AN ENERGY-MOMENTUM INTEGRATION SCHEME FOR THE NONL

COMPUTATIONALMECHANICSNewTrendsandApplicationsS.Idelsohn,E.O~nateandE.Dvorkin(Eds.)cCIMNE,Barcelona,Spain1998ANENERGY-MOMENTUMINTEGRATIONSCHEMEFORTHENONLINEARDYNAMICSOFSHELLS.APPLICATIONSTOCHAOTICANDFREELARGEOVERALLMOTIONC.Sansour,P.Wriggers,andJ.SansourDarmstadtUniversityofTechnology,FachgebietMaschinenelementeundAkustikMagdalenenstr.4,64289Darmstadt,Germany,e-mail:[e-mail]DarmstadtUniversityofTechnology,InstitutfuerMechanikHochschulstr.1,64289Darmstadt,Germany,e-mail:[e-mail],[e-mail]:shelltheory,nonlineardynamics,integrationschemes,niteelementsAbstract.Thepaperisconcernedwithadynamicalformulationofshellsandthedevelop-mentofacorrespondingrobustenergy-momentumintegrationscheme.Energy-momentumschemespreserve,bydesign,specicfeaturesofthecontinuoussystemsuchasconserva-tionofmomentum,angularmomentum,andenergywhenthesystemandtheappliedforcesallowto.Afundamentalaspectoftheschemeproposedinthispaperisitsapplica-bilitytoanyshelltheorywhateverthenonlinearitiesinthestrain-dispalcementrelationsmaybe.ThisstandsincontrasttocorrespondingformulationsduetoSimo&Tarnow.Accordingly,theschemeappliestoshellformulationswhichincludearotationtensoraswellastothosewithoutsuchaninclusion.Inthispaperashellformulationispresentedwhichiscapabletocatchnitedeformationsandfallswithintheclassofgeometricallyexacttheories.Itischaracterizedbysevendegreesoffreedomandthenonlinearitiesinthestrain-displacementrelationsareofcubicnature.Astress-hybridniteelementfor-mulationisusedforthecomputationofvariousexamplesofnonlinearshelldynamicsincludinglargeoverallandchaoticmotion.1C.Sansour,P.Wriggers,andJ.Sansour1INTRODUCTIONNonlineardynamicsofstructures,especiallychaoticandfreelargeoverallmotion,hasre-centlybeengivenconsiderableattentionintheliterature.Asfarastheanalysisofchaoticmotionisconcerned,theshellandrodmodelsusedarebased,ingeneral,onsimpliedtheorieswithsimpliedassumptionswhere,nearlyalways,onlysimplenonlinearitiesaretakenintoaccount.Moreover,forthepostcriticalbehaviouroftheconsideredstruc-turesitisoftenassumedthatonlytheso-calledactivemodesarerelevantfortheanalysis(Galerkinansatzfortheequationofmotion).Thenonlineardynamicsofgeometricallyexactshellsandrodswherethewholenonlinearityistakenintoaccountwastreatedonlybyfewauthors,anddissipativeandchaoticmotioninsuchmodelsisnotconsideredatall.Shellandrodtheoriesareessentiallydimensionallyreducedtheoriesofthree-dimensionalcontinua.Theycanbeclassiedaccordingtotheconceptsofdimensionreductionun-derlined.Ingeneral,twodierentapproachesexistforthederivationofdimensionallyreducedtheories:1)thethree-dimensionalapproachand2)thedirectapproach.Withintherstmethod,thethree-dimensionalequationsoftheclassicalcontinuumcanbere-ducedtotwo-orone-dimensionalshellorrodtheories,respectively,bymakinguseofspecicassumptionsregardingthedisplacementeldorotherphysicalquantitiesrelatedtotheproblemunderconsideration.Manygeometricallyexactshellformulationsinvolveexplicitlyatwo-parametricrota-tiontensor(seee.g.Basaretal.,1Gruttmannetal.,5Ramm,9Simo&Fox,16Wriggers&Gruttmann20)oratree-parametricrotationtensor(seee.g.Criseld,4Chroscielewskietal.,3Ibrahimbegovic&Frey,7Sansour&Bednarczyk,11,12Sansour&Bufler13).Alter-natively,someformulationscircumventtheuseoftherotationtensorbytakingthicknesschangeintoaccount.ThemodelsduetoBuechteretal.2andSansour10arebestexam-ples.WhenusingtheGreenstraintensorasstrainmeasureandwhentheuseofarotationtensoriscircumvented,thenonlinearitiesinthestrain-displacementrelationsaremuchsimplerthanthosewhenarotationtensorisincorporated.OfspecialinterestisthemodelofSansour10duetothefactthatthenonlinearitiesinthestrain-displacementrelationsarenomorequadraticbutofcubicnature;afactwhichplaysafundamentalroleinthedesignofintegrationschemes.Inthispaperwefocusonsocalledenergy-momentummethodsfortheintegrationofthedynamicalsystem.Thebasicideaofthesemethodsisthedesignofmechanicalintegratorswhichconservemomentum,angularmomentum,andenergywhenevertheexternalloadingandtheconstitutivebehaviourallowsfor.ArstformulationinthisdirectionwasreportedbySimoandTarnow17andappliedbythemtoshelldynamicsinReference.18InKuhl&Ramm,8theseconservationpropertiesweretakenassideconditionsandenforcedinanoverallmannerbytheinclusionintheniteelementformulationwiththehelpofLagrangeparameters;ideasgoingbacktoHughesetal.,6amongothers.Anyhow,adrawbackofsuchanapproachisthefactthatthesesideconditionscannotbeformulated2C.Sansour,P.Wriggers,andJ.Sansourwhendissipationisapartofthephysicalsystemunderconsideration.Theconceptofenergy-momentummethodsisquiteinterestingsinceitenhancesthestabilitypropertiesofthemechanicalintegrator.Ontheothersidethestabilityinlongtermdynamicsisaveryimportantfeatureinordertofollowthedynamicsofthesystemforlongtimesespeciallywhenspecicphenomenalikechaoticbehaviourisofinterest.ThemethodofSimo&Tarnow17appliesforelasticsystemswhenthenonlinearityinthestress-displacementrelationsisaquadraticone.Accordingly,themethoddoesnotapplyforsystemswh