Surface profiles with zero and finite adhesion force and adhesion instabilities

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1 Surfce profiles with zero d fiite dhesio force d dhesio istbilities Vleti L. Popov Techische Uiversität Berli, Str. des 7. Jui 35, 063 Berli, Germy Abstrct. A simple but geerl lysis of the stbility of xis-symmetric dhesive cotcts is provided. Adhesio is cosidered i the JKR-pproximtio. Depedig o the shpe of the cotctig bodies, vrious scerios re possible, icludig vishig dhesive force, complete cotct s well s trsitios betwee these extremes.. Itroductio Neutrl bodies re kow to ttrct ech other vi v-der-wls forces. These forces led to fiite work which is eeded to detch two surfces from ech other, which we will cll the work of dhesio. The work of dhesio per uit re of cotctig bodies is deoted s γ. The dhesive cotct problem hs bee solved i the clssicl pper by Johso, Kedll d Roberts (97) for prbolic profiles d by Kedll (97) for flt cylidricl ideter. Alysis of dhesive cotcts mostly cocetrtes o fidig the "force of dhesio" correspodig to ustble cofigurtio, fter which o further equilibrium exists d the dhesive cotct "breks dow". However, i dhesive systems other kid of istbility is possible, which till ow did ot ttrct much ttetio: ustble trsitio to the stte of complete cotct. Johso (995) hs cosidered this istbility for the cse of slightly wvy surfces. He hs show tht there exist criticl idettio depths t which the cotct becomes ustble d the dhesive cotct propgtes util the surfces come ito complete cotct. Here we preset simple geerl discussio of both kids of istbilities for xilly symmetric cotcts. The simplest solutio of the dhesive cotct for rbitrry rottiolly symmetric shpes with compct cotct re is provided by the Method of Dimesiolity Reductio [Popov ud Heß (05), (04)]. Let us shortly recpitulte the MDR solutio. We cosider frictioless dhesive cotct betwee two elstic bodies with Youg's moduli d d Poisso umbers ν d ν d differetil profile z = f() r, where r is the polr rdius i the cotct ple. The MDR procedure cosists of the followig steps: where First, the three-dimesiol profile z = f() r is replced by equivlet MDR profile x f () r. (.) 0 x r gx ( ) = x dr The elstic bodies re replced by elstic foudtio cosistig of idepedet sprigs plced with smll spcig x d hvig the orml stiffess k = x, (.) z ν ν = +. (.3) The profile gx ( ) is pressed ito the elstic foudtio with the orml force F N. The sprigs t the boudry of the cotct re detched whe their elogtio reches the criticl vlue l ( ) = π γ (.4)

2 (rule of Heß, [Heß (00)]). The theorems of the MDR stte tht the depedecies of the three qutities ( FN, d, ) (orml force, idettio depth d cotct rdius) i the equilibrium stte reproduce exctly the solutio of the origil three-dimesiol problem. From the described procedure it is esy to see tht the detchmet criterio for the outer sprigs i the MDR model reds The orml force is determied by the equtio. Discussio of qutio (.5) d = g( ) l( ) (.5) d 0 ( ) FN = d g x x. (.6) qutios (.5) d (.6) solve the dhesive cotct problem. qutio (.5) c be reorgized s If the iequlity is fulfilled, the the cotct rdius will decrese. I the opposite cse g( ) d = l( ). (.7) g( ) d > l( ) (.8) g( ) d < l( ) (.9) it will icrese. For the cse of d = 0, this is illustrted i Fig.. Let us stress tht i the followig lysis we cofie ourselves to the cse of cotrolled idettio: it is ssumed tht the chges i o-equilibrium cotct rdius do ot chge the idettio depth. The opposite cse of cotrolled force c be cosidered similrly. Fig. This figure illustrtes the q. (.7) d the iequlities (.8) d (.9). () The cse d = 0. If the curret cotct rdius is, the l > g d the rdius will icrese. If the system strts with the rdius, the l < g the rdius decreses. Thus, the poit O correspods to stte of stble equilibrium. (b) Illustrtes the cse of o-vishig either positive or egtive idettio depth. Idettio shifts the curve of g( ) either dowwrds (for positive idettio depths) or upwrds (for egtive idettio depths). The poit P is the lst oe for which there exists equilibrium stte of the system. I the geerl cse of rbitrry idettio, the bove iequlities (.8) d (.9) re illustrted i Fig. b. If the profile is ideted ito the medium, the cross-sectio poit is shifted b

3 to the right d the cotct rdius icreses. If it is pulled out of the medium, the rdius of the equilibrium stte shriks. However, equilibrium oly exists for seprtios smller th the criticl seprtio d c. The poit P c be determied from the coditio d g ( ) d l ( ) π = =. (.0) d d Let us discuss i more detil the "geeric cse" of some itermedite seprtio for which g d l hve the form show i Fig.. the curves ( ) d ( ) Fig. A digrm for the cse of egtive but still subcriticl idettio depth. Arrows idicte the directio of chge of o-equilibrium cotct rdius for the give idettio depth. Oe c see tht the poit P correspods to ustble equilibrium d the poit P to stte of stble equilibrium. Now oe c esily see tht for rdii >, l ( ) < g ( ) d d the cotct will shrik. eq, The sme is vlid for the regio to the left of eq,. Betwee eq, d eq,, l ( ) > g ( ) dd the cotct expds. Thus, the equilibrium poit P correspods to ustble equilibrium d P to stble equilibrium. 3. No-dhesio d full cotct sttes The bove cosidertio is qulittively correct for usul profile shpes which c be vguely chrcterized s "covex profiles". Cocve profiles my hve differet properties. To this ctegory belog i prticulr shrp poited profiles hvig the shpe f ( r) = Cr, with 0 < < /. (.) The digrm { l ( ); g ( ) d} hs ow the qulittive form show i Fig. 3. For d = 0, the curves hve oly oe itersectio poit O t o-vishig cotct rdius O. This oly equilibrium poit is ustble: If the iitil rdius is smller th O, the the rdius shriks to zero d if it is lrger th O, the it expds to ifiity. This remis true for y egtive idettio depth. This mes tht the dhesio force is i this cse exctly zero. However, for smll positive idettios, there exists fiite cotct rdius which is differet from tht of the o-dhesio problem. I this sese, the whole problem still remis "dhesive" despite the vishig dhesive force. For lrge eough idettio depth, the criticl stte is 3

4 chieved (poit P ) fter which there is o further equilibrium stte d the cotct rdius expds ifiitely. The coditio for this istbility coicides with (.0). As result of the istbility, the whole shpe comes ito complete cotct. Fig. 3 Adhesio properties of shrp poited profiles. A power-lw fuctio f() r /4 r ws used for illustrtio. 4. xmple of jumpig to complete cotct for power-lw profile We cosider xilly-symmetric profile i the form of power fuctio give by (.). The MDR-trsformed shpe is, ccordig to (.), qutio (.5) hs the form ( ) = κ ( ) = κ with κ = g x f x Cx + Γ π Γ( ). (.) ( ) π d = g ( ) ( ) = κc (.3) d the istbility coditio (.0) reds κ C Resolvig this equtio with respect to gives For the orml force we get c π γ =. (.4) π γ = κc. (.5) FN( ) = d g ( x) dx = κc 8π. (.6) Substitutio of the criticl vlue (.5) for provides the criticl orml force F Nc, : F Nc, + 3 ( + ) ( ) π γ = + cκ. (.7) 4

5 This force hs to be pplied to the cotct to produce the istbility of spoteous trsitio to the stte of complete cotct. I the specil cse = /, we hve ( ) g 3/ / π /C κ/ = κ, with =,3, 4 Γ(3 / 4) / π ( ) = I the stte of icipiet cotct, d = 0, / / (.8) π I. g( ) > l ( ), if C> 0,768 (.9) / π II. g( ) < l ( ), if C< 0,768 I the first cse, the rdius decreses util it vishes. I the secod cse, it icreses util complete cotct is chieved. We ote oce gi, tht the whole lysis i this pper is vlid uder coditios of cotrolled idettio depth. 5. Discussio The coditio for dhesive istbilities c be simply treted grphiclly by presetig the g d l i the sme grph. The oly prerequisite for the pplicbility depedecies ( ) d ( ) of this procedure is the kowledge of the MDR-trsformed profile g. ( ) The coditio for the istbility is just the coditio of touchig of the curves g( ) d d l( ). Depedig o whether the touchig is from the ier side or from outer side of the depedecy l( ), this leds to jump-like icrese or decrese of the dhesive cotct rdius. We cosider umber of simple cses. Of course, more complicted cses re possible, s e.g. the cse of prbolic ideter with wviess first cosidered by Guduru (007), see Fig. 4. Fig. 4 Istbility lysis for prbolic ideter with slight wviess. Let us ssume tht iitilly there existed cotct with rdius 0 t d = 0. If we ow pull the ideter, the the first touchig of the curves will occur i the poit deoted s "first poit of istbility". At this momet, the cotct rdius jumps from to. The secod touchig occurs 5

6 whe the cotct rdius is 3 d the lst oe t the rdius 4. After this, the cotct rdius jumps to the zero, thus the cotct is broke dow. 6. Refereces Guduru, P.R.: Detchmet of rigid solid from elstic wvy surfce: Theory. Jourl of the Mechics d Physics of Solids, 55(3), (007) Heß, M.; Über die exkte Abbildug usgewählter dreidimesioler Kotkte uf Systeme mit iedrigerer räumlicher Dimesio, Disserttio der Techische Uiversität Berli, 00. Johso, K.L.; Kedll, K; Roberts, A.D.: Surfce ergy d the Cotct of lstic Solids. Proceedigs of the Royl Society of Lodo, Series A, 34, (97) Johso, K. L.: The dhesio of two elstic bodies with slightly wvy surfces. It. J. Solids Structures, 3 (3/4), (995) Kedll, K.: The dhesio d surfce eergy of elstic solids. Jourl of Physics D: Applied Physics, 4(8), (97) Popov, V.L.; Heß, M.: Method of Dimesiolity Reductio i Cotct Mechics d Frictio. Spriger, Heidelberg, ISBN (05) Popov, V.L.; Heß, M.: Method of dimesiolity reductio i cotct mechics d frictio: user's hdbook. I. Axilly-symmetric cotcts. Fct Uiversittis, series Mechicl gieerig, (), -4 (04) 6

Graphing Review Part 3: Polynomials

Graphing Review Part 3: Polynomials Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)