TRIB HEIGHT- DEPENDENT ASPERITY RADII OF CURVATURES IN A CONTACT AND FRICTION MODEL

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1 Proceeings of 4 ASME/STLE Interntionl Joint Tribology Conference Long Bech, Cliforni USA, October 4-7, 4 TRIB HEIGHT- DEPENDENT ASPERITY RADII OF CURVATURES IN A CONTACT AND FRICTION MODEL George G. Ams Mechnicl n Inustril Engineering Deprtment Northestern University, Boston MA 115 Emil: ms@neu.eu Sinn Müftü Mechnicl n Inustril Engineering Deprtment Northestern University, Boston MA 115 Emil: smuftu@coe.neu.eu ABSTRACT The effect of height-epenent sperity rius of curvture is ccounte for in recently evelope scleepenent moel of contct n friction. The contct n friction moel inclues the effects of hesion, using the Mugis moel, n of scle-epenent friction, using the Hurto n Kim single sperity friction moel. This multisperity moel hs been moifie to inclue the effect of noncontcting sperities. The results inicte the types of conitions uner which the effects of height-epenent sperity rius of curvture ffects friction. INTRODUCTION Contct n friction ffect the opertion of mny mchines n tools tht we use every y, s well s some of the most bsic ctivities in nture. Exmples rnge from belt rives, brkes, tires, n clutches in utomobiles n in other mchines; gers, berings n sels in vriety of mechnicl systems; electricl contcts in motors; slier-isk interctions in computer isk rive; vrious MEMS evices; robotic mnipultor joint; the motion of humn knee-joint (nturl or rtificil); n wlking/running. The friction force F is the tngentil force resisting the reltive motion of two surfces which re presse ginst ech other with norml force P. Amontons, in 1699, n Coulomb in 1785, evelope our phenomenologicl unerstning of ry friction between two contcting boies. Amontons-Coulomb friction sttes tht the rtio of the friction force (uring sliing) to the norml force is constnt clle the coefficient of kinetic friction. Similrly the coefficient of sttic friction is the rtio of the mximum friction force F tht the surfces cn sustin, without reltive motion, to the norml force. These friction lws cn be summrize by efining the coefficient of friction s F (1) P without istinguishing between sttic n low-spee sliing friction. Although Eq. (1) provies n extrorinrily simple phenomenologicl friction lw, the nture of the friction force is not well-unerstoo. Contct moeling is n essentil prt of ny friction moel. It consists of two relte steps. First, the equtions representing the contct of single pir of sperities re etermine. For nnometer scle contcts the effect of hesion on the contct re is importnt. Secon, the cumultive effect of iniviul sperity contcts is etermine. Such contct moels re uncouple n represent surfce roughness s set of sperities, often with sttisticlly istribute prmeters. The effect of ech iniviul sperity contct is locl n consiere seprtely from the other sperities; the cumultive effect is the sum of the ctions of the iniviul sperities (e.g. the well-known Greenwoo n Willimson moel, [1]). For sufficiently smll size contcts, the hesion forces between the surfces ffect the contct conitions. Vrious hesion moels, typiclly between n elstic sphere n flt, hve been introuce. The moel by Johnson, Kenll n Roberts (JKR) ssumes tht the ttrctive intermoleculr surfce forces cuse elstic eformtion beyon tht preicte by the Hertz theory, thereby proucing subsequent increse of the contct re []. The moel by Derjguin, Muller n Toporov (DMT), on the other hn, ccounts for the hesive stress outsie of the contct re, but ssumes tht the contct stress profile remins the sme s in the Hertz theory []. Due to the ssumptions involve, the JKR/DMT moels re most suitble when the rnge of surfce forces is smll/lrge 1 Copyright 4 by ASME

2 f /G /b Figure 1. Reltionship between friction stress n contct rius ccoring to HK moel. compre to the elstic eformtions, s pointe out by Tbor [4]. Another moel, introuce by Mugis [5], escribes continuous trnsition between the JKR n DMT moels. Contct n friction moels which el with hesion in multi-sperity contcts hve lso been evelope. In the first of series of ppers Chng, Etsion n Bogy (CEB) [6] evelope n elstic-plstic multi-sperity contct moel for norml loing bse on volume conservtion of plsticlly eforme sperity control volume. In [7], the effect of hesion ws inclue by using the DMT moel for contcting sperities n the Lennr-Jones potentil between non-contcting sperities. Finlly moel for clculting the coefficient of friction ws given in [8]. It ssume tht once plstic yieling is initite in pir of contcting sperities, no further tngentil force cn be sustine. Fuller n Tbor [9] evelope theoreticl moel which use the JKR moel of hesion long with Gussin istribution of sperity heights ws evelope. Stnley, Etsion n Bogy (SEB) [1] evelope moel for the hesion of two rough surfces, ffecte by subbounry lyer lubriction, in n elstic-plstic multi-sperity contct. Polycrpou n Etsion [11] use the SEB moel to preict the sttic friction coefficient. The tngentil lo ws foun using the sme proceure s in the CEB moel [8] for soli-soli contct. Kogut n Etsion evelope multi-sperity contct [1] n friction moels [1] which inclue the effects of plstic eformtion. The mximum sher lo tht n sperity cn sustin is limite by the combine norml n sher lo which cuses the plstic eformtion zone to rech the surfce. Thus the friction nlysis [1] preicts higher friction thn the relte work in [8]. Yu, Pergne, n Polycrpou [14] extene the CEB moel to inclue n symmetric istribution of sperity heights. Experiments in the low n high norml lo regimes hve shown tht the friction coefficient epens on the mgnitue of the norml force. In prticulr, these experiments hve shown tht increses with ecresing norml lo (Rbinowicz n Kymrm [15]; Etsion n Amit, [16]). The scle-epenence of the friction stress for single sperity contcts hs recently been investigte by Hurto n Kim (HK), [17,18]. They presente micromechnicl isloction moel of frictionl slip between two sperities for wie rnge of contct rii. Accoring to the HK moel, if the contct rius is smller thn criticl vlue, the sperities slie pst ech other in concurrent slip process where the hesive forces re responsible for the sher stress; hence the sher stress remins t high constnt vlue. On the other hn, if the contct rius is greter thn tht criticl vlue, the sher stress ecreses for incresing vlues of contct rius until it reches secon constnt, but lower vlue. The reltionship between the non-imensionl friction stress ( / G ) n the non-imensionl contct rius ( / b) is pproximte in Fig. 1. The contct rius is normlize by the Burgers vector b n the friction stress is normlize by the effective sher moulus G =G 1 G /(G 1 +G ) where G 1 n G re the sher mouli of the contcting boies. Ams, Müftü, n Moh Azhr (AMM), [19] incorporte the HK moel n the hesion contct moel of Mugis, into sttisticl multi-sperity moel for contct n friction. The reltionship between the friction force n the norml lo between two rough surfces uring slip process ws etermine. Three key imensionless prmeters representing the surfce roughness, the friction regime of the contcts, n the surfce energy of hesion, were seen to influence the vlue of the friction coefficient. In [], Ams n Müftü inclue the effect of n symmetric istribution of sperity heights using Weibull function to ccount for skew n kurtosis. In this pper the effect of height-epenent sperity rius of curvture on the AMM scle-epenent contct n friction moel [19] is etermine. Such height-epenence is likely to occur in, for exmple, polishing opertions in which higher sperities re polishe to greter extent n therefore hve lrger rii of curvture thn o the shorter sperities. On the other hn, from the point-of-view of rnom surfce profile, Whitehouse n Archr [1] foun tht higher sperities hve smller curvtures. This result is consequence of the ssume rnomness of surfce in which the neighboring points bout high pek re more likely to hve height which evites from the pek height by greter mount thn woul the points nerby lower pek. Extening the work of [1], Onions n Archr [] foun tht such istribution of sperity curvtures increses the contct pressures mking plstic eformtion more likely to occur. Rther thn obtining n nlyzing t for vriety of surfce finishes, in this pper we investigte the egree of height-epenence neee to significntly influence friction. It is lso note tht moern mnufcturing techniques llow greter egree of control over surfce topogrphy thn existe t erlier times. In ition, the AMM moel [19] hs been moifie to ccount for the effect of non-contcting sperities. DEVELOPMENT OF THE MODEL Contct Moel The scle-epenent multi-sperity contct n friction moel evelope by Ams, Müftü, n Moh Azhr [19] will be extene to inclue height-epenent sperity rius of curvture. For two rel surfces seprte by istnce (efine from the men of sperity heights) the number of contcting sperities n is n N ( z) z () Copyright 4 by ASME

3 where N is the totl number of sperities, is the stnr evition of sperity pek heights, z z is the imensionless height coorinte mesure from the men of sperity heights, (z) is the probbility ensity of sperity peks, n is the non-imensionl seprtion between the two surfces (Fig. ). z The height-epenent rius of curvture is ssume to be in the form r Rg( z), g( z) 1 tnh( z) () where R is the verge rius of curvture. The choice of the function g (z) is somewht rbitrrily chosen so tht the sperity rius of curvture vries smoothly between R(1-) n R(1+). For positive higher sperities hve greter rii wheres for negtive the higher sperities hve smller rii. For lrge the vrition in rius with z occurs most bruptly in smll region surrouning the verge sperity height, wheres for smll the vrition is more grul. Although vriety of surfce height istributions my be consiere, Gussin istribution of sperity peks, which gives the following probbility ensity function, is chosen, i.e. u Men of sperity heights z 1 exp 1 z / / (4) Flt surfce of upper boy The reltion between the norml lo P n eformtion u z of two contcting sphericl sperities with hesion is given by the Mugis moel [5]. In tht moel, uniform tensile stress exists between the contcting sperities just outsie the contct zone, r c, where c is the ril extent of the hesion zone. The seprtion between the two surfces t r = c is equl to the prescribe mximum hesion istnce h. Thus the work of hesion is given by w h. In [5] the following non-imensionl reltions mong the sperity contct rius ( / b), the sperity contct force Men of surfce heights Figure. Contct of rough surfce (lower boy) with flt surfce (upper boy). x ( P P / Gb ), n the sperity eformtion ( u u / ) were obtine b 1 m 1m tn m 1 hg z (5) 4 b 1 m 1tn m 1m1 1 h 8 P 1gz 4 b m 1m tn m 1 h 1 1 u z m 1 (6) 1 b g z h (7) where m is the non-imensionl hesion rius given by c m, 1/ b 9 h in which is the non-imensionl Mugis hesion prmeter. In (5)-(7) there re three key prmeters, n which re efine by R 1/ 1 / R, (8) w,. (9) b E b A physicl interprettion of the surfce prmeters n is provie by noting tht in simple verticl scling of the surfce by fctor k, the stnr evition of sperity heights is scle by k but the sperity rius of curvture R is scle by 1/k. Thus, is scle by k, but remins constnt. Hence is representtion of the surfce roughness, n is referre to s the surfce roughness prmeter. The prmeter escribes the rtio of the contct rius (ue to n sperity penetrtion equl to ) to the Burgers vector length. Thus smll re expecte to be inictive of nno-scle sperity contcts n progressively lrger vlues of correspon to trnsition n lrger vlues of the contct rius (Fig. ). Therefore is referre to s the friction regime prmeter. The surfce energy prmeter represents the rtio of the hesive stress to the prouct of the composite Young s moulus n the Burgers vector. It is further note tht for the cse consiere here, in which one of the surfces is ssume to be rigi n flt, G =G n the composite Young's moulus is given by E E ( 1 ). Furthermore the reltion G=E/(1+) hs been use. The simultneous solution of Eqs. (5)-(7) gives the reltions mong m, P,, n u for given vlues of the surfce roughness prmeter, the friction regime prmeter, Copyright 4 by ASME

4 the surfce energy of hesion prmeter, the rtio (b/h), the Poisson s rtio (), n the height-epenent rius prmeters (,). An expression for the totl non-imensionl norml force cting on the nominl contct re is obtine by integrting the norml force on iniviul sperities, resulting in the totl norml force ( P ) T P N P ( z) z T (1) It is note tht ue to hesion uring the unloing process, sperities my remin in contct even if the sperity overlp u is negtive. This effect hs been inclue in the evlution of the integrls in Eqs. () n (1) by vrying the hesion rius rtio m in Eqs. (5)-(7) when evluting the force n contct re. Thus the lower limits of the integrls in Eqs. () n (1) re effectively slightly less thn. However, when n sperity breks free of its mting surfce uring unloing, its uneforme loction my still bring it within the istnce h in which there re ttrctive forces using the Mugis moel of hesion. The effect of these ttrctive forces on the pplie norml force ws neglecte in [19]-[], but re ccounte for here. Consier n elstic sphere in close proximity to n elstic hlf-spce. If the minimum seprtion istnce is greter thn the hesion istnce h, then the interction force vnishes. However if the seprtion istnce is less thn h, uniform tensile stress of mgnitue cts in circulr re of rius c. The surfce norml elstic isplcement in the center n long the periphery of the circle of interction for such loing re given in [] by u c 4 c, u C (11) E E respectively. At r = c, the seprtion fter eformtion must be equl to the hesion seprtion (h) resulting in c R 4 E 4 E h R (1) It is note, however, tht this solution is only vli if the elstic eformtion t the center of the circulr re is insufficient to bring these boies into contct. The norml force between the two boies is given by P c, which becomes 64 P 1 9(1 ) (1 / h) (1) Thus Eq. (1) nees to be moifie to ccount for the forces exerte by non-contcting sperities by incluing Eq. (1) for vlues of z greter thn h but less thn the vlue of z which cuses the sperity to seprte from the surfce. For vlues of >.655, when ny sperity seprtes it will return to its uneforme position which is outsie the Mugis rnge of hesion. Thus no correction for non-contcting sperities is neee for those cses. Friction Moel Although hesion ffects the reltionship between the norml force n the contct rius, it oes not ffect the reltion between the friction force n contct rius. From Fig. 1, the imensionless sher stress is function of the imensionless contct rius n is pproximte by log 1, 1 log M log B, log, 1 where the left n right limits of region- re (, 1 1) n (, ) respectively. The constnts of Eq. (14) re given in [19] where M n B re, respectively, the slope n y-intercept of the line in region- of the log-log plot of Fig. 1. The friction force F cting on single sperity cn be etermine from Eq. (14) by using F or F (15) The totl sher force F cting on the nominl contct re cn be clculte by integrting the sher forces cting on ech sperity ginst the probbility ensity function, i.e. F N F ( z) z T (14) (16) It is note tht for vlues of the pplie tngentil force ( F T ) less thn tht given by Eq. (16), the istribution of tngentil n norml forces mong the sperities my cuse some sperities to slip while others continue to stick. However when F reches the vlue given in Eq. (16) ll contcting sperities will slie resulting in globl slip. Thus, the coefficient of friction µ for two rel surfces seprte by istnce, cn be obtine from the rtio of Eqs. (16) n (1), where (1) hs been moifie to ccount for non-contcting sperities. RESULTS AND DISCUSSION In Fig. is shown the effect of non-contcting sperities on the friction coefficient. For given pplie force, the hesive force on these sperities serves to increse the contct force n hence increse the contct re n friction force, thereby giving lrger friction coefficient. As iscusse previously, for >.655 the effect of non-contcting sperities vnishes. Fig. shows results with n without non-contcting sperities for =.1, =.1, n with = 1 ( =.5) s well s with = 5 ( =.415). These cses represent the lowest vlues of use in [19]. The mximum effect on the friction coefficient is.1% for =.5 n.8% for =.415. As 4 Copyright 4 by ASME

5 ws expline in the bove, this mximum ifference correspons to the smllest pplie lo.. CoefficientofFriction, = 1 = 5.5 w/ non-contcting sperities w/o non-contcting sperities Nonimensionl Norml Force (P T /NGb ) Fig.. Effect of non-contcting sperities on the friction coefficient for =.1, =.1, n with = 1 ( =.5) n = 5 ( =.415). Fig. 4 shows the vrition of the friction coefficient with the pplie force, ech for five ifferent vlues of the prmeter (1.,.5,, -.5, -.95) n ll with =.5. A positive/negtive vlue of correspons to tller sperities which hve greter/smller rius of curvture thn o the shorter ones. For = 1., the tllest sperities hve twice the rius s the verge, wheres s for = -.95 the tllest sperities hve rius equl to.5 of the verge. It is interesting to note tht in ll three cses shown (Figs. 4,b,c), n of -.95 ecreses friction n rmticlly reuces the lo-epenence of friction which ws preicte in [19]-[] for constnt rii of curvture. Similrly for = 1., friction is highest n so is its epenence on norml lo. These results occur becuse for given norml force, the contct re increses monotoniclly with the rius of curvture. Thus negtive give smller friction thn o positive. Tht tren is less pronounce for lrger los in which greter portion of the sperities re in contct. CONCLUSIONS The scle-epenent contct n friction moel of Ams, Müftü n Moh Azhr [19] hs been moifie in two wys. First the effect of non-contcting sperities ws inclue. This effect ws foun to be smll, minly becuse the originl moel lrey inclue the effect of sperities while they remine in contct prior to seprtion. Seconly the effect of heightepenent sperity rius of curvture ws inclue. This effect cn be significnt. For rii of curvture which increse/ecrese with height, it ws foun tht the friction coefficient increses/ecreses s oes its epenence on lo. CoefficientofFriction, ) CoefficientofFriction, b) CoefficientofFriction, c) Nonimensionl Norml Force (P T /NGb ) =1 =.5 = =-.5 =-.95 1, 1, 1, 1/ Nonimensionl Norml Force (P T /NGb ) =1 =.5 = =-.5 =-.95 1, 5 1, 1, 1/ =1 =.5 = =-.5 = Nonimensionl Norml Force (P T /NGb ) 1, 1, 1, 1/ Fig. 4,b,c. The vritions of the friction coefficient with pplie force, ech for five vlues of the prmeter n ll with =.5. 5 Copyright 4 by ASME

6 REFERENCES [1] Greenwoo, J.A., n Willimson, J.B.P., 1966, Contct of Nominlly Flt Surfces, Proceeings of the Royl Society of Lonon, A95, pp [] Johnson, K.L., Kenll, K., n Roberts, A.D., 1971, Surfce Energy n the Contct of Elstic Solis, Proceeings of the Royl Society of Lonon, A4, pp [] Derjguin, B.V., Muller, V.M., n Toporov, Y.P., 1975, Effect of Contct Deformtions on the Ahesion of Prticles, Journl of Colloi n Interfce Science, 5, pp [4] Tbor, D., 1976, Surfce Forces n Surfce Interctions, Journl of Colloi n Interfce Science, 58, pp. -1. [5] Mugis, D., 199, Ahesion of Spheres: The JKR-DMT Trnsition Using Dugle Moel, Journl of Colloi n Interfce Science, 15, pp [6] Chng, R.W., Etsion, I., Bogy, D.B., 1987, An Elstic- Plstic Moel for the Contct of Rough Surfces, ASME Journl of Tribology, 19, pp [7] Chng, R.W., Etsion, I., Bogy, D.B., 1988, Ahesion Moel for Metllic Rough Surfces, ASME Journl of Tribology, 11, pp [8] Chng, R.W., Etsion, I., Bogy, D.B., 1988, Sttic Friction Coefficient Moel for Metllic Rough Surfces ASME Journl of Tribology, 11, pp [9] Fuller, K.N.G n Tbor, D., 1975, The Effect of Surfce Roughness on the Ahesion of Elstic Solis, Proceeings of the Royl Society of Lonon, A45, pp [1] Stnley, H.M., Etsion, I. n Bogy, D.B., 199, Ahesion of Contcting Rough Surfces in the Presence of Sub-Bounry Lubriction, ASME Journl of Tribology, 11, pp [11] Polycrpou, A.A., n Etsion, I., 1998, Sttic Friction of Contcting Rel Surfces in the Presence of Sub-Bounry Lubriction, ASME Journl of Tribology, 1, pp [1] Kogut, L., n Etsion, I.,, A Finite Element Bse Elstic-Plstic Moel for the Contct of Rough Surfces, Tribology Trnsctions, 46, pp [1] Kogut, L., n Etsion, I.,, A Sttic Friction Moel for Elstic-Plstic Contcting Rough Surfces, ASME Journl of Tribology, 16, pp [14] Yu, N., Pergne, S.R., n Polycrpou, A.A., 4, Sttic Friction Moel for Rough Surfces With Asymmetric Distribution of Asperity Heights, ASME Journl of Tribology,16, pp [15] Rbinowicz, E., n Kymrm, F., 1991, On the Mechnism of Filure of Prticulte Rigi Disks, Tribology Trnsctions, 4, pp [16] Etsion, I., n Amit, M., 199, "The effect of smll norml los on the sttic friction coefficient for very smooth surfces," Journl of Tribology, 115, pp [17] Hurto, J.A,. n Kim, K.-S., 1999, Scle Effects in Friction of Single Asperity Contcts: Prt I; From Concurrent Slip to Single-Disloction-Assiste Slip, Proceeings of the Royl Society of Lonon, A455, pp [18] Hurto, J.A., n Kim, K.-S.,1999, Scle Effects in Friction in Single Asperity Contcts: Prt II; Multiple- Disloction-Cooperte Slip, Proceeings of the Royl Society of Lonon, A455, pp [19] Ams, G.G., Müftü, S., n Moh Azhr, N.,, A Scle-Depenent Moel for Multi-Asperity Moel for Contct n Friction, ASME Journl of Tribology, 15, pp [] Ams, G.G., n Müftü, S.,, Asymmetric Asperity Height Distributions in Scle-Depenent Moel for Contct n Friction, Proceeings of STLE/ASME Joint Interntionl Tribology Conference, Pointe Ver Bech, Flori, October, pper TRIB-58 on CD-ROM. [1] Whitehouse, D.J., n Archr, J.F., 197, The Properties of Rnom Surfces of Significnce in Their Contct, Proceeings of the Royl Society of Lonon, A 16, pp [] Onions, R.A n Archr, J.F., 197, The Contct of Surfces Hving Rnom Structure, Journl of Physics D: Applie Physics, 6, pp [] Johnson, K.L., 1985, Contct Mechnics, Cmbrige University Press, Cmbrige, Unite Kingom. 6 Copyright 4 by ASME

Conservation Law. Chapter Goal. 6.2 Theory

Conservation Law. Chapter Goal. 6.2 Theory Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the