Analytical solution of adhesion contact for a rigid sinusoidal surface on a semi-infinite elastic body

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1 Cmputer Metds nd Experimentl Mesurements VIII 45 nlyticl slutin f de cntct fr rigid usidl surfce n semi-infinite elstic bdy R. R.. Sriwijy, K. Tksi & K. Jtmik Deprtment f Interntinl Develpment Engineering, Tky Institute f Tecnlgy, Jpn bstrct n nlyticl slutin f de cntct fr rigid usidl surfce n semi-infinite elstic bdy is presented. Te slutin fr n equilibrium cnditin f te system fr cmbintin f te wrk f Jnsn [Interntinl Jurnl f Slids nd Structures, 3(3 4), pp , 995] nd Zilbermn nd Perssn [Slid Stte Cmmunictins, 3(3 4), pp , ; Jurnl f Cemicl Pysics, 8(4), pp , 3] under zer externl pressure is btined. Te interfcil term f te ttl energy is clculted by cnsidering te curvture f te cntct re fllwing te pprc f Zilbermn nd Perssn rter tn te strigt line f te cntct re s Jnsn. Our results gree wit bt te nlyticl result f Jnsn fr sligtly wvy surfce nd te numericl results f Zilbermn nd Perssn fr lrgely wvy surfce t te limittins f teir ssumptins. Te equilibrium cntct widt is clerly expressed nd te effect f te surfce rugness is discussed. Keywrds: nlyticl slutin, equilibrium cnditin, criticl wrk f de, usidl surfce, semi-infinite elstic bdy. Intrductin Te cntct prblems f semi-infinite elstic bdy wit flt r wvy surfce ve been investigted by sme resercers. Jnsn et l. [] investigted smt cntct prblem f n elstic bdy wit sligtly wvy surfce in cntct wit rigid bdy wit flt surfce. Tey btined reltin between te pplied externl pressure nd te mplitude f rugness. WIT Trnsctins n Engineering Sciences, Vl 55, 7 WIT Press ISSN (n-line) di:.495/secm74

2 46 Cmputer Metds nd Experimentl Mesurements VIII Jnsn [] extended is wrk [] by cnsidering te de effect, nd slved it nlyticlly. Hwever, is slutin cn be pplied nly t wvy cntct wit de wit smll mplitude rugness. Zilbermn nd Perssn [3, 4] investigted n de cntct f lrgely wvy surfce nd slved it numericlly. Tey cnsidered te curvture rter tn te strigt line f te cntct re in te clcultin f interfcil term f te ttl energy. Hwever, lcl minimum s well s lcl mximum f te system cnnt be determined directly frm teir slutin. Cnsidering te limittins f te wrk f Jnsn [] nd Zilbermn nd Perssn [3, 4], te present wrk is intended t btin n nlyticl slutin fr n equilibrium cnditin f te system fr cmbintin f teir wrks under zer externl pressure. In dditin, te effect f te termdynmic wrk f de s well s te effect f te surfce rugness n te system is investigted. nlyticl metd. Pressure distributin nd displcement n te surfce semi-infinite elstic bdy wit initilly flt surfce subjected t usidl rigid surfce is cnsidered. It is ssumed tt te elstic bdy is mgeneus nd istrpic, nd te frictinless cntct presents t te interfce. Te surfce pressure distributin nd te surfce displcement f te de cntct re te resultnt f te surfce pressure distributin nd te surfce displcement f tw deless cntcts. Te first is semi-infinite elstic bdy subjected t usidl rigid surfce wile te secnd is semiinfinite elstic bdy pulled by flt rigid surfce. In fct, te secnd deless cntct cn be represented s crck prblem [5]. In te present wrk, te surfce pressure distributins nd surfce displcements f Westergrd [6] nd Kiter [7] re used. Te net surfce pressure distributin, p (x), upn te elstic bdy witin te s c cntct regin is given by [5], i.e. p( x) p ( x) + p ( x), were p s (x) is te surfce pressure distributin reltes t te usidl rigid surfce, btined by [6] s x p cs s ( ) x, () p x nd p c (x) is te surfce pressure distributin reltes t te flt rigid surfce, btined by [7] cs c c ( ), () p x p x cs WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press

3 Cmputer Metds nd Experimentl Mesurements VIII 47 s c were p is te men pressure s in [6], p is te men pressure s in [7], nd is te semi-cntct widt. In te sme mnner s te net surfce pressure distributin, p (x), te net s c men pressure is given by [], i.e. p p + p. s Jnsn et l. [] btined n expres fr te men pressure, p, in ne perid, i.e. s p ( E* ) ( ), were nd re te mplitude f rugness nd te wvelengt f usidl rigid prfile, respectively, nd E * is te plne strin mdulus f te elstic semi-infinite bdy. In te cse f rigid bdy in cntct wit n elstic bdy, te elstic mdulus, E * is given by E * E υ, were E nd υ re Yung s mdulus nd Pissn s rti f te elstic bdy, respectively. Rigid bdy Cntct re () Elstic bdy Figure : Gemetry f te cntct prblem f rigid bdy in cntct wit semi-infinite elstic bdy. Te surfce prfile f te rigid bdy is expressed by z x) cs ( x ) ( (see Fig. ). Te net surfce displcement n te elstic bdy witin te cntct regin is s c given by [3], [4], i.e. uz ( x) uz ( x) + uz ( x), were u s z (x) is te surfce displcement reltes t te usidl rigid surfce, btined by [6] ( υ ) s s p x, (3) uz cs E nd u c z (x) is te surfce displcement reltes t te flt rigid surfce, btined by [7] c c ( υ ) p uz ln. (4) E Here, u c z (x) witin cntct regin is nt zer, wic is different frm [3, 4]. WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press

4 48 Cmputer Metds nd Experimentl Mesurements VIII. Ttl energy f te present system.. Elstic term in te ttl energy Te ttl free energy f te present system cnsists f te elstic term nd te interfcil term. Te elstic term is induced by te pplied surfce pressure distributins witin te cntct regin. Te ttl pressure distributin cnsists f, ver te wle semi- Eqs. () nd (). Te ttl elstic energy term, infinite elstic bdy in ne perid is btined by Ettl E ttl p( x) uz ( x) d, (5) were te prmeters is te nminl cntct re (i.e. ). Wit Eqs. ()-(4), Eq. (5) gives E ttl s s c p p p cs + ln * 4 * E E s c c p p p - cs + ln. * * E E (6) Since we ve n externl pressure in te present system, te net men pressure is equl t zer ( p ), Eq. (6) cn be represented s E ttl E* 4. (7) 4.. Interfcil term in te ttl energy Te interfcil term, (i.e. energy cnge frm te surfce t te interfce I witin te cntct regin [8]), f te system in ne perid is determined by cnsidering te curvture f te rigid surfce, given by I s, were is te sme prmeter s in Eq. (5), nd is te termdynmic wrk f de, given by +, were nd re te surfce energies f te rigid bdy nd te elstic bdy, respectively, nd is teir interfcil energy, nd s is te surfce lengt f te cntct re. Cnsidering te curvture f te surfce rugness, s cn be expressed by WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press

5 + dx x s. (8) Te interfcil term, I, is + I dx x. (9) Since Eq. (9) cntins n elliptic integrl f te secnd kind, cnsequently it is clculted by numericl metds...3 Ttl energy f te system Te ttl energy f te system, ttl, (i.e. Gibbs free energy) in ne perid is given by I E ttl ttl +. () Substituting Eqs. (7) nd (9) int Eq. () gives te ttl energy f te system in ne perid, i.e. + ttl dx x E 4 4 *. () Eq. () cn be rerrnged t, 4 * 4 + ttl dx x - E () were is te nrmlized termdynmic wrk f de, given by ( ) / * E..3 Equilibrium f te system Te equilibrium f te system is given by minimizing te ttl energy, ttl, wit respect t te semi-cntct widt,. Terefre, te equilibrium cntct widt cn be btined by. cs * E ttl (3) 7 WIT Press WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) Cmputer Metds nd Experimentl Mesurements VIII 49

6 5 Cmputer Metds nd Experimentl Mesurements VIII Eq. (3) cn be represented by te nrmlized wrk f de, i.e. 3 cs +. (4) Tis equtin presents necessry cnditin fr equilibrium f te system. 3 Results nd discus We ve cnfirmed ur results wit te ttl energy clculted by Zilbermn nd Perssn s equtin [4] nd te equilibrium cnditin clculted by Jnsn s equtin []. It is swn tt ur results cnfrm t tse f Zilbermn nd Perssn nd Jnsn t te limittins f teir ssumptins. Figs. () nd (b) re pltted by Eq. () wit te mplitude f te rugness,.5 nd te wvelengt, 5 Å, respectively. Fig. () sws te reltin between te nrmlized ttl energy, ttl /( E * / ), nd te nrmlized cntct widt, fr nrmlized wrk f de,.. Fig. (b) sws te reltin between te nrmlized ttl energy, ttl /( E * / ), nd te nrmlized cntct widt,, fr severl nrmlized wrk f de,.6,.8,., 3., 4.6 nd 5.4. It is swn tt te nrmlized ttl energy 6 decreses s te nrmlized wrk f de increses. In Fig. (), it sws tt te nrmlized ttl energy curve s lcl minimum nd lcl mximum. Tis suggests tt wen te elstic bdy cntcts t te rug rigid bdy, te nrmlized cntct widt immeditely snp int te lcl minimum, pint. nd, wen te lcl mximum, pint B is reced, te nrmlized cntct widt immeditely snp int cmplete cntct. In Fig. (b), ec curve fr.6 -. s lcl minimum (i.e. pints - 4 ) nd lcl mximum (i.e. B -B 4 ), wile te curve fr.6 5 s rizntl inflectin (i.e. pint C). On te ter nd, te curve fr.4 6 s n rizntl inflectin, neiter lcl minimum nr lcl mximum. In te sme mnner wit Fig. (), tis suggests tt wen te elstic bdy cntcts t te rug rigid bdy, te nrmlized cntct widt immeditely increses t te lcl minimum <.6 5, wile fr.6 te nrmlized cntct widt immeditely increses t te cmplete cntct. Te sme mnner cn ls be explined fr.4. ll f te lcl minim re stble equilibrium 6 pints, weres, ll f te lcl mxim nd te rizntl inflectin re unstble pints. WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press

7 Cmputer Metds nd Experimentl Mesurements VIII () Snp int cmplete cntct energy ttl Nrmlized cntct widt ( ) Snp int prtil cntct B Hrizntl inflectin pint Curve f lcl minim B Nrmlized 3 Curve f lcl mxim B B 3 (b) 4 B 4 C Snp frm initil cntct int Figure : Te reltin between te nrmlized ttl energy nd te nrmlized cntct widt. () curve is clculted fr. nd.5, (b) curves re clculted fr. nd severl. Fig. 3 is pltted by Eq. (4) fr.5. It sws te reltin between te nrmlized wrk f de, /( E * / ), nd te nrmlized cntct widt,. Te curve f stble equilibrium pints crrespnds t te curve f lcl minim in te Fig. (b), wile te curve f unstble pints crrespnds t te curve f lcl mxim. Te criticl nrmlized wrk f de,, crit crrespnds t te rizntl inflectin pint. In te sme mnner, pints - 4, B -B 4 nd C in Fig. 3 crrespnd t pints - 4, B -B 4 nd C in Fig. (b). Te cntct widt f te stble equilibrium pints nd te unstble pints cn be btined frm te curves in Fig. 3 fr given nrmlized wrk f de. If WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press

8 5 Cmputer Metds nd Experimentl Mesurements VIII we culd give suc nrmlized cntct widt lrger tn te curve f unstble pints under zer externl pressure cnditin, te nrmlized cntct widt immeditely increses t snp int cmplete cntct..6. Curve f stble equilibrium pints Curve f unstble pints 4 C B 4 3 B 3 Nrmlized wrk f de.8 B B Figure 3: Te reltin between te nrmlized wrk f de nd te nrmlized cntct widt. Te equilibrium curve is pltted fr. 5 nd severl. Fig. 4 is pltted by Eq. (4) in te sme mnner s Fig. 3 fr severl te cse f te nrmlized mplitude f rugness is clse t zer, (i.e.. In ), te present slutin grees wit te nlyticl slutin f Jnsn [] fr sligtly wvy surfce. On te ter nd, if te nrmlized mplitude f rugness is lrge enug, te slutin grees wit te numericl slutin f Zilbermnn nd Perssn [3,4] fr lrgely wvy surfce. Te criticl wrk f de,, fr ec crit is given in Fig. 4. If vlue f te nrmlized wrk f de is lrger tn te, te nrmlized cntct widt crit immeditely increses t snp int cmplete cntct directly fter initil cntct becuse tere is n equilibrium pint witin te system. 4 Cnclus n nlyticl slutin f de cntct fr rigid usidl surfce n semi-infinite elstic bdy is presented. Te slutin fr n equilibrium cnditin f te system fr cmbintin f Jnsn s nd Zilbermn-Perssn s wrks under zer externl pressure is btined. Te interfcil term f te ttl energy is clculted by cnsidering te curvture f te cntct re in te sme wy s WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press

9 Cmputer Metds nd Experimentl Mesurements VIII 53 Zilbermn nd Perssn. Our results gree wit bt te nlyticl result f Jnsn nd te numericl results f Zilbermnn nd Perssn t te limittins f teir ssumptins. Te equilibrium cntct widt is clerly expressed nd te effect f te surfce rugness is discussed..4 Curve f Nrmlized wrk f de Figure 4: Nrmlized cntct widt ( ) Te reltin between te nrmlized wrk f de nd te nrmlized cntct widt. Te re pltted fr severl. crit cknwledgements First utr is deeply grteful t NSeed-Net/JIC fr teir finncil supprt t im during study t Tky Institute f Tecnlgy, Jpn. Te utrs tnk Prf. Sigeki Sit fr is vluble cmments, Dr. Silviu Zilbermn fr is numericl surce cde nd Mr. Hemtvy Psmpne fr is elp during prepring tis mnuscript. References [] Jnsn, K.L., Greenwd, J.. & Higginsn, J.G., Te cntct f elstic regulr wvy surfces, Interntinl Jurnl f Mecnicl Sciences, 7(6), pp , 985. WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press

10 54 Cmputer Metds nd Experimentl Mesurements VIII [] Jnsn, K.L., Te de f tw elstic bdies wit sligtly wvy surfces, Interntinl Jurnl f Slids nd Structures, 3(3-4), pp , 995. [3] Zilbermn, S. & Perssn, B.N.J., de between elstic bdies wit rug surfces, Slid Stte Cmmunictins, 3(3-4), pp ,. [4] Zilbermn, S. & Perssn, B.N.J., Nn de f elstic bdies: Rugness nd temperture effects, Jurnl f Cemicl Pysics, 8(4), pp , 3. [5] Kendl, K., n de prdx, Jurnl f de, 5, pp , 973. [6] Westergrd, H.M., Bering pressure nd crcks, Jurnl f pplied Mecnics, 6(), pp , 939. [7] Kiter, W.T., n infinite rw f clliner crcks in n infinite elstic seet, Ingenieur-rciv, 8(7), pp. 68-7, 959. [8] Tksi, K., Mizun, R. & Onzw, T., Influence f te stiffness f te mesurement system n te elstic del cntct, Jurnl de Science nd Tecnlgy, 9(), pp , 995. WIT Trnsctins n Engineering Sciences, Vl 55, ISSN (n-line) 7 WIT Press