Finite-element-method-for-piezoelectric-vibration1

INTERNATIONALJOURNALFORNUMERICALMETHODSINENGINEERING,VOL.2,151-157(1970)FINITEELEMENTMETHODFORPIEZOELECTRICVIBRATIONHENNOALLIKANDTHOMASJ.R.HUGHESGeneralDynamics,ElectricBoatDivision,Groton,ConnecticutSUMMARYAfiniteelementformulationwhichincludesthepiezoelectricorelectroelasticeffectisgiven.Astronganalogyisexhibitedbetweenelectricandelasticvariables,anda‘stiffness’finiteelementmethodisdeduced.Thedynamicalmatrixequationofelectroelasticityisformulatedandfoundtobereducibleinformtothewell-knownequationofstructuraldynamics.Atetrahedralfiniteelementispresented,implementingthetheoremforapplicationtoproblemsofthree-dimensionalelectroelasticity.INTRODUCTIONTheequationsofpiezoelectricityaresufficientlycomplextoprecludeclosedformsolutionforallbutthesimplestcases.Thisisunfortunatesincethepiezoelectriceffectplaysanimportantroleinthefieldsofcrystalphysicsandtransducertechnology.Recently,variationalprincipleshavebeenderivedwhichserveasthebasisofapproximatesolutiontechniques,suchasthepowerfulRayleigh-Ritzmethod.NoteworthycontributionsalongtheselineshavebeenmadeinthepapersofLewis’andHollandandEerNis~e.~~Althoughtheseimportantdevelopmentshaveopenedthewaytoawiderclassofproblems,theyarenotsufficientlygeneralinthemselvestobeconsideredauniversalmethodofpiezoelectricanalysis.Forinstance,asignificantdeficiencyoftheRayleigh-Ritztechniqueisthenecessityofselectingtrialfunctions,whichoftenbecomesintractableforcomplexgeometries.Thepresentpaperconcernsitselfwiththedevelopmentofageneralmethodofelectroelasticanalysisbyincorporatingthepiezoelectriceffectinafiniteelementformulation.Thetheorypre-sentedis,essentially,anoutgrowthofthevariationalprincipleenunciatedinthepapersofHollandandEerNisse,casthereinmatrixfashion.Thedynamicalmatrixequationderivedforlinearpiezoelectricityisfoundtobereducible,inform,totheordinarymatrixequationencounteredinstructuraldynamics.Theelectroelasticmatricesforasimplex‘displacement-potential’finiteelementforthree-dimensionalanalysisarepresented,illustratingthemethod.Wheneverpossible,thenotationprescribedintheJ.R.E.‘Standardsonpiezoelectriccrystal^'^isemployed.VARIATIONALPRINCIPLEThevariationalprinciplewhichservestoincorporatethepiezoelectriceffectisessentiallythatduetoHollandandEerNi~se:~whoarrivedatitbya‘trial-and-error’proceduretoyieldasEulerequationsknownrelationsofelectroelasticity.Itispertinenttomentionthatthepresentauthorshavefounditpossibletoderivethesametheorem,strictlybyapplyingtheprincipleofvirtualdisplacementstoacontinuumundertheinfluenceofelectricalandmechanicalforces.ThestartingpointforsuchaderivationisthedefinitionofvirtualworkdensityReceived25December1968RevisedI0March196901970byJohnWiley&Sons,Ltd.151152HENNOALLIKANDTHOMASJ.R.HUGHESwhere{u}denotesdisplacement,4theelectricpotential,{F}themechanicalforcedensity,CJthechargedensityand6avirtualquantity.Thisequationalsoservestopointouttheveryusefulanalogybetweenelectricalandmechanicalvariables.Hence,chargeandpotentialmaybeincludedinthenotionsofgeneralizedforceandgeneralizeddisplacement,respectively,whichmakesforaclearunderstandingofthenatureofthesequantitiesinafiniteelementformulation.AsshowninTableI,thisanalogousbehaviourofelectricalandmechanicalvariablesmaybeextendedtoincludeelectricfluxdensity{D}andstress{T},andelectricfield{E}andmechanicalstrain{S},whichforlinearmaterialbehaviourarerelatedthroughthematrixformofthecon-stitutiveequations(2)where[c]denotestheelasticstiffnesstensorevaluatedatconstantelectricfield,[elthepiezo-electrictensorand[E]thedielectrictensorevaluatedatconstantmechanicalstrain.Notethatallelectricalquantitiesareonetensorialranklowerthanthecorrespondingmechanicalquantities.I{TI=[CI{S}-[el{El{D}=[elT{S}+[El{E}TableI.AnalogybetweenmechanicalandelectricalquantitiesMechanicalElectrical..ForcedensityF,(vector)Displacementu,(vector)StressTt,(second-ordertensor)StrainS,,(second-ordertensor)Chargedensityu(scalar)Potential+(scalar)FluxdensityD,(vector)ElectricfieldE,(vector)Withoutfurtherelaborationtheresultingvariationalprinciplemaybestatedinmatrixnotationas[Jv[{as}T[c]{S}-{[el{E}-{SE}2'[elT{S}-{6E}[E]{E}-{Su}'{P}wherethefollowingprescribedquantitiesaredefinedas{F}thebodyforce,{T}thesurfacetraction,{P}thepointforce,0thebodycharge,0'thesurfacechargeandQthepointcharge.InadditionVdenotesthevolumeofthebody,S,thatpartoftheboundarywheretractionisprescribed,S,thatpartoftheboundarywherechargeisprescribedandpthedensity.Finally,notethatthevariationalprinciplecanbethoughtofasageneralizationoftheelasticityprincipleofminimumpotentialenergy,sincedeletionoftheelectricalquantitiesinequation(3)resultsinthelatterprinciple'sfirstvariation.FINITEELEMENTFORMULATIONTogeneratetheelectroelasticmatrixrelationsforafiniteelement,thecontinuousdisplacementandpotentialareexpressedintermsofinodalvaluesviainterpolationfunctionsf,,andf4FINITEELEMENTMETHODFORPIEZOELECTRICVIBRATION153Itisassumedherethattheinterpolationfunctionspossesstherequisitepropertiesforconvergencetothecorrectsolutionwithdiminishingelementsize.6Inasimilarmanner,theprescribedbodyandsurfaceforce(charge)distribu