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 steadystate 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 realtime 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 onedimensional. 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 elasticplastic. 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, elasticplastic 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 h_{0}.
In these two figures, the initial wavelength of the contact surface roughness
is l_{w} 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 ridgeplane 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 loadcarrying capacities of the ridgeplane 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 elasticplastic 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:
where F is the force exerted on the whole area of this contact and k is constant.
Differentiating Eq. 1 gives that:
While, according to Fig. 1c, the variation of the direct surface contact area between a single ridge and the smooth plane is expressed as:
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:
Integrating Eq. 4 gives:
where c_{0} is an integral constant. From the boundary condition h_{F = 0} = h_{0}, it is solved from Eq. 5 that c_{0} = h_{0}. Thus, Eq. 5 is reexpressed as:
where w_{A0,single} is the load per unit contact length on the direct surface contact area between a single ridge and the smooth plane and w_{A0,single} = F/l.
Let w_{A0,single} = w_{max,e} when h = h_{cr,e}; Here, h_{cr,e} is the (critical) value of the film thickness h when the ridge is in the maximum elastic deformation and w_{max,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 h_{cr,e}<h<h_{0}, the ridge is in elastic deformation; While, when h≤h_{cr,e}, the ridge is in plastic deformation.
From the boundary condition w_{A0,single}_{h = hcr,e} = w_{max,e}, it is solved from Eq. 6 that:
Thus, when h_{cr,e}<h<h_{0}, in the condition of the ridge elastic deformation, the film thickness expression is:
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:
In the present analysis, the average pressure p_{av} formed on the direct surface contact area between the ridge and the smooth plane is used for judging whether the ridgeplane direct contact is in elastic or plastic deformations. When p_{av}<p_{y}, the ridgeplane direct contact is in elastic deformation; While, when p_{av} = p_{y}, it is in plastic deformation; Here, p_{y} is the compressive yielding strength of the ridge. According to this principle, on the inception of the ridge plastic deformation, there is:
Determine the parameters w_{max,e} and h_{cr,e}: Solving
the coupled Eq. 7 and 10 gives the expressions,
respectively for w_{max,e} and h_{cr,e} as follows:
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:
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:
Integrating Eq. 14 gives the following relation equation:
where c_{1} is an integral constant. By using the boundary condition W_{A0, single} h = h_{cr,e} = W_{max,e}, it is solved from Eq. 15 that c_{1} = h_{cr,e} + W_{max,e} cos θ/(2p_{y}) Thus, when h≤h_{cr,e}, in the condition of the ridge plastic deformation, the film thickness expression is:
Based on Eq. 10 and 16 can actually be
further simplified as:
It can be found that in the condition of the ridge plastic deformation, the average pressure formed on the ridgeplane direct contact area is always equal to the compressive yielding strength of the ridge p_{y}.
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:
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 ridgeplane 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):
where f is the friction coefficient of the ridgeplane direct contact area, k_{m} is the heat conduction coefficient of the rough surface, ρ_{m} is the density of the rough surface and c_{m} 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:
where A_{org} is the initial direct contact area between a single ridge and the smooth plane and A_{org} = 2b_{0 }l (b_{0} 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:
Integrating Eq. 21, using the boundary condition h_{F=0} = h_{0} and rearranging gives:
Similar as described earlier let w_{A0,single} = w_{max,e} when h = h_{cr,e}. The definitions of w_{max,e} and h_{cr,e} are same as described above. Thus, when h_{cr,e}<h<h_{0}, the surface ridge undergoes elastic deformation; While, when h≤h_{cr,e}, the surface ridge undergoes plastic deformation.
From the boundary condition , it is solved from Eq. 22 that:
Substituting Eq. 23 into Eq. 22 gives the film thickness expression in the condition of the ridge elastic deformation when h_{cr,e}<h<h_{0} as follows:
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:
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:
Determine the parameters w_{max,e} and h_{cr,e}: Solving
the coupled Eq. 23 and 26 gives the expressions,
respectively for w_{max,e} and h_{cr,e} as follows:
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≤h_{cr,e}, in
the condition of the ridge plastic deformation, the film thickness expression
can be further simplified as follows, according to Eq. 16:
It can be found that for the present case, in the condition of the ridge plastic deformation, the average pressure formed on the ridgeplane direct contact area is always equal to the compressive yielding strength of the ridge p_{y}.
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:
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:
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 (l_{1}) of the zone A shown in Fig. 2a is thus equated as l_{1} = 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 subzones, which are, respectively named as zone A_{1} and zone A_{2}. In the zone A_{1} occurs conventional hydrodynamic lubrication because of the relatively high lubricant film thickness and the continuum lubricant film lubrication there; While, in the zone A_{2} 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 ridgesmooth 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:
Analysis for the lubrications, respectively occurring in the zone A_{1} and the zone A_{2} shown in Fig. 2b are demonstrated as follows.
Lubrication in the zone A_{1}: In the analysis of the lubrication
in the zone A_{1}, the following assumptions are used:
• 
The lubricant in the zone A_{1} 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 A_{1 }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 
where p is the lubricant film pressure, x is the coordinate shown in Fig. 2b, h_{x} 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 A_{1} is equal to zero i.e., q = 0, Eq. 33 can be further simplified as:
Substituting (Barus, 1893) and h_{x} = x cot θ into Eq. 34 and rearranging gives:
where Ψ = 6uη_{0}/cot^{2}θ.
Integrating Eq. 35 and using the boundary condition gives
the lubricant film pressure in the zone A_{1}:
where x_{be} is the x coordinate of the lubricant film entrance in the zone A_{1} and x_{be} = h tanθ. The lubricant film pressure at the boundary x = x_{bl} (Fig. 2b) is:
where x_{bl }is the x coordinate of the boundary between the zone A_{1} and the zone A_{2} and x_{bi} = h_{cr, ncf} tanθ; Here, h_{cr,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 A_{1} on a single surface ridge is derived as:
Lubrication in the zone A_{2}: Since the lubricant film thickness
in the zone A_{2} is of molecular scale, the zone A_{2} 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 A_{2} is equated as q = q_{conv}θ_{v},
where q_{conv} is the lubricant mass flow through the contact in the
zone A_{2 }when the boundary adsorbed layer in the zone A_{2}
is treated as the conventional continuous lubricant and θ_{v} is
the correction factor of the lubricant mass flow through the contact in the
zone A_{2} 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 A_{2}
vanishes, q_{conv} = 0. In the zone A_{2}, 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 A_{2} 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 surfaceboundary adhering layer
interface is assumed as large enough. In these conditions, the rheological model
of the boundary adhering layer in the zone A_{2 } is simplified as (Zhang
et al., 2003; Zhang, 2004b):
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 A_{2} 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 A_{2} is derived as (Zhang et al.,
2003):
where is the effective density of the boundary adhering layer. Since q_{conv}
= 0, Eq. 40 can be further simplified as:
In Eq. 41, the viscosity is equated as (Zhang et al.,
2003):
where and r approaches zero.
Substituting Eq. 42 and h_{x} = xcotθ into
Eq. 41 and rearranging gives:
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 A_{2}
as follows:
where
The load per unit contact length carried by the lubricant film in the zone
A_{2} on a single surface ridge is expressed as:
where x_{r }satisfies r = h_{x}(x_{r})/h_{cr,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:
The value of R_{w} 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:
where N is the number of the surface ridge in the whole contact.
Let w_{A0, total} = N w_{A0, single}, w_{A1, total} = N w_{A1, single} and w_{A2, total} = N w_{A2, single}. The parameters w_{A0, total}, w_{A1, total }and w_{A2, 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 w_{ul, total} = w_{total}/L,
where L is the initial width of the whole contact (before loading) (in the x
coordinate direction, Fig. 2b”. According to N = L/l_{w},
l_{w} = 2h_{0} tan θ and Eq. 47, the
parameter w_{ul, total} is equated as:
The parameter w_{ul, 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 h_{0 }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:
Let Stf_{A0,total} = NΔw_{A0,single}/Δh, Stf_{A1,total} = NΔw_{A1,single}/Δh, and Stf_{A2,total} = NΔw_{A2,single}/Δh. The parameters Stf_{A0,total}, Stf_{A1, total} and Stf_{A2,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, Δw_{A0, single}/Δh is expressed
by Eq. 14, 18 or 30.
The other two contact stiffness are, respectively expressed as follows:
where:
and
CONCLUSIONS
The present study presents an analysis of an engineering mixed lubrication contact. In this contact, one contact surface is taken as rough, elasticplastic and moving periodically imposed with regular ridgesisosceles 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 metalmetal 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 loadcarrying 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 
A_{org} 
= 
Initial direct contact area between a single truncated triangle ridge
and the smooth plane, 2b_{0 }l 
b_{0} 
= 
Initial half width of the direct contact between a single truncated triangle
ridge and the smooth plane 
c_{0}, c_{1} 
= 
Integral constants 
c_{m} 
= 
Specific heat of the rough surface 
C_{y} 
= 
Function of film thickness, Eq. 42 
f 
= 
Friction coefficient of the ridgeplane 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 
h_{cr,e} 
= 
(critical) value of the film thickness h when the ridge is in the maximum
elastic deformation 
h_{cr,ncf} 
= 
Critical thickness of the physical adsorbed layer boundary lubrication
film 
h_{x} 
= 
Lubricant film thickness at a coordinate x 
h_{0} 
= 
Initial height of the ridge 
k 
= 
Constant 
k_{m} 
= 
Heat conduction coefficient of the rough surface 
l 
= 
Contact length of the ridge 
l_{w} 
= 
Initial wavelength of the contact surface roughness 
l_{1} 
= 
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 
p_{av} 
= 
Average pressure formed on the direct surface contact area between the
ridge and the smooth plane 
p_{y} 
= 
Compressive yielding strength of the ridge 
q 
= 
Lubricant mass flow through the contact 
q_{conv} 
= 
Lubricant mass flow through the contact in the zone A_{2 }shown
in Fig. 2b when the boundary adsorbed layer in that zone
is treated as the conventional continuous lubricant 
R_{w} 
= 
Ratio of the carried load by the lubricant film lubricated area to that
by the whole contact in the present modeled contact 
Stf_{A0, total} 
= 
NΔw_{A0, single}/Δh 
Stf_{A1, total} 
= 
NΔw_{A2, single}/Δh 
Stf_{A2, total} 
= 
NΔw_{A2, single}/Δh 
Stf_{total} 
= 
Contact stiffness of the whole contact in the present model 
u 
= 
Sliding speed between the contact surfaces 
w_{A0,single} 
= 
Load per unit contact length on the direct surface contact area between
a single ridge and the smooth plane, F/l 
w_{A1,single} 
= 
Load per unit contact length carried by the lubricant film in the zone
A_{1} on a single surface ridge shown in Fig. 2b 
w_{A2,single} 
= 
Load per unit contact length carried by the lubricant film in the zone
A_{2} on a single surface ridge shown in Fig. 2b 
w_{max,e} 
= 
Maximum load on the direct surface contact area between a single ridge
and the smooth plane when the ridge undergoes elastic deformation 
w_{total} 
= 
Load on the whole contact in the present contact 
w_{ul, total} 
= 
Load per unit contact width in the whole contact 
x 
= 
Coordinate shown in Fig. 2b 
x_{be} 
= 
x coordinate of the lubricant film entrance in the zone A_{1 }shown
in Fig. 2b 
x_{bl} 
= 
x coordinate of the boundary between the zone A_{1} and the zone
A_{2} shown in Fig. 2b 
θ 
= 
Half ridge angle 
θ_{v} 
= 
Correction factor of the lubricant mass flow through the contact in the
zone A_{2} 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 
ρ_{m} 
= 
Density of the rough surface 
η 
= 
Lubricant viscosity 
η_{0} 
= 
Viscosity of continuum lubricant at ambient condition 
α 
= 
Viscositypressure index of lubricant 
τ 
= 
Shear stress 
Ψ 
= 
6uη_{0}/cot^{2}θ 
ΔT 
= 
Temperature rise of the direct contact area between the ridge and the
smooth plane 
Stf_{A0, total}, Stf_{A1, total},
Stf_{A2, 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 
w_{A0, total}, w_{A1, total}, w_{A2,
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 