2.1. The Proposed Deterministic Model
Based on the mixed EHL model reported by Zhu and Hu [
40,
41], a deterministic mixed lubrication model for parallel surfaces is proposed, with its basic equations and main features addressed in this section. More information about all the symbols used below is given in the Nomenclature.
As nominally parallel surfaces are considered, the nominal fluid pressure between the surfaces should be equal to the flooded lubricant pressure,
p0. Furthermore, the nominal parallelism and corresponding constant nominal pressure distribution will not change by ignoring the bulk deformation. Such constant nominal pressure distribution provides convenience in establishing a deterministic mixed lubrication model for parallel rough surfaces. The mixed lubrication performance of the parallel rough surfaces can be represented by a portion of the mating surfaces with roughness. It means that the solution domain does not need to cover the whole nominal contact area. A representative solution domain with equivalent surface roughness but a far smaller extent than the nominal contact area is used to conduct the mixed lubrication simulation. By reducing the solution domain, a large amount of computation is avoided, and the techniques in the mixed EHL model reported by Zhu and Hu [
40,
41] become applicable. In the meantime, such simplifications can highlight the micro-EHL effects at asperities in mixed lubrication of nominal parallel rough surfaces, which are ignored in previous studies. The boundary condition of the reduced solution domain is equal to the constant nominal pressure value.
The above discussions on reducing the solution domain are equivalent to implementing a two-scale modeling strategy. The nominal parallel rough surface is the macro-scale, and the roughness is the micro-scale. The macro-scale pressure distribution is constant due to the nominal parallelism and ignoring the bulk deformation. Thus, only the micro-scale roughness needs to be considered with a reduced solution domain. It should be noted that the size of the reduced solution domain could affect the simulation results. However, determining an appropriate size of the reduced solution domain needs systematic studies on the surface topographies, which are beyond the scope of the current study. Thus, the current work does not discuss the influence of reduced solution domain size. The size of the reduced solution domain is assumed according to the measurement equipment of roughness and the need for computational efficiency.
Fluid pressure in the reduced solution domain is obtained by solving the iso-thermal steady-state Reynolds equation [
40,
41],
where the
x coordinate is in the direction of the relative motion of the two surfaces and the
y coordinate is in the perpendicular direction. As discussed above, the pressure boundary condition can be assumed as the constant nominal pressure value
p0, which gives
p(
x0,
y) =
p(
xe,
y) =
p(
x,
y0) =
p(
x,
ye) =
p0.
As suggested by Zhu and Hu [
40,
41], when the lubricant film thickness,
h, is approaching zero (
h→0), asperity contact is assumed to occur and the Reynolds Equation (1) reduces to [
40,
41]
Recently, Wang et al. [
46] performed a rigorous study utilizing both a dry contact solver and the reduced Reynolds Equation (2) and found that the predictions from both solvers were identical. Thus, it is reasonable to use Equation (2) to solve for asperity contact pressures, and the solution of Equation (2) becomes a subset of the overall solution procedure for solving Equation (1). The boundary condition between the asperity contact and fluid regimes,
pl =
pc, is automatically satisfied during the solution process.
It is essential to mention that, although the Reynolds Equation (1) and its reduced form (2) can be used together to solve a mixed lubricated contact, using Equation (2) as an independent equation to describe the asperity contacts is not physically relevant. From the numerical simulation perspective, solving Equation (1) with h approaching zero provides contact pressure values identical to the pressure values obtained using a dry contact solver based upon contact mechanics principles. Therefore, Equation (1) is used to solve the fluid and asperity contact regimes in this work.
Another point to note is the treatment of possible inter-asperity cavitation in the current work. The mass-conservation cavitation algorithm proposed by Elrod [
50] is the most widely used cavitation boundary condition in lubrication analysis. Several researchers have used it in deterministic mixed lubrication models. The specific models are either conformal contacts without interaction between asperities and the micro-EHL effects at asperities (see Minet et al. [
17]) or non-conformal contacts (see Pu et al. [
51]). The current work is different from those previous deterministic mixed lubrication models. It is the first time the micro-EHL effects at asperities are considered in modeling mixed lubrication of parallel rough surfaces. Thus, whether the Elrod algorithm can be used in the proposed model still needs detailed investigation. In order to keep the brevity of this paper and focus on the influence of the micro-EHL effect, it is better to use a simple boundary condition to deal with the cavitation. Therefore, the simple Reynolds boundary condition is used in the current study to deal with the inter-asperity cavitation. When the pressure drops below zero at specific nodes, the corresponding pressure value is set to zero.
The film thickness equation is [
40,
41]
where
h0 is adjusted to obtain the load balance in the solution procedures,
δ1 is the rough surface data of the stationary surface, the moving surface is assumed to be smooth, and
ν (
x,
y) is the elastic deformation [
40,
41]:
Equation (4) is the Boussinesq solution based on the half-infinite assumption commonly used in simulating the dry contact of conformal contact surfaces [
52,
53]. Its detailed derivation can be found in reference [
54], chapter 2. For the macro-scale, parallel surfaces configuration, the validity of the half-infinite assumption becomes questionable. However, the elastic deformation mainly occurs at the asperities for the small solution domain considering roughness. Therefore, the solid contact points will share most of the applied load. This condition is similar to the dry contact, and Equation (4) can be used in the current work. Equation (4) is an integral equation linking the surface elastic deformation with the pressure distribution applied on the surface. No isolated asperities are defined or used to calculate the surface elastic deformation. The interaction of asperities is automatically implemented in solving the integral equation.
As the calculated pressure distribution involves the fluid pressure and asperity contact pressure, directly integrating the pressure distribution results in the load-carrying capacity, which should equal the applied load. Hence the load balance equation is [
40,
41]
where the applied load,
w, corresponds to the reduced solution domain.
The lubricant properties are represented by the Barus [
55] viscosity-pressure equation,
One point to note is that the viscosity law, Equation (6), can be replaced with other kinds of viscosity laws. The current study is not for discussing the influences of different viscosity laws. Thus, the most widely used Barus law is used.
The Dowson-Higginson [
56] equation is used to represent the density-pressure relationship of lubricant:
where
p is the pressure in GPa. It is worth mentioning that the viscosity-temperature and density temperature equations are not involved as the isothermal condition is assumed in this work.
The friction force in the mixed lubrication regime consists of asperity friction force and fluid shear force. Generally, the asperity friction force, Fc, is calculated by estimating the load carried by asperities, wc,, and a constant dry friction coefficient, fc, according to the equation: Fc = fc × wc.
In the mixed lubrication regime, the fluid film thickness is very thin, resulting in a very high shear rate. Therefore, although the Newtonian viscosity law, like Equation (6), is widely used, the non-Newtonian fluid model is still used to calculate the shear stresses in previous studies [
41,
47]. It should be noted that such incompatibility between the viscosity law used in lubrication calculation and shear stress calculation introduces errors. However, considering the focus of this work is not on studying the influence of viscosity laws used, the same manner as previously published works is employed. It means that the Newtonian viscosity law, Equation (6), is used to simulate lubrication and the non-Newtonian fluid model is used to calculate shear stresses. The shear stress model reported by Bair et al. [
57] is used to calculate the hydrodynamic traction force as,
where
where
τ is the shearing stress,
is the shearing rate,
τL is the limiting shear stress of lubricant,
τL0 is the initial shear stress of the lubricant,
γL is the pressure coefficient corresponding to the maximum friction coefficient in hydrodynamic lubrication, and
ph is the fluid pressure. According to Bair et al. [
57], for oil lubricant,
τL0 = 2 MPa and
γL = 0.05. The total friction can be written as
F =
Fc +
Fh. The coefficient of friction can be calculated as
f =
F/w.
The model proposed above involves the interaction between asperities and the micro-EHL effect at the asperity scale. In order to further illustrate the influence of the micro-EHL effect, a schematic diagram comparing models with and without the micro-EHL effect is shown in
Figure 1.
Figure 1a shows the geometry of the lubricated surfaces without the micro-EHL effect, which is only determined by the dry contact of rough surfaces. In
Figure 1b, the lubricating gap is also influenced by fluid pressure, showing the micro-EHL effect.
In the current work, a simple model without the micro-EHL effect is also used to directly illustrate the influence of the micro-EHL effect on the lubrication of parallel rough surfaces. In this simple model, the load shared by asperities solves the dry contact problem. The deformed gap between the rough surfaces is then used to calculate the hydrodynamic pressure. Next, the load carried by the fluid pressure is calculated. The total load-carrying capacity is the sum of the load shared by asperities and fluid and it should be equal to the applied load. If the load balance condition is not satisfied, the load shared by asperities is adjusted.