Representative volume element size determination for viscoplastic properties in polycrystalline materials

Science Arts & Métiers (SAM)

is an open access repository that collects the work of Arts et Métiers Institute of

Technology researchers and makes it freely available over the web where possible.

This is an author-deposited version published in:

https://sam.ensam.eu

Handle ID: .

http://hdl.handle.net/10985/13840

To cite this version :

Shaobo YANG, Justin DIRRENBERGER, Eric MONTEIRO, Nicolas RANC - Representative

volume element size determination for viscoplastic properties in polycrystalline materials

-International Journal of Solids and Structures p.1-10 - 2018

Any correspondence concerning this service should be sent to the repository

Administrator :

archiveouverte@ensam.eu

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 xxx 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-calRVEsizedeterminationmethodisappliedtoaVoronoi tessellation-basedpurecopper microstruc-ture. Thedefinition ofRVEhas remained problematic inthe literature for propertiesrelated to non-linearviscoplasticbehavior, e.g.apparent viscoplasticparameter, intrinsicplastic dissipation. Computa-tionalhomogenizationforelasticandplasticpropertiesisperformedwithinacrystalplasticityfinite ele-mentframework,overmanyrealizationsofthestochasticmicrostructuralmodel,usingperiodicboundary conditions.Thegenerateddataundergoesstatisticaltreatment,fromwhichRVEsizesareobtained.The methodusedfor determiningRVEsizeswasfound tobeoperational,even forviscoplasticity.The mi-croscaleanalysisofthe full-fieldsimulationresultsrevealsmicrostructure-relateheterogeneitieswhich shednewlightontheproblemofRVEsizedeterminationfornonlinearproperties.

1. Introduction

Inthepastdecades,full-fieldnumericalsimulationof polycrys-tallinematerialsbasedonfiniteelementanalysishasbeenwidely developed to investigate the mechanical behavior, allowing the analysisofstress andstrainfieldsata scalethat isnoteasily as-sessable experimentally (Barbe et al., 2001; Roters et al., 2011). Mostof the authorsin theliterature dedicated to thesimulation of polycrystals usually consider a population of virtual polycrys-tallinesamplesmadeofseveralhundredgrains,validatingthis ar-bitrarychoicebyanalyzingthemeanvalueandstandarddeviation foragivenpropertycomputedonsuch population(Shenoyetal., 2007,2008;Robertetal.,2012;Martinetal.,2014;Sweeneyetal., 2015; Cruzado etal., 2017, 2018). Nevertheless, the development offull-fieldsimulationofpolycrystallinematerialsresultsin shed-dingnewlightontherelationshipbetweenthemicrostructural de-scriptionatthedislocationorgrainscaleandthelocalmechanical behavior (Cailletaud etal., 2003a). Thehomogenizedmacroscopic response of a polycrystallinematerial samplewill depend on its size, hence yielding thequestion ofrepresentativity forsuch vir-tual samples.Tobring ananswerto thisquestion, onemust have aproperdefinitionofwhatanRVEis.

Inhomogenizationmethods,theconceptofRVEwasfirstly pro-posedby Hill(1963)as“a samplethatis thestructurally typicalof thewholemicrostructurefora givenmaterial,i.e.containinga

suffi-∗ _{Corresponding author.}

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

cientlylarge numberofheterogeneities, whilebeingsmallenoughto beconsideredhomogeneousfromacontinuummechanicsviewpoint”. The quantification of RVE size has been problematic until vari-ousstatisticalapproacheswereproposed inthepasttwo decades (Gusev, 1997; Terada et al., 2000; Kanit et al., 2003; Gitman etal., 2007). Based on severalstatistical hypotheses, Kanit et al. (2003)proposed a statisticalapproach to determinethe minimal RVEsizeforaconsideredproperty,inwhich,theRVEsizecouldbe associatedwithagivenprecisionoftheestimatedoverallproperty andthenumberofrealizationswithagivenvolume sizeV. Prac-tically,itisapplicableinordertodeterminetheminimal number ofrealizationstoconsiderforagivenvolumesize,inorderto esti-matetheeffectivepropertywithagivenprecision(Bironeauetal., 2016).

Usingthisapproach,Kanitetal.(2003)studiedtheRVEsizesof atwo-phase 3DVoronoimosaicforlinearelasticity,thermal con-ductivityand volume fraction,underuniform displacement, trac-tionandperiodicboundaryconditions(PBC)(Micheletal., 1999). The results showed that the PBC held an advantage of conver-gencerateofthemeanapparentpropertiesincomparisontoother boundary conditions,due to the vanishing of boundary layer ef-fects. A slow rate of convergence for the considered properties wouldyieldalargeRVEsizes(Dirrenbergeretal.,2014).Also con-sideringthe largecalculation costinthecaseofcrystalplasticity, itis preferabletorely onPBC inorderto optimizethe computa-tionstrategy,asitwasdoneinotherinvestigations(Pelissouetal., 2009andJeanetal.,2011).

The statisticalmethodofKanit etal.(2003) wasimplemented fortheestimationofRVEsize,notonlyforlinearmechanical prop-ertiesandmorphologicalproperty,butalsoforplasticproperties: Madi etal. (2006) evaluated the RVE size for2D/3D viscoplastic compositematerials. Intheirstudy,themacroscopicstrainrateof the2D/3Dmaterialwasmodeled usingaNortonflowrule.Based onthevonMisescriterion,anapparentviscoplasticparameterP_vapp

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

As amatteroffact,theconceptofRVEhasoftenbeenusedin investigationsassociatedwiththeaveragemechanicalresponse of 2Dand3D polycrystallinematerial.ThedefinitionofRVEsizecan stemfrom finite element meshing considerations or convergence ofmeanvaluesforaconsideredproperty.Forinstance,Barbeetal. (2001) described the RVE fora cubic polycrystallinemesh as an equilibriumbetweenthe numberofgrains(238)andtheaverage numberofintegrationpointspergrain(660)attainablewithin typ-icalcomputationalmeans.Morerecently,Sweeneyetal.(2015) es-timatedtheenergeticparameterofCoCrstentmaterialinhigh cy-clefatigue by averaging in5 RVEs with138–140 grains. Cruzado etal.(2017,2018)simulatedthecyclicdeformationofmetallic al-loyswith20RVEsandasizeof300grains,whichshowedanerror lessthan10% forelastoviscoplastic properties.Similar determina-tionofmaterialRVEsizecanalsobefoundinShenoyetal.(2007, 2008), Martinetal. (2014),Gillner andMu¨nstermann (2017), Te-ferraandGraham-Brady(2018).IntheseRefs,RVEsize isdefined asafewrealizationswithafewhundredgrainswhichcanrealize aconvergenceofmeanproperties.However,theseanalysesdonot allowforarigorousstatisticaldefinitionoftheRVEsize.

Ratherthanrelyingsolelyontheconvergenceofmean proper-ties,themethodproposedinKanitetal.(2003)makesuseofthe rateofconvergenceoftheensemblevarianceofthemean proper-tieswithrespecttothe volumesize, thusenablingthe definition andestimation ofastatisticalRVEsize foreachconsidered prop-erty. However, to the knowledge ofthe authors, noone ever as-sessedtheRVEsize forpolycrystallinematerial intheframework ofCPFEMwiththestatisticalRVEmethod.

Additional considerationhas to be maderegarding the appar-entpropertiestobeconsideredascriteriaforRVEsize determina-tioninviscoplasticity.Thefirstoneshouldbethedefinitionof in-trinsicdissipationwithinthecontextofcrystalplasticity.Secondly, thedefinitionofthe apparentviscoplastic parameterP_vapp willbe consideredinthecrystalplasticityframework. Meanwhile,forthe crystalplasticitybehavior,materialheterogeneityismainlydueto thelocalgrainorientation,whichcanintroducestrongstress con-centrations,leadingtoearlyonsetofplasticity.Bothgrain orienta-tionandthe choice ofcrystalplasticbehavior are likely to influ-encedirectlythevalueofRVEsizeformechanicalproperties,asit willbediscussedinthepaper.

Overall,inthispaper,theRVEsizeofpolycrystallinepure cop-perwillbe studiedinframework ofCPFEM.Inthefollowing sec-tions,the materialbehavior andconstitutivemodel arediscussed first. Then, the periodic mesh generation and the computational homogenizationmethodaredescribedindetail,alongsidewiththe variousloadingcases correspondingto thedifferentpropertiesto beconsideredforestimatingtheRVEsizes.Resultsfrom computa-tionandstatisticalanalysisarethenpresentedanddiscussed. Cal-culationsfor RVEare performed inthe last section of the paper, andcomparisonismadeforthedifferentRVEsizesdependingon theconsideredproperty.

Inthefollowing,vectorsareunderlinedfaceandwrittenin mi-nuscule, e.g. x.Second-rank are bold andslant face,e.g. x andX, andfourth-ranktensors arebold andstraight facecapitals,e.g. X. Othersarescalar.

2. Crystalplasticityconstitutivemodel

The material involved in this paper was pure polycrystalline copper.Bothanisotropiccrystalelasticityandplasticitywere con-sideredforits behavior. Thecubic elasticityis characterizedby 3 independentelastic constants,takenfromMusienko etal.(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–Cailletaud model waschosen forits ability to account for kinematichardening.Thismodelispopularwithinthecrystal plas-ticity communityandhas beenused inmany previous workson 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, 12slipsystemsare associ-atedwiththecalculation, i.e.4slipplanesgs oftype {111}and3 directionsofslipls oftype



110



,whichdepend ontheEuler an-glesof thegrains. The Schmidtensor ms is usedto compute the resolvedshearstress

τ

s andtheplasticstrainrate

ε

˙p .Foreachslip system, thesliprate

γ

˙s isdefinedasapower-lawfunction of

τ

s . TheparametersKandnrelatetothesensitivityofmaterialstothe strainrate.

υ

s representstheaccumulatedplasticstrainfortheslip systems.Thecouplesofthermodynamicalforceandstatevariable, (Xs_,

_α

s_{) and(R}s_,qs_{), respectivelyare associated withthekinematic}

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

Finally,the materialparameters fora high-puritycopper, K,n, R₀ ,Q,bandhsr ,aregiveninTable1.

3. Computationalapproach

3.1. Definitionofapparentelasticproperties

Themicromechanicallinearelasticbehaviorateachintegration pointinthefiniteelementsimulationisdescribedbyHooke’slaw usingthefourth-ranklinearelasticitytensorC,suchthat:

σ

(

x

)

=C

(

x

)

:

ε

(

x

)

(2)

3

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



andEarethemacroscopicstressandstrainsecond-rank tensors.

ForanelementaryvolumeVlargeenough(V>V_RVE ),the appar-ent propertiesdo not depend on the boundary conditions(Huet, 1990;Sab, 1992)andequaltotheeffectivepropertiesofthe con-sideredmaterial,sothat:

Capp =Ceff (4)

The following two macroscopic strain conditions Eμ and Ek

were used inthe elastictests, aiming atcomputingthe apparent 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 definedin Madi etal. (2006) for characterizing 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, thetotalplastic deformation isequalto thesumofmicro 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 equivalent uniaxial tensile stress, which is associated with J₂ (S_ij ) the second invariant ofthe deviatoricpartS_ij ofthemacroscopic stress tensor



.Ev _{isthemacroscopic viscoplasticstraintensor. p} is theequivalentaccumulatedviscoplasticstrain. Localquantities, suchastheresolvedshearstressandplasticsliprateoneach slip systems are computedin the local material frame, andthen ex-pressedinthemacroscopicframebeforeaveraging.

(2)The apparentglobalplasticstrainratecanbealso 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 asdefined by 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)

Theprocedure fordeterminingP_vapp isasfollows: foreach re-alization, a uniaxial tensile test is carried out under PBC with prescribedmacroscopicstrainratecontrol.Aspolycrystallinepure copperisaratherstrainrateinsensitivematerialatroom tempera-ture(CarrekerandHibbard,1953),onlyonestrainrateof10−3 s−1 isconsidered.Thedurationofthetestis0.1s.Therefore,the max-imummacrostrain is10−4 ,whichisinthescopeofthepractical usefortheMeric–Cailletaudmodel(Mericetal.,1991;Cailletaud, 1992).At theendoftensiletest,output values(



3J2

(

Si j

)

,e˙p ) are

computed.Usingtheleastsquarefittingmethod,thevalueofP_vapp

isidentified asthe coefficient of the fittedpower law, asshown inFig. 1. The best fit isobtained by increasing the volume size:

R2 ≥ 0.9for8gr, 0.92for27gr, 0.94for 64gr,0.96for 125gr,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 isapp 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, for an er-godicstationary random function Z(x), one can compute the en-semblevariance D2 _Z

(

V

)

ofitsaverage value ¯Z

(

V

)

overthevolume

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

D2 _Z

(

V

)

=D2 Z



A₃ V



(14)

whereD2 _Z isthe pointvariance ofZ(x)involume VandA₃ is the integralrangeoftherandomfunctionZ(x),definedas:

A3 = 1 D2 _Z  R 3 ¯ W2

(

h

)

dh (15)

whereh isa two-point segment,andW¯₂

(

h

)

is thecentered 2nd order correlation function such that, for a prescribed property Z

andforx_∈V:

¯

W2

(

h

)

=



Z

(

x+h

)

− ¯Z



Z

(

x

)

− ¯Z



(16)

In case of 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

)

=D2_Z



A∗₃ V



γ (17)

A∗₃isalsohomogeneoustoavolumeofmateriallikeA₃ andcan readilybe used todetermine RVEsizes. But thereis not a direct definitionasEq.(15),anymore.

Inthecaseofatwo-phasematerial,Kanitetal.(2003)assumed that ¯Zwasequaltothearithmetic averageofconsideredproperty withZ₁ forphase1andZ₂ forphase 2.Thepoint varianceD2 _Z of therandomvariableZwasgivenby:

D2 _Z =P

(

1− P

)

(

Z₁ − Z2

)

2 (18)

whereP is thevolume fractionofphase 1 involume V.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 difference oforientation.Eq.(18)can then beextended to multi-phasemicrostructures,asfollows:

D2 _Z =

T



i

P_i



Z_i − ¯Z



2 (19)

whereTisthenumberofgrainsorphasenumber.

Finally,inorderto determinethe RVEsize,Dirrenbergeretal. (2014) reformulatedEq.(17)to reducethe informationofD2 _Z and A∗₃ ,asfollows:

D2 _Z

(

V

)

=GV−γ (20)

withG=D2 _Z A∗₃ γ,since A∗₃ cannot be deducedindependently. Thus onlytwoparametersGand

γ

areneededtoidentifyfromthe sta-tisticaldataobtainedbylinearizationasfollows:

logD2 _Z

(

V

)

=logG−

γ

logV (21)

FollowingthemethodproposedinKanitetal.(2003),the rela-tivesamplingerrorintheeffectivepropertiesarises:



rel =2D_¯ZZ √

(

V

)

n ⇒



2 rel= 4G ¯Z2 _nVγ (22)

henceyielding the definitionofthe RVEsize, fora givenrelative 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 andstatistical work ofmesh 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, needed forPBC. Forthe sake of simplic-ity andcomparison with resultsfrom the literature (Madi et al., 2006), an isotropic morphological and crystallographic texture is considered.Inordertoobtaintheisotropicdistributionofthegrain orientation foraperiodic sample, thearbitraryshaft pointsof all crystalsinthesamplemustspanuniformlythesurfaceofasphere, assuggestedby Nédaetal.(1999).Forthe sake ofachievingthe condition, the three Euler angles (

α

,

β

,

γ

) inthe Z–X–Z type of eachgeneratedgraincellwillbegivendifferentdistributionrules:

α

and

γ

are generated randomlywith a uniformdistribution 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 randomly inthe range[0, 1]. Afterwards,the microstructurewas meshed with lineartetrahedral elementsas shownin Fig.2b. At least 300elements were used foreach grain, which is a reason-ablevalue, consideringthe usualpractice inthefull-field simula-tionliterature(Cailletaudetal.,2003a;Rotersetal.,2011;Cruzado etal.2015).Thegrainsizeinthegeneratedmicrostructurefollows a normal distribution function with a mean value of 20μm and a standarddeviationof 13.5μm. Fig.2crepresentsan exampleof equivalentvonMisesstressdistributionatamacroscopicstrainof 0.01%,andthestresswithingrainsmicrostructurewasobservedby slicingperpendicularlytoZaxis,asshowninFig.2d.

5

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

Accordingtotheapproach describedhereinabove,several real-izationsweregeneratedforstatisticalanalysis,withdifferent num-berofgrains,rangingfrom8to343,aslistedinTable2.The gen-eratedmicrostructure hasa meangrainsize(diameter) of20μm. Thus, the actualvolume size isdirectly relatedto thenumberof grains. For the sake of simplicity,we used thenumber of grains inplace ofthevolumesize (V).Generally,the numberof realiza-tionsNshouldbedifferentforeachdomainsizeinordertoachieve a similar measurement error for all sizes considered. Using Eq. (22) themeasurement error forall domain sizeswere controlled at under1% for elasticproperties, under3% for intrinsic dissipa-tion. On the contrary, the measurement error remains high, just under 120%for apparent viscoplasticparameter P_vapp ,mostly due tothelargeintrinsicvariabilityoftheproperty.Toaccomplishthe

generationandmeshing ofmicrostructures, acomputer equipped withan Intel Core i7-4750HQ CPU @ 2.0GHz and8GBRAM was employed.The consumedtime ofmeshing, aswell aselasticand plasticcalculationsforone realizationis alsopresentedfor 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 maynot exhibit an isotropic behavior. Therefore, it is necessary to check whether the gener-atedpolycrystal is isotropic ornot. Forthat purpose, six compu-tations are necessary forfinding the 21 components ofapparent elastictensorCapp oneach realization,using PBC.Fromaveraging all the realizations, the obtainedfull elastic moduli tensors with theintervals ofconfidencecorresponding toplus andminus 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

standarddeviations (±2D_Z ) are giveninTable3 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 222 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-uesof μapp and kapp increase gradually and stabilize respectively at49,031± 1085MPaand130,972± 2385MPaon the343volume size,whichare consistentwithcommonvaluesforpolycrystalline pure copper. Similar convergence behavior can be observed for

dtension ₁ .Itsmeanvalue reaches53J/m3 withthecorresponding in-tervalsofconfidenceof4.3J/m3 onthe343volumesize,whichis inthesame magnitudelevelasprevious 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 in magnitude. Forbetterpresenting, theorder ofmagnitude log

(

P_vapp

)

wasdrawn vs. the volume size in Fig. 3 asthe substi-tute ofP_vapp . However, the relative error, floating around 9%, isa littlehigherthanthose ofelasticpropertiesandintrinsic dissipa-tionbecauseoffewerrealizations.Whenthevolumesizeincreases to 343,the fluctuation tends to weaken, and the averaged value oflog

(

P_vapp

)

stabilizes at−10.74, witha fluctuationrangeof±0.8. Interestingly,thisvalue isalso approximatelythe combination of thetwoparametersofthesinglecrystalconstitutivelaw,K=5and

n=10,producingthevalueof10−10.7 asthedefinitionofP_v= 1 K n. Consideringa largeenoughvolumeofpolycrystallinepurecopper

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,witha relativeerrorofabout3% forboth

Kandn.

4.3. FluctuationofapparentpropertiesandRVEsizes

As specified inEq. (23), in orderto compute RVE sizes, three variables(D_Z , ¯Z,

γ

)havetobeestimated.Alinearfittingwasdone according to Eq.(21) in the logarithm scale for the four proper-ties,asshowninFig.4.The slopevaluesofeachlinecorresponds 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 apparent viscoplasticparameterhasa muchfasterstatistical con-vergenceratethantheotherinvestigatedproperties.Theintercept termbcanbeusedtoworkoutthevariables,G.

Inordertoexplainthediscrepanciesobservedfor

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

and μapp , respectively for 343volume size). For D_P app

v , this value

can reachatover 100, becauseofthe nonlinearityandhigh sen-sitivityof thisparameter.This impliesthat theorientation distri-bution seems to have a stronger impact on the plastic behavior variabilityincomparisontoitseffectonelasticproperties.Also,it seems thatthe high

γ

exponentforD2

P app

v

(

V

)

could be correlated

withthenonlinearityandsensitivityoftheviscoplasticparameter. One ofthe advantagesofrelying onmicrostructural computa-tionistheabilitytoconsidereach grainorphaseindividually.For thispurpose, Fig.6 illustrates the impact of grain orientation on crystalplasticity,byanalyzingthelocalheterogeneitiesofthe plas-ticbehavior inarealizationwith343grains, includingvonMises

7

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-tieswere drawn foreachgrain andthewhole volume.As shown inthevonMisescurves,grainsinthevolumeholddifferent equiv-alent stress, ranging from6MPa to 16MPa. Some grains actually remain in the elastic regime during the tensiletest. Meanwhile, grainsyield heterogeneous plastic powerdependingon the grain orientation for the same equivalent macroscopic uniaxial tensile stress, asshown on the righthand 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 associated variables in Eq.(23) forfour appar-entpropertieswereobtainedandlisted inTable4.Theminimum domainsizesthatare necessarytoreach agivenprecision 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,incomparisontotheother threeproperties.Inpracticaluse, fora 5% relative error, a volume size of 135 grainscan be con-sidered for elastic properties, while for intrinsic dissipation and apparent viscoplasticparameter, 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).

than1010 grainsand 1445 grains, respectively. Ifa higher preci-sionisrequired,morerealizationscanbealsoconsidered,suchas foraprecision of1% withn=50realizations,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 forthe sameproperty P_vapp used by Madi etal. (2006) was foundsmallerthan fortheisotropicelastic ones. 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-ityhighlydependsonthe grainorientation, asdifferentactivated

slipsystems produce different plasticdeformations. The effectof anisotropicelasticityappearstoyieldlowerstatistical heterogene-itythan thecrystalplasticitybehavior,i.e.yielding a smallerRVE sizeincomparison.

BasedonFig.6,itappearsthatplasticdeformationtakesplace inmostgrainsduringthetensiletest, butlocalizationoperatesin only a minority ofgrains, i.e. lessthan 10% of them. Thisplastic strainlocalizationandstressconcentrationphenomenoncould ex-plaina high

γ

exponent forD2

P app

v

(

V

)

. Asa matter offact, a

pat-ternoflocalizationwillformforanynumberofgrainsduetothe morphological andmaterial anisotropies. Therefore,a large num-berofgrainsisnotneededforensemblevarianceconvergenceon

9

theaveraged valueoftheviscoplasticparameter. ResultsfromEq. (23) andTable 5might lead to another conclusion: the RVEsize forP_vapp isratherlargeandonly decreasesslowlywithincreasing the relativeerror. Thisis duetothe highintrinsicpoint variance ofP_vapp,i.e.GinEq.(22).Thisvariabilityisrelatedto the nonlin-earnatureofthelocalizationphenomena.Thislargepointvariance counterbalancestheeffectofafastensemblevarianceconvergence. Onecould arguethatthepointvarianceofP_vappislikelytobe re-latedtoastrongmaterialheterogeneity asinducedinthepresent workbyelasticanisotropyandcrystalplasticity.

5. Conclusionsandprospects

Aiming at computing the intrinsic dissipation of pure cop-perbasedonacrystalplasticityframework,virtualpolycrystalline samples were generated based on EBSD microstructural analysis, andtheRVEsizeforvariousmechanicalpropertieswasestimated. Astatisticalanalysismethod wasused todetermine theRVEsize 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 sizesobtained are smaller forelastic properties than for plasticproperties,likelyduetotheanisotropic elastoviscoplas-ticmodelchosenforthematerialbehavior,aswellasthe poly-crystalline nature ofthe samples, both inducing stronger het-erogeneitiesin themechanicalfields.Thisconclusion is oppo-sitetotheonesmadebyMadietal.(2006),whileconsidering anisotropicelastoplasticbiphasicmaterial.

3. Thecomputational microstructuralstudyallowedto character-ize the local heterogeneities associated with plasticity, hence givinganinsightonthemicrostructuralbehaviorexplainingthe statisticalmacroscopictrendsobserved.

4. The viscoplastic parameter is related to the nonlinear phe-nomenonofplasticlocalization, inducing stronglocal variabil-ityforP_v .Thisinherentintrinsicheterogeneity leadstoa high pointvariance,whichitselfwillinvariablyyieldlargerRVEsize incomparisontolinearproperties.

The statistical analysis provided in this paper can be applied tootherpolycrystallinematerialsforvariousproperties,giventhat a microstructuralmorphologicalmodel isavailable forgenerating avirtual statisticalpopulationofsamples.Introducingmore infor-mation aboutthemicrostructureofmaterials appearsasa neces-sityforimprovingthepredictivecapabilityofsuchstatistical tech-niques.Furtherworkwillinvolvetheextensionofthepresent ap-proachtothe fatigueofmetallicpolycrystallinematerials. The fa-tigue strengthofpolycrystallinematerials isa longstanding prob-leminmechanicaldesign,especiallyintheveryhighcyclefatigue regime, where thestress level ismuch lower than traditional fa-tigue limit (Bathias and Paris, 2004; Stanzl-Tschegg et al., 2007; Phungetal.,2014;Torabianetal., 2016a,2016b,2017a,2017b).In ordertoreduce experimentduration time,manyauthorshave re-sorted to themethod ofself-heating tests, inwhich the thermo-mechanicalresponse ofthematerial duringcyclicloadingis ana-lyzed andtheintrinsicdissipation istakenasthe fatiguedamage indicatortoevaluatefatiguestrengthatvariousstresslevelsinthe high and very highcycles 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;Guo etal.,2015).Physically,coming fromthe irreversibility of the plastic deformation, intrinsicdissipation is assumed to be relatedtothemicroplasticdeformationingigacyclefatigueregime,

i.e.the crystalslipping behavior atthe grain scale. The approach developed inthe present work could thus be used to determine theRVEsizeassociatedwithintrinsicdissipationduringvery-high cyclefatigue.Relying onfull-field crystalplasticityfiniteelement analysiscouldthusfurtherourunderstandingofthedamage 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 .