Tribology International 60 (2013) 233–245
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Tribology International
journal homepage: www.elsevier.com/locate/triboint
A model to predict scuffing failures of a ball-on-disk contact
S. Li a,n, A. Kahraman a, N. Anderson b, L.D. Wedeven c
a
b
c
The Ohio State University, 201 W. 19th Avenue, Columbus, OH 43210, USA
Pratt & Whitney, 400 Main Street, East Hartford, CT 06108, USA
Wedeven Associates, Inc., 5072 West Chester Pike, Edgmont, PA 19028, USA
a r t i c l e i n f o
abstract
Article history:
Received 5 April 2012
Received in revised form
2 August 2012
Accepted 8 November 2012
Available online 19 November 2012
This study aims at establishing the criterion for scuffing to occur. A point contact thermal elastohydrodynamic lubrication (EHL) model is combined with a heat transfer model to predict the friction
coefficient, surface bulk temperature and surface local temperature distribution. An experimental
program is conducted to generate ball-on-disk scuffing failures under different rolling and sliding
speeds and normal loads. Simulating these experiments, the model predictions are shown to be in good
agreement with the measured ones. The probability distributions of the instantaneous maximum
surface temperature under different operating conditions are constructed and observed to be bell
shaped normal distributions. It is found the distribution median can be used as the measure to establish
the scuffing limit for a lubricant-material (steel) combination.
& 2012 Elsevier Ltd. All rights reserved.
Keywords:
Thermal EHL
Mixed lubrication
Scuffing
Ball-on-disk
1. Introduction
Scuffing is a catastrophic surface failure mainly caused by
extreme local surface temperatures, which are dictated by the
contact pressure and sliding velocity. Under full film EHL condition, rough surface contacts can experience high hydrodynamic
pressure spikes due to surface irregularities. The pressure spikes
become even more severe when the lubrication condition is
mixed-EHL type where the fluid film vanishes locally and the
instantaneous asperity contacts take place. The sliding action
between the mating surfaces generates frictional heat that
increases the surface bulk temperature when the heat cannot be
effectively removed through lubrication. Instantaneous flash
temperature along the contact adds up to the elevated bulk
temperature to result in the instantaneous surface local temperature distribution whose maximum can be above a certain critical
value. In such a case, the outcome is scuffing in the form of solid
welding of the surfaces, which are torn apart afterwards due to
the difference in the surface velocities.
The onset of scuffing is closely related to the surface local
temperature, which is the sum of the surface bulk temperature
and the instantaneous temperature rise (flash temperature)
caused by the local frictional heat flux. In an early study, Blok
[1] proposed a closed-form flash temperature formula by assuming smooth contact surfaces and uniform heat flux. Ling [2]
showed that even a limited number of asperity contacts can
n
Corresponding author. Tel.: þ1 614 247 8688; fax: þ 1 614 292 3163.
E-mail address: [email protected] (S. Li).
0301-679X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.triboint.2012.11.007
largely influence the surface temperature. To investigate the
roughness effect on the instantaneous temperature rise, deterministic thermal mixed EHL models have been proposed in the
recent years. Uncoupling the thermal analysis from the mixed
lubrication analysis, Lai and Cheng [3] and Qiu and Cheng [4]
evaluated the temperature rise induced by simulated surface
roughness. Cioc et al. [5] solved the energy equations together
with the EHL governing equations iteratively to predict the flash
temperature for line contacts having very limited asperity interactions. Zhu and Hu [6] and Wang et al. [7] introduced a reduced
Reynolds equation into the thermal mixed EHL formulations,
successfully eliminating the numerical instabilities under the
severe asperity contact condition. Deolalikar et al. [8] treated
the fluid regions and the asperity contact regions separately
considering computer generated surface roughness profiles. In
these studies the frictional heat generation was determined
through assumed friction coefficients instead of using the surface
traction predicted by the EHL model itself. Additionally, the bulk
temperatures of the contact surfaces were assumed to be known.
The other factors that may contribute to scuffing failure include
wear or fatigue debris in the lubricant, wear out of the protective
tribo-film, lubricant degradation, etc. [9]. In this work, the
temperature is considered to be the dominant cause of scuffing.
In regards to the experimental studies on scuffing failure, fourball [10], ball-on-disk or twin-disk type of set-up [11–15] has
been widely used due to the relatively easy and accurate control
of the contact parameters. These studies focused on investigating
the influence of lubricants [11,12], surface finish characteristics
[13,14], and coating [15] on the scuffing performance of lubricated contacts. The commonly used scuffing test procedure is to
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S. Li et al. / Tribology International 60 (2013) 233–245
Nomenclature
a, b
Hertzian contact half width in x and y directions,
respectively
cf, cs
Fluid and solid specific heat, respectively
E1, E2
Young’s modulus of contact bodies 1 and 2,
respectively
0
E
Equivalent
Young’s
modulus,
E0 ¼ 2½ 1u21 =E1 þ 1u22 =E2 1
fx, fy
Flow coefficients in x and y directions, respectively
g0
Separation between the contact surfaces before
deformation
h0
Reference film thickness
h
Film thickness
kf, ks
Fluid and solid thermal conductivity, respectively
Lx
Length of the EHL computational domain in x direction, Lx ¼NxDx
Lt
Length of the time period of the EHL analysis, Lt ¼NtDt
Nx, Ny
Number of elements of the EHL computational
domain in x and y directions, respectively
Nt
Number of time steps
p
Pressure
q
Shear stress
Q
Total frictional heat
Q1, Q2
Heat flux going into surfaces 1 and 2, respectively
r1
Ball radius
S1, S2
Roughness heights of surfaces 1 and 2, respectively
SR
Slide-to-roll ratio SR ¼ us =ur
t
Time
Dt
Time increment
T1, T2
Local temperature distributions of surfaces 1 and 2,
respectively
T 1max
Instantaneous maximum temperature of surfaces 1
T 1max
Median of the T 1max population
T ‘1max
Minimum of the T 1max population
T u1max
Maximum of the T 1max population
DT1, DT2 Local temperature rise (flash temperature) of surfaces
1 and 2, respectively
Tamb
Ambient temperature
increase the load stepwise while maintaining the surface
velocities (rolling and sliding) constant until the scuffing
failure occurs. The measurements during the test are usually
limited to the friction force and the bulk temperature of the
contacting surfaces as the localized maximum surface temperatures are not feasible to measure. As such, the critical scuffing
temperature was estimated theoretically in the works such as Lai
and Cheng [3].
This study focuses on the prediction of scuffing failure of a
ball–disk contact interface under high-speed and high-sliding
conditions. As shown in Fig. 1, the proposed methodology to
predict the critical scuffing temperature includes a point contact
thermal mixed EHL model and a heat balance model. These two
models are used iteratively such that the heat generated at the
contact interface predicted by the thermal mixed EHL model is an
input for the heat balance model whose surface bulk temperature
prediction is fed back into the thermal EHL analysis. Several ballon-disk scuffing experiments using a WAM machine (Wedeven
Associates, Inc.) are performed. The measurements of the friction
and surface bulk temperature are compared to the model predictions to assess its accuracy. Due to the surface roughness
variation as the roughness profiles passes through the contact,
the maximum local surface temperature varies with time.
Tb1, Tb2
Tf
Tf0
DTf
Tm
u
u1, u2
ur
us
V
W
x
Dx
y
Dy
z
a1, a2
b
w
F
g_
j
ks
Z
Z0
Znx
m
W
r
r0
rs
t0
u1, u2
Ball and disk surface bulk temperature
Fluid temperature
Inlet fluid temperature
Fluid temperature rise
Mean fluid temperature across the film
Fluid velocity
Velocities of surfaces 1 and 2 in the direction of
rolling
Rolling velocity, ur ¼ 12ðu1 þ u2 Þ
Sliding velocity, us ¼ u1 u2
Total elastic deformation
Normal load
Coordinate along the rolling direction
EHL computational domain mesh size in x direction
Coordinate perpendicular to the rolling direction x
EHL computational domain mesh size in y direction
Film thickness coordinate, pointing from surface 1 to
surface 2
Pressure viscosity coefficients for low and high pressure ranges, respectively
Thermal expansion coefficient
Oil volume ratio in the surrounding air–oil mixture
Convective heat transfer coefficient
Shear strain rate
Temperature viscosity coefficient
Solid thermal diffusivity
Lubricant viscosity
Lubricant viscosity at ambient pressure and inlet
temperature
Effective viscosity in x direction
Friction coefficient
Heat partition coefficient
Lubricant density
Lubricant density at ambient pressure and inlet
temperature
Solid density
Lubricant reference shear stress
Poisson’s ratio of contact bodies 1 and 2, respectively
Probability density distributions are constructed to find the
appropriate statistical quantity to represent the critical scuffing
temperature.
Operating Conditions
Lubricant Properties
Surface Roughness
Measurements
Point Contact
Heat Flux
Heat Balance
Thermal Mixed
EHL Model
Model
Bulk
Temperature
Local Surface
Temperature
Distributions
Statistical Analysis
for Scuffing
Characteristics
Fig. 1. Methodology used for modeling of scuffing failures.
S. Li et al. / Tribology International 60 (2013) 233–245
235
For a point contact operating under the mixed lubrication
condition, the transient flow of fluid film can be described by the
Reynolds equation as
where x and y coincide with the axes of the contact ellipse, t
represents time, and the pressure, thickness and density of the
fluid are denoted as p, h and r, respectively. The rolling velocity in
the x direction is defined as ur ¼ 12ðu1 þ u2 Þ where u1 is the velocity
of surface 1 (ball in this case) and u2 is the velocity of surface 2
(disk). The flow coefficients for an Eyring fluid in the x and y
directions are approximated, respectively as [16]
@ ur rh
@
@p
@
@p
@ðrhÞ
fx
þ
fy
¼
þ
@x
@x
@y
@y
@t
@x
fx ¼
2. Model formulations
2.1. Point contact thermal mixed EHL model
ð1Þ
Ball thermal
couple
Oil supply tube
rh3
tm
rh3 =ð12ZÞ
tm
cosh
sinh
, fy ¼
12Z
t0
tm =t0
t0
ð2a; bÞ
ω2
A
Disk thermal
couples
Ball
B
Disk
ω1
r (Y )
Y
C
Oil slinger
D
Disk
Contact Zone
X
S1 [μm]
Fig. 2. (a) Ball-on-disk scuffing test set-up and (b) the schematic view of the ball-on-disk contact.
Ax
ion
irect
ing d
Roll [mm]
S2 [μm]
ial
d
[m irect
m] ion
Ax
ial
di
[m rectio
m]
n
n
ectio
g dir
]
[mm
in
Roll
Fig. 3. (a) A ball specimen with an example three-dimensional roughness profile, and (b) a disk specimen with an example three-dimensional roughness profile.
236
S. Li et al. / Tribology International 60 (2013) 233–245
where Z is the lubricant viscosity, t0 is the lubricant reference
stress, and tm is the viscous shear stress determined by tm =t0 ¼
1
sinh ½Zðu2 u1 Þ=ðt0 hÞ.
In the local areas where the film thickness is extremely thin
(on the order of several nanometers), the Reynolds equation fails
as the assumption of a continuum fluid is no longer valid. Within
these boundary lubrication (asperity interaction) areas, it is
assumed that the film thickness is constant (h¼ 0), such that
@h
¼0
@x
ð3aÞ
At the borders between the fluid areas and the asperity contact
areas, the local film shape is assumed to preserve and travel at the
rolling velocity [17] such that
@h
@h
¼
@x
ur @t
ð3bÞ
As first proposed by Hu and Zhu [18] and used later successfully
by the others (for instance [7,16]), the unified numerical system
defined by Eq. (1) and (3) describes the lubrication behavior of the
entire contact zone robustly.
Table 1
Rolling velocity and sliding conditions of ball-on-disk scuffing test.
Test condition
ur [m/s]
SR
Test result
I
II
III
IV
V
VI
10
10
10
20
20
20
0.25
0.75
1
0.25
0.75
1
No failure
Scuffed
Scuffed
No failure
Scuffed
Scuffed
The film thickness distribution for an elastic contact is defined
as
hðx,y,t Þ ¼ h0 ðtÞ þ g 0 ðx,yÞ þ V ðx,y,t ÞS1 ðx,y,t ÞS2 ðx,y,t Þ
ð4Þ
where h0 is the reference film thickness, S1 and S2 are the
instantaneous roughness heights of the two surfaces, and g0 is
the separation between the two
surfaces
before any elastic
deformation occurs. Here g 0 ¼ x2 þ y2 =ð2r 1 Þ in the case of a
ball-on-disk contact with the ball radius r1. The total elastic
deformation V of the mating surfaces is determined using Boussinesq’s half space formula [19] as
ZZ
V¼
K ðxx0 ,yy0 Þpðx0 ,y0 ,t Þdx0 dy0
ð5Þ
G
where G is the computational
domain
and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi of the contact,
the
influence function K ¼ 2= pE0 x2 þ y2 . Here E0 ¼ 2½ 1u21 = E1 þ
1u22 =E2 1 with ui and Ei representing the Poisson’s ratio and
the Young’s modulus of contact body i. Fast Fourier transform
(FFT) technique is used to accelerate the deformation computation [20].
The two-slope pressure–viscosity relationship [5,16] is modified to include the effect of fluid temperature rise DTf (from inlet
temperature Tf0) on lubricant viscosity as
8
>
Z exp a1 pjDT f ,
p o pa
>
< 0
ð6Þ
Z ¼ Z0 exp c0 þ c1 p þ c2 p2 þ c3 p3 jDT f , pa r p r pb
>
>
: Z0 exp½a1 pt þ a2 ppt jDT f ,
p 4 pb
where a1 and a2 are the pressure–viscosity coefficients for the
low (po pa) and high (p4pb) pressure ranges, respectively, and pt
is the transition pressure between these two ranges. The constants ci (iA[1, 4]) are determined such that both Z and @Z=@p are
continuous at p ¼pa and p¼pb. The temperature–viscosity coefficient j describes the slope of lnðZÞ versus the temperature rise.
Ball
Disk
Fig. 4. Example scuffing failure images for (a) test III in Table 1 (ur ¼10 m/s) and (b) test VI in Table 1 (ur ¼ 20 m/s). Both have the highest sliding condition of SR¼ 1.
S. Li et al. / Tribology International 60 (2013) 233–245
The compressibility of the lubricant under thermal condition is
approximated as [5]:
r ¼ r0
ð1 þ l1 pÞ 1bDT f
ð1 þ l2 pÞ
ð7Þ
where l1 ¼2.266 10 9 Pa 1, l2 ¼ 1.683 10 9 Pa 1 and b is the
thermal expansion coefficient.
Equating the total contacting force due to the distributed
pressure over the entire contact zone to the normal load applied,
the load balance equation reads
ZZ
W¼
pðx,y,t Þdxdy
ð8Þ
G
Eq. (8) is used as the check for the load balance convergence of
the solution. The reference film thickness h0(t) in Eq. (4) is
adjusted within a load iteration loop until Eq. (8) is satisfied.
A simplified form of the fluid energy equation that neglects the
heat convection across the fluid film, the heat conduction along
237
the rolling direction and the compressive heating/cooling is used
here as
kf
@2 T f
@T
@T
_ ¼ rc f u f þ f
þ
t
g
@x
@t
@z2
ð9Þ
where kf, cf and Tf are the thermal conductivity, specific heat and
temperature of the fluid, respectively. The z axis points from surface
1 (z¼0) to surface 2 (z¼h), representing the position along the film
thickness. When shear flow dominates, the fluid velocity varies
linearly along the z axis such that u ¼ u1 ðhzÞ=h þu2 z=h. g_ ¼ t0 =Z
sinh t=t0 is the shear strain rate for an Erying fluid. It is also
assumed the temperature distribution across the fluid film can be
approximated as the parabolic shape of [21]
z 2
z
ð4T 1 þ 2T 2 6T m Þ þ T 1
T f ¼ ð3T 1 þ 3T 2 6T m Þ
h
h
ð10Þ
where Tm is the mean fluid temperature across the film, and T1 and
T2 are the temperatures of the bounding surfaces.
Fig. 5. Flowchart of the thermal mixed EHL computation.
238
S. Li et al. / Tribology International 60 (2013) 233–245
The energy equation for the bounding solids describes the
instantaneous local surface temperature rise as [22]
(
)
Z
Z
½ðxx0 Þui ðtt 0 Þ2 þ ðyy0 Þ2
DT i ðx,y,tÞ ¼ dt0 exp 4ks ðtt 0 Þ
t
G
105
23°C
104
50°C
103
2
100°C
where ks, rs and cs are the thermal diffusivity, density and specific
heat of the solid. Qi is the heat flux going into surface i (i¼1, 2),
both of which constitute the total local frictional heat Q in the
way of
101
165°C
10
0
Q 1 ¼ WQ ,
Q 2 ¼ ð1WÞQ
ð12a; bÞ
Here, W is the heat partition coefficient and is determined
according to [3]
10−1
Equation (6)
10−2
T 1 T 2 ¼
Measurements [23]
10−3
ð11Þ
0
0.5
1
1.5
p [GPa]
Fig. 6. Pressure–viscosity relationship of Mil-PRF-23699 lubricant considered in
this study.
h
ð12WÞQ
2kf
ð13Þ
Wherever the hydrodynamic fluid film breaks down (h ¼0),
Eq. (13) reduces to T1 ¼T2 implying a continuous temperature
transition at the interface. Ignoring the rolling component, the
heat generation Q becomes the product of the sliding viscous
shear q ¼ Znx ðu1 u2 Þ=h and the sliding velocity us ¼u1 u2 for any
0.1
700
ur = 10 m/s
SR = −0.25
0.075
ur = 20 m/s
SR = −0.25
600
500
400
0.05
300
200
0.025
100
0
0
0.1
700
ur = 10 m/s
SR = −0.75
0.075
μ
600
ur = 20 m/s
SR = −0.75
500
400
0.05
300
200
0.025
W [N]
η [Pas]
10
Q i ðx0 ,y0 ,t 0 Þdx0 dy0
4rs cs ½pkðtt 0 Þ3=2
100
0
0
700
0.1
ur = 20 m/s
SR = −1
ur = 10 m/s
SR = −1
0.075
600
500
400
0.05
300
Measured μ
Predicted μ
W
0.025
0
0
5
10
15
20
5
25
30 0
Test duration [min]
10
15
20
25
200
100
0
30
Fig. 7. Comparison of the predicted and measured m for tests (a) I, (b) II, (c) III, (d) IV, (e) V and (f) VI defined in Table 1.
S. Li et al. / Tribology International 60 (2013) 233–245
in Fig. 2(b), whose heat transfer is dictated by
fluid region [23]
u2s
ð14Þ
hðx,y,t Þ
where the effective viscosity in the x direction Znx ¼ Z=cosh tm =t0
considering an Eyring fluid. For the areas where h¼0, the surface
shear q¼ mbp and Q¼ mbp9us9. The boundary lubrication friction
coefficient mb is assumed to be 0.1 [3,23] due to the lack of the
measurements of mb for the specific lubricant additive-steel
combination used in this work. For the entire contact, the friction
RR
coefficient m can be found as m ¼ G qdxdy=W.
2.2. Heat transfer model
For the ball-on-disk contact problem as shown in Fig. 2, the disk
surface bulk temperature is controlled at the oil inlet temperature,
while the ball bulk temperature is dependent on the frictional
heating generated from the contact, the convective cooling provided
by the ambient oil and air mixture and the operating time. The heat
transfer formulation is thus only devised for the ball to estimate its
surface bulk temperature. As the ball rotates against the disk, the
contact produces a circumferential contact track on the ball surface.
It is assumed the frictional heat is evenly distributed along this
track, such that the three-dimensional heat balance problem can be
reduced to a two-dimensional problem of the shaded semicircle as
@2 T
@X
2
þ
@2 T
@Y 2
¼
ð15Þ
The boundary conditions along the diameter AB, the arc of the
contact zone CD and the arcs of the convective cooling zones AC
and BD are given as
@T
¼ 0,
@X
ks
@T
¼ Q~ 1 þ FðT amb T Þ,
@X
ks
FðY Þ ¼ 0:0665
2=3
km Zm cm 1=3 2o1 rm r 2 ðYÞ
rðYÞ
km
Zm
Zm ¼ 1w Za þ wZ0 , rm ¼ 1w ra þ wr0
ur = 20 m/s
SR = −0.25
175
600
500
400
150
300
125
200
100
100
75
0
700
225
ur = 10 m/s
SR = −0.75
200
ur = 20 m/s
SR = −0.75
175
600
500
400
150
300
125
200
100
100
0
75
225
700
ur = 10 m/s
SR = −1
200
ur = 20 m/s
SR = −1
175
600
500
400
150
125
100
0
5
10
15
20
25
30 0
ð16a cÞ
ð17Þ
for a circumferential surface with radius of r(Y) rotating at angular
velocity o1 (Fig. 2(b)). Denoting the volume ratio of oil in the
surrounding medium (air–oil mixture) as w, the thermal conductivity, specific heat, viscosity and density of the air–oil mixture
are approximated as
km ¼ 1w ka þ wkf , cm ¼ 1w ca þ wcf
ð18a; bÞ
700
ur = 10 m/s
SR = −0.25
200
75
@T
¼ FðT amb T Þ
@X
respectively. Here, ks is the solid thermal conductivity, Tamb is the
ambient temperature and F is the convective heat transfer
coefficient which can be estimated as [24]
225
T1b [°C]
1 @T
ks @t
W [N]
Q ðx,y,t Þ ¼ Znx ðx,y,t Þ
239
5
10
300
Measured Tb1
200
Predicted Tb1
100
W
0
15 20 25 30
Test duration [min]
Fig. 8. Comparison of the predicted and measured Tb1 for tests (a) I, (b) II, (c) III, (d) IV, (e) V and (f) VI defined in Table 1.
ð18c; dÞ
240
S. Li et al. / Tribology International 60 (2013) 233–245
respectively, where ka, ca, Za and ra are the thermal conductivity,
specific heat, viscosity and density of air at the ambient temperature.
The evenly distributed heat flux along the circumferential
contact track Q~ 1 is defined as
Lx =r 1
WQ ðY Þ
Q~ 1 ðY Þ ¼
2p
3. Ball-on-disk scuffing tests
The scuffing tests were performed on a ball-on-disk WAM
machine as shown in Fig. 2(a). During the test, the disk temperature was controlled by a thermal module located below the disk
and set at the oil inlet temperature. Two disk thermocouples were
used to measure and confirm the disk surface temperature. The
contacting ball of diameter of 20.64 mm was held by a hollow
shaft (to minimize the heat conduction through the shaft) and
pushed against the disk in the normal direction of the disk
surface. One thermocouple was devised to touch the ball surface
near its contact track to measure its bulk temperature. Although
this thermocouple was positioned on the other side of the contact,
this measurement still provided a good estimate of the ball
surface bulk temperature. Here, the frictional heat produced
between the thermal couples and the specimen surfaces are
considered to be negligible in comparison to that produced within
the ball–disk contact. Two drives of the machine controlled the
disk and ball rotational speeds independently. The rotational axis
of the ball was on the vertical plane that was along the disk radial
direction, such that the ball and disk surface velocities were in the
same direction (perpendicular to the disk radial direction), representing the contact condition of spur and helical gears.
In this experiment, a fully formulated turbine oil Mil-PRF23699 was used as the lubricant. The polyol ester formulation
ð19aÞ
where Lx is the length of the EHL computational domain in x
direction and Q is the average frictional heat produced along the
contact arc CD that is defined as
Z
Z
1 1
Q ðY Þ ¼
dt Q ðx,Y,t Þdx
ð19bÞ
Lt Lx t
x
with Lt denoting the length of the time period of the EHL analysis.
The average heat partition coefficient W satisfies
T b1 T b2 ¼
havg 12W Q
2kf
ð19cÞ
here, havg is the average film thickness within the Hertzian zone,
and Tb1 and Tb2 are the surface bulk temperatures of the ball and
disk, respectively. Instantaneous flash temperatures (Eq. (11)) are
added to these temperatures to find the transient local surface
temperature distributions as
T 1 ¼ T b1 þ DT 1 ,
T 2 ¼ T b2 þ DT 2
ð20a; bÞ
[GPa]
0.3
0.2
[ oC ]
5.52
520
4.14
420
2.76
320
1.38
220
0
120
0.55
520
0.4125
420
0.275
320
0.1375
220
0
120
0.1
0
-0.1
-0.2
-0.3
0.3
y [mm]
0.2
0.1
0
-0.1
-0.2
-0.3
520
0.3
0.2
420
0.1
0
320
-0.1
220
-0.2
-0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
120
x [mm]
Fig. 9. Instantaneous (a) p, (b) q, (c) asperity contact pattern, (d) Tf, (e) T1 and (f) T2 for test I at the last loading stage of ph ¼ 2.47 GPa.
S. Li et al. / Tribology International 60 (2013) 233–245
241
operating conditions leading to the onset of scuffing. Typical
scatter for these tests was on the order of three loading steps.
The representative measurements are simulated in the next
section using the model proposed. Table 1 also specifies which
tests were scuffed. The example digital images of the scuffed
surfaces are shown in Fig. 4 for the cases with the highest slideto-roll ratio of SR ¼ 1.
includes an anti-wear additive, tricresyl phosphate (TCP). The oil
was supplied into the oil slinger and pushed through the small
radial holes in the oil slinger towards the contact track by the
centrifugal force as shown in Fig. 2(a).
Examples of the measured three-dimensional roughness profiles of the ball and disk specimens are shown in Fig. 3. The disk
surfaces were textured in the radial direction with the intention
of simulating actual ground gear surface roughness (allowing the
surface velocities to be perpendicular to the roughness lay
direction), while the ball surfaces had a smoother, isotropic
texture. The composite root–mean–square (RMS) roughness
amplitude of the ball–disk pair shown in Fig. 3 is Rq ¼0.53 mm.
The test conditions in terms of the rolling velocity and the slideto-roll ratio SR ¼ us =ur are listed in Table 1. The inlet oil temperature was set at 121 3 C . For a complete scuffing test, the normal
load was increased incrementally from 18 N (corresponding to
Hertzian pressure of ph ¼0.76 GPa) to 623 N (corresponding to
Hertzian pressure of ph ¼2.47 GPa) in 30 constant load steps with
one minute of test for each loading step. The scuffing failures
were detected through the sudden increase in the measured
friction coefficient due to the start of surface welding. The tests
were run until scuffing failure occurred or the specimens survived
the whole loading range without any sign of scuffing. Twenty two
tests were conducted over a range of SR and ur to identify the
4. Simulation of ball-on-disk scuffing test
The ball-on-disk scuffing tests described above and listed in
Table 1 are simulated using the model proposed in Section 2. The
dimensions of the EHL computational domain is defined as Lx ¼ dxa in
the x direction and Ly ¼ dyb in the y direction, where a and b are the
half Hertzian widths, and the coefficients dx 43.25 and dy 42.5. The
origin is selected such that 0.625Lx rxr0.375Lx and 0.5Ly ryr0.5Ly. This computational domain is discretized into Nx and Ny
elements in the x and y directions and the mesh sizes of Dx¼ Lx/Nx
and Dy¼Ly/Ny are set fixed at Dx¼2.2 mm and Dy¼3.5 mm by
varying dx and Nx, and dy and Ny under different loading conditions
(Nx and Ny are the powers of 2). This mesh resolution corresponds to
that of the three-dimensional roughness measurements. The time
increment of Dt¼ Dx/ur is used with Nt ¼500 time steps (to take into
[GPa]
5.52
[°C]
520
0.2
0.1
4.14
420
0
2.76
320
1.38
220
0
120
0.55
520
0.4125
420
0.275
320
0.1375
220
0
120
-0.1
-0.2
y [mm]
0.2
0.1
0
-0.1
-0.2
520
0.2
0.1
420
0
320
-0.1
220
-0.2
120
-0.2
-0.1
0
0.1
0.2
-0.2
-0.1
0
0.1
0.2
x [mm]
Fig. 10. Instantaneous (a) p, (b) q, (c) asperity contact pattern, (d) Tf, (e) T1 and (f) T2 for test II at the last loading stage of ph ¼1.68 GPa.
242
S. Li et al. / Tribology International 60 (2013) 233–245
of a1 ¼11.3 GPa 1 and a2 ¼5.29 GPa 1. The transition pressure for
the two-slope pressure–viscosity relationship of Eq. (6) is curvefitted as pt ¼ 79.0163þ0.91401Tf0 with the units of pt in MPa and
temperature Tf0 in degrees Celsius. The low and high pressure range
thresholds were taken to be pa ¼0.7pt and pb ¼1.4pt, respectively.
With these parameter values, Eq. (6) agrees well with the measured
pressure–viscosity data [25] as shown in Fig. 6. The temperature–
viscosity coefficient and the thermal expansion coefficient are
assumed to be j ¼0.03 and b ¼ 7.42 10 4, respectively. The
pressure dependence of the Eyring fluid reference stress is defined
the same way as in Ref. [26]. For the convective cooling of the ball,
the oil volume ratio in air is assumed to be w ¼0.05 and the ambient
temperature is estimated as T amb ¼ 110 3 C .
The six tests (2 no-failure tests and 4 scuffed tests) operating
under the conditions as defined in Table 1 are analyzed with the after
run-in surface roughness profiles shown in Fig. 3 to minimize the
effect of roughness profile change due to mild surface wear. The
predicted friction coefficient m time histories are compared with the
measurements in Fig. 7 for all six tests. Similarly, the predicted ball
surface bulk temperature Tb1 time histories are compared to the
measured ones in Fig. 8. It is seen the measured and predicted m and
Tb1 values agree well for all test conditions. Comparing the two speed
levels of ur ¼10 m/s (left column in Figs. 7 and 8) and ur ¼20 m/s
(right column in Figs. 7 and 8), the higher ur leads to earlier scuffing
failure (at a lower loading level) for the same SR since us is
also increased in the process to raise the frictional heat generation.
y [mm]
account the effect of surface roughness variations). The detailed
discretization and linearization of the governing equations can be
found in Ref. [16].
The numerical method for the point contact thermal mixed
EHL problem is illustrated in Fig. 5. Starting with the guesses of
the initial contact pressure (Hertzian) and the initial oil temperature (inlet temperature), the surface elastic deformation and film
thickness can be determined. According to the film thickness
distribution, the asperity contact spots are located and the fluid
viscosity and density are computed for the areas where h 40. The
unified equation system of Eq. (1) and (3) is then solved for p over
the entire contact zone. The pressure solution is checked for both
the load balance convergence and pressure convergence. The
converged p and h are used to find the temperature distributions
of the bounding surfaces as well as the fluid. A thermal iteration
loop is utilized to ensure the temperature distribution convergence. The converged solutions are then used for the initial
guesses of the next time step.
For the heat transfer analysis of the ball, the semicircle in
Fig. 2(b) is discretized into NX ¼20 and NY ¼ 40 elements, which is
sufficient for the estimation of the ball bulk temperature distribution. Explicit finite difference method of second order is used
to solve Eq. (15).
At the oil inlet temperature of T f 0 ¼ 121 3 C , the lubricant (MilPRF-23699) has an ambient density of r0 ¼935.2 kg/m3, viscosity
of Z0 ¼ 0.00273 Pa s and two-slope pressure–viscosity coefficients
[ oC ]
0.2
[GPa]
5.52
0.1
4.14
420
0
2.76
320
-0.1
1.38
220
-0.2
0
120
0.2
0.55
520
0.1
0.4125
420
0.275
320
-0.1
0.1375
220
-0.2
0
120
0
520
0.2
520
0.1
420
0
320
-0.1
220
-0.2
-0.2
-0.1
0
0.1
0.2
-0.2
-0.1
0
0.1
0.2
120
x [mm]
Fig. 11. Instantaneous (a) p, (b) q, (c) asperity contact pattern, (d) Tf, (e) T1 and (f) T2 for test III at the last loading stage of ph ¼ 1.49 GPa.
S. Li et al. / Tribology International 60 (2013) 233–245
The increase of the film thickness due to the rise of the rolling
velocity is found to be limited due to the significant thermal effect.
Under the condition of ph ¼2.47 GPa (highest loading step) and
SR¼ 0.25, the average film thickness within the nominal Hertzian
zone only increases from 0.373 mm to 0.406 mm when ur is raised
from 10 m/s to 20 m/s. At the last loading stages of the scuffed tests,
the measured ball surface bulk temperatures are about the same, all
approaching 180 3 C . However, the scuffed tests III and VI have
slightly lower Tb1 (at the scuffing load) than that of the no-failure test
IV (at the end of the test) to suggest that scuffing failure is not
defined solely by the surface bulk temperature. It is also observed in
Fig. 7 that the measured m exhibits a sudden increase at the onset of
scuffing to trigger the test to stop.
Figs. 9 to 11 display the snapshots of the transient thermal
mixed EHL solutions at an arbitrary time instant during the last
loading stages of tests I–III, respectively. For test I, the Hertzian
pressure is relatively high (ph ¼2.47 GPa). The surface asperity
interactions shown in Fig. 9(c) as the black regions induce
numerous local pressure peaks in Fig. 9(a), some of which reaching the level of the material hardness. In the areas where the fluid
film breaks down (i.e., asperity contact occurs), the surface shear
also peaks as shown in Fig. 9(b). However, because of the
relatively low sliding velocity of us ¼2.5 m/s in this test, the
maximum local surface temperatures of the ball T 1max and disk
T 2max do not exceed 275 3 C at this instant as seen in Fig. 9(e and
f). Here, for the asperity contact areas where hydrodynamic film
does not exist, the fluid temperature is set to be the average of the
corresponding ball and disk local temperatures, primarily for the
purpose of displaying the lubricant temperature distribution in
Fig. 9(d).
For the tests II and III, the Hertzian pressures at the scuffing
loading stages are much lower with ph ¼1.68 and 1.49 GPa,
respectively. In spite of this, the local p values at the asperity
contact areas are still very high as shown in Figs. 10(a) and 11(a).
With the higher sliding velocities of us ¼ 7.5 and 10 m/s for tests II
and III, the maximum local surface temperatures approach
475 3 C and 500 3 C , respectively, pointing to the cause of scuffing
failure. Comparing the predicted T1 and T2 in Figs. 9–11, it is seen
that although the slower ball has higher temperature within the
fluid regions, the local temperatures of the contact surfaces are
close to each other at the asperity contact spots where the heat
partition coefficient is determined assuming a continuous temperature transition.
When the surface roughness profiles move across the contact,
the transient solutions such as those displayed in Figs. 9–11 vary
in time. The maximum local surface temperature at one time
instant may not reflect the critical scuffing temperature Tc for the
steel material and lubricant considered. As seen in Fig. 12, the
populations of T 1max for the last load stages of the six tests reveal
probability distributions close to the normal shape. The standard
120
100
80
60
40
20
0
120
Number of Occurrence
243
100
80
60
40
20
0
120
100
80
60
40
20
0
250 300 350 400 450 500 550 600 250 300 350 400 450 500 550 600
T1max [°C]
Fig. 12. Probability density distributions of T 1max at the last loading stages of test (a) I, (b) II, (c) III, (d) IV, (e) V and (f) VI defined in Table 1.
244
S. Li et al. / Tribology International 60 (2013) 233–245
550
450
350
250
150
T1max [°C]
550
450
350
250
150
550
450
l
T1max
350
T1max
250
150
u
T1max
0
5
10
15
20
5
25
30 0
Test duration [min]
10
15
20
25
30
Fig. 13. Minimum, median and maximum of T 1max under the operating conditions of (a) I, (b) II, (c) III, (d) IV, (e) V and (f) VI defined in Table 1.
deviations of these distributions are observed to increase with us.
Although for the scuffed tests, T 1max can exceed 550 3 C at some
instants, the probability of the occurrence of such a high temperature is very low. This suggests that the maximum of the T 1max
population (denoted as T u1max ), might not represent Tc properly.
The same can be said for the minimum of the T 1max population
(denoted as T ‘1max ). Since the medians of these distributions
(denoted as T 1max ) correspond to the highest probability density,
T 1max is used here as the measure of the scuffing failure. In Fig. 13,
the predicted T 1max is plotted together with T ‘1max and T u1max for all
the load stages tested. At the last stages, T 1max ¼ 303 and 362 3 C
for tests I and IV (no-failure), respectively, while T 1max ¼ 458, 457,
494 and 461 3 C for the scuffed tests of II, III, V and VI, respectively. From these observations, it can be concluded that the
critical scuffing temperature Tc for the tests presented is within
the range of 450 to 500 3 C .
distribution. Asperity interactions and fluid non-Newtonian effect
were included. This model was used to simulate a set of ball-on-disk
scuffing experiments performed under various Hertzian pressure,
sliding speed and rolling speed conditions. The predicted surface
traction and surface bulk temperature were compared to the measurements, demonstrating the accuracy of the proposed model. At the
end, the probability distributions of the instantaneous maximum
surface temperatures were used to show that the median of the
distribution can be used as a measure to establish the scuffing limit
for a lubricant-material (steel) combination.
Acknowledgements
Authors thank Pratt & Whitney for sponsoring this research
activity.
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5. Conclusion
In this study, a thermal mixed EHL model for point contacts was
combined with a heat transfer model to predict the friction coefficient, surface bulk temperature and instantaneous local temperature
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