Representative volume element size determination for viscoplastic properties in polycrystalline materials

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Justin DIRRENBERGER, Eric MONTEIRO, Nicolas RANC - Representative volume element size

determination for viscoplastic properties in polycrystalline materials - International Journal of

Solids and Structures - Vol. 158, p.210-219 - 2019

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Representative

volume

element

size

determination

for

viscoplastic

properties

in

polycrystalline

materials

S.

Yang,

J.

Dirrenberger

∗

,

E.

Monteiro,

N.

Ranc

PIMM laboratory, Arts et Métiers ParisTech, Cnam, CNRS, Paris 75013, France

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 26 February 2018 Revised 18 July 2018

Available online 11 September 2018

Keywords:

Representative volume element Computational homogenization Crystal plasticity finite element method Apparent viscoplastic parameter Intrinsic dissipation

a

b

s

t

r

a

c

t

Thesizeofrepresentativevolumeelement(RVE)for3Dpolycrystallinematerialisinvestigated.A statisti-calRVEsizedeterminationmethodisapplied toaVoronoi tessellation-basedpurecopper microstruc-ture. The definition ofRVE hasremained problematic inthe literature for properties relatedto non-linearviscoplasticbehavior, e.g.apparentviscoplastic parameter,intrinsicplastic dissipation. Computa-tionalhomogenizationforelasticandplasticpropertiesisperformedwithinacrystalplasticityfinite ele-mentframework,overmanyrealizationsofthestochasticmicrostructuralmodel,usingperiodicboundary conditions.Thegenerateddataundergoesstatisticaltreatment,fromwhichRVEsizesareobtained.The methodused fordeterminingRVEsizeswas foundtobeoperational, evenfor viscoplasticity.The mi-croscaleanalysis ofthe full-fieldsimulationresultsrevealsmicrostructure-relateheterogeneities which shednewlightontheproblemofRVEsizedeterminationfornonlinearproperties.

1. Introduction

Inthepastdecades,full-fieldnumericalsimulationof polycrys-tallinematerials basedonfiniteelementanalysishasbeenwidely developed to investigate the mechanical behavior, allowing the analysisofstress andstrain fieldsata scalethat isnot easily as-sessable experimentally (Barbe et al., 2001; Roters et al., 2011). Mostofthe authorsin the literature dedicatedto thesimulation of polycrystals usually consider a population of virtual polycrys-tallinesamplesmadeofseveralhundredgrains,validatingthis ar-bitrarychoicebyanalyzingthemeanvalueandstandarddeviation fora givenproperty computedonsuchpopulation(Shenoyetal., 2007,2008;Robertetal.,2012;Martinetal.,2014;Sweeneyetal., 2015; Cruzado et al., 2017, 2018). Nevertheless, the development offull-fieldsimulationofpolycrystallinematerialsresultsin shed-dingnewlightontherelationshipbetweenthemicrostructural de-scriptionatthedislocationorgrainscaleandthelocalmechanical behavior(Cailletaudetal., 2003a). Thehomogenized macroscopic response of a polycrystalline material sample will depend on its size,hence yielding the questionof representativityforsuch vir-tualsamples.Tobringan answertothisquestion,one musthave aproperdefinitionofwhatanRVEis.

Inhomogenizationmethods,theconceptofRVEwasfirstly pro-posedby Hill(1963)as“asample thatis thestructurallytypicalof thewholemicrostructurefora givenmaterial,i.e.containinga

suffi-∗ _{Corresponding author.}

E-mail address: Justin.DIRRENBERGER@ensam.eu (J. Dirrenberger).

cientlylarge numberof heterogeneities,whilebeingsmallenoughto beconsideredhomogeneousfromacontinuummechanicsviewpoint”. The quantification of RVE size has been problematic until vari-ousstatisticalapproacheswere proposedinthepasttwo decades (Gusev, 1997; Terada et al., 2000; Kanit et al., 2003; Gitman et al., 2007). Based on several statisticalhypotheses, Kanit etal. (2003) proposed a statisticalapproach to determine the minimal RVEsizeforaconsideredproperty,inwhich,theRVEsizecouldbe associatedwithagivenprecisionoftheestimatedoverallproperty andthenumberofrealizationswitha givenvolume sizeV. Prac-tically,itisapplicableinorderto determinetheminimalnumber ofrealizationstoconsiderforagivenvolumesize,inorderto esti-matetheeffectivepropertywithagivenprecision(Bironeauetal., 2016).

Usingthisapproach,Kanitetal.(2003)studiedtheRVEsizesof atwo-phase 3D Voronoimosaicforlinearelasticity,thermal con-ductivity andvolume fraction,under uniformdisplacement, trac-tionandperiodicboundaryconditions(PBC)(Michel etal., 1999). The results showed that the PBC held an advantage of conver-gencerateofthemeanapparentpropertiesincomparisontoother boundary conditions, dueto the vanishing of boundary layer ef-fects. A slow rate of convergence for the considered properties wouldyieldalargeRVEsizes(Dirrenbergeretal.,2014).Also con-sideringthelargecalculation costinthe caseofcrystalplasticity, itis preferableto relyon PBCinorder tooptimizethe computa-tionstrategy,asitwasdoneinotherinvestigations(Pelissouetal., 2009andJeanetal.,2011).

The statisticalmethodofKanitetal.(2003)wasimplemented fortheestimationofRVEsize,notonlyforlinearmechanical prop-ertiesandmorphologicalproperty, butalsoforplasticproperties: Madi et al.(2006) evaluated the RVE size for2D/3D viscoplastic compositematerials.Intheir study,themacroscopicstrain rateof the2D/3D materialwasmodeledusinga Nortonflowrule.Based onthevonMisescriterion,anapparentviscoplasticparameterP_vapp

wasfirstlydefinedasthecouplingoftwoparameters ofthe Nor-tonflowruleKandn,i.e.P_vapp =1/Kn .Theauthorsshowedthatthe valueofP_vapp convergedtowardsaconstantvaluewithan increas-ingvolumeofsimulationandthattheRVEsizeforP_vapp wasfound tobesmallerthantheonesforelasticmoduli.Inthepresentwork, wewillrelyonthisdefinitionoftheapparentviscoplastic param-eter, as it is adapted for describing the nonlinear behavior of a macroscopicallyisotropicpolycrystallineviscoplasticmaterial.

Asamatter offact,theconceptofRVEhasoftenbeenusedin investigations associatedwiththeaveragemechanicalresponseof 2D and3Dpolycrystallinematerial.ThedefinitionofRVEsizecan stem fromfinite element meshing considerations orconvergence ofmeanvaluesforaconsideredproperty.Forinstance,Barbeetal. (2001) describedthe RVE for a cubic polycrystalline mesh as an equilibriumbetweenthe numberofgrains(238)andtheaverage numberofintegrationpointspergrain(660)attainablewithin typ-icalcomputationalmeans.Morerecently,Sweeneyetal.(2015) es-timatedtheenergeticparameterofCoCrstentmaterialinhigh cy-cle fatigueby averaging in5 RVEs with138–140 grains. Cruzado etal.(2017,2018)simulatedthecyclicdeformation ofmetallic al-loyswith20RVEsandasizeof300grains,whichshowedanerror lessthan 10% forelastoviscoplastic properties.Similar determina-tionofmaterialRVEsizecanalsobefoundinShenoyetal.(2007, 2008), Martin etal.(2014), Gillnerand Mu¨nstermann (2017), Te-ferra andGraham-Brady(2018).In theseRefs, RVEsizeis defined asafewrealizationswithafewhundredgrainswhichcanrealize aconvergenceofmeanproperties.However,theseanalysesdonot allowforarigorousstatisticaldefinitionoftheRVEsize.

Ratherthanrelyingsolelyontheconvergenceofmean proper-ties, themethodproposed inKanit etal.(2003)makesuseofthe rateofconvergenceoftheensemblevarianceofthemean proper-tieswithrespect tothe volumesize,thus enablingthedefinition andestimationofa statisticalRVEsizeforeach considered prop-erty. However, to theknowledge of the authors,no one ever as-sessedthe RVEsize forpolycrystallinematerialinthe framework ofCPFEMwiththestatisticalRVEmethod.

Additional consideration has tobe maderegarding the appar-entpropertiestobeconsideredascriteriaforRVEsize determina-tioninviscoplasticity.Thefirstoneshouldbethedefinitionof in-trinsicdissipationwithinthecontextofcrystalplasticity.Secondly, the definitionoftheapparent viscoplasticparameter P_vapp will be consideredinthecrystalplasticityframework.Meanwhile, forthe crystalplasticitybehavior,materialheterogeneityismainlydueto thelocalgrainorientation,whichcanintroducestrongstress con-centrations,leadingtoearlyonsetofplasticity.Bothgrain orienta-tion andthe choiceof crystalplasticbehavior are likelyto influ-encedirectlythevalueofRVEsizeformechanicalproperties,asit willbediscussedinthepaper.

Overall,inthispaper,theRVEsizeofpolycrystallinepure cop-per willbe studiedinframeworkofCPFEM.In thefollowing sec-tions, thematerial behavior andconstitutivemodelare discussed first. Then, the periodic mesh generation and the computational homogenizationmethodaredescribedindetail,alongsidewiththe various loadingcasescorresponding to thedifferentpropertiesto beconsideredforestimatingtheRVEsizes.Resultsfrom computa-tionandstatisticalanalysisarethenpresentedanddiscussed. Cal-culations forRVE are performedin the last section of thepaper, andcomparisonismadeforthedifferentRVEsizesdependingon theconsideredproperty.

Inthefollowing,vectorsareunderlinedfaceandwrittenin mi-nuscule, e.g. x.Second-rank are bold andslantface, e.g. x andX, andfourth-rank tensorsare boldandstraight facecapitals,e.g. X. Othersarescalar.

2. Crystalplasticityconstitutivemodel

The material involved in this paper was pure polycrystalline copper.Bothanisotropiccrystalelasticityandplasticitywere con-sideredforits behavior.The cubic elasticityischaracterized by 3 independentelasticconstants, takenfromMusienkoetal.(2007). The crystal plasticity model considered in the present work was introduced and implemented by Meric et al. (1991) and Cailletaud (1992) in the finite element code ZeBuLoN/ZSet.1 _The

Meric–Cailletaudmodel was chosen forits ability to account for kinematichardening.Thismodelispopularwithinthecrystal plas-ticity communityandhasbeen used inmanyprevious works on computationalmechanicsforpolycrystallinematerial(Barbeetal., 2001; Cailletaud et al., 2003b; Diard et al., 2005; Gérard et al., 2009).

The model is briefly summarized here. The constitutive rela-tionsaredefinedhereafter:

˙

γ

s ₌



|

τ

s _{− X}s

_|

_{− R} 0 − Rs K



n sign

(

τ

s − Xs

)

=

v

˙s sign

(

τ

s − Xs

)

Xs =C

α

s ; ˙

α

s =

γ

˙s − D

α

s

v

˙s , with

α

s

(

t=0

)

=0 Rs =bQ r hsr qr =Q  r hsr



1− e−b vr



˙ qs =

(

1− bqs

)

v

˙s =

v

· e−b vs ms =



gs ls +ls gs



/2

τ

s _=ms _:

_σ

_{; ˙}

_ε

p ₌ s ˙

γ

s _ms ₍₁₎

whereforpolycrystallinepure copper,12 slipsystemsare associ-atedwiththecalculation,i.e.4 slipplanes gs oftype {111}and3 directionsofslipls oftype



110



,whichdependonthe Euler an-glesofthe grains. TheSchmid tensorms is usedto compute the resolvedshearstress

τ

s andtheplasticstrainrate

ε

˙p .Foreachslip system,the sliprate

γ

˙s isdefinedasa power-lawfunction of

τ

s . TheparametersKandnrelatetothesensitivityofmaterialstothe strainrate.

υ

s representstheaccumulatedplasticstrainfortheslip systems.Thecouplesofthermodynamicalforceandstatevariable, (Xs_,

_α

s_{) and(R}s_,qs_{),respectively are associatedwiththe kinematic}

andisotropichardening.Aseriesofmaterialparametersisusedto definethekinematichardening(C,D),andtheisotropichardening (R₀ ,Q,b).Thesymmetricinteractionmatrixh_sr describestheeffect ofslipon systemson theshearresistanceofslipsystemr,as il-lustratedbyMericetal.(1991).Thisincludesself-hardening(s=r) andlatenthardening(s=r).

Finally,thematerial parameters forahigh-purity copper, K,n, R₀ ,Q,bandhsr ,aregiveninTable1.

3. Computationalapproach

3.1. Definitionofapparentelasticproperties

Themicromechanicallinearelasticbehaviorateachintegration pointinthefiniteelementsimulationisdescribedbyHooke’slaw usingthefourth-ranklinearelasticitytensorC,suchthat:

σ

(

x

)

=C

(

x

)

:

ε

(

x

)

(2)

Table 1

The parameters of pure copper cubic elasticity and plasticity ( Musienko et al., 2007 ).

Elasticity Flow Isotropic hardening Kinematic hardening Interaction slip

Cubic Norton Nonlinear Nonlinear hsr∗

C11 = 159.3 GPa n = 10 R0 = 1.8 MPa D = 600 h 1 = 1, h 2 = 4.4, C12 = 121.9 GPa K = 5 MPa s 1/n _{Q = 6 C = 4500 h 3 = 4.75, h 4 = 4.75,} C44 = 80.9 GPa b = 15 h 5 = 4.75, h 6 = 5 ∗ _{Note: h 1 = h sr (s = r); h 2 = h 21 , h 31 , h 32 , h 54 , h 64 , h 65 , h 87 , h 97 , h 98 , h 11,10 , h 12,10 , h 12,11 ; h 3 = h 52 , h 63 , h 75 , h 83 ,} h91 , h 94 , h 10,6 , h 10,8 , h 11,1 , h 11,4 , h 12,2 , h 12,7 ; h 4 = h 41 , h 72 , h 86 , h 10,3 , h 11,9 , h 12,5 ; h 5 = h 42 , h 43 , h 51 , h 61 , h 71 , h 73 , h 76 , h82 , h 84 , h 85 , h 92 , h 96 , h 10,1 , h 10,2 , h 10,5 , h 10,9 , h 11,3 , h 11,5 , h 11,7 , h 11,8 , h 12,3 , h 12,4 , h 12,6 , h 12,9 ; h 6 = h 53 , h 62 , h 74 , h 81 , h93 , h 95 , h 10,4 , h 10,7 , h 11,2 , h 11,6 , h 12,1 , h 12,8 .

ForagivenvolumeV,thefourth-ranktensorofapparent mod-uliCapp canbedefinedbythemacroscopicrelations:



=

σ

=_V1 

V

σ

dV =Capp :E (3)

where



andEarethemacroscopicstress andstrainsecond-rank tensors.

ForanelementaryvolumeVlargeenough(V>V_RVE ),the appar-entproperties donot dependon the boundary conditions(Huet, 1990;Sab,1992)andequaltotheeffectivepropertiesofthe con-sideredmaterial,sothat:

Capp =Ceff (4)

The following two macroscopic strain conditions Eμ and Ek

wereused inthe elastic tests,aiming atcomputing theapparent shearmodulusμapp andapparentbulkmoduluskapp :

E_μ=



_{0 1/2 0} 1/2 0 0 0 0 0

; Ek=



_{1/9 0 0} 0 1/9 0 0 0 1/9

(5)

Thentheapparentshearandbulkmoduluscanbedefinedfrom theelasticstrainenergydensityforthemacroscopicstraingivenin Eq.(6)usingtheHill–Mandelcondition(Hill,1967),suchthat:

kapp =



: Ek=Tr

(

)

μ

app ₌

_

_{: Eμ=}

_

₁₂ (6)

3.2.Definitionofapparentplasticproperty

In this work, the notion of apparent viscoplastic parameter is considered as defined in Madi et al.(2006) forcharacterizing viscoplasticity.For the concerned polycrystalline copper, two hy-pothesesaremade:

(1)ThevonMisescriterionisdefinedforisotropicmaterial be-havior.Inthepresentcase,althoughthelocalmaterialbehavioris anisotropic,themacroscopicbehaviorisconsideredisotropic,since thegrainshavebeengeneratedwithastatisticallyisotropic distri-butionoforientation.Therefore,thevonMisescriterionissuitable formacroscopicnumericalconsiderations.BasedonvonMises cri-terion,thetotal plasticdeformation isequal tothesumof micro sheardeformationonallactivatedslipsystemsforeachelementof volume,asdescribedinthefollowingequations:

˙ ep =



: E˙v = 12  s

τ

s

_γ

_˙s ₌

_σ

_{¯ · ˙p,}

_σ

_{¯ =}

_3J 2



Si j



(7)

where,e˙p is the macroscopic plastic energy rate.

σ

¯ denotes the equivalentuniaxial tensile stress, which is associated with J₂ (S_ij ) thesecond invariant ofthe deviatoricpartS_ij ofthe macroscopic stresstensor



.Ev_{is themacroscopicviscoplasticstrain tensor.p} istheequivalentaccumulatedviscoplasticstrain. Localquantities, suchastheresolvedshearstressandplasticsliprateoneachslip systems are computed in the local material frame, and then ex-pressedinthemacroscopicframebeforeaveraging.

(2)Theapparentglobalplasticstrainratecanbealso approxi-matedbyasimpleNortonflowrule:

˙ p=

⎛

⎝

3J2



Si j



Kapp

⎞

⎠

n app (8)

Generally, this assumption can be valid, only if all local materials have the same parameters n and K, as stated by Rougieretal.(1993)incaseofcreep.ThesameKandnparameters areusedforeachslipsysteminthepresentwork,henceallowing ustorelyonEq.(8)forpurepolycrystallinecopper.

CombiningEqs.(7)and(8),onecanobtain:

˙ ep =



: E˙v = 12  s

τ

s

_γ

_˙s ₌

_3J 2



Si j



·

⎛

⎝

3J2



Si j



Kapp

⎞

⎠

n app (9)

Forthesakeofcomparison,werelyontheconceptofapparent viscoplastic parameter P_vapp as definedby Madi et al. (2006) for isotropicviscoplasticbehavior,suchthat:

P_vapp = 1

Kapp n app (10)

ThentheEq.(9)becomes:

˙ ep =



: E˙v = 12  s

τ

s

_γ

_˙s _=Papp v



3J2



Si j



n app₊₁ (11)

The procedurefordetermining P_vappis asfollows:foreach re-alization, a uniaxial tensile test is carried out under PBC with prescribedmacroscopicstrain ratecontrol.As polycrystallinepure copperisaratherstrainrateinsensitivematerialatroom tempera-ture(CarrekerandHibbard,1953),onlyonestrainrateof10−3 s−1 isconsidered.Thedurationofthetestis0.1s.Therefore,the max-imummacrostrainis10−4 ,whichisinthescope ofthepractical usefortheMeric–Cailletaudmodel(Mericetal.,1991;Cailletaud, 1992).At theendoftensiletest, outputvalues(



3J2

(

Si j

)

,e˙p ) are

computed.Usingtheleastsquarefittingmethod,thevalueofP_vapp

is identified asthe coefficient of the fittedpower law, asshown in Fig.1. The best fit is obtainedby increasing the volume size:

R2 ≥ 0.9for 8gr, 0.92for 27gr,0.94 for64gr,0.96 for125gr,0.98 for216grand0.99for343gr,R2 beingthestatisticalcoefficientof determination.

Furthermore, in order to determine the RVE size of polycrys-talline copper for viscoplasticity, the intrinsic dissipation energy densityduringtensiletestwasalsochosenasoneplasticproperty toestimatetheRVEsize.Foreachslipsystem,theintrinsic dissipa-tionpowerisdefinedastheplasticpowerminusthestoredpower associatedwithisotropicandkinematichardening,asproposedby Chrysochoosetal.(1989).

˙

Fig. 1. A fitting result of ˙ ep _{v s} _{3 J 2 ( S}

i j) , under 343 number of grains; P v is app 1.168e −11 and the corresponding values of n app and K app are 7.45 and 29.29, re- spectively.

Thespatiallyintrinsicdissipationenergydensityduringthe ten-siletestiscomputedasfollows:

dtension ₁ =  t 1 V  V  s ˙ ds ₁ dVdt (13)

3.3. DeterminationofRVEsize

Based on mathematical morphology considerations, foran er-godic stationaryrandom function Z(x), one can compute the en-semble variance D2 _Z

(

V

)

ofitsaveragevalue ¯Z

(

V

)

overthevolume

V(Matheron,1971;Cailletaudetal.,1994;Kanitetal.,2003):

D2 _Z

(

V

)

=D2 Z



A₃ V



(14)

whereD2 _Z isthepoint varianceofZ(x) involumeV andA₃ isthe integralrangeoftherandomfunctionZ(x),definedas:

A3 = 1 D2 _Z  R 3 ¯ W2

(

h

)

dh (15)

where his a two-point segment, andW¯₂

(

h

)

isthe centered 2nd order correlation function such that, for a prescribed property Z

andforx_∈V:

¯

W2

(

h

)

=



Z

(

x+h

)

− ¯Z



Z

(

x

)

− ¯Z



(16)

In caseof the Voronoi mosaic model, the value of A₃ is de-termined as a constant of 1.179 given by Gilbert (1962). Us-ing a modified scaling law with exponent

γ

, as proposed by Lantuéjoul(1990),thevariancecanberewrittenasfollows:

D2 _Z

(

V

)

=D2Z



A∗₃ V



γ (17)

A∗₃isalsohomogeneoustoavolumeofmateriallikeA₃ andcan readily be usedto determine RVEsizes. Butthere is not adirect definitionasEq.(15),anymore.

Inthecaseofatwo-phasematerial,Kanitetal.(2003)assumed that ¯Zwasequaltothearithmeticaverageofconsideredproperty withZ₁ forphase 1andZ₂ forphase 2.Thepoint varianceD2 _Z of therandomvariableZwasgivenby:

D2 _Z =P

(

1− P

)

(

Z₁ − Z2

)

2 (18)

whereP isthe volume fractionofphase 1in volumeV.When it comes to polycrystalline material, we can suppose that the mi-crostructurecontainsalargenumberofphases,i.e.grains.Foreach phase or grain, the material behavior is different because of the differenceoforientation. Eq.(18)can thenbe extendedto multi-phasemicrostructures,asfollows:

D2 _Z =

T



i

P_i



Z_i − ¯Z



2 (19)

whereTisthenumberofgrainsorphasenumber.

Finally,inorderto determinetheRVEsize, Dirrenbergeretal. (2014)reformulatedEq.(17)toreduce theinformationof D2 _Z and A∗₃ ,asfollows:

D2 _Z

(

V

)

=GV−γ (20)

withG=D2 _Z A∗₃ γ,sinceA∗₃ cannot bededuced independently.Thus onlytwoparametersGand

γ

areneededtoidentifyfromthe sta-tisticaldataobtainedbylinearizationasfollows:

logD2 _Z

(

V

)

=logG−

γ

logV (21)

FollowingthemethodproposedinKanitetal.(2003),the rela-tivesamplingerrorintheeffectivepropertiesarises:



rel = 2D_¯Z√Z

(

V

)

n ⇒



2 rel= 4G ¯Z2 _nVγ (22)

henceyielding the definitionofthe RVEsize,for agiven relative error

εrel

V_RVE = γ



4G



2 rel ¯Z2 n (23)

3.4.Periodicthree-dimensionalmeshgeneration

In this paper, a methodology is employed for generating and meshing 3D random polycrystals. The associated mesh optimiza-tion approach andstatisticalwork of mesh quality are fully pre-sented in the reference paper by Quey et al. (2011). The corre-spondingalgorithmsareimplementedanddistributedinan open-source software package: Neper.2 _{Thanks to the self-contained}

codes in Neper, the Voronoi tesselation can be constructed with a periodicityconstraint, neededfor PBC.For thesake of simplic-ityand comparison withresults from the literature (Madi etal., 2006), an isotropic morphological and crystallographic texture is considered.Inordertoobtaintheisotropicdistributionofthegrain orientation foraperiodic sample,the arbitraryshaftpointsofall crystalsinthesamplemustspanuniformlythesurfaceofasphere, assuggestedby Nédaetal.(1999).Forthe sakeof achievingthe condition, the three Euler angles (

α

,

β

,

γ

) in the Z–X–Z type of eachgeneratedgraincellwillbegivendifferentdistributionrules:

α

and

γ

are generatedrandomly witha uniform distribution on [0, 2

π

], while,

β

is chosen randomly in the range [0,

π

] with a weighted distribution and the weight factor sin(

β

) should be randomlyin the range[0, 1].Afterwards, themicrostructure was meshed withlinear tetrahedral elementsasshown inFig. 2b. At least300 elementswere usedfor each grain, which isa reason-ablevalue, consideringtheusual practicein thefull-field simula-tionliterature(Cailletaudetal.,2003a;Rotersetal.,2011;Cruzado etal.2015).Thegrainsizeinthegeneratedmicrostructurefollows a normal distribution function with a mean value of 20μm and astandard deviationof13.5μm.Fig. 2crepresentsan exampleof equivalentvonMisesstressdistributionatamacroscopicstrainof 0.01%,andthestresswithingrainsmicrostructurewasobservedby slicingperpendicularlytoZaxis,asshowninFig.2d.

Fig. 2. (a) 3D periodic generation of a polycrystalline sample (343 grains); (b) periodic meshing with tetrahedral elements; (c) The von Mises stress distribution with strain of 0.1% after tensile test; (d) XY workplace of (c) on half Z.

Table 2

Number of realizations N used for all considered domain sizes.

Domain size/Gran number (V) 8 27 64 125 216 343 Mesh generation time 15 s 1.5 min 6 min 13 min 22 min 35 min Elastic calculation time 1.2 s 3 s 6.7 s 15 s 27 s 42 s Plastic calculation time 2 min 6.7 min 18 min 37.6 min 97.6 min 194.2 min Number of realizations for u app _{514 140 56 28 16 10} Number of realizations for k app _{466 112 41 19 15 10} Number of realizations for d tension

1 352 90 40 28 14 8

Number of realizations for P app

v 56 38 26 16 12 6

4. Resultsanddiscussion

According totheapproachdescribedhereinabove,several real-izationsweregeneratedforstatisticalanalysis,withdifferent num-berofgrains,rangingfrom8to343,aslistedinTable2.The gen-eratedmicrostructurehasameangrainsize(diameter) of20μm. Thus,the actual volumesize is directlyrelatedto thenumber of grains. Forthe sake of simplicity, we used the numberof grains inplaceofthevolume size(V). Generally,thenumberof realiza-tionsNshouldbedifferentforeachdomainsizeinordertoachieve a similar measurement error for all sizes considered. Using Eq. (22)the measurementerror forall domain sizeswere controlled atunder1% forelastic properties,under3% forintrinsic dissipa-tion. Onthe contrary, the measurement error remains high, just under120%for apparent viscoplastic parameter P_vapp , mostlydue tothelargeintrinsicvariabilityoftheproperty.Toaccomplishthe

generation andmeshing ofmicrostructures, acomputer equipped withan Intel Core i7-4750HQ CPU @ 2.0GHz and8GBRAM was employed. Theconsumed timeof meshing,aswell aselasticand plasticcalculationsfor onerealizationis alsopresented for refer-enceinTable2.

4.1. Isotropyofmeanapparentmoduli

The microstructures used in tensile tests in framework of CPFEM are expected to be macroscopically isotropic. If a small volume element V is considered, it may not exhibit an isotropic behavior. Therefore, it is necessary to check whether the gener-ated polycrystal isisotropic ornot. For that purpose,six compu-tations are necessary for finding the 21 components of apparent elastictensorCapp oneach realization, usingPBC.From averaging all the realizations, the obtained full elastic moduli tensors with the intervalsofconfidence corresponding to plusand minus two

Table 3

The mean values and variance of components of C app .

Capp with V = 27, N = 140 177,772 ± 21,975 83,450 ± 17,823 83,492 ± 19,330 150 ± 8234 −437 ± 10,666 700 ± 11,035 – 176,163 ± 24,036 85,471 ± 15,358 379 ± 11,011 218 ± 8101 282 ± 13,208 – – 176,090 ± 24,364 969 ± 12,566 385 ± 11,306 −174 ± 7793 – – – 4 8,64 8 ± 9537 −207 ± 7245 287 ± 7467 – – – – 46,647 ± 11,579 10 ± 7240 – – – – – 46,723 ± 10,862 Capp with V = 125, N = 28 192,435 ± 10,455 95,170 ± 7999 95,911 ± 6895 35 ± 4256 467 ± 6260 −667 ± 4638 – 189,378 ± 8427 98,150 ± 8934 187 ± 5717 124 ± 3503 344 ± 5591 – – 188,587 ± 5096 −555 ± 5096 −580 ± 5366 107 ± 3604 – – – 50,776 ± 5729 132 ± 3515 154 ± 3574 – – – – 48,361 ± 4418 94 ± 3864 – – – – – 47,623 ± 4940 Capp with V = 343, N = 10 195,062 ± 5184 98,622 ± 4276 98,900 ± 1214 −217 ± 2640 656 ± 3267 234 ± 3345 – 193,190 ± 3821 101,367 ± 3177 −53 ± 3122 −85 ± 2160 −544 ± 3445 – – 192,721 ± 5921 1 ± 2780 510 ± 2207 −159 ± 2019 – – – 50,830 ± 2715 −146 ± 1824 −212 ± 2358 – – – – 48,241 ± 1757 56 ± 2700 – – – – – 48,023 ± 3837

standard deviations(±2D_Z ) aregiven inTable3 fordifferent vol-umesizes(componentsinMPa).Thestandarddeviationdecreases withincreasingvolumesize.Theaveragedtensorcomponents ob-tainedfor343grainsarecharacteristicofisotropicelasticitysince

C₁₁ ≈ C22 ≈ C33 and C12 ≈ C13 ≈ C23 with a maximal error of 5%,and

C₄₄ ≈ C55 ≈ C66 are approximately equal to C 11−C 2 22 with a

maxi-malerrorof10%.Theremainingcomponentsshouldvanishinthe isotropiccase,andhere,theytakeuplessthan1%ofC₁₁ .Itcanbe alsoobservedthattheelasticmodulitensorcomponentshavenot reachedtheirconvergedvaluesforsmallervolumes,whichislikely duetoabiasofrepresentativelyasstudiedby(HazanovandHuet, 1994;Huet,1997;Hazanov,1998)

4.2. Theapparentelasticandplasticproperties

After confirmingtheisotropyofthegeneratedmicrostructures, elastic properties kapp and μapp are investigated, as well as the intrinsic dissipation and the apparent viscoplastic parameter for all realizations. Fig. 3 illustrates the changes of the four appar-entparameterswithrespecttovolumesize,includingmeanvalue andstandarddeviation.Increasingthevolumesize,themean val-ues of μapp andkapp increase gradually andstabilize respectively at49,031± 1085MPaand130,972± 2385MPa onthe 343volume size,whichareconsistent withcommonvaluesforpolycrystalline pure copper. Similar convergence behavior can be observed for

dtension ₁ .Itsmeanvaluereaches53J/m3 withthecorresponding in-tervalsofconfidenceof4.3J/m3 onthe343volume size,whichis in thesamemagnitude levelasprevious studies formetals,such asChrysochoosandMartin(1989)andChrysochoosetal.(2009).

ThedefinitionofP_vapp consistsoftwoparameters,Kapp andnapp . The fluctuation of both parameters may yield a large change of

P_vapp inmagnitude. Forbetter presenting,the orderof magnitude log

(

P_vapp

)

was drawn vs. the volume size in Fig. 3as the substi-tute of P_vapp . However, the relative error,floating around 9%, is a little higherthanthose ofelasticpropertiesandintrinsic dissipa-tionbecauseoffewerrealizations.Whenthevolumesizeincreases to 343, the fluctuation tends to weaken, andthe averaged value oflog

(

P_vapp

)

stabilizesat−10.74,withafluctuation rangeof±0.8. Interestingly, thisvalue is also approximatelythe combinationof thetwoparametersofthesinglecrystalconstitutivelaw,K=5and

n=10,producingthe valueof10−10.7 asthedefinitionofP_v= 1 K n. Consideringalarge enoughvolumeofpolycrystallinepurecopper

atthestrainrateof10−3 s−1 ,thefollowingequationcanbe possi-blysatisfied: P_vapp= 1 Kapp n app =P e f f v = 1 Kn (24)

Nevertheless, the apparent parameters Kapp _{and n}app _{are not}

necessarilyequaltotheactualvaluesofKandnforthesingle crys-talbehaviorlaw.Heretheyconvergeat29.06and7.32for343 vol-umesizes, respectively,witharelative errorofabout3% forboth

Kandn.

4.3.FluctuationofapparentpropertiesandRVEsizes

As specifiedin Eq.(23), inorder to compute RVE sizes, three variables(D_Z , ¯Z,

γ

)havetobeestimated.Alinearfittingwasdone accordingto Eq. (21) in the logarithm scale for the four proper-ties,asshowninFig.4.Theslopevaluesofeachlinecorresponds to different values of

γ

, which are close to 1 for μapp , kapp and

dtension ₁ . P_vapp exhibits a

γ

value of 4.9 meaning that the defined apparentviscoplastic parameterhasamuch fasterstatistical con-vergenceratethantheotherinvestigatedproperties.Theintercept termbcanbeusedtoworkoutthevariables,G.

Inordertoexplain thediscrepanciesobservedfor

γ

,thepoint deviationD_Z ineachrealizationisalsocomputedforthefour prop-ertiesusingEq.(19),andtheresultsarespecifiedinFig.5.On343 volume size, D_μapp, D_k app, and D_d tension

1

have converged at 11,713, 4774,and34,withsmallfluctuations.Bynormalizingthepoint de-viationD_Z overtheconvergedproperty ¯Z(D_Z /¯Z),onecanfindthat theintrinsicdissipationenergyhashigherinnerdeviationthanthe apparentelasticproperties(68.4%fordtension

1 ,3.8%and23%forkapp

andμapp , respectively for343 volume size). For D_P app

v , thisvalue

canreach atover 100,because ofthe nonlinearityandhigh sen-sitivityofthisparameter. Thisimplies thatthe orientation distri-bution seems to have a stronger impact on the plastic behavior variabilityincomparisontoitseffectonelasticproperties.Also,it seemsthat thehigh

γ

exponentforD2

P app

v

(

V

)

could be correlated

withthenonlinearityandsensitivityoftheviscoplasticparameter. Oneof theadvantagesof relying onmicrostructural computa-tionistheabilitytoconsidereachgrainorphaseindividually.For thispurpose, Fig.6 illustrates the impact ofgrain orientation on crystalplasticity,byanalyzingthelocalheterogeneitiesofthe plas-ticbehaviorina realizationwith343grains,includingvon Mises

Fig. 3. The mean values and variances for the four apparent properties with increasing volume size. The error bar means plus and minus two standard deviations ( ± 2 D z ).

Fig. 4. Linear fitting for variances of four apparent properties vs . volume sizes.

stressandplasticenergyrateevolutionwithrespecttoequivalent macroscopicuniaxialtensilestress.Thecurvesforthetwo proper-tiesweredrawn foreachgrain andthewhole volume.As shown inthevonMisescurves,grainsinthevolumeholddifferent equiv-alentstress, ranging from 6MPa to 16MPa.Some grainsactually remain in the elastic regime duringthe tensile test. Meanwhile, grainsyield heterogeneousplastic power dependingonthe grain orientation for the same equivalent macroscopic uniaxial tensile stress, as shownon the right hand side in Fig. 6. These

hetero-Table 4

The associated variables in Eq. (23) for four apparent properties. Variables μapp _kapp _dtension

1 Papp v

γ 1.07 1.17 1.12 4.90

G 2.82E8 2.29E9 4073.80 9.12E −10

¯

Z 49,031.18 130,972.25 53.03 2.19E −11

Table 5

RVE sizes estimated from computation with n = 1. Relative error 1% 2% 5% 10% Vμapp 2756 753 135 37 Vkapp 1504 461 96 29 Vdtension 1 17,894 5190 1010 293 VPapp v 2788 2101 1445 1089

geneitiesarerelatedtothelocalgrainorientation,localanisotropic elasticityandanisotropiccrystalplasticityframework.

Finally, all the associatedvariables in Eq. (23)for four appar-entpropertieswereobtainedandlistedinTable4.Theminimum domainsizesthat arenecessaryto reacha givenprecision are fi-nallyshownforfourapparentproperties,inTable5.Ingeneral,for agivenmaterial,theRVEsizedependsonthespecificinvestigated property.Forpolycrystallinepurecopper,asTable5shows,the in-equalityexists:V_k app<V_μapp<V_d tension

1 , VP

app

v .Furthermore,theRVE

size for P_vapp does not depend too much on the chosen preci-sion,incomparisontotheotherthreeproperties.Inpracticaluse, for a 5% relative error, a volume size of 135 grains can be con-sidered for elastic properties, while for intrinsic dissipation and apparent viscoplastic parameter,the volume sizesmust be larger

Fig. 5. The point deviation of four apparent properties for each volume size, in which the error bar represents plus and minus two standard deviations on the variance.

Fig. 6. The heterogeneities of von Mises stress vs . macroscopic strain (left) and plastic energy rate evolution with respect to the equivalent uniaxial tensile stress (right) for each grain in one volume with 343 grains (black lines represent each grain; red lines are for the whole volume).

than 1010 grainsand 1445grains, respectively. Ifa higher preci-sionisrequired,morerealizationscanbealsoconsidered,such as fora precision of1% withn=50 realizations,V_μRV appE =71,V_k RV appE =54,

VRV E d tension 1 =545andVRV E P app v =1254.

However, in the case of isotropic plasticity behavior, the RVE size for the same property P_vapp used by Madi et al. (2006) was found smallerthan forthe isotropicelasticones. Adifferent con-clusion in the present results is likely due to the difference of material behavior, i.e. anisotropic elasticity and crystal plastic-ity. As discussed before about the deviation D_Z , crystal plastic-ityhighlydependsonthegrain orientation,asdifferentactivated

slipsystems produce differentplastic deformations.The effect of anisotropicelasticityappearstoyieldlowerstatistical heterogene-itythanthe crystalplasticitybehavior, i.e.yielding a smallerRVE sizeincomparison.

BasedonFig.6,itappearsthatplasticdeformationtakesplace inmostgrainsduringthetensiletest,butlocalizationoperatesin onlya minority of grains, i.e. lessthan 10% of them.This plastic strainlocalizationandstressconcentrationphenomenoncould ex-plaina high

γ

exponentforD2

P app

v

(

V

)

.As a matter offact,a

pat-ternoflocalizationwillformforanynumberofgrainsduetothe morphologicaland material anisotropies. Therefore,a large num-berofgrainsisnotneededforensemblevarianceconvergenceon

theaveragedvalue ofthe viscoplasticparameter. ResultsfromEq. (23)andTable 5 mightlead to another conclusion: the RVEsize forP_vapp isratherlarge andonlydecreasesslowlywithincreasing therelative error.Thisis dueto thehighintrinsicpoint variance ofP_vapp,i.e.GinEq.(22).Thisvariabilityisrelatedto the nonlin-earnatureofthelocalizationphenomena.Thislargepointvariance counterbalancestheeffectofafastensemblevarianceconvergence. Onecouldarguethatthepointvariance ofP_vapp islikelytobe re-latedtoastrongmaterialheterogeneityasinducedinthepresent workbyelasticanisotropyandcrystalplasticity.

5. Conclusionsandprospects

Aiming at computing the intrinsic dissipation of pure cop-perbasedonacrystalplasticityframework,virtual polycrystalline samples were generated based on EBSD microstructural analysis, andtheRVEsizeforvariousmechanicalpropertieswasestimated. Astatisticalanalysismethodwasusedto determinetheRVEsize of polycrystalline pure copper for four properties, including two isotropicelasticproperties:shearandbulkmodulus,the viscoplas-ticparameterandtheintrinsicdissipationenergydensityduringa tensiletest.Themainconclusionsarelistedbelow:

1. The statistical RVE method developed by Kanit et al. (2003), is applicable to viscoplastic polycrystalline materials modeled withinacrystalplasticityfiniteelementframework.

2. RVE sizes obtainedare smaller for elastic properties than for plasticproperties,likelyduetotheanisotropic elastoviscoplas-ticmodelchosenforthematerialbehavior,aswellasthe poly-crystalline nature of thesamples, both inducing stronger het-erogeneities inthemechanical fields.Thisconclusionis oppo-sitetotheonesmadebyMadietal.(2006),whileconsidering anisotropicelastoplasticbiphasicmaterial.

3. The computationalmicrostructuralstudyallowed to character-ize the local heterogeneities associated with plasticity, hence givinganinsightonthemicrostructuralbehaviorexplainingthe statisticalmacroscopictrendsobserved.

4. The viscoplastic parameter is related to the nonlinear phe-nomenonofplastic localization,inducing stronglocal variabil-ityforP_v .Thisinherent intrinsicheterogeneity leadsto ahigh pointvariance,whichitselfwillinvariablyyieldlargerRVEsize incomparisontolinearproperties.

The statistical analysis provided in this paper can be applied tootherpolycrystallinematerialsforvariousproperties,giventhat amicrostructuralmorphological modelisavailable for generating avirtualstatisticalpopulationofsamples.Introducingmore infor-mationaboutthemicrostructureofmaterials appearsasa neces-sityforimprovingthepredictivecapabilityofsuchstatistical tech-niques.Furtherworkwillinvolvetheextension ofthepresent ap-proachto thefatigueofmetallic polycrystallinematerials.The fa-tiguestrengthofpolycrystallinematerials isa longstanding prob-leminmechanicaldesign,especiallyintheveryhighcyclefatigue regime,where thestress level is muchlower than traditional fa-tigue limit (Bathias and Paris, 2004; Stanzl-Tschegg et al., 2007; Phungetal.,2014;Torabianetal.,2016a,2016b,2017a,2017b).In ordertoreduce experimentdurationtime,manyauthorshave re-sortedto the methodof self-heatingtests, inwhich the thermo-mechanicalresponseofthe materialduringcyclicloading is ana-lyzedandtheintrinsicdissipation istakenasthefatiguedamage indicatortoevaluatefatiguestrengthatvariousstresslevelsinthe highand very high cycles fatigue domain (Luong 1995; La Rosa et Risitano 2000; Boulanger et al., 2004; Doudard et al., 2010; Chrysochoos et al., 2008; Connesson et al., 2011; Blanche et al., 2015; Guoetal., 2015). Physically,comingfromtheirreversibility ofthe plastic deformation, intrinsic dissipation is assumedto be relatedtothemicroplasticdeformationingigacyclefatigueregime,

i.e.the crystalslipping behavior at thegrain scale. The approach developed in the presentwork could thus be used to determine theRVEsizeassociatedwithintrinsicdissipationduringvery-high cyclefatigue. Relyingon full-fieldcrystalplasticityfinite element analysiscould thusfurtherourunderstandingofthedamage phe-nomenatakingplaceatthemicroscale.

Acknowledgments

Theauthorswouldliketoacknowledgethefinancialsupportfor doctoralstudentShaoboYANGfromChinaScholarshipCouncil.

References

Barbe, F. , Decker, L. , Jeulin, D. , Cailletaud, G. , 2001. Intergranular and intragranu- lar behavior of polycrystalline material. Part 1: FE model. Int. J. Plasticity 17, 513–536 .

Bathias, C. , Paris, P.C. , 2004. Gigacycle Fatigue in Mechanical Practice, vol. 185. CRC Press .

Bironeau, A. , Dirrenberger, J. , Sollogoub, C. , Miquelard-Garnier, G. , Roland, S. , 2016. Evaluation of morphological representative sample sizes for nanolayered poly- mer blends. J. Microsc. 264, 48–58 .

Blanche, A. , Chrysochoos, A. , Ranc, N. , Favier, V. ,2015. Dissipation assessments dur- ing dynamic very high cycle fatigue tests. Exp. Mech. 55, 699–709 .

Boulanger, T. , Chrysochoos, A . , Mabru, C. , Galtier, A . , 2004. Calorimetric analysis of dissipative and thermoelastic effects associated with the fatigue behavior of steels. Int. J. Fatigue 26, 221–229 .

Cailletaud, G. , Jeulin, D. , Rolland, P. , 1994. Size effect on elastic properties of random composites. Eng. Comput. 11, 99–110 .

Cailletaud, G. , 1992. A micromechanical approach to inelastic behaviour of metals. Int. J. Plasticity 8, 55–73 .

Cailletaud, G. , Chaboche, J.L. , Forest, S. , Remy, L. ,2003a. On the design of single crys- tal turbine blades. Revue de Metallurgie 165–172 .

Cailletaud, G. , Forest, S. , Jeulin, D. , Feyel, F. , Galliet, I. , Mounoury, V. , Quilici, S , 2003b. Some elements of microstructural mechanics. Comput. Mater. Sci. 27, 351–374 . Carreker, R.P. , Hibbard Jr., W.R. , 1953. Tensile deformation of high-purity copper as a function of temperature, strain rate and grain size. Acta Metall. 1, 656–663 . Chrysochoos, A. , Berthel, B. , Latourte, F. , Pagano, S. , Wattrisse, B. , Weber, B. , 2008.

Local energy approach to steel fatigue. Strain 44, 327–334 .

Chrysochoos, A. , Martin, G. , 1989. Tensile test microcalorimetry for thermomechan- ical behaviour law analysis. Mater. Sci. Eng. A 108, 25–32 .

Chrysochoos, A. , Wattrisse, B. , Muracciole, J.M. , Kaim, Y.E. , 2009. Fields of stored energy associated with localized necking to steel. J. Mech. Mater. Struct. 4, 245–262 .

Chrysochoos, A. , Maisonneuve, O. , Martin, G. , Caumon, H. , Chezeaux, J.C. , 1989. Plas- tic and dissipated work and stored energy. Nucl. Eng. Des. 114, 323–333 . Connesson, N. , Maquin, F. , Pierron, F. , 2011. Experimental energy balance during the

first cycles of cyclically loaded specimens under the conventional yield stress. Exp. Mech. 51, 23–44 .

Cruzado, A. , Gan, B. , Jiménez, M. , Barba, D. , Ostolaza, K. , Linaza, A. , Molina-Al- dareguia, J.M. , Llorca, J. , Segurado, J. , 2015. Multiscale modeling of the mechani- cal behavior of IN718 superalloy based on micropillar compression and compu- tational homogenization. Acta Materialia 98, 242–253 .

Cruzado, A. , LLorca, J. , Segurado, J. , 2017. Modeling cyclic deformation of inconel 718 superalloy by means of crystal plasticity and computational homogeniza- tion. Int. J. Solids Struct. 122–123, 148–161 .

Cruzado, A. , Lucarini, S. , LLorca, J. , Segurado, J. , 2018. Microstructure-based fatigue life model of metallic alloys with bilinear Coffin-Manson behavior. Int. J. Fatigue 107 (2018), 40–48 .

Diard, O. , Leclercq, S. , Rousselier, G. , Cailletaud, G. , 2005. Evaluation of finite ele- ment based analysis of 3d multicrystalline material plasticity: application to crystal plasticity model identification and the study of stress and strain fields near grain boundaries. Int. J. Plasticity 21, 691–722 .

Dirrenberger, J. , Forest, S. , Jeulin, D. , 2014. Towards gigantic RVE sizes for 3D stochastic fibrous networks. Int. J. Solids Struct. 51, 359–376 .

Doudard, C. , Calloch, S. , Hild, F. , Roux, S. , 2010. Identification of heat source fields from infrared thermography: determination of ‘self-heating’ in a dual-phase steel by using a dog bone sample. Mech. Mater. 42, 55–62 .

Gérard, C. , N’Guyen, F. , Osipov, N. , Cailletaud, G. , Bornert, M. , Caldemaison, D. , 2009. Comparison of experimental results and finite element simulation of strain lo- calization scheme under cyclic loading. Comput. Mater. Sci. 46, 755–760 . Gilbert, E.N. , 1962. Random subdivisions of space into crystals. Ann. Math. Stat. 33,

958–972 .

Gillner, K. , Münstermann, S. , 2017. Numerically predicted high cycle fatigue proper- ties though representative volume elements of the microstructure. Int. J. Fatigue 105, 219–234 .

Gitman, I.M. , Askes, H. , Sluys, L.J. , 2007. Representative volume: Existence and size determination. Eng. Fract. Mech. 74, 2518–2534 .

Guo, Q. , Guo, X. , Fan, J. , Syed, R. , Wu, C. , 2015. An energy method for rapid evalu- ation of high cycle fatigue parameters based on intrinsic dissipation. Int. J. Fa- tigue 80, 136–144 .

Gusev, A. , 1997. Representative volume element size for elastic composites: a nu- merical study. J. Mech. Phys. Solids 45, 1449–1459 .

Hazanov, S. , 1998. Hill condition and overall properties of composites. Arch. Appl. Mech. 68, 385–394 .

Hazanov, S. , Huet, C. , 1994. Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume. J. Mech. Phys. Solids 42, 1995–2011 .

Hill, R. , 1967. The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15, 79–95 .

Hill, R. , 1963. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 .

Huet, C. , 1990. Application of variational concepts to size effects in elastic hetero- geneous bodies. J. Mech. Phys. Solids 38, 813–841 .

Huet, C. , 1997. An integration micromechanics and statistical continuum thermody- namics approach for studying the fracture behavior of microcracked heteroge- neous materials with delayed response. Eng. Fract. Mech. 58, 459–556 . Jean, A. , Willot, F. , Cantournet, S. , Forest, S. , Jeulin, D. , 2011. Large-scale computa-

tions of effective elastic properties of rubber with carbon black fillers. Int. J. Multiscale Comput. Eng. 9 (3), 271–303 .

Kanit, T. , Forest, S. , Galliet, I. , Mounoury, V. , Jeulin, D. , 2003. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40, 3647–3679 .

La Rosa, G. , Risitano, A. , 20 0 0. Thermographic methodology for rapid determination of the fatigue limit of materials and mechanical components. Int. J. Fatigue 22, 65–73 .

Lantuéjoul, C.H. , 1990. Ergodicity and integral range. J. Microsc. 161, 387–403 . Luong, M. , 1995. Infrared thermographic scanning of fatigue in metals. Nucl. Eng.

Des. 158, 363–376 .

Madi, K., Forest, S., Jeulin, D., Boussuge, M., 2006. Estimating RVE sizes for 2D/3D viscoplastic composite material. Matériaux 2006, Dijon, France. xx, 12 p., 2006.

< hal-0014 4 481>

Martin, G. , Ochoa, N. , Saï, K. , Hervé-Luanco, E. , Cailletaud, G. , 2014. A multiscale model for the elastoviscoplastic behavior of Directionally Solidified alloys: ap- plication to FE structural computations. Int. J. Solids Struct. 51 (5), 1175–1187 . Matheron, G. , 1971. The Theory Of Regionalized Variables and its Applications,

vol. Fascicule 5 of Les Cahiers du Centre de Morphologie Mathématique de Fontainebleau. Ecole des Mines de Paris .

Meric, L. , Poubanne, P. , Cailletaud, G. , 1991. Single crystal modeling for structural calculations. Part 1: model presentation. J. Engng. Mat. Technol. 113, 162–170 . Michel, J.-C. , Moulinec, H. , Suquet, P. , 1999. Effective properties of composite ma-

terials with periodic microstructure: a computational approach. Comput. Meth. Appl. Mech. Eng. 172, 109–143 .

Musienko, A. , Tatschl, A. , Schmidegg, K. , Kolednik, O. , Pippan, R. , Cailletaud, G. , 2007. Three-dimensional finite element simulation of a polycrystalline copper speci- men. Acta Mater. 55, 4121–4136 .

Néda, Z. , Florian, R. , Brechet, Y. , 1999. Reconsideration of continuum percolation of isotropically oriented sticks in three dimensions. Phys. Rev. E 59, 3717–3719 . Pelissou, C. , Baccou, J. , Monerie, Y. , Perales, F. , 2009. Determination of the size of

the representative volume element for random quasi-brittle composites. Int. J. Solids Struct. 46, 2842–2855 .

Phung, N.L. , Favier, V. , Ranc, N. , Vales, F. , Mughrabi, H. , 2014. Very high cycle fatigue of copper: evolution, morphology and locations of surface slip markings. Int. J. Fatigue 63, 68–77 .

Quey, R. , Dawson, P.R. , Barbe, F. , 2011. Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput. Methods Appl. Mech. Engrg. 200, 1729–1745 .

Robert, C. , Saintier, N. , Palin-Luc, T. , Morel, F. , 2012. Micro-mechanical modelling of high cycle fatigue behaviour of metals under multiaxial loads. Mech. Mater. 55, 112–129 .

Roters, F. , Eisenlohr, P. , Bieler, T.R. , Raabe, D. ,2011. Crystal Plasticity Finite Element methods: in Materials Science and Engineering. John Wiley & Sons .

Rougier, Y. , Stolz, C. , Zaoui, A. , 1993. Représentation spectrale en viscoélasticité linéaire des matériaux hétérogènes. Comptes. Rendus de l’Académie des Sci- ences Série II 316, 1517–1522 .

Sab, K. , 1992. On the homogenization and the simulation of random materials. Eur. J. Mech., A/Solids 11, 585–607 .

Shenoy, M. , Tjiptowidjojo, Y. , McDowell, D. , 2008. Microstructure-sensitive modeling of polycrystalline IN 100. Int. J. Plast. 24 (10), 1694–1730 .

Shenoy, M. , Zhang, J. , Mcdwell, D.L. , 2007. Estimating fatigue sensitivity to polycrys- talline Ni-base superalloy microstructures using a computational approach. Fa- tigue Fract. Eng. Mater. Struct. 30, 889–904 .

Stanzl-Tschegg, S. , Mughrabi, H. , Schoenbauer, B. , 2007. Life time and cyclic slip of copper in the VHCF regime. Int. J. Fatigue 29, 2050–2059 .

Sweeney, C.A. , O’Brien, B. , Dunne, F.P.E. , McHugh, P.E. , Leen, S.B. , 2015. Micro-scale testing and micromechanical modelling for high cycle fatigue of CoCr stent ma- terial. J. Mech. Behav. Biomed. Mater. 46, 244–260 .

Teferra, K. , Graham-Brady, L. , 2018. A random field-based method to estimate con- vergence of apparent properties in computational homogenization. Comput. Methods Appl. Mech. Engrg. 330, 253–270 .

Terada, K. , Hori, M. , Kyoya, T. , Kikuchi, N. , 20 0 0. Simulation of the multi-scale con- vergence in computational homogenization approaches. Int. J. Solids Struct. 37, 2285–2311 .

Torabian, N. , Favier, V. , Dirrenberger, J. , Adamski, F. , Ziaei-Rad, S. , Ranc, N. , 2017a. Correlation of the high and very high cycle fatigue response of ferrite based steels with strain rate-temperature conditions. Acta Mater. 134, 40–52 . Torabian, N. , Favier, V. , Ziaei-Rad, S. , Adamski, F. , Dirrenberger, J. , Ranc, N. , 2016a.

Self-Heating Measurements for a Dual-Phase Steel under Ultrasonic Fatigue Loading for stress amplitudes below the conventional fatigue limit. Proc. Struct. Eng. 2, 1191–1198 .

Torabian, N. , Favier, V. , Ziaei-Rad, S. , Dirrenberger, J. , Adamski, F. , Ranc, N. , 2016b. Thermal response of DP600 dual-phase steel under ultrasonic fatigue loading. Mater. Sci. Eng. 677, 97–105 .

Torabian, N. , Favier, V. , Ziaei-Rad, S. , Dirrenberger, J. , Adamski, F. , Ranc, N. , 2017b. Calorimetric studies and self-heating measurements for a dual-phase steel un- der ultrasonic fatigue loading. Fatigue and Fracture Test Planning, Test Data Ac- quisitions and Analysis. ASTM International .