Surprisingly, when two lubricated solids are heavily loaded against one another and when a motion is applied to the solids surfaces, these two solids may not be touching one another (see Figure 1, left)! This interesting fact of nature was discovered in the early 20^{th} century, when lubricated systems designed to last a few years happened to last much longer.

You may also have experienced something similar while driving fast on a flooded road (see Figure 1, centre) or while walking on a smooth and wet surface (see Figure 1, right). In both cases, you may have faced a scary slip. If this did not happen to you, no worries, it is better this way. You obviously did not experience floating but the presence of a liquid film that separated your vehicle or body from the ground, literally lifting up your mass and reducing dramatically the friction that usually allows you to control your vehicle or body. This film separation is the result of hydrodynamic pressure* and can occur between two rigid bodies; it is hydrodynamic lubrication (HL). However, the lubricated systems mentioned earlier (your car tyres and your foot/sole) have something in common: they deform under load and when this deformation has a similar order or magnitude to the film thickness, it is called elastohydrodynamic lubrication (EHL) and it also results in a film separation between bodies. This full separation even under load is often something looked after to design almost wear-less systems with reduced friction.

Figure 1 – Diagram of steel-steel oil lubricated contact (left), wheel on road aquaplaning (centre) and human foot slipping on a wet smooth floor (right) – liquid film thickness are oversized for clarity purposes.

However, the film thickness is not a constant value all over the contact area and predicting it is not an easy task. Indeed, elastohydrodynamic theory predicts that the contact exit and the contact sides have a smaller film thickness than the centre. The Figure 2 (bottom) represents the raw picture of a EHL film thickness measurement through white light interferometry method. One can observe that the film thickness at the centre () is larger than on the sides where the minima () can be found. Moreover, the minimum film thickness along the central line () is also smaller than (), but in the present case it is larger than (). In steel-steel contacts the gap is typically in the 10 nm – 1 µm range, which makes it at least about 50 times smaller than a human hair diameter!

Figure 2 – Film thickness in and around a torus-on-plane elastohydrodynamic contact with lubricant inlet on the left – numerical prediction (top) and experimental measurement (bottom) comparison (picture from Wheeler thesis ^{1})

Another film thickness variation comes from the surface roughness. Roughness is often the result of the device machining and finishing operation. Even with the most advance processes, perfectly flat surfaces do not exist. When the average roughness has a similar amplitude with the gap within the contact, solid to solid contact may occur. This contact may lead to surface damage, wear and ultimately to an unwanted breakdown of the mechanism.

# How to measure it?

Because of the damage risk it is very important to know the actual distance separating the solid surfaces. However, things become tricky when one wants to measure sub-micrometric distances that are literally hiding between high velocity heavily loaded strong components. In laboratories, it is possible to achieve such challenge, and a wide range of methods have been used as presented by Albahrani et al. ^{2}. Some have measured the electrical resistance ^{3} of the contact or the capacitance ^{4} formed by the two steel surfaces separated by the oil film. The results were reliable but not sufficiently accurate and could not distinguish the minimum film thickness from the central film thickness. The other way to measure such distance is to use electromagnetic radiation-based methods. Apart from a few uses of X-Rays and infra-red technics, optical interferometry is the most widely spread method to measure the gap between two solids. After a first description in 1962 by Kirk ^{5}, the experimental process has been improved over the years and it now allows a measurement precision of a few nanometres ^{6,7}! This method is the one used in Figure 2. However, this accuracy is only possible in laboratory conditions and does not prevail in industrial contacts. Tracking the lubrication quality in an industrial machinery through a real-time film thickness measurement is unattainable.

Thankfully, the above-mentioned measurements have been wisely used in the last decades to assess the accuracy of physics-based film thickness calculations.

# How to calculate it?

In the late 60s, Dowson and Higginson ^{8} were the first to publish a complete numerical solution of an EHD contact problem that was valid under some assumption (mainly an infinitely wide contact). In the mid-70s, this work has been extended to circular and elliptical contact (known as point contacts) by Hamrock and Dowson ^{9–11}. Other works have since then stretched the limits of the cases solved and removed hypothesis on the solids shape, the thermal and/or rheology effects ^{12}. But because the computations are still time consuming and require hardware resources, different studies proposed semi-analytical formulae based on dimensionless numbers. These formulae have physics-based equation shape, whereas the coefficients are obtained by minimizing the equation result when comparing them to the computation result. The oldest and most widely spread formulae for central and minimum film thickness are Hamrock and Dowson’s ^{9–11}:

(1)

with the dimensionless speed parameter, the dimensionless material parameter, the dimensionless load parameter, the reduced radius of curvature in the entrainment direction, the reduced radius of curvature perpendicular to the entrainment direction, the lubricant dynamic viscosity, the mean entrainment velocity, the velocity in the direction of surface , the reduced modulus of elasticity, and respectively the Young modulus and the Poisson ratio of the solid , the reciprocal asymptotic iso-viscous pressure according to Blok [I] and the contact normal load.

One can also mention the work of Evans & Snidle^{13}, Moes&Venner who proposed a very handy pair of dimensionless numbers ^{14}, Nijenbanning et al. ^{15} and Masjedi & Khonsari ^{16}. However, it appears that the formulae proposed by Chittenden et al. ^{17} are versatile and interestingly accurate ^{18} despite its lower range of numerical cases:

(2)

For more precision, different formulae should be used depending on the use cases ^{19}. This means that it is nowadays possible to establish relevant film thickness predictions at an early step of a device design provided that the engineer has the knowledge of the properties of the solids and lubricant. A little disclaimer is relevant at this stage: the formulae reported here are valid under isothermal, fully flooded contact inlet, Newtonian lubricant, isotropic elastic solid, smooth surfaces, and bulk lubricant assumptions. There are many assumptions here, and they are hardly ever respected altogether which is very frustrating. As a result, they should be used with care even if they remain very useful.

Now arises the question: what should you do with such a prediction of the film thickness? At which threshold can you consider that the wear will become negligible? For instance, is sufficient?

# How to use it?

When one knows the minimum film thickness of a contact, it should be at least confronted to the roughness of the mating solids. Wear and lubrication can obviously not be summarized by a ratio, but the value of the dimensionless film parameter is an indicator of the lubrication regime and according to Hamrock et al. ^{20} it reads:

(3)

with the root mean square roughness of the solid .

This dimensionless film parameter approximately distinguishes 4 lubrication regimes ^{20}:

- Boundary lubrication,
- Partial lubrication,
- Elastohydrodynamic lubrication,
- Hydrodynamic lubrication,

A large leads to low occurrence of solid to solid contact and to almost no wear. However, the targeted lubrication regime is often the elastohydrodynamic lubrication as it encounters a minimum wear but still a small film thickness. The latter shows that the oil has been well selected: viscous enough to provide a sufficient hydrodynamic lift, but not too viscous as it would increase friction forces.

Figure 3 – Lubrication regimes: hydrodynamic (left), elastohydrodynamic (centre) and partial to boundary lubrication regimes (right)

Once again, there are many assumptions in the calculation of . Among them, the surfaces are supposed to be smooth which is obviously not true in the partial and boundary lubrication regimes.

The warnings presented here should not discourage the reader even if the outcome can only be considered as a qualitative result. However, the method proposed in this blog is relevant, widely used and provides very useful predictions. As noted after comparing the different semi-analytical formulae ^{19} it is often mandatory to run dedicated numerical studies to obtain quantitative predictions.

### Notes

* I think the role of water hydrodynamic pressure in human foot slipping is clearly questionable, at least at the beginning of the motion. This example is maintained for pedagogical purposes as it derives from daily life and includes film separation, relative motion, and body deformation.

# Bibliography

1. Wheeler, J.-D. Non-Elliptical Point Contacts: the Torus-on-Plane Conjunction. (INSA de Lyon, 2016). http://theses.insa-lyon.fr/publication/2016LYSEI131/these.pdf

2. Albahrani, S. M. B., Philippon, D., Vergne, P. & Bluet, J. M. A review of in situ methodologies for studying elastohydrodynamic lubrication. *Proc IMechE Part J J Eng. Tribol.* (2015) doi:10.1177/1350650115590428.

3. Lane, T. B. & Hughes, J. R. A study of the oil-film formation in gears by electrical resistance measurements. *Br. J. Appl. Phys.* **3**, 315 (1952).

4. Crook, A. W. The lubrication of rollers. *Philos. Trans. R. Soc. London* **250**, 387–409 (1958).

5. Kirk, M. T. Hydrodynamic lubrication of ‘Perspex’. *Nature* **194**, 965–966 (1962), https://doi.org/10.1038/194965a0.

6. Hartl, M. *et al.* Thin Film Colorimetric Interferometry. *Tribol. Trans.* **44 (2)**, 270–276 (2001).

7. Wheeler, J.-D. *et al.* A Generalized Differential Colorimetric Interferometry Method: Extension to the Film Thickness Measurement of Any Point Contact Geometry. *Tribol. Trans.* **61**, (2018), https://doi.org/10.1080/10402004.2017.1386809.

8. Dowson, D. & Higginson, G. R. A numerical solution to the elasto-hydrodynamic problem. *J. Mech. Eng. Sci.* **1**, 6–15 (1959).

9. Hamrock, B. J. & Dowson, D. Isothermal elastohydrodynamic lubrication of point contacts – Part 1 – Theoretical Formulation. *J. Lubr. Technol.* **98**, 223–228 (1976).

10. Hamrock, B. J. & Dowson, D. Isothermal elastohydrodynamic lubrication of point contacts – Part 2 – Ellipticity parameter results. *J. Lubr. Technol.* **98**, 375–383 (1976).

11. Hamrock, B. J. & Dowson, D. Isothermal elasto-hydrodynamic lubrication of point contacts – Part 3 – Fully Flooded Results. *ASME J. Tribol.* **99**, 264–276 (1977).

12. Habchi, W., Eyheramendy, D., Vergne, P. & Morales-Espejel, G. E. A full-system approach to the elastohydrodynamic line/point contact problem. *J. Tribol.* **130(2)**, 21501–21510 (2008).

13. Evans, H. P. & Snidle, R. W. The Isothermal Elastohydrodynamic Lubrication of Spheres. *J. Lubr. Technol.* **103(4)**, 547–557 (1981).

14. Moes, H. Optimum similarity analysis with applications to elastohydrodynamic lubrication. *Wear* **159**, 57–66 (1992).

15. Nijenbanning, G., Venner, C. H. & Moes, H. Film thickness in elastohydrodynamically lubricated elliptic contacts. *Wear* **176**, 217–229 (1994).

16. 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**, 228–244 (2015).

17. 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. London* **A397**, 271–294 (1985).

18. 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. Inst. Mech. Eng., Part J J. Eng. Tribol.* **223(8)**, 1143–1163 (2009).

19. 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**, 369–379 (2016), https://doi.org/10.1007/s40544-016-0133-7.

20. Hamrock, B. J., Schmid, S. R. & Jacobson, B. O. *Fundamentals of Fluid Film Lubrication*. (2004).

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