Micro-scale adhesive contact of a spherical rigid punch on a. piezoelectric half-space

Transcription

1 Mico-scle dhesive contct of spheicl igid punch on piezoelectic hlf-spce Z.R. Chen, S.W. Yu * Deptment of Engineeing Mechnics, Tsinghu Univesity, Beijing 84, P.R. Chin Abstct The mico-scle dhesive contct behvio of spheicl igid punch on piezoelectic hlf-spce is investigted in this ppe. The govening equtions descibing the dhesive contct behvio e obtined by using the Mugis-Dugdle (M-D) elstic dhesive contct theoy [J. Colloid Intefce Sci. 5 (99) 43]. The solutions fo indenttion of piezoelectic mteils nd fo elstic dhesive contct cn be educed fom the pesent esults in the bsence of dhesion effect nd piezoelectic effect, espectively. The effect of the electic lod on the dhesive contct behvio is nlyzed. It is shown tht the piezoelectic effect hs significnt effect on the dhesive contct behvio of piezoelectic mteils. The obtined esults will be helpful to undestnd the mico-indenttion popeties of piezoelectic mteils. Keywods: Adhesion; Mico-contct; Piezoelectic mteils; Sufce effect; Size effect; couple; indenttion * Coesponding utho. Tel.: , fx: E-mil ddess: yusw@mil.tsinghu.edu.cn (S.W. Yu).

2 Intoduction Piezoelectic mteils hve found incesingly widen pplictions in the Mico-Electo-Mechnicl Systems (MEMS) due to thei intinsic electo-mechnicl coupling effect. With the decesing of the chcteistic dimensions of mico mechnicl ssemblies to the scle of micons, the components scled with the sque of the chcteistic dimension, such s the sufce foce nd sufce enegy, become incesingly impotnt o even dominnt in some cses, comping with those scled with the cubic of the chcteistic dimension, such s body foce. The sufce nd intefce effects hve plyed n impotnt ole in mico-contct nd mico-tibology. Due to the combintion of the electo-mechnicl coupling nd sufce effect, the contct behvio of piezoelectic mteils becomes much moe complicted. The indenttion technique, which is used in mteils chcteiztion, hs lso been used to mesue the piezoelectic popeties, to ssess the mechnicl nd electic stength of piezoelectic mteils. The indenttion of piezoelectic mteils hs been investigted theoeticlly o expeimentlly by mny eseches. Mtysik (985) investigted the poblem of pessing igid conducting punch into piezoelecto-elstic hlf spce. Using the sme method, Ginnkopoulos nd Suesh (999) pesented genel theoy fo the xisymmetic indenttion of piezoelectic solids within the context of fully coupled, tnsvesely isotopic models. Fn, Sze nd Yng (996) investigted the stess nd electicl field distibutions in piezoelectic hlf-plne unde contct lod t the sufce by using Stoh's fomlism. Wng nd Zheng (995), Ding, Hou nd Guo () obtined the genel solution fo the

3 indenttion of hlf-spce piezoelectic using the potentil function method. Some expeimentl studies on the popeties of piezoelectic mteils using indenttion technique hve been epoted (see, e.g. Sigl et l., 999; Rmmuty et l., 999; Sidh et l., ). All these investigtions e concened with the mco behvio of piezoelectic mteils. With the intoduction of the Atomic Foce Micoscope (AFM) by Binnig et l. (986), the mico-indenttion nd nno-indenttion bsed on the AFM technique hve been exploed to investigte the physicl popeties of mteils in mico-scle o nno-scle. Piezoelectic mteils e used in some contct-pone ppliction, wheein indenttion nlysis would povide the bsic foundtions fo developing n undestnding of the mechnics of dhesive contct. Chistmn et l. (998) hve mesued the mgnitude of the effective longitudinl piezoelectic constnt of thin films using n AFM, nd pointed out tht significnt chllenge of AFM piezoelectic mesuements is tht the tip motion cn be due to combintion of piezoelecticity, electostiction, nd electosttic intections between the tip nd electic field pesent. In this ppe, the dhesive contct of piezoelectic mteils is investigted using the Mugis-Dugdle elstic dhesive contct theoy. Consideing the complexity of this poblem, only the dhesive foce nd piezoelecticity e tken into ccount.. Bsic Equtions of Axisymmetic Piezoelectic Body In the bsence of body nd ineti foces, the equilibium equtions fo the xisymmetic poblems of piezoelectic body cn be expessed in the cylindicl 3

4 coodintes {,, z} θ s σ τ z σ σ + + θθ = z, () τz σ zz τz + + = z nd the Mxwell electosttic eqution, in the bsence of volume electic chges, is D D Dz + + =, () whee σ, τ nd D e the noml stess, she stess, nd electic displcement, espectively. The constitutive equtions of line, tnsvesely isotopic piezoelecticity e σ c c c3 e3 ε σ c c c3 e θθ 3 ε θθ σ zz c3 c3 c33 e 33 ε zz =, (3) τ z c44 e5 γ z D e5 ε E D z e3 e3 e33 ε33 Ez whee c ij, e ij nd ε ij e elstic, piezoelectic nd dielectic constnts, espectively. The poling diection is ssumed to be long the z-diection. The stins nd electic fields e given by the geometic equtions nd the Guss equtions, espectively, s u u w ε =, ε θθ =, ε zz =, z φ φ E =, E z =, z u w γ z = +, z (4) whee u nd w e displcements in nd z diections, espectively, nd φ is electic potentil. Substituting equtions (3) nd (4) into equtions () nd (), we obtin the govening equtions in tems of the pincipl quntities { uwφ,, } : 4

5 u u u u w φ c + c ( c3 + c44 ) + ( e5 + e3) =, z z z u z w w w z φ φ φ z = ( c + c ) + c + + c + e + + e u z w w w z φ φ φ ε 33. z = ( e + e ) + e + + e + ε (5) The Hnkel integl tnsfom pi is defined s ( ξ, ) = (, ) ( ξ ) F z F z J k d, (6) F z F ξ z ξj ξ dξ (, ) = (, ) k, whee J k is the fist kind Bessel s function of ode k. Appling the Hnkel tnsfom ( ξ ) on the fist, ( ξ ) J d two of Eq. (5), espectively, we obtin whee c3 c44 u c33w c44 w e 33 e5 e5 e3 u e33w e5 w 33 J d on the othe c u c ξ u c + c ξw e + e ξφ = + ξ + ξ + φ ξ φ =, (7) + ξ + ξ ε φ + ε ξ φ = ( ξ, ) (, ) ( ξ ) u z = u z J d ( ξ, ) (, ) ( ξ ) w z = w z J d (, ) (, ) φ ξ z = φ z J ξ d = z Let us wite the solution of Eq. (7) fo semi-infinite spce ( z ) which stisfies (8) the condition emission t infinity, i.e. u, w, φ, nd hence s z + +, s 5

6 ( ξ, ) = ˆ ( ξ) ( ξ, ) = ˆ ( ξ) (, z) = ˆ φ( ξ) u z u e w z w e φ ξ kξ z e kξ z kξ z. (9) Hee k e oots with positive el pt fo the chcteistic eqution whee c44k c ( i j ) det =, =,,3, () =, = = ( c + c ) k, 3 44 ij = = e + e k, = c k c, = = e k e, = ε ε33k. The chcteistic eqution (), bi-cubic eqution with el coefficients, in genel, hs two el oots ± k nd fou complex conjugte oots δ iω Define the following pmetes ± ± k, δω>,. α =, β = ( + ), γ = +. Then, the solutions of Eq. (7) cn be epesented s with u α α + iα ( δ+ iω) ξz w A e i A ia e, () kξ z β ( ξ) Re β β = + + ( ξ) + 3( ξ) φ γ γ+ iγ α = α( k ), β = β ( k ), γ γ ( k ) =, + i = ( + i ), β + iβ = β ( δ + iω), γ iγ γ ( δ iω) α α α δ ω The functions A ( ξ ), A ( ξ ) nd 3 + = +. A ξ hve to be detemined fom the boundy conditions. Applying Hnkel integl tnsfom, the genel solution of the system of eqution (5) stisfying the condition emission t infinity cn be given in the fom kξ z δξ z + ( ) cos δξ z + ( α A ( ξ) + αa3 ( ξ) ) e sin ( ωξz) α A ξ e α A ξ α A ξ e ωξz u(, z) = J ( ξ) d 3 ξ, 6

7 kξ z δξ z + ( ) cos δξ z + ( β A ( ξ) + βa3 ( ξ) ) e sin ( ωξz) kξ z δξ z + ( ) cos δξ z + ( γa ( ξ) + γ A3 ( ξ) ) e sin ( ωξz) βa ξ e βa ξ βa3 ξ e ωξz w(, z) = J ( ξ) dξ, γ A ξ e γ A ξ γ A ξ e ωξz φ ( z, ) = J( ξd ) ξ 3, Substituting Eq. () into (4), nd then into (3), we find the solution t the sufce ( z = ) in the genel fom s whee (,) = α ( ξ) + α ( ξ) α ( ξ) ( ξ ) u A A A3 J d (,) = β ( ξ) + β ( ξ) β ( ξ) ( ξ ) w A A A3 J d (,) = + φ γa ξ γ A ξ γ A3 ξ J ξ d m m 3 3 (,) δ + m ω m δ m = + ω σ zz k A ξ A ξ A3 ξ ξj ξ dξ δ + ω δ + ω (,) = + σ z ma ξ ma ξ m3a3 ξ ξj ξ dξ m4 m5δ+ m6ω m6δ m5ω (,) = + Dz k A ξ A ξ A3 ξ ξj ξ dξ δ + ω δ + ω m = e γ c kα + β, 5 44 m = e γ c δα ωα + β, 5 44 m = e γ c δα + ωα + β, m m m = ε γ e kα + β, 4 5 = ε γ e δα ωα + β, 5 5 = ε γ e δα + ωα + β. 6 5 To this end we hve obtined the genel solutions fo the xisymmetic semi-infinite piezoelectic solids. The genel esults fo the xisymmetic indenttion of () (3) 7

8 piezoelectic solids hve been given by Ginnkopoulos nd Suesh (999). In the following, the solutions fo n xisymmetic extenl cck in n infinite piezoelectic body, which will be used in the nlysis of dhesive contct of piezoelectic solids, e solved. 3. An xisymmetic extenl cck in n infinite piezoelectic body Conside n xisymmetic extenl cck in n infinite piezoelectic body s shown in Fig.. The cck sufce is subjected to n xisymmetic pessue p ( ) nd chge q s. The solution fo the coesponding elstic poblem ws given by Lowengub nd Sneddon (965). As esult of the symmety, we need to conside semi-infinite spce ( z ). Applying the cck sufce nd symmety conditions, i.e., the mechnicl boundy conditions: zz ( ) w, =, σ, = p, >, (4) τ, =, nd the electic boundy conditions: z φ φ, =,, (5) D, = q, > z s nd substituting Eq. (3) into Eqs. (4) nd (5), we obtin two pis dul integl equtions 8

9 ( ξ) ( ξ ) ξ A J d = f, i ξa ξ J ξ dξ = g, > i i i ( i =, ), (6) whee M φ Mφ = MM 4 MM3 = MM 4 MM3 f, f, M8p Mq 6 s Mq 5 s M7p = MM 5 8 MM 6 7 = MM 5 8 MM 6 7 g, g. Accoding to Noble (963), the solutions of Eq. (6) cn be given s whee ξ = cos( ξ) + cos ( ξ), ( =, ) (7) A x F x dx x G x dx i i i i i x d = dx i = i ( ) x Fi x f x d. G x g x d In the cse of the cck sufce unde n unifom pessue = ( ) > p p H c c, ( qs, φ ) = = (8) whee H( x ) is Heviside step function. Substituting Eq. (8) into Eq. (7), we cn obtin M8 c c c ξ = M cos 5M8 M6M7 π ξ A p t t H t dt, (9) A p t t H t dt M7 c c c ξ = M cos 5M8 M6M7 π ξ nd σ c c π, = p tn, zz MM 8 MM min, 7 (,) c c y w = M 5M8 M6M p 7 π c dy, > y c y () whee m m M = β β, M = β β, M = γ γ, m m3 m3 3 m3 9

10 m m3δ mω m M = γ γ, M =, m δ+ m 3 ω m 3 δ+ m ω m M =, M m 4 m3 7 m4 m6δ m5ω m k δ + ω m3 5 k δ + ω m δ ω δ ω =, m 5 δ m 6 ω m 6 δ m 5 ω m M = m m, M = α α, M = α α δ ω δ ω m3 m3 9 m3 m3 The stess is not self-equilibted becuse the integtion ( ) d p ( c ) p σ zz π = π c c, cos c does not equilibium the foce p ( c ) π pplied inside the cck. This is due to the implicit ssumption of no elstic displcement t z. Keeping the dius constnt, pplying t infinity tction foce P p = c c cos c leds to flt punch displcement shown in Figue, nd intoducing in the neck Boussinesq stess distibution p c c σ, zz = cos c, π = π < P h = = MM 4 MM3 P MM 4 MM3 p c c MM 5 4 MM MM 5 4 MM 6 3 cos c, MM 4 MM3 P (,) sin w = π M5M4 M6M3 4 MM 4 MM3 p c c = M cos sin, 5M4 M6M3 π c > (,) = (,) w h w MM 4 MM3 p c c = M cos cos, 5M4 M6M3 π c > () Adding the two stess distibutions nd defomtions given by Eqs. () nd (), we obtin σ w p ( ) c, = + cos tn T c c zz π c T = K m p c tn, π π m π (,) cos = + MM 4 MM3 K p c MM 8 MM7 M5M4 M6M3 π M5M8 M6M7 c c dy, > p min ( c, ) y π y c y ()

11 p whee c cos K = c +. m π c 4. Adhesive contct of piezoelectic hlf-spce unde spheicl punch Accoding to Johnson et l. (97), the dhesive contct cn be solved in the following wy. As shown in Fig. 3, sttes A ( P,, h ) nd B (,, ) P h coespond to the hlf-spce unde the ction of lods P nd P without the dhesion effect, espectively. Stte C (,, ) P h coesponds to the hlf-spce unde ction of P with dhesion effect. We cn lod to stte B fist, then keeping the contct dius constnt, nd unlod to P which equilibtes the dhesive foce. So the poblem of the dhesive contct of piezoelectic body unde spheicl punch cn be decomposed into the following thee poblems: Hetz piezoelectic contct, Boussinesq piezoelectic contct nd extenl cck in n infinite piezoelectic body (see Fig. 4). The esults fo the bove-mentioned thee poblems e: () Hetz piezoelectic contct 3K σ, =, < zz π R P =, 3 K R h = + MM 6 MM5 R M5M4 M6M3 φ, (,) w h R, = ( h ) sin π R ( ) πr ( ) ( ( ) ) sin + + ( ), >

12 (,) = + (,) w f h w π MM 6 MM5 = + cos cos, > R R M5M4 M6M3 π π φ (3) 8 M 4M5 M3M6 whee K =. 3 MM4 M3M (b) Boussinesq piezoelectic contct ( φ = ) σ zz P, =, π h = MM 4 MM 3 P, M 4M5 M3M6 4 (,) w h, = hsin ( π ), > (4) (c) Axisymmetic extenl cck in n infinite piezoelectic body Accoding to Mugis-Dugdle (M-D) elstic dhesive contct theoy, the dhesive foce is ssumed to distibute unifomly long the dhesive zone ( c). Let p = σ in Eq. (), we obtin the esults duo to the ction of the dhesive foce σ σ ( ) c, = + cos tn c c zz π c = + K m σ c tn, π π m π σ (,) cos w = MM 4 MM3 K + c MM 8 MM7 M5M4 M6M3 π M5M8 M6M7 σ min ( c, ) y π y c y c c dy, > (5) Then we cn obtin the JKR esult fo piezoelectic dhesive contct:

13 σ P P K (,) = JKR 3 zz π πr K 3K π π R =, < 4 3 = π + + JKR MM M M, π M5M4 M6M3 sin πr sin w K + φ sin, > MM 6 MM5 π M5M4 M6M3 h = + φ + φ JKR MM 6 MM5 MM 4 MM3 P P MM 6 MM5 R M5M4 M6M3 M5M4 M6M3 4 M5M4 M6M3 = + + ( MM M M ) φ P 6 5 3R 3 K M, 5M4 M6M3 MM 4 MM3 K π (,) cos M cos 5M4 M6M3 π πr πr MM 6 MM5 φ cos, > JKR = + + w π M5M4 M6M3 3 8 M4M5 M3M6 whee K ( 3R M M M M P) =. π 4 3 (6) Note the stess given by Eq. (6) is singul t 3 =, which is inconsistent with the physicl fct. Consideing the effect of the dhesive zone, we cn obtin the M-D esults fo piezoelectic dhesive contct σ ( ) M-D K 3 K K m σ c zz R π π π π, = + + tn, < Let K = Km, the stess intensity fcto due to the intenl dhesive foce cncels the stess intensity fcto due to the extenl pplied lod so tht the singulities disppe. Thus, we obtin σ 3 M-D K σ c zz πr π, = + tn, < 4 3 = π π + π + MM 6 MM5 MM 4 MM3 + φ sin K πcos w, K sin sin M-D MM M M MM MM R π MM 5 4 MM 6 3 π MM 5 4 MM 6 3 m MM min (, ) 8 MM7 σ c y MM 5 8 MM 6 7 π y c y c c dy + c cos, > MM4 MM3 σ MM 5 4 MM 6 3 π (7)

14 MM 4 MM3 σ (,) cos cos w = + + c M-D πr πr M5M4 M6M3 π MM 8 MM7 σ min c, y M5M8 M6M7 π y c y c c dy + φ cos, > MM 6 MM5 π M5M4 M6M3 h = c + φ, M-D 8σ MM6 MM5 R 3K M5M4 M6M3 δ =, M-D t w c, whee (8) δ t is the cck opening displcement (COD), i.e., the cck opening fo = c M-D t the end of the cohesive zone. When, the stess σ zz (,) = σ, thus the continuity of stess is ensued. Accoding to the line elstic fctue mechnics, the enegy elese te in Dugdle model is simply G = σ δ (9) t Let the dhesion wok W equl to the enegy elese te G, we hve W = σ δ t σ 4 3 cos M M M M πr m M cos 5M4 M6M3 m σ π W = m + m + m, σ ( m ) φ cos MM8 MM7 MM6 MM5 MM 5 8 MM 6 7 π MM 5 4 MM 6 3 m whee m = c. Intoducing the following dimensionless pmetes πwr P π W R πwk ( K ) πwr ( ) λ σ K ( R ) M M 3 ( M4M5 M3M6)( MM8 MM7) ( W KR), k =, P =, h = h, =, φ = φ π = MM 6 5 MM 4 MM3 M4M M3M M5M8 M6M7 then we obtin the esults fo piezoelectic dhesive contct in the dimensionless fom (3) 4

15 3 λ ( m ) P = m + m cos, 4 8 = 3λ + 3φ h m ( 4λ ) ( m m) λ m, m + m cos + m cos k m λφ cos =, Let λ = in Eq. (3), we obtin the dimensionless fom fo Hetz piezoelectic contct (3) P = 3 ( ) h = + φ. 8 3, (3) Let φ = nd k = in Eq. (3), which coesponds to the isotopic cse, we cn obtin the M-D elstic dhesion contct esults. Fig. 5 though Fig. 8 show the vition cuves of pmetes. It is shown tht the dhesive contct behvio becomes moe complicted due to the coupling between the piezoelectic effect nd the dhesion effect. Fig. 5 shows the effects of λ nd φ on the c cuve. It is shown tht s λ deceses, the effect of sufce enegy nd the cohesive zone incese, hence the dhesion effect becomes stonge. These cuves in Fig. 5 clely show the size effect. Fo given λ, the effect due to the pplied electic lod φ shows esemblnce to tht of λ. Fig. 6 though Fig. 8 show the vitions of cuves h, P, nd P h with the pplied electic lod φ fo vious λ. The mximum pull-off foce nd the contct dius coesponding to the zeo pplied lod, nd thei vitions with φ, cn be diectly obtined fom the cuves of P h, nd P, espectively. As shown in Fig. 8, the dheence t fixed lod coesponding to hoizontl tngent nd the dheence t fixed gips to 5

16 veticl tngent, if they exit, nd seption occus with sudden jump. Othewise seption occus t =. This behvio becomes much moe complicted due to the coupling between the dhesive effect nd the piezoelectic effect. 5. Conclusion The dhesive contct of semi-infinite piezoelectic body unde the spheicl conducting punch is investigted by using the M-D elstic dhesive contct theoy. As peliminy, n xisymmetic extenl cck in n infinite piezoelectic spce is solved using the integl tnsfom method with the id of dul integl eqution technique. Then the dimensionless fomul fo the dhesive contct of piezoelectic mteils is deived. The genel esults fo Hetz piezoelectic contct nd M-D elstic dhesive contct cn be deduced fom the pesent esults by neglecting the dhesive effect nd piezoelectic effect, espectively. The effect of the pplied electic lod on the dhesive contct behvio is investigted in detil. It is shown tht the dhesive contct behvio becomes much moe complicted due to the coupling between the dhesive effect nd the piezoelectic effective. The obtined esults will be vey helpful to explin the elted expeiments on piezoelectic mteils by AFM. Acknowledgements The uthos gteful cknowledge the suppot by Ntionl Ntul Science Foundtion of Chin unde gnt No. 7 nd No. 95, nd the suppot by Sino-Gemn Cente unde gnt GZ5/. 6

17 Refeences Binnig, G., Qute, C.F., Bebe, Ch., 986. Atomic foce micoscope. Phys. Rev. Lett. 56 (9), Chistmn, J.A., Woolcott, R.R., Kingon, J.A.I., Nemnich, R.J., 998. Piezoelectic mesuements with tomic foce micoscopy. Appl. Phys. Lett. 73 (6), Ding, H.J., Hou, P.F., Gou, F.L.,. The elstic nd electic fields fo thee-dimensionl contct fo tnsvesely isotopic piezoelectic mteils. Int. J. Solids Stuct. 37 (3), Fn, H., Sze, K.Y., Yng, W., 996. Two-dimensionl contct on piezoelectic hlf-spce. Int. J. Solids Stuct. 33 (9), Ginnkopoulos, A.E., Suesh, S., 999. Theoy of indention of piezoelectic mteils. Act Mte. 47 (7), Johnson, K.L., Kendll, K., Robets, A.D., 97. Sufce enegy nd the contct of elstic solids. Poc. Roy. Soc. Lond. A34 (558), Lowengub, M. Sneddon, I.N., 965. The distibution of stess in the vicinity of n 7

18 extenl cck in n infinite elstic solid. Int. J. Eng. Sci. 3, Mtysik, S., 985. Axisymmetic poblem of punch pessing into piezoelectoelstic hlfspce. B. Pol. Acd. Sci. Tech. 33 (-), Mugis, D., 99. Adhesion of sphees: the JKR-DMT tnsition using Dugdle model. J. Colloid Intef. Sci. 5 (), Rmmuty, U., Sidh, S., Ginnkopoulos, A.E., Suesh, S., 999. An expeimentl study of spheicl indenttion on piezoelectic mteils. Act Mte. 47 (8), Sigl, A., Ginnkopoulos, A.E., Pettemnn, H.E., Suesh, S., 999. Electic esponse duing indenttion of piezoelectic cemic-polyme composite. J. Appl. Phys. 86 (), Sidh, S. Ginnkopoulos, A.E., Suesh, S.,. Mechnicl nd electicl esponse of piezoelectic solids to conicl indenttion. J. Appl. Phys. 87 (), Wng, Z.K., Zheng, B.L., 995. The genel solution of thee-dimensionl poblem in piezoelectic medi. Int. J. Solids Stuct. 3 (),

19 Cptions of Figues Fig.. An xisymmetic extenl cck in n infinite piezoelectic body. Fig.. Defomtion unde P. Fig. 3. Illusttion of dhesive contct (fte Johnson et l., 97). Fig. 4. Decomposition of the poblem of dhesion contct of piezoelectic body: () Hetz piezoelectic contct, (b) Boussinesq piezoelectic contct, (c) An xisymmetic extenl cck in n infinite piezoelectic body. Fig. 5. Vition of c with the nomlized dius of contct. Fig. 6. Reltion between contct dius nd penettion h. Fig. 7. Vition of dius of contct with lod P. Fig. 8. Reltion between lod P nd penettion h. 9

20 o z Fig.. An xisymmetic extenl cck in n infinite piezoelectic body. P h w w Fig.. Defomtion unde P.

21 P P P P B P A C h h h h Fig. 3. Illusttion of dhesive contct (fte Johnson et l., 97).

22 () P φ (b) P (c) p ( ) Fig. 4. Decomposition of the poblem of dhesion contct of piezoelectic body: () Hetz piezoelectic contct, (b) Boussinesq piezoelectic contct, (c) An xisymmetic extenl cck in n infinite piezoelectic body.

23 λ =. λ =.3 λ =.5 λ =. c/ * ( φ = ) φ * =. φ * =.6 φ * =. φ * = -. c/ * ( λ =. ) φ = -. φ =. φ =.6 φ =. c/ * ( λ =. ) Fig. 5. Vition of c with the nomlized dius of contct. 3

24 * λ =. λ =.3 λ =.5 λ =. λ = 3. λ = 5. Non-dhesion h * ( φ = ) φ =. φ =. φ =. φ =-. φ =-. * h * ( λ =. ) * φ * =-. φ * =. φ * =.6 φ * = h * ( λ =. ) Fig. 6. Reltion between contct dius nd penettion h. 4

25 * λ =. λ =.3 λ =.5 λ =. λ = 3. Non-dhesion P * ( φ * = ) φ * =. φ * =. φ * =. φ * = -. φ * = -. Non-dhesion * P * ( λ =. ) * φ * =-. φ * =. φ * =.6 φ * = P * ( λ =. ) Fig. 7. Vition of dius of contct with lod P. 5

26 P * λ =. λ =.3 λ =.5 λ =. λ = 3. λ = 5. Non-dhesion h * ( φ * = ) - P * 4 φ =. φ =. φ =. φ =-. φ = h * ( λ =. ) - 4 P * h * ( λ =. ) φ * =-. φ * =. φ * =.6 φ * =. Fig. 8. Reltion between lod P nd penettion h. 6