Paper

A Life Estimation Method of Peeling in Rolling
Bearings Under Mixed Lubrication Conditions

Naoya HASEGAWA *Takumi FUJITA **Michimasa UCHIDATE ***Masayoshi ABO ****

Peeling, which consists of spalls and cracks with the size of about 10 µm, is one of the failures of rolling
bearings under boundary and mixed lubrication conditions. In the past, we introduced a life estimation methodof peeling which is applicable to boundary lubrication and rolling conditions without slip. In this report, we
introduce an updated life estimation method which is applicable to mixed lubrication conditions. The updated
method applies a contact analysis based on the load-sharing theory of Johnson et al. This enables us toestimate peeling lives with consideration of the effect of oil film on supporting the load under mixed lubrication
conditions. We evaluated the accuracy of the updated method by rolling contact testing. The accuracy of life
estimation by the updated method was improved as compared with that of the previous method.

1. Introduction
International efforts toward carbon neutrality are
expected to further accelerate the use of low viscosity
lubricating oil for reducing friction in machineryand automobiles. Low viscosity lubricating oils will
increase opportunities for use in boundary lubricationof rolling bearings (hereafter “bearings”) and mixed
lubrication conditions (hereafter “severe lubrication
conditions”). Improving the reliability of bearings
under these conditions is thus expected to be an
important technical issue in the future.
Peeling is a typical example of bearing failure that
occurs under dilute lubrication conditions. It refers toa dense area of spalling and cracks of about 10 µm in
size.1) Peeling tends to occur under certain conditions,such as severe lubrication conditions, where the rollingsurface oil film parameter Λ (the ratio of the minimum
film thickness of the rolling area determined by EHL
theory to the square root of the root-mean-squareroughness of the two surfaces) is low. The cause isthought to be cyclic stress acting on the direct contact
area of the surface roughness (hereafter “the realcontact area”). In addition, the authorsʼ research2, 3)
shows that the initial notches of peeling occur
because of notch formed by plastic deformation due
to the action of the cyclic stress mentioned above.
To examine the reliability of bearings under dilutelubrication conditions, it is necessary to estimate
the life as affected by peeling (hereafter the “peelinglife”). If the peeling life can be estimated, the surface
roughness and lubricant of the bearing can be selected
appropriately. In addition, it can also be applied to thedesign of the surface roughness and lubricating oil
viscosity of the bearing necessary to balance reliability

and low torque, which can also help reduce friction
for bearing users. In the previous report, the authors
introduced a peeling life estimation method5) thatthey independently developed,4) but its application
was limited to boundary lubrication conditions. In thispaper, we present a peeling life estimation method
applicable to mixed lubrication conditions, along withthe results of validating its estimation accuracy.6)
The term “peeling” is customarily used by Japanese
domestic bearing manufacturers, while the term
“micropitting” is commonly used outside Japan.

2. Conventional peeling life estimation
method5)
2.1 Procedure for life estimation

Fig. 1 shows the procedure for estimating peelinglife. In Step 1, rolling fatigue tests are carried out
under various operating conditions to obtain a historyof peeling life (hereafter “stress history”) and thestress acting on the rolling surface (von Mises stressacting on the depth of 0.5 µm directly below thetrue contact area, hereafter “surface stress”). Surface
stress is reduced during operation by decreasing thesurface roughness of the rolling surface and generating
compression residual stress (hereafter “running-in”).
The surface stress is thus estimated using the results ofcontact stress analysis and residual stress measurement
using the surface roughness measured for eachconstant load.
The von Mises stress at a depth of 0.5 µm was used
for life estimation as the von Mises stress tended tobe highest at a depth of 0.5 µm regardless of the test
conditions.5) In Step 2, the peeling S-N curve (StressNumber of cycles to failure) was obtained from the* Advanced Technology R&D Center ** Product and Business Strategic Planning Dept.
*** Faculty of Science and Engineering, Iwate University **** School of Engineering, University of Hyogo

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85

A Life Estimation Method of Peeling in Rolling Bearings Under Mixed Lubrication Conditions[Step 2 Create S-N curve][Step 1 Collect data for peeling life and stress history]
Stress-life data of experiment 1

Stress-life data of experiment 2
Stress-life data of experiment 3

StressPeeling

Regression
analysis

StressStress-life data of experiment 4…

Number of cycles[Step 3 Preliminary test under the conditionsof the subject for life estimation]
Obtain stress history data up to
the limited loading cycles Nex
StressAssume constant stress after Nex

Peeling life

[Step 4 Life calculation]

Map with S-N curve(Apply Minerʼs rule)

Nex

Estimated peeling life

Number of cycles

Fig. 1 Flow of the estimation method of peeling life4)
peeling life and stress history data obtained in Step 1
by regression analysis. In Step 3, a rolling fatigue test
simulating the operating conditions to be estimated(hereafter the “preliminary test”) is conducted toestimate the peeling life under arbitrary conditions, andthe stress history under those conditions is obtained.
The preliminary test is conducted up to the number ofloading cycles when the flexibility stops, Nex (about 104
cycles). It is assumed that the surface stress does notchange after that time. Finally, in Step 4, the peelinglife is estimated using the stress history and S-N curve
obtained in Step 3.
In Steps 2 and 4 of this procedure, the degree of
fatigue (the ratio of the total number of loads to thelifetime) is calculated using Minerʼs law from the stresshistory. The concept adopted is that peeling occurs
when the degree of fatigue reaches 100 %.
For surface stress, the average value of von Mises
stress acting at a depth of 0.5 µm below the individual
real contact area is used. To obtain this, the normal
stress component σ̂j and the shear stress component τ̂jkof the triaxial stress acting substantially at a depth of 0.5µm below the true contact area are first determined by
the following equations (1) and (2).

σ̂j = σj, con + σj, res(1)τ̂ jk = τjk, con + τjk, res(2)( j = x, y, z, k = x, y, z, j ≠ k, τjk = τkj )Here, the subscript “con” indicates the contact
stress obtained by contact stress analysis, and “res”
indicates the residual stress obtained by residual
stress measurement. In addition, x represents thecircumferential direction, y the axial direction, and z
the depth direction in the rolling surface. The triaxial
stress components obtained in equations (1) and (2)86

NTN TECHNICAL REVIEW No.89

are then substituted into equation (3) below to obtainthe von Mises stress at a depth of 0.5 µm at the realcontact area, and the surface stress is obtained as theaverage of these values.

σvm=√ {(σ − σ ) + (σ − σ ) + (σ − σ ) + 6 (τ + τ + τ )}
1
2

x

y

2

y

z

2

z

x

2

2
xy

2
yz

2
zx

(3)

Although this estimation method requires preliminary
testing, the life can be estimated in a few hours,
including preliminary testing. By using the data fromthe preliminary tests, it is possible to perform a life
estimation that accurately reflects running-in behavior
under a wide variety of conditions that are difficult tosimulate. Another feature of this method is that it takes
into account the effect of residual stress on the life. The
S-N curves should be prepared for each steel grade andheat treatment of rolling parts.

3. Improved method of contact stress analysis
for application to mixed lubrication conditions
3.1 Overview of contact stress analysis

Under mixed lubrication conditions, the effect ofload support by the oil film cannot be ignored. In thispaper, to apply the peeling life estimation method
described above to mixed lubrication conditions, thecontact stress analysis for estimating surface stress
was enhanced to a method7) that can consider theeffect of load support by the oil film. In this analysis,
the load sharing theory8) of Johnson et al. is applied.
The load on the rolling parts is considered to be sharedby the oil film and the true contact area at a certain
ratio, as shown in Fig. 2. The pressure distribution on
the rolling surface is therefore obtained as the sum of

the pressure distributions of the oil film and the truecontact area. The pressure distributions of the oil film
and the true contact area are determined separately
by convergence calculations with the load sharing
ratio of the oil film as an unknown quantity. The
convergence condition here is that the mass of theoil film in the clearance between two rough surfaces
is equal to the mass of oil film formed between
two smooth surfaces when the sharing load of oil isapplied.
Hereafter, the procedure of this analysis is describedwith reference to Fig. 3. In Step 1, the load sharing
ratio of the oil film to the load W on the rolling area
is set as α. Then, the central oil film thickness hc,
the distribution of oil film pressure Pf (x, y), and theelastic deformation of the two surfaces are calculated
under the condition in which the sharing load of oil
αW is applied to smooth surfaces. An arbitrary initialvalue is set for α in this calculation. The central oil
film thickness hc is calculated with the formula9) ofcentral oil film thickness of Chittenden et al, and thecorrection coefficient 10) due to shear heating. The oilfilm pressure Pf (x, y) and the elastic deformation are
assumed to be equal to them when the two surfaces
contact with the load αW, and are obtained by thecontact calculation program11) using the boundary
element method. This is based on the fact thatthe spike of oil film pressure under high pressure
conditions (several GPa) is small and negligible.12)
Besides, the lubrication regime of rolling bearings is
generally the E-V (Elastic-Variable viscosity) regime
in which the viscosity of the oil increases with the risein pressure and the deformation of the rolling surface
due to hydraulic pressure cannot be ignored. However,under conditions where α is close to 0, the lubrication
regime is not E-V regime. In that case, the Chittenden
et al. formula cannot be applied. Nevertheless, theerror of analysis due to applying the formula ofChittenden et al. may be small as the influence of theoil film on the load support is small when α is close to
0. In Step 2, the three-dimensional roughness dataof the rolling surface actually measured in advanceis added to the shape of the two surfaces after theelastic deformation obtained in Step 1, and the roughsurface shape of the deformed state due to the actionof hydraulic pressure (hereafter, the rough surface

Oil film pressure Pf (x, y) Lubricating oil

shape oftwo surfaces
after elastic
deformation(Step 1 results)

Pre-measured
threedimensionalroughnessα

Load

Oil Truefilm contact
area

Axis

Surfaceroughness

Oil
film
True contact area

Fig. 2 Concept of load sharing theory
shape after hydraulic action) is created.
In Step 3, the remaining load (1 − α)W is applied
to the rough surface shape after the hydraulic action,and the distribution of pressure Pa (x, y) at the truecontact area and the average clearance hm, rough of thetwo surfaces are calculated using the same contact
calculation program as in Step 1. The average clearance
hm, rough is calculated by the following equations.
(4)hm, rough = ∫A hrough (x, y) dxdy / A

In the formula (4), A is the area of the region where
the oil film pressure Pf (x, y) is greater than 0 in Step
1. In addition, the clearance at the true contact
area is set to 0. In Step 4, we examine whether thecentral oil film thickness hc found in Step 1 is equal
to the average clearance hm, rough found in Step 3. If it
is not equal, we return to Step 1 and correct α. Thisis the convergence condition of the contact stressanalysis, and is based on the idea that the mass of
the lubricating oil drawn in between the two surfaces
does not change regardless of the surface roughnessof the two surfaces (law of conservation of mass). In
Step 5, the pressure distribution P (x, y) on the rollingsurface when the total load W is applied is determinedas the sum of Pf (x, y) and Pa (x, y), then the elasticity
solution of Boussinesq13) is applied to the obtainedpressure distribution. In this way, the distribution ofinternal stresses (triaxial stress components) below
the rolling surface is calculated.

3.2 Validity of contact stress analysis

The value of the load sharing ratio α and surface
stress of the oil film obtained by this analysis method
was verified to match the index of oil film formability
obtained in the rolling fatigue test and the residual stress

True contact area pressure
Pa (x, y)

Clearance: hrough (x, y)

hc = hm, rough ?

No

Yes
α

W : Load on rolling parts
α : Oil film load sharing ratio(0 < α < 1)

when rolling under αW・elastic deformation
・oil film pressure Pf (x, y)・center oil film thickness hc

Enlargementof rolling area

arecalculated

Rough surface shapeafter oil hydraulics action

After the oil hydraulics action, the remaining load isapplied to the rough surface shape andare・True contact area pressurePa (x, y)・Average clearance of the two surfaces hm, rough calculatedhm, rough =∫A hrough (x, y) dxdy / A
A: Area of oil film pressure Pf (x, y) > 0

・Calculate the pressure P (x, y) on the rollingsurface when the total load W is applied
P (x, y) = Pf (x, y) + Pa (x, y)・Apply elasticity solution to P (x, y) andcalculate internal stress distribution

correction of αFig. 3 Contact Stress Analysis Procedure7)

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A Life Estimation Method of Peeling in Rolling Bearings Under Mixed Lubrication Conditions
change of the test piece.7) Although details are omitted
in this paper, it has been confirmed that the analyticalvalues of α and surface stress are generally consistent
with the test data. Fig. 4 shows, as an example of
the results of the study, the relationship between theestimated value of the surface stress of the rollingsurface and the measured value of the residual stress in
the rolling fatigue test.
The rolling fatigue tests were conducted in a tworoller testing machine, as described below, with a
gradually decreasing rotational speed starting at
900 min-1. The test was interrupted each time when
the rotational speed was changed, and the threedimensional roughness and residual stress of the rollingsurface were measured at each interruption. From thesemeasurements, the surface stress during the test at
each rotational speed was estimated. Fig. 4 shows thatthe surface stress (estimated value) during operation
exceeded the yield stress of the test piece at a rotational
speed of 125 min-1. The residual stress (measured value)
increased during operation at a rotational speed of 125
min-1. These results are consistent with the predictionof the contact analysis that the rolling surface shows
plastic deformation at 125 min-1 and the validation of theresidual stress measurements.
Rotational speed min-1

Residual stress(measurement value) GPa

0.65

250

125

0.625

3

Surface stress(right shaft)

Residual
0.6
stress(left shaft)

2.5

2

0.575

Yield stress
1.5

0.55
?:4.5〜3.2
0.525

0

200

?:2.1
400

?:1.4
600

800

1 000

Surface stress(estimation value) GPa

900〜
500

1

Overall test time s

Fig. 4 Relationship between surface stress estimatedby contact stress analysis and measured
residual stress7)

4. Life estimation overall accuracy verification
4.1 Rolling fatigue test

To verify the accuracy of the new life estimation
method, rolling fatigue tests were carried out with thetwo-roller testing machine shown in Fig. 5. The drive
roller and the driven roller were cylindrical with anouter diameter of φ 40 mm and a thickness of 12 mm.
A radius of 60 mm was applied to the drive roller onlyin the axial direction. A felt pad impregnated withlubricating oil was used to lubricate the roller test pieceby contacting it from below. The material of the roller
test piece was SUJ2, which was subjected to standard
quench-hardening and tempering so that the surface
hardness was about 63 HRC. Table 1 shows the test
conditions. Testing was carried out under varying
conditions in terms of the surface roughness of therolling surface, the rotation speed, the maximum
hertzian contact surface pressure, and the lubricating
oil type. The Λ of Nos. 6 and 7 tests is more than 0.5,
which corresponds to the mixed lubrication condition.
Since it is easy for peeling to occur in the driven
roller, which has low surface roughness, the life wasdetermined from the occurrence of the peeling on thedriven roller. In principle, each test was interrupted
at 102 cycles, 103 cycles, 5 × 103 cycles, 104 cycles,
and 105 cycles at load. At these interruptions andbefore the test, the three-dimensional roughness
measurement of the drive roller and the driven roller
and the residual stress measurement of the driven
roller were performed. The stress history of thedriven roller was obtained from the collected data. In
addition, the occurrence status of the peeling of thedriven roller was observed with an optical microscopeat the time of interruption. The total number of loads
when the ratio of the area of the occurrence of micro
spalling and cracks to the area of the observation fieldof view (hereinafter referred to as the peeling area
ratio) reached 0.5 % or more at six different places in
the rolling surface was taken as the actual life Lact ofthe peeling. If the peeling area ratio was less than
0.5 % at 105 cycles, the test was continued until 0.5 %was reached.

Table 1 Test conditions of the two-roller test6)
Test
No.
1
2
3
4
5
6
7
8
9
10
11

88

Surface roughness (Ra) µm
Drive roller

Driven roller

2 000

0.75
0.50
0.75
0.40
0.35

NTN TECHNICAL REVIEW No.89

Rotational
speed
min-1

0.02

500
2 000
2 000
500
1 000
2 000

Maximum
Hertzian contactpressure GPa
2.3
1.6

LubricantSynthetic oil
(PAO)

2.3
Mineral oil
2.3

Synthetic oil(PAO)

Dynamic
viscosity(40 ℃ ) mm2/s

Oil film
parameter
Λ

0.11

32.6
47.7

0.12
0.06
0.17
0.52
0.87

6.8

0.15

6.2

0.17
0.24

6.2

Drive roller

Load

Felt pad for
lubrication
Driven roller

MotorFig. 5 Two-roller testing machine2)
4.2 Method for verification of overall estimation accuracy
Peeling life was estimated for the two-roller testingmachines Nos. 1 to 11 shown in Table 1 by theprocedure shown in Fig. 1. The life was estimated withand without considering the load sharing theory in thecontact stress analysis, and the accuracy of the life
estimation was compared between the two cases. The
estimated peeling life Lest for each test was calculated
from the stress history up to 104 cycles. For the life
estimation with load sharing theory, S-N curves weregenerated using the test data except for Nos. 7 and
8, which were used for the calculation of Lest. In thecase of life estimations where the load sharing theory
was not considered, Lest was calculated using the S-N
curve that had already been created in a previous
report.5) The overall accuracy of the life estimation
was examined using the life ratio (Lact / Lest) between Lestand the actual life of the peeling, Lact , which was finallyobtained in the test.

also shown in the figure. In the case of Fig. 6 (a)without consideration of load sharing theory, theoverall accuracy of life estimation for tests Nos. 6 and7, which were conducted under mixed lubrication
conditions, was lower than that of the other tests,
and the error in estimated life relative to actual life(the error is calculated as the largest value or thereciprocal of the minimum value of Lact / Lest Lact /Lest) was 20 times. On the other hand, when the loadsharing theory is taken into account in Fig. 6 (b), theerror in the estimated overall accuracy for tests Nos. 6
and 7 is less than five times, indicating that the overall
accuracy of life estimation under mixed lubrication
conditions is improved. Considering that the life ofbearings generally varies by a factor of 10 or more
under the same conditions14) , the overall accuracyof the life estimation in the case of the load sharing
theory is considered to be at a sufficiently high levelfor practical use.

6. Summary
This article has presented an improved peeling life
estimation method applicable to mixed lubrication
conditions and the results of validation of the overall
accuracy of the life estimation.
1) In this estimation method, a method of contact
stress analysis that applies load sharing theory is
employed, and the effect of lubrication conditions
on peeling life is considered.
2) The maximum error of the peeling life obtainedby this estimation method is less than 5 times of
the actual life, and the accuracy of the peeling life
estimation under mixed lubrication conditions is
improved compared to the conventional method.
The overall accuracy of the above is sufficiently
high for practical use as a peeling life estimation
method for bearings.

5. Results
Fig. 6 shows the relationship between the estimatedlife Lest and the actual life Lact. The median, minimum,and maximum values of the life ratio (Lact / Lest) are

10

10

Test No. 1

1

11

5
9

2

3

Lact / Lest

0.1
8

0.01
0.01

4

0.1

7

6

Median value: 1.07
Minimum value: 0.39
Maximum value: 20.00

1

10Life estimation Lest’×106 times

Life estimation Lest’×106 times

10

11

1

5
8

0.01
0.01

9

6
10

7

4
0.1

2

3

0.1

Test No. 1

Lact / LestMedian value: 1.07
Minimum value: 0.20
Maximum value: 2.22

110Actual life Lact’×10 times

Actual life Lact’×10 times(a) Load sharing theory not considered(b) Load sharing theory considered66

Fig. 6 Relationship between estimated peeling lives and actual peeling lives6)
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89

A Life Estimation Method of Peeling in Rolling Bearings Under Mixed Lubrication Conditions
It should be noted that the scope of application ofthis estimation method is currently limited to purerolling conditions. Going forward, we will work tofurther improve the accuracy of the estimation andextend it to conditions where there is slip on therolling surface. This estimation technique will be used
to provide customer value in the form of improved
bearing reliability and higher machine efficiency.
This paper summarizes the content of thepresentation “Estimation Method of Micropitting
Life from S-N Curve Established by Residual StressMeasurements and Numerical Contact Analysis, 2ndReport” in the Proceedings of Tribology Congress2021 Spring Tokyo, which was organized by theJapanese Society of Tribologists and the presentation“Validation of Contact Stress Analysis of Rolling
Surface under Mixed Lubrication” in the Proceedingsof Tribology Congress 2022 Spring Tokyo. (Somefigures and tables have been translated into Japaneseand rearranged.) We thank the Japanese Society of
Tribologists for their kind permission to publish thispaper.
References
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Photo of authors

90

Naoya HASEGAWA

Takumi FUJITA

Michimasa UCHIDATE

Masayoshi ABO

Advanced
Technology R&D
CenterProduct andBusiness Strategic
Planning Dept.

Faculty of Scienceand Engineering,
Iwate University

School ofEngineering,
University of Hyogo

NTN TECHNICAL REVIEW No.89