Multiscale modeling of the elastic behavior of architectured and nanostructured Cu–Nb composite wires

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TANG GU, Olivier CASTELNAU, E. HERVE-LUANCO, F LECOUTURIER, H PROUDHON, L

THILLY - Multiscale modeling of the elastic behavior of architectured and nanostructured Cu–Nb

composite wires - International Journal of Solids and Structures - Vol. 121, p.148-162 - 2017

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Multiscale

modeling

of

the

elastic

behavior

of

architectured

and

nanostructured

Cu–Nb

composite

wires

T.

Gu

a,b

_,

_O.

_Castelnau

a,∗

_,

_S.

_Forest

b

_,

_E.

_{Hervé-Luanco}

c,b

_,

_F.

_Lecouturier

d

_,

_H.

_Proudhon

b

_,

L.

Thilly

e

a PIMM, CNRS UMR 8006, Arts et Métiers ParisTech, CNAM, 151 Bd de l’Hôpital, Paris 75013, France b Centre des matériaux, CNRS UMR 7633, Mines ParisTech, BP 87, Evry Cedex 91003, France

c Université de Versailles - Saint-Quentin en Yvelines, 45 Avenue des Etats-Unis, Versailles Cedex F-78035, France

d Laboratoire National des Champs Magnétiques Intenses, UPR 3228 CNRS-UPS-INSA-UJF, 143 avenue de Rangueil, Toulouse 31400, France

e Institut Pprime, UPR 3346, CNRS, University of Poitiers, ISAE-ENSMA, SP2MI, Boulevard Marie et Pierre Curie, BP 30179, Futuroscope Chasseneuil Cedex

86962, France

a

b

s

t

r

a

c

t

Nanostructuredandarchitecturedcopperniobiumcompositewiresareexcellentcandidatesforthe gen-erationofintensepulsedmagneticfields(>90T)astheycombinebothhighstrengthandhighelectrical conductivity.Multi-scaledCu–Nbwires arefabricatedbyaccumulativedrawingand bundling(asevere plasticdeformationtechnique),leadingtoamultiscale,architectured,andnanostructuredmicrostructure exhibitingastrongfibercrystallographictextureandelongatedgrainshapesalongthewireaxis.This pa-perpresentsacomprehensivestudyoftheeffectiveelasticbehaviorofthiscompositematerialbythree multi-scalemodelsaccountingfordifferentmicrostructuralcontents:twomean-fieldmodelsanda full-fieldfiniteelementmodel.Asthespecimensexhibitmanycharacteristicscales,severalscaletransition stepsarecarriedoutiterativelyfromthegrainscaletothemacro-scale.Thegeneralagreementamong the model responses allowssuggestingthe best strategyto estimatethe effective behavior ofCu–Nb wiresandsavecomputationaltime.Theimportanceofcrystallographicalandmorphologicaltexturesin variouscasesisdiscussed.Finally,themodelsarevalidatedbyavailableexperimentaldatawithagood agreement.

1. Introduction

Inrecentyears,therehasbeenanincreasingdemandfor next-generation structural nano-materials that exhibit extraordinarily highstrength, electrical conductivity, hardness, ductility, thermal stabilityandradiationdamagetolerance.Twotypesoffilamented andmultilayered nano-composites composed of copper and nio-bium(i.e. Cu–Nb nano-composite wires andlaminates) are high-lighted among them (Misra and Thilly, 2010). These two Cu–Nb nano-composites are fabricated respectively by two different se-vere plastic deformation techniques: Accumulative Drawing and Bundling(Dupouy etal., 1996; Thilly etal., 2002b) and Accumu-lativeRollBonding(LimandRollett,2009;Beyerleinetal.,2014).

AseriesofCu–Nbnano-compositewiresisillustrated inFig.1, referred to as co-cylindrical structure in Dubois (2010): a

mul-∗ _{Corresponding author.}

E-mail address: olivier.castelnau@ensam.eu (O. Castelnau).

tiscale Cu matrix embedding parallel Nb nano-tubes filled with Cunano-filaments.Thesenano-compositeconductorsareexcellent candidatesforgenerationofintensepulsedmagneticfields(>90T). According to Spencer et al. (2004), Béard et al. (2013), Halperin etal.(2013),Frydman(2014),intensemagneticfieldsarebecoming essential experimentalandindustrialtools.Togeneratethem,the conductorsforthewindingcoilsmustcombinebothhigh mechan-icalstrengthandhighelectricalconductivity.InThilly(2000),Vidal etal.(2007),a conductorpresentinganUltimate TensileStrength aslarge as1.9GPa at77Kis obtainedtogether withan electrical conductivityof1.72

μ



−1cm−1.

To predict the behavior of such a composite, the main chal-lengeremainsintheunderstandingofthecomplexinteraction be-tween thedifferent materialphases, andthe architecture,in par-ticularwhen theCu–Nb compositeis fabricatedby severe plastic deformationswherethe elementaryphysical deformation mecha-nismsaremodifiedbygrainsizes.Inthisfield,combiningmaterial characterizationand multi-scalemodeling ismandatory.The pre-vious studies on theCu–Nb nano-composite wires andlaminates

Fig. 1. Successive section views of the Cu/Nb/Cu nano-composite wires ( Dubois, 2010; Dubois et al., 2012 ). (a) Scanning Transmission Electron Microscope image showing details of the Cu/Nb/Cu elementary long fiber sections; (b)–(d) Scanning Electron Microscope images showing intermediate scales and a macroscopic view of the conductor. In (a)–(c), Cu appears in dark gray and Nb in light gray; in (d), it is the reverse, due to the low magnification. The diameter of specimen (d) is re- duced to 0.506 mm in this work by supplementary cold-drawing. The smaller scale for pure polycrystalline Cu or Nb (i.e. scale H0) is not shown in this figure. See

Section 2.2 for scale conventions and their notations.

Table 1

The abbreviations used in this work. SSC Standard Self-Consistent GSC Generalized Self-Consistent FEM Finite Element Method PH Periodic Homogenization HEM Homogeneous Equivalent Medium RVE Representative Volume Element

deal withdeformation mechanisms (Thilly etal., 2001), hardness (Thillyetal.,2002a),Cu–Nbinterfaces(Beyerleinetal.,2014; Car-penter etal., 2015), UltimateTensile Strength(Vidaletal., 2007), plastic flowstability(MisraandHoagland, 2007), Bauschinger ef-fect (Thillyetal., 2007; Zhanget al., 2013;Badinier etal., 2014), yieldcriterion(Thillyetal.,2009),textureevolution(Limand Rol-lett, 2009; Leeetal., 2012;Hansen etal., 2013), thermalstability andinternalstresses(Duboisetal.,2010;2012),andelectrical con-ductivity(Thilly,2000;Dubois,2010;Guetal.,2015).

The present work concentrates on the multiscalemodeling of theanisotropicelasticbehaviorofarchitecturedandnanostructured Cu–Nb compositewires.Three multi-scalemethods will be intro-duced:twomean-fieldhomogenizationmodels(Standardand Gen-eralized Self-Consistent schemes, denoted, respectively, SSC and GSC hereafter, see Table 1 for the abbreviations) and a full-field Finite Element Method (FEM) with periodic boundary conditions (denoted PH, for Periodic Homogenization). These models essen-tiallydiffer by themicrostructuralinformationthey are basedon fortheestimationoftheeffectivebehavior. Here,theSSCscheme willbe usedtodescribetheelasticresponseofpolycrystals made ofCu orNbgrains, butalsofortheestimationofa random mix-ture ofCu andNb phases. The GSCscheme takesinto account a specificfilament/nanotubeCu/Nb/Cumicrostructure.PHassumesa periodic microstructure, and its response will be compared with theonesobtainedbySSCandGSCapproaches.

The SSC scheme is known as a homogenization theory well adaptedto estimate themechanicalbehavior ofpolycrystals. This mean-field homogenization method is based on a statistical de-scription of the microstructure of polycrystalline aggregates. The underlying microstructure, described by Kröner (1978), corre-spondstoperfectdisorderwithinfinitegraduationofsize.The de-velopmentoftheSSCmodelforheterogeneouselasticitygoesback to(Hill, 1965;Budiansky, 1965;Kneer,1965;Willis,1977;Kröner, 1978). Later on, the model has been extended to visco-plastic, elasto-plastic,andelasto-visco-plasticproperties,e.g.see Molinari etal.(1987),Ponte-CastañedaandSuquet(1998),Lebensohnetal. (2011), Yoshida et al. (2011), Vu et al. (2012). Analysis of the intraphase stress and strain heterogeneity obtained by the SSC scheme,andits comparisonwithfull-field referencecalculations, canbe found e.g. inPonte-Castañeda andSuquet (1998),Brenner etal.(2004),Lebensohnetal.(2011).

TheGSCschemeisanothermean-fieldhomogenizationmethod takinginto account particular morphologies where multi-layered fibers are considered. This kind of morphology has been first studied by Hashin (1962) who has developed variational bound-ing methods applied to a Composite Sphere Assembly made of an arrangement of homothetic two-layers spheres. Then, Hashin andRosen(1964) have appliedthese variationalbounding meth-ods to exhibit bounds forthe five independent elastic moduli of thetwo-dimensionalanalogueoftheCompositeSpheresAssembly.

ChristensenandLo(1979)havethenderivedanestimationforthe elastic behavior of such sphere or fiber-reinforced composite by consideringarepresentativetwo-layersconcentricsphere/cylinder embeddedin a fictitioushomogeneous medium representing the HomogeneousEquivalentMedium(HEM). Theirmethodis known asthe “three-phase model”. Hervé and Zaoui (1993); 1995) have then extended Christensen and Lo’s (1979) approach to multi-coatedsphereorfiber-reinforcedcompositethankstothe“(n₊1 )-phase model”. This model is used here to study the elastic be-havior of the Composite Cylinders Assembly present at different scales (Fig. 1). The GSCscheme hasalsobeen extendedto visco-elastic behavior (Beurthey and Zaoui, 2000), nonlinear behavior (Zaoui, 1997), diffusion (Caré and Hervé, 2004), thermal conduc-tivity(Hervé,2002). Interphaseeffects havealso been takeninto accountinHashin(2002)andHervé-Luanco(2014).Applicationto thediffusion phenomenaispresented inGuetal.(2015), Hervé-LuancoandJoannès(2016),JoannèsandHervé-Luanco(2016).

With the increase in the computational performance and the number of available numerical software products, computational full-field homogenization methods have gained attention. Unlike mean-field approaches(e.g.,SSC andGSC schemes), the full-field methods(e.g.basedonFEM)appliedtoRepresentativeVolume El-ement (RVE) can describe the detailed experimental microstruc-tureandprovidethecomplexstress/strainfieldsinside the differ-entphasesattheexpenseofincreasedcomputationaltime.Some full-fieldmethodsforpolycrystallineaggregatesweredevelopedby

Ghoshetal.(1995);1996) intermsof aspecialclass offinite el-ement based on Voronoï cells. Making use of such full-field ho-mogenizationmodels, severallinearmaterial behaviorswere ana-lyzed:theeffectivethermalconductivity(Flaqueretal.,2007),the effective thermoelastic properties and residual stresses (Wippler etal., 2011), andthe effectiveelastic propertieswitha statistical description(Kanit etal., 2003; Fritzen et al., 2009; Böhlkeetal., 2010). More complex non-linear mechanical behaviors are also studied for Face-Centered Cubic and Body-Centered Cubic poly-crystallineaggregatesinmanyaspects,such asinCailletaudetal. (2003), Böhlke etal. (2009), Schneider et al. (2010), Fritzen and Böhlke(2011),Klusemannetal.(2012).Fortakingtheanisotropic morphological textures into account, three-dimensional Voronoï mesh generation techniques have been proposed (Barbe et al., 2001;Fritzen etal., 2009; Fritzen andBöhlke, 2011). In addition,

Yaguchi and Busso (2005), Fritzen et al. (2009), Böhlke et al. (2010) havealso compared full-field modelresults withthe pre-viouslymentionedmean-fieldonesforpolycrystallineaggregates.

The full-field FEM can also consider a specific periodic mi-crostructure,such asCompositeSpheres Assemblyin Llorcaetal. (2000),MichelandSuquet(2009)andCompositeCylinders Assem-bly in Guet al. (2015). In this homogenization method, the mi-crostructureunitcellissubjectedtoperiodicboundaryconditions (Besson et al., 2009), therefore it is named asPH model in this work. Llorca et al. (2000) compared the three above mentioned modelsSSC,GSC,andPHforisotropicelasticbehaviorsofthe Com-positeSpheresAssemblyarchitecture.

Despitea wealthofliterature worksonpolycrystalline elastic-ity,we havefound that the following threepointsare still miss-ing: 1. No systematic analysis of the simultaneous contributions ofmorphological and crystallographic textures to the anisotropic elasticpropertiesforCuandNb,especiallyforthespecific crystal-lographictextures encountered inCu–Nbwires. 2.No application ofcurrenthomogenization methods tothe complex architectures (i.e.CompositeCylindersAssembly)ofrecentCu–Nbcomposites.3. Noexperimentalcomparisonwithin-situX-ray/neutrondiffraction data(Thillyetal.,2006;2007;2009)forCu–Nbwires.

Therefore,theobjectivesofthispaperarethreefold:1.providea homogenizationmodelforCupolycrystalsandNbpolycrystals tak-ingthe crystallographicand morphologicaltextures intoaccount; 2. providea multi-scale homogenization procedure to modelthe architecturedandnanostructuredCu–Nbcompositewires; 3. pro-videaquantitativecomparisonwithelasticstrain measuredby X-ray/neutron diffraction during uniaxial loading tests. The outline ofthearticleisasfollows.Thearchitecture andnano-structureof Cu–NbcompositewiresaredescribedinSection 2.Inorderto re-producetheeffectiveelasticbehaviorofthismaterial,three multi-scale methods, i.e. SSC, GSC and PH are presented in Section 3. InSection4,theCu polycrystalsandtheNbpolycrystals are con-sideredseparately. Then in Section 5, several scale transitions of architecturedCu–Nb composites are performedto determine the effective elastic behavior of Cu–Nb wires up to macro-scale. In

Section6,theimportanceofmicro-parametersandthebest model-ingstrategiesarediscussedforvarioustexturesandmaterial prop-erties.Finally,themodelresultsarevalidatedbycomparisonwith availableexperimentaldata.

Throughout this work, the following notation is used: x for scalars,xforvectors,x

∼for2nd-ordertensors,≈xfor4th-order

ten-sors,· forsingle contraction,:for double contraction, for tensor product,xforeffective(orhomogenized)propertyand ¯x₌



x



for volumeaverage.

2. Materialdescription

2.1.Fabricationprocess

Cu–Nbnano-compositewiresarefabricatedviaasevereplastic deformationprocess,basedonAccumulativeDrawingandBundling (series of hot extrusion, cold drawing and bundling stages) ac-cordingtoDupouy etal.(1996),Thillyetal.(2002a), Thillyetal. (2002b), Dubois (2010): a Cu wire isinserted into a Nbtube, it-selfinsertedintoaCutube.Thestructureisextrudedanddrawn, thencutinto85smallerpieceswithhexagonalcrosssection.These piecesarethenbundledandinsertedintoanewCutube.Thenew composite structure is againextruded and drawn. And so on. In thepresentwork,AccumulativeDrawingandBundlingisrepeated threetimes,leadingtocopperbasedarchitecturedand nanostruc-turedcompositewires (so-called “co-cylindricalCu/Nb/Cu” wires) which are composed of a multi-scale Cu matrix embedding 853

Nbnanotubescontaining853_{Cunanofibers,asillustratedinFig.1.}

TheusedCuisOxygen-Free HighConductivity(OFHC). Itisnoted

that,unlikeCu,Nbtubesareintroducedonlyattheveryfirst fab-ricationstage.Therefore,Nbnanotubes(denotedNb-tinFig.1(a)) arealldeformedtogetherduringtheiterativeAccumulative Draw-ing and Bundling, and they exhibit thesame microstructure and similar characteristic sizes. The Cu-fand Cu-0 regions (Fig. 1(a)) areintroducedatthebeginningoftheprocess,whiletheCu-1, Cu-2,Cu-3(Fig.1(b)–(d))areintroducedsuccessivelyduringthethree stepsofAccumulativeDrawingandBundling;different microstruc-turearethusexpectedforthedifferentCu-iregions (i= f,0,1,2 and3).

2.2. Scales

For a wire with a final diameter1 _{of 0.506 mm, Nb}

nan-otubes(averagewallthickness

δ

Nb-t=88nmandtotalvolume

frac-tion XNb-t=20.8%) are filled withCu-f copper filaments(diameter

δ

Cu-f=130 nm andvolume fraction XCu-f=4.5%), separated by the

finestCu-0copperchannels(width

δ

Cu-0=93nmandvolume

frac-tion XCu-0=17.7%); groups of 85Cu/Nb/Cu elementary long fibers

areseparatedby Cu-1copperchannels(width

δ

Cu-1=360nmand XCu-1=9.6%). The widthof Cu-2 copper channel is

δ

Cu-2=3.9

μ

m

(XCu-2=19.9%).Finally,thegroupof853elementarypatternsis

em-bedded in an external Cu-3 copper jacket (

δ

Cu-3=21.1

μ

m and XCu-3=27.5%).

For multiscale modeling of the effective elastic behavior of theseCu–Nbwires, thefollowing scaleconventionswill be used: (1)homogenizationatthehighestmagnificationscale,looking di-rectlyateachpolycrystallineCuorNbphase,islabeledasH0 (Ho-mogenization0);(2)Then,homogenizationoftheassemblyof851

elementaryCu-f/Nb-t/Cu-0longfibersislabeledasH1;(3)Iterative homogenizationoftheeffectiveCu–NbcompositezoneofH(n− 1) embeddedintheCu-(n− 1)matrix(withn=2or3),islabeledas Hn,i.e.Hnprovidestheeffectivebehaviorofanassemblyof85n

el-ementarypatterns.Theeffectivestiffnesstensorsdenoted

(

C ≈

)

H0at

thescaleH0ofpolycrystallineaggregates(i.e.CuandNb polycrys-tals),andthosedenoted

(

C

≈

)

Hi (i=1,2,3)atscalesHiofassembly

of85i_{elementarylongfibers,will beobtainedfromthe}

homoge-nizationmodels.

Finally,the scale S3 (see Fig.1(d)) is definedhereas a single cylinder-shaped structure withtwo layers:effective Cu–Nb com-positezoneofH3(containing853_{elementarypatterns)surrounded}

by the external Cu-3 jacket. The structural problem S3 will be solved byFEMto computethestiffness

(

C

≈

)

S3.Then, thiseffective

stiffnesswillbecomparedwiththeavailableexperimentaldata. 2.3. Morphologicalandcrystallographictextures

The microstructuralstateof Cu–Nbwires hasbeenstudied by ScanningElectronMicroscope,ScanningTransmissionElectron Mi-croscopeandX-ray diffraction by Vidal etal.(2007),Thilly etal. (2009),Duboisetal.(2010),Dubois(2010),Duboisetal.(2012).In the presentwork, we define as the averagegrain length along the longitudinal wire axis x1 and d asthe average grain

diame-ter in the transverse x2 − x3 cross-section. The previous studies

have shownthat the morphologicaltexture exhibits highly elon-gated grainsalong x1 becauseofiterativesevere plasticextrusion

anddrawing,therefore dforCuandNbphases.

Because ofthe multi-scalestructure, differenttypes ofcopper matrixchannelsarepresentinthematrix: (i)Cuchannelswitha width

δ

Cu-i(i=2,3)largerthanafewmicrometers(so-called“large”

Cuchannels)aremainlycomposedofgrainswithatransversesize

1 Following ( _{Vidal et al., 2007; Thilly et al., 2009 ), all dimensions are given in} the x 2 − x 3 cross-section, i.e. perpendicular to the wire axis x 1 , see Fig. 1 for the coordinate system.

Table 2

Overall crystallographic textures and corresponding Full Width at Half Maximum (FWHM) of individual components, for Cu and Nb polycrystals. These fiber textures (symmetry axis x 1 ) of the Cu–Nb compos- ites were determined by X-ray diffraction ( Dubois, 2010 ). In addition, Cu and Nb contain also 5% and 1% random components, respectively.

Fiber Volume FWHM (in deg )

textures fractions (%)  1 Cu  100  37 10.1 11.5

 111  58 8.3 10.2

Nb  110  99 6.2 6.8

d=200-400nm(atypicalmicrostructureofcold-workedmaterial); (ii) On the other hand

δ

Cu-i (i=f,0,1) lies in the sub-micrometer

range(so-called“fine” Cuchannels),withonlyafewgrainslocated betweentheCu-Nbinterfaces:inthiscase,grainwidthdCu-ivaries

from

δ

Cu-i/3to

δ

Cu-i.Inaddition,thegrainsizeofNbtubesdNb-tis

comparablewiththetubewidth(d_Nb-t≈

δ

_Nb-t).

The overall crystallographic texture of a Cu–Nb co-cylindrical compositesampleatadiameterof3.5mmhasbeenestimatedby X-raydiffractioninDubois(2010).Thespecimenswithsmaller di-ameter0.506mmconsidered inthiswork arebelievedtodisplay very similarcrystallographic texture,asconfirmedby preliminary EBSD results. X-raydiffraction has shown that Cu phases exhibit strong



111



fiber texture with the remnant



100



fiber, while a single-component



110



fibertextureisobservedinNbphases.Due toextrusionanddrawingalongthewiredirection(x1) inthe

fab-rication process, x1 is also the symmetry axis of thesefibers. In

Cu,thevolumefractionsof



100



and



111



componentsarefound tobe37%and58%,respectively,(with5%ofan additionalrandom component).InNbphase,thevolumefractionof



110



fiberis99% (with1% randomcomponent).The associatedtexturespread(Full Width atHalfMaximum) ofindividual components are indicated inTable2.

In thefabrication process, polycrystalline Nbtubes are always deformedsimultaneously,thustheyalldisplaythesame crystallo-graphic texture. However,the Cu polycrystals areintroduced suc-cessivelyatthethreestepsofAccumulativeDrawingandBundling, therefore the crystallographictextures are different foreach Cu-i (i= f, 0, 1, 2 and 3). Local textures at the very fine scale need to becharacterized by EBSD,awork currentlyinprogress.Inthe presentwork,forthesakeofsimplicity,weconsiderthesameCu crystallographic textures, asdetermined fromX-ray diffraction in

Table2,atall scalesoftheCu–Nbwires.Inother words,effective elasticbehaviorsofalltheCu-iareassumedtobeidentical.Itwill beshownthatthisapproximationissufficienttopredicttheelastic behaviorforCu–Nbwiresbasedontheavailablemechanicaldata.

FromtheOrientationDistributionFunctiondescribedabove,we used the software LaboTex2 _{to generate two sets of 40,000}

dis-creteorientationseach(oneforCuandoneforNb)thathavebeen used to generate the microstructure in the threescale transition models. Fig. 2 shows the obtainedPole Figuresand Inverse Pole FiguresforbothCuandNb,using300orientationsrandomly cho-senamongeach ofthelargersetsof40_,000orientations.Thefull set of40,000 orientationshasbeen usedwithin theSSC scheme toestimatetheeffectivebehavior(Section3.1),whereassubsetsof 1000 (respectively 100)orientations randomlychosen amongthe 40,000havebeen usedforparallelepipedic (resp.Voronoï) tessel-lations(Section3.3).

2 Software for crystallographic textures –_{http://www.labosoft.com.pl/ .}

Fig. 2. {111} Pole Figure and x 1 axis Inverse Pole Figure of Cu polycrystals and Nb polycrystals. Grain orientations were generated by LaboTex based on experimental X-ray textures. Central stereographic projections.

Table 3

Cubic elastic constants of Cu and Nb single crys- tals, expressed in the crystal lattice (Voigt con- vention).

Single C11 C12 C44 Z

crystal (GPa) (GPa) (GPa) Cu 167.20 120.68 75.65 3.25 Nb 245.60 138.70 29.30 0.55

2.4.Anisotropicelasticproperties

The Cu–Nb wires are made of Face-Centered Cubic Cu and Body-CenteredCubicNbgrains. According toEpstein andCarlson (1965), YosioandGranato(1966),Carroll (1965),thecubic elastic constantsCij (Voigtconvention) ofCu andNb singlecrystals,

ex-pressed inareferenceframe attachedto thecubic crystallattice, areprovided inTable3,andthey willbe usedin H0for predict-ingtheeffectiveelasticbehaviorofpolycrystallineaggregates.The ZeneranisotropyfactorZ(Zener,1948),definedas

Z= 2C44

(

C11− C12

)

,

(1)

isameasureoftheelasticanisotropy,havingZ=1foranisotropic material. As shownin Table 3, Cu andNb single crystalsexhibit strong anisotropy.Young’s modulus in a



111



direction is about three times higher (respectively, half smaller) than the one in a



100



directioninCu(respectively,Nb)singlecrystals.

The components C_{i jkl}(r) of the single crystal elastic moduli C ≈

(r)

expressedinagenericglobalcoordinatesystemcanbe calculated fromthecomponentsCmnpq oftheindependentelastic stiffnessC ≈

inthelatticecoordinatesystem

C_{i jkl}(r)=Q_mi(r)Q_{n j}(r)Q(_pkr)Q_ql(r)Cmnpq (2)

whereQ_mi(r) arethecomponentsoftherotationmatrixQ(r)

associ-atedwiththecrystalorientationr.ThecorrespondingEulerangles (

ψ

,

θ

,

φ

)(r) _{areneededtodetermineQ}(r) _{(Slaughter,2002).}

Due to the material processing, the architecture of Cu–Nb wires,morphologicalandcrystallographictexturesare axisymmet-ricwithrespect to axisx1.As a result, theeffective material

be-havior at scale Hi (i₌0_, 1, 2, 3) is expected to be transversely isotropic.Anisotropic elasticityis thenexpressedby five indepen-dentconstants(Hervé andZaoui,1995):longitudinalYoung’s mod-ulus E1, Poisson’s ratio under longitudinal load

ν

12, longitudinal

shear modulus

μ

12, transverse shear modulus

μ

23, plane-strain

bulkmodulus K23. Incidentally, fortransverse isotropy,transverse

Young’s moduli E2≡ E3. In this work, they will be notedas E2,3

andcanalsobederivedfromtheotherparameters:

 E2,3= 4E1K23

μ

23 4K23

μ

23

ν

2₁₂+E1



_ K23+

μ

23



(3)

It is also worth noting that C11=E1+4

ν

122K23, C22=C33=K23+



μ

23, C12=C13=2

ν

12K23, C44=

μ

23, C55=C66=

μ

12, C23=K23−



μ

23 (with C23=C22− 2C44 in the case of transverse isotropy),

whereC_{i j} denotes the components ofC

≈ making use ofthe Voigt

notation.

3. Homogenizationstrategies

The homogenization strategy used inthis work, includingthe correspondingscalesandmodelused,isillustratedinFig.3. 3.1.Mean-fieldstandardself-consistentscheme

Themean-fieldSSCschemeisusedinthepresentworkto esti-matetheeffectiveelasticpropertyC

≈ofindividualCuandNb

poly-crystals.Itisalsousedtoestimatethebehaviorofafictitious ma-terialinwhichCuandNbgrainswouldberandomlymixedall to-gether,withvolumefractionsandtexturesintroducedabove,in or-dertochecktheimpactoftheparticulararchitectured microstruc-tureofourspecimens.

Owing to therandom character ofthemicrostructure withall grainsplayinggeometricallysimilarroles,theSSCschemeis espe-cially suited forpolycrystals (Kröner, 1978; Castelnau, 2011). The SSC scheme relies on specific microstructure exhibiting a suffi-cientlyirregular mixture ofgrainsandinfinite sizegraduation as illustrated in Fig. 4(a). As in Lebensohn et al. (2011), Castelnau (2011),onedefinesamechanicalphase(r)asdenotingthesetofall grainsinthemicrostructurethatsharethesameelasticproperties, that isexhibiting thesame crystalorientation; thosegrains have howeverdifferentshapesandenvironment.Thematerialcanthus bestatisticallydescribedasanequivalentaggregatefilledwith me-chanicalphasesofdifferentsizeandsurrounding,anddistributed randomly. It is supposed that the phases exhibit, on average, a spheroidalshape,andthereforethe mean stressandstrain inside thosephasescanbeestimatedwiththeEshelbyinclusionproblem; butnote that thisdoesnot meanthat stress andstrain are uni-forminsideindividualphases,seee.g.Ponte-CastañedaandSuquet (1998),Brenneretal.(2004).Here,spheroidscorrespondingtothe meangrain shape are elongated along the longitudinal direction x1,withlengthandwidthd,asinFig.4(c).Theassociated“grain

aspectratio”_/dstatistically definesthemorphological textureof thepolycrystallineaggregate.

Forelasticpolycrystals,localandeffectiveconstitutiverelations read,respectively:

σ

_∼

(

x

)

=C

≈

(

x

)

:

ε

∼

(

x

)

,

σ

∼=C≈:

ε

∼ (4)

withC

≈theeffectivestiffnesstensor

C

≈=<C≈

(

x

)

:A≈

(

x

)

> (5)

Cu or Nb grain Morphological and crystallographic textures

PH SSC

Similar results obtained Pure polycrystalline Cu or Nb Scale H0 ( C∼_∼)H0-Cu/Nb GSC PH 851_{elementary patterns} Scale H1 ( C∼_∼)H1 GSC PH 852_{elementary patterns} Scale H2 ( C∼_∼)H2 GSC PH 853_{elementary patterns} Scale H3 ( C∼_∼)H3 FEM Cylinder-shaped structure Scale S3 ( C∼_∼)S3

Fig. 3. Overview chart of the iterative scale transition steps. The considered scales and scale transition models are mentioned, together with the obtained effective be- havior.

whereA

≈

(

x

)

isthestrainlocalizationtensordefinedas:

ε

∼

(

x

)

=A≈

(

x

)

:

ε

∼, (6)

and

ε

∼beingthemacroscopicappliedstrain.Inelasticpolycrystals,

thelocal stiffnesstensoris auniform propertyinside grains. The quantity C

≈

(

x

)

in Eq. (4) can therefore be replaced by the

corre-sponding homogeneous values C ≈

(r) _{of the considered mechanical} phase (r) definedpreviously.Similar substitution can be madein

Eq.(5)leadingto:

 C ≈= r  n=1 f(r)C ≈ (r)_:A ≈ (r) ₍₇₎

where.(r) _{indicates theaverageover thevolumeofphase (r), e.g.} A ≈ (r)₌



_A ≈

(

x

)



(r),

andf(r) _{denotesvolumefractionofphase(r).}

To estimate the phase-average stress and strain, phase (r) is treated in the SSC scheme as an ellipsoidal inclusion embedded inanhomogeneousequivalentmediumwhosebehaviorrepresents thatofthepolycrystal.AccordingtoHill(1965),Budiansky(1965), tensorA ≈ (r)_{inphase(r)reads} A ≈ (r)₌



I ≈+S≈Esh:

C ≈

−1 :



C ≈ (r)_{− C} ≈



−1 , (8) with I

≈ the fourthorder unit tensor.The Eshelby tensor S≈Esh

de-pends onC

≈ and on the aspect ratio /d. Here, the Eshelby

Fig. 4. Schematic geometry of the SSC model: (a) perfectly disordered mixture of the grains, being similar with the real morphological texture in polycrystals. (b) Statistically equivalent medium composed of randomly mixed mechanical phases . A mechanical phase is denoted by the set of grains having the same crystal orientation. (c) The overall (mean) shape of all mechanical phases is taken spheroidal, elongated along direction x 1 and with aspect ratio  / d .

Eqs. (5) and (8) lead to an implicit equation for C

≈ that can be

solved with a simple fixed-point method (Kröner, 1978). Finally, from Eq. (7), it can be observed that the sole knowledge of the mean (phase average) values A

≈

(r) _{is sufficient to estimate the} overall behaviorC

≈.Therefore,computationof thismean-fieldSSC

methodisveryfast,withouthavingtoknowthecompletefieldof

A ≈

(

x

)

.

3.2. Mean-fieldgeneralizedself-consistentscheme

The “(n+1)-phase” GSCscheme (Hervé and Zaoui, 1995) has beendeveloped toestimate theoveralleffectiveelastic moduliof multi-coatedfiber-reinforced compositeswithrandomfiber distri-bution; this model can also be used to estimate the overall ef-fective elastic moduli ofthe studied materials becausethe mod-uli contrastbetweenCu andNbisweak(Beichaetal.,2016).The GSC schemewasdevelopedby considering atfirstthe “n-layered cylindricalinclusionproblem”:ann-layeredcylindricalinclusionis embedded in an infinite matrix (i.e. phase n₊1);each phase is homogeneous,linearlyelastic,transverselyisotropicwiththe sym-metryaxisalongthefiberdirectionx1.Inaddition,perfectbonding

isassumedrequiringthecontinuityofthestressvectorandofthe displacementfieldattheinterfacesofdifferentphases.The above-definedspecimenissubjectedtoseveraldifferentremoteboundary conditions in Hervé and Zaoui (1995) (so-called in-plane hydro-staticmode,normaltensilemode,in-planetransversesharemode andanti-planelongitudinalshearmode)aimingtoderivethe elas-ticstrainandstressfieldsineachphase.Theinfinitematrix,phase n₊1_,hasbeenthenreplaced byan unknownHEMcharacterized by theeffectiveelastic tensorC

≈.This tensorisfinally determined

thanks toaself-consistentenergyconditionandtothepreviously solved “n-layered cylindricalinclusion problem” (Christensen and Lo,1979;Hervé and Zaoui,1995).Inthe present work,the effective elasticbehaviorofthestudiedCu–NbCompositeCylinders Assem-blyiscomputedbytheGSCschemeforH1andH2,respectively,as illustratedinFig.5(c)and(e).

3.3. Full-fieldperiodicmodels

Inadditiontomean-fieldSSCandGSCschemes,afull-fieldFEM PH is proposed here to homogenize the effective elastic behav-ior ofCu–Nbwires atallscales (Fig. 3).Anelementary volumeV made of heterogeneous material is considered forpolycrystalline aggregates (scaleH0) in Section 3.3.1 andfor a specific

architec-ture(scalesH1–H3)inSection 3.3.2.Periodicboundaryconditions are prescribed atits boundary

∂

V.The displacement field u inV takesthefollowingform:

u

(

x

)

=

ε

¯

∼· x+

v

(

x

)

∀

x∈V (9)

where the fluctuation v is periodic, i.e. it takes the same values at two homologous points on opposite faces of V. Furthermore, thetraction vector

σ

∼·ntakes opposite values at two homologous

pointsonopposite facesofV(nistheoutwardsnormalvector to

∂

Vatx_∈V).

Using the Voigt notation, stress and strain fields

σ

∼ and

ε

∼are

expressedas6-dimensionalvectors:

σ

=(

σ

11,

σ

22,

σ

33,

σ

23,

σ

13,

σ

12)and

ε

=(

ε

11,

ε

22,

ε

33,2

ε

23,2

ε

13,2

ε

12).Inordertodetermine

the symmetric anisotropic tensorC

≈, six computations are

neces-sary for each statistical realization of the volume element V to find the 21 elastic coefficients (Kanit et al., 2003). Here, we im-posesuccessivelysixmacroscopicnormalandshearstrain bound-ary conditions over V as follows:



ε



= e_i where e_i denotes the 6-dimensionalunit vector, andi variesfrom1to 6.Then six ho-mogenizedstress tensors

σ

¯ can be determined by numerical ho-mogenizationleading to the effective elastic stiffnessC

≈ by using Eq.(4).

3.3.1. PHadaptedforpolycrystallineaggregates

Unlike mean-field models in which the microstructure is de-scribed statistically, FEM PH can account for the real experi-mental microstructure at scale H0, and it provides the full-field stress/strainfieldsoverV.Inthepresentwork,PHisusedtostudy theeffectofa specific morphological/crystallographictextureand to compare the results with the ones obtained with mean-field methods.

AsillustratedinFig.6(a),thepolycrystallineaggregateis repre-sentedbyaregularcubicgridmadeofparallelepipedicgrainswith theaspectratio/d=1.Alongeach edgeofthisthree-dimensional tessellation(finiteelementmeshusingc3d20),3 _10grainsare

con-sidered,leadingto10× 10× 10=1000grainspermesh.Theused grain orientations (i.e. crystallographic texture) and single crys-tal properties of Cu and Nb are those given in Sections2.3 and

2.4,andthediscreteorientationsarespatiallyrandomlydistributed amongthegrainsoftheparallelepipedictessellation.This elemen-taryvolumeis subjectedtoperiodicboundaryconditions,Eq.(9),

Fig. 5. Multiscale modeling of effective elastic behaviors of Cu–Nb composite wires (Orange stands for Cu, gray for Nb, and brown for Cu–Nb composite). (a,b) Cu and Nb polycrystals at the effective scale H0; the effective elastic tensor is obtained by the SSC scheme. (c) At scale H1, using the GSC scheme, the Cu–Nb 3-layered cylindrical inclusion is surrounded by an infinite Homogeneous Equivalent Medium; the Cu–Nb Composite Cylinders Assembly is assumed to exhibit a random fiber distribution in this model. (d) Using the FEM PH model at scale H1, a periodic distribution is assumed ( # denotes periodic boundary conditions). (e) At scale H2, GSC scheme assumes a 2-layered cylindrical inclusion surrounded by a HEM. (f) Using PH at scale H2, the distribution is periodic. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

inordertotakeadvantageofasmallerRVEthantheonewith ho-mogeneousboundaryconditions(Kanitetal.,2003).

Voronoï tessellationinFig.6(b)isalsousedtoevaluatethe im-pactoftheover-simplifiedparallelepipedicgrainshapeinthe pre-viousproblem.Thisaggregate(finiteelementmeshusingc3d4)4 _is

subjectedto periodicboundary conditions,and 100Voronoï cells aredistributedrandomlyinspace.Thegrainaspectratio/dis sta-tistically1.00 witha 95% confidenceinterval of ± 0.34. Although more realistic than parallelepipedic grains, Voronoï tessellations stillprovideasmallergrainsizedistributionthanrealpolycrystals (Lebensohnetal.,2005;Gérardetal.,2013).

Due to the fabrication process, Cu and Nb grains are highly elongated along x1. In order to take this morphological texture

into account, a series of parallelepipedic tessellations (c3d20) is alsoconsidered.ThecaseL_{/D =/d=}2,whereLandDdenotethe meshlengthandwidthrespectively,isillustrated inFig.6(c). Sev-eralmesheselongatedalongthewiredirectionx1 areusedinthis

worktoconsidervariousgrain aspectratios: _/d₌2,5,10and20. The case /d → ∞ is represented by a slice-shaped parallelepi-pedictessellation composed of31× 31=961 grainssubjected to periodic boundary conditions (Fig. 6(e)). Furthermore, the mesh densitythroughoutthis workhasalso beencheckedforensuring adaptednumericalaccuracy.

When a single realization over the elementary volume V is used,a relatively limited numberof grain orientations andgrain

4 c3d4: linear tetrahedrons with 4 nodes per element.

neighborhoods are accounted for. Thisleads to a bias in the es-timation of the effective properties as explained in Kanit et al. (2003).TheRVE(Hill,1963)mustcontainasufficientlylarge num-ber of heterogeneities (e.g., grains, inclusions, fibers ...) for the macroscopic propertiesto be independenton the boundary con-ditionsappliedtothisvolume.Kanitetal.(2003)proposeda sta-tisticalstrategytodeterminetheRVEsize.Followingthisapproach, weconsiderNrealizationsofthemicrostructureinavolumewith given size. This volume size is then increased to investigate the asymptoticelasticbehavior:

Z= 1 N N  i=1  Zi, D2Z= 1 N N  i=1

(

_{Zi− Z}

₎

2 ₍₁₀₎

whereZ_i isanapparent elasticmodulusobtainedforone realiza-tion andZ is its meanvalue overN realizations. Inaddition, the varianceD2

Z expressesthefluctuationsofZi.

The number of realizations N is chosen so that the obtained mean value Z does not vary any longer up to a given precision whenNisincreased. Thisprecision (i.e.relativeerror

ε

rela)ofthe

estimationoftheeffectivepropertyZisrelatedtothestandard de-viationDZandthenumberofrealizationsNby:

ε

rela=

2DZ

Z√N. (11)

Conventionally,when

ε

rela ≤ 1%, we supposethat thenumber

ofrealizationsNissufficientlylargeforbeingstatistically represen-tativeofheterogeneoustextures.TheRVEisthendetermined,and

Fig. 6. Meshes of PH models for polycrystalline aggregates: (a) parallelepipedic tes- sellation of (10 × 10 × 10) 10 0 0 grains with the aspect ratio  / d = 1; (b) Voronoï tessellation of 100 grains with  / d = 1.00 ± 0.34; (c) parallelepipedic tessellation of (10 × 10 × 10) 10 0 0 grains with L/D = /d= 2, L and D being defined as the length and width of the volume element, respectively; tessellations of L/D = /d= 1,2,5,10 and 20 are considered in this work; (d) Individual columnar grain for parallelepi- pedic tessellations in the case of  / d = 2, elongated along the wire direction x 1 ; (e) parallelepipedic tessellation of (31 × 31) 961 grains with  / d → ∞ . Periodic bound- ary conditions, denoted #, are considered.

theoveralleffectiveelasticpropertyisdefinedbythemeanvalue over N realizations, Z.In addition, the95% confidenceinterval is givenby[Z− 2DZ,Z+ 2DZ].

ForCupolycrystals,theparallelepipedictessellationsrequireat least N=3 realizations forensuring

ε

_rela ≤ 1%forall of thefive independentcomponentsoftheelasticmodulus.Inthiswork,the effectiveelastictensorof

(

C

≈

)

H0-CuaredeterminedbyusingN=10

random realizations,i.e.a total of10,000 (10 × 1000) crystal-lographic orientationsconsidered. This hasbeen done to reach a narrowerconfidenceinterval, whichisausefulpropertyfor com-paring resultswith those ofthe SSC scheme.We alsochoose 10 realizationsfortheslice-shapedtessellationwith/d→∞.

Ontheotherhand,eachrealizationoftheusedVoronoï tessel-lationcontainsasmallernumberofgrains(100) than parallelepi-pedictessellations.ForCupolycrystal,itwasfoundthat30random realizations are necessary to ensure statistical representativityof the results. Therefore,the number of used Voronoï cells is 3000 (30× 100).ThesamenumberofrealizationwasusedforNb poly-crystals.

3.3.2. PHadaptedforcompositecylindersassemblies

When usingtheanalytic mean-fieldGSC schemetodetermine theeffectiveelasticmoduliatscalesH1toH3,thelongfibersare assumedtobedistributedrandomly.Inordertotakeinto

consider-Table 4

Effective transversely isotropic moduli of Cu polycrystals and Nb polycrystals at scale H0 and their 95% confidence intervals. Experimental crystallographic textures and various morphological textures are considered for the SSC scheme and PH. Both parallelepipedic (para) and Voronoï (Voro) tessellations are used for PH.

Model SSC PH (para) PH (Voro) SSC PH (para)

 / d 1 1 1.00 ± 0.34 100 ∞

H0-Cu with a double fiber  100  and  111 

 E1 (GPa) 130.68 132.63 ± 3.72 133.90 ± 8.0 141.45 141.30 ± 2.33  ν12 0.340 0.338 ± 0.006 0.336 ± 0.012 0.327 0.327 ± 0.004  K23 (GPa) 152.37 152.79 ± 1.08 152.95 ± 2.39 153.88 153.91 ± 0.82  μ12 (GPa) 47.65 48.30 ± 0.80 49.40 ± 1.80 46.43 46.30 ± 0.52  μ23 (GPa) 48.20 48.11 ± 0.61 49.17 ± 1.14 46.86 46.80 ± 0.64 H0-Nb with a single fiber  110 

 E1 (GPa) 96.66 96.05 ± 0.70 96.56 ± 1.16 95.85 95.15 ± 0.79  ν12 0.408 0.410 ± 0.004 0.409 ± 0.005 0.408 0.411 ± 0.003  K23 (GPa) 185.75 184.54 ± 1.22 184.78 ± 2.28 185.65 184.48 ± 1.28  μ12 (GPa) 39.60 38.36 ± 0.54 38.69 ± 1.23 39.48 38.32 ± 0.62  μ23 (GPa) 36.88 38.27 ± 0.54 38.44 ± 1.30 37.04 38.26 ± 0.62

ationthequasi-periodicfiberdistributionobservedexperimentally (Fig.1)andtoinvestigatetheeffectofthisparticulardistribution, FEM PH will be also carried out for scales transitions H1 to H3 (Fig. 3). The section views ofthe unit cell ofH1 andH2 are re-spectivelyindicated in Fig. 5(d)and (f).Unit cellscontain all in-formationaboutthemorphologicalRVEattheeffectivescales H1 andH2.Theyarecomposed oftwo equivalentlong fibers(1+4× 1/4 fibers) which are arranged in an hexagonal lattice, andthey representthe(idealized)multi-scaledexperimentalmicrostructure oftheCu–Nbwires.Calculationsrequirea2Danalysiswith gener-alizedplane strain conditionsin orderto allowforhomogeneous straininthethirddirectionandalsoforoutofplaneshearing.For thatpurpose,a3Dmeshwithonesingleelementinthethickness (c3d20elements)isused, togetherwithsuitable boundary condi-tionstokeepflatupperandlowerplanes.

4. HomogenizationresultsatscaleH0

Inthissection,theeffectiveelasticbehaviorofCuandNb poly-crystals,i.e.atscale H0,is considered.Results ofthe various ho-mogenization schemes presented above will be compared with eachother.

As explained above, the anisotropic effective stiffness tensors

(

C

≈

)

H0-Cu and

(

C≈

)

H0-Nb obtained by the SSC scheme and PH are

foundtobetransverselyisotropicifwecomputeastatistically suffi-cientlylargeequivalentvolume(i.e.RVE).Itshouldbenoticedthat, unlikethehomogenizedresultsofPH,theSSConesdisplay negli-giblescatteringbecauseoftheverylargenumberofgrain orienta-tionsconsidered.

Five independent effective moduli were used to describe the overall anisotropic elastic behavior. Results for two morphologi-cal textures (/d=1, and highlyelongated grains) are shown in

Table4.In thecaseofequiaxedgrains(_/d=1), itisremarkable thatall theelastic moduliobtainedforthethreemicrostructures, i.e.SSC, parallelepipedic, and Voronoï tessellations, are in a per-fect match, witha percentagedifference5 _{smaller than 4%.There}

aresome visibledifferencesofconfidenceintervalsforPH homog-enizations,theones forparallelepipedic tessellations beingabout halfthose forVoronoï tessellations; thismight be mainly caused bythecomputednumberofgrainorientationsconsideredinthese calculations.ItcanalsobeobservedthatresultsoftheSSCscheme fallwithin the confidenceinterval ofPH, except for

μ

˜12 and

μ

˜23

forNbpolycrystalswhereslightlylargerdifferencesarefound.

5 Percentage difference (in %): the absolute difference between two values di- vided by their average.

Fig. 7. Effective longitudinal and transverse Young’s moduli (  E1 and  E2, 3 , respec- tively) in terms of the grain aspect ratio  / d . The fiber textured Cu polycrystal and Nb polycrystal are homogenized by the SSC scheme and PH (using parallelepipedic tessellations).

Fig.7furtherillustratestheSSCandPH(usingparallelepipedic tessellations) predictions of longitudinal and transverse Young’s moduli (E1 and E2,3) as a function of the grain aspect ratio /d,

forboth Cu andNb polycrystals. Correspondingnumerical values ofthefiveindependenteffectivemoduliareprovidedinTable4for /d=100(SSC scheme)and/d→∞ (PH).Again,the agreement betweenallresults isexcellent. Itcan howeverbe noticedthat a largerdiscrepancyisobservedforthetransversemodulusE2,3

ob-tainedforNbpolycrystals. Additionalnumericaltestswillbe per-formedinSection6.1todiscussthemainfactorsthatcontributeto this(small)difference.

BothSSCscheme andPHdemonstrate remarkablythat,forCu polycrystals with the double



100



−



111



fiber texture, aggre-gates with elongated grains display stiffer effective longitudinal Young’s moduli than the ones with equiaxed grains. For /d > 20,thistendencybecomessaturated,andtheelongationofgrains along the wire direction x1 has no more effect. In contrast, for

Nb polycrystals with the single



110



fiber texture, the effective Youngmoduli do not dependon themorphological textures.The

effectsofmorphologicalandcrystallographictexturesonthe effec-tiveelasticbehaviorwillbefurtherdiscussedinSection6.2.

5. EffectivebehaviorsatscalesH1toH3ofarchitectured

Cu–Nbcomposites

5.1. ResultsforscaleH1

The SSC estimate is often advocated to be a good model for polycrystallineaggregatesfortworeasons:(i)SSCschemeassumes a perfectly disordered mixture of grains which is similar to the realmorphologicaltextures inpolycrystals;(ii)SSCmodelcan be appliedtoastatisticallylargeheterogeneousvolumecomposedof a very large number of grain orientations without costing much CPU time. This has been confirmed by our results of the previ-oussection.Therefore,thehomogenizedanisotropicelastictensors

(

C

≈

)

H0-Cuand

(

C≈

)

H0-NbforCupolycrystalsandNbpolycrystals

deter-mined bytheSSCschemewillbe takenhereaslocalconstitutive behaviorsfortheupperscale transitions(Fig.3andFig.5(a) and (b)).SincegrainsinCu–Nbwiresarehighlyelongatedwith_{ d}, we considerwithinthe SCCschemeagrain aspectratio/d=100, believedto be a good approximation of thecolumnar grains ob-served inthe real microstructure (a larger aspect ratio does not changesignificantlytheelasticproperties).

Wenowproceedtothehomogenizationoftheassemblyof851

elementary continuum long fibers,i.e.at scaleH1. BothGSCand PHmodelsareappliedtothespecificCompositeCylinders Assem-blymadeoftheco-cylindricalpatternswiththreelayers:Cu-f /Nb-t/Cu-0, withproperties atscales H0provided by the SSCscheme asdetailedabove.AspresentedpreviouslyinFig.5(c)and(d),GSC scheme andPHassume that theCu–Nb Composite Cylinders As-semblyexhibitsarandomandaperiodicfiberdistribution, respec-tively.

Besides,forthesakeofcomparison,onecanalsouseforscale H1 thesimple SSC scheme,thus assuming a microstructure con-sistingofthesolerandommixtureofCuandNbgrains,i.e.without considerationanymoreofthespecificarchitectureofthereal spec-imen (Fig.3). The volumefractionof CuandNb phasesbecomes 51.6% and48.4%, respectively.Thiscorresponds to thenormalized volumefractionof10.5%,48.4%,and41.1%forCu-f,Nb-t,andCu-0, respectively(seeSection2).

Thetransverselyisotropiceffectivemoduli

(

C

≈

)

H1ofthe

assem-blyof851 _{elementarylong fibers(scaleH1)are giveninTable5.}

It is remarkablethat the GSCscheme andPH providevery close results,andresultsoftheSSCschemeare alsoinaperfect agree-ment. The percentagedifference betweenthe prediction ofthese threemodelsislessthanonly2%.

5.2. IterativescaletransitionprocessuptoscaleH3

At the effective scale H2, we suppose that the 85continuum cylindersarecomposedoftwolayers:(1)thenano-compositeCu– Nb zonescontaining851 _{elementary long fibers; (2)the}

embed-ding matrix Cu-1. In this work, an iterative process is proposed. The effective tensor of the inner layer for scale H2,

(

C

≈ (1)

₎

H2, is

givenby the effectivetensor

(

C

≈

)

H1 obtainedforscale H1.Onthe

other hand, the effective behavior of the second layer for H2,

(

C ≈

(2)

₎

H2, isassociated withtheeffectivebehavior ofCu

polycrys-tals,

(

C

≈

)

H0-Cu. The scale transitionis then performedby GSC and

PHapproaches,leadingtotheeffectivetensor

(

C

≈

)

H2forthe

assem-blyof852_{elementarylongfibers.}

Thesameiterativeprocess willberepeatedup toscaleH3 us-ing GSC andPH approaches, allowing to estimate

(

C

≈

)

H3, the

Table 5

Effective transversely isotropic moduli of Cu–Nb wires at scales H1, H2, and H3 (i.e. homogenization of the assembly of 85 1 , 85 2 and 85 3 elementary long fibers, respectively), obtained by mean-field SSC (  / d = 100) and GSC schemes, and by full-field PH.

Scale H1 H2 H3 Model SSC GSC PH SSC GSC PH SSC GSC PH  E1 (GPa) 117.39 119.62 117.44 123.07 123.64 121.86 128.51 128.56 127.27  ν12 0.371 0.367 0.371 0.361 0.360 0.363 0.351 0.351 0.353  K23 (GPa) 169.56 168.07 169.56 165.67 165.33 166.51 162.06 162.06 162.89  μ12 (GPa) 42.53 42.93 42.58 43.40 43.55 43.26 44.27 44.32 44.11  μ23 (GPa) 41.21 41.77 41.33 42.44 42.64 42.29 43.68 43.75 43.49

Cu-2.Moreover,asmentionedinSection5.1,regardlessofthe spe-cificfilament/nanotubemicrostructure,theSSCschemecanalsobe usedtopredict

(

C

≈

)

H2and

(

C≈

)

H3, considering thetrue volume

frac-tions ofCu–Nb phasesforscales H2and H3,thecrystallographic andmorphological (/d=100) texture, butdiscarding thematerial architecture.

TheeffectivemoduliofH2andH3areindicatedinTable5.As before, SSC, GSC, andPH homogenizations exhibit very close re-sultsatalltheeffectivescalesconsidered,themaximalpercentage difference amongthem beingassmallas ∼ 1.5%.Thisresultwill receivefurtherattentioninSection6.3.

6. Discussion

6.1. SSCandPHpredictionsatscaleH0

Inthissection,weinvestigatethefactorsthatcontributetothe deviationoftheSSCschemewithrespecttoPHforCupolycrystals andNbpolycrystals,i.e.consideringscaleH0.Then inSection6.2, therole ofmorphologicalandcrystallographic texturesonthe ef-fectiveelasticbehaviorwillbediscussed.

The effectiveelastic tensors

(

C

≈

)

H0-Cu and

(

C≈

)

H0-Nb were

deter-minedinSection4forthedouble



100



−



111



fibertexturedCu polycrystal and the single



110



Nb polycrystal. Both mean-field SSCschemeandfull-fieldPHwereapplied,andanexcellent agree-ment ofmodelresponses were found.However, a larger percent-age difference was found for E2,3 in the case of Nb polycrystals

(seeFig.7),adeviationthatseemslargerthanforallother investi-gatedmoduli.Forabetterunderstandingofthedeviationbetween theSSCschemeandPH,additionalnumericaltestshavebeen per-formed.

First of all, we exchanged the crystallographic textures : the experimental



110



fiber texture of Nb is taken as a fictitious crystallographic texture for Cu polycrystals. Similarly, the double



100



₋



111



fibercomponents of Cu are takento build a ficti-tious polycrystal of Nb. The predictions for the longitudinal and transverseYoung’smoduli(E1 andE2,3respectively)areplottedin Fig.8asafunctionofthegrainaspectratio/dforthesefictitious textures.

As before, it is observed that, despite the large difference of the Zener anisotropy factor Z between Cu and Nb, SSC and PH models provide very similar results, for all the considered grain aspect ratio /d andcrystallographic textures. However, one sees that alargerdiscrepancyisnowobservedfortheE2,3modulusof

Cu polycrystals, withthe fictitious



110



fibertexture. These dif-ferences mainly arise because thesemodels do not take into ac-count exactly thesame grain topology: perfectlydisordered mix-ture of grains for the SSC scheme and regular parallelepipedic grains of PH. Grain size graduation is also infinite for the SSC scheme,whereasgrainsall havethesamesizeforthe parallelepi-pedictessellations;grain sizedistributionoftheVoronoï tessella-tionisnarrow.CombiningTable4,Figs.7and8,itcanbeconcluded thattheeffectivepropertiesofsharp



110



fibertexturesaremore

Fig. 8. Effective longitudinal and transverse Young’s moduli (  E1 and  E2, 3 , respec- tively) in terms of the grain aspect ratio  / d , obtained by SSC and PH (using paral- lelepipedic tessellations). Cu polycrystal and Nb polycrystal are textured by fictitious

 110  and  100  − 111  fibers, respectively.

sensitivetomicrostructure detailsthandouble



100



−



111



tex-tures.

6.2.Anisotropyinducedbymorphologicalandcrystallographic textures

In the precedingsection, the SSCscheme has been shown to provide almost identicalresults than PH forvarious

morphologi-Fig. 9. Effective longitudinal and transverse Young’s moduli (  E1 and  E2, 3 , respec- tively) in terms of the grain aspect ratio  / d predicted by the SSC scheme. Cu poly- crystal and Nb polycrystal with experimental fiber textures are compared with the ones with a random (isotropic) texture.

calandcrystallographictextures.Thankstoits highnumerical ef-ficiency,theSSC schemewillnow be usedto explorethe role of micro-parameters,suchasmorphologicalandcrystallographic tex-tures.

Fig.9showspredictionsoftheSSCschemefortheeffective lon-gitudinalandtransverseYoung’smoduli(E1 andE2,3,respectively)

asfunctionsofthegrain aspect ratio_/d,forCu polycrystals and Nbpolycrystals witha random(i.e.isotropic)crystallographic tex-ture.Resultsarecomparedwiththeonesobtainedfor experimen-talfibertextures ofFig.7.ForNbpolycrystals,itcanbe observed thatgrainmorphologyonlyhasasmalleffectontheeffective be-havior,forbothrandomand



110



crystallographictextures.ForCu, theeffectofgrainmorphologydependsonthetexture.Ithasonly asmallinfluencefora randomtexture, butitaffectssignificantly (byabout10%)E1fortheexperimental



100



−



111



texture.

Tobemorequantitative,Thomsencoefficients(Thomsen,1986) areused for characterizingthe transverse isotropy. These dimen-sionless parameters are a combination ofthe components ofthe

Fig. 10. Thomsen parameters in terms of the grain aspect ratio  / d , obtained with the SSC scheme. Experimental fiber textures are used for Cu and Nb polycrystals.

elasticstiffnessmatrix



=C33− C11 2C11 ,

δ

=

(

C13+C66

)

2−

(

C11− C66

)

2 2C11

(

C11− C66

)

,

γ

=C44− C66 2C66 , (12)

whereindex1indicatesthesymmetryaxis(x1).Forisotropic

elas-ticity,thethree Thomsenparameters arestrictly equalto 0. Con-versely, the elastic mechanicalbehavior exhibits moreanisotropy with larger absolute values of



,

δ

and

γ

. Note also that the absolute value of these parameters is usually much less than 1.

Fig. 10 illustrates these parameters obtained for

(

C

≈

)

H0 using the

SSC schemein terms of/d forthe Cu and Nbpolycrystals with experimentalfibertextures.It canbeagainobservedthat the be-havior of the



110



fiber textured Nb is much less sensitive to grain morphology than the



100



−



111



Cu. Forthe latter one, the anisotropy isweak forequiaxed grainshape (/d=1),and in-creasessignificantlywiththegrainaspectratio.

The Cu polycrystals and Nb polycrystals with double



100



−



111



fiber textures have been studied previously. We now

pro-Fig. 11. Effective longitudinal and transverse Young’s moduli (  E1and  E2,3, respec- tively) of perfect single-component  100  and perfect single-component  111  fiber textured Cu by the SSC scheme in terms of  / d .

ceed to predict the effective elastic properties of perfect6



₁₀₀



fiber textured aggregates and perfect



111



ones separately for the propose of comparison. An analytic solution for this ho-mogenization problem has been derived by Walpole (1985). For Cu polycrystalline aggregates, one get E1 =66.03GPa, E2,3∈

[87.54,105.79]GPafora



100



fibertexture,andE1 = 191.49GPa,



E2,3∈[129.82,160.93]GPa) for



111



. Fig. 11 illustrates the

effec-tive Young’smoduli(E1 andE2,3) obtainedbytheSSCschemefor

Cupolycrystalswithvariousmorphologicaltextures.Theseresults, ofcourse,areconsistentwiththeanalyticalsolutionofWalpole.It can be noted that E1 and E2,3 are significantly different forboth

texture components.Moreover, it can be observed that, although 

E1isinsensitivetothegrainaspectratioforbothindividual



100



and



111



texture components, it become sensitive to the aspect

6 A texture component is perfect when its spread FWHM vanishes, i.e. here all {100} planes lie exactly parallel or perpendicular to x 1 axis.

ratiowhen the two componentsare mixedtogether (see Fig. 7). In contrastE2,3 decreases withthe aspect ratio forboth

individ-ualtexturecomponents,butbecomesratherinsensitivetoitwhen theyaremixedtogether.

For sake of comparison, we have also computed the ef-fective elastic parameters for Nb polycrystal with the per-fect



100



fiber texture and the perfect



111



one: E1 =

145.14GPa, E2,3 = 117.69GPa for



100



and E1 = 83.35GPa,



E2,3 =95.89GPafor



111



.Itisfoundthattheseresults(whichare

inagreement with Walpole (1985)estimations: E1 = 145.48GPa,



E2,3∈[113.62,121.38]GPa for



100



and E1 = 83.24GPa, E2,3∈

[93.21,98.28]GPafor



111



)areonlyslightlysensitivetothegrain aspectratio.Notealso that thedifference inE1 andE2,3 forboth

texturecomponentsislessthanforCu,indicatingasmaller effec-tiveanisotropy.

ForCupolycrystals,botheffectiveYoung’smoduli(E1andE2,3)

andThomsenparametersdepend onthe grainaspect ratio_/din therangeof1≤ /d<20 irrespective ofthefibertextures. How-ever,thisdependencysaturateswithincreasinglyelongatedgrains along the wire directionx1. A similar feature has also been

ob-served for Ni alloy directionally solidified polycrystalline aggre-gates in Yaguchi and Busso (2005).In contrast, the effect of the grainaspectratio_/donthestiffnesscoefficientsofNbpolycrystal isnon-zerobutrathersmall.

Concerningthe effectof crystallographictexture, we have ob-tained different E1 values for different textures,for Cu

polycrys-tals:66.03GPa forperfectsinglefiber



100



, ∼ 146GPafor



110



(meanvaluebetween/d=1and/d→∞),and191.49GPafor per-fect



111



.Thepercentagedifferencesbetweenthefirsttwomoduli andthelasttwoonesaresignificant,i.e.75%and26%,respectively. ForNb polycrystals, E1 wasfound to be 145.14GPa for a perfect

single



100



texture, ∼ 96GPa forsingle



110



,and83.35GPafor perfectsingle



111



.Thus,thepercentagedifferencesbetweenthe firsttwoonesandthelasttwoonesarelessthanforCu(40%and 15%,respectively),butstillsignificant.

In summary, it was found that crystallographic textures play an important role in the effective elastic moduli of Cu and Nb polycrystals. Thecombination ofcrystallographic and morpholog-icaltextureeffects isnotstraightforward. Forexample,grains av-eragemorphology hasstrictly noeffectonE1forCuwitha



100



or a



111



perfecttexture, but is responsible for a 10% variation forthesharp



100



−



111



experimentaltexture. Withthe exper-imental textures,grain morphologyhas amuch smallereffect on NbthatonCupolycrystals.Theseresultshighlightthenecessityof accountingforthecorrectgrainmorphologyandorientationwhen modellingtheeffectivebehavioratthedifferentscales.Acom par-isontoexperimentaldataisprovidedintheSection6.4.

6.3.ModelingstrategiesforCu–Nbwires

InSection5,threehomogenizationmodelswereappliedto per-formthescaletransitionsuptoscaleH3.TheSSCschemeassumes arandommixture ofCuandNbphases;theGSCschemeandPH bothtake intoaccountthespecific CompositeCylindersAssembly microstructurewithrandomandperiodicdistribution,respectively. AsshowninTable5,thethreemodelsprovideverycloseresultsat H1,H2,andH3scales,inspiteoftheverydifferentapproximations ofgeometry.

Thisgoodmatchislikelyduetotherelativelysmallelastic con-trastbetweenCu andNbelastic behavior.Forinstance,the effec-tivelongitudinalYoung’smoduliratiobetweenCupolycrystalsand Nb polycrystals,

(

E1

)

H0-Cu/

(

E1

)

H0-Nb, is about 1.5 (Table 4).

Con-versely,asshownbyLlorcaetal.(2000),Beichaetal.(2016), con-siderabledeviationsbetweenSSC,GSCandPHmodelsareobtained whenthecontrastisenlarged.ForinstanceinLlorcaetal.(2000), Young’smodulusobtainedbytheSSCschemecanbetwicestiffer