Tribology International 64 (2013) 148–154

Contents lists available at SciVerse ScienceDirect

Tribology International

0301-67http://d

n CorrManageTel.: þ3

E-m

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

A multiscale analysis of elastic contacts and percolation thresholdfor numerically generated and real rough surfaces

Carmine Putignano a, Luciano Afferrante a,b,n, Giuseppe Carbone a,b, Giuseppe P. Demelio a,b

a TriboLAB, Department of Mechanics, Mathematics and Management (DMMM), Politecnico di Bari, v.le Japigia 182,70126 Bari, Italyb Centre of Excellence for Computational Mechanics (CEMeC), Politecnico di Bari, via Re David 200,70124 Bari, Italy

a r t i c l e i n f o

Article history:Received 8 January 2013Received in revised form12 March 2013Accepted 15 March 2013Available online 23 March 2013

Keywords:Self-affine rough surfacesAdhesionless contactNumber of wavelengthsPercolation

9X/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.triboint.2013.03.010

espondig author at: Department of Mechanicment (DMMM), Politecnico di Bari, v.le Japigi9 080 596 3793.ail address: l.afferrante@poliba.it (L. Afferrant

a b s t r a c t

In this paper, we present numerical investigation of the contact between an elastic solid and a randomlyrough surface. In agreement with recent results, we find that the contact area vs load relation depends onthe statistical parameters only through the root mean square slope of the heights distribution. Suchresult extends to contact pressure regimes where the area/load relation is non-linear.

Moreover, we show that fractal self-affine surfaces give a good representation of real surfaces fromboth topographical and contact mechanics points of view.

Finally, we investigate how the network of non-contact areas evolves as the real contact area isincreased, finding that the percolation threshold is smaller than the one predicted by Bruggeman'stheory.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Contact mechanics between surfaces has wide implications inengineering systems: lots of technologically important aspects,like contact stiffness and electrical [1,2] or thermal contactresistance [3–8], contact dissipation [9,10], are strongly influencedby the micromechanical characteristics of the contact process.Also, the wide diffusion of seals as technical devices to isolatechambers and to minimize flow between them makes their designimportant. Calculating leakage accurately is not a simple matterbecause the surface roughness at the seal interface spans a widerange of length scales [11]. In all these applications it is crucial toevaluate with accuracy the real contact area and the effective gapdistribution between the contacting surfaces.

The earliest pioneering attempt to investigate the multiscalenature of elastic contact between rough surfaces is that of Archardin 1956 [12]. Afterwards, Greenwood and Williamson introducedasperity-based models [13–16], where the distribution of contact-ing asperities is replaced by a distribution of Hertzian asperitieswith equivalent height and curvature. Later models have usedrandom process theory to make the asperity curvature dependingon their heights or have resorted to an apparently differentapproach that uses fractal theories to recognize more directlythe multiscale nature of most real surfaces [17]. Although the

ll rights reserved.

s, Mathematics anda 182,70126 Bari, Italy.

e).

results obtained with asperity contact theories are of practicalinterest, the multi-asperity models neglect interactions and coa-lescing of contact spots. This problem becomes more and moreimportant when approaching full contact conditions. For thisreason, in [18] a new approach has been proposed, in which thesummits interaction and the coalescing of asperities are taken intoaccount and a very good agreement with the results obtained byfully numerical approaches is found.

Persson [19] has proposed a different approach, which gives theexact solution in full contact and an approximate but accurate solutionin partial contact situations. Starting from the full contact solution,where the surfaces are assumed smooth, roughness is progressivelyadded, thus generating contact pressure random fluctuations and gapformations. Persson argues that when adding an increment of rough-ness corresponding to an increment of magnification, the probabilitydensity function of the contact pressure must satisfy a diffusionequation. This allows us to take into account the interaction betweenthe contact spots and to provide exact predictions in full-contactconditions. In the case of partial contact, the initial version of thetheory provides approximate predictions [19]. However, furtherdevelopments of the theory [20] give almost exact solution, providedthat a universal correcting factor is introduced properly in Persson'scalculation of elastic energy stored at the interface. It is worth noticingthat Persson's theory predicts linearity between contact area and loadon a larger range of applied loads in agreement with experiments,although the slope of the relation differs from the predictions ofmulti-asperity contact theories and from numerically calculated values(see e.g. [21,22]).

C. Putignano et al. / Tribology International 64 (2013) 148–154 149

In this paper, starting from a numerical method recentlydeveloped by the authors [23], we investigate the effect, on thecontact between a linear elastic half-space and a rough self-affinefractal rigid surface, of the wavelength numbers used to describethe surface, in order to analyze the contact behavior at differentscales. Moreover, we show that self-affine fractal surfaces give avery good representation of real surfaces from the point of view ofcontact mechanics, and some preliminary calculations aboutpercolation related phenomena, involving multiscale aspects, arealso presented.

In this respect, if two surfaces in contact had exactly the sameshape, e.g. if there was no roughness, no fluid would leak throughtheir interface; actually, because of the presence of roughness, thecontact between sealing surfaces is imperfect, and the fluid canfind a percolating path leaking between two chambers at differentpressures. Many theories have been developed to predict the rateof percolating flow [24–29]. However, despite recent progresses,traditional approaches and industrial procedures for seal systemsare still based on approximate treatments which usually neglectthe effect of elastic deformation on the contacting surfaces[30–32]. Such approaches usually predict a percolation thresholdlying at a relative contact area An=A0 ¼ 0:5, overestimating the realvalue. In fact, there are many simulations [33–37] suggesting thatelastic contacts may percolate below An=A0 ¼ 0:5, in agreementwith the results presented in this paper.

2. Mathematical formulation

In this section we present only the fundamental aspects of themathematical formulation for the problem sketched in Fig. 1,i.e. the adhesionless contact between a periodic numericallygenerated isotropic randomly rough rigid surface and a linearelastic half-space. The interested reader is referred to Ref. [23] formore details.

The elastic displacement in any point of the half-space can beconsidered as a sum of two terms: the first is equal to the averagedisplacement umðzÞ and the second vðx,zÞ ¼ uðx,zÞ−umðzÞ, where xis the in-plane position vector, is the additional displacementcaused by the asperities induced deformation of the roughsurfaces. In particular, if we focus only at the interface z¼0, wewill notice that the maximum value of the normal displacement

Fig. 1. The figure shows a cross section of the contact between an elastic half-spaceand a rigid rough surface. Δ represents the penetration of the rigid rough surfaceinto the elastic solid, um is the average displacement and λ is the size of the squarecells constituting the periodic domain.

vðxÞ ¼ vzðx,z¼ 0Þ is the penetration depth Δ of the rough surfaceinto the half-space.

The rough surface is modeled as a self-affine fractal surface withfractal dimension Df ¼ 3−H, H being the Hurst exponent. To gen-erate the rough surface, we have employed the spectral methoddescribed in [23]. We recall that the power spectral density (PSD) ofan isotropic self-affine fractal surface can be related to the wavevector q≡ðqx,qyÞ through the following power law:

CðqÞ ¼ C0qq0

� �−2ð1þHÞð1Þ

where q¼ jqj and q0 is the roll-off wave vector. In particular, aperiodic surface with Fourier components up to the value q1 ¼Nq0has been utilized to perform calculations, N being the number ofwavelengths.

To avoid border effects, a periodic domain constituted of squarecells D¼ ½ð−λ=2,λ=2Þ � ð−λ=2,λ=2Þ� is considered. Assuming a linearbehavior, the elastic displacement field at the interface can bewritten in terms of the interfacial normal stress sðxÞ ¼ szzðx,z¼ 0Þ

uzðxÞ ¼ZDd2s Gðx−sÞsðsÞ, x∈D ð2Þ

where GðxÞ is the Green function. However, we observe that, in thecase of infinite-half space under periodic load conditions, themean displacement um of the elastic body at the interface isunbounded, and therefore only the term vðxÞ ¼ uzðxÞ−um is finite.Since our focus is just on vðxÞ, we reformulate the problem sothat only the quantities sðxÞ and vðxÞ appear. To this endconsider that from Eq. (2) one obtains um ¼ R

Dd2s GmsðsÞ, where

Gm ¼ λ−2RDd

2x GðxÞ, so that one can write

vðxÞ ¼ZDd2s Lðx−sÞsðsÞ, x∈D ð3Þ

where LðxÞ ¼ GðxÞ−Gm is equal to the elastic displacement at theinterface uzðxÞ caused by a periodically applied self-balancednormal stress distribution sðxÞ ¼ δðxÞ−λ−2.

To solve the contact problem, we control penetration depth Δand discretize the domain D in small squares of non-uniform size.The unknown stress acting on each single square is assumed to beuniformly distributed on it. The discretized version of Eq. (3)becomes

vi ¼ Lijsj ð4Þwhere vi is the central displacement of each square, si is theuniform stress on the square and Lij is the elastic response matrix(i.e. the matrix associated with the discretized version of Eq. (3)),caused by the self-balanced applied load. Matrix Lij can beobtained by recalling the Love solution (see [38]) giving the elasticdisplacement a uniform pressure acting on a rectangular area, andsumming up the contribution of each elementary cell D to takeinto account the periodicity of the problem. More details aboutthis calculation, the iterative solution procedure and the relatedanalysis of convergence can be found in [23].

Here, we remind that we have implemented an adaptive non-uniform mesh strategy, allowing to have a high number ofelements only where it is necessary: at the borders of the contactregions, where the stress and strain gradients significantlyincrease. In this way, we are able to guarantee the numerical fullconvergence and the efficiency of the method.

3. Results

3.1. Effect of scales number on contact area

Fig. 2 shows the effect of the wavelengths number N on thedependence of the real contact area A, normalized with respect to

C. Putignano et al. / Tribology International 64 (2013) 148–154150

the nominal one A0, on the dimensionless contact pressure s0=En,

where En ¼ E=ð1−ν2Þ is the composite Young's modulus of thematerial. Results refer to self-affine fractal surfaces with spectralcomponents q ranging between q0 ¼ 2π � 102 m−1 and q1 ¼Nq0.For each surface seven different realizations have been considered

Fig. 2. The variation of the normalized contact area A=A0 with the dimensionlessaverage contact pressure s0=E

n for different wavelength numbers (H¼0.85,hrms ¼ 13 μm).

Fig. 3. Contact regions (white areas) for a constant dimensionless separation s=hrms

(b) N¼32, A=A0 ¼ 0:42; (c) N¼64, A=A0 ¼ 0:39; (d) N¼128, A=A0 ¼ 0:37.

and the ensemble average of the calculated results has been taken.The half-space is assumed linear elastic with Young's modulusE¼0.4 GPa and Poisson's ratio ν¼ 0:45. The area vs. load relation islinear at small loads, according to our previous results [23] andother analytical theories [15,17,19,39–41] and numerical investi-gations [22,42–46]. However, at high loads (s0=E

n≳ffiffiffiffiffiffiffiffiffiffiffiffim2=2

p), when

the asperities with a larger radius of curvature get into contact,linearity is lost. Moreover, Fig. 2 seems to suggest that the slope ofthe lines is affected by the number of wavelengths. In particular, atgiven applied load, the contact area decreases as the number ofscales is increased. In fact, the number of asperities of the surfacestrongly grows as N increases, thus the contact splits into smallerand smaller parts and the contact area becomes smaller and morejagged. This behavior is made clear in Fig. 3, where we show howthe contact regions change with the number of scales of thesurface PSD.

However, both multiasperity [13,17] and Persson's [19] theoriespredict a contact area vs. load relation depending only on theelastic properties of the material and the rms slope m2 ¼ ⟨∇h2⟩=2of the heights distribution of the rough surface. In fact, if onekeeps q0 constant, the number of scales indirectly affects thecontact area through the moment m2, as shown in Fig. 4, wherethe dimensionless contact area A=A0 is shown as a function of thequantity s0=ðEn

ffiffiffiffiffiffiffiffiffiffi2m2

pÞ. Such figure makes clear that the depen-

dence of the contact area on the number of scales noticed in Fig. 2,is just a result of the change of m2. The present findings are inagreement with other numerical investigations [21,47]. Notice thatthis conclusion is also true in the non-linear range of the area/loadrelation.

¼ 0:1, H¼0.85 and for different wavelength numbers N: (a) N¼16, A=A0 ¼ 0:45;

Fig. 4. The variation of the normalized contact area A=A0 with the quantitys0=ðEn

ffiffiffiffiffiffiffiffiffiffi2m2

pÞ. Notice the negligible effect of N (H¼0.85, hrms ¼ 13 μm).

Fig. 5. Interfacial stress probability distribution PðsÞ for a fractal surface withH¼0.85, N¼64, hrms ¼ 13 μm and different mesh refinements.

Fig. 6. Coefficient of proportionality κ for different values of the discretizationparameter M. Notice that full convergence is achieved when M416N (H¼0.85,hrms ¼ 13 μm, N¼64).

C. Putignano et al. / Tribology International 64 (2013) 148–154 151

Focusing our attention on the range where the contact area is alinear function of the contact pressure, according to the equationA=A0 ¼ κðs0=ðEn

ffiffiffiffiffiffiffiffiffiffi2m2

pÞÞ, we know that the coefficient of propor-

tionality κ is ≈2 and is not affected by the Hurst coefficient H,the wavelengths number N and the root mean square roughnesshrms [21,48]. Different conclusions are given in other works[43,44,46,49], where a dependence of κ on the statistical para-meters of the surface is found. Such different results, in ouropinion, are probably due to insufficient mesh refinement at theborder of the contacts, or insufficient number of mesh points ineach contact spot (in the case of uniform grid), so the convergencein such cases may really be questioned. Indeed, Campana andMuser [47] have shown, by employing an approach based on theso-called Green's function molecular dynamics, that a non-sufficiently fine mesh provokes a distorted solution and preventsthe stress probability distribution PðsÞ ¼ ððA0−AÞ=A0ÞδðsÞþ ðA=A0ÞpðsÞ, δðsÞ being the δ-peak at s¼ 0, from vanishing linearly withdecreasing s, where pðsÞ is the probability density function of the

stress in the contact area. This behavior of PðsÞ must, instead, benecessarily observed as analytically demonstrated in [50–52]. Inthis respect, for a fractal surface with H¼0.85 and N¼64, Fig. 5shows the stress probability function Pðs=EnÞ for different values ofthe discretization parameter M, that identifies the size A0=ðM �MÞof the smallest cell of the discretized domain. If M is not largeenough, the numerical method is unable to compute a correcttrend of the probability distribution. Therefore, for the method toachieve the numerical convergence a sufficiently fine mesh needsto be defined into the domain of calculation. This is confirmed alsoin Fig. 6, where the coefficient of proportionality κ is shown as afunction of M. Notice that the estimate of κ is affected by anincreasing error as M decreases.

3.2. Numerical calculations on real surfaces

We wonder if the results obtained for an isotropic self-affinefractal surface, characterized by the PSD defined through Eq. (1),can be considered as a representative of the behavior of a realrough surface. To answer this question, first of all, we need toshow whether it is possible to describe the roughness h(x) of a realsurface by means of a self-affine geometry. To this end, Fig. 7shows that the PSD (dashed lined) calculated from the measuredconcrete surface (shown in Fig. 8) is well fitted by Eq. (1). Thetopography of the road surface has been measured with contact-less optical methods [53]. Moreover, as shown in Fig. 9, theprobability density function of the concrete road surface heightsis relatively well fitted by an ideal Gaussian distribution. Thelimited asymmetry of the measured height distribution (solid line)is simply due to the fact that the summit of the road surface hasalready undergone a wear process due to the interaction withtires. Thus, from a topographical point of view, we can reasonablyassume that a fractal self-affine surface, with PSD defined as inEq. (1), is a representative of a real rough surface.

A comparison between the results obtained on the concreteroad surface of Fig. 8 and those obtained on a numericallygenerated self-affine surface with the PSD represented with redline in Fig. 7, is demonstrated in Figs. 10 and 11. In particular,Fig. 10 shows the variation of the normalized contact area A=A0

with the quantity s0=ðEnffiffiffiffiffiffiffiffiffiffi2m2

pÞ, whereas Fig. 11 shows the varia-

tion of the dimensionless separation s=hrms with the dimensionless

Fig. 8. A concrete road surface provided by Pirelli and Baustoffprüfung-Aachen(Germany). The topography was measured with contact-less optical methods usinga chromatic sensor with two different optics produced by Fries Research &Technology GmbH (Bergisch Gladbach, Germany).

Fig. 9. Comparison between the probability density functions of the heights of thereal surface shown in Fig. 8 (solid line) and a Gaussian fitting (dashed line).

Fig. 7. The surface roughness power spectrum C(q), as a function of the wavevectorq (log–log scale). Dashed line corresponds to the experimental data, solid line is thefit obtained by Eq. (1). Notice that the slope of the spectral density is −3.4. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

C. Putignano et al. / Tribology International 64 (2013) 148–154152

average contact pressure s0=En. In both cases, we notice an almost

perfect agreement.

3.3. Percolation threshold

Seals are usually used in mechanical devices to prevent fluidleakage between regions at different pressures. However, despitethe progresses done in this field, many questions remain open. Forexample, the traditional approaches to seals are based on approx-imate treatments, characterized by uncontrolled assumptions. Inparticular, to derive the gap distribution, the simple cumulativeheight distribution function of the undeformed surfaces, i.e. thebearing area, is usually used as input. This approach disregards theelastic deformation of the material and this can lead to erroneousresults about mean gap, relative contact area and contact pressure.

Very recently Persson and Yang [34] and Dapp et al. [35] haveshown that elastic contacts may percolate below the ‘traditional’

Fig. 10. The variation of the normalized contact area A=A0 with the quantitys0=ðEn

ffiffiffiffiffiffiffiffiffiffi2m2

pÞ. Comparison between the results relative to the real surface of Fig. 8

and the numerically generated self-affine one.

Fig. 11. The variation of the dimensionless separation s=hrms with the dimension-less average pressure s0=E

n . Comparison between the results relative to the realsurface of Fig. 8 and the numerically generated self-affine one.

C. Putignano et al. / Tribology International 64 (2013) 148–154 153

value of the relative contact area An=A0 ¼ 0:5, corresponding to thepercolation threshold predicted by Bruggeman's effective-mediumtheory [31].

Fig. 13. Contact and non-contact patches at the percolation threshold. White areas are cpercolating non-contact areas. Calculations are carried out for different wavelength num

Fig. 12. Variation of the limit contact area An=A0 corresponding to the percolationthreshold with the number of wavelengths: N¼16 (A); N¼32 (B); N¼64 (C);N¼128 (D). Dashed lines denote the extreme values and the blue rectangles denotevalues from 25 to 75 percentiles (H¼0.85, hrms ¼ 13 μm). (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

Here, we present some preliminary calculations which corro-borate results presented in [34,35].

Fig. 12 shows the value of An=A0 at which contact-areapercolation occurs, for different N and H¼0.85. Results areaveraged on seven different realizations of the surface. Investiga-tions with other fractal dimensions (H¼0.6 and H¼0.7) providedsimilar results. The limit contact area corresponding to thepercolation threshold is almost constant and seems to be negli-gibly affected by the wavelengths number N. We find a slightlylower value of An=A0 with respect to that given in [35]. This smalldifference is probably due to finite size effects in simulation cellswith linear size of 1024 grid points (4096 grid points are used in[35], so covering a wider range of length scales).

Finally, in Fig. 13 the contact and non-contact patches areshown just at a small step before contact area percolation occurs,i.e. a small step before the continuity of percolated non-contactchannel is interrupted. Calculations have been carried out forH¼0.85. White regions correspond to contact areas, black regionscorrespond to percolating non-contact areas, and grey regionscorrespond to non-percolating non-contact areas. Calculationshave been performed for different scale numbers N of thesurface PSD.

4. Conclusions

We have investigated the contact between a linear elastic solidand isotropic randomly rough surfaces. We have found that thestatistical properties of self-affine rough surfaces affect the contact

ontact areas, black areas are percolating non-contact areas, and grey area are non-bers N: (a) N¼16; (b) N¼32; (c) N¼64; (d) N¼128 (H¼0.85, hrms ¼ 13 μm).

C. Putignano et al. / Tribology International 64 (2013) 148–154154

area vs load relation only through the moment m2. We find alsothat this holds true at high contact pressures, where the contactarea/load relation is non-linear. Our calculations also show that,due to the very large number of roughness length scales, whichmay cover even 6 orders of magnitude of spatial frequencies, theachievement of numerically exact (converged) results is not aneasy task. A crucial parameter to prevent distorted solutions andguarantee the correct prediction of the stress probability distribu-tion PðsÞ is the size of the smallest grid element used to discretizethe smallest contact area and the smallest roughness wavelength.Lack of doing so leads to a bad estimation of stress probabilitydistribution at the interface and to an erroneous evaluation of thecontact area vs. load coefficient of proportionality at small loads.

Furthermore, we have shown that, in terms of contactmechanics, the topography of real rough surfaces is well capturedby self-affine fractal surfaces and the results obtained for suchtype of surfaces give the correct contact behavior.

Finally, preliminary calculations show that percolation of non-contact areas cannot occur above a certain threshold ðA=A0Þth≈0:36,which is smaller than the value predicted by Bruggeman's effective-medium theory, in agreement with other recent results of theliterature.

Acknowledgement

The authors thank the financial support of the Italian Ministry ofEducation, Universities and Research, within the project PON0102238.

References

[1] Barber JR. Bounds on the electrical resistance between contacting elasticrough bodies. Proceedings of the Royal Society of London A 2003;459:53–66.

[2] Ciavarella M, Dibello S, Demelio G. Conductance of rough random profiles.International Journal of Solids and Structures 2008;45:879–93.

[3] Afferrante L, Ciavarella M. Instability of thermoelastic contact for two half-planes sliding out-of-plane with contact resistance and frictional heating.Journal of the Mechanics and Physics of Solids 2004;52(7):1527–47.

[4] Afferrante L, Ciavarella M. Frictionally-excited thermoelastic instability (TEI) inthe presence of contact resistance. Journal of Strain Analysis for EngineeringDesign (Special Issue) 2005;39(4):1–7.

[5] Afferrante L, Ciavarella M. The thermoelastic Aldo contact model withfrictional heating. Journal of the Mechanics and Physics of Solids 2004;52(3):617–40.

[6] Ciavarella M, Johansson L, Afferrante L, Klarbring A, Barber JR. Interaction ofthermal contact resistance and frictional heating in thermoelastic instability.International Journal of Solids and Structures 2003;40:5583–97.

[7] Barber JR, Ciavarella M, Afferrante L. Influence of thermal contact resistance onfrictionally excited thermoelastic instability (TEI). In: Proceedings of theASME/STLE international joint tribology conference (PART A), art. no.TRIB2004-64367; 2004. p. 123–6.

[8] Afferrante L, Ciavarella M, Barber JR. Sliding thermoelastodynamic instability.Proceedings of the Royal Society of London A 2006;462:2161–76.

[9] Putignano C, Ciavarella M, Barber JR. Frictional energy dissipation in contact ofnominally flat rough surfaces under harmonically varying loads. Journal of theMechanics and Physics of Solids 2011;59(12):2442–54.

[10] Barber JR, Ciavarella M, Afferrante L, Sackfield A. Effect of small harmonicoscillations during the steady rolling of a cylinder on a plane. InternationalJournal of Mechanical Sciences 2008;50(9):1344–53.

[11] Persson BNJ, Albohr O, Tartaglino U, Volokitin AI, Tosatt E. On the nature ofsurface roughness with application to contact mechanics, sealing, rubberfriction and adhesion. Journal of Physics: Condensed Matter 2005;17:R1.

[12] Archard JF, Hirst W. The wear of metals under unlubricated conditions.Proceedings of the Royal Society of London A 1956;236:397.

[13] Greenwood JA, Williamson JBP. Contact of nominally flat surfaces. Proceedingsof the Royal Society of London A 1966;295:300–19.

[14] Greenwood JA, Putignano C, Ciavarella MA. Greenwood & Williamson theoryfor line contact. Wear 2011;270:332–4.

[15] Carbone G. A slightly corrected Greenwood and Williamson model predictsasymptotic linearity between contact area and load. Journal of the Mechanicsand Physics of Solids 2009;57(7):1093–102.

[16] Ciavarella M, Delfine V, Demelio G. A “re-vitalized” Greenwood and William-son model of elastic contact between fractal surfaces. Journal of the Mechanicsand Physics of Solids 2006;54:2569–91.

[17] Bush AW, Gibson RD, Thomas TR. The elastic contact of a rough surface. Wear1975;35:87–111.

[18] Afferrante L, Carbone G, Demelio G. Interacting and coalescing Hertzianasperities: a new multiasperity contact model. Wear 2012;278–279:28–33.

[19] Persson BNJ. Theory of rubber friction and contact mechanics. Journal ofChemical Physics 2001;115:3840–61.

[20] Almqvist A, Campana C, Prodanov N, Persson BNJ. Interfacial separationbetween elastic solids with randomly rough surfaces: comparison betweentheory and numerical techniques. Journal of the Mechanics and Physics ofSolids 2011;59:2355–69.

[21] Putignano C, Afferrante L, Carbone G, Demelio G. The influence of thestatistical properties of self-affine surfaces in elastic contact: a numericalinvestigation. Journal of the Mechanics and Physics of Solids 2012;60:973–82.

[22] Carbone G, Scaraggi M, Tartaglino U. Adhesive contact of rough surfaces:comparison between numerical calculations and analytical theories. EuropeanPhysical Journal E 2009;30:65–74.

[23] Putignano C, Afferrante L, Carbone G, Demelio G. A new efficient numericalmethod for contact mechanics of rough surfaces. International Journal ofSolids and Structures 2012;49(2):338–43.

[24] Lorenz B, Persson BNJ. On the dependence of the leak rate of seals on theskewness of the surface height probability distribution. Europhysics Letters90-3 Article; 2010:38002.

[25] Lorenz B, Persson BNJ. Leak rate of seals: effective-medium theory andcomparison with experiment. European Physical Journal E 2010;31(2):159–67.

[26] Bottiglione F, Carbone G, Mangialardi L, Mantriota G. Leakage mechanism inflat seals. Journal of Applied Physics 2009;106:104902.

[27] Bottiglione F, Carbone G, Mantriota G. Fluid leakage in seals: an approachbased on percolation theory. Tribology International 2009;42(5):731–7.

[28] Sahlin F, Larsson R, Almqvist A, Lugt PM, Marklund P. A mixed lubricationmodel incorporating measured surface topograph, Part 1: theory of flowfactors. Proceedings of the IMechE Part J: Journal of Engineering Tribology2009;224:335.

[29] Sahlin F, Larsson R, Marklund P, Almqvist A, Lugt PM. A mixed lubricationmodel incorporating measured surface topography, Part 2: roughness treat-ment, model validation and simulation. Proceedings of the IMechE Part J:Journal of Engineering Tribology 2009;224:353.

[30] Abbott EJ, Firestone FA. Specifying surface quality: a method based on accuratemeasurement and comparison. Mechanical Engineering 1933;55:569.

[31] Bruggeman DAG. Berechnung verschiedener physikalischer Konstanten vonheterogenen Substanzen. Annals of Physics (Leipzig) 1935;24:636–79.

[32] Flitney R. Seals and sealing handbook. Elsevier; 2007.[33] Sahlin F. PhD thesis, Lulea University of Technology, 2008.[34] Persson BNJ, Yang C. Theory of the leak-rate of seals. Journal of Physics:

Condensed Matter 2008;20:315011.[35] Dapp WB, Lücke A, Persson BNJ, Müser MH. Self-affine elastic contacts:

percolation and leakage. Physical Review Letters 2012;108:244301.[36] Persson BNJ, Prodanov N, Krick BA, Rodriguez N, Mulakaluri N, Sawyer WG,

et al. Elastic contact mechanics: percolation of the contact area and fluidsqueeze-out. European Physical Journal E 2012;35(1) [art. 5].

[37] Stauffer D, Aharony A. An introduction to percolation theory. 2nd ed. BocaRaton, FL: CRC Press; 1991.

[38] Johnson KLJ. Contact mechanics. Cambridge University Press; 1985.[39] Thomas TR. Rough surfaces. New York: Longman Group Limited; 1982 [Chapter 8].[40] Greenwood JA. A simplified elliptic model of rough surface contact. Wear

2006;261:191–200.[41] Carbone G, Bottiglione F. Asperity contact theories: do they predict linearity

between contact area and load?. Journal of the Mechanics and Physics ofSolids 2008;56(8):2555–72.

[42] Yang C, Tartaglino U, Persson BNJ. A multiscale molecular dynamics approachto contact mechanics. European Physical Journal E 2006;19(1):47–58.

[43] Hyun S, Pei L, Molinari JF, Robbins MO. Finite-element analysis of contactbetween elastic self-affine surfaces. Physical Review E 2004;70:026117.

[44] Borri Brunetto M, Chiaia B, Ciavarella M. Incipient sliding of rough surfaces incontact: a multi-scale numerical analysis. Computer Methods in AppliedMechanics and Engineering 2001;190:6053–73.

[45] Yang C, Persson BNJ. Molecular dynamics study of contact mechanics: contactarea and interfacial separation from small to full contact. Physical ReviewLetters 2008;100:024303.

[46] Hyun S, Robbins MO. Elastic contact between rough surfaces: effect ofroughness at large and small wavelengths. Tribology International 2007;40:413–1422.

[47] Campana C, Muser MH. Contact mechanics of real vs. randomly roughsurfaces: a Green's function molecular dynamics study. Europhysics Letters2007;77(3):38005.

[48] Pastewka L, Sharp TA, Robbins MO. Seamless elastic boundaries for atomisticcalculations. Physical Review B 2012;86:075459.

[49] Paggi M, Ciaveralla M. The coefficient of proportionality κ between realcontact area and load, with new asperity models. Wear 2010;268:1020–9.

[50] Persson BNJ. Contact mechanics for randomly rough surfaces. Surface ScienceReports 2006;61:201–27.

[51] Persson BNJ, Bucher F, Chiaia B. Elastic contact between randomly roughsurfaces: comparison of theory with numerical results. Physical Review B2002;65:184106.

[52] Manners W, Greenwood JA. Some observations on Persson's diffusion theoryof elastic contact. Wear 2006;261:600–10.

[53] Lorenz B, Carbone G, Schulze C. Average separation between solids in roughcontact: comparison between theoretical predictions and experiments. Wear2010;268(7–8):984–90.