ELASTIC-PLASTIC CONTACT OF A ROUGH SURFACE WITH WEIERSTRASS PROFILE. Yan-Fei Gao and A. F. Bower

Transcription

1 ELASTIC-PLASTIC CONTACT OF A ROUGH SURFACE WITH WEIERSTRASS PROFILE a-fei Gao ad A. F. Bower Divisio of Egieerig, Brow Uiversity, Providece, RI 9, USA The classical approach to modelig cotact betwee rough surfaces follows a model developed by Greewood ad Williamso (966), i which surface roughess is idealized as a collectio of discrete asperities, with a Gaussia height distributio, ad (i the simplest model) idetical curvatures. Despite the widespread acceptace of this model, it has proved difficult to apply to realistic surfaces, whose roughess resembles a self-affie fractal over a wide rage of legth scales. For a perfectly fractal surface, the Greewood-Williamso model breaks dow completely, because it is impossible to detect idividual asperities o the surface, ad to determie their curvature. Several recet models of rough surface cotact have therefore bee based o a fractal descriptio of the surface geometry: the Weierstrass profile is oe such example. Ciavarella et al () have recetly reported a rigorous aalysis of elastic cotact of the Weierstrass profile. Our goal is to exted their aalysis to the more realistic case of a elastic-perfectly plastic solid. To this ed, we cosider a isotropic, elastic perfectly plastic solid with a rough surface, which is characterized by the two-dimesioal Weierstrass profile. The solid is ideted by a smooth frictioless cylider. We attempt to compute quatities such as the umber of cotact spots, their size (or more precisely, their size distributio); the true cotact pressure actig o the cotact spots, ad the total true area of cotact. I additio, we calculate critical coditios that will iitiate plastic flow i the asperities, ad determie the extet of this plastic flow as a fuctio of the applied loadig, the material properties of the deformig solid, ad the surface roughess. A importat coclusio of our study is that a perfectly fractal descriptio of surface roughess appears to lead to uphysical predictios of the true cotact size ad umber of cotact spots, for both elastic ad elastic-plastic solids. Possible approaches to resolvig these difficulties are discussed. Keywords: self-affie fractal, cotact mechaics, rough surface plasticity. Itroductio Whe two rough surfaces are pressed ito cotact, they meet oly at the highest poits o the two surfaces. To predict the size of the resultig cotact spots, the true cotact pressure, ad the true area of cotact is a veerable problem i cotact mechaics, ad is clearly of cosiderable practical importace, sice pheomea such as frictio, wear, ad cotact fatigue are cotrolled by the ature of asperity cotacts. Nevertheless, a fully satisfactory solutio remais elusive. The essetial character of surface roughess, ad the implicatios o the ature of cotact betwee rough surfaces, were recogized i the pioeerig work of Archard (957). His approach was to model a rough surface as a sphere, which has a array of smaller spherical bumps o its surface. These spheres i tur have still smaller spheres o their surfaces. If this geometric process is cotiued idefiitely, oe costructs what would today be regarded as a

2 surface with fractal geometry. Archard aalyzed elastic cotact betwee such a surface ad a flat ideter, ad predicted the umber of cotact spots, the cotact size, ad cotact pressure for each successive scale of spheres. Oe importat result of this calculatio is that if the surface roughess cotais a fiite umber of scales of spheres, the true area of cotact is proportioal to the load, thereby providig a elegat explaatio for Amoto s laws of frictio. For the limit of a perfectly fractal surface (with a ifiite umber of scales of spheres), oe fids that the true cotact area cosists of a ifiite umber of cotact spots with zero size, which are subjected to ifiite cotact pressure. Archard s model cotais a excellet qualitative descriptio of a rough surface, but caot easily be related to actual measuremets of real surfaces. Greewood ad Williamso (966) therefore devised a alterative approach to characterize surface roughess, ad to model cotact betwee two rough surfaces. I their origial work, Greewood ad Williamso idealized a rough surface as a collectio of asperities with spherical tips of equal radius, but with a Gaussia distributio of heights. They eglected iteractios betwee eighborig asperities, ad so were able to relate the behavior of the surface to the force-displacemet relatio for a idividual asperity ad the asperity height distributio fuctio. This approach predicts a well-defied umber of total cotact spots, cotact size, cotact pressure ad cotact area, which deped o the curvature of the asperity tips, the asperity height distributio, ad the elastic ad plastic properties of the two cotactig surfaces. The Greewood-Williamso approach has bee refied ad exteded by a umber of authors (e.g. Whitehouse & Archard 97; Nayak 97; Bush et al 975; McCool 985; Greewood 984), ad is widely used i may applicatios. Sice the early work of Greewood ad Williamso, a great deal of work has also bee doe to characterize the statistical properties of surface roughess. Various approaches have bee developed, icludig through its autocorrelatio fuctio (Whitehouse & Archard 97), power spectral desity (Sayles & Thomas 978) or as a semi-determiistic fractal (Majumdar & Tie 99; Majumdar & Bhusha 99, 99). There has bee some debate as to the most accurate descriptio of realistic surfaces, but all models agree that surfaces have a approximately selfaffie fractal character over a wide rage of legth scales, ad usually cotiue to exhibit fractal behavior dow to the limits of the resolutio of stadard surface profilometers. Ideed, Majumdar ad Bhusha (99, 99) show data that suggests that roughess spectra for hard disk drives remai fractal dow to wavelegths of a few hudred aometers. This behavior leads to some serious problems with Greewood ad Williamso s approach, because it is difficult to detect idividual asperity tips o a surface, ad eve more difficult to measure their curvature. Surface profiles are measured by samplig the surface at discrete itervals. I two dimesios, asperity tips are idetified by fidig poits that are higher tha their eighbors. Fidig summits i three dimesios is more difficult, but coceptually similar. The curvature of each asperity tip ca be determied by fittig a curve (i two dimesios) or surface (i three dimesios) through the poits surroudig the peak. For a differetiable surface, this procedure coverges to idetify a well-defied set of asperities as the samplig iterval is reduced. Whe this procedure is applied to experimetally measured surfaces, however, the umber of asperities ad their curvature appear to icrease without limit eve as the samplig iterval is reduced to extremely small values. This, of course, is a cosequece of the

3 fractal ature of surface roughess. For a perfectly fractal surface, Greewood ad Williamso s procedure breaks dow completely, sice such a surface does ot cotai well-defied summits. May recet studies of rough surface cotact have therefore abadoed the Greewood- Williamso descriptio of surface geometry i favor of oe that captures i some way the fractal character of surface roughess. Various approaches have bee proposed, icludig ad-hoc procedures for adaptig Greewood ad Williamso s calculatios to fractal surfaces (Majumdar & Bhusha 99, 99), direct umerical simulatios of experimetally measured profiles (e.g. Sayles, 996; Polosky & Keer a, b; Borri-Bruetto et al 999; Hyu et al 4; Pei et al 5), or hybrid schemes which combie umerical simulatios with Greewood ad Williamso s approach (see Mihailidis et al for a review). Attempts to adapt Greewood- Williamso to a fractal surface appear to suffer from the same difficulties as the origial Greewood-Williamso aalysis: there is o uambiguous way to cout, or to characterize, asperities o a fractal surface. Numerical simulatios provide cosiderable isight ito the deformatio fields associated with rough surface cotact, but there is a limit to the spatial resolutio of ay umerical scheme, which artificially trucates the roughess spectrum. This agai itroduces problems very similar to those associated with the Greewood-Williamso model: for a fractal surface, predictios of the cotact size, umber of cotacts, ad true cotact area will ot coverge as the resolutio of the umerical scheme is refied. Great care must be take to esure that this trucatio does ot lead to spurious predictios (Borri-Bruetto et al 999). A sigificat step towards a satisfactory model of the elastic cotact betwee rough fractal surfaces was made recetly by Ciavarella et al () ad by recet work of Persso (, 5). The approximate approach of Ciavarella et al () will be followed closely i our work, ad so will be described i more detail i Sectios ad 3 of this paper. Briefly, they chose to approximate surface roughess usig the two-dimesioal Weierstrass fuctio give i Eq. below. The Weierstrass fuctio cosists of a sum of siusoidal profiles, with progressively decreasig wavelegths Λ m =Λ γ m where m =,, ad Λ ad γ are ( D ) m geometrical parameters. The amplitude of each term i the series varies with m as G = Gγ, ad for < D < the fuctio is a self-affie fractal with dimesio D. It is difficult to aalyze cotact of rough surfaces with realistic values of γ, but Ciavarella et al show that much progress ca be made by takig γ >>. Uder these coditios, the wavelegths i the Weierstrass profile are widely separated, ad each term ca be aalyzed idepedetly of the rest. The surface the resembles Archard s early model, ad ca be treated i a similar way: istead of attemptig to idetify actual summits o the surface, ad to predict the true cotact size, pressure ad area for these summits, Ciavarella et al compute the apparet cotact size, pressure ad total area for each successive scale of roughess m. They formally take the limit m, ad recover Archard s origial predictio that the true cotact area cosists of a ifiite umber of cotact spots, with zero size, ifiite pressure, ad vaishig total area (Ciavarella et al ; Ciavarella & Demelio ). This is a purely formal mathematical result, of course: i practice pheomea such as plastic flow or adhesio might alter these predictios, ad i ay evet o real surface is perfectly fractal. There are some applicatios, such as modelig frictio, where the true cotact size ad area are of great iterest, ad predictig these quatities remais a uresolved problem. For modelig the compliace of the surface, or its electrical cotact resistace, ad possibly 3

4 physical pheomea such as plastic flow, wear, ad the iitiatio ad growth of cotact fatigue cracks, the behavior of very short wavelegth roughess is less importat, sice damage evets occur at sub-micro legth scales rather tha aometer legth-scales. Oe advatage of Archard s model, as exteded by Civarella et al, is that the behavior of log-wavelegth roughess scales ca be predicted i terms of measurable surface parameters without eedig to idetify real asperities o the surface. This work has provided a importat step towards uderstadig cotact betwee surfaces with a fractal character. Their predictios leave several issues ope, however. Their aalysis shows clearly that for elastic cotact betwee perfectly fractal surfaces, the true cotact area cosists of a ifiite umber of cotact spots with vaishig size, which are subjected to ifiite cotact pressure. The total true cotact area also vaishes. This is clearly a uphysical predictio. Of course, real surfaces differ from the idealized model i several respects, which iclude: (i) real surfaces are ot perfectly elastic; (ii) the surface roughess of a real surface caot be perfectly fractal the spectrum must be trucated at some very short legth-scale; (iv) adhesive forces may act betwee real surfaces at very short legth-scales; ad (v) Ciavarella et al s model is restricted to surfaces with widely separated scales of roughess. Weierstrass profiles with large values of real γ clearly do ot accurately characterize real surface roughess, eve if oe accepts that the Weierstrass represetatio with γ is a realistic model of a surface. The questio is, which of these effects plays the most importat role i cotrollig the true cotact area i real solids? These are importat issues for further research. I this paper, our goal is to exted the calculatios of Ciavarella et al to accout for plastic deformatio. The problem to be solved is illustrated i Fig.. We cosider a isotropic, elastic perfectly plastic solid with oug s modulus E, Poisso s ratio ν ad yield stress σ. The solid has a rough surface, which is characterized by the two-dimesioal Weierstrass profile give i Eq.. The solid is ideted by a smooth cylider with radius R, which is subjected to a force P per uit legth out of the plae. The goal of our aalysis is to estimate quatities such as the umber of cotact spots, their size (or more precisely, their size distributio); the true cotact pressure actig o the cotact spots, ad the total true area of cotact. It is of particular iterest to calculate critical coditios that will iitiate plastic flow i the asperities, ad to determie the extet of this plastic flow as a fuctio of the applied loadig, the material properties of the deformig solid, ad the surface roughess. The remaider of this paper is orgaized as follows. I the ext sectio, we give a more careful statemet of the problem to be solved, the assumptios uderlyig our calculatios, ad the results that will be calculated. Sectio 3 gives a brief review of the idetatio respose of a elastic-perfectly plastic solid with siusoidal roughess, as the first step towards aalyzig the Weierstrass profile. I Sectio 4, we review the procedure proposed by Ciavarella et al to aalyze cotact of the Weierstrass profile, ad devise a simplified, approximate procedure that ca be used to derive aalytical results for both elastic ad plastic cotacts. Sectio 5 presets detailed results, ad sectio 6 cotais coclusios. The pricipal coclusio of our calculatio is that for a perfectly fractal elastic-plastic surface, the true cotact size cosists of a ifiite umber of cotact spots with zero size, which are subjected to a fiite cotact pressure. This limitig cotact pressure is approximately twice the 4

5 material hardess, due to lateral iteractios betwee eighborig cotacts. I additio, the total area of cotact betwee the two surfaces is fiite. Thus, plasticity resolves some, but ot all, of the sigular behavior associated with elastic cotact betwee ideally fractal surfaces. Determiig the true cotact spot size, ad the umber of true cotacts, remai ope questios. S(ω) (µm 3 ) ω (rad/µm) (a) (b) Fig. (a) A typical rough surface topography of sigle crystal copper, measured by the atomic force microscope (courtesy of B.C.P. Burke). (b) Fractal properties ca be deduced from the power spectrum. The straight lie is by least squares fittig, givig rise to D Rough Surface Model Majumdar ad Bhusha (99) suggest that the two dimesioal Weierstrass fuctio ( D ) m m γ π γ m= hx ( ) = G cos( x / Λ ) () with γ.5 ad D.4 provides a good qualitative descriptio of may practical surface profiles. As a represetative example, we show i Fig (a) a three dimesioal profile z( x, y ) of a surface (i this case a copper sigle crystal) obtaied with a atomic force microscope. To compare the surface profile with the represetatio i Eq.(), we take a two-dimesioal trace zxy (, = µ m) of the surface i the x directio. We the compute the Fourier trasform zˆ( ω) of the two-dimesioal trace, ad plot the power spectral desity S ( ω) i Fig. (b). The power spectrum for the ideal Weierstrass profile is a series of Dirac delta fuctios, but Berry ad Lewis (98) argue that the Weierstrass distributio ca be used to approximate a cotiuous power spectrum by esurig that the itegral of the cotiuous power spectral desity matches that of the Weierstrass fuctio. The resultig cotiuous approximatio is give by GΛ S( ω ) = Λ ω () log ( ) ( ) D 5 γ 5

6 ad would correspod to a straight lie with slope D-5 o Fig (b). The Weierstrass fuctio is clearly ot a fully accurate descriptio of real surface roughess. The actual power spectrum clearly appears to be better approximated by a cotiuous fuctio tha a series of Dirac delta fuctios. I additio, roughess wavelegths i the rage 5 µ m - mm are strogly sesitive to processig ad i geeral are ot well characterized by eve the cotiuous approximatio to the Weierstrass fuctio. Similarly, the Weierstrass profile becomes uphysical at very short wavelegths: as m, the surface becomes eedle-like, with roughess amplitude much greater tha its wavelegth. Ay departure from ideal behavior at log wavelegth (low m) will lead to quatitative, but ot qualitative chages i our predictios ad could readily be icorporated i the computatios. The departure from ideal behavior at short wavelegths (large m) has a critical ifluece o the true size of the cotact spots, ad for ideally elastic surfaces, o the total true area of cotact ad cotact pressure. It will ot, however, have a sigificat ifluece o the behavior of lower scales of roughess. Experimets suggest that γ.5 ad D.4 i Eq. () give the best qualitative fit to real surfaces. Uder these coditios, roughess scales with successive values of m do ot clearly decouple, however, ad aalyzig the cotact betwee such surfaces is difficult. Ciavarella et al () show that calculatios are greatly simplified by approximatig a fractal surface with a larger value of γ > 5. The surface the cosists of a series of harmoic profiles, with widely separated wavelegth, each of which ca be aalyzed idepedetly. Although the resultig surface does ot resemble ay actual surface, it does capture the self-affie fractal character of real surfaces, ad so provides a ivaluable qualitative picture of the behavior of two cotactig fractal surfaces. We ow cosider a elastic perfectly-plastic solid, with oug s modulus E, Poisso s ratio ν ad yield stress σ, which has a surface roughess defied by Eq. (). The solid is ideted by a smooth cylider with radius R as illustrated i Fig.. The cylider is loaded by a total force (per uit out of plae legth) P, to iduce a cotact pressure distributio p( x ). Of course, the two surfaces actually meet oly at the tips of asperities. We shall fid that for a elastic-perfectly plastic solid with a ideal Weierstrass profile, the true cotact cosists of a ifiite umber of cotact spots of zero size, which are all subjected to the same, uiform cotact pressure. This is a purely formal result, however, ad does ot characterize the loadig or deformatio of the solid i a useful maer. We obtai a better picture by regardig the loadig ad deformatio to cosist of a series of separate scales, =,,..., as illustrated i Fig.. The lowest scale, = correspods to the omial cotact. Successive terms =,,... characterize the behavior of the solid at legth scales comparable to successive wavelegths i the Weierstrass profile. The fields associated with each term iteract weakly through the requiremet that the pressure at scale must be statically equivalet to the pressure at scale +. 6

7 Scale - R P p - (x) a - Scale p p (x) a λ Scale p p (x) a λ Figure : Idealized two-dimesioal model of cotact betwee a smooth cylider ad a surface with Weierstrass profile with widely separated scales. Successive scales iteract through the coditio that the omial pressure actig o scale is equal to the true pressure actig o scale -. The omial cotact area ad pressure (scale =-) ca be computed usig results for smooth cotacts (see e.g. Johso 985). It is helpful to review these results briefly. The severity of the omial load is coveietly characterized by a dimesioless load parameter PE * / σ R. For * PE / σ R < 3., the cotact is elastic, ad the cotact width ad omial cotact pressure distributio follow as where 4PR a = * π E 4 p ( x) = p x / a π mea mea p P/a.4σ (3) = (4) 7

8 * is the average omial cotact pressure. For PE / σ R > 4, the cotact is fully plastic, ad the cotact width ad stress distributio ca be approximated usig the rigid plastic slip-lie field solutio, which gives a = P/( Hσ ) (5) p ( x) = H σ where H.9 is the ratio of the plae strai hardess of the solid to its tesile yield stress. I * the itermediate regime 3. < PE / σ R < 4, the cotact width ad pressure distributio would eed to be computed usig umerical simulatios. This will ot be attempted here: we shall focus istead o the limitig cases of perfectly elastic ad fully plastic bulk cotacts. It is importat to ote that the omial cotact geometry itroduces a atural ad well defied log wavelegth cut-off for the roughess spectrum. Roughess wavelegths greater tha a clearly do ot ifluece the cotact pressure. The roughess compoet with wavelegth closest to a should be regarded as part of the omial cotact geometry (its effect is equivalet to modifyig the shape of the ideter). The first compoet of the profile to ifluece the cotact pressure distributio has idex m = + log( Λ / a )/logγ, ad has wavelegth ad amplitude ( ) ( D ) λ = a / γ g = G γλ / a (6) Heceforth, we will characterize the surface roughess withi the area of cotact with a trucated Weierstrass profile give by ( D ) γ π γ λ = hx ( ) = g cos( x / ) (7) where gad λ represet the amplitude ad wavelegth of the first roughess compoet that has wavelegth less tha the omial cotact width. It will be importat to keep i mid that while γ, D, G ad Λ are properties of the surface geometry oly, gad λ deped also o the omial cotact geometry as show i Eq. (6). We shall retur to this issue whe presetig our results i Sectio 5. If we ow examie a small regio of the omial cotact, with legth λ << dx<< a ad cetered at some poit x withi the cotact area, we observe a siusoidal surface with wavelegth λ i cotact with a omially flat ideter. The ideter is subjected to a uiform omial cotact pressure p ( ) = p x, which iduces a cotact width a ad pressure p () x o the zero-th roughess scale. We ca the examie more closely a regio with size λ << dx << λ withi oe of these cotacts. Here, we observe a siusoidal surface with wavelegth λ, subjected to omial cotact pressure p = p ( x), iducig a cotact size a, ad cotact pressure p ( ) x. This process ca evidetly be cotiued idefiitely. The goal of our aalysis is to determie, for each roughess scale, the total umber of cotact spots N, the apparet size of the cotact spots a, ad the apparet cotact pressure distributio p ( x ). Calculatios are complicated by the fact that the quatities of iterest vary 8

9 with positio o the surface. It is possible (but exceedigly cumbersome) to calculate this spatial variatio. I practice, it is preferable to quatify the variatio by computig distributio fuctios, which, for example, specify the fractio of asperity cotacts with size betwee a ad a + da, or the fractio of the omial cotact area that is subjected to pressure betwee p + dp, for each scale =.... p ad The distributio fuctio for the pressure plays a particularly importat role i subsequet calculatios. For each scale of roughess, we defie a dimesioless fuctio q ( p/ σ ) such that q( p/ σ)( dp / σ ) is the fractio of the omial cotact (which has total width a ) that is subjected to a dimesioless pressure betwee p / σ ad ( p + dp )/ σ. As a example, the distributio fuctio q ( / ) p σ characterizig the omial cotact ca easily be computed. The cumulative distributio Q( p ) (the proportio of the cotact over which the pressure exceeds p) ca be computed from the cotact pressure distributio as Q ( p) = xp ( )/ a. The distributio fuctio q ( p) the follows as q ( p) = Q p. For example, if the omial cotact is elastic, we have The distributio fuctio follows as ( π ) Q ( p) = x( p)/ a = p/4p mea (8) q dq ( ξ) = σ = dp mea ( /4ξ ) ξ π ( πξ /4 ξ ) mea mea where ξ = p σ, ξ = p / σ. Similarly, for fully plastic cotact, the distributio fuctio is evidetly q ( ξ) = δξ ( H) () where δ deotes the Dirac delta distributio ad H. 9 is the ratio of the hardess to the tesile yield stregth of the solid, as before. Similar distributios will characterize the pressure distributio for each higher roughess scale. Although it is ot possible to write dow a simple expressio relatig the cotact pressure p( x ) to q( p ) for, Ciavarella et al () show that this is ot ecessary. Their procedure for computig q will be reviewed i detail i Sectio 4. mea (9) The distributio fuctio q( p ) provides a coveiet descriptio of the cotact pressure actig o the th roughess scale, but is difficult to iterpret physically. To provide a more direct measure of the deformatio of each roughess scale, we itroduce the cotact size distributio fuctio S ( a ) defied such that d = NS ( ada ) () with N the total umber of cotact spots at the th scale, gives the umber of cotacts with size betwee a ad a+ da. Similarly, as a more direct measure of the loadig applied to the cotacts i the th scale, we defie the mea cotact pressure for a idividual cotact spot as 9

10 p mea a = pxdx ( ) () a a mea We the itroduce the cotact pressure distributio P ( p ) such that ( mea mea d = NP p ) dp (3) gives the umber of cotacts i the th scale that are subjected to mea pressure betwee mea p mea mea ad p + dp. Naturally, sice there is a direct relatioship betwee the cotact size ad mea cotact pressure for idividual cotact spots, there will be a similar direct relatioship mea betwee S( a ) ad P ( p ). mea Although we shall formally compute expressios for S( a ) ad P ( p ) for our model surface, we readily cocede that i view of the rather crude descriptio of surface topography that forms the basis for our calculatios, the exact distributio fuctios may have little practical sigificace. I may cases the average values of cotact size ad pressure for each scale of mea cotact are sufficiet. These are formally related to S( a ) ad P ( p ) through mea = = I practice, however, it is simpler to compute (4) a S ( aada ) p P ( ppdp ) a ad mea p directly, rather tha via the distributio fuctios. The total cotact area A = N a (5) ad the omial cotact pressure om p = P/ A (6) are also of iterest. We shall calculate these quatities usig the elegat procedure developed by Ciavarella et al () to model cotact betwee elastic solids with a Weierstrass profile. A key assumptio i their work is that the behavior of a particular roughess scale at a poit o the surface ca be calculated from the solutio to a siusoidal surface ideted by a flat, frictioless puch. Successive roughess scales iteract oly through the requiremet that the omial pressure p that acts o scale is equal to the true pressure p ( ) x for the precedig scale -. This assumes that o iteractio occurs betwee the deformatio fields associated with successive roughess scales. For a elastic solid, this assumptio ca easily be justified usig Sait Veat s priciple. For elastic-plastic materials, o rigorous proof exists. We speculate, however, that provided the bulk cotact (at scale -) cosists of a sigle, isolated cotact area, with a size much smaller tha ay characteristic structural dimesio, the same decouplig ca be applied for elastic-plastic solids. As a extreme example, cosider the limitig case of a rigid-perfectly plastic solid. The slip-lie field solutios for cotact betwee a semi-ifiite solid ad a sigle ideter ivariably show a rigid regio uder the cotact area (Hill 95; also reproduced i Johso 985). Cosequetly, the deformatio associated with the first roughess scale i a Weierstrass surface (scale ) may be embedded withi the rigid regio associated with the omial cotact. The behavior of the

11 first scale of roughess is therefore equivalet to that of a siusoidal surface, with rigid boudary coditios at ifiity. Our umerical solutio to this problem shows that a rigid regio forms uder each cotact area. Cosequetly, the secod roughess scale ca similarly be embedded withi this regio, ad this process ca be cotiued for all higher roughess scales. Sice the deformatio field costructed i this maer forms a kiematically admissible collapse mechaism, our solutio should strictly be regarded as a upper boud to the collapse load, ad a lower boud to the cotact size ad cotact area. We preset some further support for our calculatios i the Appedix A, where the aalytical calculatios are compared with the predictios of full-field fiite elemet computatios for up to three roughess scales. It is importat to recogize, however, that if the deformatio field for the bulk cotact (scale -) is ot cotaied, a rigid regio may ot form below the bulk cotact i the fully plastic limit. I this case, the first scale of roughess is ot cotaied, ad this behavior will propagate upwards throughout all subsequet scales of roughess. Such ucostraied flow is particularly likely to occur i metal formig operatios. 3. Idetatio respose of a siusoidal surface Clearly, establishig the respose of a simple siusoidal surface to idetatio loadig is a key step towards solvig the cotact problem outlied i the precedig sectio. The idetatio respose of a elastic-perfectly plastic solid with a harmoic roughess was aalyzed i detail by Gao et al (5). Oly a few of their results are eeded here: these are reviewed briefly below. We focus attetio o a sigle roughess scale with wavelegth / λ = λ γ (7) ad amplitude ( D ) g = gγ (8) as illustrated i Fig.. The substrate is a elastic-perfectly plastic solid with oug s modulus E, Poisso s ratio ν ad yield stress σ. Gao et al (5) show that the stregth of the surface is coveietly characterized by a material ad geometric parameter give by * ge ψ = (9) λσ E. I the discussio to follow, the value of ψ for the zero-th roughess scale (=) plays a cetral role i characterizig the resistace of the rough surface to plastic flow. I terms of this parameter, we have * g E ψ = λσ, ( D ) ψ = ψ γ () Note that the resistace to plastic flow varies i iverse proportio to ψ : a rough surface with a low yield stress has a high value of ψ, while a smooth, hard surface would have a low ψ. I * with = E ( ν ) additio, ote that ψ icreases with, so we aticipate that roughess scales with low wavelegth will be elastic, but all scales beyod a critical value of will deform plastically.

12 The solid is ideted by a flat, rigid surface, uder the actio of a omial pressure p, as illustrated i Fig.. The loadig iduces a cotact width a, pressure distributio p( x ) ad mea mea pressure p give by fuctioal relatioships p g a / λ = A, ψ, σ λ p ( x) x p g =Φ,, ψ, σ λ σ λ mea p mea p g =Φ, ψ, σ σ λ mea Here, we have show that formally that A, Φ ad Φ deped o both ψ ad g / λ. I fact, if the surface remais elastic, exact calculatios show that the solutio depeds oly o ψ ad is idepedet of g / λ. If the surface deforms plastically, umerical simulatios show that the fuctios deped oly weakly o g / λ. It would ot be difficult to iclude this variatio i our computatios, but for simplicity we have chose to eglect it. ().5.4 Iteractio betwee eighborig asperities.3 a/λ.. Elastic Elastic-Plastic Fully-Plastic - ψ=e * g/λσ Fig. 3: A map illustratig the regimes of behavior of a elastic-perfectly plastic solid with siusoidal surface whe ideted by a rigid flat puch. Fially, we must also compute the pressure distributio fuctio q ( p/ σ ), which is defied so that q ( p/ σ )( dp / σ ) is the fractio of the siusoidal surface that is subjected to ormalized

13 pressure betwee p / σ ad ( p + dp)/ σ. This distributio fuctio ca be determied from the cotact pressure, followig the procedure give for the omial cotact i equatios (8) ad (9). The result will be specified by the fuctioal relatioship p p p q = q,, ψ () σ σ σ mea We have calculated approximatios to the fuctios A, Φ, Φ ad q by fittig curves to the umerical simulatios reported i Gao et al (5). To specify the results, it is ecessary to distiguish betwee four geeral regimes of behavior: (i) Elastic; (ii) Elastic-plastic; (iii) Fully plastic, with o iteractio betwee eighborig asperities ad (iv) Fully plastic, with iteractio betwee eighborig asperities. The distictio betwee these regimes is discussed i detail i Gao et al (5), ad will be summarized oly briefly here. Elastic behavior is observed at low values of applied load; while the elastic-plastic regime occurs for loads that slightly exceed the elastic limit. For sufficietly large values of load, the cotact pressure distributio becomes approximately uiform. This coditio defies fully plastic behavior. I the fully plastic regime, if the cotact spots are small compared with the wavelegth of the roughess, the asperity cotacts are isolated ad the cotact pressure is equal to the plae strai hardess of the solid (.9σ ). If the cotact width approaches the roughess wavelegth, the plastic zoes uder eighborig cotact spots begi to iteract. Uder these coditios, the cotact pressure remais approximately uiform, but its magitude icreases with applied load, ad approaches values up to twice the material hardess as a λ. Gao et al (5) show that these four regimes of behavior ca be coveietly displayed o a map with a / λ ad ψ as parameters. The map is show i Fig. 3: the elastic limit is give approximately by the coditio ψa / λ =., while ψa / λ = defies the trasitio to fully plastic behavior. I the sectios below, we give expressios for the fuctios defied i Eqs () ad () i each of the four regimes. 3. Elastic Regime < ξ < πψ si ( π / ψ) I the elastic regime, A, Φ ad si ξ, ξ < πψ A( ξψ, ) = π πψ, ξ πψ ξ cos( πη) { si ( πa /) si ( πη) } / ξ < πψ Φ ( ηξψ,, ) = si ( π A /) ξ + πψ cos( πη) ξ > πψ mea Φ are give by the Westergaard solutio (Westergaard 939) (3) (4) ( ξψ, ) πξ mea Φ = si ( ξ / πψ ) ξ πψ ξ ξ πψ (5) 3

14 η / π η < / q( ηξψ,, ) {( πψ ξ) η }{ ξ( πψ ξ ( πψ ξ) η ) η } = η > πψξ πψξ (6) Note that the elastic solutio depeds oly o the ormalized omial pressure ξ = p ad surface property ψ, ad is idepedet of g / λ. The critical load ξ = p / σ = πψ correspods to full cotact ( a/ λ = ). This suggests a secod physical iterpretatio for the parameterψ : a surface with a low value for ψ ca easily be flatteed. This solutio is valid below the elastic limit: Gao et al (5) suggest that a coveiet (but approximate) estimate for the yield poit is give by the coditio ψa / λ <., which requires that ξ = / σ < πψ si ( π / ψ). p / σ 3. Elastic-Plastic regime πψ π ψ ξ ψ si ( / ) < < H / The elastic-plastic regime of behavior is observed for applied loads that slightly exceed yield. Uder these coditios, the elastic ad plastic strais uder the ideter are comparable, ad mea exact expressios for A, Φ ad Φ caot be foud. Detailed umerical results are give i Gao et al (5). For the problem at had, however, we fid that at most a sigle scale of roughess is i the elastic-plastic regime, so it is ot ecessary to characterize this regime i mea detail. Cosequetly, we have costructed simple fuctios for A, Φ ad Φ by usig a selfcosistet iterpolatio betwee the elastic ad fully plastic regimes. The iterpolatio is costructed to predict correctly: (i) the elastic limit, (ii) the critical load to trasitio to fully plastic behavior; (iii) the cotact pressure distributio at the elastic limit; ad (iv) the cotact pressure at the fully plastic limit. I additio, we iterpolate the cotact size betwee the elastic ad fully plastic limits as follows.9 ( ξ ξp ) ( ξ ξe) A( ξψ, ) = Ae + A.9 p (7) ( ξ ξ ) ξ ξ where ( ) e p p e π H ξe = πψ ξ ψ = = ψ A = /(5 ψ) A = / ψ e si p H.8 p defie dimesioless values of the critical omial pressure at yield; the critical omial pressure at full plasticity, ad the ormalized cotact widths at yield ad the fully plastic limit, respectively. The cotact pressure distributio is approximated as αξψ (, )cos( πη) Φ ( ηξψ,, ) = si ( πa/) si ( πη) + βξψ (, ) (9) si ( π A/) { } / where the scalig factors α ad β are itroduced to esure that the cotact pressure has the correct average ad maximum values. This requires (8) 4

15 max ( ) Aξ ξ si( πa/) max ξ ξ si( π A/) αξψ (, ) = βξψ (, ) = (3) A si( π A/) A si( πa/) where the maximum cotact pressure is iterpolated betwee its values at yield ad full plasticity as where ξ ( p ) ( p) ( ) A A ξ H A A = + A A A A A max ( p ) ( ξ / ) + ( ξ / πψξ ) Ae H e Ae Ap e Ae e A = H πψξe is a costat chose to esure that the peak cotact pressure has the correct value max ξ = πψξ at the elastic limit ( A= A, ξ = ξ ). The mea cotact pressure follows as e e mea Φ ( ξψ, ) = ξ / A (33) I additio, the cotact pressure distributio fuctio may be computed as ( η βh)/ π q( ηξψ,, ) = / { ω + ( η βh) }{ α( ω+ ω + ( η βh) ) ( η βh) } (34) ω = αcot ( πa/) 3.3 Fully plastic regime o-iteractig asperities H/ ψ < ξ < H/3 I the fully plastic regime, the plastic strais uder the ideter greatly exceed the elastic strais, ad the stress fields approach the rigid-plastic limit (give e.g. by slip lie field solutios). Here, it is ecessary to distiguish betwee two distict types of behavior. As log as the cotact spots are small (umerical results suggest a/ λ <.67 ), there is o iteractio betwee mea eighborig asperity cotacts, ad A, Π ad Π ca be estimated from slip-lie field solutios (or umerical simulatios) for a flat puch idetig a rigid-plastic half-space as A( ξ) = ξ / H Φ ( x) = H ( x / a< ) mea Φ = H where H.9 is the hardess of the material ormalized by its tesile yield stress. Numerical simulatios ad slip- lie field solutios show that H varies weakly with g / λ, but this variatio is isigificat for our preset purposes. Note that i this regime the cotact pressure distributio is perfectly uiform, ad the solutio is idepedet of ψ. Fially, the pressure distributio fuctio follows as q( ηξψ,, ) = A( ξψδη, ) ( H) (36) where δ ( x) deotes the Dirac delta distributio. e (3) (3) (35) 5

16 3.4 Fully plastic regime iteractig asperities H /3< ξ Whe the size of the cotact spots becomes comparable to the wavelegth, the plastic strai fields associated with eighborig asperities begi to iteract. Numerical simulatios show that i this regime, the cotact pressure distributio remais approximately uiform, but the magitude of the pressure icreases substatially, ad i some cases reaches twice the hardess of the material. This behavior has a dramatic ifluece o the behavior of fie roughess scales. As i Sectio 3. we have devised approximate aalytical approximatios to umerical simulatios to simplify our calculatios. It is simplest to specify the relatioships betwee ξ, ψ, A ad mea Φ parametrically as ( ( ψ ) )( 3 ) 3 mea Φ = H + H H A (37) mea ξ = AΦ (38) where H ( ) ψ is the limitig value of the ormalized cotact pressure px ( )/ σ at full cotact. The followig fuctio gives a approximate fit to the results show i Fig. 8 of Gao et al (4) H 3/ ( ψ ) = 5.8 6/(6 + ψ ) (39) It is straightforward to solve these equatios to determie the fuctios defied i Eqs (). The results ivolve the solutio to a cubic equatio, however, ad so are too legthy to be recorded here. Fially, the cotact pressure distributio fuctio i the iteractig asperity regime has the same form as that give i Eq. (36). Normalized cotact pressure p/σ Approximatio Numerical ψ= ψ=. ψ=5 ψ= ψ= Cotact fractio a/λ Mea cotact pressure p mea /σ ψ=. Approximatio Numerical ψ= Cotact fractio a/λ Fig. 4: The relatioship betwee omial pressure ad mea pressure ad cotact fractio for a siusoidal surface i cotact with a rigid, flat surface. The symbols show the results of fiite elemet computatios, while the solid lies show the approximate aalytical expressios used i our rough surface model. ψ=5 ψ= ψ=3.5 6

17 Our approximate curves are compared with the results of fiite elemet simulatios i Fig. 4, which shows a parametric plot of the omial ad mea cotact pressures as a fuctio of the cotact size as the load icreases, for various values of ψ. 4. Cotact mechaics for a geeral Weierstrass profile We proceed to aalyze cotact of the Weierstrass profile. Our aalysis follows exactly the work of Ciavarella et al () ad so will be reviewed oly briefly. We will first preset a fully geeral procedure for computig distributio fuctios for cotact size ad pressure for each scale. Subsequetly, we suggest a simple approximatio that ca be used to obtai closed-form estimates for the umber of cotact spots, their size, ad mea pressure for each scale of loadig. 4. Results for geeral surface cotact Although it is difficult to compute the spatial cotact pressure distributio for the Weierstrass surface, it is straightforward to determie the probability distributio fuctio q ( p/ σ ) defied i Sectio 3. The calculatio begis by computig the distributio fuctio q ( / )( / ) p σ dp σ, which specifies the proportio of the omial cotact area that is subjected to a pressure betwee p + dp σ. The result is give i Eq. (9). p σ ad ( ) The distributio q ( ) p for the zero-th roughess scale may therefore be computed as follows. Recall that we itroduced the dimesioless fuctio q, such that q( p/ σ, ξψ, )( dp / σ ) specifies the proportio of the siusoidal surface that is subjected to pressure betwee p σ ad ( p + dp) σ whe loaded by a ormalized omial pressure ξ. The zeroth roughess scale is subjected to a distributio of omial pressure, as quatified by q ( / ) p σ. Summig over all possible values of the cotact pressure for the omial cotact, we fid that q p/ σ = q p/ σ, ξψ, q ( ξ) dξ (4) ( ) ( ) The upper limit o the itegral here is purely formal, of course: the omial cotact pressure does ot exceed p = P/ πa, so q ( ξ ) = for ξ > P/ πa σ. This result ca be applied recursively, to see that q p/ σ = q p/ σ, ξψ, q ( ξ) dξ (4) ( ) ( ) Oce the hierarchy of distributio fuctios q( p/ σ ) has bee computed, various quatities of iterest ca be deduced. For example, to calculate the total (apparet) area of cotact for the th scale, ote that the area of cotact for oe wavelegth of the th scale subjected to pressure p is ( ) a = λa p σ, ψ. I additio, there are a / λ such wavelegths withi the omial area of cotact. The total cotact area therefore follows as ( ) A = a A ξψ, q ( ξ) dξ (4) 7

18 The total umber of cotact spots at the -th scale withi the omial cotact is A N = (43) λ whereupo the average size of the cotacts at the th scale follows as A A a = = λ (44) N A The average cotact pressure ca be foud by a similar approach, with the result mea a mea p = Φ ( ξψ, ) q ( ξ) dξ A Fially, the omial cotact pressure for the th scale follows as (45) mea P mea p = = p A( ξψ, ) q ( ξ) dξ A (46) We ca also fid expressios for the cotact size ad mea pressure distributio fuctios S ( a ) mea ad P ( p ) defied i Eqs. () ad (3). To compute the distributio fuctio for the cotact size, deote the pressure required to iduce a cotact size a is give by the fuctioal relatioship p/ σ = A ( a / λψ, ), where A deotes the iverse of the fuctio A defied i equatio (). Give the distributio fuctio for the cotact pressure at ( ) -th scale, we ca easily compute the probability distributio fuctio of the cotact fractio for scale as follows. For a sigle cotact, a / λ = A( p σ, ψ), so that let / p σ = A ( a / λ, ψ). The probability of fidig a cotact fractio i betwee a / λ ad ( a + da )/ λ is S ( a ) ( a λ ) ( A ( a λ ψ )) da λ = q, (47) d 4. Simplified aalysis the uiform cotact pressure assumptio The results listed i the precedig sectio ca be simplified dramatically if the cotact pressure distributio for each scale i the cotact (icludig the omial cotact area) is uiform. This is a excellet approximatio whe each scale of cotact is fully plastic. At lighter loads, the omial cotact, ad some log-wavelegth roughess scales, will remai elastic, ad for these scales the distributio of cotact pressure is ot uiform. Surprisigly, however, we have foud that averaged quatities such as the total area of cotact, the average cotact size, ad the average cotact pressure ca be computed remarkably accurately eve for elastic surfaces by replacig this o-uiform cotact pressure distributio with a statically equivalet, uiform pressure. Proceedig alog these lies, therefore, we suppose that the etire omial cotact area is subjected to the same cotact pressure. For this approximatio, the distributio fuctio q ( / ) p σ, which specifies the fractio of the omial cotact area (scale -) subjected to pressure betwee p + dp σ, must therefore be give by p σ ad ( ) 8

19 q ( ξ) = δξ ( p / σ ) (48) mea mea where δ ( x) deotes the Dirac delta distributio ad p = P/a. The distributio fuctio for the zero-th scale of roughess follows from Eq. (4) as mea mea σ = σ ξψ δξ σ ξ = σ σ ψ q ( p/ ) q( p/,, ) ( p / ) d q( p/, p /, ) (49) mea Recall ow that the fuctio q( p/ σ, p / σ, ψ ) specifies the fractio of a siusoidal surface that is subjected to pressure betwee p + dp σ, whe ideted by a omial cotact pressure p mea distributio, with magitude p σ ad ( ). If the pressure is approximated by a statically equivalet uiform mea p, it follows that ( ( )) q( p/ σ, p / σ, ψ) = A( p / σ, ψ) δ p/ σ Φ p / σ, ψ, (5) mea mea mea mea mea where the fuctios A ad Φ are defied i equatios (), ad give specific forms i Eqs. (3)--(39). Repeatig this procedure for subsequet roughess scales gives the followig result for the distributio fuctios q ( p ) mea mea mea ( / σ) { ( j / σ, ψ j) } δ ( / σ ( / σ, ψ) ) j= q p = Π A p p Φ p (5) Here, the mea cotact pressure at each scale ca be determied from mea mea mea p / σ =Φ ( p / σ, ψ ) (5) mea where the sequece is started with p give i Eq.(48). mea Give p for each scale, ad the distributio fuctios q( p/ σ ), we ca determie the remaiig quatities of iterest usig Eqs. (4)-(46). The total area of cotact at the th scale follows as mea = ( ξψ) ξ ξ = Π j σ ψ j j= A a A, q ( ) d a Ap ( /, ) (53) where the fuctio A is defied i Eq. (). The umber of cotacts follows as mea λ j σ ψ j j= N = ( a / ) Π Ap ( /, ) (54) ad the average cotact size is give by a mea = λ Ap ( / σ, ψ ) (55) Fially, equatio (46) provides a additioal way to compute { ( /, ) j= } mea mea mea j j mea p as p = p Π Ap σ ψ (56) from which we also recover the expected relatioship betwee the mea pressure at successive scales mea mea mea p = p Ap ( / σ, ψ ) (57) 9

20 4.3 Approximate aalytical expressios for average cotact size, pressure, umber of cotacts, ad total cotact area for elastic cotact If the cotacts remai elastic, the procedure outlied i the precedig sectio ca be used to derive simple aalytical expressios for the average cotact size, average cotact pressure. Uder these coditios we may approximate ξ ξ A( ξψ, ) = si π πψ π πψ mea πξ π Φ ( ξψ, ) = si ( ξ / πψ ) whereupo Eq. (5) reduces to p π p π p σ σ σ This recursio relatio is satisfied by πξψ (58) mea mea mea ( D ) = πψ = πψγ (59) ( + ) mea mea p π D ( D ) p = ( γ ) πψγ (6) σ σπψ The average cotact size may be estimated usig Eq. (55) as ( + ) mea π D p a = λ ( γ ) (6) σπψ Similarly, the total area of cotact at the th scale follows as ( + ) mea ( D )( π + ) p A = a γ (6) σπψ The total umber of cotacts at the th scale ca be estimated as ( ) N = A γ + ( a ) (63) For large, these results may be approximated by their asymptotes p σ mea π = πψ γ ( D ) (64) ( D) < a >= λγ (65) mea π ( D ) p π P A = a γ = σπψ σπψ N = mea π D ( D ) p γ σπψ As expected, equatio (66) shows that the total cotact area at the th scale (for large ) is proportioal to the cotact load P ad idepedet of the omial cotact size a or the ideter geometry (other tha through its effect of settig the wavelegth of the zeroth roughess (66) (67)

21 scale). This result also provides a opportuity to check the predictios obtaied usig the costat pressure approximatio (Sect 4.) with the exact solutio. Ciavarella et al. () fid that A =.39 P/( σπψ ), givig a coefficiet that is withi 4% of the approximate value i Eq. (66). A more detailed compariso of the approximate ad exact calculatios i sectio 5 shows similar agreemet uder all coditios. Note also that Eq. (64) shows that the mea cotact pressure actig o each roughess scale becomes progressively more severe as icreases. Cosequetly, for a elastic-plastic solid, the short wavelegth roughess scales will always deform plastically. We also see that the formal limit for a perfectly elastic solid cosists of a ifiite umber of cotact spots, with zero size, which are subjected to ifiite cotact pressure. The total true cotact area vaishes. 4.4 Deformatio ad loadig for the most heavily loaded regio at each scale (elastic cotact) To calculate the coditios ecessary to iitiate yield at each successive scale of roughess, or to idetify coditios that will just begi to brig a roughess scale ito complete cotact with the ideter, it is of iterest to determie the cotact size ad cotact pressure for the most heavily loaded asperities at each successive scale of roughess. We assume that all roughess scales, as ( D ) well as the omial cotact, remai elastic; ad assume also that γ >. Equatio (3) shows that the maximum cotact pressure exerted by the omial cotact is p PE = (68) σ max * πrσ The most severely loaded asperities o the zeroth roughess scale are therefore subjected to a max omial cotact pressure p =. Equatio (4) gives the maximum cotact pressure for a p siusoidal surface that is subjected to a ormalized omial pressure ξ = p as p max ( ξψ, )/ σ = πξψ (69) whece the maximum cotact pressure o the zeroth roughess scale is p / σ = π p ψ / σ. Similarly, the maximum cotact pressure o successive scales of max max roughess follows as p max max / σ / σ = π p ψ / σ. The recursio relatio is satisfied by ( + ) max max p D ( D ) p = ( ) ( γ ) πψγ (7) σ σπψ Other tha a umerical factor, this result is idetical to eq (6), which estimates the average cotact pressure o each roughess scale. However, Eq. (7) does ot assume isolated Hertztype cotacts ad so is valid for all values of ψ, provided that o scale reaches full cotact.

22 4.5 A graphical approach to calculatig cotact size, pressure ad area for roughess scales i the fully plastic regime If the bulk cotact is fully plastic (which requires PE / σ R > 4 ), ad the surface roughess * is such that ψ > 6, the the procedure outlied i sectio 4. gives exact results, sice all roughess scales are i the fully plastic regime, so the cotact pressure distributio at each scale is uiform. I additio, uder these coditios, the method ca be illustrated usig a simple graphical procedure, origially suggested by Childs (973, 977) i a discussio of the persistece of asperities uder cotact loadig. We begi by simplifyig the behavior of the siusoidal surface for the fully plastic regime. Cosider a sigle scale of roughess with wavelegth λ, which is ideted by a rigid plate. The surface is subjected to a omial cotact pressure (force per wavelegth) p, which iduces a cotact size a. Each cotact spot is subjected to a uiform cotact pressure with magitude mea p. For values of ψ > 6, the relatioship betwee the mea ormalized cotact pressure ad cotact size for the fully plastic regimes give i Eqs. (35)-(39) ca be approximated by /3 mea mea H A< p / σ =Φ = 3 (7) H + ( H( ψ ) H)( 3A ) A> /3 where A= a/ λ, H.9 ad H ( ψ ) = 5.8. I additio, the ormalized omial cotact mea pressure follows as ξ = p/ σ = AΦ. Both these fuctios are plotted i Fig. 5. Cotact pressure p/σ p mea /σ p - mea /σ = p /σ p/σ p mea /σ = p /σ p mea /σ = p /σ Cotact fractio a /λ a /λ a /λ a/λ Fig. 5: A graphical techique for calculatig the cotact size ad cotact pressure for successive scales of a rigid-perfectly plastic surface with Weierstrass profile. The solid lies show the mea mea pressure p / σ ad omial pressure p / σ for a rigid plastic siusoidal surface as a fuctio of cotact fractio. Successive scales of roughess are related by the coditio that mea p / σ = p / σ.