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Synergy of Viscosity Wedge and Squeeze Under Zero Entrainment Velocity in EHL Contacts

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Abstract

In an elastohydrodynamic lubricated (EHL) contact under Zero Entrainment Velocity (ZEV) condition, surfaces cannot be separated by hydrodynamic lift. In this work, two other phenomena responsible for a film thickness build-up in ZEV contacts are studied using a numerical model. First, the thermal effect called “viscosity wedge” is investigated in steady-state conditions. Second, the “squeeze” effect is described in an environment where dynamic (time dependent) loads are considered. Then, both the viscosity wedge and squeeze effects are considered together. For each one of the two mechanisms, a characteristic time is considered. The ratio of these two times allows the identification of a dominant effect. Depending on this ratio, a prediction is attempted using semi-analytical models describing each effect. For an ideal set of parameters, it is shown that the combination of squeeze and viscosity wedge in EHL contact under ZEV allows for an enhanced performance.

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Fig. 1
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Fig. 6

adapted from Wang et al. [19] (see Eqs. 21 and 22). The numerical values are listed in Table 6

Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

adapted from Wang (see Eqs. 21 and 22) are given as references in red and blue, respectively. The black line corresponds to a composite model (Eq. 27) accounting for three regimes named “Slow”, “Intermediate” and “Fast” loading

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Abbreviations

A :

Steady-state thermal conditions

B :

Transient isothermal conditions

C :

Transient thermal conditions

*:

Semi-analytical formula

1,2:

Solids 1 and 2, respectively

c :

Central—film thickness

f :

Fluid

m :

Minimum—film thickness

min:

Minimum—overtime

a (m):

Dry contact radius(Hertz)

\({a}_{v}\) (K−1):

Parameter for the Murnaghan density formula

\({a}_{\mathrm{C}\mathrm{Y}}\) (−):

Parameter for the Carreau–Yasuda non-Newtonian viscosity formula

\({A}_{1}\) (K):

Coefficient for the WLF viscosity correlation

\({A}_{2}\) (Pa−1):

Coefficient for the WLF viscosity correlation

\({B}_{1}\) (Pa−1):

Coefficient for the WLF viscosity correlation

\({B}_{2}\) (−):

Coefficient for the WLF viscosity correlation

\({C}_{1}\) (−):

Coefficient for the WLF viscosity correlation

\({C}_{2}\) (−):

Coefficient for the WLF viscosity correlation

\({C}_{p}\) (J k g−1 K−1):

Heat capacity

\(E\) (Pa):

Young modulus

\({E}^{^{\prime}}\) (Pa):

Material parameter

F (−):

Variable for the WLF viscosity correlation

\({G}_{\mathrm{C}\mathrm{Y}}\) (Pa):

Parameter for the Carreau–Yasuda non-Newtonian viscosity formula

h (m):

Film thickness

h0 (m):

Rigid body separation

\({h}_{c}^{{\mathrm{W}\mathrm{a}\mathrm{n}\mathrm{g}}^{*}}\) (m):

Transient semi-analytical formula for the prediction of \({h}_{c}\)

\({h}_{m}^{A*}\) (m):

Steady-state semi-analytical formula for the prediction of \({h}_{m}\)

k (W m−1 K−1):

Thermal conductivity

\({K}_{00}\) (−):

Parameter for the Murnaghan density formula

\({K}_{M}\) (−):

Parameter for the Murnaghan density formula

\({K}_{M}^{^{\prime}}\) (−):

Parameter for the Murnaghan density formula

L (−):

Dimensionless Moes parameter

M (−):

Dimensionless Moes parameter

\({n}_{\mathrm{C}\mathrm{Y}}\) (−):

Parameter for the Carreau–Yasuda non-Newtonian viscosity formula

p (Pa):

Pressure

ph (Pa):

Hertz contact pressure

Q (W m−3):

Total heat source

R (m):

Radius of curvature

Req (m):

Equivalent radius of curvature

t (s):

Time

\({t}_{\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}\) (s):

Time at the end of calculation

\({t}_{\mathrm{L}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}}\) (s):

Time at which the reference load is reached, the loading time

\({t}_{T}\) (s):

Characteristic thermal time

\(\overline{t}\) (−):

Dimensionless time

\(\overline{{t_{{C|m,\,{\text{min}}}} }}\) (−):

Dimensionless instant at which the minimum value of the minimum film thickness is observed in transient thermal conditions

\(\overline{t} _{T}\) (−):

Characteristic thermal–transient ratio

T (K):

Temperature

\({T}_{0}\) (K):

External temperature

\({T}_{g}\) (K):

Glass transition temperature

\({T}_{g0}\) (K):

Glass transition temperature at ambient pressure

\({T}_{R}\) (K):

Reference temperature

u (m s−1):

Surface velocity

\({U}_{DH}\) (−):

Dimensionless velocity parameter

w (N m−1):

Load per unit length

\({w}_{i}\) (N m−1):

Initial load per unit length

\({w}_{\mathrm{r}\mathrm{e}\mathrm{f}}\) (N m−1):

Reference load per unit length

\(x, y, z\) (m):

Coordinates

\({\alpha }^{*}\) (Pa−1):

Reciprocal asymptotic isoviscous pressure–viscosity coefficient

\({\beta }_{k}\) (K−1):

Temperature–density parameter

\(\delta\) (m):

Equivalent elastic surface displacement of both solids

\(\eta\) (m):

Dynamic viscosity

\({\eta }_{e}\) (k g s−1 m−2):

Generalised viscosity parameter

\({\eta }_{e}^{^{\prime}}\) (k g s−1 m−3):

Generalised viscosity parameter

\({\lambda }^{{Wang}^{*}}\) (−):

Parameter for the transient central film thickness prediction formula

\({\mu }_{g}\) (Pa  s):

Viscosity at glass transition

v (−):

Poisson coefficient

\(\rho\) (k g  m−3):

Density

\({\rho }_{e}\) (k g  m−3):

Generalised density parameter

\({\rho }_{e}^{^{\prime}}\) (k g s):

Generalised density parameter

\(\rho _{{e^{\prime\prime}}}\) (k g m s):

Generalised density parameter

\({\rho }_{R}\) (k g  m−3):

Reference density

\({\rho }^{*}\) (k g  m−3):

Generalised density parameter

\({\left(\frac{\rho }{\eta }\right)}_{e}\) (m s):

Generalised density/viscosity ratio

\({\tau }_{\mathrm{z}x}\) (Pa):

Shear stress along the x-axis

\({\tau }_{\mathrm{e}}\) (Pa):

Shear stress norm

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Acknowledgements

This work was funded by the “Lubricated Interfaces for the Future” research chair established by INSA Lyon and the SKF company. The authors want to thank the rich discussions with Dr. Jonas Stahl from SKF PS&E during the preparation of the manuscript.

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Authors and Affiliations

  1. Univ Lyon, INSA-Lyon, CNRS UMR5259, LaMCoS, 69621, Lyon, France

    Bilel Meziane, Nicolas Fillot & Guillermo E. Morales-Espejel

  2. SKF Research and Technology Development, 3992 AE, Houten, The Netherlands

    Guillermo E. Morales-Espejel

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  1. Bilel Meziane
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Correspondence to Nicolas Fillot.

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Appendix: Semi-analytical Formula for the Prediction of the Minimum Film Thickness in Steady-State ZEV Contacts

Appendix: Semi-analytical Formula for the Prediction of the Minimum Film Thickness in Steady-State ZEV Contacts

As described in Eqs. 19 and 20, a semi-analytical model was employed to attempt a prediction of the minimum film thickness under steady-state ZEV condition. The given formula was established using the experimental and numerical data measured in [23]. The corresponding study focused on the minimum film thickness in contacts between a sapphire disk and a steel barrel under ZEV condition. With \(293.15 \mathrm{K}\) boundary temperature, the relative difference between the model and the experimental results stays below \(16\%\). This study showed that in the present range of conditions, steel on steel contacts are similar to sapphire on steel ones and their respective minimum film thicknesses are close enough. Complementary calculations were conducted at lower loads in order to expand the range of the study. For material data on the lubricant and steel, refer to Tables 1 and 2, respectively. The material data for sapphire are listed in Table 10. The remaining input conditions are listed in Table 11.

Table 10 Material properties for the sapphire disk used in [23]
Full size table
Table 11 Input data for the experimental and numerical campaign initially used in [23]. For each configuration, experimental and numerical results were obtained, except for the two lower loads noted (*) for which only numerical results were obtained
Full size table

Using these values, a curve-fit was established using the method of least squares, as given in Eqs. 28 and 29. This model is confronted to the numerical and experimental results in Fig. 

Fig. 12
figure 12

Ratio of the minimum film thickness over the value of \({h}^{\mathrm{*}}\) as a function of the ratio of the velocity over \({u}^{\mathrm{*}}\)

Full size image

12. The coefficient of determination for the experimental dataset is \(0.96\); for the numerical dataset, it is \(0.98\).

$$\frac{{h}_{m}^{A*}\left(u,w\right)}{{h}^{*}}={w}^{0.146}\times (1-\mathrm{e}\mathrm{x}\mathrm{p}(-u/{u}^{*}))$$
(28)
$$\frac{{u}^{*}}{{u}_{0}}=1+\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{w}{{w}_{0}}\right)$$
(29)

where \(w\) is expressed in \(\mathrm{N}\,{\mathrm{m}}^{-1},\)\({h}^{*}=69.8\, \mathrm{n}\mathrm{m},\)\({u}_{0}=2.74\, \mathrm{m} {\mathrm{s}}^{-1},\)\({w}_{0}=14695\, \mathrm{N} {\mathrm{m}}^{-1}\).

The factor \({\left(\frac{{R}_{\mathrm{e}\mathrm{q}}}{{R}^{*}}\right)}^{-0.42}\) found in Eq. 19 was determined by the method of least squares, using the simulation results of Case A and the partial model given in Eqs. 28 and 29. Because the exponent \(-0.42\) was determined with only two radii, and because these radius values are so close (\(0.01\,\mathrm{m}\) and \(0.0128\,\mathrm{m}\)), extreme precaution should be taken when using the formula in Eq. 19 if one would apply it to very different radius values.

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Meziane, B., Fillot, N. & Morales-Espejel, G.E. Synergy of Viscosity Wedge and Squeeze Under Zero Entrainment Velocity in EHL Contacts. Tribol Lett 68, 74 (2020). https://doi.org/10.1007/s11249-020-01311-y

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