Effective behavior of long and short fiber-reinforced viscoelastic composites

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Applications in Engineering Science

journalhomepage:www.elsevier.com/locate/apples

Effective behavior of long and short fiber-reinforced viscoelastic composites

O.L. Cruz-González

, A. Ramírez-Torres

, R. Rodríguez-Ramos

, J.A. Otero

, R. Penta

, F. Lebon

aAix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France

bSchool of Mathematics and Statistics, Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow G12 8QQ, UK

cFacultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Vedado, La Habana CP 10400, Cuba

dEscuela de Ingeniería y Ciencias, Tecnológico de Monterrey, Campus Estado de México, Atizapán de Zaragoza EM CP 52926, Mexico

a r t i c le i n f o

Keywords:

Homogenization Viscoelasticity

Fiber reinforced composites Power-law model Transverse isotropy Finite elements

a b s t r a ct

Westudythehomogenizedpropertiesoflinearviscoelasticcompositematerialsinthreedimensions.Thecompos- itesareassumedtobeconstitutedbyanon-aging,isotropicviscoelasticmatrixreinforcedbysquareorhexagonal arrangementsofelastictransverselyisotropiclongandshortfibers,thelatterbeingcylindricalinclusions.The effectivepropertiesofthesekindofmaterialsareobtainedbymeansofasemi-analyticalapproachcombining theAsymptoticHomogenizationMethod(AHM)withnumericalcomputationsperformedbyFiniteElements(FE) simulations.Weconsidertheelastic-viscoelasticcorrespondenceprincipleandwederivetheassociatedlocaland homogenizedproblems,andtheeffectivecoefficientsintheLaplace–Carsondomain.Theeffectivecoefficients arecomputedfromthemicroscalelocalproblems,whichareequippedwithappropriateinterfaceloadsarising fromthediscontinuitiesofthematerialpropertiesbetweentheconstituents,fordifferentfibers’orientationsin thetimedomainbyinvertingtheLaplace–Carsontransform.WecompareourresultswiththosegivenbytheLo- callyExactHomogenizationTheory(LEHT),andwithexperimentalmeasurementsforlongfibers.Indoingthis, wetakeintoconsiderationBurger’sandpower-lawviscoelasticmodels.Additionally,wepresentourfindingsfor shortfiberreinforcedcompositeswhichdemonstratesthepotentialofourfullythreedimensionalapproach.

1. Introduction

Materials characterized by both a viscoelastic response and a composite-likegeometrical arrangement arefoundin several biolog- icalcontextsdrivenbynaturalevolution,see,e.g., (Atthapreyangkul etal., 2021; Ojanen etal., 2017; Sherman etal., 2017). Especially, longandshortfiber-reinforcedcomposites arebeingincreasinglyex- ploitedin avariety of engineering andmanufacturingprocesses be- causeoftheircapabilityofoptimisingpropertiessuchaslightweight, stiffness,andstrength.Ontheonehand,highperformancecomposites aretypicallymadeoflongcontinuous fibresembeddedinapolymer matrixandexhibitviscoelasticproperties(see,e.g.,OrnaghiJr.etal., 2020;Wangetal.,2020).Ontheotherhand,reinforcementviashort fibers(i.e.,fiber-shapedinclusions)canprovideavaluablealternative inthemodellingoffailuresappearingincompositesreinforcedbylong fibers,see,e.g.,Cepero-Mejíasetal.,2020;NonatoDaSilvaetal.,2020. Materialcompositesreinforcedbyshortfiberscanalsoprovidesignifi- canteconomicalandmanufacturingadvantagesovercontinuousfiber- reinforcedcompositeswithoutcompromisinghighperformance,aslong

∗Correspondingauthor.

E-mailaddress:Raimondo.Penta@glasgow.ac.uk(R.Penta).

astheaspectratioishighenoughtosupportloadtransfer(Huangand Huang,2020;WangandSmith,2019;Yuetal.,2014).

The multiscale modellingof viscoelastic composites hasbeen in- creasinglyaddressedinrecentcontributions.Inthisrespect,microme- chanicalmodelsareparticularlysuitablewhenevertheaimistodeter- minetheeffectiveresponseofmaterialsonthebasisofindividualcon- stituents’properties,suchasviscoelasticmoduliandfiberspropertiesin termsofgeometricalarrangement,volumefractionandorientation.For instance,inSevostianovetal.(2016),theeffectiveviscoelasticproper- tiesofshortfiberreinforcedcompositesareinvestigatedbymeansof thefraction-exponentialoperatorsof ScottBlair-Rabotnov.Moreover, inKernetal.(2019)afrequency-domainfiniteelementsimulationsare consideredtodeterminetheeffectivemoduliofviscoelasticcoatedfiber- reinforcedcomposites.Theinvestigationofthepolymeralignmentwith theaidofdirectnumericalsimulationsoftheturbulentchannelflow ofaviscoelasticFENE-PfluidisconductedinPereiraetal.(2020).In addition,theeffectiveviscoelasticcreepbehaviorofalignedshortfiber compositesisobtainedinWangandSmith(2019)viaaRVE-basedFi- niteElementalgorithm.InOrnaghiJr.etal.(2020),theAuthorseval-

https://doi.org/10.1016/j.apples.2021.100037 Received30December2020;Accepted5February2021 Availableonline17February2021

2666-4968/© 2021TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

uatethecreep,recovery,andviscoelasticpropertiesofunidirectional carbon/epoxyfilament woundcomposite laminatesunder controlled stress,time,andtemperature.Recently,aprobabilisticmicromechan- icsdamageframeworktopredictthemacroscopicstress-strainresponse andprogressivedamageinunidirectionalglass-reinforcedthermoplastic polymercompositeshasbeenproposedinChenetal.(2021).

Amongthemostusedtechniquesaddressingthecalculationofthe effectivepropertiesofviscoelasticheterogeneousstructureswefindthe AsymptoticHomogenization Method(AHM).Forexample,analytical closedformexpressionsfortheeffectivecoefficientsoffibrousviscoelas- ticcomposites areobtainedin Rodríguez-Ramosetal.(2020), Otero etal. (2020)bymeansof thetwo-scaleAHM.The theoreticalbases ofthemethodarefoundinthecontributionsofseveralauthors(see, e.g., Allaire andBriane,1996; Auriault etal., 2009; Bakhvalov and Panasenko,1989;Bensoussanetal.,1978;Sanchez-Palencia,1980).In general,thetechniquetakesadvantageofthescalesseparationassump- tionfordecouplingthespatialvariablesintoamicroscopicandamacro- scopicone.Thus,thesolutionoftheoriginalproblematthemacroscale isapproximatedbyconsideringthesolutionofthecorrespondinglocal problemsonaperiodiccellandofthehomogenizedproblem.Thisper- mitstodecreasethecomputationalcomplexityoftheproblemathand, andtoencodetheinformationatthemicroscaleintotheeffectivecoef- ficientsofthemacroscalemodel.

ThemaindisadvantageoftheAHMisthattheanalyticalsolutionof thelocalproblemscanbederivedforafewofsimplecompositestruc- tureswhichcanbereducedtooneortwodimensionsanalysis(asthose forlaminatedcompositesandlongfibers).Forinstance,instudiesre- latedwithviscoelastic composites forlaminatedandfiberreinforced composites(Cruz-Gonzálezetal.,2020a;Oteroetal.,2020).Forthis reason,inordertohandlemorecomplexmicrostructures,numericalap- proachesbasedontheFiniteElements(FE)providearobustalternative tosolvethelocalproblemsformoregeneralmicrostructures.Basedon theseconsiderations,inthepresentwork,weaimtostudytheeffec- tivepropertiesofviscoelasticcompositesbymeansofthecombination oftheAHM andtheFE method,thelatter allowsustofind thenu- mericalsolutionofthemicroscaleperiodiclocalproblemfordifferent three-dimensionalarrangements.Thisapproachprovidesanewandef- ficientcomputationalplatformforcomputingtheeffectivepropertiesof viscoelasticcompositesinthreedimensions.

Themainaimofthisworkistocalculatetheeffectivepropertiesof non-aging,linearviscoelasticcompositesinthreedimensions.Forour purposes,weemploythemodelingapproachintroducedinourprevious work(Cruz-Gonzálezet al.,2020b),wherein acombinedframework basedontheoreticalandcomputationaltechniquesforcomputingthe effectivepropertiesofviscoelasticcompositesisemployed.Specifically, inCruz-Gonzálezetal.(2020b),weanalyzedseveraltypesofcompos- itestructuresreinforcedbyfibers andinclusions.Here,however,we gofurtherinourinvestigationsandconsiderhexagonalperiodiccells andunidirectionallyalignedshortfibers.Furthermore,inthepresent framework,weconsiderthefibers tobe transverselyisotropic, while inCruz-Gonzálezetal.(2020b),thestudywasfocusedoncomposites withisotropicconstituents.Thisextensionrequiresasuitablegenerali- sationofthecomputationalsetupoftheproblemfound,forexample, inPentaandGerisch(2016)forelasticcomposites.Inparticular,the interfaceloadsrelatedtotheauxiliarylocalcellproblemsareobtained formoregeneralorthotropicmaterials,andthenspecialisedtotrans- verseisotropicconstituentsinourcalculations.Anotherextensionofthe presentworkwithrespecttoCruz-Gonzálezetal.(2020b)isthat,inthe presentstudy,wetakeintoconsiderationdifferentfiber’sorientations.

Wefurthernoticethat, tothebestof ourunderstanding,thema- jornovelty of this work withrespect to othersexistent in theliter- ature and focused on thestudy of viscoelastic composites (see, e.g.

Amiri-Radetal., 2019; Ornaghi Jr.etal., 2020; Oteroet al., 2020;

Rodríguez-Ramosetal.,2020;Sevostianovetal.,2016;TangandFe- licelli,2015;Tranetal.,2011;WangandPindera,2016a;Wangand Smith,2019;YanceyandPindera,1990;Yietal.,1998) isthathere,

Fig.1.(a)Macroscale:viscoelasticheterogeneousmaterialwith(b)square(i,ii) orhexagonal(iii,iv)arraysofnon-overlappinglongandshortfibers,respec- tively.(c)𝜀-structurallevel.Microscale:periodiccellforlongandshortfibers inclusionsthatdonotintersecttheboundaries.

usingasemi-analyticalapproach,wearecapabletocomputetheeffec- tivepropertiesofthree-dimensional,non-aging,viscoelasticcomposite materials.Particularly,itshouldbepointedoutthatalthoughwecon- sidered someof theresultsgivenin YanceyandPindera(1990)and WangandPindera(2016a,b)forcomparisonwithours,ourmethodol- ogyprovidesfurtherdevelopmentsbecauseofthepossibilityofconsid- eringmorecomplexcellgeometries.Indeed,wecalculatetheeffective propertiesforaviscoelasticcompositereinforcedwithperfectlyaligned shortfibersandthisgeometricalconfigurationisnottreatableunderthe two-dimensionalformulationreportedinWangandPindera(2016a,b). The manuscriptis organizedas follows.In Section2, we present thegeometricaldescriptionofthemodelandweformulatethelinear viscoelastic heterogeneousproblem. InSection3,we applythe two- scaleAHM toobtainthemacroscalefunctionalform oftheeffective viscoelasticcoefficients.InSection4,wecalculatetheeffectiveprop- ertiesbysolvingappropriatelocal(cell)problemsinthreedimensions.

Thetheoreticalframeworkisillustratedingeneralbyconsideringthat bothphasesareviscoelastic,althoughtheresultsarethenpresentedby consideringpurelyelasticfibersembeddedinaviscoelasticmatrixfor thesakeofcomparisonagainstpreviousanalyticalresultsandrelevant experiments.InSection5,wecompareourresultsagainstalternative homogenisation techniquesinthecaseof longfibers,andemphasise thepotentialofournewapproachbyillustratingtheresultsinthecase ofcompositesreinforcedbyshortfibers(i.e.,cylindricalinclusions).Fi- nally,inSection6,wesummarizeourfindingsandhighlightthelimi- tationsofthecurrentmodelandpossiblefurtherdevelopmentsofthe work.

2. Modeldescription

Weidentifytheheterogeneous,linearviscoelasticmaterialwithan open,boundedsetΩ⊂ℝ³(seeFig.1(a)).Inparticular,weconsiderΩ asatwo-constituentcompositemadeofamatrixreinforcedbysquare (Fig.1(b)(i,ii))orhexagonal(Fig.1(b)(iii,iv))arraysofunidirectional andperiodicallydistributedlongandshortfibersinΩ(seeFig.1(b)).

Furthermore,weconsidertheexistenceoftwodistinct,well-separated lengthscales𝓁and𝐿whicharerelatedwiththecharacteristicsizeofthe periodicmicro-structureandthatofthewholecomposite,respectively (seeFig.1).Inthisframework,weintroducethedimensionlessscaling parameter𝜀asfollows,

𝜀= 𝓁

𝐿≪1, (1)

andthemicroscopicspatialvariable 𝑦=𝑥

𝜀, (2)

where𝑥issaidtobethemacroscopicspatialvariable.

Inparticular,wesetΩ =Ω^𝜀₁∪ Ω^𝜀₂ withΩ^𝜀₁∩ Ω^𝜀₂=Ω^𝜀₁∩ Ω^𝜀₂=∅,and whereΩ^𝜀₂ denotesthe matrix andΩ^𝜀₁=∪^𝑁_𝑖=1𝑖Ω^𝜀₁ represent the inclu- sionswith𝑁∈ℕ.Additionally,theinterfacebetweenΩ^𝜀

1andΩ^𝜀

2 (see Fig.1(b))isdenotedbyΓ^𝜀.Referring toFig.1(c),theunitaryperi- odiccell𝑌 isconsideredtobeconstitutedbyafiber(longorshort)𝑌1

inthematrix𝑌2sothattheperiodiccellisgivenby𝑌 =𝑌1∪𝑌2with 𝑌1∩𝑌2=𝑌1∩𝑌2=∅.Atthisscale,theinterfacebetween𝑌1and𝑌2 is denotedbyΓ_𝑌,seealsoDiStefanoetal.(2020)foranillustrationof variousperiodiccellarrangementinthecontextofelectro-activecom- posites.

2.1. Statementoftheproblem

Forthesakeofsimplicity,weneglectinertiaandexternalvolume forcesinthemodel,andimposecontinuityconditionsfordisplacements andtractionsontheinterfaceΓ^𝜀,i.e.,thematrixandthesub-phasesare inidealcontact.Therefore,thebalanceoflinearmomentuminΩ⧵Γ^𝜀 togetherwiththeinterfaceconditionsread

∇⋅𝝈^𝜀(𝑥,𝑡)=𝟎 in(Ω⧵Γ^𝜀)×ℝ, (3a)

𝒖^𝜀(𝑥,𝑡)=𝟎 onΓ^𝜀×ℝ, (3b)

𝝈^𝜀(𝑥,𝑡)𝒏^(𝑦)=𝟎 onΓ^𝜀×ℝ, (3c)

where𝝈^𝜀 represents thesecond-order stresstensorand𝑢𝑢𝑢^𝜀 is thedis- placementfield.Moreover,𝑛𝑛𝑛^(𝑦)denotestheoutwardunitvectortothe interfaceΓ^𝜀,andtheoperator𝜙^𝜀describesthejumpof𝜙^𝜀acrossthe interfaceΓ^𝜀betweenΩ^𝜀₁ andΩ^𝜀₂.Noticethatthesuperscript𝜀isused toindicatethenotation 𝜙^𝜀(𝑥,𝑡)=𝜙(𝑥,𝑦,𝑡)(refer toDiStefano etal., 2020foracomprehensivediscussionregardingthisnotation).Thesys- temofEqs.(3a)–(3c)hastobesupplementedwithboundaryconditions on𝜕Ω ×ℝandinitialconditionsinΩ × {0}.However,theseconditions donotplayaroleinthederivationoftheeffectivecoefficients,andthey aretypicallytobespecifiedexplicitlyonlywhentheaimistoobtaina specificsolutionofthemacroscalesystemofhomogenizedPDEs,which isnotthecasehere.

Inthepresentframework,thecompositebehavesasanon-agingvis- coelasticmaterialsothat(Christensen,1982)

𝜎𝜎𝜎^𝜀(𝑥,𝑡)=

∫

𝑡 0

^𝜀(𝑥,𝑡−𝜏)∶𝜕𝜉𝜉𝜉(𝑢𝑢𝑢^𝜀(𝑥,𝜏))

𝜕𝜏 𝑑𝜏, (4)

where^𝜀isthefourth-ordertensorofrelaxationmoduli,whichhereis scale-dependent.Thediscontinuitiesofthepropertiesbetweenthehost mediumandthesubphasesareencodedinthetensor^𝜀and,thus,its componentsareassumedtobesmoothfunctionsof𝑥in(Ω⧵Γ^𝜀)×ℝ, butdiscontinuousonΓ^𝜀×ℝ.Furthermore,wenoticethat,inEq.(4),𝜉𝜉𝜉 denotesthesecond-orderstraintensorforsmalldisplacements,namely 𝜉𝜉𝜉(𝑢𝑢𝑢^𝜀(𝑥,𝑡))=1

2

(∇𝑢𝑢𝑢^𝜀(𝑥,𝑡)+(∇𝑢𝑢𝑢^𝜀(𝑥,𝑡))^𝑇)

, (5)

andwerequirebothminorandmajorsymmetrypropertiesfor,i.e.,

^𝜀_{𝑖𝑗𝑘𝑙}=^𝜀_{𝑗𝑖𝑘𝑙}=^𝜀_{𝑖𝑗𝑙𝑘}=^𝜀_{𝑘𝑙𝑖𝑗}. (6) TheintegralinEq.(4),standingforthestress-strainrelationshipfor non-aging,viscoelasticmaterials,canbemanipulatedbymeansofin- tegraltransforms.Inparticular,theLaplace–Carsontransform,whichis givenby

̂𝜙^𝜀(𝑥,𝑝)=𝑝∫

∞

0 𝑒^−𝑝𝑡𝜙^𝜀(𝑥,𝑡)𝑑𝑡, ∀𝑡≥0, (7)

where𝑝isthevariableintheLaplace–Carsonspace,reduces(4)toan algebraic equationrepresenting theconstitutiverelationsin classical elasticitytheory(see,forinstance,Lakes,2009).Thismethodologyorig- inallyproposedbyHashin(1965)andknownastheelastic-viscoelastic

correspondenceprinciple,continuestogaininterestinthescientificlit- erature (seeLiu etal., 2020; Vilchevskayaet al., 2019; Yangetal., 2019andreferencestherein).Hence,basedontheaboveconsiderations, theoriginalsystem(3a)–(3c)writtenintheLaplace–Carsondomainis givenby

∇⋅[̂^𝜀(𝑥,𝑝)∶𝜉𝜉𝜉(

̂𝑢 𝑢𝑢^𝜀(𝑥,𝑝))]

=000 in(Ω∖Γ^ε)× [0,+∞) (8a)

𝑢𝑢̂𝑢^𝜀(𝑥,𝑝)=000 onΓ^𝜀× [0,+∞) (8b) [̂^𝜀(𝑥,𝑝)∶𝜉𝜉𝜉(

̂𝑢 𝑢𝑢^𝜀(𝑥,𝑝))

]𝑛𝑛𝑛^(𝑦)=000 onΓ^𝜀× [0,+∞). (8c) 3. Asymptotichomogenizationapproach

Inthissection,wesummarizethemethodologydescribedinCruz- Gonzálezetal.(2020b)andwritethelocalandhomogenizedproblems resultingfromtheapplicationoftheAHMtoEqs.(8a)–(8c).

Beforegoingfurther,weremarkthatusingthechainrulethefollow- ingrelationforthespatialderivativesholds

𝜕̂𝜙^𝜀_𝑖(𝑥,𝑝)

𝜕𝑥_𝑗 = 𝜕̂𝜙_𝑖(𝑥,𝑦,𝑝)

𝜕𝑥_𝑗 +1 𝜀

𝜕̂𝜙_𝑖(𝑥,𝑦,𝑝)

𝜕𝑦_𝑗 . (9)

Moreover,Eq.(5)becomes, 𝜉_𝑘𝑙(̂𝜙^𝜀(𝑥,𝑝))=𝜉_𝑘𝑙(̂𝜙(𝑥,𝑦,𝑝))+1

𝜀𝜉_𝑘𝑙^(𝑦)(̂𝜙(𝑥,𝑦,𝑝)), (10) wherewehaveintroducedthenotation

𝜉_𝑘𝑙^(𝑦)(̂𝜙(𝑥,𝑦,𝑝))=1 2

(𝜕̂𝜙_𝑘(𝑥,𝑦,𝑝)

𝜕𝑦_𝑙 +𝜕̂𝜙_𝑙(𝑥,𝑦,𝑝)

𝜕𝑦_𝑘 )

. (11)

TheAHM(BakhvalovandPanasenko,1989;CioranescuandDonato, 1999)proposesthesolutionoftheviscoelasticheterogeneousproblem (8a)-(8c)asaformalseriesexpansioninpowersof𝜀.IntheLaplace–

Carsondomainitreads

̂𝑢 𝑢 𝑢^𝜀(𝑥,𝑝)=

∑+∞

𝑖=0

𝜀^𝑖𝑢𝑢̂𝑢^(𝑖)(𝑥,𝑦,𝑝), (12)

wherethecoefficients𝑢𝑢̂𝑢^(𝑖)(𝑥,𝑦,𝑝)areassumedtobeperiodicinthemicro- scopicvariable𝑦.Thus,followingthestandardprocedureinasymptotic homogenization(seeBakhvalovandPanasenko,1989;Cioranescuand Donato,1999),aftersubstitutionoftheseriesexpansion(12)intheorig- inalproblem(8a)–(8c)andbyequatingtheresultinthesamepowers of𝜀,weobtainthat,inthelimit𝜀→0,

̂𝑢^𝜀_𝑚(𝑥,𝑝)=̂𝑢⁽⁰⁾_𝑚(𝑥,𝑦,𝑝)+̂𝑢⁽¹⁾_𝑚(𝑥,𝑦,𝑝)𝜀+𝑜(𝜀)

=̂𝑣_𝑚(𝑥,𝑝)+ ̂𝜒_𝑘𝑙𝑚(𝑦,𝑝)𝜉_𝑘𝑙(𝑣𝑣̂𝑣(𝑥,𝑝))𝜀+𝑜(𝜀), (13) where𝑣𝑣̂𝑣isasmoothvectorfunctionof𝑥and𝑝,andthethirdordertensor

̂𝜒_𝑘𝑙𝑚isthesolutionofthe𝜀-localproblemgivenby

𝜕𝑦𝜕_𝑗

[̂_{𝑖𝑗𝑠𝑞}(𝑦,𝑝)𝜉_𝑠𝑞^(𝑦)(̂𝜒𝜒𝜒_𝑘𝑙(𝑦,𝑝))+̂_{𝑖𝑗𝑘𝑙}(𝑦,𝑝) ]

=0 in(𝑌 ⧵Γ_𝑌)× [0,+∞), (14a) ̂𝜒_𝑘𝑙𝑚(𝑥,𝑦,𝑝)=0 onΓ_𝑌× [0,+∞), (14b) ̂𝑖𝑗𝑠𝑞(𝑦,𝑝)𝜉⁽_𝑠𝑞^𝑦)(̂𝜒𝜒𝜒_𝑘𝑙(𝑦,𝑝))+̂𝑖𝑗𝑘𝑙(𝑦,𝑝)]𝑛⁽_𝑗^𝑦)=0 onΓ_𝑌× [0,+∞), (14c)

̂𝜒_𝑘𝑙𝑚(𝑦,𝑝)=0 in𝑌× {0}, (14d)

where

𝜉_𝑠𝑞^(𝑦)(̂𝜒𝜒𝜒_𝑘𝑙(𝑦,𝑝))= 1 2

(𝜕̂𝜒_𝑘𝑙𝑠(𝑦,𝑝)

𝜕𝑦_𝑞 +𝜕̂𝜒_𝑘𝑙𝑞(𝑦,𝑝)

𝜕𝑦_𝑠 )

. (15)

Theuniquenessofthesolutionofthelocalproblem(14a)–(14d)isguar- anteedbyenforcingeither,thecondition ⟨̂𝜒_𝑘𝑙𝑚⟩_𝑦=0orbyfixingthe valueof ̂𝜒𝑘𝑙𝑚 atonepointofthereferenceperiodiccell𝑌 (seePenta andGerisch,2016;PentaandGerisch,2017).Inparticular,inthesub- sequentsectionswewillusethelatterbecauseofitsadvantagewhen dealingwithnumericalsimulations.Notethat,the𝜀-localproblemhas tobesupplementedwithaninitialconditionin𝑌× {0}.

Forcompletenessinouranalysis,wereportthehomogenizedequa- tionatthemacroscaleintheLaplace–Carsonspace,whichisobtained afterequatinginthesamepowersof𝜀⁰.Specifically,thiscanbewritten as

̂^(∗)_{𝑖𝑗𝑘𝑙}(𝑝) 𝜕

𝜕𝑥𝑗𝜉_𝑘𝑙(𝑣𝑣̂𝑣(𝑥,𝑝))=0 inΩ^ℎ× [0,+∞), (16a) where

̂^(∗)_{𝑖𝑗𝑘𝑙}(𝑝)∶=⟨̂_{𝑖𝑗𝑘𝑙}(𝑦,𝑝)+̂_{𝑖𝑗𝑚𝑛}(𝑦,𝑝)𝜉_𝑚𝑛^(𝑦)(

̂𝜒𝜒𝜒_𝑘𝑙(𝑦,𝑝))⟩

𝑦. (17)

istheeffectiverelaxationmodulusintheLaplace–Carsonspace.InEq.

(17),thenotation⟨𝜙⟩_𝑦denotesthecellaverageoperatorandisdefined bytheexpression

⟨𝜙⟩_𝑦= 1

|𝑌|∫_𝑌𝜙 𝑑𝑦, (18)

with|𝑌|beingthevolumeoftheperiodiccell𝑌. 4. Calculationoftheeffectiveproperties

Forsimplicityinourcalculations,weconsiderthattherelaxation modulus,̂_{𝑖𝑗𝑘𝑙},is𝑦-constantineachconstituentoftheperiodiccell𝑌, i.e.

̂𝑖𝑗𝑘𝑙(𝑦,𝑝)=

{̂⁽¹⁾_{𝑖𝑗𝑘𝑙}(𝑝), if𝑦∈𝑌1,

̂⁽²⁾_{𝑖𝑗𝑘𝑙}(𝑝), if𝑦∈𝑌2, (19) wherethesuperscriptindicatethecorrespondingconstituent,“(1)” for thematrixand“(2)” fortheinclusion(seeFig.(1)(c)).Then,thelocal problem(14a)–(14d)isrewrittenasfollows,

𝜕𝑦𝜕_𝑗

[̂⁽¹⁾_{𝑖𝑗𝑠𝑞}(𝑝)𝜉_𝑠𝑞^(𝑦)(

̂𝜒𝜒𝜒⁽¹⁾_𝑘𝑙(𝑦,𝑝))]

=0 in𝑌1× [0,+∞), (20a)

𝜕

𝜕𝑦𝑗

[̂⁽²⁾_{𝑖𝑗𝑠𝑞}(𝑝)𝜉_𝑠𝑞^(𝑦)(

̂𝜒𝜒𝜒⁽²⁾_𝑘𝑙(𝑦,𝑝))]

=0 in𝑌2× [0,+∞), (20b)

̂𝜒_𝑘𝑙𝑚⁽¹⁾(𝑦,𝑝)= ̂𝜒_𝑘𝑙𝑚⁽²⁾(𝑦,𝑝) onΓ_𝑌× [0,+∞), (20c) [̂⁽¹⁾_{𝑖𝑗𝑠𝑞}(𝑝)𝜉⁽_𝑠𝑞^𝑦)(

̂𝜒𝜒𝜒⁽¹⁾_𝑘𝑙(𝑦,𝑝))]

𝑛^(y)_𝑗

−[

̂⁽²⁾_{𝑖𝑗𝑠𝑞}(𝑝)𝜉_𝑠𝑞^(𝑦)(

̂𝜒𝜒𝜒⁽²⁾_𝑘𝑙(𝑦,𝑝))]

𝑛^(y)_𝑗 =𝑓_𝑖𝑘𝑙^(𝑦)(𝑝) onΓ_𝑌× [0,+∞), (20d)

̂𝜒_𝑘𝑙𝑚(𝑦,𝑝)=0 in𝑌× {0}. (20e)

Wenoticethat,inEq.(20d),thestressjumpconditionsleadtotheoc- currenceofinterfaceloads,i.e.,

𝑓_𝑖𝑘𝑙^(𝑦)(𝑝)=[

̂⁽²⁾_{𝑖𝑗𝑘𝑙}(𝑝)−̂⁽¹⁾_{𝑖𝑗𝑘𝑙}(𝑝)]

𝑛^(y)_𝑗 , (21)

whichariseasaconsequenceofthediscontinuitiesinthecoefficients of̂ betweenthehostmediumandthesub-phases,andrepresentthe drivingforcetoobtainnontrivialsolutionsofthesixelastic-typelocal problems((𝑘,𝑙),𝑘≥𝑙)(seePentaandGerisch,2016;PentaandGerisch, 2017).Inparticular,whenthematrixandsubphasesareorthotropic materialsandconsideringVoigt’snotation,theinterfaceloads𝑓_𝑖𝑘𝑙^(𝑦)read

𝑓𝑓𝑓⁽₁₁^𝑦)(𝑝)=[̂⁽²⁾₁₁(𝑝)−̂⁽¹⁾₁₁(𝑝)]𝑛⁽₁^𝑦)𝑒𝑒𝑒1+[̂⁽²⁾₂₁(𝑝)−̂⁽¹⁾₂₁(𝑝)]𝑛⁽₂^𝑦)𝑒𝑒𝑒2

+[̂⁽²⁾₃₁(𝑝)−̂⁽¹⁾₃₁(𝑝)]𝑛⁽₃^𝑦)𝑒𝑒𝑒3, (22a) 𝑓 𝑓

𝑓⁽₂₂^𝑦)(𝑝)=[̂⁽²⁾₁₂(𝑝)−̂⁽¹⁾₁₂(𝑝)]𝑛⁽₁^𝑦)𝑒𝑒𝑒1+[̂⁽²⁾₂₂(𝑝)−̂⁽¹⁾₂₂(𝑝)]𝑛⁽₂^𝑦)𝑒𝑒𝑒2

+[̂⁽²⁾₃₂(𝑝)−̂⁽¹⁾₃₂(𝑝)]𝑛⁽₃^𝑦)𝑒𝑒𝑒3, (22b)

𝑓𝑓

𝑓⁽₃₃^𝑦)(𝑝)=[̂⁽²⁾₁₃(𝑝)−̂⁽¹⁾₁₃(𝑝)]𝑛⁽₁^𝑦)𝑒𝑒𝑒1+[̂⁽²⁾₂₃(𝑝)−̂⁽¹⁾₂₃(𝑝)]𝑛⁽₂^𝑦)𝑒𝑒𝑒2

+[̂⁽²⁾₃₃(𝑝)−̂⁽¹⁾₃₃(𝑝)]𝑛⁽₃^𝑦)𝑒𝑒𝑒3, (22c)

𝑓 𝑓

𝑓⁽₂₃^𝑦)(𝑝)=𝑓𝑓𝑓⁽₃₂^𝑦)(𝑝)=[̂⁽²⁾₄₄(𝑝)−̂⁽¹⁾₄₄(𝑝)]𝑛⁽₃^𝑦)𝑒𝑒𝑒2+[̂⁽²⁾₄₄(𝑝)−̂⁽¹⁾₄₄(𝑝)]𝑛⁽₂^𝑦)𝑒𝑒𝑒3, (22d) 𝑓𝑓

𝑓⁽₁₃^𝑦)(𝑝)=𝑓𝑓𝑓⁽₃₁^𝑦)(𝑝)=[̂⁽²⁾₅₅(𝑝)−̂⁽¹⁾₅₅(𝑝)]𝑛⁽₃^𝑦)𝑒𝑒𝑒1+[̂⁽²⁾₅₅(𝑝)−̂⁽¹⁾₅₅(𝑝)]𝑛⁽₁^𝑦)𝑒𝑒𝑒3, (22e) 𝑓

𝑓

𝑓⁽₁₂^𝑦)(𝑝)=𝑓𝑓𝑓⁽₂₁^𝑦)(𝑝)=[̂⁽²⁾₆₆(𝑝)−̂⁽¹⁾₆₆(𝑝)]𝑛⁽₂^𝑦)𝑒𝑒𝑒1+[̂⁽²⁾₆₆(𝑝)−̂⁽¹⁾₆₆(𝑝)]𝑛⁽₁^𝑦)𝑒𝑒𝑒2, (22f) where{𝑒𝑒𝑒_𝑖}³_𝑖=1representsthestandardvectorbasis.

4.1. Numericalapproach

Atthispoint,itispossibletosolvenumericallythesetofelastic- typelocalproblems(20a)–(20e)intheLaplace–Carsonspaceandthen, tocomputetheeffectiveviscoelasticpropertiesbyusing(17).Forthis purpose,weusethefiniteelementsoftwareCOMSOLMultiphysics®in conjunctionwithLiveLink^TMforMatlab^®scripting(seeCruz-González etal.,2020b;PentaandGerisch,2016;PentaandGerisch,2017).Par- ticularly,once̂^(∗)isknown,theeffectivecreepcompliance̂^(∗)inthe Laplace–Carsonspacecanbecomputedthroughtherelationship

̂^(∗)_{𝑖𝑗𝑚𝑛}(𝑝)̂_{𝑚𝑛𝑘𝑙}^(∗) (𝑝)=𝐼𝑖𝑗𝑘𝑙, (23) where𝐼_{𝑖𝑗𝑘𝑙}denotesthecomponentsofthefourth-orderidentitytensor (seeHashin,1972).

Theinversionoftheeffectivecoefficientstotheoriginaltimedomain isperformedbyemployingtheMATLAB’sfunctionINVLAP(seeJuraj, 2020;ValsaandBranĉik,1990andreferredheretoasValsa’smethod.

Thestepsaresummarizedasfollows,

(a) Discretizethetimeinterval𝑡=[𝑡1,𝑡2,...,𝑡𝑁]

(b) Foreach 𝑡_𝑖,obtainthecomponents𝑝_𝑗∶=𝛼_𝑗∕𝑡_𝑖and𝐵_𝑗∶=𝛽_𝑗∕𝑡_𝑖for 𝑗=1,...,(𝑛𝑠+𝑛𝑑+1),where𝑛𝑠and𝑛𝑑areimplicitparameters,and 𝛼and𝛽aredefinedinValsa’smethod.

(c) Calculate̂^(∗)(𝑝_𝑗)and̂^(∗)(𝑝_𝑗)for𝑗=1,...,(𝑛𝑠+𝑛𝑑+1).

(d) UsethelaststepofValsa’smethodtodeterminetheeffectivecoeffi- cientsinthetimedomain

^(∗)(𝑡𝑖)=

𝑛𝑠+∑𝑛𝑑+1 𝑗=1

Re[𝐵𝑗̂^(∗)(𝑝𝑗)∕𝑝𝑗] for𝑖=1,...,𝑁, (24a)

^(∗)(𝑡_𝑖)=

𝑛𝑠+∑𝑛𝑑+1 𝑗=1

Re[𝐵_𝑗̂^(∗)(𝑝_𝑗)∕𝑝_𝑗] for𝑖=1,...,𝑁, (24b)

whereReindicatestherealpartofacomplexvariable.

Moreover, totake intoaccountthedifferent orientationsthatthe unidirectionalviscoelasticcompositesmayhave,werotatetheeffective viscoelastictensors^(∗)and^(∗)byanangle𝜃andobtain^(∗)_𝜃 and_𝜃^(∗). Inthisrespect,thefollowingtransformationsareuseful,

^(∗)_𝜃 =^(∗)^𝑇, (25a)

_𝜃^(∗)=(⁻¹)^𝑇^(∗)⁻¹, (25b)

Fig.2. MethodologysketchofAHMFE.

where

[]=

⎡⎢

⎢⎢

⎢⎢

⎢⎢

⎣

²₁₁ ²₁₂ ²₁₃ 21213 21311 21112

²₂₁ ²₂₂ ²₂₃ 22223 22321 22122

²₃₁ ²₃₂ ²₃₃ 23233 23331 23132

2131 2232 2333 2233+2332 2331+2133 2132+2231

3111 3212 3313 3213+3312 3311+3113 3112+3211

1121 1222 1323 1223+1322 1321+1123 1122+1221

⎤⎥

⎥⎥

⎥⎥

⎥⎥

⎦

and𝑖𝑗(𝑖,𝑗=1,2,3)arethecoefficientsoftheorthogonalrotationten- sor(seeRamírez-Torresetal.,2018;Ting,1996).

Toconcludewiththissection,itisworthtoremarkthatsteps(a)- (d)intheinversionprocessareequivalentstothestage(IV)inCruz- Gonzálezetal.(2020b). Here,weillustratethroughaflowchart(see Fig.2) themethodology describedin (I)-(IV)ofCruz-González etal.

(2020b)withmore details. Inthefollowing sections, werefer toas AHMFEthe semi-analytical approachproposed in thepresentwork, whichcombinestheAsymptoticHomogenisationMethod(AHM)and FiniteElements(FE)simulations.

5. Results

5.1. Instantelasticresponse

Tobeginwithouranalysis,inthissection,wecomputetheeffective instantelasticresponseofacompositewithahexagonalarrangement oftransversely,purelyelasticisotropiclongfibers(seeFig.1(b)-(iii)).

Althoughthetheoreticalframeworkintroducedintheprevioussection holds forviscoelasticfibersaswell,wefocuson purelyelasticfibers forthesakeofcomparisonofourresultswithalternativeanalytictech-

Fig.3.MeshdiscretizationsA,BandCforthehexagonalperiodiccellwith𝑉𝑓=0.6.

Table1

Elasticpropertiesoftheconstituents.

Materials 𝐸 _𝐴(GPa) 𝐸 _𝑇(GPa) 𝜇_𝐴(GPa) 𝜇_𝑇(GPa) 𝜈_𝐴

AS4 graphite fiber 225 15 15 7 0.20

3501-6 epoxy 4.2 4.2 1.567 1.567 0.34

E-glass fiber 69.0 69.0 28.28 28.28 0.22

Boron fiber 420 420 175 175 0.20

Aluminum 69.0 69.0 25.94 25.94 0.33

niques.Inthiscontext,theelasticlimitcaseisreachedbyconsidering 𝑡=0inEq.(4),whichimpliesthat^(∗)_{𝑖𝑗𝑚𝑛}becomestheeffectivestiffness tensor.

Here, we determine the instant elastic effective response of a graphite/epoxysystemwithhexagonalarchitectureanddifferentfiber volumefractions.Itisworthnoticingthatthefollowingresultsdiffer fromtheonesobtainedinCruz-Gonzálezetal.(2020b)sincethereinit wasconsideredasquarearrayofinclusionsinthematrixphasecom- prisingisotropicconstituents.Thevaluesoftheparametersreportedin Table1areobtainedfromWangandPindera(2016b).Thenotation𝐸_𝐴 (𝐸_𝑇)and𝜇_𝐴 (𝜇_𝑇)isusedfortheaxial(transverse)Young’sandshear moduli,respectively,and𝜈_𝐴representstheaxialPoisson’sratio.

Beforeweproceedwiththeresultsoftheeffectivecoefficients,we gatherinformationontwomainfeaturesrelatedwiththecomputational approach.Thesolution’sconvergencebehavior,throughthreetypesof meshdiscretization,andtheexecutiontimesrequiredforthesecalcula- tions.Withthispurpose,weonlyprovidetheresultsofthecomputation oftheeffectivetransverseYoung’smodulus𝐸_𝑇^(∗)sincethoserelatedwith thetransversePoisson’sratio𝜈_𝑇^(∗)andtheaxialshearmodulus𝜇_𝐴^(∗)show similarcharacteristics.Furthermore,withregardstotheanalysisofthe solution’sconvergence,weusethemeshesA,BandCreportedinFig.3. InFig.4,weshow theeffectivetransverseYoung’smodulus𝐸^(∗)_𝑇 , normalizedbythecorrespondingmatrixYoung’smodulus𝐸_𝑚.Specifi- cally,inFig.4(a)theeffectivetransverseYoung’smodulusiscomputed, foreachofthemeshesdiscretizationsA,BandC,asafunctionofthe volumefraction𝑉_𝑓.Moreover,inFig.4(b)wecomparetheresultsob- tainedwiththepresentmodel(AHMFE)withthoseproducedbyWang andPindera(2016b)(seeFig.7inWang andPindera,2016b) using thefinite-volumedirectaveragingmicromechanics(FVDAM)theory.A closerlookatthedatainFig.4(b)revealsthattherelativeerrorbe- tweenthesolutionsthepresentapproach(AHMFE)andthatinWang andPindera(2016b)(FVDAM),namely

𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝐸𝑟𝑟𝑜𝑟(𝜙)∶=𝜙^{(𝐹 𝑉𝐷𝐴𝑀)}−𝜙^{(𝐴𝐻𝑀𝐹 𝐸)}

𝜙^{(𝐹 𝑉𝐷𝐴𝑀)} × 100%, (26)

Table2

Maximumrelativeerror(%).

𝐴𝐻 𝑀 𝐹 𝐸 →𝐿𝐸𝐻 𝑇 𝐸 ^(∗)_𝑇∕ 𝐸 _𝑚 𝜈^(∗)_𝑇 𝜇^(∗)_𝐴∕ 𝜇_𝑚 glass/epoxy 0.5844 0.3969 0.0767 graphite/epoxy 0.1971 0.1311 0.0604 boron/aluminum 0.3476 0.2088 0.0604 aluminum/porosity 1.4575 1.1796 0.0677

reacheshismaximumvaluewhenthecoarsermesh(MeshA)isconsid- eredandislessthan1.3%.Ontheotherhand,therelativeerrorcom- putedwiththemeshesdiscretizationsBandCislessthan0.2%.Then,we canconcludethatoursimulationsprovideagoodagreementwithFV- DAM,andthat,inthiscase,ourresultsdonotundergolargevariations aftersubsequentmeshdiscretizations.

Tocontinue withouranalysis, in Fig.4(c)weprovide thetotal executiontimesneededtoobtaintheentiresetofeffectivemoduliin relationtothethree meshdiscretizations.Thesimulationsaresetup totakeintoaccountafinitenumberoffibervolumefractionsranging from𝑉_𝑓=0.05to𝑉_𝑓=0.7withanincrementof0.05,andtheywere performedusingamachinerunningWindows10Professional64-bitop- eratingsystem,with32.0GBRAMandIntel(R)Core(TM)i5-8350UCPU 1.70GHz.AsillustratedinFig.4(c),theperformanceusingthemeshC couldbeconsideredinefficientintermsoftime,inpartduetothefact thatthemeshCissignificantlyfinercomparedtothemeshesAandB. Particularly,charts(d),(e)and(f)inFig.4offermoreprecisedetails onthecomputingtimeforeachvolumefractionandeachmesh.The differencesin theresultsareduetothefactthatdifferentvolumetric fractionsmodifythegeometryoftheperiodiccellforthecorrespond- ingfixedmesh.Wealsohighlightthecontrastinrelationtothemean computingtime.Takingintoaccountboththerelativeerrorsandthe executiontimes,weconcludethatthemeshBisthebestpossiblechoice foroursimulations.

Finally,tocomplete ouranalysis, in Fig.5, weshow thenumer- ical values of the effective moduli 𝐸_𝑇^(∗)∕𝐸𝑚, 𝜈_𝑇^(∗) and 𝜇^(∗)_𝐴 ∕𝜇𝑚 for a hexagonal array of unidirectional longfibers anddifferent contrasts in the constituents. In particular, we study the pairs glass/epoxy, boron/aluminum,aluminum/porosityandgraphite/epoxy.Wereferto Table1forthematerialpropertiesoftheseconstituents.Byreferringto Table 2, inwhichsummarizemaximumrelativeerrors, wenotethat our numerical results arein agreementwith theresults provided in WangandPindera(2016b)usingthelocallyexacthomogenizationthe- ory(LEHT).AsobservedinTable2themaximumrelativeerrorisless than1.46%.

Fig.4. Chart(a)showsthecomputationofthenormalizedeffectiveYoung’smodulusofagraphite/epoxysystemwithhexagonalarchitecture.Chart(b)displays therelativeerrorofAHMFEinrelationtoFVDAM.Chart(c)showsthetotalcomputingtimeandthethreecharts(d),(e)and(f)providethespecificcomputingtime foreachmesh,respectively.

Fig.5.Calculationoftheeffectivemoduliforunidirectionalcompositeswithdifferentcontrastintheconstituentsandhexagonalperiodiccell.Thecomparisonsare performedwithFig.9ofWangandPindera(2016b).

5.2. Viscoelasticresponse

Inthissection,we computetheeffectiveproperties oflinearvis- coelasticcompositematerials.Particularly,inSection5.2.1wecompute theeffectivepropertiesforacompositematerialmadeofisotropiccon- stituentswheretheviscoelastic behaviour ofthematrix isdescribed by means of a Burger’s model. We considera long fiber reinforce- mentwithsquareandhexagonalgeometricalarrangements,andcon- siderthecaseinwhichPoisson’sratioorthebulkmodulusofthevis- coelasticmatrixareconstants.Furthermore,inSection5.2.2,wedeal

withtransverseisotropiclongfiberswithdifferentorientationsandcon- sider thepower-law modelgiven in Yancey andPindera (1990)for the characterizationof thecreep compliance of theviscoelastic ma- trix.Indoingthis,weobtainedgoodagreementswithboththeLETH andexperimentalresults.Finally,inSection5.2.3,weaddressthecal- culationoftheeffectivepropertiesforacompositemadeofperfectly aligned shortfibers embeddedintoa viscoelastic matrix withtrans- verselyisotropicbehaviour.So,weshowthepotentialofourapproach inthesolutionoffullythree-dimensionalproblemsinvolvinginclusions, which cannot be addressedby meansof analyticalmethods suchas

Table3

Maximumrelativeerror(%)inthetimeintervalunderstudy.

Constant Poissons ratio Constant bulk modulus AHMFE →LEHT  ₂₂^(∗)  ₄₄^(∗)  ₆₆^(∗)  ₂₂^(∗)  ₄₄^(∗)  ₆₆^(∗) Hexagonal cell 0.2840 0.2605 0.2622 0.2707 0.2345 0.2901 Square cell 1.7953 0.0994 0.3588 2.2884 0.2034 0.3230

theLEHT.Wementionthatintheupcomingsimulations,weemploy MeshB.

5.2.1. Comparisonwithlocally-exacthomogenizationtheory

Inthissection,weanalyzetheinfluenceofthesquareandhexag- onalarrangementoflongfibers (seeFig.1) inthecalculationofthe effectivecreepcomplianceinpolymericmatrixcomposites.Inaddition, asinWangandPindera(2016a),weconsiderthateitherPoisson’sratio (𝜈_𝑚)or thebulk modulus(𝐾_𝑚)of theviscoelasticmatrixisconstant.

This assumption isjustified bythe necessityof modelingthepoten- tiallytime-independent responseofpolymeric matricesunderhydro- staticloadingWang andPindera(2016a).Ontheone hand,we take 𝜈_𝑚=0.38asthevalueforthematrixandthereforethetime-dependent bulkmodulusisgivenby𝐾_𝑚(𝑡)=𝐸_𝑚(𝑡)∕(3(1−2𝜈_𝑚)),where𝐸_𝑚(𝑡)stands fortheone-dimensionalrelaxationmodulus.Ontheotherhand,ifweas- sumeaconstantbulkmodulusforthematrix,weusetheequation𝐾_𝑚= 𝐸0∕(3(1−2𝜈_𝑚))todetermineits value,andthen,thetime-dependent Poisson’sratio𝜈_𝑚(𝑡)arisesfromtheequation𝜈_𝑚(𝑡)=1∕2−𝐸_𝑚(𝑡)∕(6𝐾_𝑚). Here,𝐸0representstheinstantaneouselasticrelaxationmodulus.

In the following, we investigate the effective properties of unidirectionally-reinforcedglass/epoxycompositeswithlinearisotropic constituents,whereelasticglassfibersareembeddedintoaviscoelastic polymericmatrix(see,e.g.CavalcanteandMarques,2014;Chenetal., 2017;Cruz-Gonzálezetal.,2020b;WangandPindera,2016a).Here,in contrastwithCavalcanteandMarques,2014;Chenetal.,2017;Cruz- Gonzálezetal.,2020b,weconsiderhexagonalperiodiccellsinthecom- putations.ThemechanicalpropertiesofthefibersaregivenbyYoung’s modulus𝐸_𝑓=68.77GPaandPoisson’sratio𝜈_𝑓 =0.21.Furthermore,we describetheviscoelasticmatrixbyassumingtherelaxationrepresenta- tionofthefour-parametermodelorBurger’smodel,i.e.twoMaxwell elementssetinparallel(seeMainardiandSpada(2011)forfurtherde- tails).Specifically,theexpressionoftherelaxationmodulusisgivenas follows,

𝐸𝑚(𝑡)=𝐺1exp (

− 𝑡 𝜂_𝜎,1

) +𝐺2exp

(

− 𝑡 𝜂_𝜎,2

)

, (27)

where the material properties are taken from Wang and Pindera (2016a)bymeansofthescalarformof(23)(seeParkandKim,1999) andsome algebraic transformations.Thedata setis reported asfol- lows,𝐺1=1.12511GPa, 𝐺2=2.14489GPa, 𝜂_𝜎,1=6999.34h and𝜂_𝜎,2= 58.2551h, where 𝐺_𝑛 (𝑛=1,2) represents the elasticmodulus of the springand𝜂𝜎,𝑛is arelaxationtimeMainardiandSpada(2011). Itis worthtoremarkthatfromeq.(27),weobtain𝐸0=𝐺1+𝐺2.

InFig.6,weshowthecurvescorrespondingtotheeffectivecreep compliances₂₂^(∗),₄₄^(∗)and₆₆^(∗)forconstantPoissonsratio(leftcharts) andconstantbulkmodulus(rightcharts).Specifically,inFig.6,wecom- pareourresultswiththoseobtainedinWangandPindera(2016a)via theLEHT.Inthesecomparisons,hexagonalandsquarearraysoffibers arestudiedforafibervolumefractionequalto0.6.Asitcanbenoticed, thereisagoodagreementbetweenthetwoapproaches,whichisfurther evidencedbythemaximumrelativeerrorsprovidedinTable3.

5.2.2. Power-lawmodel.Comparisonwithexperiments

Inthissection,wefollow theanalysisadoptedinWang andPin- dera (2016a), and compareour results with theexperimental creep measurements obtained in Yancey and Pindera (1990) for off-axis graphite/epoxy specimens, and with the numerical results given in

Table4

ElasticpropertiesofthetransverselyisotropicT300graphitefiberat room(22^◦𝐶)andelevated(121^◦𝐶)temperature.

Temperature 𝐸 _𝐴(GPa) 𝐸 _𝑇(GPa) 𝜇_𝐴(GPa) 𝜈_𝐴 𝜈_𝑇 22 ^◦𝐶 202.82 25.30 44.12 0.443 0.05 121 ^◦𝐶 214.33 14.82 68.18 0.450 0.05

WangandPindera(2016a).InYanceyandPindera(1990),theauthors observed thatat22^◦𝐶 and121^◦𝐶 theT300 graphitefiberpresentan elasticbehavior,whereasthecreepresponseofthe934epoxymatrixis fittedbythepower-law

𝑆_𝑚(𝑡)= 1 𝐸0

+𝐶𝑡^𝑛, (28)

where𝐶and𝑛areexperimentallymeasuredparametersand𝐸0isthe instantaneouselasticrelaxationmodulus(YanceyandPindera,1990).

Beforeproceeding,itisworthmentioningthat,eventhoughinthe presentmodelwedonotdealwithfractionalviscoelasticity(werefer theReadertoAtanacković etal.,2016;Beltempoetal.,2019;Bouras etal.,2018;Mainardi,2010;MainardiandSpada,2011andtherefer- encestherein),wefindconvenienttousesomeoftheresultsgivenin theseworksforthecalculationoftherelaxationmodulus ̂𝐸_𝑚(𝑝)which isneededinoursimulations.

Withthispurpose,weintroducethenotation𝜇=𝐸0∕2,𝜏=𝛽^−1∕𝑛, 𝛽=𝐸0𝐶Γ(1+𝑛)and𝛼=𝑛,sothatEq.(28)canbeequivalentlyrewrit- tenas

𝑆_𝑚(𝑡)=𝑆_𝑀(𝑡)= 1 2𝜇

[

1+ (𝑡∕𝜏)^𝛼 Γ(1+𝛼)

]

, (29)

whereΓdenotestheGammafunction,𝛼∈ ]0,1],and𝑆_𝑀(𝑡)represents thefractionalcreepcompliance (Mainardi andSpada,2011).Hence, byreferringtotheresultsobtainedinMainardiandSpada(2011),the fractionalrelaxationmodulusisgivenbytheexpression

𝐸_𝑀(𝑡)=2𝜇_𝛼(−(𝑡∕𝜏)^𝛼), (30)

where_𝛼denotestheMittag–Lefflerfunctionoforder𝛼,whichisdefined bytheexpression(Gorenfloetal.,2014)

_𝛼(−(𝑡∕𝜏)^𝛼)=

∑∞ 𝑝=0

(−1)^𝑝 (𝑡∕𝜏)^𝛼𝑝

Γ(1+𝛼𝑝), 0<𝛼 <1, 𝜏 >0, (31)

andfor𝛼=1reducesto_𝛼((−𝑡∕𝜏)^𝛼)=exp(−t∕τ).Wenoticethat,from Eq.(30)andbyvirtueoftheparameteridentificationsestablishedfor obtainingEq.(29),obtain

𝐸_𝑚(𝑡)=𝐸0_𝑛(

−𝐸0𝐶Γ(1+𝑛)𝑡^𝑛)

. (32)

However,Eq.(32)doesnotprovide,inadirectway,anincremental solutionfortheprobleminthetimedomain.Therefore,werelyonthe expressionfortherelaxationmodulus ̂𝐸_𝑚(𝑝),expressedwithrespectto theLaplace–Carsondomain,forthecomputationoftheeffectiveproper- ties.Thus,byemployingtheresultsgiveninMainardiandSpada(2011), theLaplace–CarsontransformofEq.(30)is

̂𝐸_𝑀(𝑝)= 2𝜇(𝑝𝜏)^𝛼

1+(𝑝𝜏)^𝛼, (33)

whichimpliesthat

̂𝐸_𝑚(𝑝)= 𝐸0𝑝^𝑛

𝑝^𝑛+𝐸0𝐶Γ(1+𝑛). (34)

Inthisway,theLaplace–Carsontransformfortherelaxationmodulus canbeanalyticallyobtainedinanexplicitform,whichhighlyreduces thenumericalcomplexityoftheproblem.

Inthefollowing,weconsideracompositewithhexagonalarrange- mentof longfibers wherethepropertiesof theconstituents,i.e. the elasticfibers andtheviscoelasticmatrixgivenin Wang andPindera (2016a)aresummarizedinTables4and5,respectively.

Fig.6. Charts(a)-(c)showtheresultsunder theconsiderationsofconstantPoisson’sratio whilecharts(d)-(f)assumeconstantbulkmod- ulus.We consider square andhexagonalar- raysoffibers.Ourresultsarecomparedwith Fig.7ofWangandPindera(2016a).

Table5

Materialpropertiesoftheepoxymatrixatroom(22^◦𝐶)andel- evated(121^◦𝐶)temperature.

Temperature 𝐸 0(GPa) 𝜈 𝐶(1/(GPa ×min)) 𝑛

22 ^◦𝐶 4.51 0.311 0.0135 0.17

121 ^◦𝐶 3.36 0.317 0.0250 0.20

Fig.7shows theeffectivecreepresponseof theviscoelastic com- positematerialgivenbythecoefficients₁₁^(∗)and₂₂^(∗).Similarlytothe previoussection,weconsiderthecasesofconstantPoisson’sratioand constantbulkmodulus.Inaddition,weanalyzetheinfluenceoftwodif-

ferenttemperatures,i.e.22^◦𝐶(room)and121^◦𝐶(elevated).Theresults correspondtocompositeswithfibersthatarerotated10^◦and90^◦coun- terclockwiseaboutthe𝑦2-axis.Inthiscase,thefibervolumefraction ofthefiberisfixedto𝑉_𝑓=0.6.Acomparisonofourresultswiththose obtainedinWangandPindera(2016a)usingLEHTshowsagoodagree- mentbetweenbothapproaches.Additionally,thequalitativebehavior ofthecurvesisverysimilartotheexperimentaldata.InTable6,we providethemaximumrelativeerrorsbetweentheresultsobtainedvia twomethods.

5.2.3. Modelingofshortfiberreinforcement

Thelocally-exacthomogenizationtheory(LEHT)isbasedonatwo- dimensionalformulationwhichisonlycapableoftakingintoaccount

Fig.7. Calculationoftheeffectivecreepresponseinviscoelasticcompositeswithhexagonalarrayoftransverselyisotropicfiberwith10^◦and90^◦off-axisspecimens aboutthe𝑥2-axis.Weconsidertwodifferenttemperatures,i.e.22^◦𝐶(room)and121^◦𝐶(elevated).Inaddition,Charts(a)-(b)showsthecasesofconstantPoissons ratioandcharts(c)-(d)theconstantbulkmodulus.

Table6

Maximumrelativeerror(%)inthetimeintervalunderstudy.

Constant Poissons ratio Constant bulk modulus Hexagonal cell 𝜃= 10 ^◦ 𝜃= 90 ^◦ 𝜃= 10 ^◦ 𝜃= 90 ^◦ AHMFE →LEHT  ₁₁^(∗)  ₂₂^(∗)  ₁₁^(∗)  ₂₂^(∗)

22 ^◦𝐶 0.0798 0.3287 0.0870 0.3799

121 ^◦𝐶 0.1340 0.3847 0.1781 0.1666

AHMFE →Experiments  ₁₁^(∗)  ₂₂^(∗)  ₁₁^(∗)  ₂₂^(∗)

22 ^◦𝐶 4.2505 3.3196 4.5503 5.2098

121 ^◦𝐶 5.6338 4.7281 6.7843 1.7558

longcylindricalfibers(see,e.g.Chenetal.,2017).Inthissection,we showthepotentialoftheAHMFEapproachinthemodelingofviscoelas- ticcompositesforthree-dimensionalgeometricalconfigurations.Partic- ularly,weconsideraviscoelasticcompositematerialwithsquareand hexagonalarrangementofperfectlyalignedshortfibers(seeFig.1(b) (ii,iv))represented bycylindrical inclusions.In addition,weassume

thattheconstituentsbehaveasthoseinSection5.2.2,thatis,wecon- sideraviscoelasticmatrix(934epoxy)withpower-lawcreepcompli- anceasgivenin(28),andreinforcedbytransverselyisotropicelastic fibers(T300graphite).

In this context we define the parameters𝛾1∶=ℎ1∕𝐻1 and𝛾2∶=

ℎ2∕𝐻2 whichrelatethelengthmeasurementofthefiberandthema- trixinthesquareandhexagonalperiodiccell,respectively(seeFig.8 (a)).Inaddition,weassumethefibers tobecenteredin theperiodic cells.Therefore,wenoticethat0≤𝛾1,𝛾2≤1,whereazerovaluerep- resentsahomogeneousmaterialmadeonlywiththematrixandaone valuereproducesthecaseoflongfibersasparticularcaseofthisap- proach.Moreover,asobservedinFig.8(a),westudycounterclockwise uniformrotationsoftheshortfibersabout𝑦2-axisof0≤𝜃 <𝜋.

InFig.8(b)and(c),weshowourfindingsinthecalculationofthe effectivemoduli𝐸₁^(∗)and𝜈^(∗)₃₂ forsquareandhexagonalperiodiccell.

Inparticular, we assumetheroom temperature(22^◦𝐶) andthecon- stantbulkmodulusapproachdiscussedintheSection5.2.2.Thefiber volumefractionandtheratio𝛾 arefixedto𝑉_𝑓=0.1and𝛾=𝛾1=𝛾2=