asperity contact

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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 56 (2008) 2555– 2572

0022-50

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jmps

Asperity contact theories: Do they predict linearity between contactarea and load?

G. Carbone �, F. Bottiglione

DIMeG, Politecnico di Bari, v.le Japigia 182, 70126 Bari, Italy

a r t i c l e i n f o

Article history:

Received 28 November 2007

Received in revised form

11 March 2008

Accepted 20 March 2008

Keywords:

Contact mechanics

Surface roughness

Asperity contact

Area of contact

Tribology

96/$ - see front matter & 2008 Published b

016/j.jmps.2008.03.011

responding author. Tel.: +39 080 596 2746; f

ail address: [email protected] (G. Carbone).

a b s t r a c t

During the last few years, the scientific community has been debating about which theory

of contact between rough surfaces can be considered as the most accurate. The authors

have been attracted by such a discussion and in this paper try to give their personal

thought and contribution to this debate. We present a critical analysis of the principal

contact theories of rough surfaces. We focus on the multiasperity contact models (which

are all based on the original idea of Greenwood and Williamson (GW) [1966. Proc. R. Soc.

London A 295, 300]), and also briefly discuss a relatively recent contact theory developed

by Persson [2001. J. Chem. Phys. 115, 3840]. For small loads both asperity contact models

and Persson’s theory predict a linear relation between the area of true contact and the

applied external load, but the two theories differ for the constant of proportionality.

However, this is not the only difference between the two approaches. Indeed, we show

that the fully calculated predictions of asperity contact models very rapidly deviates from

the predicted linear relation already for very small and in many cases unrealistic

vanishing applied loads and contact areas. Moreover, this deviation becomes more and

more important as the PSD breadth parameter a (as defined by Nayak) increases.

Therefore, the asymptotic linear relation of multiasperity contact theories turns out to be

only an academic result. On the contrary, Persson’s theory is not affected by a and shows a

linear behavior between contact area and load up to 10–15% of the nominal contact area,

i.e. for physical reasonable loads. The authors also prove that, at high separation, all

multiasperity contact models, which take into account the influence of summit curvature

variation as a function of summit height, necessarily converge to a (slightly) improved

version of the GW model, which, therefore, remains one of the most important milestones

in the field of contact mechanics of rough surfaces.

& 2008 Published by Elsevier Ltd.

1. Introduction

Interaction upon contact between two solids plays a major role in a large number of physical phenomena andengineering applications, e.g. structural adhesives, protective coating, friction of tires, lubrication, wear and seals, but stillis not completely understood. The well-known Amontons–Coulomb’s friction law, which states that the friction force isproportional to the load, is commonly believed to be a consequence of a direct proportionality between the applied loadand the area of real contact. The explanation, given by Bowden and Tabor (1939), was straightforward, they assume that theasperities of the rough solid surfaces undergo a plastic deformation as soon as they are in contact, and, since the yield stress

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G. Carbone, F. Bottiglione / J. Mech. Phys. Solids 56 (2008) 2555–25722556

is taken to be constant, they end up with a direct proportionality between the load and the area of intimate contact.However, the argument of Bowden and Tabor cannot be applied in many engineering situations. As an example, wearexperiments suggest that contact of asperities is formed under elastic rather than plastic deformations: when machine partsslide against each other for perhaps millions of cycles, the protuberances may be plastically deformed at the beginning, but,after, they reach a steady-state condition in which the load is supported elastically, thus avoiding serious damage to themachine. In these cases plastic deformations are very limited in amount, and elastic deformations play a major role (Archardand Hirst, 1956). Even more, in the case of rubber materials (e.g. tires and seals), plastic deformations often do not occur andthe area of intimate contact is therefore determined by the viscoelastic properties of the material (Persson et al., 2004; Huiet al., 2000), the surface energies of the contacting bodies and their roughness (Persson, 2002a,b; Carbone et al., 2004). In allsuch situations the simple explanation given by Bowden and Tabor does not work. This shows that the relationship betweenthe area of real contact, the external applied load and the surface morphology is still an open question, and it is notsurprising that a great deal of theoretical work has been devoted to the comprehension of this very complicatedphenomenon. However, the interest for this subject is not limited to the speculative level. Very recently this topic has gaineda renewed interest as a consequence of the strong effort to produce increasingly smaller mechanical and electrical devicesdown to the micro- and nano-scales (Bushan, 2003). As an example, in the case of microelectromechanical systems (MEMS),large contact areas often causes the stiction of the moving components, and, therefore, the failure of the system (Zhao et al.,2003). Electrostatic micromotors, for instance, might need complicated and costly antistiction layers to prevent permanentadhesion and reduce drag forces (Sundararajan and Bhushan, 2001). In view of the above considerations it is clear thatnumerous factors may contribute to determine the extension of the real contact area between two bodies in contact. But,among these, perhaps, the most important is the interfacial roughness. In fact the direct proportionality between the loadand the area of intimate contact seems to be a very robust property, independent of the rheological properties of thematerials (elastic, elastoplastic, plastic, or viscoelastic), which instead affect only the constant of proportionality. Thus, oneconcludes that this direct proportionality should be a consequence of the statistical properties of the surfaces roughness.Numerical studies (Hyun et al., 2004; Borri-Brunetto et al., 2001; Campana and Muser, 2007; Yang et al., 2006) have shownthat each time an elastic body is brought in contact with a rough surface, the number of contact areas increaseproportionally to the load, whereas the mean contact area in each contact region remains almost constant. This propertyseems not to depend on the particular model used to describe the problem, so that different theories may give similarresults. However, the scientific community is still debating about which theory gives the more accurate results. Indeed, thereare some numerical investigations (Hyun et al., 2004; Borri-Brunetto et al., 2001; Campana and Muser, 2007; Yang et al.,2006) aiming to understand what is the most accurate between the two approaches available in literature: (i) themultiasperity contact models, all based on the original idea by Greenwood and Williamson (GW) (1966), and (ii) thetheoretical approach by Persson (2001). However, till now no definitive answer to the question has been found.

In this paper the authors discuss these two main approaches and show novel aspects, which to the best of theirknowledge have never been discussed and noticed before. The multiple asperity contact approach is a generalization of theHertz contact theory to take account of the surface statistics. The very first idea of multiasperity contacts was proposed byArchard (1957), however, a profound refinement of this idea was due to GW (1966), who modeled the roughness of thesurface as an ensemble of identical spherical asperities, the summits, with randomly distributed heights. The ultimatedevelopment of this idea was due to Bush, Gibson and Thomas (BGT) (1975), who, moving from Longuet-Higgins(1957)–Nayak (1971) statistical theory of isotropic randomly rough surfaces, modeled the asperities as paraboloids withtwo different radius of curvature. The statistics of asperity height, of asperity curvatures, etc. were completely taken intoaccount. Of course, multiasperity contact theories break down as the contact moves toward full contact conditions, i.e.these theories are believed to hold true only for small loads and contact areas. The second approach was proposed byPersson (2001). This approach gives instead exact solution for full contact conditions. The theory, indeed, assumes that thepower spectral density (PSD) FðkÞ of the deformed surface of the elastic half-space is well approximated by that of theunderlying rigid rough substrate, which the elastic body is in contact with. This assumption of course is correct in fullcontact conditions but less accurate as we move towards small contact area and small loads (this is, indeed, in our opinionthe only very heavy approximation introduced by Persson). Amazingly, both asperity based models and Persson theorypredict, in the limit of small load, a linear relation between the contact area and the applied load Ac ¼ LðE0

ffiffiffiffiffiffiffiffiffiffi2m2

pÞ�1F,

where m2 is the mean square slope of the surface heights, the effective elastic modulus is E0 ¼ E=ð1� n2Þ and the numericalfactor L ¼ LBGT ¼

ffiffiffiffiffiffi2pp

for multiasperity contact theories (actually we show in the paper that the factorffiffiffiffiffiffi2pp

does not holdonly for the BGT theory but also for all those asperity contact theories which account in some way for the statisticaldistribution of summit curvatures) and L ¼ LP ¼

ffiffiffiffiffiffiffiffi8=p

pfor the Persson’s theory. However, we show that the linear relation

predicted by multiasperity contact theories restricts its validity to a very small and in many cases unrealistic vanishingrange of applied loads and contact areas. This instead does not occur for the Persson’s theory where the linear predictionholds true up to contact areas of about 10–15% of the nominal contact area as predicted by several numerical calculations(Hyun et al., 2004; Borri-Brunetto et al., 2001; Campana and Muser, 2007; Yang et al., 2006). Moreover, as the load isincreased, all multiasperity models predict a contact area vs. load curve which rapidly bends below the linearapproximation. Thus, it follows that in almost all practical situations multiasperity contact theories actually do not predicta linear relation between the area of true contact and the load as, instead, commonly believed. In addition it is alsofound that, in many cases, these models give an area vs. load curve which rapidly falls below that predicted by thePersson’s theory.

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G. Carbone, F. Bottiglione / J. Mech. Phys. Solids 56 (2008) 2555–2572 2557

We also prove that those asperity contact theories which take into account the effect of asperity curvature change as afunction of summit height, have all the same asymptotic behavior at large separations. The GW model, not taking intoaccount the variation of curvature with the summit height, cannot indeed predict a linear relation between the contact areaand the load. However, the asymptotic analysis we carry out in Appendix A shows that a very simple slightly improvedversion of the GW model exists which corrects for this, thus suggesting that the GW model should be preferred to the othermore complicated multiasperity contact models, because, despite its simplicity, gives very similar results.

2. Multiasperity contact models

Within the framework of multiasperity contact models, the determination of area of contact and load as a function ofthe distance between the two approaching bodies moves from the knowledge of the joint probability distribution per unitarea Pðx1; k1; k2Þ of summits heights x1, and summits curvature k1 and k2 (where k1o0 and k2o0 to account for stationarypoints with negative curvature, i.e. the summits). Recalling the Longuet-Higgins (1957) and Nayak (1971) analysis ofsurface statistics it is possible to show that for a isotropic surface the joint probability distribution Pðx1; k1; k2Þ is given by

Pðx1; k1; k2Þ ¼

ffiffiffiffiffiffi27p

ð4pÞ21

m2m4ffiffiffiffiffiffiffiffiffiffiffiffiffim0m4p C1=2

1 exp �C1x1

m1=20

þ3

2ffiffiffiap

k1 þ k2

2m1=24

!224

35jk1 � k2j

�k1k2 exp �3

16m4½3ðk1 þ k2Þ

2� 8k1k2�

� �(1)

where the quantities m0, m2, and m4 simply represent the moments of the PSD FyðsÞ of a profile obtained by theintersection of the random rigid surface with a plane perpendicular to the mean plane of the rough surface itself, in thedirection y. Thus, we have

mp ¼

Z þ1�1

duFyðuÞup (2)

Because of isotropy the PSD FyðuÞ does not depend on the orientation of the plane intersecting the surface, therefore onecan, for instance, choose the plane ky ¼ 0 and write, FyðkxÞ ¼ Fy¼0ðkxÞ ¼

RdkyFðkx; kyÞ, with Fðkx; kyÞ ¼ FðkÞ being the

surface PSD. In Eq. (1) the breadth parameter (as defined by Nayak) is a ¼ m0m4=m22 and C1 ¼ a=ð2a� 3Þ.

In order to determine the area of contact and load between the rough rigid surface and a initially flat elastic half-space,multiasperity contact theories make use of the Hertz’s theory to calculate, for a given penetration d ¼ x1 � d (where d is thedistance of the approaching rigid plane from the mean plane of the rough surface and x1 is the height of summit), thecontact area and load upon contact between an elastic half-space and a single rigid asperity (Fig. 1 shows a schematic of anHertzian contact between a rigid sphere and an elastic initially flat plane). Hertz’s theory states that, besides the elasticproperties of the contacting bodies, the contact area and load depends, for a non-conformal contact, only on thepenetration d and the principal radii of curvature of the contacting asperities. In particular it is shown that the contact areais an ellipse, whose semi-axes a1 and a2 depend on, d, k1, and k2 by means of implicit relations. For the sake of simplicity itsuffices to consider a1ðd; k1; k2Þ, a2ðd; k1; k2Þ, smðd;k1; k2Þ where sm is the mean pressure acting on the contact area. Thus,one can calculate the fraction of area in contact and the mean pressure in the nominal contact area F=A0 as

Ac

A0¼

Z þ1d

Z 0

�1

Z 0

�1

dx1 dk1 dk2pa1a2Pðx1; k1;k2Þ (3)

Fig. 1. A rigid sphere in contact with an elastic half-space. The sphere is displaced of a constant quantity d against the substrate.

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F

A0¼

Z þ1d

Z 0

�1

Z 0

�1

dx1 dk1 dk2pa1a2smPðx1; k1; k2Þ (4)

Eqs. (3) and (4) constitute the basis of the BGT theory, although in BGT (1975) the derivation of the area of contact andmean pressure was indeed slightly different, and, of course, involves full calculation of elliptic integrals. Within theframework of multiasperity contact theory the BGT model is commonly considered as the most accurate. However, as wewill show later it does not differ significantly from other simpler multiasperity contact models, which can be easilyobtained by Eqs. (3) and (4) through approximate relations involving the curvature of summits. In what follows we brieflydescribe the main asperity contact models, starting from the one by GW (1966) as improved by Mc Cool (1986), where thesummits are supposed to be spheres with the same radius of curvature. We will refer to this model as the GW–Mc Coolmodel. We also briefly describe the BGT model and also a simplified version, given by Greenwood (2006), where thesummits are spheres but with a distribution of curvatures, each curvature being kG approximated by the square root of theGaussian curvature of the summit, i.e. kG ¼

ffiffiffiffiffiffiffiffiffiffik1k2p

. A different model is also presented where the curvature of spheres isinstead given by the arithmetic mean curvature of summits, i.e. kA ¼ ðk1 þ k2Þ=2. This last model has been partiallypresented by Thomas (1982), and referred to as the Nayak model (although Nayak has never published and or presentedsuch a model). However, in Thomas (1982), only very partial results are shown, and figures also are very incomplete: thereader has no clear indication of what kind of diagram is reported there (log–log, linear–log or linear–linear). We prefer torefer to this model as to the Nayak–Thomas (NT) model.

In this paper we report full calculations of NT model and show that its main results negligibly differ from theGreenwood 2006 model and the BGT one. Moreover, in Section 2.2, full calculations and results from the BGT model are alsogiven. We show that the original load vs. separation calculations of the BGT original paper (see BGT, 1975, Table 1) areactually wrong, as already inferred by Greenwood (2006). We also stress that the three theories, i.e. Greenwood 2006, NTand BGT, have the same asymptotic linear load–area relation, but also that this asymptotic behavior is obtained only forunrealistic very small loads and contact areas.

Besides asperity models, in Section 3 we also discuss the Persson’s theory, which instead predicts a linear relationbetween the load and contact area up to about 10–15% of the nominal contact area.

2.1. The GW– Mc Cool model

GW theory deals with the contact between a deformable flat plane and a rigid randomly rough indenter. They assumedthat roughness of the rigid indenter can be regarded as a spatial distribution of identical spherical summits with radius ofcurvature R and randomly distributed heights. Let d be the separation between the flat deformable plane and the summitmean plane, and x1 the summit height. It is assumed that the Hertz contact theory can be utilized to determine the area ofcontact and the load on each single asperity higher than d, as a function of the penetration d ¼ x1 � d. In case of sphericalindenter the Hertzian theory allows us to calculate the radius a of the circular contact area as

a ¼ ðRdÞ1=2 (5)

and the mean pressure in the contact area as

sm ¼4

3pE0

dR

� �1=2

(6)

where E0 ¼ E=ð1� n2Þ, R�1¼ �k1 ¼ �k2 and a1 ¼ a2 ¼ a. Fig. 2 shows a schematic of the GW model of rough surface. In this

case the probability distribution Pðx1; k1; k2Þ simply becomes

PGWðx1; k1; k2Þ ¼ Psumðx1Þdðk1 þ R�1Þdðk2 þ R�1

Þ (7)

mean plane

rough rigid body

elastic body

R

h d

Fig. 2. The Greenwood and Williamson model of contact of rough surfaces. The roughness of the rigid surface is modeled as an ensemble of spherical

asperities of equal radius R. The elastic substrate is displaced against the rigid rough surface up to a distance d, referred to as separation, far from the mean

plane of the rough surface.

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where dð�Þ is the Dirac delta function and Psumðx1Þ is the probability distribution per unit area of summits height (seeAppendix A). Using Eqs. (7), (3) and (4) one obtains

Ac

A0¼

Z þ1d

dx1pa2Psumðx1Þ (8)

F

A0¼

Z þ1d

dx1pa2s0Psumðx1Þ (9)

GW made also the additional assumption to consider that the summits heights were distributed according to a Gaussianprobability distribution with a standard deviation ss with respect to the summit mean plane. Note that this is not a verystrict assumption since many real surfaces present very large values of the parameter a (even greater than 100), which, inturn, implies (as shown by Nayak, 1971) that the summits distribution is indeed a normal distribution. Given theseassumptions, the estimated area of real contact and the mean pressure on the nominal contact area become

Ac

A0¼ pssRDsum

Z 1d�

dxðx� d�ÞfðxÞ (10)

F

A0¼

4

3E0R1=2s3=2

s Dsum

Z 1d�

dxðx� d�Þ3=2fðxÞ (11)

with x ¼ x1=ss, d� ¼ d=ss, and fðxÞ ¼ sspsumðssxÞ ¼ ð2pÞ�1=2 expð�x2=2Þ. In order to carry out numerical calculations weneed to estimate the values of R, ss and the number of summits per unit area Dsum, which all depend on the real surfacetopography. Here we follow (Mc Cool, 1986) who, referring to Nayak (1971) and BGT (1976) gives

1

R¼

8

3

m4

p

� �1=2

s2s ¼ 1�

0:8968

a

� �m0 (12)

Observe that for a!1, s2s ! m0, i.e. the mean square deviation of the summits height equals that of surface heights.

Using the given expressions, the factor ssRDsum in Eq. (10) becomes

ssRDsum ¼1

48

3

pða� 0:8968Þ

1=2

(13)

which shows that the estimated extension of the contact area as a function of d� ¼ d=ss also depend on the parameter a.

2.2. The BGT model

BGT (1975) developed the most complete theory of contact mechanics within the framework of multiasperity contactmodels. They made use of Eqs. (3) and (4) but in a different form. Instead of focusing on the radii of curvature of theasperities (which in agreement with the Hertz theory were treated as a paraboloidal asperities) they developedcalculations by referring to the ellipse of contact. In this case Eqs. (3) and (4) become

Ac

A0¼

Z þ1d

Z þ10

Z þ10

dx1 da1 da2pa1a2PBGTðx1; a1; a2Þ (14)

F

A0¼

Z þ1d

Z þ10

Z þ10

dx1 da1 da2pa1a2smPBGTðx1; a1; a2Þ (15)

where PBGTðx1; a1; a2Þ ¼ jqðk1; k2Þ=qða1; a2ÞjPðx1; k1; k2Þ, and where 0oa1oþ1 and 0oa2oþ1 to account only forstationary points with negative curvatures (the summits). Since most of the relations can only be given in implicit form, thecalculation of the above written integrals Eqs. (14) and (15) required a change of variables and a lot of long calculations. Werefer the reader to the original paper by BGT (1975) for a detailed description of the model. Here we only want to presentsome results which differ from those reported in BGT (1975). Indeed, by following the same path outlined by Bush, Gibsonand Thomas, we have recalculated the integrals Eqs. (14) and (15), and checked the BGT results. Our analysis has shownthat the results reported in BGT (1975) contain some errors. In particular in BGT (1975, Table 2), which shows the calculatedvalues of contact area and load, reports wrong load values, thus affecting the BGT representation of the full calculatedcontact area vs. load curve. The full calculated contact area vs. load relation should, indeed, approach the BGT asymptoticlinear relation

Ac ¼1

E0p

m2

� �1=2

F (16)

so that the quantity F=ðAcOÞ, where O ¼ E0ðm2=pÞ1=2, must approach the unit value as the separation d between the rigid

rough surface and the elastic body increases. But we do not observe this trend in the original values reported in BGT (1975).Indeed by looking at Table 1, for a ¼ 2, and Table 2 for a ¼ 10 it is clear that our calculations correctly give a constant trendfor the quantity F=ðAcOÞwhich constantly decreases towards unity as the dimensionless separation t ¼ d=m1=2

0 is increased.

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Table 2Comparison between the original BGT (1975) results and our BGT calculations

t Recalculated BGT Original BGT

Ac

A0

F

A0OF

AcOA

A0

Fc

A0OFc

AO

a ¼ 10

0.0 0.2040 0.3680 1.804 0.2040 0.2936 1.44

0.5 0.1174 0.1950 1.661 0.1174 0.1779 1.52

1.0 0.05809 0.08940 1.539 0.05809 0.09981 1.72

1.5 0.02415 0.03471 1.437 0.02415 0.04723 1.96

2.0 0.008269 0.01120 1.354 0.008269 0.01770 2.14

2.5 0.002297 0.002956 1.287 0.002297 0.005104 2.22

3.0 0.0005120 0.0006308 1.232 0.0005119 0.001121 2.19

The numerical value of the contact area Ac=A0, dimensionless load F=ðA0OÞ, and dimensionless mean pressure in the contact area F=ðAcOÞ are reported for

a ¼ 10. The quantity O ¼ ðm2=pÞ1=2E0.

Table 1Comparison between the original BGT (1975) results and our BGT calculations

t Recalculated BGT Original BGT

Ac

A0

F

A0OF

AcOAc

A0

F

A0OFc

AcO

a ¼ 2

0.0 0.1644 0.2270 1.380 0.1645 0.1811 1.10

0.5 0.1080 0.1368 1.266 0.1080 0.1190 1.10

1.0 0.06039 0.07102 1.176 0.06039 0.06541 1.08

1.5 0.02750 0.0360 1.113 0.02750 0.02942 1.07

2.0 0.009940 0.01066 1.073 0.009940 0.01054 1.06

2.5 0.002825 0.002962 1.049 0.002825 0.002957 1.05

3.0 0.0006301 0.0006519 1.035 0.0006300 0.0006534 1.04

The numerical value of the contact area Ac=A0, dimensionless load F=ðA0OÞ, and dimensionless mean pressure in the contact area F=ðAcOÞ are reported for

a ¼ 2. The quantity O ¼ ðm2=pÞ1=2E0.


However, the original data, especially for a ¼ 10 (see Table 2), do not have this trend thus showing that the original BGTload calculations must contain some errors.

It is also very interesting to note that the correct values of F=ðAcOÞ as reported in Tables 1 and 2, although given up toseparation d ¼ 3m1=2

0 , are still far from the unit value, they take in the asymptotic limit. This, in turn, suggests that theasymptotic linear relation of the BGT theory can be obtained only for almost unrealistic small loads and contact areas. Thisresults will indeed be confirmed in Section 4. Besides the wrong values reported in BGT (1975, Table 2), we also found somemisprint in the final two equations of the BGT paper at page 110. Here we report the correct relation between the load F andthe dimensionless separation t

FðtÞ

A0¼

8ffiffiffi3p

a5=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipð2a� 3Þ

p OZ þ1

t

Z þ10

Z p=4

0dz dw dYPðy;YÞðz� tÞ3=2w7=2 exp

�az2 þ 3a1=2zw� Cw2

2a� 3

!dydY

(17)

where O has been already defined before as O ¼ E0ðm2=pÞ1=2, and the correct expression for the quantity Pðy;YÞ is

Pðy;YÞ ¼sin3 y cos2 y cos 2y

ðtan2 yþ tan2 YÞ1=2Kðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� tan2 Y

pÞ

(18)

The reader is referred to BGT (1975) for the meaning of the several symbols reported in Eqs. (17) and (18).

2.3. The Greenwood 2006 model

In a recent paper, Greenwood (2006) proposed a simplification of the BGT model based on the observation that manyasperities are only mildly ellipsoidal and that, recalling results shown in Greenwood (1985), a good approximation toHertzian elliptical contacts is obtained by considering an equivalent spherical contact with the curvature given bygeometric mean curvature of the surface, i.e. by the square root of the Gaussian curvature: R�1

¼ kG ¼ ðk1k2Þ1=2. To

compute the area of contact and the load we still can use Eqs. (3) and (4) but we need to change the integration variables.

ARTICLE IN PRESS


Thus, let us consider the new variables

kG ¼ ðk1k2Þ1=2

w ¼k1 � k2

2(19)

we can invert the above written equations to calculate k1 and k2 as a function of kG and w. Thus we get

k1 ¼ w� ðk2G þ w2Þ

1=2

k2 ¼ �w� ðk2G þ w2Þ

1=2 (20)

now observing that

qðk1;k2Þ

qðkG; wÞ

�� ¼ 2kG

ðk2G þ w2Þ

1=2(21)

we can rewrite Eqs. (3) and (4) as

Ac

A0¼

Z þ1d

Z þ10

dx1 dkGpa2PGðx1;kGÞ (22)

F

A0¼

Z þ1d

Z þ10

dx1 dkGpa2smPGðx1; kGÞ (23)

where

PGðx1;kGÞ ¼

Z þ1�1

dw2kG

ðk2G þ w2Þ

1=2Pðx1; k1; k2Þ (24)

and the contact radius a, and the pressure p0 are calculated on the basis of Hertzian formulas Eqs. (5)–(6). By defining thefollowing quantities:

t ¼ d=m1=20 ; x ¼ x1=m1=2

0 ; g ¼ kG=m1=24

and carrying out calculations then one obtains

AcðtÞ

A0¼

3a

4½6pða� 1Þ�1=2

Z 1t

Z 10

dxdg erfc m 3g �xa1=2

a� 1

� � exp

1

23g2 �

ax2

a� 1

!" #g2ðx� tÞ (25)

FðtÞ

A0¼ O

a5=4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ða� 1Þ

p Z 1t

Z 10

dxdg erfc m 3g �xa1=2

a� 1

� � exp

1

23g2 �

ax2

a� 1

!" #g5=2ðx� tÞ3=2 (26)

where m ¼ ½ða� 1Þ=ð4a� 6Þ�1=2.

2.4. Nayak– Thomas (NT) model

In order to complete the panoramic view on asperity contact models, in this section we also report a simpler modelwhich considers the summits as rigid spherical asperities with a curvature equal to the arithmetic mean curvature, i.e.

R�1¼ kA ¼

k1 þ k2

2(27)

The approach differs from that of Greenwood (2006), in that here the arithmetic mean curvature is considered in place ofthe geometric mean curvature. This model was reported for the first time in chapter 8 of a book edited by Thomas (1982).Bush, who was the author of the chapter, referred to this model as Nayak’s model, although Nayak has never presented orpublished such a model. However, in that chapter only very incomplete results were reported, which we have furtherdeveloped to show final results of contact area and load. In this case we can again move from Eqs. (3) and (4) and use thenew quantities

kA ¼k1 þ k2

2

w ¼k1 � k2

2(28)

from which it follows jqðk1; k2Þ=qðkA; wÞj ¼ 2. Using the new variables the integrals reported in Eqs. (3) and (4) become

Ac

A0¼

Z þ1d

Z 0

�1

dx1 dkApa2PAðx1; kAÞ (29)

ARTICLE IN PRESS


F

A0¼

Z þ1d

Z 0

�1

dx1 dkApa2smPAðx1; kAÞ (30)

where

PAðx1;kAÞ ¼ 2

Z �kA

kA

dwPðx1; kA þ w; kA � wÞ (31)

and kAowo� kA in order to guarantee that k1 and k2 are always negative. Now let us define k�A ¼ kA=m1=24 , t ¼ d=m1=2

0 ,

z ¼ x1=m1=20 , C1 ¼ a=ð2a� 3Þ, and C2 ¼ C1ð12=aÞ1=2, we get

AcðtÞ

A0¼ �

1

2pa

16C3=21

Z 0

�1

dk�A3k�2A þ 2 expð�3k�2A =2Þ � 2ffiffiffi

3p

k�Aexp �

2C1t þffiffiffi3p

C2k�A2

t �3

2C1k�2A

" #

� 4C1=21 �

ffiffiffipp

expð4C1t þ

ffiffiffi3p

C2k�AÞ2

16C1

" #ð4C1t þ

ffiffiffi3p

C2k�AÞerfc4C1t þ

ffiffiffi3p

C2k�A4C1=2

1

!( )(32)

FðtÞ

A0¼

4ffiffiffi3p

3

E

1� n2

ðm2C1Þ1=2

ð2pÞ2a3

� �3=4Z 0

�1

Z 1t

dk�A dzðz� tÞ3=2

ð�ffiffiffi3p

k�AÞ1=2

�½3k�2A þ 2 expð�3k�2A =2Þ � 2� exp½�C1z2 � ð3C1k�2A þffiffiffi3p

C2k�AzÞ=2� (33)

Table 3 shows numerical values of the area of contact and load as resulted from the last three theories BGT, Greenwood2006, and NT. Two values of a ¼ 2 and 10 have been considered. It is clear that the results of the three models becomecloser and closer as the separation is increased, this suggests the theories have the same asymptotic behavior for very smallloads, as indeed we show in Appendix A.

3. Persson’s theory of contact mechanics

Persson’s theory removes the assumption, which is implicit in the multiasperity contact theories, that the area of realcontact is small compared to the nominal contact area. On the contrary, the Persson’s approach moves from the limitingcase of full contact conditions (where it gives the exact solution) between a rigid rough surface and an initially flat elastichalf-space, and accounts for partial contact by requiring that, in case of adhesionless contacts (as those discussed in thispaper) the stress probability distribution vanishes when the local normal surface stress s vanishes. The theory assumes thePSD FðkÞ of the deformed (initially flat) elastic surface being the same as that of the underlying rough surface. Thisassumption of course is correct in full contact conditions but less and less accurate as we move towards small contact areasand small loads. For small squeezing force it also predicts the area of real contact being proportional to the applied load,whereas as the squeezing force is increased it predicts the area of real contact Ac to approach continuously to the nominal

Table 3Contact area, load and contact pressures calculated by means of BGT, Greenwood 2006 and NT models, for two different values of a

t BGT Greenwood 2006 NT

Ac

A0

F

A0OF

AcOAc

A0

F

A0OF

AcOAc

A0

F

A0OF

AcO

a ¼ 2

0.0 0.1644 0.2270 1.380 0.1674 0.2218 1.325 0.1449 0.2089 1.442

0.5 0.1080 0.1368 1.266 0.1098 0.1339 1.219 0.09593 0.1266 1.320

1.0 0.06039 0.07102 1.176 0.06129 0.06967 1.137 0.05425 0.06624 1.221

1.5 0.02750 0.0360 1.113 0.02785 0.03010 1.081 0.02507 0.02881 1.149

2.0 0.009940 0.01066 1.073 0.01004 0.01052 1.048 0.009210 0.01014 1.101

2.5 0.002825 0.002962 1.049 0.002848 0.002931 1.029 0.002657 0.002844 1.070

3.0 0.0006301 0.0006519 1.035 0.0006341 0.0006463 1.019 0.0006003 0.0006307 1.051

a ¼ 10

0.0 0.2040 0.3680 1.804 0.2089 0.3588 1.726 0.1785 0.3361 1.883

0.5 0.1174 0.1950 1.661 0.1195 0.1902 1.591 0.1030 0.1785 1.732

1.0 0.05809 0.08940 1.539 0.05912 0.08730 1.477 0.05121 0.08209 1.603

1.5 0.02415 0.03471 1.437 0.02456 0.03393 1.382 0.02139 0.03198 1.495

2.0 0.008269 0.01120 1.354 0.008403 0.01096 1.304 0.007367 0.01036 1.406

2.5 0.002297 0.002956 1.287 0.002333 0.002896 1.241 0.002069 0.002745 1.333

3.0 0.0005120 0.0006308 1.232 0.0005194 0.0006188 1.191 0.0004617 0.0005883 1.274

The quantity O ¼ ðm2=pÞ1=2E0.

ARTICLE IN PRESS


contact area A0 (asperity contact models are not able to predict such a behavior). The theory yields very simple formulasand needs as inputs only the surface PSD FðkÞ and the elastic properties of the contacting bodies.

Now let z ¼ k=kL ¼ L=l be the magnification at which we are observing the surface, where L is the lateral size of thesample and l is the wavelength we are focusing on, k ¼ 2p=l and kL ¼ 2p=L. Also let us use the symbol s to denote thenormal pressure at the interface, and pðs; zÞ the pressure probability distribution in the contact area at the magnification z.Theory of elasticity, as Persson shows, implies that, for a Gaussian surface, under full contact conditions, the stressprobability distribution at the interface must satisfy a diffusion type equation (Persson, 2001)

qpðs; zÞqz

¼ f ðzÞq2pðs; zÞ

qs2(34)

with the initial and boundary conditions

pðs;1Þ ¼ dðs� F=A0Þ (35)

pð�1; zÞ ¼ 0 (36)

pð1; zÞ ¼ 0 (37)

where f ðzÞ ¼ G0ðzÞðF=A0Þ2 is a function of the magnification, and GðzÞ is

GðzÞ ¼p4

A0

F

� �2

E02Z zkL

kL

dkk3FðkÞ (38)

from which it follows

f ðzÞ ¼p4

E02kLk3FðkÞ (39)

It can be easily shown that, if the surface roughness is isotropic (as in our case), the quantity pR zkL

kLdkk3FðkÞ in Eq. (38) is

just the second profile PSD moment m2, i.e., pR zkL

kLdkk3FðkÞ ¼ m2ðzÞ.

Eq. (34) is supposed to hold true also in partial contact conditions with f ðzÞ still given by Eq. (39). However, in order toaccount for partial contact, the boundary condition Eq. (36) is replaced by

pð0; zÞ ¼ 0 (40)

If AcðzÞ denotes the apparent area of contact (projected on xy-plane) when the system is observed at the magnificationz ¼ L=l ¼ k=kL, then one can easily show that

AðzÞA0¼

Z 10

pðs; zÞds (41)

Solving Eq. (34) with the conditions Eqs. (35), (37) and (40), one obtains

Ac

A0¼ erf

1

m1=22 E0

F

A0

!(42)

Eq. (42) gives the contact area as a function of the magnification z and applied load F. The true area of contact Ac can becalculated by choosing the maximum magnification zmax. It is very important to stress that Eq. (42) does not contain anydependence on the a parameter, i.e. the contact area–load relation as calculated by Persson’s theory depends only onm2 ¼ hrh2

i=2, this is indeed the only geometrical quantity related to surface roughness appearing in the final formulas ofthe theory. From this point of view, we believe this constitutes a fundamental qualitative difference in comparison toasperity contact theories which also involve the geometrical parameter a.

Eq. (42) for small loads gives a linear relation between the area of real contact Ac and the squeezing force F, it suffices toexpand the erf function to the first order to get

Ac ¼2

p1

E0p

m2

� �1=2

F (43)

These results have to be compared with the asymptotic result of the BGT theory [see Eq. (16)]. Indeed, the functionaldependence of the contact area from the applied load F is exactly the same except for a constant numerical factor 2=p,showing that, at least in the linear asymptotic approximation the Persson’s theory predicts a smaller contact area. However,as we shall see in Section 4 this is not actually true since the fully calculated curves of the BGT theory or analogue modelsbased on the asperity contact idea, rapidly move far below the BGT asymptotic linear approximation, already for very small,unrealistically small contact areas and loads. This trend is so strong that fully calculated predictions of asperity contactmodels give contact areas even smaller that those predicted by the Persson’s theory.

4. Discussion

In this section we discuss and compare the main results of the multiasperity contact models, and also discuss thesemodels in respect to the Persson’s theoretical predictions. Figs. 3(a)–(f) show the contact area and load as a function of the

ARTICLE IN PRESS

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

t

Ac

/ A0

Ac

/ A0

Ac

/ A0

BGT

Greenwood 2006

NT

GW - McCool

� = 2

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

t

F /

(A0Ω

)F

/ (A

0Ω)

F /

(A0Ω

)

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

t0 0.5 1 1.5 2 2.5 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

t

BGT

Greenwood 2006

NT

GW - McCool

BGT

Greenwood 2006

NT

GW - McCool

BGT

Greenwood 2006

NT

GW - McCool

BGT

Greenwood 2006

NT

GW - McCool

BGT

Greenwood 2006

NT

GW - McCool

� = 2

� = 10� = 10

� = 100� = 100

Fig. 3. Contact area Ac=A0 and load F=ðA0OÞ as calculated by means of multiasperity contact models as a function of the non-dimensional separation

t ¼ d=m1=20 , for three different values of a, (a) and (b) a ¼ 2; (c) and (d) a ¼ 10; (e) and (f) a ¼ 100. The quantity O ¼ ðm2=pÞ

1=2E0.


ARTICLE IN PRESS


dimensionless separation t ¼ d=m1=20 as predicted by the four different asperity based contact theories above described.

Calculation has been carried out for three different values of the parameter a ¼ 2;10;100. The first two values are thosealready considered in BGT (1975), whereas the value a ¼ 100 has been considered to take into account that many realsurfaces have roughness on many length scales spanning even more than 6 order of magnitudes, thus implying very highvalues of a. The first striking property that is observed by looking at the figures is that the three curves (BGT, Greenwood2006 and NT) run almost parallel to each other and follow the same trend. In particular the pioneering model by GW (1966)as improved by Mc Cool (1986) gives results which are very similar to those of the other refined models. This is surprising ifone consider the very simple assumptions which the GW–Mc Cool theory relies on, and also suggests that, at least becauseof its simplicity, the GW–Mc Cool model should be preferred among the other multiasperity contact theories. Moreinteresting are the diagrams of the contact area fraction Ac=A0 as a function of the dimensionless load F=ðA0OÞ, whereO ¼ ðm2=pÞ

1=2E0. Figs. 4(a)–(c) show, indeed, that the BGT theory always lies in between the Greenwood 2006 curve and NTmodel predictions. Thus, although, especially for a ¼ 2 [see Fig. 4(a)], the Greenwood 2006 curve seems to run a bit closerto the BGT curve if compared to the NT predictions, this is less and less evident at higher values of a ¼ 10 and 100 [seeFigs. 4(b) and (c)]. Thus, given also the uncertainty of BGT theory accuracy, it appears impossible to say which of the models(NT or Greenwood 2006) gives the better approximation. Also, Fig. 4 shows that the GW–Mc Cool model, despite itssimplicity, gives an estimation of contact area as a function of applied load which is still relatively close to the BGTpredictions for all values of the parameter a. Thus, the complications introduced by the BGT and other theories are actuallynot fully justified if compared to the GW–Mc Cool model. Of course, one can argue that the GW–Mc Cool model does nothave the same asymptotic (linear) behavior of the other multiasperity contact theories (see below in the text), but, as weshow in Appendix A, a ‘minimal’ improvement of the model does exist to correct for this. For now, observe that none of the

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

F / (A0Ω) F / (A0Ω)

F / (A0Ω)

Ac

/ A0

Ac

/ A0

Ac

/ A0

BGT

Greenwood 2006

NT

GW - McCool

BGT - Asymptotic

� = 2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

� = 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

� = 100

BGT

Greenwood 2006

NT

GW - McCool

BGT - Asymptotic

BGT

Greenwood 2006

NT

GW - McCool

BGT - Asymptotic

Fig. 4. Contact area Ac=A0, obtained by multiasperity contact theories, as a function of the non-dimensional applied load F=ðA0OÞ, for (a), a ¼ 2;

(b), a ¼ 10; (c), a ¼ 100. The solid black line is the asymptotic prediction of the BGT (1975) theory. The quantity O ¼ ðm2=pÞ1=2E0.

ARTICLE IN PRESS


asperity contact theories, actually, predict a linear dependence of the contact area Ac on the applied load F, as insteadcommonly believed. Indeed, the asymptotic linear prediction of the BGT theory restricts its validity to continuouslydecreasing values of the contact area as the parameter a is increased. Already for a ¼ 2 the linearity is lost for contact areaAc larger than 0:01% (see below in the text) of the nominal contact area A0, whereas for a ¼ 100, all curves very significantlydeviates from the asymptotic linear relation Eq. (16) already for actually vanishing (and therefore non-physical) loads andcontact areas. This is of course less evident in log–log plots, as those reported in Fig. 5, where a straight line is simply thegraphical representation of a more general power law, the exponent of which is given by the slope of the straight line. Thelog–log plots of Fig. 5 show all asperity contact theories almost fall on the same master straight line, which is not parallel tothe curve representing the BGT asymptotic approximation. This again shows that the contact area vs. load relation, given byasperity contact theories, may be approximated by a power law but not by a linear relation. Also observe that this masterstraight line shifts down as a is increased, thus showing that the slope of the contact area–load curve (in a linear–linearplot) becomes smaller and smaller as a increases. This is even more clear in Figs. 6(a)–(c) where the non-dimensional slopeOqAc=qF is shown as a function of the non-dimensional contact area in a log–linear plot. Of course such a slope is constantand equal to the unit value for the asymptotic linear relation given by the BGT theory, as shown by Eq. (16). However, Fig. 6clearly shows that BGT, NT and Greenwood 2006 models all have dimensionless slopes which are significantly smaller thanunity for all physical reasonable values of the contact area, and all converge to unity only when the area of contact isreduced to very negligible values. For a ¼ 2 this happens for Ac=A0o10�4 [see Fig. 6(a)], whereas for higher values of a [seeFigs. 6(b)–(c)] this is not even visible in the diagrams and actually the unit value is achieved only for much smaller,unrealistic contact areas Ac=A0o10�10. However, since BGT, NT and Greenwood 2006 curves converge to a single master curve

10-4 10-3 10-2 10-1 10010-4

10-3

10-2

10-1

100

F / (A0Ω)

Ac

/ A0

10-4 10-3 10-2 10-1 10010-4

10-3

10-2

10-1

100

F / (A0Ω)

Ac

/ A0

10-4 10-3 10-2 10-1 10010-4

10-3

10-2

10-1

100

F / (A0Ω)

Ac

/ A0

BGT

Greenwood 2006

NT

GW - McCool

BGT - Asymptotic

� = 2 � = 10

� = 100

BGT

Greenwood 2006

NT

GW - McCool

BGT - Asymptotic

BGT

Greenwood 2006

NT

GW - McCool

BGT - Asymptotic

Fig. 5. Log–log plot of contact area Ac=A0, obtained by multiasperity contact theories, as a function of the non-dimensional applied load F=ðA0OÞ, for (a),

a ¼ 2; (b), a ¼ 10; (c), a ¼ 100. The solid black line is the asymptotic prediction of the BGT (1975) theory. The quantity O ¼ ðm2=pÞ1=2E0.

ARTICLE IN PRESS

10-8 10-6 10-4 10-20.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ac / A0

10-8 10-6 10-4 10-2

Ac / A0

10-8 10-6 10-4 10-2

Ac / A0

Ω∂

Ac

∂ F

Ω∂

Ac

∂ F

Ω∂

Ac

∂ F

BGT

Greenwood 2006

NT

GW - McCool

� = 2

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

BGT

Greenwood 2006

NT

GW - McCool

� = 10

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

BGT

Greenwood 2006

NT

GW - McCool

� = 100

Fig. 6. Dimensionless slope O�1qAc=qF of the area–load curve as a function of the contact area fraction Ac=A0; (a) a ¼ 2; (b) a ¼ 10; (c) a ¼ 100. It is clearly

shown that BGT, Greenwood 2006 and NT models all converge to unit value of the dimensionless slope (i.e., gives a linear relation between area and load)

as the area fraction is decreased to zero. The original GW at higher relatively large contact areas agrees with the other theories, but does not converge to

unity as the contact area is decreased. However, notice that, BGT, Greenwood 2006 and NT predict linearity between area and load (constant

dimensionless slope O�1qAc=qF) only for vanishing small contact areas. Already for a ¼ 2 linearity is obtained for Ac=A0o10�4, which appear a very

unrealistic value. For higher values of a the linearity is not yet reached even for contact areas smaller then Ac=A0 ¼ 10�8. The quantity O ¼ ðm2=pÞ1=2E0.


as the contact area is continuously reduced to zero, this also means that the asymptotic linear approximation, given by the BGTtheory, can be also obtained by expanding to the first order, other simpler models as the NT and Greenwood 2006 ones. Indeed,in Appendix A we prove this to happen because at large separation, on each single asperity, the Hertzian contact solutionconverges to that of the spherical case, i.e. summits behaves as they were almost perfectly spherical. One may object that theoriginal GW–Mc Cool model does not show this trend. However, in Appendix A, we also show that the crucial factor to achievethe same asymptotic behavior as in the BGT theory, is simply that of taking into account, although in a very simple way, thatthe mean curvature of the asperity changes as the separation is augmented. Indeed a very simple, slightly improved version ofthe GW–Mc Cool model does exist which gives the correct asymptotic behavior. Fig. 6 also shows that the deviation of thedimensionless slope OqAc=qF from unity becomes faster and faster as a grows, thus resulting in contact areas that forreasonable values of the applied load will lie below the Persson’s results (see below). Fig. 7 shows the dimensionless pressurein the contact area F=ðAcOÞ as a function of separation, for three values of a ¼ 2;10;100. As already observed by Greenwood(2006) the ratio F=ðAcOÞ takes the unit values only asymptotically for very large separation higher than six times the standarddeviation m1=2

0 of the surface height profile, this is of course very suspect since at so large separations it is hardly believed thata realistic height distribution can still follow a Gaussian distribution.

Given that all the multiasperity contact theories, which consider the influence of the summit curvature distribution,give very similar results we can refer to the BGT theory to carry out a comparison with the Persson’s theory. In Fig. 8

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0 1 2 3 4 5 60.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

F /

(AcΩ

)

F /

(AcΩ

)

F /

(AcΩ

)

BGT

Greenwood 2006

NT

GW - McCool

� = 2

0 1 2 3 4 5 60.8

1

1.2

1.4

1.6

1.8

2

t

� = 10

0 1 2 3 4 5 61.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

t

� = 100

BGT

Greenwood 2006

NT

GW - McCool

BGT

Greenwood 2006

NT

GW - McCool

Fig. 7. The mean contact pressure F=ðAcOÞ as a function of dimensionless separation t ¼ d=m1=20 for different values of a ¼ 2;10;100. Notice that the unit

value (which is the asymptotic value of the BGT theory) is reached only for very high separation in many cases larger than six times the surface rms. The

quantity O ¼ ðm2=pÞ1=2E0.


predictions of the contact area vs. dimensionless load F=ðOA0Þ are plotted for different values of the parameter a for thePersson’s and BGT theories. It is clearly shown that, even for the smallest value of a ¼ 2, the asperity contact theories do notfollow a straight line, and that for higher values of a the contact area predicted by multiasperity contact theories lies belowthe Persson’s predictions. Persson’s model in turn, as already observed, is not affected by a values and perfectly follows astraight line up to Ac=A0 ¼ 0:1–0.15, this is in agreement with some numerical simulations (Hyun et al., 2004; Borri-Brunetto et al., 2001; Campana and Muser, 2007; Yang et al., 2006), although the slope of the straight line may differ fromthe numerical calculated one. In view of these results, the authors wonder whether the asymptotic behavior of BGT andsimilar theories has actually a physical meaning or rather represents only an academic result. They also wonder whether itis not more correct to assert that there is only one theory which actually predicts a linear relation between the contact areaand the applied load. Of course this theory, due to Persson (2001), presents some defects and heavy approximations thatneeds an experimental or theoretical confirmation.

5. Conclusions

In this paper the authors present a critical analysis of asperity based theories, and also briefly discuss Persson’s theory ofcontact mechanics. They show that those multiasperity contact theories which take into account the statistical distributionof the summits curvatures give similar results, and also prove that such theories necessarily have the same asymptoticbehavior as the contact area is continuously reduced to zero. Also, it is shown that the original model by Greenwood

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

F / (ΩA0)

Ac

/ A0

Persson

Bush � = 2

Bush � = 10

Bush � = 100

Persson Asymptotic

Bush Asymptotic

Fig. 8. The comparison between the BGT model and Persson’s theory of rough contact for three values of a (a ¼ 2, 10, 100). Observe that the Persson’s

theory does not depend on the a values, and that it follows a linear relation even for contact areas larger than 10% of the nominal contact area A0. BGT full

calculation instead shows a very different trend. In particular the BGT asymptotic linear relation highly overestimates the actual predicted area of contact

for all a values. BGT full calculated curves show indeed a much smaller average slope that for large a values is much smaller than that predicted by the

Persson’s theory. The quantity O ¼ ðm2=pÞ1=2E0.


and Williamson (1966) as improved by Mc Cool (1986), although it does not present the same asymptotic behavior asmore advanced asperity contact theories do have, gives results which closely match those of the other asperity contacttheories. We also show that the GW–Mc Cool model can be slightly improved to correctly capture the asymptotic behaviorof other theories, thus making the GW–Mc Cool model preferable to more complicated asperity contact theories as BGT,Greenwood 2006 and NT. However, the asymptotic linear relation of these theories turns out not to be more than anacademic result. Indeed, we show, that fully calculated area vs. load predictions very fast deviate from their asymptoticlinear behavior, and therefore seem not to represent correctly the predicted area–load curve for physically reasonableloads. The linearity predicted by BGT and similar models is obtained only for very high separation (much more than sixtimes the rms of the surface profile), i.e. only for unrealistic small contact areas Ac=A0510�4 and loads. The authors alsoshown that the Persson’s predictions of contact area vs. load curve does not depend on the parameter a, which insteadaffects asperity contact theories, and that the fully calculated Persson’s area–load curve actually follows a linear behaviorup to contact areas of about 10–15% of the nominal one. Thus, although the validity of Persson’s theory must be stillconfirmed, and despite several and maybe heavy approximations, the results presented in this paper suggest this theory tobe the only one to predict a linear relation between contact area and load for physical reasonable loads and contact areas.

Acknowledgments

The authors would like to thank Dr. J.A. Greenwood for useful comments and very welcome discussions.

Appendix A. On the asymptotic behavior of multiasperity contact theories

In this section we carry out an asymptotic analysis of the general model given by Eqs. (3) and (4), on which allmultiasperity contact theories rely, to show that at high separation it converges to a very simple, but slightly corrected, GWtype model which then represents the simplest conceivable multiasperity contact theory predicting a linear relationbetween the area of contact and load at high separation. This also explains why BGT, Greenwood 2006 and NT have all thesame asymptotic behavior.

Let us observe that the Hertzian contact area pa1a2 ¼ AHðx1; k1; k2Þ, the load pa1a2sm ¼ FHðx1; k1k2Þ, and the probabilitydensity function per unit area Pðx1;k1; k2Þ [see Eq. (1)] satisfy the following symmetry relations

AHðx1; k1; k2Þ ¼ AHðx1; k2; k1Þ

FHðx1; k1; k2Þ ¼ FHðx1;k2; k1Þ

Pðx1; k1; k2Þ ¼ Pðx1; k2; k1Þ (A.1)

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and can be written in terms of the commutative quantities kA ¼ ðk1 þ k2Þ=2 and k2G ¼ k1k2 ¼ k2

Að1� r2Þ. Thus, if we definex ¼ x1=m1=2

0 , k�A ¼ kA=m1=24 , r2 ¼ ðw=kAÞ

2 and write AHðx1; k1; k2Þ ¼AHðx; k�A; r2Þ, FHðx1; k1; k2Þ ¼FHðx; k�A; r

2Þ and

Pðx; k�A; r2Þ ¼

qðx1; k1; k2Þ

ðx; kA; r2Þ

��Pðx1; k2; k1Þ

¼3ffiffiffi3p

8p2

m4

m2C1=2

1 exp �C1 xþ3k�A2ffiffiffiap

� �2" #

k�4A ð1� r2Þ exp �3

4k�2A ð1þ 2r2Þ

(A.2)

where

qðx1; k1; k2Þ

qðx; k�A; r2Þ

�� ¼ m1=2

0 m4jk�Ajffiffiffiffiffir2

p (A.3)

it follows that Eqs. (3) and (4) can be rephrased as

AcðtÞ

A0¼

Z þ1t

Z 0

�1

Z 1

0dxdk�Adðr2ÞAHðx;k�A; r

2ÞPðx; k�A;r2Þ (A.4)

FðtÞ

A0¼

Z þ1t

Z 0

�1

Z 1

0dxdk�Adðr2ÞFHðx; k�A;r

2ÞPðx; k�A; r2Þ (A.5)

where t ¼ d=m1=20 . Observe, that in the above written integrals Eqs. (A.4) and (A.5) the limits of integration on r2 and k�A

have been chosen in order to guarantee that k1o0 and k2o0. Now, let us write

Pðx; k�A; r2Þ ¼ pðk�Ajx; r

2Þpðr2jxÞPsumðxÞ (A.6)

where the conditional probability

pðk�Ajx; r2Þ ¼

Pðx; k�A; r2Þ

Pðx; r2Þ(A.7)

with

Pðx; r2Þ ¼

Z 0

�1

dk�APðx; k�A; r

2Þ (A.8)

and

pðr2jxÞ ¼Pðx; r2Þ

PsumðxÞ(A.9)

The quantity

PsumðxÞ ¼Z 1

0dðr2ÞPðx; r2Þ (A.10)

simply represents the probability density function per unit area of summits heights, given by (see for example Greenwood,2006)

PsumðxÞ ¼1

6pffiffiffi3p

m4

m2

3

2pð2a� 3Þ1=2

ax expð�C1x

2Þ þ

33=2

2ð2pÞ1=2

1

aðx2� 1Þ expð�x2=2Þð1þ erfbÞ

(

þa1=2

ð2pÞ1=2ða� 1Þ1=2

expð�ax2=½2ða� 1Þ�Þð1þ erfgÞ

)(A.11)

with b ¼ f3=½2ð2a� 3Þ�g1=2x and g ¼ fa=½2ð2a� 3Þða� 1Þ�g1=2. Since we want to discuss what happens at very high separationt, this means that also x will be very large. Under these conditions it is relatively easy to show that for very high x theconditional probability distribution pðr2jxÞ becomes

pðr2jxÞ ! 2dðr2Þ (A.12)

where dð�Þ is the Dirac delta function. Eq. (A.12) simply states that as x!þ1 the value of r2 converge to zero, i.e. summitsof very large heights behave as they were almost perfectly spherical. Using Eq. (A.12) we get

AcðtÞ

A0¼

Z þ1t

Z 0

�1

dxdk�AAHðx; k�A;0Þpðk�Ajx;0ÞPsumðxÞ (A.13)

FðtÞ

A0¼

Z þ1t

Z 0

�1

dxdk�AFHðx; k�A;0Þpðk�Ajx;0ÞPsumðxÞ (A.14)

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also it can be shown that for x!1 the conditional probability distribution pðk�Ajx;0Þ becomes a Gaussian distributioncentered at k�AðxÞ ¼

R 0�1

dk�Ak�Apðk�Ajx;0Þ with

k�AðxÞ ! �1

a1=2x (A.15)

hðk�A � k�AÞ2i !

2a� 3

3a(A.16)

from which it follows that hðk�A � k�AÞ2i=k�2A / x�2

! 0 as x!1. Thus, for very large x values, it is possible to write

pðkA�jx;0Þ � d½kA � k�AðxÞ� (A.17)

where again dð�Þ is the Dirac delta function. Eqs. (A.13) and (A.14) then become

AcðtÞ

A0¼

Z þ1t

dxAHðx; k�AðxÞ;0ÞPsumðxÞ (A.18)

FðtÞ

A0¼

Z þ1t

dxFHðx; k�AðxÞ;0ÞPsumðxÞ (A.19)

Observe that Eq. (A.15) confirms the result, given for the first time by Onions and Archard (1973), that as the height of theasperities is increased their curvature increases too. The just above written integrals Eqs. (A.18) and (A.19) are relativelysimple to calculate if one observes that at large separation and therefore at large x the probability density function PsumðxÞ,given by Eq. (A.11), can be approximated by

PsumðxÞ �1

2pffiffiffiffiffiffi2pp

m4

m2

1

ax2 expð�x2=2Þ (A.20)

so that using Eq. (A.15) and recalling Hertz’s formulas for the case of a spherical asperity of radius R ¼ k�AðxÞ�1 one obtains

AHðx; k�AðxÞ;0Þ ¼ pm0

m4

� �1=2

a1=2 x� t

x(A.21)

FHðx; k�AðxÞ;0Þ ¼4

3E0a1=4 m3=4

0

m1=44

ðx� tÞ3=2

x1=2(A.22)

and

AcðtÞ

A0¼

1

2ffiffiffiffiffiffi2pp

Z þ1t

dxðx� tÞx expð�x2=2Þ (A.23)

FðtÞ

A0¼

ffiffiffi2p

3E0

m1=22

pffiffiffipp

Z þ1t

dx½ðx� tÞx�3=2 expð�x2=2Þ (A.24)

The first integral Eq. (A.23) can be easily calculated by parts to give

AcðtÞ

A0¼

1

4erfc

tffiffiffi2p

� �(A.25)

which also shows that a general results of multiasperity contact theories is that, at large separations, the contact area Ac isnecessarily half of the bearing area Ab ¼ ð

12Þ erfcðt=

ffiffiffi2pÞ. This is not surprising, if one notices that at high separations the

asperity behaves as they where spherical and considers that the Hertz theory states, in case of a spherical contact, that thearea of contact is just the half of the bearing area.

The second integral Eq. (A.24) gives instead

FðtÞ

A0¼ E0

m2

p

� �1=2 1

4erfc

tffiffiffi2p

� �(A.26)

Notice also that, by noting that ðp=2Þ1=2erfcðt=ffiffiffi2pÞ ¼ t�1 expð�t2=2Þ at very large t, one concludes that Eqs. (A.25) and (A.26)

are the same as those given by the BGT theory. Eqs. (A.25) and (A.26) give

F ¼ E0m2

p

� �1=2

Ac (A.27)

which is exactly the linear asymptotic results of the BGT theory. In view of these results, one necessarily concludes that all

multiasperity contact theories, that take in account the influence of the summit curvature variation as a function of summit

height, give similar results and have all the same asymptotic behavior. Also it follows that a minimal improvement of the

GW– Mc Cool model does exists which yields the same asymptotic behavior of other and more complicated multiasperity contact

theories. This minimally improved GW–Mc Cool model is simply represented by Eqs. (A.18) and (A.19) with PsumðxÞ given byEq. (A.11). Simply speaking, this improved version of the GW–Mc Cool model still considers a distribution of spherical

ARTICLE IN PRESS


asperities, but this time the spheres do not have all the same radius. Indeed the radius of curvature is taken to be a functionk�1

A ðxÞ of the asperity height.

References

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