1962 - Wave Propagation in an Infinite Piezoelectric Plate

8/3/2019 1962 - Wave Propagation in an Infinite Piezoelectric Plate

http://slidepdf.com/reader/full/1962-wave-propagation-in-an-infinite-piezoelectric-plate 1/6

THE JOURNAL O1, THE A•_OU•TICAL SOCIETY ()F AMERICA VOLUME 35, NUMBER 2 FEBRUARY 1963

Wave Propagation in an Infinite Piezoelectric Plate

H. F. TIERSrEar

Bell Telephone aboratories,nc., Whippany,New Jersey

(Received 26 June 1962)

The problem f wavepropagationn an infinitepiezoelectriclatebelongingo crystallographiclasssituatedbetween horted lectrodes,s rigorously nalyzed or two special rientations f the sixfoldaxis.The solutionsderived sing he inearpiezoelectricquations. he analysis howshat [or a given requencyand wave number n the propagationdirection here are three independent olutionsof the differentialequations, nd, furthermore,hat these hree solutions ouple t the traction-free oundaries f the plate.The dispersion urves an be calculated rom the resulting ranscendentalquations.t haspreviously eenshown hat the resonant requenciest infinitewavelength f a plate, n which the electromechanicalou-pling factorsare high,deviateconsiderablyrom thoseof the purely elasticsolution. hus the dispersionspectrum alculatedrom this analysiswill deviateconsiderablyrom that of the purely elasticanalysis.It is shown n an appendix hat the solution or the plate of infinitewidth is alsoapplicablen the easeofthe plate of infinitesimalwidth.

INTRODUCTION

AVE ropagationna homogeneousnfinitelaslicplate has been he subjectof very muchinvestigationover the years.Rayleigh gave he solutionfor an isotropicelastic plate, and Ekstein ' gave thesolution or a particularanisotropic monoclinic) lasticplate. In recent years, much work has been done incalculating he roots of Rayleigh's -• and Ekstein's-•'•transcendentalrequencyequations n order to deter-mine the resultingdispersion elationsand solvehigh-frequency vibration problems. These solutions and

calculations re applicable o purely elasticplates andto plates which obey the linear piezoelectric quations,but have low electromechanical oupling factors. How-

t Lord Rayleigh, Proc. Loud. Math. Soc. 20, 225 (1888-1889).a H. Eksteln, Phys. Rev. 68, 11 (1045).a H. Lamb, Proc. Roy. Soc.A93, 114 0917).• A. N. Holden,Bell System ech. J. 30, 956 (1951).• R. D. Mindlin, "An Introduction to the Mathematical Theory

of Vibrations of Elastic Plates," U.S. Army SignalCorps Eng. Lab.,Fort Monmouth,N.J. (1955).

• M. Ouoe,"A Study of the Branches f the Velocity-DispersionEquations of Elastic Plates and Rods," Rept. Joint Committeeon Ultrasonics of the Inst. of Electr. Commun. Eng. and theAcoust.Soc. of Japan (1955).

* I. Tolstoy and E. Usdin. J. Acoust.Soc.Am. 29, 37 (1957).a E.G. Newman and R. D. Mindlin, J. Acoust. Soc. Am. 29,1206 (1957).

0 R. K. Kaul and R. D. Mindlin, J. Acoust. Soc. Am. 34, 1895(1962).

• R. K. Kaul and R. D. Mindlin, J. Acoust.Soc. Am. 34, 1902(1%2).

ever, certain polarized erroelectric era•nics e.g.,B,T,Oa, PZT-5, PZT-7) which obey the linear piezo-electric quations avehigh electromechanicalouplingfactors;consequently,he aforementionedolutions reinadequateo describe avepropagationn theseplates.The failure of the elasticequationswhen the electro-mechanical ouplingactorsare high s deafly pointedout n a previous aper which reats he special ase fpure thickness ibrations.

In this paper, the rigoroussolutionof the problemof wavepropagationn an infinitepiezoelectriclate sderived from the linear piezoelectric quations.Thesurfaces f the plate are traction-free nd completelycoated with electrodes which are shorted. No restriction

on the relativemagnitudeof the elastic,piezoelectric, rdielectric onstantss imposed.Consequently,he solu-tion will be valid for materials with high electromechani-cal couplingactors.Two special ases re considered,one with the ferroelectricceramic polarized in thethickness irection, nd the otherpolarizedn the planeof theplate. n each nstance transcendentalrequencyequation s derived, he rootsof whichdetermine hemodes f propagation nd the dispersionelationsn an

unbounded late and may be used to determine hemodesof vibration in a boundedplate. The modesoffree vibration in a boundedpiezoelectricplate may be

n H. F. Tiersten, "ThicknessVibrations of PiezoelectricPlates,"J. Acoust.Soc. Am. 35, 53 (1963).

234

d 14 Jul 2011 to 132.210.72.142. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info



WAVE PROPAGATION IN INFINITE PIEZOELECTRIC PI. ATE 23.5

determine:ldirectly from the solutionpresented erebymeans of various approximatemethods,such as onesuggestedy Mindlin andanotheremployed y Onoe?The soluton shouldalso prove useful n the develop-ment of approximate two-dimensional heories 4a5ofvibration •f suchplates. n addition, this solutionmayreadilybe combinedwith the thickness olutiongiven na previous apern to obtainan approximate teady-statesolution or a boundedplate forcedelectrically.Thisforced solarion will be discussed in detail in a forth-

comingpaper.It is shown n the Appendix hat the solution or the

plate of in,hire width is alsoapplicable n the caseof theplate of irfinitesimal width, provided certain modifica-tions n th • elastic,piezoelectric, nd dielectricconstantsare made.

I. GENERAL EQUATIONS

A rectaugularCartesian oordinate ystem •, i= 1, 2,3 is chosenwith xa=:hh defining he facesof the platewhen t is poled n the thickness irection,and x•= =[:hwhen t is poled n its plane. An indexpreceded y acommadenotesdifferentiationwith respect o a spacecoordinate, i.e.,

Uc• Our/Oxi.

The summation onvention' for repeated ensor ndicesis employed,as is the dot notation for differentiation

with resp(ct o time.The systemof equations overninghe behaviorof

this plate consists f the following:

The stressequationsof motion:

T,¾a=p•i. (1)

The chargeequationof electrostatics:

D•a=O. (2)

The stralt[mechanical isplacementelations:

= « (3)

The electric ield-electric otential relations:

Ea= -- •,•. (4)

The linear,piezoelectric onstitutive elations:

Tij •-•- ijklES l - e ij]• k,(s)

•2R. D. ldindlin, "Investigations n the Mathematical Theoryof Vibratior s of Anisotropic odies,"CU-4-56-SC-64687-CE, :inal

Rept., U.S. Army Signal Corps Eng. Lab., Fort Monmouth,N.J. (1956•.n M. Once,J. Acoust.Soc.Arm 30, 1159 (1958).•4R. D. Mindlin, Quart. Appl. Math. 19, 51 (1961).• H. F. Tiersten and R. D. Mindlin, Quart. Appl. Math. 20,

107 (1962).• H. Jeffreys,CartesianTensors CambridgeUniversity Press,

New York, 1931, reprinted1952).

where, in the above, To', ui, D;, S;s, E• are the com-ponentsof stress,mechanical isplacement,lectricdis-placement, train and electric ield, respectively; and• are the massdensityand the electric otential, espec-tivdy; and c•i•/, e•c;, $ are the elastic,piezoelectric,and dielectricconstants, espectively.

The facesof the infinite plate are traction-freeandcompletelycoated with electrodeswhich are shorted.Under these circumstances,he boundary conditionsbecome

Ta•=0 at xa=4-h,(6)

½=0 at xa=-bh,

for the plate poled n the thicknessdirection; and

Ti•=0 at x•=4-h,(7)

½=0 at

for the plate poled n its plane.Note that Eqs. (3) and(4) may be substitutedn (5), whichmay then be sub-stituted n (1) and (2) to obtain our equationsn thefour dependent ariablesu• and •.

The arrays ?of the elastic,piezoelectric,nd dielectricconstants or a ferroelectricceramic poled in the xsdirection are:

ei•=!l•11

•ij S= 0o

c•=•a0

n c•a 0 0

c•a caa 0 0

0 0 c. 0 , (8)0 0 0 6440 0 0 0 c•

=

0 0 0e•00 e•s 0 0

ea• eaa 0 0 0

011

o

wherep or q= 1, 2, 3, 4, 5, 6 may be replaced y 11, 22,

33, 23 or 32, 31 or 13, 12 or 21, respectivdy. Uponsubstitution f (8), (3), and (4), Eqs. (5) become

T•=

T2= c•2uL•q- nu2,2q-c•aua,aq-ea•,a

T6: 644u3,1+443tl,+ e15½,, (9)

Da= eatUl,x+ ea•U=,= eaaUa,a•

•7 "I.R.E. Standards on Piezoelectric Crystals: Measurementsof Piezoelectric Ceramics, 1961," Proc. IRE 49, 1161-1169(July 1961).

14 Jul 2011 to 132.210.72.142. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/inf



WAVE PROPAGATIOX IN INFINITE PIEZOELECTRIC PLATE 237

a point on _he ispersionpectrum ndyieldsamplituderatios (C(:•:Ca•:C o•) when substituted n (19). Inprinciple, his completeshe analyticalsolution.

For puq>osesf calculation,t is conveniento writethe pertinentequationsnvolved n the calculationn

termsof certaindimensionlessuantities. he pertinentequationsre (15), (16), (19), (20), and (21).They maybe written respectivelyn the followingway: (22)

(e,•-y*-n- •aot*---(•44-[- 13)•O/

(•la-•- 44)•0t(c•4• + ca• - (•,•,-+O•a •) = O,

-- (•n•,2+ •a •)(23)

3 7r

• C(øL, ø sin•odo0,

3 •I'

Y] C(•)L, i) cos at') • (24)

3 7F

• C")/• (ø sin-a(o=0,•-t 2

(25)

M (u cot--a½øq-M-*) ot a(-ø)q-.•(a)cot-a(a)=0, (26)2 2 2

where he following elationshave beenused:

•/= 2•h/r, a= 2•lh/*r, •= B(•aa/c•O,

ev•=cv,,./c,a, •i•,=eiv/(c,e•a) , •q=•o/eaa, (27)

[•=•1•, &=(•/2h)(c44/p)t;and

]I • =L•(o(L,(• • -L, •

M(•)=L•("(L,(')•,(•)-L•(•)•,(')), (28)

The rehtiou between and T, i.e., the dispersionrelation, s wanted.This relationmay be obtained romEqs. (22)- (28) by numericalmethods. straightforwardprocedureo• calculating he dispersionelationbeginsby selecting valueof % The furtherchoice f • permitsthe calculition f the •(o from (23) and he •?) from(22),so hat theL] (o and a(o maybecomputedrom(25). The M (ø may now be determined rom (28). Ifthe values husdetermined atisfyEq. (26), the selectedvaluesof • and • constitutea point on the dispersioncu•e. If Eq. (26) is not satisfied,epeat he calculation

for different valuesof • until Eq. (26) i• salishcdand aroot has ')een obtained. When a su•cicnt number of

valuesof-, and • satisfyingEq. (26) havebeenobtained,the dispersion urvesmay be plotted.

The aforementioned et of solutionsand dispersioncu•es completelydetermines he longitudinal modesfor this p•ezoe]ecLriclate. Another ndependent et of

solutions an be obtainedby interchangingost/xa ndsint/xan Eqs. (14) and (18), sinn(øh nd cos•/(øhnEqs. 19)and 21),simra(ø/2ndcosrra(i)/2n Eq. (24),

andreplacingotangenty tangentn Eq. (26). Thissetof solutionsields ispersionurves hich re nde-pendent f those f the othersetanddeterminehe11exuralmodes or the plate.

III. SOLUTION AT NONZERO FREQUENCYAND INFINITE WAVELENGTH

The solution it infinite wavelength an bc obtained

directly rom the propagatingolution resentednSec.I simply y setting=0. However,heprocedureis not quiteas straightforwards onemightexpect.

Consequently,hedetailswillbepresentedere.When} is set equal o zero, Eqs. (15) and (16) become,respectively,

(c.,:m-- p•)A a+ eaan•B0, (29)

eaa•A s-- •aa• B = 0,

j o =o. (30)eaa• • eaa•-

FromEq. (30) one eadilyseeshat the threen(• are

givenby

(. ",n )= (31)

whereeaa=caa+eaae/eaa.im substitution f the three•(o successivelynto Eqs. (29) yields

10 (32)where he column efers o the •o and the row to the

ui or •. Equations 19) nowbecome

CO•c4• •o sin•O)h= 0,C • ess•m)cos• x h

(33)+C (a) caa+eaff/eaa)•a)cos•(a)h=O,

C (s•sin•(•h+C (a• eaa/eaa)inetach= .

The last two of Eqs. (33) sho•v that as ,/(a)__•O,

d 14 Jul 2011 to 132.210.72.142. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info



238 H. F. TIERSTEN

Ce-) -, oosince in,/Ca)hnd cosneathannotequalzerosimultaneously. owever, C (ø-) nd n(•) alwaysoccurasan ndete•inateroduct.ence, qs. 335maybewritten

Keaa+Cta}eaa. ) cos.(a'•h 0, (34)Kh+C ta) eaa/eaa)in,{a)h 0,

where K•C(•), (•). Equations (34) yield a nontrivialsolution when

c4•) in•mh 0eaa eaa•a•cos•{a•h =0. (35)I 0 h (eaa/eaa)in•mh

Equation (35) yields two transcendental quations,eachof which determinesa set of roots. The two equa-tions may be written

sinutab=0, tangtaCh=(aJk./kt", (36)

wherekt = (eaa•/eascaa).

Equation 36)s s denticalwith Eq. (4.9) ofapreviouspaper. Thus t is seen hat oneset of modes t infinitewavelengthgiven by the solution presentedhere isidentical with the piezoelectricmodesgiven by thethicknessolution resentedhere.Equation 36) yieldsmodes that are identical with the purely elastic,symmetric, hickness-shear odeswhich were not con-sidered n the aforementionedpaper since they corre-spond o ui even, and cannotbe excitedelectrically nan nfiniteplate with infiniteelectrodesnbothsurfaces.

IV. PLATE POLED IN ITS PLANE

The solution of this problem follows the analysispresentedn Sec. I step or step. n fact, onecanobtainthis solution directly from the solution presented nSec. I by performing he followingoperations:

1. Interchange and • in Eqs. (15) and (16).2. Interchangex• and Xa n Eqs. (14) and (18).

3. ReplaceEqs. (20) by

(37)

4. Interchange and a in Eqs. (22) and (23).5. ReplaceEqs. (25)by

(0 C O) (t) (,) (0 ( )

All equations ther han thosementionedemainexactlyas they appear n Sec. I. The solution t infinitewave-lengthcan be obtained rom the aboveby followingprocedure irectly analogouso the one outlined nSec. III.

The above mentioned set of solutions dete•ines the

flexural modes or the plate poled in its plane. The

operations rescribedt the endof Sec. I may be per-formedon the solution resentedn this section o obtaina set of solutionswhich are independent f the above,and which determine the longitudinalmodes for theplate.

V. SUMMARY

The analytical solutionof the problemof wave propa-gation in an infinite piezoelectricplate belonging ocrystallographic lassC6• locatedbetweenshortedelec-trodes has been given. Two special cases were con-sidered; one with the sixfold axis in the thicknessdirection, and the other with the sixfold axis in the

planeof the plate.The solution resented erehasbeenrigorously derived from the linear, piezoelectricequa-tions, and consequentlys valid even when the electro-mechanical oupling s high. The analysisshows hatfor a given requency nd wave number n the propaga-tion direction there are three solutions of the differential

equations, achof which s givenby a root of a cubicequation.The analysis urther shows hat all threesolutions re required o satisfy he traction-free ndelectricpotential-free onditions t the surfaces f theplate. The three solutionsare coupled through atranscendentalquation, he rootsof which determinethe dispersionelation.The two setsof modes t infinitewavelength given by this solution have been deter-mined, and one set is identical with the piezoelectricthicknessmodesobtained n a previouspaper? In that

paper, the resonant requencies f thickness ibrationof a plate having high piezoelectriccouplingwereshown o depart markedly rom thosepredictedby thepurely elastic heors'.Hence, it is clear that the dis-persion spectrum resulting from this analvsis willdeviate considerablyrom that predictedby the purelyelasticanalysis.

ACKNOWLEDGMENTS

The author would like to thank Professor R. D.

Mindlin of Columbia University and Dr. M. Onoe for

stimulatingdiscussions,nd Dr. J. E. May, Jr., Dr. A.

H. Meitzler, and G. A. Coquin or carefully eading hetext and offeringvaluablecriticisms nd suggestions.

APPENDIX

This paper has consideredhe propag•rtion f wavesalong he lengthof a plate of infinite width and infinitelength.The solutions resented re alsoapplicablewhenwaves are propagating along the length of a plate ofinfinitesimalwidth (i.e., reD, narrow) and infinitelength,providedsomechangesn the elastic,piezoelec-tric, and dielectric constantsare made. These changeswill be derivedand explained n this appendix.

Since all the coordinate systems utilized in theanalyses lace he x• coordinate xis n the width direc-tion, the additional faces of the plate are taken atx.,.=d=w. t canbe reasonedhat the properassumptions

ed 14 Jul 2011 to 132.210.72.142. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/



WAVE PROPAGATION IN INFINITE PIEZOELECTRIC PLATE 239

for a plate in whichw/h¾<lare

To_= 4= T•= D2= 0, CA1)

alongwith

z'/2= . CA2)

Assumption A1)• permits the eliminationof uo_.o_romthe constitutive relations (9) with a consequent e-combination of elastic, piezoelectric, and dielectricconstants. t seemsappropriate to refer to these re-combined onstantsasplate constantsnasmuch s theyreduce o Voigt'selastic onstantsvhen he piezoelectricconstants anish;consequentlyhey will be denotedbythe supers:riptP. When the assumptionsA1)•, (A1)a,and (A1)• are also introduced nto the constitutiverelations, neobtains n placeof (9)

T1: 611PNl,l•ClaPNa.

Oa: ealP• ,• eaaPz•a,a

T2= T4- T• - Ds = 0,

wNcFc

611P-- • 11--6122/611),

ea• = (ea•--Clo_ea•/c•O,

eaa = (eaa--claeal/cn),

Cla : (618--612618/611),

eaa -- eaa-•ealo_/611.

Inserting Eqs. CA3) in (1) and (2) and consideringassumption A2),oneobtains n placeof (10) and (11)

611Pltl,1i-c laPlta,al-}-a1Pf alq-C14z•a,la

-J-½441gl,aa-}-lS<f,a=

5441t3,11•-C441•1a1+e•a•11• c•aPzgL1a

elsUaAl+ lauLa•--

+eaa•ua.aa- eaa•,aa= 0. CA6)

The bouudarv conditions (12) or (13), whichever s tobe satisfied, emain the same; whereas he boundaryconditionson the additional faces,x== •w, are satisfied

identically y virtue of the assumptionsA1). nasmuchas the solution presented n Sec. II takes u==0 andu•, ua,and • independent f x=, he effective quationssatisfied y the solutiongiven n Sec. I are identical,except or the modification f the constants,with thesystemof equationsobtained n this appendix or theextensionalmotion of thin piezoelectric lates. Con-sequently, he analysis given in Sees. I IV applieswithout change o the thin plate, provided the cp•,e,.• e replaced espectively y the %•, eipP, eijP definedin CA4).

It shouldbe noted that assumptions A1)and CA2)can also be used to obtain equationsof extensionalmotion (neglectinghe coupling etxveen xtension ndflexure) of thin, arbitrarily anisotropic,piezoelectricplates wi[hout electrodes n the large surfaces. heresulting quations re the piezoelectric eneralizations

of the Cauchy-Voigt equationsof extensionalmotionof elasticplates. t shouldalso be noted that, for theanalysis resentedn this appendixo be applicable,hefrequencymust be well below the lowestwidth fre-quency y virtueof the assumptionsA1) nd CA2).

14 Jul 2011 to 132 210 72 142 Redistribution subject to ASA license or copyright; see http://asadl org/journals/doc/ASALIB home/inf

1962 - Wave Propagation in an Infinite Piezoelectric Plate

Documents

Transcript of 1962 - Wave Propagation in an Infinite Piezoelectric Plate