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Research ArticlePartial Contact of a Rigid Multisinusoidal Wavy Surface withan Elastic Half-Plane
Ivan Y Tsukanov
Ishlinsky Institute for Problems in Mechanics RAS Prospekt Vernadskogo 101-1 Moscow 119526 Russia
Correspondence should be addressed to Ivan Y Tsukanov ivanyutsukanovgmailcom
Received 7 May 2018 Revised 24 July 2018 Accepted 4 October 2018 Published 18 October 2018
Academic Editor Patrick De Baets
Copyright copy 2018 Ivan Y Tsukanov This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The interaction effects arising at partial contact of rigid multisinusoidal wavy surface with an elastic half-plane are considered inthe assumption of continuous contact configuration The analytical exact and asymptotic solutions for periodic and nonperiodiccontact problems for wavy indenters are derived Continuous contact configuration appearing at small ratios of amplitude towavelength for cosine harmonics leads to continuous oscillatory contact pressure distribution and oscillatory relations betweenmean pressure and a contact length Comparison of periodic and nonperiodic solutions shows that long-range elastic interactionbetween asperities does not depend on a number of cosine wavelengths
1 Introduction
Real rough surfaces are three-dimensional and multiscaleBesides fully random rough surfaces [1] there are natural andtechnical surfaces having quasiregular character of asperitieson several scales (eg periodic anisotropic waviness) [2] Forthese surfaces the geometric model of two-dimensional wavy2D profile can be applied as a first approximation Also insome fields of engineering the wavy textures of differentshapes are used (eg in optical devices and MEMS) [3]Considering the elastic contact processes occurring for softmaterials (polymers and biological materials) the variousanalytical methods of plane elasticity can be applied Inthe case of full contact when the gap between surfaces isfilled the problem can be easily solved by Fourier transformmethod [4] However the very high applied pressure isrequired to reach the full contact condition even for softmaterials so partial contact is the more often case Partialcontact between wavy surfaces is a problem with mixedboundary conditions which was solved by different mathe-matical techniques
The classic periodic contact problem in plane elasticityis an old problem [5 6] Concerning geometry of a wavysurface considered in the previous studies the cosine [7 8]the squared cosine [9] and evenly spaced parabolic or wedge
stamps [10 11] were generally used The analytical solutionof the pointed problem for those surface geometries wasobtained by different methods For the cosine profile theyare complex stress function [7] dual series equation [12]intercontact gaps method [13] variable transform method[11] and fracture mechanics approach [14]
Taking into account simple wavy geometry (cosine orsquared cosine) the contact problems with more complicatedboundary conditions were studied sliding problem withfriction [15 16] with a fluid lubricant [17] with a partial slip[11 18] with adhesion and sliding friction [19] for viscoelasticmaterial [20] for Winkler model of viscoelastic material andadhesion [21] for elastic layer with presence of friction andwear [22] and dynamic problem for anisotropic half-plane[23]
The normal elastic problem for a two-dimensional non-sinusoidal wavy profile where a shape of a waveform iscontrolled by a parameter was solved analytically [24]It was established that the pressure distribution is highlysensitive to the shape of a wavy surface especially at largeloads
The presence of several scales of a wavy surface leads ingeneral to a multizone periodic contact problem [9 25] Theasymptotic approximate solution for initial contact for a two-scale wavy surface was obtained [24] It was shown that even
HindawiAdvances in TribologyVolume 2018 Article ID 8431467 8 pageshttpsdoiorg10115520188431467
2 Advances in Tribology
pinfin
2Δ1
a
0
z
x
E ]
1
p(x)
(a) (b) (c)
Figure 1 Contact of a cosine wavy profile having one (a) two (b) and three (c) harmonics with an elastic half-plane
for initial contact the interplay between harmonics existsConsidering the problem in a wide range of applied loadsit is necessary to use a numerical procedure Such studieswere performed by different techniques Fourier series andcotangent transform [26] full contact solution and iterationprocedure [27] FFT and variational principle [28] nonlinearboundary integral equation [29] boundary element method[30] and finite element method [31] The equations for inter-nal stresses for sinusoidal pressure distributions in 2D and3D cases were also derived [32] The results of these studiesshow that multiscale character of a wavy surface at partialcontact with an elastic half-plane leads to multiple peaks ofhigh pressureThe pressure distribution is jagged in this caseand a load-area dependence tends to proportionality at largenumber of harmonics [31]
In cases considered in the previous studies the contactwas partial at all scales because the amplitudes of differ-ent harmonics were comparable This situation leads to adiscontinuous (discrete) contact configuration [33] Besidesnumerical methods the other way to solve these problemsis usage of multiasperity contact models Based on thenature of the surfaces models can be deterministic andstatistical Review of statistical models based on individualasperity contact in comparison with the Perssonrsquos model andnumerical simulations is performed in [34] For the nearlycomplete contact case when the ratio of the real area ofcontact to the nominal contact area approaches unity thestatistical model based on fracture mechanics approach wasdeveloped [35] For deterministic multiscale surfaces (egmultisinusoidal self-affine surfaces) the Archardrsquos approachwas successfully implemented [36 37]
However if at a certain scale amplitude of the cosineharmonic is much smaller than its period full contacton this scale occurs and a continuous oscillating pressuredistribution on a larger scale will be observed [33 38] Fordistinguishing these cases the Johnson parameter couplingan amplitude a period of cosine harmonic and a reducedmodulus of elasticity with Hertzian pressure at the pointwhere maximum pressure occurs is used [33] In the presentstudy the continuous contact configuration observed at
small amplitudes of subsequent cosine harmonics is analyzedanalytically for periodic and nonperiodic multisinusoidalrigid indenters in contact with an elastic half-plane
2 Problem Formulation and Assumptions
The general scheme of the problem on the single period 1205821for one two and three cosine harmonics profile is presentedin Figure 1
The wavy surface is assumed to be rigid and the elastichalf-plane is an isotropic semi-infinite body with two elasticconstants Youngrsquos modulus 119864 and Poissonrsquos coefficient ]Also the plain strain condition is applied The amplitudes ofcosine harmonics are much smaller than their periods (Δ 119894 ≪120582119894 where i = 1 2 N is a harmonic sequence number)This condition makes it possible to apply the linear elasticitytheory The Johnson parameter 120594 = 120587EΔ 1198942pℎ120582119894 (where 119901ℎ =2pinfin120582119894120587a pinfin is an applied mean pressure and a is a contacthalf-width) should be120594 lt 1 [33] for preserving the continuouscontact configuration
The two different problems with similar geometry of arigid surface are considered For the problem with periodicboundary conditions the integral equation with Hilbertkernel is used [11]
1198642 (1 minus ]2) 120597ℎ (119909)120597119909 = 12120587 int119886
minus119886119901 (120585) cot 119909 minus 1205852 119889120585 (1)
where h(x) is an initial gap between surfaces and p(x) isa contact pressure distribution
For a nonperiodic indenter the integral equation withCauchy kernel is used [10]
1198642 (1 minus ]2) 120597ℎ (119909)120597119909 = 12120587 int119886
minus119886
119901 (120585)119909 minus 120585119889120585 (2)
Choosing for simplicity the largest wavelength 1205821 = 2120587one can write the expression for the gap function derivativefor the 119894th cosine harmonic
120597ℎ119894 (119909)120597119909 = 120597120597119909 (120575 minus Δ 119894 (1 minus cos 119899119894119909)) = minusΔ 119894119899119894 sin 119899119894119909 (3)
Advances in Tribology 3
where 120575 is a contact approach and 119899119894 = 1205821120582119894 In the givenformulation of the problem 119899119894 isin N
So on the basis of the superposition principle and takinginto account the assumed continuous contact configurationthe total contact pressure distribution can be obtained as asum of distributions of separate cosine harmonics
119901 (119909) = 1198642 (1 minus ]2)119873sum119894=1
119901119894 (119909) (4)
where 119873 is a number of wavelengths and 119901119894(x) is acomponent of pressure distribution for the 119894th wavelength
The vertical elastic displacements can be obtained via thefollowing expression [9 10]
119906119911 (119909) = 2 (1 minus ]2)120587119864 intinfin
minusinfin119901 (120585) ln 1003816100381610038161003816119909 minus 1205851003816100381610038161003816 d120585 + 119862
= 2 (1 minus ]2)120587119864
119873sum119894=1
119906119911119894 (119909) + 119862(5)
where 119862 is a constant depending on the selected datumpoint and 119906119911119894(119909) are the surface vertical displacements for the119894th wavelength
The mean (nominal) pressure pinfin is determined byinvoking the equilibrium equation
119901infin = 119864(1 minus ]2)119873sum119894=1
119901infin119894 = 14120587 119864(1 minus ]2)119873sum119894=1
int119886minus119886119901119894 (119909) 119889119909 (6)
where pinfin119894 is a component ofmean pressure corresponding tothe 119894th wavelength
3 Solutions of the Problem forthe 119894th Harmonic
31 Solution for a PeriodicWavy Surface Following equations(1) and (3) the integral equation for the 119894th harmonic is asfollows
minusΔ 119894119899119894 sin 119899119894119909 = 1120587 int119886
minus119886119901119894 (120585) cot 119909 minus 1205852 119889120585 (7)
The analytical solution for the contact pressure distribu-tion for the 119894th harmonic can be obtained via the reductionof equation (7) to the integral equation with Cauchy kernelusing the following variable transform [11]
119906 = tan1205852 V = tan1199092 120572 = tan1198862
(8)
Considering the symmetry of the profile the integralequation (7) is reduced to
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1 (1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (9)
where 119880119899119894 ndash is a Chebyshev polynomial of the second kindwith a degree 119899119894 [39]
Taking into account the considered assumptions thesolution of equation (9) can be obtained by means of theChebyshev expansion of the left side and the known spectralrelations for the Chebyshev polynomials (Appendix) Ininitial variables the contact pressure distribution for the 119894thharmonic is determined by
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (10)
where
119860 119894119895 = 2120587 int1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (11)
120593119894 (119904) = tan (1198862) 1199041 + (tan (1198862) 119904)2119880119899119894minus1 (
1 minus (tan (1198862) 119904)21 + (tan (1198862) 119904)2) (12)
where 119879119895 ndash is a Chebyshev polynomial of the first kindwith a degree j [39]
The total pressure distribution is obtained by usingequation (4) For numerical calculations it is necessary tohold finite terms of the infinite series in equation (10) For thearbitrary period the variables 119909 and 119886 in equations (10) and(12) should be multiplied on 21205871205821 The mean pressure pinfin119894and the vertical displacements 119906119911119894(119909) can be obtained usingnumerical integration in equations (5) and (6)Themaximumpressure is determined as the pressure at the point x = 0
32 Solution for a Wavy Rigid Nonperiodic Indenter Follow-ing equations (2) and (3) the integral equation for the 119894thharmonic is
minusΔ 119894119899119894 sin 119899119894119909 = 1120587 int119886
minus119886
119901119894 (120585)119909 minus 120585 119889120585 (13)
The solution of the equation (13) can be obtained usingan inversion without singularities on both endpoints [8 11]and the Chebyshev expansion of the left side [39] which canbe written explicitly
119901119894 (119909)= Δ 119894119899119894radic1 minus (119909119886)
2 infinsum119895=0
(minus1)minus119895 J2119895+1 (119886119899119894) 1198802119895 (119909119886) (14)
where J119895(t) is the Bessel function of the first kind of theinteger order 119895 and the argument t [39]
The displacements within the contact zone 119909 isin [minus119886 119886]can be determined analytically using equation (5) and the
4 Advances in Tribology
relations for Chebyshev polynomials [40] For the 119894th har-monic the final relation is
119906119911119894 (119909) = minusΔ 119894119899119894119886[[J1 (119886119899119894) (11990921198862 minus 12 minus ln 2119886)
+ infinsum119895=3
(minus1)(1minus119895)2 J119895 (119886119899119894)
sdot (119879119895+1 (119909119886)119895 + 1 minus 119879119895minus1 (119909119886)119895 minus 1 )]]
(15)
where signsum identifies the sum of terms with odd 119895 onlyAccording to equation (6) the mean pressure for the 119894th
harmonic pinfin119894 is calculated by integration of equation (14) andresulting in a simple expression
119901infin119894 = 025Δ 119894119886119899119894J1 (119886119899119894) (16)
The approximate close-form relation for the contactpressure distribution can be obtained assuming that thelargest values of pressure are concentrated near the point x= 0Then equation (14) can be represented as
119901119894 (119909) = Δ 119894119899119894radic1 minus (119909119886)2 120597120597119909
sdot infinsum119895=0
(minus1)(1minus119895)2 119886J119895 (119886119899119894) 119879119895 (sin (119909119886))119895 (17)
where signsum identifies the sum of terms with odd 119895 onlyUsing the known relations for theChebyshev polynomials
[41] the following expression can be written
119901119894 (119909)= Δ 119894119899119894radic1 minus (119909119886)
2 infinsum119895=0
J2119895+1 (119886119899119894) cos ((2119895 + 1) (119909119886)) (18)
With the use of an approximate relation between zeros ofBessel functions of integer order [42] the following expres-sion can be written
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [[J1 (119886119899119894)
+ infinsum119895=2
J2119895 (119886119899119894 + 1) cos (2119895 (119909119886))]]
(19)
Then applying the Jacobi-Anger expansion [42] theclose-form relation is
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [J1 (119886119899119894)
+ J2 (119886119899119894 + 1) cos (2 (119909119886))+ 05 cos((119886119899119894 + 1) sin (119909119886)) minus J0 (119886119899119894 + 1)]
(20)
The close-form integral relation for a maximum pressure(x = 0) can be determined exactly from equation (18)
119901119894max = Δ 119894119899119894 infinsum119895=0
J2119895+1 (119886119899119894) = 05Δ 119894119899119894 int1198861198991198940
J0 (119905) 119889119905 (21)
4 Results and Discussion
The evolution of the dimensionless contact pressure dis-tribution p(x)plowast (plowast = 120587EΔ 11205821) for a periodic problem(equations ((6) and (10)-(12)) 1205821 = 2120587 Δ 1 = 05) for variouscontact lengths (2a) and two different profiles f (x) is shownin Figure 2
The exact (solid lines equation (14)) and the approximate(dotted lines equation (20)) graphs of the dimensionless con-tact pressure p(x)plowast for different profiles of a nonperiodicwavy indenter are shown in Figure 3
Figures 2 and 3 illustrate that with increasing the num-ber of harmonics the pressure distribution becomes morecomplex and themaximumpressure grows significantly For asingle-scale periodic cosine profile (Figure 2(a)) the Wester-gaardrsquos solution is recovered For a single-scale nonperiodicindenter (Figure 3(a)) the Hertz solution is observed asthe cosine function is very close to the quadratic parabolaThereby the distributions presented in Figures 3(b) and3(c) correspond to wavy cylinder problem at small waviness[38] Comparison of the exact and the approximate values ofpressure for a single indenter (Figure 3) shows that equation(20) satisfactorily describes the behavior of the pressuredistribution
Comparing the periodic and the nonperiodic solutionsthe elastic interaction effect is of interest The mean pressurendash contact length curves for two profiles calculated fromperiodic and nonperiodic solutions are presented in Figure 4
Figure 4 shows that at small contact lengths (2a lt0251205821) the solutions are closeThepiece of graphs agreementdoes not depend on profile geometry With increase of loadthe periodic solution gives the smaller contact length due toelastic interaction on the largest scale For the profile withtwo cosine harmonics and continuous contact configurationpresented in this study the oscillations of mean pressure ndashcontact length curves are observed (Figure 4(b)) Curves inFigure 4(a) correspond to Westergaardrsquos (curve 1) and Hertz(curve 2) solutions recovered for profilewith onewavelength
Graphs of the mean and the maximum pressures versuscontact length on the interval 2alt 0251205821 for different profilesof a wavy nonperiodic indenter are shown in Figure 5
Advances in Tribology 5
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(a)
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(b)
Figure 2 Evolution of contact pressure distribution for wavy periodic profile (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 031205821 (3) 2a = 051205821 (4) 2a = 081205821(5)
minus025 025
xa
0
1
23
0
05
10
p(x)plowast
(a)
minus025 0250
xa
12
3
0
05
10
p(x)plowast
(b)
0
05
10
minus025 0250
xa
12
3
p(x)plowast
(c)
Figure 3 Evolution of contact pressure distribution for a nonperiodic wavy indenter (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (c) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 0251205821 (3)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(a)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(b)
Figure 4 Graphs of dimensionless mean pressure as a function of dimensionless contact length (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a)f (x) = Δ 1 cos(119909)+002Δ 1cos (11x) (b) 1 ndash periodic solution 2 ndash nonperiodic solution
6 Advances in Tribology
0 007 013 019
015
010
005
pinfinplowast
2a1
1
23
(a)
05
1
pmaxplowast
0 007 013 019 2a1
12
3
(b)
Figure 5 Dimensionless mean pressure (a) and maximum pressure (b) as a function of dimensionless contact length for profileswith different numbers of cosine harmonics (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (1) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
05
10
pmaxplowast
0 01 02 03
1
2
3
pinfinplowast
Figure 6 Dimensionless maximum pressure as a function ofdimensionless mean pressure for different profiles (1205821 = 2120587 Δ 1 =05) f (x) =Δ 1 cos(119909) (1) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x)= Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
Figure 5 shows that the maximum pressure depends onprofile geometry stronger than the mean pressure Howeveradding the third harmonic leads to insignificant change ofthe graphs character Continuous contact configuration at thepresence of several cosine wavelengths leads to oscillatorycharacter of the mean and the maximum pressure graphsCombining these two graphs numerically one can obtain thedependence of peak pressure frommean pressure (Figure 6)
Figure 6 shows that dependences of the maximum pres-sure from the mean pressure are not oscillatory for theprofiles with two and three wavelengths and additionalcosine harmonics change the graph considerably in value butnot in character This statement can be useful in the analysisof contact surfaces fracture processes [33]
5 Conclusions
The continuous contact configuration is one of the twopossible configurations arising at indentation of a multi-sinusoidal 2D wavy surface into an elastic half-plane Thisconfiguration leads to continuous oscillatory contact pressuredistribution Comparison of the derived periodic and nonpe-riodic solutions shows that the long-range elastic interactionbetween asperities does not depend on a number of cosinewavelengths and can be neglected at small loads (contactlengths) for arbitrary wavy profile geometry The assumptionof neglecting the long-range periodicity leads to exact equa-tions for determining the remote and themaximumpressuresfrom the contact length described by oscillatory functionsHowever the dependences of the maximum pressure fromthe mean pressure are not oscillatory for the profiles with twoand three wavelengths and resemble those for a simple cosineprofile of indenter The influence of the additional cosineharmonics on the maximum pressure is significantly largerthan on the mean pressure for the same contact zone lengthThe derived equations can be used at the analysis of contactcharacteristics of deterministic profiles of arbitrary geometryand also at the validation of more complex numerical modelsof rough surfaces contact
Appendix
Derivation of Contact Pressure Distributionfor the Periodic Problem
Themain integral equation of the considered contact problemfor the 119894th harmonic in transformed variables (8) is
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1(1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (A1)
Advances in Tribology 7
where 119880119899 is a Chebyshev polynomial of a second kindwith a degree 119899119894
The appropriate inversion of this integral equation has tobe nonsingular on both endpoints [11]
119901119894 (V) = minusΔ 119894119899119894120587 radic1205722 minus V2 int120572minus120572
V1 + V2119880119899119894minus1(1 minus V21 + V2
)sdot 1radic1205722 minus 1199062
1119906 minus V119889119906
(A2)
By introducing the new variables
119903 = V120572119904 = 119906120572
(A3)
the expression (A2) can be written in the following form
119901119894 (119903) = minusΔ 119894119899119894120587 radic1 minus 1199032 int1minus1120593119894 (119904) 1radic1 minus 1199042
1119904 minus 119903119889119904 (A4)
where the function 120593119894(s) is120593119894 (119904) = 1205721199041 + 12057221199042119880119899119894minus1(1 minus 120572
211990421 + 12057221199042) (A5)
Since the integrand function is defined on the interval [-11] and satisfies the Holder condition it can be represented asan expansion in Chebyshev polynomials of the first kind [41]
120593119894 (119904) = 119860 11989402 + 119860 11989411198791 (119904) + 119860 11989421198792 (119904) + (A6)
where119879119895 is a Chebyshev polynomial of the first kind witha degree j [39]
The coefficients 119860 119894119895 in equation (A6) are defined by thefollowing expression [43]
119860 1198940 = 0119860 119894119895 = 2120587 int
1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (A7)
With the use of integral relation between the Chebyshevpolynomials of the first and the second kind [41]
1120587 int1
minus1
119879119895 (119904)radic1 minus 11990421119904 minus 119903119889119904 = 119880119895minus1 (119903) 119895 = 0 1 2 (A8)
and equation (A4) the expression for the contact pressuredistribution for the 119894th harmonic is
119901119894 (119903) = minusΔ 119894119899119894radic1 minus 1199032 infinsum119895=1
119860 119894119895119880119895minus1 (119903) (A9)
Returning to the original variables and bearing in mindpositive pressures notation one can obtain
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (A10)
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Theresearchwas supported byRSF (project no 14-29-00198)
References
[1] B N J Persson ldquoTheory of rubber friction and contactmechanicsrdquoThe Journal of Chemical Physics vol 115 no 8 pp3840ndash3861 2001
[2] A C Rodrıguez Urribarrı E van der Heide X Zeng and MB de Rooij ldquoModelling the static contact between a fingertipand a rigid wavy surfacerdquo Tribology International vol 102 pp114ndash124 2016
[3] A S Adnan V Ramalingam J H Ko and S Subbiah ldquoNanotexture generation in single point diamond turning usingbackside patterned workpiecerdquo Manufacturing Letters vol 2no 1 pp 44ndash48 2013
[4] Y Ju and T N Farris ldquoSpectral Analysis of Two-DimensionalContact Problemsrdquo Journal of Tribology vol 118 no 2 p 3201996
[5] M A Sadowski ldquoZwiedimensionale probleme der elastizitat-shtheorierdquo ZAMMmdashZeitschrift fur Angewandte Mathematikund Mechanik vol 8 no 2 pp 107ndash121 1928
[6] N I Muskhelishvili Some Basic Problems of the MathematicalTheory of Elasticity Springer Dordrecht Netherlands 1977
[7] H M Westergaard ldquoBearing pressures and cracksrdquo Journal ofApplied Mechanics vol 6 pp 49ndash53 1939
[8] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1987
[9] I Y Schtaierman Contact Problem of Theory of ElasticityGostekhizdat Moscow Russia 1949
[10] L A Galin Contact problems Springer Netherlands Dor-drecht 2008
[11] J M Block and L M Keer ldquoPeriodic contact problems in planeelasticityrdquo Journal of Mechanics of Materials and Structures vol3 no 7 pp 1207ndash1237 2008
[12] J Dundurs K C Tsai and L M Keer ldquoContact between elasticbodies with wavy surfacesrdquo Journal of Elasticity vol 3 no 2 pp109ndash115 1973
[13] A A Krishtafovich R M Martynyak and R N Shvets ldquoCon-tact between anisotropic half-plane and rigid body with regularmicroreliefrdquo Journal of Friction and Wear vol 15 pp 15ndash211994
[14] Y Xu and R L Jackson ldquoPeriodic Contact Problems inPlane Elasticity The Fracture Mechanics Approachrdquo Journal ofTribology vol 140 no 1 p 011404 2018
[15] E A Kuznetsov ldquoPeriodic contact problem for half-planeallowing for forces of frictionrdquo Soviet Applied Mechanics vol12 no 10 pp 1014ndash1019 1976
[16] M Nosonovsky and G G Adams ldquoSteady-state frictionalsliding of two elastic bodies with a wavy contact interfacerdquoJournal of Tribology vol 122 no 3 pp 490ndash495 2000
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
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2 Advances in Tribology
pinfin
2Δ1
a
0
z
x
E ]
1
p(x)
(a) (b) (c)
Figure 1 Contact of a cosine wavy profile having one (a) two (b) and three (c) harmonics with an elastic half-plane
for initial contact the interplay between harmonics existsConsidering the problem in a wide range of applied loadsit is necessary to use a numerical procedure Such studieswere performed by different techniques Fourier series andcotangent transform [26] full contact solution and iterationprocedure [27] FFT and variational principle [28] nonlinearboundary integral equation [29] boundary element method[30] and finite element method [31] The equations for inter-nal stresses for sinusoidal pressure distributions in 2D and3D cases were also derived [32] The results of these studiesshow that multiscale character of a wavy surface at partialcontact with an elastic half-plane leads to multiple peaks ofhigh pressureThe pressure distribution is jagged in this caseand a load-area dependence tends to proportionality at largenumber of harmonics [31]
In cases considered in the previous studies the contactwas partial at all scales because the amplitudes of differ-ent harmonics were comparable This situation leads to adiscontinuous (discrete) contact configuration [33] Besidesnumerical methods the other way to solve these problemsis usage of multiasperity contact models Based on thenature of the surfaces models can be deterministic andstatistical Review of statistical models based on individualasperity contact in comparison with the Perssonrsquos model andnumerical simulations is performed in [34] For the nearlycomplete contact case when the ratio of the real area ofcontact to the nominal contact area approaches unity thestatistical model based on fracture mechanics approach wasdeveloped [35] For deterministic multiscale surfaces (egmultisinusoidal self-affine surfaces) the Archardrsquos approachwas successfully implemented [36 37]
However if at a certain scale amplitude of the cosineharmonic is much smaller than its period full contacton this scale occurs and a continuous oscillating pressuredistribution on a larger scale will be observed [33 38] Fordistinguishing these cases the Johnson parameter couplingan amplitude a period of cosine harmonic and a reducedmodulus of elasticity with Hertzian pressure at the pointwhere maximum pressure occurs is used [33] In the presentstudy the continuous contact configuration observed at
small amplitudes of subsequent cosine harmonics is analyzedanalytically for periodic and nonperiodic multisinusoidalrigid indenters in contact with an elastic half-plane
2 Problem Formulation and Assumptions
The general scheme of the problem on the single period 1205821for one two and three cosine harmonics profile is presentedin Figure 1
The wavy surface is assumed to be rigid and the elastichalf-plane is an isotropic semi-infinite body with two elasticconstants Youngrsquos modulus 119864 and Poissonrsquos coefficient ]Also the plain strain condition is applied The amplitudes ofcosine harmonics are much smaller than their periods (Δ 119894 ≪120582119894 where i = 1 2 N is a harmonic sequence number)This condition makes it possible to apply the linear elasticitytheory The Johnson parameter 120594 = 120587EΔ 1198942pℎ120582119894 (where 119901ℎ =2pinfin120582119894120587a pinfin is an applied mean pressure and a is a contacthalf-width) should be120594 lt 1 [33] for preserving the continuouscontact configuration
The two different problems with similar geometry of arigid surface are considered For the problem with periodicboundary conditions the integral equation with Hilbertkernel is used [11]
1198642 (1 minus ]2) 120597ℎ (119909)120597119909 = 12120587 int119886
minus119886119901 (120585) cot 119909 minus 1205852 119889120585 (1)
where h(x) is an initial gap between surfaces and p(x) isa contact pressure distribution
For a nonperiodic indenter the integral equation withCauchy kernel is used [10]
1198642 (1 minus ]2) 120597ℎ (119909)120597119909 = 12120587 int119886
minus119886
119901 (120585)119909 minus 120585119889120585 (2)
Choosing for simplicity the largest wavelength 1205821 = 2120587one can write the expression for the gap function derivativefor the 119894th cosine harmonic
120597ℎ119894 (119909)120597119909 = 120597120597119909 (120575 minus Δ 119894 (1 minus cos 119899119894119909)) = minusΔ 119894119899119894 sin 119899119894119909 (3)
Advances in Tribology 3
where 120575 is a contact approach and 119899119894 = 1205821120582119894 In the givenformulation of the problem 119899119894 isin N
So on the basis of the superposition principle and takinginto account the assumed continuous contact configurationthe total contact pressure distribution can be obtained as asum of distributions of separate cosine harmonics
119901 (119909) = 1198642 (1 minus ]2)119873sum119894=1
119901119894 (119909) (4)
where 119873 is a number of wavelengths and 119901119894(x) is acomponent of pressure distribution for the 119894th wavelength
The vertical elastic displacements can be obtained via thefollowing expression [9 10]
119906119911 (119909) = 2 (1 minus ]2)120587119864 intinfin
minusinfin119901 (120585) ln 1003816100381610038161003816119909 minus 1205851003816100381610038161003816 d120585 + 119862
= 2 (1 minus ]2)120587119864
119873sum119894=1
119906119911119894 (119909) + 119862(5)
where 119862 is a constant depending on the selected datumpoint and 119906119911119894(119909) are the surface vertical displacements for the119894th wavelength
The mean (nominal) pressure pinfin is determined byinvoking the equilibrium equation
119901infin = 119864(1 minus ]2)119873sum119894=1
119901infin119894 = 14120587 119864(1 minus ]2)119873sum119894=1
int119886minus119886119901119894 (119909) 119889119909 (6)
where pinfin119894 is a component ofmean pressure corresponding tothe 119894th wavelength
3 Solutions of the Problem forthe 119894th Harmonic
31 Solution for a PeriodicWavy Surface Following equations(1) and (3) the integral equation for the 119894th harmonic is asfollows
minusΔ 119894119899119894 sin 119899119894119909 = 1120587 int119886
minus119886119901119894 (120585) cot 119909 minus 1205852 119889120585 (7)
The analytical solution for the contact pressure distribu-tion for the 119894th harmonic can be obtained via the reductionof equation (7) to the integral equation with Cauchy kernelusing the following variable transform [11]
119906 = tan1205852 V = tan1199092 120572 = tan1198862
(8)
Considering the symmetry of the profile the integralequation (7) is reduced to
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1 (1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (9)
where 119880119899119894 ndash is a Chebyshev polynomial of the second kindwith a degree 119899119894 [39]
Taking into account the considered assumptions thesolution of equation (9) can be obtained by means of theChebyshev expansion of the left side and the known spectralrelations for the Chebyshev polynomials (Appendix) Ininitial variables the contact pressure distribution for the 119894thharmonic is determined by
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (10)
where
119860 119894119895 = 2120587 int1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (11)
120593119894 (119904) = tan (1198862) 1199041 + (tan (1198862) 119904)2119880119899119894minus1 (
1 minus (tan (1198862) 119904)21 + (tan (1198862) 119904)2) (12)
where 119879119895 ndash is a Chebyshev polynomial of the first kindwith a degree j [39]
The total pressure distribution is obtained by usingequation (4) For numerical calculations it is necessary tohold finite terms of the infinite series in equation (10) For thearbitrary period the variables 119909 and 119886 in equations (10) and(12) should be multiplied on 21205871205821 The mean pressure pinfin119894and the vertical displacements 119906119911119894(119909) can be obtained usingnumerical integration in equations (5) and (6)Themaximumpressure is determined as the pressure at the point x = 0
32 Solution for a Wavy Rigid Nonperiodic Indenter Follow-ing equations (2) and (3) the integral equation for the 119894thharmonic is
minusΔ 119894119899119894 sin 119899119894119909 = 1120587 int119886
minus119886
119901119894 (120585)119909 minus 120585 119889120585 (13)
The solution of the equation (13) can be obtained usingan inversion without singularities on both endpoints [8 11]and the Chebyshev expansion of the left side [39] which canbe written explicitly
119901119894 (119909)= Δ 119894119899119894radic1 minus (119909119886)
2 infinsum119895=0
(minus1)minus119895 J2119895+1 (119886119899119894) 1198802119895 (119909119886) (14)
where J119895(t) is the Bessel function of the first kind of theinteger order 119895 and the argument t [39]
The displacements within the contact zone 119909 isin [minus119886 119886]can be determined analytically using equation (5) and the
4 Advances in Tribology
relations for Chebyshev polynomials [40] For the 119894th har-monic the final relation is
119906119911119894 (119909) = minusΔ 119894119899119894119886[[J1 (119886119899119894) (11990921198862 minus 12 minus ln 2119886)
+ infinsum119895=3
(minus1)(1minus119895)2 J119895 (119886119899119894)
sdot (119879119895+1 (119909119886)119895 + 1 minus 119879119895minus1 (119909119886)119895 minus 1 )]]
(15)
where signsum identifies the sum of terms with odd 119895 onlyAccording to equation (6) the mean pressure for the 119894th
harmonic pinfin119894 is calculated by integration of equation (14) andresulting in a simple expression
119901infin119894 = 025Δ 119894119886119899119894J1 (119886119899119894) (16)
The approximate close-form relation for the contactpressure distribution can be obtained assuming that thelargest values of pressure are concentrated near the point x= 0Then equation (14) can be represented as
119901119894 (119909) = Δ 119894119899119894radic1 minus (119909119886)2 120597120597119909
sdot infinsum119895=0
(minus1)(1minus119895)2 119886J119895 (119886119899119894) 119879119895 (sin (119909119886))119895 (17)
where signsum identifies the sum of terms with odd 119895 onlyUsing the known relations for theChebyshev polynomials
[41] the following expression can be written
119901119894 (119909)= Δ 119894119899119894radic1 minus (119909119886)
2 infinsum119895=0
J2119895+1 (119886119899119894) cos ((2119895 + 1) (119909119886)) (18)
With the use of an approximate relation between zeros ofBessel functions of integer order [42] the following expres-sion can be written
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [[J1 (119886119899119894)
+ infinsum119895=2
J2119895 (119886119899119894 + 1) cos (2119895 (119909119886))]]
(19)
Then applying the Jacobi-Anger expansion [42] theclose-form relation is
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [J1 (119886119899119894)
+ J2 (119886119899119894 + 1) cos (2 (119909119886))+ 05 cos((119886119899119894 + 1) sin (119909119886)) minus J0 (119886119899119894 + 1)]
(20)
The close-form integral relation for a maximum pressure(x = 0) can be determined exactly from equation (18)
119901119894max = Δ 119894119899119894 infinsum119895=0
J2119895+1 (119886119899119894) = 05Δ 119894119899119894 int1198861198991198940
J0 (119905) 119889119905 (21)
4 Results and Discussion
The evolution of the dimensionless contact pressure dis-tribution p(x)plowast (plowast = 120587EΔ 11205821) for a periodic problem(equations ((6) and (10)-(12)) 1205821 = 2120587 Δ 1 = 05) for variouscontact lengths (2a) and two different profiles f (x) is shownin Figure 2
The exact (solid lines equation (14)) and the approximate(dotted lines equation (20)) graphs of the dimensionless con-tact pressure p(x)plowast for different profiles of a nonperiodicwavy indenter are shown in Figure 3
Figures 2 and 3 illustrate that with increasing the num-ber of harmonics the pressure distribution becomes morecomplex and themaximumpressure grows significantly For asingle-scale periodic cosine profile (Figure 2(a)) the Wester-gaardrsquos solution is recovered For a single-scale nonperiodicindenter (Figure 3(a)) the Hertz solution is observed asthe cosine function is very close to the quadratic parabolaThereby the distributions presented in Figures 3(b) and3(c) correspond to wavy cylinder problem at small waviness[38] Comparison of the exact and the approximate values ofpressure for a single indenter (Figure 3) shows that equation(20) satisfactorily describes the behavior of the pressuredistribution
Comparing the periodic and the nonperiodic solutionsthe elastic interaction effect is of interest The mean pressurendash contact length curves for two profiles calculated fromperiodic and nonperiodic solutions are presented in Figure 4
Figure 4 shows that at small contact lengths (2a lt0251205821) the solutions are closeThepiece of graphs agreementdoes not depend on profile geometry With increase of loadthe periodic solution gives the smaller contact length due toelastic interaction on the largest scale For the profile withtwo cosine harmonics and continuous contact configurationpresented in this study the oscillations of mean pressure ndashcontact length curves are observed (Figure 4(b)) Curves inFigure 4(a) correspond to Westergaardrsquos (curve 1) and Hertz(curve 2) solutions recovered for profilewith onewavelength
Graphs of the mean and the maximum pressures versuscontact length on the interval 2alt 0251205821 for different profilesof a wavy nonperiodic indenter are shown in Figure 5
Advances in Tribology 5
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(a)
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(b)
Figure 2 Evolution of contact pressure distribution for wavy periodic profile (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 031205821 (3) 2a = 051205821 (4) 2a = 081205821(5)
minus025 025
xa
0
1
23
0
05
10
p(x)plowast
(a)
minus025 0250
xa
12
3
0
05
10
p(x)plowast
(b)
0
05
10
minus025 0250
xa
12
3
p(x)plowast
(c)
Figure 3 Evolution of contact pressure distribution for a nonperiodic wavy indenter (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (c) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 0251205821 (3)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(a)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(b)
Figure 4 Graphs of dimensionless mean pressure as a function of dimensionless contact length (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a)f (x) = Δ 1 cos(119909)+002Δ 1cos (11x) (b) 1 ndash periodic solution 2 ndash nonperiodic solution
6 Advances in Tribology
0 007 013 019
015
010
005
pinfinplowast
2a1
1
23
(a)
05
1
pmaxplowast
0 007 013 019 2a1
12
3
(b)
Figure 5 Dimensionless mean pressure (a) and maximum pressure (b) as a function of dimensionless contact length for profileswith different numbers of cosine harmonics (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (1) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
05
10
pmaxplowast
0 01 02 03
1
2
3
pinfinplowast
Figure 6 Dimensionless maximum pressure as a function ofdimensionless mean pressure for different profiles (1205821 = 2120587 Δ 1 =05) f (x) =Δ 1 cos(119909) (1) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x)= Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
Figure 5 shows that the maximum pressure depends onprofile geometry stronger than the mean pressure Howeveradding the third harmonic leads to insignificant change ofthe graphs character Continuous contact configuration at thepresence of several cosine wavelengths leads to oscillatorycharacter of the mean and the maximum pressure graphsCombining these two graphs numerically one can obtain thedependence of peak pressure frommean pressure (Figure 6)
Figure 6 shows that dependences of the maximum pres-sure from the mean pressure are not oscillatory for theprofiles with two and three wavelengths and additionalcosine harmonics change the graph considerably in value butnot in character This statement can be useful in the analysisof contact surfaces fracture processes [33]
5 Conclusions
The continuous contact configuration is one of the twopossible configurations arising at indentation of a multi-sinusoidal 2D wavy surface into an elastic half-plane Thisconfiguration leads to continuous oscillatory contact pressuredistribution Comparison of the derived periodic and nonpe-riodic solutions shows that the long-range elastic interactionbetween asperities does not depend on a number of cosinewavelengths and can be neglected at small loads (contactlengths) for arbitrary wavy profile geometry The assumptionof neglecting the long-range periodicity leads to exact equa-tions for determining the remote and themaximumpressuresfrom the contact length described by oscillatory functionsHowever the dependences of the maximum pressure fromthe mean pressure are not oscillatory for the profiles with twoand three wavelengths and resemble those for a simple cosineprofile of indenter The influence of the additional cosineharmonics on the maximum pressure is significantly largerthan on the mean pressure for the same contact zone lengthThe derived equations can be used at the analysis of contactcharacteristics of deterministic profiles of arbitrary geometryand also at the validation of more complex numerical modelsof rough surfaces contact
Appendix
Derivation of Contact Pressure Distributionfor the Periodic Problem
Themain integral equation of the considered contact problemfor the 119894th harmonic in transformed variables (8) is
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1(1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (A1)
Advances in Tribology 7
where 119880119899 is a Chebyshev polynomial of a second kindwith a degree 119899119894
The appropriate inversion of this integral equation has tobe nonsingular on both endpoints [11]
119901119894 (V) = minusΔ 119894119899119894120587 radic1205722 minus V2 int120572minus120572
V1 + V2119880119899119894minus1(1 minus V21 + V2
)sdot 1radic1205722 minus 1199062
1119906 minus V119889119906
(A2)
By introducing the new variables
119903 = V120572119904 = 119906120572
(A3)
the expression (A2) can be written in the following form
119901119894 (119903) = minusΔ 119894119899119894120587 radic1 minus 1199032 int1minus1120593119894 (119904) 1radic1 minus 1199042
1119904 minus 119903119889119904 (A4)
where the function 120593119894(s) is120593119894 (119904) = 1205721199041 + 12057221199042119880119899119894minus1(1 minus 120572
211990421 + 12057221199042) (A5)
Since the integrand function is defined on the interval [-11] and satisfies the Holder condition it can be represented asan expansion in Chebyshev polynomials of the first kind [41]
120593119894 (119904) = 119860 11989402 + 119860 11989411198791 (119904) + 119860 11989421198792 (119904) + (A6)
where119879119895 is a Chebyshev polynomial of the first kind witha degree j [39]
The coefficients 119860 119894119895 in equation (A6) are defined by thefollowing expression [43]
119860 1198940 = 0119860 119894119895 = 2120587 int
1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (A7)
With the use of integral relation between the Chebyshevpolynomials of the first and the second kind [41]
1120587 int1
minus1
119879119895 (119904)radic1 minus 11990421119904 minus 119903119889119904 = 119880119895minus1 (119903) 119895 = 0 1 2 (A8)
and equation (A4) the expression for the contact pressuredistribution for the 119894th harmonic is
119901119894 (119903) = minusΔ 119894119899119894radic1 minus 1199032 infinsum119895=1
119860 119894119895119880119895minus1 (119903) (A9)
Returning to the original variables and bearing in mindpositive pressures notation one can obtain
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (A10)
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Theresearchwas supported byRSF (project no 14-29-00198)
References
[1] B N J Persson ldquoTheory of rubber friction and contactmechanicsrdquoThe Journal of Chemical Physics vol 115 no 8 pp3840ndash3861 2001
[2] A C Rodrıguez Urribarrı E van der Heide X Zeng and MB de Rooij ldquoModelling the static contact between a fingertipand a rigid wavy surfacerdquo Tribology International vol 102 pp114ndash124 2016
[3] A S Adnan V Ramalingam J H Ko and S Subbiah ldquoNanotexture generation in single point diamond turning usingbackside patterned workpiecerdquo Manufacturing Letters vol 2no 1 pp 44ndash48 2013
[4] Y Ju and T N Farris ldquoSpectral Analysis of Two-DimensionalContact Problemsrdquo Journal of Tribology vol 118 no 2 p 3201996
[5] M A Sadowski ldquoZwiedimensionale probleme der elastizitat-shtheorierdquo ZAMMmdashZeitschrift fur Angewandte Mathematikund Mechanik vol 8 no 2 pp 107ndash121 1928
[6] N I Muskhelishvili Some Basic Problems of the MathematicalTheory of Elasticity Springer Dordrecht Netherlands 1977
[7] H M Westergaard ldquoBearing pressures and cracksrdquo Journal ofApplied Mechanics vol 6 pp 49ndash53 1939
[8] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1987
[9] I Y Schtaierman Contact Problem of Theory of ElasticityGostekhizdat Moscow Russia 1949
[10] L A Galin Contact problems Springer Netherlands Dor-drecht 2008
[11] J M Block and L M Keer ldquoPeriodic contact problems in planeelasticityrdquo Journal of Mechanics of Materials and Structures vol3 no 7 pp 1207ndash1237 2008
[12] J Dundurs K C Tsai and L M Keer ldquoContact between elasticbodies with wavy surfacesrdquo Journal of Elasticity vol 3 no 2 pp109ndash115 1973
[13] A A Krishtafovich R M Martynyak and R N Shvets ldquoCon-tact between anisotropic half-plane and rigid body with regularmicroreliefrdquo Journal of Friction and Wear vol 15 pp 15ndash211994
[14] Y Xu and R L Jackson ldquoPeriodic Contact Problems inPlane Elasticity The Fracture Mechanics Approachrdquo Journal ofTribology vol 140 no 1 p 011404 2018
[15] E A Kuznetsov ldquoPeriodic contact problem for half-planeallowing for forces of frictionrdquo Soviet Applied Mechanics vol12 no 10 pp 1014ndash1019 1976
[16] M Nosonovsky and G G Adams ldquoSteady-state frictionalsliding of two elastic bodies with a wavy contact interfacerdquoJournal of Tribology vol 122 no 3 pp 490ndash495 2000
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
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Advances in Tribology 3
where 120575 is a contact approach and 119899119894 = 1205821120582119894 In the givenformulation of the problem 119899119894 isin N
So on the basis of the superposition principle and takinginto account the assumed continuous contact configurationthe total contact pressure distribution can be obtained as asum of distributions of separate cosine harmonics
119901 (119909) = 1198642 (1 minus ]2)119873sum119894=1
119901119894 (119909) (4)
where 119873 is a number of wavelengths and 119901119894(x) is acomponent of pressure distribution for the 119894th wavelength
The vertical elastic displacements can be obtained via thefollowing expression [9 10]
119906119911 (119909) = 2 (1 minus ]2)120587119864 intinfin
minusinfin119901 (120585) ln 1003816100381610038161003816119909 minus 1205851003816100381610038161003816 d120585 + 119862
= 2 (1 minus ]2)120587119864
119873sum119894=1
119906119911119894 (119909) + 119862(5)
where 119862 is a constant depending on the selected datumpoint and 119906119911119894(119909) are the surface vertical displacements for the119894th wavelength
The mean (nominal) pressure pinfin is determined byinvoking the equilibrium equation
119901infin = 119864(1 minus ]2)119873sum119894=1
119901infin119894 = 14120587 119864(1 minus ]2)119873sum119894=1
int119886minus119886119901119894 (119909) 119889119909 (6)
where pinfin119894 is a component ofmean pressure corresponding tothe 119894th wavelength
3 Solutions of the Problem forthe 119894th Harmonic
31 Solution for a PeriodicWavy Surface Following equations(1) and (3) the integral equation for the 119894th harmonic is asfollows
minusΔ 119894119899119894 sin 119899119894119909 = 1120587 int119886
minus119886119901119894 (120585) cot 119909 minus 1205852 119889120585 (7)
The analytical solution for the contact pressure distribu-tion for the 119894th harmonic can be obtained via the reductionof equation (7) to the integral equation with Cauchy kernelusing the following variable transform [11]
119906 = tan1205852 V = tan1199092 120572 = tan1198862
(8)
Considering the symmetry of the profile the integralequation (7) is reduced to
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1 (1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (9)
where 119880119899119894 ndash is a Chebyshev polynomial of the second kindwith a degree 119899119894 [39]
Taking into account the considered assumptions thesolution of equation (9) can be obtained by means of theChebyshev expansion of the left side and the known spectralrelations for the Chebyshev polynomials (Appendix) Ininitial variables the contact pressure distribution for the 119894thharmonic is determined by
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (10)
where
119860 119894119895 = 2120587 int1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (11)
120593119894 (119904) = tan (1198862) 1199041 + (tan (1198862) 119904)2119880119899119894minus1 (
1 minus (tan (1198862) 119904)21 + (tan (1198862) 119904)2) (12)
where 119879119895 ndash is a Chebyshev polynomial of the first kindwith a degree j [39]
The total pressure distribution is obtained by usingequation (4) For numerical calculations it is necessary tohold finite terms of the infinite series in equation (10) For thearbitrary period the variables 119909 and 119886 in equations (10) and(12) should be multiplied on 21205871205821 The mean pressure pinfin119894and the vertical displacements 119906119911119894(119909) can be obtained usingnumerical integration in equations (5) and (6)Themaximumpressure is determined as the pressure at the point x = 0
32 Solution for a Wavy Rigid Nonperiodic Indenter Follow-ing equations (2) and (3) the integral equation for the 119894thharmonic is
minusΔ 119894119899119894 sin 119899119894119909 = 1120587 int119886
minus119886
119901119894 (120585)119909 minus 120585 119889120585 (13)
The solution of the equation (13) can be obtained usingan inversion without singularities on both endpoints [8 11]and the Chebyshev expansion of the left side [39] which canbe written explicitly
119901119894 (119909)= Δ 119894119899119894radic1 minus (119909119886)
2 infinsum119895=0
(minus1)minus119895 J2119895+1 (119886119899119894) 1198802119895 (119909119886) (14)
where J119895(t) is the Bessel function of the first kind of theinteger order 119895 and the argument t [39]
The displacements within the contact zone 119909 isin [minus119886 119886]can be determined analytically using equation (5) and the
4 Advances in Tribology
relations for Chebyshev polynomials [40] For the 119894th har-monic the final relation is
119906119911119894 (119909) = minusΔ 119894119899119894119886[[J1 (119886119899119894) (11990921198862 minus 12 minus ln 2119886)
+ infinsum119895=3
(minus1)(1minus119895)2 J119895 (119886119899119894)
sdot (119879119895+1 (119909119886)119895 + 1 minus 119879119895minus1 (119909119886)119895 minus 1 )]]
(15)
where signsum identifies the sum of terms with odd 119895 onlyAccording to equation (6) the mean pressure for the 119894th
harmonic pinfin119894 is calculated by integration of equation (14) andresulting in a simple expression
119901infin119894 = 025Δ 119894119886119899119894J1 (119886119899119894) (16)
The approximate close-form relation for the contactpressure distribution can be obtained assuming that thelargest values of pressure are concentrated near the point x= 0Then equation (14) can be represented as
119901119894 (119909) = Δ 119894119899119894radic1 minus (119909119886)2 120597120597119909
sdot infinsum119895=0
(minus1)(1minus119895)2 119886J119895 (119886119899119894) 119879119895 (sin (119909119886))119895 (17)
where signsum identifies the sum of terms with odd 119895 onlyUsing the known relations for theChebyshev polynomials
[41] the following expression can be written
119901119894 (119909)= Δ 119894119899119894radic1 minus (119909119886)
2 infinsum119895=0
J2119895+1 (119886119899119894) cos ((2119895 + 1) (119909119886)) (18)
With the use of an approximate relation between zeros ofBessel functions of integer order [42] the following expres-sion can be written
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [[J1 (119886119899119894)
+ infinsum119895=2
J2119895 (119886119899119894 + 1) cos (2119895 (119909119886))]]
(19)
Then applying the Jacobi-Anger expansion [42] theclose-form relation is
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [J1 (119886119899119894)
+ J2 (119886119899119894 + 1) cos (2 (119909119886))+ 05 cos((119886119899119894 + 1) sin (119909119886)) minus J0 (119886119899119894 + 1)]
(20)
The close-form integral relation for a maximum pressure(x = 0) can be determined exactly from equation (18)
119901119894max = Δ 119894119899119894 infinsum119895=0
J2119895+1 (119886119899119894) = 05Δ 119894119899119894 int1198861198991198940
J0 (119905) 119889119905 (21)
4 Results and Discussion
The evolution of the dimensionless contact pressure dis-tribution p(x)plowast (plowast = 120587EΔ 11205821) for a periodic problem(equations ((6) and (10)-(12)) 1205821 = 2120587 Δ 1 = 05) for variouscontact lengths (2a) and two different profiles f (x) is shownin Figure 2
The exact (solid lines equation (14)) and the approximate(dotted lines equation (20)) graphs of the dimensionless con-tact pressure p(x)plowast for different profiles of a nonperiodicwavy indenter are shown in Figure 3
Figures 2 and 3 illustrate that with increasing the num-ber of harmonics the pressure distribution becomes morecomplex and themaximumpressure grows significantly For asingle-scale periodic cosine profile (Figure 2(a)) the Wester-gaardrsquos solution is recovered For a single-scale nonperiodicindenter (Figure 3(a)) the Hertz solution is observed asthe cosine function is very close to the quadratic parabolaThereby the distributions presented in Figures 3(b) and3(c) correspond to wavy cylinder problem at small waviness[38] Comparison of the exact and the approximate values ofpressure for a single indenter (Figure 3) shows that equation(20) satisfactorily describes the behavior of the pressuredistribution
Comparing the periodic and the nonperiodic solutionsthe elastic interaction effect is of interest The mean pressurendash contact length curves for two profiles calculated fromperiodic and nonperiodic solutions are presented in Figure 4
Figure 4 shows that at small contact lengths (2a lt0251205821) the solutions are closeThepiece of graphs agreementdoes not depend on profile geometry With increase of loadthe periodic solution gives the smaller contact length due toelastic interaction on the largest scale For the profile withtwo cosine harmonics and continuous contact configurationpresented in this study the oscillations of mean pressure ndashcontact length curves are observed (Figure 4(b)) Curves inFigure 4(a) correspond to Westergaardrsquos (curve 1) and Hertz(curve 2) solutions recovered for profilewith onewavelength
Graphs of the mean and the maximum pressures versuscontact length on the interval 2alt 0251205821 for different profilesof a wavy nonperiodic indenter are shown in Figure 5
Advances in Tribology 5
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(a)
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(b)
Figure 2 Evolution of contact pressure distribution for wavy periodic profile (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 031205821 (3) 2a = 051205821 (4) 2a = 081205821(5)
minus025 025
xa
0
1
23
0
05
10
p(x)plowast
(a)
minus025 0250
xa
12
3
0
05
10
p(x)plowast
(b)
0
05
10
minus025 0250
xa
12
3
p(x)plowast
(c)
Figure 3 Evolution of contact pressure distribution for a nonperiodic wavy indenter (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (c) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 0251205821 (3)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(a)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(b)
Figure 4 Graphs of dimensionless mean pressure as a function of dimensionless contact length (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a)f (x) = Δ 1 cos(119909)+002Δ 1cos (11x) (b) 1 ndash periodic solution 2 ndash nonperiodic solution
6 Advances in Tribology
0 007 013 019
015
010
005
pinfinplowast
2a1
1
23
(a)
05
1
pmaxplowast
0 007 013 019 2a1
12
3
(b)
Figure 5 Dimensionless mean pressure (a) and maximum pressure (b) as a function of dimensionless contact length for profileswith different numbers of cosine harmonics (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (1) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
05
10
pmaxplowast
0 01 02 03
1
2
3
pinfinplowast
Figure 6 Dimensionless maximum pressure as a function ofdimensionless mean pressure for different profiles (1205821 = 2120587 Δ 1 =05) f (x) =Δ 1 cos(119909) (1) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x)= Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
Figure 5 shows that the maximum pressure depends onprofile geometry stronger than the mean pressure Howeveradding the third harmonic leads to insignificant change ofthe graphs character Continuous contact configuration at thepresence of several cosine wavelengths leads to oscillatorycharacter of the mean and the maximum pressure graphsCombining these two graphs numerically one can obtain thedependence of peak pressure frommean pressure (Figure 6)
Figure 6 shows that dependences of the maximum pres-sure from the mean pressure are not oscillatory for theprofiles with two and three wavelengths and additionalcosine harmonics change the graph considerably in value butnot in character This statement can be useful in the analysisof contact surfaces fracture processes [33]
5 Conclusions
The continuous contact configuration is one of the twopossible configurations arising at indentation of a multi-sinusoidal 2D wavy surface into an elastic half-plane Thisconfiguration leads to continuous oscillatory contact pressuredistribution Comparison of the derived periodic and nonpe-riodic solutions shows that the long-range elastic interactionbetween asperities does not depend on a number of cosinewavelengths and can be neglected at small loads (contactlengths) for arbitrary wavy profile geometry The assumptionof neglecting the long-range periodicity leads to exact equa-tions for determining the remote and themaximumpressuresfrom the contact length described by oscillatory functionsHowever the dependences of the maximum pressure fromthe mean pressure are not oscillatory for the profiles with twoand three wavelengths and resemble those for a simple cosineprofile of indenter The influence of the additional cosineharmonics on the maximum pressure is significantly largerthan on the mean pressure for the same contact zone lengthThe derived equations can be used at the analysis of contactcharacteristics of deterministic profiles of arbitrary geometryand also at the validation of more complex numerical modelsof rough surfaces contact
Appendix
Derivation of Contact Pressure Distributionfor the Periodic Problem
Themain integral equation of the considered contact problemfor the 119894th harmonic in transformed variables (8) is
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1(1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (A1)
Advances in Tribology 7
where 119880119899 is a Chebyshev polynomial of a second kindwith a degree 119899119894
The appropriate inversion of this integral equation has tobe nonsingular on both endpoints [11]
119901119894 (V) = minusΔ 119894119899119894120587 radic1205722 minus V2 int120572minus120572
V1 + V2119880119899119894minus1(1 minus V21 + V2
)sdot 1radic1205722 minus 1199062
1119906 minus V119889119906
(A2)
By introducing the new variables
119903 = V120572119904 = 119906120572
(A3)
the expression (A2) can be written in the following form
119901119894 (119903) = minusΔ 119894119899119894120587 radic1 minus 1199032 int1minus1120593119894 (119904) 1radic1 minus 1199042
1119904 minus 119903119889119904 (A4)
where the function 120593119894(s) is120593119894 (119904) = 1205721199041 + 12057221199042119880119899119894minus1(1 minus 120572
211990421 + 12057221199042) (A5)
Since the integrand function is defined on the interval [-11] and satisfies the Holder condition it can be represented asan expansion in Chebyshev polynomials of the first kind [41]
120593119894 (119904) = 119860 11989402 + 119860 11989411198791 (119904) + 119860 11989421198792 (119904) + (A6)
where119879119895 is a Chebyshev polynomial of the first kind witha degree j [39]
The coefficients 119860 119894119895 in equation (A6) are defined by thefollowing expression [43]
119860 1198940 = 0119860 119894119895 = 2120587 int
1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (A7)
With the use of integral relation between the Chebyshevpolynomials of the first and the second kind [41]
1120587 int1
minus1
119879119895 (119904)radic1 minus 11990421119904 minus 119903119889119904 = 119880119895minus1 (119903) 119895 = 0 1 2 (A8)
and equation (A4) the expression for the contact pressuredistribution for the 119894th harmonic is
119901119894 (119903) = minusΔ 119894119899119894radic1 minus 1199032 infinsum119895=1
119860 119894119895119880119895minus1 (119903) (A9)
Returning to the original variables and bearing in mindpositive pressures notation one can obtain
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (A10)
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Theresearchwas supported byRSF (project no 14-29-00198)
References
[1] B N J Persson ldquoTheory of rubber friction and contactmechanicsrdquoThe Journal of Chemical Physics vol 115 no 8 pp3840ndash3861 2001
[2] A C Rodrıguez Urribarrı E van der Heide X Zeng and MB de Rooij ldquoModelling the static contact between a fingertipand a rigid wavy surfacerdquo Tribology International vol 102 pp114ndash124 2016
[3] A S Adnan V Ramalingam J H Ko and S Subbiah ldquoNanotexture generation in single point diamond turning usingbackside patterned workpiecerdquo Manufacturing Letters vol 2no 1 pp 44ndash48 2013
[4] Y Ju and T N Farris ldquoSpectral Analysis of Two-DimensionalContact Problemsrdquo Journal of Tribology vol 118 no 2 p 3201996
[5] M A Sadowski ldquoZwiedimensionale probleme der elastizitat-shtheorierdquo ZAMMmdashZeitschrift fur Angewandte Mathematikund Mechanik vol 8 no 2 pp 107ndash121 1928
[6] N I Muskhelishvili Some Basic Problems of the MathematicalTheory of Elasticity Springer Dordrecht Netherlands 1977
[7] H M Westergaard ldquoBearing pressures and cracksrdquo Journal ofApplied Mechanics vol 6 pp 49ndash53 1939
[8] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1987
[9] I Y Schtaierman Contact Problem of Theory of ElasticityGostekhizdat Moscow Russia 1949
[10] L A Galin Contact problems Springer Netherlands Dor-drecht 2008
[11] J M Block and L M Keer ldquoPeriodic contact problems in planeelasticityrdquo Journal of Mechanics of Materials and Structures vol3 no 7 pp 1207ndash1237 2008
[12] J Dundurs K C Tsai and L M Keer ldquoContact between elasticbodies with wavy surfacesrdquo Journal of Elasticity vol 3 no 2 pp109ndash115 1973
[13] A A Krishtafovich R M Martynyak and R N Shvets ldquoCon-tact between anisotropic half-plane and rigid body with regularmicroreliefrdquo Journal of Friction and Wear vol 15 pp 15ndash211994
[14] Y Xu and R L Jackson ldquoPeriodic Contact Problems inPlane Elasticity The Fracture Mechanics Approachrdquo Journal ofTribology vol 140 no 1 p 011404 2018
[15] E A Kuznetsov ldquoPeriodic contact problem for half-planeallowing for forces of frictionrdquo Soviet Applied Mechanics vol12 no 10 pp 1014ndash1019 1976
[16] M Nosonovsky and G G Adams ldquoSteady-state frictionalsliding of two elastic bodies with a wavy contact interfacerdquoJournal of Tribology vol 122 no 3 pp 490ndash495 2000
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
4 Advances in Tribology
relations for Chebyshev polynomials [40] For the 119894th har-monic the final relation is
119906119911119894 (119909) = minusΔ 119894119899119894119886[[J1 (119886119899119894) (11990921198862 minus 12 minus ln 2119886)
+ infinsum119895=3
(minus1)(1minus119895)2 J119895 (119886119899119894)
sdot (119879119895+1 (119909119886)119895 + 1 minus 119879119895minus1 (119909119886)119895 minus 1 )]]
(15)
where signsum identifies the sum of terms with odd 119895 onlyAccording to equation (6) the mean pressure for the 119894th
harmonic pinfin119894 is calculated by integration of equation (14) andresulting in a simple expression
119901infin119894 = 025Δ 119894119886119899119894J1 (119886119899119894) (16)
The approximate close-form relation for the contactpressure distribution can be obtained assuming that thelargest values of pressure are concentrated near the point x= 0Then equation (14) can be represented as
119901119894 (119909) = Δ 119894119899119894radic1 minus (119909119886)2 120597120597119909
sdot infinsum119895=0
(minus1)(1minus119895)2 119886J119895 (119886119899119894) 119879119895 (sin (119909119886))119895 (17)
where signsum identifies the sum of terms with odd 119895 onlyUsing the known relations for theChebyshev polynomials
[41] the following expression can be written
119901119894 (119909)= Δ 119894119899119894radic1 minus (119909119886)
2 infinsum119895=0
J2119895+1 (119886119899119894) cos ((2119895 + 1) (119909119886)) (18)
With the use of an approximate relation between zeros ofBessel functions of integer order [42] the following expres-sion can be written
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [[J1 (119886119899119894)
+ infinsum119895=2
J2119895 (119886119899119894 + 1) cos (2119895 (119909119886))]]
(19)
Then applying the Jacobi-Anger expansion [42] theclose-form relation is
119901119894 (119909) asymp Δ 119894119899119894radic1 minus (119909119886)2 [J1 (119886119899119894)
+ J2 (119886119899119894 + 1) cos (2 (119909119886))+ 05 cos((119886119899119894 + 1) sin (119909119886)) minus J0 (119886119899119894 + 1)]
(20)
The close-form integral relation for a maximum pressure(x = 0) can be determined exactly from equation (18)
119901119894max = Δ 119894119899119894 infinsum119895=0
J2119895+1 (119886119899119894) = 05Δ 119894119899119894 int1198861198991198940
J0 (119905) 119889119905 (21)
4 Results and Discussion
The evolution of the dimensionless contact pressure dis-tribution p(x)plowast (plowast = 120587EΔ 11205821) for a periodic problem(equations ((6) and (10)-(12)) 1205821 = 2120587 Δ 1 = 05) for variouscontact lengths (2a) and two different profiles f (x) is shownin Figure 2
The exact (solid lines equation (14)) and the approximate(dotted lines equation (20)) graphs of the dimensionless con-tact pressure p(x)plowast for different profiles of a nonperiodicwavy indenter are shown in Figure 3
Figures 2 and 3 illustrate that with increasing the num-ber of harmonics the pressure distribution becomes morecomplex and themaximumpressure grows significantly For asingle-scale periodic cosine profile (Figure 2(a)) the Wester-gaardrsquos solution is recovered For a single-scale nonperiodicindenter (Figure 3(a)) the Hertz solution is observed asthe cosine function is very close to the quadratic parabolaThereby the distributions presented in Figures 3(b) and3(c) correspond to wavy cylinder problem at small waviness[38] Comparison of the exact and the approximate values ofpressure for a single indenter (Figure 3) shows that equation(20) satisfactorily describes the behavior of the pressuredistribution
Comparing the periodic and the nonperiodic solutionsthe elastic interaction effect is of interest The mean pressurendash contact length curves for two profiles calculated fromperiodic and nonperiodic solutions are presented in Figure 4
Figure 4 shows that at small contact lengths (2a lt0251205821) the solutions are closeThepiece of graphs agreementdoes not depend on profile geometry With increase of loadthe periodic solution gives the smaller contact length due toelastic interaction on the largest scale For the profile withtwo cosine harmonics and continuous contact configurationpresented in this study the oscillations of mean pressure ndashcontact length curves are observed (Figure 4(b)) Curves inFigure 4(a) correspond to Westergaardrsquos (curve 1) and Hertz(curve 2) solutions recovered for profilewith onewavelength
Graphs of the mean and the maximum pressures versuscontact length on the interval 2alt 0251205821 for different profilesof a wavy nonperiodic indenter are shown in Figure 5
Advances in Tribology 5
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(a)
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(b)
Figure 2 Evolution of contact pressure distribution for wavy periodic profile (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 031205821 (3) 2a = 051205821 (4) 2a = 081205821(5)
minus025 025
xa
0
1
23
0
05
10
p(x)plowast
(a)
minus025 0250
xa
12
3
0
05
10
p(x)plowast
(b)
0
05
10
minus025 0250
xa
12
3
p(x)plowast
(c)
Figure 3 Evolution of contact pressure distribution for a nonperiodic wavy indenter (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (c) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 0251205821 (3)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(a)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(b)
Figure 4 Graphs of dimensionless mean pressure as a function of dimensionless contact length (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a)f (x) = Δ 1 cos(119909)+002Δ 1cos (11x) (b) 1 ndash periodic solution 2 ndash nonperiodic solution
6 Advances in Tribology
0 007 013 019
015
010
005
pinfinplowast
2a1
1
23
(a)
05
1
pmaxplowast
0 007 013 019 2a1
12
3
(b)
Figure 5 Dimensionless mean pressure (a) and maximum pressure (b) as a function of dimensionless contact length for profileswith different numbers of cosine harmonics (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (1) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
05
10
pmaxplowast
0 01 02 03
1
2
3
pinfinplowast
Figure 6 Dimensionless maximum pressure as a function ofdimensionless mean pressure for different profiles (1205821 = 2120587 Δ 1 =05) f (x) =Δ 1 cos(119909) (1) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x)= Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
Figure 5 shows that the maximum pressure depends onprofile geometry stronger than the mean pressure Howeveradding the third harmonic leads to insignificant change ofthe graphs character Continuous contact configuration at thepresence of several cosine wavelengths leads to oscillatorycharacter of the mean and the maximum pressure graphsCombining these two graphs numerically one can obtain thedependence of peak pressure frommean pressure (Figure 6)
Figure 6 shows that dependences of the maximum pres-sure from the mean pressure are not oscillatory for theprofiles with two and three wavelengths and additionalcosine harmonics change the graph considerably in value butnot in character This statement can be useful in the analysisof contact surfaces fracture processes [33]
5 Conclusions
The continuous contact configuration is one of the twopossible configurations arising at indentation of a multi-sinusoidal 2D wavy surface into an elastic half-plane Thisconfiguration leads to continuous oscillatory contact pressuredistribution Comparison of the derived periodic and nonpe-riodic solutions shows that the long-range elastic interactionbetween asperities does not depend on a number of cosinewavelengths and can be neglected at small loads (contactlengths) for arbitrary wavy profile geometry The assumptionof neglecting the long-range periodicity leads to exact equa-tions for determining the remote and themaximumpressuresfrom the contact length described by oscillatory functionsHowever the dependences of the maximum pressure fromthe mean pressure are not oscillatory for the profiles with twoand three wavelengths and resemble those for a simple cosineprofile of indenter The influence of the additional cosineharmonics on the maximum pressure is significantly largerthan on the mean pressure for the same contact zone lengthThe derived equations can be used at the analysis of contactcharacteristics of deterministic profiles of arbitrary geometryand also at the validation of more complex numerical modelsof rough surfaces contact
Appendix
Derivation of Contact Pressure Distributionfor the Periodic Problem
Themain integral equation of the considered contact problemfor the 119894th harmonic in transformed variables (8) is
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1(1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (A1)
Advances in Tribology 7
where 119880119899 is a Chebyshev polynomial of a second kindwith a degree 119899119894
The appropriate inversion of this integral equation has tobe nonsingular on both endpoints [11]
119901119894 (V) = minusΔ 119894119899119894120587 radic1205722 minus V2 int120572minus120572
V1 + V2119880119899119894minus1(1 minus V21 + V2
)sdot 1radic1205722 minus 1199062
1119906 minus V119889119906
(A2)
By introducing the new variables
119903 = V120572119904 = 119906120572
(A3)
the expression (A2) can be written in the following form
119901119894 (119903) = minusΔ 119894119899119894120587 radic1 minus 1199032 int1minus1120593119894 (119904) 1radic1 minus 1199042
1119904 minus 119903119889119904 (A4)
where the function 120593119894(s) is120593119894 (119904) = 1205721199041 + 12057221199042119880119899119894minus1(1 minus 120572
211990421 + 12057221199042) (A5)
Since the integrand function is defined on the interval [-11] and satisfies the Holder condition it can be represented asan expansion in Chebyshev polynomials of the first kind [41]
120593119894 (119904) = 119860 11989402 + 119860 11989411198791 (119904) + 119860 11989421198792 (119904) + (A6)
where119879119895 is a Chebyshev polynomial of the first kind witha degree j [39]
The coefficients 119860 119894119895 in equation (A6) are defined by thefollowing expression [43]
119860 1198940 = 0119860 119894119895 = 2120587 int
1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (A7)
With the use of integral relation between the Chebyshevpolynomials of the first and the second kind [41]
1120587 int1
minus1
119879119895 (119904)radic1 minus 11990421119904 minus 119903119889119904 = 119880119895minus1 (119903) 119895 = 0 1 2 (A8)
and equation (A4) the expression for the contact pressuredistribution for the 119894th harmonic is
119901119894 (119903) = minusΔ 119894119899119894radic1 minus 1199032 infinsum119895=1
119860 119894119895119880119895minus1 (119903) (A9)
Returning to the original variables and bearing in mindpositive pressures notation one can obtain
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (A10)
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Theresearchwas supported byRSF (project no 14-29-00198)
References
[1] B N J Persson ldquoTheory of rubber friction and contactmechanicsrdquoThe Journal of Chemical Physics vol 115 no 8 pp3840ndash3861 2001
[2] A C Rodrıguez Urribarrı E van der Heide X Zeng and MB de Rooij ldquoModelling the static contact between a fingertipand a rigid wavy surfacerdquo Tribology International vol 102 pp114ndash124 2016
[3] A S Adnan V Ramalingam J H Ko and S Subbiah ldquoNanotexture generation in single point diamond turning usingbackside patterned workpiecerdquo Manufacturing Letters vol 2no 1 pp 44ndash48 2013
[4] Y Ju and T N Farris ldquoSpectral Analysis of Two-DimensionalContact Problemsrdquo Journal of Tribology vol 118 no 2 p 3201996
[5] M A Sadowski ldquoZwiedimensionale probleme der elastizitat-shtheorierdquo ZAMMmdashZeitschrift fur Angewandte Mathematikund Mechanik vol 8 no 2 pp 107ndash121 1928
[6] N I Muskhelishvili Some Basic Problems of the MathematicalTheory of Elasticity Springer Dordrecht Netherlands 1977
[7] H M Westergaard ldquoBearing pressures and cracksrdquo Journal ofApplied Mechanics vol 6 pp 49ndash53 1939
[8] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1987
[9] I Y Schtaierman Contact Problem of Theory of ElasticityGostekhizdat Moscow Russia 1949
[10] L A Galin Contact problems Springer Netherlands Dor-drecht 2008
[11] J M Block and L M Keer ldquoPeriodic contact problems in planeelasticityrdquo Journal of Mechanics of Materials and Structures vol3 no 7 pp 1207ndash1237 2008
[12] J Dundurs K C Tsai and L M Keer ldquoContact between elasticbodies with wavy surfacesrdquo Journal of Elasticity vol 3 no 2 pp109ndash115 1973
[13] A A Krishtafovich R M Martynyak and R N Shvets ldquoCon-tact between anisotropic half-plane and rigid body with regularmicroreliefrdquo Journal of Friction and Wear vol 15 pp 15ndash211994
[14] Y Xu and R L Jackson ldquoPeriodic Contact Problems inPlane Elasticity The Fracture Mechanics Approachrdquo Journal ofTribology vol 140 no 1 p 011404 2018
[15] E A Kuznetsov ldquoPeriodic contact problem for half-planeallowing for forces of frictionrdquo Soviet Applied Mechanics vol12 no 10 pp 1014ndash1019 1976
[16] M Nosonovsky and G G Adams ldquoSteady-state frictionalsliding of two elastic bodies with a wavy contact interfacerdquoJournal of Tribology vol 122 no 3 pp 490ndash495 2000
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Advances in Tribology 5
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(a)
p(x)plowast
165
110
055
0minus08 minus05 minus03 0 03 05 08
xa
1
2
3
4
5
(b)
Figure 2 Evolution of contact pressure distribution for wavy periodic profile (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 031205821 (3) 2a = 051205821 (4) 2a = 081205821(5)
minus025 025
xa
0
1
23
0
05
10
p(x)plowast
(a)
minus025 0250
xa
12
3
0
05
10
p(x)plowast
(b)
0
05
10
minus025 0250
xa
12
3
p(x)plowast
(c)
Figure 3 Evolution of contact pressure distribution for a nonperiodic wavy indenter (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (b) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (c) 2a = 0051205821 (1) 2a = 021205821 (2) 2a = 0251205821 (3)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(a)
0 02 04 06
08
06
04
02
pinfinplowast
2a1
1
2
(b)
Figure 4 Graphs of dimensionless mean pressure as a function of dimensionless contact length (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (a)f (x) = Δ 1 cos(119909)+002Δ 1cos (11x) (b) 1 ndash periodic solution 2 ndash nonperiodic solution
6 Advances in Tribology
0 007 013 019
015
010
005
pinfinplowast
2a1
1
23
(a)
05
1
pmaxplowast
0 007 013 019 2a1
12
3
(b)
Figure 5 Dimensionless mean pressure (a) and maximum pressure (b) as a function of dimensionless contact length for profileswith different numbers of cosine harmonics (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (1) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
05
10
pmaxplowast
0 01 02 03
1
2
3
pinfinplowast
Figure 6 Dimensionless maximum pressure as a function ofdimensionless mean pressure for different profiles (1205821 = 2120587 Δ 1 =05) f (x) =Δ 1 cos(119909) (1) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x)= Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
Figure 5 shows that the maximum pressure depends onprofile geometry stronger than the mean pressure Howeveradding the third harmonic leads to insignificant change ofthe graphs character Continuous contact configuration at thepresence of several cosine wavelengths leads to oscillatorycharacter of the mean and the maximum pressure graphsCombining these two graphs numerically one can obtain thedependence of peak pressure frommean pressure (Figure 6)
Figure 6 shows that dependences of the maximum pres-sure from the mean pressure are not oscillatory for theprofiles with two and three wavelengths and additionalcosine harmonics change the graph considerably in value butnot in character This statement can be useful in the analysisof contact surfaces fracture processes [33]
5 Conclusions
The continuous contact configuration is one of the twopossible configurations arising at indentation of a multi-sinusoidal 2D wavy surface into an elastic half-plane Thisconfiguration leads to continuous oscillatory contact pressuredistribution Comparison of the derived periodic and nonpe-riodic solutions shows that the long-range elastic interactionbetween asperities does not depend on a number of cosinewavelengths and can be neglected at small loads (contactlengths) for arbitrary wavy profile geometry The assumptionof neglecting the long-range periodicity leads to exact equa-tions for determining the remote and themaximumpressuresfrom the contact length described by oscillatory functionsHowever the dependences of the maximum pressure fromthe mean pressure are not oscillatory for the profiles with twoand three wavelengths and resemble those for a simple cosineprofile of indenter The influence of the additional cosineharmonics on the maximum pressure is significantly largerthan on the mean pressure for the same contact zone lengthThe derived equations can be used at the analysis of contactcharacteristics of deterministic profiles of arbitrary geometryand also at the validation of more complex numerical modelsof rough surfaces contact
Appendix
Derivation of Contact Pressure Distributionfor the Periodic Problem
Themain integral equation of the considered contact problemfor the 119894th harmonic in transformed variables (8) is
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1(1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (A1)
Advances in Tribology 7
where 119880119899 is a Chebyshev polynomial of a second kindwith a degree 119899119894
The appropriate inversion of this integral equation has tobe nonsingular on both endpoints [11]
119901119894 (V) = minusΔ 119894119899119894120587 radic1205722 minus V2 int120572minus120572
V1 + V2119880119899119894minus1(1 minus V21 + V2
)sdot 1radic1205722 minus 1199062
1119906 minus V119889119906
(A2)
By introducing the new variables
119903 = V120572119904 = 119906120572
(A3)
the expression (A2) can be written in the following form
119901119894 (119903) = minusΔ 119894119899119894120587 radic1 minus 1199032 int1minus1120593119894 (119904) 1radic1 minus 1199042
1119904 minus 119903119889119904 (A4)
where the function 120593119894(s) is120593119894 (119904) = 1205721199041 + 12057221199042119880119899119894minus1(1 minus 120572
211990421 + 12057221199042) (A5)
Since the integrand function is defined on the interval [-11] and satisfies the Holder condition it can be represented asan expansion in Chebyshev polynomials of the first kind [41]
120593119894 (119904) = 119860 11989402 + 119860 11989411198791 (119904) + 119860 11989421198792 (119904) + (A6)
where119879119895 is a Chebyshev polynomial of the first kind witha degree j [39]
The coefficients 119860 119894119895 in equation (A6) are defined by thefollowing expression [43]
119860 1198940 = 0119860 119894119895 = 2120587 int
1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (A7)
With the use of integral relation between the Chebyshevpolynomials of the first and the second kind [41]
1120587 int1
minus1
119879119895 (119904)radic1 minus 11990421119904 minus 119903119889119904 = 119880119895minus1 (119903) 119895 = 0 1 2 (A8)
and equation (A4) the expression for the contact pressuredistribution for the 119894th harmonic is
119901119894 (119903) = minusΔ 119894119899119894radic1 minus 1199032 infinsum119895=1
119860 119894119895119880119895minus1 (119903) (A9)
Returning to the original variables and bearing in mindpositive pressures notation one can obtain
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (A10)
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Theresearchwas supported byRSF (project no 14-29-00198)
References
[1] B N J Persson ldquoTheory of rubber friction and contactmechanicsrdquoThe Journal of Chemical Physics vol 115 no 8 pp3840ndash3861 2001
[2] A C Rodrıguez Urribarrı E van der Heide X Zeng and MB de Rooij ldquoModelling the static contact between a fingertipand a rigid wavy surfacerdquo Tribology International vol 102 pp114ndash124 2016
[3] A S Adnan V Ramalingam J H Ko and S Subbiah ldquoNanotexture generation in single point diamond turning usingbackside patterned workpiecerdquo Manufacturing Letters vol 2no 1 pp 44ndash48 2013
[4] Y Ju and T N Farris ldquoSpectral Analysis of Two-DimensionalContact Problemsrdquo Journal of Tribology vol 118 no 2 p 3201996
[5] M A Sadowski ldquoZwiedimensionale probleme der elastizitat-shtheorierdquo ZAMMmdashZeitschrift fur Angewandte Mathematikund Mechanik vol 8 no 2 pp 107ndash121 1928
[6] N I Muskhelishvili Some Basic Problems of the MathematicalTheory of Elasticity Springer Dordrecht Netherlands 1977
[7] H M Westergaard ldquoBearing pressures and cracksrdquo Journal ofApplied Mechanics vol 6 pp 49ndash53 1939
[8] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1987
[9] I Y Schtaierman Contact Problem of Theory of ElasticityGostekhizdat Moscow Russia 1949
[10] L A Galin Contact problems Springer Netherlands Dor-drecht 2008
[11] J M Block and L M Keer ldquoPeriodic contact problems in planeelasticityrdquo Journal of Mechanics of Materials and Structures vol3 no 7 pp 1207ndash1237 2008
[12] J Dundurs K C Tsai and L M Keer ldquoContact between elasticbodies with wavy surfacesrdquo Journal of Elasticity vol 3 no 2 pp109ndash115 1973
[13] A A Krishtafovich R M Martynyak and R N Shvets ldquoCon-tact between anisotropic half-plane and rigid body with regularmicroreliefrdquo Journal of Friction and Wear vol 15 pp 15ndash211994
[14] Y Xu and R L Jackson ldquoPeriodic Contact Problems inPlane Elasticity The Fracture Mechanics Approachrdquo Journal ofTribology vol 140 no 1 p 011404 2018
[15] E A Kuznetsov ldquoPeriodic contact problem for half-planeallowing for forces of frictionrdquo Soviet Applied Mechanics vol12 no 10 pp 1014ndash1019 1976
[16] M Nosonovsky and G G Adams ldquoSteady-state frictionalsliding of two elastic bodies with a wavy contact interfacerdquoJournal of Tribology vol 122 no 3 pp 490ndash495 2000
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
6 Advances in Tribology
0 007 013 019
015
010
005
pinfinplowast
2a1
1
23
(a)
05
1
pmaxplowast
0 007 013 019 2a1
12
3
(b)
Figure 5 Dimensionless mean pressure (a) and maximum pressure (b) as a function of dimensionless contact length for profileswith different numbers of cosine harmonics (1205821 = 2120587 Δ 1 = 05) f (x) = Δ 1 cos(119909) (1) f (x) = Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
05
10
pmaxplowast
0 01 02 03
1
2
3
pinfinplowast
Figure 6 Dimensionless maximum pressure as a function ofdimensionless mean pressure for different profiles (1205821 = 2120587 Δ 1 =05) f (x) =Δ 1 cos(119909) (1) f (x) =Δ 1 cos(119909)+002Δ 1 cos(11119909) (2) f (x)= Δ 1 cos(119909)+002Δ 1 cos(11119909)+00015Δ 1 cos(40119909) (3)
Figure 5 shows that the maximum pressure depends onprofile geometry stronger than the mean pressure Howeveradding the third harmonic leads to insignificant change ofthe graphs character Continuous contact configuration at thepresence of several cosine wavelengths leads to oscillatorycharacter of the mean and the maximum pressure graphsCombining these two graphs numerically one can obtain thedependence of peak pressure frommean pressure (Figure 6)
Figure 6 shows that dependences of the maximum pres-sure from the mean pressure are not oscillatory for theprofiles with two and three wavelengths and additionalcosine harmonics change the graph considerably in value butnot in character This statement can be useful in the analysisof contact surfaces fracture processes [33]
5 Conclusions
The continuous contact configuration is one of the twopossible configurations arising at indentation of a multi-sinusoidal 2D wavy surface into an elastic half-plane Thisconfiguration leads to continuous oscillatory contact pressuredistribution Comparison of the derived periodic and nonpe-riodic solutions shows that the long-range elastic interactionbetween asperities does not depend on a number of cosinewavelengths and can be neglected at small loads (contactlengths) for arbitrary wavy profile geometry The assumptionof neglecting the long-range periodicity leads to exact equa-tions for determining the remote and themaximumpressuresfrom the contact length described by oscillatory functionsHowever the dependences of the maximum pressure fromthe mean pressure are not oscillatory for the profiles with twoand three wavelengths and resemble those for a simple cosineprofile of indenter The influence of the additional cosineharmonics on the maximum pressure is significantly largerthan on the mean pressure for the same contact zone lengthThe derived equations can be used at the analysis of contactcharacteristics of deterministic profiles of arbitrary geometryand also at the validation of more complex numerical modelsof rough surfaces contact
Appendix
Derivation of Contact Pressure Distributionfor the Periodic Problem
Themain integral equation of the considered contact problemfor the 119894th harmonic in transformed variables (8) is
minusΔ 119894119899119894 2V1 + V2119880119899119894minus1(1 minus V21 + V2
) = 1120587 int120572
minus120572
119901119894 (119906)V minus 119906 119889119906 (A1)
Advances in Tribology 7
where 119880119899 is a Chebyshev polynomial of a second kindwith a degree 119899119894
The appropriate inversion of this integral equation has tobe nonsingular on both endpoints [11]
119901119894 (V) = minusΔ 119894119899119894120587 radic1205722 minus V2 int120572minus120572
V1 + V2119880119899119894minus1(1 minus V21 + V2
)sdot 1radic1205722 minus 1199062
1119906 minus V119889119906
(A2)
By introducing the new variables
119903 = V120572119904 = 119906120572
(A3)
the expression (A2) can be written in the following form
119901119894 (119903) = minusΔ 119894119899119894120587 radic1 minus 1199032 int1minus1120593119894 (119904) 1radic1 minus 1199042
1119904 minus 119903119889119904 (A4)
where the function 120593119894(s) is120593119894 (119904) = 1205721199041 + 12057221199042119880119899119894minus1(1 minus 120572
211990421 + 12057221199042) (A5)
Since the integrand function is defined on the interval [-11] and satisfies the Holder condition it can be represented asan expansion in Chebyshev polynomials of the first kind [41]
120593119894 (119904) = 119860 11989402 + 119860 11989411198791 (119904) + 119860 11989421198792 (119904) + (A6)
where119879119895 is a Chebyshev polynomial of the first kind witha degree j [39]
The coefficients 119860 119894119895 in equation (A6) are defined by thefollowing expression [43]
119860 1198940 = 0119860 119894119895 = 2120587 int
1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (A7)
With the use of integral relation between the Chebyshevpolynomials of the first and the second kind [41]
1120587 int1
minus1
119879119895 (119904)radic1 minus 11990421119904 minus 119903119889119904 = 119880119895minus1 (119903) 119895 = 0 1 2 (A8)
and equation (A4) the expression for the contact pressuredistribution for the 119894th harmonic is
119901119894 (119903) = minusΔ 119894119899119894radic1 minus 1199032 infinsum119895=1
119860 119894119895119880119895minus1 (119903) (A9)
Returning to the original variables and bearing in mindpositive pressures notation one can obtain
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (A10)
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Theresearchwas supported byRSF (project no 14-29-00198)
References
[1] B N J Persson ldquoTheory of rubber friction and contactmechanicsrdquoThe Journal of Chemical Physics vol 115 no 8 pp3840ndash3861 2001
[2] A C Rodrıguez Urribarrı E van der Heide X Zeng and MB de Rooij ldquoModelling the static contact between a fingertipand a rigid wavy surfacerdquo Tribology International vol 102 pp114ndash124 2016
[3] A S Adnan V Ramalingam J H Ko and S Subbiah ldquoNanotexture generation in single point diamond turning usingbackside patterned workpiecerdquo Manufacturing Letters vol 2no 1 pp 44ndash48 2013
[4] Y Ju and T N Farris ldquoSpectral Analysis of Two-DimensionalContact Problemsrdquo Journal of Tribology vol 118 no 2 p 3201996
[5] M A Sadowski ldquoZwiedimensionale probleme der elastizitat-shtheorierdquo ZAMMmdashZeitschrift fur Angewandte Mathematikund Mechanik vol 8 no 2 pp 107ndash121 1928
[6] N I Muskhelishvili Some Basic Problems of the MathematicalTheory of Elasticity Springer Dordrecht Netherlands 1977
[7] H M Westergaard ldquoBearing pressures and cracksrdquo Journal ofApplied Mechanics vol 6 pp 49ndash53 1939
[8] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1987
[9] I Y Schtaierman Contact Problem of Theory of ElasticityGostekhizdat Moscow Russia 1949
[10] L A Galin Contact problems Springer Netherlands Dor-drecht 2008
[11] J M Block and L M Keer ldquoPeriodic contact problems in planeelasticityrdquo Journal of Mechanics of Materials and Structures vol3 no 7 pp 1207ndash1237 2008
[12] J Dundurs K C Tsai and L M Keer ldquoContact between elasticbodies with wavy surfacesrdquo Journal of Elasticity vol 3 no 2 pp109ndash115 1973
[13] A A Krishtafovich R M Martynyak and R N Shvets ldquoCon-tact between anisotropic half-plane and rigid body with regularmicroreliefrdquo Journal of Friction and Wear vol 15 pp 15ndash211994
[14] Y Xu and R L Jackson ldquoPeriodic Contact Problems inPlane Elasticity The Fracture Mechanics Approachrdquo Journal ofTribology vol 140 no 1 p 011404 2018
[15] E A Kuznetsov ldquoPeriodic contact problem for half-planeallowing for forces of frictionrdquo Soviet Applied Mechanics vol12 no 10 pp 1014ndash1019 1976
[16] M Nosonovsky and G G Adams ldquoSteady-state frictionalsliding of two elastic bodies with a wavy contact interfacerdquoJournal of Tribology vol 122 no 3 pp 490ndash495 2000
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Advances in Tribology 7
where 119880119899 is a Chebyshev polynomial of a second kindwith a degree 119899119894
The appropriate inversion of this integral equation has tobe nonsingular on both endpoints [11]
119901119894 (V) = minusΔ 119894119899119894120587 radic1205722 minus V2 int120572minus120572
V1 + V2119880119899119894minus1(1 minus V21 + V2
)sdot 1radic1205722 minus 1199062
1119906 minus V119889119906
(A2)
By introducing the new variables
119903 = V120572119904 = 119906120572
(A3)
the expression (A2) can be written in the following form
119901119894 (119903) = minusΔ 119894119899119894120587 radic1 minus 1199032 int1minus1120593119894 (119904) 1radic1 minus 1199042
1119904 minus 119903119889119904 (A4)
where the function 120593119894(s) is120593119894 (119904) = 1205721199041 + 12057221199042119880119899119894minus1(1 minus 120572
211990421 + 12057221199042) (A5)
Since the integrand function is defined on the interval [-11] and satisfies the Holder condition it can be represented asan expansion in Chebyshev polynomials of the first kind [41]
120593119894 (119904) = 119860 11989402 + 119860 11989411198791 (119904) + 119860 11989421198792 (119904) + (A6)
where119879119895 is a Chebyshev polynomial of the first kind witha degree j [39]
The coefficients 119860 119894119895 in equation (A6) are defined by thefollowing expression [43]
119860 1198940 = 0119860 119894119895 = 2120587 int
1
minus1
120593119894 (119904) 119879119895 (119904)radic1 minus 1199042 119889119904 119895 = 1 2 (A7)
With the use of integral relation between the Chebyshevpolynomials of the first and the second kind [41]
1120587 int1
minus1
119879119895 (119904)radic1 minus 11990421119904 minus 119903119889119904 = 119880119895minus1 (119903) 119895 = 0 1 2 (A8)
and equation (A4) the expression for the contact pressuredistribution for the 119894th harmonic is
119901119894 (119903) = minusΔ 119894119899119894radic1 minus 1199032 infinsum119895=1
119860 119894119895119880119895minus1 (119903) (A9)
Returning to the original variables and bearing in mindpositive pressures notation one can obtain
119901119894 (119909)= Δ 119894119899119894radic1 minus ( tan (1199092)tan (1198862))
2 infinsum119895=1
119860 119894119895119880119895minus1 ( tan (1199092)tan (1198862)) (A10)
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Theresearchwas supported byRSF (project no 14-29-00198)
References
[1] B N J Persson ldquoTheory of rubber friction and contactmechanicsrdquoThe Journal of Chemical Physics vol 115 no 8 pp3840ndash3861 2001
[2] A C Rodrıguez Urribarrı E van der Heide X Zeng and MB de Rooij ldquoModelling the static contact between a fingertipand a rigid wavy surfacerdquo Tribology International vol 102 pp114ndash124 2016
[3] A S Adnan V Ramalingam J H Ko and S Subbiah ldquoNanotexture generation in single point diamond turning usingbackside patterned workpiecerdquo Manufacturing Letters vol 2no 1 pp 44ndash48 2013
[4] Y Ju and T N Farris ldquoSpectral Analysis of Two-DimensionalContact Problemsrdquo Journal of Tribology vol 118 no 2 p 3201996
[5] M A Sadowski ldquoZwiedimensionale probleme der elastizitat-shtheorierdquo ZAMMmdashZeitschrift fur Angewandte Mathematikund Mechanik vol 8 no 2 pp 107ndash121 1928
[6] N I Muskhelishvili Some Basic Problems of the MathematicalTheory of Elasticity Springer Dordrecht Netherlands 1977
[7] H M Westergaard ldquoBearing pressures and cracksrdquo Journal ofApplied Mechanics vol 6 pp 49ndash53 1939
[8] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1987
[9] I Y Schtaierman Contact Problem of Theory of ElasticityGostekhizdat Moscow Russia 1949
[10] L A Galin Contact problems Springer Netherlands Dor-drecht 2008
[11] J M Block and L M Keer ldquoPeriodic contact problems in planeelasticityrdquo Journal of Mechanics of Materials and Structures vol3 no 7 pp 1207ndash1237 2008
[12] J Dundurs K C Tsai and L M Keer ldquoContact between elasticbodies with wavy surfacesrdquo Journal of Elasticity vol 3 no 2 pp109ndash115 1973
[13] A A Krishtafovich R M Martynyak and R N Shvets ldquoCon-tact between anisotropic half-plane and rigid body with regularmicroreliefrdquo Journal of Friction and Wear vol 15 pp 15ndash211994
[14] Y Xu and R L Jackson ldquoPeriodic Contact Problems inPlane Elasticity The Fracture Mechanics Approachrdquo Journal ofTribology vol 140 no 1 p 011404 2018
[15] E A Kuznetsov ldquoPeriodic contact problem for half-planeallowing for forces of frictionrdquo Soviet Applied Mechanics vol12 no 10 pp 1014ndash1019 1976
[16] M Nosonovsky and G G Adams ldquoSteady-state frictionalsliding of two elastic bodies with a wavy contact interfacerdquoJournal of Tribology vol 122 no 3 pp 490ndash495 2000
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
8 Advances in Tribology
[17] Y A Kuznetsov ldquoEffect of fluid lubricant on the contactcharacteristics of rough elastic bodies in compressionrdquo Wearvol 102 no 3 pp 177ndash194 1985
[18] M Ciavarella ldquoThe generalized Cattaneo partial slip plane con-tact problem II Examplesrdquo International Journal of Solids andStructures vol 35 no 18 pp 2363ndash2378 1998
[19] G Carbone and L Mangialardi ldquoAdhesion and friction of anelastic half-space in contact with a slightly wavy rigid surfacerdquoJournal of the Mechanics and Physics of Solids vol 52 no 6 pp1267ndash1287 2004
[20] N Menga C Putignano G Carbone and G P Demelio ldquoThesliding contact of a rigid wavy surface with a viscoelastic half-spacerdquoProceedings of the Royal Society AMathematical Physicaland Engineering Sciences vol 470 no 2169 pp 20140392-20140392 2014
[21] I G Goryacheva and Y Y Makhovskaya ldquoModeling of frictionat different scale levelsrdquo Mechanics of Solids vol 45 no 3 pp390ndash398 2010
[22] I A Soldatenkov ldquoThe contact problem for an elastic stripand a wavy punch under friction and wearrdquo Journal of AppliedMathematics and Mechanics vol 75 no 1 pp 85ndash92 2011
[23] Y-T Zhou and T-W Kim ldquoAnalytical solution of the dynamiccontact problem of anisotropic materials indented with a rigidwavy surfacerdquoMeccanica vol 52 no 1-2 pp 7ndash19 2017
[24] I Y Tsukanov ldquoEffects of shape and scale in mechanics ofelastic interaction of regular wavy surfacesrdquo Proceedings of theInstitution of Mechanical Engineers Part J Journal of Engineer-ing Tribology vol 231 no 3 pp 332ndash340 2017
[25] I GGoryachevaContactmechanics in tribology vol 61 KluwerAcademic Publishers Dordrecht 1998
[26] W Manners ldquoPartial contact between elastic surfaces withperiodic profilesrdquo Proceedings of the Royal Society vol 454 no1980 pp 3203ndash3221 1998
[27] O G Chekina and LMKeer ldquoA new approach to calculation ofcontact characteristicsrdquo Journal of Tribology vol 121 no 1 pp20ndash27 1999
[28] H M Stanley and T Kato ldquoAn fft-based method for roughsurface contactrdquo Journal of Tribology vol 119 no 3 pp 481ndash4851997
[29] F M Borodich and B A Galanov ldquoSelf-similar problemsof elastic contact for non-convex punchesrdquo Journal of theMechanics and Physics of Solids vol 50 no 11 pp 2441ndash24612002
[30] M Ciavarella G Demelio and C Murolo ldquoA numericalalgorithm for the solution of two-dimensional rough contactproblemsrdquo Journal of Strain Analysis for Engineering Design vol40 no 5 pp 463ndash476 2005
[31] M Paggi and J Reinoso ldquoA variational approach with embed-ded roughness for adhesive contact problemsrdquo 2018 httpsarxivorgabs180507207
[32] J H Tripp J Van Kuilenburg G E Morales-Espejel and P MLugt ldquoFrequency response functions and rough surface stressanalysisrdquo Tribology Transactions vol 46 no 3 pp 376ndash3822003
[33] C Paulin F Ville P Sainsot S Coulon and T LubrechtldquoEffect of rough surfaces on rolling contact fatigue theoreticaland experimental analysisrdquo Tribology and Interface EngineeringSeries vol 43 pp 611ndash617 2004
[34] R L Jackson and I Green ldquoOn the modeling of elastic contactbetween rough surfacesrdquo Tribology Transactions vol 54 no 2pp 300ndash314 2011
[35] Y Xu R L Jackson and D B Marghitu ldquoStatistical modelof nearly complete elastic rough surface contactrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1075ndash10882014
[36] M Ciavarella G Murolo G Demelio and J R Barber ldquoElasticcontact stiffness and contact resistance for the Weierstrassprofilerdquo Journal of the Mechanics and Physics of Solids vol 52no 6 pp 1247ndash1265 2004
[37] R L Jackson ldquoAn Analytical solution to an archard-type fractalrough surface contact modelrdquo Tribology Transactions vol 53no 4 pp 543ndash553 2010
[38] Hills D A Nowell D and Sackfield A Mechanics of elasticcontacts Butterworth-Heinemann Oxford UK 1993
[39] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier 8th edition 2015
[40] A Arzhang H Derili and M Yousefi ldquoThe approximatesolution of a class of Fredholm integral equations with alogarithmic kernel by using Chebyshev polynomialsrdquo GlobalJournal of Computer Sciences vol 3 no 2 pp 37ndash48 2013
[41] J C Mason and D C Handscomb Chebyshev PolynomialsChapman and Hall London UK 2003
[42] K Oldham J Myland and J Spanier An Atlas of FunctionsSpringer New York NY USA 2nd edition 2008
[43] D Elliott ldquoThe evaluation and estimation of the coefficients inthe Chebyshev series expansion of a functionrdquo Mathematics ofComputation vol 18 pp 274ndash284 1964
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom