimum film thickness can be hard to accurately measure, and it is usually poorly estimated by classical film thickness formulae. For this, an advanced finite element-based numerical model is used to quantify variations of the central-to-minimum film thickness ratio with operating conditions, under isothermal Newtonian pure-rolling conditions. An ensuing analytical expression is then derived and compared to classical film thickness formulae and to more recent similar expressions. The comparisons confirmed the inability of the former to predict the minimum film thickness, and the limitations of the latter, which tend to overestimate the ratio of central-to-minimum film thickness. The proposed approach is validated against numerical results as well as experimental data from the literature, revealing an excellent agreement with both. This framework can be used to predict minimum film thickness in circular elastohydrodynamic contacts from knowledge of central film thickness, which can be either accurately measured or rather well estimated using classical film thickness formulae.
Similar content being viewed by others
Availability of Data and Material (Data Transparency)
The authors declare that all data supporting the findings of this study are available within the article.
Abbreviations
- \(a\) :
-
Dry or Hertzian contact radius (m)
- \({A}_{1}, {A}_{2}\) :
-
Parameters in the modified Yasutomi-WLF model
- \({b}_{1}, {b}_{2}\) :
-
Parameters in the modified Yasutomi-WLF model
- \({C}_{1}, {C}_{2}\) :
-
Parameters in the modified Yasutomi-WLF model
- \({E}_{1}, {E}_{2}\) :
-
Young modulii of solids 1 and 2 (Pa)
- \(E{^{\prime}}\) :
-
Reduced modulus of elasticity (Pa) \(2/E{^{\prime}} = (1 - {\nu }_{1}^{2} )/{E}_{1} + (1 - {\nu }_{2}^{2} )/{E}_{2}\)
- \(F\) :
-
Contact external applied load (N)
- \(G\) :
-
Dimensionless material parameter (Hamrock and Dowson) \(= {\alpha }^{*}.E{^{\prime}}\)
- \({h}\) :
-
Film thickness (m)
- \({h}_{c}\) :
-
Central film thickness (m)
- \({h}_{m}\) :
-
Minimum film thickness (m)
- \({h}_{0}\) :
-
Rigid body separation (m)
- \({H}_{c}\) :
-
Dimensionless central film thickness (-) \(={h}_{c}/({R}_{x}.{U}^{0.5})\)
- \({H}_{m}\) :
-
Dimensionless minimum film thickness (-) \(={h}_{m}/({R}_{x}.{U}^{0.5})\)
- \({K}_{0}^{'}, {K}_{00}\) :
-
Parameters of the Murnagham equation of state
- \(L\) :
-
Dimensionless material parameter (Moes) \(= G{.(2U)}^{0.25}\)
- \(\tilde{L }\) :
-
Natural logarithmic value of \(L\)
- \(M\) :
-
Dimensionless load parameter (Moes) for point contact \(= W/{(2U)}^{0.75}\)
- \(\tilde{M }\) :
-
Natural logarithmic value of \(M\)
- \({p}_{H}\) :
-
Hertzian pressure (GPa)
- \({R}_{x}\) :
-
Reduced radius of curvature (m)
- \(T\) :
-
Temperature (°C)
- \({T}_{g0}\) :
-
Glass transition temperature at ambient pressure (°C)
- \({u}_{e}\) :
-
Mean entrainment velocity (m/s) \(= ({u}_{1} + {u}_{2})/2\)
- \({u}_{1},{u}_{2}\) :
-
Velocity in the \(x\) -direction of surfaces 1 and 2 (m/s)
- \(U\) :
-
Dimensionless speed parameter (Hamrock and Dowson) \(= \mu .{u}_{e}/({E}^{^{\prime}}.{R}_{x})\)
- \(w\) :
-
Normal load (N)
- \(W\) :
-
Dimensionless load parameter (Hamrock and Dowson) \(= w/({E}^{^{\prime}}.{R}_{x}^{2})\)
- \({\alpha }^{*}\) :
-
Reciprocal asymptotic isoviscous pressure, according to Blok [35] (Pa−1)
- \({\alpha }_{film}\) :
-
General pressure viscosity coefficient for film forming, according to Bair [37] (Pa−1)
- \({\beta }_{K}\) :
-
Parameter of the Murnagham equation of state
- δ :
-
Combined normal surface deformation of contacting solids (m)
- \({\nu }_{1}, {\nu }_{2}\) :
-
Poisson coefficient of solids 1 and 2
- \(\mu\) :
-
Lubricant dynamic viscosity (Pa s)
- \({\mu }_{0}\) :
-
Lubricant dynamic viscosity at ambient pressure (Pa s)
- \({\mu }_{g}\) :
-
Dynamic viscosity at the glass transition (Pa s)
- \({\rho }\) :
-
Lubricant density (kg m−3)
- \({\rho }_{0}\) :
-
Lubricant density at ambient pressure (kg m−3)
References
Albahrani, S., Philippon, D., Vergne, P., Bluet, J.: A review of in situ methodologies for studying elastohydrodynamic lubrication. Proc. Inst. Mech. Eng. J. 230(1), 86–110 (2016)
Luo, J., Wen, S., Huang, P.: Thin film lubrication part I: study on the transition between EHL and thin film lubrication using a relative optical interference intensity technique. Wear 194, 107–115 (1996)
Molimard J.: Etude expérimentale du régime de lubrification en film mince - application aux fluides de laminage, PhD thesis (in French). Institut National des Sciences Appliquées de Lyon. N° d’ordre: 99ISAL0121 (1999)
Ma, L., Luo, J.: Thin film lubrication in the past 20 years. Friction 4(4), 280–302 (2016)
Cusseau, P., Vergne, P., Martinie, L., Philippon, D., Devaux, N., Briand, F.: Film forming capability of polymer-base oil lubricants in elastohydrodynamic and very thin film regimes. Tribol. Lett. 67(2), 45 (2019)
Koye, K.A., Winer, W.O.: An experimental evaluation of Hamrock and Dowson minimum film thickness equation for fully flooded EHD point contacts. J. Lubr. Technol. 103(2), 284–294 (1981)
Hamrock, B.J., Dowson, D.: Isothermal elastohydrodynamic lubrication of point contacts Part III–fully flooded results. J. Lubr. Technol. 99(2), 264–276 (1977)
Smeeth, M., Spikes, H.A.: Central and minimum elastohydrodynamic film thickness at high contact pressure. J. Tribol. 117, 291–296 (1997)
Venner C.H.: Multilevel solution of the EHL line and point contact problems. PhD thesis, Twente University (1991)
Chaomleffel, J.P., Dalmaz, G., Vergne, P.: Experimental results and analytical predictions of EHL film thickness. Tribol. Int. 40(10–12), 1543–1552 (2007)
Nijenbanning, G., Venner, C.H., Moes, H.: Film thickness in elastohydrodynamically lubricated elliptic contacts. Wear 176, 217–229 (1994)
Chevalier F.: Modélisation des conditions d'alimentation dans les contacts élastohydrodynamiques ponctuels. PhD thesis in French, INSA de Lyon, France, n° 96 ISAL 0124 (1996)
van Leeuwen, H.: The determination of the pressure–viscosity coefficient of a lubricant through an accurate film thickness formula and accurate film thickness measurements. Proc. IMechE Part J. 223(8), 1143–1163 (2009)
van Leeuwen, H.: The determination of the pressure–viscosity coefficient of a lubricant through an accurate film thickness formula and accurate film thickness measurements. Part 2: high L values. Proc. IMechE. J. 225(6), 449–464 (2011)
Wheeler, J.D., Vergne, P., Fillot, N., Philippon, D.: On the relevance of analytical film thickness EHD equations for isothermal point contacts: qualitative or quantitative predictions? Friction 4(4), 369–379 (2016)
Evans, P., Snidle, R.: The isothermal elastohydrodynamic lubrication of spheres. J. Lubr. Technol. 103, 547–557 (1981)
Chittenden, R.J., Dowson, D., Dunn, J.F., Taylor, C.M.: A theoretical analysis of the isothermal elastohydrodynamic lubrication of concentrated contacts - Part 2: General case, with lubricant entrainment along either principal axis of the Hertzian contact ellipse or at some intermediate angle. Proc. R. Soc. Lond. A 397, 271–294 (1985)
Masjedi, M., Khonsari, M.M.: On the effect of surface roughness in point-contact EHL: formulas for film thickness and asperity load. Tribol. Int. 82(A), 228–244 (2015)
Morales-Espejel, G.E., Dumont, M.L., Lugt, P.M., Olver, A.V.: A limiting solution for the dependence of film thickness on velocity in EHL contacts with very thin films. Tribol. Trans 48(3), 317–327 (2005)
Glovnea, R.P., Olver, A.V., Spikes, H.A.: Experimental investigation of the effect of speed and load on film thickness in elastohydrodynamic contact. Tribol. Trans. 48(3), 328–335 (2005)
Venner, C.H.: EHL film thickness computations at low speeds: risk of artificial trends as a result of poor accuracy and implications for mixed lubrication modelling. Proc. IMechE J. 219, 285–290 (2005)
Vergne, P., Bair, S.: Classical EHL versus quantitative EHL: a perspective - Part I: real viscosity-pressure dependence and the viscosity-pressure coefficient for predicting film thickness. Tribol. Lett. 54(1), 1–12 (2014)
Bair, S., Martinie, L., Vergne, P.: Classical EHL versus quantitative EHL: a perspective part II - super-Arrhenius piezoviscosity, an essential component of elastohydrodynamic friction missing from classical EHL. Tribol. Lett. 63(3), 37 (2016)
Habchi, W.: Finite Element Modeling of Elastohydrodynamic Lubrication Problems. Wiley, Chichester (2018)
Wu, S.R.: A penalty formulation and numerical approximation of the Reynolds-Hertz problem of elastohydrodynamic lubrication. Int. J. Eng. Sci. 24(6), 1001–1013 (1986)
Habchi, W., Eyheramendy, D., Vergne, P., Morales-Espejel, G.E.: Stabilized fully-coupled finite elements for elastohydrodynamic lubrication problems. Adv. in Eng. Softw. 46, 4–18 (2012)
Deuflhard, P.: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer, Berlin (2004)
Habchi, W., Issa, J.S.: An exact and general model order reduction technique for the finite element solution of elastohydrodynamic lubrication problems. J. Tribol. 139(5), 051501 (2017)
Moes, H.: Optimum similarity analysis with applications to elastohydrodynamic lubrication. Wear 159, 57–66 (1992)
Venner, C.H., Bos, J.: Effects of lubricant compressibility on the film thickness in EHL line and circular contacts. Wear 173, 151–165 (1994)
Habchi, W., Bair, S.: Quantitative compressibility effects in thermal elastohydrodynamic circular contacts. J. Tribol. 135(1), 011502 (2013)
Murnaghan, F.D.: The compressibility of media under extreme pressures. Proc. Natl. Acad. Soc. USA 30, 244–247 (1944)
Wheeler, J.-D., Molimard, J., Devaux, N., Philippon, D., Fillot, N., Vergne, P., Morales-Espejel, G.E.: A generalized differential colorimetric interferometry method: extension to the film thickness measurement of any point contact geometry. Tribol. Trans. 61(4), 648–660 (2018)
Bair, S., Mary, C., Bouscharain, N., Vergne, P.: An improved Yasutomi correlation for viscosity at high pressure. Proc. IMechE J. 227(9), 1056–1060 (2013)
Blok H.: Inverse problems in hydrodynamic lubrication and design directives for lubricated flexible surfaces. In: Proceedings of the International Symposium on Lubrication and Wear, Houston, pp. 1–151. McCutchan Publishing Corporation, Berkeley (1963)
Sperka, P., Krupka, I., Hartl, M.: Analytical formula for the ratio of central to minimum film thickness in a circular EHL contact. Lubricants 6, 80 (2018)
Bair, S., Liu, Y., Wang, Q.J.: The pressure–viscosity coefficient for Newtonian EHL film thickness with general piezoviscous response. J. Tribol. 128, 624–631 (2006)
Jubault, I., Molimard, J., Lubrecht, A.A., Mansot, J.-L., Vergne, P.: In situ pressure and film thickness measurements in rolling/sliding lubricated point contacts. Tribol. Lett. 15(4), 421–429 (2003)
Funding
This work receives no funding.
Author information
Authors and Affiliations
Contributions
WH and PV conceived the project. WH performed the simulations. PV prepared the bibliography and found data for the comparisons. WH and PV wrote the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Habchi, W., Vergne, P. A Quantitative Determination of Minimum Film Thickness in Elastohydrodynamic Circular Contacts. Tribol Lett 69, 142 (2021). https://doi.org/10.1007/s11249-021-01512-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11249-021-01512-z