有效介质理论 Effective Medium Theory

23EffectiveMediumTheoriesW.Kreher23.1IntroductionPredictingmacroscopicpropertiesofmaterialsinvolvestwotasks:1.Projectionofmacroscopicloadingconditionsontothemicrostructurallevelwithsufficientandadjusteddetails(resultinginfluctuatinginter-nalfields)2.Averagingthelocalreactionofmaterialtoobtainthemacroscopicre-sponsetotheloadingsForpiezoelectric/ferroelectricpolycrystallinematerials,therelevantmicro-structurallevelispresentedbyspecialdomainconfigurationswithingrainswhichhavedifferentorientations.Thelocalmaterialresponsecomprisestheelastic,dielectric,andintrinsicpiezoelectriceffectaswellastherestructuringofdomainconfigurationbymovementofdomainwallsorcompletereorienta-tionofthespontaneouspolarization,i.e.,domainswitching.Otherprocessesasmovingfreechargesand/ordefectdipolesarenormallynotconsideredinpresentmicro–macromodels.Theyappearintheformofalreadyaveragedlocalproperties,e.g.,asaneffectivefrictionalforceoractivationenergyfordomainwallmovement.Obviously,exactthree-dimensionalsolutionsfortheinternalfieldsandprocessesareimpossible,soseveralsimplificationsareintroducedinthemod-elsdependingontheaim(i.e.,whetherlinear,nonlinear,orcompleterepolar-izationbehaviorisconsidered).Typicalapproximationsare:•Simplificationofdomainconfigurationeitherbyaparticulardeterministicdomainarrangement(e.g.,stackofplate-likedomains)orbyassumptionofasetofnoninteractingrepresentativedomains•Simplificationoftheinteractionsbetweenneighboringgrains(e.g.,assump-tionofhomogeneousfields,smoothingbytheeffectivemediumapproach)Thepresentchapterdescribesthesemodelingapproachesinsomedetail,puttingemphasisonpartiallyexactsolutionsandpracticalapplicability.518W.KreherSection23.2describestherelevantmodelingtools,namelytheexactsolutionsforalamellardomainstackandforanellipsoidalinclusioninahomogeneousmatrix.Onthisbasis,theeffectivemediumapproachforapolycrystallinesolidisintroduced.Section23.3extendsthetheorytotheextrinsicpropertieswhicharedeterminedbythe(reversible)movementofdomainwalls.Finally,Sect.23.4givesashortoverviewaboutmodelingthenonlinearbehaviorwhichisconnectedwithextensivedomainreconfiguration.Torepresentthetheoreticalbackgroundinacompactandcomprehensibleform,itisnecessarytointroduceashortformnotationforthepiezoelec-tricmateriallaws.Theseso-calledconstitutiveequations,relatingmechanicalstressandelectricdisplacementtostrainandelectricfield,areexpressedbyTp=cEpqSq−elpEl+TEp,Dk=ekqSq+εSklEl+DSk,p,q=1,2,...,6,k,l=1,2,3.(23.1)(WemakeuseoftheabbreviatedVoigtnotationandtheEinsteinsumma-tionconvention.)Here,TEpisthestressstatewhicharisesifasolid,whichisclamped(Sq=0)andshortcircuited(El=0),undergoesaferroelec-tric/ferroelastictransformation,andDSkistheelectricdisplacementobservedunderthesameconditions.Ofcourse,theseremnantpropertiesmaybeex-pressedbythecorrespondingquantitiesunderstress-freeconditions(i.e.,transformationstrainandspontaneouspolarization);yettherepresentationadoptedhereisespeciallysuitedforthederivationofmaterialpropertieswithintheeffectivemediumapproximation.Thoughwedonotexplicitlyconsiderpyroelectricandthermoelasticef-fects,itshouldbementionedthattheseeffectsarecomprisedbyTEandDS,too.Thus,anytheoreticalresultshownbelowcanbeutilizedtomakepredictionsalsoforthesethermalproperties.Combiningmechanicalandelectricalfieldquantitiesinto9-dimvectorsgivesthecompactnotation:ji=Lijfj+jsi,i,j=1,2,...,9(23.2)withj=TpDk,L=cEpqetplekq−εSkl,f=Sq−El,js=TEpDSk.(23.3)Here,thesuperscripttmeansthetranspositionofthe3×6matrixofpiezoelec-tricconstants.ThematrixLrepresentsthecompletesetofthelinearelastic,piezoelectric,anddielectricconstants;itisasymmetric9×9matrix,i.e.,L=Lt.(23.4)Notethatthissymmetryisaconsequenceofthespecialrepresentationoffwithanegativesignfortheelectricfield.23EffectiveMediumTheories519Allprecedingequationsfitintothestandardvector–matrixcalculuswhichisespeciallyconvenientforthenumericalevaluation.Subsequently,uppercaselettersdenote9×9matricesandlowercaselettersdenote9×1vectors.So,inmostcases,wecanavoidwritinganyindices.Inparticular,wefindfortheinversedmateriallawf=Mj+fs.(23.5)Again,thematrixMofinverselinearmaterialconstantsisasymmetric9×9matrixandfsrepresentstheremnantpropertiesunderfixedstressandchargeconditions:M=L−1=sDpqgtplgkq−βTkl,fs=−Mjs=SDp−ETk.(23.6)Caremustonlybetakenwhenatransformationtoalocalcoordinatesystem(e.g.,fixedtothecrystalordomainaxes)hastobeperformed.Thismustbedoneexclusivelyintheoriginaltensorrepresentationofthefieldquantities.Othercombinednotationsofmechanicalandelectricalfieldvariables(e.g.,theBarnett–Lothenotation)arealsopossible–yetthevariantpresentedhereisbestsuitedforthecompactdescriptionofeffectivemediummodels.23.2LinearIntrinsicPropertiesWeconsiderhereapiezoelectricceramicconsistingofgrainswithagivenrandomortexturedorientationdistribution.Eachgrainconsistsofacertainarrangementofdomainsformingatwinnedandhierarchicallyorderedstruc-ture.Incontrasttotheextrinsicproperties(seeSect.23.3),herethedomainstructureisassumedtobefixed.Thepolingstateisdescribedbyaparticularorientationdistributionofdomains.Anexactsolutionfortheinternalfieldsandtheeff