NANO-SCALE EFFECTS IN THE ADHERENCE, SLIDING AND ROLLING OF A CYLINDER ON A SUBSTRATE

Transcription

1 Nno-Sle Effets in Clindril Contts Sri et l. NANO-SCALE EFFECTS IN THE ADHERENCE, SLIDING AND ROLLING OF A CYLINDER ON A SUBSTRATE Ö. T. Sri, G. G. Adms, S. Müftü Mehnil Engineering Deprtment Northestern Universit Boston, MA, 05, USA tln@oe.neu.edu, dms@oe.neu.edu, smuftu@oe.neu.edu Abstrt The behvior of nno-sle lindril bod (e.g. fiber), ling on substrte nd ted upon b ombintion of norml nd tngentil fores, is the subjet of this investigtion. As the sle dereses to the nno level, dhesion beomes n importnt issue in this ontt problem. Thus this investigtion trets the two-dimensionl plne strin elsti deformtion of both the linder nd the substrte during rolling/sliding motion, inluding the effet of dhesion using the Mugis model. Due to dhesion, the ontt re inreses nd finite fore (i.e. pull-off fore) is needed to seprte the bodies. For the initition of sliding, the Mindlin pproh is used, wheres for rolling, the Crter pproh is utilized. Eh se is modified for nno-sle effets using the dhesion theor of frition for the frition stress. Results re given for the norml nd tngentil loding problems, inluding the pull-off fore, the initition of sliding nd rolling in terms of dimensionless quntities representing dhesion, linder size, nd pplied fores. Ke words: Clinder; Nno-sle; Adhesion; Contt; Mugis Model; Rolling; Sliding; Pull-off Fore.

2 Nno-Sle Effets in Clindril Contts Sri et l. Introdution At the mro-sle, dhesion between ontting bodies hs negligible effet on surfe intertions. However, s the ontt size dereses, dhesion between the bodies beomes signifint, espeill for len surfes nd lightl loded sstems. As tehnolog dvnes, the sle of some devies derese nd dhesion beomes n importnt issue, espeill in suh pplitions s informtion storge devies nd miroeletromehnil sstems (MEMS). In ontting bodies, toms of mterils re seprted b n equilibrium distne z 0 of few Angstroms. At tht seprtion, vn der Wls fores re dominnt over eletrostti fores []. The derivtive of the Lennrd-Jones potentil expresses the surfe pressure in terms of the seprtion z between two prllel surfes 8w z 0 9 z0 3 p ( z) ( ) ( ) 3z () z z o where w is the work of dhesion wγ +γ -γ. Here the surfe energies of the ontting bodies re γ nd γ, nd the interfe energ of the two surfes is γ. The Lennrd- Jones potentil inludes ttrtive fores, whih t over long-rnge nd repulsive fores, whih t over short-rnge. If the seprtion is less thn the equilibrium sping, the repulsive (i.e., ontt) fores re dominnt, wheres if the seprtion exeeds the equilibrium sping there will be net ttrtive fore (i.e., the dhesion fore). The Lennrd-Jones pressure vnishes t the equilibrium sping. Numerous studies hve been onduted on the dherene of spheril bodies. Brdle used the ttrtive fore between two moleules nd, b integrting through the

3 Nno-Sle Effets in Clindril Contts Sri et l. whole bodies, found the ohesive fore between two rigid spheres of rdii R nd R []. The pull-off fore F required to seprte two rigid spheril bodies ws found to be where R R R /( R + R ) is the equivlent rdius of urvture. F πwr () Johnson, Kendll nd Roberts (JKR) presented theor on the dherene of deformble elsti bodies [3]. In the JKR pproximtion the dhesion outside the ontt region is ssumed to be zero. The blne between the elsti, surfe nd potentil energies ws used, nd the resulting stresses t the edges were infinite. It ws found tht the ontt re is lrger thn the ontt re predited b Hertz, nd the pull-off fore ws found to be F (3/ ) πwr. (3) In 975, Derjguin, Muller nd Toporov (DMT) presented nother theor on the dhesion of deformble elsti bodies [4]. In the DMT theor the dhesion fore is tken into ount outside the ontt re, but the form of the stress distribution in the ontt is ssumed to be unffeted. The pull-off fore ws found to be the sme s the Brdle reltion. It is seen tht for both the JKR nd DMT theories, the pull-off fore is independent of the elsti properties of the mterils. Although the JKR nd DMT theories seem to be ompetitive, it ws shown b Tbor tht these two theories represent the extremes of prmeter µ, whih is the rtio of elsti deformtion to the rnge of dhesive fores [5], i.e. where E is the omposite Young s modulus. / 3 Rw µ 3 (4) E z0 3

4 Nno-Sle Effets in Clindril Contts Sri et l. Thus lrge omplint bodies re in the JKR regime, wheres stiff nd smll solids orrespond to the DMT regime. Greenwood onstruted n dhesion mp tht overs these regimes [6]. Mugis presented model for the trnsition between the JKR nd DMT theories [7]. Similr to the Dugdle model of rk, the Mugis model ssumes onstnt tensile surfe stress σ 0 in regions where the surfes re seprted b distne less thn h, where h is obtined from the work of dhesion through the reltion wσ 0 h. When this work of dhesion is set equl to tht for the Lennrd-Jones potentil, the mximum seprtion distne for dhesion h is found to be 0.97 z 0 where σ 0 is tken s the theoretil strength. In this pper the dhesion of ontting linders or equivlentl linder in ontt with hlf spe t the nno-sle is onsidered. If the linder is subjeted to ombined tngentil nd norml loding it m remin t rest, roll, slide or undergo omplex motion depending on the mgnitudes nd the pplition points of the loding. The elsti behvior of the lindril bod nd hlf-spe with dhesion re investigted using the plne strin theor of elstiit. This nno-sle nlsis differs from the orresponding mro-sle problem in two importnt ws. First, due to the smll sle of the ontt re, dhesion beomes importnt. The Mugis model is used to pproximte the dhesive stress outside the ontt region. Seond, in the mro sle Coulomb s frition, whih sttes tht fritionl fore is proportionl to the norml lod, is onsidered vlid. Due to surfe roughness in the ontt of two nominll flt surfes, the bodies touh t disrete number of ontts, so the rel ontt re beomes muh smller thn the pprent ontt re. If sttistil distribution for the sperit peks is ssumed, it turns out tht the rel re of 4

5 Nno-Sle Effets in Clindril Contts Sri et l. ontt is pproximtel proportionl to the norml fore. This result is onsistent with the dhesion theor of frition. Aording to the dhesion theor of frition, ontting sperities form strong dhesive juntions. In order to hve reltive motion between the ontting bodies, these juntions need to be broken whih ours when the interfil sher stress rehes the sher strength of the weker mteril. Therefore onstnt sher stress ours in the rel re of ontt during the reltive motion of ontting bodies. Thus onstnt sher stress long with ner proportionlit between norml lod nd rel ontt re gives frition fore pproximtel proportionl to the norml lod, i.e. Coulomb frition. At the nno-sle, however, in whih the ontt rdius is in the order of 0 nm or less, it seems more resonble to ssume single rel ontt re, i.e. onstnt sher stress during sliding. Adhesion of lindril bodies on substrte is enountered in nno-wires, rbon nno-tubes nd nno-fibers, nd in different fields suh s mirobiolog, miroeletronis nd MEMS/NEMS devies. Determintion of the pull-off fore nd the fores neessr to roll or slide lindril bod on the substrte re importnt quntities to know in these pplitions. In some ses it is importnt to mnipulte these single fibers to form struture wheres in other instnes the sliding nd rolling motions re importnt in ontmintion removl proesses. Theor nd Disussion of the Results The ontt of lindril bod of rdius R with flt surfe is investigted. The results re equll vlid for the ontt of two linders b using the equivlent rdius of urvture previousl defined. Liner plne strin elstiit is used throughout, 5

6 Nno-Sle Effets in Clindril Contts Sri et l. whih implies tht the fores re given per unit length. Norml loding, the initition of sliding, sted sliding nd sted rolling re investigted. Norml Loding Consider norml loding in whih norml lod F is pplied to linder when tngentil fore T is zero. As shown in Fig., there will be entrl ontt zone (-< x <) surrounded b two dhesion zones (< x <) in whih the seprted surfes re under onstnt tensile stress s desribed b the Mugis model [7]. The tensile dhesive stress is effetive up to seprtion h, beond whih it vnishes. The geometri reltion for the deformtions in the norml diretion (u () () nd u for bod nd respetivel) t the ontt interfe is u () () x u δ 0 +, -<x< (5) R in the ontt zone, where δ 0 is the mximum linder penetrtion whih ours t the enter of the ontt zone. Aording to plne strin liner elstiit (e.g. Brber [9]), the derivtive of the surfe norml displements n be written in terms of norml nd sher stresses s d( u () u dx () ) A 4π p ( ξ) dξ x ξ B px( x) 4 x R, -<x< (6) where p x is the tngentil trtion in x diretion nd the ontt pressure p is onsidered positive in ompression. The mteril prmeters A nd B re given b 4( υ ) 4( υ A + G G ) 8, E B 4υ 4υ, G G E υ υ +, Ei G i ( + υi ) E E (7) 6

7 Nno-Sle Effets in Clindril Contts Sri et l. where G, G re the sher moduli, υ, υ re the Poisson's rtios nd E, E re the moduli of elstiit of bodies nd respetivel. Eqn. (6) n be simplified for either identil mterils (B0) or for the fritionless se (p x (x)0). Even if the mterils re not identil, the effet of the onstnt B is usull smll [8] nd is often negleted. Thus norml/sher stresses do not produe tngentil/norml reltive displements. For mthemtil ese we let p ( x) p ( x) σ 0, x (8) where p ( x) 0 for < x < resulting in p ( ξ) x σ dξ + ξ x R Eπ 0 Eπ x ln( ) + x, -< x < (9) The singulr integrtion in Eqn. (9) is bounded t the end points. This eqution n be onverted to sstem of liner equtions using the numeril ollotion tehnique of Erdogn, Gupt, nd Cook [0], given in ppendix A. Sine the ontt hlf-width nd the dhesion hlf-width re both unknown, two extr equtions re needed. The geometril reltion for the seprtion of the two bodies t x nd x n be used long with Eqn. (6), nd leds to p ( ξ ) dξ () () ( u u ) dx h Eπ x R ξ (0) If the order of the integrtion is hnged nd Eqn. (8) is substituted into Eqn. (0), it tkes the form σ p h 0 π R Eπ E ξ ( ξ)ln dξ ξ x ln dx + x () 7

8 Nno-Sle Effets in Clindril Contts Sri et l. The lst integrl in Eqn. () ws evluted nltill. Using fore equilibrium in the -diretion, the totl pplied lod F n be relted to the ontt hlf-width (), i.e. p 0 ( ξ ) dξ σ F () Equtions () nd () n be onverted into two lgebri equtions nd long with Eqn. (9) solved using the Erdogn, Gupt nd Cook reltions. If the integrl eqution (9) is solved for zero dhesion, nd the fore blne eqution (Eqn. ()) is used, two-dimensionl Hertz ontt solution is obtined nd is given s [8] π E F (3) 4R The solution with dhesion inluded, shows tht the pressure distribution is ompressive in the entrl prt of the ontt zone but tensile t the ends of the ontt zone. In Figs. nd 3 the dimensionless pplied norml lod (F/Eh) is given s funtion of ontt hlf-length (/R) for different dhesion stresses (σ 0 /E). Fig. orresponds to R/h00 nd Fig. 3 to R/h00. Note tht for smll dhesion stress, the Hertz solution is pprohed. It n be seen tht s the pplied lod inreses, the ontt width lso inreses monotonill. For lrger vlues of σ 0 /E, the lod rehes lol minimum, whih orresponds to snp-off from the surfe, t finite fore nd ontt rdius is observed. A negtive lod is lws observed s the ontt width pprohes either zero, or lol minimum, whih orrespond to pull-off fore. The vlue of this pull-off fore inreses with inresing dhesion stress. On the other hnd, s the two bodies pproh eh other, the n be expeted to ttrt eh other beuse of the tensile dhesive 8

9 Nno-Sle Effets in Clindril Contts Sri et l. stress, resulting in snp-on. It is noted tht the pull-off fore inreses slightl s the fiber rdius doubles (Figs. nd 3). However this inrese in surfe fore is muh less thn the orresponding inrese in bod fore (i.e., the weight). Figs. 4 nd 5 show the vrition of the dimensionless dhesion width (/R) with the dimensionless ontt width (/R) for different dhesion stress vlues (σ 0 /E). As the pplied lod inreses, the ontt width nd the dhesion width inrese. It m seem ounter-intuitive tht for given /R, stronger dhesion gives smller /R. However, rell tht for stronger dhesion smller fore is needed to obtin tht vlue of /R. An lgebri reltion n be found for the fore orresponding to zero ontt rdius b ombining Eqns. () nd () nd tking the limit s the ontt zone shrinks to zero, giving F Eh 8 ln R σ 0 σ 0 σ 0 π h + π h E E E 8 ln R (4) Unless the dhesion stress (σ 0 /E) is suffiientl high, the vlue of F from Eqn. (4) orresponds to the pull-off fore. Otherwise the pull-off fore will be lrger thn tht given b Eqn. (4) due to the snp-off s disussed previousl. This eqution shows tht in the se of dhesionl ontt of linder with flt surfe, the pull-off fore strongl depends on the omposite modulus of elstiit (E) in ontrst to the JKR nd DMT theories for spheres. Fig. 6 shows the vrition of the zero ontt length fore (from Eqn. (4)) with dhesion stress for different linder rdii (R/h). 9

10 Nno-Sle Effets in Clindril Contts Sri et l. Initition of Sliding Tngentil fores n be trnsmitted b frition in ontting bodies. Consider linder in ontt with hlf-spe, whih re ompressed b norml fore F, nd ted upon b tngentil fore T. The problem is solved for the unoupled se (B0) in Eqn. (6). Thus the ontt re remins in stte of stik during the pplition of the norml lod nd the ontt re remins onstnt during the pplition of the tngentil fore. For the initition of sliding, the tngentil fore is grdull inresed. Mindlin studied the initition of mro-sle sliding of linder using Coulomb frition without dhesion [3]. He first ssumed tht stik previls everwhere during the tngentil loding phse. But when tht problem is solved, singulrit exists for the sher stress distribution t the edges x, wheres the norml stress is bounded. Hene the fritionl inequlit must be violted; so it ws dedued tht slip will our t the edges of the ontt zone. Thus, ording to Mindlin s theor there is stik zone in the entrl region nd slip zones smmetrill loted in both the leding nd triling edges. In nno-sle ontts, dhesion between the two bodies will ffet the ontt width. This effet in lindril ontts is desribed in the previous setion. In the mro-sle Coulomb frition is used, wheres t the nno-sle the frition stress in the slip zone is ssumed to be onstnt, i.e. the dhesion theor of frition s disussed previousl. Aording to the plne strin liner elstiit formultions (e.g. Brber [9]), the reltion between the reltive deformtions of the bodies in the tngentil diretion to the boundr stresses is: 0

11 Nno-Sle Effets in Clindril Contts Sri et l. () d( ux u dx () x ) A 4π px ( ξ) dξ x ξ B p 4 ( x), -<x< (5) where p x, the sher stress distribution in the x-diretion, is equl to the onstnt τ 0 in the slip zone. For mthemtil onveniene we tke p x τ 0 -τ in the entire ontt region. Prior to the initition of globl sliding motion, the horizontl shift (i.e. u () x - u () x ) of the bodies in the stik zone (-d<x<d) will be onstnt. Note tht the reltion d < holds for the limits of the stik zone. Then the eqution for tngentil deformtions beomes d τ ( ξ ) dξ τ 0 π ξ x π d ln + x, -d<x<d (6) x Note tht the ontt width is found s desribed previousl in the norml loding problem. The integrl Eqn. (6) is singulr nd the solution is bounded t the end points. The numeril method of Erdogn, Gupt nd Cook [0] is used to obtin set of lgebri equtions, whih re solved numerill. The fore blne in the horizontl diretion n be used to find the reltion between the hlf-length of the stik zone (d) nd the pplied sher fore (T). This proedure gives d τ 0 τ( ξ) dξ T (7) d whih n lso be evluted using the method in [0]. In Fig. 7 the reltion between the non-dimensionl tngentil fore (T/E) nd the non-dimensionl hlf-length of the stik zone (d/) is shown for vrious vlues of onstnt sher stress in the slip zone. As it n be lerl seen from this figure, the slip

12 Nno-Sle Effets in Clindril Contts Sri et l. zone vnishes in the limit s the pplied tngentil fore pprohes zero. As the tngentil fore rehes ritil vlue, stte of omplete slip (i.e. globl sliding) ours. Pure Sliding If the ontting linder hs reltive sliding motion with respet to the plne it is in ontt with, there need not be smmetr due to the nonliner nture of dhesion. The origin of the oordinte sstem will be hosen in the enter of the ontt region. An eentriit e will indite the vlue of x orresponding to the pex of the linder. The leding dhesion zone will be strip (<x< ) nd the triling dhesion zone will be nother strip (- <x<-) s shown in Fig. 8. The geometril reltion between the deformtions of the bodies in the -diretion inside the ontt is u () ( x e) u δ 0 + (8) R ( ) The sme elstiit formultion used for the norml ontt n be used here i.e., d u u ) A dx 4π ( p ( ξ ) dξ xξ B x e px ( x) 4 R, -<x< (9) If the stress distribution given b Eqn. (8) is substituted into Eqn. (9), it tkes the form Eπ p ( ξ) dξ ξ x x + R e R + σ 0 Eπ ln( x ) + x, -< x < (0) The geometril reltions for the seprtion of the two bodies n be used whih is in the sme form with Eqn. (0), but with different integrtion limits due to non-smmetr.

13 Nno-Sle Effets in Clindril Contts Sri et l. 3 When the linder is in stte of sted sliding, the dhesion effet in the triling edge is ssumed to be lrger thn in the leding edge. This ssumption is onsidered vlid beuse dhesion between surfes is ffeted b surfe ontmintion. Sine the surfe will be prtill lened due to the sliding motion of the ontting surfes, dhesion in the triling edge is expeted to be lrger thn in the leding edge. This effet is ounted for b tking the seprtion in the triling edge (h ) lrger thn in the leding edge (h ), wheres σ 0 is ssumed unhnged. Due to the differene between h nd h the dhesion width vlues will not be the sme in the leding nd the triling edges. Sine there re two more unknowns, the seprtion equtions must be written for both the leding edge nd the triling edge. On the leding edge this gives + 0 () () ln ) ( ) ( )ln ( ) ( x x E h R e e d p E u u π σ ξ ξ ξ ξ π () nd the triling edge reltion is () () ln ) ( ) ( )ln ( ) ( x x E h R e e d p E u u π σ ξ ξ ξ ξ π () The Erdogn, Gupt nd Cook reltions [0] re used nd Eqns. (0-) re solved numerill. In Figs. 9 nd 0 the vrition of the differene of the dhesion widths between the triling nd leding edges s funtion of /R re given for mximum dhesion seprtion rtios (h /h ) for given linder rdius (R/h 00) nd dhesion stress (σ 0 /E0.0 in Fig. 9 nd σ 0 /E0.04 in Fig. 0). As the h /h rtio inreses, the differene between the dhesion widths of the leding nd the triling edges lso inreses due to smmetri loding. For h h the results re smmetri.

14 Nno-Sle Effets in Clindril Contts Sri et l. The pplied norml fore n be found from the fore equilibrium in the - diretion. Due to the smmetr with h /h, resultnt moment will t on the upper bod due to the smmetri norml stress distribution. If moment equilibrium is written with respet to the enter of ontt, the resultnt moment (lokwise diretion tken to be positive) is found s shown in Figs. nd. As the mximum dhesion seprtion rtio inreses or s the dhesion stress inreses, the resultnt moment lso inreses s shown lerl in these figures. Rolling The problem of sted stte rolling of n elsti linder on n elsti hlf-spe (or equivlentl one linder rolling on nother) with Coulomb frition ws solved b Crter []. Aording to Crter s solution the leding edge of the ontt zone (d<x<) is in stte of stik nd the triling edge (-<x<d) is in stte of slip. In Crter s pproh, x x Vt is oordinte moving to the right with the speed of the ontt region, nd x represents the sttionr oordinte sstem. As in the pplition to loomotive wheel [], the upper bod is the driving linder. During this rolling motion, the liner veloit is not extl equl to the ωr, where ω is the ngulr veloit. Similrl, for two rollers, the ngulr veloities of the rollers will not be inversel proportionl to their rdii. Due to slip, the driving roller veloit will hve greter mgnitude thn the veloit of the driven roller. The reep veloit represents this veloit differene. 4

15 Nno-Sle Effets in Clindril Contts Sri et l. At the nno-sle the sher stress (frition stress) in the slip zone is ssumed to be onstnt, s previousl disussed. As with sliding, dhesion ffets the reltion between the norml fore nd ontt width. s The tngentil reltive displement (shift) between the bodies n be expressed () () s( x, t) u ( x,0, t) u ( x,0, t) C( t) (3) x x + where C represents the rigid bod motion of the upper bod reltive to the lower bod. In the stik zone the time derivtive of the shift in the moving oordinte sstem is zero. The stik ondition n be written s d s ( x, t) V ( u u ) + C x x 0 (4) dx where Ċ (t) is the onstnt rigid bod slip (or reep) veloit. From liner elstiit, the reltion for the deformtions of the bodies in the tngentil diretion n be used (Eqn. (5)). The sher distribution is now written s x d τ ( x) τ 0 + τ ( x), d<x< (5) d in the stik zone. It hs vlue of t 0 t xd, vlue of zero t x. Furthermore τ( x ) τ, -<x<d 0 in the slip zone, nd so the unknown funtion τ (x) is zero t the end points of the stik zone. In the stik zone the time derivtive of the shift is zero. If Eqn. (5) is substituted into Eqn. (4) for the unoupled se (B0) 5

16 Nno-Sle Effets in Clindril Contts Sri et l. A V 4π is obtined, whih beomes τ ( x) dξ + C x ξ 0, d<x< (6) VA ( ) τ ( x) VA x d x dξ + C τ ln x ln ( x d ) ln ( + x) 4 x 4 0, d<x< π ξ π d d d The integrl eqution is gin singulr nd the solution is bounded t the end points. If fore equilibrium is written in the x-diretion, the pplied sher fore n be relted to the ontt width prmeter (d) b whih long with Eqn. (5) beomes (7) T τ ( x) dx (8) 3 + d T τ ( ξ) dξ + τ 0 (9) d Eqns. (7) nd (9) re solved numerill using the Erdogn, Gupt nd Cook reltions [0]. The problem is not smmetri so the middle eqution nnot be negleted. Insted it provides the extr eqution needed for the dditionl unknown, i.e. the reep veloit. Fig. 3 shows the vrition of the dimensionless pplied sher fore (T/E) nd Fig. 4 is the vrition of the dimensionless reep veloit, both with respet to the slip zone length prmeter (d/). As the pplied sher fore pprohes τ 0, the length of the stik zone pprohes zero. At tht point there is omplete sliding nd the reep veloit ttins its mximum mgnitude. As the pplied tngentil fore pprohes zero there is full stik zone nd the reep veloit beomes zero. The mening of negtive reep veloit is tht ωr for the driving wheel is greter thn the veloit of the ontt zone. 6

17 Nno-Sle Effets in Clindril Contts Sri et l. Also s the vlue of the onstnt sher stress in the slip region inreses, the mgnitude of the pplied sher fore lso inreses s expeted. The onstnt sher stress hs the sme effet on the mgnitude of the reep veloit s n be lerl seen from the figure. Note tht s the stik zone vnishes (d? ) the reep veloit pprohes smll frtion of the sliding veloit (V). Thus this stte orresponds to omplete sliding long with rigid bod rottion. Conlusion As the sle of ontting bodies dereses, dhesion effets beome signifint espeill for smooth surfes nd lightl loded sstems. The behvior of linder (e.g., fiber) whih dheres to substrte nd whih is subjeted to ombintion of norml nd tngentil fores is the subjet of this investigtion. In the presene of tngentil lod, the linder might slide, roll or m undergo omplex motion. This pper trets the two-dimensionl elsti ontt problem of the linder on the substrte during rolling/sliding motion nd inludes the effet of dhesion using the Mugis model. In ontrst to the spheril se, the pull-off fore for linder on flt surfe is shown to depend strongl on the ombined modulus of elstiit of the ontting bodies. Mindlin s lssi investigtion of the initition of sliding of linder on hlfplne used Coulomb frition. As the sle dereses, two effets should be inluded. First, dhesion inreses the ontt width, espeill under light norml loding. Seond, ording to the dhesion theor of frition for single rel re of ontt, the sher stress n be ssumed to be onstnt in the slip regions. These effets re inluded 7

18 Nno-Sle Effets in Clindril Contts Sri et l. for the unoupled se in whih norml/sher loding does not produe reltive tngentil/norml displements. During the initition of sliding, there is entrl stik zone surrounded b slip regions in the leding nd triling edges. As the pplied tngentil lod inreses, the lengths of the slip zones inrese, until, t ertin vlue of the tngentil fore, there is omplete slip. During sted sliding the brsive tion of the sher stress n be expeted to prtill len the surfe, resulting in different leding nd triling edge dhesive properties. This effet is inluded in the model of sted nno-sle sliding. Crter investigted the rolling of linder on substrte t the mro-sle. Similrl to the sliding se, s the dimensions pproh the nno-sle, dhesion under norml loding nd onstnt sher stress in the slip regions under tngentil loding re ssumed. The vritions of the reep veloit nd the length of the stik zone with the pplied sher fore re determined for vrious vlues of the frition stress. As the trtion fore inreses, the stik zone length dereses nd the reep veloit inreses eventull leding to pure slip with rottion. Future work should inlude the nlsis of the trnsition from sliding to rolling nd pplitions to tehnologil problems. It should lso ddress the nlsis of the three-dimensionl problem (spheril se) for sliding nd rolling motions. 8

19 Nno-Sle Effets in Clindril Contts Sri et l. Referenes. D. S. Rimi nd D. J. Quesnel, 00, Fundmentls of Prtile Adhesion, Polmer Surfes nd Interfes Series, Globl Press.. R. S. Brdle, 93, The ohesive fore between solid surfes nd the surfe energ of solids Phil. Mg., 3, pp K. L. Johnson, K. Kendll nd A. D. Roberts, 97, Surfe energ nd the ontt of elsti solids, Pro. Rol So. London Ser. A, 34, pp B. V. Derjguin, V. M. Muller nd Y. P. Toporov, 975, Effet of ontt deformtions on the dhesion of prtiles, J. Coll. nd Interfe Si., 67, pp D. Tbor, 976, Surfe fores nd surfe intertions, J. Coll. nd Interfe Si., 58, pp Greenwood, J.A., 997, Adhesion of Elsti Spheres, Pro. R. So. Lond.A, 453, pp D. Mugis, 99, Adhesion of spheres: the JKR-DMT trnsition using Dugdle model J. Coll. nd Interfe Si., 50, pp K. L. Johnson, 985, Contt Mehnis, Cmbridge Universit Press. 9. J. R. Brber, 99, Elstiit, Kluwer Ademi Publishers. 0. F. Erdogn, G. D. Gupt nd T. S. Cook, 979, Numeril Solutions of Singulr Integrl Equtions, in Mehnis of Frture, edited b G.G. Shih, Vol., Noordhoff Interntionl, Leden. 9

20 Nno-Sle Effets in Clindril Contts Sri et l.. O. T. Sri, August 003, Mster of Siene Thesis, Mehnil Engineering Dept., Northestern Universit, Boston.. F. W. Crter, 96, On the tion of loomotive driving wheel, Pro. Ro. So. (London), A, pp R. D. Mindlin, 949, Compline of elsti bodies in ontt, J. Appl. Meh., 7, pp List of Figures Figure. Contt width () nd the dhesion width () for the ontt of linder with hlf-spe. Figure. The vrition of the dimensionless pplied norml lod (F/Eh) with the dimensionless ontt hlf-width (/R) for dimensionless linder rdii R/h00. Figure 3. The vrition of the dimensionless pplied norml lod (F/Eh) with the dimensionless ontt hlf-width (/R) for dimensionless linder rdii R/h00. Figure 4. The vrition of the dimensionless dhesion hlf-width (/R) with the dimensionless ontt hlf-width (/R) for dimensionless linder rdii R/h00. Figure 5. The vrition of the dimensionless dhesion hlf-width (/R) with the dimensionless ontt hlf-width (/R) for dimensionless linder rdii R/h00. Figure 6. The vrition of the dimensionless zero ontt length fore (F/Eh) with the dimensionless dhesion stress (s 0 /E). Figure 7. The vrition of the dimensionless tngentil fore (T/E) with the nondimensionl hlf-length of the stik zone (d/) inside the ontt region during the initition of sliding. 0

21 Nno-Sle Effets in Clindril Contts Sri et l. Figure 8. Contt width () nd the dhesion widths in triling nd leding edges for the ontt of linder with hlf-spe in the presene of sliding motion. Figure 9. The differene between the dimensionless dhesion hlf-widths of triling nd leding edges vs. dimensionless ontt hlf-width (/R) for R/h 00 nd σ 0 /E0.0. Figure 0. The differene between the dimensionless dhesion hlf-widths of triling nd leding edges vs. dimensionless ontt hlf-width (/R) for R/h 00 nd σ 0 /E0.04. Figure. The dimensionless resultnt moment (M/ER ) vs. dimensionless ontt hlf-width (/R) for R/h 00 nd σ 0 /E0.0. Figure. The dimensionless resultnt moment (M/ER ) vs. dimensionless ontt hlf-width (/R) for R/h 00 nd σ 0 /E0.04. Figure 3. The vrition of the dimensionless stik zone prmeter (d/) with dimensionless tngentil fore pplied (T/E) during rolling motion. Figure 4. The vrition of the dimensionless reep veloit (Ċ /V) with the dimensionless stik zone prmeter (d/) during rolling motion.

22 Nno-Sle Effets in Clindril Contts Sri et l. Appendix A The Erdogn, Gupt nd Cook reltions [0] n be used to obtin solution of singulr integrl eqution. Consider π P( t) dt t x f ( x) (A) in whih solution bounded t both ends, is to be found. The solution is P ( ξ ) w( ξ ) q( ξ ), w ( ξ ) ξ (A) where g(?) is smooth funtion whih is bounded nd non-zero t the end points. The singulr integrl eqution (A) n be pproximted b n i ti g( ti) n + t i x k f ( x k ) (A3) where t i iπ os, i,,..,n (A4) n + re the qudrture points, nd x k π k os, k,,..,n+ (A5) n + re the ollotion points. In this mnner the singulr integrl eqution (A) is repled with n+ liner lgebri equtions. For the smmetri problems, the middle eqution (orresponding to x k 0 for n even) is ignored [0].