A Mortar Finite Element Formulation for Large Deformation Lubricated Contact Problems with Smooth Transition Between Mixed, Elasto-Hydrodynamic and Full Hydrodynamic Lubrication

This work proposes a novel model and numerical formulation for lubricated contact problems describing the mutual interaction between two deformable 3D solid bodies and an interposed fluid film. The solid bodies are consistently described based on nonlinear continuum mechanics allowing for finite deformations and arbitrary constitutive laws. The fluid film is modelled as a quasi-2D flow problem on the interface between the solids governed by the (thickness-)averaged Reynolds equation, which relates pressure to velocity and film thickness, with the latter two fields being provided by the averaged surface velocity and gap profile of the interacting solid bodies. The averaged Reynolds equation accounts for surface roughness utilizing spatially homogenized, effective fluid parameters and for cavitation through a positivity constraint imposed on the pressure field. In contrast to existing approaches, the proposed model accounts for the co-existence of frictional contact tractions and hydrodynamic fluid tractions at every local point on the contact surface of the interacting bodies and covers the entire range from boundary lubrication to mixed, elastohydrodynamic, and eventually to full film hydrodynamic lubrication in one unified modelling framework with smooth transition between these different regimes. Critically, the model relies on a recently proposed regularization scheme for the mechanical contact constraint combining the advantages of classical penalty and Lagrange multiplier approaches by expressing the mechanical contact pressure as a function of the effective gap between the solid bodies while at the same time limiting the minimal gap value occurring at the (theoretical) limit of infinitely high contact pressures. From a methodological point of view, this is the key ingredient to regularize the pressure field in the averaged Reynolds equation, i.e., to avoid the pressure field’s singularity in the limit of vanishing fluid film thickness, and thus to enable a smooth transition between all relevant lubrication regimes. From a physical point of view, this approach can be considered as a model for the elastic deformation of surface asperities, with a bounded magnitude depending on the interacting solids’ surface roughness. The finite element method is applied for spatial discretization of the 3D solid-mechanical problems and the 2D interface effects, consisting of the averaged Reynolds equation governing the fluid film and the non-penetration constraint of the mechanical contact problem, the latter relying on variationally consistent mortar methods for contact traction discretization. A consistent and accurate model behavior is demonstrated and validated by employing several challenging and practically relevant benchmark test cases. The ability of the model to accurately represent the velocity-dependent friction coefficient of the different lubrication regimes (i.e. mixed, elasto-hydrodynamic and full film lubrication) according to the well-known Stribek curve is also demonstrated via test cases. Eventually, a parametric study is performed to analyze the effect of regularization parameter choice.