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Table of Contents
1. Friction - The Present State of Our Understanding ………….….3-13
2. Contact of nominally flat surfaces ……….…………………….14-33
3. On the plastic contact of rough surfaces ….……………………34-48
4. An Elastic-Plastic Model for the Contact of Rough Surfaces….49-55
5. Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat…56-61
6. A Finite Element Based Elastic-Plastic Model for the Contact
of Rough Surfaces …………………………………………….62-69
7. Adhesion and Micromechanical Properties of Metal Surfaces...70-94
8. Adhesion Model for Metallic Rough Surfaces………………..95-101
9. Adhesion of Contacting Rough Surfaces in the Presence of
Sub-Boundary Lubrication ………………………………….102-108
10. Adhesion in elastic-plastic Spherical microcontact ………....109-115
11. Static Friction Coefficient Model for Metallic
Rough Surfaces……………………………………………...116-122
12. A Semi-Analytical Solution for the Sliding Inception of a
Spherical Contact ……………………………………………123-130
13. A Static Friction Model for Elastic-Plastic Contacting Rough
Surfaces ……………………………………………………..131-137
14. The Effect of Small Normal Loads on the Static Friction
Coefficient for Very Smooth Surfaces ……………………...138-142
herewithfullydi-
ssure,sticodelsnces.
L. Kogute-mail: [email protected]
I. Etsione-mail: [email protected]
Fellow ASME
Department of Mechanical Engineering,Technion,
Haifa 32000, Israel
Elastic-Plastic Contact Analysis ofa Sphere and a Rigid FlatAn elastic-plastic finite element model for the frictionless contact of a deformable sppressed by a rigid flat is presented. The evolution of the elastic-plastic contactincreasing interference is analyzed revealing three distinct stages that range fromelastic through elastic-plastic to fully plastic contact interface. The model providesmensionless expressions for the contact load, contact area, and mean contact precovering a large range of interference values from yielding inception to fully plaregime of the spherical contact zone. Comparison with previous elastic-plastic mthat were based on some arbitrary assumptions is made showing large differe@DOI: 10.1115/1.1490373#
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IntroductionThe elastic-plastic contact of a sphere and a flat is a fundam
tal problem in contact mechanics. It is applicable, for exampleproblems such as particle handling~@1#!, or sealing, friction, wear,and thermal and electrical conductivity between contacting rosurfaces. Indeed, an impressive number of works on the contarough surfaces, that were published so far~see review by Liuet al. @2#!, are based on the contact behavior of a single spherasperity~Bhushan@3#! in a statistical model of multiple asperitcontact ~Bhushan@4#!. Some of these works are restrictedmainly pure elastic deformation of the contacting sphere, e.g.,pioneering work of Greenwood and Williamson@5#, which isbased on the Hertz solution for a single elastic sphere~e.g., Ti-moshenko and Goodier@6#!. Other works are restricted to purplastic deformation of the contacting sphere, based on the mof Abbott and Firestone@7#, which neglects volume conservatioof the plastically deformed sphere.
The works on either pure elastic or pure plastic deformationthe contacting sphere overlook a wide intermediate range of inest where elastic-plastic contact prevails. An attempt to bridgegap was made by Chang et al.@8# ~CEB model!. In this model thesphere remains in elastic Hertzian contact until a critical interence is reached, above which volume conservation of the sptip is imposed. The contact pressure distribution for the plasticdeformed sphere was assumed to be rectangular and equalmaximum Hertzian pressure at the critical interference. The Cmodel suffers from a discontinuity in the contact load as well asthe first derivatives of both the contact load and the contact arethe transition from the elastic to the elastic-plastic regime. Thdeficiencies triggered several modifications by other researchEvseev et al.@9# suggested a uniform pressure distribution, eqto the maximum Hertzian pressure at the critical interferencethe central portion of the contact area, and an elliptical Hertzdistribution outside this portion starting from the maximum presure and approaching zero at the contact boundary. The auconcluded their paper with a recommendation to find a more geral model for the elastic-plastic regime. Chang@10# used an ap-proximate linear interpolation for the elastic-plastic regimeconnecting the value of the contact load at yielding inceptionthat at the beginning of the fully plastic regime. Zhao et al.@11#used mathematical manipulation to smooth the transition of
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OFMECHANICAL ENGINEERSfor publication in the ASME JOURNAL OF APPLIED ME-CHANICS. Manuscript received by the ASME Applied Mechanics Division, August2001; final revision, December 14, 2001. Associate Editor: E. Arruda. Discussiothe paper should be addressed to the Editor, Prof. Robert M. McMeeking, Dement of Mechanical and Environmental Engineering University of California–SaBarbara, Santa Barbara, CA 93106-5070, and will be accepted until four moafter final publication of the paper itself in the ASME JOURNAL OF APPLIEDMECHANICS.
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contact load and contact area expressions between the elastielastic-plastic deformation regimes. Kucharski et al.@12# solvedthe contact problem of a deformed sphere by the finite elemmethod~FEM! and developed empirical proportional expressiofor the contact load and the contact area. Although the authintended to describe elastic-plastic contact, their results contrated on the behavior of the sphere deep into the plastic regSurprisingly, the mean contact pressure in@12# was, in somecases, higher than the indentation hardness and thereunreasonable.
The work in@1# employed the finite element method to analythe contact of two identical spheres, which by symmetryequivalent to that of one sphere in contact with a frictionless riplane. The analysis in@1# was restricted to an aluminum sphereradiusR50.1 m loaded with a mean contact pressure that neexceeded 2.3 times the material’s yield strength.
As can be seen from the literature survey, accurate generalutions for the elastic-plastic contact of a deformable sphere anrigid flat are still missing. The existing elastic-plastic solutiosuffer from several deficiencies caused mainly by assuming sarbitrary contact pressure distribution or an arbitrary evolutionthe plastic region inside the sphere. The few existing finite ement method solutions are too restricted in terms of materigeometry, and loading.
It should be noticed here that much research has also been~mostly by utilizing the finite element method! on the indentationproblem of a half-space by a rigid sphere, e.g.,@13–16#. However,from the results provided by Mesarovic and Fleck@17# for both asphere pressed by a rigid flat and a half-space indented by asphere, deep into the fully plastic regime, it seems that the beior of these two cases is different. Intuitively, one can see thathe indentation case the radius of the rigid spherical indentermains constant whereas the curvature of a deformable spchanges continuously during the deformation. Moreover, theplaced material in the indented half-space is confined by the rindenter and the elastic bulk of the half-space. This is quiteferent from the situation where the displaced material of theformable sphere is free to expand radially as shown schematicin Fig. 1.
The present research offers an accurate finite element mesolution for the elastic-plastic contact of a deformable spherea rigid flat by using constitutive laws appropriate to any modedeformation, be it elastic or plastic. It also offers a general dimsionless solution not restricted to a specific material or geome
Theoretical BackgroundFigure 1 presents a deformable hemisphere, with a radiuR,
pressed by a rigid flat. The solid and dashed lines show the s
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ation after and before the deformation, respectively. The interence,v, and contact area with a radius,a, ~see Fig. 1! correspondto a contact load,P.
The critical interference,vc , that marks the transition from thelastic to the elastic-plastic deformation regime~i.e., yielding in-ception! is given by~e.g., Chang et al.@8#!
vc5S pKH
2E D 2
R. (1)
The hardness,H, of the sphere is related to its yield strengthH52.8Y ~@18#!. The hardness coefficient,K, is related to the Pois-son ratio of the sphere by~Chang et al.@19#! K50.45410.41n.E is the Hertz elastic modulus defined as
1
E5
12n12
E11
12n22
E2
whereE1 , E2 andn1 , n2 are Young’s moduli and Poisson’s ratioof the two materials, respectively. In the case of the rigid flatE2→`.
The Hertz solution for the elastic contact of a sphere and aprovides the contact load,Pe , and contact area,Ae , for v<vc inthe form
Pe54
3ER1/2v3/25PcS v
vcD 3/2
(2)
Ae5pRv5Ac
v
vc(3)
where Pc and Ac are the contact load and contact area, resptively, at v5vc . Note thatPe andAe can be normalized byPcandAc , respectively, to obtain simple exponential functions of tdimensionless interference,v/vc . These functions are independent of the material properties and sphere radius.
Using Eqs.~1!–~3! the mean contact pressure,pe5Pe /Ae , forv<vc is
pe52
3KHS v
vcD 1/2
5pcS v
vcD 1/2
(4)
wherepc is the mean contact pressure atv5vc .For v.vc the contact is elastic-plastic and a numerical so
tion is required to find the relation betweenv/vc , the contactload, contact area, and mean contact pressure. The finite elemethod~for example, Refs.@20# and @21#! is commonly used forsuch a numerical solution where the contact between the spand the flat is detected by special contact elements~@22#!. A yield-ing criterion should be adopted in solving elastic-plastic probleIn the present analysis the von Mises criterion, which correlawell with experiments~see Bhushan@3#! was selected as the preferred criterion. A recent example for the finite element methsolution to an elastic-plastic contact problem can be found inet al. @23#.
Fig. 1 A deformable sphere pressed by a rigid flat
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Finite Element Model „FEM …
A commercial ANSYS 5.7 package was used to solve the ctact problem. The hemisphere, shown in Fig. 2, was modeledquarter of a circle, due to its axisymmetry. The rigid flat wmodeled by a line. The material of the sphere was assumelastic-perfectly plastic with identical behavior in tension acompression. Although the model can easily accommodate shardening the simpler behavior was selected to allow comparwith existing previous models. A static, small-deformation anasis type was used and justified by comparison with the resultslarge-deformation analysis. The von Mises yielding criterion wused to detect local transition from elastic to plastic deformati
The finite element method numerical solution requires asinput some specific material properties and sphere radius~see@1#,for example!. However, in order to generalize the present solutand eliminate the need for a specific input, the numerical reswere normalized with respect to their corresponding critical vues at yielding inception,vc , similar to Eqs.~2! and ~3!. Thenormalization of the mean contact pressure,p, was done withrespect to the yield strength,Y, of the sphere material. The validitof this normalization was tested by solving the problem for seral different material properties (100,E/Y,1000,n50.3) andsphere radii (0.1 mm,R,10 mm). The dimensionless results oP/Pc , A/Ac , and p/Y versus the dimensionless interferencv/vc , were always the same regardless of the selection of mrial properties and sphere radius.
The finite element mesh consisted of 225 eight-node quadreral axisymmetric elements comprising a total of 714 nodHigh-order elements were selected to better fit the curvature ofsphere. The sphere was divided into two different mesh denzones. Zone I, within a 0.1R distance from the sphere tip~see Fig.2!, contained 87% of the nodes and had extremely fine mesbetter handle the high stress gradients in this zone and to achgood discretization for accurate detection of the contact areadius, a. For this reason the typical mesh size was 0.03ac whereac5(Rvc)
1/2. Zone II, outside the 0.1R distance, had graduacoarser mesh at increasing distance from the sphere tip.model also contained a single two-dimensional target elementing on the flat and 16 two-dimensional surface-to-surface conelements on the sphere surface in zone I.
The boundary conditions are presented in Fig. 2. The nodethe axis of symmetry of the hemisphere cannot move in the radirection. Likewise the nodes on the bottom of the hemisphcannot move in the axial direction due to symmetry. Restrictalso the radial motion of these nodes did not affect the result
Fig. 2 Model description
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the finite element analysis~FEA! since this boundary is very faaway from the contact zone and therefore has very little effecthe contact results.
The numerical model was first verified by comparing its outpwith the analytical results of the Hertz solution in the elasticgime, i.e., forv,vc . The verification included the contact loacontact area radius, and stress distribution in the contact areaalong the axis of symmetry. The difference between the numerand analytical results was always less than 2.8%. Another vecation of the model was done in the elastic-plastic regime~for 1,v/vc,110! by increasing the mesh density to 2944 nodes acomparing the results with these obtained with the original 7nodes. The largest differences in the contact load and contactwere only 1% and 3%, respectively. These two verificationstablish the validity of the numerical model with the original meto study the behavior of the sphere in the elastic-plastic regim
Results and DiscussionFigure 3 presents the evolution of the plastic region inside
sphere~within the dashed line frame shown in Fig. 2! for increas-ing interference values up tov/vc5110. The elastic-plasticboundary at each interference is determined by all the nodesequivalent total strain larger than the yield strain,«Y . The axialand radial coordinates in Fig. 3 are normalized by the criticontact radius,ac . It is interesting to note the larger axial penetration of the plastic region compared to its radial spread.v/vc5110, for example, the plastic region penetrates about 3acbelow the contact surface and reaches only about 18ac on thesphere surface.
The evolution of the plastic region at its earlier stages,v/vc<11, is shown in more details in Fig. 4. Up tov/vc56 theplastic region is completely surrounded by elastic material.v/vc56 the plastic region first reaches the sphere surfaceradius of about 2.7ac . At this point an elastic core remains lockebetween the plastic region and the sphere surface. As the inteence increases abovev/vc56 and the plastic region grow, thelastic core gradually shrinks as shown in Fig. 5. The shrinkrate is very small belowv/vc530 and rapidly increases thereater. The surface of the sphere at the contact region is now divinto three subregions as follows:~I! an inner circular elastic subregion extending radially from the center of the contact until
Fig. 3 Evolution of the plastic region in the sphere tip for 12ÏvÕvcÏ110
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edge of the elastic core;~II ! an intermediate annular plastic subregion between the edge of the elastic core and the outer fronthe plastic region, and~III ! an outer elastic subregion thereafteThe evolution of these three subregions on the sphere surfacv/vc>6 is demonstrated in Fig. 6 that shows the radial locatioof the inner and outer elastic-plastic boundaries normalized bycontact area radius,a, as a function of the dimensionless interfeencev/vc . The horizontal dashed line atr /a51 indicates thecircular boundary of the contact area. From the figure it caneasily seen that belowv/vc56 the sphere surface is fully elasticAt v/vc56 the plastic region reaches the sphere surface for
Fig. 4 Evolution of the plastic region in the sphere tip for 1ÏvÕvcÏ11
Fig. 5 Dimensionless radial location, r Õac c , of the innerelastic-plastic boundary on the sphere surface showing itsshrinkage for 6 ÏvÕvcÏ68
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first time. This occurs very close to the boundary of the contarea, atr /a50.94. For 6<v/vc<56 the annular plastic subregion remains within the contact area. Its outer boundary, whfirst reaches the edge of the contact area atv/vc56.2, coincideswith that of the contact area while its inner boundary graduamoves towards the contact center as the elastic core shown in4 shrinks. Forv/vc.56 the outer boundary of the annular plastsubregion somewhat exceeds the boundary of the contactwhile the inner elastic core continues to shrink and disappecompletely atv/vc568. From there on the entire contact zoneplastic and the rate of its radial expansion increases substant
From the above discussion it can be seen that the evolutiothe elastic-plastic contact can be divided into three distinct staThe first one for 1<v/vc<6 where the plastic region developbelow the sphere surface and the entire contact area is elasticsecond one for 6<v/vc<68 where the contact area is elastiplastic containing an annular plastic subregion confined by inand outer elastic ones. The third stage forv/vc.68 correspondsto a fully plastic contact area.
Figure 7 presents the results of the mean contact pressurep/Yas a function of the interference,v/vc , that were obtained by thepresent finite element analysis along with the results fromCEB model~@8#! and from Zhao et al.@11#. When the discretenumerical results of the finite element analysis were curve fittebecame evident that a distinct transition point exists atv/vc56.This is clearly observed in Fig. 7 by the discontinuity in the sloof the finite element analysis results atv/vc56. Apparently, thetransition from fully elastic to elastic-plastic contact area, whioccurs when the expanding plastic region first reaches the spsurface, changes the behavior of the mean contact pressuresimilar transition or change was found atv/vc568 that marks theinception of fully plastic contact area when the central elastic cis completely eliminated. The empirical expressions obtainfrom the curve fitting for the mean contact pressure in the stathat were discussed above are
Fig. 6 Radial location of inner and outer elastic-plastic bound-aries on the sphere surface for 6 ÏvÕvcÏ110
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S p
YD1
51.19S v
vcD 0.289
for 1<v/vc<6 (5)
S p
YD2
51.61S v
vcD 0.117
for 6<v/vc<110. (6)
From Fig. 7 it can be seen that the dimensionless mean conpressure of the finite element analysis atv/vc5110 approachesthe valuep/Y52.8. This is identical to the ratio between thhardness and yield strength found experimentally for many mrials as indicated by Tabor@18#. Hence, the value ofp at this pointis that of the material hardness,H, and, hence,v/vc5110 marksthe inception of the fully plastic regime where the mean contpressure assumes a constant value equals to the material hard
The CEB model~@8#! predicts a constant mean contact pressuwhich largely underestimates the finite element analysis resexcept for a small range,v/vc<3, where it largely overestimatethe finite element analysis results. This is one of the limitationsthis model as discussed by Evseev et al.@9#.
Zhao et al. model@11# predictsp/Y values that are fairly closeto the finite element analysis results. The largest deviationabout 9% occurs atv/vc554, which was selected in Ref.@11#,based on the work of Johnson@24#, as the lowest possible inception of fully plastic regime wherep/Y52.8. Actually the fullyplastic regime starts atv/vc5110 as can be seen from the finielement analysis results in Fig. 7.
The results obtained by Kucharski et al.@12# cover the range of175<v/vc<2800 that is very deep into the fully plastic regimand therefore outside the range of interest of the present anal
The change in the slope of the mean contact pressure atransition pointv/vc56 is somewhat similar to a typical stresstrain curve where a change of slope occurs at the elastic limithe spherical contact problem the valuev/vc56 is analogous tothe critical strain, which corresponds to yielding inception. Thpoint marks the elastic limit of the spherical contact interfaFrom there on the resistance of the material to increasing stdecreases and eventually disappears atv/vc5110.
The finite element analysis results for the dimensionless conarea and contact load are presented in Figs. 8 and 9, respectalong with the results of Refs.@8# and @11#. The correspondingempirical expressions obtained from curve fitting of the finiteement analysis numerical results in the various stages of thelution of the elastic-plastic contact are
Fig. 7 Dimensionless mean contact pressure, p ÕY, as a func-tion of the dimensionless interference, vÕvc , in the elastic-plastic regime
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vcD 1.146
for 6<v/vc<110. (8)
The accuracy of the curve fitting for Eqs.~7! and ~8! was betterthan 97% throughout the range ofv/vc .
From Fig. 8 it is clear that the contact area obtained by the Cmodel~@8#! overestimates the finite element analysis results. Thlargest difference is 56% atv/vc54. This difference diminishesas the interference increases and forv/vc5110 it becomes less
Fig. 8 Dimensionless contact area, A ÕA c , as a function of thedimensionless interference, vÕvc , in the elastic-plastic regime
Fig. 9 Dimensionless contact load, PÕPc , as a function of thedimensionless interference, vÕvc , in the elastic-plastic regime
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than 7%. The reason for the larger deviation at smaller interences is that the CEB model assumes volume conservation oentire sphere tip forv/vc>1. This in fact is equivalent to assuming fully plastic regime of the entire sphere tip as soon ascritical interference is reached. From Fig. 3 it is clear thatplastic region develops gradually with increasing interferenceonly for very large interferences the entire asperity tip is placally deformed.
The Zhao et al.@11# results underestimate the finite elemeanalysis ones by up to 18% atv/vc510 and overestimate themby up to 20% atv/vc551. The Zhao et al. model assumes fulplastic sphere tip atv/vc554. From this point on the contact areis calculated from the geometrical intersection of the flat withoriginal profile of the sphere according to Abbott and Firesto@7#. This is also true for the CEB model, which therefore predithe same results at large interferences.
At v/vc5110 the contact area based on the Abbott and Festone approximate calculation is only 7% higher than the maccurate result of the finite element analysis. It seems therethat the Abbott and Firestone model is a relatively fair appromation for the contact area in the fully plastic regime.
Figure 9 presents the contact loadP/Pc versus the interferencev/vc . The contact load obtained by the CEB model~@8#! clearlydiffers from the finite element analysis results. It overestimatesfinite element analysis results at small interferences, by up to 6at v/vc52, and underestimates these results by up to 38%v/vc5110. This is due to a combination of the very inaccuraassumption of constant mean pressure and too large contactin @8# as shown in Figs. 7 and 8. Contrary to the CEB model,contact load obtained by Zhao et al. underestimates the finiteement analysis results at small interferences~21% at v/vc57!and overestimates these results at large interferences~about 30%at v/vc552!.
Since the model is general enough to accommodate matbehavior other than elastic-perfectly plastic, various levels ofear isotropic strain hardening were also investigated. In thetreme case of a very large tangent modulus that is 0.1E, the dif-ference in the results, compared to the present elastic-perfeplastic case, was less than 20%. In fact forv/vc<20 the maxi-mum difference was less than 4.5%. For most practical matethe tangent modulus is less than 0.05E hence, the difference in theresults is much smaller and the present case can be considegeneral elastic plastic one.
It is interesting to compare some features of the present conproblem of a deformable sphere and a rigid flat with these ofhalf-space indented by a rigid sphere. The fully plastic regimeindentation starts atA/Ac5113.2 according to Francis@25#, and atP/Pc>360 according to Johnson@24#. The corresponding finiteelement analysis results for fully plastic deformable spherev/vc5110 areA/Ac5205 andP/Pc5534. Clearly the two prob-lems exhibit different behavior. The indented half-space yiemore easily than the pressed sphere. This is probably due togreater resistance to radial expansion that is imposed on theflected material in the case of the indented half-space as compto the case of the deformable sphere.
ConclusionThe elastic-plastic contact problem of a deformable sphere
a rigid flat was solved by the finite element method considerthe actual constitutive laws for the relevant regime of deformtion. Hence, the present model is much more accurate than pous ones that relied on unrealistic assumptions regarding thetact pressure distribution or evolution of the plastic region abothe critical interference. By properly normalizing the contact loacontact area, and mean contact pressure, the present modevides simple analytical expressions that extend the classical Hsolution up to a fully plastic contact.
It was found that the evolution of the elastic-plastic contact cbe divided into three distinct stages. The first one for 1<v/vc
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<6 where the plastic region develops below the sphere surand the entire contact area is elastic. The second one fo<v/vc<68 where the contact area is elastic-plastic, and the tstage forv/vc.68 corresponds to a fully plastic contact area.
The numerical results of the present finite element analwere normalized in a way that allowed a general solution thaindependent of specific material and radius of the sphere. Dimsionless expressions for the mean contact pressure, contactand contact area were derived for a large range of interferevalues up tov/vc5110.
A change in the behavior of the mean contact pressureobserved atv/vc56, which marks the elastic limit of the contacarea. The interferencev/vc5110 marks the inception of fullyplastic regime where the mean contact pressure becomes eqthe material hardness.
A comparison of the present results with the results of previelastic-plastic models as well as with these of indentation moshowed substantial differences.
AcknowledgmentThis research was supported in parts by the Fund for the
motion of Research at the Technion and by the German-IsrProject Cooperation~DIP!.
Nomenclature
a 5 radius of contact areaA 5 contact areaE 5 Hertz elastic modulus
E1,2 5 Young’s moduliH 5 hardness of the sphereK 5 hardness factor, 0.45410.41nP 5 contact loadp 5 mean contact pressure,P/AR 5 radius of the sphereY 5 yield strength of the spheren 5 Poisson’s ratio of the sphere
n1,2 5 Poisson’s ratiov 5 interference
Subscripts
c 5 critical valuese 5 elastic contact
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References@1# Vu-Quoc, L., Zhang, X., and Lesburg, L., 2000, ‘‘A Normal Force
Displacement Model for Contacting Spheres Accounting for Plastic Deformtion: Force-Driven Formulation,’’ ASME J. Appl. Mech.,67, pp. 363–371.
@2# Liu, G., Wang, Q. J., and Lin, C., 1999, ‘‘A Survey of Current Models foSimulating the Contact between Rough Surfaces,’’ Tribol. Trans.,42, pp. 581–591.
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@5# Greenwood, J. A., and Williamson, J. B. P., 1966, ‘‘Contact of Nominally FSurfaces,’’ Proc. R. Soc. London, Ser. A,295, pp. 300–319.
@6# Timoshenko, S. P., and Goodier, J. N., 1970,Theory of Elasticity, 3rd Ed.,McGraw-Hill, New York.
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@9# Evseev, D. G., Medvedev, B. M., and Grigoriyan, G. G., 1991, ‘‘Modificatioof the Elastic-Plastic Model for the Contact of Rough Surfaces,’’ Wear,150,pp. 79–88.
@10# Chang, W. R., 1997, ‘‘An Elastic-Plastic Contact Model for a Rough SurfaWith an Ion-Plated Soft Metallic Coating,’’ Wear,212, pp. 229–237.
@11# Zhao, Y., Maietta, D. M., and Chang, L., 2000, ‘‘An Asperity MicrocontaModel Incorporating the Transition From Elastic Deformation to Fully PlasFlow,’’ ASME J. Tribol., 122, pp. 86–93.
@12# Kucharski, S., Klimczak, T., Polijaniuk, A., and Kaczmarek, J., 1994, ‘‘FinitElements Model for the Contact of Rough Surfaces,’’ Wear,177, pp. 1–13.
@13# Hardy, C., Baronet, C. N., and Tordion, G. V., 1971, ‘‘The Elasto-Plasticdentation of a Half-Space by a Rigid Sphere,’’ Int. J. Numer. Methods Eng3,pp. 451–462.
@14# Kral, E. R., Komvopoulos, K., and Bogy, D. B., 1993, ‘‘Elastic-Plastic FiniElement Analysis of Repeated Indentation of a Half-Space by a Rigid SpheASME J. Appl. Mech.,60, pp. 829–841.
@15# Komvopoulos, K., and Ye, N., 2001, ‘‘Three-Dimensional Contact AnalysisElastic-Plastic Layered Media with Fractal Surface Topographies,’’ ASMETribol., 123, pp. 632–640.
@16# Giannakopoulos, A. E., 2000, ‘‘Strength Analysis of Spherical IndentationPiezoelectric Materials,’’ ASME J. Appl. Mech.,67, pp. 409–416.
@17# Mesarovic, S. D., and Fleck, N. A., 2000, ‘‘Frictionless Indentation of Dissimlar Elastic-Plastic Spheres,’’ Int. J. Solids Struct.,37, pp. 7071–7091.
@18# Tabor, D., 1951,The Hardness of Metals, Clarendon Press, Oxford, UK.@19# Chang, W. R., Etsion, I., and Bogy, D. B., 1988, ‘‘Static Friction Coefficie
Model for Metallic Rough Surfaces,’’ ASME J. Tribol.,110, pp. 57–63.@20# Reddy, J. N., 1993,An Introduction to the Finite Element Method, 2nd Ed.,
McGraw-Hill, New York.@21# Owen, D. R. J., and Hinton, E., 1980,Finite Elements in Plasticity: Theory
and Practice, Pineridge Press, Swansea, UK.@22# Zhong, Z. H., 1993,Finite Element Procedures for Contact Impact Problem,
Oxford University Press, New York.@23# Liu, G., Zhu, J., Yu, L., and Wang, Q. J., 2001, ‘‘Elasto-Plastic Contact
Rough Surfaces,’’ Tribol. Trans.,44, pp. 437–443.@24# Johnson, K. L., 1985,Contact Mechanics, Cambridge University Press, Cam
bridge, UK.@25# Francis, H. A., 1976, ‘‘Phenomenological Analysis of Plastic Spherical Ind
tation,’’ ASME J. Eng. Mater. Technol.,98, pp. 272–281.
Transactions of the ASME
nless ex-interferenceeir effectthe regimesproximate
Journal of Colloid and Interface Science 261 (2003) 372–378www.elsevier.com/locate/jcis
Adhesion in elastic–plastic spherical microcontact
Lior Kogut1 and Izhak Etsion∗
Department of Mechanical Engineering, Technion, Haifa 32000, Israel
Received 29 May 2002; accepted 14 January 2003
Abstract
An elastic–plastic adhesion model for a metallic deformable sphere pressed by a rigid flat is presented. Analytical dimensiopressions for the local separation outside the contact area and for the adhesion force are provided covering a large range ofvalues from a point contact to a fully plastic contact. Two main dimensionless parameters of the problem are identified and thon the adhesion is investigated. The significance of the adhesion in the elastic–plastic contact force balance is discussed andwhere adhesion is important or negligible are pointed out. A comparison of the present results with a previously published apelastic–plastic model shows substantial differences in the local separation and in the adhesion force. 2003 Elsevier Science (USA). All rights reserved.
Keywords: Adhesion force; Microcontact; Elastic–plastic spherical contact
hert isca-on-
lish-aniche-on
e di-as
astic. Thedel,er,
[7],alltionstica o
ity of
tiveand
rfacefromliessiontressels
f anreted
ofs ofe.g.,0].
erendd ans inr theealt19].eenfew
las-de-hoandhat
1. Introduction
The adhesion force between a deformable smooth spand a rigid smooth flat is a fundamental problem thatreated both analytically and experimentally. It is applible, for example, in calculating the adhesion between ctacting rough surfaces [1], postchemical mechanical poing cleaning [2], and biochemical processes in the humbody [3]. In recent years, the possibility of using atomforce microscope (AFM)-based techniques to study adsion of a soft polymer sphere under large loads was demstrated [4] and later extended to even larger loads wherrect evidence of plastic deformation in a microcontact wpresented [5].
Two basic adhesion models for contact between an elsphere and a flat have been proposed in the literaturefirst model by Johnson et al. [6], known as the JKR mois more suitable for large-radius compliant solids like rubbfor example. The second model by Derjaguin et al.known as the DMT model, seems more suitable for smstiff spheres. The JKR model is based on the assumpthat attractive intermolecular surface forces result in eladeformation of the sphere and, thus, increase the are
* Corresponding author.E-mail address: [email protected] (I. Etsion).
1 Present address: Department of Mechanical Engineering, UniversCalifornia, Berkeley, CA 94720.
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights rdoi:10.1016/S0021-9797(03)00071-7
e
-
f
contact beyond that predicted by Hertz theory. The attracforces, however, are confined to the area of contactare zero outside. The DMT model assumes that the suforces do not change the deformed profile of the spherethat predicted by Hertz theory. The attractive forces alloutside the contact area and are balanced by the comprein the contact area (this compression having the Hertz sdistribution). Tabor [8] suggested that these two mod(JKR and DMT) are appropriate in opposite extremes oadhesion dimensionless parameter that may be interpas the ratio of the elastic deformation to the rangeaction of the adhesive forces. For intermediate valuethis parameter numerical computations were presented,Muller et al. [9] and in much greater detail Greenwood [1Analytical solutions, for various interaction potentials, woffered by Maugis [11], Barthel [12], and Greenwood aJohnson [13]. Johnson and Greenwood [14] presenteadhesion map for the elastic regime that demonstratewhich regions of the dimensionless adhesion parameteavailable adhesion models are valid. Adhesion is also dwith in several review papers and books, e.g., Refs. [15–
Despite the large volume of work that has so far bpublished on adhesion of contacting spheres only astudies have dealt with the elastic–plastic and fully ptic contact regimes. Several articles deal with plasticformation, e.g., Pollock [20]; Pethica and Tabor [21], walso studied the effect of an oxide layer on adhesion;Maugis and Pollock [22]. In these articles it is shown t
eserved.
L. Kogut, I. Etsion / Journal of Colloid and Interface Science 261 (2003) 372–378 373
de-thengel,stlyeri-onof ade-
tic–cal-
andingated
genble
ic–ingsticva
on
enblerigidtivesticstictionssin
s
bma-eo-,
e
is:
ughs soface
allf an
ntact
the
ur-rityals
stic.
27]
.3–
foree.d in
the
surface forces alone are sufficient to cause local plasticformation. Roy Chowdhury and Pollock [23] extendedwork of Pollock [20] to multiasperity rough surface usithe model of Greenwood and Williamson [24]. This modhowever, is limited to the case where the contact is moelastic; i.e., the percentage of plastically deformed aspties is very small (about 2%). An improved DMT adhesimodel was used by Chang et al. [25] to study adhesionsingle sphere pressed by rigid flat in the elastic–plasticformation regime. Due to the lack of an accurate elasplastic contact solution some approximation was used toculate the local separation between a deformed sphereflat. Mesarovic and Johnson [26] dealt with the unloadprocess of a sphere in the fully plastic regime and calculthe force to separate the surfaces.
As can be seen from the literature survey, an accurateeral solution for the elastic–plastic adhesion of a deformasphere and a rigid flat is still missing. The existing elastplastic solution for adhesion has the main deficiency of usan equation for the local separation that is valid in the elaregime. An adequate theoretical model can be used in ariety of practical implementations, including AFM adhesimeasurements.
The present research offers an accurate finite elemanalysis (FEA) for the local separation of a deformaelastic, perfectly plastic smooth sphere pressed by asmooth flat in the elastic–plastic regime by using constitulaws appropriate to any mode of deformation be it elaor plastic. The adhesion of the sphere in the elastic–plaregime is calculated by using this accurate local separain the DMT model. It also offers a general dimensionlesolution not restricted to a specific material or geometrythe elastic and elastic–plastic regimes.
2. Analysis
Figures 1a and 1b show a deformable sphere of radiuR
pressed by a rigid flat. The contact loadP between the twocontacting surfaces is the sum of the external loadF andthe adhesion forceFs . The solid and dashed lines in Fig. 1show schematically the situation after and before defortion, respectively, as perceived by Chang et al. [25]. The gmetrical parameters of the problem, i.e., the interferenceω,the contact area with radiusa, and the local separation,Z(r),outside the contact area wherer > a are all dependent on thcontact load,P .
The following assumptions are adopted for the analys
1. The dimensions of the contact zone are small eno(microcontact) compared with the surface roughnesthat such roughness can be neglected and the surcan be treated as perfectly smooth.
2. The DMT model prevails; i.e., the attractive forceslie outside the contact area and, in the absence o
a
-
-
t
s
(a)
(b)
Fig. 1. (a) Forces acting between a sphere and a rigid flat. (b) The coof a deformable sphere and a rigid flat under normal loading.
external load, are balanced by the compression incontact area.
These assumptions limit the analysis to small radii of cvature ranging, for example, from values typical of aspesummits in rough surfaces to AFM tips and to stiff materilike metals.
3. The sphere material behavior is elastic–perfectly pla
The attractive pressure outside the contact region,p(Z),according to the Lennard-Jones interaction potential is [
(1)p(Z) = 8
3
γ
ε
[(ε
Z
)3
−(
ε
Z
)9],
whereε is the intermolecular distance, which is about 00.5 nm, and γ is the energy of adhesion given by
(2) γ = γ1 + γ2 − γ12.
γ1 andγ2 are the surface energies of the two surfaces becontact andγ12 is the surface energy of their interfacValues of surface energy for various metals can be founRabinowicz [28].
According to Muller et al. [27], the adhesion force,Fs ,is equal to the sum of all molecular interactions outsidecontact area only; i.e.,
(3)Fs = 2π
∞∫a
p(Z)r dr.
At point contact, i.e.,ω = 0, the adhesion force,Fs0, is [27]
(4)Fs0 = 2πR γ.
374 L. Kogut, I. Etsion / Journal of Colloid and Interface Science 261 (2003) 372–378
nstic
l
io
’sase
ce
),ial
rcen
ualsisom
e
toointhatcanver,
a-ins--term
etryhatof
nce,
n it
dAFM
ess
in
en a
e12)ss
an
The local separation,Z(r), depends on the deformatioof the sphere, which may be elastic or elastic–pladepending on the sphere material and contact loadP . Elasticdeformation corresponds to an interference,ω, smaller thana critical value,ωc, at yielding inception. This criticainterference is given (e.g., Chang et al. [25]) by
(5)ωc =(
πKH
2E
)2
R.
The hardness coefficient,K, is related to the Poisson ratof the sphere byK = 0.454+ 0.41ν. E is the Hertz elasticmodulus defined as
(6)1
E= 1− ν2
1
E1+ 1− ν2
2
E2,
whereE1, E2 andν1, ν2 are Young’s moduli and Poissonratios of the two materials, respectively. In the present cof a rigid flat E2 → ∞ and E is very close toE1. ThehardnessH relates to the softer sphere. The radiusac andthe contact loadPc corresponding to the critical interferenare
(7)ac = (ωcR)1/2,
(8)Pc = (2/3)KHπωcR.
Substituting Eq. (1) in Eq. (3), dividing by Eq. (4substitutingR from Eq. (7), using the dimensionless radcoordinater̄ = r/a, and normalizingε andZ by ωc, a di-mensionless form of the adhesion force is obtained:
(9)Fs
Fs0= 8
3
(a/ac)2
ε/ωc
∞∫1
[(ε/ωc
Z/ωc
)3
−(
ε/ωc
Z/ωc
)9]r̄ dr̄.
It is interesting to examine the ratio of the adhesion foto the contact load,Fs/P . For a small ratio the adhesioforce may be negligible and the contact load simply eqthe external normal loadF , otherwise the contact loadP = F + Fs (see Fig. 1a). This ratio can be obtained frthe dimensionless adhesion force in the form
(10)Fs
P= Fs
Fs0
Fs0/Pc
P/Pc
.
The ratioFs0/Pc is obtained from Eqs. (4) and (8) in thform
(11)Fs0
Pc
= 12
π2
γ
RKH
(E
KH
)2
.
WhenFs0/Pc = 1 the adhesion force alone is sufficientinitiate plastic deformation once the sphere comes to a pcontact with a rigid flat. From Eq. (11) it can be seen tvery small sphere radius combined with small hardnessresult in such a situation. In most practical cases, howethe ratioFs0/Pc is less than 0.1.
As will be shown later the local dimensionless deformtion, Z/ωc , consists of two terms. The first term contathe dimensionless interference,ω/ωc , and the dimensionless radial coordinate,̄r = r/a, and is independent of material properties and radius of the sphere. The second
is the dimensionless intermolecular distance,ε/ωc , which isa function of both material properties and sphere geomthroughωc (see Eq. (5)). It should, however, be noted tsince the ratioH/E is about 0.01 for a large spectrummetallic elements and metal alloys [28],ωc and, hence,ε/ωc
are not very sensitive to the material of the sphere. Hethe dimensionless intermolecular distance,ε/ωc, is mainlya measure of the sphere size where higherε/ωc representssmaller sphere radius. For order of magnitude estimatiocan be seen from Eq. (5) thatωc is of order 10−4R. If the sizeof the sphere is limited so that its radius is of order 10−3 mat most, thenε/ωc > 10−3. The upper limit is determineby the smallest relevant sphere that can represent antip whereR is of order 10−8 m and, hence,ε/ωc < 100.
Sincea/ac and P/Pc are functions ofω/ωc only, thedimensionless adhesion force,Fs/Fs0, and the ratioFs/P
in Eqs. (9) and (10) are functions of two dimensionlparameters, namely, the dimensionless interference,ω/ωc,and the dimensionless intermolecular distance,ε/ωc . Theratio Fs/P also depends on the ratioFs0/Pc that includesseveral properties of the contact problem as shownEq. (11).
2.1. Elastic contact
Muller et al. [27] presented the local separation,Z,outside the contact area for an elastic contact betwesphere and a flat in the form
(12)
Z = 1
πR
[a(r2 − a2)1/2
− (2a2 − r2) tan−1
(r2
a2 − 1
)1/2]
+ ε,
where the radius,a, of the contact area is given by
a = (ωR)1/2,
and the intermolecular distance,ε, is added to adjust thcontact concept to the molecular scale. Dividing Eq. (by the critical interference,ωc , and using the dimensionleradial coordinate,̄r = r/a, and the relationa2 = ωR gives
(13)Z
ωc
= 1
π
ω
ωc
f (r̄) + ε
ωc
,
where
(14)f (r̄) = (r̄2 − 1
)1/2 − (2− r̄2) tan−1(r̄2 − 1
)1/2.
Using Eq. (9) with the ratio(a/ac)2 = ω/ωc gives the
dimensionless adhesion force,Fs/Fs0, for elastic contact inthe form
(15)Fs
Fs0= 8
3
ω/ωc
ε/ωc
∞∫1
[(ε/ωc
Z/ωc
)3
−(
ε/ωc
Z/ωc
)9]r̄ dr̄.
Analytical integration of Eq. (15) is complex; however, it cbe performed numerically to obtainFs/Fs0 and then curve
L. Kogut, I. Etsion / Journal of Colloid and Interface Science 261 (2003) 372–378 375
mes
stic,
enttedstic
tivestic
as
e
ues
tact.n bent
nctEt-t
de-nt
tinctdi-
-
ilarress
on-u-
er-os ofur-
et-hen
mstic
as a
theirre-cu-nge
-
fitting used to express this ratio as a function ofω/ωc andε/ωc .
By use of Eq. (10) withP/Pc = (ω/ωc)3/2 the ratio of the
adhesion force over contact load for elastic contact beco
(16)Fs
P= Fs
Fs0
Fs0/Pc
(ω/ωc)3/2 .
2.2. Elastic–plastic contact
For ω/ωc � 1 the contact region becomes elastic–plaand no analytical solution exists for the local separationZ.A certain approximation forZ was offered in [25] but anaccurate solution requires a more rigorous finite elemanalysis (FEA). Recently, Kogut and Etsion [29] treanormal loading of a spherical contact in the elastic–plaregime where 1� ω/ωc � 110. Following the same FEAnumerical procedure of [29] the local separation,Z, in theelastic–plastic regime can be found by using constitulaws appropriate to any mode of deformation be it elaor plastic.
Kogut and Etsion [29] found that the ratios(a/ac)2
and P/Pc in elastic–plastic contact can be expressedsimple functions of the critical interference,ω/ωc having theforms
(17a)
(a
ac
)2
= 0.93
(ω
ωc
)1.136
for 1 � ω/ωc � 6,
(17b)
(a
ac
)2
= 0.94
(ω
ωc
)1.146
for 6 � ω/ωc � 110,
(18a)P
Pc
= 1.03
(ω
ωc
)1.425
for 1 � ω/ωc � 6,
(18b)P
Pc
= 1.40
(ω
ωc
)1.263
for 6 � ω/ωc � 110.
The numerical results forZ obtained from the FEA havto be supplemented by the intermolecular distance,ε [seeEq. (12)] for completeness. Normalization of these valwith respect to the critical interference,ωc , provides thelocal dimensionless separation for elastic–plastic conA dimensionless local separation expression can thefound by curve fitting the numerical results for differevalues ofω/ωc in the range of 1� ω/ωc � 110.
As can be seen from Eqs. (17) and (18) two distielastic–plastic subregions were found by Kogut andsion [29]. In the first whereω/ωc � 6 the entire contacarea is still elastic, while in the second whereω/ωc > 6 thecontact area is either partially or completely plasticallyformed. Therefore, it is anticipated that, likewise, differedimensionless expressions prevail in each of these dissubregions for the dimensionless local separation. Themensionless adhesion force,Fs/Fs0, and the ratio of adhesion force to contact pressure,Fs/P , can be found by usingEqs. (17) and (18) in Eqs. (9) and (10), respectively. Simto the elastic contact case these parameters can be exp
edby appropriate curve fitting as functions of the dimensiless interference,ω/ωc , and the dimensionless intermoleclar distance,ε/ωc .
3. Results and discussion
A very wide range of values of the dimensionless intmolecular distance, 0.005� ε/ωc � 100, were selected tinvestigate the effect of this parameter. The smaller valueε/ωc correspond to larger sphere radii typical of rough sface asperity summits. The higher values ofε/ωc correspondto smaller sphere radii typical of AFM tips, for example. Bter curve fitting of the numerical results was obtained wtwo subregions, 0.005� ε/ωc � 0.5 and 0.5 < ε/ωc � 100,were used separately.
3.1. Elastic contact
Curve fitting of the numerical results obtained froEq. (15) for the dimensionless adhesion force in the elacontact regime,ω/ωc < 1, yields
Fs
Fs0= 0.979
(ε
ωc
)−0.29(ω
ωc
)0.298
(19a)for 0.005� ε/ωc � 0.5,
Fs
Fs0= 1.001+ 0.192ω/ωc
ε/ωc
(19b)for 0.5< ε/ωc � 100.
Figure 2 presents the dimensionless adhesion forcefunction ofω/ωc for the lower values ofε/ωc (larger sphereradius). The numerical results, shown as squares, andapproximation by Eq. (19a), shown as solid lines, are psented forε/ωc values of 0.005 and 0.05. The average acracy of the curve fitting presented by Eq. (19a) in the ra
Fig. 2. Dimensionless adhesion force,Fs/Fs0, as a function of the dimensionless interference,ω/ωc , in the elastic regime, for lowε/ωc values.
376 L. Kogut, I. Etsion / Journal of Colloid and Interface Science 261 (2003) 372–378
of
forcepro-s-che
lat-
ical)e oft ofce
ell
ic
e
lorcee-
vidu
ly9].ic–
etion
bylts o
r the
r the
with
pro-en-
ossd byrig-pro-for
ape.the
antwoion,
0.1� ω/ωc < 1 is 93%. This accuracy deteriorates asω/ωc
approaches zero whereFs/Fs0 should equal 1 regardlessε/ωc . Hence, Eq. (19a) is valid in the range 0.1� ω/ωc � 1.As can be seen from Fig. 2 the dimensionless adhesioncan reach values that are substantially larger than unityvided thatε/ωc is very small. Asε/ωc increases, its effect awell as that ofω/ωc on Fs/Fs0 diminishes and the dimensionless adhesion force becomes uniform and approaunity. Indeed as can be seen from Eq. (13) asε/ωc increaseswith respect to the first term on the right-hand side, theter becomes less and less significant andZ/ωc approachesε/ωc , namely, a point contact condition.
The accuracy of Eq. (19b) in describing the numerresults for the higher values ofε/ωc (smaller sphere radiuswas found to be better than 99.5% over the entire rangits relevancy. As can be seen from Eq. (19b) the effecboth ε/ωc and ω/ωc on the dimensionless adhesion forfor ε/ωc � 1 is minimal. At ε/ωc = 1, for example, themaximum value ofFs/Fs0 is less than 1.2. Hence, it may bconcluded that forε/ωc > 1, namely, for medium and smacurvaturesR, the effect ofε/ωc on the adhesion in elastcontact can be neglected and the approximationFs = Fs0 isvalid.
From Eq. (16) it is clear thatFs/P can become very largas ω/ωc approaches zero. Atω/ωc = 1 we haveFs/P =(Fs/Fs0)(Fs0/Pc). Since the sphere radiusR has oppositeeffects onFs/Fs0 andFs0/Pc it is hard to draw a generaconclusion regarding the significance of the adhesion fas ω/ωc approaches 1. Certainly it cannot be totally nglected and each specific case should be evaluated indially.
3.2. Elastic–plastic contact
For an elastic–plastic contact where 1� ω/ωc � 110, thedimensionless separation,Z/ωc, was calculated numericalby using a FEA like that described in Kogut and Etsion [2Curve fitting of the numerical results in the two elastplastic subregions resulted in the expressions
Z
ωc
= 0.951
π
(ω
ωc
)1.153
f (r̄) + ε
ωc
(20a)for 1 � ω/ωc � 6,
Z
ωc
= 0.457
π
(ω
ωc
)1.578
f (r̄) + ε
ωc
(20b)for 6 � ω/ωc � 110,
where the functionf (r̄) is that given by Eq. (14). Thesexpressions are quite different from the approximapresented in the CEB adhesion model [25] in the form
(21)Z
ωc
= 1
π
ω
ωc
(2− ωc
ω
)f (r̄) + ε
ωc
for ω/ωc > 1.
It is clearly seen that the approximation representedEq. (21) always underestimates the more accurate resu
s
-
f
Fig. 3. Elastic–plastic deformed profiles of the sphere at and neacontact,ω/ωc = 4.
Fig. 4. Elastic–plastic deformed profiles of the sphere at and neacontact,ω/ωc = 110.
Eqs. (20a) and (20b). This underestimation increasesincreasing interference.
Figures 3 and 4 present the elastic–plastic deformedfiles of the sphere at and near the contact for two dimsionless interference values ofω/ωc = 4 andω/ωc = 110,respectively. Because of symmetry only half of the crsection is presented. The numerical results, representeEqs. (20a) or (20b), are compared with the undeformed oinal sphere profile and the approximated correspondingfiles resulting from Eq. (21). The rigid flat is also showna reference. Note the different normalization of theZ andr coordinates, which somewhat distorts the spherical shAlso note the intersection of the deformed profile withrigid flat which provides the dimensionless radius,a/ac, ofthe contact area.
Different behavior of the deformed sphere profile cbe observed in Figs. 3 and 4, which are typical of thedifferent elastic–plastic subregions. In the first subreg
L. Kogut, I. Etsion / Journal of Colloid and Interface Science 261 (2003) 372–378 377
nal(see
dialsureorebe
onoralunsle,rate
daladii
to
as a
22),efor
-n-with
tionrcetic–
-
-
-ac-
as a
t
1 � ω/ωc � 6, the deformed profile approaches the origione from the inside as the radial location increasesFig. 3). In the second subregion, 6� ω/ωc � 110, thedeformed profile departs from the original one as the ralocation increases (see Fig. 4). Since the attractive presoutside the contact region [see Eq. (1)] diminishes mrapidly at larger separations this different behavior mayreflected in the adhesion force in these two subregions.
The deformed profile resulting from the approximatiin [25] intersects the rigid flat either inside (Fig. 3)at (Fig. 4) the intersection of this flat with the originundeformed sphere profile. Immediately thereafter it ralmost parallel to, but on the outside of, the original profiresulting in a substantial underestimation of the acculocal separationZ.
The dimensionless adhesion force,Fs/Fs0, in the elastic–plastic contact regime, 1� ω/ωc � 110, can be calculatewith Eqs. (9), (17), and (20). Curve fitting of the numericresults in this contact regime yields for the larger sphere r0.005� ε/ωc � 0.5,
�Fs
�Fs0= 0.792
(ε
ωc
)−0.321(ω
ωc
)0.356
(22a)for 1 � ω/ωc � 6,
�Fs
�Fs0= 1.193
(ε
ωc
)−0.332(ω
ωc
)0.093
(22b)for 6 � ω/ωc � 110,
and for the smaller sphere radii 0.5< ε/ωc � 100,
�Fs
�Fs0= 0.961+ 0.157
ε/ωc
+ 0.261 ln(ω/ωc)
ε/ωc
(23a)for 1 � ω/ωc � 6,
�Fs
�Fs0= 1.756−
(0.516− 0.303
ε/ωc
)ln
ω
ωc
+ 0.052
(ln
ω
ωc
)2
(23b)for 6 � ω/ωc � 110.
The accuracy of the above four curve fittings was foundbe better than 90%.
Figure 5 presents the dimensionless adhesion forcefunction ofω/ωc for the lower values ofε/ωc (larger sphereradius). The numerical results, represented by Eq. (are shown for twoε/ωc values of 0.005 and 0.05. Thresults of the CEB adhesion model [25] are also showncomparison.
The behavior ofFs/Fs0 in the first elastic–plastic contact regime, 1� ω/ωc � 6, resembles that in the elastic cotact regime (see Fig. 2), showing increased adhesionincreasing interference. The rate of increase inFs/Fs0 di-minishes with increasingε/ωc , which is again similar to thebehavior in the elastic contact regime. A sharp transioccurs atω/ωc = 6 and the dimensionless adhesion foremains practically uniform throughout the second elasplastic subregion. Its average value forε/ωc = 0.005 is
Fig. 5. Dimensionless adhesion force,Fs/Fs0, as a function of the dimensionless interference,ω/ωc , in the elastic–plastic regime, for lowε/ωc
values.
Fig. 6. Dimensionless adhesion force,Fs/Fs0, as a function of the dimensionless interference,ω/ωc, in the elastic–plastic regime, for highε/ωc
values.
about 8, and forε/ωc = 0.05 it is about 4. The approximation offered in [25] substantially overestimates the morecurate FEA results and, hence, should be avoided.
Figure 6 presents the dimensionless adhesion forcefunction of ω/ωc for the higher values ofε/ωc (smallersphere radius). Atε/ωc = 1 the behavior is similar to thashown in Fig. 5 but the maximum value ofFs/Fs0 is onlyabout 1.8. Atε/ωc = 10, Fs/Fs0 varies from about 1.0 toabout 0.7 over the entire range ofω/ωc . Hence, as in theelastic contact case it can be concluded that forε/ωc > 1,namely, for medium and small curvaturesR, the effect
378 L. Kogut, I. Etsion / Journal of Colloid and Interface Science 261 (2003) 372–378
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of ε/ωc on the adhesion in elastic–plastic contact canneglected and the approximationFs = Fs0 is valid. Here toothe approximation of [25] is inappropriate.
The ratioε/ωc may have a wider physical meaning thjust a measure of the sphere radius. Values ofε/ωc higherthan 1 are obtained whenωc , the sphere deformationplastic inception, is smaller than 0.5 nm. Such deformatioextremely small and remains negligible even asω increasesto values ofω/ωc much larger that 1. Deformation of thorder leaves the original sphere profile almost unchanthroughout the entire contact regimes, resulting in adheforce that is very close to that of a point contact.
From Eqs. (10) and (18) it is clear thatFs/P decreaserapidly asω/ωc increases. Since in most practical caFs0/Pc is less than 0.1 and, as can be seen from Figs. 5 athe dimensionless adhesion forceFs/Fs0 is less than 10the ratioFs/P becomes less than 0.1 already atω/ωc = 6.Hence, it is safe to conclude that the adhesion force cacompletely neglected in the second elastic–plastic regwhereω/ωc > 6.
4. Conclusion
The adhesion force of an elastic–plastic spherical micontact was calculated. The local separation was founa FEA considering the actual constitutive laws for theevant regime of deformation. Hence, the present modmuch more accurate than a previous one that relied on atain approximation regarding the local separation outsidecontact area above the critical interference. By properlymalizing the separation and the adhesion force the premodel provides simple analytical expressions that extendclassic DMT solution up to a fully plastic contact wheω/ωc = 110.
The dimensionless separation and adhesion forcefound to be functions of the dimensionless interferenω/ωc , and the dimensionless intermolecular distance,ε/ωc .The latter was shown to be a certain measure of the spsize with a range of interest 0.005� ε/ωc � 100 and atransition at aboutε/ωc = 1 where lower values are moappropriate to asperities of rough surfaces and higher vato smaller spheres typical of an AFM tip.
For a critical interference smaller thanε, namely,ε/ωc > 1, the adhesion force is about that for a point ctact throughout the entire deformation regimes.
For critical interference larger thanε the adhesion forcincreases withω as long as the contact area remains elathat is,ω/ωc � 6. From there on, the adhesion force remaalmost uniform throughout the remaining range of elasplastic deformation.
The adhesion force may be significant with respec
,
-
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e
s
the contact load for typical asperities of rough surfacethe elastic regime and in early stages of the elastic–plregime. It becomes negligible as the contact becomes mand more plastic.
A comparison of the present results with those opreviously published elastic–plastic model shows substadifferences in the local separation and in the adheforce.
Acknowledgments
This research was supported in part by the Fund forPromotion of Research at the Technion and by the GermIsraeli Project Cooperation (DIP).
References
[1] K.N.G. Fuller, D. Tabor, Proc. R. Soc. London Ser. A 345 (1975) 3[2] K. Cooper, A. Gupta, S. Beaudoin, J. Colloid Interface Sci. 234 (20
284.[3] S. Lorthois, P. Schmitz, E. Angles-Cano, J. Colloid Interface Sci.
(2001) 52.[4] S. Biggs, G. Spinks, J. Adhes. Sci. Technol. 12 (1998) 461.[5] M. Reitsma, V. Craig, S. Biggs, Int. J. Adhe. Adhes. 20 (2000) 445[6] K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc. London S
A 324 (1971) 301.[7] B.V. Derjaguin, V.M. Muller, Y.P. Toprov, J. Colloid Interface Sci. 6
(1975) 314.[8] D. Tabor, J. Colloid Interface Sci. 68 (1976) 1.[9] V.M. Muller, V.S. Yushenko, B.V. Derjaguin, J. Colloid Interfac
Sci. 77 (1980) 91.[10] J.A. Greenwood, Proc. R. Soc. London Ser. A 452 (1997) 1277.[11] D. Maugis, J. Colloid Interface Sci. 150 (1992) 243.[12] E. Barthel, J. Colloid Interface Sci. 200 (1998) 7.[13] J.A. Greenwood, K.L. Johnson, J. Phys. D 31 (1998) 3279.[14] K.L. Johnson, J.A. Greenwood, J. Colloid Interface Sci. 192 (19
326.[15] K.L. Johnson, Proc. Inst. Mech. Eng. 196 (1982) 363.[16] K.L. Johnson, Contact Mechanics, Cambridge Univ. Press, C
bridge, 1985.[17] D. Tabor, Tribol. Int. 28 (1995) 7.[18] G.G. Adams, M. Nosonovsky, Tribol. Int. 33 (2000) 431.[19] K.L. Johnson, Tribol. Int. 31 (1998) 413.[20] H.M. Pollock, J. Phys. D 11 (1978) 39.[21] J.B. Pethica, D. Tabor, Surf. Sci. 89 (1979) 182.[22] D. Maugis, H.M. Pollock, Acta Metall. 32 (1984) 1323.[23] S.K. Roy Chowdhury, H.M. Pollock, Wear 66 (1981) 307.[24] J.A. Greenwood, J.B.P. Williamson, Proc. R. Soc. London Ser. A
(1966) 300.[25] W.R. Chang, I. Etsion, D.B. Bogy, J. Tribol. Trans. ASME 110 (19
50.[26] S.D. Mesarovic, K.L. Johnson, J. Mech. Phys. Solids 48 (2000) 2[27] V.M. Muller, B.V. Derjaguin, Y.P. Toporov, Colloids Surf. 7 (1983
251.[28] E. Rabinowicz, Friction and Wear of Materials, 2nd ed., Wile
Interscience, New York, 1995.[29] L. Kogut, I. Etsion, J. Appl. Mech. Trans. ASME 69 (2002) 657.
rigidtan-
cep-yieldrestic-ichions
he
Lior Kogut1
Mem. ASMEe-mail: [email protected]
Izhak EtsionFellow ASME
e-mail: [email protected]
Dept. of Mechanical Engineering,Technion, Haifa 32000,
Israel
A Semi-Analytical Solutionfor the Sliding Inceptionof a Spherical ContactA finite element analysis, for an elastic perfectly plastic sphere normally loaded by aflat, is combined with an approximate analytical solution to evaluate the maximumgential load (static friction) that can be supported by the spherical contact at the intion of sliding. Sliding inception is treated as a failure mechanism based on plasticrather than a Coulomb friction law with a certain friction coefficient. Two different failumodes are identified, either on the contact area or below it, depending on the elaplastic status of the normal preloading. A limiting normal preload is found above whthe contact cannot support any additional tangential load. Simple analytical expressfor an ‘‘internal static friction coefficient’’ are presented for both the elastic and telastic-plastic regimes.@DOI: 10.1115/1.1538190#
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IntroductionThe elastic-plastic contact of spheres under combined nor
and tangential loading is a fundamental problem in contactchanics. It is applicable, for example, in particle handling@1#, orfriction between contacting rough surfaces@2#. It can be found ina variety of technologies, both new and traditional as describeTichy and Meyer@3#. Indeed, the number of works on sphericcontact under steady state sliding~see reviews by Bhushan@4# andby Adams and Nosonovsky@5#! and under static or rolling conditions ~see review by Johnson@6#! that were published so far iimpressive. Issues such as normal frictional loading, tangenloading and sliding inception of contacting elastic bodies are wdescribed@7#. However, in spite of their elastic-plastic nature, tmathematical complexity involved with these frictional contaproblems@8# restricted their solutions to mostly linear elasticity
Combined loading begins with a normal component that pduces the contact area, which can then support an added tangcomponent and, hence, should be treated as a frictional coproblem.
Frictional normal loading of elastic spheres differs from tclassical frictionless contact problem of the Hertz theory~e.g.,@9#!and was treated by several researchers,@10–13#. However, whenthe contacting sphere materials have the same elastic constan~orare incompressible! the mutual contact pressure produces identitangential displacements that eliminate any tendency of interfaslip. Hence, no shear stress is developed in the interface anHertz theory is still valid. Johnson@6# concluded that even withdissimilar realistic materials the local shear traction is an ordemagnitude smaller than the corresponding local contact presand, hence, may be neglected~except for brittle materials!.
Historically, the treatment of combined loading of sphericcontact started with the pioneering works by Mindlin@14# andMindlin and Deresiewicz@15# ~in fact an earlier paper from 193in Italian is due to Cattaneo, see@7#!. They showed that the contact of two spheres under combined loading consists of a cestick region surrounded by a slip zone. As the tangential foincreases, the size of the stick region gradually vanishes untilsliding begins when the Coulomb friction law is satisfied. Tnormal loading was assumed to be frictionless and the resu
1Currently Post Doctoral Fellow, Dept. of Mechanical Engineering, UC [email protected]!.
Contributed by the Tribology Division for publication in the ASME JOURNAL OFTRIBOLOGY. Manuscript received by the Tribology Division May 16, 2002; revismanuscript received September 12, 2002. Associate Editor: G. G. Adams.
Copyright © 2Journal of Tribology
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contact area and contact pressure distribution remained thesthe Hertz theory, even when the tangential force was added.
For the case of full stick Mindlin@14# presented a surface shetraction that causes uniform tangential displacement in the inface. This shear traction becomes infinite around the contact e~see@7# and@14#! and, according to Mindlin, would require unrealistic infinite friction coefficient to resist sliding. To overcomthis problem Mindlin applied a local Coulomb friction law and sthe product of a certain friction coefficient and the local contpressure as the upper limit for the local shear traction. At full sthe shear traction at the contact interface was assumed to reaupper limit over the entire contact area. At partial slip the shtraction in the slip region is identical to this of full slip while inthe central stick region it consists of a superposition of two trtion distributions that together produce a uniform tangential dplacement. The dimension of the central stick region dependsthe loads and the value of a selected friction coefficient~@7,14#!. Itshould be noticed that the full slip is a limiting case of a partslip, where the dimension of the central stick region is zero.
Most of the researchers following Mindlin used his approacha local Coulomb friction law to obtain an upper limit for the shetraction. Bryant and Keer@16# obtained the surface and subsurfastresses of two geometrically and elastically identical rough bies under combined loading. They considered the general casan elliptical contact area and partial slip. Sackfield and Hills@17#deduced the entire stress field for an elliptical contact under cbined loading for the simplest case of full slip. Hamilton@18#presented explicit formulas for the stresses beneath a slidingmally loaded Hertzian contact based on an earlier work of Hamton and Goodman@19#, where the stress field was expressed inimplicit form. Hamilton @18# used the approach of Mindlin@14#for the case of full slip as a boundary condition for his solutioHe presented the location of yielding inception that can be beor on the surface depending on the friction coefficient. Contraryall previous results for combined loading, the stress field obtaiby Hamilton @18# was no longer axisymmetric. However, photgraphs of the surface attrition obtained in the experimental wby Johnson@20# contrast Hamilton’s results and suggest that tyielding region, and therefore the stress field, is axisymmetric
Chang, Etsion and Bogy@21# ~CEB friction model! used theexplicit formulas of Hamilton@18# to calculate the maximum tangential force that a sphere, normally preloaded by a rigid flat,resist before sliding inception. The CEB approach differs frothat of Mindlin @14# since it relates the start of full slip to materiaproperties rather than to a local Coulomb friction law with a ctain friction coefficient. Sliding inception was interpreted by CE
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as the first yielding of a single material point. This, howevunderestimates the real permissible tangential force since thefailed point is still surrounded by considerable volume of elasmaterial. Roy Chowdhury and Ghosh@22# used the same CEBapproach with another limitation on the maximum tangential lothat considers adhesion. Students and Rudzitis@23# found an ap-proximate relationship between the maximum equivalent stfor a sliding elliptical contact and the geometric parameters ofcontact area, the friction coefficient and maximum pressure incenter of the contact area.
Finite Element Method~FEM! can handle elastic-plastic frictional contact problems~see@4#!. Mijar and Arora@24# reviewedelastostatic frictional contact solutions that employ both vational inequality and variational equality formulations. More iformation on the use of FEM for elastic-plastic problems canfound in @25#. Most of the works that incorporate FEM to solvfull sliding frictional problems~e.g.,@26–32#! rely on the Mindlinapproach of a local Coulomb friction law with a certain frictiocoefficient for the upper limit of the local shear traction.
The friction coefficient has long been used in science andgineering. It is easy to define on a global scale, but difficultunderstand on a fundamental level. The tabulated friction cocient values have their own limitations, as indicated by Blau@33#.The Mindlin approach of using a local Coulomb friction law to sthe upper limit for the local shear traction has some deficiencFor large contact pressures or large friction coefficients, an ualistic stress state can be obtained with an upper limit for the straction higher than the shear strength of the material. Moreothe artificial coupling of local normal and tangential stresses bcertain proportionality factor is not physically sound.
Another approach to combined loading problems claiequivalence between fracture mechanics and contact mechaGiannakopoulos et al.@34# showed an analogy between the geoetry of the near-tip regions of cracked specimens and the shedged contact region of two contacting surfaces. In the case ostick Johnson@35# treated an elastic sphere under combined loing as mixed-mode interfacial fracture. Similar treatment of staand sliding friction as well as of the interaction between frictiand adhesion can be found in@36#.
As can be seen much research has been done so far ocombined loading of a spherical contact. However, an accugeneral solution for the sliding inception is still missing. The eisting analytical and FEM solutions suffer from deficiencicaused by applying a local Coulomb friction law with some artrary friction coefficient to set an upper limit for the local shestress regardless of the material strength.
The present research attempts a different approach for solthe sliding inception of spherical contacts. It is based on usconstitutive laws appropriate to any mode of deformation beelastic or plastic. Friction is modeled by considering physicasound material properties, rather than unrealistic coupling of nmal and tangential stresses by some arbitrary friction coeffici
Theoretical BackgroundConceptually, sliding inception can be considered as a cer
mode of failure of small junctions that are formed between tcontacting bodies. The CEB static friction model@21# applied thisconcept to analyzing static friction between contacting rough sfaces. The model determines the maximum tangential load,Q* ,that an elastic spherical contact can support in the presencenormal preload,P, just before the first appearance of a plastic sin the stress field of@18#. Depending on the magnitude of thnormal preload,P, the first plastic event can take place eitherthe contact interface or beneath it. According to the CEB frictmodel if the normal load,P, reaches a critical value,Pc , suffi-cient to cause the first plastic yielding by itself, the contactunable to support any additional tangential load. The CEB failcriterion used in@21# results in an underestimation of the re
500 Õ Vol. 125, JULY 2003
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value forQ* . This is because at the first event of plastic yieldithe single plastic spot is still surrounded by an elastic regionmay support additional tangential load.
The critical normal load,Pc , corresponds to a critical interference,vc , that marks the transition from the elastic to the elastplastic deformation regime and is given by~e.g.,@21#!
vc5S pKH
2E D 2
R (1)
In Eq. ~1! R is the sphere radius,H is the hardness of the spherK is a hardness coefficient related to the Poisson ratio ofsphere by:K50.45410.41v, andE is the Hertz elastic modulusdefined as
1
E5
12v12
E11
12v22
E2
whereE1 , E2 andv1 , v2 are Young’s moduli and Poisson ratioof the two contacting materials, respectively.
The Hertz theory provides analytical expressions for the conload,P, and contact area,A, in the elastic regime of the sphericacontact where the dimensionless interference isv/vc<1.
Kogut and Etsion@37# treated normal loading of a sphericacontact in the elastic-plastic regime wherev/vc.1. In the range1<v/vc<6 the plastic region, which starts atv/vc51 as asingle point beneath the center of the contact, develops gradutowards the sphere surface while completely surrounded by elamaterial. Atv/vc56 this plastic region first reaches the sphesurface close to the circular edge of the contact area. An elacore, which contains the central zone of the contact area,becomes locked between the plastic region and the contact iface. As the dimensional interference increases abovev/vc56and the plastic region grows the elastic core gradually shrinks
It will be reasonable to postulate that the spherical contactder a normal loadP is capable of supporting additional tangentiload only as long asv/vc,6 and the entire contact interfacestill elastic. As soon as the plastic region reaches the sphereface the central elastic core becomes afloat on it like an ‘‘islanThis floating elastic core will simply slide out under the smalletangential load. Hence, a realistic upper limit of the normal loadPfor nonzero tangential loadsQ* is that corresponding tov/vc56 which is quite different from the upper limit in@21#.
The method of obtainingQ* for a given normal preloadPshould, in principle, be similar to the approach of the CEB frictimodel@21# with a few important exceptions. The equivalent stredistribution in the contact region as a result of applying a tangtial load Q to the preloaded sphere should be determined. Tequivalent stress distribution should then be tested against aure criterion and the loadQ gradually increased until the contacresistance to increased tangential load is exhausted.
This approach calls for the use of FEM since a realistic failucriterion will consist of a transition from elastic to an elastiplastic and then to a plastic regime. Unfortunately, the currexisting versions of commercial finite element packages aresuitable for solving frictional contact problems based onmethod that was outlined above. They all suffer from a balimitation regarding the relation between plastic deformation athe stick/slip status switch of their contact elements. Similar toMindlin approach@14#, these commercial packages switch the stus of a contact element from stick to slip if the local shear strat this element exceeds the product of the local normal presand a certain selected friction coefficient regardless of the lostrength of the material~a local Coulomb friction law approach!.Another limitation is the lack of flexibility to change the statuscontact elements from slip, during the normal~frictionless! load-ing, to stick when the tangential~frictional! loading begins. Thefollowing analysis is an attempt to overcome these limitationsresorting to a semi-analytical solution.
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One last point that deserves mentioning in this TheoretBackground is the Mindlin@14# shear traction distribution thafulfils the requirement of uniform tangential displacement as loas the contact is in full stick. This shear traction has the form
txz5q0~12r 2/a2!21/2 (2)
where r is a radial coordinate measured from the center ofcontact andq0 is a constant equal to the shear traction at tcenter. The shear tractiontxz is axisymmetric in its value andeverywhere parallel to the direction of the applied tangenforce. As can be seen from Eq.~2! the shear traction becomeinfinitely large asr approaches the edge of the contact area whr 5a. This of course is physically impossible and a realistic uplimit should be set that takes into account material propertiesthe yield strength.
AnalysisFigure 1 presents the deformable sphere pressed by a rigid
The solid and dashed lines show the situation after and beforedeformation, respectively. The interference,v, and the contactarea with a radius,a, correspond to a normal contact load,Pacting in thez direction at the origin of the coordinate systemxyzthat is located in the center of the contact area. Followingnormal loading a tangential loadQ is added in thex direction.
In the following it will be assumed that the addition of thtangential loadQ has no effect on the contact dimensions, and tit does not modify the contact pressure distribution due tonormal loading.
Johnson@7# has shown that for purely elastic contact the effeof tangential force on the contact area and contact pressugenerally small, particularly when the tangential and normal loare of the same order of magnitude. Therefore, as long ascontact interface is elastic, the stress distribution componentsto the normal and the tangential forces are assumed to be ipendent of each other. It can be shown~see Appendix! that theabove main assumptions also hold for elastic-plastic contact.
The sphere material is assumed to be elastic perfectly plawith identical behavior in tension and compression. The yistrength of the material,Y, is related to the hardness,H, ~Tabor@38#! by H52.8Y. For most metals the Poisson ratio is aboutv50.3 hence, this value is used in the present model. The cosponding value of the hardness coefficient isK50.577.
The von Mises criterion that correlates well with experime~see@4#! is adopted as the preferred yielding criterion. The equilent von-Mises stress is calculated by~e.g., Hodge@39#!
Fig. 1 The contact of a deformable sphere and a rigid flat un-der combined loading
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2@~sx2sy!21~sy2sz!
21~sz2sx!2
16~txy2 1tyz
2 1txz2 !#J 1/2
(3)
and local yielding~neglecting strain hardening! corresponds toseq5Y.
Sliding inception is interpreted as a failure of the contact butwill be shown, the failure mechanism in the case of elastic normloading,v/vc<1, is somewhat different from that in the elastiplastic regime, 1,v/vc<6.
„a… Elastic Preloaded Contact, vÕvcÏ1. It is assumedthat in the absence of plastic deformation inside the sphere, thwhen P,Pc , yielding of the spherical contact due to an addtional tangential force,Q, will begin at the contact interface anwill be confined to it. This is because the shear stresstxz due toQis the highest at the contact interface and its decay is very rapthe z direction. For purely elastic normal loading the classicHertz normal stress distributionsx , sy , sz at the contact inter-face prevails@18#. The only two additional shear stress compnents at the contact interface aretyz andtxz . With Q acting in thex directiontyz is negligible compared totxz , @7#, and hence, theonly significant shear stress at the contact interface istxz . Super-posing the two stress fields and substituting in Eq.~3! with tyz5txy50 and seq5Y, gives the maximum allowed local sheastress,txz* , corresponding to the local yield, at any point~x, y! onthe contact surface, in the form:
txz* 5H 2Y22@~sx2sy!21~sy2sz!21~sz2sx!
2#
6 J 1/2
(4)
Integration oftxz* over the entire contact area gives the maximutangential load,Q* , that can be carried by this contact when itnormally pre-stressed with a loadP in the elastic regime.
Q* 5E0
a
2pr txz* dr5a2E0
1
2pr
atxz* dS r
aD (5)
Using the relations@21#: Pc52/3KHpac2, (a/ac)
25v/vc alongwith K50.577 andH52.8Y gives:
Q*
Pc51.857
v
vcE
0
1 r
a
txz*
YdS r
aD (6)
It can be seen from Eq.~6! that Q* /Pc is a function of thedimensionless interferencev/vc and of the ratiotxz* /Y whichitself is a function ofv/vc only. Hence, the ratioQ* /Pc in Eq.~6! is independent of the radius or material properties ofsphere and represents a general solution for the failure modthe surface.
„b… Elastic-Plastic Preloaded Contact, 1ÏvÕvcÏ6. Fornormal loadingP.Pc a plastic region does exist beneath tcontact area~this was shown for the first time in 1917 by Belyaesee@7#! and it will gradually grow with an increasing additionatangential load. Since a tangential load produces an axisymmshear traction distribution at the contact area~see@7# and Appen-dix! it is assumed that the growing plastic region also progresaxisymmetrically towards the sphere surface. For a certain sciently large tangential load the plastic region eventually reacthe sphere surface~see Fig. 2~a!!. It is assumed that this will firstoccur on the circle whereseq due to the normal loadP aloneattains its largest value. This circle consists of the weakest surpoints, with the smallest value oftxz* .
The shear stress distributiontxz* can be calculated from Eq.~4!as was done in the case of the elastic normal preloading, butthe normal stress distribution on the contact surfacesx , sy , szmust be obtained from a finite element solution like that descri
JULY 2003, Vol. 125 Õ 501
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in @37# for example. Oncetxz* is found for the entire contact areathe circular locus of the weakest points can be identified. Tcircle has a radiusr 5c wherec/a,1 ~see Fig. 2!.
It should be emphasized that when the plastic region fireaches the sphere surface atr 5c, the contact inside this circle(r /a,c/a) is still in elastic state and, hence, ‘‘sticks’’ to the rigiflat. The annular area outsider 5c (c/a,r /a,1) is also in elas-tic state. As the tangential load further increases this outer annbecomes progressively plastic and finally yields completely~seeFig. 2~b!!. The central elastic core becomes fully afloat on tplastic region beneath it and any attempt to further increasetangential load will trigger sliding of this central elastic core ‘‘iland’’ over the elastic-plastic interface around it. This ‘‘under tsurface’’ failure mode was conservatively considered in the Cfriction model@21# to occur at the inception of plastic yield belothe contact namely, whenv/vc51. Such a conservative approacobviously results in underestimation of the real value ofQ* .
In the following it will be assumed that the shear traction dtribution suggested by Mindlin@7,14# for purely elastic preloadedcontact, see Eq.~2!, also holds for the contact interface inside tcircle of radiusc under elastic-plastic preloading condition~seeAppendix!. In our case the possibility of the shear stress toproach infinite values is prevented by truncating the Mindlin dtribution at r 5c where the shear stress attains the previoufound minimum value oftxz* . To fulfill this condition the constantq0 should be:
q05~txz* !min~12c2/a2!1/2 (7)
The maximum tangential load,Q* , that can be carried by thecontact, when it is preloaded by a normal load,P, in the elastic-plastic regime, can now be obtained from integrating the shstresstxz of Eqs. ~2! and ~7! over the central contact area from
Fig. 2 Elastic and Plastic zones in a sphere under combinedloading for vÕvcÌ1: „a… plastic region just reaching the con-tact interface; and „b… plastic region at sliding inception.
502 Õ Vol. 125, JULY 2003
,his
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EB
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e
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r 50 to r 5c and adding the integration of thetxz* distributionover the annular area fromr 5c to r 5a. NormalizingQ* by Pc
as before ~see Eq. ~6!! and using the relation (a/ac)2
50.93(v/vc)1.136 that was found in Kogut and Etsion@37# for 1
<v/vc<6 gives:
Q*
Pc51.727S v
vcD 1.136F E
0
c/a r
a
~txz* !min
Y S 12c2/a2
12r 2/a2D 1/2
dS r
aD1E
c/a
1 r
a
txz*
YdS r
aD G (8)
Results and Discussion
„a… Elastic Preloaded Contact. Figure 3 presents the dimensionless shear stress distribution,txz* /Y, at sliding inception~see Eq.~4!! as a function of the dimensionless radial coordinar /a for v/vc values 0.1, 0.5 and 1. It can be seen thattxz* /Y isalmost constant along the radial span of the contact area. It hminimum atr /a50 and then rises to a certain maximum closethe edge of the contact before decreasing to its lowest valur /a51. The shear stress distribution in Fig. 3 represents an uplimit at sliding inception and is based on material strength conserations. It is completely different from the upper limit of Mindli@14#, which is simply the product of a certain friction coefficienand the local Hertzian pressure and completely ignores matproperties.
The values oftxz* /Y are somewhat lower for higher interferencvalues however; the maximum variation is less than 6 percNote that values oftxz* /Y over the entire range shown in Fig. 3 avery close to the theoretical normalized shear strength,ts /Y50.577, of the material. This is due to the very small contributiof the normal stresses to the equivalent stress,seq , on the contactarea in the elastic regime.
An ‘‘internal static friction coefficient,’’m, can be defined as theratio between the tangential load,Q* , required to cause slidinginception under the normal load,P. This ‘‘internal static frictioncoefficient’’ differs from the actual static friction coefficient duethe neglecting of the adhesion force, which should be subtra
Fig. 3 Dimensionless maximum shear stress, txz* ÕY, as func-tion of the dimensionless radial location, r Õa, in the elastic re-gime
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from P to obtain the actual external load@21#. In the following theterm ‘‘internal’’ will be dropped for simplicity. This static frictioncoefficient can be expressed in the form:
m5Q*
P5
Q* /Pc
P/Pc(9)
By substituting the results from Eq.~6! and the Hertzian relation
P/Pc5~v/vc!3/2 (10)
into Eq. ~9! the static friction coefficient in the elastic regime cabe obtained as a function ofv/vc . The results are presentedFig. 4, showing thatm decreases as the interference increasescurve fitting, an expression form in the elastic regime,v<vc ,can be obtained in the form:
m50.516S v
vcD 20.518
(11)
A simple analytical solution can be formulated by noting, froFig. 3, that the dimensionless shear stress distribution is almindependent ofr /a or v/vc and is very close to the material shestrength namely,txz* /Y50.577. By substituting this constant valuin Eq. ~6! and following the steps that lead to Eq.~11! the staticfriction coefficient is:
m50.536S v
vcD 20.5
(12)
The maximum difference betweenm values obtained from Eqs~11! and~12! corresponds tov/vc51 and is less than 4 percenHence, the latter expression can serve as a very good approxtion of the static friction coefficient in the elastic preloading rgime.
Also shown in Fig. 4 are the results of the CEB friction mod@21#. This model is based on shear stress distribution@18#, whichbasically originated from the Mindlin approach@14#. In additionthe CEB model failure criterion is too conservative, relating sling inception to the first event of plastic yield of a single poeither on or below the surface. This, as pointed out in the Thretical Background, leads to underestimation of the tangential lrequired to initiate sliding, as is very well demonstrated in Fig.In the range of 0.1<v/vc<0.9 the present results ofm are about100 percent higher than the CEB model prediction. The discrancy introduced by the CEB model becomes very large in
Fig. 4 Static friction coefficient, m, as function of the dimen-sionless interference, vÕvc , in the elastic regime
Journal of Tribology
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range of 0.96<v/vc<1 where the CEB model predicts failurbelow the surface and assumesm50 at v/vc51.
„b… Elastic-Plastic Preloaded Contact. Figure 5 presentsvalues oftxz* /Y that were obtained from Eq.~4! in conjunctionwith the stress distribution from a finite element solution, suchin @37#, versusr /a for a range ofv/vc values in the elastic-plasticregime. Here, differently from the previous elastic case, the mamum value oftxz* /Y is at r /a50. As r /a increases the dimensionless shear stress decreases until a minimum is reached at a cvalue r /a5c/a that depends onv/vc and indicates the dimensionless radius of the weakest circle on the contact interface.r /a.c/a where the entire depth below the surface is still elas~outside the internal plastic zone!, the shear stress increases inway that reminds its behavior in the elastic preloading regi~Fig. 3!. Also shown by dashed lines in Fig. 5 is the Mindlin stredistribution obtained from Eqs.~2! and ~7! for r /a,c/a.
As v/vc increases the weakest circle moves towards the ctact edge and the value oftxz* /Y on it becomes smaller. Atv/vc56 the weakest circle is very close to the contact edge,c/a50.94, andtxz* /Y on this circle actually vanishes. This marks thfirst occasion of the internal growing plastic region reachingcontact interface due to increase of the normal preload aloneelastic plastic boundary has now been formed between an ielastic core and an outer elastic annulus on the contact area~simi-lar to the schematic description in Fig. 2~a!!. A further increase ofthe normal preload forms an expending plastic annuluspushes the elastic annulus outward until it completely disapp~similar to the description in Fig. 2~b!!. This happens atv/vc56.2 and from this point on, the preloaded contact is unablesupport any additional tangential component. Note that inrange 6<v/vc<6.2 the radiusc remains practically constan@37#.
The values of (txz* /Y)min , its correspondingc/a, and the distri-bution oftxz* /Y from c/a outward, as shown in Fig. 5, can be usein Eq. ~8! to determine the value ofQ* /Pc for any givenv/vc inthe elastic plastic preloading regime. The values ofQ* /Pc cannow be used in Eq.~9! together with the relevant relation~seeKogut and Etsion@37#!:
P/Pc51.03~v/vc!1.425 (13)
Fig. 5 Dimensionless maximum shear stress, txz* ÕY, as func-tion of the dimensionless radial location, r Õa, in the elastic-plastic regime
JULY 2003, Vol. 125 Õ 503
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a
qn
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to obtain the static friction coefficient,m, in the elastic-plasticregime.
The results are presented in Fig. 6 showing similar behavioin the elastic regime wherem decreases with increasing valuesv/vc . By proper curve fitting an empirical expressions form isobtained, for the elastic-plastic regime, in the form:
m520.007S v
vcD 3
10.085S v
vcD 2
20.389v
vc10.822 (14)
Eq. ~14! holds for the rangevc,v<6.2vc . At v/vc51 Eq.~14!yields m50.511 which is in very good agreement withm50.516 that results from Eq.~11! for the same interference.
For practical purposes it is more convenient to have static ftion coefficient values as function of the normal load rather ththe interference. By using the appropriate relations betweenP/Pcandv/vc from Eqs.~10! and ~13! in Eqs.~11! and ~14!, respec-tively, the following expressions are obtained:
m50.516S P
PcD 20.345
(15)
for 0,v/vc<1, and
m520.007S P
PcD 2.104
10.083S P
PcD 1.405
20.380S P
PcD 0.701
10.822
(16)
for 1,v/vc<6.2.These results are plotted in Fig. 7 together with the correspo
ing tangential load that initiates sliding,Q* /Pc which is basicallythe static friction force, for the full range of the relevant dimesionless normal loads. It is interesting to note that according topresent model, and its assumptions, the spherical contact casupport additional tangential load when the normal load exceabout 14 times the critical Hertzian load,Pc . This is an order ofmagnitude higher than the unrealistic low limiting normal loP5Pc suggested in the CEB friction model@21#. The dimensionalcritical load Pc can be easily related to material properties aradius of the contacting sphere by:
Pc5~pK !3
6 S H
E D 2
HR2 (17)
Hence, knowing the sphere radius and the material propertiethe contacting bodies, it is possible with the help of Fig. 7 or E~15! to ~17! to find the static friction coefficient and static frictio
Fig. 6 Static friction coefficient, m, as function of the dimen-sionless interference, vÕvc , in the elastic-plastic regime
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force for any normal loadP. A very interesting observation is thdecrease of the friction force after reaching a maximum of ab1.4Pc when the normal load is about 9Pc . This behavior demon-strates well the effect of normal preloading on the contstrength.
ConclusionThe maximum tangential load that can be supported by a n
mally preloaded elastic perfectly plastic spherical contact atinception of sliding was analyzed using an approximate seanalytical solution. Sliding inception was treated as a failumechanism considering physically sound principals rather tunrealistic coupling of local normal and tangential stress comnents and assuming a local Coulomb friction law.
Two different failure modes were identified depending on tnature of the normal preloading. When the normal load is lthan the Hertzian critical load the failure occurs on the contarea. If the normal load exceeds that critical one the failure occbelow the contact area. A limiting normal preloading was fouabove which the contact cannot support any additional tangeload. This limiting normal load is about 14 times the Hertzicritical load.
By properly normalizing the various relevant loads the presmodel, although assisted by a numerical FEM solution, providsimple analytical expressions that are valid for any specific mrials and radius of the sphere. Simple expressions for the sfriction coefficient were presented for both the elastic andelastic-plastic regimes.
A comparison of the present results with these of the previCEB friction model showed substantial differences of over 10in the friction coefficient and an order of magnitude in the limitinnormal preload.
AcknowledgmentThis research was supported in parts by the Fund for the P
motion of Research at the Technion, J. and S. Frankel ReseFund and by the German-Israeli Project Cooperation~DIP!.
Nomenclature
a 5 contact area radiusc 5 radius of central stick regionE 5 Hertz elastic modulus
Fig. 7 Static friction coefficient, m, and dimensionless forceQ* ÕPc versus the dimensionless contact load, PÕPc , in theelastic and elastic-plastic regimes
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H 5 hardness of the sphereK 5 hardness factor, 0.45410.41vP 5 normal load
Pc 5 Hertz critical loadQ 5 tangential load
Q* 5 maximum tangential load at sliding inceptionr 5 radial coordinateR 5 radius of the sphereY 5 yield strength of the spherem 5 internal static friction coefficient,Q* /Pv 5 Poisson’s ratio
seq 5 equivalent von-Mises stressts 5 shear strength
txz 5 shear traction in the central stick regiontxz* 5 maximum allowed shear stress at sliding inceptionv 5 interference
vc 5 Hertz critical interference
AppendixA three-dimensional finite element model was used@40# to in-
vestigate the validity of the main assumptions made in this paA commercial ANSYS 5.7 package was utilyzed for this pupouse. Due to symmetry about thexz plane~see Fig. 1! only halfof the hemisphere was treated using a mesh that consists often-node tetrahedral three-dimensional structural solid elemcomprising a total of 6884 nodes. A three-dimensional singleget element for the rigid flat and three-dimensional 622 surfato-surface contact elements for the sphere surface were usedcontact zone to model the contact between these two bodieorder to be able to deal with the problem with reasonablesources a higher density mesh of the solid elements was usethe very close vicinity of the contact zone where most ofdrama takes place, and a much coarser mesh was used at ining distance from the contact zone. In addition, material properand sphere radius were specifically selected so as to result invalues forPc and critical interference. This allows to capture moof the interesting events at a small region near the top ofsphere and makes the total of 4513 elements sufficient up toP/Pcvalues deep into the plastic contact regime.
As indicated in the Theoretical Background all current existversions of commercial finite element packages suffer frommain limitations regarding the stick/slip status switch of their cotact elements. Hence, accurate solution of the combined loadinimpossible and only approximation approaches to the investtion of the effect of adding a tangential load to a normally elasplastic preloaded contact can be used.
In a first such approach normal loads of up toP/Pc514 wereapplied under stick condition. Subsequently a gradually increatangential load was added to the rigid flat while allowing possichanges in the normal displacement and contact area to take pAt each step of the increased tangential loading the status of econtacting element was changed from stick to slip wheneveshear stress reached the virgin shear strength limit,Y/), of thesphere material. The maximum additional tangential load cosponded to the case where all contacting nodes were slippingthis instance an unrestricted tangential displacement of the rflat occurred indicating sliding inception. The corresponding ctact area and normal stress distribution on it remained, howethe same as at the end of the normal preloading prior to the ation of any tangential load. Hence, no effect of the tangential loon the contact area or on the normal stresses that were originfrom the normal preloading was observed, similar to Johnsoobservation for an elastic contact@7#.
In a second approach a normal load ofP/Pc51.5 was appliedunder full slip condition. Subsequently all the contacting nodwere uniformly displaced in thex direction, simulating increasedtangential loading under full stick. This was done while allowipossible changes in the normal displacement and contact artake place. The above uniform tangential displacement could
Journal of Tribology
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increased until plastic deformation first appeared at the coninterface and invalidated the requirement of full stick and unifotangential displacement over the entire contact area. The tantial load at each displacement step was obtained from the intetion of the shear stress distributiontxz /Y over the contact area. Itsvalue corresponding to the maximum possible displacementfound to be 0.27Pc . Here again no change in the contact areathe normal stresses that were originated from the elastic-planormal preloading was obtained.
The normalized shear stresstxz /Y, that corresponds to theabove maximum tangential displacement was found to be apprmately axisymmetric in its value and very close to the shear stdistribution of Mindlin @14# ~see Eq.~2!! over the relevant portionof the contact area wherer ,c. The numerical results along threradii coinciding with the1x, the 2x and they directions arepresented in Fig. 8 and compared with the shear stress distribuof Mindlin. As can be seen forr /a,0.9, txz /Y is practicallyaxisymmetric and follows the Mindlin profile. Hence, the assumtions made regarding the behavior oftxz /Y under elastic-plasticnormal preloading seem to be valid.
It should be noted here that at higher normal preloads the vof txz /Y over the elastic portion of the contact area wherer ,cbecomes smaller and smaller~see Fig. 5!. Hence, its actual behavior is less important at the advanced stages of the elastic-planormal preloading.
References@1# Vu-Quoc, L., and Zhang, X., 1999, ‘‘An Accurate and efficient Tangent
Force-Displacement Model for Elastic Frictional Contact in Particle-FloSimulations,’’ Mech. Mater.,31, pp. 235–269.
@2# Ogilvy, J. A., 1992, ‘‘Numerical Simulation of Elastic-Plastic Contact BetweAnisotropic Rough Surfaces,’’ J. Phys. D,25, pp. 1798–1809.
@3# Tichy, J. A., and Meyer, D. M., 2000, ‘‘Review of Solid Mechanics in Triboogy,’’ Int. J. Solids Struct.,37, pp. 391–400.
@4# Bhushan, B., 1996, ‘‘Contact Mechanics of Rough Surfaces in TriboloSingle Asperity Contact,’’ Appl. Mech. Rev.,49, pp. 275–298.
@5# Adams, G. G., and Nosonovsky, M., 2000, ‘‘Contact Modeling—Forces,’’ Tbol. Int., 33, pp. 431–442.
@6# Johnson, K. L., 1982, ‘‘One Hundred Years of Hertz Contact,’’ Proc. InMech. Eng.,196, pp. 363–378.
@7# Johnson, K. L., 1985,Contact Mechanics, Cambridge University Press, Cambridge.
@8# Barber, J. R., and Ciavarella, M., 2000, ‘‘Contact Mechanics,’’ Int. J. SolStruct.,37, pp. 29–43.
@9# Timoshenko, S. P., and Goodier, J. N., 1970,Theory of Elasticity, 3rd Edition,McGraw-Hill Inc., New York.
Fig. 8 Dimensionless shear stress, txz ÕY, as function of thedimensionless radial location, r Õa, in the elastic-plastic regime
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inct,’’
nteredJ.
ntn an
ta-n,
ofnc-l.,
ntal,
t,’’
ofeticalta
nd
tic
ere
of
@10# Goodman, L. E., 1962, ‘‘Contact Stress Analysis of Normally Loaded RouSpheres,’’ ASME J. Appl. Mech.,29, pp. 515–522.
@11# Spence, D. A., 1975, ‘‘The Hertz Contact Problem with Finite Friction,’’Elast.,5, pp. 297–319.
@12# Hills, D. A., and Sackfield, A., 1987, ‘‘The Stress Field Induced by NormContact Between Dissimilar Spheres,’’ ASME J. Appl. Mech.,54, pp. 8–14.
@13# Kosior, F., Guyot, N., and Maurice, G., 1999, ‘‘Analysis of Frictional ContaProblem Using Boundary Element Method and Domain DecomposiMethod,’’ Int. J. Numer. Methods Eng.,46, pp. 65–82.
@14# Mindlin, R. D., 1949, ‘‘Compliance of Elastic Bodies in Contact,’’ ASME JAppl. Mech.,16, pp. 259–268.
@15# Mindlin, R. D., and Deresiewicz, H., 1953, ‘‘Elastic Spheres in Contact UnVarying Oblique Forces,’’ ASME J. Appl. Mech.,20, pp. 327–344.
@16# Bryant, M. D., and Keer, L. M., 1982, ‘‘Rough Contact Between Elasticaand Geometrically Identical Curved Bodies,’’ ASME J. Appl. Mech.,49, pp.345–352.
@17# Sackfield, A., and Hills, D. A., 1983, ‘‘Some Useful Results in the TangentiaLoaded Hertzian Contact Problem,’’ J. Strain Anal.,18, pp. 107–110.
@18# Hamilton, G. M., 1983, ‘‘Explicit Equations for the Stresses Beneath a SlidSpherical Contact,’’ Proc. Inst. Mech. Eng., Part C: Mech. Eng. Sci.,197C, pp.53–59.
@19# Hamilton, G. M., and Goodman, L. E., 1966, ‘‘The Stress Field Created bSliding Circular Contact,’’ ASME J. Appl. Mech.,33, pp. 371–376.
@20# Johnson, K. L., 1961, ‘‘Energy Dissipation at Spherical Surfaces in ConTransmitting Oscillating Forces,’’ J. Mech. Eng. Sci.,3, pp. 362–368.
@21# Chang, W. R., Etsion, I., and Bogy, D. B., 1988, ‘‘Static Friction CoefficieModel for Metallic Rough Surfaces,’’ ASME J. Tribol.,110, pp. 57–63.
@22# Roy Chowdhury, S. K., and Ghosh, P., 1994, ‘‘Adhesion and Adhesional Ftion at the Contact Between Solids,’’ Wear,174, pp. 9–19.
@23# Students, E., and Rudzitis, J., 1996, ‘‘Contact of Surface Asperities in WeTribol. Int., 29, pp. 275–279.
@24# Mijar, A. R., and Arora, J. S., 2000, ‘‘Review of Formulations for ElastostaFrictional Contact Problems,’’ Structural and Multidisciplinary Optimizatio20, pp. 167–189.
@25# Owen, D. R. J., and Hinton, E., 1980,Finite Elements in Plasticity: Theoryand Practice, Pineridge Press LTD., Swansea.
@26# Tian, H., and Saka, N., 1991, ‘‘Finite Element Analysis of an Elastic-PlaTwo-Layer Half-Space: Sliding Contact,’’ Wear,148, pp. 261–285.
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@27# Anderson, I. A., and Collins, I. F., 1995, ‘‘Plane Strain Stress DistributionDiscrete and Blended Coated Solids Under Normal and Sliding ContaWear,185, pp. 23–33.
@28# Kral, E. R., and Komvopoulos, K., 1996, ‘‘Three-Dimensional Finite ElemeAnalysis of Surface Deformation and Stresses in an Elastic-Plastic LayMedium Subjected to Indentation and Sliding Contact Loading,’’ ASMEAppl. Mech.,63, pp. 365–375.
@29# Kral, E. R., and Komvopoulos, K., 1997, ‘‘Three-Dimensional Finite ElemeAnalysis of Subsurface Stress and Strain Fields Due to Sliding Contact oElastic-Plastic Layered Medium,’’ ASME J. Tribol.,119, pp. 332–341.
@30# Tworzydlo, W. W., Cecot, W., Oden, J. T., and Yew, C. H., 1998, ‘‘Computional Micro-and Macroscopic Models of Contact and Friction: FormulatioApproach and Applications,’’ Wear,220, pp. 113–140.
@31# Stephens, L. S., Liu, Y., and Meletis, E. I., 2000, ‘‘Finite Element Analysisthe Initial Yielding Behavior of a Hard Coating/Substrate System With Futionally Graded Interface Under Indentation and Friction,’’ ASME J. Tribo122, pp. 381–387.
@32# Faulkner, A., and Arnell, R. D., 2000, ‘‘The Development of a Finite ElemeModel to Simulate the Sliding Interaction Between Two, Three-DimensionElastoplastic, Hemispherical Asperities,’’ Wear,242, pp. 114–122.
@33# Blau, P. J., 2001, ‘‘The Significance and Use of the Friction CoefficienTribol. Int., 34, pp. 585–591.
@34# Giannakopoulos, A. E., Lindley, T. C., and Suresh, S., 1998, ‘‘AspectsEquivalence Between Contact Mechanics and Fracture Mechanics: TheorConnections and a Life-Prediction Methodology for Fretting-Fatigue,’’ AcMater.,46, pp. 2955–2968.
@35# Johnson, K. L., 1996, ‘‘Continuum Mechanics Modeling of Adhesion aFriction,’’ Langmuir, 12, pp. 4510–4513.
@36# Johnson, K. L., 1997, ‘‘Adhesion and Friction Between a Smooth ElasSpherical Asperity and a Plane Surface,’’ Proc. R. Soc. London, Ser. A,453,pp. 163–179.
@37# Kogut, L., and Etsion, I., 2002, ‘‘Elastic-Plastic Contact Analysis of a Sphand a Rigid Flat,’’ ASME J. Appl. Mech.,69, pp. 657–662.
@38# Tabor, D., 1951,The Hardness of Metals, Clarendon Press, Oxford.@39# Hodge, P. G., 1959,Plastic Analysis of Structures, McGraw-Hill Book Com-
pany, New York.@40# Kogut, L., 2002, ‘‘An Elastic-Plastic Model for the Contact and Friction
Rough Surfaces,’’ Ph.D. thesis, Technion-Israel Institute of Technology.
Transactions of the ASME
s isfor theticall forcelawstatic
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Lior Kogut1
Mem. ASMEe-mail: [email protected]
Izhak EtsionFellow ASME
e-mail: [email protected]
Dept. of Mechanical Engineering,Technion, Haifa 32000,
Israel
A Static Friction Model forElastic-Plastic Contacting RoughSurfacesA model that predicts the static friction for elastic-plastic contact of rough surfacepresented. The model incorporates the results of accurate finite element analyseselastic-plastic contact, adhesion and sliding inception of a single asperity in a statisrepresentation of surface roughness. The model shows strong effect of the externaand nominal contact area on the static friction coefficient in contrast to the classicalof friction. It also shows that the main dimensionless parameters affecting the sfriction coefficient are the plasticity index and adhesion parameter. The effect of adhon the static friction is discussed and found to be negligible at plasticity index valarger than 2. It is shown that the classical laws of friction are a limiting case ofpresent more general solution and are adequate only for high plasticity index and ngible adhesion. Some potential limitations of the present model are also discusseding to possible improvements. A comparison of the present results with those obtfrom an approximate CEB friction model shows substantial differences, with the lseverely underestimating the static friction coefficient.@DOI: 10.1115/1.1609488#
Keywords: Friction Modeling, Contacting Rough Surfaces, Static Friction, Adhesio
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IntroductionIt is well known from everyday experience that to displace o
body relative to another when the bodies are subjected to a cpressive force necessitates the application of a specific tangeforce, known as the static friction force, and until the requirforce is applied the bodies remain at rest. Accurate predictionthe static friction force may have an enormous impact on a wrange of applications such as bolted joint members@1#,workpiece-fixture element pairs@2#, static seals@3#, clutches@4#,compliant electrical connectors@5# magnetic hard disks@6,7#, andMEMS devices@8,9#, to name just a few.
Static friction was considered by the pioneers of frictionsearch: Leonardo da Vinci, Guillame Amontons, Leonard EuCharles Augustin de Coulomb, George Rennie, Arthur-JuMorin, Robert Hooke and others@10#. In early experimental workit was observed that the proportionality of the force opposrelative motion to the force holding the bodies together seemebe constant over a range of conditions. Amontons, for examplremembered for his two laws of friction:
1. The force of friction is directly proportional to the applieload.
2. The force of friction is independent of the nominal areacontact.
A common method for calculating the static friction force~Cou-lomb friction law! was drawn from these two basic laws by mutiplication of the normal applied load by a proportionality costant, known as the static friction coefficient, taken froengineering handbooks as a function of the contacting materStatic friction coefficients are conveniently tabulated and incorrated into engineering handbooks for at least 300 years. Howethese tabulated values represent average coefficients of fric
1Currently Post-Doctoral Fellow, Dept. of Mechanical Engineering, UC [email protected]!
Contributed by the Tribology Division of THE AMERICAN SOCIETY OF ME-CHANICAL ENGINEERSfor presentation at the STLE/ASME Joint International Trbology Conference, Ponte Vedra, FL October 26–29, 2003. Manuscript receivethe Tribology Division January 24, 2003; revised manuscript received June 10, 2Associate Editor: G. G. Adams.
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determined over a broad spectrum of test conditions. While thnumbers provide a general guideline of the sensitivity of theefficient of friction to the materials in contact, they may not neessarily be representative of the coefficient of friction that wresult between actual contact pairs. The friction coefficient is prently recognized as both material- and system-dependent@11# andis definitely not an intrinsic property of two contacting materia
Blau @11# in his review paper indicated that the friction coefficient is an established, but somewhat misunderstood, quantithe field of science and engineering. While friction coefficientsrelatively easy to determine in laboratory experiments, the funmental origins of sliding resistance are not so clear, and hencis extremely important to understand the process involved in ftion. Indeed, a great deal of progress has been made sincepioneering work of Amontons in 1699 and Coulomb in 1785, asevident from recent works that consider both atomistic pointview ~e.g., @12–14#! and continuum mechanics principals~e.g.,@15–17#!.
Tabor@18# in his general critical picture of friction understanding pointed out three basic elements that are involved in the ftion of dry solids:
1. The true area of contact between mating rough surfaces2. The type and strength of bond formed at the interface wh
contact occurs.3. The way in which the material in and around the contact
regions is sheared and ruptured during sliding.
The importance of these three elements can be easily undersfrom the definition of the friction coefficient,m:
m5Qmax
F5
Qmax
P2Fs(1)
where Qmax is the tangential force needed to fail the junctiocreated between the contacting surfaces, andF, the external force,~see Fig. 1! is the balance between the actual contact load,P, inthe true area of contact and the amount of the intermolecforces or the adhesion,Fs , acting between the surfaces in contaThe right hand side of Eq.~1! contains all the three elementmentioned above. The contact load,P, is related to the true area o
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contact. The adhesion,Fs , is related to the strength of the bonformed at the interface. The maximum tangential load,Qmax, isrelated to the failure of the contact.
Chang et al.@19# presented a model~CEB friction model! forpredicting the static friction coefficient of rough metallic surfacbased on the three elements indicated by Tabor@18#. The CEBfriction model uses a statistical representation of surface rouness@20# and calculates the static friction force that is requiredfail all of the contacting asperities, taking into account their nmal preloading. This approach is completely different from tclassical Coulomb friction law and shows that the latter is a liiting case of a more general behavior where static friction coecient actually decreases with an increasing applied load orcreasing nominal contact area. The CEB friction model actutreats the static friction as a plastic yield failure mechanism cresponding to the first occurrence of plastic deformation incontacting asperities. This can severely underestimate frictionefficient values for contacting rough surfaces since it neglectsability of an elastic-plastic deformed asperity to resist additioloading before failure occurs as was demonstrated recentlyKogut and Etsion@21#.
Roy Chowdhury and Ghosh@22# followed the same approacof the CEB friction model with additional adhesion related restrtion on the maximum tangential load that can be carried bsingle asperity. Etsion and Amit@23# demonstrated experimentally, for small normal loads and relatively smooth surfaces tthe static friction coefficient decreases with increasing normloads as predicted by the CEB friction model. Polycarpou aEtsion @24# extended the original CEB friction model to includthe presence of sub-boundary lubrication. In a following pa@25# they compared their model prediction@24# with publishedexperimental results and found good agreement. Liu et al.@26#, inyet another extension of the CEB friction model, developedstatic friction model for the case of rough surfaces in the preseof thin metallic films and compared their theoretical results wexperimental data in@27#.
The original CEB friction model@19# as well as its followingextensions@24# and@26# calculate the static friction coefficient b
Fig. 1 The forced acting between contacting rough surfaces
Journal of Tribology
d
es
gh-tor-
he-
ffi-de-llyor-heco-thealby
ic-a
-atal
ndeer
anceith
using Eq.~1! where the contact load,P, and adhesion force,Fs ,are obtained from previous approximate models of Chang e@28# and @29#, respectively. However, as was shown in a serrecent works,@21,30,31#, that are based on finite element analysthe previous approximate models@19,28,29# produce large dis-crepancies on the single asperity level.
As can be seen from the above literature survey, available talated values of static friction coefficient do not account for suimportant parameters as surface roughness, surface energychanical properties and contact load that have strong effect onfriction. An adequate theoretical model will eliminate the curreneed for extensive empirical work and will shed more lightunderstanding the dominant parameters affecting the static friccoefficient. The aforementioned approximate models for stfriction coefficient assume failure of a contacting asperity as sas the first plastic point appears, and hence, underestimateactual friction force. These models also rely on approximate ctact and adhesion solutions for a single asperity, that present ldiscrepancies with respect to recent finite element solutions.present work relies on these finite element solutions for contadhesion and friction, and hence, should improve the accuracthe original CEB friction model. This remains to be verified bcomparison with controlled experiments that will hopefully bpresented in subsequent works.
AnalysisFigure 2 describes schematically the geometry of the contac
rough surfaces. The two rough surfaces of Fig. 1 are replaceda single equivalent rough surface in contact with a flat. The baassumptions of Greenwood and Williamson@20# regarding theshape and statistical distribution of the asperities along withtransformation to the more practical surface height distribut~see Nayak@32#! are adopted in the present analysis.R is theuniform asperity radius of curvature,z andd denote the asperityheight and separation of the surfaces, respectively, measuredthe reference plane defined by the mean of the original aspeheights. The separationh is measured from the reference plandefined by the mean of the original surface heights.f(z) is theasperity height probability density function, assumed toGaussian:
f~z!51
A2pss
expF20.5S z
ssD 2G (2)
wheress is the standard deviation of asperity heights. The intference is defined as:
v5z2d (3)
and only asperities with positive interference are in contact.During loading, the contact load,P̄, adhesion force,F̄s , and
the static friction force,Q̄max, of each individual asperity depenonly on its own interference,v, assuming there is no interactiobetween asperities. The dependence ofP̄, F̄s andQ̄max on v mustbe determined by the asperity mode of deformation, which canelastic, elastic-plastic or fully plastic. Once these expressions
Fig. 2 Contact model of rough surfaces
JANUARY 2004, Vol. 126 Õ 35
Table 1 The values of a, b , and c for the various deformation regimes in Eqs. „10… to „12…
Deformation regime
Eq. ~10! Eq. ~11! Eq. ~12!
a b a b c i ai bi
Fully elastic,v/vc,1 1 1.5 0.98 0.298 20.290 1 0.52 0.9821st elastic-plastic, 1.03 1.425 0.79 0.356 20.321 1 20.01 4.4251<v/vc<6 2 0.09 3.425
3 20.40 2.4254 0.85 1.425
2nd elastic-plastic, 6<v/vc<110 1.40 1.263 1.19 0.093 20.332 N/AFully plastic,v/vc.110 3/K 1 N/A N/A
s
f
a
t
ity
con-
aterialicity
-
i.e.,
.in the
3–
le
ofsent
known, the total contact load,P, adhesion force,Fs , and staticfriction force, Qmax, are obtained by summing the individual aperity contributions using a statistical model:
P5hAnEd
`
P̄~z2d!f~z!dz (4)
Fs5hAnE2`
`
F̄s~z2d!f~z!dz (5)
Qmax5hAnEd
d16vc
Q̄max~z2d!f~z!dz (6)
whereAn is the nominal contact area andh is the area density othe asperities. The integrals in Eqs.~4!–~6! are solved in parts forthe different deformation regimes of the contacting asperitiesshould be noted that while the contact load,P, and static frictionforce, Qmax, are calculated for contacting asperities only, thehesion force,Fs , is calculated also for non-contacting asperitieand hence, the difference in the lower limit of the integral in E~5!. The upper limit of the integral in Eq.~6! is due to the obser-vation in @21# that preloaded asperities are unable to supportditional tangential load if their interference is larger than 6vc . Itshould also be noted that Eqs.~4!–~6! are general in terms of theasperity height probability density functionf(z). Other non-Gaussian distributions can be used in these equations~see e.g.,@33#!.
The critical interference,vc , that marks the transition fromelastic to elastic-plastic deformation is given by~see e.g., Changet al. @28#!
vc5S pKH
2E D 2
R (7)
whereH is the hardness of the softer material andK, the hardnesscoefficient, is related to the Poisson’s ratio of the softer mateby ~see CEB friction model@19#!:
K50.45410.41n
E is the Hertz elastic modulus defined as:
1
E5
12n12
E11
12n22
E2
whereE1 , E2 andn1 , n2 are Young’s moduli and Poisson’s ratioof the contacting surfaces, respectively.
All length dimensions are normalized by the standard deviaof the surface heights,s, and the dimensionless values are dnoted by* . Hence,ys* is the difference betweenh* andd* ~Bushet al. @34#! which, after some algebra becomes:
ys* 5h* 2d* 51
A48pb(8)
whereb is a surface roughness parameter defined as
b5hRs
36 Õ Vol. 126, JANUARY 2004
-
. It
d-s,q.
ad-
rial
s
ione-
f* (z* ) is the dimensionless asperity heights probability densfunction obtained from Eq.~2! by substituting the normalizedlength dimensionsz/s andss /s.
The dimensionless critical interference,vc* , is another form ofthe plasticity index,C>(vc* )21/2, that was first introduced byGreenwood and Williamson@20#. It was shown in@35# that C isthe most important dimensionless parameter in elastic-plastictact problems of rough surfaces. It has the form:
C5S vc*s
ssD 20.5
52E
pKH S ss
R D 0.5
(9)
and as can be seen it depends on surface roughness and mproperties. Rougher and softer surfaces have higher plastindex.
Kogut and Etsion@30# found that the entire elastic-plastic contact regime of a single asperity extends over the range 1<v/vc,110 with a transition atv/vc56 that divides it into two sub-regions. Dimensionless contact parameters of a single asperityP̄/ P̄c , F̄s /F̄s0 and Q̄max/P̄c were presented in@30,31#, and@21#,respectively, whereP̄c5(2/3)KHpvcR is the critical contactload at yielding inception (v5vc), F̄s052pRDg is the adhesionforce at point contact (v50) andDg is the energy of adhesionThese dimensionless contact parameters can be expressedgeneral form:
P̄
P̄c
5aS v
vcD b
(10)
F̄s
F̄s0
5aS v
vcD bS «
vcD c
~contacting asperities,v/vc.0!
(11)
F̄snc
F̄s0
54
3F S v/vc
v/vcD 2
20.25S «/vc
v/vcD 8G
~non-contacting asperities,v/vc,0! (11a)
Q̄max
P̄c
5(i
aiS v
vcD bi
(12)
where« is the intermolecular distance that is typically about 0.0.5 nm. The constantsa, b, andc for the elastic, elastic-plastic~inthe two sub-regions!, and plastic regimes are summarized in Tab1. Note that Eq.~12! is not applicable forv/vc.6 ~see@21#!, andEq. ~11! is not applicable forv/vc.110 ~see@31#!. The analysesin Refs. @30#, @31#, and @21# are all based on an assumptionelastic perfectly-plastic material behavior and hence, the premodel is also adequate for such materials.
The dimensionless contact loadP* , is obtained from Eqs.~4!and ~10! ~see@35#! in the form:
Transactions of the ASME
fs
d
:
g
-an
ef-
ins-is
qs.it
e
-i-
-
t
the
m-tofs the
be
alrnal
een
tic
P* 5P
AnH5
2
3pbKvc* S E
d*
d* 1vc*I 1.511.03E
d* 1vc*
d* 16vc*I 1.425
11.4Ed* 16vc*
d* 1110vc*I 1.2631
3
K Ed* 1110vc*
`
I 1D (13)
whereI b is a general form of the integrand:
I b5S z* 2d*
vc*D b
f* ~z* !dz* (14)
The four integrals in Eq.~13! and their corresponding limits ointegration represent the contribution of the asperities in elaelastic-plastic~in the two sub-regions! and fully plastic contact,respectively. This methodology will be maintained in thfollowing.
Multiplying Eqs.~11! and (11a) by F̄s0 and using the values inTable 1, the adhesion force,F̄s , of a single contacting asperity inthe elastic and elastic-plastic regimes as well asF̄snc for a singlenoncontacting asperity, can be obtained. Substituting in Eq.~5!,and using the dimensionless form of Eq.~3! one can obtain thedimensionless adhesion force,Fs* , between rough surfaces in thform:
Fs* 5Fs
AnH52pbuS E
2`
d*Jnc10.98E
d*
d* 1vc*J20.29
0.298
10.79Ed* 1vc*
d* 16vc*J20.321
0.356 11.19Ed* 16vc*
d* 1110vc*J20.332
0.093 D (15)
whereJnc accounts for the contribution of the noncontacting aperities and has the form:
Jnc54
3 F S «*
d* 2z* D 2
20.25S «*
d* 2z* D 8Gf* ~z* !dz* (16)
and Jcb is a general form of the integrands accounting for t
contribution of contacting asperities:
Jcb5S z* 2d*
vc*D bS «*
vc*D c
f* ~z* !dz* (16a)
The dimensionless adhesion parameter,u, is:
u5Dg
sH(17)
Note that contribution of fully plastic asperities (v/vc>110) wasnot included in Eq.~15! in accordance with the observation main Ref. @31#.
Multiplying Eq. ~12! by P̄c and using the values in Table 1, thstatic friction force,Q̄max, of a single asperity in the elastic anfirst elastic-plastic sub-region can be obtained. Substituting in~6!, and following the same procedure that have lead to Eq.~15!,The dimensionless static friction force is obtained in the form
Qmax* 5Qmax
AnH5
2
3pbKvc* F0.52E
d*
d* 1vc*I 0.982
1Ed* 1vc*
d* 16vc*~20.01I 4.42510.09I 3.42520.4I 2.425
10.85I 1.425!G (18)
whereI b is defined in Eq.~14!.Equations~13!, ~15!, and ~18! can be transformed, by usin
Eqs.~8! and ~9!, to presentP* , Fs* andQmax* as functions of the
Journal of Tribology
tic,
e
e
s-
he
e
edEq.
more practical parametersh* andC. Also, the dimensionless external forceF* ~see Fig. 1! as a function of these parameters cbe obtained in the form:
F* 5F
AnH5P* 2Fs* (19)
The static friction coefficient as defined in Eq.~1! may be ex-pressed in the form:
m5Qmax*
F*5
Qmax*
P* ~12Fs* /P* !(20)
It should be noted here that other definitions for the friction coficient are found, e.g.,@36#. However, Eq.~1! seems to be themore practical definition.
Some insight regarding the role of the plasticity indexC in thestatic friction problem can be gained by following the analysis@35# for the contact problem. With a Gaussian distribution of aperity heights the maximum practical height of a given asperityz* >3. Therefore, the integrals for the contacting asperities in E~13!, ~15! and~18! are practically zero whenever their lower limis higher than 3. Using the approximationvc* >C22 and notingthat the relevant limits of integration have the general formd*1kC22 the condition for meaningful contribution of any of thesintegrals is:
C.S k
32d* D 1/2
(21)
It is clear from Eq.~13! for example, that the contribution of itslast three integrals~wherek>1) vanish for anyd* >0 wheneverC,1/). Therefore,C50.6 can be defined as the plasticity index value below which the contact problem is fully elastic. Simlarly, the last integral in Eq.~13! ~wherek5110) becomes appreciable only ifC.6. Hence, as was shown in@35#, C>8 indicatesa fully plastic contact.
Following the same reasoning the last integral of Eq.~15!~where k56) becomes increasingly significant asC becomeslarger than&. Since it was found in@31# that the adhesion forceof asperities withv/vc.6 is negligible compared to their contacload, it is reasonable to expect negligibly small effect ofFs* /P* inEq. ~20! whenC increases above 1.4.
Results and DiscussionIn accordance with the findings of@35# a wide range of plastic-
ity index values fromC50.5 to C58, was covered to analyzethe effect of surface roughness and material properties onstatic friction of contacting rough surfaces. A value ofb50.04was selected according to the finding of Greenwood and Williason@20#. A constant value ofK50.577 was used correspondinga typical Poisson’s ratio,n50.3, for metals. For typical values oadhesion energy, material hardness and surface roughnesrange of the adhesion parameter,u, is 1024<u<0.01 where theupper limit corresponds to very high adhesion energy that canobtained with very clean surfaces under vacuum conditions.
The numerical results of Eqs.~13!, ~15!, and ~18! to ~20! forany givenh* and C, can be cross-plotted to provide a practicpresentation of the relevant parameters vs. the known exteapplied forceF* . Following the reasoning of@35# only the resultsfor the range of practical engineering interest namely, 0<h* <3andF* <0.1, will be presented. Lowerh* and higherF* valuesmay invalidate the basic assumptions of no interaction betwneighboring asperities and no bulk deformation, respectively~see@20#!.
From Eq.~20! it is clear that the effect of adhesion on the stafriction coefficient depends on the ratioFs* /P* and this effectbecomes negligible whenFs* /P* !1.
JANUARY 2004, Vol. 126 Õ 37
r
y
t
t
-
i
a
i.
i
thege
ar
toon-
ersthesher
xlify-
-rain
ntactandsulttialtichese
tionnnex,der-c
Figure 3 presents the ratioFs* /P* vs. the dimensionless extenal force,F* , for the range of the plasticity index,C and for arelatively high value of the adhesion parameteru50.003. Thishigh value ofu was selected to facilitate the distinction of theffect of C at its higher values whereFs* /P* may become verysmall. Note that the ratioFs* /P* depends linearly onu ~see Eq.~15!! and, hence, it can be easily deduced for values ofu differentthan 0.003 from the results shown in Fig. 3.
As can be seen from Fig. 3 the ratioFs* /P* decreases sharplwith increasing plasticity index. ForC>2, Fs* /P* becomes lessthan 0.11 throughout the range ofF* even for the high value ofu50.003. Hence, forC>2 and more practical~smaller! values ofthe adhesion parameteru, it can be concluded thatP* 5F* is areasonable approximation~see Eq.~19!! and the effect of adhesionon the static friction coefficient is negligible. In contrast, the raFs* /P* is significant at low plasticity index,C50.5, over most ofthe range ofF* , and becomes small enough only at the upplimit of F* . For C51 the ratioFs* /P* becomes less than 0.1 foexternal force higher than a threshold value ofF* 50.01. It can,therefore, be concluded that the effect of adhesion is imporonly in purely elastic contacts whereC,0.6, or in lightly loadedcontacts with plasticity index up toC51 and high adhesion parameteru.0.001. Hence, wheneveru,0.001 orC.2 the effectof adhesion on the static friction can be safely neglected.
Figure 4 presents the dimensionless static friction force,Qmax* ,versus the dimensionless external force,F* , for various values ofthe plasticity index,C, whenu50.003. It can be seen that atgiven external force, the static friction force decreases withcreasing plasticity index. At higher plasticity index the contactmore plastic and a larger population of the contacting asperundergo interference in the rangev/vc>6, where according tothe finding in@21# they are unable to support any tangential loand hence, do not contribute to the static friction. Increasingexternal force at a given plasticity index also increases the numof such high interference asperities but at the same time brinto contact more asperities that were initially noncontactingturns out that the latter effect is more dominant, and, hencauses an increase of the static friction force with increasingternal force. This behavior of the static friction force is differefrom that in the case of a single asperity@21# where the staticfriction force first increases with increasing normal load, reachemaximum and than starts decreasing.
Reducing the adhesion parameteru reduces somewhat the statfriction at C50.5 and low external force but otherwise has ve
Fig. 3 Dimensionless force ratio, Fs* ÕP* , as a function of thedimensionless external force, F* , for various values of theplasticity index, C at uÄ0.003
38 Õ Vol. 126, JANUARY 2004
-
e
io
err
ant
ain-is
ties
dtheberngsIt
ce,ex-nt
s a
cry
little effect on the results shown in Fig. 4, in accordance withnegligible effect of the adhesion over most of the practical ranof F* as was shown in Fig. 3.
Note the log/log scale used in Fig. 4 showing almost linerelation having the general form:
Qmax* 5C~F* !m (22)
This relation differs from the classical law of friction,Qmax*5mF* , whenevermÞ1. Indeed in Fig. 4 the powerm is less than1 and varies fromm50.82 atC52 to m50.86 atC50.5, indi-cating a smaller rate of increase of the friction force comparedthat of the external force as more asperities are brought into ctact.
As shown in Fig. 4 at the highest plasticity index,C58, thestatic friction force is extremely small being between 3 to 4 ordof magnitude smaller than the external force. This is a result ofcontact becoming fully plastic, see@35#, where large percentageof the contacting asperities undergo interferences much higthan v/vc56. Such small friction force at high plasticity indeseems unreasonable. It may be attributed to some of the simping assumptions made in Ref.@30# namely, an elastic perfectlyplastic behavior of the materials that neglects more realistic sthardening effects. In addition, Mesarovic and Fleck@37# presenteda finite element analysis that shows a decrease in the mean copressure of a single asperity under very high normal loadsextreme interference deep into the fully plastic regime. As a resuch an asperity may regain its ability to resist a finite tangenload and thereby contribute to the static friction of highly plascontacting rough surfaces. The present model, not showing teffects, may be invalid at large plasticity index values.
Figure 5 presents the static friction coefficient,m, ~see Eq.~20!!versus the dimensionless external force,F* , for low and mediumvalues of the plasticity index,C, and u50.003. Increasing theplasticity index, at a given external force, decreases the friccoefficient, similar to the behavior of the friction force as showin Fig. 4. However, in contrast to the behavior of the frictioforce, increasing the external force, at a given plasticity inddecreases the static friction coefficient. This can be easily unstood from substituting Eq.~22! in the expression for the statifriction coefficientm5Qmax* /F* , which results in:
m5C~F* !m215C~F* !2n (23)
Sincem is less than 1 we can see from Eq.~23! that m decreaseswith increasing external force. Etsion and Amit@23# investigated
Fig. 4 Dimensionless static friction force, Qmax* , as a functionof the dimensionless external force, F* , for various values ofthe plasticity index, C at uÄ0.003
Transactions of the ASME
oi0
e
n
se
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y1
r
r
d
nn
e
andy-ntthe
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experimentally the effect of external load on the static fricticoefficient between aluminum alloy pins and a nickel coated dThey found for plasticity index values ranging from 0.67 to 1.that the powern in Eq. ~23! has values between 0.102 and 0.13respectively. The corresponding values ofn obtained from Fig. 5for the range of plasticity index values between 0.5 and 2between 0.09 to 0.13 foru50.001. This is a fair agreement considering the unknown exactu value in the experiment. It should bnoted that asF* approaches zero the static friction coefficiemay become very large and this too was observed in@23#.
Also shown in Fig. 5 in dashed lines are the results obtaifrom the original CEB friction model@19#. As can be seen thisapproximate model substantially underestimates the static friccoefficient and already atC52 predicts unrealistic small valuesThis is due to a restrictive assumption made in@19#, that asperitieswith v/vc>1 are unable to support any tangential load, caussevere underestimation of the static friction coefficient at plasity index values above 0.6. Another assumption that was madthe CEB friction model@19# is that the elastically preloaded aperities havingv/vc,1 cannot support tangential loads highthan that causing the onset of plasticity. This assumption canverely underestimate the maximum tangential load that cansupported by these asperities as was demonstrated in@21#, and isresponsible for the lower static friction coefficient that is predicby the CEB model even atC50.5
Figure 6 shows the effect of the adhesion parameter,u, on thestatic friction coefficient,m. It can be seen that for a low plasticitindex,C50.5, reducingu from the high value of 0.003 to 0.00~a three fold reduction of the adhesion force! reduces substantiallythe static friction coefficient at a given external force at the lowend of that force. This effect diminishes as the external foincreases and becomes negligibly small at the upper limit ofexternal force. A further reduction of the adhesion parameteu50.0001 has a much smaller effect since the adhesion beconegligible anyway~see discussion of Fig. 3!. The effect ofu on mdisappears at the higher plasticity indexC52, since in this casetoo the adhesion force is negligible. High adhesion forcecreases the separation,h* , at a given external force and bringmore asperities into contact especially when the external forcsmall, thus enabling to support larger tangential force, and, hethe static friction force and friction coefficient increase with icreasingu.
From Figs. 5 and 6 it can be seen that the friction coefficidepends on the dimensionless external force,F* , i.e. on the ex-
Fig. 5 Static friction coefficient, m, as a function of the dimen-sionless external force, F* , for various values of the plasticityindex, C at uÄ0.003
Journal of Tribology
nsk.10,
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tion.
ingtic-e in-rse-be
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ercetheto
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ternal force as well as on the nominal contact area~see Eq.~19!!.This later dependency is due to the effect ofAn on the separationd* ~see@35#! that appears in the integrals of Eqs.~13!, ~15!, and~18!, which are then substituted in Eq.~20!. Additionally, thestatic friction coefficient depends on mechanical propertiessurface roughness~through C! and on the adhesion energ~throughu!. This is quite different from the classical laws of friction. As the plasticity index increases the static friction coefficiebecomes much less sensitive to these parameters, similar toteaching of the classical laws of friction. Hence, the classical Clomb friction law ~which was obtained some 300 years ago psumably for highC and lowu values! can be regarded as a limiting case of the more general model presented in this work.
ConclusionA model that predicts the static friction for elastic-plastic co
tact of rough surfaces was presented. It incorporates the resuaccurate finite element analyses for the elastic-plastic contacthesion and sliding inception of a single asperity in a statistirepresentation of surface roughness. Strong effect of the exteforce and nominal contact area on the static friction coefficiwas found in contrast to the classical laws of friction. The madimensionless parameters affecting the static friction coefficare the plasticity indexC and adhesion parameteru.
The effect of adhesion on the static friction was found tonegligible at plasticity index values larger than 2 throughoutpractical external force range that was investigated regardlesu. At plasticity index values lower than 1 adhesion may be imptant if u.0.001 and the external force is not too large.
The present model that assumes elastic perfectly-plastic mrial behavior may be invalid at high plasticity index values whethe contact approaches fully plastic state. Unreasonably sstatic friction was found under this condition and an improvmodel that considers strain hardening effects and possible juncgrowths may be required.
It was shown that the classical laws of friction are a limitincase of the present more general solution and are adequatefor high plasticity index and negligible adhesion.
A comparison of the present results with those obtained fromapproximate CEB friction model showed substantial differenwith the latter severely underestimating the static fricticoefficient.
Fig. 6 Static friction coefficient, m, as a function of the dimen-sionless external force, F* , for various values of the plasticityindex, C, and the dimensionless adhesion parameter, u
JANUARY 2004, Vol. 126 Õ 39
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s
o
a
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02,.
-
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t,’’
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ta-n,
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AcknowledgmentThis research was supported in parts by the Fund for the
motion of Research at the Technion, by the J. and S. FraResearch Fund and by the German-Israeli Project Coopera~DIP!.
Nomenclature
d 5 separation based on asperity heightsd* 5 dimensionless separation,d/sE 5 Hertz elastic modulusF 5 external force
F* 5 dimensionless external force,F/AnHFs 5 adhesion forceFs* 5 dimensionless adhesion force,Fs /AnHH 5 hardness of the softer materialh 5 separation based on surface heights
h* 5 dimensionless separation,h/sK 5 hardness factor, 0.45410.41nP 5 contact load
P* 5 dimensionless contact load,P/AnHQ 5 friction force
Q* 5 dimensionless friction force,Q/AnHR 5 asperity radius of curvature
ys* 5 h* 2d*z 5 height of an asperity measured from the mean of a
perity heightsz* 5 dimensionless height of an asperity,z/sb 5 surface roughness parameter,hRs
Dg 5 energy of adhesionf* 5 dimensionless distribution function of asperity heigh
h 5 area density of asperitiesm 5 static friction coefficientn 5 Poisson’s ratio of the softer materialu 5 dimensionless adhesion parameter,Dg/sHs 5 standard deviation of surface heights
ss 5 standard deviation of asperity heightsv* 5 dimensionless interferencevc 5 critical interference at the inception of plastic defor-
mationvc* 5 dimensionless critical interference,vc /sC 5 plasticity index, Eq.~9!
Subscripts
c 5 yielding inception
Superscripts
— 5 single asperity
References@1# Karamis, M. B., and Selcuk, B., 1993, ‘‘Analysis of the Friction Behavior
Bolted Joints,’’ Wear,166, pp. 73–83.@2# Xie, W., De Meter, E. C., and Trethewey, M. W., 2000, ‘‘An Experiment
Evaluation of Coefficients of Static Friction of Common Workpiece-FixtuElement Pairs,’’ Int. J. Mach. Tools Manuf.,40, pp. 467–488.
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