Some Analytical Approximations and Numerical Solutions of EHL Problems for Non-Newtonian Fluids Fatigue

Plane steady isothermal and non-isothermal EHL problems for non-Newtonian lubricants in a line contact are considered. Two cases of lubricant rheologies are studied. Namely, lubricants for which (a) the shear stress is any explicitly given nonlinear function of the shear strain and (b) the shear strain is any explicitly given nonlinear function of the shear stress are analyzed. The isothermal EHL problem is reduced to solution of a nonlinear equation for the sliding shear stress f, generalized Reynolds equation for pressure p, and the equation for gap h as well as the classic balance and boundary conditions. The non-isothermal EHL problem is reduced to solution of the above mentioned nonlinear equations and the nonlinear equations for temperatures of the lubricant T and of the contact surfaces Tw1 and Tw2. The EHL problems are considered in the case of heavily loaded contact when the rolling shear stress in lubricant is much smaller than the lubricant sliding shear stress [17]. Therefore, the problems contain a small parameter represented by the ratio of the characteristic rolling and sliding shear stresses. That leads to the opportunity to use the perturbation methods for simplifying the EHL problem formulations. In the isothermal case using the perturbation methods an analytical two-term asymptotic expression for the sliding shear stress f as a function of pressure p, gap h, lubricant temperature T, and lubrication exit film thickness he is obtained [2-6]. In the non-isothermal case using the perturbation methods analytical two-term solutions for the sliding shear stress f and the lubricant temperature T are obtained [2,3,7]. These two-term solutions for the temperature T are used to simplify the integral equations for the surface temperatures Tw1 and Tw2 and to obtain the corresponding two-term perturbation solutions for Tw1 and Tw2. Furthermore, the two-term solutions for f and T are used to simplify the generalized Reynolds equation and to reduce it to the form similar to the one in the isothermal case [2,3,7]. The simplified EHL problems that include the reduced Reynolds equations and the gap equation are analyzed asymptotically [3-8] and solved numerically for some specific lubricant rheologies such as the Newtonian, the Ostwald-de Waele (“power law”), and the Reiner-Philippoff-Carreau models. The asymptotic analysis of the simplified EHL problems is based on the fact that the simplified EHL problem formulation contains a small dimensionless parameter such as the reciprocal of the product of the viscosity pressure coefficient and the Hertzian pressure and/or dimensioless speed-loading parameter. The unified method of obtaining formulas for lubricant film thickness for different loading regimes including starved and fully flooded conditions is proposed. Some analytical formulas for lubricant film thickness and shear stress in a heavily loaded contact are obtained [8]. These formulas depend on both the lubricant rheology and the dependence of the lubricant viscosity on pressure and temperature. The discussion of the way the pressure viscosity coefficient should be chosen is provided. These methods and formulas are specified for some examples of lubricants with nonNewtonian rheologies. The dependence of the film thickness on lubricant rheology and problem parameters is analyzed.