# A bit of context

Let’s say you want to design a new car but that you are a new car manufacturer. You can’t really step back and look at your previous experiences, so everything you design in this car has to be carefully calculated… and you will still make mistakes! Any resemblance to a real car manufacturer is purely coincidental.

So you are designing a new car! You want it to be sufficiently reliable and long lasting but not too heavy. At some point you will have to carefully choose your bearings and other lubricated contacts. To predict their lifespan you will need to identify their load cycles. From this stage onwards, your approach will be multi-scale. Your primary concerns will be the car components (meter scale) and their loads, the driving conditions and the way they are applied to the contacts (millimeter scale in the contact plan). Knowing the contacting bodies properties, you can determine the contact pressure through Hertz theory. Thanks to a Weibull analysis you can then start making your first predictions about the lifespan.

But it may not be sufficient: pressure distribution within lubricated contacts may be very different from Hertz distribution in hydrodynamically or elasto-hydrodynamically lubricated contacts. To take into account this new pressure distribution which may be locally much larger, you will need to include the flow contribution to the pressure and its effect on the elastic bodies: your approach will now be multi-physic. Still, this approach may not be sufficient: the roughness of the surfaces have an influence on the flow and therefore on the pressure distribution. The latter can vary drastically within the contact and affect the tribological system lifespan. Unfortunately, the roughness requires an additional scale: the micrometer scale. To be accurate, each bump and each depression will demand a computation cost similar to the whole contact. Within a single contact you may find several hundreds of those. It’s just not yet possible to include them!

# The homogenisation method

Different homogenisation methods have been developed in the past years and the approach developed by Checo et al. [2] is presented here.

## Introducing the differentiation operator

Checko et al. [2] based their work on the introduction of a multi-scale differentiation operator and they consider periodical roughness:

where is the scale ratio, is the roughness wavelength, is the contact size, is the space variable describing the contact scale (also called slow variable) and is the space variable describing the roughness scale (also called fast variable).

## Deriving the multi-scale Reynolds equations

This differentiation operator is used in the Reynolds equation and replaces the classical operator. As the Poiseuille, the Couette and the squeeze terms expand, different scales start to emerge in the equation: they are distinguished by the coefficients with . This leads to different partial derivative equations and other conditions. The two may outcomes of this development are:

1. a contact scale equation which takes into account the homogenised roughness scale hydrodynamic contributions through the microscopic scale average operator :   with a constitutive parameter, the entrainment velocity, the lubricant density, the lubricant film gap, the homogenised pressure and the microscopic pressure.
2. a roughness scale equation which describes the flow at the vicinity of a single roughness:

The two equations described here are interdependent but together they allow for computing:

• the homogenised pressure and homogenised lubricant film gap
• but also the local microscopic pressure and solid deformation.

The averaged terms in the macroscopic equation stemming from the microscopic equation can be compared to flow factors. They have been used in other approaches, but in the study presented in this blog a formal method is introduced to compute them.

## Relocalisation

The homogenisation theory also introduces the relocalisation. It means that the full hydrodynamic pressure and film thickness can be reconstructed:

Even if the contact is not discretized at a sufficient scale to describe the roughness, it is possible to obtain a rather correct prediction of the actual pressure.

# Method assessment

Checo et al. ran different cases and compared the result of the reconstructed film gap and pressure with the fully discretized rough contact (known as reference). The comparison is made on cases where the latter case can also be computed. The roughness considered is a sine wave. More details about the method evaluation can be found in the full article [2] but the agreement in Figure 2 is impressive. The reconstructed solution closely matches the reference solution.

# To conclude

The method developed by Checo et al. certainly makes roughness contribution more accessible to elastohydrodynamic modelling approach. Previously, the computation cost of such a contribution was unaffordable most of the time. Of course, this paper doesn’t mean it becomes straightforward to design new elastohydrodynamically lubricated contacts, it provides new tools for the engineer to better predict their behaviour. Together with experimental approaches, it is still possible to innovate in the field of mechanics and automotive industry!

[1] Bearing photo by Georg Eiermann on Unsplash

[2] Hugo M. Checo, David Dureisseix, Nicolas Fillot, Jonathan Raisin, A homogenized micro-elastohydrodynamic lubrication model: Accounting for non-negligible microscopic quantities, Tribology International, Volume 135, 2019, Pages 344-354,

ISSN 0301-679X,