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Study of an Engineering Mixed Contact: Part I-Theoretical Analysis



Yongbin Zhang
 
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ABSTRACT

The present study presents theoretical analysis of an engineering mixed lubrication contact in one-dimensional case between a deformable plane of regular roughness and an ideally smooth rigid plane. The roughness of the plane, respectively take the forms of triangle and truncated triangle ridges which are evenly distributed on the plane surface. The rough plane is assumed as elastic-plastic. The contact between the tip of the ridge on the plane and the other smooth plane is assumed as in dry contact or oxidized chemical boundary layer contact, while the regions between other parts of the ridge and the smooth plane are, respectively in physical adsorbed boundary layer contact and continuum film hydrodynamic contact. The lubricant in the continuum film hydrodynamic contact is assumed as Newtonian. The whole contact between these two planes is obtained by periodically distributing the ridge on the plane. Theoretical analysis are developed for this mode of contact, respectively for triangle and truncated triangle surface ridges for wide operational parameters when the plane ridge, respectively undergoes elastic and plastic deformations and the hydrodynamic contact areas between the planes are varied. The present study demonstrates the theoretical analysis. In the following parts will be demonstrated the results obtained from the present analysis.

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  How to cite this article:

Yongbin Zhang , 2007. Study of an Engineering Mixed Contact: Part I-Theoretical Analysis. Journal of Applied Sciences, 7: 1249-1259.

DOI: 10.3923/jas.2007.1249.1259

URL: https://scialert.net/abstract/?doi=jas.2007.1249.1259

INTRODUCTION

Mixed lubrication contact in mechanical engineering was conventionally defined as the hydrodynamic contact in which the contact surface roughness effect is considerable. The analysis of this contact should incorporate the contact surface roughness effect. In the past time, the type of the analysis of mixed lubrication contact can be classified as two kinds i.e., stochastic approach and deterministic approach. Stochastic approach studies the statistical effect of the contact surface roughness in mixed lubrication contact by considering the stochastic and random properties of the contact surface roughness. The stochastic approach to mixed lubrication contact only needs a few characterization parameters of the contact surface roughness which are the same for a batch of hydrodynamic components. It can give the average hydrodynamic pressure and the average hydrodynamic film thickness in mixed lubrication contact for a batch of rough hydrodynamic components which have the same values of the contact surface roughness characterization parameters. However, since the contact surface roughness in mixed lubrication contact practically essentially causes the time dependent i.e., transient performance of the contact, the stochastic approach has the shortcoming of being unable to give the instant film pressure and thickness distributions in the contact, the variations of the film pressure and thickness with time and the characteristic values of both the film pressure and thickness during these variations such as the maximum hydrodynamic pressure, the local fluid cavitation (i.e., the minimum hydrodynamic pressure), the maximum hydrodynamic pressure gradient and the minimum hydrodynamic film thickness, which are important for the failures of the mixed lubrication contact. Therefore, the stochastic mixed lubrication contact results are not able enough to describe the performance of the contact. It is mainly used in determination of the condition of this kind of contact. The studies of mixed lubrication contact in the references of Christensen and Tonder (1969), Patir and Cheng (1979) and Hughes and Bush (1993) belong to this type of approach.

To give more detailed and precise mixed lubrication contact results and to better describe the performance of this contact, the deterministic approach was used. In the past time, the contact surface roughness was either artificially defined or measured for one defined couple of hydrodynamic components when applying the deterministic approach to mixed lubrication contact. The steady-state or transient results for mixed lubrication contacts were usually numerically solved for a given operating condition based on this contact surface roughness by the deterministic approach. These results give the real-time hydrodynamic film pressure and thickness during the operation. Therefore, these results detailed and precisely describe the performance of mixed lubrication contact for a given operating condition.

The deterministic approach is more capable of exploring the mechanism of the contact surface roughness effect in mixed lubrication contact than the stochastic approach. In the past time, it was popularly used in the mixed lubrication contact analysis. The studies in the references of Goglia et al. (1984) and Lubrecht et al. (1988) and Greenwood and Johnson (1992) belong to this type of approach.

Before 1990s, the theoretical analysis of mixed lubrication contact was mainly limited to the study of the mixed lubrication contact where the hydrodynamic film was fully distributed in the whole contact and the hydrodynamic film was relatively thick and thus continuum across the film thickness. In Jiang et al. (1999) presented an analysis of mixed lubrication contact considering the direct contact between the coupled surface asperities. Their study was substantially progressive than the earlier theoretical study of mixed lubrication contact. In Holmes et al. (2003) pointed out that the direct surface asperity contact should not occur in the way Hu and Zhu (2000) described in their analysis (Holmes et al., 2003), which was very similar to the analysis by Jiang et al. (1999). They suggested that fluid side leakage have a predominant effect on the direct asperity contact occurrence in this contact. However, Zhang (2005a) pointed out that the direct surface asperity contact may be not directly caused by fluid side leakage but directly caused by the surface pressure effect.

In the review of the research on mixed lubrication contact, Zhang (2004a) classified the modes of mixed lubrication contact in the theoretical analysis in the past time as three kinds i.e., the classical mode of mixed lubrication contact, the modern mode of mixed lubrication contact and the future mode of mixed lubrication contact. The classical mode of mixed lubrication contact refers to the contact where the hydrodynamic film is fully distributed in the whole contact area and the hydrodynamic film is relatively thick and thus continuum across the film thickness. The modern mode of mixed lubrication contact refers to the contact where the hydrodynamic film between the rough contact surfaces is molecularly thin on some separate locations of the contact, while in the other zones of the contact the hydrodynamic film is relatively thick and thus continuum across the film thickness. The future mode of mixed lubrication contact refers to the contact where the physical adsorbed boundary layer contact may occur on some separate locations of the whole contact, at the same time the oxidized chemical boundary layer contact and the fresh metal to metal dry contact between the opposing asperities of the contact surfaces may occur, respectively on some other separate locations of the whole contact, while in the other zones of the whole contact the hydrodynamic film is relatively thick and thus continuum across the film thickness. Zhang (2004a) pointed out that the future mode of mixed lubrication contact is the most advanced. These arguments were also mentioned in his other papers (Zhang, 2006a,b).

The present study presents an analysis of an engineering mixed lubrication contact, which belongs to the future mode of mixed lubrication contact. The analyzed contact is one-dimensional. One contact surface is taken as the rough contact surface where the triangle or truncated triangle ridges are periodically imposed. This contact surface is treated as elastic-plastic. The other contact surface is taken as ideally smooth and rigid. Dry contact or oxidized chemical boundary layer contact occur between the tip of the ridge and the smooth plane. Between other parts of the ridge and the smooth plane, respectively occur physical adsorbed boundary layer contact and continuum film hydrodynamic contact, according to the formed film between the mated contacts. During the loading of the contact, the surface ridge is compressed and undergoes elastic or plastic deformations and the direct contact area between the two contact surfaces formed on the tip of the ridge is increased; The physical adsorbed boundary layer contact area and the continuum film hydrodynamic area between the two contact surfaces are also, respectively varied with the loading of the contact. The load partitions in different types of contact areas between the two contact surfaces are also changed with the operating condition. This mode of contact is believed to follow the engineering contact, according to the experiments (Begelinger and Gee de, 1974, 1976; Tabor, 1981). In the present study is presented analysis of this contact. In the following parts will be presented the results obtained from the present analysis for this contact.

MODEL

The analyzed contact models are, respectively shown in Fig. 1a and Fig. 1b. In Fig. 1a and Fig. 1b, the upper contact surface is rough, elastic-plastic and moving with the speed u; The lower contact surface is ideally smooth, rigid and stationary. In Fig. 1a and b, the contact surface ridges are, respectively isosceles triangle and isosceles truncated triangle both with the half ridge angle θ and their initial heights both are h0. In these two figures, the initial wavelength of the contact surface roughness is lw and L is the initial contact width of the rough surface (before loading).

Fig. 1:
Analyzed contact. (a) The surface ridge is isosceles triangle; (b) The surface ridge is isosceles truncated triangle and (c) Example of the contact under loading

In Fig. 1, when the contact is loaded and then the ridge and the ideally smooth plane are in direct contact carrying a load, the ridge is compressed and undergoes elastic or plastic deformations; In the zone A of the contact, the hydrodynamic pressure is built up, while in the zone B of the contact, the fluid is cavitated and the hydrodynamic pressure vanishes. The load of the contact is carried by both the ridge-plane direct contact and the hydrodynamic lubricated area formed in the contact. Figure 1c schematically shows an example of the analyzed contact under loading.

In the following sections will be, respectively derived the load-carrying capacities of the ridge-plane direct contact area and the hydrodynamic lubricated area in the present model when the contact surface ridge is, respectively isosceles triangle and isosceles truncated triangle.

ANALYSIS

Ridge elastic-plastic deformations
For isosceles triangle surface ridge
Elastic deform ation
Basic equations:
For isosceles triangle surface ridges as shown in Fig. 1a, according to the theory of elasticity (Timoshenko and Goodier, 1951), the area of the elastic contact formed between a single ridge and the ideally smooth rigid plane is expressed as:

(1)

where F is the force exerted on the whole area of this contact and k is constant.

Differentiating Eq. 1 gives that:

(2)

While, according to Fig. 1c, the variation of the direct surface contact area between a single ridge and the smooth plane is expressed as:

(3)

where h is the film thickness between the root of the ridge and the smooth plane and l is the contact length of the ridge.

Substituting Eq. 2 into Eq. 3 yields:

(4)

Integrating Eq. 4 gives:

(5)

where c0 is an integral constant. From the boundary condition h|F = 0 = h0, it is solved from Eq. 5 that c0 = -h0. Thus, Eq. 5 is re-expressed as:

(6)

where wA0,single is the load per unit contact length on the direct surface contact area between a single ridge and the smooth plane and wA0,single = F/l.

Let wA0,single = wmax,e when h = hcr,e; Here, hcr,e is the (critical) value of the film thickness h when the ridge is in the maximum elastic deformation and wmax,e is the maximum load on the direct surface contact area between a single ridge and the smooth plane when the ridge undergoes elastic deformation. Therefore, when hcr,e<h<h0, the ridge is in elastic deformation; While, when h≤hcr,e, the ridge is in plastic deformation.

From the boundary condition wA0,single|h = hcr,e = wmax,e, it is solved from Eq. 6 that:

(7)

Thus, when hcr,e<h<h0, in the condition of the ridge elastic deformation, the film thickness expression is:

(8)

Integrating Eq. 3 and using the boundary condition gives the formed direct surface contact area between a single ridge and the smooth plane as follows:

(9)

In the present analysis, the average pressure pav formed on the direct surface contact area between the ridge and the smooth plane is used for judging whether the ridge-plane direct contact is in elastic or plastic deformations. When pav<py, the ridge-plane direct contact is in elastic deformation; While, when pav = py, it is in plastic deformation; Here, py is the compressive yielding strength of the ridge. According to this principle, on the inception of the ridge plastic deformation, there is:

(10)

Determine the parameters wmax,e and hcr,e: Solving the coupled Eq. 7 and 10 gives the expressions, respectively for wmax,e and hcr,e as follows:

(11)

(12)

Plastic deformation: When the ridge undergoes plastic deformation, the variation of the direct surface contact area between a single ridge and the smooth plane is expressed as:

(13)

where ΔF is the variation of the force exerted on the whole area of this contact.

Substituting Eq. 3 into Eq. 13 and rearranging gives:

(14)

Integrating Eq. 14 gives the following relation equation:

(15)

where c1 is an integral constant. By using the boundary condition WA0, single |h = hcr,e = Wmax,e, it is solved from Eq. 15 that c1 = hcr,e + Wmax,e cos θ/(2py) Thus, when h≤hcr,e, in the condition of the ridge plastic deformation, the film thickness expression is:

(16)

Based on Eq. 10 and 16 can actually be further simplified as:

(17)

It can be found that in the condition of the ridge plastic deformation, the average pressure formed on the ridge-plane direct contact area is always equal to the compressive yielding strength of the ridge py.

Contact stiffness: In the condition of the ridge elastic deformation, the contact stiffness of the direct surface contact area between a single ridge and the smooth plane is obtained as:

(18)

In the condition of the ridge plastic deformation, this stiffness is expressed as Eq. 14.

Contact surface temperature rise: Considering the frictional heating in the ridge-plane direct contact area is the main heat source of the whole mixed lubrication contact and the contact surface temperature rise is mainly represented by the temperature rise of the direct contact area between the ridge and the smooth plane, the temperature rise of the direct contact area between the ridge and the smooth plane due to the frictional heating is expressed as (Blok, 1937):

(19)

where f is the friction coefficient of the ridge-plane direct contact area, km is the heat conduction coefficient of the rough surface, ρm is the density of the rough surface and cm is the specific heat of the rough surface.

For isosceles truncated triangle surface ridge: The present study also studies the mixed lubrication performance when the isosceles triangle ridge on the rough plane is shaved in different degrees i.e., when the rough plane ridge is isosceles truncated triangle. The contact model for this case is shown in Fig. 1b. Contact analysis for the coupling of the isosceles truncated triangle surface ridge and the smooth plane shown in Fig. 1b is demonstrated as follows.

Elastic deformation
Basic equations:
According to the theory of elasticity (Timoshenko and Goodier, 1951), the infinitesimal variation of the area of the direct contact between a single ridge and the ideally smooth rigid plane shown in Fig. 1 b which is elastic is expressed as:

(20)

where Aorg is the initial direct contact area between a single ridge and the smooth plane and Aorg = 2b0 l (b0 is the initial half width of the direct contact between a single ridge and the smooth plane), F is the force exerted on the whole area of this contact, ΔF is its infinitesimal variation and k is same as shown in Eq. 1.

Substituting Eq. 3 into Eq. 20 and rearranging gives:

(21)

Integrating Eq. 21, using the boundary condition h|F=0 = h0 and rearranging gives:

(22)

Similar as described earlier let wA0,single = wmax,e when h = hcr,e. The definitions of wmax,e and hcr,e are same as described above. Thus, when hcr,e<h<h0, the surface ridge undergoes elastic deformation; While, when h≤hcr,e, the surface ridge undergoes plastic deformation.

From the boundary condition , it is solved from Eq. 22 that:

(23)

Substituting Eq. 23 into Eq. 22 gives the film thickness expression in the condition of the ridge elastic deformation when hcr,e<h<h0 as follows:

(24)

Integrating Eq. 3 and using the boundary condition gives the formed direct surface contact area between a single ridge and the smooth plane as follows:

(25)

Using the same principles of judging the ridge elastic and plastic deformations as presented earlier on the inception of the ridge plastic deformation, there is:

(26)

Determine the parameters wmax,e and hcr,e: Solving the coupled Eq. 23 and 26 gives the expressions, respectively for wmax,e and hcr,e as follows:

(27)

(28)

Plastic deformation: Equation 13 through 16 are still valid for the isosceles truncated triangle surface ridge in the present analysis. Based on Eq. 26, when h≤hcr,e, in the condition of the ridge plastic deformation, the film thickness expression can be further simplified as follows, according to Eq. 16:

(29)

It can be found that for the present case, in the condition of the ridge plastic deformation, the average pressure formed on the ridge-plane direct contact area is always equal to the compressive yielding strength of the ridge py.

Contact stiffness: In the condition of the ridge elastic deformation, the contact stiffness of the direct surface contact area between a single ridge and the smooth plane for the present case is obtained as:

(30)

In the condition of the ridge plastic deformation, this stiffness is expressed as Eq. 14.

Contact surface temperature rise: Similar as described earlier the temperature rise of the ridge surface in the direct contact area between the ridge and the smooth plane due to the frictional heating is derived as:

(31)

Lubricant film lubrication: Now here is analyzed the lubricant film lubrication occurring in the present mode of mixed lubrication. Figure 1a and b show that in the present modeled contact the lubricant film lubrication occurs in the zone A of the contact shown in these figures, while the fluid in the zone B of the contact is cavitated and no lubricant film pressure should be built there because of the divergent gaps between the contact surfaces formed there (Pinkus and Sternlicht, 1961). Figure 2a shows the geometry of the zone A of the present contact for both isosceles triangle and isosceles truncated triangle surface ridges when the contact is loaded.

In the present analysis, the half ridge angle θ is assumed as constant when the contact is loaded, the width (l1) of the zone A shown in Fig. 2a is thus equated as l1 = h tan θ and the film thickness h in Fig. 2a is varied with the load of the contact. Figure 2b more clearly shows an elementary cell of the zone A and its geometries in the present analysis. As Fig. 2b shows, the zone A of the present contact can actually be divided into two sub-zones, which are, respectively named as zone A1 and zone A2. In the zone A1 occurs conventional hydrodynamic lubrication because of the relatively high lubricant film thickness and the continuum lubricant film lubrication there; While, in the zone A2 occurs physical adsorbed layer boundary lubrication because of the very low lubricant film thickness and the physical adsorption and ordering of the lubricant film to the contact surfaces there. Therefore, in the present mixed lubrication are mixed three contact regimes which are, respectively the ridge-smooth plane direct contact, the conventional hydrodynamic lubrication and the physical adsorbed layer boundary lubrication and, respectively occur in different areas of the contact. The present mode of mixed lubrication belongs to the future mode of mixed lubrication defined by Zhang (2004b, 2006a, b) and is believed to be more engineering than the classical mode of mixed lubrication. The lubricant film pressure boundary conditions in the zone A shown in Fig. 2b are:

(32)

Analysis for the lubrications, respectively occurring in the zone A1 and the zone A2 shown in Fig. 2b are demonstrated as follows.

Lubrication in the zone A1: In the analysis of the lubrication in the zone A1, the following assumptions are used:

The lubricant in the zone A1 is Newtonian;
The lubricant viscosity follows the Barus viscosity equation (Barus, 1893);
The operating condition is isothermal.

According to these assumptions, the compressible lubricant lubrication Reynolds equation in the zone A1 is (Pinkus and Sternlicht, 1961):

Fig. 2:
The zone A in the present contact. (a) The geometry of the zone A for both types of surface ridges when the contact is loaded and (b) An elementary cell of the zone A and its geometries
(33)

where p is the lubricant film pressure, x is the coordinate shown in Fig. 2b, hx is the lubricant film thickness at a coordinate x, ρ is the lubricant density, η is the lubricant viscosity and q is the lubricant mass flow through the contact.

Since the lubricant mass flow through the contact in the zone A1 is equal to zero i.e., q = 0, Eq. 33 can be further simplified as:

(34)

Substituting (Barus, 1893) and hx = x cot θ into Eq. 34 and rearranging gives:

(35)

where Ψ = 6uη0/cot2θ.

Integrating Eq. 35 and using the boundary condition gives the lubricant film pressure in the zone A1:

(36)

where xbe is the x coordinate of the lubricant film entrance in the zone A1 and xbe = h tanθ. The lubricant film pressure at the boundary x = xbl (Fig. 2b) is:

(37)

where xbl is the x coordinate of the boundary between the zone A1 and the zone A2 and xbi = hcr, ncf tanθ; Here, hcr,ncf is the critical thickness of the physical adsorbed layer boundary lubrication film. The load per unit contact length carried by the lubricant film in the zone A1 on a single surface ridge is derived as:

(38)

Lubrication in the zone A2: Since the lubricant film thickness in the zone A2 is of molecular scale, the zone A2 is in physical adsorbed layer boundary lubrication. According to the physical adsorbed layer boundary lubrication theory developed by Zhang et al. (2003), Zhang and Lu (2005) and Zhang (2006c), the lubricant mass flow through the contact in the zone A2 is equated as q = qconvθv, where qconv is the lubricant mass flow through the contact in the zone A2 when the boundary adsorbed layer in the zone A2 is treated as the conventional continuous lubricant and θv is the correction factor of the lubricant mass flow through the contact in the zone A2 due to the lubricant film discontinuity and inhomogeneity effects across the lubricant film thickness there. In the present analysis, since the lubricant mass flow q through the contact in the zone A2 vanishes, qconv = 0. In the zone A2, due to the interaction between the lubricant molecule and the contact surfaces and the pressure within the lubricant film, the boundary adhering layer is solidified. In the present analysis, the shear modulus of elasticity and the Eyring stress of the boundary adhering layer (Zhang et al., 2003) in the zone A2 are not considered and the boundary adhering layer is assumed not to slip at the contact surfaces i.e., the shear strength at the contact surface-boundary adhering layer interface is assumed as large enough. In these conditions, the rheological model of the boundary adhering layer in the zone A2 is simplified as (Zhang et al., 2003; Zhang, 2004b):

(39)

where τ is shear stress, is shear strain rate and is the effective viscosity of the boundary adhering layer which is dependent on the pressure within and the thickness of the boundary adhering layer.

In the present analysis, the lubricant in the zone A2 is taken as compressible and the operating condition is taken as isothermal. In these conditions and according to Eq. 39, the Reynolds equation for the boundary adhering layer in the zone A2 is derived as (Zhang et al., 2003):

(40)

where is the effective density of the boundary adhering layer. Since qconv = 0, Eq. 40 can be further simplified as:

(41)

In Eq. 41, the viscosity is equated as (Zhang et al., 2003):

(42)

where and r approaches zero.

Substituting Eq. 42 and hx = xcotθ into Eq. 41 and rearranging gives:

(43)

Integrating Eq. 43 and using both the boundary condition formulated by Eq. 32 and the boundary pressure formulated by Eq. 37 gives the lubricant film pressure in the zone A2 as follows:

(44)

where

The load per unit contact length carried by the lubricant film in the zone A2 on a single surface ridge is expressed as:

(45)

where xr satisfies r = hx(xr)/hcr,ncf.

Performance parameters
Load partition in the contact: Define the ratio of the carried load by the lubricant film lubricated area to that by the whole contact in the present modeled contact as:

(46)

The value of Rw shows the load partition on the lubricant film lubricated area (or on the direct surface contact area between the two planes) in the present contact.

Total load: The load on the whole contact in the present contact is expressed as:

(47)

where N is the number of the surface ridge in the whole contact.

Let wA0, total = N wA0, single, wA1, total = N wA1, single and wA2, total = N wA2, single. The parameters wA0, total, wA1, total and wA2, total are the total loads, respectively on the direct surface contact area, the conventional hydrodynamic lubricated area and the physical adsorbed layer boundary lubrication area in the whole contact of the present model.

Load per unit contact width in the whole contact: Define the load per unit contact width in the whole contact as wul, total = wtotal/L, where L is the initial width of the whole contact (before loading) (in the x coordinate direction, Fig. 2b”. According to N = L/lw, lw = 2h0 tan θ and Eq. 47, the parameter wul, total is equated as:

(48)

The parameter wul, total can be used to study the influence of the initial rough contact surface geometry structure i.e., the surface ridge geometry shape parameter θ and the initial surface ridge height parameter h0 on the mixed lubrication performance in the present model.

Total contact stiffness: The contact stiffness of the whole contact in the present model is expressed as:

(49)

Let StfA0,total = NΔwA0,single/Δh, StfA1,total = NΔwA1,single/Δh, and StfA2,total = NΔwA2,single/Δh. The parameters StfA0,total, StfA1, total and StfA2,total are, respectively the contact stiffness of the total direct surface contact area, the total conventional hydrodynamic lubricated area and the total physical adsorbed layer boundary lubrication area in the whole contact.

In Eq. 49, ΔwA0, single/Δh is expressed by Eq. 14, 18 or 30. The other two contact stiffness are, respectively expressed as follows:

(50)

where:

and

(51)

 

CONCLUSIONS

The present study presents an analysis of an engineering mixed lubrication contact. In this contact, one contact surface is taken as rough, elastic-plastic and moving periodically imposed with regular ridges-isosceles triangle ridges or isosceles truncated triangle ridges; The other contact surface is taken as ideally smooth, rigid and stationary. The whole contact in the present model is composed of three types of contact areas i.e., the direct surface contact area between the contact surfaces, the conventional hydrodynamic lubricated area and the physical adsorbed boundary layer contact area; The direct surface contact area between the contact surfaces can be the oxidized chemical boundary layer contact area or the fresh metal-metal dry contact area. In this contact thus occurs mixed contact regimes. The present modeled contact is believed to be more engineering and realistic according to the experimental and theoretical research results.

Analysis are, respectively derived for the direct contact between the two planes, the conventional hydrodynamic lubricated area and the physical adsorbed boundary layer lubricated area in the present modeled contact, respectively for isosceles triangle surface ridges and isosceles truncated triangle surface ridges when the contact is loaded. The load-carrying capacity and contact stiffness of the present contact are obtained. In the following parts will be presented in detail the results obtained from the present analysis, respectively for isosceles triangle surface ridges and isosceles truncated triangle surface ridges.

Nomenclature

A = Area of the contact formed between a single ridge and the ideally smooth rigid plane
Aorg = Initial direct contact area between a single truncated triangle ridge and the smooth plane, 2b0 l
b0 = Initial half width of the direct contact between a single truncated triangle ridge and the smooth plane
c0, c1 = Integral constants
cm = Specific heat of the rough surface
Cy = Function of film thickness, Eq. 42
f = Friction coefficient of the ridge-plane direct contact area
F = Force exerted on the whole area of the contact between a single ridge and the ideally smooth rigid plane
h = Film thickness between the root of the ridge and the smooth plane
hcr,e = (critical) value of the film thickness h when the ridge is in the maximum elastic deformation
hcr,ncf = Critical thickness of the physical adsorbed layer boundary lubrication film
hx = Lubricant film thickness at a coordinate x
h0 = Initial height of the ridge
k = Constant
km = Heat conduction coefficient of the rough surface
l = Contact length of the ridge
lw = Initial wavelength of the contact surface roughness
l1 = Width of the zone A shown in Fig. 2a
L = Initial contact width of the rough surface (before loading)
N = Number of the surface ridge in the whole contact
p = Lubricant film pressure
pav = Average pressure formed on the direct surface contact area between the ridge and the smooth plane
py = Compressive yielding strength of the ridge
q = Lubricant mass flow through the contact
qconv = Lubricant mass flow through the contact in the zone A2 shown in Fig. 2b when the boundary adsorbed layer in that zone is treated as the conventional continuous lubricant
Rw = Ratio of the carried load by the lubricant film lubricated area to that by the whole contact in the present modeled contact
StfA0, total = NΔwA0, single/Δh
StfA1, total = NΔwA2, single/Δh
StfA2, total = NΔwA2, single/Δh
Stftotal = Contact stiffness of the whole contact in the present model
u = Sliding speed between the contact surfaces
wA0,single = Load per unit contact length on the direct surface contact area between a single ridge and the smooth plane, F/l
wA1,single = Load per unit contact length carried by the lubricant film in the zone A1 on a single surface ridge shown in Fig. 2b
wA2,single = Load per unit contact length carried by the lubricant film in the zone A2 on a single surface ridge shown in Fig. 2b
wmax,e = Maximum load on the direct surface contact area between a single ridge and the smooth plane when the ridge undergoes elastic deformation
wtotal = Load on the whole contact in the present contact
wul, total = Load per unit contact width in the whole contact
x = Coordinate shown in Fig. 2b
xbe = x coordinate of the lubricant film entrance in the zone A1 shown in Fig. 2b
xbl = x coordinate of the boundary between the zone A1 and the zone A2 shown in Fig. 2b
θ = Half ridge angle
θv = Correction factor of the lubricant mass flow through the contact in the zone A2 shown in Fig. 2b due to the lubricant film discontinuity and inhomogeneity effects across the lubricant film thickness (Zhang and Lu, 2005; Zhang, 2006)
ρ = Lubricant density
ρm = Density of the rough surface
η = Lubricant viscosity
η0 =

Viscosity of continuum lubricant at ambient condition

α = Viscosity-pressure index of lubricant
τ =

Shear stress

Ψ = 6uη0/cot2θ
ΔT = Temperature rise of the direct contact area between the ridge and the smooth plane
StfA0, total, StfA1, total, StfA2, total = Contact stiffness of the total direct surface contact area, the total conventional hydrodynamic lubricated area and the total physical adsorbed layer boundary lubrication area in the whole contact
wA0, total, wA1, total, wA2, total = Total loads, respectively on the direct surface contact area, the conventional hydrodynamic lubricated area and the physical adsorbed layer boundary lubrication area in the whole contact of the present model
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