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A Quantitative Determination of Minimum Film Thickness in Elastohydrodynamic Circular Contacts


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.

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Availability of Data and Material (Data Transparency)

The authors declare that all data supporting the findings of this study are available within the article.


\(a\) :

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] (Pa1)

\({\alpha }_{film}\) :

General pressure viscosity coefficient for film forming, according to Bair [37] (Pa1)

\({\beta }_{K}\) :

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 m3)

\({\rho }_{0}\) :

Lubricant density at ambient pressure (kg m−3)


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Author information




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

Correspondence to Philippe Vergne.

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Appendix A: Dimensionless Central and Minimum Film Thicknesses

See Tables 3 and 4.

Table 3 Central dimensionless film thickness (\({\mathrm{H}}_{\mathrm{c}})\) as a function of the dimensionless parameters M and L
Table 4 Minimum dimensionless film thickness \({(\mathrm{H}}_{\mathrm{m}})\) as a function of the dimensionless parameters M and L

Appendix B: Published \({{\varvec{h}}}_{{\varvec{c}}}/{{\varvec{h}}}_{{\varvec{m}}}\) Ratios

Chevalier Table [12]

See Table 5.

Table 5 Variations of the \({\mathrm{h}}_{\mathrm{c}}/{\mathrm{h}}_{\mathrm{m}}\) ratio as a function of \(\mathrm{M}\) and \(\mathrm{L}\) according to Chevalier [12]

Sperka, Krupka and Hartl [36]

See Table 6.

Table 6 Variations of the \({\mathrm{h}}_{\mathrm{c}}/{\mathrm{h}}_{\mathrm{m}}\) ratio as a function of \(\mathrm{M}\) and \(\mathrm{L}\) according to Sperka, Krupka and Hartl [36], limited to the case where \({\alpha}_{\mathrm{film}}=20.6 {\mathrm{GPa}}^{-1}\) and for \(\mathrm{M}\ge 10\) and \(\mathrm{L}\le 20\)

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Habchi, W., Vergne, P. A Quantitative Determination of Minimum Film Thickness in Elastohydrodynamic Circular Contacts. Tribol Lett 69, 142 (2021).

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  • Minimum film thickness
  • Elastohydrodynamic lubrication (EHL)
  • Circular contacts
  • Central to minimal film thickness ratio