The size-dependent elastohydrodynamic lubrication contact of a coated half-plane with non-Newtonian fluid

Based on the couple-stress theory, the elastohydrodynamic lubrication (EHL) contact is analyzed with a consideration of the size effect. The lubricant between the contact surface of a homogeneous coated half-plane and a rigid punch is supposed to be the non-Newtonian fluid. The density and viscosity of the lubricant are dependent on fluid pressure. Distributions of film thickness, in-plane stress, and fluid pressure are calculated by solving the nonlinear fluid-solid coupled equations with an iterative method. The effects of the punch radius, size parameter, coating thickness, slide/roll ratio, entraining velocity, resultant normal load, and stiffness ratio on lubricant film thickness, in-plane stress, and fluid pressure are investigated. The results demonstrate that fluid pressure and film thickness are obviously dependent on the size parameter, stiffness ratio, and coating thickness.


Introduction
Coatings on the contact surface can reduce mismatching stresses, resist contact damage, improve the wear resistance and corrosion resistance, and possess thermal shielding functions. Therefore, coatings are commonly applied in various devices, such as turbines, transducers, actuators, bearings, gears, furnace liners, sensors, micro-electronics, and micro-power generators [1][2] . In addition, coatings can also be used to improve surface performance of elastohydrodynamic lubrication (EHL) contact. In general, lubricants and coatings are employed together to improve contact damage and wear of the material surface. Hence, the size-dependent EHL line contact of a homogeneous coated half-plane. The viscosity and density of non-Newtonian fluid lubricant are dependent on the fluid pressure. The generalized stress function method is used to calculate the normal displacement on the surface for size-dependent dry contact of a coated half-plane. Subsequently, an iterative method is used to handle flow rheology equation, film thickness equation, load balance equation, and Reynolds equation. Influences of punch radius, size parameter, coating thickness, slide/roll ratio, entraining velocity, resultant normal load, and stiffness ratio on EHL contact behaviors are illustrated.
This paper makes the first attempt to analyze the size effect on the EHL line contact between a coated half-plane and a rigid cylindrical punch. We highlight new aspects as follows. (i) A size-dependent EHL contact model is first developed in this paper. This model can be used to evaluate the effect of size parameter on the fluid pressure, in-plane stress, and film thickness. (ii) The effect of non-Newtonian fluid is considered in size-dependent EHL contact for the first time. By using the Ree-Eyring model, the viscosity loss effect of non-Newtonian fluid can be described. The influence of slide/roll ratio on fluid pressure, frictional coefficient, and shear stress at coating surface is illustrated. Figure 1 illustrates a homogeneous elastic coated half-plane for the size-dependent EHL line contact. The lubricant is supposed to be the non-Newtonian fluid. P and R are the normal load and punch radius. The homogeneous elastic half-plane is perfectly bonded to the coating with the thickness H. Sliding velocities are V 1 and V 2 for punch and half-plane, respectively.

Theoretical formulations
Homogeneous elastic half-plane

Couple-stress elasticity theory
The geometric equations between strain components (ε xxi , ε yyi , ε xyi , and ε yxi ) and displacement components (u xi and u yi ) can be found in Eq. (2) of Ref. [23] within the two-dimensional linearized couple-stress elasticity theory in the plane stain state. The geometric equations for the homogeneous half-plane and coating about the rotation ω i = ω zi are expressed as where superscripts i = 1, 2 represent the coating (0 < y H) and half-plane (y 0), respectively; k xzi and k yzi are the curvature tensor components.

Reynolds equation, load balance equation, and rheology equation
A Ree-Eyring fluid with the nonlinear viscous model is used as the lubricant between two solids. The rheological law of the Ree-Eyring fluid is described as [32,36] ∂V ∂y where V is the velocity component along the x-direction; ∂V ∂y is the velocity gradient (i.e., shear strain rate); τ = τ x is the shear stress component; η is the fluid viscosity. The Eyring stress τ 0 is irrelevant to the temperature and fluid pressure, and indicates the transformation from Newtonian fluid to the non-Newtonian fluid. When τ 0 is infinite, the Ree-Eyring fluid is changed to the Newtonian fluid.
The Reynolds equation was modified to describe the non-Newtonian fluid by Yang and Wen [38] . The generalized Reynolds equation of the EHL line contact is obtained, i.e., d dx where V 0 = (V 1 + V 2 )/2, p, ρ, and h are the entraining velocity, fluid pressure, fluid density, and film thickness, respectively. We have the following relations: is the slide-roll ratio, changing from 0 (pure rolling conditions) to 2 (pure sliding conditions). For the Newtonian fluid, i.e., η * = η, Eq. (6) is reduced to the one given by Dowson [39] . For the Ree-Eyring fluid, Yang and Wen obtained the equivalent viscosity η * [38] , where τ 1 is the shear stress at coating surface, The Dowson-Higginson pressure-fluid density relation and Roelands pressure-viscosity relation (i.e., rheology equation) can be found in Eqs. (12) and (15) of Ref. [40]. In these equations, fluid viscosity η 0 and fluid density ρ 0 are constants at the ambient temperature and pressure; Z = α/(5.1 × 10 −9 (9.67 + ln η 0 )) is the pressure-viscosity index with α being the pressure-viscosity coefficient [40] .
Boundary conditions for p are [38] p(x out ) = p(x in ) = dp dx (x out ) = 0, where x in and x out are fluid inlet and outlet of the EHL contact region, respectively. In general, the EHL contact region (x in , x out ) includes the dry contact region (−a, a). The fluid inlet x in is generally assumed to be known. x in 4a is chosen to guarantee sufficient oil supply [41] . At this time, x in has little influence [41] . We choose x in = −4.759 8a in this work [40] . But the outlet x out should be solved. For the EHL contact, the load balance equation requires

Film thickness equation
To derive the film thickness equation, we should first obtain displacement u y1 (x, H) of the coating [40] . Figure 2 shows the size-dependent dry contact with contact region −a x a.
Homogeneous elastic half-plane Using the generalized stress function method, the surface displacement u y1 (x, H) and surface in-plane stress σ xx1 (x, H) of coated half-plane are obtained in the following integral forms [23] : where [14,23] ; s is the Fourier integral transform variable; p 0 (x) is the normal dry contact pressure; the expression of m 21 (s, H) and m 11 (s, H) can be found in Eq. (44) of Ref. [23]. For the classical elastic dry contact (l = 0), we get f 21 = (1 − ν 1 )/µ 1 and f 11 = (1 − 2ν 1 )/(2µ 1 ) [42] . According to the relationship [43] ∞ 0 Eq. (9) can be simplified as where )/sds. For the EHL contact, the fluid pressure is denoted as p(x). By using the parabolic approximation for a cylindrical punch, and according to the surface displacement u y1 (x, H) of dry contact in Eq. (12), the lubricant film thickness could be obtained as [40,42] h where γ 0 is the rigid displacement at x = 0.

Methodology
3.1 Computing lubricant film thickness By using uneven mesh discrete nodes x in = x 0 < x 1 < · · · < x M = x out and Eq. (13), the film thickness is discretized as [40] where with M being an even integer. The matrices Q and Λ are determined by computing (13), respectively. By applying the interpolation function to approximate fluid pressure [45] , δ k is transformed as (21) where 'deformation matrix' Q can be found in Eq. (79) of Ref. [46] by replacing f 1 /(2µ) with f 21 .

Comparison and convergence study
By ignoring the lubricant effect, the present EHL contact is reduced to the dry contact with couple stress. Figure 4 shows dry contact pressure p 0 (x) for P = 20 N/m and H = 0.1 µm, as well as the results by Song et al. [23] . The two results are observed to be in good agreement.
By neglecting the effect of the size parameter, coating, and non-Newtonian fluid, the present EHL contact is reduced to the EHL contact of a homogeneous half-plane in the classical elasticity theory. Figure 5 illustrates the film thickness and fluid pressure with l = 0, H = 0, R = 20 mm, V 0 = 0.6 m/s, P = 80.8 kN/m, S = 0, µ 2 = 41 GPa, and ν 2 = 0.34. Meanwhile, the results by Yang and Wen [40] are shown for comparison. Again, both results are found to be in good agreement. The influence of the discrete node number M on h(x) and p(x) is analyzed in Fig. 6. With the increase of M , the results of h(x) and p(x) tend to converge. h(x) and p(x) are almost the same for M = 160, M = 180, and M = 200. Thus, the discrete node number is chosen as M = 180 in this paper. Figure 7 illustrates the variation of h(x) and p(x) with different materials. Three different materials, i.e., steel (µ 2 = 80.7 GPa and ν 2 = 0.3), copper (µ 2 = 41 GPa and ν 2 = 0.34), and aluminum (µ 2 = 27.3 GPa and ν 2 = 0.3), are considered in this example. Obviously, the change of material type has a significant effect on fluid pressure at the whole contact region. However, it has slight effect on h(x) at the contact center. The pressure spike is sensitive to shear modulus of materials. It is found that aluminum (steel) has the minimum (maximum) pressure spike.

EHL contact analysis
The effect of size parameter on p(x), h(x), and σ xx1 (x, H) is analyzed in Fig. 8. In-plane stress σ xx1 (x, H) in Fig. 8(b) is taken at the surface of the coating, i.e., y = H. As expected, h(x), p(x), and σ xx1 (x, H) are significantly affected by l. Moreover, Fig. 8 also plots the classical result for l = 0. It is clearly observed that the fluid pressure at the whole EHL contact region is enlarged distinctly due to the appearance of the size effect. Therefore, neglect of the size effect could result in the underestimation of the results at micro-/nano-scale EHL. Pressure spike emerging close to the contact outlet enlarges with the increase of l. Size effect reduces h(x) at the inlet, but enlarges it at center and outlet. h(x) gets small at the inlet and center with l increasing from 0.3 µm to 2.0 µm, while it is slightly affected at the outlet. In particular, the outlet with minimum film thickness is prone to severe wear. For the classical elasticity (l = 0), the in-plane stress σ xx1 (x, H) reaches its maximum near the contact center, and it is compressive at all locations. For couple-stress elasticity, σ xx1 (x, H) is still compressive at the whole lubricant region for a small l. But, with the increase of l, compressive stress gradually transforms to tensile stress at the center, and maximum tensile stress appears at the outlet. Because different asymptotic values f 11 are obtained in couple-stress elastic contact (−1/(µ 1 (6 − 4ν 1 ))) [14,23] and classical elastic contact ((1 − 2ν 1 )/(2µ 1 )) [42] . These two values have opposite signs. This explains the change of σ xx1 (x, H) from compressive ones to tensile ones in couple-stress elastic contact with l increasing.  Figure 9 shows the effect of the stiffness ratio µ 1 /µ 2 on h(x) and p(x). When µ 1 /µ 2 changes from 1/1.4 to 1.4, i.e., the coating becomes stiffer and stiffer, p(x) becomes greater at the whole EHL contact region. Particularly, with µ 1 /µ 2 increasing, the pressure spike increases distinctly at the outlet and moves forward slightly. The effect of µ 1 /µ 2 is pronounced on fluid pressure, but slight on film thickness. The increase of µ 1 /µ 2 reduces h(x) at the inlet, and slightly enlarges it at the center and outlet. These results suggest that distributions of film thickness and fluid pressure can be changed by modifying stiffness ratio. Hence, coatings can be effectively employed to improve contact damage of material surfaces in EHL contact.
The effect of coating thickness H on h(x) and p(x) is plotted in Fig. 10. The presence of the coating yields a remarkable increase in fluid pressure, while reduces film thickness at inlet of EHL contact region. However, the coating has little effect on h(x) at center and outlet. With the increase of H from 0 µm to 5 µm, the fluid pressure shows an increasing trend.  Figure 11 illustrates the effect of entraining velocity V 0 on p(x), h(x), and σ xx1 (x, H). As V 0 increases, pressure spike tends to move forward and becomes large, and h(x) increases at the EHL contact region. σ xx1 (x, H) is compressive at the whole lubricant region, and maximum compressive stress appears at the contact center. V 0 has little effect on in-plane stress σ xx1 (x, H). Figure 12 gives the effect of slide/roll ratio S on p(x), h(x) and σ xx1 (x, H). The slide-roll ratio S is very essential in this study. S = 0 implies pure rolling condition, i.e., V 1 = V 2 , while S = 2 for pure sliding condition, i.e., V 1 = 0. Moreover, results of Newtonian fluid are also presented. It is found that results of Newtonian and non-Newtonian fluid models are identical for S = 0. S has a great effect on pressure spike at the outlet, whereas it has little effect on film thickness and in-plane stress. An increase of S yields a remarkable decrease of pressure spike. Interestingly, the similar influencing trend of S can also be found in the study of the classical EHL contact [48][49] .  Figure 13 illustrates the variation of h(x) and p(x) with the resultant normal load P . As P increases, the fluid pressure increases at center, but pressure spike moves backward and decreases. h(x) decreases with P increasing at center and outlet. Figure 14 discusses the effect of punch radius R on h(x) and p(x). With R increasing, h(x) increases but p(x) decreases at the EHL contact region. The increase of R results in the forward movement of pressure spike. Figure 15 plots the effect of slide/roll ratio S on shear stress τ 1 (x) at the coating surface y = H for non-Newtonian fluid. When S changes from 0 to 2, the shear stress τ 1 (x) becomes greater at the whole EHL contact region, and its maximum value appears near the outlet. Note that the numerical results in Figs. 8-10 and 14 are obtained for a constant normal load P . Therefore, the integral of p(x) over the contact region should be a constant (i.e., xout xin p(x)dx = P ). However, an apparent violation is observed in these figures due to the use of normalized contact region (x in /a x/a x out /a), other than the actual contact region (x in x x out ). As we know, a changes for different size parameter l, stiffness ratio µ 1 /µ 2 , coating thickness H or punch radius R when P is fixed. We choose the normalized transverse coordinate as x/a in order to compare the pressure spike and its location.
In the above analysis, the size parameters of the coating and the half-plane are considered as the same value. However, the size parameters of the coating and the half-plane may be different. Therefore, we need to consider the effect of the size parameter ratio on the fluid pressure and film thickness. Figure 17 shows the effect of the size parameter ratio l 1 /l 2 on p(x) and h(x). When l 1 /l 2 changes from 0.5 to 2.0, the fluid pressure p(x) becomes greater at the whole EHL contact region. h(x) gets small at the inlet with l 1 /l 2 increasing, while it is slightly affected at the center and outlet. It should be pointed out that the present paper focuses on the EHL contact of the cylindrical punch based on the couple stress theory. Our results in Fig. 8 show that the fluid pressure increases as the size parameter increases from 0 µm to 0.2 µm. However, Song et al. [23] investigated the dry contact problem for both cylindrical punch and flat punch based on the couple stress theory. Their results for the cylindrical punch have the similar trend with that of the present paper, but the trend for the flat punch is different. For the flat punch, the pressure distribution first departs from and then approaches the classic result as the size parameter increases from 0 µm to 0.2 µm. Similar phenomenon was also observed by Zisis et al. [14] for flat punch and cylindrical punch.

Conclusions
We analyze size-dependent EHL line contact by employing the couple-stress elasticity theory. A deformable homogeneous coated half-plane and the rigid cylindrical punch are separated by a non-Newtonian lubricant. An iterative method is conducted to calculate film thickness, in-plane stress, and fluid pressure. The following results can be obtained.
(I) When the size effect is considered, the fluid pressure is enlarged at the whole EHL contact region, and the film thickness is enlarged at center and outlet, but reduced at the inlet. With the increase of the size parameter, in-plane stress gradually transforms from the compressive to the tensile at the center. The film thickness is slightly affected at the outlet, but decreases at inlet and center as the size parameter increases.
(II) As the stiffness ratio increases, fluid pressure becomes greater at the whole EHL contact region and pressure spike moves forward slightly, and film thickness increases slightly at center and outlet.
(III) The presence of the coating enlarges the fluid pressure at the whole EHL contact region but decreases the film thickness at inlet, and has little influence on film thickness at outlet and center. With the increase of coating thickness, the fluid pressure shows an increasing trend.
(IV) An increase in the slide/roll ratio results in a significant decrease of pressure spike at the outlet, whereas it has little influence on the film thickness.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.