JiashiYangDepartmentofEngineeringMechanics,UniversityofNebraska,Lincoln,NE68588-0526e-mail:jyang1@AReviewofaFewTopicsinPiezoelectricityThisisareviewarticleonafewspecialtopicsinpiezoelectricity:gradientandnonlocaltheories,fullydynamictheorywithMaxwellequations,piezoelectricsemiconductors,andmotionsofrotatingpiezoelectricbodies.Theyallrequiresomeextensionoftheclassicaltheoryofpiezoelectricity.Theyarerelativelynew,moreadvanced,andgrowingsubjectswithapplicationsorpotentialapplicationsinvariouselectromechanicaldevices.Thearticlecontains209references.(InmemoryofRaymondD.Mindlin(1906–1987.?DOI:10.1115/1.2345378?1IntroductionElectroelasticmaterialsexhibitelectromechanicalcoupling.Theyexperiencemechanicaldeformationswhenplacedinanelec-tric?eld,andbecomeelectricallypolarizedundermechanicalloads.Strictlyspeaking,piezoelectricityreferstolinearelectrome-chanicalcouplingsonly.Electrostrictionmaybethesimplestnon-linearelectromechanicalcouplinginwhichthemechanical?eldsdependontheelectric?eldsquadraticallyinthesimplestdescrip-tion.Piezoelectricmaterialshavebeenusedforalongtimetomakemanyelectromechanicaldevices.Examplesincludetrans-ducersforconvertingelectricenergytomechanicalenergyorviceversa,resonatorsand?ltersforfrequencycontrolandselectionfortelecommunicationandprecisetimekeepingandsynchronization,andacousticwavesensors.Thesearemostlyresonantdevicesoperatingwithaparticularmodeofavibratingpiezoelectric.Vi-brationsofapiezoelectricbodywerereviewedinDokmeci?1?.Areviewonthehigher-ordertheoriesofpiezoelectricplatesforhighfrequencyvibrationsandapplicationsinpiezoelectricresonatorswasgivenbyWangandYang?2?.Effectsofinitial?eldsinelec-troelasticmaterialswithapplicationsinfrequencystabilityofpi-ezoelectricresonatorsandacousticwavesensorsweresumma-rizedanddiscussedinYangandHu?3?.Therelativelyrecentdevelopmentofsmartstructurespresentsmanynewapplicationsofpiezoelectricmaterials.Afewreviewarticlesontheapplicationofpiezoelectricmaterialsinsmartstructureshavealreadyap-peared?RaoandSunar?4?,SunarandRao?5?,Cheeetal.?6?,andTanietal.?7??.2GradientandNonlocalEffectsItiswellknownthatgradienttheoriescandescribesizeeffectswhichareimportantinsmallscaleproblems.Theyalsohaveim-portantconsequencesinproblemswithsingularitieslikeconcen-tratedsourcesanddefects,andcandescribesurfaceandboundarylayerphenomena.Gradienttheoriesareclosertomicroscopictheorieslikelatticedynamicsthanclassicalcontinuumtheories.Theyarestillapplicablewhenthecharacteristiclengthofaprob-lemissosmallthatclassicalcontinuumtheoriesbegintofail.Straingradienttheoriesaretheoldestgradienttheoriesandarenotourmaininteresthere.Thissectionismainlyoneffectsofgradi-entsofelectricvariables.Thedevelopmentofnewtechnologyresultsinverythinelectromechanical?lmsandverysmallelec-tronicdevices.Thestudyofthesesmalldevicespresentsnewproblemsthatclassicaltheoriesmaynotbeabletodescribe.Theo-TransmittedbyAssoc.EditorS.Adali.AppliedMechanicsReviewsNOVEMBER2006,Vol.59/335Copyright?2006byASMErieswithgradienteffectsofelectricvariablesmayallowustogofurtherintheseproblemsthantheclassicaltheories.2.1PolarizationGradient.Forelasticdielectricstherearetwoformulations.Oneusestheelectricpolarizationvectorastheindependentelectricconstitutivevariable?Toupin?8??.Theotherusestheelectric?eldvector?Tiersten?9??.Mindlin?10?extendedthelinearversionofthepolarizationformulationbyallowingthestoredenergydensity?todependonthepolarizationgradientPi,j,inadditiontothepolarizationvectorPiandthestraintensorSij:??ui,Pi,??=?V???Sij,Pi,Pj,i??12?0?,i?,i+?,iPi?dVSij=?ui,j+uj,i?/2?1?whereuiisthedisplacementvector,?theelectricpotential,and?0thepermittivityoffreespace.Thesummationconventionforrepeatedtensorindicesandtheconventionthatanindexfollowingacommadenotespartialdifferentiationwithrespecttothecoor-dinateassociatedwiththeindexarefollowed.Thestationarycon-ditionoftheabovefunctionalforindependentvariationsofui,?andPiisTji,j=0??0?,ii+Pi,i=0Ei+Eji,j??,i=0?2?whereTij=???Sij,Ei=????Pi,Eij=???Pj,i?3?Equation?2?representssevenequationsforuiPi,and?.IfthedependenceonPi,jinEq.?1?isdropped,onlyEq.?2?1,2willresult,whicharetheequationsoflinearpiezoelectricitywiththeintroductionofDi=??0?,i+Pi.WhatmotivatedMindlintostudytheeffectsofpolarizationgradientwasthecapacitanceofaverythindielectric?lm.Experi-mentsshowedthatthecapacitanceofaverythin?lmissystem-aticallysmallerthantheclassicalprediction.Usinghispolariza-tiongradienttheory,Mindlin?11?showedthatwhenthe?lmthicknessbecomescomparabletoamaterialparameterinthegra-dienttheory,whichhasthedimensionofalengthandcanberelatedtoamicroscopicinteractiondistance,thegradientsolutioncancapturethetrendofthedeviationfromtheclassicalpredic-tion.Healsoshowedthatthepolarizationgradientsolutionofthin?lmcapacitanceagreeswiththepredictionfromlatticedynamics.Mindlin?14?alsostudiedtheelectricpotentialofapointcharge.Theclassicalsolutiondivergesatthepointcharge.Thisisbecauseatapointveryclosetothecharge,thechargecannolongerbeconsideredasapointchargeanditsdistributionhastobetakenintoconsideration.Thegradienttheoryyieldsasolutionthatdif-fersfromtheclassicalsolutiononlyatthecloserangeofthesourcepoint,andisvalidatacloserdistancetothesourcepointthantheclassicalsolution.Mindlin?15?showedthatinamaterialwithcentrosymmetrywithoutpiezoelectriccoupling,linearelec-tromechanicalcouplingcanstillexistduetopolarizationgradient.Healsostudiedthepolarizationgradienteffectandelectromag-netic?eldsindiatomicdielectrics?Mindlin?16??,andelectromag-neticradiationfromavibratingbody?Mindlin?17??.Withthepolarizationgradienttheory,Askaretal.?18?studiedsurfaceeffectsandcrackproblems.Schwartz?19?developedstressfunctionsandstudied?eldsduetoaconcentratedforce.ChowdhuryandGlockner?20,21?studiedapointchargeinahalfspaceandaconcentratedforceonahalfspace?theBoussinesqproblem?.Collet?22?andDost?23?analyzedaccelerationwaves.ShockwaveswereinvestigatedbyCollet?24?.YangandBatra?25?derivedconservationlawsforthepolarizationgradienttheoryfromtheinvarianceofthevariationalintegral.Mindlin’spolarizationgradienttheorywasextendedinseveraldifferentdirections.Thenonlinearversionofthetheorywas?rstgivenbySuhubi?26?.ThermalcouplingwasincludedinthetheorybyChowdhuryetal.?27?,andChowdhuryandGlockner?28,29?.FullyelectromagneticcouplingwithcompleteMaxwellequationswasconsideredinTierstenandTsai?30?whichisaverygeneraltheoryalsoincludingmagnetization.Instudyingcertainphenomenainferroelectriccrystals,polarizationinertia?Maugin,?31,32??needstobeconsidered.Atheoryincludingbothpolariza-tiongradientandinertiawasgivenbyMauginandPouget?33?,whichhassupportfromlatticedynamics?Askaretal.?34?,Pougetetal.?35,36??,andhasbeenusedtostudyvariousmodesandwavesinferroelectrics?PougetandMaugin?37–39?,Collet?40??.AmoregeneraltheoryincludingpolarizationgradientandinertiaaswellasstraingradientwasgivenbySahinandDost?41?.Atheoryincludingpolarizationgradientandinertiaeffectsindi-atomicelasticdielectricswasderivedbyDemirayandDost?42?.AlatticedynamicsapproachofdiatomicdielectricscanbefoundinAskarandLee?43?.Asystematicpresentationofthepolariza-tiongradienttheorycanbefoundinMaugin?44?.PolarizationgradientcanalsobeincludedintheLandau-Ginsburgfunctionalusedsuchafunctionaltostudyelectroelasticcomposites.Considerthefollowingfunctional?Yangetal.?57??:??ui,??=?V???Sij,Ei,Ei,j??12?0EiEi?dVSij=?ui,j+uj,i?/2,Ei=??,i?4?Inthelinearcase,onecantake??Sij,Ei?=12cijklSijSkl?12?0?ijEiEj?eiklEiSkl?12?0?ijklEi,jEk,l?5?where?0?ijklEi,jEk,l/2istheonlyadditionaltermtoclassicallin-earpiezoelectricity.ThenthestationaryconditionofEq.?4?re-sultsinthefollowingequations:cijkluk,lj+ekij?,kj=0eikluk,li??ij?,ij+?0?ijkl?,ijkl=0?6?336/Vol.59,NOVEMBER2006TransactionsoftheASMEwherecijklaretheelasticconstants,eijklthepiezoelectriccon-stants,and?ij=?0??ij+?ij?theelectricpermittivitytensor.Theterms?ijklarenewmaterialconstantsduetotheintroductionoftheelectric?eldgradientintheenergydensityfunction;?ijklhasthedimensionof?length?2.Physicallytheymayberelatedtochar-acteristiclengthsofatomicormicrostructuralinteractionsofthematerial.When?ijkl=0,Eq.?6?reducestotheclassicaltheoryofpiezoelectricity.Inthecaseofanti-planeproblemsofpolarizedceramics,Eq.?6?reducestothefollowingverysimpleformwhichallowsmoreinsightintotheeffectsofelectric?eldgradients:c?2u+e?2?=0e?2u???2?+?0??2?2?=0?7?2.3StrainGradient.Forelasticdielectrics,thegradientofmechanical?eldsandelectricalvariablescanbebothincludedintoconstitutiverelations?SahinandDost?41?,KalpakidesandTij?x?=?V?cijkl?x,x??Skl?x???ekij?x,x??Ek?x???dV?x??Fig.1Capacitanceofathindielectric?lmFig.2ElectricpotentialofalinesourceFig.3Dispersionofshortwavesduetoelectric?eldgradientFig.4DispersionofshortwavesbystraingradienttheoriesandlatticesdynamicsAppliedMechanicsReviewsNOVEMBER2006,Vol.59/337Di?x?=?V??ij?x,x??Ej?x??+eikl?x,x??Skl?x???dV?x???8?Equation?8?resultsinintegral-differentialequationswhensubsti-tutedintotheequationsofmotionandelectrostatics.Moregeneralnon-localtheoriesforelectromagneticelasticsolidscanbefoundin?69?.Thepotential?eldtoapointchargewasobtainedinEringen?68?,whichdiffersfromtheclassicalCoulomb?eld.Eringen?68?alsoshowedthedispersionofshortplanewaves.Yang?70?obtainedanonlocalsolutionforthin?lmcapacitanceshowingsimilarbehaviorsasinFig.1whichwasfromanelectric?eldgradienttheory.3DynamicEffectsFromMaxwellEquationsThetheoryofpiezoelectricityisbasedonaquasi-staticap-proximation?Nelson?51??.Thisapproximationcanbeconsideredasthelowestorderapproximationofaperturbationprocedurebasedonthefactthattheacousticwavespeedismuchsmallerthanthespeedoflight.Asaresultofthisapproximation,inthetheoryofpiezoelectricity,althoughthemechanicalequationsaredynamic,theelectromagneticequationsarestaticandtheelectric?eldandthemagnetic?eldarenotdynamicallycoupled.There-fore,itdoesnotdescribethewavebehaviorofelectromagnetic?elds.Formanyapplicationsofpiezoelectricacousticwavede-vicesthequasi-statictheoryissuf?cient,buttherearesituationsinwhichfullelectromagneticcouplingneedstobeconsidered.Forexample,electromagneticwavesgeneratedbymechanical?eldsneedtobestudiedinthecalculationofradiatedelectromagneticpowerfromavibratingpiezoelectricdevice.FullMaxwellequa-tionsalsoneedtobeconsideredindevicesinwhichacousticwavesproduceelectromagneticwavesorviceversa.Whenelec-tromagneticwavesareinvolved,thecompletesetofMaxwellequationsneedstobeused,coupledtothemechanicalequationsofmotion.Suchafullydynamictheoryhasbeencalledpiezoelec-tromagnetismbysomeresearchers.3.1GoverningEquations.Forapiezoelectricbutnon-magnetizabledielectricbodythethree-dimensionalequationsoflinearpiezoelectromagnetismconsistoftheequationsofmotionandMaxwellequationsTji,j=?u¨i?ijkEk,j=?B˙i,?ijkHk,j=D˙iBi,i=0,Di,i=0?9?aswellasthefollowingconstitutiverelations:Tij=cijklSkl?ekijEkDi=eijkSjk+?ijEjBi=?0Hi?10?whereBiisthemagneticinduction,andHithemagnetic?eld.Theterm?0isthemagneticpermeabilityoffreespace,?ijkisthepermutationtensor.FromEqs.?9?and?10?onecanobtaincijkluk,lj?ekijEk,j=?u¨i?1?0?ijk?kmnEn,mj=eijku¨j,k+?ijE¨j?11?whichclearlyshowsthewavenatureoftheelectromagnetic?elds.Mindlin?71?derivedavariationalprincipleforpiezoelectro-magnetisminacompoundcontinuumrepresentingadiatomicma-terial.Lee?72?gaveavariationalformulationforthe?eldsinsideandoutsidea?nitebodywithcontinuityconditionsattheinter-facebetweenthebodyandfreespace.Ageneralizedvariationalprinciplewithallmechanicalandelectromagnetic?eldsasinde-pendentvariableswasgivenbyYang?73?.Yang?74?andYangandWu?75?alsoobtainedvariationalprinciplesandgeneralizedvariationalprinciplesfortheeigenvalueproblemoffreevibrationsofapiezoelectromagneticbody.3.2DynamicSolutions.Earlysolutionsfromthedynamictheorybeganwiththepropagationofplanewavesinanun-boundedmediumbyKyame?76?.Inadditiontowavesthatareessentiallyacoustic,therearealsowavesthatareessentiallyelec-tromagnetic.Thesetwogroupsofmodesarecoupledbypiezo-electriceffects.EffectsofviscosityandconductivityonplanewaveswerestudiedinKyame?77?.LaterPailloux?78?andHruska?79,80?alsostudiedthepropagationofplanewaves.TsengandWhite?81?andTseng?82?obtainedsolutionsforsurfacewavesinhexagonalcrystals.SpaightandKoerber?83?analyzedsurfacewavesinlithiumniobate.ForwavesinplatesMindlin?84?solvedtheproblemofme-chanicallyforcedthickness-shearvibrationofanAT-cutquartzplateandcalculatedradiatedelectromagneticpower.Lee?85?studiedthickness-shearvibrationofanAT-cutquartzplateunderlateral?eldexcitationandcalculatedradiatedelectromagneticpower.Leeetal.?86?latergaveathoroughanalysisofelectro-magneticradiationfrommechanicallyforcedvibrationsofapi-ezoelectricplate.Radiationfroma?niteplatewascalculatedinCampbellandWeber?87?.Fullelectromagneticcouplingandra-diationinthepolarizationgradienttheorywereconsideredinMindlin?16,17?.Foranti-planeproblemsofpolarizedceramicsEqs.?9?and?10?canbewrittenasthefollowingtwosimplewaveequations:vT2?2u3=u¨3c2?2H3=H¨3?12?wherevTisanacousticwavespeedandcthespeedoflight.FromtheelectromagneticpointofviewEq.?12?describestheso-calledtransversemagneticwaves.Electromechanicalcouplingcomesintoplaywhencalculatingtheelectric?eldfromthedisplacementandthemagnetic?eld.Couplingscanalsobecausedatbound-aries.Transientsurfacewavesinaceramichalfspaceundersur-faceloadwereanalyzedbySedovandSchmerrJr.?88?,andSchmerrJrandSedov?89?.Li?90?obtainedanti-planeshearhori-zontal?SH?surfaceandinterfacewavesolutionsinpolarizedce-ramicsusingthescalarandvectorpotentialformulationofelec-tromagnetic?elds.ToandGlaser?91?studiedSHwavesinplatesusingthepotentialformulation,andfoundthatthequasi-statictheorymaypredictmodesthatarenotapproximationsofthemodespredictedbythefullydynamictheory.ForSHwavesthescalarandvectorpotentialformulationresultsinfourequations?Li?90??.Twooftheseequationsarecoupled,andtheothertwoareone-waycoupled.Inaddition,agaugeconditionneedstobeimposed.SHsurfacewavesolutionsweregiveninYang?92?us-offullelectromechanicalcouplingontheacousticsurfacewavespeedisverysmall,whichsupportsthequasi-staticapproxima-tion.ItwasalsonoticedinYang?92?thatinthelimitwhenthespeedoflightgoestoin?nity,thewavespeedpredictedbypiezo-electromagnetismreducestothatpredictedbyquasi-staticpiezo-electricity.Lovewavesinahalfspacecarryingalayerofadif-ferentmaterialwereanalyzedinYang?93?,SHwavesinaplatewereobtainedinYang?94?.Sinceelectric?eldscanexistinfreespace,certainpiezoelectricdeviceshaveairgapsinthemandacousticwavesonbothsidesofagapcanstillinteractthroughtheelectric?eldinthegap.PiezoelectromagneticgapwaveswereanalyzedinYang?95?.Yang?96,97?alsoobtained?eldsassoci-atedwithamovingsemi-in?nitecrackandamovingdislocation.Fieldsassociatedwitha?nitecrackwereobtainedinLiandYang?98?.ElectromagneticradiationfromavibratingceramiccylinderwasanalyzedbyYang?99?.338/Vol.59,NOVEMBER2006TransactionsoftheASME4PiezoelectricSemiconductorsPiezoelectricmaterialscanbeeitherdielectrics?insulators?orsemiconductors.Anacousticwavepropagatinginapiezoelectriccrystalisusuallyaccompaniedbyanelectric?eld.Whenthecrys-talisalsosemiconducting,theelectric?eldproducescurrentsandspacechargeresultingindispersionandacousticloss.Theinter-actionbetweenatravelingacousticwaveandmobilechargesinpiezoelectricsemiconductorsiscalledtheacoustoelectriceffect?HustonandWhite?100??whichisaspecialcaseofamoregen-eralphenomenonwhichmaybecalledthewave-particledragsemiconductionineachcomponentphase.Itwasalsofoundthatanacousticwavetravelinginapiezoelectricsemiconductorcanbeampli?edbytheapplicationofadcelectric?eld?White?102??.Inadditiontoacousticwaveampli?ers,theacoustoelectriceffectcanalsobeusedtodesignexperimentsformeasuringchargemo-bilityandmakedevicesforchargetransferdrivenbyacousticwaves.4.1GoverningEquations.Thebasicbehaviorofpiezoelec-tricsemiconductorsandtheacoustoelectriceffectcanbede-scribedbyalinearphenomenologicaltheory?HustonandWhite?100?,White?102??.Considerahomogeneous,one-carrierpiezo-electricsemiconductorunderauniformdcelectric?eldEˉj?equa-tionsforamulti-carriersemiconductorcanbefoundinFischler?103??.ThesteadystatecurrentisJˉi=qnˉ?ijEˉj,whereqisthecarriercharge,nˉisthesteadystatecarrierdensitywhichproduceselectricalneutrality,and?ijisthecarriermobility.Whenanacousticwavepropagatesthroughthematerial,perturbationsoftheelectric?eld,thecarrierdensityandthecurrentaredenotedbyEj,n,andJi.Thelineartheoryforsmallsignalsconsistsoftheequationsofmotion,Gauss’slawofelectrostatics?thechargeequation?,andtheconservationofchargeTji,j=?u¨iDi,i=qnqn˙+Ji,i=0?13?Theaboveequationsareaccompaniedbythefollowingconstitu-tiverelations:Tij=cijklSkl?ekijEkDi=eijkSjk+?ijEjJi=qnˉ?ijEj+qn?ijEˉj?qdijn,j?14?wheredijarethecarrierdiffusionconstants.WithsubstitutionsfromEq.?14?,Eq.?13?canbewrittenas?veequationsforu,?,andncijkluk,lj+ekij?,kj=?u¨ieikluk,li??ij?,ij=qnn˙?nˉ?ij?,ij+?ijEˉjn,i?dijn,ij=0?15?4.2DeviceModeling.Theacoustoelectriceffectandtheacoustoelectricampli?cationofacousticwaveshaveledtothedevelopmentofacoustoelectricampli?ersofacousticwaves.ASAWacoustoelectricampli?erisshowninFig.5.Typicalbehav-ioroftheimaginarypartofthewavespeedversusthebiasingelectric?eldinanacousticwaveampli?erisshowninFig.6.Whenthebiasing?eldreachesacriticalvalue,i.e.,whenthecarrierdriftspeedunderthebiasingelectric?eldisequaltotheacousticwavespeed,theimaginarypartofthewavespeedchangesitssign,indicatingthetransitionfromadampedwavetoagrowingwaveorwaveampli?cation.Duetomulti-?eldcouplingandanisotropy,devicemodelingpresentscomplicatedmathematicalproblems.SurfacewavesoverahalfspaceorahalfspacecarryingalayerofadifferentmaterialwerestudiedbyLakinandShaw?104?,Ramakrishna?105?,Inge-brigtsen?106?,KinoandReeder?107?,Kino?108?,Wangetal.?109?,andGangulyandPal?110?.WauerandSuherman?111?obtainedsolutionsfortheone-dimensionalproblemofthicknessvibrationsofplates.Propagatingwavesinsemiconductorplatesorpiezoelectricplatescarryingsemiconductor?lmswereanalyzedinFischler?112,113?,Dietzetal.?114?,andJosse?115?.Itshouldbenotedthatsomeofthesurfaceandplatewavestructureshaveairgaps,whicharemorecommoninpiezoelectricsemiconductordevicesthaninotherpiezoelectricdevices.Multi-layeredstruc-tureswerestudiedbyPalmaandDas?116,117?.PalanichamyandSingh?118?studiedtheeffectofnon-uniformelectric?elds.Theaboveanalyseswerebasedonthethree-dimensionalequa-equationsofcoupledextensional,?exural,andthickness-shearmotionsofapiezoelectricsemiconductorplate.Theequationswerespecializedtocrystalsof6mmsymmetryandweresimpli-?edbyanapproximationforthickness-shearwaves.Thesimpli-?edequationswereusedtostudytheampli?cationofthickness-shearwavesinaplate.ThelowestorderplateequationsderivedinYangandZhou?119?areofgeneralizedplane-stresstypeforex-tensionalmotionsoftheplate.Theseequationsforextensionwereusedtomodelsurfacewavesoverapiezoelectrichalfspacecar-ryingasemiconductor?lm?YangandZhou?120??,gapwavesbetweena?lmandahalfspace?YangandZhou?121??,andinterfacewavesbetweentwohalfspaceswitha?lm?YangandZhou?122??.Yangetal.?123,124?alsoderivedequationsforlaminatedplatesandshellsofpiezoelectricsemiconductorsandstudiedwavesinthem.Foranti-planeproblemsofpiezoelectriccrystalsof6mmsym-metrywhichincludesquiteafewwidelyusedsemiconductors,Eq.?15?takesthefollowingsimpleformwhichallowstheoreticalanalysis:c?2u+e?2?=?u¨Fig.5Asurfaceacousticwaveampli?erFig.6TransitionfromdampedwavestogrowingwavesAppliedMechanicsReviewsNOVEMBER2006,Vol.59/339e?2u???2?=qnn˙?nˉ??2???Eˉ·?n?d?2n=0?16?UsingEq.?16?,Yang?125?alsoanalyzedtheelectromechanical?eldsaroundasemi-in?nitecrackinasemiconductor.TheresultsinYang?125?showedthatasemi-in?nitemodeIIIcrackstillhastherootrsingularity,butthecoef?cientofthesingular?eldismodi?edbysemiconduction.Theaboveanalysesonsemiconduc-torsareallanalytical.Thereseemtobenoreportednumericalresultsby,e.g.,the?niteelementmethod.4.3MoreSophisticatedTheories.DeLorenziandTiersten?126?usedamacroscopicphysicalmodeltoderiveamoresophis-ticatednonlineartheoryfordeformablesemiconductors.Themodelconsistsofseveralinteractingcontinuarepresentingthelattice,electrons,andholes,etc.Basiclawsofphysicsareappliedtoeachcontinuum.Theresultingequationsarecombinedtoob-tainamacroscopicdescriptionofthematerial.TheseequationsarealsogiveninTiersten?127?.AnconaandTiersten?128?usedtheseequationsinthestudyofSi–SiO2interface,andmetal-insulator-semiconductorstructures?AnconaandTiersten?129??.McCarthyandTiersten?130,131?andMcCarthy?132?analyzedthepropagationofaccelerationwavesandshockwaves.MauginandDaher?133?,andDaherandMaugin?134?alsodevelopedanonlinearphenomenologicaltheoryfordeformablesemiconductors.Wavepropagationunderabiasing?eldwasana-lyzedbyDaher?135?,andDaherandMaugin?136,137?.Vermaetal.?138?studiedradialvibrationsofacylindricaltube.5MotionsofRotatingPiezoelectricBodiesVoltagesensitivityasafunctionofthedrivingfrequencyisplottedinFig.8fora?xed?,where?0isanormalizingfre-quency.Zistheimpedanceoftheoutputcircuit.Z0isanormal-izingimpedance.Itisseenthatnearthetworesonantfrequenciesthesensitivityassumesmaximalvalues.Therefore,thedeviceshouldbeoperatingatafrequencynearresonance.Thedependenceofsensitivityontherotationrate?isshowninFig.9fora?xeddrivingfrequencynearresonanceandfordiffer-entvaluesofZ.When?ismuchsmallerthan?0,whichistrueinmostapplicationsofpiezoelectricgyroscopes,therelationbe-tweenthesensitivityand?isessentiallylinear.Therefore,intheanalysesofpiezoelectricgyroscopes,veryoftenthecentrifugalforcewhichrepresentshigherordereffectsof?isneglectedandthecontributiontosensitivityiscompletelyfromtheCoriolisforcewhichislinearin?.RotationinducedfrequencyshiftsinSAWorBAWpiezoelec-tricresonatorscanalsobeusedtomeasureangularrates.Whenapiezoelectricresonatorwithresonantfrequency?0isattachedtoabodyrotatingatanangularrate?,theresonantfrequencyoftheresonator?orequivalentlythewavespeedwhenapropagatingwaveisused?changesduetorotation.Foraproperlydesignedresonator,thisfrequencyshiftisproportionalto?andcanbeusedtomeasureit.Figure10showsthefrequencyshiftsofcertainwavesinaceramicplaterotatingaboutitsnormal.Theslopesofthecurvesattheoriginrepresentsensitivitytorotation.Forcer-tainwavesthesensitivityiszero.Thesewavesareusefulwhenfrequencystabilityinadeviceunderrotationisdesired.Fig.7AsimplepiezoelectricgyroscopeFig.8Outputvoltageversusthedrivingfrequency?Fig.9Outputvoltageversustherotationrate?340/Vol.59,NOVEMBER2006TransactionsoftheASMETheliteratureonpiezoelectricgyroscopesisgrowing.Manypublicationshaveappearedafterthetwoearlierreviewarticles?Burdessetal.?139?,Soderkvist?140??.ThetworecentPh.D.dissertations?Loveday?141?,Fang?142??containsomerecentreferences.5.1GoverningEquations.Thebasicbehaviorsofapiezo-electricgyroscopearegovernedbytheequationsofarotatingpiezoelectricbody,whichconsistoftheequationsoflinearpiezo-electricitywithrotationrelatedCoriolisandcentrifugalaccelera-tions.Apiezoelectricgyroscopeisinsmallamplitudevibrationinareferenceframerotatingwithit.Theequilibriumstateintherotatingreferenceframeiswithinitialdeformationsandstressesduetothecentrifugalforce.Therefore,anexactdescriptionofthemotionofapiezoelectricgyroscoperequirestheequationsforsmall,dynamic?eldssuperposedonstaticinitial?elds?Baum-hauerandTiersten?143??duetothecentrifugalforce,whichhastobeobtainedfromthenonlineartheoryofelectroelasticity?Tier-sten?144??.ThegoverningequationsintherotatingframecanbewrittenasTji,j?2??ijk?ju˙k????i?juj??j?jui?=?u¨iDi,i=0Tij=cijklSkl?ekijEk,Di=eijkSjk+?ijE?17?wherethetermsrelatedto?jrepresentthesumoftheCoriolisandcentrifugalforces.Termsduetotheinitial?eldsareignored.Sincepiezoelectricgyroscopesareverysmall?oftheorderof10mm?,theiroperatingfrequency?0isveryhigh,usuallyoftheorderoftensofkHzorhigher.Piezoelectricgyroscopesareusedtomeasureanangularrate?muchsmallerthanitsresonantfre-quency?0.Inthiscasethecentrifugalforceduetorotation,whichisproportionalto?2,ismuchsmallercomparedtotheCoriolisforcewhichisproportionalto?0?.Therefore,theeffectofrota-tiononmotionsofpiezoelectricgyroscopesisdominatedbyCo-riolisforce.Thisisfundamentallydifferentfromtherelativelywellstudiedsubjectofvibrati