The modified couple stress functionally graded Tim

ThemodifiedcouplestressfunctionallygradedTimoshenkobeamformulation——2021Themodi?edcouplestressfunctionallygradedTimoshenkobeamformulationM.Asghari?,M.Rahaeifard,M.H.Kahrobaiyan,M.T.AhmadianSchoolofMechanicalEngineering,SharifUniversityofTechnology,Tehran,IranarticleinfoArticlehistory:Received14April2021Accepted31August2021Availableonline25September2021Keywords:F.ElasticbehaviorA.FunctionallygradedmaterialsE.MechanicalpropertiesabstractInthispaper,asize-dependentformulationispresentedforTimoshenkobeamsmadeofafunctionallygradedmaterial(FGM).Theformulationisdevelopedonthebasisofthemodi?edcouplestresstheory.Themodi?edcouplestresstheoryisanon-classiccontinuumtheorycapabletocapturethesmall-scalesizeeffectsinthemechanicalbehaviorofstructures.Thebeampropertiesareassumedtovarythroughthethicknessofthebeam.Thegoverningdifferentialequationsofmotionarederivedfortheproposedmodi?edcouple-stressFGTimoshenkobeam.Thegenerallyvalidclosed-formanalyticexpressionsareobtainedforthestaticresponseparameters.Ascasestudies,thestaticandfreevibrationofthenewmodelarerespectivelyinvestigatedforFGcantileverandFGsimplysupportedbeamsinwhichpropertiesarevaryingaccordingtoapowerlaw.Theresultsindicatethatmodelingbeamsonthebasisofthecouplestresstheorycausesmorestiffnessthanmodelingbasedontheclassicalcontinuumtheory,suchthatforbeamswithsmallthickness,asigni?cantdifferencebetweentheresultsofthesetwotheoriesisobserved.ó2021ElsevierLtd.Allrightsreserved.1.IntroductionFunctionallygradedmaterials(FGMs)areproducedfrommix-ingoftwodifferentmaterials.Thistypeofmaterialsprovidesthespeci?cbene?tsofbothoftheconstituents.Theycanbede?nedasinhomogeneouscompositeswhicharemadefromamixtureoftwodifferentmaterials,usuallyametalandaceramic,withade-siredcontinuousvariationofpropertiesasafunctionofpositionalongcertaindimension(s).Thecontinuouslycompositionalvaria-tionoftheconstituentsinFGMsalongdifferentdirectionsisthegreatbene?tofFGMs,becausethispropertyoffersasolutiontotheproblemofappearinghighmagnitudeshearstressesthatmaybeinducedinlaminatedcomposites,wheretwomaterialswithgreatdifferencesinpropertiesarebonded.Nowadays,struc-turesmadeofFGMshaveagreatpracticalroleinengineeringandindustrial?elds.SomeworkshavebeenperformedbyresearchersonthestaticanddynamicbehaviorofbeamsandplatesmadeofFGMs.Asgharietal.[1]havementionedsomeinstancesoftheseworks,includingRefs.[2–7].Asanotherinstance,thethermalsnappingoffunction-allygradedplateshasbeeninvestigatedbyPrakashetal.[8].Also,Jomehzadehetal.[9]presentedananalyticalapproachforthestressanalysisoffunctionallygradedannularsectorplates.More-over,analyticalmodelingofthermalresidualstressesinsomefunctionallygradedmaterialsystemshasbeenpresentedbyBou-chafaetal.[10].Itisnotedthatthesesampleworksarebasedontheclassicalcontinuumtheory,whiletheformulationpresentedinthisworkisbasedonanon-classicalcontinuumtheory,themodi?edcouplestresstheory,whichisdiscussedindetailinthefollowing.Inrecentyears,theapplicationofFGmaterialshasbroadlybeenspreadinmicroandnanostructuressuchasthin?lmsintheformofshapememoryalloys[11,12],micro-andnano-electromechani-calsystems(MEMSandNEMS)[13,14]andalsoatomicforcemicroscopes(AFMs)[15].BeamsusedinMEMS,NEMSandAFMs,havethethicknessintheorderofmicronsandsub-microns,sothatthesmallscaleeffectsintheirbehaviorisconsiderable.Thesize-dependentstaticandvibrationbehaviorinmicroscalesareexperimentallyvalidated(seeforexample[16–19]).Consideringexperimentalobservations,itiswell-knownthatsize-dependentbehaviorisaninherentpropertyofmaterialswhichappearsforabeamwhenthecharacteristicsizesuchasthicknessordiameterisclosetotheinternalmateriallengthscaleparameter[20].Theclassicalcontinuummechanicstheoriesarenotcapableofpredictionandexplanationofthesize-dependentbehaviorswhichoccurinmicron-andsub-micron-scalestructures.However,non-classicalcontinuumtheoriessuchashigher-ordergradienttheo-riesandthecouplestresstheoryareacceptablyabletointerpretthesize-dependencies.In1960ssomeresearchersintroducedthecouplestresselastic-itytheory[21–23].Intheconstitutiveequationofthistheory,somehigher-ordermateriallengthscaleparametersappearinadditiontothetwoclassicalLameconstants.Yangetal.[24]arguedthatinadditiontotheclassicalequilibriumequationsofforcesandmomentsofforces,anotherequilibriumequationshouldbeconsideredforthematerialelements.Thisadditionalequationis0261-3069/$-seefrontmatteró2021ElsevierLtd.Allrightsreserved.doi:10.1016/j.matdes.2021.08.046Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:asghari@(M.Asghari).theequilibriumofmomentsofcouples.Then,theyconcludedthatthisadditionalequilibriumequationimpliesthesymmetryofthecouplestresstensor.Accordingly,theymodi?edtheconstitutiveequationsofthecouplestresstheoryandpresentthenewconsti-tutiveequations.Utilizingthemodi?edcouplestresstheory,ParkandGao[25]analyzedthestaticmechanicalpropertiesofanEuler–Bernoullibeam.Recently,Kongetal.[20]derivedthegov-erningequation,initialandboundaryconditionsofanEuler–Ber-noullibeambasedonthemodi?edc