research article contact mechanics of rough spheres...

Hindawi Publishing CorporationAdvances in TribologyVolume 2013 Article ID 974178 4 pageshttpdxdoiorg1011552013974178

Research ArticleContact Mechanics of Rough Spheres Crossover fromFractal to Hertzian Behavior

Roman Pohrt and Valentin L Popov

Technische Universitat Berlin Straszlige des 17 Juni 135 10623 Berlin Germany

Correspondence should be addressed to Roman Pohrt romanpohrttu-berlinde

Received 4 June 2013 Accepted 2 July 2013

Academic Editor Dae-Eun Kim

Copyright copy 2013 R Pohrt and V L Popov This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We investigate the normal contact stiffness in a contact of a rough sphere with an elastic half-space using 3D boundary elementcalculations For small normal forces it is found that the stiffness behaves according to the law of PohrtPopov for nominally flatself-affine surfaces while for higher normal forces there is a transition to Hertzian behavior A new analytical model is deriveddescribing the contact behavior at any force

1 Introduction

Since Bowden and Tabor [1] it has been known that surfaceroughness plays a decisive role in contact adhesion frictionand wear The main understanding of the contact mechanicsof nominally flat rough surfaces was achieved in the middleof the 20th century due to works by Archard [2] andGreenwood and Williamson [3] In the last years contactmechanics of rough surfaces has once again become a hottopic [4ndash6] Most of the previous work was devoted toinvestigation of nominally flat surfaces Formany tribologicalapplications however the contact properties of rough bodieswith macroscopically curved surfaces are of great interestA first analysis of the contact problem including a curvedbut rough surface was given by Greenwood and Tripp [7]They applied the GreenwoodWilliamson (GW) model [3]of independent asperities with a Gaussian distribution to aparabolic shape In this model the roughness can be seen asan additional compressible layer They calculated the meanpressures as a function of the radius and for low loads founda reduction in the maximum pressure and an enlargement ofthe apparent area of contact For high loads the indentationbehavior found was Hertzian

In the present paper we will investigate the indentationof a rough sphere into an elastic half-space without therestrictions stemming from the GWmodel We calculate the

incremental normal stiffness of the contact which not onlydetermines the dynamic properties of a tribological systembut also its electrical and thermal conductivity [8 9] In theinterest of purity of results we assume the bodies to be elasticat all scales and confine ourselves to self-affine roughnesswithout cutoff Contact stiffness of such surfaces has beenrecently studied in detail numerically and analytically [10 11]We show that there is a pronounced crossover from thebehavior which is typical for fractal surfaces [12 13] to Hertz-like behavior [14] similar to GT [7] Furthermore we derivean analytical approximation for the entire range of forces

2 Methods

We consider a rigid rough spherical indenter with the radius119877 which is approximated by a superposition of a parabolicshape 119911119901(119909 119910) = (119909

2+ 1199102)(2119877) and a nominally flat random

self-affine roughness 119911119903(119909 119910) with the Hurst Exponent 119867(Figure 1)Thepower spectrum1198622D(119902)of the randomly roughself-affine statistically isotropic surface has the form

1198622D (119902) = 1198620(119871119902)minus2119867minus2

(1)

where 119871 is the system size and 119902 is the absolute value of thewave vector 119902 The surface topography was calculated with

2 Advances in Tribology

(a) (b)

Figure 1 Graphical representations of a fractal rough surface with119867 = 06 (a) grayscale representation and (b) a cut through the surface

the help of the power spectrum according to

ℎ () = sum

119902

1198612D ( 119902) exp (119894 ( 119902 sdot + 120601 ( 119902))) (2)

with

1198612D ( 119902) =2120587

119871radic1198622D ( 119902) = 1198612D (minus 119902) (3)

and the phases 120601( 119902) = minus120601(minus 119902) which are randomlydistributed on the interval [0 2120587) All samples were generatedon a grid with 513 times 513 discrete evenly spaced pointsA typical example of a self-affine surface with the Hurstexponent119867 = 06 is shown in Figure 1 while Figure 2 showsa sample of the superposition of both sphere and roughness

The indenter was pressed into an elastic half-space withthe normal force 119865 The indentation depth and the configu-ration of the contact were calculated using the boundary ele-mentmethodwith an iterativemultilevel algorithm similar to[15 16] The incremental normal stiffness 119896 was calculated byevaluating the differential quotient of force and indentationdepth All values were obtained by ensemble averaging of 50surface realizations having the same power spectrum

3 Results and Discussion

31 Numerical Results Figure 3 shows the resulting depen-dency (red points) For small normal forces the system isdominated by the roughness and the stiffness approachesthe asymptotic dependence with the slope (119867 + 1)

minus1 char-acteristic for nominally flat fractal surfaces [12] For highernormal forces the influence of the roughness vanishes andthe system behaves like a smooth spherical Hertzian indenter(slope 13)

32 Analytical Estimation In [12] it was shown that forrandomly rough self-affine surfaces the contact stiffness is apower function of the normal force of the form

119896 prop 1198651(119867+1)

(4)

Figure 2 Numerically generated rough sphere119867 = 07

with 0 lt 119867 lt 1 In [17] it was shown that this relationremains valid for Hurst exponents 0 lt 119867 lt 2 and that theonly property needed for the validity of the scaling law (4) isthe self-affinity of the surface which means that the surfaceappears undistinguishable from the original when viewedunder an arbitrary magnification 120595 [15]

1199111015840

119903(119909 119910) = 120595

119867119911119903 (

119909

120595119910

120595) (5)

Here it is not important whether the surface is randomlyrough or just a simple axisymmetric profile with the samescaling properties [18]

119911119904 (119903) = 1198763D sdot |119903|119867 (6)

Thus with respect to the contact stiffness the randomlyrough fractal and self-affine surface can be replaced equiva-lently with a simple axisymmetric form (6) Of course thisequivalence only holds for the average values of multiplerandom rough realizations In [15 17] it was shown thatfor the indentation of a rigid indenter having the shape (6)

Advances in Tribology 3

Asymptotical fractalbehavior

Hertz

Analytical prediction3D BEM calculations

100

10minus1

10minus2

10minus3

10minus6 10minus4 10minus2

k

LElowast

FN

L2Elowast

Figure 3 Normalized normal force 119865119873 versus normal stiffness 119896for a spherical indenter superimposed with a random roughness(119877 = 05 119867 = 07 119871 = 1 ℎrms = 9 sdot 10

minus3) The red-dottedline is the result of a 3D boundary element study The black lineis the theoretical prediction according to (16) Furthermore theasymptotical behavior of the plain roughness (11) and the smoothHertzian case (9) are shown The slope of the asymptotes is 13 inthe Hertzian case and (119867 + 1)

minus1= 05882 for the roughness The

crossover occurs at 119865tr(1198712119864lowast) = 8 times 10

minus5 see (17)

zp

(a)

zs

(b)

zc

(c)

Figure 4 (a) Parabolic shape 119911119901 (b) indenter shape 119911119904 according to(6) with119867 = 07 (c) superposition 119911119888 of both

into an elastic half-space the following relation of the normalcontact stiffness 119896 and the normal force 119865 is valid

119896 =120597119865

120597119889= 2119864lowast(

(119867 + 1) 119865

21198763D119864lowast119867120581 (119867)

)

1(119867+1)

prop 1198651(119867+1)

(7)

with

120581 (119867) = 119867int

1

0

120585119867minus1

radic1 minus 1205852119889120585 =

radic120587

2

119867Γ (1198672)

Γ (1198672 + 12) (8)

For example the Hertzian contact is one particular case ofself-affine surfaces with119867 = 2 and 1198763D = 1(2119877) giving theclassical result [14]

119896

119864lowast= (6119877

119865

119864lowast)

13

(9)

or resolved with respect to the force

119865

119864lowast=

1

6119877(

119896

119864lowast)

3

(10)

The same is true for fractal rough surfaces [12 15]

119896

119864lowast119871= 120577 (119867) (

119865

119864lowastℎ119871)

1(119867+1)

prop 1198651(119867+1) (11)

or

119865

119864lowast= ℎ119871(

119896

120577 (119867) 119864lowast119871

)

119867+1

(12)

The constant 120585(119867)must be (until now) found fromnumericalcalculations Latest studies suggest that

120585 (119867) asymp17

119867 + 1 (13)

Comparison of (7) and (11) shows that the contact prop-erties of a self-affine roughness and a corresponding axis-symmetric profile can be made identical if the followingprefactor 1198763D in (6) is chosen

1198763D = (2

120577119871)

119867+1(119867 + 1) 119871ℎ

2119867120581 (119867) (14)

Using this analogy between randomly rough surfaces andaxis-symmetric profiles we suggest the rough sphere to bemodeled as a superposition of the parabolic shape 119911119901 with theequivalent rotationally symmetric shape 119911119904

119911119888 (119903) =1199032

2119877+ (

2

120577119871)

119867+1(119867 + 1) 119871ℎ

2119867120581 (119867)|119903|119867 (15)

Figure 4(c) shows a cut through the resulting three-dimensional rotationally symmetric indenter shapes

As the contact stiffness depends only on the currentcontact radius [17] the superposition principle is valid forforces (10) and (12) at the given contact stiffness

119865 (119896) = 119864lowast[1

6119877(119896

119864)

3

+ 119871ℎ(119896

120577119864119871)

119867+1

] (16)

For small values of the normal force or the contact stiffnesswe expect to see behavior according to (11) while at higherforces the Hertzian behavior (9) is predictedThe crossover isexpected to take place at the intersection of both asymptoticcurves

119865tr119864lowast

= [120577(119867)3119867+3

ℎminus31198713119867(6119877)minus119867minus1

]1(119867minus2)

(17)

The dependence (16) is shown in Figure 3 together withthe numerical results of the Boundary Element MethodThe solid line is the theoretical prediction according to(16) Furthermore the asymptotical behavior of the plainroughness (11) and the smooth Hertzian case (9) are shownThe slope of the asymptotes is 13 in the Hertzian case and1(119867 + 1) for the roughness

4 Conclusions

We investigated the normal contact problem of a sphere witha self-affine roughness by means of the BEM and suggested


a simple analytical method of calculating the contact stiffnessfor arbitrary normal forcesThemethod can be applied with-out changes to indentation of arbitrary bodies of revolutionwith self-affine roughness with any Hurst exponent in therange 0 lt 119867 lt 2

In the present paper we considered the contact partnersto be elastic at all scales and confined ourselves to self-affineroughness without cutoff For this class of contact problemswe have shown the validity of the superposition principle forprofiles The consideration of roughness covering all scalesincluding the system size means no real restriction of theproposed approach as a generalization to amore general caseof surfaces having multifractal roughness or arbitrary powerspectrum is straightforward

In [17] it has been shown that the GW model is valid forsurfaces with 119867 le 0 We can thus compare the results fromGreenwood and Trip to the current findings by assuming119867 = 0 In this case (16) predicts a linear dependence betweennormal force and contact stiffness just likeGT although in thelatter this is not stated explicitly The transition to Hertzianbehavior occurs at


= 120582ℎ32

11987712 (18)

with 120582 asymp 11 (17) or 120582 asymp 141 (GT [7]) respectivelyAnother insight can be gained from the comparison of

both results In the original GT paper [7] the apparentcontact area was found to grow larger with added roughnesscompared to the smooth Hertzian case Bearing in mind thatthe contact radius is proportional to the contact stiffness inconcentrated contacts one could expect the contact stiffnessto grow larger as well However the current results show thatthis is not the case For loads smaller than the characteristictransition force (17) the added roughness will always lead toa decrease in the contact stiffness compared to the smoothHertzian case see Figure 3

The assumption of elasticity on all scales does not playa crucial role in the current findings The reason is thatthe contact stress at any particular scale has the order ofmagnitude 120590 asymp 119864

lowastnabla1199112 wherenabla119911 is the rms surface gradient

corresponding to this scale [19]The breakdown of the elasticbehavior occurs therefore at the smallest scales correspond-ing to the largest rms surface gradient The contact stiffnesson the contrary is determined by the large wavelength in thepower spectrum of the surface roughness and is not sensitiveto the occurrence on the atomic scale

Acknowledgments

The authors are grateful for a discussion with Lars PastewkaThis material is based upon work supported by the DeutscheForschungsgemeinschaft (DFG Grant no PO 81024-1)

References

[1] F P Bowden and D Tabor The Friction and Lubrication ofSolids Clarendon Press Oxford UK 1986

[2] J F Archard ldquoElastic deformation and the laws of frictionrdquoProceedings of the Royal Society A vol 243 no 1233 pp 190ndash205 1957

[3] J A Greenwood and J B P Williamson ldquoContact of nominallyflat surfacesrdquo Proceedings of the Royal Society A vol 295 no1442 pp 300ndash319 1966

[4] S Akarapu T Sharp and M O Robbins ldquoStiffness of contactsbetween rough surfacesrdquo Physical Review Letters vol 106 no20 Article ID 204301 4 pages 2011

[5] R Buzio C Boragno F Biscarini F Buatier De Mongeot andU Valbusa ldquoThe contact mechanics of fractal surfacesrdquo NatureMaterials vol 2 no 4 pp 233ndash236 2003

[6] C Campana B N J Persson and M H Muser ldquoTransverseand normal interfacial stiffness of solids with randomly roughsurfacesrdquo Journal of Physics vol 23 no 8 Article ID 0850012011

[7] J A Greenwood and J H Tripp ldquoThe elastic contact of roughspheresrdquo ASME Journal of Applied Mechanics vol 34 no 1 pp153ndash159 1967

[8] J R Barber ldquoBounds on the electrical resistance betweencontacting elastic rough bodiesrdquoProceedings of the Royal SocietyA vol 459 pp 53ndash66 2003

[9] J R Barber ldquoIncremental stiffness and electrical contact con-ductance in the contact of rough finite bodiesrdquo Physical ReviewE vol 87 no 1 Article ID 013203 5 pages 2013

[10] R Pohrt and V L Popov ldquoNormal contact stiffness of elasticsolids with fractal rough surfacesrdquo Physical Review Letters vol108 no 10 Article ID 104301 4 pages 2012

[11] B N J Person ldquoContact mechanics for randomly roughsurfacesrdquo Surface Science Reports vol 61 no 4 pp 201ndash2272006

[12] R Pohrt V L Popov and A E Filippov ldquoNormal contactstiffness of elastic solids with fractal rough surfaces for one- andthree-dimensional systemsrdquo Physical Review E vol 86 no 2Article ID 026710 7 pages 2012

[13] R Pohrt and V L Popov ldquoInvestigation of the dry normal con-tact between fractal rough surfaces using the reductionmethodcomparison to 3D simulationsrdquo PhysicalMesomechanics vol 15no 5-6 pp 275ndash279 2012

[14] H Hertz ldquoUeber die beruhrung fester elastischer korperrdquoJournal Fur Die Reine Und AngewandteMathematik vol 92 pp156ndash171 1881

[15] V Popov ldquoMethod of reduction of dimensionality in contactand friction mechanics a linkage between micro and macroscalesrdquo Friction vol 1 no 1 pp 41ndash62 2013

[16] I A Polonsky and L M Keer ldquoA numerical method forsolving rough contact problems based on the multi-level multi-summation and conjugate gradient techniquesrdquoWear vol 231no 2 pp 206ndash219 1999

[17] V L Popov and M Heszlig Methode Der Dimensionsreduktion inDer Kontaktmechanik Und Reibung Springer 2013

[18] N Sneddon ldquoThe relation between load and penetration inthe axisymmetric boussinesq problem for a punch of arbitraryprofilerdquo International Journal of Engineering Science vol 3 no1 pp 47ndash57 1965

[19] S Hyun and M O Robbins ldquoElastic contact between roughsurfaces effect of roughness at large and small wavelengthsrdquoTribology International vol 40 no 10ndash12 pp 1413ndash1422 2007

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DistributedSensor Networks



(a) (b)

Figure 1 Graphical representations of a fractal rough surface with119867 = 06 (a) grayscale representation and (b) a cut through the surface

the help of the power spectrum according to

ℎ () = sum

119902

1198612D ( 119902) exp (119894 ( 119902 sdot + 120601 ( 119902))) (2)

with

1198612D ( 119902) =2120587

119871radic1198622D ( 119902) = 1198612D (minus 119902) (3)

and the phases 120601( 119902) = minus120601(minus 119902) which are randomlydistributed on the interval [0 2120587) All samples were generatedon a grid with 513 times 513 discrete evenly spaced pointsA typical example of a self-affine surface with the Hurstexponent119867 = 06 is shown in Figure 1 while Figure 2 showsa sample of the superposition of both sphere and roughness

The indenter was pressed into an elastic half-space withthe normal force 119865 The indentation depth and the configu-ration of the contact were calculated using the boundary ele-mentmethodwith an iterativemultilevel algorithm similar to[15 16] The incremental normal stiffness 119896 was calculated byevaluating the differential quotient of force and indentationdepth All values were obtained by ensemble averaging of 50surface realizations having the same power spectrum

3 Results and Discussion

31 Numerical Results Figure 3 shows the resulting depen-dency (red points) For small normal forces the system isdominated by the roughness and the stiffness approachesthe asymptotic dependence with the slope (119867 + 1)

minus1 char-acteristic for nominally flat fractal surfaces [12] For highernormal forces the influence of the roughness vanishes andthe system behaves like a smooth spherical Hertzian indenter(slope 13)

32 Analytical Estimation In [12] it was shown that forrandomly rough self-affine surfaces the contact stiffness is apower function of the normal force of the form

119896 prop 1198651(119867+1)

(4)

Figure 2 Numerically generated rough sphere119867 = 07

with 0 lt 119867 lt 1 In [17] it was shown that this relationremains valid for Hurst exponents 0 lt 119867 lt 2 and that theonly property needed for the validity of the scaling law (4) isthe self-affinity of the surface which means that the surfaceappears undistinguishable from the original when viewedunder an arbitrary magnification 120595 [15]

1199111015840

119903(119909 119910) = 120595

119867119911119903 (

119909

120595119910

120595) (5)

Here it is not important whether the surface is randomlyrough or just a simple axisymmetric profile with the samescaling properties [18]

119911119904 (119903) = 1198763D sdot |119903|119867 (6)

Thus with respect to the contact stiffness the randomlyrough fractal and self-affine surface can be replaced equiva-lently with a simple axisymmetric form (6) Of course thisequivalence only holds for the average values of multiplerandom rough realizations In [15 17] it was shown thatfor the indentation of a rigid indenter having the shape (6)



Hertz


100

10minus1

10minus2

10minus3


k

LElowast

FN

L2Elowast





minus5 see (17)

zp

(a)

zs

(b)

zc

(c)



119896 =120597119865

120597119889= 2119864lowast(

(119867 + 1) 119865

21198763D119864lowast119867120581 (119867)

)

1(119867+1)

prop 1198651(119867+1)

(7)

with

120581 (119867) = 119867int

1

0

120585119867minus1

radic1 minus 1205852119889120585 =

radic120587

2

119867Γ (1198672)

Γ (1198672 + 12) (8)


119896

119864lowast= (6119877

119865

119864lowast)

13

(9)


119865

119864lowast=

1

6119877(

119896

119864lowast)

3

(10)


119896

119864lowast119871= 120577 (119867) (

119865

119864lowastℎ119871)

1(119867+1)

prop 1198651(119867+1) (11)

or

119865

119864lowast= ℎ119871(

119896

120577 (119867) 119864lowast119871

)

119867+1

(12)


120585 (119867) asymp17

119867 + 1 (13)


1198763D = (2

120577119871)

119867+1(119867 + 1) 119871ℎ

2119867120581 (119867) (14)


119911119888 (119903) =1199032

2119877+ (

2

120577119871)

119867+1(119867 + 1) 119871ℎ

2119867120581 (119867)|119903|119867 (15)



119865 (119896) = 119864lowast[1

6119877(119896

119864)

3

+ 119871ℎ(119896

120577119864119871)

119867+1

] (16)



= [120577(119867)3119867+3


]1(119867minus2)

(17)


4 Conclusions







= 120582ℎ32

11987712 (18)






Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Hertz


100

10minus1

10minus2

10minus3


k

LElowast

FN

L2Elowast





minus5 see (17)

zp

(a)

zs

(b)

zc

(c)



119896 =120597119865

120597119889= 2119864lowast(

(119867 + 1) 119865

21198763D119864lowast119867120581 (119867)

)

1(119867+1)

prop 1198651(119867+1)

(7)

with

120581 (119867) = 119867int

1

0

120585119867minus1

radic1 minus 1205852119889120585 =

radic120587

2

119867Γ (1198672)

Γ (1198672 + 12) (8)


119896

119864lowast= (6119877

119865

119864lowast)

13

(9)


119865

119864lowast=

1

6119877(

119896

119864lowast)

3

(10)


119896

119864lowast119871= 120577 (119867) (

119865

119864lowastℎ119871)

1(119867+1)

prop 1198651(119867+1) (11)

or

119865

119864lowast= ℎ119871(

119896

120577 (119867) 119864lowast119871

)

119867+1

(12)


120585 (119867) asymp17

119867 + 1 (13)


1198763D = (2

120577119871)

119867+1(119867 + 1) 119871ℎ

2119867120581 (119867) (14)


119911119888 (119903) =1199032

2119877+ (

2

120577119871)

119867+1(119867 + 1) 119871ℎ

2119867120581 (119867)|119903|119867 (15)



119865 (119896) = 119864lowast[1

6119877(119896

119864)

3

+ 119871ℎ(119896

120577119864119871)

119867+1

] (16)



= [120577(119867)3119867+3


]1(119867minus2)

(17)


4 Conclusions







= 120582ℎ32

11987712 (18)






Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














= 120582ℎ32

11987712 (18)






Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article contact mechanics of rough spheres...

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