Anisotropic plasticity and viscoplasticity [Elektronische Ressource] / von David Schick
107 pages
Deutsch
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Anisotropic plasticity and viscoplasticity [Elektronische Ressource] / von David Schick

Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus
107 pages
Deutsch

Description

Anisotropic Plasticity andViscoplasticityVom Fachbereich Mechanikder Technischen Universit at Darmstadtzur Erlangung des Grades einesDoktor Ingenieurs(Dr.-Ing.)genehmigteDissertationvonDipl.-Ing. David Schickaus IchenhausenHauptreferent: Prof. Dr.-Ing. Ch. TsakmakisKorreferent: Prof. F. GruttmannTag der Einreichung: 31.10.2003Tag der mundlic hen Prufung: 07.01.2004Darmstadt 2004D 17Die vorliegende Arbeit entstand w ahrend meiner T atigkeit als wissenschaftlicher Mitarbeiteram Institut fur Mechanik der Technischen Universit at Darmstadt.Herrn Prof. Dr.-Ing. Ch. Tsakmakis m ochte ich herzlich fur die hervorragende wissenschaftlicheund au erorden tlich freundschaftliche Betreuung, sowie fur die Ubernahme des Hauptreferatesdanken. Herrn Prof. Dr.-Ing. F. Gruttmann danke ich fur das Interesse an dieser Arbeit undfur die freundliche Ubernahme des Korreferates.Fur die fachlichen Diskussionen und Anregungen m ochte ich mich bei meinen Kollegen, ins-besondere bei Herrn Dr. rer. nat. P. Grammenoudis, bedanken.Darmstadt, im Januar 2004 David SchickZusammenfassungAnisotropie, gekoppelt mit inelastischem Flie en spielt in vielen Bereichen der Materialtheorieeine wichtige Rolle. Beispiele dafur sind Sto gesetze zur Kristallplastizit at, zur Beschreibungvon Texturen in Blechen usw. Im ersten Teil der vorliegenden Arbeit werden die konstitu-tiven Materialgleichungen fur die Materialantwort bei Orthotropie und kubischer Anisotropieentwickelt.

Sujets

Informations

Publié par
Publié le 01 janvier 2004
Nombre de lectures 98
Langue Deutsch
Poids de l'ouvrage 2 Mo

Exrait

AnisotropicViscoplasticitPlasticityyand

VomFachbereichMechanik
derTechnischenUniversit¨atDarmstadt
einesGradesdesErlangungzur

IngenieursDoktor(Dr.-Ing.)

genehmigte

Dissertation

onv

Dipl.-Ing.DavidSchick

t:Hauptreferent:KorreferenTTagagderderm¨Einreicundlichhenung:Pr¨ufung:

henhausenIcaus

2004Darmstadt

17D

Prof.Prof.Dr.-Ing.Dr.-Ing.Ch.F.GruttmannTsakmakis
31.10.200307.01.2004

DieamvInstitutorliegendef¨urMecArbeithanikendertstandTecw¨hniscahrendhenUnivmeinerersit¨T¨atatigkeitDarmstadt.alswissenschaftlicherMitarbeiter

HerrnProf.Dr.-Ing.Ch.Tsakmakism¨ochteichherzlichf¨urdiehervorragendewissenschaftliche
¨unddanken.außerordenHerrntlicProf.hfreundscDr.-Ing.F.haftlicheGruttmannBetreuung,dankesoicwiehff¨¨ururdiedasInUbteresseernahmeandesdieserArbHauptreferateseitund
f¨urdiefreundliche¨UbernahmedesKorreferates.

bF¨uresonderediefacbeihlichenHerrnDr.Diskussionenrer.nat.undP.AnregungenGrammenoudis,m¨ocbhteedankichen.mich

imDarmstadt,2004uarJan

Kollegen,meineneib

khicScvidDa

ins-

Zusammenfassung

Anisotropie,gekoppeltmitinelastischemFließenspieltinvielenBereichenderMaterialtheorie
einewichtigeRolle.Beispieledaf¨ursindStoffgesetzezurKristallplastizit¨at,zurBeschreibung
vonTextureninBlechenusw.ImerstenTeildervorliegendenArbeitwerdendiekonstitu-
entivtenwickelt.MaterialgleicZuhdiesemungenZwf¨ecurkdiewirddasMaterialanintwortTsakmakisbei[106Orthotropie]vundorgestelltekubiscthermoherdynamiscAnisotropieh
konsistentekonstitutiveMaterialmodellf¨urPlastizit¨atundViskoplastizit¨atbeigroßenDefor-
Wichmationentigef¨urdieseBestandteilebeidenderF¨alleTheoriedersindAnisotropiediemwultiplikeiterativausgefe¨uhrt.ZerlegungdesDeformationsgradi-
engenannteninteneinenPostulatselastiscvonhenIlund’iushininelastiscf¨urhenPlastizit¨Anteilat.soEswiewirddiesowAnnahmeohleinederG¨anisotropultigkeeitdeskinema-so-
tischeVerfestigungalsaucheineallgemeineGestalt¨anderungderFließfl¨acheber¨ucksichtigt.
DieStarrk¨Theorieorpisterrotationenph¨inanomenologiscderhplastiscformhenuliertZwiscundhenkinvarianonfigurationtgegen¨unduberderbeliebigenMomen¨tankuberlagertenonfigura-
tion.DieAnisotropiewirdmitHilfesogenannterStrukturtensoreninderfreienEnergiefunktion
undderFließfunktionformuliert.F¨urdenFallderkubischenAnisotropiewurdeeinBrinell
NickKugeleindrucelbasislegierungkversuchsim(CMSX4)uliertvunderglichen.qualitativmitdemExperimentaneinereinkristallinen

Beieinemanf¨anglichisotropenMaterialkanndurchdieplastischeDeformationeineAnisotropie
induziertwerden,wassichinsbesonderebeiMetallendurcheineVerschiebung,Rotationund
Verzerrung(formativeVerfestigung)derFließfl¨acheausdr¨uckt.Dieswurdeauchdurchver-
schiedeneexperimentelleUntersuchungenunabh¨angigvonderDefinitiondesFließbeginnsbe-
st¨atigt.ImzweitenTeilderArbeitwirdeineinfaches,thermodynamischkonsistentesMa-
terialmodellf¨urkleineDeformationenentwickelt,dasdieEvolutionderAnisotropieinder
Fließfl¨achebeschreibt.DasModellerf¨ullthinreichendeBedingungenf¨urdiesogennanteDissi-
pationsungleichung.AbschließendwirddieEvolutionderFließfl¨achef¨urverschiedeneVorbelas-
tungensimuliertundmitdenExperimentenvonIshikawaanSUS304EdelstahlRohrproben
hen.erglicvqualitativ

i

ii

Abstract

Plasticanisotropyeffectsmaybedescribedinaphenomenologicalmodelbyemployinginthe
constitutivetheoryasetofinternalvariables,whicharedefinedsuitably.Thesevariableshave
tomodelthehardeningresponseofthematerialunderconsiderationtodescribee.g.thero-
tationofsomesymmetryaxes.Suchaxesareimaginedtoberelatedwiththedevelopmentof
thematerialsubstructureassumed,or,correspondingly,withthestatevariablescharacterizing
thisdevelopment.Theobjectiveofthefirstpartofthisworkistodeveloptheconstitutive
equationsgoverningthematerialresponseforthecaseoforthotropicandcubicanisotropy.
Thereforethethermodynamicallyconsistenttheoryforplasticity(andviscoplasticity),recently
publishedbyTsakmakis[106],whichaccountsforanisotropyeffectsispresentedandextended
fortheaforementionedcasesofanisotropy.
Importantfeaturesofthetheoryaretheuseofthemultiplicativedecompositionofthedefor-
mationgradienttensoraswellastheassumptionofthevalidityofIl’iushin’spostulateinthe
caseofplasticity.Forsimplicity,apartfromkinematichardeningeffects,onlyorientational
evolutionoftheunderlyingsubstructureisregarded.Careistakenthatthetheoryisinvariant
withrespecttorigidbodyrotationssuperposedtoboth,thecurrentandtheso-calledplastic
configuration.termediateinAnisotropyeffectsareelaboratedinthefreeenergyandtheyieldfunctionbymeansofstruc-
turaltensors.ForthecaseofcubicmaterialsymmetryaBrinellhardnessindentationtesthas
beensimulatedandiscomparedqualitativelywiththeexperimentforacommerciallyavailable
single-crystalnickel-basedsuperalloy(CMSX4).

Inelasticdeformationsinduceanisotropyinthematerialresponse,evenifthisisinitially
isotropic.Formetallicmaterials,deformationinducedanisotropyisreflected,aboveall,by
translation,rotationanddistortionoftheyieldsurface.Thishasbeenconfirmedbyseveral
exppartoferimenthistalwinorkvaestigationssimple,indepthermoendentdynamicallyofthewayconsistenthetyieldmopdeloinistispropdefined.osed,Indescribingthesecondthe
evolvinganisotropyoftheyieldsurface.Themodelisfirsttheoreticallyestablished,based
onasufficientconditionforthedissipationinequalitytobesatisfied.Then,itisappliedto
exppredicterimenthetallybsubsequenyIshikatyieldwaforsurfaces,SUS304aftervstainlessarioussteel.prestressings,whichhavebeenobserved

iii

iv

tenConts

1ductiontroIn11.1Objectiveofthework.................................1
1.2Outlineofthethesis.................................4
1.3Notation........................................5
1.4GlossaryofSymbols.................................7

relationskinematicalBasic2

11

3Modellingofanisotropic(Visco-)Plasticity14
3.1Secondlawofthermodynamics...........................15
3.1.1LocalformoftheClausius-Duheminequality..............16
3.2Elasticitylawanddissipationinequality......................17
3.3FlowruleforplasticityandthepostulateofIl’iushin...............19
3.4Flowruleforviscoplasticity.............................25
3.5Kinematichardeningandyieldfunction.......................25
3.6Constitutivemodelfororthotropicanisotropy...................30
3.6.1PlasticSpins.................................30
3.6.2Elasticitylaw.................................32
3.6.3Kinematichardeningrule..........................34
3.6.4Yieldfunction–flowrule..........................37
3.7Constitutivemodelforcubicanisotropy.......................42
3.7.1Elasticitylawforcubicanisotropy......................42
3.7.2Kinematichardeningruleforcubicanisotropy...............43
3.7.3Yieldfunctionandflowruleforcubicanisotropy..............43

4FinitecrystalelemenNi-basetsimsuperalloulationyofa(CMSX4),Brinellorienhardnesstedinindentation[001]-directiontestofasingle-45
4.24.1ExpComparisonerimentalofnproumericalcedure-withMaterialexperimenparameterstalresults...................................4845

5Phenomenologicalmodeltodescribeyieldsurfaceevolutionduringplastic
flowforsmalldeformations54
5.1SubsequentYieldSurfacesofStainlessSteel....................54
5.2ProposedConstitutiveModel............................55
5.2.1BasicRelations................................55
5.2.2YieldFunction-FlowRule.........................56
5.2.3HardeningRules...............................57
5.3ComparisonwithExperiments-ConcludingRemarks...............60

Summary6

v

81

vi

A

B

Transformationsunderrigidbodyrotationssuperposed

theandplastic

formsReduced

yBibliograph

configurationtermediatein

for

the

specific

free

energy

function

ψ

e

on

oth,b

CONTENTS

the

actual

83

85

86

1Chapter

ductiontroIn

1.1Objectiveofthework

Ametalscloser–aviewlotonofunclearanisotropicissuesplasticandandunsolvedviscoplasticproblems.materialbRealisticehaviormaterialreveals–propespertieseciallyinputfor
spiterepresenoftssomeoneveryofthesubtlemajortheoreticallimitationstreatmenintscomputerofplasticstressdeformationanalysisinsucthehasforplasticexamplerange.theIn
approachtodislocationdynamicsbasedonanatomisticunderstandingofcrystaldefectsand
theirgatestomovtheemensliptorbehatheviorcrystalinsingleplasticitycrystals,approacthehso-calledrelatingthebehaphenomenologicalviorofptheoryolycrystallineofplasticitaggre-y
remainsthetheoryusedextensivelyinstressanalysisproblems.

Inthefirstpartofthepresentworknewaspectsofathermodynamicconsistentconstitutive
modelforsinglecrystalsandlargedeformation,basedonrecentworksofTsakmakis[106]
andH¨ausleretal.[43]willbepresented.Herematerialsareconsideredthathaveasub-
structurewhichmaymacroscopicallybeaccountedforbyemployingasetofinternalstate
variables.Theconstitutivemodelsdealtwitharerate-dependentandrate-independentplas-
ticitylawsexhibitinganisotropyeffectsrelatedtokinematicandorientationalhardening.(For
simplicityisotropichardeninganddistortionalhardeningisnotregarded).Suchplasticitylaws
haveextensivelybeendiscussedbyDafalias(seethecomprehensivestudyinDafalias[32]and
thereferencescitedherein)intheframeworkofconstitutiveandrelatedplasticspinconcepts.
Physically,themechanicalresponsedescribedmaybeassignedtoinitiallyanisotropicmaterials
ase.g.rolledplates,singlecrystalsormaterialsinstructuralgeology.Also,suchconstitutive
lawsmaybeviewedasthefirststeptowardsdescribingthematerialbehaviourofpolycrys-
tallinematerialsindicatinganisotropyeffectsofbothorientationalanddistortionaltype.

Generally,inallplasticanisotropymodelssomecharacteristicdirectionsareattachedtothe
materialwhichmayrotateduetothedeformationprocess.Thespinofthisrotationisrelated
tosomeoneoftheso-calledplasticspinconcepts.Thelatterareoftendefinede.g.byexam-
iningbasickinematicalaspectsofthedeformationorbyconsideringthephysicalmechanisms
ofinelasticflowatthecrystallevel.Publicationsconcerningthissubjectare,amongothers
theworksofAsaroandRice[8],Asaro[7],Loret[71],Dafalias[26],[29],[31],Dafaliasand
Rashid[28],DafaliasandAifantis[30],LoretandDafalias[72],ChoandDafalias[20],Aravas
andAifantis[3],Aravas[4],[5],NingandAifantis[84],vanderGiessen[37],[38],Tu˘gcuand
Neale[107]aswellasTu˘gcuetal.[108].Themaindifferencesbetweentheseworksandthe
presentoneisintheconstitutiveequationsandtherelatedplasticspinissuesgoverningthe
modelresponseandinparticularthekinematichardeningrule.

1

2

CHAPTERODUCTIONINTR1.

Teryobemoreadmissiblespproecific,cessaisplasticitpresenyted.theoryFollowhicwinghasatisfiesproposalthebyDsecondaflaaliaswof[31]thermo(cf.alsodynamicsDafinaliasev-
[32]),variousconstitutivespinsareintroduced,responsibleforrotationsofaxesofsymmetry
e.g.respectivassoely.ciatedInthewithcasetheofelasticitcrystalylaw,plasticittheytheyieldfunctionassumptionandofthetheexistencekinematicofhardeningdifferentaxesrule,
offactthatsymmetrythee.g.latticeforisthedisturbelasticitedloycallyandbtheydislokinematiccationsorhardeningsomelaotherwmakindsybeofjustifieddefects.byThtheus
differentaxesofsymmetrycanbeattributedtodifferentkindsofphysicalmechanisms.Several
depimpendenortanttflofeaturesw),theoftheexistencetheoryofaareyieldthesurfaceconstancywhicofhvolumedesignatesduringtheplasticstressflostatewat(pressuretheonsetin-
ofandplastictheassoflow,ciatedtheflowhardeningrulerulerelatingthedescribingplasticthecstrainhangerateinthewithyieldtheyieldsurfacefunction.withplasticThefloneww
aspectsherebyaretheusedtransformationbehaviorofso-calledstructuraltensors,describing
wtheellevastheolutionofconditionsanisotropforytheinthematerialelasticityparameterslaw,thefortheyieldcaseoffunctionorthotropicandtheandhardeningcubicrule,materialas
symmetry,thatareworkedoutexplicitly.

Inthesecondpart,theaspectofdeformationinducedanisotropyoftheyieldsurfaceafter
variouspreloadingsiselaboratedforsmalldeformations.Theapproachhereisidenticalto
thatonepresentedrecentlyinDafaliasetal.[34].Animportantfeatureintheconstitutive
theoryofrate-independentplasticityandrate-dependent(visco-)plasticityistheassumptionof
theexistenceofayieldsurfaceinthestressorstrainspace,whichseparatespurelyelasticstates
fromelastic-plasticstates(seee.g.KhanandHuang[64],Naghdi[82]).Closelyrelatedto
theyieldsurfacearealsotheso-calledloadingconditions,whichdecidewhetherornotinelastic
flowhastobeinvolved.Theseconditionsaresatisfiedforthecaseofworkhardeningplasticity
iftheactualstrainorstressstateisontheyieldsurfaceandtheimposedstrainorstress
incrementpointsoutwardfromtheyieldsurface(seee.g.CaseyandNaghdi[15],Dafalias
andPopov[23]).Ontheotherhand,whenviscoplasticityisconcerned,loadingconditions
aredefinedcommonlytobefulfilledifanon-vanishing,so-calledoverstressapplies.Thenotion
overstresshasbeenintroducedbyKrempl[67]andPerzyna[87]andisdefinedasascalar
valuedfunctionofastressstatewhichisoutsideoftheareaenclosedbytheyieldsurfacein
stressspace(formoredetailsseeTsakmakis[102]).
Also,theconceptofyieldsurfaceplaysacrucialroleiftheplasticstrainissupposedtoobey
anassociatednormalityrule,i.e.iftheplasticstrainrateispositiveproportionaltotheouter
normalattheyieldsurface.Suchevolutionequations,termed”flowrules”,mayoftenbeob-
tained,atleastforisotropicmaterialresponse,fromsomeoverallworkpostulates(alonglist
ofpapersdealingwithworkpostulatesinplasticityisgiveninTsakmakis[103]).

Conformityoftheyieldsurfaceconceptwithexperimentalresultshasbeenexaminedinseveral
works.Agoodoverviewofthisisgiven,amongothersintheworksofHecker[44],[45],
Hellingetal.[46],Henshalletal.[47],Ikegami[58],Ishikawa[59],Ishikawa
andSasaki[60],[61],KhanandWang[63],Kowalewskiand´Sliwowski[65],Mi-
astkowski[79],MiastkowskiandSzczepi´nski[80],Phillips[88],PhillipsandDas
[89],PhillipsandMoon[90],PhillipsandTang[91],Stoutetal.[97],Trampczyn-
ski[98],WilliamsandSvensson[110],[111].Generallytherearesomedifferencesinthe
approachesemployedtomeasureyieldsurfaces.Forexample,thedefinitionofplasticyielding
isnotunique.Customary,themethodofdeparturefromthelinearity(proportionallimit),the
methodofbackwardextrapolationandthestressatastrainoffsetbyagivensmallamountare

1.1.OBJECTIVEOFTHEWORK

3

utilizedtodeterminetheinitialyieldsurfaceaswellassubsequentyieldsurfacesafterpreload-
ing.Thefirstmethodisusede.g.inMiastkowski[79],MiastkowskiandSzczepi´nski
[80],PhillipsandDas[89],PhillipsandMoon[90],PhillipsandTang[91],thesecond
onee.g.inKhanandWang[63],Stoutetal.[97],whiletheoffsetcriterionhasbeenem-
ployede.g.inHellingetal.[46],Ishikawa[59],IshikawaandSasaki[60],[61],Khan
andWang[63],Kowalewskiand´Sliwowski[65],Miastkowski[79],Miastkowski
andSzczepi´nski[80],Trampczynski[98],WilliamsandSvensson[110],[111].Fur-
therreferencesonexperimentaldeterminationofyieldsurfacescanbefoundinthereview
papersHenshalletal.[47],Ikegami[58],Phillips[88].Asitcanbeseenfromthese
works,theassumeddefinitionofyieldingaffectstheidentifiedyieldsurfacecrucially.Similarly,
theformofthemeasuredyieldsurfacesdependsheavilyontheloading-unloading-reloading
pathschosen.Essentially,afterpreloadingsthesubsequentyieldsurfacesmaytranslate,rotate
anddistort,evenifaninitiallyisotropicyieldsurfacehasbeenrecorded.Insomecases,when
theoffsetstrainsareverysmall,thesubsequentyieldsurfaceshavebeenobservedtoexhibit
asharpeninginthedirectionofpreloadingandaflatteningontheoppositeside.However,
whentheyieldsurfacesaremeasuredbypartialunloadingfromtheactualstressstatetothe
assumedcenteroftheyieldsurface,theyieldlocireferredtoplanestressloadingshaveturned
outtoformratherellipses(seeIshikawa[59],IshikawaandSasaki[60],Trampczynski
]).[98

floSomew(seeefforte.g.hasthebeenliteraturemadetogivendescribineWegenertheoreticallyandtheevSchlegelolutionof[109yield]).surfacesBecauseofduringtheirplasticsim-
theplicity,stressyieldtensorfunctionsareverywhichattractivcontaine.aThisisfourth-orderthecasestatee.g.fortensortheandareconstitutivequadraticmodelsfunctionsproposedof
byBackhaus[9],BaltovandSawczuk[10],Ishikawa[59],Rees[93],Williamsand
Svensson[110],[111],Wuetal.[112]andYoshimura[113].Itisworthnoting,thatall
theseworksareformulatedinapurelymechanicalcontext.

Hereathermodynamicconsistenttheoryisformulated,whichisachievedbyestablishingsuf-
ficientconditionsforthesatisfactionoftheso-calleddissipationinequality.Forthesakeof
simplicity,theproposedmodelisoutlinedforyieldsurfaceswhichareinitiallyisotropicand
theinitialyieldsurfacemaybeapproximatedwithsufficientaccuracybyavonMisesyield
function.Thisreferstoe.g.experimentsbyIshikawa[59],whichwillbeusedinorderto
discussthecapabilitiesofthemodel.

4

thesistheofOutline1.2

ODUCTIONINTR1.CHAPTER

Afterintroducingsomedefinitionsandthenotation,thebasickinematicrelations,usedinthis
work,willbepresentedinChapter2.Thestartingpointofthetheoryisthemultiplicative
essarydecompstrainositionandofthestressdeformationmeasuresaregradienalsotdefined.tensorinThetoanevelasticolutionandanequationsinelasticdeveloppart.edmAllustnec-be
invariantunderarbitraryrigidbodyrotations,superposedonboththeactualandtheso-called
plasticintermediateconfiguration.Theformulationoftheconstitutivetheoryiscompletely
configuration.thistoerelativ

InChapter3,athermodynamicallyconsistentconstitutivemodelforanisotropic,largede-
formationplasticityandviscoplasticityisoutlinedasproposedinTsakmakis[106].The
thermodynamicconsistencyisrequiredwithrespecttotheClausius-Duheminequality.Asa
result,ananisotropicelasticitylawaswellasadissipationinequalityarederived.Makinguse
oftheso-calledpostulateofIl’iushin,ayieldconditionandanormalityruleareobtained.Vis-
coplasticityofoverstresstypeisassumedtoapply.Forthesakeofsimplicity,isotropichardening
willbedroppedandonlykinematichardeningisconsidered.Theyieldfunctionissupposedto
exhibit,besidesofkinematichardening,orientationalanisotropicbehaviour.Followingtheout-
linedtheory,twospecialcasesofanisotropyarediscussed.Thefirstonedescribesorthotropic
anisotropy,applicableforanorthorhombiccrystalstructure.So-calledstructuraltensorsof
second-orderareintroducedthatrepresentlocalaxesofsymmetryintheelasticitylaw,the
kinematichardeningandtheyieldfunction(cf.Boehler[12],Liu[70]).These,togetherwith
therepresentationtheoremsforisotropictensorfunctions(cf.Spencer[95],Zheng[114])are
usedinformulatinge.g.theconstitutiveequationsforthefreeenergyandtheyieldfunction.
Thesecondcaseaddressescubicmaterialsymmetry,whichcanbetreatedasaspecialcaseof
orthotropicsymmetry(cf.BillingtonandTate[11]).InChapter4thecapabilitiesofthe
presentedconstitutivemodelforcubicanisotropywillbedemonstrated.Experimentalfindings
ofaBrinell-hardnessindentationtestforanickel-basedsingle-crystalsuperalloy(CMSX4)
arecomparedwithafiniteelementsimulationoftheindentationtest,usingthefiniteelement
].[1QUSABAprogram

Chapter5focusesattentiononthedescriptionofsmallelastic-viscoplastic(rate-dependent)de-
formationsofpolycrystallinematerials.Hereaphenomenologicalmodel,previouslypresented
inDafaliasetal.[34],isdiscussed,whichshowshowdeformationinducedanisotropyofthe
yieldsurfacemaybeformulatedinathermodynamicallyconsistentmanner.Then,itisapplied
topredictthesubsequentyieldsurfaces,aftervariousprestressings,ofcommerciallyavailable
SUS304stainlesssteel,whichhavebeenmeasuredexperimentallybyIshikawa[59].

5TIONANOT1.3.Notation1.3Onlyisothermaldeformationswithauniformtemperaturedistributionwillbeconsidered.We
writeϕ˙(t)forthematerialtimederivativeofafunctionϕ(t),wheretisthetime.Anexplicit
referencetospacewillbedroppedthroughoutthework,sincedeformationsarenotaffectedby
abyspacethedepsameendencysymb.ol.AsIfusual,differenatfunctionrepresenandtationsthevofaluetheofsamethatfunctionfunctionsatareapoinused,tarethesymdescribbolsed
forthatfunctionwillalsovary.Forrealx,xdenotesthefunction
0xifx≥x:=0ifx<0.(1.1)
Vectorsandsecond-ordertensorsaredenotedbybold-faceletters,whereasfourth-ordertensors
areproductdenotedandbythebtensorold-faceproductcalligraphicofthevletters.ectorsInaandbparticular,,respaectiv∙bely.anda⊗bdenotetheinner
Forsecond-ordertensorsAandB,trA,detAandATiswrittenforthetrace,thedeterminant
andthetransposeof√A,respectively,whileA∙B=trABTistheinnerproductbetweenA
andBandA=A∙AistheEuclideannormofA.Further,
1=δijei⊗ej,(1.2)
i,j=1,2,3,representstheidentitytensorofsecond-order,whereδijistheKronecker-
deltaand{ei}isanorthonormalbasisinthethree-dimensionalEuclideanvectorspaceDin
Awhic−h1(trtheA)1materialforthebodydeviatorunderofAconsiderationandAT−1is=p(A−ostulated1)T,topromovidedve.A−Also,1theexists,arenotationsused.A=
LetK3,Pbetwofourth-ordertensors,Aasecond-ordertensorandvavector.Withrespect
totheorthonormalbasis{ei},thefollowingapplies.IfK,P,Aandvarerepresentedby
K=Kijklei⊗ej⊗ek⊗el,P=Pijklei⊗ej⊗ek⊗el,A=Aijei⊗ej(oftenuseismadeofthe
notationAij=(A)ij)andv=viei,respectively,then
KP=KijmnPmnklei⊗ej⊗ek⊗el,(1.3)
KT=Kijklek⊗el⊗ei⊗ej,(1.4)
K[A]=KijmnAmnei⊗ej,(1.5)
A2=AA=AijAjkei⊗ek,A−2=A−1A−1,(1.6)
Av=Aijvjei.(1.7)
Thus,forsecond-ordertensorsA,B,
A∙K[B]=B∙KT[A].(1.8)
Inaddition,Iiscalledthefourth-orderidentitytensor,
I=δimδjnei⊗ej⊗em⊗en,(1.9)
whichsatisfiestheproperty
I=E+J,(1.10)
with1E=Eimjnei⊗em⊗ej⊗en=2(δijδmn+δinδmj)ei⊗em⊗ej⊗en,(1.11)
J=Jimjnei⊗em⊗ej⊗en=21(δijδmn−δinδmj)ei⊗em⊗ej⊗en.(1.12)

6

1.CHAPTERODUCTIONINTR

Hence,denotedforbytheASandsymmetricAA,respandectivtheskely,folloew-symmetricwspartofanarbitrarysecond-ordertensorA,

AS=E[A],AA=J[A],

while

I[A]=A.

Theinnerproductbetweentwofourth-ordertensorsKandPisgivenby

(1.13)

(1.14)

(1.15)

K∙P=KijklPijkl,(1.15)
whereKijkl,PijklarethecomponentsofKandP,respectively,relativetotheorthonormal
basis{ei}.

1.4.GLOSSARYOFSYMBOLS7
1.4GlossaryofSymbols
SymbolNamePlaceofdefinition
ccurrenceofirstorBoldfacearabicnumbers
0zerovector,zerotensor(3.107)
1identitytensor(1.2)
letterslatincapitalBoldface(3.135)tensorstrainAlmansiABleftCauchy-Greenstraintensor(2.12)
CrightCauchy-Greenstraintensor(2.14)
DsymmetricpartofthevelocitygradienttensorL(2.8)
EGreenstraintensor(2.14),(2.17)
Fdeformationgradienttensor(2.3)
Lvelocitygradienttensor(2.7)
(3.113)tensorstructuralMˆNtensordefiningtheoutwardnormalontheyieldsurface(3.59)
PˆMandelstresstensor(2.24)
PiTransformationmatrix(3.212)
RQproprigiderbodyorthogonalrotationrotationtensor(2.4)(3.21)
SweightedCauchystresstensor(2.21)
TCauchystresstensor(2.21)
TˆSecondPiola-Kirchhoffstresstensorrelativeto
theplasticintermediateconfiguration(2.22)
T˜SecondPiola-Kirchhoffstresstensorrelativeto
(2.23)configurationreferencetheVUsymmetricsymmetricppositivositiveedefinitedefinitestretcstretchhtensortensor(2.4)(2.4)
Wskew-symmetricpartofthevelocitygradienttensorL(2.8)
¯XpositionvectorinthereferenceconfigurationRR(2.1)
XYinternal,configuration,symmetricinverseofx¯second-orderstraintensor(2.2)(3.64)
Zinthermoternal,dynamicalsymmetricconjugatestresstotensor,Y(3.65)
letterslatinsmallBoldfaceeiorthonormalbasisinEuclideanvectorspace(1.2)
hˆsetofinternalstatevariablesinstressformulation(3.36)
miunitvector,representinglocalaxesofsymmetry(3.110)
npositiveunitnormalvector(3.4)
qheatfluxvector(3.4)
xvpveloositioncityvvectorectorininthethecurrencurrenttconfigurationconfigurationRRt(2.1)(2.7)
tx¯configuration,one-to-onemapping(2.1)

8

ODUCTIONINTR1.CHAPTER

letterslatinCapitalsurfaceAABCDsmallstraincycle
parametersmaterialBiCs[t0,te]smallstraincycle
DdiameterofsteelballinBrinellhardnesstest
Foverstress/yieldfunction
ytropenHI(t0,te)integraloverthestresspower
factorloadingLPloadinBrinellhardnesstest
complianceelasticSolumevVWp(ef)effectiveinelasticstresspower
letterslatinSmallparametersmaterialbiecispmaterialecificinnerparametersenergy
functionyieldfhspecificentropy
k0materialparameterrepresentingconstantyieldstress
parametersmateriallismplasticviscosityarclengthparameter
timetparametersmaterialvilettersgreekcapitalBoldfaceΔrotationtensorinthekinematichardeninglaw
Φrotationtensorintheelasticitylaw
Γˆinternalstraintensor
measuredeformationΛΠrotationtensorintheflowrule
Θproperorthogonaltensor,representingeitherΦorΔorΠ
ˆspinplasticΩlettersgreeksmallBoldfaceξσbaceffectivkstressestresstensor
lettersGreekΘΦsurfaceabsolutedensittempyerature
parametersmaterialαiγspecificentropyproduction
δijKroneckersymbol
ζnormingfactorinthenormalityrule

(3.3)3.2Fig.(5.35)(3.42)(4.1)(5.9)(3.1)(3.42)(3.39)(4.1)(4.2)(3.2)(3.31)

(3.142)(3.140),,(5.34)(5.22)
(3.13)(5.10),(3.36)(3.2)(5.9),(3.169)(3.154)(3.59)(3.63),,(5.18)(5.19)
(3.169)3.1Fig.3.1Fig.(2.11)(3.43)3.1Fig.(3.110)(3.111),(3.26)(3.166)(5.9),(3.66)

(3.3)(3.4)(5.25)(3.126)(3.5)(1.2)(5.17),(3.200)

SYMBOLSOFYGLOSSAR1.4.

parameteryviscositηfunctionexvconχλ,µLame´constants
energyfreeecificspψϕconstitutivefunctiondescribingtheevolutionofbackstress
,Rmassdensityinthecurrentandreferenceconfiguration
φangleinBrinellhardnesstest
ydensitolumevξletterscalligraphicBoldfaceA,Ajfourth-ordertensor(s)inyieldfunction
B(k)symmetric,positivedefinitefourthordertensor
C(e)fourth-orderelasticitytensor
C(k)fourth-ordertensorinkinematichardeningrule
DjthermodynamicalconjugateofAj
IofpartsymmetricEH,H0fourth-ordertensorinyieldfunction
Ifourth-orderidentitytensor
Jskew-symmetricpartofI
Kfourth-ordertensorinyieldfunction
Lfourth-ordertensorinkinematichardeningrule
M(k)fourth-ordertensor,inverseofC(k)
lettersCalligraphicDBinmaterialternalbodydissipation
intconfigurationactualRRRt,Rtactualconfiguration
Rˆt,Rˆtactualconfiguration
eratorsOptdeterminandetdivdivergenceoperatorwithrespecttotheactualconfiguration
Divdivergenceoperatorwithrespecttothereferenceconfiguration
gradgradientoperatorwithrespecttotheactualconfiguration
Gradgradientoperatorwithrespecttothereferenceconfiguration
tracetrIndices(∙˜)quantityinthereferenceconfiguration
(∙ˆ)quantityintheactualconfiguration
(∙˙)materialtimederivative
(∙)Oldroydtimederivative
(∙)−1inverseofatensor
(∙)Ddeviatorofatensor
(∙)Ttransposeofatensor
(∙)(e)quantityrelatedtoelasticity

(5.19),(3.63)(3.107)(5.4)(5.2),(3.14)(5.14)(3.16)(3.2),(4.1)(3.3)

(5.26),(5.11)(3.102)(5.3),(3.128)(3.83)(5.25)(1.11)(5.11)(5.9),(1.9)(1.12)(3.169)(3.155)(3.90)

(3.29)2.1Fig.Fig.2.12.1Fig.

9

10

((∙∙))((ky))
((∙∙))AS
)(e∙((∙∙))pR

quantityrelatedtokinematichardening
quantityrelatedtotheyieldfunction
tensoraofpartsymmetric

tensoraofpartsymmetricskew-symmetricpartofatensor
tensoraofpartelastictensoraofpartinelastic

inelasticytitquan

ofparttensoratheinreference

configuration

CHAPTER

1.

ODUCTIONINTR

2Chapter

relationskinematicalBasic

(2.3)

LetusconsideramaterialbodyBinthethree-dimensionalEuclideanspaceE,thatoccupies
attimet=0thespatialareaRR,alsocalledreferenceconfiguration.Afterchoosingafixed
origininE,everymaterialpointP∈Bmaybeidentifiedbyapositionvector(orreferenced
position)XofpointXrelativetothefixedorigin.xdescribesthepositionvector(orcurrent
position)foranassociatedpointx,occupiedbythesamematerialpointPattimetinthe
actualconfigurationRt.AmotionofthecontinuumbodyBinEisaoneparameterfamily
ofconfigurations,wheretimetistheparameter,
x¯:(X,t)→x=x¯(X,t),(2.1)
andwhichisuniquelyinvertibleatfixedtimetthrough
X=X¯(x,t).(2.2)
Further,itisassumedthatthemotionpossessescontinuousderivativeswithrespecttospace
andtime,asdesired.Thedeformationgradienttensorconnectedtomotion(2.1)isdefined
throughF=F(X,t)=∂x¯=Gradx¯.(2.3)
X∂SincedetF>0isassumed,auniquepolardecomposition
F=RU=VR(2.4)
exists,withtheproperorthogonaltensorRandthesymmetric,positivedefinitestretchtensors
UandV.Themultiplicativedecompositionofthedeformationgradienttensorintoanelastic
part,plasticaandF=FeFp,(2.5)
issupposedtoapply.Assumingplasticincompressibility,
detFp=1.(2.6)
ThematerialtimederivativeofthedeformationgradienttensordefinestheEulerianvelocity
:LtensortgradienL=∂v=∂x¯˙(X,t)∂X=∂∂x¯(X,t)∂X=F˙F−1,(2.7)
∂x∂X∂x∂t∂X∂x
11

(2.5)

(2.7)

12

RR

Fp

wherev(x,t):=x˙and

CHAPTER2.BASICKINEMATICALRELATIONS

F

QRt

RtFeˆRtˆRtQ

Figure2.1:Decompositionofthedeformation

L=D+W,D=21(L+LT),W=21(L−LT).

(2.8)

Themultiplicativedecomposition(2.5)introducesaso-calledplasticintermediateconfiguration
Rˆt(cf.alsoFig.2.1),whichingeneralisnotcompatibleandthereforenotanEuclidean
one(forfurtherreferencesdealingwith(2.5)seealsoLeeandLiu[68],Lubliner[74],
Maugin[78]).Quantitiesreferredtotheplasticintermediateconfigurationwillbedenotedby
asuperposed(∙ˆ)-symbol,whileasuperposed(∙˜)-symbolrepresentsaquantityinthereference
configurationRR.TheplasticvelocitygradientLˆpisgiventhrough

1−Lˆp=F˙pFp=Dˆp+Wˆp,

with

Dˆp=1(Lˆp+LˆpT),Wˆp=1(Lˆp−LˆpT).
22

(2.9)

(2.10)

13Byusing(2.5),thefollowingkinematicalrelationscanbeobtained(seealsoAppendixA):
Γˆe=21Cˆe−1,Cˆe=FeTFe=ˆUe2,(2.11)
Γˆp=211−Bˆp−1,Bˆp=FpFpT=Vˆp2,(2.12)
Γˆ=Γˆe+Γˆp,(2.13)
E=1(C−1)=FpTˆΓFp,C=FTF,(2.14)
2Ee=FpTΓˆeFp,(2.15)
Ep=1(Cp−1)=FpTΓˆpFp,Cp=FpTFp,(2.16)
2E=Ee+Ep.(2.17)
Here,CˆeandBˆparetheelasticrightCauchy-GreenandtheplasticleftCauchy-Green
tensors,respectively.Also,thetensorsΓˆeandΓˆparecalledtheelasticGreenandtheplastic
Almansistraintensorswithrespecttotheplasticintermediateconfiguration.Onthebasisof
theserelationsitcanbeseenthat
Dˆp=Γˆp=FpT−1E˙pFp−1,(2.18)
Γˆ=FpT−1˙EFp−1=FeTDFe,(2.19)
withXˆ=Xˆ˙+LˆpTXˆ+XˆLˆp(2.20)
forasecond-ordertensorXˆrelativetotheplasticintermediateconfiguration.As(2.18)indi-
cates,DˆpmaybeinterpretedasaparticularOldroydderivativeofΓˆp(seealsoTsakmakis
]).[100Weˆ˜designatebyTtheCauchystresstensor,byStheweightedCauchystresstensorand
byT,TthesecondPiola-Kirchhoffstresstensorrelativetotheplasticintermediateand
thereferenceconfiguration,respectively:
S=(detF)T,(2.21)
ˆT=Fe−1SFeT−1,(2.22)
T˜=Fp−1ˆTFpT−1=F−1SFT−1.(2.23)
Anotherstresstensor,relatedtotheplasticdissipation,istheso-calledMandelstresstensor
])[73Lubliner(cf.Pˆ:=FeTSFeT−1=GˆΓˆe,Tˆ=1+2ΓˆeTˆ=CˆeTˆ,(2.24)
whichisreferredtotheplasticintermediateconfiguration.Themultiplicativedecomposition
ofplastictheindeformationtermediategradienconfigurationtisunique(seeFig.except2.1forandacf.rigidCaseybodyandrotationNaQghdi,sup[14],erposedGreenonandthe
Naghdi[39]).Undersuchrotations,thedeformationandstressfieldstransformaccordingto
theequationsgiveninAppendixA.

Chapter3

Modelling(Visco-)Plasticitofyanisotropic

Inthischapterathermodynamicallyconsistentmodelforanisotropic(visco-)plasticity,de-
rivedfromthesecondlawofthermodynamics,ispresented.Itisbasedonrecentpublications
byticitHy¨law,ausleretnonlinearal.[43]anisotropicandTsakmakiskinematic[106hardening].Theandmodelanconsistsanisotropicofanflowrule.anisotropicForelas-the
derivinneredasscalar-andsufficienttensor-vconditionsaluedtostatefulfillvtheariables,seconddescribinglawofthermohardening,dynamicsconstitutivinetheformequationsoftheare
Clausius-Duheminequality.Todescriberotationsoftheaxesofanisotropyintheelastic-
initytrolaw,duced,therespkinematicectively(cf.hardeningFig.3.1).andtheflowrule,threerotationtensors,Φ,ΔandΠare

RR

Fp

F

ΠΔΦ,,

ˆRt

Fe

Figure3.1:Representationoftheaxesofanisotropy

14

Rt

3.1.SECONDLAWOFTHERMODYNAMICS

15

3.1Secondlawofthermodynamics
Physicalphenomenahaveoftenthetendencyto”move”inonedirection.Forexample,heat
tendstoflowfromthe”warmer”tothe”colder”regionofabody.Theseeffectscanbeanalyzed
correctlythroughtheintroductionofanewquantity,theso-calledentropyH.Theentropyis
supposedtobeascalarvaluedfunction,

(3.1)

(3.2)

H=H(Rt,t).(3.1)
Thenaspecificentropyh=hˆ(x,t)exists,with
H=RthdV,(3.2)
andforhappliesabalancerelation
H˙=∂RtΦdA+RtξdV,(3.3)
whereΦandξaresurface-andvolume-densities,respectively.Inordertofulfilltherequired
dissipationproperty(irreversiblebehavior),theassumptionismadethatH˙consistsoftwo
parts,onebeingresponsibleforthesupplyofentropyfromthesurrounding,
−q∙ndA+rdV,Θ:absolutetemperature,r:radiationterm,(3.4)
∂RtΘRtΘ
andγdV,γ:specificentropyproduction.(3.5)
RtHere,qistheso-calledheatfluxvectorandnisthepositiveunitoutwardnormalon∂Rt.
From(3.3),(3.4)and(3.5)followsthat
qΦ=−Θ∙n,(3.6)
ξ=r+γ,(3.7)
ΘandrqH˙=∂R−Θ∙ndA+R(Θ+γ)dV.(3.8)
ttTheirreversiblecharacteristakenintoaccountbydemandingthat
RtγdV≥0.(3.9)
ThisisalsocalledthesecondlawofthermodynamicsinformoftheClausius-Duheminequal-
.yit

(3.8)

(3.9)

16CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

(3.10)

(3.12)

3.1.1LocalformoftheClausius-Duheminequality
,(3.2)romFH˙=h˙dV.(3.10)
RtOnusing(3.10),(3.9)canberewrittenas
qrRt(h˙−Θ)+divΘdV≥0,(3.11)
andonapplyingthelocalizationtheoremthelocalforoftheClausius-Duheminequality
asreadsh˙−r+1divq≥0.(3.12)
ΘΘTogetherwiththefirstlawofthermodynamics,whichstates
11divq=r+T∙D−e˙,(3.13)
whereeisthespecificinternalenergy,andthedefinitionofthespecificfreeenergyψ,
ψ:=e−Θh,(3.14)
(3.12)canberecastedinto
11RS∙D−ψ˙−hΘ˙−Θq∙gradΘ≥0.(3.15)
Inviewofisothermaldeformationswithauniformtemperaturedistribution,assumedinthis
work,theClausius-Duheminequalityreads(cf.ColemanandGurtin[18],Haupt[41],
TruesdellandNoll[99])
S∙D−Rψ˙≥0.(3.16)
Byvirtueof(2.19)and(2.22),(3.16)canberewrittenintheform
Tˆ∙Γˆ−Rψ˙≥0.(3.17)
Thespecificfreeenergyψisassumedtobeadditivelydecomposedintoanelasticandaplastic
part,ψeandψp,respectively,
ψ(t)=ψe(t)+ψp(t).(3.18)
Hence,Tˆ∙Γˆ−Rψ˙e−Rψ˙p≥0.(3.19)

(3.16)

(3.18)

(3.19)

3.2.ELASTICITYLAWANDDISSIPATIONINEQUALITY

17

3.2Elasticitylawanddissipationinequality
Inthiswork,theelasticresponseofamaterialissupposedtoexhibitanorientationaltype
ofanisotropy.Dafalias[27],[32]assumesforsuchakindofanisotropy,thatsomeaxesof
anisotropymayrotate,therateofrotationbeingspecifiedbytheplasticspinconcept.The
mainconcernoftheseworksistheplasticspingoverningtherateofrotation.However,these
worksarewritteninapurelymechaniccontext.Inopposite,ourworkisembeddedina
thermodynamicalframework,asdescribedinTsakmakis[106].Accordingtothis,theelastic
partofthefreeenergyfunctionψeisassumedtobeafunction,besidesofFe,ofΦ(t),the
rotationoftheaxesofanisotropyintheelasticitylaw:
ψe(t)=ψe(Fe(t),Φ(t)),ΦT=Φ−1.(3.20)
ThetensorΦisdefinedtorotatevectorsfromthereferenceconfigurationtotheplasticin-
termediateconfigurationandtosatisfytransformationpropertiesunderarbitraryrigidbody
rotationsQsuperposedontheplasticintermediateconfigurationsimilartothoseforRp(see
A):endixAppΦ→Φ=QΦ.(3.21)
Fromamorephysicalpointofview,Φisassumedtorotatesomeaxescharacteristicforthe
elasticanisotropyoftheunderlyingsubstructure.Ifnoplasticflowoccursduringaloading
process,theseaxeshavetoremainfixed.SoΦisakinematicalquantity,whichinadditionto
Fpcharacterizestheplasticdeformationprocess.
Itshouldberemarkedhere,thatintheterminologyofDafalias[32],therateΦ˙ΦTisthe
constitutivespin.Itcanbeshown(seeAppendixB),thatrequiringfromtheelasticfreeenergy
ψetoremainunalteredunderarbitraryrigidbodyrotationssuperposedonboththecurrentand
theplasticintermediateconfigurationisequivalenttorequirefromψetopossessrepresentations
oftheform
ψe=ψeΓˆe,Φ=ψ˜eΓ˜e,Γ˜e:=ΦTΓˆeΦ.(3.22)
Γ˜edenotesastrainmeasureinthereferenceconfigurationwith
Φ∂ψ˜eΦT=∂ψe,2ΓˆeΦ∂ψ˜eΦT=∂ψeΦT.(3.23)
∂Γ˜e∂Γˆe∂Γ˜eA∂ΦA
˜˜From(3.22)follows
ψ˙e=Φ∂˜ψeΦT∙Γˆ˙e+tr2ΓˆeΦ∂˜ψeΦTΦΦ˙T
∂Γe∂Γe
=Φ∂ψ˜eΦT∙Γˆe−2Φ∂ψ˜eΦT∙ΓˆeLˆp+tr2ΓˆeΦ∂ψ˜eΦTΦΦ˙T
∂Γ˜e∂Γ˜e∂Γ˜e
=Φ∂ψ˜eΦT∙Γˆe−2ΓˆeΦ∂ψ˜eΦT∙Lˆp+2ΓˆeΦ∂ψ˜eΦT∙˙ΦΦT
∂Γ˜e∂Γ˜e∂Γ˜e
=Φ∂ψ˜eΦT∙Γˆ−Φ∂ψ˜eΦT∙Dˆp−2ΓˆeΦ∂ψ˜eΦT∙Lˆp+2ΓˆeΦ∂ψ˜eΦT∙˙ΦΦT
∂˜Γe∂Γ˜e∂Γ˜e∂Γ˜e
=Φ∂ψ˜eΦT∙Γˆ−1+2ΓˆeΦ∂ψ˜eΦT∙Dˆp−2ΓˆeΦ∂ψ˜eΦT∙Wˆp−˙ΦΦT.(3.24)
∂Γ˜e∂Γ˜e∂Γ˜e

(3.23)

18CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

(3.25)

˜Here(2.9),(2.13)and(2.18)havebeenused.Itisworthremarkingthatthetensor∂˜ψe(and
Γ∂ethereforethetensorΦ∂ψ˜eΦTtoo)issymmetric,whilethetensor˙ΦΦTisskew-symmetric.On
˜substituting(3.24)into∂Γe(3.19),
Tˆ−RΦ∂ψ˜eΦT∙Γˆ+1+2ΓˆeRΦ∂˜ψeΦT∙Dˆp
˜˜
˜∂Γe∂Γe(3.25)
+2ΓˆeRΦ∂˜ψeΦT∙Wˆp−˙ΦΦT−Rψ˙p≥0.
Γ∂eIntheterminologyofDafalias[32],Wˆpistheplasticmaterialspin,while
Ωˆ(e):=Wˆp−˙ΦΦT(3.26)
denotestheplasticspinassociatedwiththeelasticitylaw.Underrigidbody(e)rotationssu-
perposedontheplasticintermediateconfigurationtheskew-symmetrictensorΩˆtransforms
accordingto(seeAppendixA,(3.21))
Ωˆ(e)→Ωˆ(e)∗=QΩˆ(e)QT.(3.27)
Theplasticpartofthefreeenergy,ψpisassumedtodependoninternalstatevariablesdescribing
thehardeningresponseandtheCauchystressTˆintheplasticintermediateconfigurationis
definedtobeafunctionofstatevariables(butnotoftheirrates).Forthecaseofrate-
dependentplasticity(alsocalledviscoplasticity)theevolutionoftheinternalstatevariables
dependsonstatevariablesonly,whichmeansthatDˆp,Wˆp−˙ΦΦTandψ˙parefunctionsofstate
variablesonly.Thus,followingsimilarargumentsasusedinColemanandGurtin[18],for
viscoplasticitytherelations(cf.(3.23))
Tˆ=R∂ψe=RΦ∂ψ˜eΦT,(3.28)
∂Γˆe∂Γ˜e
Dint:=1+2ΓˆeRΦ∂ψ˜eΦT∙Dˆp
∂Γ˜e(3.29)
+2ΓˆeRΦ∂ψ˜eΦT∙Wˆp−˙ΦΦT−Rψ˙p≥0
˜Γ∂ecanbeproventobenecessaryandsufficientconditionsinorderforinequality(3.25)tobevalid
ineveryadmissibleprocess.Sothistheoryofviscoplasticityfallsinthegeneralframeworkof
ColemanandGurtin’s[18]thermodynamicswithinternalstatevariables(cf.alsoKra-
tochvilandDillon[66]).Inequality(3.29)isknownastheinternaldissipationinequality.
Inthecaseofrate-independentplasticity(alsojustcalledplasticity)theevolutionofinternal
statevariablesisdefinedtodepend,besidesonthestatevariables,onthedeformationrate.As
aconsequence,therelations(3.28),(3.29)arenecessaryandsufficientfor(3.25)tobevalidin
everypurelyelasticadmissibleprocess.If(3.28)and(3.29)arealsoassumedtoapplyalong
loadingpathswhereinelasticflowisinvolved,theninthecaseof(rate-independent)plasticity
lawstheserelationsaregenerallyonlysufficientconditionsforthevalidityof(3.25)inevery
cess.proadmissible

(3.28)

3.3.FLOWRULEFORPLASTICITYANDTHEPOSTULATEOFIL’IUSHIN19
Forplasticityaswellasviscoplasticitylaws,itfollowsfrom(3.28),(3.29),togetherwith(2.9),
(2.24),that
Dint=PˆS∙Dˆp+PˆA∙Wˆp−˙ΦΦT−Rψ˙p=Pˆ∙Lˆp−˙ΦΦT−Rψ˙p≥0.(3.30)
Equation(3.28)representsageneralelasticitylawcharacterizingmaterialswithanisotropyof
orientationaltype.Theterm
Wp(ef):=Pˆ∙Lˆp−˙ΦΦT(3.31)
isinterpretedtodescribeaneffectiveinelasticstresspower.From(3.21),
Φ˙ΦT=Q˙ΦΦTQT+Q˙QT,(3.32)
sothat(cf.AppendixA)
Lˆp−Φ˙ΦT=QLˆp−Φ˙ΦTQT,(3.33)
Wp(ef)=Pˆ∙Lˆp−Φ˙ΦT=Pˆ∙Lˆp−Φ˙ΦT=Wp(ef).(3.34)
ThatmeanstheeffectiveplasticstresspowerWp(ef)remainsunalteredunderarbitraryrigid
bodyrotationssuperposedontheplasticintermediateconfiguration.Therateoftheplastic
partofthefreeenergy,ψ˙p,ispostulatedtodescribethepowerrelatedtotheenergystoredinthe
onmaterialtheplasticandisinalsotermediaterequiredtobconfiguration.eunalteredSinceunder=arbitraryremainsrigiduncbodyhanged,itrotationsfollowssupthaterposedthe
RinternaldissipationisunalteredunderarbitraryrigidbRodyrotationssuperposedontheplastic
intermediateconfigurationaswell:
Dint=Dint.(3.35)
3.3FlowruleforplasticityandthepostulateofIl’iushin
byThemanpyostulateauthorsofIlase.g.’iushinCaseyhasbandeeninvTsengestigated[17],inDafthealiascontext[24],ofFosdickrate-indepandendenVtolkmannplasticity
[36],Hill[49],HillandRice[50],LinandNaghdi[69],Lubliner[73],Lucchesiand
Silhavy[75],Srinivasa[96],aswellasTsakmakis[103],[104].Variousaspectsofthepos-
ruletulateforhaLˆvpebcannoteenbediscussedderivedingenerallyLubliner.Ho[73],wever,whereinitthehascasebeenofanshoinwnvertiblethataisotropicuniquenormalitelasticityy
lawapropernormalityruleforDˆpcanbeestablished(cf.Tsakmakis[103]andTsakmakis
andWilluweit[105]).Thissectionwilldealwithrate-independentplasticityonly.Forthis
forcasetheandfordeformationelasticrateanisotropLˆp−yΦ˙ofΦTorienwillbetationalderivtypede,asasadescribsufficienedtinconditionSect.3.2,forathepnormalitostulate.yrule
Letassumetheexistenceofayieldfunctioninastressspaceformulationwithrespecttothe
plasticintermediateconfigurationoftheform
f(t)=fˆPˆ,hˆ=f¯PˆS,PˆA,hˆ.(3.36)
Here,hˆstandsforasetofinternalstatevariableshˆi,1≤i≤M,whicharescalarsorcompo-
nentsoftensorsreflectinghardeningproperties.Itisassumedthat(3.36)mayberewrittenin
astrainspaceformulationwithrespecttothereferenceconfigurationintheform
f(t)=g˜(E,Ep,q),(3.37)

20CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY
whereqdenotesasetofˆinternalstatevariablesqj,1≤j≤N,associatedinsomewaywith
thehardeningvariableshi.Theequation
f=fˆPˆ,hˆ=g˜(E,Ep,q)=0(3.38)
iscalledyieldcondition.Itdescribesforfixedvaluesofhˆaso-calledyieldsurfaceinthespace
oftheMandelstresstensorsPˆ,andforfixedvaluesofEpandqayieldsurfaceinthespace
ofthestraintensorsE.Forsimplicity,theseyieldsurfacesareassumedtobesmooth.(A˜
discussionaboutyieldsurfacesinthestrainandstressspaces,expressedintermsofEandT,
respectively,isgiveninCaseyandNaghdi[16]).
Forusing,rate-indepinsteadofendentimett,plasticitascalary,loadingparameterproscessesdenotinginvolvingaplasticplasticarcflowlength.maybIteispdescribostulatededby
thatfors=constallinternalstatevariableshavetoremainconstantaswell.Further,itis
convenienttointroduceaso-calledloadingfactorL(t),
L:=f˙s=const.(3.39)
Then,themodelresponseischaracterizedasfollows(adiscussionaboutloadingconditionsis
giveninTsakmakis[101]):
f<0⇔elasticrange,(3.40)
<0elasticunloading
0>loadingplasticf=0&L=0⇔neutralloading.(3.41)
Plasticflowisdefinedtooccuronlywhenconditionsforplasticloadingaresatisfied.Itis
importanttoremarkthatnoteverytensorPˆ,satisfyingtherelationfˆPˆ,hˆ≤0,mustbean
aaccessiblenonlinearMandelsystemofstressequationsstate.forThisΓˆandfollowsitmafromyhapptheenfactthatthatnoforgivsolutionsenPˆexist.(2.24)represents
Now,cyclesinthespaceoftheGreenestraintensorsEareconsidered.Notethatacyclein
thevicevspaceersa.ofFaollostrainwingmeasureLucchesiimpliesandaSilhacyclevyin[75the],spacestrainofancyclesyarefurtherdenotedstrainasmeasuresmall(butand
notnecessarilyinfinitesimalsmall)onesifthefollowingconditionissatisfied.Duringthecyclic
process,theinitialstrainstateisalwaysonorinsidetheyieldsurfacesg˜=0corresponding
totheprocess.Inotherwords,theinitialstrainstateliesalwaysintheintersectionofallthe
forelasticasmallrangescycle,surroundedwhichbbyeginstheatyieldtimet0surfacesandg˜ends=at0timeduringte.theApromaterialcess.isCs[t0,definedte]istowrittensatisfy
thepostulateofIl’iushinforsmallcycles,ifforfixedmaterialparticle
tete
I(t0,te):=S(t)∙D(t)dt=T˜(t)∙E˙(t)dt≥0foreveryCs[t0,te].(3.42)
tt00Itisworthemphasizingthat(3.42)istheisothermalversionofageneraldissipationpostulate,
whichhasbeenproposedbyLucchesiandSilhavy[75]asanon-isothermalgeneralizationof
theclassicalpostulateofIl’iushinforarbitraryisothermalstraincycles.Inequality(3.42)has
bsmalleenalsopuresuppelasticosedbrange.yHoTsakmakiswever,the[104]inconditiondiscussingthattheelastic-plasticcyclesshouldmaterialsbesmallwithisvimpanishingosed
inLucchesiandSilhavy[75]inordertomakethepostulateofIl’iushin”derivablefrom

3.3.FLOWRULEFORPLASTICITYANDTHEPOSTULATEOFIL’IUSHIN21

(3.45)

(3.46)

somesufficientconditions(thenormalityrule)”,whichisratheramathematicalpointofview.
Ontheotherhand,theconditionofsmallcycleswasassumedinTsakmakis[104]inorderto
obtaina”stabilitycondition”formaterialresponse,whichisnottoorestrictivewhenmodeling
theobservedbehaviorofvariousmaterials.Thisisratheraphysicalpointofview.
Forderivingsomeconsequencesfrominequality(3.42),itisconvenienttoexpressthestress
tensorT˜intermsoftheGreenstraintensor.TothisendthedeformationmeasureΛinthe
referenceconfigurationisdefinedby
Λ:=Fp−1Φ,(3.43)
hwhicfromΛ˙:=−Fp−1Lˆp−˙ΦΦTΦ.(3.44)
Thus,from(3.22)2,(2.13)–(2.17),
˜Γe=ΛTEeΛ=ΛTEΛ−ΛTEpΛ,(3.45)
sothat,byvirtueof(3.22)1,
ψe=ψ˜eΛTEΛ−ΛTEpΛ=:ψ˜˜e(E,Ep,Λ).(3.46)
Henceψ˙e=∂ψ˜˜e∙E˙+∂ψ˜˜e∙E˙p+∂ψ˜˜e∙Λ˙
∂E∂Ep∂Λ
=Fp∂ψ˜˜eFpT∙Γˆ+Fp∂ψ˜˜eFpT−FpT−1∂ψ˜˜eΦT∙Lˆp−˙ΦΦT.(3.47)
∂E∂Ep∂Λ
Ontheotherhand,ontakingthematerialtimederivativeof(3.22)1,weget,aftersomerear-
terms,oftrangemenψ˙e=∂ψ˜e∙Γ˜˙e
˜Γ∂e˜=∂ψ˜e∙ΛTEeΛ∙
Γ∂e=Λ∂ψ˜eΛT∙E˙−Λ∂ψ˜eΛT∙E˙p+2∂ψ˜e∙ΛTEeΛ˙
∂Γ˜e∂Γ˜e∂Γ˜e
=1Tˆ∙Γˆ−1Tˆ∙Dˆp−2ΓˆeTˆ∙Lˆp−˙ΦΦT
RRR
=1Tˆ∙Γˆ−1Pˆ∙Lˆp−˙ΦΦT.(3.48)
RROncomparing(3.47)with(3.48),
Pˆ∙Lˆp−˙ΦΦT=−RFp∂ψ˜eFpT−FpT−1∂ψ˜eΦT∙Lˆp−Φ˙ΦT(3.49)
˜˜
Λ∂E∂p

(3.48)

(3.49)

22CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

(3.50)

and˜˜Tˆ=RFp∂∂ψEeFpT,(3.50)
hwhicfrom˜T˜=R∂ψ˜e.(3.51)
E∂Now,(3.42)isasssumedtoapplyandasmallstraincycleABCD(Fig.3.2)isconsidered,

g˜E,Ep(B),q(B)=0


DA

B

Cg˜E,Ep(C),q(C)=0


Figure3.2:AsmallstraincyclewithplasticflowoccurringbetweenBandConly.

whichisparameterizedbytimet.WedenotebyX(P)thevalueofsomequantityXatthe
pointP.Thus,thetimesassociatedwithpointsA,B,C,Daret(A),t(B),t(C),t(D),respectively
(t(A)<t(B)<t(C)<t(D)).ThestraincyclebeginsandendsatE=E(A)=E(D),whileplastic
flowoccursbetweenBandConly.Since(3.28),andtherefore(3.51)too,isassumedtohold
duringplasticloadingaswell,itcanbeshownthat(cf.(3.42))

It(A),t(D)=t1T˜(t)∙E˙(t)dt
(D)
Rt(A)R
=t(D)∂ψ˜˜e(E(t),Ep(t),Λ(t))∙E˙(t)dt
t(A)∂E(t)
=ψ˜˜eE(A),Ep(C),Λ(C)−ψ˜˜eE(A),Ep(B),Λ(B)
)C(−t∂ψ˜˜e(E(t),Ep(t),Λ(t))∙E˙p(t)dt
t(B)∂Ep(t)
−t(C)∂ψ˜˜e(E(t),Ep(t),Λ(t))∙Λ˙(t)dt
t(B)∂Λ(t)

3.3.FLOWRULEFORPLASTICITYANDTHEPOSTULATEOFIL’IUSHIN

23

(3.52)

t(C)∂ψ˜˜eE(A),Ep(t),Λ(t)˙
=t(B)∂Ep(t)∙Ep(t)
+∂ψ˜˜eE(A),Ep(t),Λ(t)∙Λ˙(t)dt
)t(Λ∂−∂ψ˜e(E(t),Ep(t),Λ(t))∙E˙p(t)dt
t(C)˜
t(B)∂Ep(t)
−t(C)∂ψ˜˜e(E(t),Ep(t),Λ(t))∙Λ˙(t)dt
t(B)∂Λ(t)
t(C)∂ψ˜˜eE(A),Ep(t),Λ(t)
=t(B)∂Ep(t)
−∂ψ˜˜e(E(t),Ep(t),Λ(t))∙E˙p
∂Ep(t)
∂ψ˜˜eE(A),Ep(t),Λ(t)(3.52)
+∂Λ(t)
˜−∂ψ˜e(E(t),Ep(t),Λ(t))∙Λ˙(t)dt≥0.
)t(Λ∂ByusingTaylor’stheorem,
It(A),t(D)/R
t(C)→limt(B)t(C)−t(B)=
=∂ψ˜eE(A),Ep(t),Λ(t)∙E˙p(t)−∂ψ˜e(E(t),Ep(t),Λ(t))∙E˙p(t)(3.53)
˜˜
∂Ep(t)∂Ep(t)t=t(B)
+∂ψ˜˜eE(A),Ep(t),Λ(t)∙Λ˙(t)−∂ψ˜˜e(E(t),Ep(t),Λ(t))∙˙Λ(t)≥0.
∂Λ(t)∂Λ(t)t=t(B)
SincethepointBcanbechosenrandomlyontheyieldsurface,theindext(B)in(3.53)maybe
droppedtoget,asanecessaryconditionfor(3.42),theinequality
˜˜−∂ψ˜e(E,Ep,Λ)∙E˙p−∂ψ˜e(E,Ep,Λ)∙Λ˙≥
∂Ep∂Λ(3.54)
−∂ψ˜˜eE(A),Ep,Λ∙E˙−∂ψ˜˜eE(A),Ep,Λ∙Λ˙,
∂Epp∂Λ
whereEdenotesastrainstateontheyieldsurface,thevariablesEp,Λbeingassociatedwith
thisstate.E(A)isastrainstateonorinsidetheyieldsurface,i.e.g˜E(A),Ep,q≤0,where
theinternalstatevariablesqareassociatedwiththestrainstateE.Conversely,(3.54)isa
sufficientconditionfor(3.42).Thiscanbeexaminedbytakingtheintegralof(3.54)alonga
straincycleasshowninFig.3.1.For(3.54)toremainvalidduringthisstraincycle,E(A)must
alwayslieintheintersectionofalltheelasticrangesduringthestraincycle,whichinturn
impliesthatthestraincycleABCDissmall.Then,followingstepssimilartothosein(3.52),
itisastraightforwardmattertoarriveat(3.42).

24CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

(3.55)

Using(3.44),(3.49),itisreadilyseenthat
∂ψe∙Λ˙=−∂ψe∙E˙p−1Pˆ∙Lˆp−˙ΦΦT.(3.55)
˜˜˜˜
∂Λ∂EpR
Byvirtueofthisresultinequality(3.54)isequivalentto
Pˆ∙Lˆp−˙ΦΦT≥Pˆ(A)∙Lˆp−˙ΦΦT,(3.56)
orPˆS∙Dˆp+PˆA∙Wˆp−Φ˙ΦT≥PˆS(A)∙Dˆp+Pˆ(AA)∙Wˆp−Φ˙ΦT.(3.57)
Recallthat(3.56)isequivalenttotheinequality(3.54)andthereforeto(3.42)aswell.In-
equality(3.56)expressestheso-calledprincipleofmaximumplasticstresspower.Indeed,with
respecttoapurelymechanicalformulationofthetheory,accordingto(3.31),inequality(3.56)
statesthat,forgiveneffectiveplasticdeformationrateLˆp−˙ΦΦT,amongalladmissiblestress
tensorsPˆ(A),theactualtensorPˆmaximizestheeffectiveplasticstresspowerWp(ef).The
termadmissiblestresstensordenotesanaccessibleMandelstresstensorwhichisonorinside
theyieldsurfacefˆ=0(cf.(3.38)).Inthecaseofisothermaldeformationswithauniform
temperaturedistribution,thisworkdealswith,theinternaldissipationisgivenby(cf.(3.30))
DintPˆ,ˆLp−˙ΦΦT,ψ˙p=Pˆ∙Lˆp−˙ΦΦT−Rψ˙p.(3.58)
Then,(3.56)states,thatforgiveninternalstatevariablesandtheirrates,i.e.forgivenLˆp−˙ΦΦT
andψ˙p,amongalladmissiblestresstensorsPˆ(A),theactualonePˆmaximizestheplastic
dissipationDint.Itcanbeshown(seee.g.Lubliner[74],Sect.3.2.2)foratreatmentinthe
contextofsmalldeformations),Tthatconvexityofthelevelset{Pˆ|fˆ(Pˆ,hˆ)≤0,hˆ=fixed}anda
normalityruleforLˆp−˙ΦΦaresufficientconditionsforinequality(3.56).Thetermnormality
rulemeansthatLˆp−˙ΦΦThastobedirectedalongtheoutwardnormalontheyieldsurface,
whichhasbeenassumedtobesmooth:
ˆf∂Lˆp−Φ˙ΦT=3s˙Nˆ,Nˆ:=∂Pˆ,(3.59)
ˆP∂2∂fˆ
wheres˙isapositivescalarforplasticloading.Ofcourse,(3.59)1canbedecomposedintoits
symmetricandskew-symmetricpart:
∂f¯∂fˆ
33ˆ3∂Pˆ
Dˆp=2s˙NˆS=2s˙∂∂Pf¯S=2s˙ˆS,(3.60)
∂f
∂Pˆ∂Pˆ
ˆ∂f¯∂fˆ
Ωˆ(e):=Wˆp−˙ΦΦT=3s˙NˆA=3s˙∂Pˆ¯A=3s˙∂PA,(3.61)
22∂f2∂fˆ
∂Pˆ∂ˆP

(3.59)

(3.60)

(3.61)

3.4.FLOWRULEFORVISCOPLASTICITY

25

withNˆSandNˆAbeingthesymmetricandtheskew-symmetricpartofNˆ,respectively.(3.59)
(respectively(3.60),(3.61))representstheflowruleforrateindependentplasticity,wheres˙has
dtobedeterminedfromtheso-calledconsistencyconditiondtfˆ=0.Moreover,itisreadilyseen
thats˙=2Lˆp−˙ΦΦT∙Lˆp−Φ˙ΦT
3
=2Dˆp∙Dˆp+Wˆp−˙ΦΦT∙Wˆp−Φ˙ΦT.(3.62)
3Itisperhapsofinteresttoremarkthat,onthebasisof(3.61),theplasticspinrelatedtothe
elasticityˆlawvanishesidenticallyifandonlyiftheyieldfunctionisdependentonthesymmetric
likparteofthatPofonlyrigid.boAlso,diestheapplies,approachinpracticallyNa,ghditotheandpresenTrappttheory[83]asforwell.discussinglimitingcases

3.4Flowruleforviscoplasticity
Forfromthethepurpplasticitosesyofonesthisbwyorkitadoptingsufficesalltotheconcenconstitutivtrateeonequationsviscoplasticitexceptymofromdelsthewhicevholutionarise
equationfors.Thisisnowdefinedintermsofaso-calledoverstress.Notethatwhereasfor
rate-independentplasticitytheyieldfunctionalwayssatisfiestheconditionf=fˆPˆ,hˆ≤0,
inonlythesuccasehoffunctionsviscoplasticitfareynoadmitted,suchforwhicrestrictionshtheonlevfelareset{impPˆ|fˆ(osed.Pˆ,hˆ)Ho≤w0ev,hˆer,=forfixed}isviscoplasticitconvex.y
Apositivevalueoffiscalledoverstress,sothats˙issupposedtobegivenasafunctionoffˆ.
Especially,anevolutionequationoftheform
mˆfs˙=η≥0(3.63)
isassumedtohold,wheremandηarepositivematerialparameters.

(3.63)

3.5Kinematichardeningandyieldfunction
ψForissuppsimplicitosedy,toinwhatconsistfolloonlywsoftheisotropiccontributionhardeninginarisingthefromyieldkinematicfunctionisnothardeningregarded.effects.InSo
panalogytoψe(cf.(3.22)),ψpisassumedtobeoftheform
ψp=ψpYˆ,Δ=ψ˜pY˜,Y˜:=ΔTYˆΔ,ΔT=Δ−1.(3.64)
inHere,termediateYˆisaninconfiguration,ternalsymmetricwhilethepropsecond-ordererorthogonalstraintensor,tensorΔwhic(thwopo-poineratestintensorthefield)plasticis
definedreferencetotorotatethesomeplasticinsymmetrytermediateaxes,relatedconfiguration.tothePhysicallykinematic,plastichardeningflowrespcausesonse,thefromplasticthe
deformationgradientFp,therelatedplasticstrainwithrespecttotheplasticintermediate
configurationand(ef)therelatedeffectiveplasticstresspower,ΓˆpandWp(ef),respectively.However,
onlyapartofWpmaybedissipatedasheat,andtheremainderwillbestoredinthematerial

26CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY
mainlybychangesinthedensityandarrangementofdislocations(measurementsofenergy
ˆstorageinthematerialcanbefounde.g.inOliferuketal.[86]).ThestrainYˆisapartof
ΓthepandlaterisbineingterpretedRψ˙pto(furtherrepresentremarksthestrainaboutrelatedthistotopicthecanstressbepowfounderinstoredinTsakmakisthematerial,and
Willuweit[105]).WedenotebyZˆtheinternalstresstensor,whichisthermodynamically
conjugatetoYˆ.Inanalogyto(3.28)
Zˆ:=R∂ψp=RΔ∂ψ˜pΔT(3.65)
∂Yˆ∂Y˜
isset.Evidently,thetensorZˆissymmetricandoperatesintheplasticˆintermediateconfigu-
pration.ossestheKinematicstructureofahardeningMandelisdescribstressedtensor.bytheFbacormally,k-stressξˆcanbtensoreξdefined,,whicinhisanalogyptoostulated(2.24)to,
throughξˆ=1+2YˆZˆ=ξˆS+ξˆA,(3.66)
whereξˆS=Zˆ+YˆZˆ+ZˆYˆ,(3.67)
ξˆA=YˆZˆ−ZˆYˆ,(3.68)
orξˆA=RΔY˜∂ψ˜p−∂ψ˜pY˜ΔT.(3.69)
∂Y˜∂Y˜
Itcanbeshown,thatunderarbitraryrigidbodyrotationssuperposedontheplasticinterme-
diateconfigurationξˆtransformsaccordingto
ξˆ∗=QξˆQT.(3.70)
AlsothesameistrueforξˆSandξˆA.Moreo˜∗ver,˜Y˜isreferredtothereferenceconfiguration
boanddysatisfiesrotationsthesuperptransformationosedonthepropplasticertyYin=Ytermediate,sothattheconfigurationinvariancedoesnotwithresprestrictectψ˜toptorigidbe
anisotropictensorfunctionofY˜.Therefore,
Y˜∂ψ˜p=∂ψ˜pY˜(3.71)
∂Y˜∂Y˜
ˆTheandξAyieldin(3.69)functionwillisnotsuppvosedanishtogenerallyexhibit,.besideskinematic,alsoorientationalhardening:
f=fPˆ,ξˆ,Π,ΠT=Π−1,(3.72)
thewhereyieldΠ,likecondition.ΦandΔ,Inequalitisany(3.30)orthogonalwillbetensor,inspwhicectedhnext,rotateswhicsomehnowsymmetrytakestheaxesformrelatedto
Dint=Pˆ−ξˆ∙Lˆp−˙ΦΦT+ξˆ∙Lˆp−˙ΦΦT−Rψ˙p

=23s˙Pˆ−ξˆ∙Nˆ+Zˆ∙Dˆp+2YˆZˆ∙Dˆp+ξˆA∙Ωˆ(e)−Rψ˙p≥0.(3.73)


3.5.KINEMATICHARDENINGANDYIELDFUNCTION

27

Becauseoftheconvexityofthelevelset{Pˆ|f(Pˆ,ξˆ,Π)=const,ξˆ=fixed,Π=fixed},itcan
beshownthatPˆ−ξˆ∙Nˆ≥0andtheinequality(3.73)issatisfied,providedtheinequality
Zˆ∙Dˆp+2YˆZˆ∙Dˆp+ξˆA∙Ωˆ(e)−Rψ˙p≥0(3.74)
holds.From(3.64),(3.65),
˜ψ˙p=∂ψ˜p∙Δ˙TYˆΔ+ΔTYˆ˙Δ+ΔTYˆΔ˙=1Zˆ∙Yˆ˙+1ξˆA∙˙ΔΔT,(3.75)
∂YRR
where˙ΔΔTdenotesaskew-symmetrictensor(theso-calledconstitutivespin).Onsubstitut-
,(3.74)in(3.75)ingZˆ∙Dˆp+2YˆZˆ∙Dˆp+ξˆA∙Ωˆ(e)−Zˆ∙Yˆ˙−ξˆA∙Δ˙ΔT≥0.(3.76)
Sufficientconditionsforthevalidityofthisinequalitycanbeestablishedappropriatelyby
prescribingtherateofYˆ(strainspaceformulation)ortherateofZˆ(stressspaceformulation).
Thisapproachisdemonstratedforthestressspaceformulation.Tothisend,ψ˜pY˜isassumed
tobegivenby
ψp=ψ˜pY˜=1Y˜∙C˜(k)Y˜,(3.77)
2RwhereC˜(k):=∂2ψ˜p(3.78)
∂Y˜∂Y˜
denotesatimeindependentsymmetricpositivedefinitefourth-ordertensor.Thelatersatisfies
ertiesproptheC˜ij(k)kl=C˜j(k)ikl=C˜ij(k)lk=C˜kl(k)ij,(3.79)
withrespecttoanorthonormalbasis{ei},andremainsunalteredifarbitraryrigidbodyrota-
tionsQaresuperposedontheplasticintermediateconfiguration:
C˜(k)→C˜(k)∗=C˜(k).(3.80)
From(3.64),(3.65)and(3.77),
ψp=ψpYˆ,Δ=1ˆY∙Cˆ(k)(Δ)Yˆ,(3.81)
2RZˆ=Cˆ(k)Yˆ,(3.82)
wherethefourth-ordertensorCˆ(k)isgivenby
2Cˆ(k):=∂ψp(3.83)
∂Yˆ∂Yˆ
andsatisfies,(k)with(k)respecttoanorthonormalbasis{ei},propertiesoftheform(3.79).Thetwo
tensorsC˜andCˆarerelatedby
Cˆ(k)Xˆ=ΔC˜(k)ΔTˆXΔΔT,(3.84)


(3.82)

28CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

(3.87)

(3.89)

Xˆbeinganarbitrarysecond-ordersymmetrictensorrelativetotheplasticintermediatecon-
figuration.Withrespecttoanorthonormalbasis{ei},(3.84)yields
Cˆij(k)mn=ΔirΔjsΔmpΔnqC˜r(k)spq.(3.85)
UnderarbitraryrigidbodyrotationsQsuperposedontheplasticintermediateconfiguration,
Δisdefinedtotransformaccordingto(cf.(3.21))
Δ→Δ∗=QΔ,(3.86)
fromwhich,byvirtueof(3.80)and(3.84)
Cˆ(k)→Cˆ(k)∗,(3.87)
withQTCˆ(k)∗QXˆQTQ=Cˆ(k)Xˆ.(3.88)
HereˆXisgivenasin(3.84)andisdefinedtotransformaccordingto
Xˆ→Xˆ∗=QˆXQT.(3.89)
LetM˜(k)beafourth-ordertensorwith
M˜(k)C˜(k)=C˜(k)M˜(k)=E.(3.90)
NoteinpassingthattheexistenceofM˜(k)impliestheexistenceofMˆ(k)with
Mˆ(k)Cˆ(k)=Cˆ(k)Mˆ(k)=E.(3.91)
Ofcourse,M˜(k)andMˆ(k)satisfypropertiesoftheform(3.79).Inaddition,˜M(k)andMˆ(k)
obeytransformationrulesoftheforms(3.80)and(3.88),respectively.
Now,inviewof(3.77),
R∂ψ˜p=C˜(k)Y˜,(3.92)
˜Y∂)(3.65)(cf.orY˜=M˜(k)R∂ψ˜p=M˜(k)ΔTˆZΔ.(3.93)
˜Y∂Usingtheaboverelationsitcanbeseen,aftersomealgebraicmanipulations,that
Zˆ∙Yˆ˙=Yˆ∙Zˆ˙−2ξˆA∙Δ˙ΔT.(3.94)
hand,othertheOnYˆ∙Zˆ˙=M˜(k)ΔTˆZΔ∙ΔTˆZΔ+2YˆZˆ∙Dˆp+ξˆA∙Wˆp,(3.95)


(3.92)

(3.94)

(3.95)

3.5.KINEMATICHARDENINGANDYIELDFUNCTION

29

(3.96)(3.97)

whereZˆ:=Zˆ˙−LˆpZˆ−ZˆLˆpT.(3.96)
Oninserting(3.95),(3.96)in(3.76),aftersomerearrangementofterms,
ΔTˆZΔ∙ΔTDˆpΔ−M˜(k)ΔTˆZΔ+ξˆA∙Ωˆ(e)−Ωˆ(k)≥0,(3.97)

wheretheskew-symmetrictensorΩˆ(k)isgivenby
Ωˆ(k):=Wˆp−˙ΔΔT.(3.98)
Δ˙ΔTistheconstitutivespinrelatedtothekinematichardeningandΩˆ(k)thecorresponding
plasticspin.From(3.86)andAppendixAitcanbededucedthat
Ωˆ(k)→Ωˆ(k)∗=QΩˆ(k)QT.(3.99)
,ClearlyΔTˆZΔ∙ΔTDˆpΔ−M˜(k)ΔTˆZΔ≥0,(3.100)
ξˆA∙Ωˆ(e)−Ωˆ(k)≥0(3.101)
aresufficientconditionsfor(3.97).SinceZˆ=QZˆQT,itfollowsthatinequalities(3.100)
and(3.101)remainunalteredifarbitraryrigidbodyrotationsaresuperposedontheplastic
intermediateconfiguration.
Inordertofulfill(3.100),ΔTDˆpΔ−M˜(k)ΔTˆZΔisassumedtobegivenby
ΔTDˆpΔ−M˜(k)ΔTˆZΔ=s˙M˜(k)B˜(k)ΔTZˆΔ,(3.102)

whereB˜(k)representsasymmetric,positivesemi-definitefourth-ordertensor,whichunderrigid
bodyrotationssuperposedontheplasticintermediateconfigurationobeysatransformationrule
(3.80).formtheof,(3.102)romFZˆ=ΔC˜(k)ΔTDˆpΔΔT−s˙ΔB˜(k)ΔTˆZΔΔT,(3.103)

orZˆ=Cˆ(k)Dˆp−s˙Bˆ(k)Zˆ.(3.104)




30CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

Here,thesymmetricpositivesemi-definitefourth-ordertensorBˆ(k)=Bˆ(k)(Δ)isdefinedby
Bˆ(k)Xˆ:=ΔB˜(k)ΔTˆXΔΔT,(3.105)
whereXˆdenotesatensorasin(3.84).
Itisreadilyshownthatunderˆarbitrary(k)rigidbodyrotationssuperposedontheplasticinter-
mediateconfiguration,thetensorBtransformsaccordingto
QTBˆ(k)QXˆQTQ=Bˆ(k)Xˆ,(3.106)
ˆAwithsimpleXobeyingsufficienthetconditiontransformationfor(3.101)(3.89).maybeconstructedbyassumingtheexistenceofa
functionχ=χˆ(ξˆA,...)≥0,(3.107)
whichisconvexwithrespecttoξˆA,remainsunalteredifrigidbodyrotationsQaresuperposed
ontheplasticintermediateconfigurationandsatisfiesthepropertyχˆ(0,...)=0.Then,the
conditionΩˆ(e)−Ωˆ(k)=s˙∂ˆχˆ(3.108)
ξ∂Aissufficientfor(3.101).
From(3.108)and(3.61),
Ωˆ(k)=23s˙NˆA−s˙∂ˆχˆ.(3.109)
ξ∂AHerehardening(3.104)andand(3.108)(3.109)indicatesrepresentthatthetheevplasticolutionspinequationsrelatedgotovtheerningelasticittheyresplawonseisofalwayskinematicequal
totheplasticspinrelatedtothekinematicalhardeningruleprovidedthetensor∂ˆχˆvanishes
identically.∂ξA

(3.108)

3.6Constitutivemodelfororthotropicanisotropy
SpinsPlastic3.6.1Beforeanyinelasticdeformationhasoccurred,thematerialissupposedtoexhibitorthotropic
anisotropyintheelasticitylaw,thekinematichardeningruleandtheyieldfunction.Wedenote
bym˜i(e),m˜i(k)andm˜i(y),i=1,2,3,thetemporarilyconstantunitvectorsinthereference
configuration,representingthethreelocalaxesofsymmetryintheelasticitylaw,thekinematic
hardeningruleandtheyieldfunction,respectively.Withevolvingplasticdeformationthe
constitutivepropertiesareassumedtoremainorthotropic,withmˆi(e),mˆi(k)andmˆi(y),i=1,2,3,
denotingthecorrespondingaxesofsymmetryintheplasticintermediateconfiguration.The
vectorsmˆi(e),mˆi(k)andmˆi(y)areassumedtoemergebyrotationfromm˜i(e),m˜i(k)andm˜i(y),
respectively,thecorrespondingrotationsbeingΦ,ΔandΠ.Inthefollowingthenotationm˜i

3.6.CONSTITUTIVEMODELFORORTHOTROPICANISOTROPY31
isadoptedforanyoneofthevectorsm˜i(e),m˜i(k)andm˜i(y)andmˆidenotesthecounterpartof
m˜iintheplasticintermediateconfiguration.Thus
mˆi=Θm˜i,(3.110)
withΘbeingeitherΦorΔorΠ,dependingonwethertheelasticitylaworthekinematic
hardeningruleortheyieldfunction(e)is(k)regarded,(y)respectively.InthesamesenseΩˆiswritten
foranyoneoftheplasticspinsΩˆ,ΩˆandΩˆ:
Ωˆ:=Wˆp−˙ΘΘT.(3.111)
Thecounterpartofm˜iintheactualconfigurationisassumedtobe
mi=Remˆi=ReΘm˜i.(3.112)
Thevectorsm˜i,mˆiandmiareusedtointroducethestructuraltensorsM˜i,MˆiandMi:
M˜i:=m˜i⊗m˜i,Mˆi:=mˆi⊗mˆi=ΘM˜iΘT,Mi:=mi⊗mi=ReMˆiReT.(3.113)
,ClearlyM˜iM˜i=M˜i,trM˜i=1(3.114)
and,becauseΘisaproperorthogonaltensor,
MˆiMˆi=Mˆi,trMˆi=1.(3.115)
ItisworthnotingthatMisatisfiespropertiesoftheform(3.114)or(3.115),thisbeingthe
motivationforthedefinition(3.113)3.Thevectorm˜iisassumedtoremainunalteredwhenever
asrigidthebodytransformationrotationsQruleareΘsup→erpΘosed∗=onQtheΘ(seeplasticinTsakmakistermediate[106]),configuration.renderM˜iThis,andasMˆiwellto
toaccordingtransformM˜i→M˜i∗=M˜i,Mˆi→Mˆi∗=QMˆiQT.(3.116)
Bytakingthematerialtimederivativeof(3.113)2andsinceM˜iareconstant,
Mˆ˙i=˙ΘΘTMˆi−Mˆi˙ΘΘT(3.117)
orMˆ˙i=WˆpMˆi−MˆiWˆp−ΩˆMˆi+MˆiΩˆ,(3.118)
orMi:=M˙i+LTMi+MiL
=DMi+MiD−(W−R˙eReT−ReWˆpReT+ReˆΩReT)Mi
+Mi(W−R˙eReT−ReˆWpReT+ReˆΩReT).(3.119)
AusefulrelationbetweenLandR˙eReTcanbederivedfrom(2.4),(2.5)and(2.8):
L=V˙eVe−1+VeR˙eReTVe−1+VeReLˆpReTVe−1(3.120)

32CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY
orR˙eReT=(Ve−1LVe)A+(V˙eVe−1)A−ReWˆpReT.(3.121)
Onsubstituting(3.121)andthedefinition
Ω:=ReˆΩReT(3.122)
findsone,(3.119)inMi=DMi+MiD−{W−(Ve−1LVe)A−(V˙eVe−1)A+Ω}Mi
+Mi{W−(Ve−1LVe)A−(V˙eVe−1)A+Ω},(3.123)
ˆwithlutionΩequationdenotinggovanerningEuleriantherespcounonseofterpartMof.theThesumplasticofspintheΩ.three(3.123)structuralrepresentensorststheevequalso-
iytitidenM˜1+M˜2+M˜3=1,(3.124)
soonlytwooutofthethreestructuraltensorsarenecessarytodescribeorthotropyandinthe
followingtheindexi=3willbedroppedandonlyM˜1,M˜2(respectivelyMˆ1,Mˆ2orM1,M2)
used.ebwillwlayElasticit3.6.2Second-orderstructuraltensorsmaybeusedtodescribeanisotropicconstitutiveproperties(see
e.g.Boehler[12],H¨ausler˜[42(e)],TLiu[70]).IfΦisassumedtoenterintotheelasticpartof
thefreeenergyintheformΦMΦ,then
ψe=ψe(Γˆe,Φ)=ψˆe(ˆΓe,Mˆ(e))(3.125)
withrespectto(3.113)2.Sinceψeisrequiredtobeunalteredunderrigidbodyrotations
superposedontheplasticintermediateconfiguration,itfollows,from(3.115)2andAppendix
A,thatψˆeisanisotropictensorfunctionofΓˆe,Mˆ1(e)andMˆ2(e).Hence,ψˆemayberepresented
byusingthetheoremsonisotropictensorfunctionsoutlinedinSpencer[95],Zheng[114].
Toderivealinearelasticitylaw,ψeisrepresented(cf.Aravas[4],aswellasSpencer[94],
Chapter6and[95])intheform
Rψˆe(Γˆe,Mˆ1(e),Mˆ2(e))=α1trΓˆe2+α2tr(Γˆe2Mˆ1(e))+α3tr(Γˆe2Mˆ2(e))
+α4{trΓˆe}2+α5{tr(ΓˆeMˆ1(e))}2+α6{tr(ΓˆeMˆ2(e))}2
+α7(trΓˆe)tr(ΓˆeMˆ1(e))+α8(trΓˆe)tr(ΓˆeMˆ2(e))+α9(trΓˆeMˆ1(e))tr(ΓˆeMˆ2(e)),(3.126)
withαi,i=1,...,9beingmaterialparameters.(3.28)2,(3.125)and(3.126)thenyield
Tˆ=R∂ψˆe(Γˆe,Mˆ1(e),Mˆ2(e))
ˆΓ∂e={2α1Γˆe+α2(ΓˆeMˆ1(e)+Mˆ(1e)Γˆe)+α3(ΓˆeMˆ2(e)+Mˆ2(e)Γˆe)
+{2α4trΓˆe+α7tr(ΓˆeMˆ1(e))+α8tr(ΓˆeMˆ2(e))}1
+{α7trΓˆe+2α5tr(ΓˆeMˆ1(e))+α9tr(ΓˆeMˆ2(e))}Mˆ1(e)
+{α8trΓˆe+α9tr(ΓˆeMˆ1(e))+2α6tr(ΓˆeMˆ2(e))}Mˆ2(e).(3.127)

3.6.CONSTITUTIVEMODELFORORTHOTROPICANISOTROPY33

(3.130)

(3.131)

Thisrepresentsalinearorthotropicelasticitylawrelativetotheplasticintermediateconfigu-
ration,Tˆ=Cˆ(e)[ˆΓe],(3.128)
or,withrespecttoanorthonormalbasissystem,
Tˆij=Cˆij(ek)l(Γˆe)kl.(3.129)
Thenotationisusuallysimplifiedbyusinganorthonormalbasissystem{ei},i=1,2,3,with
e1=mˆ1(e)ande2=mˆ2(e)beingthetwonecessaryaxestodescribeorthotropicanisotropy.With
Tˆ11Tˆ1
respecttothisbasissystemthefollowingabbreviation(Voightnotation)maybeused
Tˆ22Tˆ2
ˆˆ
Tˆij→Tˆi=ˆTˆ33→Tˆ3,(3.130)
T12T4
Tˆ23Tˆ6
Tˆ13Tˆ5
(Γˆe)11(Γˆe)1
andanalogous
(Γˆe)22(Γˆe)2
(Γˆe)33(Γˆe)3
(Γˆe)ij→(Γˆe)i=ˆ(Γˆe)12→(Γˆe)4,(3.131)
(Γˆe)23(Γˆe)6
(Γˆe)13(Γˆe)5
sothat(3.129)takestheform
Tˆi=(Cˆ(e))ij(Γˆe)j.(3.132)
Thecomponents(Cˆ(e))ijaregivenby
c11c12c13000
c12c22c23000
(Cˆ(e))ij=ˆc13c23c33000,(3.133)
55000000c044c000
00000c66
wherec11=2(α1+α2+α4+α5+α7),
c22=2(α1+α3+α4+α6+α8),
c33=2(α1+α4),
c12=2α4+α7+α8+α9,
c13=2α4+α7,
c23=2α4+α8,
c44=2α1+α2+α3,
c55=2α1+α2,
c66=2α1+α3.(3.134)

(3.132)

(3.133)

(3.134)

34CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

Toevaluatetheelasticitylawnumericallyitisnecessarytotransform(3.127)fromtheplastic
intermediatetotheactualconfiguration.ThereforetheAlmansistraintensorsinthecurrent
configurationA=1(1−FT−1F−1),(3.135)
2Ae=1(1−FeT−1Fe−1)=1(1−Be−1)=FeT−1ΓˆeFe−1,(3.136)
22Ap=A−Ae=1(FeT−1Fe−1−FT−1F−1)=FeT−1ΓˆpFe−1(3.137)
2areintroduced(cf.(2.17)).Then,observingthetransformation(3.113)3,(3.127)canberewrit-
astenS=2α1(Ve2AeVe2)+α2(Ve2AeVeM1(e)Ve+VeM1(e)VeAeVe2)
+α3(Ve2AeVeM2(e)Ve+VeM(2e)VeAeVe2)
+{2α4tr(Ve2Γˆe)+α7tr(VeΓˆeVeMˆ1(e))+α8tr(VeΓˆeVeMˆ2(e))}Ve2
+{α7tr(Ve2Γˆe)+2α5tr(VeΓˆeVeMˆ1(e))+α9tr(VeΓˆeVeMˆ2(e))}VeM1(e)Ve
+{α8tr(Ve2Γˆe)+α9tr(VeΓˆeVeMˆ1(e))+2α6tr(VeΓˆeVeMˆ2(e))}VeM2(e)Ve
=:C(e)[Ae].(3.138)

rulehardeningKinematic3.6.3Followingstepsquitesimilartothoseinthelastsectionwearriveat
ψp=ψp(Yˆ,Δ)=ψˆp(Yˆ,Mˆ1(k),Mˆ2(k))=21Yˆ∙Cˆ(k)[Yˆ],
RRψˆp(Yˆ,Mˆ1(k),Mˆ2(k))=c1trYˆ2+c2tr(Yˆ2Mˆ1(k))+c3tr(Yˆ2Mˆ2(k))
+c4{trYˆ}2+c5{tr(YˆMˆ1(k))}2+c6{tr(YˆMˆ2(k))}2
+c7(trYˆ)tr(YˆMˆ1(k))+c8(trYˆ)tr(YˆMˆ2(k))+c9(trYˆMˆ1(k))tr(YˆMˆ2(k)),
(k)(k)
Zˆ=R∂ψˆp(Yˆ,Mˆ1,Mˆ2)=Cˆ(k)[Yˆ]
ˆY∂=2c1Yˆ+c2(YˆMˆ1(k)+Mˆ1(k)Yˆ)+c3(YˆMˆ2(k)+Mˆ2(k)Yˆ)
+{2c4trYˆ+c7tr(YˆMˆ1(k))+c8tr(YˆMˆ2(k))}1
+{c7trYˆ+2c5tr(YˆMˆ1(k))+c9tr(YˆMˆ2(k))}Mˆ1(k)
+{c8trYˆ+c9tr(YˆMˆ1(k))+2c6tr(YˆMˆ2(k))}Mˆ2(k),
whereci,i=1,..(k).,9arematerialparameters.
ForthetensorBˆin(3.104)itisnaturaltoset,inanalogyto(3.141),
Bˆ(k)[Zˆ]=2b1Zˆ+b2(ˆZMˆ1(k)+Mˆ1(k)Zˆ)+b3(ZˆMˆ2(k)+Mˆ2(k)Zˆ)
+{2b4trZˆ+b7tr(ZˆMˆ1(k))+b8tr(ZˆMˆ2(k))}1
+{b7trZˆ+2b5tr(ZˆMˆ1(k))+b9tr(ZˆMˆ2(k))}Mˆ1(k)
+{b8trZˆ+b9tr(ZˆMˆ1(k))+2b6tr(ZˆMˆ2(k))}Mˆ2(k),

(3.138)

(3.139)(3.140)

(3.141)

(3.142)

(3.145)

(3.146)(3.147)(3.148)(3.149)

3.6.CONSTITUTIVEMODELFORORTHOTROPICANISOTROPY35
bi,i=1,...,9beingmaterialparameters.Ofcourse,thetensorsCˆ(k)andBˆ(k)satisfy,with
respecttoanorthonormalbasissystem{ei},i=1,2,3withe1=mˆ1(k)ande2=mˆ2(k),
propertiesoftheform(3.133),(3.134).ToobtainEuleriancounterpartsof(3.82)and(3.104),
thestresstensorZandthestraintensorYareintroduced,with
Z:=FeˆZFeT,(3.143)
Y=FeT−1YˆFe−1.(3.144)
Notethat(cf.Tsakmakis[103])
Z:=Z˙−LZ−ZLT=FeˆZFeT(3.145)
andΓˆp:=ˆ˙Γp+LˆpTΓˆp+ΓˆpLˆp=Dˆp,(3.146)
Ap:=A˙+LTAp+ApL=FeT−1DˆpFe−1,(3.147)
A:=A˙+LTA+AL=D.(3.148)
Hence,(3.141)leadsto(cf.also(3.138))
Z=C(k)[Y]:=Fe(Cˆ(k)[FeTYFe])FeT(3.149)
withC(k)[Y]=2c1Ve2YVe2+c2(Ve2YVeM1(k)Ve+VeM1(k)VeYVe2)
+c3(Ve2YVeM2(k)Ve+VeM2(k)VeYVe2)
+{2c4tr(Ve2Y)+c7tr(VeYVeM1(k))+c8tr(VeYVeM2(k))}Ve2
+{c7tr(V2eY)+2c5tr(VeYVeM1(k))+c9tr(VeYVeM2(k))}VeM1(k)Ve
+{c8tr(Ve2Y)+c9tr(VeYVeM1(k))+2c6tr(VeYVeM2(k))}VeM2(k)Ve,(3.150)
whichmaybesolvedforY:
Y=M(k)[Z],M(k)C(k)=C(k)M(k)=E.(3.151)
Eqs.(3.149),(3.150)areEuleriancounterpartsof(3.82),thecorrespondingcounterpartfor
eingb(3.104)Z=C(k)[Ap]−s˙B(k)[Z],(3.152)
whereB(k)isgivenby
B(k)[Z]=2b1Z+b2(ZVe−1M1(k)Ve+VeM1(k)Ve−1Z)
+b3(ZVe−1M2(k)Ve+VeM2(k)Ve−1Z)
+{2b4tr(Ve−2Z)+b7tr(Ve−1ZVe−1M(1k))+b8tr(Ve−1ZVe−1M2(k))}Ve2
+{b7tr(Ve−2Z)+2b5tr(Ve−1ZVe−1M1(k))+b9tr(Ve−1ZVe−1M2(k))}VeM1(k)Ve
+{b8tr(Ve−2Z)+b9tr(Ve−1ZVe−1M1(k))+2b6tr(Ve−1ZVe−1M2(k))}VeM2(k)Ve.
(3.153)

(3.152)

36CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY
Toaccomplishthekinematichardeningruleitremainstoprecise(3.109).Itisconvenientto
assumeχ=χˆ(ξˆA,Mˆ(1k),Mˆ2(k)).Thefunctionχˆisquadraticinthestressesand,onapplying
thetheoremsforisotropictensorfunctionsandtakingintoaccount(3.116),
χ=χˆ(ξˆA,Mˆ1(k),Mˆ2(k))=−l21trξˆ2A−l22tr(ξˆ2AMˆ1(k))−l23tr(ξˆA2Mˆ2(k)):=21ξˆA∙Lˆ[ξˆA],(3.154)
whereli,i=1,2,3arematerialparameters.Thefourth-ordertensorLˆisgivenby
2χˆ∂Lˆ=∂ξˆA∂ξˆA,(3.155)
∂ˆχˆ=Lˆ[ξˆA]=l1ξˆA+l22(ξˆAˆM1(k)+Mˆ1(k)ξˆA)+l23(ξˆAMˆ2(k)+Mˆ2(k)ξˆA)(3.156)
ξ∂Aandtransforms,underrigidbodyrotationsQsuperposedontheplasticintermediateconfigu-
toaccordingration,QT(Lˆ∗[QξˆAQT])Q=Lˆ[ξˆA].(3.157)
Withrespecttoanorthonormalbasissystem{ei},i=1,2,3withe1=mˆ1(k)ande2=mˆ2(k)
relationstheLˆijkl=−Lˆjikl=−Lˆijlk=Lˆklij,(3.158)
lLˆijkl=l1Iijkl+42{δikMˆ1jl−δilMˆ1jk+Mˆ1ikδjl−Mˆ1ilδjk}
+l43{δikMˆ2jl−δilMˆ2jk+Mˆ2ikδjl−Mˆ2ilδjk}(3.159)
ˆapplyandkeepinginmindthatξAisadeviator,
χ=l1+l22+l23(ξˆA)212+l1+l22(ξˆA)213+l1+l23(ξˆA)223.(3.160)
Thelatterimpliesχ≥0providedl1+l2+l3≥0,l1+l2≥0andl1+l3≥0.After
inserting(3.156)in(3.109),2222
Ωˆ(k)=Ωˆ(e)−s˙l1ξˆA+l22(ξˆAMˆ1(k)+Mˆ1(k)ξˆA)+l23(ξˆAMˆ2(k)+Mˆ2(k)ξˆA).(3.161)
Withrespecttotheactualconfiguration,(3.161)takestheform(cf.(3.122)):
Ω(k)=Ω(e)−s˙Rel1ξˆA+l2(ξˆAMˆ1(k)+Mˆ1(k)ξˆA)+l3(ξˆAMˆ2(k)+Mˆ2(k)ξˆA)ReT.(3.162)
22Itisassumedthat,inviewof(3.68),
ReξˆAReT=VeYZVe−1−Ve−1ZYVe,(3.163)
sothat(cf.Eqs.(3.113)3,(3.122))
Ω(k)=Ω(e)−s˙N(Ak),(3.164)
withN(Ak):=l1(VeYZVe−1−Ve−1ZYVe)
+l2{(VeYZVe−1−Ve−1ZYVe)M1(k)+M1(k)(VeYZVe−1−Ve−1ZYVe)}
2+l23{(VeYZVe−1−Ve−1ZYVe)M2(k)+M2(k)(VeYZVe−1−Ve−1ZYVe)}.(3.165)

3.6.CONSTITUTIVEMODELFORORTHOTROPICANISOTROPY37

(3.166)

(3.167)

3.6.4Yieldfunction–flowrule
TherotationΠin(3.72)1canbereplacedbythestructuraltensorsMˆ1(y)andMˆ(2y),usingsimilar
argumentsasinSect.3.6.2.Further,Pˆandξˆisassumedtoenterintotheyieldfunctionin
termsofaeffectivestressσˆ,
σˆ:=(Pˆ−ξˆ)D.(3.166)
Thus,from(3.72)1(cf.Aravas[4])
f=f(Pˆ,ξˆ,Π)=fˆ(σˆS,σˆA,Mˆ1(y),Mˆ2(y))(3.167)
whereσˆS=(PˆS−ξˆS)D,σˆA=PˆA−ξˆA(3.168)
denotethesymmetricandskew-symmetricpartoftheeffecivestressσˆ,respectively.Inthe
ensuinganalysis,theyieldfunctionisassumedtobequadraticinthestresses.Then,on
applyingtherepresentationtheoremsforisotropictensorfunctions(seeSpencer[95]),
f=fˆ(σˆS,σˆA,Mˆ1(y),Mˆ2(y))
(y)(y)
={v1trσˆS2+v2tr(σˆSMˆ1σˆS)+v3tr(σˆSMˆ2σˆS)
+v4(tr(σˆSMˆ1(y)))2+v5(tr(σˆSMˆ2(y)))2+v6tr(σˆSMˆ1(y))tr(σˆSˆM2(y))
+v7trσˆ2A+v8tr(σˆAMˆ1(y)σˆA)+v9tr(σˆAMˆ2(y)σˆA)
+v10tr(σˆAMˆ1(y)σˆS)+v11tr(σˆAMˆ2(y)σˆS)+v12tr(σˆAMˆ(1y)σˆSMˆ2(y))}21−k0
=:σˆ∙Kˆ[ˆσ]−k0,(3.169)
wherek0,vi,i=1,...,12arematerialparameters.Withrespecttoanorthonormalbasis
system{ei},i=1,2,3withe1=mˆ1(y)ande2=mˆ2(y),(3.169)takestheform
f=σˆmKˆmnσˆn−k0(3.170)
with

σˆ11σˆ12
σˆ13
σˆ21
σˆij→σˆm=ˆσˆ22,
σˆ23
σˆ32σˆ31
σˆ33

(3.171)

38CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

andkeepinginmindthattheeffectivestresstensorisadeviator,
K1111000K1122000K1133
0K12120K122100000
00K1313000K133100
0K12210K212100000
Kˆijkl→Kˆmn=ˆK1122000K2222000K2233,
00000K23230K23320
00K1331000K313100
K1133000K2233000K3333
00000K23320K32320
(3.172)where111111111
K1212=2v1+4v2+4v3−2v7−4v8−4v9−4v10+4v11−4v12,(3.173)
K1221=2v1+v2+v3+2v7+v8+v9,(3.174)
11111K1313=2v1+4v2−2v7−4v8−4v10,(3.175)
K1331=2v1+v2+2v7+v8,(3.176)
111111111
K2121=2v1+4v2+4v3−2v7−4v8−4v9+4v10−4v11+4v12,(3.177)
11111K2323=2v1+4v3−2v7−4v9−4v11,(3.178)
K2332=2v1+v3+2v7+v9,(3.179)
11111K3131=2v1+4v2−2v7−4v8+4v10,(3.180)
11111K3232=2v1+4v3−2v7−4v9+4v11,(3.181)
4K1111−4K1122−4K1133+K2222+2K2233+K3333
=6v1+4v2+v3+4v4+v5−2v6,(3.182)
−4K1111+10K1122−2K1133−4K2222−2K2233+2K3333
=−6v1−4v2−4v3−4v4−4v5+5v6,(3.183)
−4K1111−2K1122+10K1133+2K2222−2K2233−4K3333
=−6v1−4v2+2v3−4v4+2v5−v6,(3.184)
K1111−4K1122+2K1133+4K2222−4K2233+K3333
=6v1+v2+4v3+v4+4v5−2v6,(3.185)
2K1111−2K1122−2K1133−4K2222+10K2233−4K3333
=−6v1+2v2−4v3+2v4−4v5−v6,(3.186)
K1111+2K1122−4K1133+K2222−4K2233+4K3333
=6v1+v2+v3+v4+v5+v6.(3.187)

3.6.CONSTITUTIVEMODELFORORTHOTROPICANISOTROPY39

Forgivenv1,...,v12thisisaninhomogeneouslinearsystemconsistingof15equationswith15
unknownsK1111,...,K3333.From(3.182)-(3.187),itcanbeseenthattherankofthecoefficient
matrixKandtheaugmentedmatrixKVareequal,sothatthissystemhassolutions.

4−4−412110−2001
−410−2−4−2201−10−11
(K):=−4−2102−2−4=ˆ0001−21(3.188)
1−424−41000000
2−2−2−410−4000000
12−41−44000000
4−4−41216v1+4v2+v3+4v4+v5−2v6
−410−2−4−22−6v1−4v2−4v3−4v4−4v5+5v6
(KV):=−4−2102−2−4−6v1−4v2+2v3−4v4+2v5−v6
1−424−416v1+v2+4v3+v4+4v5−2v6
2−2−2−410−4−6v1+2v2−4v3+2v4−4v5−v6
12−41−446v1+v2+v3+v4+v5+v6
10−20012v1+v2+v4
01−10−11v1+21v6
=ˆ0001−212v1+v3+v5(3.189)
0000000
0000000
0000000

(3.189)

However,sincetherankis3,threeoutofthesixequationsarelineardependentand(3.182)-
toreduce(3.187)

K1111−2K1133+K3333=2v1+v2+v4,
1K1122−K1133−K2233+K3333=v1+2v6,

K2222−2K2233+K3333=2v1+v3+v5.

(3.190)(3.191)

(3.192)

freelyThereforeandthethreeofremainingthecocoefficienefficientsKts1111will,Kbe1122,Kdetermined1133,Kb2222y,Ksolving2233and(3.173)K3333-(3.181)can,bec(3.190)hosen-
(3.192).Ontheotherhand,ifKˆmnaregiven,thenv1,...,v12maybedetermineduniquelyby
solving(3.173)-(3.181),(3.190)-(3.192).Inordertoensuretheconvexitypropertiestheyield
functionfhastosatisfy,itmustbenotedthatthelevelset{σˆm|σˆmKˆmnσˆn≤k0}isidentical
tothelevelset{σˆm|σˆmKˆmnσˆn≤k02}.ThelaterrepresentsaconvexsetifthematrixKˆmn
ispositivesemi-definite.Byusingsomestandardalgebraicsolveritcanbeproventhatthe

40CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

eigenvaluesofKˆmnarenonnegativeprovidedthefollowinginequalitieshold:
10≤v1+v2+v4−v6,
210≤v1+v3+v5−v6,
210≤v1+v6,
2111111
0≤v1+v2+v3−v7−v8−v9
244244
1+{v210+v211+v212+2(−v10v11+v10v12−v11v12)+16(v22+v32+v82+v92)
4+32(v8v9+v2v3+v2v8+v2v9+v3v8+v3v9)+128v1v7
1+64(v12+v1v2+v1v3+v1v8+v1v9+v2v7+v3v7+v72+v7v8+v7v9)}2,
11111222
0≤v1+v3−v7−v9+{v11+16(v3+v9)+32v3v9
242441
22+64(v1+v1v3+v1v9+v3v7+v7+v7v9)+128v1v7}2,
111110≤v1+v2−v7−v8+{v210+16(v22+v82)+32v2v8
242441
22+64(v1+v1v2+v1v8+v2v7+v7+v7v8)+128v1v7}2.
Theflowrule(3.60)and(3.61)canberewrittenas
ˆs˙∂fˆˆ(e)s˙∂fˆ
Dp=,Ω=,
ζ∂PˆSζ∂PˆA
with2∂f
P∂ζ=3ˆ,
andˆ1f∂=2v1σˆS+v2(σˆSMˆ1(y)+Mˆ1(y)σˆS)+v3(σˆSMˆ2(y)+Mˆ2(y)σˆS)
∂PˆS2(f+k0)
+v10σˆAMˆ1(y)−Mˆ1(y)σˆA+v11σˆAMˆ2(y)−Mˆ2(y)σˆA
11
221+v12Mˆ2(y)σˆAMˆ1(y)−Mˆ(1y)σˆAMˆ2(y)
21−(2v2+2v4+v6)tr(Mˆ1(y)σˆS)(2v3+2v5+v6)tr(Mˆ2(y)σˆS)1
31+2v4tr(Mˆ1(y)σˆS)+v6tr(Mˆ2(y)σˆS)Mˆ1(y)
3+2v5tr(Mˆ2σˆS)+v6tr(Mˆ1σˆS)Mˆ2,
1(y)(y)(y)
3ˆ1f∂=−2v7σˆA−v8(σˆAMˆ1(y)+Mˆ1(y)σˆA)−v9(σˆAMˆ2(y)+Mˆ2(y)σˆA)
∂PˆA2(f+k0)
11+v10(σˆSMˆ1(y)−Mˆ1(y)σˆS)+v11(σˆSMˆ2(y)−Mˆ2(y)σˆS)
221+v12(Mˆ2(y)σˆSMˆ1(y)−Mˆ1(y)σˆSMˆ2(y)).
2

(3.193)(3.194)(3.195)

(3.196)(3.197)(3.198)(3.199)

(3.200)(3.201)(3.202)

3.6.CONSTITUTIVEMODELFORORTHOTROPICANISOTROPY41

Inordertorewritetheyieldfunctionwithrespecttotheactualconfigurationtheeffectivestress
tensorσanditssymmetricandskew-symmetricparts,σS,σA,respectively,aredefinedby
TTTσ:=ReσˆRe,σS=ReσˆSRe,σA=ReσˆARe.(3.203)
Using(3.66),(3.135),(3.138),(3.143),(3.144)and(3.166)
σ=Re(Pˆ−ξˆ)DReT={Ve−1(S−Z)Ve−1+2Ve(AeS−YZ)Ve−1}D.(3.204)
Asaconsequenceof(3.113)3,(3.203)2and(3.203)3,theyieldfunction(3.169)canberewritten
intheactualconfigurationasfollows:
2(y)(y)
f={v1trσS+v2tr(σSM1σS)+v3tr(σSM2σS)
+v4(tr(σSM1(y)))2+v5(tr(σSM2(y)))2+v6tr(σSM1(y))tr(σSM2(y))
(y)(y)
+v7trσ2A+v8tr(σAM1σA)+v9tr(σAM2σA)
(y)(y)(y)(y)1
+v10tr(σAM1σS)+v11tr(σAM2σS)+v12tr(σAM1σSM2)}2−k0
=:σ∙K[σ]−k0.(3.205)
Inasimilarfashionitcanbeshown,byusingamongothers(3.122),(3.147)and(3.200),that
anEulerianformatof(3.199)1,2couldbe
s˙Ap=NS(y),(3.206)
ζΩ(e)=s˙N(y),(3.207)
Aζwith

ˆNS(y):=FeT−1∂fFe−1
ˆP∂S1=Ve−12v1σS+v2(σSM1(y)+M1(y)σS)+v3(σSM2(y)+M2(y)σS)
2(f+k0)
+1v10σAM1(y)−M1(y)σA+1v11σAM2(y)−M2(y)σA
22+1v12M2(y)σAM1(y)−M1(y)σAM2(y)
2−1(2v2+2v4+v6)tr(M1(y)σS)(2v3+2v5+v6)tr(M2(y)σS)1
3+12v4tr(M1(y)σS)+v6tr(M2(y)σS)M1(y)
3+12v5tr(M(y)σS)+v6tr(M(y)σS)M(y)V−1,
e2123ˆf∂N(Ay):=ReReT
ˆP∂A=1−2v7σA−v8(σAM(y)+M(y)σA)−v9(σAM(y)+M(y)σA)
2(f+k0)1122
+1v10(σSM1(y)−M1(y)σS)+1v11(σSM2(y)−M2(y)σS)
22+1v12(M2(y)σSM1(y)−M1(y)σSM2(y)),
2

(3.208)

(3.209)

42CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY
andˆˆˆˆ
∂PS∂PS∂PA∂PA
ζ=32∂ˆf∙∂ˆf+∂ˆf∙∂ˆf=32(Ve2NS(y)Ve2∙NS(y)+N(Ay)∙N(Ay)).(3.210)
ItremainstospecifytheevolutionofM(1y)andM2(y),or,whichisthesame(cf.(3.122)),the
equationgoverningtheresponseofΩ(y).However,sincenoexperimentalevidenceisavailable
yet,thefollowingansatzwillbeused:
Ω(y)=Ω(e).(3.211)
3.7Constitutivemodelforcubicanisotropy
Inthissectionaconstitutivemodelexhibitingcubicsymmetryisproposed,whichhasbeen
derivmaterials,edfromintrotheducedconstitutivabove.emoCubicdelforsymmetryviscoplasticitrepresenytswithaspecialkinematiccaseofhardeningorthotropicforsymmetryorthotropic,
withthreeorthogonalplanesofsymmetryandadditionallythreeextraaxesofsymmetrywhich
canbetakenasrotationsthrough90◦abouttheX1,X2andX3axisoforthotropy,respectively
(cf.BillingtonandTate[11]).Afourth-ordertensorKthenmustbeinvariantunderthe
transformations100001010
P1:=001,P2:=010,P3:=−100.(3.212)
0−10−100001
Allrelationsoftheprevioussectionstillhold,withadditionalrestrictionsoutlinedinthefol-
wing.lo3.7.1Elasticitylawforcubicanisotropy
Thecomponents(Cˆ(e))ijofthetensorCˆ(e)intheelasticitylawforcubicanisotropyaregiven
by(cf.BillingtonandTate[11])
c11c12c12000
c12c11c12000
44(Cˆ(e))ij=ˆc012c012c011c00000,(3.213)
00000c44
0000c440
wherec11=c22=c33,c12=c13=c23,c44=c55=c66andthethreeremainingindependent
constantsare(cf.(3.134))
c11=2(α1+α2+α4+α5+α7)=2(α1+α3+α4+α6+α8)=2(α1+α4),
c12=2α4+α7+α8+α9=2α4+α7=2α4+α8,
c44=2α1+α2+α3=2α1+α2=2α1+α3.(3.214)
Resolving(3.214)1−3leadstotheadditionalrelations
α2=α3=0,α5=α6=−α7=−α8=α9,(3.215)

3.7.CONSTITUTIVEMODELFORCUBICANISOTROPY43

withα1,α4andα9beingthethreeindependentmaterialparametersforcubicanisotropy.
Then(3.138)canberewrittenas
S=2α1(Ve2AeVe2)
+{2α4tr(Ve2Ae)−α9tr(VeAeVeMˆ1(e))−α9tr(VeAeVeMˆ2(e))}Ve2
+{−α9tr(Ve2Ae)+2α9tr(VeAeVeMˆ1(e))+α9tr(VeAeVeMˆ2(e))}VeM1(e)Ve
+{−α9tr(Ve2Ae)+α9tr(VeAeVeMˆ1(e))+2α9tr(VeAeVeMˆ2(e))}VeM2(e)Ve
=:C(e)[Ae].(3.216)

3.7.2Kinematichardeningruleforcubicanisotropy
ThetensorsCˆ(k)andBˆ(k)satisfy,withrespecttoanorthonormalbasissystem{ei},i=1,2,3
withe1=mˆ1(k)ande2=mˆ2(k),propertiesoftheform(3.213)-(3.215),so(3.150)and(3.153)
readwnoC(k)[Y]=2c1Ve2YVe2
+{2c4tr(Ve2Y)−c9tr(VeYVeM1(k))−c9tr(VeYVeM2(k))}Ve2
+{−c9tr(Ve2Y)+2c9tr(VeYVeM1(k))+c9tr(VeYVeM2(k))}VeM1(k)Ve
+{−c9tr(Ve2Y)+c9tr(VeYVeM1(k))+2c9tr(VeYVeM2(k))}VeM2(k)Ve,
(3.217)

and

B(k)[Z]=2b1Z
+{2b4tr(Ve−2Z)−b9tr(Ve−1ZVe−1M1(k))−b9tr(Ve−1ZVe−1M2(k))}Ve2
+{−b9tr(Ve−2Z)+2b9tr(Ve−1ZVe−1M1(k))+b9tr(Ve−1ZVe−1M2(k))}VeM1(k)Ve
+{−b9tr(Ve−2Z)+b9tr(Ve−1ZVe−1M1(k))+2b9tr(Ve−1ZVe−1M2(k))}VeM2(k)Ve.
(3.218)

3.7.3Yieldfunctionandflowruleforcubicanisotropy
Bymakinguseof(3.212)1−3,thefourth-ordertensorKˆin(3.172)nowbecomes
K1111000K1122000K1122
0K12120K122100000
00K01221K01313K013130000K012210000
Kˆmnˆ=K1122000K1111000K1122,
00000K12120K12210
0000K012210000K01221K01212K0131300
K1122000K1122000K1111
(3.219)

44CHAPTER3.MODELLINGOFANISOTROPIC(VISCO-)PLASTICITY

where111111111
K1212=2v1+4v2+4v3−2v7−4v8−4v9−4v10+4v11−4v12,(3.220)
11111K1212=v1+v3−v7−v9−v11,(3.221)
44242K1212=1v1+1v2−1v7−1v8+1v10,(3.222)
44242K1221=2v1+v2+2v7+v8,(3.223)
K1221=2v1+v2+v3+2v7+v8+v9,(3.224)
K1221=2v1+v3+2v7+v9,(3.225)
111111111
K1313=2v1+4v2+4v3−2v7−4v8−4v9+4v10−4v11+4v12,(3.226)
11111K1313=v1+v2−v7−v8−v10,(3.227)
4424211111K1313=2v1+4v3−2v7−4v9+4v11,(3.228)
6K1111−6K1122=6v1+4v2+v3+4v4+v5−2v6,(3.229)
−6K1111+6K1122=−6v1−4v2−4v3−4v4−4v5+5v6,(3.230)
−6K1111+6K1122=−6v1−4v2+2v3−4v4+2v5−v6,(3.231)
6K1111−6K1122=6v1+v2+4v3+v4+4v5−2v6,(3.232)
−6K1111+6K1122=−6v1+2v2−4v3+2v4−4v5−v6,(3.233)
6K1111−6K1122=6v1+v2+v3+v4+v5+v6.(3.234)
Fromtheserelationsitcanbeseenthatcubicsymmetryimpliesthefollowingconditionsfor
parameters:materialthe,0=v2,0=v3v4=v5=v6,
,0=v8,0=v9v11=−v10,
v12=−3v10.(3.235)

(3.235)

4Chapter

FinitehardnesselemenindentsimtationulationtestofofaaBrinell
(CMSX4),single-crystalorienNi-basetedinsuperallo[001]-directiony

Thedetermineindenthetationmectesthanicalwithapropsphere,ertiesoffirstmetallicproposedbmaterials,yJ.A.whenevBrinellerthein1900,standardisoftenmethouseddsliktoe
thetension-andtorsion-testarenotfeasible.Duetothelocalrestrictionofthedeformation
alsoverysmallvolumesofmaterialcanbeexamined(cf.Dieter[13]).Basicinvestigations,
showingthatthesphericalformoftheindenterenablestheidentificationofalargepartof
thestress-strain-responsethroughthedepth-loadresponseoftheindentationtesthavebeen
performedforelastic-plasticmaterialsbyHuber[57](seealsotheliteraturecitedherein).
toTheidenaimtifyofthismaterialandfurtherparameters.materialForatestingconstitutivproeceduresmodellikewithe.g.thelinearnano-indenisotropicterishardeninggenerallythis
isandshownSteinin[76]Huberandforetal.plasticit[52y],forwithnonlinearviscoplasticityisotropicwithandnonlinearkinematichardeninghardeningininMahnkenHuber
[57],Huberetal.[53],[54],[55],[56].
Indebated.thiswork,Rather,thewexpewillensiveaddressprocedurequalitativofepropdeterminingertiesofmaterialsphericalindenparameterstationwillwhennotbtheetestingfurther
materialexhibitscubicanisotropy.

4.1Experimentalprocedure-Materialparameters

IntheBrinellhardnesstestthesurfaceofaspecimenisindentedwithaballatacertainload.
Theloadisappliedforastandardtimeandthediameteroftheindentationismeasuredwitha
lowpowermicroscopeafterremovaloftheload.Theaverageoftworeadingsofthediameterof
theimpressionatrightanglesshouldbemade.Thesurfaceonwhichtheindentationismade
shouldbesmoothandfreefromdirtorscale.TheBrinellhardnessnumber(BHN)is
asexpressed

(4.1)

PBHN=(π2)D2(1−cosφ),(4.1)
wherePistheappliedload,Dthediameteroftheballand2Φtheangleincludedbythe
indentationandthecenteroftheball.Fig.4.1showsasketchofthebasicparametersused

45

46CHAPTER4.SIMULATIONOFABRINELLHARDNESSINDENTATIONTEST

todescribetheBrinellhardnessindentationtestanditcanbeseenthatthediameterofthe
indentationdisgivenbyd=Dsinφ.
P

D

φ2

d

Figure4.1:BasicparametersinBrinelltest

TheexperimentaldataofaBrinellhardnessindentationtestofanickel-basedsingle-crystal
ofsuptheeralloyMaterials(CMSX4),ResearcthathdisplaInstituteysatcubicDarmstadtsymmetryUnivwereersityofsuppliedTecbyhnology,Dipl.-Ing.GermanK.yWinandtricareh
showninTable4.1.

appliedforceP490.35[N](50[kg])
elasticmodulusofthesteelball210[GPa]
diameterDoftheball1.25[mm]
durationoftheindentationtest10[s]
averagediameterdofindentation4.95×10−1[mm]
calculateddepthofindentation5.11×10−2[mm]
Table4.1:Parametersoftheindentingexperiment(courtesyofK.Wintrich,TU-Darmstadt)

Advancedindustrialgasturbinesmustoperatedatincreasinghightemperaturestoimprove
theefficiencyandtoenhancepoweroutput,buttheyalsohavetoretainatechnicallyuseful
strengthattheseelevatedtemperatures.Inordertomeettherequirementsforthisapplication,
turbinebladesaremanufacturedfrommonocrystallinealloys,likee.g.thenickel-basedCMSX4
superalloywithγ3-(Ni3(Al,Ti))precipitates,showingastrongcubicanisotropy.Forthequan-
titativedescriptionoftheelasticbehaviorofthisanisotropicmaterialtheelasticsinglecrystal
constantsareneeded.Thedeterminationoftheelasticmodulic11,c12andc44wasperformed
withtwoindependentmeasuringmethods(surfaceBrinellscatteringandresonantfrequency
measurementbyCominsetal.[21]andHermannetal.[48],respectively)andshowa
verygoodagreement.

4.1.EXPERIMENTALPROCEDURE-MATERIALPARAMETERS47

Cominsetal.[21]Hermannetal.[48]
elasticmodulusc11243±2[GPa]245[GPa]
elasticmodulusc12153±2[GPa]155[GPa]
elasticmodulusc44128±1[GPa]129[GPa]
Table4.2:ElasticmoduliforCMSX4atambienttemperature

Thevaluesinrow3oftable4.2werecalculatedfromtheelasticcompliancesS,measured
inHermannetal.[48]forCMSX4superalloyatambienttemperaturewiththeformulae
(cf.Dieter[13],Hermannetal.[48])
S+S1211c11=(S11−S12)(S11+S12),(4.2)
S12−c12=(S11−S12)(S11+S12),(4.3)
1c44=S44(4.4)
structure.crystalcubicafor

α1=64.0[GPa]α2=0.0[GPa]α3=0.0[GPa]
α4=57.5[GPa]α5=-38.0[GPa]α6=-38.0[GPa]
α7=38.0[GPa]α8=38.0[GPa]α9=-38.0[GPa]
b1=17.5[-]b2=0.0[-]b3=0.0[-]
b4=0.0[-]b5=0.0[-]b6=0.0[-]
b7=0.0[-]b8=0.0[-]b9=0.0[-]
c1=2.0[GPa]c2=0.0[GPa]c3=0.0[GPa]
c4=0.0[GPa]c5=0.0[GPa]c6=0.0[GPa]
c7=0.0[GPa]c8=0.0[GPa]c9=0.0[GPa]
v1=1.0[-]v2=0.0[-]v3=0.0[-]
v4=0.0[-]v5=0.0[-]v6=0.0[-]
v7=0.0[-]v8=0.0[-]v9=0.0[-]
v10=0.0[-]v11=0.0[-]v12=0.0[-]
m=4.0[-]η=3.0×104[MPams]k0=200.0[MPa]
l1=100.0[MPa−1]l2=0.0[MPa−1]l3=0.0[MPa−1]
Table4.3:Materialparametersforcubicanisotropy

Totaskanddetermineisbeytheondthematerialscopeofparametersthisgothesis.verningDevisingthesuitablehardeningexpresperimenonseistalaproveryceduresdifficultfor

48CHAPTER4.SIMULATIONOFABRINELLHARDNESSINDENTATIONTEST

oneofconstitutivtheemostmodelscandhallengingidentaskstifyinginfromtoday’stheseexpmaterialerimentsresearch.materialAgoodparametersimpressionisofprobablythe
Howdifficultiesever,togoencounanyteredfurtherisgivween,mayamongusetheothersinmaterialHuberparameters[57],cHuberhosenetinHal.¨[52ausler]-[56].[42],but
adjustedforcubicanisotropy,asshowninTable4.3.Itisemphasizedthatthesematerial
elasticitparametersymoare,duliexcept(4.2)-(4.4)forthewithelastic(3.214),oneshypwhicotheticalhhavvebalueseensothatcalculatedthefromobtainedtheresultsmeasuredcan
havequalitativemeaningonly.

4.2Comparisonofnumericalwithexperimentalresults

(4.5)

TheconstitutivetheoryhasbeenimplementedintheUMATsubroutineofthefiniteelement
codeABAQUS.Detailsofthenumericaltimeintegrationaswellasthetangentoperatorare
publishedforexampleinDiegeleetal.[35],H¨ausler[42],Jansohn[62]andareomitted
here.Fig.4.2showsthefiniteelementmeshusedforsphericalindentation,consistingof3000solid
continuumelements(C3D8)and3555nodes.Themodelhasbeenmeshedinawaythat
accountsfortheindentationprocess,leavingtheouterareasrelativelycoarseandrefiningthe
inner,stronglydeformedpartattheindentation.Theinitialaxisofanisotropyofthecubic
anisotropicconstitutivemodelarealignedalongtheglobalaxisx,yandzofthefiniteelement
model,asdepictedin4.2,withthetwoinitialstructuraltensorsbeing
100000
000000
M1:=000,M2:=010(4.5)
intheelasticitylaw(3.216),thekinematichardening(3.217),(3.218)andtheyieldfunc-
tion(3.219),respectively.
Largedeformationtheoryhasbeenusedintheanalysis,togetherwiththeassumptionofsmall
elasticstrains,whichimpliesVe≈1(seealsoH¨ausleretal.[43]).Althoughthecalcula-
tiontimeincreasessignificantly,Huber[51]hasshownthatevenforsmallindentationslarge
deformationtheorymustbeappliedinordertoobtainaconsistentresult.InFig.4.3the
finiteelementmodelisshownafterthefullindentationwithasphericalrigidbodysurface,
representingthesteelballindenter.Therigidbodysurfaceisnotshownhereandinanysub-
sequentfiguresinordertoshowthedeformedunderlyingfiniteelementmesh.Togiveabetter
impressionofthebulgingoftherimoftheindentation,Fig.4.4displaysthefiniteelement
modelinthesectionalview.Clearly,thebulgeshowstherelativelycoarsestructureofthe
finiteelementmodel,butsinceanyfurtherrefinementofthesolidmeshwouldresultinan
intolerableincreaseofcalculationtime,thechosenmeshgeometryseemstobeadequate.The
topviewoftheindentedmeshinFig.4.5letsanticipatetheanisotropicdeformation,especially
displayedintheouterrimoftheindentation.
TheresultofaBrinellhardnessindentationexperiment,performedbyK.Wintrich,isshown
inFig.4.6.Thecubicsymmetryofthetestedmaterialmanifestsitselfinthesquareformof
theindentation.Italsoshows,thatananisotropicdeformationtookplaceduringtheprocess
ofindenting.Obviouslytheindentingsteelballwasalsodeformedelasticallyanisotropic,since
theelasticmodulusoftheballissignificantlylowerthanc11oftheCSMX-4specimen.But
sincenoexperimentaldataisavailablethiswasnotfurtherconsideredintheanalysis.The
smallelasticspringback,displayedinthesectionalcutinFig.4.7depictsitsminorinfluence
ontheanisotropicbehavior.

4.2.

ARISONCOMP

Figure

4.3:

OF

NUMERICAL

Figure

Finite

4.2:

telemen

Finite

delmo

WITH

telemen

after

full

EXPERIMENTAL

model

eforeb

tation,inden

TSRESUL

tationinden

rigid

dyob

indenter

not

wnsho

49

50

CHAPTER

Figure

4.4:

Figure

4.

TIONSIMULA

Sectional

4.5:

opT

view

view

after

after

OF

full

full

A

BRINELL

tation,inden

tation,inden

HARDNESS

rigid

rigid

dyob

dyob

TIONAINDENT

terinden

terinden

not

not

wnsho

wnsho

TEST

4.2.

COMPARISONOFNUMERICALWITHEXPERIMENTALRESULTS

Figure

4.6:

Photographof(courtesy

4.7:Figure

ofK.

theWintricindenh,tationexpTU-Darmstadt)eriment,

after

the

terinden

Sectionalcutatmaximumindentation(grey)andafter

has

eenb

51

edvremo

kspringbac

52

CHAPTER

Figure

4.8:

4.

TIONSIMULA

tsEnhancemen

of

Figs.

OF

4.5

A

BRINELL

and

4.6,

HARDNESS

circles

added

for

INDENTTIONA

clarification

purp

TEST

ose

4.2.COMPARISONOFNUMERICALWITHEXPERIMENTALRESULTS

53

Generally,theconstitutivemodelshowstheexpecteddistinctdifferenceoftheindentation
fromacircularformduetoanisotropy.ThiscanbeseeninFig.4.8,whichcomparesthe
numericalresult(enhancementofFig.4.5)withtheexperiment(enhancementofFig.4.6).
Fshoorwsanclarification,indentationcirclesthathaveresembeenblesdraawnsquarearoundwiththefourindenroundtations.edges,ThewhereasBrinellthefiniteexpelemenerimentt
calculationproducedanellipticalindentation.Currentlyitisstillanopenquestion,whether
justotherdifferenmaterialtmaterialfunctions,asparametersforwexampleouldabedifferencapabletyieldtoprofunction,ducemoreshouldbesatisfyingchosenresultstoobtainorif
outcome.etterba

Chapter5

Phenomenologicalmodeltodescribe
yieldsurfaceevolutionduringplastic
wflodeformationssmallfor

TheanalysisinthischaptercorrespondsessentiallytothatonegiveninDafaliasetal.
[34]andaddressessomefeaturesofthedescriptionofsubsequentyieldsurfacesaftervarious
preloadings.Theaimofthisworkistoshowhowdeformationinducedanisotropyofthe
yieldsurfacemaybeformulatedinathermodynamicallyconsistentmanner.Thisisachieved
byestablishingsufficientconditionsforthesatisfactionoftheso-calleddissipationinequality.
Forreasonsofsimplicity,theproposedmodelisoutlinedforyieldsurfaceswhichinitially
areisotropicandtheinitialyieldsurfacemaybeapproximatedwithsufficientaccuracybya
vonMisesyieldfunction.Thisreferstoe.g.theexperimentsbyIshikawa[59],whichwill
beusedinordertodiscussthecapabilitiesofthemodel.Abriefsummaryoftheexperimental
resultsbyIshikawa[59],willbegiveninthefollowing.

5.1SubsequentYieldSurfacesofStainlessSteel

ThespecimensusedinIshikawa’s[59]investigationsaredrawingtubesoftypeAISISUS304
stainlesssteel,subjectedtosolutionheattreatment.Stresscontrolleddeformationprocesses,
ings.withaYieldingconstantwasstressdefinedratebofya4.3vonMPa/sMiseshavebeffectiveeneimpstrainosed,of50µconsistingm/m,ofwhichisaxial-torsionalsmallenoughload-
tomineddetectbyyieldpartiallysurfacesunloadinginstressthespspaceecimenbyusingfromathesingleactualspstressecimen.stateThetoyieldthecensurfacesterofarethedeter-yield
path.surface,Morethelopreciselycation,ofthewhiccenhterhasofbtheeenapprosubsequentximatedyieldduringsurfacetheinthepreloadingexperimenparttisofsimtheulatedstress
btheyimpusingosedthestressconstitutivpaths,eandmodelthegivenresultinginyieldIshikawasurfaces,andwithSasakirespect[60].totheFigs.σ-√35.1-5.7τ-coillustrateordinate
system(usedstressspace),whereσ,τaretheaxialandshearstresscomponents,respectively.
Forcircle.allspTheseecimens,approtheximatedinitialinitialyieldloyieldcusmacirclesybeareapprodenotedximatedbybrokwellenbylines,avontheMisescorrespond-yield
bingeapproradiibeingximatedtakwenellbyfromTellipses,able1asinshownIshikainwathe[59].figuresAllbydetectedsolidlines.subsequenDuringtyieldproplociortionalmay
loadingtheellipsesaretranslatedandcompressedinthedirectionoftheprestress.During
Thenon-propapproortionalximationsloadingofthethesubsequensubsequenttyieldyieldlociellipsesindicatedtranslate,inFigs.rotate5.1-5.7andarechangeconstructedtheirshapfrome.

54

5.2.PROPOSEDCONSTITUTIVEMODEL55
theexperimentaldatabyusingafittingprocedure.Notethat,becauseoftimeeffects(rate-
dependence)inthematerialbehavior,thesubsequentyieldlocidogenerallynotcontainthe
ingisprestressnotpoinpresents.t.TheInfact,experimenexistencetalofresultsisotropice.g.inFigs.hardening5.1,5.2(softening)indicatewouldthatimplyisotropicsubsequenharden-t
yieldloci,afterradialloading,whicharebroader(smaller)inthedirectionperpendicularto
thepreloadingpath.Butsuchbehaviorisnotobserved(seeFigs.5.1c,5.2c).Therefore,the
experimentalresultsinFigs.5.1-5.7areinterpretedastothatisotropichardeningisgenerally
absent,themeanvalueoftheconstantyieldstressbeingk0=194MPa.
5.2ProposedConstitutiveModel
RelationsBasic5.2.1Attentionisconfinedtosmallelastic-viscoplastic(rate-dependent)deformationsandbyEand
TthetheformlinearizedulationisstrainnottensoraffectedbandyathespaceCadepuchyendence,stressantensorexplicitaredenoted,referenceresptoectivspaceely.willSincebe
dropped.Onlyisothermaldeformationswithhomogeneoustemperaturedistributionwillbe
considered,sothatatemperaturedependencewillbedroppedaswell.Asusually,thestrain
tensorisassumedtosatisfythedecomposition
E=Ee+Ep,(5.1)
whereexistenceEeofandaspEpecificarethefreeelasticenergyψandistheassumed,plasticwithstrainaparts,corresprespondingectivelyelastic.Fandurthermore,plasticde-the
(3.18))(cf.asositioncompψ=ψe+ψp.(5.2)
Forsimplicity,ψeissupposedtobeanisotropicfunctionofEe,oftheform
1ψe=ψˆe(Ee)=2Ee∙C[Ee],(5.3)
C=2µE+λ1⊗1.(5.4)
Here,isthemassdensityandµ,λaretheelasticityparameters.LetCijklbethecomponentsof
thefourth-ordertensorCwithrespecttotheorthonormalbasis,ei.ThenChastheproperties
Cijkl=Cklij=Cjikl.(5.5)
TEq.According(5.5)1tostatesthethattheassumptionstensorCmade,isthesymmetric,secondlai.e.wCof=Cthermo.dynamics,intheformofthe
ˆClausius-Duhem-inequality,reads
ψ∂DC−D=T∙E˙−ψ˙=T−∂Eee∙E˙−ψ˙p≥0.(5.6)
Usingstandardarguments,itcanbeshownthattherelations
T=∂ψˆe=C[Ee],(5.7)
E∂eDd:=T∙E˙p−ψ˙p≥0(5.8)
aredependennecessarytand(visco-)plasticitsufficienty,conditionsconsideredforintheinequalitpresenyt(5.6)work.tobeRelationsatisfied(5.8)inistheknocasewnofasrate-the
.yinequalitdissipation

MODELPHENOMENOLOGICAL5.CHAPTER565.2.2YieldFunction-FlowRule
Ishikawa’sexperimentalresultsareinterpretedasfollows.Kinematichardeningeffectsare
wellpresenbyt,ellipses.whereasThisisotropicmeans,afterhardeningisprestressing,absent.Allthesubsequensubsequenttyieldyieldlolocicimaytranslatebeapproandximateddistort.
Forsimplicity,rotationofthesubsequentyieldˆlociisnotassumedexplicitly.Tomodelthese
phenomena,theexistenceofayieldfunctionF(T,ξ,H)isassumedintermsofafourth-order
formtheofHtensorF=Fˆ(T,ξ,H)=fˆ(T,ξ,H)−k0,(5.9)
f=fˆ(T,ξ,H):=(T−ξ)D∙H[(T−ξ)D],(5.10)
withH=H0+ϕA(5.11)
andwhereyieldoccurswhenF=0.
Intheseequations,ξistheso-calledbackstresstensorandk0isamaterialparameterrepre-
sentingconstantyieldstress.H0andAarefourth-ordertensorsmodellingdistortion,butnot
represenexplicitlytingrotationtheofinitialthevaluesubsequenofH,twhileyieldAevsurfaces.olvesInwithparticular,plasticfloHw0isfromassumeditsinitialtobezerovconstanalue,t,
representingtheanisotropicdevelopment,andsatisfieshomogeneousinitialconditions.The
scalarϕwillbediscussedsubsequently.TheexperimentsbyIshikawasuggestmodellingof
theinitialyieldsurfacebyusingthevonMisesyieldfunction.Therefore,
H0=23(E−311⊗1),(5.12)
issettobedeviatoric.Withrespecttotheorthonormalbasisei,H0exhibitstheproperties
(5.5)aswellastheproperty
(H0)iikl=0.(5.13)
compAccordinglyonents,offromAAwithijklresptheectproptotheerties(5.5),orthonormal(5.13)basisaree.requiredNoticeaswthatell,(5.10)whereAijtogetherklarewiththe
inormalityE˙∼∂f/∂T,satisfiestheincompressibilityconditiontrE˙=0withoutanyre-
strictiononpH.Butsincethereareonlyfiveindependentcomponenptsof(T−ξ)D,the21
componentsofHmustreduceto15independentones.Thisisinfactachievedbythesix
Theequationsexperimen(5.13)tsforbillustratedothH0inandFigs.A,5.5,hence,5.6forshowHthat(cf.Dtheafyieldaliaslo[25ci]).shrinkafterprestressing
andexpandafterunloadingtothestatewherethebackstresstensorisnearlyvanishing.To
incorporatesuchphenomenaintotheconstitutivemodel,ϕisassumedtobeaconstitutive
functionofξ.FortherangeofexperimentalresultsbyIshikawa,theassumption
ϕ=1+ϕ0(1−eϕ1ξ)(5.14)
seemstobeappropriateforthepurposesofthepresentwork,withϕ0andϕ1beingmaterial
ts.constanNow,inelasticflowispostulatedtooccur,ifapositiveoverstressapplies.Consequently,on
definingoverstresstobegivenbyF,inelasticflowoccursonlyifF>0.Moreover,foran
associatedflowruletheyieldfunctionservesalsoasaplasticpotential,thus
ˆf∂ˆE˙p=s˙23∂∂Tfˆ=ζs˙∂∂Tf,(5.15)
T∂

5.2.PROPOSEDCONSTITUTIVEMODEL57
∂fˆ=1(H0+ϕA)[(T−ξ)D],(5.16)
∂Tf
T∂3ζ:=2∂fˆ,(5.17)
s˙:=32E˙p∙E˙p.(5.18)
Itiscommonuseintheframeworkofunifiedviscoplasticitytoassumes˙asafunctionofthe
overstressF.Inparticular,
m˙s=F(5.19)
ηisset,wherem,ηarepositivematerialparameters,andthefunctionxisdefinedby
xifx≥0
x:=0ifx<0,(5.20)
.xrealallforRulesHardening5.2.3Considerthecasethatkinematichardeninganddistortionalhardeningarenotcoupled.This
maybetakenintoaccountbyanadditivedecompositionofψpintotwoparts,ψpkinandψpdist,
whicharerelatedtothekinematichardeningandthedistortionalhardeningeffects,respectively,
ψp=ψpkin+ψpdist.(5.21)
Duringinelasticflow,apartoftheplasticpowerT∙E˙pwillbedissipatedintoheat,whilethe
ψpkinremainderandwillψpbdistestoredrepresenintthejustthematerialenergyinformstoredofininternalthematerialstructureduetorearrangemenkinematicts.Thehardeningparts
anddistortionalhardening,respectively.FollowingChabocheetal.[19],theexistenceof
internal,second-orderstraintensorsYi,i=1,..,kisassumed,sothatψpkinisafunctionofthe
strainsYi.Forsimplicity,ψpkinisassumedtobeanisotropicfunctionoftheform
kψpkin=ψˆpkin(Yi)=ciYi∙Yi.(5.22)
2=1iTheinternalstresstensors,thermodynamicallyconjugatetothestrainsYiare
inkˆξi:=∂∂ψYp=ciYi(nosumoni).(5.23)
iAccordingtothisapproach,thebackstresstensorξisgivenbytheformula
kξ=ξi.(5.24)
=1i

58

MODELPHENOMENOLOGICAL5.CHAPTER

Inasimilarfashion,theexistenceofinternal,symmetricfourth-ordertensorsDj,j=1,..,dis
andassumed,dψpdist=ψˆpdist(Dj)=αjDj∙Dj.(5.25)
2=1jByAjthefourth-ordertensors,whicharethermodynamicallyconjugatetoDj,aredenoted
distˆψ∂Aj:=∂Dp=αjDj(nosumonj),(5.26)
jandAisdefinedby
dA=Aj.(5.27)
=1jInrelations(5.22),(5.25),ci,αjarenon-negativematerialparameters.Toaccomplishthe
hardeninglaws,evolutionequationsfortheinternalvariablesYiandDjmustbeformulated,
whichmustbecompatiblewiththedissipationinequality.
(5.8),in(5.21)-(5.27)usingAfterdkDd=T∙E˙p−ξi∙Y˙i−Aj∙D˙j
=1j=1idk=(T−ξ)∙E˙p+ξi∙(E˙p−Y˙i)−Aj∙D˙j≥0,(5.28)
=1j=1ior,byvirtueof(5.15)-(5.18),
Dd=s˙(T−ξ)D∙H0[(T−ξ)D]
ζfd+Aj∙s˙ϕ(T−ξ)D⊗(T−ξ)D
ζf=1jdk+ξi∙(E˙p−˙Yi)−Aj∙D˙j
=1j=1i=Dd(0)+Dd(kin)+Dd(dist)≥0,(5.29)

whereDd(0)=s˙(T−ξ)D∙H0[(T−ξ)D],
ζfkDd(kin)=ξi∙(E˙p−Y˙i),
=1idDd(dist)=Aj∙s˙ϕ(T−ξ)D⊗(T−ξ)D−D˙j.
ζf=1j

(5.28)

(5.29)

(5.30)(5.31)(5.32)

5.2.PROPOSEDCONSTITUTIVEMODEL59
SinceH0ispositivedefiniteforalldeviatoricsecond-ordertensors,Dd(0)≥0willbeset.
Therefore,(5.29)willalwaysbesatisfiedif
Dd(kin)≥0,Dd(dist)≥0.(5.33)
Clearlytherelations(nosumoni,j)
E˙p−Y˙i=s˙biξi,
s˙ϕ(T−ξ)D⊗(T−ξ)D−D˙j=˙sfBjAj,
ζforY˙i=E˙p−s˙biξi,(5.34)
D˙j=s˙fϕζ(T−ξ)D⊗(T−ξ)D−fBjAj(5.35)
aresufficientconditionsfor(5.33)tohold,withbi,Bjbeingnon-negativematerialparameters.
byEqs.(5.34),Chabochetogetheretal.with[19].F(5.23),ork=(5.24),1,onerepresenobtainstthethekinematicso-calledArmstrhardeninglawinong-Fredericktroduced
kinematichardeningmodel(seeArmstrongandFrederick[6]).Marquis[77]showedthat
theArmstrong-Frederickrulemaybederivedinapurelymechanicalcontextbyusinga
tmowdelo-surfaceofmoChabochedel(seeetalsoal.Daf[19]aliasfromaandso-calledPopovm[22]).ultisurfaceLater,model.TsakmakisGenerally[100,]thederivedconceptthe
ofmultisurfaceplasticityhasbeenintroducedbyMroz[81].DafaliasandPopov[22],[23]
werethefirsttointroducethetwo-surface(yieldandbounding)modelinordertodescribe
cyclicloadingprocesses.Today,alargenumberofsimilarapproachescanbefoundinthe
literature.Theequationsgoverningtheresponseofdistortionalhardeningaregivenby(5.26),(5.27)and
(5.35).Alternatively,(5.35)mayberewrittenas(nosumonj)
A˙j=z˙Θj(T−ξ)D⊗(T−ξ)D−αjBjAj,(5.36)
Θj:=ϕf2αζj,z˙:=˙sf,(5.37)
inviewof(5.26).Onassuminghomogeneousinitialconditions,Ajmaybeintegratedtoget
)jonsum(nozAj(z)=e−αjBj(z−z)Θj(z)(T(z)−ξ(z))D⊗(T(z)−ξ(z))Ddz.(5.38)
0Fromthis,itisnotdifficulttoseethateveryAjsatisfiestherequiredproperties(5.5),(5.13),
o.toAthereforeand

60

CHAPTERMODELPHENOMENOLOGICAL5.

5.3ComparisonwithExperiments-ConcludingRemarks
Figs.5.8-5.14illustrateyieldlocipredictedbytheproposedmodelfortheloadingpathsgiven
inFigs.5.1-5.7,respectively.Ishikawa’sexperimentaldataarealsodisplayedinFigs.5.8-5.14.
ThepredictedresponsesarecalculatedbyassumingξandAtoconsistoftwoparts,respectively,
i.e.ξ=ξ1+ξ2,A=A1+A2.ThematerialparametersarechosenasshowninTable5.1.

µ=7.88×104[MPa]λ=1.18×105[MPa]
m=2.25[-]η=1.50×107[MPams]
k0=1.94×102[MPa]
ϕ0=1.00[-]ϕ1=1.80×10−2[MPa−1]
α1=4.50[MPa−1]B1=1.60[MPas−1]
α2=0.80[MPa−1]B2=0.50[MPas−1]
c1=4.50×104[MPa−1]b1=3.30×10−2[-]
c2=9.00×103[MPa−1]b2=0[-]
parametersMaterial5.1:ableT

Itisemphasizedthatthesevaluesarechosenbasedontrialanderror.Asystematicidentifica-
tionofmaterialparametersbyusingestablishedoptimizationalgorithmsisbeyondthescope
ofthework.Therefore,comparisonofthepredictedresponseswiththeexperimentaldatahas
.onlymeaningequalitativFormonotonoustension,Fig.5.8bindicatesthatthetranslationoftheyieldlocus,controlled
byspacetheinbacFig.kstress5.8cξsho,iswswaellgoodpredicted.agreemenThetbetwsubsequeneentexpyielderimenlotalcus,andplacedinpredictedtheoriginresultsofforstressthe
Figs.distortion5.9b,of5.9c).theFigs.yield5.10b,surface.5.10cTherevealsamethatisalsotranslationtrueforandmonotonousdistortionoftorsionaltheyieldloadinglocus(seeare
inessencewellpredictedforthecaseofmonotonousradialloadingconditions.However,from
Figs.recognized,5.10bandwhich5.10c,arisesomefromadifferencesmissingbetexplicitweentherotationpredictedintheandexptheoreticalerimenmotaldel.resultsInthecancasebe
ofacombinedtension-torsionloadinghistory,thenumericalresultsinFigs.5.11band5.11c
showaverygoodagreementwiththeexperimentforthetensilepartA,butlessgoodforthe
followingtorsionalpartCduetotheaforementionedmissingexplicitrotationinthemodel.
forDuringthetensionuniaxialandcycliccompressionloading,thephasemodel(seepredictsFigs.the5.12b,beha5.12c,viorof5.12d),thebutyieldlesssurfacegoodveryforwtheell
reloadingtensilepart(seeFig.5.12e).Thismaybeduetothechosenvaluesofthematerial
thatparametersthepresenϕ0tandmoϕdel1,isornotduetoamplifiedthecbhosenyfurtherconstitutivinternalevfunctionariablesϕitselfdescribingorduecyclictotheloadingfact
effects.Morecomplexloadinghistories,includingtensileloadingfollowedbytorsionalloading,
aredisplayedinFigs.5.13and5.14.Itcanbeseenthattheinitialtensileloadingbehavioris
describedwell,whereasthesubsequentyieldsurfacesafterthetorsionalloadingpartarenot
verywellpredicted,whichmaybeinterpretedtobecausedbythemissingexplicitrotationin
del.moeconstitutivtheFromthisdiscussion,itcanbeconcludedthatthemodelisgenerallyabletopredictthe
experimentallymeasuredyieldloci.Fortheobserveddeviationsfromtheexperimentalresults

5.3.COMPARISONWITHEXPERIMENTS-CONCLUDINGREMARKS

61

inessencetworeasonsarelikelytoaccountfor.Ontheonehand,sincethemodelishighly
nonlinear,itisverydifficulttochoosethematerialparametersappropriately.Thismeans,one
mayassumethatothervaluesforthematerialparameters,whichwillbeidentifiedbyusing
Ontheestablishedotherhand,optimizationonlyprotranslationcedures,andmayfurnishdistortionbofettertheyieldagreementsurfacewithcanthebeexpdescriberimenedtalbydata.the
proposedconstitutivetheory.Themeasuredyieldloci,however,translate,distortandrotate,
dependingontheimposedloadinghistory.Therefore,onemayexpectimprovedpredicted
results,explicitlyif.Ttheoclarifyconstitutivthispeointtheorywillbwillethebesubjectamplifiedoftofuturemowdelork.rotationsoftheyieldsurface

MODEL

1a)PHENOMENOLOGICAL

5.

CHAPTER

1c)

1b)

cuslo

yield

tSubsequen

5.1c)

A.

at

prestressing

after

cuslo

Yield5.1b)

osed.imp

path

Loading

5.1a)

loading.

tensile

Monotonous

5.1:

Figure

spacestressthe

oforigin

origintheinplaced

placed5.1b)from

62

63

Figure

5.2:

locusMonotonousfrom5.2b)torsionalplacedinloading.theorigin5.2a)

oftheLoadingstresspathspaceimposed.

5.2b)

Yield

locus

after

prestressing

at

A.

5.2c)

Subsequent

yield

2b)

2c)

REMARKS

CONCLUDING

-

2a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

3a)PHENOMENOLOGICAL

5.

CHAPTER

3c)

3b)

cuslo

yield

tSubsequen

5.3c)

A.

at

prestressing

after

cuslo

Yield5.3b)

osed.imp

path

Loading

5.3a)

loading.

radial

Monotonous

5.3:

Figure

spacestressthe

oforigin

origintheinplaced

placed5.3b)from

64

65

Figure

5.4:

ComSubsequenbinedtyieldtension-torsionlocifrom5.4b)loadingplacedhistoryin.

the5.4a)originLoadingofthepathstressimpspaceosed.

5.4b)

Yield

loci

after

prestressing

at

A

and

C.

5.4c)

4b)

4c)REMARKS

CONCLUDING

-

4a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

5a)PHENOMENOLOGICAL

5.

CHAPTER

5c)

5b)

tSubsequen

5.5c)

D.

and

C

B,

A,

at

prestressing

after

cilo

Yield

5.5b)spaceosed.stressimpthepathoforiginLoadingthein5.5a)placedloading.5.5b)cyclicfrom

ciloUniaxial,yield

5.5:

Figure

66

67

6b)

6c)

REMARKS

CONCLUDING

-

6a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

PHENOMENOLOGICAL

5.

CHAPTER

6e)

6d)

5.6c)5.6e)

B.D.

andand

AC

atat

prestressingprestressing

afterafter

cicilolo

YieldYield5.6d)5.6b)

space.osed.spaceimppathstresstheofstresstheof
Loading5.6a).origintheinoriginthein
historyloadingplaced5.6b)5.6d)placed
fromfromtension-torsionciloyieldtciloyieldt
binedSubsequenSubsequenCom

5.6:

Figure

68

69

7b)

7c)

REMARKS

CONCLUDING

-

7a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

PHENOMENOLOGICAL

5.

CHAPTER

7e)

7d)

5.7c)5.7e)

C.G.

andand

AE

atat

prestressingprestressing

afterciloaftercilo
YieldYield5.7d)5.7b)

space.osed.spaceimppathstresstheofstresstheof
Loading5.7a).origintheinoriginthein
historyloadingplaced5.7b)5.7d)placed
fromfromtension-torsionciloyieldtciloyieldt
binedSubsequenSubsequenCom

5.7:

Figure

70

71

Figure

5.8:

afterComparisonprestressing

ComparisonprestressingofatpredictedA.5.8c)responsesSubsequenwithtexpyieldloerimencustalfromdata,b)

displaplacedyined

intheFig.origin5.8a)oftheLoadingstresspathspace

pathspaceimposed.

5.8b)

Yield

locus

8b)

8c)REMARKS

CONCLUDING

-

8a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

9a)PHENOMENOLOGICAL

5.

CHAPTER

9c)9b)

cuslo

Yield

5.9b)

osed.impspacepathstressLoadingtheof5.9a)originFig.the

inin

edyplaceddispla5.9b)data,fromtalcuserimenloyieldexptwithSubsequen

withSubsequenonsesresp5.9c)A.predictedatofprestressingComparisonafter

5.9:

Figure

72

73

Figure

5.10:

afterComparisonprestressing

ComparisonprestressingofatpredictedA.5.10c)responsesSubsequen

onsesSubsequenwithtexpyielderimenlocustal

data,fromdispla5.10b)inplacedinedyFig.
the5.10a)originofLoadingthepathstress

pathstressimpspaceosed.

osed.

5.10b)

Yield

locus

10b)

10c)

REMARKS

CONCLUDING

-

10a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

11a)PHENOMENOLOGICAL

5.

CHAPTER

11c)11b)cilo

Yield

5.11b)spaceosed.stressimpthepathofLoadingorigin

Loadingthe5.11a)inedyFig.inplaced5.11b)
displafrom

displafromcidata,lotalyieldterimenexpSubsequenwith5.11c)onsesC.respandApredictedatofprestressingComparisonafter

5.11:

Figure

74

75

12b)

12c)

REMARKS

CONCLUDING

-

12a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

PHENOMENOLOGICAL

5.

CHAPTER

12e)12d)

ciloYieldYield5.12d)5.12b)spacespace.osed.imppathstresstheofstresstheof
originLoading5.12a)theinoriginthein
placedplaced

5.12a)placedplacedFig.5.12d)in5.12b)edyfromcidispladata,fromcilolo
yieldyieldttalterimenSubsequenSubsequenexpwith5.12e)D.onsesrespB.5.12c)andC
andatApredictedatprestressingofprestressingafterComparisonciafterlo

5.12:

Figure

76

77

13b)

13c)

REMARKS

CONCLUDING

-

13a)

EXPERIMENTS

WITH

ARISONCOMP

5.3.

MODEL

PHENOMENOLOGICAL

5.

CHAPTER

13e)13d)

ciloYieldYield5.13d)5.13b)spacespace.osed.imppathofstressthestresstheof
originLoading5.13a)origintheinthein
placedplaced

5.13a)placedplacedFig.5.13d)in5.13b)edyfromcidata,displafromcilolo
yieldyieldttalterimenSubsequenSubsequenexpwith5.13e)D.onsesresp5.13c)B.Cand
andatApredictedatprestressingofprestressingafterComparisonciafterlo

5.13:

Figure

78

79

14b)

14c)

REMARKS

CONCLUDING

-

14a)

EXPERIMENTS

WITH

COMPARISON

5.3.

MODEL

PHENOMENOLOGICAL

5.

CHAPTER

14e)14d)

ciYieldloYield5.14d)5.14b)space.osed.stressimpthepathoforiginLoadingthe5.14a)inyFig.inedplaced5.14b)
displafromdata,talyieldtcilo
erimenspaceSubsequenexpwith5.14e)5.14c)G.onsesrespC.andAandEat
predictedatprestressingofComparisonprestressingafteraftercilo

5.14:

Figure

80

6Chapter

Summary

Thisconstitutivworkeismodelbasicallydescribingdividedinplastictwoparts.anisotropInytheatfirstlargeone,adeformationsthermohasdynamicallybeenadjustedconsistentot
basedpredictonthethemecmhanicalultiplicativrespeonsedecompofsingleositionofcrystalthealloysdeformationshowinggradiencubicttensorsymmetryinto.anTheelasticmodelandis
aninelasticpartandisinvariantunderarbitraryrigidbodyrotationssuperposedonboththe
actualandtheplasticintermediateconfiguration.Inordertodescribethehardeningresponse,
aninternalbackstresstensorofMandeltypeisintroducedandevolutionequationsarederived
asStructuralsufficienttensors,conditionsrepresenforthetingvloaliditcalyofaxestheofdissipationsymmetryareinequalitusedytoinevdescriberyeadmissibleanisotropicprocon-cess.
ofstitutivtheepropsymmetryerties.axes.TheThisisconstitutivpossibleebytheoryincorpallowsoratingtopredictrotationatensors,deformationfollowinginducedDafrotationalias
[26][31]andDafaliasandRashid[28],whichcorrespondtotheindependentevolutionof
anisotropyintheelasticitylaw,thekinematichardeningandtheflowrule.Fortherotation
tensorsintheelasticitylawandthekinematichardening,evolutionequationsarederivedas
sufficientconditionsforthesecondlawofthermodynamicsinformoftheClausius-Duhem
dualinequalitvyariables.The(seeHaconstitutivupteandmodelwTsakmakisasdevelop[40])edandforwlargeasformdeformationsulatedintheusingthestress-freeconceptplasticof
configuration.termediateinForsmallelasticstrains,theanisotropicplasticitymodelwasimplementedinthefiniteelement
codeABAQUSthroughtheusersubroutineUMAT(seeABAQUSStandardUser’sMan-
ualkinematic[2],24.2.30).hardeningNoandafurtherhyperelasticitsimplificationylawhaswbereeenused.made,Thebutsystemtheofcompletedifferentialexpressionequationsfor
ishassolvbeened,resolvusingedamidpthroughointanruleopanderator-splitintheprosecondcedure.opIneratorthethefirstopinelasticeratorparttheissolvelasticedpartvia
animplicit-Euler-procedure.ABAQUSrequirestheso-calledmaterialJacobianmatrix,
whichhasbeencomputednumerically,sinceananalyticalsolutionofthiswouldbeveryfault
sensitiveandinflexibletoanychangeintheconstitutiveequations.Herethedearestpartofthe
orthotropcomputationy),isforthewhichsolutionnoofreallytheefficiensystemtofalgorithmnonlinearexists.equationsTheinfollothewingsecondquotationoperator(50expressesfor
ell:wquitethis

for”Wesolvingmakeansystemsextreme,ofmorebutthanwhollyonedefensible,nonlinearstatemenequation”(t:TherePressareetnoal.goo[92d,],cgeneralhaptermetho9.6).ds

Totationtestdemonstratewassimtheulatedcapabilitiesandofcomparedthepropwithosedexperimenconstitutivtalemoresults.del,aItcouldBrinellbeshohardnesswn,thatinden-the

81

82

YSUMMAR6.CHAPTER

constitutivemodelisabletopredictthephysicalbehaviorofthematerialcorrectly,althoughan
explicitdeterminationofmaterialparameters,describingtheexperiment,wasnotdone.The
questionstillremains,whetheritispossibletoobtainbetterresultsthroughasimplechange
ofmaterialparameters,orifothermaterialfunctions,namelyadifferentyieldfunction,would
findings.realisticmoretolead

Inthetheevolutionsecondofpartofanisotropthisywinorktheayieldthermosurfacedynamicallyforpconsistenolycrystallinetmodelmaterials.waspropTheosed,modelisdescribingbased
onthederivationofsufficientconditionsfortheso-calleddissipationinequality.Attentionwas
confinedtosmallelastic-viscoplasticdeformations,forwhichtheequations,constitutingthe
materialmodel,werederived.AstheexperimentalresultsbyIshikawa[59]suggested,an
initialisotropicyieldsurfaceusingthevonMisesyieldfunctionwassupposedandisotropic
ovhardeningerstresscouldapplies.beTheneglected.hardeningInelasticruleswfloerewwformaspulatedostulatedinawtoayothatccur,whenevkinematiceraphardeningositive
anddistortionalhardeningwereconsideredseparately,whereasrotationalhardeningwasnot
involvconstitutived.emoComparisondelshowbedetwitseenthecapabilitiesexptoerimentalpredictresultsmaterialofrespIshikaonsewaw[59ell]forandthethecasepropoften-osed
sileandtorsionalloadingalone,butalsorevealeditsshortcomingsinthecaseofcombined
tensileandtorsionalloadingduetothelackofboth,asuitabledescriptionoftherotationof
theyieldsurfaceandanappropriatesetofmaterialparameters.

AendixAppTransformationsunderrigidbody
rotationssuperposedonboth,the
actualandtheplasticintermediate
configuration

Itcanbeseen(cf.CaseyandNaghdi[14],[16],GreenandNaghdi[39])thatunderrigid
bodyrotationsQ=Q(t)superposedontheactualconfiguration,andrigidbodyrotations
Q=Q(t)superposedontheplasticintermediateconfigurationsimultaneously,thefollowing
transformationsforthedeformationandstresstensorsapply.
tensor:tgradienDeformationTF→F=QF=QFeQQFp.(A.1)
tensors:deformationElasticTFe→Fe=QFeQ,(A.2)
TQe→Qe=QQeQ,(A.3)
TUˆe→Uˆe=QUˆeQ,(A.4)
TVe→Ve=QVeQ,(A.5)
TCˆe→ˆC=QCˆeQ,(A.6)
eTΓˆe→Γˆ=QˆΓeQ.(A.7)
etensors:deformationPlasticFp→Fp=QFp,(A.8)
Qp→Qp=QQp,(A.9)
Up→Up=Up,(A.10)
TVˆp→Vˆp=QVˆpQ,(A.11)
TBˆp→Bˆp∗=QBˆpQ,(A.12)
TΓˆp→Γˆp=QΓˆpQ.(A.13)
83

83

84APPENDIXA.TRANSFORMATIONSUNDERRIGIDBODYROTATIONS

Plasticvelocitygradients:

TTLˆp→Lˆ=QLˆpQ+Q˙Q,
pTTWˆp→Wˆp=QWˆpQ+Q˙Q,
TDˆp→Dˆp=QDˆpQ.
tensors:Stress

TTˆ→ˆT=QTˆQ,
TPˆ→Pˆ=QPˆQ.

(A.14)(A.15)(A.16)

(A.17)(A.18)

Thetransformationrulesunderrigidbodyrotationssuperposedonlyontheactualoronlyon
theplasticintermediateconfigurationareobtainedbysettingintherelationsaboveQ=1or
Q=1,respectively.

BendixApp

Reducedformsforthespecificfree
ψfunctionenergye

Letψebegivenby(3.20).Thenthefollowingapplies.
Theorem:Thefreeenergyψeisunalteredunderarbitraryrigidbodyrotationssuperposedonboththeactual
andtheplasticintermediateconfiguration,i.e.
ψe=ψe(Fe,Φ)=ψe(Fe,Φ),(B.1)
ifandonlyifψeobeystherepresentations
ψe=ψe(Fe,Φ)=ψeΓˆe,Φ=ψ˜eΦTΓˆeΦ.(B.2)


(B.4)

of:ProFirstitcanbeshownthat(B.1)implies(B.2).TherelationsinAppendixAand(3.21)willbe
obtaintousedψe=ψe(Fe,Φ)=ψeQFeQT,QΦ.(B.3)
Here,Q=QReTisset,sothat
ψe=ψeQReTReUˆeQT,QΦ=ψeQUˆeQT,QΦ.(B.4)
ForQ=1itcanbeshown
ψe=ψeCˆe,Φ=:ψeΓˆe,Φ,(B.5)
whichconfirms(B.2)2.Ontheotherhand,choosingQ=ΦTinEq.(B.4),then
ψe=ψeΦTUˆeΦ,1.(B.6)
thatNote2ΦTUˆeΦ=ΦTCˆeΦ=2ΦTΓˆeΦ+1,(B.7)
85

(B.5)(B.6)

(B.7)

86APPENDIXB.REDUCEDFORMSFORTHESPECIFICFREEENERGYψE

whichindicatesthatΦTUˆeΦmaybeTexpressedintermsofΦTΓˆeΦ.Thus,followingfrom(B.6),
ψecanberecastedasafunctionofΦΓˆeΦ,whichimpliestherepresentation(B.2)3.Inorder
toproofthat(B.2)leadsto(B.1),itshouldbeobservedthat(B.2)implies

ψe(Fe,Φ)=ψ˜eΦTΓˆeΦ,(B.8)
or,byvirtueofthepropertyΦTΓˆeΦ=ΦTΓˆeΦ,followingfromtherelationsinAppendixA
,(3.21)and

ψe(Fe,Φ)=ψ˜eΦTΓˆeΦ.(B.9)
Inviewof(B.2),thelastequationtakestheform(B.1),whichcompletestheproofofthe
theorem.

yBibliograph

[1]ABAQUS,version6.3,2003.

[2]ABAQUSStandardUser’sManual,version6.3,2003.

[3]N.Aravas,E.C.Aifantis.Onthegeometryslipandspininfiniteplasticdeformation.
InternationalJournalofPlasticity,7:141–160,1991.

[4]N.tionalAravas.JournalFiniteofSolidselastoplasticandStructures,transformations29:2137–2157,oftransv1992.ersallyisotropicmetals.Interna-

[5]N.Aravas.Anisotropicplasticityandtheplasticspin.ModellingSimulationinMaterial
ScienceandEngineering,2:483–504,1994.

[6]P.BauscJ.hingerArmstrong,effect.C.GenerO.alFEleredericctrick.AGeneratingmathematicalBoard,reprepresenortRD/B/Ntationof731,the1966.multiaxial

[7]R.CrystalsJ.Asaro.andPAdvancesolycrystals,inpp.Applied1-115.MecAcademichanics,VPress,olume:san23,Diego,Chapter:1983.Micromechanicsof

[8]R.J.Asaro,J.R.Rice.Strainlocalizationinductilesinglecrystals.JournaloftheMe-
chanicsandPhysicsofSolids,6:309–338,1977.

[9]G.Backhaus.ZurFließgrenzebeiallgemeinerVerfestigung.ZAMM,48:99–108,1968.

[10]A.Baltov,A.Sawczuk.Aruleofanisotropichardening.ActaMechanica,1:81–92,1964.

[11]E.W.Billington,A.Tate.ThePhysicsofDeformationandFlow,McGraw-Hill,NewYork,
1981.

[12]J.P.Boehler.Representationsforisotropicandanisotropicnon-polynomialtensorfunc-
tions.InApplicationsoftensorfunctionsinsolidmechanics,J.P.Boehler.CISMcourses
andlectures,No292,31–53,Springer,Wien–NewYork,1987.

[13]G.E.Dieter.MechanicalMetallurgy,SImetericedition,McGraw-Hill,London,1988.

[14]J.Casey,P.Naghdi.AremarkontheuseofthedecompositionF=FeFpinplasticity.
JournalofAppliedMechanics,47:672–675,1980.

[15]J.Casey,P.Naghdi.Onthecharacterizationofstrain-hardeninginplasticity.Journalof
AppliedMechanics,48:285–295,1981.

[16]J.tationalCasey,P.significance.Naghdi.AJournalcorrectofAppliedefinitiondofMeelasticchanics,and48:983–985,plastic1981.deformationanditscompu-

87

88

BIBLIOGRAPHY

[17]J.Casey,M.Tseng.Aconstitutiverestrictionrelatedtoconvexityofyieldsurfacesin
plasticity.Zeitschriftf¨urangewandteMathematikundPhysik,35:478–496,1984.
[18]B.D.Coleman,M.E.Gurtin.Thermodynamicswithinternalstatevariables.Journalof
ChemistryinPhysics,47:597–613,1967.
[19]J.L.Chaboche,K.Dang-Van,G.Cordier.Modelizationofthestrainmemoryeffecton
thecyclichardeningof316stainlesssteel.SMIRT-5,DivisionL,Berlin,1979.
[20]H.Cho,Y.F.Dafalias.Distortionalandorientationalhardeningatlargeviscoplasticde-
formations.InternationalJournalofPlasticity,12:903–925,1996.
[21]J.D.Comins,A.G.Every,P.R.Stoddart,W.Wang,X.Zang.NDEofsolidsurfaces
andthinsurfacecoatingsbymeansofsufacebrillouinscatteringoflight.PresentedatThe
NDE(NonDestructiveEvaluation)workshop,CapeTown,April2002.
[22]Y.F.Dafalias,E.P.Popov.Amodelofnonlinearlyhardeningmaterialsforcomplex
loading.ActaMechanica,21:173–192,1975.
[23]Y.F.Dafalias,E.P.Popov.Plasticinternalvariablesformalismofcyclicplasticity.Journal
ofAppliedMechanics,98:645–651,1976.
[24]Y.F.Dafalias.Il’iushin’spostulateandresultingthermodynamicconditionsonelastic-
plasticcoupling.InternationalJournalofSolidsandStructure,13:239–251,1977.
[25]Y.F.Dafalias.Anisotropichardeningofinitiallyorthotropicmaterials.ZAMM,59:437–
1979.446,[26]Y.F.Dafalias.Theplasticspinconceptandasimpleillustrationofitsroleinfiniteplastic
transformations.MechanicsofMaterials,3:223–233,1984.
[27]Y.F.Dafalias.Issuesontheconstitutiveformulationatlargeelastoplasticdeformations.
PartI:Kinematics.ArchivesofMechanics,69:119–138,1987.
[28]Y.F.Dafalias,M.M.Rashid.Theeffectofplasticspinonanisotropicmaterialbehaviour.
InternationalJournalofPlasticity,5:227–246,1989.
[29]Y.F.Dafalias.Theplasticspininviscoplasticity.InternationalJournalofSolidsand
1990.26(2):149–163,,esStructur[30]Y.F.Dafalias,E.C.Aifantis.Onthemicroscopicoriginoftheplasticspin.Archivesof
1990.82:31–48,,chanicsMe[31]Y.F.Dafalias.Onmultiplespinsandtexturedevelopement.Casestudy:Kinematicand
orthotropichardening.ArchivesofMechanics,100:171–194,1993.
[32]Y.F.Dafalias.Plasticspin:Necessityorredundancy.InternationalJournalofPlasticity,
1998.14:909–931,[33]Y.F.Dafalias.Orientationalevolutionofplasticorthotropyinsheetmetals.Journalof
MechanicsandPhysicsofSolids,48:2231–2255,2000.
[34]Y.F.Dafalias,D.Schick,Ch.Tsakmakis.Asimplemodelfordescribingyieldsurface
evolutionduringplasticflow.InDeformationandfailureinmetallicmaterials,Eds.K.
HutterandH.Baaser,Spinger,Berlin,169–201,2003.

BIBLIOGRAPHY

89

[35]E.Diegele,W.Jansohn,Ch.Tsakmakis.Finitedeformationplasticityandviscoplastic-
itylawsexhibitingnonlinearhardeningrules;PartI:Constitutivetheoryandnumerical
integration.ComputationalMechanics,25:1–12,2000.

[36]R.Fosdick,E.Volkmann.Normalityandconvexityoftheyieldsurfaceinnon-linearplas-
ticity.QuarterlyJournalofAppliedMathematics,51:117–127,1993.

[37]E.v.d.Giessen.Continuummodelsforlargedeformationplasticity.PartI:Largedeforma-
tionplasticityandtheconceptofanaturalreferencestate.EuropeanJournalofMechanics
1989.8:15–34,,A/Solids

[38]E.v.d.Giessen.Continuummodelsforlargedeformationplasticity.PartII:Akinematic
hardeningmodelandtheconceptofaplasticallyinducedorientationtensor.European
JournalofMechanicsA/Solids,8:89–108,1989.

[39]A.E.InternationalGreen,P.JournalNaghdi.ofSomeEngineeringremarksonSciencthee,9:1219–1229,elastic-plastic1971.deformationsatfinitestrains.

[40]P.Haupt,Ch.Tsakmakis.Ontheapplicationofdualvariablesincontinuummechanics.
ContinuumMechanicsandThermodynamics,1:165–196,1989.

[41]Pon.Haupt.ContinuumFMeoundationchanicsofConintinEnviruumonmentMechanics.SciencInesIUTandAMGeophysicsInternational.Udine,SummerJune1992.School

[42]O.6351,H¨Fausler.orschungszenAnisotroptrumesplastiscKarlsruhehesGmFließenbH,beiInstitutgroßenf¨urDeformationen.Materialforschung,Ph.D1999.thesis.FZKA

[43]O.H¨ausler,D.SchickandCh.Tsakmakis.Descriptionofplasticanisotropyeffectsatlarge
deformations.PartII:Thecaseoftransverseisotropy.InternationalJournalofPlasticity,
2004.20:199–223,

[44]S.S.Hecker.Yieldsurfacesinprestrainedaluminumandcopper.MetallurgicalTransac-
1971.2:2077–2086,,tions

[45]S.aluminS.Hecumker.andcoppInfluenceer.ofMetallurdeformationgicalTrhistoryansactionson,yieldlo4:985–989,cusand1973.stress-strainbehaviourof

[46]D.E.Helling,A.K.Miller,M.G.Stout.Anexperimentalinvestigationoftheyieldloci
of1100-0aluminum,70:30brass,andanoveraged2024aluminumalloyaftervarious
prestrains.JournalofEngineeringMaterialsandTechnology,ASME,108:313–320,1986.

[47]G.A.Henshall,D.E.Helling,A.K.Miller.ImprovementsintheMATMODequationsfor
modelingsoluteeffectsandyield-surfacedistortion.InUnifiedConstitutivelawsofplastic
deformation,Eds.A.S.Krausz,K.Krausz,AcademicPress,NewYork,153–227,1996.

[48]W.Hermann,H.G.Sockel,J.Han,A.Bertram.Elasticpropertiesanddeterminationof
Eds.elasticR.D.constantsKissinger,ofnickD.J.el-baseDeye,supD.eralloL.ysAnbyton,aA.freeD.beamCetel,tecM.V.hnique.Nathal,InSupT.eralM.loysPollo1996ck,,
D.A.Woodford,TheMinerals,Metals&MaterialsSociety,1996.

[49]R.Hill.Onconstitutiveinequalitiesforsimplematerials-II.JournaloftheMechanicsand
PhysicsofSolids,16:315–322,1968.

90

BIBLIOGRAPHY

[50]R.Hill,J.R.Rice.Elasticpotentialsandthestructureofinelasticconstitutivelaws.SIAM,
1973.25:448–461,[51]N.Ph.DHuber.thesis.ZurFZKABestimm5850,ungFvorsconhmecungszenhanisctrumhenEigenscKarlsruhehaftenGmbH,mitdemInstitutEindrucf¨urkvMaterial-ersuch.
1996.ung,hforsc[52]N.denHubtation.er,D.JournalMunz,ofCh.MaterialsTsakmakis.Research,Determination12:2459–2469,ofY1997.oung’smodulusbysphericalin-

[53]N.Huber,Ch.Tsakmakis.Determinationofconstitutivepropertiesfromsphericalindenta-
tiondatausingneuralnetworks.PartI:Thecaseofpurekinematichardeninginplasticity
laws.JournaloftheMechanicsandPhysicsofSolids,47:1569–1588,1999.

[54]N.Huber,Ch.Tsakmakis.Determinationofconstitutivepropertiesfromsphericalindenta-
tiondatausingneuralnetworks.PartII:Plasticitywithnonlinearisotropicandkinematic
hardening.JournaloftheMechanicsandPhysicsofSolids,47:1589–1607,1999.

[55]N.Huber,A.Konstantinidis,Ch.Tsakmakis.DeterminationofPoisson’sratiobyspherical
indentationusingneuralnetworks.PartI:Theory.JournalofAppliedMechanics,68:218–
2001.223,[56]N.Huber,Ch.Tsakmakis.DeterminationofPoisson’sratiobysphericalindentationusing
neuralnetworks.PartII:Identificationmethod.JournalofAppliedMechanics,ASME,68:
2001.224–229,[57]N.HubHabilitationsscer.Anwhrift.endungenFZKA6504,NeuronalerForschNetzeungszenbeitrumnichtlinearenKarlsruheGmProblemenbH,InstitutderfMec¨urhanik.Mate-
2000.ung,hrialforsc[58]K.Ikegami.Experimentalplasticityontheanisotropyofmetals.InProceedingsofthe
EuromechColloquium115,Ed.J.P.Boehler,´EditionsduCentreNationaldelaRecherche
Scientifique,Paris,201–227,1982.

[59]H.Ishikawa.Subsequentyieldsurfaceprobedfromitscurrentcenter.InternationalJournal
1997.13:533–549,Plasticityof

[60]H.Ishikawa,K.Sasaki.YieldsurfaceofSUS304undercyclicloading.JournalofEngineer-
ingMaterialsandTechnology,ASME,110:364–371,1988.
[61]H.Ishikawa,K.Sasaki.Deformationinducedanisotropyandmemorizedbackstressin
constitutivemodel.InternationalJournalofPlasticity,14:627–646,1998.
[62]W.DeformationenJansohn.FinormderulierungundThermoplastizit¨Inattegrationundvon-viskoplastizit¨Stoffgesetzenat.Ph.DzurBescthesis.hreibungFZKAgroßer6002,
ForschungszentrumKarlsruheGmbH,Institutf¨urMaterialforschung,1997.
[63]A.S.Khan,X.Wang.Anexperimentalstudyonsubsequentyieldsurfaceafterfiniteshear
prestraining.InternationalJournalofPlasticity,9:889–905,1993.
[64]A.S.Khan,S.Huang.Continuumtheoryofplasticity.Wiley,NewYork,1995.
´[65]loZ.L.w-alloKoywsteel.alewski,M.InternationalSliwowski.JournalEffectofofMecyclicchanicalloadingScienconesyield,39:51–68,surfaceev1997.olutionof18G2A

BIBLIOGRAPHY

91

[66]J.Kratochvil,O.W.Dillonjr.Thermodynamicsofelastic-plasticmaterialsasatheory
withinternalstatevariables.JournalofAppliedPhysics,40:3207–3218,1969.
[67]E.Krempl.Modelsofviscoplasticity-somecommentsonequilibrium(back)stressand
dragstress.ActaMechanica,69:25–42,1987.
[68]E.H.Lee,D.T.Liu.Finitestrainelastic-plastictheoryparticularyforwaveanalysis.
JournalofAppliedPhysics,38:19,1967.
[69]H.C.Lin,M.P.Naghdi.Necessaryandsufficientconditionsforthevalidityofaworkin-
equalityinfiniteplasticity.TheQuarterlyJournalofMechanicsandAppliedMathematics,
1989.42:13–21,[70]I.-S.Liu.Onrepresentationsofanisotropicinvariants.InternationalJournalofEngineering
1982.20:1099–1109,,esScienc[71]B.Loret.Ontheeffectofplasticrotationinthefiniteelementdeformationofanisotropic
elastoplasticmaterials.MechanicsofMaterials,2:287–304,1983.
[72]B.Loret,Y.F.Dafalias.Theeffectofanisotropyandplasticspinonfoldformations.
JournaloftheMechanicsandPhysicsofSolids,40:417–439,1992.
[73]J.Lubliner.Normalityrulesinlarge-deformationplasticity.MechanicsofMaterials,5:
1986.29–34,[74]J.Lubliner.Large-deformationplasticity.InPlasticityTheory,chapter8,438–470,Macmil-
lanPublishingCompany,NewYork,1990.
[75]M.Lucchesi,M.Silhavy.Il’iushin’sconditionsinnon-isothermalplasticity.Archivesof
RationalMechanicsandAnalysis,113:121–163,1991.
[76]R.Mahnken,E.Stein.Theidentificationofparametersforvisco-plasticmodelsviafinite-
elementmethodsandgradientmethods.ModellingandSimulationinMaterialScienceand
1994.2:597–616,,eringEngine[77]D.Marquis.Mod´elisationetidentificationdel’´ecrouissageanisotropedesm´etaux.General
ElectricGeneratingBoard,reportRD/B/N731,1979.
[78]G.Maugin.InTheThermodynamicsofPlasticityandFracture,chapter5,Cambridge
UniversityPress,NewYork,1992.
[79]J.Miastkowski.Analysisofthememoryeffectofplasticallyprestrainedmaterial.Archiwum
MechanikiStosowanej,20:257–276,1968.
[80]J.Miastkowski,W.Szczepi´nski.Anexperimentalstudyofyieldsurfacesofprestrained
brass.InternationalJournalofSolidsandStructures,1:189–194,1965.
[81]Z.Mroz.Onthedescriptionofanisotropicworkhardening.JournaloftheMechanicsand
PhysicsofSolids,15:163ff,1967.
[82]P.M.Naghdi.Stress-strainrelationsinplasticityandthermoplasticity.InProc.Second
Symp.NavalStructuralMechanics,Eds.E.H.LeeandP.S.Symonds.Pergamon,Oxford,
1960.121–169,

92

BIBLIOGRAPHY

[83]ofP.M.yieldNaghdi,surfacesJ.inA.Tplasticitrapp.y.OntheJournalnatureofofAppliednormalitMeychanicsof,plastic42,E,strain1:61–66,rateand1975.convexity

[84]J.Ning,E.C.Aifantis.Anisotropicyieldandplasticflowofpolycrystallinesolids.Inter-
nationalJournalofPlasticity,12:1221–1240,1996.

[85]R.W.Ogden.Non-linearelasticdeformation.EllisHarwoodLtd.,Chichester,1984.
[86]W.Oliferuk,W.´Swi¸atnicki,M.Grabski.Rateofenergystorageandmicrostructureevolu-
tionA161:55–63,duringthe1993.tensiledeformationofausteniticsteel.MaterialsScienceandEngineering,

[87]P.Perzyna.Theconstitutiveequationsforratesensitiveplasticmaterials.Quarterlyof
AppliedMathematics,20:321–332,1963.

[88]A.Phillips.Thefoundationsofplasticity.Experiments.Theoryandselectedapplications,
1979.189–271,Udine,,CISM

[89]A.Phillips,P.K.Das.Yieldsurfacesandloadingsurfacesofaluminumandbrass:an
experimentalinvestigationatroomandelevatedtemperatures.InternationalJournalof
1985.1:89–109,,Plasticity

[90]A.Phillips,H.Moon.Anexperimentalinvestigationconcerningyieldsurfacesandloading
surfaces.ActaMechanica,27:91–102,1977.

[91]A.Phillips,J.L.Tang.Theeffectofloadingpathontheyieldsurfaceatelevatedtemper-
atures.InternationalJournalofSolidsandStructures,8:463–474,1972.
[92]W.fortran,H.Press,secondS.A.edition.TeukCamolsky,bridgeW.T.UnivVersityetterling,Press,B.P1992..Flannery.Numericalrecipesin

[93]D.W.A.Rees.Yieldfunctionsthataccountfortheeffectsofinitialandsubsequentplastic
anisotropy.ActaMechanica,43:223–241,1982.
[94]A.J.M.Spencer.Deformationoffibre-reinforcedmaterials.OxfordPress,Clarendon,
1972.

[95]A.J.M.Spencer.Isotropicpolynomialinvariantsandtensorfunctions.Applicationsof
tensorfunctionsinsolidmechanics,J.P.Boehler.CISMcoursesandlectures,No292,
31–53,Springer,Wien–NewYork,1987.
[96]A.R.Srinivasa.Onthenatureoftheresponsefunctionsinrate-independentplasticity.
InternationalJournalofNon-LinearMechanics,32:103–119,1997.

[97]M.G.Stout,P.L.Martin,D.E.Helling,G.R.Canova.Multiaxialyieldbehaviourof1100
aluminumfollowingvariousmagnitudesofprestrain.InternationalJournalofPlasticity,
1985.1:163–174,[98]W.Trampczynski.Theexperimentalverificationoftheunloadingtechniquefortheyield
surfacedetermination.ArchivesofMechanics,44:171–190,1992.
[99]vC.TolumeIruesdell,II/3,W.Ed.S.Noll.Fl¨Theugge.nonlinearSpringer,fieldBerlin–Heidelbtheoriesinmecerg–Newhanics.YInork,1965.HandbuchderPhysik,

BIBLIOGRAPHY

93

[100]Ch.Tsakmakis.¨UberinkrementelleMaterialgleichungenzurBeschreibunggroßerin-
elastischerDeformationen.Ph.Dthesis.TechnischeHochschuleDarmstadt,Institutf¨ur
1987.hanik,Mec

[101]Ch.Tsakmakis.Ontheloadingconditionsandthedecompositionofdeformation.In
Anisotropyandlocalizationofplasticdeformation,Eds.J.-P.Boehler,A.S.Khan335–
356.ElsevierAppliedScience,Springer,London–NewYork,1991.
[102]Ch.Tsakmakis.Formulationofviscoplasticitylawsusingoverstress.ActaMechanica,115:
1996.179–202,[103]Ch.Tsakmakis.Kinematichardeningrulesinfiniteplasticity.PartI:Aconstitutiveap-
proach.ContinuumMechanicsandThermodynamics,8:215–231,1996.
[104]Ch.Tsakmakis.RemarksonIl’iushin’spostulate.ArchivesofMechanics,49:677–695,
1997.

[105]Ch.deformations.Tsakmakis,A.InternationalWilluweit.JournalAofcomparativNon-LineestudyarMeofchanicskinematic,inpress.hardeningrulesatfinite

[106]Ch.Tsakmakis.Descriptionofplasticanisotropyeffectsatlargedeformations.PartI:
RestrictionsimposedbythesecondlawandthepostulateofIl’iushin.InternationalJournal
2004.20:167–198,,Plasticityof

[107]P.Tuˇgcu,K.W.Neale.Ontheimplementationofanisotropicyieldfunctionsintofinite
strainproblemsofsheetmetalforming.InternationalJournalofPlasticity,15:1021–1040,
1999.

[108]P.Tuˇgcu,P.D.Wu,K.W.Neale.Finitestrainalalysisofsimpleshearusingrecent
anisotropicyieldcriteria.InternationalJournalofPlasticity,15:939–962,1999.

[109]K.Wegener,M.Schlegel.Suitabilityofyieldfunctionsfortheapproximationofsubse-
quentyieldsurfaces.InternationalJournalofPlasticity,12:1151–1177,1996.

[110]J.F.Williams,N.L.Svensson.Locusof1100-Faluminum.JournalofStrainAnalysis,
1970.5:128–139,[111]J.F.Williams,N.L.Svensson.Effectoftorsionalprestrainontheyieldlocusof1100-f
aluminum.JournalofStrainAnalysis,6:263–272,1971.
[112]H.C.Wu,H.K.Hong,J.K.Lu.Anendochronictheoryaccountedfordeformation
inducedanisotropy.InternationalJournalofPlasticity,11:145–162,1995.
[113]Y.strainYoshimhistoryura..AerHyponauticotheticalalResetheoryarchofInstitute,anisotropyandUniversitytheofBauscTokyohinger,repeffectortNo.dueto349,plastic221-
1959.247,[114]Q.-S.Zheng.Theoryofrepresentationsfortensorfunctions–Aunifiedinvariantapproach
toconstitutiveequations.AppliedMechanicsReview,ASME,47:221-247,1994.

94

BIBLIOGRAPHY

Geburtstag:Geburtsort:

hSculbildung:

1980–197619801990–05/1990

enslaufLeb

khicScvidDa51aDellenBrokstedt24616

22.01.1970hlandDeutschenhausen,Ic

GrundschuleUnterkn¨oringen
Dossenberger-GymnasiumG¨unzburg
Abitur

AkademischerundberuflicherWerdegang:

1993–199019931999–05/1999

2000–1999

2003–05/2000

06/2003seit

StudiumderPhysikanderUniversit¨atAugsburg
AbscStudiumhluss:derPhysikDiplom-IngenieuranderTfec¨urhniscPhysikhenanUnivderersit¨TecathniscDarmstadthen
Darmstadtatersit¨UnivSimDiplomarbulationveit:on”Entkurzfaservwicklungerst¨vonaktenMaterialmoThermoplasten”.dellenzur
DeutscBetreuer:hesDr.-Ing.Kunststoffinstitut,T.PflammDarmstadt

MitarbeiterderFa.TecosimGmbH,R¨usselsheim

MecWissenschanik,AhaftlicGherKontinMitarbuumsmeceiteramhanikInstitutf¨ur(Materialtheorie),
TecBetreuer:hnischeProf.Universit¨Dr.-Ing.atCh.Darmstadt.Tsakmakis.

MitarbeiterderFa.MenckGmbH,Kaltenkirchen

Hiermit

hereersicv

ich,

Dissertationorliegendev

andigselbst¨

und

urn

dass

onv

mit

Hilfsmittelnenenangegeb

fertigt

wurde.

Darmstadt,

den

13.

uarJan

2004

die

mir

den

ange-

..................................................

(Dipl.-Ing.

vidDa

Sck)hic

  • Accueil Accueil
  • Univers Univers
  • Livres Livres
  • Livres audio Livres audio
  • Presse Presse
  • BD BD
  • Documents Documents