Mechanism of Crack Growth in Lubricated Rolling- Sliding Contact

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Mechanism of Crack Growth in Lubricated Rolling/

Sliding ContactM. Kaneta

a, H. Yatsuzuka

b& Y. Murakami

c

aKyushu Institute of Technology, Tobata, Kitakyushu, Japan

bHitachi Ltd., Kokubu, Hitachi, Japan

cKyushu University, Hakozaki, Fukuoka, Japan

Published online: 25 Mar 2008.

To cite this article: M. Kaneta , H. Yatsuzuka & Y. Murakami (1985): Mechanism of Crack Growth in Lubricated Rolling/Slidi

Contact, A S L E Transactions, 28:3, 407-414

To link to this article: http://dx.doi.org/10.1080/05698198508981637

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Mechanism of Crack Growth inLubricated RollingISliding Contact@M. KANETA (Member, ASLE)

Kyushu Institute of Technology, Toba ta, Kitakyushu, JapanH. YATSUZUKA

Hitachi Ltd., Kokubu, Hitachi, Japanan d

Y. MURAKAMIKyushu University, Hakozaki, Fukuoka, Japan

In order to explain the mechanism of rolling-contact atigue crack This paper describes the mechanism of a three-dimen-growth a nalytically, fracture mechanics are applie d to a semi- sional surface crack propagation, qualitatively and synthet-circular suqace crack inclined at an angle to the elastic half-space ically, referring Keer's treatment. th e Pro-loaded by Hertzian stresses. cess of rolling-contact fatigue involves plastic flow below thecontact, its effect is ignored.It is shown that the surface traction is the controllin g factor fo rlubricant seepag e into the crack and for shear mode crack growthrat e. It is also clarijied that the gene ration of pits results from tensile METHOD OF ANALYSISnzode crack growth mainly due to the oil hydraulic pressure action. Analytical Model and Method of AnalysisINTRODUCTION

Although a great deal of investigation has been expendedin elucidating the mechanism of rolling-contact fatigue phe-nomenon (pitting, flaking, spalling, etc.) the subject is stillgreatly debated. It is quite adequate to utilize fracture me-chanics as a method to understand rolling-contact fatiguephenomenon, because the phenomenon is concerned withthe growth of cracks.

In the previous paper ( I ) , the authors discussed the growthbehavior of cracks formed on lubricated rollinglsliding con-tact surfaces by focusing their at tention to the relationshipbetween the m agnit ude of stress intensity factor K,,, whichprescribes the intensity of the tensile stress field near thetip of the crack and the magni tude of the threshold stressintensity factor A&,,, of the actual materials. As a result, ithas been shown from the viewpoint of the fatigue crackpropagation that the oil hydraulic pressure effect pointedou t by Way (2) in 1935 may be accepted as a possible mech-anism of surface crack growth. Moreover, it has been an-alytically confirmed that the crack on the follower surfacepropagates more easily than that on the driver surface, andthat the surface traction, i.e. the tangential force, becomesan important factor for the crack propagation in the pointthat it controls lubricant seepage into the crack. However,Keer and his coworkers (3) , 4) have suggested, throughtheir analysis of stress intensity factors for two-dimensionalsurface crack du e to Hertzian loading, that the crack growthdepends upon the shearing mode, i.e. mode 11, and theyhave given negative view with respect to the tensile modecrack growth.

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A lubricated rollingJsliding contact system which containscracks on their surfaces can he simulated by the systemshown in Fig. 1. The elastic half-space con taining a surfacecrack inclined at an angle (90" - a) o the half-space surfaceis loaded by Hertzian contact pressure . A cycle of rollingcan be viewed by shifting Hertzian contact pressure acrossthe su rface of the cracked half-space in a direction oppositeto that of rolling. Traction force proportional to Hertziancontact pressure appears also within the contact region. Thesenormal and tangential stresses are defined as follows:

where f is mean coefficient of surface traction. A surfacecrack is assumed to be a semicircle with radius a , and to bein they* - z* plane.

The effect of lubricant penetrated into the crack is mod-eled by assuming tha t the fluid lubricant transmits the no r-mal Hertzian contact pressure applied at the mouth of thecrack, though the value of pressure is assumed to decreaselinearly from crack mouth to crack front. That is, taking eas the distance between the crack mouth and the center ofHertzian contact, the fluid pressure applied at the crackfaces is represented as

Th e three-dinlensional stress intensity factors lor the sur-face crack were obtained with the aid of the body force


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408 M. KANETA, . YATSUZUKAND Y. MURAKAMImethod. The semicircular crack was divided into 72 trian-gular subregions as shown in Fig. 2 and the values of weight-ing functions indicating the intensity of the pair of the bodyforces were determined so as to satisfy the boundary con-clitions for he crack uncler the contact stress field. I n thisproceclure, each triangular subregion was subdivided intofour smaller triangles with the same size in orde r to improve(h e accuracy ofanalysis. T he mean value of four weightingf~~nctionsvaluated at the centers of gravity of the fourtriatngles was regarded as the representative value for sat-isfying the boundary condition. The values of weightingf ~ ~ n c ~ i o n setermined at the triangular subregions at thecrack contour correspond to the din~ensionless tress inten-sicy liictors F I ,FI I , nd FIII .The stress intensity factorsKl, KII , nd Kill are related to the dimensionless stress in-tensity factors as follows:

KI = FI p,G, [41KII= FIIp,6 nd Kill = Fill po6

In cliscussing Way's hypothesis as t he possible mechanismof pitting plienonienon, the values of KI, KI I, nd Kill wereevaluacecl by using he experimental data of Ref. ( 5 ) , .e.the maxim~rmHertzian pressurep , = 1.1 GPa and the half-contact wiclth c = 0.2 mni. Poisson's ratio was assumed tobe 0.3.Conditions of Fatigue Crack Growth

- 7I he stress intensity factors which represent the intensitiesof the fields of shearing stress r,e and tensile stress cre near(.he crack tip a re expressed by the following formulas:

1 0= - cos - [KIsinO + KII 3cosO - l) ]2 2

0 0 3K,,(O) = = cos - [K1cos2- - - KllsinO] [6]2 2 2where ( r ,0) indicates the polar coordinate systems the originof which is located at crack tip (see Fig. I).

Fig. l-Analytlcal model and coordinate systema = radius of semi-circular crackc = half width of Hertzian contactu = crack incllnatlon angle(r,0) = polar coord inates with origln at crack tipp = angle from deepest crack tip of crack front

Fig. 2-Mesh pattern of crack face

Otsuka and his coworkers (6) have shown through theirfractographical considerations that the critical condition ofshear mode fatigue crack growth is given by K,(0),, =

(consl.), and that of tensile m ode fatigue crack growthby K,(O),, = M o t h (const.), approximately an d show alsothat K,(O) is equal to AK,,,,t the transition from shear modegrowth to tensile mode growth. Therefore, in this study, itis assumed that the shear mode growth starts macroscopi-cally in the direction along which KT@)has the maximumvalue, when KT,, is larger than the threshold stress intensityfactor AK,th. Th e angle 0, which gives the maximum valueof K,(O) is obtained by putting the derivative of K,(0) withrespect to 0 equal to zero, i.e.

3 0 1 0 7 0 1tan - - - tan2- - - an - + - 0, y = KIIIKI [7]2 y 2 2 2 2 yConcerning the tensile mode growth, the criterion given byErdogan and Sih (7 ) is adop ted . The angle 0, of crack growthby tensile mode is given by one of two roots of the followingequation obtained by putting the derivative of K,(0) withrespect to 0 equal to zero.

0tan = (1 t -)My, y = KIIIKI [8 ]2where we must choose the angle 0, which makes K,(O) themaximum. T he critical condition in both the tensile modefatigue crack growth and the transition from shear modeto tensile mode is given by K,(0),,zhK,th.

Otsuka el al (6) have experimentally determined thethreshold stress intensity factors as AKslh=1.5M ~ a . m l / ~ndM u I h = 6MPa.ml'' for low carbon steels. Many experimenta lresults have commonly shown that most values of M a t h infatigue fracture range only from 3 to 7.5 ~Pa . m ' / ' , r re -spective of the kinds of materials.

RESULTS AND DISCUSSIONVariations of Stress intensity Factors Caused byMovement of Contact

Figures 3 and 4 show the variations of dimensionless stressintensity factors for semicircular inclined crack caused bythe movement of Hertzian contact pressure. F I and FIIarethe values for crack tip at the deepest position. In thesecalculations, it is assumed that the crack faces receive fluidpressure due to lubricant penetrated into the crack. Theabscissa shows the normalized distance, elc, between the crackmouth, which is at origin , and the center of Hertzian contactpressure. In the region of (elc(


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Mechanism of Crack Growth in Lubricated RollinglSliding Contact 409responding to Eq . [3] acts on the crack faces. The relation-ship between the inclined angle of crack and the directionof he movement of contact pressure differs depending onthe direction of surface traction force. So, this relation isrealized by changing the direction of traction force underthe condition where the direction of inclination of crack is

- 0 2 ' 3 I I I I I I-2 - 1 0 1 e / c 3(b)

Fig. 3--Varlatlons of stress intensity tactors due to moveme nt of contact(a) f = -0.1( b ) f = 0 .1

fixed to the positive x direction in Fig. 1. When Hertziancontact moves from left of the crack mouth to right overthe semi-infinite surface, the conditions f > 0 and f < 0correspond to the driver surface, i.e. positive sliding surface,and the follower surface, i.e. negative sliding surface, re-spectively [see Fig. 5(A) ] . On the contrary, when Hertziancontact moves from right to left, the conditions f > 0 andf < 0 correspond to the driver and the follower surfaces,respectively [see Fig. 5(B)] .

Fig. 4--Varletlons of stress intensity factors due to movement of contact(a) f = -0.3(b) f = 0.3


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410 M. KANETA.H. YATSUZUKA ND Y. MURA KAMIWhen the right end of Hertzian contact pressure is at theleft of the crack mouth, i.e. elc < - 1, as described in the

previous paper (I ), the crack on the driver surface (f > 0)is closed (FI < 0) and on the follower surface (f < 0) thecrack is open (FI > 0) at least at the mouth of the crack.Since lubricant cannot penetrate into the crack interior ifthe mouth of the crack is closed, it may be quite reasonableto consider that lubricant seepage into the crack on theclriver side is almost impossible. It can be seen from thecomparison of Figs. 3 and 4 that the magnitude of lFll inthe region of lelcl>l increases with increasing the surfacetraction, i.e. the magnitude of Ifl.Furthermore, the effectof surface traction increases with decreasing crack size.

Contrary to the above discussion, in the case where Her t-ziiill contact pressure moves over the inclined crack surfacefrom right to left, the crack on the driver (f < 0) is closedwhen the left end of Hertzian contact pressure is at theright of the crack mouth. The mouth of the crack on thefollower (f > 0) is open if the crack size is small or them:~gnitudeof the surface traction is large. However, lubri-cant penetrated into the crack in this region may be ex-cluded because the crack is closed from its deepest positionLO crack mouth as the contact moves at the region of e/c >1 in the neighborhood of elc= 1 as described in the previouspape r (1). In case of pur e rolling (f = O), the crack is alwaysclosed in the region of le/c(hl.

Consequently, it can be concluded that lubricant may verypossibly penetrate into the crack, only when it is inclinedso as to deepen in the direction of the movement of Hertziancontact, on the follower surface. In other cases involvingthe pure rolling, lubricant seepage into the crack interiormay be impossible or seems to be limited.When Hertzian contact pressure is applied over the mouthof the crack uncler the existence of lubricant in the crackinterior, the crack is likely to open (FI > 0) clue to the effectof the Iluid pressure as shown in Figs. 3 and 4. However,if the crack size is small, FI has negative value because thestresses in the neighborhood of Hertzian contact region arecompressive states. It should be noted that the magnitudeof positive value of FI is not remarkably influenced by thesurface traction.Posslbillty of She ar Mode Fatigue Crack Growth

It has been founcl out in the previous paper (I), hat ift.here is no fluicl pressure effect brought by the lubricant

A ( a ) F o l l o w e r s u r f a c e A ( b ) D r i v e r s u r fa c e

B ( a ) F o l l o w e r s u r f a c e B ( b ) D r i v e r s u r f a ceFlg. 5--Relatlonshlp amon g dlrectlons of crack Incllnatlon, movement ofcontact pressure and surface traction.

seepage into the crack interior, it is difficult to explain thetensile mode crack growth from the viewpoint of fatiguecrack propagation. However, as pointed ou t by Keer et al( 3 ) , 4 ) , he shear mode crack growth may occur. The shearmode fatigue crack growth will be remarkably influencedby frictional force between the contacting crack faces if thecrack is closed. Therefore, we will discuss the possibility ofthe shear mode fatigue crack growth under the conditionswhere the re is no fluid pressure and the parts of crack facesare capable of transmitting compressive stresses.

The boundary condition with respect to the shearing stressT,, which acts on the crack faces, was assumed as

In the above, at and T* are the compressive and shearingstresses, respectively, which act on the crack faces imaginedin semi-infinite body with no crack subjected to the rollingcontact pressure,f, is the coefficient of friction between thecontacting crack faces, and sgn(r*) s the sign function, de-fined as follows:

I I I I I I- 3 -2 -1 0 1 e/c 3

(b )Fig. &Effec t of frlctlonal force between crack faces for small crack(a) f, = 0.2, alc = 0. 1

(b) f, = 0.5, alc = 0.1


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Mechanism of Crack Growth in Lubricated RollingISliding Contact 41 1

their experimental facts with respect to the occurrence O Fpitting.

The above results may arrive at a conclusion that in thecase of pure rolling (f= 0) or when the contact surfacesare separated completely by EHL film (If < 0.1), the pos-sibility of the propagation of microcrack is extremely lowexcept for the case that the maximum contact pressure be-comes several times as large as p, = 1.1 GPa assumed in. 1 0 1 e/ c 3 this analysis.

Equation [9] is defined only when a*< 0 on the closed partsof the crack. Figures 6(a) and 6(b) are plots of KT,, at thedeepest position of the crack front, for normalized smallcrack size alc= 0.1 and for f , = 0.2 and 0.5.

IK,,,l achieves the maximum value at the near points ofelc= + 1, where the right a nd left edges of Hertzian contactpressure a re just on the crack mouth. Dur ing a passage ofHertzian contact over the crack, i.e. lelcl < 1,KT,, becomeszero over the most part of the interval - < elc


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412 M. ANETA,H. YATSUZUKA N D Y. MURAKAMIpropagation rate of crack is accelerated, especially on theIbllower si~rfacewhere the direction of surface traction isthe siuiie as that of rolling (the inverse direction to themovement of the contact pressure); that is, the fatigue lifeis longer Ibr the drive r surface than for the follower surface.'1-he differ ence of crack growth rate du e to the dif ferenceof the direction of surface traction will be explained byconsidering that the shear n~ocle rack growth is affectedby the frictional force between the crack faces and also thetensile mode crack growth is influenced by the fluid pressureclue to lubricant seepage into the crack.

As clescribecl in the previous section, when the crack isinclined as shallowing to the direction of movement of thecontact pressure as shown in Fig. 5 (B) , lubricant seepageinto the crack interior is difficult for both cases o f f> 0 and

f < 0. Thus, the propagation of such kind of crack is unlikelyto occur or its growth ra te is very slow. On the oth er ha nd,when the crack is inclined as deepening to the direction ofmovement of the contact pressure as shown in Fig. 5(a),lubricant seepage is easier forf < 0 (i.e. the follower or thenegative sliding surface) than forf > 0. As a resul t, it shouldbe emphasized that forf < 0 the mean frictional coefficient,f,, between the crack faces is smaller in magni tude and theeffect of fluid pressure is also greate r as compared with thatfor f > 0.

Figures 6 and 7 indicate that the value of KT,, whichcontrols the shear mode crack growth is greater for f > 0than for/ < 0. However, from the above discussion, f, forf < 0 is supposed to be smaller than that for f > 0. Forexample, in comparing the case off = 0.3 with that off =

Fig. 8-Variations of and K,mar due to rnovernant of contact(a) Kgvrnar (f = -0.1)(b) K7rn.x (f = -0.1)(c) K,,rnax (f = 0.1)(d) K m a x (f = 0.1)


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Mechanism of Crack Growth in Lubricated Rolling/Sliding Contact 413

Fig. +Variations of Kmmrand K,msr due to m ovement of contact(a ) KOm.. (f = f 0.3)(b)KTmax (f = f0.3)

- 0.3 n Fig. 6, he maximum range of K , fo r f = 0.3an d f, = 0.5 s 2.7 M ~ a . m ' / ~ ,nd that forf = -0.3 an df,= 0.2 s 2.8 M ~ a . m ' / ~ .ventually there is very little differ-ence between them. The experimental result obtained bySoda and Yamamoto (5)shows that the growth rate of mi-crocracks formed on the driver surface (f> 0) s slightlyhigher than that of microcracks formed on the followersurface (f< 0) r both rates a re almost the same. This resultmay be consistent with the present result.

As the crack extends, the growth mode is altered fro111the shear mode to the tensile mode. This is due to theinfluence of the fluid pressure by lubricant penetrated intothe crack. In this regime, the maximum value of K,,, isgreater for f > 0 than for f < 0. However, this result isobtained by assuming that both cases have the same fluidpressure effect. In an actual situation, we must consider thepossibility of lubricant seepage. As a result, it is expectedthat the cracks on t he surfaces are mor e likely to propagateat higher rate on the follower side (f< 0) han on the driverside (f> O), nd also that the crack growth in the case ofpure rolling (f= 0) s unlikely to occur.

The crack growth angle, 00 , where K,,, becomes themaximum value is indicated in Figs. 8 and 9. Notice thatthe growth angle has the following tendencies. It decreases

as the crack size increases but it always has positive value,and its value is greater forf> 0 han forf< 0. urthermore,as (f increases, the growth angle has increasing tendencyforf > 0, ut it has decreasing tendency for f < 0.

CONCLUDING REMARKSIn this study, the problem of fatigue crack propagation

und er lubricated rolling/sliding contact by utilizing the frac-ture mechanics has been discussed. Th e possible mecha-nisms concerning the crack growth and the occurrence ofpit will be summarized as follows.

Th e crack mouth op ening depends on the magnitude ofsurface traction. U nder the condition of pu re rolling or thatof full film lubrication, the surface traction is zero or verysmall, and, accordingly, it is difficult for the lubricant topenetrate into the crack interior and then the crack growthhardly occurs. This indicates that as regards the crack growth,the action of surface traction plays an important role.

Tiny cracks form ed on t he rolling/sliding contact surfacestake the shear mode fatigue growth at the first stage. Localincrease in the surface traction induced by local collapse ofEHL film and the decrease in the frictional coefficient be-tween the crack faces due to the lubricant seepage are con-sidered to be the necessary conditions for the crack growth.Since the contact pressure gives rise to large compressivestress field in the neighborhood of the contact region, thecrack growth is unlikely to occur without sufficient surfacetraction and small frictional force between the crack faces.Although the relationship between the lubricant additivesand the rolling-contact fatigue is not yet proved experi-mentally, the following may be predicted from the view-point of the fatigue crack propagation. The additives, whichimprove the characteristics of boundary lubrication, retardthe crack growth in the point t hat they bring on the decreasein the surface traction; yet, since they give rise to the de-crease in the coefficient of friction between the crack faces,the crack growth rate may be accelerated.

The shear mode fatigue crack growth makes the crackextend along the original crack plane. Ther efo re, the tran-sition from a crack to a pit is not induced by the shear modecrack growth. T he crack growth by tensile mode is necessaryfor the occurrence of pit. The tensile mode crack growthdominates as the crack extends under the fluid pressureeffect. T h e relationship between the direction of the crackinclination and the surface traction is an important factorfor the crack growth in the point that it governs lubricantseepage into the crack. The crack inclined as deepening tothe direction of movement of the contact pressure is mostlyapt to propagate if the direction of the surface traction isopposite to that of movement of the contact pressure. Thisindicates that the crack propagation rate in this situation ishigher for the crack formed on the follower surface thanfor that on the driver surface. Since, in this case, the growthangle becomes smaller as the surface traction increases inmagnitude, it can be predicted that the higher the surfacetraction, the greater the pit in size. Moreover, if a crackformed on the driver surface propagates by the fluid pres-sure, a shallow pit may be produced as compared with a pit


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414 M. KANETA, H. ATSUZUKAAND Y . MURAKAMIproclitced on the follower surface because of the greatergl-owth angle .

Since the crack cannot p ropag ate in the opposite directionto (lie initial inclination a nd also a tiny crack pr opaga tesstraight by the shearing mechanism, we can conclude thatthe d irection of a p i t ting crack would be de term ined a t avery early stage, i .e. during the prepropagation stage.

Finally, we must make an additional remark. We haverecently found analytically that the lubricant entering thecrack in adva nce of the contact is shu t up in th e crack in-terior Ij y the mouth of the crack which is sealed as themovement of the contact . This phenomenon may further; iccelera~ehe tensi le mode crack growth. T h e detai ls of th isanalysis are expected to app ear in the nea r fu ture.

ACKNOWLEDGMENT'I'l~is resea rch was su pp or te d financially by the SaneyoshiScholarship Founclation.

REFERENCES( I ) Murakami, Y., Kaneta , M., and Yatsuzuka, H., "AnalysisofSurface CrackPropagation in Lubricated Rolling Contact," to be published in ASLE

T r a m .(2) Way, S., "Pitting Due to Rolling Contact," J . Appl. Mech., Trans. ASME.,

2, pp A49-A58 (1935).(3) Keer. L. M., Bryant, hi. D. and Haritos, G . K., "Subsurface and SurfaceCracking Due to Hertzian Contact,"J . Lubr. Tech., Trans. ASME, 104, pp347-351 (1982).(4) Keer, L. M. and Bryant, M. D., "A Pit ting Model for Rolling ConvactFatigue," J . Lubr. Tech., Trans. ASME., 105, pp 198-205 (1983).(5) Soda, N. an d Yarnarnoto, T. , "Effect of Tangential Traction and Rough-ness on Crack InitiationlPropagation During Rolling Contact," ASLE

Trans., 25, 2, pp 198-206 (1982).(6) Otsuka, A,, Mori, K., and Miyata, T., "The Condition of Fatigue CrackGrowth in Mixed Mode Condition," Eng. Fracfure Mech.. 7 , pp 429-439

(1975).(7 ) Erdogan. F. and Sih, G . C., "On the Crack Extension in Plates UnderPlane Loading and Transverse Shear," J. Basic Eng., Trans. ASME, 85,pp 519-527 (1963).(8) Ichirnaru, K. , Nakajirna, A., and Hirano, F., "Effect of Asperity Inter-action on Pitting in Rollers and Gears," j. Mech. Design, Trans. ASME,

103, pp 482-491 (1981).(9) Soda, N. and Yarnarnoto, T., "Effects of Tangential Tract ion and SurfaceRoughness of Mating Roller of Cr-Mo Steel on Rolling-Fatigue Life of0.45 percent Carbon Steel," Proc. JSLE-ASL E I nf. Lub. Conf., Tohyo, pp

458-465 (1975).

Mechanism of Crack Growth in Lubricated Rolling- Sliding Contact

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