Peridynamic beams A non-ordinary, state-based model

InternationalJournalofSolidsandStructures51(2014)3177–3183ContentslistsavailableatScienceDirectInternationalJournalofSolidsandStructuresjournalhomepage:/locate/ijsolstrPeridynamicbeams:Anon-ordinary,state-basedmodelJamesO’Grady1,JohnFoster⇑TheUniversityofTexasatSanAntonio,OneUTSACircle,SanAntonio,TX78249,UnitedStatesarticleinfoabstractThispaperdevelopsanewperidynamicstatebasedmodeltorepresentthebendingofanEuler–delisnon-ultipleperidynamicmaterialmodelscapturethebehaviorofsolidmaterials,thisisthefificientlyhomogeneousanddifferentiabledisplacements,themodelisshowntobeequivalenttoEringen’eridynamichorizonapproaches0,itreducestotheclassicalEuler–testcasesdemonstratethemodel’sperformance.Óehistory:Received24February2014Receivedinrevisedform18April2014Availableonline23May2014Keywords:uctionAgoalofmanymechanicalengocessessuchasfracturearemodeled,thepartial-differentialequationsofclassicalmechanicsareill-defiynamicformulationofcontinuummechanicscastsmaterialbehaviorintermsofintegralfunctionsofdisplace-ment(asopposedtogradientsofdisplacement),sothasperidynamicmaterialmodelscapturethedeformationbehaviorof3-dimensionalsolidobjects(Sillingetal.,2007;SillingandAskari,2005;Gerstleetal.,2007),butwouldbeveryexpensivetoimple-mentforathinplateorbeam,asthethru-thicknessdiscretizationrequirementtoproperlycaptureresistancetobendingwouldbeprohibitivelyexpensiveinacomputationalsettingforalong,eridynamicmodelscapturetensionandcompressionin1Dbars(Sillingetal.,2003)and2Dmembranes(SillingandBobaru,2005),tpaperbyTaylorandSteigmann(2013)reducesabondbased3Dplatetotwodimensionswithanintegralthroughtheplate’eatesamodelthatcanrepresentthinstructuresandincludesabendingterm,elislimitedtothe3Dbond-basedPoissonratiom¼1,thoughthesametechperpresentsaperidynamicequivalenttoanEuler–Bernoullibeam,alongwithamethodologyforrepresentingnon-uniformcross-sections,plasticbehavior,manycontinuumbeamtheoriesthatderivenewequationsofmotion(suchasfourthorderPDE’s)fromthe3Delasticconstitutivemodel,thenewmodelisnotderivedfrompriorordinaryperidynamicmodelsbasedonbondextension,butisamaterialmodelthatdirectlyresistsbendingdeformationwhtiontodirectlymodelingabeaminbending,thesimplebeamcaselaysthetheoreticalframeworkformorecomplexperidynamicbeam,plate,emanyanalysesofinterestarepartlyorwhollycomprisedofthesetypesoffeatures,theirdevelopmainderofthisintroductionreviewsothernonlocalworkandprovidesabriefintroductiontoperidynamics,n2presentsthestatebasedbeammodelanddemonstratesequivalencetoclassicalEuler–nBdemonstratesthemodel’srelationshiptoEringen’albeammodelsNonlocalelasticitygenerallyallowsforforcesatapointthataredependentonthematerialconfigurationofanentirebody,ratherthantheconfigurationatthatpoint(EringenandEdelen,1972).Whilelong-rangeforcesareobviousatthemolecularmodel,mate-rialatlargerscalesisconventionallymodeledasthoughinternalforcesarelocalorcontactforces(Kröner,1967).Theresultofsuch⇑Correspondingauthor.E-mailaddresses:jogrady@().1Principalcorrespondingauthor.(J.O’Grady),@p:///10.1016/tr.2014.05.0140020-7683/Óhtsreserved.

3178J.O’Grady,/InternationalJournalofSolidsandStructures51(2014)3177–3183approximationisaccuratefordeformationsthatarehomogeneous,butintroducessomeinaccuracyforinhomogtodistinguishbetweenhomogeneousandinhomogeneoutressinclassicalelasticityisafunctionofthe(first)gradi-entofdeformation,Eringen’sformulationofanonlocalmodulusinEringen(1983)approximatesaweightedsumofthefitroducesalengthscaletothemodelandhastheeffectofsmearingoutlocaldeformationinhomogene-itiesoverthesurroundingmaterial,wusworkinthenonlocalmechanicsmelandWangdemonstrateinChallamelandWang(2008)thatEringennonlocalelasticitycannotreproducethescalestiffening,butthatstiffeningdoesresultfromotheeallofthesemodelsincorporatehigher-ordergradientsofdeformation,theyimposestrongercontinuityrequirementsthanclassicalelasticity,ethegradientsareevaluatedlocally,workbyPaolaetal.(2014)developsadisplacement-basedbeaminwhichrelativeaxialdisplacement,sheardisplacement,androtationofnon-adjacentbeamsegmentsareresistedbythreekindsofnonlocalspring,eappropriatenonlocalstiff-nesses,namicsThetermperidynamicalludestothefactthattheforceatapointisaffectedbynearbymaterialconfigurationandwascoinedbySillingtodescribethenewformulationofcontinuummechanicshedevelopedinSilling(2000).Incontrasttogradientmodels,theperidynamicmodelisstronglynonlocalandcastsmaterialbehavioratapointastheintegralequationqðxÞ€uðxÞ¼Zfðx;qÞdVqþbðxÞdofthedivergenceofstress,wehavetheintegralofa‘‘force’’frcefunctionalmaydependonx;q,theirdeformedpositions,theoriginalanddeformedpositionsofotherpointsinX,history,tutivemodelingofawidevarietyofmaterialhesimplestforcefunctionsrecreateaone-parameterlinearelasticsolidmaterial(Silling,2000),otherforcefunctionscanbeusedtomodelnonlinearelasticity,plasticity,damage,andotherbehaviors(SillingandBobaru,2005).Todescribeforcefunctionalsthatincorporatethebehaviorofatotalityofpointsinthenearbymaterial(notjustxandq),ucedbySillingetal.(2007),statesarefutcommonstatesarescalar-statesandvector-stateswhicharescalarandvectorvalued,asecondordertensor,whichcanonlymapvectorslinearlytoothervectors,antpropertiesofstatesaremagnitudeanddirection,whileimportantoperationsincludetheadditionanddecompositionofstates,innerandtensorproducts,andtheFréchetderivativeofafunctionwithrespecttoastate(Sillingetal.,2007).Conservationoflinearmomentuminthestate-basedperidy-namicformulationresultsintheequationofmotion,qðxÞ€uðxÞ¼ZðT½x