机械专业毕业设计外文翻译相关外文文献

JiashiYangDepartmentofEngineeringMechanics,UniversityofNebraska,Lincoln,NE68588-0526e-mail:jyang1@AReviewofaFewTopicsinPiezoelectricityThisisareviewarticleonafewspecialtopicsinpiezoelectricity:gradientandnonlocaltheories,fullydynamictheorywithMaxwellequations,piezoelectricsemiconductors,andmotionsofrotatingpiezoelectricbodies.Theyallrequiresomeextensionoftheclassicaltheoryofpiezoelectricity.Theyarerelativelynew,moreadvanced,andgrowingsubjectswithapplicationsorpotentialapplicationsinvariouselectromechanicaldevices.Thearticlecontains209references.(InmemoryofRaymondD.Mindlin(1906–1987.͓DOI:10.1115/1.2345378͔1IntroductionElectroelasticmaterialsexhibitelectromechanicalcoupling.Theyexperiencemechanicaldeformationswhenplacedinanelec-tricfield,andbecomeelectricallypolarizedundermechanicalloads.Strictlyspeaking,piezoelectricityreferstolinearelectrome-chanicalcouplingsonly.Electrostrictionmaybethesimplestnon-linearelectromechanicalcouplinginwhichthemechanicalfieldsdependontheelectricfieldsquadraticallyinthesimplestdescrip-tion.Piezoelectricmaterialshavebeenusedforalongtimetomakemanyelectromechanicaldevices.Examplesincludetrans-ducersforconvertingelectricenergytomechanicalenergyorviceversa,resonatorsandfiltersforfrequencycontrolandselectionfortelecommunicationandprecisetimekeepingandsynchronization,andacousticwavesensors.Thesearemostlyresonantdevicesoperatingwithaparticularmodeofavibratingpiezoelectric.Vi-brationsofapiezoelectricbodywerereviewedinDokmeci͓1͔.Areviewonthehigher-ordertheoriesofpiezoelectricplatesforhighfrequencyvibrationsandapplicationsinpiezoelectricresonatorswasgivenbyWangandYang͓2͔.Effectsofinitialfieldsinelec-troelasticmaterialswithapplicationsinfrequencystabilityofpi-ezoelectricresonatorsandacousticwavesensorsweresumma-rizedanddiscussedinYangandHu͓3͔.Therelativelyrecentdevelopmentofsmartstructurespresentsmanynewapplicationsofpiezoelectricmaterials.Afewreviewarticlesontheapplicationofpiezoelectricmaterialsinsmartstructureshavealreadyap-peared͑RaoandSunar͓4͔,SunarandRao͓5͔,Cheeetal.͓6͔,andTanietal.͓7͔͒.2GradientandNonlocalEffectsItiswellknownthatgradienttheoriescandescribesizeeffectswhichareimportantinsmallscaleproblems.Theyalsohaveim-portantconsequencesinproblemswithsingularitieslikeconcen-tratedsourcesanddefects,andcandescribesurfaceandboundarylayerphenomena.Gradienttheoriesareclosertomicroscopictheorieslikelatticedynamicsthanclassicalcontinuumtheories.Theyarestillapplicablewhenthecharacteristiclengthofaprob-lemissosmallthatclassicalcontinuumtheoriesbegintofail.Straingradienttheoriesaretheoldestgradienttheoriesandarenotourmaininteresthere.Thissectionismainlyoneffectsofgradi-entsofelectricvariables.Thedevelopmentofnewtechnologyresultsinverythinelectromechanicalfilmsandverysmallelec-tronicdevices.Thestudyofthesesmalldevicespresentsnewproblemsthatclassicaltheoriesmaynotbeabletodescribe.Theo-TransmittedbyAssoc.EditorS.Adali.AppliedMechanicsReviewsNOVEMBER2006,Vol.59/335Copyright©2006byASMErieswithgradienteffectsofelectricvariablesmayallowustogofurtherintheseproblemsthantheclassicaltheories.2.1PolarizationGradient.Forelasticdielectricstherearetwoformulations.Oneusestheelectricpolarizationvectorastheindependentelectricconstitutivevariable͑Toupin͓8͔͒.Theotherusestheelectricfieldvector͑Tiersten͓9͔͒.Mindlin͓10͔extendedthelinearversionofthepolarizationformulationbyallowingthestoredenergydensity⌺todependonthepolarizationgradientPi,j,inadditiontothepolarizationvectorPiandthestraintensorSij:⌫͑ui,Pi,␾͒=͵V͚ͫ͑Sij,Pi,Pj,i͒−12␧0␾,i␾,i+␾,iPiͬdVSij=͑ui,j+uj,i͒/2͑1͒whereuiisthedisplacementvector,␾theelectricpotential,and␧0thepermittivityoffreespace.Thesummationconventionforrepeatedtensorindicesandtheconventionthatanindexfollowingacommadenotespartialdifferentiationwithrespecttothecoor-dinateassociatedwiththeindexarefollowed.Thestationarycon-ditionoftheabovefunctionalforindependentvariationsofui,␾andPiisTji,j=0−␧0␾,ii+Pi,i=0Ei+Eji,j−␾,i=0͑2͒whereTij=ץ͚ץSij,Ei=−ץ͚ץPi,Eij=ץ͚ץPj,i͑3͒Equation͑2͒representssevenequationsforuiPi,and␾.IfthedependenceonPi,jinEq.͑1͒isdropped,onlyEq.͑2͒1,2willresult,whicharetheequationsoflinearpiezoelectricitywiththeintroductionofDi=−␧0␾,i+Pi.WhatmotivatedMindlintostudytheeffectsofpolarizationgradientwasthecapacitanceofaverythindielectricfilm.Experi-mentsshowedthatthecapacitanceofaverythinfilmissystem-aticallysmallerthantheclassicalprediction.Usinghispolariza-tiongradienttheory,Mindlin͓11͔showedthatwhenthefilmthicknessbecomescomparabletoamaterialparameterinthegra-dienttheory,whichhasthedimensionofalengthandcanberelatedtoamicroscopicinteractiondistance,thegradientsolutioncancapturethetrendofthedeviationfromtheclassicalpredic-tion.Healsoshowedthatthepolarizationgradientsolutionofthinfilmcapacitanceagreeswiththepredictionfromlatticedynamics.Mindlin͓14͔alsostudiedtheelectricpotentialofapointcharge.Theclassicalsolutiondivergesatthepointcharge.Thisisbecauseatapointveryclosetothecharge,thechargecannolongerbeconsideredasapointchargeanditsdistributionhastobetakenintoconsideration.Thegradienttheoryyieldsasolutionthatdif-fersfromtheclassicalsolutiononlyatthecloserangeofthesourcepoint,andisvalidatacloserdistancetothesourcepointthantheclassicalsolution.Mindlin͓15͔showedthatinamaterialwithcentrosymmetrywithoutpiezoelectriccoupling,linearelec-tromechanicalcouplingcanstillexistduetopolarizationgradient.Healsostudiedthepolarizationgradienteffectandelectromag-neticfieldsindiatomicdielectrics͑Mindlin͓16͔͒,andelectromag-neticradiationfromavibratingbody͑Mindlin͓17͔͒.Withthepolarizationgradienttheory,Askaretal.͓18͔studiedsurfaceeffectsandcrackproblems.Schwartz͓19͔developedstressfunctionsandstudiedfieldsduetoaconcentratedforce.ChowdhuryandGlockner͓20,21͔studiedapointchargeinahalfspaceandaconcentratedforceonahalfspace͑theBoussinesqproblem͒.Collet͓22͔andDost͓23͔analyzedaccelerationwaves.ShockwaveswereinvestigatedbyCollet͓24͔.YangandBatra͓25͔derivedconservationlawsforthepolarizationgradienttheoryfromtheinvarianceofthevariationalintegral.Mindlin’spolarizationgradienttheorywasextendedinseveraldifferentdirections.ThenonlinearversionofthetheorywasfirstgivenbySuhubi͓26͔.ThermalcouplingwasincludedinthetheorybyChowdhuryetal.͓27͔,andChowdhuryandGlockner͓28,29͔.FullyelectromagneticcouplingwithcompleteMaxwellequationswasconsideredinTierstenandTsai͓30͔whichisaverygeneraltheoryalsoincludingmagnetization.Instudyingcertainphenomenainferroelectriccrystals,polarizationinertia͑Maugin,͓31,32͔͒needstobeconsidered.Atheoryincludingbothpolariza-tiongradientandinertiawasgivenbyMauginandPouget͓33͔,whichhassupportfromlatticedynamics͑Askaretal.͓34͔,Pougetetal.͓35,36͔͒,andhasbeenusedtostudyvariousmodesandwavesinferroelectrics͑PougetandMaugin͓37–39͔,Collet͓40͔͒.AmoregeneraltheoryincludingpolarizationgradientandinertiaaswellasstraingradientwasgivenbySahinandDost͓41͔.Atheoryincludingpolarizationgradientandinertiaeffectsindi-atomicelasticdielectricswasderivedbyDemirayandDost͓42͔.AlatticedynamicsapproachofdiatomicdielectricscanbefoundinAskarandLee͓43͔.Asystematicpresentationofthepolariza-tiongradienttheorycanbefoundinMaugin͓44͔.PolarizationgradientcanalsobeincludedintheLandau-Ginsburgfunctionalusedsuchafunctionaltostudyelectroelasticcomposites.Considerthefollowingfunctional͑Yangetal.͓57͔͒:⌫͑ui,␾͒=͵V͚ͫ͑Sij,Ei,Ei,j͒−12␧0EiEiͬdVSij=͑ui,j+uj,i͒/2,Ei=−␾,i͑4͒Inthelinearcase,onecantake͚͑Sij,Ei͒=12cijklSijSkl−12␧0␹ijEiEj−eiklEiSkl−12␧0␣ijklEi,jEk,l͑5͒where␧0␣ijklEi,jEk,l/2istheonlyadditionaltermtoclassicallin-earpiezoelectricity.ThenthestationaryconditionofEq.͑4͒re-sultsinthefollowingequations:cijkluk,lj+ekij␾,kj=0eikluk,li−␧ij␾,ij+␧0␣ijkl␾,ijkl=0͑6͒336/Vol.59,NOVEMBER2006TransactionsoftheASMEwherecijklaretheelasticconstants,eijklthepiezoelectriccon-stants,and␧ij=␧0͑␦ij+␹ij͒theelectricpermittivitytensor.Theterms␣ijklarenewmaterialconstantsduetotheintroductionoftheelectricfieldgradientintheenergydensityfunction;␣ijklhasthedimensionof͑length͒2.Physicallytheymayberelatedtochar-acteristiclengthsofatomicormicrostructuralinteractionsofthematerial.When␣ijkl=0,Eq.͑6͒reducestotheclassicaltheoryofpiezoelectricity.Inthecaseofanti-planeproblemsofpolarizedceramics,Eq.͑6͒reducestothefollowingverysimpleformwhichallowsmoreinsightintotheeffectsofelectricfieldgradients:cٌ2u+eٌ2␾=0eٌ2u−␧ٌ2␾+␧0␣ٌ2ٌ2␾=0͑7͒2.3StrainGradient.Forelasticdielectrics,thegradientofmechanicalfieldsandelectricalvariablescanbebothincludedintoconstitutiverelations͑SahinandDost͓41͔,KalpakidesandTij͑x͒=͵V͓cijkl͑x,xЈ͒Skl͑xЈ͒−ekij͑x,xЈ͒Ek͑xЈ͔͒dV͑xЈ͒Fig.1CapacitanceofathindielectricfilmFig.2ElectricpotentialofalinesourceFig.3DispersionofshortwavesduetoelectricfieldgradientFig.4DispersionofshortwavesbystraingradienttheoriesandlatticesdynamicsAppliedMechanicsReviewsNOVEMBER2006,Vol.59/337Di͑x͒=͵V͓␧ij͑x,xЈ͒Ej͑xЈ͒+eikl͑x,xЈ͒Skl͑xЈ͔͒dV͑xЈ͒͑8͒Equation͑8͒resultsinintegral-differentialequationswhensubsti-tutedintotheequationsofmotionandelectrostatics.Moregeneralnon-localtheoriesforelectromagneticelasticsolidscanbefoundin͓69͔.ThepotentialfieldtoapointchargewasobtainedinEringen͓68͔,whichdiffersfromtheclassicalCoulombfield.Eringen͓68͔alsoshowedthedispersionofshortplanewaves.Yang͓70͔obtainedanonlocalsolutionforthinfilmcapacitanceshowingsimilarbehaviorsasinFig.1whichwasfromanelectricfieldgradienttheory.3DynamicEffectsFromMaxwellEquationsThetheoryofpiezoelectricityisbasedonaquasi-staticap-proximation͑Nelson͓51͔͒.Thisapproximationcanbeconsideredasthelowestorderapproximationofaperturbationprocedurebasedonthefactthattheacousticwavespeedismuchsmallerthanthespeedoflight.Asaresultofthisapproximation,inthetheoryofpiezoelectricity,althoughthemechanicalequationsaredynamic,theelectromagneticequationsarestaticandtheelectricfieldandthemagneticfieldarenotdynamicallycoupled.There-fore,itdoesnotdescribethewavebehaviorofelectromagneticfields.Formanyapplicationsofpiezoelectricacousticwavede-vicesthequasi-statictheoryissufficient,buttherearesituationsinwhichfullelectromagneticcouplingneedstobeconsidered.Forexample,electromagneticwavesgeneratedbymechanicalfieldsneedtobestudiedinthecalculationofradiatedelectromagneticpowerfromavibratingpiezoelectricdevice.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=0͑9͒aswellasthefollowingconstitutiverelations:Tij=cijklSkl−ekijEkDi=eijkSjk+␧ijEjBi=␮0Hi͑10͒whereBiisthemagneticinduction,andHithemagneticfield.Theterm␮0isthemagneticpermeabilityoffreespace,␧ijkisthepermutationtensor.FromEqs.͑9͒and͑10͒onecanobtaincijkluk,lj−ekijEk,j=␳u¨i−1␮0␧ijk␧kmnEn,mj=eijku¨j,k+␧ijE¨j͑11͒whichclearlyshowsthewavenatureoftheelectromagneticfields.Mindlin͓71͔derivedavariationalprincipleforpiezoelectro-magnetisminacompoundcontinuumrepresentingadiatomicma-terial.Lee͓72͔gaveavariationalformulationforthefieldsinsideandoutsideafinitebodywithcontinuityconditionsattheinter-facebetweenthebodyandfreespace.Ageneralizedvariationalprinciplewithallmechanicalandelectromagneticfieldsasinde-pendentvariableswasgivenbyYang͓73͔.Yang͓74͔andYangandWu͓75͔alsoobtainedvariationalprinciplesandgeneralizedvariationalprinciplesfortheeigenvalueproblemoffreevibrationsofapiezoelectromagneticbody.3.2DynamicSolutions.Earlysolutionsfromthedynamictheorybeganwiththepropagationofplanewavesinanun-boundedmediumbyKyame͓76͔.Inadditiontowavesthatareessentiallyacoustic,therearealsowavesthatareessentiallyelec-tromagnetic.Thesetwogroupsofmodesarecoupledbypiezo-electriceffects.EffectsofviscosityandconductivityonplanewaveswerestudiedinKyame͓77͔.LaterPailloux͓78͔andHruska͓79,80͔alsostudiedthepropagationofplanewaves.TsengandWhite͓81͔andTseng͓82͔obtainedsolutionsforsurfacewavesinhexagonalcrystals.SpaightandKoerber͓83͔analyzedsurfacewavesinlithiumniobate.ForwavesinplatesMindlin͓84͔solvedtheproblemofme-chanicallyforcedthickness-shearvibrationofanAT-cutquartzplateandcalculatedradiatedelectromagneticpower.Lee͓85͔studiedthickness-shearvibrationofanAT-cutquartzplateunderlateralfieldexcitationandcalculatedradiatedelectromagneticpower.Leeetal.͓86͔latergaveathoroughanalysisofelectro-magneticradiationfrommechanicallyforcedvibrationsofapi-ezoelectricplate.RadiationfromafiniteplatewascalculatedinCampbellandWeber͓87͔.Fullelectromagneticcouplingandra-diationinthepolarizationgradienttheorywereconsideredinMindlin͓16,17͔.Foranti-planeproblemsofpolarizedceramicsEqs.͑9͒and͑10͒canbewrittenasthefollowingtwosimplewaveequations:vT2ٌ2u3=u¨3c2ٌ2H3=H¨3͑12͒wherevTisanacousticwavespeedandcthespeedoflight.FromtheelectromagneticpointofviewEq.͑12͒describestheso-calledtransversemagneticwaves.Electromechanicalcouplingcomesintoplaywhencalculatingtheelectricfieldfromthedisplacementandthemagneticfield.Couplingscanalsobecausedatbound-aries.Transientsurfacewavesinaceramichalfspaceundersur-faceloadwereanalyzedbySedovandSchmerrJr.͓88͔,andSchmerrJrandSedov͓89͔.Li͓90͔obtainedanti-planeshearhori-zontal͑SH͒surfaceandinterfacewavesolutionsinpolarizedce-ramicsusingthescalarandvectorpotentialformulationofelec-tromagneticfields.ToandGlaser͓91͔studiedSHwavesinplatesusingthepotentialformulation,andfoundthatthequasi-statictheorymaypredictmodesthatarenotapproximationsofthemodespredictedbythefullydynamictheory.ForSHwavesthescalarandvectorpotentialformulationresultsinfourequations͑Li͓90͔͒.Twooftheseequationsarecoupled,andtheothertwoareone-waycoupled.Inaddition,agaugeconditionneedstobeimposed.SHsurfacewavesolutionsweregiveninYang͓92͔us-offullelectromechanicalcouplingontheacousticsurfacewavespeedisverysmall,whichsupportsthequasi-staticapproxima-tion.ItwasalsonoticedinYang͓92͔thatinthelimitwhenthespeedoflightgoestoinfinity,thewavespeedpredictedbypiezo-electromagnetismreducestothatpredictedbyquasi-staticpiezo-electricity.Lovewavesinahalfspacecarryingalayerofadif-ferentmaterialwereanalyzedinYang͓93͔,SHwavesinaplatewereobtainedinYang͓94͔.Sinceelectricfieldscanexistinfreespace,certainpiezoelectricdeviceshaveairgapsinthemandacousticwavesonbothsidesofagapcanstillinteractthroughtheelectricfieldinthegap.PiezoelectromagneticgapwaveswereanalyzedinYang͓95͔.Yang͓96,97͔alsoobtainedfieldsassoci-atedwithamovingsemi-infinitecrackandamovingdislocation.FieldsassociatedwithafinitecrackwereobtainedinLiandYang͓98͔.ElectromagneticradiationfromavibratingceramiccylinderwasanalyzedbyYang͓99͔.338/Vol.59,NOVEMBER2006TransactionsoftheASME4PiezoelectricSemiconductorsPiezoelectricmaterialscanbeeitherdielectrics͑insulators͒orsemiconductors.Anacousticwavepropagatinginapiezoelectriccrystalisusuallyaccompaniedbyanelectricfield.Whenthecrys-talisalsosemiconducting,theelectricfieldproducescurrentsandspacechargeresultingindispersionandacousticloss.Theinter-actionbetweenatravelingacousticwaveandmobilechargesinpiezoelectricsemiconductorsiscalledtheacoustoelectriceffect͑HustonandWhite͓100͔͒whichisaspecialcaseofamoregen-eralphenomenonwhichmaybecalledthewave-particledragsemiconductionineachcomponentphase.Itwasalsofoundthatanacousticwavetravelinginapiezoelectricsemiconductorcanbeamplifiedbytheapplicationofadcelectricfield͑White͓102͔͒.Inadditiontoacousticwaveamplifiers,theacoustoelectriceffectcanalsobeusedtodesignexperimentsformeasuringchargemo-bilityandmakedevicesforchargetransferdrivenbyacousticwaves.4.1GoverningEquations.Thebasicbehaviorofpiezoelec-tricsemiconductorsandtheacoustoelectriceffectcanbede-scribedbyalinearphenomenologicaltheory͑HustonandWhite͓100͔,White͓102͔͒.Considerahomogeneous,one-carrierpiezo-electricsemiconductorunderauniformdcelectricfieldE¯j͑equa-tionsforamulti-carriersemiconductorcanbefoundinFischler͓103͔͒.ThesteadystatecurrentisJ¯i=qn¯␮ijE¯j,whereqisthecarriercharge,n¯isthesteadystatecarrierdensitywhichproduceselectricalneutrality,and␮ijisthecarriermobility.Whenanacousticwavepropagatesthroughthematerial,perturbationsoftheelectricfield,thecarrierdensityandthecurrentaredenotedbyEj,n,andJi.Thelineartheoryforsmallsignalsconsistsoftheequationsofmotion,Gauss’slawofelectrostatics͑thechargeequation͒,andtheconservationofchargeTji,j=␳u¨iDi,i=qnqn˙+Ji,i=0͑13͒Theaboveequationsareaccompaniedbythefollowingconstitu-tiverelations:Tij=cijklSkl−ekijEkDi=eijkSjk+␧ijEjJi=qn¯␮ijEj+qn␮ijE¯j−qdijn,j͑14͒wheredijarethecarrierdiffusionconstants.WithsubstitutionsfromEq.͑14͒,Eq.͑13͒canbewrittenasfiveequationsforu,␾,andncijkluk,lj+ekij␾,kj=␳u¨ieikluk,li−␧ij␾,ij=qnn˙−n¯␮ij␾,ij+␮ijE¯jn,i−dijn,ij=0͑15͒4.2DeviceModeling.Theacoustoelectriceffectandtheacoustoelectricamplificationofacousticwaveshaveledtothedevelopmentofacoustoelectricamplifiersofacousticwaves.ASAWacoustoelectricamplifierisshowninFig.5.Typicalbehav-ioroftheimaginarypartofthewavespeedversusthebiasingelectricfieldinanacousticwaveamplifierisshowninFig.6.Whenthebiasingfieldreachesacriticalvalue,i.e.,whenthecarrierdriftspeedunderthebiasingelectricfieldisequaltotheacousticwavespeed,theimaginarypartofthewavespeedchangesitssign,indicatingthetransitionfromadampedwavetoagrowingwaveorwaveamplification.Duetomulti-fieldcouplingandanisotropy,devicemodelingpresentscomplicatedmathematicalproblems.SurfacewavesoverahalfspaceorahalfspacecarryingalayerofadifferentmaterialwerestudiedbyLakinandShaw͓104͔,Ramakrishna͓105͔,Inge-brigtsen͓106͔,KinoandReeder͓107͔,Kino͓108͔,Wangetal.͓109͔,andGangulyandPal͓110͔.WauerandSuherman͓111͔obtainedsolutionsfortheone-dimensionalproblemofthicknessvibrationsofplates.PropagatingwavesinsemiconductorplatesorpiezoelectricplatescarryingsemiconductorfilmswereanalyzedinFischler͓112,113͔,Dietzetal.͓114͔,andJosse͓115͔.Itshouldbenotedthatsomeofthesurfaceandplatewavestructureshaveairgaps,whicharemorecommoninpiezoelectricsemiconductordevicesthaninotherpiezoelectricdevices.Multi-layeredstruc-tureswerestudiedbyPalmaandDas͓116,117͔.PalanichamyandSingh͓118͔studiedtheeffectofnon-uniformelectricfields.Theaboveanalyseswerebasedonthethree-dimensionalequa-equationsofcoupledextensional,flexural,andthickness-shearmotionsofapiezoelectricsemiconductorplate.Theequationswerespecializedtocrystalsof6mmsymmetryandweresimpli-fiedbyanapproximationforthickness-shearwaves.Thesimpli-fiedequationswereusedtostudytheamplificationofthickness-shearwavesinaplate.ThelowestorderplateequationsderivedinYangandZhou͓119͔areofgeneralizedplane-stresstypeforex-tensionalmotionsoftheplate.Theseequationsforextensionwereusedtomodelsurfacewavesoverapiezoelectrichalfspacecar-ryingasemiconductorfilm͑YangandZhou͓120͔͒,gapwavesbetweenafilmandahalfspace͑YangandZhou͓121͔͒,andinterfacewavesbetweentwohalfspaceswithafilm͑YangandZhou͓122͔͒.Yangetal.͓123,124͔alsoderivedequationsforlaminatedplatesandshellsofpiezoelectricsemiconductorsandstudiedwavesinthem.Foranti-planeproblemsofpiezoelectriccrystalsof6mmsym-metrywhichincludesquiteafewwidelyusedsemiconductors,Eq.͑15͒takesthefollowingsimpleformwhichallowstheoreticalanalysis:cٌ2u+eٌ2␾=␳u¨Fig.5AsurfaceacousticwaveamplifierFig.6TransitionfromdampedwavestogrowingwavesAppliedMechanicsReviewsNOVEMBER2006,Vol.59/339eٌ2u−␧ٌ2␾=qnn˙−n¯␮ٌ2␾−␮E¯·ٌn−dٌ2n=0͑16͒UsingEq.͑16͒,Yang͓125͔alsoanalyzedtheelectromechanicalfieldsaroundasemi-infinitecrackinasemiconductor.TheresultsinYang͓125͔showedthatasemi-infinitemodeIIIcrackstillhastherootrsingularity,butthecoefficientofthesingularfieldismodifiedbysemiconduction.Theaboveanalysesonsemiconduc-torsareallanalytical.Thereseemtobenoreportednumericalresultsby,e.g.,thefiniteelementmethod.4.3MoreSophisticatedTheories.DeLorenziandTiersten͓126͔usedamacroscopicphysicalmodeltoderiveamoresophis-ticatednonlineartheoryfordeformablesemiconductors.Themodelconsistsofseveralinteractingcontinuarepresentingthelattice,electrons,andholes,etc.Basiclawsofphysicsareappliedtoeachcontinuum.Theresultingequationsarecombinedtoob-tainamacroscopicdescriptionofthematerial.TheseequationsarealsogiveninTiersten͓127͔.AnconaandTiersten͓128͔usedtheseequationsinthestudyofSi–SiO2interface,andmetal-insulator-semiconductorstructures͑AnconaandTiersten͓129͔͒.McCarthyandTiersten͓130,131͔andMcCarthy͓132͔analyzedthepropagationofaccelerationwavesandshockwaves.MauginandDaher͓133͔,andDaherandMaugin͓134͔alsodevelopedanonlinearphenomenologicaltheoryfordeformablesemiconductors.Wavepropagationunderabiasingfieldwasana-lyzedbyDaher͓135͔,andDaherandMaugin͓136,137͔.Vermaetal.͓138͔studiedradialvibrationsofacylindricaltube.5MotionsofRotatingPiezoelectricBodiesVoltagesensitivityasafunctionofthedrivingfrequencyisplottedinFig.8forafixed⍀,where␻0isanormalizingfre-quency.Zistheimpedanceoftheoutputcircuit.Z0isanormal-izingimpedance.Itisseenthatnearthetworesonantfrequenciesthesensitivityassumesmaximalvalues.Therefore,thedeviceshouldbeoperatingatafrequencynearresonance.Thedependenceofsensitivityontherotationrate⍀isshowninFig.9forafixeddrivingfrequencynearresonanceandfordiffer-entvaluesofZ.When⍀ismuchsmallerthan␻0,whichistrueinmostapplicationsofpiezoelectricgyroscopes,therelationbe-tweenthesensitivityand⍀isessentiallylinear.Therefore,intheanalysesofpiezoelectricgyroscopes,veryoftenthecentrifugalforcewhichrepresentshigherordereffectsof⍀isneglectedandthecontributiontosensitivityiscompletelyfromtheCoriolisforcewhichislinearin⍀.RotationinducedfrequencyshiftsinSAWorBAWpiezoelec-tricresonatorscanalsobeusedtomeasureangularrates.Whenapiezoelectricresonatorwithresonantfrequency␻0isattachedtoabodyrotatingatanangularrate⍀,theresonantfrequencyoftheresonator͑orequivalentlythewavespeedwhenapropagatingwaveisused͒changesduetorotation.Foraproperlydesignedresonator,thisfrequencyshiftisproportionalto⍀andcanbeusedtomeasureit.Figure10showsthefrequencyshiftsofcertainwavesinaceramicplaterotatingaboutitsnormal.Theslopesofthecurvesattheoriginrepresentsensitivitytorotation.Forcer-tainwavesthesensitivityiszero.Thesewavesareusefulwhenfrequencystabilityinadeviceunderrotationisdesired.Fig.7AsimplepiezoelectricgyroscopeFig.8Outputvoltageversusthedrivingfrequency␻Fig.9Outputvoltageversustherotationrate⍀340/Vol.59,NOVEMBER2006TransactionsoftheASMETheliteratureonpiezoelectricgyroscopesisgrowing.Manypublicationshaveappearedafterthetwoearlierreviewarticles͑Burdessetal.͓139͔,Soderkvist͓140͔͒.ThetworecentPh.D.dissertations͑Loveday͓141͔,Fang͓142͔͒containsomerecentreferences.5.1GoverningEquations.Thebasicbehaviorsofapiezo-electricgyroscopearegovernedbytheequationsofarotatingpiezoelectricbody,whichconsistoftheequationsoflinearpiezo-electricitywithrotationrelatedCoriolisandcentrifugalaccelera-tions.Apiezoelectricgyroscopeisinsmallamplitudevibrationinareferenceframerotatingwithit.Theequilibriumstateintherotatingreferenceframeiswithinitialdeformationsandstressesduetothecentrifugalforce.Therefore,anexactdescriptionofthemotionofapiezoelectricgyroscoperequirestheequationsforsmall,dynamicfieldssuperposedonstaticinitialfields͑Baum-hauerandTiersten͓143͔͒duetothecentrifugalforce,whichhastobeobtainedfromthenonlineartheoryofelectroelasticity͑Tier-sten͓144͔͒.ThegoverningequationsintherotatingframecanbewrittenasTji,j−2␳␧ijk⍀ju˙k−␳͑⍀i⍀juj−⍀j⍀jui͒=␳u¨iDi,i=0Tij=cijklSkl−ekijEk,Di=eijkSjk+␧ijE͑17͒wherethetermsrelatedto⍀jrepresentthesumoftheCoriolisandcentrifugalforces.Termsduetotheinitialfieldsareignored.Sincepiezoelectricgyroscopesareverysmall͑oftheorderof10mm͒,theiroperatingfrequency␻0isveryhigh,usuallyoftheorderoftensofkHzorhigher.Piezoelectricgyroscopesareusedtomeasureanangularrate⍀muchsmallerthanitsresonantfre-quency␻0.Inthiscasethecentrifugalforceduetorotation,whichisproportionalto⍀2,ismuchsmallercomparedtotheCoriolisforcewhichisproportionalto␻0⍀.Therefore,theeffectofrota-tiononmotionsofpiezoelectricgyroscopesisdominatedbyCo-riolisforce.Thisisfundamentallydifferentfromtherelativelywellstudiedsubjectofvibrati