THREE-DIMENSIONAL FUNDAMENTAL SOLUTIONS IN MULTILAYERED PIEZOELECTRIC SOLIDS

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1 Spcal Dca o Svn ra o roor Toma Ta Tng THREE-DMESOL FUDMETL SOLUTOS MULTLYERED EZOELETR SOLDS E. an Dp. o vl Engnrng T Unvr o ron ron OH U.S.. STRT T arcl Grn ncon n r-mnonal D anoropc polcrc an mllar ma. T wo-mnonal D Forr ranorm ar r appl o wo oronal varabl cangng paral rnal qaon o an ornar on. T gnral olon n ranorm oman n pr n rm o D Sro ormalm. T propagaor mar or ac lar rv rom gnral olon o a olon n Forr ranorm oman a an vrcal lvl can b mpl pr n rm o propagaor marc. T pcal oman olon alo o nmrcal aoca w nvr Forr ranorm. Fnall vral pcal ca ar c nclng D ormaon n anoropc polcrc an mllar ma an D ormaon n ranvrl oropc polcrc an mllar ma.. TRODUTO n l o appl mcanc maral anorop alwa an mporan opc. Rpon o an anoropc maral can b compll rn o a o an oropc on. n parclar anoropc maral ma prn npc ar wc col b o pcal nr o ngnrng gn []. n rcn ar on anoropc polcrc maral a bcom an acv rarc ara wr polcrc maral an nal componn n mar rcr al monorng m c. n orr o nran bac propr varo analcal olon av bn alo vlop nclng bo wo-mnonal D an r-mnonal D namnal olon.. Grn ncon o a pon orc. Unr ampon o a D ormaon So al. [4] racr problm n anoropc polcrc ma wl corrponng namnal olon n an nn-plan al-plan an bmaral plan wr b arn an Lo [5] ng an Tng [6] an [7] an R [89]. W mnon a cran n polcrc lm rcr nr D ormaon wr alo c; or ampl R. []. For a ranvrl oropc polcrc ol nr D ormaon Dng an co-worr [] an Dnn an co-worr [4] rv namnal olon n an nn-pac al-pac an bmaral pac. Wn maral gnral anorop an an co-worr rv corrponng namnal olon [58]. W alo pon o a ama an Tanma [9] oban lac placmn an lcrc ponal n an anoropc polcrc nnpac ng n Sro ormlm [5]. T arcl Grn ncon n D anoropc polcrc an mllar ma. W r appl D Forr ranorm o wo oronal varabl o cang paral rnal qaon o an ornar on. T gnral olon n Forr ranorm oman n pr n rm o D Sro ormalm mlar o Tng []. W alo rv propagaor mar or ac lar rom gnral olon o a olon n Forr ranorm oman a an vrcal lvl can b pr n rm o propagaor marc. T pcal oman olon pr b nvr Forr ranorm wr nvolv nmrcal ar c. W mnon a prn olon conan vral olon a pcal ca nclng corrponng D olon an D olon n ranvrl oropc polcrc an lar ol.. GOVERG EQUTOS OF SOTRO EZOELETR SOLD wll nown or an anoropc polcrc oca roor T n ornal o Mcanc-Sr Vol. 9 o. Marc 7

2 ol govrnng qaon con o.g. [57]: Eqlbrm qaon: σ 8 l γ l Smlarl qlbrm qaon n rm o n r can b rca no σ ; D q σ 9 Wr σ an D ar r an lcrc placmn rpcvl; an q ar bo orc an lcrc carg n wc wll b rplac lar b a concnra orc an lcrc carg. n papr lowrca pprca bcrp alwa rang rom o o 4 an mmaon ovr rpa lowrca pprca bcrp mpl. bcrp comma no paral rnaon w rpc o coorna.. or. onv rlaon: σ γ lm lm E D γ ε E Wr γ ran an E lcrc l; lm an ε ar lac mol polcrc cocn an lcrc conan rpcvl. W rmar a copl a.. prl lac an prl lcrc ormaon can b oban b mpl ng. Elac ran-placmn an lcrc l-ponal rlaon: γ ; E φ Wr an φ ar lac placmn an lcrc ponal rpcvl. bor w aop noaon nroc r b arn an Lo [5] an mo lar b an [7] a γ σ φ γ E 4 l σ D l l l εl 4 4 ; 4; n rm o oran noaon conv rlaon can b n no ngl qaon: w bng n orc n a q 4 n ollowng con w n placmn or lac placmn an lcrc ponal a n b 4 an n r or r an lcrc placmn a n b 6. Frrmor w n σ σ σ D a n racon on conan plan. n Eq. prcrp no ranpo o a mar.. GEERL SOLUTO HOMOGEEOUS LTE TRSFORMED DOM L conr an anoropc polcrc an lar al-pac ma o paralll an omogno lar lng ovr a omogno al-pac. W nmbr lar rall w lar a op bng lar an al-pac bng lar. T orgn o aran coorna a rac o lar al-pac an -a rawn own no mm. T - lar bon b nrac. Tror w av an H w H bng p o la nrac. n n pon orc a an -lvl an loang on rac can all b conr wc wll b c lar. W now appl D Forr ranorm.. or wo-pon n placmn ; ; o Eq. 8 an 9 w n orc n Eq. 9 bng ro. T n Forr ranorm oman w oban ollowng ornar rnal qaon wc mlar o prl lac conrpar []. β β β gnral olon o n placmn can b rv a [6] p ; a 4 8 T n ornal o Mcanc-Sr Vol. 9 o. Marc

3 w p an a ang ollowng gnrlaon: wr mp > p 4 p a 4 a b 4 [ a a a a4] [ b b b b4 [ Q p R R p T ] a Q n n R n m T w m 5 m 6 n n n co θ n θ m m m 7 o a a polar coorna ranorm n blow a bn co θ ; n θ 8 Eqaon 5 Sro gnrlaon or oblq plan pann b n an m n b 7. Sc a conncon bwn D an D Sro gnrlaon wa r ma b arn an Lo [5] an wa crb n al b Tng [] n a orm abl or o lar rcr. W alo mnon a b pov rqrmn on ran nrg n all gnval o Eq. 5 ar r compl or prl magnar.g. [4]. Ung now Sro gnval an gnvcor n racon vcor on conan plan an n n-plan r vcor n a σ σ σ D D 9 can b pr n Forr-ranorm oman a [6] b ] 4 c c c ] 4 [ c4 wr m an or magnar par an ovr bar or compl conga. n anal ollow w am a p ar nc an gnvcor a an b a normalaon rlaon [5] b a a b δ 5 w δ bng 4 4 roncr la n mar. W alo rmar a rpa gnval p can b avo b ng lgl prrb maral cocn w nglgbl rror [7]. n ong o mpl rcr o olon prn blow can alwa b. can b own a Eq. 5 an can b rca no a 8 8 lnar gnm wr a a b b T R Q RT R T RT 6 7 Tror a gnral olon n Forr-ranorm oman or n placmn an racon vcor n a omogno b gnral anoropc polcrc mm can b pr a p b p p p p c p wr 8 w p p p p4 ag [ p ] 9 b R pt a Q pr a p c V a wr mar V n b V w ang mmaon rom o. W now no b p m a m an b m m 8 gnval an aoca gnvcor o 5 an an orr m n c a wa o a an an ar wo 4 conan colmn marc o b rmn. 4. ROGTOR MTRX D SOLUTO OF MULTLYERED STRUTURES From Eq. 8 w can own a olon a an vrcal lvl n omogno lar.. n placmn an racon vcor n Forr ranorm oman can b pr b a a op nrac o am lar.. a wr T n ornal o Mcanc-Sr Vol. 9 o. Marc 9

4 p p p p call propagaor mar o lar [45]. L blow ar r mporan ar o propagaor mar wc can b al prov. 4 no a n orr o conrc propagaor mar n Eq. on n o n nvr o compon mar own on rg-an o Eq.. Fornal can b on al b mplong ollowng mporan orogonal rlaon [] n o polcrc ca a 5 T propagang rlaon can b rpal o a on can propaga olon rom op rac o boom nrac H o lar rcr. onqnl w av H L 6 wr cn o lar an propagaor mar o am lar. obrv a Eq. 6 a rprngl mpl rlaon. For gvn bonar conon n Forr ranorm oman on op rac along w raaon conon or olon n omogno al-pac > H nnown nvolv n Eq. 6 can b olv. n orr o oban n placmn an racon vcor n ranorm oman a an p a n lar w propaga olon rom op rac o -lvl [5].. L 7 W n placmn an racon vcor a a gvn p bng olv corrponng n-plan qan can b vala ng Eq. an. W rr mnon a n olvng Eq. 6 bonar conon gvn on op rac can b r- con- or r-p. n o ml- lar rcr ng bonar lmn mo on wll n Grn ncon olon or namnal olon n lar rcr. T anl b ollowng approac. r an nrnal orc orc carg lo- caon c. loca a lvl wn lar w arcall v lar no wo b-lar w an w an n Forr ranorm conn acro orc lvl a 8 W pon o a or a gvn pon orc Forr ranorm can all b on n an ac clo-orm. For ampl or an n pon orc appl a.. 9 δ δ δ wr 4 ar ampl o n pon orc Forr ranorm con- n n Eq. 8 ar on o b 4 gan propagang propagaor marc rom op rac o boom nrac H o lar rcr an mang o conn rlaon 9[5] w arrv a ollowng mporan qaon H L L 4 Smlarl or gvn bonar conon n Forr ranorm oman on op rac along w raaon conon or olon n omogno al-pac > H nnown nvolv n Eq. 4 can b olv. r a n placmn an racon vcor n ranorm oman a an -lvl can b on b ng Eq. 7. W rr mnon a Eq. 6 an 4 can b qall appl o mllar pla wr bonar conon on bo op an boom rac can b mpo. 5. HYSL-DOM SOLUTO ow n orr o n pcal oman olon w n o appl D nvr Forr ranorm o ranorm n placmn an r. For ampl or n placmn w av ; ; π 4 T n ornal o Mcanc-Sr Vol. 9 o. Marc

5 T ngral lm o varabl an n Eq. 4 ar rom o. lrnavl D nvr Forr ranorm can alo b carr o n polar coorna a ; θ π ; coθ n θ θ 4 wr ngral lm or rom o an or θ rom o π. T obl ngral n Eq. 4 or 4 can b on cnl ng aapv ngraon algorm vlop b Yang an an [6]. Howvr a rc mplmnaon o c an algorm ma no wor or cran aon n parclar wn -lvl o orc an l pon ar clo o ac or. n ca ngran gl ocllang an vn vrgn. n c a ca nn-pac Grn ncon [5] bmaral Grn ncon [6] an vn rmaral Grn ncon [7] n o b ncorpora no mllar Grn ncon o g r o nmrcal cl. T wll b c n a r papr [8]. nor nmrcal cl ma occr wn lar nmbr larg cran lar ar vr c or wn ngral varabl n Eq. 4 larg. n c a aon om lmn n propagaor mar ma ovrlow nc ar ponnal ncrang ncon o an. To ovrcom problm orwar an bacwar mlplcaon approac n o b appl. Ta w propaga propagaor mar rom r op rac o boom nrac or rom boom nrac o op rac pnng pon rlav -lvl o orc an l pon [59]. 6. SEL SES W pon o a prn olon conan olon or cran pcal ca. Fr olon o corrponng D ormaon o anoropc polcrc an lar rcr can b rc rom D olon prn r. w l on o oronal coorna an an on o corrponng Forr varabl or bng ro n problm wll b rc o a gnral plan-ran on n polcrc an mllar rcr. For ca onl D Forr nvr ranorm n n orr o g pcal-oman olon []. T con pcal olon or D ormaon b or maral n ac lar bng ranvrl oropc polcrc. For ca D Forr ranorm can b rplac b Hanl ranorm onl D m-nn ngral ar rqr o b carr o nmrcall. Frrmor or ca clnrcal an aran m o vcor ncon can b l [5]. On o avanag o ng c m a a-mmrc ormaon an D ormaon can all b ncl a pcal ca o gnral olon []. nor avanag a alo aoca w m a oal olon can b pr n rm o wo par: Wl on par copl w lcrc qan or on prl lac []. 7. OLUSOS n papr w Grn ncon n D anoropc polcrc an mllar ma. W r appl D Forr ranorm o wo oronal varabl rcng paral rnal qaon o an ornar on. T gnral olon or ranorm placmn an r n pr n rm o D Sro ormalm mlar o a prn n Tng []. T propagaor mar or ac lar rr rv rom gnral olon o a placmn an r n ranorm oman a an vrcal lvl can b pr n rm o propagaor marc. T pcal-oman olon can b oban b carrng o nvr Forr ranorm. Svral nmrcal aoca w nvr ranorm ar conr. Fnall wo pcal ca o prn olon.. corrponng D ormaon an D ormaon n ranvrl oropc polcrc an lar ma ar brl c. OWLEDGEMETS wol l o gv parclar an o roor Tng or conan ncoragmn an ppor. av bn a lo rom commncaon w m an rom anoropc lac boo wn rvng Grn ncon n anoropc ma. wol alo l o an roor arn. Drng m cor o Grn ncon roor arn a gvn m varo nrcv commn an ggon on Grn ncon n lac an polcrc ma n parclar on o rla o m-nn an bmaral ma. REFEREES. Tng T.. T. noropc Elac Oor Unvr r Oor Tng T.. T. Rcn Dvlopmn n noropc Elac n.. Sol Srcr 7 pp Tng T.. T. T Wonrl Worl o noropc Elac n Ecng Tm ar o V roc. 4 acc nrnaonal onrnc on ropac Scnc an Tcnolog pp. 7. T n ornal o Mcanc-Sr Vol. 9 o. Marc

6 4. So Z. o. M. arn D. M. an Wll. R. Fracr Mcanc or olcrc ramc. Mc.. Sol 4 pp arn D. M. an Lo. Dlocaon an Ln arg n noropc olcrc nlaor. Sa. Sol. b 67 pp ng M. Y. an Tng T.. T. olcrc Sol w an Ellpc nclon or Hol n.. Sol Srcr pp an E. EM nal o Fracr Mcanc n D noropc olcrc Sol Eng. nal. on. Elmn pp R. Q. Elb roblm or Two- Dmnonal olcrc nclon o rbrar Sap roc. R. Soc. Lon. 456 pp R. Q. Two-Dmnonal Elb roblm or Two on olcrc Hal-lan roc. R. Soc. Lon. 457 pp L Y. Fan H. an Yang. nal o Sar Sr Tranrr rom a arall Elcro olcrc caor o an Elac Sbra Smar Mar. Src. 9 pp Dng H.. n. an Lang. On Grn Fncon or Two-a Tranvrl oropc olcrc Ma n.. Sol Src. 4 pp Dng H.. Y. an Go F. Solon or Tranvrl oropc olcrc nn o Sm-Fn o an maral nn o Sbc o Unorm Rng Loang an arg n.. Sol Src. 6 pp Dnn M. L. an Wnc H.. Grn Fncon or Tranvrl oropc olcrc Sol n.. Sol Src. pp Dnn M. L. an Wnc H.. Hal-Spac Grn Fncon or Tranvrl oropc olcrc Sol. ppl. Mc. 66 pp an E. an Tonon F. Tr-Dmnonal Grn Fncon n noropc olcrc Sol n.. Sol Src. 7 pp an E. an Yan F. G. Tr-Dmnonal Grn Fncon n noropc olcrc maral n.. Eng. Sc. 8 pp an E. Mnln roblm or an noropc olcrc Hal Spac w Gnral onar onon roc. R. Soc. Lon. 458 pp an E. Som w Tr-Dmnonal Grn Fncon n noropc olcrc maral Elcronc ornal o onar Elmn n pr. 9. ama M. an Tanma. Grn Fncon o noropc olcrc roc. R. Soc. Lon. 45 pp Sro.. Dlocaon an rac n noropc Elac l. Mag. pp Sro.. Sa Sa roblm n noropc Elac. Ma.. 4 pp Trn H. F. Lnar olcrc la Vbraon lnm w Yor Yang. an an E. Tr-Dmnonal Grn Fncon n noropc Trmaral n.. Sol Src. 9 pp Glbr F. an ac G. ropagaor Marc n Elac Wav an Vbraon roblm Gop. pp an E. Sac Grn Fncon n Mllar Hal Spac ppl. Ma. Mollng pp Yang. an an E. Ecn Evalaon o Tr-Dmnonal Grn Fncon n noropc Elaoac Mllar ompo Eng. nal. on. Elmn 6 pp Yang. an E. an Twar V.. Tr-Dmnonal Grn Fncon n noropc olcrc Trmaral o b bm. 8. Yang. an E. an Twar V.. Tr-Dmnonal Grn Fncon n noropc olcrc Mllar Srcr bm. 9. Z. Y. Y Z. Q. Tam L. G. an Yang M. En Snon an M Solon or Mllar Elac Maral n.. Eng. Sc. 4 pp W.. an Y. T. T Elac Fl o a Dlocaon n an noropc Srp n.. Sol Srcr pp Y. T. an W.. nal or Elac Srp nr oncnra Loa. ppl. Mc. 65 pp an E. Sac Rpon o a Tranvrl oropc an Lar Hal-Spac o Gnral Srac Loa. Ear lan. nr. 54 pp an E. Sac Rpon o a Tranvrl oropc an Lar Hal-Spac o Gnral Dlocaon Sorc. Ear lan. nr. 58 pp T n ornal o Mcanc-Sr Vol. 9 o. Marc