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Tribology International 40 (2007) 687–693
Deterministic solutions and thermal analysis for
mixed lubrication in point contacts
Wen-zhong Wang, Yuan-zhong Hu, Yu-chuan Liu, Hui Wang
State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China
Available online 27 December 2005
Abstract
A deterministic numerical model has been developed for simulation of mixed lubrication in point contacts. The nominal contact area
between rough surfaces can be divided into two parts: the regions for hydrodynamic lubrication and asperity contacts (boundary
lubrication). In the area where the film thickness approaches zero the Reynolds equation can be modified into a reduced form and the
normal pressure in the region of asperity contacts can be thus determined. As a result, a deterministic numerical solution for the mixed
lubrication can be obtained through a unite system of equations and the same numerical scheme. In thermal analysis, the solution for a
moving point heat source has been integrated numerically to get surface temperature, provided that shear stresses in both regions of
hydrodynamic lubrication and asperity contacts have been predetermined. A rheology model based on the limit shear stress of lubricant
is proposed while calculating the shear stress, which gives a smooth transition of friction forces between the hydrodynamic and contact
regions. The computations prove the model to be a powerful tool to provide deterministic solutions for mixed lubrication over a wide
range of film thickness, from full-film to the lubrication with very low lambda ratio, even down to the region where the asperity contact
dominates.
r 2005 Elsevier Ltd. All rights reserved.
Keywords: Mixed lubrication; Deterministic solution; Thermal analysis; Limit shear stress
1. Introduction
Hydrodynamic or full-film lubrication means that two
surfaces in relative motion are separated completely by
liquid films, but the general situation found in mechanical
elements is mixed lubrication where the applied load is
shared by the hydrodynamic films and asperity contacts.
Asperity contacts result in significant local pressure
fluctuations and high flash temperatures, which are
believed to have a great impact on lubrication failure. Up
to now, however, there is no efficient way to measure the
local and transient variables. With the rapid development
of computational capability, it is now possible to investigate
the mixed lubrication through numerical simulations.
The early models of mixed lubrication were developed
based on stochastic approaches [1,2], which successfully
provided statistical information on lubrication, the average
film thickness and pressure distributions, but the details,
such as pressure peaks, local contacts and asperity
deformations, substantial for understanding lubrication
failure, were wiped out during the process of statistics.
The attentions have been shifted during past decades to
develop deterministic models of mixed lubrication. Chang
[3] presented a partial elastohydrodynamic lubrication
model of line contacts. Hua et al. [4] gave an analysis of
mixed lubrication in which the pressures of hydrodynamic
lubrication or asperity contact were calculated, respectively,
from Reynolds equations or Hertzian formula. In
1999, Jiang et al. [5] presented a model that successfully
solved hydrodynamic and asperity contact pressure simultaneously
by searching the actual contact domain in a trialand-
error method, but the convergent solutions presented
were limited only to the cases where film thickness is
relatively large. Recently, Hu and Zhu [6,7] proposed an
isothermal mixed lubrication model in which the pressure
was calculated from a uniform Reynolds equation system
without having to know the information of the border
between the hydrodynamic and asperity contact areas. It
allows us to study lubricated contacts for the entire regimes
ARTICLE IN PRESS
www.elsevier.com/locate/triboint
0301-679X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.triboint.2005.11.002
Corresponding author. Tel.: +8610 62788310; fax: +8610 62781379.
E-mail address: huyz@tsinghua.edu.cn (W.-z. Wang).
of lubrication, from full-film EHL to boundary lubrication,
and does not encounter much convergence problem.
While providing valuable information on film thickness
and pressure distributions, the isothermal models are
unable to predict surface temperature and related thermal
effects. A series of efforts was made to incorporate thermal
analysis into the model of mixed lubrication. Based on
isothermal solutions of mixed lubrication, Lai [8] and Qiu
[9] evaluated the surface temperature, using the moving
point heat source integration [10]. They proposed that
viscous and frictional heating would concentrate on the
middle layer of a lubricating film and temperature varied
linearly across the film. Zhai and Chang [11] deterministically
calculated temperatures by solving energy equations
for both solid and fluid subject to thermal interfacial
conditions. The key technique is to use a coordinate
transformation for mapping the time-varying solid and
fluid regions into three rectangular calculation domains.
The present authors [12] presented another thermal model
of mixed lubrication to investigate pressure–temperature
dependence.
The intention of this article is to summarize the
successive efforts recently made by the authors in modeling
the mixed lubrication. First, a deterministic numerical
model was developed under isothermal condition, and then
it was extended to incorporate thermal analysis: the
temperature rises at surfaces and thermoelastic deformations
on asperities. The emphasis is on the numerical
approaches to get solutions of mixed lubrication.
2. Isothermal model
In mixed lubrication, the nominal contact zone, O, may
be divided into two different types of areas (as in Fig. 1):
the hydrodynamic lubrication area, Ol, where two surfaces
are separated by a lubricant film, and the asperity contact
area, Os, where two surfaces are in direct contact. The
applied load is shared simultaneously by the hydrodynamic
films and contacting asperities. A deterministic solution to
the problem means to find out the film thickness and
pressure at every location within the computation domain.
A natural way for the problem is to solve the equations
that control the pressure in the two types of regions, that is,
the Reynolds equation to be solved in hydrodynamic
regions, and the equation from theory of elasticity
employed for the pressure of asperity contacts. However,
the border between the two regions is usually irregular and
time-dependent, so it requires a great skill and effort to
search for the locations where the contact takes place. For
this reason we presented a simple numerical model which
allows to determine the pressure through a uniform
equation.
ARTICLE IN PRESS
Nomenclature
a Hertzian contact radius, mm
h, H oil film thickness (unit: mm) and its dimensionless
parameter, H ¼ h=a
h0(t) distance between surfaces at x ¼ 0 without
accounting for deformation
hs(x,y) function for original geometry of contacting
bodies
c1, c2 specific heat of body 1 and 2, J/(kg K)
f(x,y,t) heat partition coefficient
Kf thermal conductivities of solid and fluid,
w/m1k
p, P pressure (unit: pa) and its dimensionless
parameter, P ¼ p=ph
pl, pc pressure in lubricated region and contact
region, respectively, pa
ph maximum Hertzian pressure, pa
Pex, Pey Pe´ clet numbers in x and y directions
q heat flux density (W/m2 or N/ms)
Q the Fourier transform of heat flux q
slip slide to roll ratio
t time
Tm, T1, T2 temperature of film on middle layer, surface
temperatures of body 1 and 2, K
Tb1, Tb2 bulk temperatures of body 1 and 2
DT1, DT2 surface temperature rises of body 1 and 2, K
u entraining speed, mm/s
u1, u2 velocities (mm/s) of body 1 and 2
up elastic deformation, mm
Up the Fourier transform of the elastic deformation
up
uT thermo-elastic deformation, mm
UT the Fourier transform of thermo-elastic
deformation uT
Vs relative sliding velocity (mm/s)
x,y x and y coordinate
a pressure coefficient, in the range of 0.03–0.15
m friction coefficient
n Poisson ratio
O solution domain
Os area where two surfaces are in direct contact
Ol area where two surfaces are separated by the
lubricant film
d(x,y,t) the roughness height of surface, mm
e constant with a very small value
Z viscosity of lubricant, pa s
Z* effective viscosity
r density of lubricant, kg/m3
r1, r2 density of body 1 and 2, kg/m3
as1, as2 heat diffusivity of body 1 and 2, m2/s
txz shear stress, pa
tl limiting shear stress of lubricant, pa
t0 the initial limiting shear stress
ox,oy angular frequencies in the frequency domain
688 W.-z. Wang et al. / Tribology International 40 (2007) 687–693
2.1. A uniform equation for mixed lubrication
In the hydrodynamic regions, the pressure, Pl, is
governed by the Reynolds equation:
q
qx
r
12Z h3 qPl
qx
 
þ
q
qy
r
12Z h3 qPl
qy
 
¼ u
qðrhÞ
qx
þ
qðrhÞ
qt
.
(1)
It is well known from computational practice that as the
film thickness decreases for perfectly smooth surfaces, the
pressure from EHL solutions will approach to a Hertzian
distribution. We believe that (1) there are thin lubricating
films between the contacting asperities, and (2) the
Reynolds equation under the constraint of h ! 0, if
solvable, will give the same solution as those from the
theory of elasticity. For this reason, it would be a more
straightforward way to apply the Reynolds equation to the
entire region of mixed lubrication, rather than to discriminate
the contact from lubrication. As a result, we
proposed a reduced form of the Reynolds equation, as
shown in the following, to be used for calculating the
pressure in case the film thickness goes below a very small
value.
u
qh
qx
þ
qh
qt
¼ 0 when h=ape, (2)
where e is a constant with very small value pre-assigned in
the computer program. Usually, e times a is less than 1 nm.
Eq. (2) was frequently quoted in the literatures to
describe EHL behavior under the heavy load or in thin
films. It makes sense from the point of view of physics that
the left hand terms of the Reynolds equation, representing
the lubricant flow due to the hydrodynamic pressure
gradient, will gradually vanish as the film thickness
approaches zero. It has to be emphasized, however, that
the use of the reduced Reynolds equation is mainly for the
considerations to avoid numerical errors when small-value
numbers are involved in computations. The essential fact
here is that we apply the Reynolds equations to the entire
region of mixed lubrication. Other equations or expressions
for describing lubricant properties, film thickness, surface
deformations, and load balance are the same as those in
Ref. [6].
2.2. Numerical approach
The discrete form of the Reynolds equation was solved
over a 256256 grid through an iterative scheme similar to
that of Ai’s [13], but a special switch was designed to make
the options as for whether or not the pressure flow should
be turned-off, that is, a very small value, e ¼ 1  106, was
set as a criterion for checking if h ¼ 0. When dimensionless
film thickness, H ¼ h=a, becomes less than e, it is
considered that the film thickness is practically zero and
the pressure flow terms in the Reynolds Equation should be
turned off, which implies the reduced Reynolds equation is
to be used for the pressure solution. An FFT method based
on the theorem of 2-D discrete convolution was applied to
speed up the calculation of elastic deformations [14,15].
2.3. Results
Simulations were performed for the cases with different
surface geometries to show the capacity of the numerical
model and the effects of surface roughness. Fig. 2a gives a
comparison of the pressure, obtained from the proposed
method at very low speed, to the solution of a Hertz
contact. At the given speed, u ¼ 0:001mm=s, the real
contact area has extended to the whole nominal contact
region, so the pressure is expected to be the same as a
Hertzian distribution. The result is in a good agreement
with the expectation, which provides verification for the
present model. Fig. 2b shows the film thickness and
pressure profiles when a curved surface superposed with
single asperity sliding over a rigid plate, from which a
pressure peak caused by the asperity contact can be clearly
seen. Figs. 2c and d show the results from two other cases
in which an artificial surface in sinusoidal form and a
real engineering surface prepared by CBN grinding are
involved. Significant pressure fluctuations and asperity
contacts induced by surface roughness are observed in both
cases, but the machining surface appears to experience
more severe stress concentration.
3. Thermal model without thermal deformation
3.1. Calculation of surface temperatures
A typical method for thermal analysis is to solve energy
equation in hydrodynamic films and heat conduction
equation in solids, simultaneously, along with the other
governing equations. In addition to the great computational
work required, the discontinuity of the hydrodynamic
films due to asperity contacts presents major
difficulty to the problems of mixed lubrication. As long
as the heat flux from friction or viscous shear is properly
ARTICLE IN PRESS
X
Y
Hydrodynamic
Zone |
Computation
domain
(x ’,y ’ )
( x ,y )
Asperity contact
zone s
Fig. 1. Schematic diagram for mixed lubrication in point contacts.
W.-z. Wang et al. / Tribology International 40 (2007) 687–693 689
estimated, however, the surface temperature, of interest in
most engineering problems, can be determined through the
integration of an analytical solution of temperature rise
caused by a moving point heat source, without having to
solve the energy equation. For two solid bodies with
velocity u1 and u2 in dry contacts, the temperature rises
at the surfaces can be predicted by the formula presented
in [10],
DT1ðx; y; tÞ ¼
Z t
0
ZZ
O
½1  f ðx0; y0; t0Þqðx0; y0; t0Þ
4r1c1½pas1ðt  t0Þ3=2
e
½ðxx0Þu1ðtt0 Þ2þðyy0 Þ2
4as1ðtt0 Þ dx0 dy0 dt0, ð3Þ
DT2ðx; y; tÞ ¼
Z t
0
ZZ
O
f ðx0; y0; t0Þqðx0; y0; t0Þ
4r2c2½pas2ðt  t0Þ3=2
e
½ðxx0Þu2ðtt0 Þ2þðyy0 Þ2
4as2ðtt0 Þ dx0 dy0 dt0, ð4Þ
where DT1 and DT2 are the temperature rises at the two
surfaces, q(x0,y0,t0) denote a point heat source at the
location (x0,y0) and time t0, and f(x0,y0,t0) is the heat
partition coefficient that describes how the heat fluxes are
assigned between the two bodies.
Eqs. (3) and (4) are valid too, in hydrodynamic regions
for calculating surface temperature, providing an assumption
is made that viscous heating concentrates on the
middle layer of lubricating films and temperature varies
linearly across the film [8]. The Fourier law of heat
conduction gives rise to the following expressions:
Kf
Tmðx; y; tÞ  T1ðx; y; tÞ
hðx; yÞ
¼ ð1  f ðx; y; tÞÞ qðx; y; tÞ, (5a)
Kf
Tmðx; y; tÞ  T2ðx; y; tÞ
hðx; yÞ
¼ f ðx; y; tÞ qðx; y; tÞ, (5b)
where Tm is the temperature of lubricant in the middle
layer, T1 and T2 are the temperatures at the solid surfaces,
the sum of bulk temperatures of Tbi and temperature rises
DTi , and Kf is the thermal conductivity of fluid. f(x,y,t) is
heat flux partition coefficient
The heat flux partition coefficient f(x,y,t) is determined
by the following condition of temperature equaling, which
is obtained by eliminating Tm from Eq. (5)
½Tb2 þ DT2ðx; y; tÞ  ½Tb1 þ DT1ðx; y; tÞ
¼ Kf ½1  2f ðx; y; tÞ hðx; yÞ qðx; y; tÞ. ð6Þ
If contacts occur at position (x,y), namely hðx; yÞ ¼ 0,
Eq. (6) is reduced to
½Tb2 þ DT2ðx; y; tÞ  ½Tb1 þ DT1ðx; y; tÞ ¼ 0. (7)
This means that the temperatures at two contacting
surfaces are the same.
In summary, the temperature rises DTi and the heat
partition coefficient f(x,y,t) can be determined, through
solving Eqs. (3), (4) simultaneously with (6) or (7), if the
heat flux q(x,y,t) has been given in advance. In the simplest
way, an estimation of heat generation can be made through
ARTICLE IN PRESS
-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
X, /a (Rolling direction)
H
U=0.001 mm/s
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Pressure
Hertzian Pressure
Film thickness
P
-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
X, /a (Rolling direction)
H
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
U=1200mm/s
P
0.0
0.5
1.0
1.5
2.0
P
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P
-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
X, /a (Rolling direction)
H
u=625mm/s
-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
X, /a (Rolling direction)
H
U=625mm/s
(a)
(c) (d)
(b)
Fig. 2. Film thickness and pressure profiles for different rough surfaces: (a) Smooth surface, pure rolling, (b) smooth surface carrying an asperity,
slip ¼ 0.5, (c) sinusoid wave surface, slip ¼ 0.6, and (d) a ground surface, slip ¼ 1.6.
690 W.-z. Wang et al. / Tribology International 40 (2007) 687–693
calculating frictional work based on the friction coefficients
assigned by the experience.
q ¼ mpVs. (8)
The friction coefficient, m, is assigned with different
values in different friction conditions. Vs is the relative
sliding velocity of two contacting bodies.
3.2. Results and discussions
Following the approach described previously, two cases
with surface geometries similar to those in Fig. 2b and c
were analyzed to show the thermal effects. The heat flux
was evaluated by Eq. (8), where the friction coefficients m
were given as 0.05 and 0.1 for the regions of hydrodynamic
lubrication and asperity contact, respectively.
In the case of a smooth surface carrying a single asperity,
a temperature peak appears clearly near the contacting
asperity (Fig. 3b), but the comparison in Fig. 3a indicates
that the pressure and film thickness are little affected by the
thermal activities. Similar phenomenon is found in Fig. 4
for a sinusoidal wavy surface sliding against a smooth
plane. This proves that the thermal effect on pressure and
film thickness is relatively small if thermo-elastic deformation
is neglected.
3.3. Heat generation estimated via shear stress
The heat generation can be estimated more accurately
from the shear stresses in both hydrodynamic and contact
regions. However, the difficulty is that the Newtonian
lubricant under high shear rate in thin films will give rise to
unreasonably large heat. In fact, lubricant cannot endure
boundless shearing so there must be a limit shear stress. In
present study, we assume that the lubricant rheology obeys
the following relation:
txz ¼
Z qu
qz ; txzptl ;
tl ¼ t0 þ aP; txz4tl ;
(
(9)
where txz is the shear stress in sliding direction and tl
defines a limit shear stress a lubricant can endure. The limit
shear stress is dependent of normal pressure.
It is also assumed that the friction in asperity contact
regions is controlled by boundary lubrication and the shear
stress in those regions is set as tl. This is based on the
consideration that as the film thickness decreases, the
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-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
x/a (Sliding direction)
H
0.00
0.25
0.50
0.75
1.00
1.25
thermal solution
isothermal solution
slip=0.5
P
-1.5 -1.0 -0.5 0.0 0.5 1.0
0
10
20
30
40
50
60
70
smooth surface1
smooth surface 2
x/a (Sliding direction)
temperature rise, K
surface 1
surface 2
(a) (b)
Fig. 3. Smooth surface with single asperity, slip ¼ 0.5, u ¼ 1200mm=s: (a) pressure and film thickness profiles, in comparison with isothermal solutions;
and (b) temperature rises, the dashed lines are solutions for smooth surfaces.
-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
Thermal solution
Isothermal solution
x/a (Sliding direction)
Film thickness(/a) 0.0
0.5
1.0
1.5
2.0
Pressure(/Ph)
-1.5 -1.0 -0.5 0.0 0.5 1.0
0
50
100
150
200
250
300
350
400
surface 1
surface 2
smooth surface 2
smooth surface 1
x/a (Sliding direction)
Temperature rise,K
(a) (b)
Fig. 4. Sinusoidal surface, u ¼ 625mm=s; Load ¼ 800 N, slip ¼ 2.0: (a) pressure and film thickness profiles, in comparison with isothermal solutions, and
(b) temperature rises, the dashed lines are solutions for smooth surfaces.
W.-z. Wang et al. / Tribology International 40 (2007) 687–693 691
hydrodynamic shear stress should evolve smoothly into the
shear strength of boundary films. As a result, the heat flux
generated between two contacting bodies can be evaluated
by the following equations:
q ¼
Z h
0
Z
qu
qz
 2
þ
qv
qz
 2
" #
dz 
ðu2  u1Þ
h
Z h
0
txz dz
in hydrodynamic regions;
q ¼ tl ju2  u1j in asperity contact regions: ð10Þ
Fig. 5 gives the surface temperatures from the simulations
when heat flux is estimated via the rheology model
proposed in this section. For a smooth surface in mixed
lubrication (Fig. 5a), a smooth transition of temperature
from lubrication region to the contact can be seen. In the
case of a smooth surface carrying an asperity (Fig. 5b), the
temperature at the middle layer of film remains a
continuous transition, but there are stepwise changes in
surface temperature caused by the discontinuity of film
thickness.
4. Thermal-mechanical model of mixed lubrication
Research [16] shows that thermo-elastic deformation has
an important impact on the contact behavior. Frictional
heating causes asperities to grow and greatly modifies the
pressure, real contact area, and subsurface stresses. In this
section, a thermal-mechanical model is proposed to
incorporate the effects of thermo-elastic deformation.
When considering thermo-mechanical behavior, an
additional term uT has to be included for describing the
changes in the gap between two surfaces. Thus, the gap
equation should be revised into the following form.
hðx; y; tÞ ¼ h0ðtÞ þ hsðx; yÞ þ d1ðx; y; tÞ
þ d2ðx; y; tÞ þ upðx; y; tÞ þ uTðx; y; tÞ ð11Þ
The thermo-elastic term in frequency domain takes the
form of [17]
UTðox;oy; 0; tÞ
Qðox;oyÞ
¼ 2i
1  exp½iðoxPex þ oyPeyÞtðw
ffiffi
t
p
Þ  wðw0 ffiffi
t
p
Þ=w0
ox Pex þ oy Pey
,
ð12Þ
where UT and Q are the Fourier transforms for the
deformation uT and heat flux q, ox and oy denote angular
frequencies in the frequency domain, w ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o2
x þ o2y
q
,
w0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffififfiffiffiffiffiffi
w2 þ iðox Pex þ ox PeyÞ
p
, and Pex,Pey are the Pe´ clet
number. The left-hand term in the above equation is
reduced to 2t when w ¼ 0.
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-1.5 -1.0 -0.5 0.0 0.5 1.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
300
350
400
450
0.000
0.005
0.010
0.015
0.020
0.025
Temperature, K
X Axis, /a X Axis, /a
T1
T2
Tm
Film thickness, /a
Film thickness, /a
300
320
340
360
380
400
420
440
460
0.000
0.005
0.010
0.015
0.020
0.025
0.030
Temperature, K
T1
T2
Tm
(a) (b)
Fig. 5. Surface temperatures when heat generation is estimated via shear stress: (a) smooth surfaces in mixed lubrication, and (b) a single asperity in
contact with a rigid plane.
-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.005
0.010
0.015
0.020
(a)
thermal solution without thermo deformation
Solution with thermo deformation
X Axis, /a (Sliding direction)
Film thickness, /a
0.0
0.3
0.6
0.9
1.2
1.5
Pressure, /Ph
-1.5 -1.0 -0.5 0.0 0.5 1.0
0.000
0.005
0.010
0.015
0.020
0.0
0.3
0.6
0.9
1.2
1.5
Film Thickness, /a
X, /a (Sliding direction)
thermal solution without thermo deformation
with thermal deformation
(b)
Pressure, /Ph
Fig. 6. A comparison of pressure and film thickness profiles, pure sliding and load ¼ 400 N: (a) smooth surface u ¼ 400mm=s, and (b) smooth surface
with a single asperity, u ¼ 600mm=s.
692 W.-z. Wang et al. / Tribology International 40 (2007) 687–693
Thermo-elastic deformation can be thus determined
numerically based on Eq. (12), using a FTT algorithms
proposed by Liu et al. [17].
Fig. 6 shows film thickness and pressure profiles taken
across symmetrical line (Y ¼ 0), when steel ball with
smooth surface (Fig. 6a), or superposed with a single
asperity (Fig. 6b), slides over a rigid plane. The thermal
solutions without thermo-elastic deformation are also
shown in the figures for a comparison. It can be seen that
when thermo-elastic deformation is included, the pressure
distribution becomes more concentrated on central region
of the contact, and the maximum pressure grows higher. A
close examination on the contacting asperity in Fig. 6b
confirms that the real contact area decreases slightly when
thermo-elastic deformation is considered.
5. Conclusions
The efforts and progresses made by the present authors
in modeling mixed lubrication are reviewed in this paper.
The Reynolds equation, or its reduced form, is applied to
the entire computation domain regardless the nature of
contact. The numerical model is so successful in computational
practices that it allows us to perform simulations in
entire lubrication regimes, from full-film EHL to boundary
lubrication.
The model has been extended to incorporate the capacity
to predict surface temperature. While providing valuable
information on temperature rises, results show that film
thickness and pressure distributions have been affected
slightly by the thermal activities if the thermo-elastic
deformations are neglected.
A thermo-mechanical model of mixed lubrication is also
established to evaluate the effects of thermo-elastic
deformation of two contacting bodies. The results show
that the thermo-elastic deformations cause higher pressure,
a slight decrease in contact area, and significant growth of
thermal stress.
Acknowledgments
The authors would like to express their appreciation for
the continuous supports from GeneralMotor Corporation,
USA.
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نویسنده : عماد موسوی ; ساعت ۱٠:٤۸ ‎ب.ظ روز یکشنبه ۱۳۸٩/۱/۱٥