Two-Dimensional Exact Solution for Multilayer Piezoelectric Cantilevers

Transcription

1 Research of Materals cece December, Volume, Issue, PP. 8 Two-Dmesoal Exact oluto for Multlayer Peoelectrc Catlevers Hu Hua #, Chuahou Do Des Brach of Cha Ralway Major Brde Eeer Co., td, Wuha 5, P.R. Cha #Emal: huahu76@6.com Abstract I our preset paper, the pla stra theory of trasversely sotropc bodes for study a peoelectrc mult-layer catlever has bee used, ad also order to fd the eeral soluto of a mult-layer peoelectrc catlever the verse method has bee obtaed. The stress ad ducto fuctos the form of polyomals as well as the eeral solutos of the beam have bee obtaed. Fally, based o the eeral soluto we have deduced the solutos of the catlever uder dfferet load codtos. Keywords: Peoelectrc; Mult-layer; Catlever; Elastcty I. INTRODUCTION ce the Cure brothers foud the peoelectrc effect 88, peoelectrc materals has bee wdely used sesor actuators. I varous eeer applcatos, may mportat achevemets have bee obtaed, such as the vestatos o peoelectrc ultrasoc motors [] vbrato cotrol of dfferet ds of structures [, ]. I eeer feld, these peoelectrc devces are usually desed as sadwch structures wth multple layers, to provde lare electrcal or mechacals outputs. However, the bod layer may crac at low temperature or peel off at hh temperature, whch wll reduce the lfe tme ad relablty of these smart materal devces []. I recet years, more ad more researchers have pad ther attetos o how to reduce the crac of these smart materals ad also someoe have made these materals used eeer more perfectly. To mprove the relablty, fuctoally radet peoelectrc materals have bee used to produce these sesors ad actuators [5]. Based o the theory of elastcty, some basc problems for fuctoally raded peoelectrc sesors ad actuators have bee solved, ad a set of exact solutos has bee obtaed [6-]. Assum a peoelectrc coeffcet d as a lear fucto the thcess drecto ad other materal parameters as costats, vestato of the BaTO FGPM actuator showed that, wth respect to a classcal b-morph, the defecto of the FGPM catlever actuators s oly slhtly smaller, whereas the teral mechacal stress s drastcally reduced []. I our preset paper we cotue to study multple layers peoelectrc catlever uder dfferet ds of loads, clud the affecto of the temperature. To et the eeral solutos, ths paper we assume the costats ad parameters of every layer s dfferet, t meas elastc complace j of layer s j, ad also the peoelectrc j electrc mpermeabltes j, ad thermal stra costat, pyro-electrc coeffcets q,are deoted by the same way. We have obtaed the stress ad ducto fuctos the form of polyomal as well as the eeral solutos of the multple layer catlevers, whch the sem-verse method s used. II.BAIC EQUATION FOR MUTI AYER PIEZOEECTRIC MATERIA The plae stra (or stress) theory of trasversely sotropc peoelectrc materals s referred to a Cartesa coordate system x ad.ett j, j, D ad E as the compoets of stra, stressucto ad electrc fled, respectvely, the two-dmesoal costtutve equatos ca be wrtte the follow forms []. x x D T x D T x x Dx - -

2 Ex x Dx () E x D qt Where j, j ad j are costats of the effectve elastc complace, peoelectrc ad delectrc mpermeabltes, respectvely. T s the temperature rse, ad q are the thermal stra costat ad pyro-electrc coeffcets of the materal. The relatos of the stra ad dsplacemet compoets (u ad w ) as well as the compoets of electrc fled potetal are the follow forms: u x x Ex w x () E u w x x Wthout cosder body chare, the equlbrum equatos are ve by: x x Fx x x F () x Dx D x Where F x ad F are the compoets of body force. Ad the compatblty equatos ca be expressed by the compoets of stress ad ducto as: x x x x () Ex E x I steady state, the temperature fled s overed by Fourer s heat codto equato T T (5) x Us equatos () ad (), the above compatblty equatos ca be expressed by the compoets of stress ad ductos as: x x x D D Dx x x T T Fx F x x x D x x x D T Fx q x x (6) - -

3 I ths case, the weht of the oly body force ad t has a potetal.thus we have V Fx ; F (7) By symmetry, the heat codto equato (5) ca be smplfed to: T T T ; (8) x x Where V V( ) ad T T are the potetal fucto. By troduc the stress fucto ad troducto fucto, the compoets of stress ad ducto ca be descrbed as: x Dx (9) V x D x x x ubsttut expresso (9) to equato(6), the compatblty equato ca be rewrtte as x x V () x x x x x Thus the soluto of a desty fuctoally radet peoelectrc materal reduces to fd a soluto of equato () that satsfes all the boudary codtos of the problem. Ⅲ. GENERA OUTION FOR A MUTI AYER PIEZOEECTRIC I eeral, the potetal fucto V ad the temperature fucto T ca be represeted by a power seres. I the follow aalyss we tae V (-) T t t (-) Where (,,,) ad t, are materal costats. Furthermore, order to fd the eeral soluto for a peoelectrc mult-layer catlever the stress ad ducto fuctos are assumed to be the form of polyomals ad be expressed as: x b b b b xc c c5 () 5 d d d d x A A xb B BB D D D 5 Where coeffcets,, b c d ad A, B, D are costats whch wll be determed later. Wthout lose of eeralty t s assumed that the N-layered (wth the same coeffcet of the thermal coductvty ) are complete cotact at the terface. Frst the cotuty codtos at the terface ( h,,, ) ca be cosdered as follows: T( ) T( ) T( ) T( ) () - - h (,,, )

4 ubsttut Eqs.(-), () the follow result ca be obtaed as: t t t t t t Where the temperature boudary ca be assumed as: at the bottom of the sadwch beam the temperature st, ad at the upper surface the temperature st, ad the to smply the problem we have studed recetly, we assume the temperature at the bottom ad the upper of beam are both statc. The we ca et the follow expresses as: T T t (5) h t T ubsttut equatos () to equato (), oe fds that equato () ca be satsfed oly f d b B d b B c a D A (6) c a D A b B b B ubsttut equato () to (9) we ca et as x x 6b bx6c cd d 6dd b b b b (7) x xb bbc cc5 Dx Ax xb BBD DD5 D xa AB B BB ubsttut equato (7) to () ad () we ca et as: x x 6b b 6c c b b b u 6 x b x d d 6dd x A A xb B B B x tt c c W A A c c c5d D D5 (8-) 6 x b b x c c b b w 5d d d d b b x A A B B B B t t bx cx dx b x A x Bx t x b x A x B x x H x x x b b b A B BB xc c c5 xd D D5 (8-) b b b q t t 5d d d d b B B B BM - - ()

5 Where,W, H ad M are the costats wll be ve later. Equato (8) s the eeral soluto of peoelectrc mult-layer catlever.the costats bcd,, ad A, B, D ths eeral soluto should be determed aalyss of cocrete case. Wthout los the eeralty, ow we deote every parameter whch has bee show above dfferetly. FIG. ANAYTICA MODE For coveece, the dsplacemet ad electrc potetals, stressucto at the upper ad the lower surfaces of layer are deoted by ( u U, U, U, U, U, U w D ) ad ( u,,,,, w D ) as show FIG. respectvely. For a sadwch catlever the dstrbutos of stress ad ducto should satsfy the boudary codtos every layer s terface. U U U ; u u ; w w (Where,, ) (9) U U U ; ; D D Here as follows aother ducto boudary at x ad x. h D d ) () (Deote x h ubsttut equatos (9) to equato (8) above all we ca et: (see appedx A) b b; b b b b b b b b b b b b b b b A A A A A A A B B b; B B b B B b B b B b B B B b B b B b B B B c c A c c c A c A c c5 c 5 A c 5 A c 5 c c 5 c c 5 d d b d ; d d b d d d b d b d b d B d t d d d d b d b d b d b d B d B d d (-) (-) (-) (-) (-5)

6 D D A D c D D A D A D c D c (-6) D5 D 5 A D 5 A D 5 c D 5 c D 5 M M bm bm b M b M B M B 5 6 M 7 d M 8 d M 9 t M t M M c c A A (-7) W W cw cw AW A W 5 D5 W H H b H b H b H b H 5 B H 6 B H 7 d H 8 d H 9 t H t H Equato () s the relatos of every costat we wat to ve later. The result boudary codto wll be expressed as: x, () w, u, w, x From equatos (8), () we ca et M d, A D5, b B b c d b A B t W 6 () b c d b H A B t b A B b c 6d b A B t b A B FIG. IMPE BENDING by a COUPE FORCE FIG. IMPE BENDING by a CONCENTRATED FORCE FIG. IMPE TENION by a CONCENTRATED FORCE FIG.5 BENDING by a UNIFORM OADING - 6 -

7 FIG.6 TENION by a UNIFORM OADING FIG.7 REPONE to a APPIED VOTAGE Ⅳ.OUTION FOR MUTI AYER PIEZOEECTRIC CANTIEVER UNDER DIFFERENT OADING CONDITION Based o the eeral soluto obtaed above, a seres of fudametal problems ca be solved, whch cludes ot oly the smple bed of a peoelectrc sadwch catlever by a couple but also the smple teso ad bed of a peoelectrc sadwch catlever by a cocetrated force. Moreover, the resposes of a peoelectrc sadwch catlever to a uform load, appled axally ad laterally o the upper surface of the beam, or to a appled voltae are also studed. To mae the expresses of the costats we wll obta below coveetly, the we deote the follow: h h h h (-) h h h h h h P h b hb P h b b (-) P h b ; P hb P5 b h A. mple Bed of a Catlever by a Couples For a catlever whch s bet by a couple, as show F., the dstrbutos of stress ad ducto should satsfy the boudary codto exactly. h h xd xd h h xd xd M (at x ) (5) h h xd xd

8 Ad the x at the bottom ad top surface of the catlever; the ducto at the top surface must D x, h ; ad as follows at the bottom surface must satsfy the boudary D x,. ubsttut equato (8) to all the boudary codto above, we ca et: A A c c c (5-) 5 satsfy the boudary codto as: codto as: b P P 5 P ; b P P P 5 P P P P P P P P (5-) b ; b The the costatsb, b, B, B have bee obtaed. B ; B (5-) Now we et the follow costats B, B, D, as the follows we deote: Use the equato (5), ow we ca et: b h b h B b B t b b h b h b B B t b d h d h 6d h b h b h b hb B h B h B hb d h d h 6d h b h b h b hb B h B h B hb (5-) 5d h h d h h h h h h 5 d h h d h h h h h h h h 5 h 5 h h h h 6 h (5-5) Now we ca et:

9 d M M ; d M d W bd 6 t b 6d t H b d t B. mple Bed of a Catlever by a Cocetrated Force (5-6) (5-7) For a catlever whch s bet by a cocetrated force, as show FIG., the dstrbuto of stress ad ducto should satsfy the boudary codto exactly. h h xd xd F h h xd xd (atx ) (6) h h xd xd Ad the x at the bottom ad top surface of the catlever; the ducto at the top surface must satsfy the boudary codto as: Dx, h ; ad as follows at the bottom surface must satsfy the boudary codto as: D x,. ubsttut equato (8) to all the boudary codto above, we ca et: A A c c c5 (7-) P5 P P P5 P P b ; b (7-) P P P P P P P P b ; b The the costats b b B B have bee obtaed. B ; B (7-) Now we et the follow costats B, B, D, as the follows w e deote: Use the equato (5), ow we ca et: F F 6 5 d ; d (7-) M d W bdt 6 (7-5) b 6d t H b d t C. mple Teso of a Catlever by a Cocetrated Force For a catlever tesoed axally by a cocetrated force, as show FIG., the dstrbutos of stress ad ducto should satsfy the boudary codto exactly.

10 h h xd xd h h xd xd h h x x - - d d F (at x ) (8) Ad the x at the bottom ad top surface of the catlever; the ducto at the top surface must satsfy the boudary codto as: Dx, h ; ad as follows at the bottom surface must satsfy the boudary codto as: D x,. ubsttut equato (8) to all the boudary codto above, we ca et: A A c5 (9-) F F c ; c (9-) P5 P P P5 P P b ; b P P P P (9-) P P P P b ; b The the costats b b B B have bee obtaed. B ; B (9-) Now we et the follow costats B, B, D, as the follows w e deote: Use the equato (5), ow we ca et: 6 5 d ; d (9-5) M d W b cd 6 t (9-6) b c 6d b t H b c d b t D. Bed by a Uform oad For a catlever whch s bet laterally by a uform load, as show FIG.5, the dstrbutos of stress ad ducto should satsfy the boudary codto exactly. h h xd xd h h xd xd () h h xd xd (at x ) Ad the x at the bottom; at the top surface of the catlever q, x ; the ducto at the top surface must satsfy the boudary codto as: D x, h ; ad as follows at the bottom surface must satsfy D x,. the boudary codto as:

11 ubsttut equato (8) to all the boudary codto above, we ca et: A A c c c (-) 5 P P q P q P 5 5 P P b ; b P P P P (-) P P P P b ; b The the costats b b B B have bee obtaed. B ; B (-) Now we et the follow costats B, B, D, as the follows w e deote: Use the equato (5), ow we ca et: E. Tes by a Uform oad d ; d M d W b cd 6 t b c 6d b t H b c d b t - - (-) (-5) For a catlever whch s tesoed axally by a uform load, as show FIG.6, the dstrbutos of stress ad ducto should satsfy the boudary codto exactly. h h xd xd h h xd xd (at x ) () h h xd xd Ad the x at the bottom; at the top surface of the catlever, x q ; the ducto at the top surface must satsfy the boudary codto as: D x, h ; ad as follows at the bottom surface must satsfy the boudary codto as: D x,. ubsttut equato (8) to all the boudary codto above,we ca et: A A (-) q q c ; c P5 P P P P b ; b P 5 P P P P (-) P P P P The the costats B, B have bee obtaed. b ; b ; B B (-) Now we et the follow costats B, B, D, as the follows w e deote: Use the equato (5), ow we ca et:

12 d ; d M d W b cd 6 t b c 6d b t H b c d b t P P P P - - (-) (-6) F. Respose of a Catlever to a Appled Voltae For a peoelectrc catlever subjected to a voltae, as show FIG.7, some boudary codtos whch used above are also vald, ad the follow boudary codtos should be obtaed as: h h xd xd h h xd xd (at x ) () h h xd xd Ad the x at the bottom ad top surface of the catlever; the ducto at the top surface must satsfy the boudary codto as: x, h V ; ad as follows at the bottom surface must satsfy the boudary codto as: D x,. ubsttut equato (8) to all the boudary codto above, we ca et: A B c5 (5-) P5 P P P5 P P b ; b (5-) P P P P b ; b I the follow equato (6) all the costats are appedx B. tc tc td td td td tc tc td td V The from the equato (6) the result costats we eed to solve are obtaed as the follows: t t t t t t t t t t V t t t t V t t c ; c t t t t t t t t t t t t t t t t t t t t (6)

13 d t t t t t t t t t t V t ; d t t t V t t t t t t t t t t t t t t t t t t t t Ⅴ.CONCUION Based o the theory of elastcty, the preset aalyss provdes a exact soluto of a peoelectrc multple layers catlever uder dfferets loads. It s foud that wth the crease of the umber of layers, the dscotuty for the ormal stress the terface of every bod ca be sfcatly reduced. It s demostrated that the solutos of the N-layered catlever are coveret to those for the case of fuctoally radet catlever. (7) REFERENCE [] Phlppe, Bouchlloux ad ej, Vcho. Combed Fte Elemet Aalyss-Geetc Alorthm Method for the Des of Ultrasoc Motors, Joural of Itellet Materals ystems ad tructures, (): [] Clar, Wllam W.. Vbrato Cotrol wth tate-swtched Peoelectrc Materals, Joural of Itellet Materals ystems ad tructures, (): 6-7 [] Corr,. R ad Clar, W. W.. Eery Dsspato Aalyss of Peoelectrc em-actve Vbrato Cotrol tructures, tructural Dyamcs ad Materal Coferece. eattle. W. A, Aprl 6-9 [] Zhu, X.H., Wa, Q. ad Me, Z.Y A Fuctoally Gradet Peoelectrc Actuator Prepared by Powder Metallurcal Process PNN-PZ-PT ystem, Joural of Materals cece etters (7): [5] Qu, J.H.etal. Fabrcato ad Hh Durablty of Fuctoally Graded Peoelectrc Bed Actuators, mart Mater. truct : 5- [6] h, Z.F.,. Geeral oluto of a Desty Fuctoally Gradet Peoelectrc Catlever ad ts Applcatos, mart Mater. truct. : -9 [7] u, T.T. ad h, Z.F.. Bed Behavor of Fuctoally Gradet Peoelectrc Catlever, Ferroelectrcs 8:-5 [8] h, Z.F. ad Che, Y. Fuctoally Graded Peoelectrc Catlever uder load, Arch. Appl. Mech. 7: 7-7 [9] Che, Y. ad h, Z.F 5 Exact olutos of Fuctoally Gradet Peo-thermal-elastc Catlevers ad Parameter Idetfcato, Joural of Itellet Materals ystems ad tructures, 6: 5-59 [] h, Z.F 5 Bed Behavor of Peoelectrc Cured Actuator, mart Mater. truct. : 85-8 [] Huae, T. Kouvatov A. tehause R. efert W. Bee, H. ahammer H.T. ad Abcht H. Bed Behavor of Fuctoally Gradet Materals, Ferroelectrcs 8: 95- [] Wu, X.H, he, Y.P., Che, W.Q. A Exact olutos for Fuctoally Graded Peothermalelectrc Cyldrcal hell as esors or Actuators, Mater, ett. 57: 5-5 [] Marcus, M.A. 98 Performace Characterstcs of Peoelectrc Polymer Flexure Mode Devces, Ferroelectrc 57: - [] Kouvatov A, tehause R, efer W. Haue T, ahammer H.T, Bee H ad Abcht H 999 Comparso betwee B-morphc ad Polymorphc Bed Devces, J.Eur. Ceram.oc.H. 9: 5-56 [5] Z.F. h, H.J. Xa, B.F. pecer, Jr 6 Exact Aalyss of Mult-layer Peoelectrc/Composte Catlever, mart Materals ad tructures 5: 7-58 APPENDIX A I ths secto all the expressos equato ca be deoted as: b h b b h (A) - -

14 b h b h j b j j h j A (A) A h B B (A) B B B h b b B B B h b b B B B B h B B h B B B h B B h B B B h B h c A A (A7) c c c A A h A A (A8) c A c5 c c h c c h c 5 c c h (A9) c5 h h c 5 h d B (A) d d B (A) d - - (A) (A5) (A6)

15 d h b B d h b B d B d d 6 d 6 (A) ( ) d B d B ( ) h d B 6d B b b 6 d B b b h ( ) d B d B ( ) h d B 6d B b b 6 d B b b h d 6 d B 6 d B h B d d 6d 6 d h d d d d h d d d h d d h d d h b j j jh j h j j D A c D (A) (A) D A c D c D c D (A5) - 5 -

16 D c D c D h D c D c D D D h 5 c D c D h c 5 5 D h c 5 5 c D c D h c D D D h c A c h c h h A h A c h c h h A h c c h h c h h c h h B B M 5 d 5 d h b B b B d d h b d B b d B h B M d d h b B b B d d h b d B b d B h B B M d d h d B d B h M d B d B h M 5 d d h B B h M 6 b b h M 7 h ; M 8 h q q q q M 9 d d h ; M h M 5 d 5 d h d d h d d h d d h (A6) (A7) (A8) - 6 -

17 W A h A h Dh Dh 6 c h c h c h c h c 5h h A h A h D h D h 6 c h c h c h c h c 5 h h W h D D h c h c h c 5h h c h c h c 5 h h W D D h D D h h c h c 5h h c h c 5 h h W D D h h h c 5 c 5 h h W 5 h H 5dh dh dh 5d h d h d h B B h b h b h h h B h B h b B h b h h h B h H dh dh dh d h d h d h B h b h B b h h h B h B B h b h b h h h B h H d h d h d h d h B B h h B h h B h H d d h H 5 d h d h h B B h H 6 h; H 7 h H 7 h; H 8 d d h H 9 h H 5d h d h d h d h 5d h d h d h d h h h h b h h h h b ah (A9) (A)

18 APPENDIX B I ths secto all the expressos equato 6 ca be deoted as: t D h D hd (B) 5 t D h D 5 (B) t h h t h h ; t h h (B) t h h ; ; (B) t h c 5 D h D hd (B5) 5 (B6) t hc 5 D hd 5 t h M ; 7 t h M (B7) 8 5d h d h d bd bd t d h d b d b d b d h d h d h d bd bd t d h d b d b d b d h b h b h b B h B h B b B b B b 8 b h b h b h b h 5d h d h q th t h M b M b M b M t M t M d bd bd bd t d h d b d b d b d b d h B h B h B b B b B b h B b B b B b h 9 (B8) (B9) (B) AUTHOR Hu Hua, male bor Chedu, chua Provce, Cha, Nov.98. Graduated from Bej Jaoto Uversty, whch s located Bej, Cha, 5.Master of cece Eeer. Maly eaed the study of brde. He s ow wor Wuha, Hube provce Des Brach of Cha Zhote Major Brde Eeer Co.,td. Chuahou Do, male bor Zhja, Hube Provce, Cha, ep.978. Graduated from Wuha Uversty of Techoloy, whch s located Whua, Cha, 5.Master of cece Eeer. Maly eaed the study of brde. He s ow wor Wuha, Hube provce Des Brach of Cha Zhote Major Brde Eeer Co.,td