Multi-layer thin films/substrate system subjected to non-uniform misfit strains

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1 Available online at Intenational Jounal of Solid and Stuctue 45 (00) Multi-laye tin fil/ubtate yte ubjected to non-unifo ifit tain X. Feng a, Y. Huang b, *, A.J. oai c a Depatent of Engineeing Mecanic, Tingua Univeity, Beijing 10004, Cina b Depatent of Civil and Envionental Engineeing and Mecanical Engineeing, Notweten Univeity, Evanton, IL 60, USA c Gaduate Aeonautical Laboatoy, Califonia Intitute of Tecnology, aadena, CA 115, USA eceived 0 June 007 Available online 1 Septebe 007 Abtact Cuent etodologie ued fo te infeence of tin fil te toug cuvatue eaueent ae tictly eticted to te and cuvatue tate tat ae aued to eain unifo ove te entie fil/ubtate yte. Tee etodologie ave ecently been extended to a ingle laye of tin fil depoited on a ubtate ubjected to te non-unifo ifit tain in te tin fil. Suc etodologie ae fute extended to ulti-laye tin fil depoited on a ubtate in te peent tudy. Eac tin fil ay ave it own non-unifo ifit tain. We deive elation between te tee in eac tin fil and te cange of yte cuvatue due to te depoition of eac tin fil. Te inteface ea tee between te adjacent fil and between te tin fil and te ubtate ae alo obtained fo te yte cuvatue. Ti povide te bai fo te expeiental deteination of tin fil tee in ulti-laye tin fil on a ubtate. Ó 007 Elevie Ltd. All igt eeved. Keywod: Multi-laye tin fil; Non-unifo ifit tain; Non-unifo wafe cuvatue; Non-local te-cuvatue elation; Intefacial ea 1. Intoduction Stoney (10) tudied a yte copoed of a tin fil of ticne f, depoited on a elatively tic ubtate, of ticne, and deived a iple elation between te cuvatue, j, of te yte and te te, (f), of te fil a follow: ðfþ ¼ E j 6 f ð1 Þ : ð1:1þ In te above te ubcipt f and denote te tin fil and ubtate, epectively, and E and ae te Young odulu and oion atio. Eq. (1.1) i called te Stoney foula, and it a been extenively ued * Coeponding auto. Tel.: E-ail adde: y-uang@notweten.edu (Y. Huang) /$ - ee font atte Ó 007 Elevie Ltd. All igt eeved. doi: /j.ijolt

2 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) in te liteatue to infe fil te cange fo expeiental eaueent of yte cuvatue cange (Feund and Sue, 004). Stoney foula wa baed on te following auption, oe of wic ave been elaxed. (i) Bot te fil ticne f and te ubtate ticne ae unifo and f, wee epeent te caacteitic lengt in te lateal diection (e.g., yte adiu own in Fig. 1). Ti auption wa ecently elaxed fo te tin fil and ubtate of diffeent adii (Feng et al., 006) and fo abitaily non-unifo fil ticne (Ngo et al., 007). Tei analytical eult ave been veified te X-ay icodiffaction expeient (Bown et al., 007). (ii) Te tain and otation of te plate yte ae infiniteial. Ti auption a been elaxed by vaiou lage defoation analye (Mate and Salaon, 13; Salaon and Mate, 15; Finot et al., 17; Feund, 000), oe of wic ave been validated by expeient (Lee et al., 001; a et al., 003). (iii) Bot te fil and ubtate ae oogeneou, iotopic, and linealy elatic. To ou bet nowledge ti auption a not been elaxed yet. (iv) Te fil te tate ae equi-biaxial (two equal te coponent in any two, utually otogonal inplane diection) wile te out-of-plane diect te and all ea tee vani. Ti auption a been elaxed fo non-equi-biaxial te tate (Sen et al., 16; Wito et al., 1a; a and Sue, 000; Feund and Sue, 004). (v) Te yte cuvatue coponent ae equi-biaxial (two equal diect cuvatue) wile te twit cuvatue vanie in all diection. Ti auption a been elaxed fo non-equi-biaxial cuvatue coponent and non-vaniing twit coponent (Sen et al., 16; Wito et al., 1b; a and Sue, 000; Feund and Sue, 004). (vi) All uviving te and cuvatue coponent ae patially contant ove te plate yte uface, a ituation tat i often violated in pactice. ecently, Huang et al. (005) and Huang and oai (005) elaxed ti auption fo te tin fil/ubtate yte ubjected to non-unifo, axiyetic ifit tain (in tin fil) and tepeatue cange (in bot tin fil and ubtate), epectively, wile Ngo et al. (006) and Huang and oai (in pe) tudied te tin fil/ubtate yte ubject to abitaily non-unifo (e.g., non-axiyetic) ifit tain and tepeatue. Tei ot ipotant eult i tat te fil tee depend non-locally on te yte cuvatue, i.e., tey depend on cuvatue of te entie yte. nt laye fil z fn 1t laye fil ubtate f1 θ Fig. 1. A ceatic diaga of ulti-laye tin fil depoited on a ubtate, owing te cylindical coodinate (,, z).

3 360 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) Depite te explicitly tated auption of patial te and cuvatue unifoity, te Stoney foula i often, abitaily, applied to cae of pactical inteet wee tee auption ae violated. Ti i typically done by applying Stoney foula pointwie and tu extacting a local value of te fo a local eaueent of te cuvatue of te yte. Ti appoac of infeing fil te clealy violate te unifoity auption of te analyi and, a uc, it accuacy a an appoxiation i expected to deteioate a te level of cuvatue non-unifoity becoe oe evee. Many tin fil/ubtate yte involve ultiple laye of tin fil. Te ain pupoe of ti pape i to extend te above analye by Huang, oai and co-woe to a yte copoed of ulti-laye tin fil on a ubtate ubjected to non-unifo ifit tain ditibution. We will elate tee in eac fil and yte cuvatue to te ifit tain ditibution, and ultiately deive a elation between te tee in eac fil and yte cuvatue tat would allow fo te accuate expeiental infeence of fil tee fo full-field and eal-tie cuvatue eaueent.. Axiyetic ifit tain We fit conide a yte of ulti-laye tin fil depoited on a ubtate ubjected to axiyetic ifit tain ditibution ðþ in te it laye (i =1,,...,n), wee i te adial coodinate, and n i te total nube of laye of tin fil (Fig. 1). Te tin fil and ubtate ae cicula in te lateal diection and ave a adiu. Te defoation i axiyetic and i teefoe independent of te pola angle..1. Govening equation Let fi ði ¼ 1; ;...nþ denote te ticne of te it tin fil (Fig. 1). Te total ticne f ¼ n f i of all n fil i uc le tan te ubtate ticne, and bot ae uc le tan, i.e. f. Te Young odulu and oion atio of te it tin fil and ubtate ae denoted by E fi, fi, E and, epectively. Te ubtate i odeled a a plate ince it can be ubjected to bending and. Te tin fil ae odeled a ebane tat ave no bending igiditie due to tei all ticne f. Teefoe tey all ave. Te te- te ae in-plane diplaceent u f () in te adial () diection. Te tain ae e ¼ du f d e in te it tin fil can be obtained fo te linea elatic contitutive odel a ðf iþ ¼ E f i du f 1 f i d þ u f f i ð1þ f i Þ ðfiþ ¼ E f i du f 1 fi f i d þ u f ð1þ f i Þ Te ebane foce in te it tin fil ae N ðf iþ ¼ fi ðf iþ ; N ðfiþ ¼ fi ðfiþ : ; : and e ¼ u f Fo non-unifo ifit tain e ðiþ ðþ, te ea te taction along te fil/fil and fil/ubtate inteface do not vani, and ae denoted by (i) () (i =1,,...,n) a own in Fig.. Te noal te taction till vani becaue tin fil ave no bending igiditie. Te equilibiu equation fo tin fil, accounting fo te effect of inteface ea te taction, becoe dn ðfiþ d þ N ðfiþ N ðfiþ ð i iþ1 Þ¼0; wee n+1 = 0 fo te taction fee uface. Subtitution of Eq. (.1) and (.) into (.3) and te uation of it left and ide yield d u f d þ 1 du f d u f E fi fi ¼ ð1þ þ Xn E fi fi d 1 f i 1 fi d : ð:4þ ð:1þ ð:þ ð:3þ

4 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) z nt laye fil τ (n) 1t laye fil ubtate τ () τ (1) τ (1) Fig.. A ceatic diaga of te non-unifo ea taction ditibution at te fil/ubtate and fil/fil inteface. Let u denote te diplaceent in te adial () diection at te neutal axi of te ubtate, and w te diplaceent in te noal (z) diection. Te foce and bending oent in te ubtate ae obtained fo te linea teo-elatic contitutive odel a N ðþ ¼ E du 1 d þ u N ðþ ¼ E du 1 d þ u M ¼ E 3 d w 1ð1 Þ d þ M ¼ E 3 1ð1 Þ ; ; d w d þ 1 dw d ; dw d : Te ea te (1) at te tin fil/ubtate inteface i equivalent to te ditibuted axial foce (1) and bending oent ð1þ applied at te neutal axi of te ubtate. Te in-plane foce equilibiu equation of te ubtate ten becoe dn ðþ d þ N ðþ N ðþ þ ð1þ ¼ 0: Te out-of-plane foce and oent equilibiu equation ae given by dm d þ M M þ Q ð1þ ¼ 0; ð:þ dq d þ Q ¼ 0; ð:þ wee Q i te ea foce noal to te neutal axi. Subtitution of Eq. (.5) into Eq. (.7) yield d u d þ 1 du d u ¼ 1 ð1þ : ð:10þ E Eliination of Q fo Eq. (.) and (.), in conjunction wit Eq. (.6), give d 3 w d þ 1 d w 3 d 1 dw d ¼ 6ð1 Þ ð1þ : E Te continuity of diplaceent aco te tin fil/ubtate inteface equie u f ¼ u dw d : Eq. (.4) and (.10), (.11), (.1) contitute fou odinay diffeential equation fo u f, u, w and (1). ð:5þ ð:6þ ð:7þ ð:11þ ð:1þ

5 36 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) We can eliinate u f, u and w fo tee fou equation to obtain te ea te (1) at te tin fil/ubtate inteface in te of te ifit tain. Fo f, (1) and te ea tee (i) (i =,3,...,n) between tin fil ðiþ ¼ Xn E fj fj de ðjþ 1 fj d : ð:13þ Ti i a eaable eult tat old egadle of bounday condition at te edge =. Teefoe, te inteface ea te i popotional to te gadient of ifit tain. Fo unifo ifit tain ðþ ¼contant, te inteface ea te vanie, i.e., (1) =0. Subtitution of te above olution fo ea te (1) into Eq. (.11) and (.10) yield odinay diffeential equation fo diplaceent w and u in te ubtate. Tei olution, at te liit of f, ae dw d ¼ E u ¼ 1 1 E Z g Xn 0 Z g Xn 0 E fi fi 1 ðgþdg þ B 1 fi E fi fi 1 fi ; ð:14þ ðgþdg þ B ; ð:15þ wee B 1 and B ae to be deteined. Te diplaceent u f in te fil i obtained fo te continuity condition (.1) aco te inteface a u f ¼ 4 1 Z 1 g Xn E fi fi E 1 ðgþdg þ B fi B 1 : ð:16þ 4 0 Te fit bounday condition at te fee edge = equie tat te net foce vani, N ðfiþ þ N ðþ ¼ 0at ¼ ; ð:17þ wic give B ¼ ð1 Þ E fi fi e ðiþ ð:1þ E 1 fi fo f, wee e ðiþ ¼ 0 geðiþ ðgþdg ¼ da i te aveage ifit tain in te it tin fil. Te econd p bounday condition at te fee edge = i vaniing of net oent, i.e., M wic give B 1 ¼ 6 ð1 Þ N ðfiþ ¼ 0at ¼ ; ð:1þ E E fi fi 1 fi : ð:0þ.. Stee in ulti-laye tin fil and yte cuvatue Te yte cuvatue ae elated to te out-of-plane diplaceent w by j ¼ d w d of tee two cuvatue i E fi fi 1 fi þ 1 þ j þ j ¼ 1 1 E Te diffeence between two yte cuvatue i eðiþ and j ¼ 1 dw. Te u d : ð:1þ

6 j j ¼ 6 1 E X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) E fi fi 1 fi Z Te u and diffeence of tee in eac tin fil ae given by 0 g ðgþdg : ð:þ ðfiþ ðfiþ þ ðfiþ ¼ E f i ; ð:3þ 1 fi ðf iþ ¼ 4 E f i 1 þ fi 1 E It i noted tat ðfiþ ðfiþ j¼1 E fj fj e ðjþ 1 fj Z 0 ge ðjþ ðgþdg i in geneal expected to be alle tan ðfiþ.3. Extenion of Stoney foula fo a ulti-laye tin fil/ubtate yte : ð:4þ þ ðfiþ fo f / 1. We extend te Stoney foula fo a ulti-laye tin fil/ubtate yte ubjected to non-unifo ifit by etabliing te diect elation between te tee in eac tin fil and yte cuvatue. Bot j j in Eq. (.) and ðfiþ ðfiþ in Eq. (.4) ae popotional to n E fi fi 1 fi 0 geðiþ ðgþdg. Eliination of ifit tain give ðf iþ ðfiþ in eac fil diectly popotional to te diffeence j j in yte cuvatue, ðfiþ ðfiþ ¼ E f i 3ð1 þ fi Þ ðj j Þ: We now focu on te u of tin-fil tee ðfiþ yte cuvatue j þ j i defined a ZZ þ ðf iþ ð:5þ and u of yte cuvatue j + j. Te aveage j þ j ¼ 1 ðj p þ j Þgdg d ¼ Z gj ð A þ j Þdg: ð:6þ 0 It can be elated to te aveage ifit tain by aveaging bot ide of Eq. (.1), i.e., j þ j ¼ 1 1 E E fi fi 1 fi : ð:7þ Te deviation fo te aveage cuvatue, j þ j j þ j, can be elated to te deviation fo te aveage ifit tain a j þ j j þ j ¼ 6 1 E E fi fi 1 fi eðiþ : ð:þ Eliination of ifit tain n E fi fi 1 fi eðiþ and aveage ifit tain n E fi fi 1 fi fo Eq. (.7), (.) and (.3) give te u of tin-fil tee in te of yte cuvatue a fi f ðfiþ þ ðfiþ E ¼ j þ j þ 1 ðj þ j j þ j Þ ; ð:þ 6ð1 Þ f 1 þ wee f ¼ n f i i te total ticne of tin fil. Eq. (.) only give te weigted u of tee in all tin fil, n fi f ðfiþ þ ðfiþ, in te of te yte cuvatue, but not tee in eac tin fil. It i clea tat te cuvatue alone fo a yte wit all n tin fil ae not ufficient to deteine te tee in all tin fil. Additional paaete tat can be eaued in expeient ae needed fo te coplete deteination of all fil tee. One poibility i te yte cuvatue j ðiþ and j ðiþ afte te fit i tin fil ae depoited, and j ðnþ ¼ j and j ðnþ ¼ j. Te yte cuvatue j ðiþ and j ðiþ can be eaued duing te

7 364 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) depoition poce, o afte te depoition poce by etcing te top n i tin fil away. Te tee in te it tin fil ae ten given by ðfiþ þ ðfiþ ¼ E 6ð1 Þ þ DjðiÞ þ 1 fi 1 þ þ DjðiÞ DjðiÞ þ ; ð:30þ wee ¼ j ðiþ jði 1Þ and ¼ jðiþ jði 1Þ ð:31þ ae te cange of yte cuvatue due to te depoition of te it tin fil, and j ð0þ ¼ 0. Te above equation i identical to it countepat fo a ingle laye of tin fil (Ngo et al., 006) except tat it cuvatue ¼ j ð0þ and ae eplaced by te cange of yte cuvatue in Eq. (.31). Eq. (.31), togete wit Eq. (.5), povide te diect elation between tee in eac tin fil and yte cuvatue. Te tin-fil tee at a point depend not only on te cange of yte cuvatue þ DjðiÞ at te ae point (local dependence), but alo on te aveage cange of yte cuvatue þ in te entie yte (non-local dependence). Te inteface te (1) between te ubtate and te fit tin fil and (i) between tin fil in Eq. (.13) can alo be given by yte cuvatue ðiþ ¼ wee j ð0þ E 6ð1 Þ ¼ j ð0þ d d Dj ðjþ þ Dj ðjþ ¼ E d 6ð1 Þ d j þ j j ði 1Þ j ði 1Þ ; ð:3þ ¼ 0. Te above equation povide a eaably iple way to etiate te inteface ea tee fo adial gadient of te two non-zeo yte cuvatue. Te ea tee ae eponible fo pooting yte failue toug debonding of tin fil. 3. Non-axiyetic ifit tain We extend te analyi in te peviou ection to a yte of ulti-laye tin fil depoited on a ubtate to abitay non-unifo ifit tain. Te analyi i alo an extenion of Ngo et al. (007) fo a ingle tin fil to ulti-laye tin fil. Te non-unifo ifit tain in te it tin fil, ð; Þ, can be expanded to te Fouie eie ð; Þ ¼X1 ¼0 wee 0c ðþ ¼ 1 p p X1 c ðþ co þ 0 ðþ in ; ð3:1þ ð; Þd, eðiþ c ðþ p ¼1 p Stee in ulti-laye tin fil and yte cuvatue ð; Þ co d and eðiþ ðþ p ¼1 p 0 ð; Þ in d ð 1Þ. Te yte cuvatue ae j ¼ o w o ; j ¼ 1 ow o þ 1 o w o ; j ¼ o o 1 ow o : ð3:þ Te u of yte cuvatue i elated to te ifit tain by j þ j ¼ 1 1 X n þ 1þ eðiþ E fi fi E 1 fi þ 1 co " 1 3þ ð þ 1Þ 0 gþ1 c ðgþdg # ; ð3:3þ þ þ in 0 gþ1 ðgþdg >;

8 wee ¼ 1 p A eðiþ ðg; uþda i te aveage ifit tain in te it tin fil, da = gdgdu, and eðiþ i alo elated to 0c ven by by eðiþ ¼ 0 geðiþ 0c ðgþdg. Te diffeence between two cuvatue, j j, and te twit j ae gi- j j ¼ 6 1 E j ¼ 3 1 E E fi fi 1 fi E fi fi 1 fi 1 1 3þ þ 1 1 Te u ðfiþ þ ðfiþ and diffeence ðfiþ to te ifit tain by 0 geðiþ þ 1 3þ 1 1 þ1 þ 1 þ1 þ þ1 þ þ1 þ 0c dg i co ð 1Þ þ in co 0 gþ1 c dg þ in 0 gþ1 0 gþ1 c dg 0 gþ1 dg dg! ; ð3:4þ ð 1Þ co g 1 c dg þ in g 1 dg >; i in ð 1Þ 0 gþ1 c dg! co 0 gþ1 dg in 0 gþ1 c dg co 0 gþ1 dg : ð3:5þ ð 1Þ in g 1 c dg co g 1 ðfiþ dg of tee and te ea te ðfiþ in te it tin fil ae elated ðfiþ þ ðfiþ ¼ E f i ; ð3:6þ 1 fi e ðjþ 0 geðjþ 0c dg ðfiþ ðfiþ ¼ 4 E f i 1 1 þ fi E j¼1 ðfiþ ¼ E f i 1 1 þ fi E j¼1 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) E fj fj 1 fj E fj fj 1 fj 1 þ1 þ þ 1 þ 3þ 1 1 þ1 þ 1 3þ 1 co 0 gþ1 e ðjþ c dg þ in 0 gþ1 e ðjþ ð 1Þ þ1 þ dg >; co g1 e ðjþ c dg þ in g1 e ðjþ i co ð 1Þ 0 gþ1 e ðjþ in 0 gþ1 e ðjþ c dg co 0 gþ1 e ðjþ ð 1Þ in þ1 þ g1 e ðjþ c dg þin 0 gþ1 e ðjþ dg dg c dg co i in ð 1Þ 0 gþ1 e ðjþ g1 e ðjþ dg c dg co 0 gþ1 e ðjþ dg dg! >; ð3:7þ : ð3:þ! >; Te ea tee ð1þ and ð1þ between te fit tin fil and ubtate and ðiþ tin fil ae elated to te ifit tain by and ðiþ (i =,3,...,n) between ðiþ ¼ Xn E fj fj oe ðjþ 1 fj o ; ðiþ ¼ Xn E fj fj 1 1 fj oe ðjþ o : ð3:þ

9 366 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) Extenion of Stoney foula fo a ulti-laye tin fil/ubtate yte We extend te Stoney foula fo a ulti-laye tin fil/ubtate yte by etabliing te diect elation between te tee in eac tin fil and yte cuvatue. We define te coefficient C and S in te of te yte cuvatue j + j by C ¼ 1 j ð p A þ j Þ g co uda S ¼ 1 j ð p A þ j Þ g in uda; ð3:10þ wee te integation i ove te entie aea A of te ubtate, and da = gdgdu. Te diffeence in tee ðfiþ ðf iþ and ea te ðf iþ in te it fil ae given in te of yte cuvatue by ðfiþ ðfiþ ¼ E f i < 4ðj j Þ = 61þ ð fi Þ 1 ð þ 1Þ i ð 1Þ : ðc co þ S in Þ; ; ð3:11þ ðfiþ ¼ E f i < 4j = 6ð1 þ fi Þ þ 1 1 ð þ 1Þ i ð 1Þ : ðc in S co Þ; ; ð3:1þ Siila to Section.3, we define te yte cuvatue j ðiþ ; jðiþ and jðiþ afte te fit i tin fil ae depoited, wic can be eaued duing te depoition poce, o afte te depoition poce by etcing te top n i tin fil away. Te cange of yte cuvatue due to te it tin fil ae ¼ j ðiþ jði 1Þ ; ¼ jðiþ i 1 j ð Þ ; ¼ jðiþ jði 1Þ ; ð3:13þ wee DC ðiþ j ðnþ and DS ðiþ ¼ j ; j ðnþ DC ðiþ ¼ 1 p A DS ðiþ ¼ 1 p A ¼ j ; j ðnþ ¼ j, and j ð0þ ¼ j ð0þ ¼ jð0þ in te of te cange of yte cuvatue þ DjðiÞ DjðiÞ þ DjðiÞ DjðiÞ þ DjðiÞ g co uda in uda: g ¼ 0. We alo define te coefficient by ð3:14þ Te u of tee in te it fil i given in te of te cange of yte cuvatue by ðf iþ þ ðfiþ ¼ E 6 6 fi ð1 Þ4 þ DjðiÞ þ 1 1þ 1 1þ 1 ð þ 1 Þ þ DjðiÞ DC ðiþ DjðiÞ þ co þ DS ðiþ in ; ð3:15þ wee þ ¼ DCðiÞ 0 ¼ 1 p A DjðiÞ þ DjðiÞ da i te aveage ove entie aea A of te ubtate. Eq. (3.11), (3.1) and (3.15) povide diect elation between tee in tin fil and yte cuvatue. It i ipotant to note tat tee at a point in te tin fil depend not only on cuvatue at te ae point (local dependence), but alo on te cuvatue in te entie ubtate (non-local dependence). Te inteface ea tee ðiþ and ðiþ can alo be diectly elated to ubtate cuvatue via

10 ðiþ ¼ E 6 1 ¼ ð E 6ð1 Þ ðiþ ¼ n E 6ð1 Þ ¼ E 6ð1 Þ X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) o Dj ðjþ n o þ Dj ðjþ 6 Þ þ ð 1Þ DC ðjþ co þ DS ðjþ 1 5 in o j o þ j j ði 1Þ j ði 1Þ ð3:16þ "!! # 1 1 ð þ 1Þ n DC ðjþ co þ n ; DS ðjþ 1 in >; 3 1 o Dj ðjþ o þ j ðjþ 6 þ þ ð 1Þ DC ðjþ in DS ðjþ 1 5 co 1 o j o þ j j ði 1Þ j ði 1Þ ð3:17þ "!! # þ 1 1 ð þ 1Þ n DC ðjþ in n : DS ðjþ 1 co >; Tee povide a way to etiate te inteface ea tee fo te gadient of yte cuvatue. Tey alo diplay a non-local dependence. 4. Concluding ea and dicuion Te Stoney foula i extended in te peent analyi fo ulti-laye tin fil depoited on a ubtate ubjected to non-unifo ifit tain. Fo ulti-laye tin fil (i = 1,,..., n) on a ubtate, te total yte cuvatue j + j only give te aveage tee in all tin fil, not tee in eac tin fil. In te peent tudy te tee in te it tin fil ae obtained in te of te cange of yte cuvatue þ DjðiÞ due to te depoition of te it tin fil. Te inteface ea tee between adjacent tin fil and between te tin fil and ubtate ae alo obtained fo te cuvatue. Ti povide te bai fo expeiental deteination of te tee in eac tin fil and inteface ea tee. Siila to a ingle laye of tin fil on a ubtate, te tee in ulti-laye tin fil ae elated to te yte cuvatue, and uc dependence i non-local ince te tee at a point on te fil depend on bot te local value of te yte cuvatue (at te ae point) and on te value of cuvatue of all ote point on te plate yte (non-local dependence). efeence Bown, M.A., oai, A.J., Feng, X., Huang, Y., Utundag, E., 007. Tin fil/ubtate yte featuing abitay fil ticne and ifit tain ditibution: at II. Expeiental validation of te non-local te-cuvatue elation. Int. J. Solid Stuct. 44, Feng, Huang, Y., Jiang, H., Ngo, D., oai, A.J., 006. Te effect of tin fil/ubtate adii on te Stoney foula fo tin fil/ ubtate ubjected to non-unifo axiyetic ifit tain and tepeatue. J. Mec. Mate. Stuct. 1, Finot, M., Blec, I.A., Sue, S., Fijioto, H., 17. Lage defoation and geoetic intability of ubtate wit tin-fil depoit. J. Appl. y. 1, Feund, L.B., 000. Subtate cuvatue due to tin fil iatc tain in te nonlinea defoation ange. J. Mec. y. Solid 4, 115. Feund, L.B., Sue, S., 004. Tin Fil Mateial; Ste, Defect Foation and Suface Evolution. Cabidge Univeity e, Cabidge, UK. Huang, Y., Ngo, D., oai, A.J., 005. Non-unifo, axiyetic ifit tain in tin fil bonded on plate ubtate/ubtate yte: te elation between non-unifo fil tee and yte cuvatue. Acta Mec. Sin. 1, Huang, Y., oai, A.J., in pe. Extenion of Stoney foula to abitay tepeatue ditibution in tin fil/ubtate yte. J. Appl. Mec. (ASME Tan.). Huang, Y., oai, A.J., 005. Extenion of Stoney foula to non-unifo tepeatue ditibution in tin fil/ubtate yte. Te cae of adial yety. J. Mec. y. Solid 53,

11 36 X. Feng et al. / Intenational Jounal of Solid and Stuctue 45 (00) Lee, H., oai, A.J., Feund, L.B., 001. Full field optical eaueent of cuvatue in ulta-tin fil/ubtate yte in te ange of geoetically nonlinea defoation. J. Appl. y., Mate, C.B., Salaon, N.J., 13. Geoetically nonlinea te deflection elation fo tin fil/ubtate yte. Int. J. Eng. Sci. 31, Ngo, D., Huang, Y., oai, A.J., Feng, X., 006. Spatially non-unifo, iotopic ifit tain in tin fil bonded on plate ubtate: te elation between non-unifo tee and yte cuvatue. Tin Solid Fil 515, 0. Ngo, D., Feng, X., Huang, Y., oai, A.J., Bown, M.A., 007. Tin fil/ubtate yte featuing abitay fil ticne and ifit tain ditibution: pat I. Analyi fo obtaining fil te fo nonlocal cuvatue infoation. Int. J. Solid Stuct. 44, a, T.-S., Sue, S., 000. Effect of line and paivation geoety on cuvatue evolution duing poceing and teal cycling in coppe inteconnect line. Acta Mate. 4, a, T.-S., Sue, S., oai, A.J., yu, J., 003. Meaueent of full field cuvatue and geoetical intability of tin fil ubtate yte toug CGS intefeoety. J. Mec. y. Solid 51, Salaon, N.J., Mate, C.B., 15. Bifucation in iotopic tin fil/ubtate plate. Int. J. Solid Stuct. 3, Sen, Y.L., Sue, S., Blec, I.A., 16. Stee, cuvatue, and ape cange aiing fo pattened line on ilicon wafe. J. Appl. y. 0, Stoney, G.G., 10. Te tenion of etallic fil depoited by electolyi. oc.. Soc. Lond. A, Wito, A., Gudundon,., Sue, S., 1a. Teoelatic analyi of peiodic tin line depoited on a ubtate. J. Mec. y. Solid 47, Wito, A., Gudundon,., Sue, S., 1b. Analyi of aveage teal tee in paivated etal inteconnect. J. Appl. y. 6,