INTRODUCTION: Surface roughness has a well-acknowledged impact on lubricated contacts, especially in those operating in severe conditions. Understanding the influence of the microgeometry of the surfaces in contact on Elastohydrodynamic (EHD) lubricated pairs is essential for the design of improved bearings. From the numerical point of view this constitutes a challenge, given the physics involved and the difference in scales imposed by the geometry of the bearing and the one of the surface roughness. This problem has been handled in two ways: a direct resolution of the problem up to the fine scale, which is called in the literature the deterministic approach, and also by means of averaged methods. Stochastic methods, averaging techniques and flow factors emerged to solve this drawback and decouple the fine scale from the coarse scale, thus reducing the computational cost. From those, the most commonly adopted are Patir and Cheng's1 flow factors (PC), which are still commonly used in EHD lubrication. All of these techniques are heuristic solutions to the formal approach which is homogenization. Averaged equations are developed with its coefficients being computed in periodic cells (the so-called local problems) with the dimensions of the roughness wavelength. Furthermore, the flow factors can be formally defined through homogenization techniques.
Even though the surface roughness can be smaller in some orders of magnitude compared to the dimensions of the contact, the asymptotic assumption –that is, that of an infinitesimal wavelength- can lead to significant differences compared to the fine-scale solution. Homogenized asymptotic models in EHD lubrication are widespread in the literature, the first results being due to Bayada et al2,3. As shown by Venner and Lubrecht4, high frequency roughness is almost undeformed by the fluid pressure, while the large wavelengths are largely affected. In an effort to incorporate the effects of a not-so-small wavelength, some authors have considered the deformations taking place in the local problems mostly through heuristic approaches5, 6 or through FE2-type homogenization techniques7. A precise definition of the homogenized EHD problem with finite-wavelength roughness and the corresponding flow factors is given by Scaraggi et al8, however, they developed their approach for low contact pressures, and hence the lubricant properties are pressure-independent. In this work we present a homogenized model for EHD lubricated contacts that takes into account piezoviscous effects and density variations with pressure, and where the size of the surface roughness is assumed to be non-infinitesimal. Developments led to non-linear local problems, and thus to a new definition of the flow factors which encompasses the classic linear one. The problem of precomputing this newly defined flow factors is addressed, thus decoupling the coarse from the fine scale in an effective manner.
PROBLEM STATEMENT AND MODEL: We aim to solve rough lubricated contacts in the EHD regime considering piezoviscous effects and density variations with pressure. Cavitation effects are to be taken into account too. The equations solved are the elastostatic equation, the Reynolds equation with cavitation effects through a penalization method and the load balance equation. The roughness on the surfaces is assumed periodic, hence homogenization techniques can be applied. The deterministic EHD formulation is presented as well as the homogenized non-asymptotic one, while differences with the classical homogenized asymptotic models are discussed.
RESULTS: The equations were discretized by means of the Finite Element method. Sensitivity tests were performed on the parameters governing the problem, showing low errors both in pressure and clearance. Figure 1 shows the solution of a case obtained with precomputed flow factors (A – solving 288 local problems, B-72 local problems), as well as the fine scale reference solution and a homogenized non-asymptotic solution solved in a FE2 manner. An excellent fit can be seen both in pressure and clearance for the solution with precomputed flow factors A.