Effects of elastic ball and coating on pure squeeze EHL motion for constant load with couple stresses using FDM method

In this paper, an elastic sphere approaching a lubricated elastic coating surface with couple stress lubricant is explored under constant load condition. The finite difference method (FDM), Gaussian elimination, and Gauss-Seidel iteration method are used to solve the transient modified Reynolds equation with its boundary condition, the elasticity deformation equations of ball and coating layer, load balance equation, and lubricant rheology equations simultaneously. The transient pressure profiles, film shapes, and elastic deformation during the pure squeeze process under various operating conditions in the elastohydrodynamic lubrication (EHL) regime are discussed.


Introduction
Many mechanical elements with contact pairs such as gear teeth, cams /followers, piston ring/cylinder, rolling element bearings, and the stretching process of metal sheets occurred elastic dimple at the center of the contact region due to the squeeze effects, reveal the elastohydrodynamic lubrication (EHL) problem.Surface coatings have long been used in industrial technologies, as they can reduce friction and wear, and thus improve the service life of mechanical components.Therefore, the characteristics of an isotropic elastic coating/elastic substrate with non-Newtonian lubricants in the EHL region need to be further investigated.
To solve the formation of the dimple in pure squeeze motion problems, many numerical solutions have been proposed, including those by Yang and Wen [1], and Chang [2].In these studies, the ball dropping case is often used since it includes all the effects of pure squeeze motion which are of interest.Dowson and Wang [3] and Larsson and Högund [4] analyzed the bouncing of an elastic sphere on an oily plate.These analyses were restricted to normal motion in the first instance in order to develop the numerical technique and to relate the overall findings to the results presented by Safa and Gohar [5].
To solve the coupled hydrodynamic equation and elasticity equation of an EHL problem, the elasticity modulus and thickness of the coating are important parameters.Many studies have treated the coating and contact surfaces as linear elastic isotropic materials.Elsharkawy and Hamrock [6] used the deformation model developed by Johnson [7] to explore the Newtonian EHL of an elastic coated surface on a rigid cylinder and rigid substrate.Jaffar [8] derived a new set of explicit expressions for the contact pressure, total load, and penetration depth for the frictionless indentation problem of a spherical punch and a bounded thick elastic layer.Liu et al. [9] developed a coating EHL model for point contacts by combining the DC-FFT algorithm for the elastic deformation of a coated surface with the unified mixed EHL model.Habchi [10] developed a full system approach to analyze a thermal EHL problem in coated circular contacts with an equivalent cube under steady state condition.Chu et al. [11] analysed a rigid sphere approaching a lubricated flat surface with a layer of elastic coating on the elastic substrate is explored under constant load conditions.But, study of EHL in pure squeeze motion on coated surface with couple stress lubricant is lacking.
In this paper, pure squeeze EHL motion of circular contacts with coating is explored under constant load condition.The finite difference method and the Gauss-Seidel iteration method are used to solve the transient modified Reynolds equation, the elasticity deformation equation, load balance equation, and lubricant rheology equations simultaneously.The transient pressure profiles and film shapes during the pure squeeze process under various operating conditions in the EHL regime are discussed.

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According to the Stokes micro continuum theory [12], the field equations of a coupled stress fluid in the absence of body forces and body couples are where V is the velocity vector, is the density, p is the pressure, is the classical viscosity coefficient, and is a new material constant with the dimension of momentum responsible for the couple stress fluid property.Since the ratio / has the dimensions of length squared, the dimension of 2 / 1 ) / ( l characterizes the material length of couple stress fluids, and l is assumed to be a material constant in the present analysis. For EHL problems, two spheres approach each other can be expressed as an equivalent sphere approaching a plane.In Figure 1, an elastic sphere of radius R is approaching an infinite rigid plate with a velocity under a constant load.The lubricant in the system is taken to be a compressible fluid.Under the usual assumption of EHL applicable to a thin film, the reduced momentum equations and the continuity equation governing the motion of the lubricant in polar coordinates can be obtained.Integrating the reduced momentum equations with the no-slip-boundary conditions, the velocity components are then obtained.Substituting velocity components into continuity equation and integrating across the film thickness with the boundary conditions of v(r,z) i.e. v(r,h)= h/ t, we can then derive the transient Reynolds equation in polar coordinates as: where or in dimensionless form as: where The radial coordinate, X, is with its origin in the center of the contact.The boundary conditions for Eq. ( 5) are: 0

Rheology equations
As the pressure increases with time, the elastic deformation and the effect of pressure on the viscosity and density cannot be neglected.This stage is denoted as the high-pressure stage.The pressure dependence of viscosity and density is an important issue of the present problems.
The viscosity of the lubricant is assumed to be the function of pressure only.The relationship between viscosity and pressure used by Roelands [13] where 0 is the viscosity under ambient pressure, and ' z is the pressure-viscosity index.According to Dowson and Higginson [14], the relationship between density and pressure is given as:

Elasticity equation
The film thickness within the EH conjunction can be written as: To calculate the static deformation of elastic sphere caused by pressure distribution, influence coefficients ij D are introduced.The deformation can thus be computed at discrete points i as a sum of the deformation contributions from all pressure points j : Within the frame of linear elasticity, the normal deformation of coating is given by [8] ICPMMT 2017 The dimensionless film thickness between two elastic bodies with coating in circular contacts can be expressed as: where the influence coefficients, ij D , are computed according to Yang and Wen [1] and Larsson and Höglund [4].

Load balance equation
The instantaneous load balance equation for a constant load condition is:

Numerical solution
The finite difference method (FDM), Gaussian elimination, and the Gauss-Seidel iteration method are used to solve the transient modified Reynolds equation with its boundary condition, the elasticity deformation equations of ball and coating layer, load balance equation, and lubricant rheology equations simultaneously.Equation (5) has to be discretized and solved at a finite number of locations along the radial coordinate axis X.The discretized form of the Equation ( 5) can be derived as: X is the mesh size, i. e., , the superscript k, denoting discretization in time, T is the time increment, i. e.,

Results and Discussions
To discuss the effects of flow rheology and elastic deformation ( ) of spheres and coating ( ) on the squeezing motion, the point contact EHL problems are discussed under the conditions of non-isoviscous, compressible lubricant, and constant load.Numerical solutions of film profiles (h) and pressure distributions (p) in pure squeeze motion are calculated using the parameters listed in Table 1 The upper limit of the computational region in the beginning is chosen as X max =16.0.The Reynolds equation is discretized by the central difference technique in the space domain and the explicit technique in the time domain.When more than half of the region was cavitated, the maximum analyzed region X max reduces to half of its initial region, and so on, until X max =2.0.The grid was made up of 401 nodes, which are evenly distributed, in every calculating domain.The Gauss-Seidel iteration method is employed to calculate the film thickness and pressure distribution at each time step.
In the case of constant load conditions, Figure 2 shows the relative change in the p and h for a flexible sphere approaching a lubricated coating surface with couple stress lubricant.It is observed from Figure 2 that the p is quite flat at relatively large h, but it becomes steeper with decreasing h.It was found that the p c gradually increases with decreasing h c from 2.6ms to 15.7ms when the h c decreases to 33nm.As time goes on, the h will decrease, and the contact region will reach the Hertz contact condition.
The p of couple stress lubricant is greater than that of Newtonian lubricant at central region.As the loading is constant, the integration of the pressure distribution over the loading area is a constant.Therefore, the p found reverse at outside central region.As seen in Figure 2, the h decreases gradually as the squeeze proceeds.The h DOI: 10.1051/matecconf/201712300007 ICPMMT 2017 of couple stress lubricant is greater than that of Newtonian.

Fig. 2. p and h distribution versus time using different models
Figure 3 shows the central pressure and film thickness versus time under constant load condition.The p c increases rapidly with time at the initial stage.Meanwhile, the p c increases quickly to a maximum.Then the p c decreases slowly to near the amplitude of the well-known Hertzian pressure with time at the final stage.This stage can be considered as the quasi-static condition.This stage keeps longer time.The h c decreases rapidly with time at the initial stage.Then, they decrease slowly with time.The p c with coating is smaller than that without coating.The h c with coating is greater than that without coating.The h min with coating is greater than that without coating at the initial stage.The h min with coating is smaller than that without coating from 14ms to 60ms.The h min with coating is greater than that without coating at the final stage.Figure 5 shows the p and h for different E in rdirection under constant load condition at t=0.1s.At central region, the greater the E, the greater the p, and the smaller the h.The film thicknesses and pressures are is found reverse at outside central region.The smaller the E, the greater the positions of minimum film thickness.Figure 6 shows the p and h for different in rdirection under constant load condition at t=0.1s.At central region, the greater the , the smaller the p, and the greater the h.The film thicknesses and pressures are found reverse at outside central region.The greater the , the greater the positions of minimum film thickness.
Figure 7 shows the relationship of the central normal squeeze velocity (v c ) and the h c using present models under constant load condition.The v c decreased rapidly with decreasing h c at the initial stage.The greater the E, the greater the v c .The greater the , the smaller the v c .
Figure 8 shows the central pressure and film thickness versus time under constant load condition for different l.It is observed from Figure 8 that the greater the l, the greater the p c , and the greater the h c and h min .DOI: 10.1051/matecconf/201712300007

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The time needed to achieve maximum p c increases with increasing l.Similarly, the time needed to achieve the Hertzian pressure also increases with increasing l.

Conclusions
In this study, a numerical method for general applications was developed to investigate the effects of the isotropic coating at pure squeeze EHL motion of circular contacts under constant load conditions using FDM.The conclusions from the main results can be summarized as follows: 1.At central region, the greater the E, the greater the p, and the smaller the h.The h and p is found reverse at outside central region.2. At central region, the greater the , the smaller the p, and the greater the h.The h and p is found reverse at outside central region.3. The greater the l, the greater the h.At central region, the greater the l, the greater the p, the p is found reverse at outside central region.4. The positions of minimum film thickness for smaller elastic modulus are greater than that for larger elastic modulus.The positions of minimum film thickness for greater coating thickness are greater than that for smaller coating thickness.5.The greater the E, the greater the v c .The greater the , the smaller the v c .
(7c) to the first three grid points at X=0.For the constant load case, the rigid separation is an unknown variable in each time step.It can be determined by solving the transient Reynolds equation with the load balance equation.

Fig. 3 .
Fig. 3. p c and h versus time with/without coating.

Figure 4 Fig. 4 .
Figure4shows the central pressure and film thickness versus time under constant load condition for different E and .It is observed from Figure4that the greater the E, the greater the p c , and the smaller the h c .

Fig. 5 .
Fig. 5. p and h versus radius with various E.

Fig. 8 .
Fig. 8. p c and h versus time with various l .

Figure 9 Fig. 9 .
Figure9shows the p and h for different l in rdirection under constant load condition at t=0.1s.At central region, the greater the l, the greater the p, and the