o. 19

Life predictionSurface roughness

o-dcro-ratial Fll racon

quantify micro-pitting failure, the micro-pitting severity index (MSI) which is dened as the cumulative

of botine gehe relily altertionali

Extensive experimental studies have been conducted in litera-ture to investigate the inuences of various potential factors on mi-cro-pitting. Using a twin-disk set-up, Tokuda et al. [4] showed thatsurface roughness was a key parameter inuencing micro-pittingeven under full lm lubrication condition. Ariura et al. [5] studiedthe roughness effect on micro-pitting for gear contacts, conrmingthe critical role of surface roughness. Webster and Norbart [6] per-

concerns in mind, various biodegradable lubricants were testedby Cardoso et al. [10] for their micro-pitting performance.

In a recent paper, these authors proposed a physics-based mi-cro-pitting prediction methodology [1]. This methodology em-ployed a mixed elastohydrodynamic lubrication (EHL) model of apoint contact [11] to determine the transient surface traction dis-tributions. These surface traction distributions were applied to aboundary element based rough surface stress prediction model tond the histories of the multi-axial stress components for all thematerial points passing through the contact. The boundary Corresponding author. Tel.: +1 614 247 8688; fax: +1 614 292 3163.

International Journal of Fatigue 48 (2013) 918

Contents lists available at

u

lsE-mail address: li.600@osu.edu (S. Li).certain operating conditions, the amount of micro-pits might stabi-lize after a certain number of loading cycles as the surface devia-tions due to micro-pitting redistribute and relieve the contactpressure. However, the continued cyclic contact can result in thefatigue failure in the form of macro-pitting, which often initiatesform the boundaries of the micro-pitted zones [2,3]. On the otherhand, the prole changes of gear teeth due to excessive micro-pit-ting activity increase the motion transmission error amplitudes tocause elevated vibration levels and dynamic tooth contact forces,further accelerating the rate of micro-pitting.

cro-pitting reduction. Ahlroos et al. [7] performed the fatigue testsusing a twin-disk machine to study the inuences of different steelmaterials, surface roughness amplitudes, surface treatments (sur-face hardness and coatings) as well as lubricants on micro-pitting.Several other experimental works focused on the effects of lubri-cant additives on micro-pitting. Brechot et al. [8] reported thatanti-wear (AW) and extreme-pressure (EP) additives typicallyaggravated micro-pitting. A study by Laine et al. [9] suggested thatfriction modier agents alleviated the occurrence of micro-pitsthrough the reduction of boundary friction. With environmental1. Introduction

Micro-pitting of contact surfacesautomotive, aerospace and wind turbjor problem that adversely impacts tprocess of micro-pitting progressivecontact surfaces, affecting the func0142-1123/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijfatigue.2012.12.003probability of fatigue failure is proposed. The micro-pitting test outcomes are compared to the predic-tions of a recently developed physics-based micro-pitting model [1] to describe the failure mechanismand assess the model accuracy.

2012 Elsevier Ltd. All rights reserved.

h gears and bearings ofarboxes has been a ma-ability of products. Thes the geometries of thety of gearboxes. Under

formed twin-disk fatigue tests and found micro-pits tended to ap-pear on the surface with negative sliding. It was also shown thatthe reduction of micro-pitting could be achieved through thereduction of slide-to-roll ratio and/or roughness amplitude. How-ever, increasing the k ratio (the ratio of lm thickness to roughnessamplitude) by simply increasing the lm thickness while keepingthe roughness amplitude unchanged had limited benets in mi-Rolling contact fatigueMicro-pitting

For the normal operating stage that follows, lower roughness amplitude, lower slide-to-roll ratio, higherrolling velocity and lower contact pressures are observed to lead to reduced micro-pitting activity. ToMicro-pitting fatigue lives of lubricated pand model validation

Sheng Li , Ahmet KahramanDepartment of Mechanical and Aerospace Engineering, The Ohio State University, 201 W

a r t i c l e i n f o

Article history:Received 19 September 2012Received in revised form 12 November 2012Accepted 1 December 2012Available online 12 December 2012

Keywords:

a b s t r a c t

This study employs the twvarious parameters on mimatrix that spans the opestructed using the Fractionrolling velocity, slide-to-roa run-in stage with higher

International Jo

journal homepage: www.ell rights reserved.int contacts: Experiments

th Avenue, Columbus, OH 43210, USA

isk rolling contact fatigue test methodology to investigate the impacts ofpitting performance of lubricated point contacts of rough surfaces. A testng ranges of the sun-planet gear pair of a wind turbine gearbox is con-actorial technique to rank the order of the inuences of contact pressure,tio, roughness amplitude and run-in process. The test results indicate thattact pressure and lower rolling velocity reduce the amount of micro-pits.

SciVerse ScienceDirect

rnal of Fatigue

evier .com/locate / i j fa t igue

the contact in the rolling (tangential) and axial directions, respec-tively. The specimens are made out of AISI 4620 low carbon gearsteel and are case hardened to a surface hardness of 6062 HRCto represent a typical gear tooth surface hardness. A special surfacenishing process was developed to simulate the roughness laydirection of the actual ground gear tooth surface which is perpen-dicular to the direction of sliding, as the previous rolling contact fa-tigue tests that simulated gear contacts found such treatment to benecessary [2]. The axially directed roughness pattern on the rollerand disk surfaces as shown in Fig. 1b are the direct results of thisspecial nishing process that ensures not only the amplitudesbut also the directionality of ground gear surface roughness canbe simulated. Two batches of specimens with the average root-mean-square (RMS) surface roughness amplitudes of 0.3 lm and0.5 lm (composite roughness amplitudes of about 0.4 lm and0.7 lm) are procured. These roughness values are representativeof roughness ranges of typical ground gear tooth surfaces.

2.2. Design of experiments test matrix and test procedure

Table 1 lists the test matrix constructed using the FractionalFactorials technique [12]. This Design of Experiment (DOE) ap-proach uses a fraction of all the combinations of levels for all thefactors considers, allowing statistically meaningful measurementswith substantially reduced number of test runs. In order to studythe inuence of run-in on micro-pitting occurrence, a run-in stageis implemented before each normal test stage. This is especiallyrelevant to wind turbine gearboxes that are put through a run-inprocess. The run-in process that consists of 0.2 million roller con-

Pneumatic cylinder

l Jouelement model included the full description of the microroughnessgeometries in the stress computation such that surface asperity in-duced local stress concentrations can be captured fully. The fatiguedamage was then evaluated using a multi-axial fatigue criterion asdescribed in Refs. [13].

This study focuses on the experimental investigation of micro-pitting with two main objectives. The rst objective is to quantifythe inuences of the contact pressure, rolling velocity, slide-to-rollratio, surface roughness amplitude and a run-in process on micro-pitting failure with the ranges of these parameters to be represen-tative of gears, establishing a statistically meaningful data set. Thesecond objective is to simulate the experiments using the micro-pitting model of Li and Kahraman [1] in order to assess the accu-racy of the model through comparing its predictions to the resultsof micro-pitting experiments.

A test matrix is dened using the Fractional Factorial techniqueto bring certain statistical meaning to the measurements with rel-atively small number of test runs [12]. The operating conditionsare dened based on the sun-planet gear mesh of a wind turbinegearbox [13]. The tests are performed on a two-disk rolling contactfatigue machine with the contact pairs whose surfaces are axiallyground in order to simulate the surface roughness textures of gearsin relation to the direction of rolling. To quantify the degree of mi-cro-pitting after a certain number of contact cycles, Nf0, the micro-pitting severity index (MSI), W, is dened as the cumulative prob-ability of micro-pit crack initiation at Nf0. The probability distribu-tions are constructed using the predicted fatigue lives [1]. NotingthatW is equivalent to the micro-pitting area percentage, the pre-dictions are allowed to compare with the measurements to showreasonably good agreement.

2. Micro-pitting experiments

2.1. Test set-up and specimens

The two-disk set-up as shown in Fig. 1a is employed in thisstudy to evaluate the inuences of various potential factors,including contact pressure, rolling velocity, slide-to-roll ratio, sur-face roughness amplitude and run-in process, on the fatigue failureof micro-pitting. The larger component of the contact pair inFig. 1a, referred as the disk, is fastened axially against its shaftshoulder using a retaining lock nut. The smaller one of the twodisks, referred as the roller, is shrink-tted onto its shaft and in-stalled into the pivoted loading arm, which is pushed against thedisk by a pneumatic cylinder. The lubricant is provided throughan overhead lubrication jet in an into-the-mesh manner. The rollerand the disk are driven independently by two AC motors at therotational speeds of x1 and x2, respectively. Given d1 and d2 asthe diameters of the roller and the disk, the tangential surfacevelocities of v1 12d1x1 and v2 12 d2x2 are adjusted through x1and x2 to create a certain degree of relative sliding at the contactinterface. Dening the rolling and sliding velocities of the contact,respectively, as

v r 12 v1 v2; 1a

v s v1 v2; 1bThe relative sliding at the contact interface is dened by the

dimensionless slide-to-roll ratio as

SR vsv r 2v1 v2v1 v2 1c

10 S. Li, A. Kahraman / InternationaThe roller specimen is designed as a simple cylinder with no ax-ial crown while the disk has a circular axial crown of 76.2 mm ra-dius in order to provide the contact of elliptical shape. Thediameters of the roller and the disk are d1 = 31.75 mm andd2 = 57.15 mm, respectively. With these diameters and axialcrown, b/a = 3.74 where a and b are the Hertzian half-widths of

Roller surface

Disk surface

(b)

1v2v

Fig. 1. (a) The twin-disk contact set-up and (b) close-up view of the disk and rollersurfaces showing the lay direction of the roughness.Loading arm

Lubrication jet

(a)

Roller Disk

rnal of Fatigue 48 (2013) 918tact cycles is followed by the normal test stage of 20 million rollercontact cycles. The same type of lubricant (A typical wind turbinegear uid, Castrol Optigear Synthetic X320) is used for both the

run-in and normal test stages. Six specic parameters are consid-ered in the experimental study as:

Hertzian pressure p0h of the run-in stage, rolling velocity v 0r of the run-in stage, Hertzian pressure ph of the normal test stage, rolling velocity vr of the normal test stage, slide-to-roll ratio SR (run-in and normal test stages have thesame SR), and

initial composite RMS surface roughness amplitudeRq

R2q1 R2q2

q, where Rq1 and Rq2 are the RMS roughness

amplitudes for roller and disk, respectively.

According to the operating condition of the intended wind tur-bine gearbox application, the low and high levels of these param-

eter are dened as ph = 1 and 1.5 GPa, vr = 3.9 and 7.8 m/s,SR = 0.2 and 0.65, and Rq = 0.4 and 0.7 lm. For the run-in stage,p0h and v 0r are determined in relation to their respective normal teststage values as p0h=ph 0:6 and 0.8, and v 0r=v r 0:5 and 1.5 for thelow and high levels, respectively. The inlet lubricant temperature ismaintained at 95 C for both the run-in and normal test stages. Ta-ble 1 lists the test matrix of a total of eight tests dened by theFractional Factorial technique.

Depending on the roller speed of each test, the testing timeranges from days to weeks. During each test, the machine is pausedperiodically to allow interim inspections of the specimens, whichinclude the circumferential surface roughness measurement usinga surface roughness proler (Taylor Hobson Form Talysurf 60), theprole measurement in the axial direction using a gear CoordinateMeasurement Machine, and the micro-pitted area measurementusing a digital microscope. The upper cutoff for the ground surfaceroughness measurements is set at 0.8 mm. These inspections areperformed for three predened locations that are positioned 120apart from each other circumferentially. For the run-in stage, suchinspections are performed before, in the middle and after the run-in process. For the normal test stage, interim inspections are car-ried out every 2 million roller contact cycles.

2.3. Micro-pitting test results

The last column of Table 1 lists a micro-pitting parameter U,which represents the average value (of the three inspection posi-

Table 1Rolling contact fatigue test matrix dened by using Fractional Factorials.

Test # p0hph

v 0rv r

ph (GPa) vr (m/s) SR Rq (lm) U (%)

1 0.6 1.5 1 3.9 0.2 0.4 0.022 0.8 0.5 1.5 3.9 0.2 0.4 0.073 0.8 0.5 1 3.9 0.65 0.7 0.414 0.6 0.5 1 7.8 0.2 0.7 3.065 0.6 1.5 1.5 3.9 0.65 0.7 28.106 0.8 1.5 1 7.8 0.65 0.4 0.007 0.6 0.5 1.5 7.8 0.65 0.4 0.008 0.8 1.5 1.5 7.8 0.2 0.7 2.71

(a)

Position I Position II Position III 1v

S. Li, A. Kahraman / International Journal of Fatigue 48 (2013) 918 11(b) (c)

Fig. 2. Micro-scope images (100magnication) of the roller surface at three differen4 million cycles, (b) 12 million cycles, and (c) 20 million cycles.t locations positioned 120 away from each other circumferentially for Test 5: (a)

Run-in stage

Normal test stage

1 0.65mqR =0 million cycles

1 0.53mqR =0.2 million cycles

1 0.64mqR =4 million cycles

1 0.72mqR =12 million cycles

1 0.66mqR =20 million cycles

1 1.5 2 2.5 3

[mm]x

2 1 0

-1 -2 -3

2 1 0

-1 -2 -3

2 1 0

-1 -2 -3

2 1 0

-1 -2 -3

2 1 0

-1 -2 -3

1R[

m]

(a)

(b)

(c)

(d)

(e)

Fig. 3. Measured roller surface roughness proles in the circumferential directionfor Test 5.

Run-in stage

Normal test stage

2 0.62 mqR =0 million cycles

2 0.58mqR =0.2 million cycles

2 0.47 mqR =4 million cycles

2 0.47 mqR =12 million cycles

2 0.42 mqR =20 million cycles

1 1.5 2 2.5 3[mm]x

2 1 0

-1 -2 -3 2 1 0

-1 -2 -3

2 1 0

-1 -2 -3

2 1 0

-1 -2 -3 2 1 0

-1 -2 -3

2R[

m]

(a)

(b)

(c)

(d)

(e)

Fig. 4. Measured disk surface roughness proles in the circumferential direction forTest 5.

(a)

Position I

(b)

(c)

Position II Position III 1v

Fig. 5. Micro-scope images (100magnication) of the roller surface at three different locations positioned 120 away from each other circumferentially for Test 4: (a)4 million cycles, (b) 12 million cycles, and (c) 20 million cycles.

12 S. Li, A. Kahraman / International Journal of Fatigue 48 (2013) 918

tions) of the micro-pitted area percentage. This area percentage isdened as the ratio of the micro-pitted area to the total inspectedarea. With the magnication level set at 100, each microscopeimage of each inspection position covers an area of(1.74)(1.30) = 2.26 mm2 with the dimension of 1.74 mm being inthe rolling direction. These inspections are performed at three pre-dened circumferential angles of 0, 120 and 240. The low andhigh loading levels as dened in Table 1 corresponds to the Hertz-ian zone size of 0.34 mm and 0.51 mm, respectively, in the rollingdirection, and 1.27 mm and 1.91 mm, respectively, in the axialdirection. Although the microscope does not capture the entireHertzian zone under the high loading condition, the area coverageof 2.26 mm2 at multiple circumferential positions are believed tobe sufcient to obtain representative measurements of U.

Micro-pits show as dark spots on the digital microscope imagesof the contact surfaces. These areas are examined under greatermagnications (200 and 500) to verify they are indeed pittedzones of micro-scale and are then highlighted in red as shown be-low to quantify the total micro-pitted area. Any circumferentialwear scratches appear as lines in the rolling direction. As such,these wear scars have no resemblance to micro-pits and can beeasily excluded from the micro-pits quantication.

In Table 1, it is observed that Test #5, which corresponds to thehigh levels of ph, SR, Rq and v 0r while the low levels of p0h and vr, isthe most severe micro-pitting case with U = 28.1%. The micro-pit-ted roller surfaces of Test #5 are shown in Fig. 2 for 4, 12 and20 million roller contact cycles of testing at three different circum-

ferential positions. In these images, the micro-pits are highlightedin red. It is seen that the severity of micro-pitting is not uniform atdifferent circumferential locations, such that it is necessary tomeasure at a number of positions and use the average value asthe measure. The measured surface roughness proles for the roll-er and the disk at Position I are shown in Figs. 3 and 4, respectively.The roughness amplitude of the roller is observed to decrease dur-ing the run-in stage. The asperity peaks are rounded off in the pro-cess while the deep valleys are preserved. For the normal teststage, however, Rq1 is found to increase in comparison to the afterrun-in roughness amplitude, which is caused by the removal ofsurface material in the form of numerous micro-pits. This observa-tion is not evident on the disk surface, which experiences positivesliding. Fig. 4 shows that Rq2 decreases in the run-in stage and isfurther reduced during the normal test stage. Due to the highslide-to-roll ratio of Test #5, the disk actually experiences morecontact cycles than the roller does. However, only very limitednumber of micro-pits are produced on the disk surface, having neg-ligible inuence on Rq2 during the normal test stage. Figs. 5 and 6are the microscope images for Tests #4 and #6 as dened in Ta-ble 1. Test #4 produces a small amount of micro-pits (U = 3.1%)while Test #6 has no sign of micro-pit (U = 0) at the completionof the 20 million roller contact cycle test. Several circumferentialwear scars that are parallel to the rolling direction are also ob-served in Fig. 6.

UsingU at the end of each test as the response, the main effectsmodel of ~U b0

P6i1bigi is constructed. Here, ~U is the estimated

(a)

Position I Position II Position III 1v

S. Li, A. Kahraman / International Journal of Fatigue 48 (2013) 918 13(b)

(c) Fig. 6. Micro-scope images (100magnication) of the roller surface at three differen4 million cycles, (b) 12 million cycles, and (c) 20 million cycles.t locations positioned 120 away from each other circumferentially for Test 6: (a)

tions and displacements along the boundary of the contact bodydetermined, the near surface multi-axial stress elds are calculatedusing the boundary elements approach. In the process, the singularand near singular behaviors involved in the boundary integralequations are eliminated through efcient numerical methods.The predicted means and amplitudes of the stress components ofevery material point that passes through the contact are used toevaluate the fatigue damages according to a multi-axial fatigue cri-terion. The detailed predictions of the mixed lubrication behavior,the near surface stress distributions and the crack nucleation fati-gue life distributions are illustrated here by using the simulation ofTest #5 in Table 1 (which is most severely micro-pitted). The pre-dicted micro-pitting severity indexes of all the eight tests are com-pared with the measured micro-pitting area percentages todemonstrate the model capabilities as well as any potentialimprovement.

The dimension of the surface computational grid is dened suchthat 1.875a 6 x 6 1.125a and 1.5b 6 y 6 1.5b, where a and brepresent the Hertzian half widths in the x (rolling) and y (axial)directions, respectively. The mesh density of 256 256 is used,which results in the mesh sizes ofDx = 3 lm andDy = 11 lm. Sincethe variation of the roughness prole in the y direction is limited

10

[P

as]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

105

104

103

102

101

100

10-1

10-2

l Journal of Fatigue 48 (2013) 918average value of the micro-pitted area percentage, and g1g6 repre-sent the low or high level of the six factors of p0h=ph, v 0r=v r , ph, vr, SRand Rq, respectively (gi = 1 for low level and gi = 1 for high level).The absolute value of the slope bi indicates the extent of impor-tance of the factor gi. From the measurements, it is determined thatb0 = 0.043, b1 = 0.035, b2 = 0.034, b3 = 0.034, b4 = 0.029,b5 = 0.028 and b6 = 0.043. The surface roughness amplitude withthe maximum absolute slope of |b6| = 0.043 is found to be the mostinuential factor for the operating condition range considered inthis study. The corresponding main effects plot of the six contactparameters is shown in Fig. 7, in which, the inuence of each factoron micro-pitting is determined by holding the other factors con-stant at zero [12]. The reference line of ~U 4:3% is the overallmean of the eight tests. It is found that a run-in stage with rela-tively large contact pressure and relatively small rolling velocityeffectively reduces the occurrence of micro-pits. The potential rea-sons include (i) gradual surface polishing, which alleviates thestress concentrations induced by local roughness interactions forthe normal test stage, (ii) the formation of a tribo-lm with lowerboundary friction that reduces the surface shear, (iii) the raise ofthe local surface hardness through work hardening, and (iv) crea-tion of local compressive residual stresses that introduce compres-sive mean stresses. For the normal operating stage, lower contact

7.5

5

2.5

0 3.9 7.8 -0.65 -0.2 0.4 0.7

rv [m/s] SR qR [m]

Fig. 7. Main effects plot for the DOE screening process using Fractional Factorials asdened in Table 1.0.6 0.8 0.5 1.5 1.0 1.5

10

7.5

5

2.5

0

h hp p r rv v hp [GPa]

[%

] 14 S. Li, A. Kahraman / Internationapressure and higher rolling velocity are observed to lead to thereduction of micro-pitting, which is due to the reduced multi-axialstress amplitudes under the lower loading condition and the re-duced asperity contact activities resulted from the thicker uidlm under the higher rolling velocity condition. Additionally, smal-ler roughness amplitude and lower sliding (smaller absolute valueof SR) are shown to suppress the occurrence of micro-pits. The re-duced roughness peaks decrease the local stress concentrations.The low sliding condition alleviates the shear thinning of lubricantlm, reducing the surface asperity interactions.

3. Simulation of rolling contact fatigue experiments and modelvalidation

The physics-based micro-pitting model proposed by Li andKahraman [1] is employed in this study to simulate the micro-pit-ting experiments presented in Section 2. This model determinesthe surface normal and tangential tractions using a mixed elasto-hydrodynamic lubrication formulation for rough surface point con-tacts [11]. The induced surface displacements are computedthrough a boundary element based formulation which fully de-scribes the local microsurface roughness geometry. With both trac-p [GPa]

Fig. 8. Pressureviscosity relationship used in this study.

0 200 400 600 800 1000

aC

0.15

0.12

0.09

0.06

0.03

0

B

A

C n

Fig. 9. Variation of asperity contact area ratio with time for Test 5.

due to the axially oriented roughness textures, this relatively largeDy is considered to be sufcient. The time increment of D# = 2Dx/vr is used for a total of N0 = 1000 time steps, such that the entireanalysis covers about 4 mm traveling distance of the roller surface.For the wind turbine gear uid (Castrol Optigear Synthetic X320)used in this study, its density is roughly measured to beq0 = 812 kg/m3 at the inlet temperature of 95 C. Due to the lackof the data of the viscosity dependence on pressure for OptigearSynthtic X320, the property of a similar lubricant (Optigear Synth-tic A320) whose ambient viscosity g0 = 0.0276 Pas and pressureviscosity coefcient a = 14.3 GPa1 is used in the simulation in-stead. The Roelands pressureviscosity relationship [14] is as-

sumed as shown in Fig. 8 because of these limited oil propertymeasurements. To take into account the effects of the surfaceroughness variation in the run-in stage (the round-off of asperitypeaks), the measured roughness proles at the end of the run-inprocess (Figs. 3b and 4b) are used in the simulation.

For the determination of the severity of asperity interactionactivities, the area ratio of asperity contact Ca, which is denedas the ratio of the total asperity contact area to the nominal Hertz-ian area is plotted as a function of time step n0 (n0 2[1,N0]) in Fig. 9(for Test #5). The average value of Ca over this time period is about6.5% (i.e. 6.5% of the Hertzian area experiences asperity contacts onaverage). For the three example time instants of n0 = 148, 506 and

S. Li, A. Kahraman / International Journal of Fatigue 48 (2013) 918 15Fig. 10. Mixed EHL predictions of normal contact pressure (left column) and tangential shand (c) n9 = 657 (C in Fig. 9) for Test 5.ear (right column) for time steps of (a) n9 = 148 (A in Fig. 9), (b) n9 = 506 (B in Fig. 9)

657 (denoted as points A, B and C in Fig. 9), which respectively rep-resent the instances of high (Ca = 13.2), median (Ca = 6.5) and low(Ca = 1.7) asperity interaction, the predicted normal contact pres-sure and tangential surface shear distributions are shown inFig. 10. The pressure uctuations are observed in the distributionsfor all the three time instants, and the surface shears are seen tospike up wherever the lm thickness breaks down (a typicalboundary friction coefcient value of 0.15 is used in asperity con-tact areas). As Ca decreases, the asperity contact friction reducesevidently. The peaks of the normal pressure, however, can still berelatively high because of the surface discontinuities. For instances,although the Ca value at point C is substantially smaller than that atpoint B, the maximum local pressures at these time instants arequite comparable.

With the surface loading condition as illustrated in Fig. 10, thesurface and near surface stress state for every material point con-sidered can be calculated using the boundary element based stressmodel as proposed in Ref. [1]. Fig. 11 shows the stress elds alongthe central vertical plane of y = 0 for point A, B and C dened in

tions on the roller surface have different surface roughness proles,resulting in the distributions that might be different from the oneshown in Fig. 13. In order to include such variation, the bootstrap-ping method is used to determine the 95% condence interval for

0.7 0.35 0 -0.35 -0.7

0.1

-0.925

-1.95

-2.975

-4

[GPa]

0 0.1 0.2 -0.2 -0.1 0 0.1 0.2

506 657n =

14

12.06

10.13

8.20

6.26

1

0

-1

-2

-3

-4

-5

-6

-7

10log ( )fN

x

z[

m]

1 1.5 2 2.5 3 3.5

16 S. Li, A. Kahraman / International Journal of Fatigue 48 (2013) 918Fig. 9. The components of rxy and ryz are negligibly small in com-parison with the others and are omitted in this gure. It is observedthat the normal stress concentrations mostly occur at the high-lands of the surface roughness, below which the orthogonal shearalternates its direction. These roughness highlands are exactlywhere the micro-pits were found in the experiments of Section 2.

With the history of the stress state of a certain material point,its fatigue life is assessed using a multi-axial fatigue criterion [13]. As the material points of the contact body pass through the con-tact zone, they experience different loading histories because ofthe different local surface roughness geometries and the sliding ac-tion (if any) which changes the transient match of the local rough-ness heights of the mating surfaces. In order to include thesevariations, the fatigue analysis is performed covering a sufcientlylong roughness prole segment of a length about 4 mm. Fig. 12 dis-plays the distribution of the micro-pitting crack nucleation fatiguelife Nf for the central vertical plane of y = 0. The most critical spots(with short lives) are observed to be on the highlands of the surfaceroughness. Many of the material points located in the roughnessvalleys are predicted to be able to survive more than (10)9 contactcycles. Comparing the fatigue lives along the depth direction z, it isconcluded that the failure is surface initiated, which is in agree-ment with the experimental observation.

Focusing on the surface layer of Fig. 12, the probability densityfunction w(Nf) of the fatigue lives is constructed in Fig. 13e for Test

(a)

0 -2 -4 -6

1 0

-1

-0.2

z[

m]

R[

m]

0 -2 -4 -6

0 -2 -4 -6

0 -2 -4 -6

-0.1 0 0.1 0.2 -0.2 -0.1

(b)

(c)

(d)

(e)

148n = n =x [mm

Fig. 11. Transient stress elds of (b) rx, (c) ry, (d) rz and (e) rxz under the transient ro#5, with the predicted w(Nf) for the other seven tests specied inTable 1 displayed in Fig. 13ad and fh. For Test #5, it is predictedthat about 29% of the surface material points have the fatigue livesthat are less than 20 million cycles while around 40% of the mate-rial points have the lives in excess of 100 million cycles. To providea measure for micro-pitting failure prediction, the micro-pittingseverity index (MSI) W(Nf0) at the cycle number of Nf0 is denedas the percentage of the material points having the fatigue livesless than Nf0, i.e. W(Nf0) is the cumulative probability function of

WNf0 Z Nf01

wNf dNf 2

Assuming that the population of these surface layer materialpoints considered is a representative pool of the entire surfacepoints, W(Nf0) is equivalent to the micro-pitting area percentageU, such that the model predictions can be compared directly tothe measurements. It is recognized, however, that different posi-

Fig. 12. Micro-pitting crack nucleation fatigue life distribution along the centralvertical plane (y = 0) for the entire 4 mm surface roughness segment considered forTest 5.]

ughness contact condition of (a) along the central vertical plane (y = 0) for Test 5.

(e)

(f)

(g)

l Jou40

30

20

10

0

[%]

40

30

20

10

0

40

30

(a)

(b)

(c)

620 10fN =

S. Li, A. Kahraman / InternationaW(Nf0). This simple technique constructs subsamples with thesame size as that of the original data set (predicted fatigue life pop-ulation) by drawing with replacement. As a result, some of thesubsamples have the data points with relatively short fatigue livesrepresenting the surface area where the asperity peaks are rela-tively intense and some of the subsamples have the opposite char-acteristics. Wi(Nf0) is then calculated for each subsample i (i = 1 to3000 in this work) and the probability density distribution is builtfor the data set of Wi(Nf0) such as in Fig. 14 for Test #5 to nd itscondence interval.

In Fig. 15, the predictedW(Nf0) (Nf0 = 20 million cycles) togetherwith its 95% condence interval are compared with the measuredaverage micro-pitting area percentage U for all the eight tests. Forthe case ofW = 0 (Tests #1, #2, #6 and #7), the condence intervalcannot be constructed and only W is plotted. It is observed inFig. 15 that the model predictions are in good agreement with mostof the measurements except for Tests #3 and #8. Comparing theoperating conditions dened in Table 1 between Tests #3 and #4only for the normal test stage, it is seen that Test #3 has lower roll-ing velocity and higher sliding which would lead to the appearanceof more micro-pits (the model predictsW = 15). However, the mea-sured average micro-pitting area percentage is only 0.41%, which ismuch lower than that of Test #4 (3%). Similarly, comparing Test #8with #4, the only difference for the normal test operating condition

log

20

10

0

40

30

20

10

0

(d)

6 7 8 9 10 11 12 13 14

Fig. 13. Predicted probability density distribution of the micro-pitting crack nucleation fTest 4, (e) Test 5, (f) Test 6, (g) Test 7 and (h) Test 8.rnal of Fatigue 48 (2013) 918 17is the higher contact pressure in Test #8, which would result inmore severe micro-pitting. However, the measurement showssmaller amount of micro-pits of 2.7% while the model predicted

10 ( )fN

(h)

6 7 8 9 10 11 12 13 14

atigue life for the material points along Z = R for (a) Test 1, (b) Test 2, (c) Test 3, (d)

0( )fN [%]22 24 26 28 30 32 34 36 38

16

14

12

10

8

6

4

2

0

Freq

uenc

y [%

]

Fig. 14. The probability density distribution ofW(Nf0) with Nf0 = 20 106 for Test 5,constructed using bootstrapping method.

possible causes include the rounding-off of the asperity peaks,the formation of low-friction tribo-lm, the local hardness increasedue to work hardening and the production of local residualstresses.

The experiments were simulated using a physics-based micro-pitting model which incorporates the roughness geometry in thestress prediction. The micro-pitting severity index was proposedas a measure of the micro-pitted area percentage. Through thecomparison between the model predictions and the measure-ments, the model was shown to be able to yield results that are

MeasurementPrediction

[%

] 35

30

25

20

15

Measurements

Predictions

18 S. Li, A. Kahraman / International Journal of Fatigue 48 (2013) 918W = 6. The possible cause for these deviations might be that certainbenets introduced by the run-in process are not captured by themodel. For instance, the high level of the contact pressure or thelow level of the rolling velocity or the combination of the two inthe run-in stage could help the formation of a low-friction tribo-lm on the contacting surfaces and/or raise the local surface hard-ness through work hardening. Although the effects of the surfaceroughness variation during the run-in stage is included in the mod-eling by using the measured roughness proles after the run-inprocess, the other potential effects of the run-in process are notconsidered in the model of Ref. [1].

4. Conclusions

Test Number

10

5

0 1 2 3 4 5 6 7 8

Fig. 15. Comparison of W(Nf0 = 20 106) between the measurements and themodel predictions.A set of rolling contact fatigue tests with the operating condi-tions that are representative of those of a gear pair from a turbinegearbox were conducted to assess the inuences of various contactparameters on micro-pitting failure. The Fractional Factorial tech-nique was used to construct the test matrix in order to reducethe number of test runs meanwhile obtaining statistical meaning-ful measurements. It was found reduced contact pressure, slide-to-roll ratio and surface roughness amplitude and increased rollingvelocity lead to the reduction of micro-pits. It was also interestingto see that relatively large contact pressure and relatively smallrolling velocity (in other words severe contact conditions) in therun-in stage can effectively reduce the amount of micro-pits. Thein good agreements with the experimental observations in termsof micro-pit crack nucleation site and fatigue life. Certain discrep-ancies between the predictions and measurements point to certainareas of potential improvements for the model, including the accu-rate modeling of the run-in mechanisms.

Acknowledgements

Author thanks Department of Energy (DOE) EERE Wind &Water Power Program for sponsoring this research activity.

References

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[2] Li S, Kahraman A. A fatigue model for contacts under mixedelastohydrodynamic lubrication condition. Int J Fatigue 2011;33:42736.

[3] Li S, Kahraman A, Klein M. A fatigue model for spur gear contacts operatingunder mixed elastohydrodynamic lubrication conditions. ASME J Mech Des2012;134(041007):11.

[4] Tokuda M, Nagafuchi M, Tsushima N, Muro H. Observations of the peelingmode of failure and surface-originated aking from a ring-to-ring rollingcontact fatigue test rig, rolling contact fatigue testing of bearing steels. ASTMSpecial Technical Publication; 1982. pp. 771.

[5] Ariura Y, Ueno T, Nakanishi T. An investigation of surface failure of surface-hardened gears by scanning electron microscopy observations. Wear1983;87:30516.

[6] Webster MN, Norbart CJJ. An experimental investigation of micropitting usinga roller disk machine. Tribol Trans 1995;38(4):88393.

[7] Ahlroos T, Ronkainen H, Helle A, Parikka R, Virta J, Varjus S. Twin discmicropitting tests. Tribol Int 2009;42(10):14606.

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[9] Laine E, Olver AV, Lekstrom MF, Shollock BA, Beveridge TA, Hua DY. The effectof a friction modier additive on micropitting. Tribol Trans2009;52(4):52633.

[10] Cardoso NFR, Martins RC, Seabra JHO. Micropiting of carburized gearslubricated with biodegradable low-toxicity oils. J Eng Tribol2009;223(3):48195.

[11] Li S, Kahraman A. A mixed EHL model with asymmetric integrated controlvolume discretization. Tribol Int 2009;42(8):116372.

[12] Dean A, Voss D. Design and analysis of experiments. New York: Springer-Verlag; 1999.

[13] Kahraman A, Li S, Houser DR. An experimental and theoretical investigation ofmicro-pitting in wind turbine gears and bearings, nal report for GrantNumber DE-EE0002735. Department of Energy; 2012. http://www.osti.gov/bridge/product.biblio.jsp?osti_id=1037344.

[14] Roelands CJA, Correlational aspects of the viscositytemperaturepressurerelationship of lubricating oils, PhD thesis, University of Technology, Delft,1966.

Micro-pitting fatigue lives of lubricated point contacts: Experiments and model validation1 Introduction2 Micro-pitting experiments2.1 Test set-up and specimens2.2 Design of experiments test matrix and test procedure2.3 Micro-pitting test results

3 Simulation of rolling contact fatigue experiments and model validation4 ConclusionsAcknowledgementsReferences