Acta mater. Vol. 47, No. 18, pp. 4653±4664, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Printed in Great Britain PII: S1359-6454(99)00312-2 1359-6454/99 $20.00 + 0.00

THE ROLE OF ADHESION IN CONTACT FATIGUE

A. E. GIANNAKOPOULOS, T. A. VENKATESH, T. C. LINDLEY{ and S. SURESH{ Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

(Received 26 May 1999; accepted in revised form 12 August 1999)

AbstractÐBy incorporating the e€ects of interfacial adhesion in the mechanics of rounded contact between two bodies, a new approach is proposed for the quantitative analysis of a wide variety of contact fatigue situations involving cyclic normal, tangential or torsional loading. In this method, conditions of ``strong'' and ``weak'' adhesion are identi®ed by relating contact mechanics and fracture mechanics theories. Invok- ing the notion that for strong and weak adhesive contact, a square-root stress singularity exists at the rounded contact edge or at the stick±slip boundary, respectively, mode I, II or III stress intensity factors are obtained for normal, sliding and torsional contact loading, accordingly. A comparison of the cyclic variations in local stress intensity factors with the threshold stress intensity factor range for the onset of fatigue crack growth then provides critical conditions for crack initiation in contact fatigue. It is shown that the location of crack initiation within the contact area and the initial direction of crack growth from the contact surface into the substrate can be quantitatively determined by this approach. This method obvi- ates the need for the assumption of an arti®cal length scale, i.e. the initial crack size, in the use of known fracture mechanics concepts for the analyses of complex contact fatigue situations involving rounded con- tact edges. Predictions of the present approach are compared with a wide variety of experimental observations. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Adhesion; Fracture; Fatigue; Wear; Surfaces and interfaces

1. INTRODUCTION the total energy of the system, while elastic defor- Functional structures, at macro or micro size-scales, mation across the interface that accomodates ad- by design or default, involve contact between simi- hesion leads to an increase in the potential energy lar or dissimilar materials, under static or cyclic of the system. The resulting thermodynamic equili- loading conditions, in a passive or aggressive, brium and extent of adhesion is then determined by thermo-chemical environment. Consequently, in the the relative dominance of these competing phenom- most general case, di€erent aspects of contact such ena. as material properties, interface chemistry and con- For monotonic contact loading, adhesion has tact mechanics can independently or interdepen- been well documented analytically and experimen- dently in¯uence the overall mechanical behavior of tally under normal loads [1±5], and to a lesser the contacting system. extent under tangential loads [6]. For cyclic loading, In most cases, the bulk mechanical behavior of a number of experimental studies have investigated the contacting system is characterized by a macro- adhesion in a variety of materials Ð metals [7, 8], scopic continuum level analysis that evaluates the glasses [9], and polymers [10], for di€erent surface uncoupled mechanics response [1]. However, a more and environmental conditions [11, 12], temperatures complete analysis requires a consideration of the [13], hardness [8], and fatigue cycles [14]. microscopic or atomic structure where coupled in- Theoretical work has demonstrated that under teractions at the contact interface become signi®- monotonic compressive loading of the contact inter- cant. These are typically analyzed within the face, adhesion induces tensile square-root singular framework of surface thermodynamics by recogniz- stress ®elds at the contact edges [2, 3]. However, ing that contacting surfaces are subject to short- connections to contact fatigue experiments speci®- range interatomic forces such as the van der Waals cally aimed at correlations between adhesion and forces. Consequently, spontaneous adhesion leading fatigue crack initiation have not been established. to the formation of a low energy interface reduces The objectives of the present work are, therefore, to develop and experimentally validate a general continuum level mechanics model that incorporates {Permanent address: Department of Materials, Imperial work of adhesion, material elastic properties, and College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BP, U.K. contact loads, for predicting key features of contact {To whom all correspondence should be addressed. fatigue crack initiation (thresholds for the onset of

4653 4654 GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE cracking, location of cracking, and initial orien- arising from the contact loads, leads to a novel tation of the crack plane with respect to the contact classi®cation whereby cases of strong and weak ad- surface) for a variety of loading conditions (cyclic hesion are recognized unambiguously{. Thus, the mode I or steady mode I with cyclic mode II or crack initiation location (contact perimeter for cyclic mode III) and contact geometry (sphere or strong adhesion and the stick±slip boundary for cylinder on a planar substrate). weak adhesion) can be predicted without uncer- This paper addresses the following key issues of tainty. contact mechanics and contact fatigue. Fifth, the present methodology conceptually fa- First, the present work extends the crack ana- cilitates the incorporation of known e€ects of sur- logue methodology developed earlier [15] for the face conditions, environment, hardness, and contact between a sharp-edged pad and a planar temperature on adhesion, through a simple modi®- substrate, where stress singularities are introduced cation of the work of adhesion. by the sharp cornered geometry. We now investi- The paper is arranged in the following sequence. gate the more general case of rounded contacts The mechanics of adhesive contacts for a variety of where stress singularities are induced by adhesion{ monotonic loading conditions are reviewed in and thus present a universal methodology that Section 2. The contact mechanics of cyclic loading enables analysis of a variety of contact problems for three-dimensional, spherical, adhesive contacts from those due to fretting fatigue in large-scale are analyzed in Section 3. The results of the ad- structures to contact fatigue in micro-scale devices, hesion model for cyclic loading for two-dimen- with adhesive or non-adhesive, sharp or rounded sional, cylindrical contacts are presented in Section geometries. 4. The range of applicability of the adhesion model Second, as the adhesion-induced, square-root as well as its signi®cance and limitations are dis- singular stress ®elds are amenable for analysis cussed in Section 5. Available experimental results within the framework of a ``crack analogue'' [15], for di€erent material systems and loading con®gur- ations are compared to model predictions in the pre-existing long crack introduced by the con- Section 6. Finally, the paper concludes with a sum- tact circumvents ``length scale'' problems inherent mary of results and potential palliatives for contact in the modeling of crack initiation based on conven- fatigue from the perspective of the adhesion model tional fracture mechanics [16], or small crack in Section 7. growth based on initial dislocation distributions [17]. Third, under conditions of small-scale yielding, 2. MECHANICS OF STATIC ADHESIVE CONTACT the e€ects of static and/or oscillatory bulk stresses (i.e. residual stresses induced by surface treatments Consider a sphere of diameter, D, elastic mod- such as shot-peening or laser shock-peening, or far- ulus, E, and Poisson ratio, n, contacting the planar ®eld applied stresses acting parallel to the contact surface of a large substrate of similar material, Fig. surface) on contact fatigue crack initiation can also 1(a), with r and z being the global radial and depth be analyzed by recognizing that these are analogous coordinates attached to the center of the contact to the T-stresses present in a simple linear elastic circle. Three loading conditions are considered as fatigue±fracture formulation. follows. Fourth, all previous analyses which are based on 2.1. Normal loading stress-based approaches to fatigue at critical points (such as those using elastic stress ®elds of a sphere For elastic, monotonic loading from zero to Pmax on a ¯at plane [18] in combination with a variety of (>0), under non-adhesive conditions, the contact fatigue strain-based, multiaxial criteria for endur- stress ®eld is non-singular [Fig. 1(b)] and the maxi- ance limits [19±22], predict contact fatigue cracking mum contact radius, a0, is given as [1] to initiate at the contact perimeter. This prediction  1=3 3D 1 n2† is contrary to many experimental results which a0 ˆ Pmax : 1† clearly indicate that cracking could initiate at either 4E the contact perimeter or the stick±slip boundary. Under adhesive conditions, when two surfaces with The present analysis, though an examination of the surface energies g1 and g2 adhere to form a new work of adhesion via-aÂ-vis the crack driving force interface of lower energy, g12, the corresponding work of adhesion is de®ned as, w=(g1+g2g12)r0. {The adhesion induced stress singularities are not For two contacting solids with the same elastic expected to modify the geometry induced stress singular- properties, which are presently considered, w=2g1. ities in the case of sharp edged contacts. For metals, w11 N/m [23]. {The conditions of strong and weak adhesion can con- The short-range forces of attraction that promote ceptually be visualized as being analogous, respectively, to the concepts of ``static'' and ``dynamic'' friction when the adhesion e€ectively increase the contact load across interfacial slip behavior across two contacting surfaces is the interface and thus increase the contact radius, considered. amax [Fig. 1(c)], to the one given as GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE 4655

Fig. 1. Schematics illustrating: (a) non-adhesive contacts and (b) the corresponding non-singular contact stress ®eld; (c) adhesive contacts and (d) the corresponding stress ®eld exhibiting tensile square-root singularity. 2 0 4Ea3 2 à max 43D 1 n †@ 3pDw P max ˆ : 3† a ˆ P ‡ 3 1 v2†D max 4E max 2 Unlike the non-adhesive case (Fig. 2), the adhesive 13 s 1=3 contact produces a tensile, square-root singular  2 3pDw A5 ‡ 3pDwP ‡ : 2† stress ®eld [24] which is asymptotically equal to the max 2 mode I crack ®eld at the contact perimeter (Fig. 1(d)). With r and y as the local polar coordinates at the contact perimeter, f the angular circumferen- * The apparent load, Pmax, required to maintain the tial coordinate, and Trr and Tff as the non-singular same contact radius, amax, without adhesion is stress terms in the radial and circumferential direc- given by Hertzian analysis [1] tions, the asymptotic stress ®eld is given by [24]

Fig. 2. Schematics illustrating: (a) sharp-edged, non-adhesive contact and (b) the corresponding crack analogue exhibiting compressive stress singularity. 4656 GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE

Fig. 3. Schematics illustrating: (a) tangential loading and (b) torsional loading of adhesive, spherical contacts.

  s K y y 3y nor 1 D 2 1 n2†pw srr ˆ p cos 1 sin sin ‡ Trr 2pr 2 2 2 rk ˆ : 6† 4 Eamax 4a† The associated energy release rate can be identi®ed   as 1 n2†K 2=E{. The maximum non-singular stres- K y y 3y 1 snor ˆ p1 cos 1 ‡ sin sin 4b† ses in the radial and circumferential directions due zz 2pr 2 2 2 to contact at maximum load are given by 1 2n†P à   max T ˆmax T ˆ max T ˆ max : K y y 3y rr ff 2pa2 snor ˆ p1 cos sin cos 4c† max rz 2pr 2 2 2 7†

nor nor nor 2.2. Combined normal and tangential loading sff ˆ n srr Trr ‡ szz †‡Tff: 4d† For elastic, monotonic, tangential loading from

The corresponding tensile stress intensity factor, K1, zero to Qmax (>0), under a constant normal load, is given as [25, 26] Pmax, an adhesion-induced, square-root singular à stress ®eld [6] is obtained at the contact perimeter P Pmax K1 ˆ p: 5† (Fig. 3(a)). The mode II ®eld, at the leading and the 2amax pamax trailing edges{, can be expressed as [15]    The size of the K-dominance around the contact tan KII y y 3y srr ˆ p sin 2 ‡ cos cos perimeter, rk, is identi®ed by the nodal positions in 2pr 2 2 2 the stress ®eld [Fig. 1(d)] and is given by 8a†

{(a) For contacting surfaces of di€erent diameters, D tan KII y y 3y can be replaced by D1D2= D1 ‡ D2†, where the indices 1 s ˆ p sin cos sin 8b† and 2 refer to the contacting bodies 1 and 2. (b) For con- zz 2pr 2 2 2 tact involving elastically dissimilar materials, (1n2)/E can be replaced by f‰ 1 n2†=E ‡ 1 n2†=E Šg=2 and the   1 1 2 2 K y y 3y energy release rate relations are preserved. However, the stan ˆ pII cos 1 sin sin 8c† form of the stress intensity factor remains invariant only if rz 2pr 2 2 2 the stress ®elds are square-root singular for which the sec- ond Dundurs' parameter, b=0 [27], i.e. stan ˆ n stan ‡ stan † 8d† E 1 ‡ n † 1 2n †E 1 ‡ n † 1 2n † yy rr zz 2 1 1 1 2 2 ˆ 0: 2 2 E2 1 n1†‡E1 1 n2† where the mode II stress intensity factor, KII,is {In general, a mixed mode II/III ®eld is obtained along given as the contact perimeter with the corresponding stress inten- Q sity factors being: max KII ˆ p: 9† 2amax pamax Qmax cos f Qmax sin f KII ˆ p , KIII ˆ p 2amax amax 2amax amax The maximum T-stresses due to contact are where f is the angle between the radius vector at the per- obtained at the maximum tangential load and are ipheral point and the direction of Qmax [29]. given as GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE 4657   P à 3mp 4 ‡ n† max T ˆ max 1 2n†‡ 10a† contacting surfaces are not perfectly clean or rr 2 2pamax 8 smooth, a direct correlation between friction and   adhesion has not been established. Hence, in P à 9mpn order to maintain generality, an a priori relation- max T ˆ max 1 2n† : 10b† ff 2 ship between friction and adhesion is not 2pamax 8 assumed [33]. Therefore, work of adhesion and 2.3. Combined normal and torsional loading coecient of friction are introduced in the analy- sis in an independent way. However, for particu- For elastic, monotonic, torsional loading from lar cases where an explicit relationship between zero to Mz (>0) (Fig. 3(b)), under a constant nor- friction and adhesion can be identi®ed, the mal load, Pmax, the adhesive asymptotic stress ®eld analysis can be modi®ed as indicated in the due to torsional loading{, is square-root singular at Appendix. the contact perimeter [28]: 4. Strong adhesion is de®ned as that for which the K y K y work needed to debond the two contacting sur- stor ˆ pIII sin , stor ˆ pIII cos 11† rz 2pr 2 yz 2pr 2 faces, Gd, is high enough to resist local debond- ing at the contact perimeter. The work of with the mode III stress intensity factor, KIII, given adhesion for receding (separating or opening) as contact, Gd, is greater than the work of adhesion p 3M for advancing (approaching or closing) contact, K ˆ stor 2p a r† ˆ pz : 12† III rz max 4a2 pa w, i.e. Gd > w: Such adhesion hysteresis is due to max max mechanical and/or chemical e€ects, e.g. micro- The maximum energy release rate, max G, is related plasticity of asperities [34, 35]. to the mode I and III stress intensity factors as 5. The e€ect of bulk and/or residual stresses in the substrate, on fatigue crack initiation, can be in- K 2 1 ‡ n† K 2 1 n2† max G ˆ III ‡ 1 13† corporated by superposition with the contact- E E induced T-stresses. where KIII is given by equation (24) and KI is given 6. Finally, when certain fatigue threshold con- by equation (26). The mode III ®elds do not pro- ditions are met (as described below), contact fati- duce any T-stresses. gue cracks are expected to initiate both in the contact pad and the ¯at substrate. As cracks in the contact pad are expected to be mirror images 3. MECHANICS OF CYCLIC ADHESIVE CONTACT of those in the substrate across the plane of con- 3.1. Model assumptions tact (i.e. they initiate at the same contact lo- cation), we discuss the stress state of the In modeling adhesive contacts under cyclic load- substrate only. ing conditions, we invoke the following assump- tions. 1. Materials are homogeneous, isotropic and linear 3.2. Three-dimensional spherical contacts elastic. 3.2.1. Normal contact fatigue. Consider a spheri- 2. In order to keep the analysis simple and to avoid cal pad in oscillatory normal contact with a ¯at complications from elastic mismatch of the con- substrate (Fig. 1), with the applied load, P, in the tacting bodies and from the strong nonlinear range P rP r0. As the load decreases from e€ects of contact geometry, only the sphere or max min its maximum, P , to its minimum value, P , the cylinder on ¯at surface of similar materials is max min strain energy release rate, G, at the contact per- analyzed. To avoid microstructural scale e€ects, imeter increases monotonically. the contact area is assumed to cover at least sev- When max G < G or P P †R3pG D=4, eral grains of the material. d max min d strong adhesion is obtained with the contact radius 3. For ideal contact conditions characterized by remaining unchanged at a . The mode I cyclic clean, smooth surfaces under inert atmospheres, max stress intensity factor at the contact perimeter is a direct relationship between adhesion and fric- given as tion can be obtained [30±32]. However, under real, ambient, atmospheric conditions, where the Pmax Pmin DKI ˆ Kmax Kmin ˆ p 14† 2amax pamax {The normal load creates a mode I stress ®eld, as described by equations (4). with the corresponding load ratio, R ˆ {It is implicitly assumed in this work that debonding min KI= max KI†ˆ Pmin =Pmax †: always occurs in a brittle and axisymmetric manner along When max G > Gd or Pmax Pmin †r3pGdD=4, the contact perimeter. Hence, the maximum value of Gd cannot be higher than the critical energy release rate for weak adhesion is obtained as the work of adhesion fracture of the bulk contacting materials under monotonic is insucient to sustain the singularity at the maxi- loading. mum contact radius, amax{. Consequently, the con- 4658 GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE tact debonds at the contact perimeter and the con- tact radius decreases. The oscillatory mode I stress intensity factor at the contact edge, amax, is given as

3pGdD DKI ˆ p 15† 8amax pamax with the corresponding load ratio, R, being P 3pG D=4† R ˆ max d : 16† Pmax As the contact conditions e€ectively imply a virtual circumferential crack with a circular crack front co- incident with the contact perimeter [15], the in- itiation of a fatigue crack at the contact edge is conceptually equivalent to the onset of propagation of this pre-existing virtual crack. Hence, a fatigue crack is expected to initiate at the contact edge, amax, for strong or weak adhesion, if DKIrDKth, where DKth is the mode I long crack initiation fati- gue threshold stress intensity range for the corre- sponding values of R and max T. Fig. 4. Schematic illustrating tangential loading of the 3.2.2. Tangential fretting fatigue. Consider a spherical contact. sphere in contact with a planar substrate under an applied normal load Pmax (Fig. 4). The maximum  tangential load, Q max , that can sustain adhesion at maximum contact radius is obtained resulting in a partial slip annulus,  cRrRa (Fig. 4). From global equilibrium [28] Q max ˆ max v " # u 2 3  3 u !2 c u 2 2n† E 3G I pD Qmax ˆ mPmax 1 ; 2  t pa3 4G II d 5 amax max d 2 3=2 19† 2 n† 1 n † 4amax  Q max < Qmax RmPmax : 17† The tangential load that is balanced in the stick I II where Gd and Gd are the critical debonding energies zone of radius, c,is under pure normal and tangential loads, respect- P ively. Qin ˆ Q m max a2 c2†3=2: 20† max max 3 max If the amplitude of the applied tangential load amax Q RQ or equivalently the maximum energy max max In this case, K =0, as the crack analogue predicts a release rate at the contact edge upon load reversal, I closed crack-tip. max G < G , then strong adhesion (stick) is d The mode II stress intensity factor at the leading obtained{. and trailing edge of the stick±slip interface is given The stress intensity factor that corresponds to the by mixed mode I (steady) crack ®eld is given by ! equation (5) and that which corresponds to mode II r Qin G IIE (oscillatory) crack ®elds at the leading and the trail- DK ˆ 2  min pmax, d 21† II 2c pc 1 n2 ing contact edges is given by

Qmax with the local load ratio R=1. DKII ˆ p 18† A fatigue crack is expected to initiate, at the con- amax pamax tact perimeter or the stick-slip boundary, for strong with the local e€ective load ratio, R=1. or weak adhesion, respectively, if the corresponding

If the amplitude of the applied tangential load DKIIrDKth where DKth is the threshold stress inten-  Qmax rQ max or equivalently the maximum energy sity range that corresponds to R=1, and under release rate at the contact edge upon load reversal, mixed constant mode I and oscillatory mode II fati- max G > Gd, then weak adhesion (stick±slip) is gue. This situation holds both for strong adhesion and under pure oscillatory mode II fatigue for weak {The critical energy release rate, G , may depend on the adhesion. d Upon initiation, the continued propagation of the mode mixity, KI/KII, and this could introduce a contact size e€ect for elastically dissimilar surfaces. crack-tip depends on the local mode I and mode II GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE 4659

contact edges is given by p 3M DK ˆ 2stor 2p a r† ˆ pz 24† III rz max 2 2amax pamax with the local e€ective load ratio, R=1. If the amplitude of the applied tangential load,  MzrM z, or equivalently, the maximum energy release rate at the contact edge upon load reversal,

max G > Gd, then weak adhesion (stick±slip) is obtained resulting in a partial slip annulus, cRrRamax : A power series approximation for the relation between c/amax and Mz=Pmax , that is accu- Fig. 5. Schematic illustrating crack initiation from the rate within 3.4% error of the exact solution is edge of the contact perimeter with the local crack-tip given by [37] mode I and II stress intensity factors being k1, k2 and   domains of local and global plasticity. M 3 1 z 1k2 1 k2 k4 ; mP a 8 64 max max 25† q k ˆ 1 c=a †2: stress intensity factors, k1 and k2, respectively (Fig. max 5). Following Cotterell and Rice [36], it is postu- lated that the crack advances in a direction along The torsional load that is balanced in the stick zone which the local mode II stress intensity factor, k , is 2 0 vanishes [or equivalently, the strain-energy release rate is maximized @=@y k2 ‡ k2†ˆ0, @ 2=@y2 k2 ‡ in 3mPmax amax @p c 1 2 1 M ˆ Mz arcsin 2 z 8 2 a k2† < 0†Š: The initial angle of crack propagation, a, max is then readily obtained from !s1    c c2 c2 1 a 3a 1 a 2 1 1 A 26† k ˆ sin ‡ sin K ‡ cos ‡ 3 a a2 a2 2 4 2 2 I 4 2 max max max  3a and the mode III cyclic stress intensity factor is cos K ˆ 0 22† 2 II given by r! 3Min G E at maximum load. DK ˆ 2  min pz, d : 27† III 2 3.2.3. Torsional fretting fatigue. Consider a sphere 4c pc 1 ‡ n in contact with a planar substrate under an applied In this case, K =0, as the crack analogue predicts a normal load P (Fig. 3). The maximum torsional I max closed crack-tip. load, M , that can sustain adhesion at maximum z A fatigue crack is expected to initiate, at the con- contact radius is v tact perimeter or the stick±slip boundary, for strong u 2 3 !2 or weak adhesion, respectively, if the corresponding u I 4amax u E 3G pD M ˆ tpa3 4G III d 5 DKIII r DKth where DKth is the threshold stress z max d 3=2 3 1 ‡ n† 4amax intensity range that corresponds to R=1 and T=0. This is valid both under mixed constant 23† mode I and oscillatory mode III fatigue for strong I III adhesion, and under pure oscillatory mode III fati- where Gd and Gd are the critical debonding ener- gies under pure normal and torsional loads, respect- gue for weak adhesion. ively. 3.3. Two-dimensional cylindrical contacts If the amplitude of the applied torsional load  3.3.1. Normal contact fatigue. For a cylinder of MzRM z, or equivalently, the maximum energy release rate at the contact edge upon load reversal, diameter, D, in adhesive contact with a ¯at sub- max G < G , then strong adhesion (stick) is strate (Fig. 6) under a normal load, Pmax, the con- d tact width, a , is given as [5] obtained{. max    The stress intensity factor that corresponds to the pE a2 4a w 1 n2† P ˆ max 2 max : mixed mode I (steady) [equation (5)] and III (oscil- max 4 1 n2† D pE latory) crack ®elds at the leading and the trailing 28†

{The critical energy release rate, G , may depend on the Under an oscillatory normal load in the range Pmax d rP r0, conditions for strong or weak adhesion mode mixity, KI/KIII, and this could introduce a contact min size e€ect for elastically dissimilar surfaces. are obtained when 4660 GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE

Table 1. Adhesion-induced stress intensity factors for the case of a cylinder in oscillatory contact with a planar substrate

Loading Adhesion Normal In-plane Out-of-plane Mode Strong-I Weak-I Strong-II Weak-II Strong-III

 r! P P 3 G 2pED 1=3 2Q Qin G E S S DK maxpmin p d pmax 2 min pmax , d maxpmin 2 2 pamax 2pamax 8 1 n † pamax pc 1 n pamax

 P 3 G 2pED 1=3 R max 1 d 1 1 1 2 Pmin 2Pmax 8 1 n †

* T-stress max Txx,maxTyy=0 max Txx=n max Tyy=(2mPmax/pamax)maxTxx,maxTyy=0

  3 pED 1=3 specimen and the applied far-®eld boundary con- P P †R or > G 2=3 ditions. Hence the precise de®nition for strong and max min 2 d 8 1 n2† weak adhesion cannot be made by the contact 29† analysis alone. For weak adhesion, however, the stick-zone respectively. The corresponding stress intensity fac- width, c, is given from global equilibrium as tors are summarized in Table 1. 3.3.2. In-plane fretting fatigue. When a cylindrical "   # c 2 pad in contact with a much larger, ¯at substrate Q ˆ mP 1 ; max max a (Fig. 6) is subjected to an oscillatory tangential line max 30† load Qmax (0RQmaxRmPmax), under a constant nor- 0 < Qmax RmPmax : mal load Pmax, the elastic energy of the cylinder becomes unbounded and the displacements are Assuming adhesion is re-established at the stick indeterminate. The solution to this problem depends zone, the tangential line load that is balanced in the critically on the overall dimensions of the fretting stick zone is given as

Fig. 6. Schematic illustrating loading con®gurations for the two-dimensional cylindrical contact geometry. GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE 4661 mP in max weaker of the two contacting bodies, s 13s , when Qmax ˆ Qmax 0 y p small-scale plasticity conditions are valid at the con- " p p c a2 c2 tact perimeter [40]. Alternately, s0 could be an  ‡ arctan max e€ective stress that combines the maximum normal 2 c2 a2 max and the frictional shear stress [40]. !# Metals show appreciable adhesion at elevated c c2 1 : 31† temperatures and prolonged contact loading times. a a2 max max Under high-frequency fretting conditions, friction produces heat that raises the surface temperature The corresponding stress intensity ranges for strong and could promote creep. Therefore, for the preced- and weak adhesion cases are summarized in Table ing analysis to be valid, the maximum surface tem- 1. Crack initiation is expected at the contact edge perature due to tangential load oscillations must be or the stick±slip boundary for strong or weak ad- less than a third of the homologous temperature, hesion cases, respectively, when the corresponding T stress intensity ranges exceed the fatigue thresholds H   for the corresponding R-ratios and T-stresses. 9 3p 4† 2 n† 1 ‡ n† mP 2 T max O < H 35† 3.3.3. Out-of-plane fretting fatigue. The case of a 2 64p EK amax 3 cylinder of diameter, D, under a constant normal load, Pmax, subjected to an oscillatory line load where O is the fretting frequency and K is the ther- with amplitude, Smax, and in adhesive contact with mal conductivity coecient [41]. a ¯at substrate (Fig. 6), can also be analyzed along Another limitation of the present analysis comes similar lines and a mode III cyclic stress intensity from the surface morphology. Surface roughness factor range can be identi®ed at the contact edge diminishes the in¯uence of adhesion if [42] for strong adhesion as summarized in Table 1.  2 3 1 n2† G 2r r100 d 0 36† 2E S3 4. RANGE OF APPLICABILITY OF THE ADHESION MODEL where S is the standard deviation of the micro- asperity heights and r0 is the average radius of cur- The preceding analysis is strictly valid under lin- vature of the asperity tips. ear elastic conditions, the limits of which are de®ned by several criteria that predict the onset of 5. EXPERIMENTAL VALIDATION OF THE local or global plasticity (Fig. 5), Noting that ad- ANALYSES hesion increases the e€ective stress under the con- tact area, global indentation induced plasticity can In comparing various experiments to model pre- be avoided [38], provided that dictions, only reasonable estimates are chosen from the empirical relationship [equation (A1)] that a E G E 2 max < 3:675, d < 7:031 32† relates the friction coecient to the debond energy, Ds 1 n2† 3 2 2 y Dsy 1 n † since experimentally measured values for the debond energy are not available. where sy is the yield strength of the softer of the two contacting bodies. Also, macroscopic plasticity is suppressed for m > 0.25, if [18, 24]  3P 1 2n†2 1 2n† 2 n†mp s3 > max ‡ y 2 2pamax 3 4

1=2 16 4n ‡ 7n2†m2p2 ‡ : 33† 64 An alternative, self-consistent, Dugdale±Barenblatt model [39] for adhesive normal contact, that assumes the maximum adhesive force intensity, s0, to be constant until debonding is reached, where- upon it falls to zero, predicts the elastic (local, small-scale plasticity) conditions for adhesion to be preserved for Fig. 7. Room temperature fretting fatigue experiments on Al-7075 T6 (with in 2 in. diameter sphere on ¯at geome- D 1 n2†2 250 try) (see Ref. [22] and current work), evaluated using the s3 > : 34† present adhesion model where in the mode II threshold 0 2 pGdE 9 stress intensity factor for an R-ratio of 1 was estimated to be11 MPa m1/2 and the work of adhesion for advan- The maximum adhesive stress, s , could be related 0 cing and receding contacts (w, Gd) being11 and 119 N/m, to the uniaxial yield strength of the cohesively respectively. 4662 GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE

Fig. 9. Room temperature fretting fatigue experiments on Al-2024 T351 (with a 0.5 in. diameter cylinder on ¯at geo- metry [50]) evaluated using the present adhesion model where in the mode II threshold stress intensity factor for an R-ratio of 1 was estimated to be 11 MPa m1/2 and the work of adhesion for advancing and receding contacts (w, Gd) being 11 and 112 N/m, respectively.

model predicts weak adhesion, in agreement with experiments where in all cases, stick±slip behavior was observed. The conditions (Fig. 7) and location Fig. 8. Fretting scar produced on the ¯at Al-7075 T6 sub- of crack initiation (stick±slip boundary, Fig. 8) are strate [22] when fretted with a spherical pad of the same also predicted well by the model. While the model material and diameter 2 in., for Pmax=20 N, Qmax=15 N and an axial stress of 59 NPa with an R-ratio=1 indicat- predicts an initial crack angle of 70.58, crack in- ing that cracks initiate near the stick±slip boundary. itiation angles ranging between 618 and 808 were observed experimentally. Some deviations from model predictions are expected as the material grain 5.1. Three-dimensional spherical contacts size of 60mm was comparable to the stick±slip zone Room temperature tangential fretting fatigue ex- size in many cases, while the model is strictly valid periments, using a set-up described in detail in Ref. under conditions of complete material homogeneity [43], were performed by Wittkowsky et al. [22] on and isotropy. 7075 T6 aluminum alloy, with mechanical proper- Also, as summarized in Table 2, fretting fatigue ties, E=71.5 GPa, n=0.33, sy=483 MPa, and experiments on a variety of other material systems 1/2 threshold DKII=1 MPa m , and the experimen- can be interpreted within the context of adhesion. tally determined friction coecient, m=1.2. Using Conditions of strong and weak adhesion can be reasonable values for the work of adhesion for identi®ed and, in a number of cases, the observed advancing and receding contacts, w=1 N/m and location of crack initiation agrees with the model

Gd=19 N/m [from equation (A1)], the adhesion predictions, as shown in Table 2.

Table 2. Experimental observations on contact fatigue crack initiationa

Material De€ Pmax Pmin Q M m Pmax (N), Adhesion DKmodel DKth Cracking Location Ref. 1/2 1/2 (mm) (N) (N) (N) (Nm) Qmax (N), (MPa m ) (MPa m ) predicted Mz (N m) (observed) Predicted Observed

En31 steel 10.16 1537 170 0 0 0.8 0.9 Weak 0.02 2±5 No (Yes) ± Surface [44] Hard steel 63.5 127,400 4900 0 0 0.8 6.0 Weak 0.004 2±5 No (Yes) ± Edge [45] SAE 52100 12.7 88 88 0 0.009 0.8 0.001 Weak 2.5 2±5 Yes (Yes) Stick±slip Surface [46] steel Al-7075 600 1000 1000 930 0 1.2 296 Weak 2.4 1 Yes (Yes) Stick±slip Edge, [20, 21] stick±slip Al-7075 1000 500 500 525 0 1.2 252 Weak 2.4 1 Yes (Yes) Stick-slip Edge, [47] T7351 stick±slip Ti±6Al±4V 300 700 700 525 0 0.8 152 Weak 2.1 2 Yes (Yes) Stick±slip Edge [47] Ti±6Al±4V 8 100 100 70 0 0.8 9 Weak 2.1 2 Yes (Yes) Stick±slip Edge [13] Nb 6.25 10.9 10.9 10 0 1.0 2.8 Weak 2.3 ± ± (Yes) ± Stick±slip [49]

a In all cases, sphere on ¯at geometry was used, except for Nb where a cylinder on cylinder point contact was established. De€:the e€ective diameter of the contacting sphere; Pmax: maximum normal load; Pmin: minimum normal load; Q: tangential load amplitude;    M:torsional load amplitude; Pmax Qmax Mz: critical normal, tangential or torsional loads above which weak adhesion is obtained; DKmodel: stress intensity factor range predicted by the adhesion model; DKth: material fatigue threshold stress intensity factor range for load ratio R=1 (estimated from Ref. [48]); Surface: contact surface; Edge: edge of the contact; Stick±slip: stick±slip boundary. GIANNAKOPOULOS et al.: ADHESION IN CONTACT FATIGUE 4663

5.2. Two-dimensional cylindrical contacts . By comparing to the material fatigue thresholds, For the tangential fretting fatigue studies on contact conditions required for crack initiation 2024±T351 aluminum reported in Ref. [50], with could be predicted. Additionally, the location and material properties, E=74.1 GPa, n=0.33, m=0.65, initial crack propagation directions could also be and using threshold DK =1 MPa m1/2, w=1 N/m predicted, in reasonable agreement with exper- II imental results. and Gd=12 N/m, the cyclic mode II stress intensity factors predicted by the model are greater than the Under conditions of small-scale yielding, the e€ects fatigue threshold for R=1, in agreement with the of static and/or oscillatory bulk stresses acting par- observation of crack initiation in all the reported allel to the contact surface (such as the far-®eld experiments (Fig. 9). While dominant fatigue cracks applied or residual stresses arising from surface were located at the contact edge, cracks were also modi®cation treatments due to shot-peening or observed at the stick±slip boundary. This indicates laser shock-peening) can be analyzed by recognizing a possible transition from an initial strong adhesion that these are analogous to the T-stresses present in stage to a ®nal weak adhesion stage. a linear elastic fatigue±fracture formulation. The In fretting fatigue studies on 0.34% carbon steel propensity for contact fatigue crack initiation could [51], adhesion was identi®ed as a precursor to the be suppressed by selecting material or environmen- contact edge cracks that initiated under very low tal combinations that e€ectively reduce adhesion in tangential load amplitude. This indicates strong ad- a contact system. hesion in the context of the present analysis. When a cylinder on cylinder line contact system AcknowledgementsÐThis work was supported by the was subjected to oscillatory oblique force loading, Multi-University Research Initiative on ``High Cycle two type of cracks, one at the contact edge and the Fatigue'', which is funded at MIT by the Air Force Oce second at the stick±slip boundary were observed of Scienti®c Research, Grant No. F49620-96-1-0278, [52]. The adhesion model can explain this phenom- through a subcontract from the University of California at Berkeley. enology by recognizing that the equivalent normal and tangential load oscillations can initiate cracks at the contact edge and the stick±slip boundary, re- REFERENCES spectively. In addition, the adhesion model can qualitatively 1. Johnson, K. L., Contact Mechanics. Cambridge University Press, Cambridge, 1985. predict the observed cracking pattern reported by 2. Johnson, K. L., Proc. R. Soc. Lond., 1954, A230, 531. Dawson [45] for normal load contact fatigue tests 3. 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