On adhesion energy estimates derived from the pressurized brittle delamination of a flexible thin film

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1 Ac Meili 55 (7) On desion enegy esimes deived fom e pessuized bile delminion of flexible in film A.P.S. Selvdui Depmen of Civil Engineeing nd Applied Mecnics, McGill Univesiy, 7 Sebooke See Wes, Monel, QC, Cnd H3A K6 Received 9 Jnuy 7; eceived in evised fom 6 Apil 7; cceped 7 Apil 7 Avilble online June 7 Absc Tis ppe exmines e poblem of e developmen of cicul delminion due o pessuizion e inefce of flexible elsic film bonded o n elsic subse. Te nlysis kes ino consideion e elsic defomions of e in film cn include pue flexue, flexue wi se defomions o lge deflecions nd e elsiciy of e subse. Te ppe pesens explici esuls could be used o esime eie e desion enegy of e inefce o e size of blise ceed duing e pessuizion. Ó 7 Ac Meili Inc. Publised by Elsevie Ld. All igs eseved. Keywods: Delminion sesses; Ple model of flexible coing; Elsic blise; Se defomions of in film; Lge deflecions of flexible coings. Inoducion Te sudy of e mecnics of delminion of flexible lye bonded o n elsic subse s pplicions o viey of poblem es in meils science nd engineeing [ 4]. Bonded flexible coings e ofen used o ennce e lod cying cpciy of n oewise sofe subse. In is egd, e inegiy of e bond beween e einfocing sufce lye nd e undelying subse is of ciicl imponce o e funcionl use of e composie elemen. Te ssessmen of desive seng, inefcil fcue enegy o desion enegy beween einfocing sufce lye nd e undelying subse s been invesiged by mny eseces using diffeen expeimenl ecniques nd eoeicl inepeions. Te pocedues include inefce pessuizion [,], scc esing [5,6], fou-poin bending [7], indenion [ ] nd sessed ovelyes [,]. In ecen sudies [3 5], eleponecod-sped buckles ve lso been used o deemine e desion enegies in in films. Tis ppe focuses on e modelling of e poblem of desion enegy esimion oug e use of n inefce pessuizion ecnique. Te modelling doped in mny E-mil ddess: pick.selvdui@mcgill.c of e sudies in is e focuses on e desion of wo egions consising of in elsic film nd subse, epesening e in film s flexible elsic ple expeiences eie smll sin flexue o smll deflecion buckling, wile eing e subse s eie igid bse o defomble elsic egion of semi-infinie exen. Te elemeny modelling s limiions wen i is pplied o exmining in lye delminion poblems wee se defomions nd lge deflecions of e in lye nd elsic defomions of e subse cn exe n influence on e mecnics of e delminion poblem. Tis cn led o eos in inepeion of e inefce desion enegy. Tis ppe eefoe exmines e poblem of e delminion of n inefce beween n elsic in film nd n elsic subse by inefce pessuizion nd develops eoeicl esimes fo e inefce desion enegy oug modelling ccommodes flexul nd se enegy of e in film, lge deflecions of e in film nd elsic enegy ssocied wi e defomions of e subse.. Modelling Te poblem is consideed of n elsic in film of lge exen is bonded o n elsic subse wi dep /$3. Ó 7 Ac Meili Inc. Publised by Elsevie Ld. All igs eseved. doi:6/j.cm.7.4.3

2 46 A.P.S. Selvdui / Ac Meili 55 (7) subsnilly lge n e ickness () of e film. A egion of e inefce beween e in film nd e subse is subjeced o pessuizion, cusing e developmen of sble delmined egion of dius. Te defomed configuion of e in film nd e elsic subse is sown in Fig.. In genel, e undeced egion of e inefce will exibi displcemens sisfy e ovell kinemic consins nd e globl equilibium condiions pplicble o e poblem. In is sudy, oweve, i is explicily ssumed e inefce beyond e delmined zone exibis zeo displcemens in e xil diecion nd fuemoe eie no se cions develop in e undeced zone o e dil displcemens e zeo ove e enie inefce. Te ssumpion eled o zeo se cions implies e desive enegy is lgely geneed by coninuiy of noml displcemens e inefce. Tis ssumpion cn be modified o include e developmen of se sesses only wiin e unbonded egion, bu is would enil ddiionl nlysis will no be consisen wi e simplificions consideed in e ovell nlysis of e poblem pesened ee, picully s i eles o e epesenion of e in film s sucul elemen. Te nlysis of e poblem consides e elsic enegy of e in film is consisen wi is defomion beviou, e elsic enegy of e subse deived oug e defomions occu wiin e pessuized egion nd e desion enegy of e delmined zone. Te objecive of e nlyicl modelling is o develop elionsip cn be used o pedic e dius of sble debonded zone () fo given pessue (p ), knowing e desion enegy (C), e ickness of e in film () nd e elsiciy cceisics of e lye nd e subse o, lenively, o use e elionsips o deemine e desive enegy of e inefce oug knowledge of e pplied pessue nd e dimensions of sble delminion. A Giffi-ype bile delminion cieion is doped o exmine e elsic delminion poblem. In ccodnce wi Giffi s eoy, e delminion will popge wen e decese in e elsic enegy of e sysem blnces e enegy needed o cee e new sufces of e delminion zone. In ode o deemine e sin enegy elesed s e delminion exends, e elsic enegy of e defomed in film nd e elsic enegy of e defomed subse wiin e delminion zone e exmined... Elsic enegy of e subse To esime e elsic enegy of e subse, e elsiciy poblem eled o lf spce egion (z P ) is subjeced o e following mixed boundy vlue poblem is consideed: zz ð; Þ ¼ p ; < < ðþ u z ð; Þ ¼; 6 < ðþ z ð; Þ ¼; < < ð3þ wee u z is e componen of e displcemen veco u i in e xil diecion z nd efeed o e cylindicl pol coodine sysem(,, z), nd ij is e Cucy sess efeed o e sme sysem. Te displcemen boundy condiion (Eq. ()) nd e cion boundy condiion (3) e ppoximions o e condiions pplicble o e bonded egion P beyond e delminion zone. An lenive o e cion boundy condiion (Eq. (3)) is o impose zeo displcemen boundy condiion ove e enie inefce, i.e., u ð; Þ ¼; < < ð4þ Te specificion of moe pecise se of displcemen nd cion condiions is peps unwned in view of e ppoximions will be invoked in modelling e in film s n elsic sucul elemen. Te soluion of e mixed boundy vlue poblem defined by Eqs. () (3) is sndd nd cn be found in exs on inegl equions, elsiciy nd fcue mecnics [6,7]. Te esul of inees ee is e deflecion of e sufce of e subse in e delmined zone, wic cn be evlued in e genelized fom u z ð; Þ ¼ fð mþp ð Þ = ð5þ pl wee f =, fo e boundy condiion fo e zeo se cion pescibed e inefce, defined by Eq. (3), f =(3 4m)/4( m), fo e zeo dil displcemen consin, pescibed by Eq. (4), ndm nd l e, especively, Poisson s io nd e line elsic se modulus of e subse meil. I my be noed, wen m = /, f, nd ee is no diffeence beween e esuls obined eie by imposing e cion boundy condiion (Eq. (3)) o e displcemen boundy condiion (Eq. (4)). Te elsic enegy elesed owing o e ceion of e delminion zone in e subse egion is given by U Subse ¼ 6fp 3 ð mþ ð6þ 3l.. Elsic enegy of e in film in ple model Fig.. Te geomey of e cicul delminion. Te mecnicl beviou of e in film cn be modelled using viey of eoies in sucul mecnics

3 A.P.S. Selvdui / Ac Meili 55 (7) ke ino ccoun mecnicl esponses, including flexul defomions, se defomions, membne effecs, ec. Te coice of picul eoy of sucul mecnics pplicble o in film cnno be decided e ouse. Fo exmple, wen e dimension of e delminion zone is muc smlle n e ickness of e in film, i is eniely inppopie o model e mecnics of e in film s sucul elemen; ee e mecnicl beviou of e in film s o be modelled s n elsic solid, wic exibis n xisymmeic se of ee-dimensionl defomion. In is ppe, enion is esiced o ee ypes of in films: Te fis cegoy consides in films wi delminion zones fo wic (/), nd e in film expeiences flexue due o e pplicion of e pessue p. I is fue ssumed e deflecions of e in film e pimily due o flexul deflecions e sisfied by e Poisson Kicoff Gemin in ple eoy [7,], i.e., D wðþ ¼pðÞ ð7þ wee D (=l 3 /6( m )) is e flexul igidiy of e ple, l nd m e, especively, e line elsic se modulus nd Poisson s io of e in film, is e ickness of e in film, p() is n biy pessue nd ¼ d d þ d ðþ d is Lplce s opeo fo dil symmey. Since i is ssumed e in film does no expeience ny flexue beyond e delminion zone, e elevn kinemic boundy condiions e wðþ ¼ d ¼ ¼ Fo e in film subjeced o unifom pessue p, e soluion of Eq. (7) wic sisfies e boundy condiions (9) kes e fom wðþ ¼ p 64D ð Þ ðþ Te elsic enegy ssocied wi e pue flexul defomions of e in film is given by Z U F TF ¼ pd ð wðþþ ð m pþ ðþ d d wðþ d ð9þ d ¼ p6 p 34D.3. Elsic enegy of e in film ick ple model ðþ In e second cegoy, i is ssumed e in film lso exibis deflecions due o se. An ddiionl coecion o e deflecion cn be incopoed oug eie e consideion of se sins in e in film o oug e use of ige-ode ple eoy. Te se sesses oug e ickness of e film vy in pbolic fsion. I is ssumed e mid-sufce of e flexible in film does no sec o conc, nd ee e no oug-ickness defomions in e in film. Te flexul momens M, M nd se foce Q in e in film cn be expessed in ems of e oion of plne noml o e mid-sufce, w. Te elevn expessions e M ¼ D d þ m w ; M ¼ D m d þ w ; Q ¼ dm þ ðm M Þ ðþ d Te diffeenil equions govening e oion w nd e displcemen w() e given by D d d ðwþ ¼ pðþ; d ¼ D j l ðwþ w ð3þ wee p() is e nsvese pessue. Te deflecion of e in film cn be obined fom elionsip (3), wee j is se coefficien depends on e disibuion of se sesses ove e ickness of e in film nd on Poisson s io of e meil. In e cse of e pessuized delminion poblem, e pplied pessue p() =p, nd e boundy condiions govening e oion w nd e deflecion w e wðþ ¼; wðþ ¼ ð4þ In ddiion, e displcemen nd oion fields sould be finie nd bounded in e domin (, ). I cn be sown e oion nd e displcemen ke e foms w ¼ p 6D ð Þ; wðþ¼ p ð Þ ð Þþ 64D 3j ð m Þ ð5þ In e cse of e exc soluion [], e se sess disibuion is pbolic nd e se coefficien j /3. Tis vlue cn be comped wi oe esimes obined fom vesions of ple eoies ccoun fo se defomions; fo exmple, e eoy poposed by Reissne [9] gives vlue of j 5/6 nd, fo e eoy poposed by Mindlin [], e vlue of j is deemined fom e soluion of e cceisic equion pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ð fj Þð j Þ ¼ð j Þ ð6þ wi f =( m )/( m ). I my be noed, wen <m < /,.76 < j <.9 nd n ppoxime vege esime coesponds o p /. Te elsic enegy ssocied wi e defomions of in film expeiences flexue nd se defomions is given by " # U FS TF ¼ p6 p 34D þ 4 j ð m Þ ð7þ.4. Elsic enegy of e in film lge deflecion ple model Te peceding esimes of e elsic enegy of e in film wee deived on e bsis of e ssumpion of in

4 46 A.P.S. Selvdui / Ac Meili 55 (7) ple ppoximion wee e deflecions of e in film wee smlle n e film ickness nd e coesponding sins in e in film sisfied Hooke s lw fo line elsic beviou. Anoe possible mode of defomion is, duing delminion of e inefce oug inenl pessuizion, e in film expeiences lge deflecions of sufficien mgniude o iniie secing of e in film. In e id cegoy, e flexue of e in film cn be exmined by consideing e secing displcemens u() of e mid-plne of e in film in ddiion o is flexul deflecion w(). Tee e numbe of pocedues cn be employed o obin e soluion o e poblem of e deflecion of ple exibis bo flexue nd in-plne exension due o lge deflecion effecs. Some of ese e given in Refs. [,]. Hee, n ppoxime nlysis is doped bsed on viionl pocedue. Fo xisymmeic lge deflecions of e in film, e flexul deflecions nd e secing defomions e ppoximed, especively, by elionsips wðþ ¼ C ; uðþ ¼ C þ C ðþ wee C, C nd C e biy consns. Te ssumed foms of e deflecion w() nd e secing of e midplne u() sisfy e condiion of flexul fixiy e ( ) FS ( )F boundy = nd e bsence of dil displcemens on bo e xis of symmey of e in film nd e boundy =. Te viionl pocedue cn be used o develop esul fo e enegy of e in film expeiences bo flexul deflecions nd in-plne exension consisen wi e ssumed foms of e defomions given by Eq. (). Te deils of e nlysis of e lge deflecion poblem e given in Appendix A. Tis povides e following esime fo e ol sin enegy of e in film undegoing lge deflecions involving bo flexue nd secing: p i pp 6 LD þv U TF ¼ ð9þ 34D 64D A genel esul fo e sin enegy of e delmined in film cn be developed, wic cn be wien in e fom U TF p6 p ð þ WÞ 34D wee e vlue of W cn be deduced fom e esuls in Eqs. (), (7) nd (9) pplicble o e in film exibis pue flexue, flexue wi se defomions o lge deflecions wi in-plne exension, especively. I my be noed, s e secondy effecs vnis, W! nd Eqs. (7) nd (9) educe o e clssicl esul in Eq. (). ( ) FS ( )F ðþ = ( ) FS ( )F ; = =.5 ; ( ) FS ( )F = ; =.5 = =.5 ; =.5 Fig.. Viion in e nomlized sufce enegy fo n elsic in film elsic subse sysem wi modul io l/l nd ickness io /. Tin film wi se defomion effecs.

5 A.P.S. Selvdui / Ac Meili 55 (7) Adesion enegy of e delminion Te desion enegy of e delmined sufces ceed cn be clculed by consideing e specific desion enegy Cnd e es e ceed oug e delminion pocess. (Fom n expeimenl poin of view, i my be necessy o pply e pessuizion ove smll dius b in ode o cee e delminion of lge dius.) In e genel cse, e desion enegy of e delminion is given by U S ¼ pð b ÞC ðþ Any cnge in e ol enegy of e sysem occus s esul of e developmen of delminion of dius is given by DU ¼ p6 p 34D ð þ WÞþ6fp 3 ð mþ pð b ÞC 3l ðþ In ems of e bile delminion cieion fo e inefce, e delminion will s o exend wen o ðduþ ¼ o ð3þ Tis condiion cn be used o deemine e sufce enegy ssocied wi e inefce delminion, i.e., C ¼ p 4fð mþ l þ 3ð m Þ þ X m l p l g 3 ; p l ð4þ wee X m ; p ¼ l >< 4j =gð m Þ >: fvð m Þ =4g gðp =l Þ ð5þ depending upon wee e in film exibis, especively, pue flexue, flexue wi se defomion o flexue wi in-plne exension. 3. Numeicl esuls Since e esul in Eq. (4) s n explici nlyicl fom, clculions cn be cied ou o esime e ee b ( ) LD ( ) F ( ) LD ( ) F p / =.5 p / =.5 c d ( ) LD ( ) F ( ) LD ( ) F p / =.5 p / =.5 Fig. 3. Viion in e nomlized sufce enegy fo n elsic in film elsic subse sysem wi modul io l /l, ickness io / nd nomlized sess p /l. Tin flexible film wi effecs of in-plne defomions (m = m = ).

6 464 A.P.S. Selvdui / Ac Meili 55 (7) possible vlues of e desion enegy C ssocied wi e ee models. Te esul in Eq. (4) cn be used moe effecively by exmining e vlues fo e sufce enegy esimes e influenced by se effecs nd in-plne secing in elion o e sufce enegy esimes e deived fom e cse wee e in film exibis pue flexue ccoding o e Poisson Kicoff Gemin in ple eoy. Tis indiecly esblises e condiions unde wic e simple pue flexue nlysis cn be used o esime e desion enegy fo e inefce duing debonding unde pessuizion. Two elive mesues e esblised: fo e cse wee e in film exibis flexue nd se defomions: CR FS ¼ þ pg j ½3pð m Þ þ 5fð mþg3 ðl =lþ ð6þ nd fo e cse wee e in film exibis flexue nd inplne exension: 3 63pvð m Þ p R CFL ¼ þ 3 4g ½3pð m Þ þ 5fð mþg ðl =lþ l ð7þ I sould be noed, if e elsiciy consns fo bo e in film (l, m) nd e subse (l, m) sisfy e e- modynmic consins l >, l >, 6 m 6 /, 6 m 6 /, e second em on e ig-nd side of bo Eqs.(6) nd (7) is posiive definie. Fom is i is concluded e esime fo e sufce enegy deived fom e in ple model undeesimes e coesponding esul is deived fom n nlysis kes ino ccoun se defomions nd effecs of in-plne exension.resuls will be pesened coesponding o e cse wee f =, epesening e cse wee e se cions e ideniclly zeo e inefce. (Anlogous esuls cn lso be developed fo e cse wee e ppopie vlue of f is pplicble fo e inexensible inefce.)fig. illuses e viion in CR FS fo specific combinions of m nd m, nd fo / = g (.,.5) nd (l/l) (, ). I is eviden, s g becomes lge nd (l/ l)!, e esime fo e sufce enegy deemined fom e elemeny in ple model esuls in n ppecible eo.i is lso noed e discepncy beween e wo esimes becomes negligible wen e se modul io l/l > 5. Tis is due o e limiing vlues of expession (6) s eie g! o (l/l)!. Figs.3 nd 4 illuse e coesponding esuls fo e viion in CR FL fo limiing combinions of m nd m, fo / = g (.,.5),(l/l) (, ) nd ee specific vlues of ( Γ ) LD ( Γ) F b ( Γ ) LD ( Γ) F p / μ =.5. p / μ =.5 c ( Γ ) LD ( Γ) F 5 d ( Γ ) LD ( Γ) F p / μ = p / μ =.5 Fig. 4. Viion in e nomlized sufce enegy fo n elsic in film elsic subse sysem wi modul io l/l, ickness io / nd nomlized sess p/l. Tin flexible film wi effecs of in-plne defomions (m = m =.5).

7 A.P.S. Selvdui / Ac Meili 55 (7) (Γ) (Γ) LD F μ μ (Γ) (Γ) LD F 6 4 ν = ν = ; p / μ =.5 ν = ν =.5 ; p / μ =.5 5. μ μ Fig. 5. Viion in e nomlized sufce enegy fo n elsic in film elsic subse sysem wi modul io l /l, ickness io / nd nomlized sess p /l. Tin flexible film wi effecs of in-plne defomions. (p /l ), wic vy by odes of mgniude. Te effecs of in-plne defomions domine, picully fo vlues of (i) p /l >.5, (l /l) (, ),g < 3 nd fo m =, m =, nd (ii) p /l >.5, (l /l) (, ), g < nd fo m = /m = /. Te pleu egion in Figs. 3c,d nd 4c,d indice C FL >C F. Modul ios l /l in e nge ve no dominn influence in educing e discepncy beween C FL nd C F. Te modul io l /l s o in subsnilly lge vlue fo e discepncy o disppe. Fig. 5 illuses e viion in C FL /C F fo g (.,.) nd (l /l) (, 5 ). Tese esuls indice end in e influence of l /l on e mplificion of C FL /C F nd confim e obsevion concening is limiing vlue s l /l!. As s been poined ou in e icles by Suo nd Hucinson [], Hucinson nd Suo [3] nd Hucinson nd Evns [4], s e elive vlues of e elsic consns cnge, e Dundus pmees nd b fo e flexible in film subse sysem, defined by ¼ l ð mþ lð m Þ l ð mþþlð m Þ ; b ¼ l ð mþ lð m Þ l ð mþþlð m Þ ðþ will conol e mode of delminion. Depending on ese pmees, e delminion pocess cn include mixed modes, wic will esul in e non-self simil exension of e delminion ino eie e in film o e subse, unless ee is pedisposiion fo e delminion o be esiced solely o e inefce. Te limis of e numeicl esuls pesened ee sould eefoe be inepeed wi consins simil o ose pesened in Refs. [,3]. 4. Conclusions Tis ppe pesens se of compc nlyicl esimes o e poblem of e bile elsic delminion of e inefce beween in flexible film nd subse. Te specificion of e exc eoy of elsiciy boundy condiions e edge of e delmined zone is mde compliced owing o e bonded conc beween flexible sucul elemen nd n elsic coninuum. A compuionl emen of e poblem, bsed on eie finie elemens o boundy elemens, will lso equie e use of specil elemens cn ccommode e exc compibiliy condiions e unbonded boundy. Te ppoc doped in e ppe fo e soluion of e delminion poblem is o impose p elxed boundy condiion includes egul = ffiffi sess singuliy e edge of e delmined zone long wi kinemic nd/o cion boundy condiions e bonded egions. Using suc n ppoc, n ppoxime nlysis cn be conduced o ccommode defomions of e elsic subse nd flexue of e in elsic film, including e influence of se defomions nd in-plne exension of e in film duing flexue. Te nlysis yields se of ppoxime elionsips cn be used o esime e desion enegy of e inefce. Te numeicl esuls pesened in e ppe illuse e imponce of e selecion of e ppopie ple model fo e flexue of e in film fo puposes of inepeing e desion enegy of e inefce. Te coice of e model fo inepeing e desion enegy sould be diced no only by e elive geomeic cceisics of e in film bu lso elive siffness cceisics of e in film o e subse nd e elive mgniude of e delminion pessue, in siuions wee e deflecions of e in film e sufficienly lge o induce in-plne exension. Acknowledgemens Te wok descibed in is ppe ws suppoed in p by e Mx Plnck Foscungspeis in e Engineeing Sciences wded o e uo by e Mx Plnck Gesellscf, Gemny. Te uo is indebed o e eviewe fo igly consucive commens nd fo ing enion o e possibiliy of e developmen of mixed-mode delminion pocesses.

8 466 A.P.S. Selvdui / Ac Meili 55 (7) Appendix A Tis Appendix pesens n ppoxime soluion o e poblem of e lge deflecion of unifomly loded in ple exibis Hooken elsic beviou. Te soluion is bsed on viionl pocedue kes ino ccoun e conibuions fom e flexul enegy of e in ple s well s e in-plne exension ccompnying lge deflecions. Fo xisymmeic lge deflecions of e in film, e flexul deflecions nd e secing defomions e ppoximed, especively, by elionsips wðþ ¼C ; uðþ ¼ C þ C ðaþ wee C, C nd C e biy consns. Te ssumed foms of e deflecion w() nd e secing of e midplne u() sisfy e condiion of flexul fixiy e boundy of e ple = nd e bsence of dil displcemens on bo e xis of symmey of e ple nd e boundy =. Te sin enegy in e in film due o flexue nd in-plne exension e given by Z " d w U B ¼ pd þ # þ m d w d d d d d ðaþ nd U E ¼ pl Z du ð m Þ d þ! þ u 4 d þm u du d þ!# d d ða3þ especively, wee D (=l 3 /6( m )) is e flexul igidiy of e ple, l nd m e, especively, e line elsic se modulus nd Poisson s io of e in ple, nd is e ple ickness. Te biy consns C nd C e fis deemined in ems of C, using e viionl consin e ol poenil enegy U E is minimum fo n equilibium condiion, i.e., ou E ¼ ou E ¼ ða4þ oc oc Te sin enegy U E consisen wi e equilibium configuion cn en be dded o e flexul enegy U B o deemine e ol sin enegy U of e in film exibis lge deflecions nd in-plne secing s follows: U ¼ 3pDC 3 " þ v C # ða5þ wee v = {755 + (45 79m )m }/35,. Te unknown consn C is deemined by mking use of e pinciple of viul displcemens, i.e., Z du dc ¼ p p dc dc d ða6þ Tis gives e esul C ¼ p 3 64D! þ vðc =Þ ða7þ Te esul (A7) is cubic in C, wic cn be solved o expess e single el oo of C explicily in ems of p.if ^pð¼ 3p ð m Þ=3l Þ <, nd g = /, e posiive el oo of Eq. (A7) is 3p ffiffiffiffiffi 36 g =3 þ ffiffiffi 3p pffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii =3 6 9 v ^p þ 6g þ ^p C ¼ 6 ffiffi 3p pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii =3 ðaþ g 9 v ^p þ 6g þ ^p Te elsic enegy of e in ple expeiencing lge deflecions cuse in-plne exension cn now be expessed in ems of p by subsiuing is vlue of C in Eq. (A5). As n ppoximion, oweve, e enegy componen in Eq. (A5) due o secing is eined, bu e vlue of C is ppoximed by e dominn em on e ig-nd side of Eqs. (A7) nd (4), wi e undesnding ems of ode v(c /) nd ige cn be negleced. Tis povides e following esime fo e ol sin enegy of e in film undegoing lge deflecions involving bo flexue nd secing: U LD TF ¼ pp 6 34D þ v p i ða9þ 64D Refeences [] Andeson GP, Benne SJ, de Vies KL. Anlysis nd esing of desive bonds. New Yok: Acdemic Pess; 977. [] Mil KL, edio. Adesion mesuemens of films nd coings. Uec, Te Neelnds: V.S.P. In. Sci.;. [3] Doene MF, Olive WC, P GM, Bozen FR, edios. Tin films: sesses nd mecnicl popeies. In: MRS symposium, vol. ; 99. [4] Buns R, edio. Hndbook of d coings. New Yok: Noyes Publicions;. [5] Wu TW. J Me Res 99;6:7. [6] Bull SJ, Rickeby DS. Suf Co Tecnol 99;4:49. [7] Lne M, Duskd RH, Vincein A, Go H. J Me Res ;5:75. [] Msll DB, Evns AG. J Appl Pys 94;:63. [9] Kiese MD, Moody NR, Gebeic WW. J Me Res 999;4:37. [] Moody NR, Hwng RQ, Venk-Rmn S, Angelo JE, Nowood DP, Gebeic WW. Ac Me 99;46:55. [] Bgci A, Evns AG. Tin Solid Films 99;6:. [] Codill MJ, B DF, Moody NR, Gebeic WW. IEEE Tns Dev Me Relib 4;4:63. [3] Moon M-W, Cung J-W, Lee K-R, O KH, Wng R, Evns AG. Ac Me. ;5:9. [4] Pund A, Nikiin E, Pekski P, Kiceim R. Ac Me 4;5:579. [5] Codill MJ, B DF, Moody NR, Gebeic WW. Me Sci Eng 7;443:5. [6] Sneddon IN. Mixed boundy vlue poblems in poenil eoy. Amsedm: No-Hollnd; 966.

9 A.P.S. Selvdui / Ac Meili 55 (7) [7] Selvdui APS. Pil diffeenil equions in mecnics. Te Bimonic equion, Poisson s equion, Vol.. Belin: Spinge;. [] Timosenko SP, Woinowsky-Kiege S. Teoy of ples nd sells. New Yok: McGw-Hill; 959. [9] Reissne E. J Appl Mec Tns ASME 945;:A [] Mindlin RD. J Appl Mec Tns ASME 95;:3. [] Ngdi PM. In: Tuesdell C, edio. Encyclopedi of pysics, vol. VI A/. Belin: Spinge; 97. p. 45. [] Suo Z, Hucinson JW. In J Fcue 99;43:. [3] Hucinson JW, Suo Z. Adv Appl Mec 99;9:63. [4] Hucinson JW, Evns AG. Suf Co Tecnol ;49:79.