A non-iterative numerical approach for visco-elasto-hydrodynamic lubrication problems

In lubrication applications, a common scenario involves a hard solid in contact with a soft viscoelastic substrate. However, most of the existing literature focuses on the case of an elastic substrate. In this paper, we introduce a numerical method that overcomes the challenges of converging iterative techniques and is specifically designed to handle viscoelastic substrates described by a Prony series. Our approach is fully automated, stable, and efficient, requiring only the solution of a linear matrix equation at each time step. We apply this method to investigate the transient squeezing of a thin liquid film between a rigid spherical indenter and a soft viscoelastic substrate. We explore intriguing differences between the EHL (Elasto-hydrodynamic lubrication) problem and three viscoelastic substrates with single and multiple relaxation times to understand how temporally evolving stiffness affects the pressure, surface displacement, and liquid film thickness. One of the key differences between the EHL and the VEHL (Visco-elasto-hydrodynamic lubrication) problem comes to light upon looking at the entrapped volume which can be held nearly constant when using a viscoelastic substrate whereas elastic substrates show exponential decay.