Strain localization analysis for planar polycrystals based on bifurcation theory

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Strain localization analysis for planar polycrystals based

on bifurcation theory

Mohamed Ben Bettaieb, Farid Abed-Meraim

To cite this version:

Mohamed Ben Bettaieb, Farid Abed-Meraim. Strain localization analysis for planar polycrystals

based on bifurcation theory. Comptes Rendus Mécanique, Elsevier Masson, 2018, 346 (8), pp.647-664.

�10.1016/j.crme.2018.06.006�. �hal-01882210�

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Mohamed BEN BETTAIEB, Farid ABED-MERAIM - Numerical investigation of localized necking

of planar polycrystals - Comptes Rendus de Mécanique - Vol. 346, p.647-664 - 2018

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Comptes Rendus Mecanique

Computational modeling of material forming processes / Simulation numérique des procédés de mise en forme

Strain localization analysis for planar polycrystals based on

bifurcation theory

Mohamed Ben Bettaieb

a

,

b

_{, Farid Abed-Meraim}

a

,

b

,

∗

a _{Arts et Métiers ParisTech, Université de Lorraine, CNRS, LEM3, 57000 Metz, France}

b _{DAMAS, Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures, Université de Lorraine, France}

a b s t r a c t

In the present paper, an efficient numerical tool is developed to investigate the ductility limit of polycrystalline aggregates under in-plane biaxial loading. These aggregates are assumed to be representative of very thin sheet metals (with typically few grains through the thickness). Therefore, the plane-stress assumption is naturally adopted to numerically predict the occurrence of strain localization. Furthermore, the initial crystallographic texture is assumed to be planar. Considering the latter assumptions, a two-dimensional single-crystal model is advantageously chosen to describe the mechanical behavior at the microscopic scale. The mechanical behavior of the planar polycrystalline aggregate is derived from that of single crystals by using the full-constraint Taylor scale-transition scheme. To predict the occurrence of localized necking, the developed multiscale model is coupled with bifurcation theory. As will be demonstrated through various numerical results, in the case of biaxial loading under plane-stress conditions, the planar single- crystal model provides the same predictions as those given by the more commonly used three-dimensional single-crystal model. Moreover, the use of the two-dimensional model instead of the three-dimensional one allows dividing the number of active slip systems by two and, hence, significantly reducing the CPU time required for the integration of the constitutive equations at the single-crystal scale. Furthermore, the planar polycrystal model seems to be more suitable to study the ductility of very thin sheet metals, as its use allows us to rigorously ensure the plane-stress state, which is not always the case when the fully three-dimensional polycrystalline model is employed. Consequently, the adoption of this planar formulation, instead of the three-dimensional one, allows us to simplify the computational aspects and, accordingly, to considerably reduce the CPU time required for the numerical predictions.

1. Introduction

Over the last decades, the manufacture industry has been facing an increasing demand from customers for lightweight design to significantly reduce energy consumption and improve cost efficiency, while extending component service life by enhancing their ductility and mechanical resistance under a wide range of mechanical loading paths. Consequently, one of

the mainchallengesinmodern metalformingprocessesistomanufacture high-strengthandlight-weightstructural com-ponents,whichalsogreatlycontributetoreducingcarbondioxideemissions.Itisalsowellknownthatductilefailureisthe mainmechanismthatlimitsmaterialformabilityduringformingprocesses.Ductilefailuremaybeduetotheaccumulation of damage during the formingprocess orto the occurrence oflocalized necking without the presence ofprior damage. The predictionofductilefailureduring sheetmetalformingprocessesusingrelevantdamagemodelshasbeenextensively studiedintheliterature.Inthisfield,wecanquotesomephenomenologicalconstitutivemodelsdevelopedtotakeinto ac-counttheeffectofthedamagephenomenonontheevolutionofplasticdeformationthroughascalarortensormacroscopic damagevariable(see,e.g.,[1–3]).Damagedescriptionhasbeenalsoconsideredinseveralmultiscaleschemes(suchasthe self-consistent scheme), wherea damage variablehasbeen introducedat acrystallographic slipsystemscale inorderto betterdescribethematerialdegradationbyinitiation,growthandcoalescenceofmicrodefectsinsidethepolycrystalline ma-terial(see,e.g.,[4,5]).Inthepresentpaper,weassumethatductilefailureintraditionalsheetmetalformingprocesses(such asdeepdrawingandrolling)issolelyduetotheoccurrenceoflocalizednecking(withoutthedevelopmentofdamage).To characterizetheoccurrenceoflocalizedneckingandhencetopredicttheductilitylimitofsheetmetals,theclassicalconcept of FormingLimitDiagram (FLD),initially introduced byKeeler andBackofen[6], hasbeenwidely usedinboth academia andindustry.ThenumericaldeterminationofFLDsrequiresthecouplingbetweenastrainlocalizationcriterionanda con-stitutivemodel,whichdescribesthemechanicalbehaviorofthestudiedmaterial.Therefore,reliablepredictionsoflocalized neckingcannotbeachievedwithoutresortingtoefficientandaccuratecomputationaltoolsbasedon:

•

theuseofadvancedconstitutivemodeling,whichpermitstoaccuratelydescribethemechanicalbehaviorofthestudied material andthe influenceofthe keyconstitutive features onthe initiationoflocalized necking.Indeed,it iswidely recognized that the ductility limit is extremely sensitive to the mechanical and physical properties. Accordingly, a wide range ofphenomenological behaviormodels hasbeen implemented inseveralnumerical toolsto studythe ef-fect ofsome constitutivefeatures on localizednecking, such asplastic anisotropy [7,8], strain-rate sensitivity [9,10], anddamage-inducedsoftening[11,12].Despitetheir goodpredictivecapabilitiesandtheir wideusefortheprediction oflocalized necking,thephenomenologicalmodelsare notrelevant enoughtoaccount forsome keyphysicalaspects ofthematerialbehavior,suchasinitialandinducedtexturesandothermicrostructure-relatedparameters(dislocation motionandaccumulation,grainmorphology,crystallographicstructures

. . .

).Toaddresstheselimitations, micromechan-ical modelinghasbeenrecentlyused inseveralcontributions topredicttheductility limitofmetalsheets.Compared to phenomenological approaches, micromechanicaldescriptions reveal to be more appropriate to accurately link the above-mentioned microstructuralparameterstotheductility limit.Inthiskindofmodeling (namelymicromechanical modeling),theconstitutiveequationsareformulatedatthesingle-crystalscale.Hence,theseconstitutiveequations ac-countforrelevantmechanismsatthisscale,suchaslatticerotation,slipofcrystallographicplanes,hardeningevolution, dislocationmotion.Theoverallbehaviorofthepolycrystallineaggregateisobtainedfromthatofitssingle-crystal con-stituentsbyusingsome scale-transitionschemes.Duetoitsgoodcapabilities,themicromechanicalmodelingislargely followedintheliteraturetopredicttheductilitylimitofthinmetalsheets.Themicromechanicalmodels,mainly formu-latedwithin afinitestrainframework, canbe classifiedintotwo mainfamilies:rate-dependentandrate-independent frameworks; within the familyof rate-dependentapproaches, one can quote, forinstance, Inalet al.[13], who have coupled the Taylormultiscale scheme withthe initial imperfection approach developedby Marciniakand Kuczynski [14].The same ductility limit approach has beencoupled withthe rate-dependent self-consistent schemein several other works(see,e.g.,[15,16]).Astothefamilyofrate-independentapproaches, fewercontributionshavebeen devel-oped,amongwhichKnoackertetal.[17],whohavecoupledtheinitialimperfectionapproachwiththeTaylormultiscale model.More recently,the rate-independentself-consistent modelhasbeen coupledin[18,19] with Rice’sbifurcation theory [20] topredicttheincipienceofplasticstrainlocalization. Predictionsoflocalizedneckingobtainedby the ini-tialimperfectionapproach havebeencomparedtothosegivenby bifurcationtheory forthecaseofrate-independent materialsin[18,21].In[22],ahomogenization-basedfiniteelementmethodhasbeencoupledwiththeinitial imperfec-tionapproachtoanalyzetheoccurrenceofplasticflowlocalizationinheterogeneousrate-independentmaterials.Inthe presentinvestigation,ourattentionisconfinedtotheductilityanalysisofmaterialswithFCCcrystallographicstructure. The mechanical behavior atthe single-crystal scale isassumed tofollow a large strain rate-independent framework. The plasticflow, which isassumedto be solelydueto shearover crystallographicslipsystems, ismodeled by using theSchmidrule[23].Theevolutionofthecriticalshearstresses,whichiscloselyrelatedtotheevolutionofthe micro-scopicyieldsurfaces,isexpressedthroughanisotropichardeninglaw.Therate-independentconstitutivemodelchosen to describethemechanicalbehavioratthesingle-crystalscale isknowntobevery accurateinthedescription ofthe deformation during cold-forming processes.Furthermore, ithasbeen shownto be themostadequate framework for thepredictionoflocalizedneckingbasedonbifurcationtheory[18,19].Toderivethemechanicalbehaviorofthe poly-crystalline aggregatefromthat ofitssingle-crystal constituents,thefull-constraintTaylormodelisusedinthiswork. Despiteits simplicity,thismultiscaleschemeisabletoaccurately predictthe textureevolution [24] andthe ductility limit[21];

•

the development of robust numerical tools that can efficiently predict strain localization. Indeed, numerical predic-tions basedonmicromechanicalmodelinggenerallyrequiresubstantialamountsofCPUtimeandmemoryspace. This is due, among other factors, to the fact that, inmost numericalcontributions devoted to the predictionof FLDs for thinsheet metalsusingmultiscaleschemes [16,18,19,21],theadopted polycrystallinemodelisformulated withinthe

generalthree-dimensionalframework(atthesingle-crystalscaleand,hence,atthepolycrystallinescale).However,the plane-stressconditioniscommonlyusedtopredicttheoccurrenceofplasticstrainlocalization.Thisassumptionisfully justifiedbytheverysmallthicknessofthestudiedsheetscomparedtotheothercharacteristicdimensions[25].Thus,to ensurefullcompatibilityandtolinkthethree-dimensionalbehaviormodeltothetwo-dimensionalformulationofthe localizationcriterion,atwo-dimensional formulationofthe polycrystallinemodelshould bederived fromtheoriginal three-dimensionalone.Thederivationofsuchatwo-dimensionalformulationfromtheoriginalone generallyrequires theapplicationofaniterativecomputationalproceduretoreachtheplane-stressstate[18–21].Themajordrawbackof usingathree-dimensionalformulationoftheconstitutive equations,embedded intothisiterativeproceduretoensure themacroscopicplane-stressassumption,isthatitrequiressubstantialamountsofCPUtimeandmemoryspace. Typ-ically,thenumericaldeterminationofFLDfora polycrystallineaggregate composedofabout1000grainsneedsmany computation hours. Of course, thiscomputation time dependson a number ofmechanical choices(total numberof slipsystemsforeach singlecrystal, scale-transitionscheme

. . .

), andnumericalparameters(the selectedincrementfor thestrain–pathratio,thesizeofthestrainstepusedtointegratetheconstitutiveequations

. . .

).Thiscomputationtime alsodependsonthenumberandspeedoftheprocessorsusedandonthelevelofparallelization,etc.Furthermore,the applicationofthe above-describediterativeproceduretoathree-dimensionalmodelallows satisfyingtheplane-stress stateatthemacroscopiclevelonly.Indeed,underathree-dimensionalformulation, themicroscopicstresscomponent normaltotheplaneofthesheetisheterogeneouslydistributedoverthepolycrystallineaggregateandisnotnecessarily equalto zeroforall single crystals (only thevolume averageof thismicroscopic stresscomponent isequal tozero). Thisobservationrevealssomeinconsistencywiththetrueplane-stresscondition,whichshouldbefulfilledateach ma-terialpointofthesheet.Consequently,thedevelopmentofrobustnumericaltoolsthat aimtoensuretheplane-stress condition atboth themicroscopicandmacroscopicscales andto considerablyreducetheCPUtime representsavery attractive andchallenging objective.One of thesolutions, which can be followed to reach thisobjective, consistsin usingatwo-dimensional idealizationofpolycrystallinemodels inthepredictionofplasticstrain localization.Inthese idealizedmodels,theconstitutiveequationsatthesingle-crystalscaleareinitiallyexpressedunderatwo-dimensional formulation(two-dimensionalslipsystems).Withinthisformulation,themicroscopicandmacroscopicplane-stress con-ditionsarenaturally ensuredwithouttheneedforresortingtoanyiterativeprocedure.Thisclass ofmodels hasbeen originallydevelopedbyAsaroandRice[26] toanalyzeplasticstrainlocalizationinductileFCCsinglecrystalsdeforming by single planar slip(instead oftwelve slipsystems in the generalthree-dimensional case). In thislatter work, the plasticflowismodeledasrate-insensitive,andstrainlocalizationisviewedasabifurcationfromahomogeneous defor-mationmodetoonethatisconcentratedonanarrow‘shearband’[20].ThepioneeringanalysisofAsaroandRice[26], restrictedtosinglecrystalswithasingleslipsystem,hasbeenextendedbyAsaro[27] tosinglecrystalsundergoingtwo symmetricslip systemsin tension.Inthe two above-mentioned works[26,27],the single-crystalconstitutive models havebeenpurposelyoverlysimplifiedinordertoallowadeepunderstandingofthebifurcationphenomenonthrough purelyanalyticaltreatment.Forthisreason,theseplanar modelshavebeenwidely usedtogainagreaterphysical in-sightintothesecomplexplasticinstabilityphenomena. Inthiscontext,thetwo-dimensionalmodelresultingfromthe double-slip mechanismdevelopedby Asaro[27] has beencoupledby IwakumaandNemat-Nasser [28] withthe Hill self-consistentapproachtoderivetheoverallbehavioroftwo-dimensionalFCCpolycrystallineaggregates. Inthislatter contribution,theself-consistent modelhasbeencoupledwiththebifurcation approach(see, e.g.,[29])to predictthe onsetofstrainlocalizationinpolycrystallinematerials.Besidestheanalysisofstrainlocalizationinsingle crystalsand polycrystals,theplanardouble-slip crystalmodelshavebeenwidelyusedtodescribethemechanicalbehaviorofthin structuresandtopredict theevolution ofcrystallographic textureduring variousdeformation processes[30–34]. The modeldevelopedbyAsaro[27] hasbeenextendedto amoregeneraltwo-dimensional crystalconstitutivedescription (see,e.g., [35,36]).Notethat,contraryto theAsaro model,thetwo-dimensionalconstitutive descriptiondevelopedin [35] and [36] provides exactly thesame predictionsasthose givenby the three-dimensionalmodel,when the load-ing is plane (biaxial strain andplane-stressstates) andapplied in one ofthe symmetry planesof the single crystal. In thisparticularcase, biaxial strain state andplane-stress state become compatible(abiaxial strain state induces a plane-stressstate andviceversa).Thistwo-dimensionalsingle-crystalmodelhasbeencoupledin[24] withtheTaylor scale-transitionschemetopredictthetextureevolutionofplanarpolycrystallineaggregates.Aplanarpolycrystalmodel, similartotheoneusedin[24],iscoupledinthecurrentcontributionwiththebifurcationapproachtopredictthe oc-currenceofstrainlocalization.Aswillbedemonstratedthroughvariousnumericalsimulations,theresultsobtainedby usingtheplanarsinglecrystalformulationareexactlythesameasthosegivenbythegeneralthree-dimensionalmodel. However,theFLDspredictedbythetwo-dimensionalpolycrystallinemodelareveryslightlydifferentfromthose deter-mined bythe morecommonlyusedthree-dimensional model.Thisslightdifferenceis duetoa slightdeviationfrom themicroscopicplane-stresscondition,whichisrigorouslysatisfiedonlyinthecaseofthetwo-dimensionalmodel.In termsofcomputationalefficiency,theuseofthetwo-dimensionalmodelinsteadofthethree-dimensionalone allows considerablyreducingthe computationalcostrequiredforthepredictionoflocalizednecking inthinsheet metals.To theauthors’ bestknowledge,thisis thefirst timethat thisplanar polycrystalmodel isusedto predictforming limit diagramsforpolycrystallineaggregates.

•

the secondsection isdevotedtothepresentationoftheconstitutiveequationscorresponding tothetwo-dimensional polycrystal model. This section also includes the algorithmic aspects relating to the numerical integration of these constitutiveequations;

•

the theoretical framework andthe numerical implementation of the bifurcation approach are presented in detail in Section3;

•

thefourthsectionisdedicatedtothepresentationofvariousnumericalresultstoillustratethesuitabilityoftheplanar polycrystalmodelinthepredictionoflocalizednecking.

2. Planarpolycrystalmodel

2.1. Constitutiveequations

Theclassicalformulationoffiniteelasto-plasticdeformationhasbeenextensivelystudiedinseveralearlyworks(see,for instance,references[37–39]).Thisformulationassumestheexistenceofaninfinitenumberofintermediateconfigurations (also calledelastically relaxed configurations), obtained fromthe currentone by elastic unloading to a stress-freestate. Underthisassumption,thetotaldeformationgradient f maybemultiplicativelydecomposedintoelasticandplasticparts, denotedfe_andfp_{,respectively:}

f

=

fe

·

fp (1)

Tensor femayitselfbemultiplicativelydecomposedintoastretchingtensorve andarotationtensorre:

fe

=

ve

·

re (2)

Rotation re _{definestheorientation oftheintermediate configurationcoordinatesystemwithrespecttothe coordinate}

systemrelatedtothedeformedconfiguration(currentconfiguration).

Due to itssuitability to describe themechanicalbehavior ofmetallic materials, an Eulerianformulationis adoptedto expresstheconstitutiveequations.CombiningEqs. (1) with(2),thevelocitygradientg canbedecomposedasfollows:

g

= ˙

f

·

f−1

= ˙

fe

· ˙

fe−1

+

fe

· ˙

fp

·

fp−1

·

fe−1

= ˙

ve

·

ve−1

+

ve

· ˙

re

·

re

·

ve−1

+

ve

·

re

· ˙

fp

·

fp−1

·

re

·

ve−1 (3)

By considering that theelastic strain isvery small comparedto theplastic strain, we can approximatethestretching tensorve_{bythesecond-orderidentitytensor:}

ve

≈

I2 (4)

Underthisapproximation,Eq. (3) ofthevelocitygradientg canbereducedto:

g

= ˙

ve

+ ˙

re

·

re

+

re

· ˙

fp

·

fp−1

·

re (5)

Tensor g canalsobedecomposedintoitssymmetricpartd andskew-symmetricpartw:

g

=

d

+

w (6)

Thetotalstrainratetensord andthetotalspintensorw areinturndecomposedasfollows:

d

=

de

+

dp

;

de

= ˙

ve

;

dp

=

re

·

˙

fp

·

fp−1



_S

·

re w

=

we

₊

_wp

_;

_we

_{= ˙}

_re

_·

_re

_;

_wp

₌

_re

_·

˙

_fp

_·

_fp−1



A

·

re

(7)

where

•

S(resp.,

•

A)denotesthesymmetric(resp.,skew-symmetric)partoftensor

•

.

Asstatedbefore,attentionisconfinedinthispapertothestudyofFCCsinglecrystals.Themechanicalbehaviorofthese crystalsisassumedtofollowafinite-strainrate-independentformulation,wheretheplasticdeformationisonlyduetothe sliponthecrystallographicslipsystems.Underthisassumption,theplasticpartofthevelocitygradientre

· (˙

fp

·

fp−1

)

·

re

canbe writtenunderthefollowingform(fortheparticularcaseofFCCsingle crystals,thenumberofcrystallographicslip systemsisequalto12): re

·

˙

fp

·

fp−1



·

re

=

12



α=1

˙

γ

αm



α

⊗ 

nα (8)

where

γ

˙

α isthealgebraicvalueofthesliprateforsystem

α

,andvectormα (resp.



nα )



representstheslipdirection(resp. the normalto the slip plane) in the current configuration.The counterparts of vectors mα and



nα in



the intermediate configuration,whicharedenotedrespectivelybym



α andn



α ,aredefinedbythefollowingrelations:



Table 1

Thenumberingofthethree-dimensionalslipsystemsforFCCsinglecrystals.

α 1 2 3 4 5 6 √ 3nα 0 1 1 1 1 1 1 1 1 1 1 1−1 1 1−1 1 1 −1 √ 2mα 0 [1−1 0] [1 0−1] [0 1−1] [1 0 1] [1−1 0] [0 1 1] α 7 8 9 10 11 12 √ 3nα0 1−1 1 1−1 1 1−1 1 −1 1 1 −1 1 1 −1 1 1 √ 2mα0 [1 0−1] [0 1 1] [1 1 0] [0 1−1] [1 1 0] [1 0 1]

Intheparticularcaseofsinglecrystals,theintermediateconfigurationischosensuchthatthevectorsm



α andn



α remain

constant,equal respectivelyto mα



₀ and



nα₀,during thedeformation. The numberingofvectorsmα



₀ andnα



₀ forFCC single crystalsisgiveninTable1.

Tosatisfytheobjectivityprinciple,thelatticeco-rotationalderivative

σ

∇ oftheCauchystresstensor

σ

isintroducedby thefollowingrelation:

σ

∇

= ˙

σ

−

we

·

σ

+

σ

·

we (10)

where

σ

˙

is the time derivative of the Cauchy stress tensor

σ

,and we is the elastic spin tensor introduced in Eq. (7)4.

Notethatthelatticeco-rotationalderivative introducedinEq. (10) isbasedontheelasticpartofthespintensorwe_.Itis

usedinsteadofthewell-knownJaumannderivativebasedonthetotalspintensorw becauseitallowsaccuratelyfollowing therotation ofthe crystallographiclattice [27,40]. Tensor

σ

∇ canbe relatedto theelastic strain ratede by theisotropic fourth-orderelasticitytensorle_:

σ

∇

=

le

:

de (11)

Forsimplicity, the elastic behavior ismodeled by an isotropiclaw, despite crystals havecubic symmetry. This choice ismotivatedby thefact thattheeffect oftheelasticbehavior onthepredictionofthe ductilitylimit (whichis themain objectiveofthecurrentcontribution)remainsverysmall.

The plasticflow ofthesingle crystalis definedbythe Schmidlaw[23]. Thislawstatesthat slipmayoccur on aslip system

α

onlywhenitsresolvedshearstress

τ

α reachesacriticalvalue

τ

α

c (criticalshearstress):

∀α

=

1

, . . . ,

12

:

 |τ

α

_{| <}

_τ

α c

⇒ ˙

γ

α

=

0

|τ

α

_{| =}

_τ

α c

⇒ | ˙

γ

α

| ≥

0 (12)

wheretheresolvedshearstress

τ

α ,actingonagivenslipsystem

α

,isdefinedastheprojectionoftheCauchystress

σ

on theSchmidtensorMα (

= 

mα

⊗ 

nα )correspondingtothatslipsystem:

∀α

=

1

, . . . ,

12

:

τ

α

=

σ

:

Mα (13)

and

τ

α

c isthecriticalshearstress.Theevolutionof

τ

cα isdefinedasafunctionof

γ

α bythefollowinghardeninglaw:

∀α

=

1

, . . . ,

12

: ˙

τ

cα

=

12



β=1

hαβ

 ˙

γ

β



(14)

whereh isthehardeningmatrix.

TakingintoaccountEq. (9),theresolvedshearstress

τ

α definedbyEq. (13) canbeexpressedasfollows:

∀

α

=

1

, . . . ,

12

:

τ

α

=

σ

:

Mα

=

σ

:

Mα

=

σ

:



m



α₀

⊗ 

nα₀



where

σ

=

re

·

σ

·

re (15)

Asinglecrystalisknowntobeathree-dimensionalanisotropicmaterial,whichmeansthatabiaxialstrainstate (g13

=

g23

=

g31

=

g32

=

0)mayinduceathree-dimensionalstress state(

σ

13

=

0,

σ

23

=

0,

σ

33

=

0), andviceversa.Nevertheless,

ifthe single crystalis submitted to in-planeloading along one ofits symmetry planes,then the biaxialstrain state and theplane-stressstatebecomecompatible,resultinginatruetwo-dimensionalcrystalmodel.Astheloadingpathsadopted intheFLDpredictionsare plane(biaxialstrain andplane-stressstates), thetwo-dimensionalcrystalmodelseemstobe a relevantmodelto describethemechanicalbehavior atthe microscopiclevel.As discussedearlierbyChenaouietal.[24], FCCsingle crystalshavetwo familiesofsymmetryplanes:

{

100

}

and

{

110

}

.Theloadingissaidtobeplane ifitisapplied alongoneofthesesymmetryplanes.Theapplicationofloadingalongoneofthesetwofamiliesofsymmetryplanesallows ustodefine two differentplanarFCC single crystalmodels: thePFCC1 forthe

{

100

}

family, andthe PFCC2forthe

{

110

}

family.

In the currentinvestigation, focus is restrictedto the PFCC1 model, withloading applied along the

{

001

}

plane. The conceptofthePFCC1modelwillbe brieflyrecalledinthefollowingdevelopments.Moredetails onthisconceptaregiven inRefs. [24,35,36].

Table 2

ThenumberingoftheplaneslipsystemsforthePFCC1singlecrystal.

Planar slip system Three-dimensional slip systems In-plane Schmid tensor

1 1↔5 X10= 1 √ 6 _{1 1} −1−1  2 2↔4 X20= 1 √ 6 _{1 1} 0 0  3 3↔6 X30= 1 √ 6 _{0 0} 1 1  4 7↔12 X40= 1 √ 6 ₁₋₁ 0 0  5 8↔10 X50= 1 √ 6 _{0 0} 1−1  6 9↔11 X60= 1 √ 6 ₁₋₁ 1 −1 

IntheparticularcaseofthePFCC1model,therotationre oftheintermediate configurationwithrespecttothecurrent onecanbereducedtothefollowingform:

re

=

⎡

⎣

cos

(

ϕ

)

sin

(

ϕ

)

0

−

sin

(

ϕ

)

cos

(

ϕ

)

0 0 0 1

⎤

⎦

(16)

Undertheplane-stresscondition,andtakingintoaccounttheform(16) oftherotationre_{,theCauchystresstensor}

_σ

_in

theintermediateconfigurationcanbeexpressedbythefollowinggenericform:

σ

=

re

·

σ

·

re

=

re

·

⎡

⎣

σ

11

σ

12 0

σ

12

σ

22 0 0 0 0

⎤

⎦ ·

re

=

⎡

⎣

σ

₁₁

σ

₁₂ 0

σ

₁₂

σ

₂₂ 0 0 0 0

⎤

⎦

(17)

Letusconsider,forinstance,thecrystallographicslipsystems1and5(seeTable1).TheSchmidtensorscorresponding tothesetwosystemscanbeexpressedintheframeoftheintermediateconfigurationasfollows:

M1

= 

m1₀

⊗ 

n1₀

=

√

1 6

⎡

⎣

1 1 1

−

1

−

1

−

1 0 0 0

⎤

⎦ ;

M5

= 

m5₀

⊗ 

n5₀

=

√

1 6

⎡

⎣

1 1

−

1

−

1

−

1 1 0 0 0

⎤

⎦

(18)

ByanalyzingtheSchmidtensorsofEq. (18),onecanconcludethat:

•

thecrystallographicslipsystems1and5haveexactlythesameresolvedshearstressundertheplane-stresscondition,

τ

1

=

σ

:

M1

=

σ

11

√

−

σ

22 6

τ

5

₌

_σ

_:

_M5

₌

σ

11

_√

−

σ

22 6

⎫

⎪

⎪

⎬

⎪

⎪

⎭

⇒

τ

1

=

τ

5 (19)

•

thecontributionsofbothslipsystemstothebiaxialstrainstatearethesame;

•

thecontributionsofbothslipsystemstothestraincomponentsinthethirddirectionareopposite.

Based on the latter analysis, one can clearly confirm that the crystallographic systems 1 and 5 are symmetric and identically loaded. Consequently,

γ

˙

1 _{is identically equal to}

_γ

_˙

5 _{during the loading. Then, the contribution of these two}

slipsystemstotheplasticdeformation

˙

fp

·

fp−1 is:

˙

γ

1m



1₀

⊗ 

n1₀

+ ˙

γ

5m



5₀

⊗ 

n5₀

= ˙

γ

1



m



1₀

⊗ 

n1₀

+ 

m5₀

⊗ 

n5₀



= ˙

γ

1



2 3

⎡

⎣

1 1 0

−

1

−

1 0 0 0 0

⎤

⎦  ˙

η

1X1₀ where

η

˙

1

=

2

γ

˙

1 and X1₀

=

√

1 6



_{1 1}

−

1

−

1



(20)

which isa plane strain state (the components13, 23,31,and32are equalto 0). Thetwo systems 1and5 canthen be degenerated inonesingle-planepseudo-slipsystemdefinedbyits in-planeSchmidtensorX1₀.Theother slipsystemsmay be analyzedinasimilar way:2

↔

4,3

↔

6,7

↔

12,8

↔

10,and9

↔

11.Therefore,the12three-dimensionalslipsystems aresymmetrizedinto6-planeslipsystems(seeTable2).

For the sake ofsimplicity, all vector andtensor fields usedin the remainder of thepaper will be expressed intheir in-planeformwithoutchangingthenotationsadopteduntilnow.Forinstance,tensor

σ

willbedefinedasfollows:

Fig. 1. Initial yield locus corresponding to the PFCC1 model in the(X1,X2,X3)space.

σ

=



σ

11

σ

12

σ

12

σ

22



(21)

Ontheotherhand,atensorvariable

•

willbedenoted

•

3D,whenitisexpressedinitsthree-dimensionalform.

Withthesedefinitionsandnotations,Eq. (8) canberewritteninitsin-planeform:

re

·

˙

fp

·

fp−1



·

re

=

6



α=1

˙

η

αXα (22)

Similarly,theisotropicfourth-order elasticitytensorle,relatingtheelasticstrainratedeto theco-rotationalderivative

σ

∇ ofthestresstensor(seeEq. (11)),isdefinedbyitsmatrixform(moreadaptedforthenumericalimplementation):

le

=

E 1

−

ν

2

⎡

⎣

1

ν

0

ν

1 0 0 0 1−₂ν

⎤

⎦

(23)

whereE and

ν

aretheYoungmodulusandthePoissonratio.

AstotheSchmidlawgiveninEq. (12),itisreformulatedasfollows:

∀α

=

1

, . . . ,

6

:

 |

τ

α

_{| <}

_τ

α c

⇒ ˙

η

α

=

0

|

τ

α

_{| =}

_τ

α c

⇒ | ˙

η

α

| ≥

0 where

τ

α

=

σ

:

Xα (24)

TheinitialyieldlocuscorrespondingtothePFCC1modelisadecahedroninthethree-dimensionallattice–stressspace, havingthecomponents X1, X2 andX3 asaxes,asshowninFig.1.Thecomponents X1, X2 andX3 aredeterminedbythe

followinglineartransformationsfromthecomponentsoftensor

σ

: X1

=

√

2

σ

₁₂

;

X2

= (σ

11

+

σ

22

)/

√

2

;

X3

= (σ

11

−

σ

22

)/

√

2 (25)

For practical reasons related to the numerical implementation of the single-crystal constitutive equations, it is more suitabletoonlyhandlepositivevaluesofthesliprates.Tothisend,eachslipsystemissplitintotwoorientedslipsystems. TheseorientedslipsystemsarecharacterizedbythefollowingSchmidtensors:

Xα

=



Xα

∀α

=

1

, . . . ,

6

−

Xα−6

_∀α

₌

₇

_{, . . . ,}

₁₂ (26)

Withthe above conventions,whichwill be followed inthe remainder ofthe paper,Eqs. (24), (14), and(22) become, respectively:

∀α

=

1

, . . . ,

12

:



_τ

α

_<

_τ

α c

⇒ ˙

η

α

=

0

τ

α

₌

_τ

α c

⇒ ˙

η

α

≥

0

∀α

=

1

, . . . ,

6

: ˙

τ

cα+6

= ˙

τ

cα

=

12



β=1 hαβ

η

˙

β (27) r

·

˙

fp

·

fp−1



·

r

=

12



α=1

˙

η

αXα

From Eq. (27)3, the plastic strain rate dp and the plastic spin rate wp can be expressed as functionsof tensors Rα

dp

=

12



α=1

˙

η

αRα

;

wp

=

12



α=1

˙

η

αSα (28)

ThecombinationofEqs. (7)1,(11),and(28)1 leadsto:

σ

∇

=

le

:

de

=

le

:



d

−

dp



=

le

:



d

−

12



α=1

˙

η

αRα



(29)

Byintroducingthe set

A

ofactiveslipsystems(

∀

α

∈

A

: ˙

η

α

>

0),andbytakingintoaccountEq. (27)1,onecaneasily

demonstratethatthefollowingconditionisfulfilled:

∀α

∈

A

: ˙

χ

= ˙

τ

α

− ˙

τ

_cα

=

0 (30)

Byusingthe definitionsof

τ

α and

τ

α

c ,aswell astheabovedevelopments,thefollowing relationcanbederived from

Eq. (30):

∀

α

∈

A

:

Rα

:

le

:

d

=



β∈A



hαβ

+

Rα

:

le

:

Rβ



η

˙

β ₍₃₁₎

Thus,theslipratesoftheactiveslipsystemsread:

∀

α

∈

A

: ˙

η

α

=



β∈A

MαβRβ

:

le

:

d (32)

whereM istheinverseofmatrixB,definedbythefollowingindexform:

∀

α

, β

∈

A

:

Bαβ

=

hαβ

+

Rα

:

le

:

Rβ (33)

ToapplythebifurcationcriteriondetailedinSection3,letusintroducethemicroscopictangentmodulusl,whichrelates thenominalstressratetensorn to

˙

thevelocitygradientg:

˙

n

=

l

:

g (34)

The expressionofn can

˙

be obtainedfromthedefinitionofthethree-dimensionalnominalstresstensorn3_D,whichis

givenasfollows:

n3D

=

j3Df−3D1

.σ

3D (35)

ThetimederivativeofEq. (35) gives:

˙

n3D

=

j3Df−_3D1

.



σ

˙

3D

+

σ

3DTr

(

d3D

)

−

g3D

.σ

3D



(36)

An updated Lagrangianformulation (i.e.f3_D

=

13_D and j3_D

=

1) isadoptedin whatfollows.Withinthisformulation,

Eq. (36) becomes:

˙

n3D

= ˙

σ

3D

+

σ

3DTr

(

d3D

)

−

g3D

·

σ

3D (37)

MakinguseofEqs. (7),(8),(10),and(11),onecandemonstrate,aftersomestraightforwarddevelopments,thatn can

˙

be expressedasafunctionofthesliprates

ηα as

˙

follows:

˙

n

=



le

+

χ σ

⊗

1



:

d

−

σ

·

w

−

d

·

σ

−



α∈A



le

:

Rα

+

Sα

·

σ

−

σ

·

Sα



η

˙

α (38)

where

χ

isdefinedbythefollowingequation:

χ

=

Tr

(

d3D

)

Tr

(

d

)

=

E Tr

(

σ

˙

)

(

1

−

2

ν

)(

d11

+

d22

)

(39)

Substituting thesliprates

ηα in

˙

Eq. (38) bytheirexpression fromEq. (32),one obtains,afterstraightforwardalgebraic manipulations,thefollowingrelation:

˙

n

=



le

+

χ σ

⊗

1



−

1_σl

−

2_σl

−



α,β∈A



le

:

Rα

+

Sα

·

σ

−

σ

·

Sα



MαβRβ

:

le



:

g (40) where 1

σ l and2σ l areforth-ordertensorscomprisingconvective termsofthe Cauchystresscomponents.Theirindexforms

1 σlijkl

=

1 2

(δjl

σ

ik

− δ

jk

σ

il)

;

2 σlijkl

=

1 2

(δik

σ

jl

+ δ

il

σ

jk) (41)

where

δ

jl denotesthe Kroneckerdelta. Finally,the analyticaltangent modulus l relating thenominalstress raten to

˙

the

velocitygradientg canbeeasilyidentifiedfromEq. (40):

l

=



le

+

χ σ

⊗

1



−

1_σl

−

2_σl

−



α,β∈A



le

:

Rα

+

Sα

·

σ

−

σ

·

Sα



MαβRβ

:

le (42)

Alltheaboveconstitutiveequations–fromEq. (1) toEq. (42) –correspondtothemicroscopiclevel(thesinglecrystalscale). Tomodelthemechanicalbehaviorofapolycrystallineaggregate(whichisassumedtoberepresentativeofthestudiedmetal sheet) from the modeling at the single-crystal scale, the Taylormultiscale scheme is used. This scheme assumes strain homogeneityatthepolycrystallinescaleandisusedtocomputethemacroscopictangentmodulusL.Thelatterrelatesthe macroscopicnominalstressrateN to

˙

themacroscopicvelocitygradientG:

˙

N

=

L

:

G (43)

Consideringthestrainhomogeneityassumption,oneobtains:

G

=

g and L

=

1

V



V

l dx (44)

whereV isthecurrentvolumeofthepolycrystallineaggregate.

2.2. Numericalintegration

Theconstitutive equationsatthesingle-crystalscale aresolved byusingan incremental algorithmover atypical time incrementI

= [

t0

,

t0

+ 

t

]

.Thisalgorithmbelongstothefamilyofultimatealgorithms,initiallyintroducedby Borjaand

Wren [41] for the case of linear constitutive equations (without material and geometric non-linearities). More recently, Ben Bettaieb et al.[42,43] have developed explicitand implicit versions ofthe ultimate algorithm to take into account the non-linear behavior (non-linear hardening, finite strain, andfinite rotation). Due to its computational efficiency, the explicitultimatealgorithmdevelopedbyAkpamaetal.[43] is usedinthispaper,aftersomesignificant improvements.In thelatterreference,ithasbeendemonstratedthattheultimatealgorithmismuchmoreefficientandmorerobustthanthe return-mappingalgorithmintroducedbyAnandandKothari[44] inthecaseofrate-independentsingle-crystalconstitutive equations.Thisultimatealgorithmisbasedontheidea ofpartitionofthetime incrementI _{intoseveralsub-increments}

Iδn

= [

_tn

_,

_tn

+1

]

(wheretn=0

=

t0). Thesize ofsub-increments

δ

tn

=

tn+1

−

tn andtheir numberare apriori unknown.The

size

δ

tn isdetermined insucha waythat theSchmidcriterion(27)1 remainsfulfilledover Iδn.Byadoptingthispartition,

theconstitutiveequationsatthesingle-crystalscalemustbeintegratedovereachtimesub-incrementIδn_{,wheretheknown} quantitiesare:thematerialparameters(elasticityandhardening),themechanicalfields

σ

,re,and

τ

α

c (for

α

=

1

,

. . . ,

6)at

tn,andthevelocity gradientg assumedto beconstant overI _{(hence,over I}δn_{). Theaimofthe incrementalalgorithmis} tocomputethevaluesof

σ

,re,

τ

α

c andl attn+1.Byanalyzingthesingle-crystalconstitutiveequations,itisclearthatthe

determinationofthesliprates

ηα for

˙

thedifferentslipsystemsallowsthecomputation oftheothermechanicalvariables overthe currenttimesub-increment Iδn_.Tocompute

ηα for

˙

_{allslipsystems,let usintroducethesetofpotentially active} slipsystems

P

attn:

P

=



α

=

1

, . . . ,

12

;

τ

α

(

tn)

−

τ

_cα

(

tn)

=

σ

(

tn)

:

Rα

(

tn)

−

τ

_cα

(

tn)

=

0



(45)

ConsideringEq. (45),theSchmidlawgivenbyEq. (27)1,reducedtothepotentiallyactiveslipsystem,canberewritten

asfollows:

∀

α

∈

P

: ˙

χ

α

(

tn)

= ˙

τ

_cα

(

tn)

− ˙

τ

α

(

tn)

≥

0

;

η

˙

α

(

tn)

≥

0

;

χ

˙

α

(

tn)

η

˙

α

(

tn)

=

0 (46)

Eq. (46) may be viewed as a non-smooth complementarity problem andcan therefore be replaced by an equivalent mathematicalsysteminvolvingthesemi-smoothFischer–Burmeisterfunction[45,46]:

∀α

∈

P

: ξ

α

₍

_tn)

₌



_χ

_˙

α

₍

_tn)



2

₊



_η

_˙

α

₍

_tn)



2

₋



_χ

_˙

α

₍

_tn)

_{+ ˙}

_η

α

₍

_tn)



₌

_{0 (47)}

whichcanbeefficientlysolvedbyusingtheiterativeNewton–Raphsonmethod,wherethemainunknownsarethesliprates

˙

ηα of

thepotentiallyactiveslipsystems.ToapplythisNewton–Raphsonmethod,aJacobianmatrix

J

shouldbecomputed ateachiteration.Thismatrixisdefinedbythefollowingindexform:

∀α

, β

∈

P

:

J

αβ

(

tn)

=

∂ξ

α

₍

_tn)

The pseudo-inversion technique (see [44]) is usedto invertthe Jacobian matrix

J

when itis singular. TheNewton– Raphsoniterationsarecarriedoutuntilreachingthefollowingconvergencecriterion:



ξ

α

(

tn)



<

10−5 (49)

It mustbe notedthat,during theglobalNewton–Raphsoniterations, noad hocassumptionsare necessaryconcerning thedeterminationoftheactive setofslipsystems.Indeed,theuseofthesemi-smoothfunction(47),insteadoftheinitial formulation(46),permitstocombinebothtasks,namelytheidentificationofthesetofactiveslipsystemsandthe compu-tationofthe correspondingsliprates.Thiscombinationallowsustosignificantly increasethe efficiencyofthedeveloped numericalscheme.

Oncethe slipratesofthepotentially activeslipsystemsare computed,thelength

δ

tn ofthesub-incrementIδn _{can be} determined.

δ

tn should be inferior orequalto



t and it isdetermined insuch a waythatthe Schmidcriterion remains

satisfied forall slipsystems.In view ofEq. (46), it isobviousthat thiscriterion isfulfilled forthepotentially active slip systems.Fortheothersystems(

∈

/

P

),thefollowingconditionmustbeverified:

∀α

∈

/

P

:

τ

α

(

tn

+ δ

tn)

≤

τ

cα

(

tn

+ δ

tn) (50)

Byusingthedefinitionsof

τ

α and

τ

α

c ,thefollowingrelationscanbeeasilyobtained:

∀α

∈

/

P

:

⎧

⎪

⎨

⎪

⎩

τ

α

(

tn

+ δ

tn)

=

τ

α

(

tn)

+ δ

tnRα

(

tn)

:

σ

∇

(

tn)

=

τ

α

(

tn)

+ δ

tnRα

(

tn)

:

le

: (

d

− ˙

η

α

(

tn)Rα

(

tn))

τ

α c

(

tn

+ δ

tn)

=

τ

cα

(

tn)

+ δ

tn



_β∈Phαβ

(

tn)

η

˙

β

(

tn) (51)

ThecombinationofEqs. (50) and(51) providesthefollowingconditionon

δ

tn:

δ

tn

=

min α∈P/





t

,

τ

α c

(

tn)

−

τ

α

(

tn) Rα

₍

_tn)

_:

_le

_{: (}

_d

_{− ˙}

_η

α

₍

_tn)Rα

₍

_tn))

₋



β∈Phαβ

(

tn)

η

˙

β

(

tn)



(52)

Oncethelength

δ

tn isdetermined,theothermechanicalvariablesareupdatedasfollows:

δ

r

=

exp

δ

tn w

−



α∈P

˙

η

α

(

tn)Sα

(

tn)

!!

re

(

tn

+ δ

tn)

= δ

re

·

re

(

tn)

σ

(

tn

+ δ

tn)

= δ

re

·



σ

(

tn)

+ δ

tnle

:

d

−



α∈P

˙

η

α

(

tn)Rα

(

tn)

!

· δ

re

∀α

=

1

, . . . ,

12

:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

τ

α c

(

tn

+ δ

tn

)

=

τ

cα

(

tn

)

+ δ

tn



β∈Phαβ

(

tn

)

η

˙

β

(

tn

)

Rα

(

tn

+ δ

tn)

=

re

(

tn

+ δ

tn)

·



Xα₀



_S

·

re

(

tn

+ δ

tn) Sα

(

tn

+ δ

tn)

=

re

(

tn

+ δ

tn)

·



Xα₀



_A

·

re

₍

_t n

+ δ

tn) (53)

Thesizeofthetimeincrement



t andtheinitialtimetn arealsoupdated:



t

← 

t

− δ

tn

;

tn

←

tn

+ δ

tn (54)

After this update procedure, the computation should be restarted with a new sub-increment Iδn_{, until reaching the} followingcondition:



t

= δ

tn (55)

OncethedifferentmechanicalvariablesareupdatedbyusingEq. (53),themicroscopictangentmodulusl

(

t0

+ 

t

)

atthe

endofthetimeincrementI _{mustbecomputed.ThemacroscopictangentmodulusL}

₍

_t

0

+ 

t

)

isthenderivedbyusingthe

averagingrelation(44)2.

3. Bifurcationtheory

Bifurcationtheory(see,e.g.,[20,29])isusedheretopredicttheoccurrenceoflocalizedneckinginpolycrystalline aggre-gates.Thistheoreticalapproachassumesthattheaggregateundergoesahomogeneousstrainstatebeforetheonsetofstrain localization. Therefore,in suchan approach, diffusenecking, whichusually occursprior tolocalizednecking, isimplicitly excluded.Furthermore,thethicknessofthestudiedsheetsisassumedtobeverysmallcomparedtotheothercharacteristic dimensions.Withthesemodelingchoices,itcanbelegitimatelyassumedthatthestrainhomogeneityandplane-stress con-ditionsaresatisfieduntiltheonsetoflocalizednecking.Thegoverningequationsandthealgorithmicaspectscorresponding tobifurcationtheoryaredetailedinSections3.1and3.2,respectively.

Fig. 2. Localization of the deformation along a narrow band. 3.1. Governingequations

IntheapproachproposedbyRiceandco-workers[15,16,19],materialinstabilitycorrespondstoabifurcationassociated withadmissible jumpsfor strain andstress ratesacross alocalization band,asillustrated in Fig. 2.Since field equations havetobesatisfied,thekinematicconditionforthestrainratejumpwrites:

J

G

K =

G+

−

G−

= ˙

C

⊗ 

N

(56)

where:

• J

G

K

isthejumpofthefieldG acrossthediscontinuityband;

• ˙

C isthejumpvector;

• 

N

istheunitvectornormaltothelocalizationband.

Ontheotherhand,thecontinuityoftheequilibriumofforcesthroughthebandisexpressedas:



N · J ˙

N

K = 

0 (57)

ThecombinationofEqs. (56) and(57) withtheconstitutivelaw(43) leadstothefollowingequation:



N · (

L

: ˙

C

⊗ 

N ) = 

0 (58)

whichisequivalentto:

( 

N ·

L

· 

N ) · ˙

C

= 

0 (59)

where

N ·



L

· 

N

istheso-calledacoustic tensor.As longasthistensorisinvertible,the jumpvector C remains

˙

equalto zero,thusprecludinganydiscontinuity(bifurcation)inthedeformation field.However,whentheacoustictensorbecomes singular,therearenon-zerojumpvectorsthatsatisfyEq. (59),andthiscanbeseenasanindicatorofeffectivebifurcation. Fromanumericalpointofview,strainlocalizationoccurswhentheacoustictensorisnolongerinvertible:

det

( 

N ·

L

· 

N ) = 

0 (60)

Eq. (60) istheso-calledRicebifurcation criterion,whichcorrespondsto thelossofellipticityofthepartialdifferential equationsgoverningtheassociatedboundaryvalueproblem.

3.2. Algorithmicaspects

To determine FLDs,proportional strain paths are applied to the polycrystalline aggregate, which are characterized by constant strain–path ratios

ρ

. The two-dimensional macroscopic velocity gradient corresponding to these strain paths is definedbythefollowinggenericexpression:

G

=

G11 0 0

ρ

G11

!

(61)

Ourmainobjectivethroughthecurrentcontributionisnottopredictforminglimitdiagramsandtoanalyzethe corre-spondingnumericalresults(readers interestedinthismattermayconsultRefs. [18,19]).Instead,weareparticularly inter-estedinestablishingacomparativestudybetweenthetwopolycrystallinemodels(two-dimensionalandthree-dimensional) whentheyareappliedtothepredictionoflocalizednecking.Accordingly,toreducethecomputationalcosts,ournumerical predictionsarerestrictedtothenegativerangeofstrainpaths(

−

1

/

2

≤

ρ

≤

0).Inthisrange,theinitialcrystallographic tex-turehasastrongeffectonthebehaviorofthepolycrystallineaggregate,whichcanbe consideredasarealheterogeneous medium.

•

foreachstrain–pathratio

ρ

varyingfrom

−

1

/

2 (uniaxialtensilestate)to0 (planestrainstate),with



ρ

=

0

.

1: – for each time step I

_{= [}

_t

0

,

t0

+ 

t

]

, apply the algorithm developed in Section 2.2 to integrate the constitutive

equations ofthe polycrystal corresponding tothe safezone (i.e. outsidethe localizationband).With thisaim,the two-dimensionalmacroscopicvelocitygradientG givenbyEq. (61) isusedasloading,withthecomponentG11fixed

to 1.Afterapplyingthisalgorithm,themacroscopic tangentmodulus L mustbe determinedatt0

+ 

t.The

deter-minant ofthe acoustic tensor

N ·



L

· 

N

is minimizedover all band orientations

θ

ranging from0◦ to 90◦.If the computedminimumisinferiororequaltozero,thenlocalizedneckingisdetected.Thecorrespondingangle

θ

isthe orientationofthelocalizationband,whilethecorrespondingprincipallogarithmicstrainisthepredictedlimitstrain

E11.Thecomputationisthenstopped.Otherwise,theintegrationiscontinuedforthenexttimeincrementI.

Itmustbenotedthatthesamealgorithmcanbeusedtopredictbifurcationwithathree-dimensionalmodel,aftersome small modifications.Indeed,a thirdnestedloop shouldbe embedded within thesecond loop (correspondingto thetime integrationovereachtimeincrement).Thisthirdloopaimstoensurethemacroscopicplane-stresscondition:

˙

Σ

3D33

=

0

⇔

N

˙

3D33

=

0 (62)

Inthislatterthree-dimensionalcase,themacroscopicvelocitygradient G3_D33isaprioriunknown, anditisiteratively

computed onthe basisof thefixed point methodto fulfillcondition (62).Aftercareful checking(based onvarious poly-crystallinesimulations), wecanconfirmthattheconvergenceofthisiterativeschemeisalwaysensured(providedthatthe timestepremainsrelativelysmall),leadingtoauniquesolutiontothemacroscopicplane-stressconditiongivenbyEq. (62). Furtherdetailsonthisiterativeprocedurecanbefoundin[18].

4. Numericalresultsanddiscussions

4.1. Materialdata

Thefollowingchoicesformaterialsandparametersareadoptedinthesubsequentsimulationsbothatthesinglecrystal andpolycrystalscales:

•

elasticity is assumedto be linearand isotropic.The Young modulus andthe Poisson ratio are takento be equalto 210 GPaand0.3,respectively;

•

theinitialcriticalshearstress

τ

0isassumedtobethesameforallslipsystems,anditistakentobeequalto40 MPa;

•

hardeningisassumedtobeisotropic.Accordingly,thedifferentcomponentsofthehardeningmatrixh areassumedto

bethesameandequaltoh.Thehardeningvariableh isdefinedbythefollowingpower-typelaw: h

=

h0 1

+

h0

Γ

τ

0n

!

n−1

;

Γ

=

6



α=1



γ

α

+

γ

α+6



(63)

whereh0andn aretwohardeningparameterssetto390MPaand0.35,respectively.

4.2. Single-crystalsimulations

The mainobjectiveofthissection isto demonstratethatthe planarsingle crystalmodelleads tothesame numerical predictions as the three-dimensional one, when the loading is applied under plane-stress conditions. To this aim, two bi-dimensional loading paths are applied to the plane single-crystal model and the three-dimensional one, respectively. Theseloadingpathsaredefinedbythefollowingmicroscopicvelocitygradients:

g

=



1 0 0

−

1

/

2



;

g3D

=

⎡

⎣

1 0 0 0

−

1

/

2 0 0 0 ?

⎤

⎦

(64)

Forthethree-dimensionalsingle-crystal model,the componentg3D33 isaprioriunknown,anditis determinedbyan

iterativeproceduresothattheplane-stressconditionissatisfied:

˙

σ

3D33

=

0 (65)

Forthesetwotypesofnumericalsimulations,theinitialvalueoftheangle

ϕ

describingtherotationoftheintermediate configurationofthesinglecrystalwithrespecttothecurrentone(seeEq. (16))issetto

π/

3.

Fig.3comparesthepredictionsobtainedbythetwosingle-crystalmodelsintermsoftheevolutionoftheslipof crys-tallographicslipsystems.Eight(resp.four)slipsystemsare activatedwhenthethree-dimensional(resp.two-dimensional) single crystalmodelis used.Byexaminingthe correspondencebetweenthe twomodels, intermsofthe crystallographic slipsystemsdetailedinTable2,andbyanalyzingthecurvesofFig.3,one canconcludethatthetwo modelsleadto the

Fig. 3. Evolution of the slip of the active slip systems: (a) three-dimensional single-crystal model; (b) two-dimensional single-crystal model.

Fig. 4. Enforcementoftheplane-stressconditionforthethree-dimensionalmodel:(a)evolutionoftheunknowncomponentg3D33;(b)evolutionofthe numberofiterationsrequiredtoenforceEq. (65).

samemechanicalresponse.Forinstance,inthethree-dimensionalmodel,systems3and6areactivewiththesame evolu-tionofslipamplitude duringloading.Ontheotherhand,inthetwo-dimensionalmodel,slipsystem3isactivewithaslip amplitudeequaltothesumoftheslipamplitudeofsystems3and6inthethree-dimensionalmodel.Theslipevolutionof theother activeslipsystemscanbeanalyzedinasimilar way(

γ

10

₌

_γ

12

3D

+

γ

19 3D

;

γ

2

₌

_γ

2 3D

+

γ

4 3D and

γ

5

₌

_γ

8 3D

+

γ

22 3D).

Consideringtheabove-describedconstitutiveequations,onecanconcludethatthesimilarityintheevolutionofslipofthe crystallographicsystemsleadstoaperfectsimilarityintheevolutionoftheothermechanicalvariables(stresstensor, criti-calshearstresses,plasticdeformation,crystallographicorientation).Thisresultisquiteexpectedandconfirmstheideathat bothsinglecrystalmodelsleadtothesamepredictionwhenabiaxialloadingisappliedunderplane-stressconditionsand withplanar crystallographicorientation. Notethat, by applyingother typesof biaxialloadingandplanar crystallographic orientations,thesametrendsandconclusionshavebeenobserved.

As previously explained, when the three-dimensional single-crystalmodel is considered, an iterative procedure is re-quiredto enforce the plane-stresscondition expressed by Eq. (65). The need forsuch an iterativeprocedure is naturally avoidedinthecaseofthetwo-dimensionalsinglecrystalmodel.Theevolutionoftheunknowncomponentg3D33isshown

inFig.4a.Thiscomponentevolvesrapidlyatthebeginningofloading(whentheamountofplasticdeformationisnotyet dominantcomparedtotheelasticcounterpart),andtendsto

−(

g3D11

+

g3D22

)

whenthedeformationbecomesimportant.

Thisresultisexpectableconsideringtheincompressibilityofplasticdeformation.Theevolutionofthenumberofiterations (denoted N I)requiredtoenforcetheplane-stressconditionisshowninFig.4b.Thisnumbervariesbetween9(atthe be-ginningofloading,when theelasticstrain isnot verysmallcomparedto theplasticstrain)and2.It mustbe notedthat thisnumberisstronglydependentonthesizeofthetimeincrement



t,whichissetto0.001sinthecurrentsimulations. ThecomparisonoftheCPUtimesrequiredby thetwodifferentsingle crystalmodels fortheabovesimulations reveals that the two-dimensional model is significantly more efficient than the three-dimensionalone. Indeed,4 s are required forthesimulationswiththetwo-dimensionalmodel,versus11sforthethree-dimensionalone.Thislargedifferenceis ex-pectableandcanbeexplainedbythesizeofthemathematicalsysteminEq. (47) tobesolvedtocomputethesliprates.This

Fig. 5. (a)EffectofthepolycrystalmodelontheevolutionofthemacroscopicstresscomponentsΣ11andΣ22asfunctionsofE11;(b)evolutionofthe

microscopicstresscomponentσ3D33asafunctionofE11fortwoarbitrarilyselectedsinglecrystals.

sizeisproportionaltothenumberofpotentiallyactiveslipsystems.Furthermore,theapplicationoftheiterativealgorithm to ensure the plane-stress condition inthe case ofthe three-dimensional modelconsiderably increasesthe computation time.Consequently,fromtheresultsreportedinFigs.3and4,andthecomparisonoftheCPUtimes,onecanconcludethat thetwo-dimensional single-crystalmodelismuchmoresuitablethanthethree-dimensionaloneforthesimulationofthe mechanicalbehaviorofsinglecrystalssubmittedtobiaxialloadingunderaplane-stressstate.

4.3. Polycrystalsimulations

Thissectionisfocusedontheinvestigationoftheeffectofthesingle-crystalmodelonthenumericalpredictionsatthe polycrystalline scale.Tothisend, weconsider apolycrystallineaggregate composedof 1000grains,whichhave thesame initialvolumefraction.Theinitialcrystallographictextureassociatedwiththisaggregateisassumedtobeisotropic,andthe initialorientation

ϕ

0

(

i

)

correspondingtotheithgrain(with1

≤

i

≤

1000)issetto

ϕ

0

(

i

)

= (

180

(

i

−

1

))/

1000.

Asforthecaseofthesinglecrystal,thepredictionsobtainedwiththethree-dimensionalpolycrystalmodelarecompared tothosegivenbythetwo-dimensionalmodel.Theloadingpathsadoptedinthesecomparisonsaredefinedbythefollowing macroscopicvelocitygradients:

G

=

1 0 0

−

1

/

2

!

;

G3D

=

⎛

⎝

1 0 0 0

−

1

/

2 0 0 0 ?

⎞

⎠

(66)

TheunknowncomponentG3_D33istakentobethesameforallthesinglecrystals(sincetheTaylormultiscaleschemeis

used),anditisdeterminedbytheenforcementofthemacroscopicplane-stresscondition(

Σ

3D33

=

0).

The evolutions ofthe macroscopicstress components

Σ

11 and

Σ

22 asfunctionsofthe macroscopicstrain component

E11 are plottedin Fig. 5a. Inthis figure, both resultsobtained withthe two-dimensional and three-dimensionalmodels

are presented. However, due to the perfect similarity of the results yielded by the two models, the curves plottedare indistinguishable.Despitetheperfectagreementintheevolutionofthemacroscopicstresscomponents,aspredictedbyboth polycrystal models,the resultsatthemicroscopic scale aredifferent. Indeed,asdemonstratedin Fig.5b,the microscopic normal stress

σ

3D33 isdifferent fromzerofor grains#10 and#50(but the magnitudeofthis normalstress component

remains smallcomparedto themagnitudeofthe macroscopicstresscomponents

Σ

11 and

Σ

22).Theseresultsrevealthat

themicroscopicplane-stressconditionisnotfulfilledforthesegrains.Notethatthesegrainshavebeenchosenarbitrarilyto illustratethisobservation.Moregenerally,thisnon-satisfactionofthemicroscopicplane-stressconditionisacommontrend forallthegrainsthatcomposethepolycrystallineaggregate.Thisobservationmotivatesthechoiceoftheplanarpolycrystal model,insteadofthethree-dimensionalone,tostudytheductilityofverythinsheetmetalsunderbiaxialloading.Indeed, the numberofgrainsthroughthethicknessforverythinsheetsdoesnot exceed5,andtheplane-stressconditionshould beobviouslyverifiedinallthegrainsthatcomposethepolycrystallineaggregate.

Attention is directed now towards the analysis of the effect of the polycrystal model on the localization bifurcation predictions. Itisnow widelyrecognized that theincremental tangentmodulus L plays amajor rolein thedetermination of the limit strains in the framework of bifurcation theory. Therefore, before analyzing the effect of the polycrystalline model(two-dimensionalorthree-dimensional)onthepredictionofplasticflowlocalization,letusfirstexaminetheeffect of the constitutive modeling onthe evolution of thecomponents of thistangent modulus. Tothis end, the evolution of the components L1111, L2222, L1122 andL1212 asa function ofthe macroscopicstrain component E11 is plottedinFig. 6