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Crack initiation at contact surfaceY.J. Xiea,*, D.A. HillsbaDepartment of Mechanical Engineering, Liaoning University of Petroleum & Chemical Technology,Fushun, LN 113001, PR ChinabDepartment of Engineering Science, Oxford University, Parks Road, Oxford OX1 3PJ, UKAbstractBy using the S-theory, the crack initiation angle from the contact surface of rectangular rigid punch and flat-surfacesubstrate has been investigated. The coefficient of friction at the edge of contact, which characterizes the asymptoticstress field, is considered as the controlling parameter in the analysis. The predicted results are in agreement with theexperimental observations. The information gained may lend insight into the different stages of damage associated withthe complex process of fretting fatigue.? 2003 Elsevier Ltd. All rights reserved.Keywords: Fracture; Fretting fatigue; Mechanics; Crack nucleation1. IntroductionContact problem covers a wide range of engi-neering problems. They may include splint jointsbetween shafts, bolted connections, and certaingeometries of gas turbine fan blade dovetails etc.These components are prone to failure by fatigue.Fretting fatigue failure can be divided into twostages: crack initiation and propagation. Life esti-mate of cracked structures may be solved by manyexisting models. But the effect of fretting contact oncrack initiation is likely to be far less amenable toanalysis because crack nucleation and initiation aremore complex as the problem may involve aknowledge of the combined interaction of load,geometry and material in addition to the use of avalid failure criterion that preferably could handlecrack nucleation, initiation and propagation.Fatigue life and endurance limit of mechanicalcomponents are reduced by cyclic fretting fatigueinvolving contact. Cyclic fretting loads tend toactivate flaw at the contact surface which will leadto the development of cracks. However, the ana-lysis of fretting fatigue is difficult because thedamage process may involve a multitude of cracks1,2. In many test, multiple cracks are often foundnear the edge of contact or near the ship-stickboundary 3,4. Conventional approach does notconsider the presence of cracks and makes use ofstress and strain in certain planes along the contactsurface that are regarded as critical 58. Thestress and strain, however, are highly elevated nearthe contact edge similar to that of a crack tip. Aslight change in the fretting conditions, can causelarge changes in the final fracture. This would leadto large scatter in the data. The present work will*Corresponding author.E-mail address: yjxie (Y.J. Xie).0167-8442/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.tafmec.2003.09.003Theoretical and Applied Fracture Mechanics 40 (2003) 279283/locate/tafmecuse contact and fracture mechanics for predictingthe direction of crack initiation from the contactsurface into the contacting body.2. Strain energy density criterionThe S-criterion can be used to describe thefracture behavior of stable and unstable crackgrowth, critical loads and crack initiation angle inmixed mode. It makes use of the strainenergydensity factor S which is related to the strain en-ergy density function dW =dVby the relationdW =dV S=r where r is the distance from thecrack tip. The critical value of dW =dV can be de-termined from the area under the uniaxial stressand strain curve 911. For the asymptotic ana-lysis S depends only on the angle h. In general, Scan depend on r. In what follows, it suffices to usethe asymptotic form of S:S a11K2I a12KIKII a22K2II a33K2III1wherea111 l8pE3 ? 4l ? cosh1 cosha121 l8pE2sinhcosh ? 1 2la221 l8pE41 ? l1 ? cosh 1 cosh3cosh ? 1?a331 l2pE2with l being the Poisson?s ratio and E the elasticmodulus.For the past several decades, S-theory has beenwidely used in analysis of crack problems by usingthe asymptotic stress field on r?1=2. However, thetheory remains valid when using the completestress field. In this case, S would depend on r andh. It should be pointed out that the relationdW =dV S=r is necessitated from introducing adistance from the crack tip to locate the site offailure. This distance is r which coincides with thesingular character of the strain energy densityfield. Moreover, r can also be used as the distancefrom the contact edge in the absence of a pre-existing crack. Hence, it can be used to predict thedirection of crack initiating from the contact.3. Asymptotic stress field in sliding contact3.1. Boundary conditionThe sliding contact of a rigid, square-endedpunch, of half width a, on a homogeneous, iso-tropic, elastic body in half plane is shown in Fig. 1.The Cartesian coordinates x1;x2, and the polarcoordinates r;h, both with the origin at left edgeof contact, are selected. Normal force P and tan-gential force Q act on the punch. The normal andshear tractions along interface have been solved inclosed form 2,12, which arepx1 ?P sinkpp2?x1a?k?1x1a?k3andqx1 fpx14where f is the coefficient of friction; the k can bedetermined bytankp 21 ? lf1 ? 2l;0 k 15with l being the Poisson?s ratio of the substrate.Eq. (3) shows that the stress state near the punchcorners varies asrij0rk?1;x1 2a0r?k;x1 0?as r ! 06When l 0:5 or with f 0, it yields k 0:5.This gives the same order of stress singularity asFig. 1. Contact between a two-dimensional rectangular punchand a substrate.280Y.J. Xie, D.A. Hills / Theoretical and Applied Fracture Mechanics 40 (2003) 279283that for a crack from Eqs. (5) and (6). However,the stress intensity factor is not related to the en-ergy release rate as in the crack problem. There-fore, the physical meaning of KIwhen used as afailure criterion is not clear. The strain energydensity theory does not have this limitation since itfocuses attention on the failure of an element ref-erenced from the vicinity of a site where failure islikely to occur. In passing, it may be emphasizedthat failure is always assumed to initiate from afinite distance away from the crack tip whose exactlocation can never be found in the material.When l 0:5, the asymptotic stress boundaryconditions of substrate in contact area next to leftcorner arer22jh0 ?Ppffiffiffiffiffiffiffi2arp7andr21jh0 ?fPpffiffiffiffiffiffiffi2arp83.2. Mode I singular stress fieldThe singular stress fields at the sharp edge of thecontact between the rectangular rigid punch andsubstrate are known from the asymptotic contactanalyses 12. Using the polar coordinates r;h,Fig. 1, the stresses at the left corner are found tovary asrIrrrIhhrIrh0B1CA ?KIffiffiffiffiffiffiffi2prpcosh21 sin2h2?cos3h2sinh2cos2h20BBBBBB1CCCCCCA9This expression is identical in form with the ModeI compressed stress field, but whereKIPffiffiffiffiffiffipap103.3. Mode II singular stress fieldFrom the solution in 12, the asymptotic stressfield of Mode II is given byrIIrrrIIhhrIIrh0B1CA ?KIIffiffiffiffiffiffiffi2prpsinh21 ? 3sin2h2?3sinh2cos2h2cosh21 ? 3sin2h2?0BBBBBBB1CCCCCCCA11whereKIIfPffiffiffiffiffiffipap123.4. Characters of the stress fieldsThe initiation of a crack depends on singularstress fields. This stress fields is equal to the Eq. (9)adding Eq. (11). Then the local (effective) kIandkII, are expressed in terms of h as 13kIh rhhhffiffiffiffiffiffiffi2prp13kIIh rrhhffiffiffiffiffiffiffi2prp14where rhhh rIhh rIIhhand rrhh rIrh rIIrh.Fig. 2 shows the typical angular variations of Eqs.(13) and (14) with angle h and f. It indicates thatkIh will be positive at certain h and f, whichmeans that the contact boundary possesses typicalfracture characteristics that can be investigated byusing the S-theory.4. Boundary cracking in complete fretting contactFor singular stress fields next to the corners, anew crack will form at a certain critical load. Thecrack initiation angles can be obtained from thestrain energy density factor criterion 911. It isassumed that the crack tends to run in the h0di-rection for which Sminprevails. That is oS=oh0 0.This leads toa011 a012f a022f2 015wherea011 sinh02l ? 1 cosh0a012 2cos2h0? 1 ? 2lcosh0? 1a022 sinh01 ? 2l ? 3cosh016Y.J. Xie, D.A. Hills / Theoretical and Applied Fracture Mechanics 40 (2003) 279283281Fig. 3 gives schematically the relationship ofnormalized cracking angle h0=p and the coefficientof friction. Experimental observations 13 indicatethat the crack initiation angle in fretting fatiguetests ranges widely from 25? to 80?. In these fret-ting tests, f varies with cycling from initial value0.20.4 to final stable value 0.71.2. Fig. 3 indi-cates that the cracking angles predicted by S-theoryforcrackinitiatingfromthecontactboundary are in agreement with the experimentalresults.5. ConclusionsThe problem of contact between a rectangularrigid punch and flat-surface substrate has been in-vestigated by using the S-theory. Cracking anglesin boundary of the substrate are theoretically pre-dicted. The present findings provide a method fordetermining crack initiation from a site that had noFig. 2. Variation of effective stress intensity factors kI, kIIwith angle h, for different f.0.00.81.01.20.0 20.0 40.0 60.0 80.0 100.0 120.0 Cracking angle o degFriction coefficient f Fig. 3. Cracking angle h0versus friction coefficient f.282Y.J. Xie, D.A. Hills / Theoretical and Applied Fracture Mechanics 40 (2003) 279283existing crack. Keep in mind that a pre-requisite forusing fracture mechanics is that the material mustbe assumed to have an existing crack.Additionally, the work in 14 studied the stressstate near the corner of complete contact subject tofretting action using an asymptotic analysis. Theirresults indicated that there are infinite combina-tions of friction coefficient f and wedge angle uthat would result for a )0.5 singularity next to thecorner. Even though the order of the stress sin-gularity from the contact boundary without acrack is the same as that for a re-existing crack, thephysical interpretation of the stress intensity fac-tors for the contact problem is different, themeaning of which is not clear. The use of the strainenergy density theory does not have this problem.Its physical meaning in terms of predicting failureby fracture remains the same. This distinctionshould be recognized.References1 J.M. Dobromirski, Variables of fretting process: are there50 of them?, in: M. Helmi Attia, R.B. Waterhouse (Eds.),Standardization of Fretting Fatigue Test Methods andEquipment, ASTM STP 1159, American Society forTesting and Materials, Philadelphia, 1992, pp. 6066.2 D.A. Hills, D. Nowell, Mechanics of Fretting Fatigue,Kluwer Academic Publishers, Netherlands, 1996.3 V. Lamacq, M.-C. Dubourg, L. Vincent, Crack pathprediction under fretting fatiguea theoretical and experi-mental approach, ASME J. Tribol. 118 (1996) 711720.4 M.H. Wharton, D.E. Taylor, R.B. Waterhouse, Metallur-gical factors in the fretting-fatigue behavior of 70/30 Brasand 0.7% carbon steel, Wear 23 (1973) 251260.5 H.K. Kim, S.B. Lee, Crack initiation growth behavior ofAl 2024-T4 under fretting fa