A model for stress and plastic strain induced nonl

INTERNATIONALJOURNALFORNUMERICALANDANALYTICALMETHODSINGEOMECHANICSInt.J.Numer.Anal.Meth.Geomech.2008;32:833–861Publishedonline16July2007inWileyInterScience().DOI:10.1002/nag.648Amodelforstressandplasticstraininducednonlinear,hyperelasticanisotropyinsoilsA.GajoandD.Bigoni∗,†DipartimentodiIngegneriaMeccanicaeStrutturale,Universit`adiTrento,ViaMesiano77,I-38050Povo,Trento,ItalySUMMARYTheexperimentalevidencethatcohesiveandgranularsoilspossessanelasticrangeinwhichtheelasticityisbothnonlinearandanisotropic—withstiffnessanddirectionalcharacteristicsstronglydependentonstressandplasticstrain(theso-called‘stresshistory’)—isgivenaformulationbasedonhyperelasticity.Thisisaccomplishedwithintheframeworkofelastoplasticcoupling,throughanewproposalofelasticpotentialsandacombineduseofaplastic-strain-dependentfabrictensorandnonlinearelasticity.Whenusedwithinasimpleelastoplasticframework,theproposedmodelisshowntoyieldveryaccuratesimulationsoftheevolutionofelasticpropertiesfrominitialdirectionalstiffeningtofinalisotropicdegradation.Withintheproposedconstitutiveframework,itisshownthatpredictionsofshearbandformationandevolutionbecomeclosertotheexistingexperimentalresults,whencomparedtomodellinginwhichelasticitydoesnotevolve.Copyrightq2007JohnWiley&Sons,Ltd.Received2November2006;Revised21May2007;Accepted21May2007KEYWORDS:elasticbehaviour;anisotropy;sand;clay;shearbands;elastoplasticityDedicatedtoProf.TomaszHueckelontheoccasionofhis60thbirthday1.INTRODUCTIONExperimentalevidenceshowsthatdeformationofcohesiveandgranularsoilsatverysmallstrains(lessthanabout0.001%forsandsand0.01%forclays)isessentiallyrecoverableandrateinsensitive[1–7].Whileinthisrangeofstrainstheelasticresponseisverystiff,atlargerstrain,whenirreversibledeformationsbecomeimportant,twoconcurrentphenomenaoccur,namely,adecreaseoftangentialstiffnessandadegradationofelasticmoduli(anexperimentalevidencerecently∗Correspondenceto:D.Bigoni,DipartimentodiIngegneriaMeccanicaeStrutturale,Universit`adiTrento,ViaMesiano77,I-38050Trento,Italy.†E-mail:[e-mail],URL:∼bigoniContract/grantsponsor:MURST-Cofin2005;contract/grantnumber:2005085973002Copyrightq2007JohnWiley&Sons,Ltd.834A.GAJOANDD.BIGONIprovidedbyKuwanoandJardine[8],andHoqueandTatsuoka[9]).Inotherwords,theelasticbehaviourishighlynonlinearandplasticstraindependent.Inaddition,theelasticityexhibitsimportantdirectionalpropertiesevolvingwithboththestressandtheplasticstrainstates.Insummary,elasticityofsoilsdependson(AppendixA):•thevoidratio,•thestressstate,•theso-called‘stresshistory’,whichisheremodelledintermsofplasticstrain.Alltheabovedependenciesregardboththeelasticstiffnessandtheelasticanisotropy.Inparticular,theanisotropy,whichisusuallyaclassoforthotropy,‡maybeinherent,thusresultingfromsoildeposition,andinducedbystressand/orplasticstrain.Theunderstandingoftheabove-describedelasticbehaviourofsoil—havingboththeoreticalimportance(sinceitprovesthatsoilsexhibitanelasticrange,thefundamentalstartingpointforplasticitytheory)andpracticalrelevancefordesignofgeotechnicalstructures—isdeemedbyAtkinson[15]as‘oneofthemajorachievementsofgeotechnicalengineeringresearchoverthepast30years’.Accordingly,astrongresearchefforthasbeendevotedtoconstitutivedescriptionoftheelasticpropertiesofsoils.Inparticular,§thestressdependenceofelasticmoduliandthestress-inducedelasticcrossanisotropyhavebeenmodelledbyVermeer[17],Boyce[18],MrozandNorris[19],Loret[20],Houlsby[21],Jardineetal.[22],LadeandNelson[23],Molekamp[24],PuzrinandBurland[25]andHoulsbyetal.[26]forclayandsand,whiletheonlyattempttodescribeplasticallyinducedanisotropyisduetoHueckeletal.[27],foroverconsolidatedclays.¶Modellingofbothstressandplasticdependencyofelasticityhasneverbeenattempted.Acomprehensivemodelcapableofdescribingtheevolutionofstressandplasticallyinduced(hyper-)elasticanisotropy,includingthedependenceofelasticstiffnessonthestress,voidratioandplasticstatesandthestiffnessdegradationoccurringatlargeappliedshearstressisstilllacking.Suchaconstitutivemodelisformulatedinthisarticleanditisshowntoagreewithavailableexperimentalresults.Inparticular,anewnonlinearhyperelasticmodelisintroduced(inSections2and3),whichcandescribeageneralevolutionofelasticanisotropyandthestressdependencyofelasticproperties.Thismodelisdevelopedwithintheframeworkofelastoplasticcoupling[30–33],inwhichtheelasticanisotropyisinducedpartlybythestressstateandpartlybytheplasticstrainstate,throughthedefinitionofafabrictensor[34,35].Elasticorthotropywithrotatingaxescanbedescribed,togetherwiththestressandthedensitydependencyofelasticstiffnessanditsdegradationatlargestressratio,observedinrecentexperimentsbyHoqueandTatsuoka[9].Duetothedoublesource‡Fororthotropy,aconvenientmeasureoftheanisotropydegreeistheratiobetweenelasticYoung’smoduliinthehorizontalandverticaldirectionsn=Eh/Ev.Innormallyconsolidatedclays,thisrationmayrangebetween0.9and1.35[10,11].Inheavilyoverconsolidatedclays,amuchlargeranisotropycanbeobservedwithtypicalvaluesofnrangingbetween1.35and2.4[12,13].Finallyinsands,nca