各向同性磁流变弹性体磁相关非线性动态力学行为的本构模型.pdf

Constitutive modeling of the magnetic-dependent nonlineardynamic behavior of isotropic magnetorheological elastomersBochaoWang1,YanLi1,HaomingPang1,ZhenbangXu2,andXinglongGong11CAS Key Laboratory of Mechanical Behavior and Design of Materials,Department of Modern Mechanics,University of Science and Techno-logy of China,Hefei 230027,China;2CAS Key Laboratory of On-orbit Manufacturing and Integration for Space,Optics System,Changchun Institute of Optics,Fine Mechanics andPhysics,Chinese Academy of Sciences,Changchun 130033,ChinaCorrespondence:XinglongGong,E-mail:2024TheAuthor(s).ThisisanopenaccessarticleundertheCCBY-NC-ND4.0license(http:/creativecommons.org/licenses/by-nc-nd/4.0/).Cite This:JUSTC,2024,54(1):0106(12pp)ReadOnlineAbstract:Isotropicmagnetorheologicalelastomers(MREs)aresmartmaterialsfabricatedbyembeddingmagnetizableparticlesrandomlyintoapolymermatrix.Underamagneticfield,itsmoduluschangesrapidly,reversibly,andcontinu-ously,offeringwideapplicationpotentialinthevibrationcontrolarea.Experimentalobservationsshowthatthereisastrongfrequency,strainamplitude,andmagneticdependenceofthedynamicbehaviorofisotropicMRE.Althoughimport-antforpotentialapplications,themagnetic-dependentnonlineardynamicbehaviorofisotropicMREhasreceivedlittlethe-oreticalattention.ToaccuratelyevaluatethedynamicperformanceofisotropicMREandtoguidethedesignofisotropicMRE-basedproducts,anewconstitutivemodelbasedoncontinuummechanicstheoryisdevelopedtodepictthemagnetic-dependentnonlineardynamicbehaviorofisotropicMRE.Subsequently,thenumericalimplementationalgorithmisde-veloped,andthepredictionabilityofthemodelisexamined.Themodelprovidesadeeperunderstandingoftheunderly-ingmechanicsofthemagnetic-dependentnonlinearviscoelasticbehaviorofisotropicMRE.Furthermore,themodelcanbeutilizedtopredictthemagnetomechanicalcouplingbehaviorofisotropicMREandthereforeservesasausefulplat-formtopromotethedesignandapplicationofisotropicMRE-baseddevices.Keywords:isotropicmagnetorheologicalelastomer;frequencydependence;strainamplitudedependence;magneticde-pendence;nonlinearviscoelasticity;constitutivemodelingCLC number:O345Document code:A1 IntroductionMagnerheologicalelastomers(MREs)aresmartmaterialsthatare often fabricated by embedding magnetizable particlesintoapolymermatrix.Underamagneticfield,duetothein-teractionbetweenparticlesandthematrix,magnetictunabil-ityofthemoduluscanbeachievedforMRE.Byutilizingthemagnetictunabilityofthemodulus,applicationsofMREinsemiactivevibrationisolators13,smarttunedmassdampers4,5and vibration absorbers6 have been explored.Meanwhile,sincethemagneticpermeabilityofMREislargerthanthatofrubberandair,astressmismatchoccursandresultsinmag-netostrictionofMREifamagneticfieldisapplied.Duetomagneto-induceddeformation,amagneticallycontrolledsur-facepattern7,8usingMREisexploited.ToguidethedesignofMRE-basedproducts,accuratechar-acterization and modeling of their dynamic behavior isneeded.Experimentally,a dynamic test of isotropic MREunder harmonic cyclic loading shows that the modulus ofMREincreases with magnetic field and frequency but de-creaseswithstrainamplitude9,10.Ontheotherhand,thestressrelaxationtestofisotropicMREshowsthatalongertimeisneededforittoreachtheequilibriumstatewhenamagneticfieldisapplied11,12.Therefore,duetothecouplingbetweenparticlesandthematrixwithintheisotropicMREundermag-netic and mechanical loadings,a complex,magnetic-dependentnonlineardynamicbehaviorisexhibited.Theoretically,magneticdipoletheory13wasinitiallyutil-izedtodepictthemodulusmagneticstiffeningeffectofMREunderthequasistaticcase.Next,thedistributionoftheparticlechain14andinelasticbehaviorofisotropicMREweretakenintoaccountthroughtheZenerviscoelasticelement15,16,fric-tionelement17,andfractionaldashpotelement18,19basedontheinfinitesimalstrainassumption.Afterwards,basedonthephenomenologicalmodelofMRE,thedynamicperformanceof MRE-based devices was modeled,e.g.,Refs.20,21.However,duetothesoftnatureofthematrix,magneticdipole-basedtheorycannotpredictthedynamicbehaviorofMREunderlargedeformation.Afinitestrainmagnetoelasticinter-actiontheorywasdevelopedbyDorfmannandOgden22.Itispostulated that Helmholtz free energy exists for isotropicMREandthatthederivativeofthefreeenergywithrespecttothedeformationgradientandthemagneticfieldstrengthres-ultsinthecorrespondingstressandmagneticfluxdensity,re-spectively.FollowingthetheoreticalpathofDorfmannandOgden22,constitutivemodelsthatincorporatethemagnetiza-tionbehaviorandthecontributionofMaxwellstresstotheArticlehttp:/Received:November 28,2022;Accepted:February 01,202301061DOI:10.52396/JUSTC-2022-0173JUSTC,2024,54(1):0106totalstresswithinisotropicMREweredeveloped23,24.Sub-sequently,the corresponding finite element implementationplatformwasdevelopedandusedtopredictthemagnetostric-tion-induceddeformationofanisotropicMREfilmbondedtoa nonmagnetic substrate2527.However,for the model de-veloped21,22,theeffectofthemagneticfieldiscoupledtotheclassicalhyperelasticmodelinanadditiveform,andthemod-ulusmagneticstiffeningeffectisonlyexhibitedindirectionswheretheMaxwellstressisnotzero.However,experimentaltestingresultsshowedthatamodulusmagneticstiffeningef-fectalsoexistedinotherdirections.Therefore,themodulusmagneticstiffeningeffectcannotberepresentedfullybythemodeldeveloped.Currently,theoreticalworkshavemainlyfocusedonpre-dictingtherate-independentbehaviorofisotropicMRE.AsanessentialcomponentofthemechanicalbehaviorofMREandwhichisofgreatimportancetotheapplicationofisotrop-icMREinthevibrationcontrolarea,thedynamicbehaviorofMREhasreceivedlittleattention.Constitutivemodelstode-pictthehysteresisbehaviorofisotropicMREmainlyfocusonthemagnetic-dependentviscoelasticbehaviorbasedonanin-finitesimalstrain assumption by adding a dashpot or vis-coelastic Zener model1519.Although a finite strain-basedZenerviscoelasticelement2830wasintroducedtoaccountfortheviscoelasticbehaviorofthepolymermatrix,theviscosityoftheZenerviscoelasticelementwasassumedtobeacon-stantvalue,irrespectiveofthemagneticfieldandstrainamp-litude.However,as observed experimentally912,the vis-coelasticbehaviorofisotropicMREishighlymagneticandstrainamplitudedependent.Therefore,thereisacertaingapbetweentheexperimentaltestingandconstitutivemodelingofisotropicMRE.To evaluate the dynamic performance of isotropic MREunderdifferentfrequencies,strainamplitudes,andmagneticfieldsandtoguidethedesignofMRE-baseddevices,anewconstitutivemodelofMREthatincorporatesmagnetic,fre-quency,andamplitudedependencyisdevelopedbasedonex-perimentaltestingandtheoreticalanalysis.Themajornov-eltyofthisstudyistodevelopanewconstitutivemodelbasedonfinitestraintheorythatthoroughlyincorporatesthemodu-lusmagneticstiffeningeffectandthenonlinearviscoelasti-cityofisotropicMRE.Thisworkcanprovideguidanceforthe accurate prediction of the magnetomechanical couplingbehaviorofisotropicMREandpromotethedesignandpos-sibleapplicationsofisotropicMRE-baseddevices.Theremainderofthispaperisorganizedasfollows.InSec-tion2,quasistatic,magnetization,and harmonic cyclic dy-namic tests of MRE are conducted.In Section 3,thecontinuummechanicsframeworkisintroduced.Afterwards,thespecificconstitutiveequationsareproposedinSection4.InSection5,theexperimentalandsimulationresultsarecom-pared;furthermore,themodelpredictionresultsarepresen-ted.Finally,theconclusionispresentedinSection6.2 Experimental resultsCarbonylironparticles(CIPs,typeCN,BASF,Germanydia-meter7monaverage),polydimethylsiloxane(PDMS),andcuring agent are used.The mass ratio for carbonyl ironparticles,PDMS,andcuringagentis75251.Thespecif-ic fabrication process is as follows.First,PDMS,curingagent,andcarbonylironparticlesaremixedfor5min.After-wards,the air bubbles within the mixture were extractedusingavacuumchamberforhalfanhour.Subsequently,themixturewaspouredintoarectangularmoldandvulcanizedat100Cforhalfanhour.Duringcuring,nomagneticfieldisapplied.Therefore,thecarbonylparticlesareassumedtobehomogenizedanddistributedwithinthematrix,andthefab-ricatedMREisisotropic.Theschematicconfigurationcorres-pondingtothefabricationprocessofisotropicMREisshowninFig.1.Formoredetailsregardingthefabricationofiso-tropicMRE,thereadersarereferredtoRef.31.3105s115%Afterfabrication,aquasistatictestoftheisotropicMREwithastrainrateofandstrainamplitudeofundermagneticfieldswithmagnitudesof0,0.2T,and0.4Tareconductedonadynamicmechanicalanalyzer.Forthedy-namictest,0.1Hz,1Hz,and10Hzareconsidered,threesetsofstrainamplitudesof5%,10%,and15%andtwosetsofmagneticfieldsof0and0.4Tareapplied.AnElectroforce3200-typedynamicmechanicalanalyzer(DMA)fromTAin-strumentswiththefunctionofapplyingamagneticfield,asshowninFig.2,isusedtotestthemagnetic-dependentmech-anicalbehaviorofisotropicMRE.Beforetesting,therelationbetweenthemagneticfluxdensityandappliedcurrentwascalibratedthroughaTeslameter.Eachcombinationoffre-quency,strainamplitude,andmagneticfieldistestedthreetimes,andthemeanvalueistakenasthefinaltestdata.TocharacterizethemagnetizationperformanceofMRE,ahysteresismeasurementofsoftandhardmagneticmaterials(HyMDCMetis,Leuven,Belgium)wasutilized.Aftertest-ing,themeasurementresultsareshowninFigs.35.TheresultsinFig.3showthatapronouncedmodulusmag-neticstiffeningeffectisexhibitedforisotropicMRE.Further-more,thequasistaticbehaviorofisotropicMREisnonlinear.Specifically,theslopeofthestressstraincurvedecreasesfirst,then reaches a linear region,and afterwards,there isanother increase in the slope.From the perspective ofFig.1.SchematicconfigurationforthefabricationofisotropicMRE.Magnetic-dependentnonlineardynamicmodelofisotropicmagnetorheologicalelastomersWangetal.01062DOI:10.52396/JUSTC-2022-0173JUSTC,2024,54(1):0106modeling,ahyperelasticmodelthatisabletodepictthenon-linearstress straincurveisneeded.Regardingthemagnetiza-tioncurve,it is found that the magnetization strength in-creaseswithincreasingmagneticfieldintensityuntilmagnet-icsaturationisreachedforisotropicMRE.Regarding the dynamic hysteresis stress strain curve,asshowninFigs.4and5,thepeakstressincreaseswithincreas-ing magnetic field,frequency,and strain amplitudes.Toquantitativelyevaluatetheinfluenceofthestrainamplitudeon the dynamic behavior of isotropic MRE,the Fouriertransform-basedmethodisappliedtoextracttheequivalentshearmodulusandlossfactorofisotropicMREinthefre-quencydomain.Mathematically,theequivalentshearmodu-lusandlossfactoroftheMREarereachedthroughG=G+jG=,(1)|G|=(G)2+(G)2,(2)and=GG,(3)GGG|G|where,anddenotethecomplex,storage,andlossmoduli,respectively.and aretheFouriertransformsoftheshearstressandshearstraininthefrequencydomain.istheequivalentshearmodulus,and istheequivalentlossfactor.AfterapplyingEqs.(1)(3),theequivalentstorageandloss modulus of the isotropic MRE is shown in Fig.6.Clearly,the equivalent modulus and the loss factor of theisotropicMREdecreasewithincreasingstrainamplitude.Mi-croscopically,this phenomenon is closely related to thedeformation-enhancedshearthinningofthepolymer.Fromaconstitutive modeling perspective,a strain amplitude andmagnetic field process-dependent nonlinear viscoelasticmodel is needed to depict the nonlinear and magnetic-dependentdynamicbehaviorofisotropicMRE.3 Continuum mechanics basis3.1 Kinematics and magnetic equationsreferencereferencecurrentreferencecurrentDuetothesoftnatureandlargedeformationthatisotropicMREmayencounterduringapplication,continuummechan-icstheoryshouldbeappliedtodepictthemagnetomechanicalcouplingbehaviorofisotropicMRE.AsshowninFig.7,theinitialconfigurationoftheisotropicMREwherenoloadingisapplied is denoted as.After loading,the isotropicMREmovesfromto.ThedeformationgradientthatconnectsandisF=(X)x,(4)XxC=FTFFvFeF=FeFvCe=(Fe)TFeCv=(Fv)TFvwhereisthemotionofaspecificmaterialpointwithco-ordinatesand inthereferenceandcurrentconfiguration,respectively.ThecorrespondingrightCauchyGreentensoris,wheresuperscriptTdenotesthetransposeofthematrix.TodepicttheviscoelasticbehaviorofisotropicMRE,following the theoretical path of finite strain viscoela-sticity3234,the deformation gradient is further multiplicat-ivelydecomposedintoaviscous()andanelasticpart()as.ThecorrespondingrightCauchyGreentensorsareand.referenceBRHRcurrentBHFurthermore,asshowninFig.7,themagneticfluxdensityand magnetic field intensity in are and.AccordingtothemagnetoelasticinteractiontheorybyDorf-mannandOgden22,theequivalentinteractionsinareandwithBR=JF1B,(5)HR=FTH.(6)BHAccordingtoRef.22,theboundaryconditionssatisfiedforandarenB=0，(7)nH=0，(8)()=()outside()MREnBHB=0H0=1.256106TmA1SwheredenotesthedifferencebetweentheoutsidesurroundingsandMRE.isthenormaldirectionattheMREandsurroundingairinterfacealongtheappliedmagneticfluxdensitydirection.Invacuum,andarecon-nectedby,whereisthemagneticpermeabilityinvacuum.Furthermore,inFig.6denotes the second PiolaKirchhoff stress in the referenceconfiguration,anditisconnectedtotheCauchystressbyFig.2.Dynamicmechanicalanalyzerwithfunctionofapplyingmagneticfield.(a)Photographofthedynamicmechanicalanalyzer.(b)Schematicconfigurationofthedynamicmechanicalanalyzer.Fig.3.QuasistaticstressstraintestresultsofisotropicMREunderdiffer-entmagneticfieldsandthemagnetizationstrength-magneticfieldintens-itycurveofisotropicMRE.Thelinesanddotsareexperimentalandsim-ulationresults,respectively.Wangetal.01063DOI:10.52396/JUSTC-2022-0173JUSTC,2024,54(1):0106Fig.4.DynamichysteresisstressstrainofisotropicMREunderdifferentfrequenciesandstrainamplitudes.Themagneticfieldis0T.ThelinesanddotsaretheexperimentalandsimulationresultsusingthemodifiedEyringviscoelasticmodel,respectively.Fig.5.DynamichysteresisstressstrainofisotropicMREunderdifferentfrequenciesandstrainamplitudes.Themagneticfieldis0.4T.ThelinesanddotsaretheexperimentalandsimulationresultsusingthemodifiedEyringviscoelasticmodel,respectively.(a)(b)Fig.6.Equivalentshearmodulus(a)andlossfactor(b)ofisotropicMREinthefrequencydomainunderdifferentstrainamplitudesandmagneticfields.Magnetic-dependentnonlineardynamicmodelofisotropicmagnetorheologicalelastomersWangetal.01064DOI:10.52396/JUSTC-2022-0173JUSTC,2024,54(1):0106=J1FSFT，(9)J=det(F)with.3.2 Thermodynamic frameworkCHRInthiswork,notemperaturechangeisconsidered.Therefore,an isotropic MRE is assumed under isothermal conditions.Temperaturehasasignificanteffectonthemechanicaldy-namicbehaviorofpolymer-basedmaterials.Thecouplingofthe temperature-dependent magnetomechanical behavior ofisotropic MRE may be taken into account in the currentmodelthroughthetimetemperatureequivalenceprinciple35ortheWilliamsLandauFerryfunction36,whichwillbeoneofthefutureresearchdirectionsfollowingthiswork.Underisothermalconditions,itispostulatedthataHelmholtzfreeenergyexistswithintheMREandisafunctionofand.Todepictthemagnetizationbehavior,themodulusmag-neticstiffeningeffect,theviscoelasticbehavioralongwiththecontributionoftheMaxwellstressonthetotalstress,isde-coupledintothreepartsas:(C,HR)=me(C,HR)+mve(Ce,HR)+m(C,HR),(10)memvemwhere,andarethemagnetohyperelastic,mag-netoviscoelastic,andpuremagneticfreeenergyparts,respect-ively.AccordingtoRef.30,theClausiusPlanckinequalityofthermodynamicsthatmustbesatisfiedisHRBR+12S:C 0，(11)(Ce,HR)(C,HR)mvewhere the dot between two vectors represents the scalarproductoperator,twodotsarethesecond-o