In this work, we are mainly interested in a particular lubrication regime known as elastohydrodynamic (EHD). In this regime, the contacting bodies are separated by a full lubricant film. The pressure generated in the film is high enough to induce a significant elastic deformation of the contacting bodies. Therefore a strong coupling between hydrodynamic and elastic effects is involved. Here we present a finite element full-system approach to model the lubricant flow in such contacts. The elastic problem is modelled by applying the classical linear elasticity equations to a structure with large enough dimensions compared to the contact size. As for the hydrodynamic problem, the Reynolds [1] equation for thin film Newtonian flows is solved on a part of the boundary of the structure corresponding to the contact area. This equation is a simplified version of the Navier-Stokes equations dedicated to thin film flows where the pressure gradient across the film thickness can be neglected. It is highly non-linear since the rheological properties of lubricants show a high dependence on pressure. The two problems are solved simultaneously by means of a Newton-like procedure. This leads to high convergence rates of the solution especially when compared to semi-system approach based models where the two problems are solved separately and an iterative procedure is established between them. At the exit of the contact, a free boundary problem arises. It is handled by applying the penalty method. For highly loaded contacts, the standard Galerkin solution of the Reynolds equation exhibits an oscillatory behaviour. This is explained by writing the Reynolds equation as a convection-diffusion equation as a function of pressure. When the load is highly increased, this equation becomes convection dominated and requires streamline upwinding techniques (Streamline Upwind Petrov-Galerkin [2] / Galerkin Least Squares [3]) in order to stabilize the solution.

Finally, the non-Newtonian behaviour of the lubricant is taken into account along with the heat generation that takes place in the lubricant film due to both shear forces and the lubricant's compressibility.

1

Reynolds, O., "On the Theory of the Lubrication and its Application to Mr Beauchamp Tower's Experiments, Including an Experimental Determination of the Viscosity of Olive Oil", Philos. Trans. R. Soc., 177, pp.157-234, 1886. doi:10.1098/rstl.1886.0005

2

Brooks, A.N., and Hughes, T.J.R., "Streamline-Upwind/Petrov-Galerkin Formulations for Convective Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations", Comp. Meth. Appl. Mech. Engnrg., 32, pp.199-259, 1982. doi:10.1016/0045-7825(82)90071-8

3

Hughes, T.J.R., Franca, L.P., and Hulbert, G.M., "A New Finite Element Formulation for Computational Fluid Dynamics: VII. The Galerkin-Least Squares Method for Advective-Diffusive Equations", Comp. Meth. Appl. Mech. Engnrg., 73, pp. 173-189, 1989. doi:10.1016/0045-7825(89)90111-4

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