Generalized interfacial energy and size effects in composites Journal of the Mechanics and Physics of Solids

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Journal of the Mechanics and Physics of Solids

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Generalized interfacial energy and size effects in composites

George Chatzigeorgiou

^a

, Fodil Meraghni

^a

, Ali Javili

^b,∗

a LEM3-UMR 7239 CNRS, Arts et Métiers ParisTech Metz, 4 Rue Augustin Fresnel Metz 57078, France

b Department of Mechanical Engineering, Bilkent University, Ankara 06800, Turkey

a r t i c l e i n f o

Article history:

Received 24 April 2017 Revised 6 June 2017 Accepted 6 June 2017 Available online 8 June 2017 Keywords:

General interface Ultimate size effect Nano-composites Capillary effect CCA

a b s t r a c t

Theobjectiveofthiscontributionisto explainthe sizeeffectincompositesduetothe interfacialenergybetweentheconstituentsoftheunderlyingmicrostructure.The gener- alizedinterfaceenergyaccountsforbothjumpsofthedeformation aswellasthestress acrosstheinterface.The cohesivezoneand elasticinterfaceareonlytwo limitcasesof thegeneralinterfacemodel.Aclosed formanalyticalsolution isderivedtocomputethe effectiveinterface-enhancedmaterialresponse.Ournovelanalyticalsolution isinexcel- lentagreementwith thenumericalresults obtainedfromthe finiteelementmethodfor abroadvarietyofparametersanddimensions.Aremarkableobservationisthattheno- tionofsizeeffectistheoreticallyboundedverifiedbynumericalexamples.Thus,thegain orloss viareducingthedimensionsofthe microstructureislimitedto certainultimate values,immediatelyrelevantfordesigningnano-composites.

© 2017ElsevierLtd.Allrightsreserved.

1. Introduction

Homogenization(Hill, 1963;1972;Ogden, 1974) isa commonlyacceptedmethodology toexplain the overall response ofcompositematerialsbased onitsconstituentsatthemicroscale.While theclassicalhomogenizationiswell-established today,theinfluenceofinterfacesatthemicroscaleremainselusiveandpoorlyunderstood.Thiscontributioninvestigateson theimpactofinterfacesatthemicroscaleontheeffectivematerialresponsethroughaninterface-enhancedhomogenization schemefrombothanalyticalandnumericalperspectives.

Theinterphasesbetweenvarious constituentsofa heterogeneousmicrostructure canplay a crucialrole ontheoverall materialresponse.Thegeneralinterfacemodelhererepresentsthefinitethicknessinterphase.Note,theideaofthegeneral interfacemodelfollowstheseminalworkofHashin(2002)wherehedistinguishesbetweenperfectandimperfectinterface models.Furthermore,McBrideetal.(2012) showthat classicalinterface modelscannot capturetheresponse ofheteroge- neousmateriallayers, seeFig.1.Emergingapplicationsofnano-materials requirebetter understandingofinterfaces since theinfluenceoflower-dimensionalmediaontheoverallmaterialresponseincreaseswithdecreasingsize.

Interfacescanbecategorizedintofourmodelsaccordingtotheir kinematicorkineticcharacteristics,asshowninFig.2. Theperfectinterfacemodeldoesnotallowforthedisplacementjumpnorthetractionjumpacrosstheinterface.Theelastic interfacemodel issemi-imperfectinthesense thatitiskinematicallycoherentbutkineticallynon-coherent.Interfaceelas- ticitytheory (DaherandMaugin,1986; Dell’Isola andRomano,1987; FriedandGurtin,2007; GurtinandMurdoch,1975;

Moeckel,1975;Murdoch,1976)endowstheinterfacewithanelasticresistancealongtheinterface.Thetractionjumpacross theinterface isduetotheinterface stress(see Chenetal., 2006; Javilietal.,2013c,among others).Interfaceandsurface

∗ Corresponding author.

E-mail addresses: georges.chatzigeorgiou@ensam.eu (G. Chatzigeorgiou),fodil.meraghni@ensam.eu (F. Meraghni),ajavili@bilkent.edu.tr (A. Javili).

http://dx.doi.org/10.1016/j.jmps.2017.06.002 0022-5096/© 2017 Elsevier Ltd. All rights reserved.

Fig. 1. Motivation for the need of a general interface model. Homogenization of heterogeneous material layers shows that even the simplest elastic response requires a general interface model to be properly captured McBride et al. (2012) . The zero-thickness interface model (left) is representative of a finite thickness interphase (right).

kinematic description

coherent non-coherent

kinetic description non-coherentcoherent

kinematic description

coherent non-coherent

kinetic description non-coherentcoherent

Fig. 2. An overview of interface models. Graphical illustration (left) and mathematical explanation (right). From the viewpoint of continuum mechanics, interfaces can be divided into four categories depending on their kinematic or kinetic characteristics. The elastic interface model does not allow for the displacement jump, but the traction may suffer a jump across the interface. The cohesive interface model on the contrary, allows for the displacement jump but assumes the continuity of the traction field across the interface. The intersection of elastic and cohesive interface model is the perfect interface model for which both the traction and displacement across the interface are continuous. This contribution formulates the general interface model which encompasses all other interface types.

elasticitytheory is a maturefield investigated inAltenbach andEremeyev (2011);Chhapadia etal. (2011); Corderoet al.

(2016);Dingreville etal.(2014);DingrevilleandQu(2008);Duanetal.(2009);FriedandTodres(2005);Gaoetal.(2014); Gurtinetal.(1998);HuangandWang(2006);Javilietal.(2013a);Liuetal.(2017);SteigmannandOgden(1999);Steinmann (2008); Wangetal.(2010a);2010b) amongothers.¹ Thecohesive interfacemodel allowsforthe displacementjumpacross theinterface,butremains kineticallycoherenthence,semi-imperfect.Thecohesiveinterfacemodeldatesbacktothesem- inalworks(Barenblatt, 1959;1962;Dugdale, 1960) andhasbeen extensivelystudied(AlfanoandCrisfield, 2001;vanden Bosch etal., 2006; Charlotteetal.,2006; Despringreetal., 2016;Dimitri etal.,2015; FagerstrandLarsson,2006; Gasser andHolzapfel,2003; MoslerandScheider, 2011;OrtizandPandolfi,1999;ParkandPaulino,2013;Park etal.,2009; Qian et al., 2017; Tijssens etal., 2000; Wu et al., 2016;Xu andNeedleman, 1994) in thepast. The perfect interface model is theintersectionofthetwosemi-perfectinterfacemodels.From theperspectiveofderiving interfacemodelsasasymptotic limitsofthininterphases,thecohesiveinterfacemodelisderivedasthelimitcaseofsoftinterphasesandistermedspring

1 An important Erratum Huang and Wang (2010) on Huang and Wang (2006) was later published by its authors. Furthermore, the same authors published a more comprehensive version of their work on interfacial energy and micromechanics with interface effect in Huang and Wang (2013) .

interfacemodel.Onthecontrary,theelasticinterfacemodelisobtainedasthelimitcaseofstiff interphasesandistermed stressinterfacemodel,seealsoWangetal.(2005).Thegeneral(imperfect)interfacemodelunifiesallvarioustypesofinter- facesandisbothkinematicallyandkinetically non-coherent.Thegeneralinterfacemodel(Hashin,2002)hasbeenstudied inBenveniste(2006);2013b);BenvenisteandMiloh(2001);BenvenisteandMilton(2010a);2010b);Bö (1994);Gu and He (2011); Guetal. (2011); Monchiet andBonnet (2010), butderived fromsimplified asymptotic limitsof thininterphases.

ParticulatecompositeswithgeneralinterfaceshavebeenexaminedinGuetal.(2014).AsillustratedinFig.1,theappropri- ateinterfacemodel todescribethe heterogeneousmateriallayer betweenthe microstructuresdependsonthe underlying microstructureoftheinterfaciallayer.Forinstance,ifthemicrostructureactssimilartoasetofparallelspringsnormalto theinterface,thecohesiveinterfacemodelissuitable.Ontheotherhand,iftheinterfaciallayershowsextremelyhighresis- tanceagainstopening,thentheelasticinterfacemodelcancapturetheresistanceagainstdeformationsalongtheinterface.

Obviously,boththeelasticinterfacemodelaswellasthecohesiveinterface modelarethelimitsofageneralizedinterface modelaccountingforbothin-planeandout-of-planeresistanceoftheinterface.

HomogenizationpioneeredbyHillandOgden,isextended inthiscontributiontoaccountfortheinterfacesatthemi- croscale. Forfurther details on homogenization, see the reviewsGeers et al. (2010); Kanout etal. (2009); Matous etal.

(2017);Muraetal.(1996);Ostoja-Starzewskietal.(2016);Saebetal.(2016)andreferencestherein.Althoughtheclassical first-orderhomogenization iswell-established today,it suffers fromthe lack ofa length-scale. In particular, theclassical homogenizationcannotcapturethesizeeffectinthematerialresponse.Accountingforinterfacesnaturallyleadstoascale dependentoverallresponsesincethearea-to-volumeratioisproportionaltotheinverseofthedimension.Scaledependent macroscopicbehavior due tolower-dimensional elasticityatthe microscale hasbeen a popular topicinthe past decade frombothanalytical(Brisardetal.,2010;Chatzigeorgiouetal.,2015;DuanandKarihaloo,2007;Duanetal.,2005b;Huang andSun,2007; Lietal., 2011;Limetal., 2006; Mogilevskayaetal., 2008; Nazarenkoetal.,2017; Sharma,2004; Sharma etal.,2003;SharmaandWheeler,2007;TianandRajapakse,2007)andcomputationalperspectives(FritzenandLeuschner, 2015;Javilietal.,2015;2013b;Monteiroetal.,2011;TuandPindera,2014;Yvonnetetal.,2008).Furthermore,itcanbear- guedthatthesizeeffectduetotheinterfaceelasticityisphysicallymeaningfulsupportedbyatomisticsimulations(Davydov etal.,2013;Elsneretal.,2017;HeandLilley,2008;LevitasandSamani,2011;OlssonandPark,2012;ParkandKlein,2007;

2008;Parketal.,2006;Yvonnetetal.,2012).²Theaforementionedcontributionsdealonlywiththesizeeffectduetoeither theelasticinterfacemodelorthecohesiveinterfacemodel.Thiscontributionelaboratesonthesizeeffectduetothegeneral interfacemodel.Obviously,thecurrentmodelreducestoboththecohesiveandelasticinterfacemodels.

Theclassificationoftheinterfacemodelsherearebasedontheir “kinematic” descriptions.Moreprecisely,theintrinsic featureoftheelasticinterfacemodelisthatitisgeometricallycoherentinthesensethatthedisplacementjumpacrossthe interfacevanishes.Onthecontrary,thecohesiveinterfacemodelisgeometricallynon-coherentandallowsforthedisplace- mentjump.Theconstitutiveresponseofeachinterfacemodelremainstobediscussedanddependsonitsenergyarguments.

Forinstance,theenergyoftheelasticinterfacemodelmaydependontheinterfacestrain,curvatureorhighergradientsof theinterfacestrain.Theinterfaceelasticenergyinthiscontributionisassumedtobeonlyafunctionoftheinterfacestrain andnot its gradientsnorthe curvature.Itcan beargued thatforcertain applicationsinbiomembranes andlipidbilayers thecurvature dependenceofthe interface maynot be negligible,see e.g. Gao etal.(2017). Accountingforthe curvature formally fitsin theframework developed here, butitalso leadstothe dependenceofthe overall responseon additional interfaceparameters.Note,theultimategoalofthiscontributionistocarryoutaninterfaceparameteridentificationviaan- alyzingthesizedependentoverallresponseofthematerial.Introducingmorecomplicatedinterfacemodelsintroducesmore parameterswithoutnecessarilyproviding additionalinsight.Eventually, onecan maketheinterface modelsocomplicated suchthatnoanalyticalsolutionfortheoverallresponsecanbeprovided,butthatcompletelydefeatstheinitialpurposeof thismanuscript.Here,forthefirsttimeweprovideananalyticalformfortheeffectivebehaviorofaheterogeneousmaterial containinggeneralizedinterfaces.

1.1. Organizationofthismanuscript

Thismanuscriptis organizedasfollows.Notation anddefinitions are shortlyintroduced andkey contributions ofthis manuscript are highlighted immediately afterward. Section 2 defines the problemof interest andbriefly formulates the associatedgoverningequations.BalanceequationsofgeneralinterfacesaregiveninSection2.1.Section2.2elaboratesonthe micro-to-macrotransitionofcontinuaaccountingforgeneralinterfaces.Analyticalsolutionsforeffectivematerialproperties of unidirectional fiber composites are obtained via the composite cylinder approach in Section 3. Section 4 investigates theinfluenceofgeneralinterfacesontheoverallbehaviorofmaterialsviaaseriesofparametricstudies.Inparticular,the proposedsemi-analyticalsolutioniscomparedagainstnumericalresultsusingthefiniteelementmethod.Furthermore,the sizeeffect duetointerfaces isdemonstrated andthenotion ofultimatesize effectis discussed.Section 5 concludesthis workanddiscussespossibleextensionsandoutlooks.

2 Alternatively, a second-order homogenization scheme can be developed ( Kouznetsova et al., 20 04; 20 02 ) to introduce a length-scale into homogeniza- tion.

1.2. Notationsanddefinitions

Quantitiesdefinedontheinterfacearedistinguishedfromthoseinthebulkbyabarplacedabovethequantity.Thatis,

{

•

}

^{referstoaninterfacevariablewithitsbulk}counterpartbeing{•}.Analogously,surfaceandcurvequantitiesaredenoted as

{

•

}

^and

{

•

}

,respectivelytobedistinguishedfromthebulkquantity{•}.Moreover,macroscalequantitiesaredifferentiated frommicroscale quantities by theleft super-script “M” placed next tothe quantity.Thatis, ^M

{

•

}

^{refers to a}macroscopic variablewithitsmicroscopiccounterpartbeing{•}.Directnotationisadoptedthroughout.Thedyadicproductoftwovectors a and b is a second-order tensor D=a b with [D]_{i j}=[a]_i[b]_j. The scalar product of two vectors a and b is denoted a· b=[a]_i[b]_i=[a b]:iwhereiisthesecond-orderidentitytensor.Thescalarproductoftwosecond-ordertensorsAand BisdenotedA:B=[A]_{i j}[B]_{i j}.Thecompositionoftwosecond-ordertensorsAandB,denotedA· B,isasecond-ordertensor withcomponents[A· B]_{i j}=[A]_im[B]_{m j}. Theaction ofasecond-order tensorAon avector ais givenby[A· a]_i=[A]_{i j}[a]_j. The averageandjump ofa quantity {•}over theinterface are definedby

{{ {

•

} }}

= ¹₂[

{

•

}

⁺+

{

•

}

^−]and[[

{

•

}

^]]=

{

•

}

⁺−

{

•

}

⁻, respectively.Theaverageandjumpoperatorsshowtheproperty[[

{

•

}

·

{

◦

}

^]]=[[

{

•

}

^]]·

{{ {

◦

} }}

+

{{ {

•

} }}

· [[

{

◦

}

^]].

Here,unliketheclassicalfirst-orderhomogenization,theterm“size” referstotheactualdimensionoftherepresentative volumeelement (RVE). Seeforinstance KhisaevaandOstoja-Starzewski (2006) forfurtherdetails on the definitionofthe

“size” oftheRVEwithin theclassicalfirst-orderhomogenizationcontext.Forthesakeofsimplicity,thiscontributiondeals withperfectlyperiodicmicrostructuressuchthattheunitcellisbydefinitionrepresentativeandhenceaRVE.

1.3. Keyaspectsandcontributions

The mainobjectiveof thiscontributionisto explain thesize effectincomposites dueto theinterfacialenergyofthe underlyingmicrostructure.Thekeyfeaturesandcontributionsofthismanuscriptare

• tocategorizeinterfacemodelsandproposeageneralizedmodel,

• toformulategeneralizedinterfaceswithinasmall-straincontinuummechanicssetting,

• toincorporatethegeneralinterfacemodelintoamicro-to-macrotransitionframework,

• toderiveananalyticalsolutionfortheeffectivepropertiesofcompositescontaininggeneralinterfaces,

• tocomparethenovelanalyticalsolutionwithnumericalresultsinaseriesofparametricstudies,

• toanalyzethesizeeffectduetointerfacesembeddedwithinthemicrostructure,

• tointroducethenotionofultimatesizeeffectandoptimalRVEsize.

2. Governingequations

Consideracontinuumbodythatoccupiestheconfiguration^MB atthemacroscaleasshowninFig.3.Theconfiguration^MB isheterogeneousandcomposedofitsRVEatthemicroscaleB.Atthemicroscale,theinterphasezonebetweentheinclusion andthematrixiscapturedby thezero-thicknessgeneralinterface model.The constitutivebehaviorofthecomponents at themicroscaleisassumedtobeknownandcollectivelyresultinginthemacroscaleresponse.Thatis,prescribingthemacro strain^M

ε

^{ontheRVEresultsinthemacrostressM}

σ

^.Homogenizationprovidesaframeworktoappropriatelyrelatethemicro quantitiesto macroquantities.Inhomogenizationofheterogeneousmedia,an underlyingassumption istheseparationof theproblemintotwoscales.Themacroscale, identifiedbyplacements ^Mx,describesthebodyasifitwashypotheticallya homogeneousmedium.Themicroscale,identifiedbyplacementsx,describesthemicrostructureofthecomposite.Homog- enizationrequires(i)themicroscaletobesignificantlysmallerthanthemacroscaleand(ii)anexistingRVEwithsufficient detailsofthemicrostructuresuchthatitcandescribetheoverallbehaviorofthecompositeinanaveragesense.Theforth- comingdiscussionsinthissectionfollowcloselytherecentcontributions oftheauthors(Chatzigeorgiouetal., 2015;Javili etal.,2017).

2.1. Generalizedinterfaces

Thepurposeofthissectionistoestablishthebalanceequationsgoverningcontinuaembeddinggeneralinterfaces.These governing equationsobviouslycorrespond tothe microscale inFig.3. Theinterface I splits the bodyB into twodisjoint subdomainsB⁻ andB⁺.Theinterface unitnormaln pointsfromtheminustotheplusside oftheinterface. Theoutward unit normalto theexternalboundaryS isdenotedn.Iftheinterface isentirelyenclosed withinthebody,itsintersection withtheexternalsurfaceS isan emptyset.However,iftheinterfaceisopenandreachestheboundaryofthemicroscale, itsintersectionwiththeexternalsurfaceS isthecurveC as

I∩S=∅ ⇔ closedinterface and I∩S=C ⇔ openinterface.

Theoutward unitnormaltothecurveC buttangenttotheinterface I isdenotednandplays animportantroletoderive thebalanceequationsforopeninterfaces.

Thedisplacementinthebodyisdenotedu.Toproceed,itproves convenienttodefinetheminusandplussidesofthe interfaceasI⁻=I∩

∂

B⁻andI⁺=I∩

∂

B⁺,respectively.Thedisplacementontheminusandplussidesoftheinterfaceare denotedu⁻andu⁺,respectively.Thedisplacementjumpacrosstheinterface[[u]]needsnotvanishforthegeneralinterface

macro-scale problem

micro-scale problem

zero-thickness general interface

model

finite thickness interphase between constituents

Fig. 3. A graphical summary of micro-to-macro transition accounting for general interfaces. The microscale corresponds to the representative volume element. Constitutive laws at the microscale are assumed to be known and the goal is to compute the effective response via homogenizing the response of the underlying microstructure. The zero-thickness interface model at the microscale replaces the finite-thickness interphase between different constituents.

model.Thus,thedisplacementoftheinterfaceitselfu remainsunknown.Acrucial,yetintuitivelymeaningfulassumption, forthegeneralinterfacemodelisthatu=

{{

^u

}}

^{.Thatis,theinterfacecoincideswiththedefinitionofthe}mid-surfaceacross itsminusandplussides.Thesymmetricstrainsinthebulkandontheinterfacearedefinedby

ε

⁼¹₂



i· gradu+[gradu]^t· i



inB and

ε

⁼¹₂



i· gradu+[gradu]^t· i



onI, (1)

inwhichgrad{•}denotestheusualgradientoperator.Theinterfacegradientoperatorgrad

{

•

}

^{isdefinedbytheprojectionof}

thegradientoperatorontotheinterfaceasgrad

{

•

}

=grad

{

•

}

· i.Note,thedefinitionofthebulkstrain(1)₁ rendersexactly thesymmetricgradientofthedisplacementanditscontractionwiththeidentitytensoridoesnotalteritsproperties.That is not the casefor the interface strain (1)₂ though. The contraction i· gradu, where i=i− n n,serves asa projection ontotheinterface.Thatis,theinterfacestrainisnotonlythesymmetricgradientoftheinterfacedisplacement,butalsoits tangentialcomponent.

Governingequationsofcontinuaembeddingthegeneralinterfacemodelinsmall-strainelasticitytheorycanbeobtained inavariationallyconsistentmanner.Todoso,thetotalenergyisminimizedviaimposingthestationarityconditionhence, settingitsvariationstozero.Thetotalenergyconsistsofthefreeenergyinthebulk^{andthefreeenergyoftheinterface}

^{togetherwithexternal}contributions.Thefreeenergiesinthebulkandontheinterfacearetheintegralsoftheirdensities

ψ

^and

ψ

,respectively,overtheircorrespondingdomains.Thefreeenergydensityforthebulkisafunctionofthestrain

ε

as

ψ

=

ψ

(

ε

)^{.Neglectingthe}scalar-valuedliquid-likeinterface tension,theinterfacefreeenergydensity

ψ

^dependsonthe

interfacestrain

ε

^andthedisplacementjump[[u]]as

ψ

=

ψ

(

ε

,[[u]])^.Constitutiverelationsinthebulkandontheinterface connectingtheircorrespondingenergyconjugatesareprovidedfromthefreeenergydensitiesas

σ

⁼

∂ψ

∂ ε

^{inB and}

σ

⁼

∂ ψ

∂ ε

^{, t=}

∂ ψ

∂

^{[[u]] onI, (2)}

inwhicht denotes theaverage traction across theinterface or moreprecisely t=

{{ σ}}

· n. Finally,the incrementsof the energydensitiesare

δψ

⁼

σ

^:

δε

^{inB and}

δψ

⁼

σ

^:

δε

^+t·

δ

^{[[u]]. (3)}

Settingthevariationsofthetotalenergyfunctionaltozeroforalladmissiblevariationsofdisplacements rendersthegov- erningequations.Inthe absenceofexternalforce densitieswithin thebulkandontheinterface, thegoverningequations

forquasi-staticproblemsread

div

σ

^{=0 inB,}

σ

^{· n=t onS,}

div

σ

+[[

σ

^]]· n=0 onI

(

^along

)

,

{{ σ}}

· n=t onI

(

^across

)

,

σ

· n=t onC,

(4)

wheret denotesthe surfacetraction on S and similarly,t isthe linetraction on C.The interface divergence operatoris definedasdiv

{

•

}

=grad

{

•

}

^{:i.Itisofparticularinterestthatthecurvatureoftheinterfaceisembeddedwithintheinterface}

divergenceoperatoritself.Furthermore,thebalanceequation alongthe interface(4)₃ recoverstheclassical Young–Laplace equation,seeAppendixA.Forthesakeofbrevity,thedetailsofderivationsofthegoverningEq.(4)areomittedhere.

Next,the material behavior inthe bulk andon the interface shall be discussed.The material response in thebulk is assumedtobestandardandisotropicelasticaccordingtothelinearrelation

σ

=E:

ε

^inwhichEistheconstitutivefourth- order tensor resulting in

σ

=2

μ ε

+

λ

^[

ε

^{:i]i in which}

μ

^and

λ

^{are the} Lamé parameters. The behavior ofthe general interfacemodelcanbeexpressedusingtheconstitutivelaws

σ

=E:

ε

=2

μ ε

+

λ

^[

ε

^:i]i , t=k· [[u]]=k[[u]]. (5) Thefourth-orderconstitutivetensorEandthesymmetricsecond-orderpositivedefinitetensorkcorrespondtotheinterface response along and across the interface, respectively. Here,

μ

^and

λ

^{denote the interface} Lamé parameters andk is the cohesiveresistanceoftheinterface.Finally,itisstraightforwardtoformulatethebulkandinterfaceenergies,respectivelyas

δψ

=

σ

^:

δε

⇒

ψ

=1

2

ε

^:E:

ε

,

δψ

=

σ

^:

δε

+t·

δ

^[[u]] ⇒

ψ

=1

2

ε

^:E:

ε

+1

2k:[[u]]_[[u]].

(6)

Obviously,theconstitutivemodel(5)corresponds toanisotropicinterfacebehaviorontheinterface tangentplane.The nature oftheLamé parameters is similarto thetwo-dimensional plane stress elasticityandthey both correlatewiththe resistanceof theinterface againstdeformationstangent toits plane.Onthe other hand,the interfacecohesive parameter kdescribestheresistanceoftheinterfaceagainstopeningandcanbeunderstoodasorthogonalresistanceoftheinterface.

Inparticularcaseoffibercompositeshere, theinterfaceparameter

λ

^{canbe settozerosinceonlyoneparametersuffices}

todefinetheinterfacebehavior.Inthiscase,theinterfaceparameter

μ

^{describestheresistanceoftheinterfaceagainstthe}

changeofitslength.

Inpassing,wementionthatthecurrentinterfacemodelneglectsthecontributionsfromtheinterfacetensionalongthe interface.Moreprecisely,westudysolelythesizedependentelasticresponseofthematerial.Includingtheinterfacetension inthecurrentframeworkdoesnotaddadditionalinsighttothefinalresultsnoritrequiresfurthereffortintheframework whileitintroducesamorecomplexnotation.Furthermore,accountingfortheinterfacetensionrequiresdealingwitheffec- tive interfaceelasticconstantsinsteadoftheLamé parameters.TheparameteridentificationinMilleretal.(2000) reveals thattheinfluenceoftheinterface tensioninsolidsisgenerallyexpectedtobe considerablysmallerthanthatoftheinter- faceelastic resistance.The interfacetensionessentially resultsina residualstressinthematerial thatresemblesthepore pressureinthehomogenizationofporousmedia.Obviously,themostgeneralframeworkfortheinterfaceelasticitytheory mustincludetheinterfacetensionaswellastheinterfaceelasticity,seeHuangandWang(2006);2013),Javilietal.among others.

2.2. Microtomacrotransition

Thegoalofthissectionistoprovideanenergeticallyconsistentmicro-to-macrotransitionframework.Thatis,toelabo- rateonhowtocalculatethemacrostress ^M

σ

^{viaaveraging themicroscaleresponsetothemacrostrainM}

ε

^{.Thegoverning}

equationsatthemacroscalearestandardandcanbesummarizedas

M

ε

=1 2



i·^Mgrad^Mu+[^Mgrad^Mu]^t· i



and ^Mdiv^M

σ

+^Mb=0 in^MB, (7)

subject to its corresponding boundary conditions on

∂

^MB. The macroscale body force density is denoted ^Mb. Similar to the microscale, the macroscaleenergy density ^M

ψ

^{can be assignedto the body. The} incremental energy densityat the macroscalecanbeidentifiedas

δ

^M

ψ

^=M

σ

^:

δ

^M

ε

^{, (8)}

Acommonlyacceptedstrategytoconnect themacroscaleandmicroscalemeasures isthevolumeaveraging ofthe mi- croscale quantitiesovertheRVE.Due tothepresenceofthenon-standardgeneralinterfacemodelatthemicroscale,such

Table 1

Summary of the micro-to-macro transition accounting for general interfaces. The macroscopic quantities are expressed as integrals at the microscale. In the absence of the jump across the interface and the elastic response along the inter- face, the integrals reduce to their familiar formats in the classical homogenization.

Bulk Along the interface Across the interface Macro strain ^Mε _V^{1 }

Bε d V + 1

2 V



I[[ u ]] n + n [[ u ]] d A Macro stress ^Mσ _V^{1 }

Bσ^{d V +} _V^{1 }

Iσ^{d A} Macro incr. energy ^Mσ: δ^Mε _V^{1 }

Bσ: δε d V + 1 V



Iσ: δε d A + 1 V



It ·δ[[ u ]] d A

classical definitions mustbe carefullyexamined andextended appropriately. LetV denotes the total volume ofthe RVE whichcanbecomputedby

V i=



S

x_ndA or _V =



B

dV=1 3



S

x· ndA. (9)

ThemacrostrainastheaverageofthesurfaceintegralovertheboundaryoftheRVEreads

M

ε

⁼_V^{1 }

S

1

2[u_n+n_u]dA. (10)

Throughappropriatedivergencetheoremsinthebulkandontheinterface,thesurfaceintegral(10)canbetransformedinto acombinationofintegralsovertheRVEB andontheinterfaceI as

M

ε

⁼_V^{1 }

B

ε

^dV+_V^{1 }

I

1

2[[[u]]_n+n_[[u]]]dA. (11)

Clearly,thesecondtermcorrespondstothedisplacementjumpacrosstheinterfaceandvanishesintheabsenceofinterfaces resultingintheclassicaldefinitionofthemacrostrain.

Next,the macrostress ^M

σ

^{should be linked to the} microscale behavior.Motivated by theaverage stress theorem, the macrostresscanbedefinedbyintegralsovertheboundaryoftheRVEas

M

σ

⁼_V^{1 }

S

t_xdA+ 1 V



C

t_xdL. (12)

Obviously,the integral over the curve C vanishes forclosed interfacesentirelyembedded within the RVE.Similar to the macrostrain,themacrostress^M

σ

^{canbeexpressed(e.g.}Chatzigeorgiouetal.,2015)intermsofintegralsoverB andI as

M

σ

⁼_V^{1 }

B

σ

^dV+_V^{1 }

I

σ

^{dA. (13)}

Thesecond integralcorresponds totheinterface stress andvanishes intheabsenceofelasticityalongthe interface.Note, theinterfaceaveragetractiont doesnotexplicitlyentersthedefinitionofthemacrostress.Table1summarizesthemicro- to-macrotransitionofcontinuaaccountingforgeneralinterfaces.

Equippedwith thedefinitions of themacro strain ^M

ε

^{andmacrostress M}

σ

^{interms ofmicroscale} quantities, thelast stepofaphysicallymeaningfulhomogenizationistoguaranteetheincrementalenergyequivalencebetweenthetwoscales.

The incremental energy equivalence betweenthe scales is often referred to as the macrohomogeneity conditionor the Hill–Mandelconditionandcanbeexpressedas

δ

^M

ψ

^=! _V^{1 }

B

δψ

^dV+_V^1

I

δψ

^{dA, (14)}

in which the sign =^! denotes the conditional equality. Inserting the incremental micro and macro energies (3) and (8), respectivelyintotheHill–Mandelcondition(14)andcollectingallthetermsononesideyields

1 V



B

σ

^:

δε

^dV+_V^1

I

σ

^:

δε

^+t·

δ

^[[u]]dA−M

σ

^:

δ

^M

ε

^{=! 0. (15)}

Finally,replacingthe macrostrain (10)andmacrostress (12)into theleft-handside ofthe expandedHill–Mandelcondi- tion(15)aftersomemathematicalstepsresultsintheidentity

1 V



B

σ

^:

δε

^dV+_V^1

I

σ

^:

δε

^+t·

δ

^[[u]]dA−M

σ

^:

δ

^M

ε

⁼ _V^{1 }

S[

δ

^u−

δ

^M

ε

^{·x]· [t−M}

σ

^{· n]dA, (16)}

forclosedinterfaces.The identity(16) isessentiallyan extendedHill’slemma accountingforthe generalinterface model.

Obviously,in theabsence of interfacesat themicroscale this identity reducesto the classical Hill’slemma. Utilizing the identity(16)intheexpandedHill–Mandelcondition(15),rendersanalternativeformatoftheHill–Mandelconditionas



S[

δ

^u−

δ

^M

ε

^{·x]· [t−M}

σ

^{· n]dA=! 0. (17)}

Fig. 4. Unidirectional fiber composite and its RVE, according to the composite cylinders method.

Thesignificanceofthecondition(17)isthatitfurnishessuitableboundaryconditionsforthemicroscaleproblem.Allbound- ary conditions that satisfy the condition(17) sufficiently fulfill the Hill–Mandel condition (14) andthus, are admissible choicesforanenergeticallyconsistenthomogenization.Threecanonicalboundaryconditionsare(1)lineardisplacements(2) constanttractionand(3)periodicdisplacementandanti-periodictraction.

3. Effectivepropertiesofunidirectionalfibercomposites

Thissection elaborateson an analyticalapproach toobtain theeffectivepropertiesof aunidirectional fibercomposite embedding generalized interfaces.The problem ofcomposites with interfaceshas been studiedextensively in thelitera- ture ofmicromechanics methods, see forinstance Benveniste andMiloh (2001); Hashin(1990) amongothers. In several approaches the interface is captured by considering a very thininterphase (Benveniste, 2013a; Hashin, 2002). For zero- thickness cohesive interfaces,Hashin hasperformed numerousstudies (Hashin, 1990; 1991;1992). In the case ofelastic interfaces,studiesbasedonparticulateorunidirectionalfibercompositesexistintheliterature(Duanetal., 2005b;2006;

SharmaandGanti,2004).Themainissuewiththeelasticinterfacesisthatthemeanfieldtheoriescanonlybeappliedfor reinforcementsof constantcurvature (spherical particlesorcylindrical fibers),dueto thefact that inthe other casesthe Eshelbyproblemdoesnot provideuniformelasticstate insidethe inclusionaspointedoutby SharmaandGanti(2004). Duan etal.(2005a) haveshown thatthe elastic interfacescausethe Eshelbytensorandthe stress concentration tensors tobenon-uniformandsizedependent.Thisrestrictionofcourseisalsoextendedinthecaseofgeneralizedinterfacesand thus,numericalapproachesareunavoidableforshortfibercompositesorcompositeswithreinforcementofellipsoidalform.

Guetal.(2014)haveprovidedanalytical formsfortheelastic moduliofsphericalparticulatecomposites withgeneralized interfaces.Chenetal.(2016)havestudiedtheeffectofresidualinterfacestressonthermo-elasticpropertiesofunidirectional fibercomposites.

Foraunidirectional fibercomposite,theRVEcanbe welldefinedandconsistsofa cylindricalfibersurroundedbythe matrixmaterial.AsHashinandRosen(1964)haveillustrated,theRVEcanbedescribedasasystemofconcentriccylinders asshowninFig.4.The radiusofthe fiberisdenotedr₁,whilethe externalradius ofthematrixsurroundingthe fiberis denoted as r₂. The form ofthe RVE makes preferable to write all thenecessary equations of the problemin cylindrical coordinates.Inthiscoordinatesystemthemainaxesare theradiusr,theangle

θ

^{andtheaxisparallel tothefiberaxisz.}

When boththe fiber(phase1) andthematrix(phase2) are linearisotropicmaterials withLameconstants

λ

^and

μ

^,the

stressesandthestrainsinbothphasesareconnectedthroughtheequations

σ

rr⁽ⁱ⁾=[

λ

⁽ⁱ⁾⁺²

μ

^(i)]

ε

⁽rrⁱ⁾+

λ

⁽ⁱ⁾

ε

_θθ⁽ⁱ⁾⁺

λ

⁽ⁱ⁾

ε

zz⁽ⁱ⁾,

σ

_θθ⁽ⁱ⁾⁼

λ

⁽ⁱ⁾

ε

⁽rrⁱ⁾+[

λ

⁽ⁱ⁾⁺²

μ

^(i)]

ε

_θθ⁽ⁱ⁾⁺

λ

⁽ⁱ⁾

ε

⁽zzⁱ⁾,

σ

zz⁽ⁱ⁾=

λ

⁽ⁱ⁾

ε

rr⁽ⁱ⁾+

λ

⁽ⁱ⁾

ε

⁽_θθⁱ⁾+[

λ

⁽ⁱ⁾+2

μ

^(i)]

ε

zz⁽ⁱ⁾,

σ

rz⁽ⁱ⁾=

σ

^zr=2

μ

⁽ⁱ⁾

ε

rz⁽ⁱ⁾,

σ

_θ⁽ⁱ_z⁾=

σ

zθ=2

μ

⁽ⁱ⁾

ε

_z⁽_θⁱ⁾,

σ

_r⁽_θⁱ⁾=

σ

θ^r=2

μ

⁽ⁱ⁾

ε

⁽_rθⁱ⁾.

(18)

whilethekinematicconditionsprovidetherelationsbetweenthestrainsanddisplacements

ε

rr⁽ⁱ⁾=

∂

^u(rⁱ⁾

∂

^{r ,}

ε

_θθ⁽ⁱ⁾=1 r

∂

^u(_θⁱ⁾

∂θ

^+u

(ⁱ) r

r ,

ε

zz⁽ⁱ⁾=

∂

^u(zⁱ⁾

∂

^{z ,}

2

ε

^rz=

∂

^u(zⁱ⁾

∂

^{r +}

∂

^u(rⁱ⁾

∂

^{z , 2}

ε

θ^z=1 r

∂

^u(zⁱ⁾

∂θ

⁺

∂

^u(_θⁱ⁾

∂

^{z , 2}

ε

rθ=

∂

^u(_θⁱ⁾

∂

^{r +}

1 r

∂

^u(rⁱ⁾

∂θ

⁻

u⁽_θⁱ⁾ r .

(19)

The index i takes thevalue 1for the fiberand 2for the matrix.For the generalinterface, asdiscussed in the previous section,thedisplacementisequaltotheaverageofthedisplacementsatthebordersofthetwomaterialphasesasu=

{{

^u

}}

^.

Fig. 5. Fiber composite RVE under radial strain conditions.

Ontheotherhand,forisotropicinterfacewithzerosurfaceLameconstant

λ

^{theinterfacestressesandstrainsareconnected}

throughtherelations

σ

θθ=2

μ ε

θθ,

σ

^zz=2

μ ε

^zz,

σ

θ^z=2

μ ε

θ^z, (20) wheretheinterfacestrainsareexpressedintermsoftheinterfacedisplacementsas

ε

θθ =1 r1

∂

^uθ

∂θ

⁺

ur

r1,

ε

^zz=

∂

^uz

∂

^{z , 2}

ε

θ^z=1 r1

∂

^uz

∂θ

⁺

∂

^uθ

∂

^{z . (21)}

TheinterfaceequilibriumEq.(4)₃ holdsontheinterfaceatr=r₁ andcanbeexpressedincylindricalcoordinatesas



div

σ 

r+[[tr]]=0,



div

σ 

θ+[[t_θ]]=0,



div

σ 

z+[[tz]]=0, (22)

oralternatively

−

σ

θθ

r1 +[[

σ

rr]]=0, 1 r1

∂ σ

θθ

∂θ

⁺

∂ σ

θ^z

∂

^{z +[[}

σ

rθ]]=0, 1 r1

∂ σ

θ^z

∂θ

⁺

∂ σ

zz

∂

^{z +[[}

σ

rz]]=0. (23) Theaveragetraction ontheinterface(4)₄ satisfiesthe generalform

{{ σ}}

· n=k[[u]].Equippedwiththeserelations,inthe nextsubsections twospecialcasesare examined.First,a dilatationalexpansionoftheRVEin thedirectionnormalto the fibersaxissecond,shearingoftheRVEinthedirectionnormaltothefibersaxis.

3.1. Dilatationalexpansion— inplanebulkmodulus

Consider that the RVEof the fibercomposite issubjected to constant radial strain ^M

ε

rr=

ε

0 andit isfixed on upper andlowersurfaces(Fig.5).Thisisawelldefinedproblemforelasticmaterialsandanalyticalsolutionsareprovidedinthe literaturewhentheinterfacesareperfect,i.e.nojumpsintractionsanddisplacements(HashinandRosen,1964).Analytical solutionsalsoexistforpureelasticityinthecaseswhereeitherjumpindisplacementsisallowed(Hashin,1991)orjumpin tractionsisallowed(Duanetal.,2006).

HashinandRosen(1964)haveshown that,undertheseboundary conditions,thedisplacement iszero atthe

θ

^andz

directions,whilethedisplacementfieldsintherdirectionareexpressedas u⁽_rⁱ⁾=D⁽₁ⁱ⁾r+D⁽₂ⁱ⁾1

r, i=1,2.

For thesedisplacement fields,the strains andstresses forthe fiber, the matrix and the general interface are expressed withthe helpoftherelations (18)–(21).Theelasticityprobleminthat caseis reducedon calculatingD₁⁽¹⁾,D⁽₂¹⁾, D⁽₁²⁾and D⁽₂²⁾.Theseconstantsareobtainedwiththehelpoftheoverallboundaryandcompatibilityconditions,aswellasthejump conditionsattheinterface:

• Finitesolutionatthecenterofthefiber

urfiniteatr=0 or D⁽₂¹⁾=0. (24)

• Boundaryconditionattheoutercylinder

u⁽_r²⁾

(

^r2

)

=

ε

0r2. (25)

• Tractionequilibriumattheinterface

−

σ

θθ

r1 +

σ

rr⁽²⁾

(

^r1

)

⁻

σ

rr⁽¹⁾

(

^r1

)

^{=0. (26)}

Fig. 6. An overview of interface models with respect to the effective bulk modulus. Explicit expression for the effective bulk modulus ^M K (left) and its degeneration to existing models (right). The superscript index take the value 1 for the fiber and 2 for the matrix.

• Theaveragetractionattheinterface

σ

rr⁽²⁾

(

^r1

)

+

σ

rr⁽¹⁾

(

^r1

)

2 =k



u⁽_r²⁾

(

^r1

)

− u⁽r¹⁾

(

^r1

) 

. (27)

ThesystemofEqs.(24)–(27) islinearandcanbesolved explicitly.Then,allthe stresses,strainsanddisplacementsare obtainedanalytically. Dueto theuniformity ofthe tractionvector attheboundary, themacroscopic radialstress is given by ^M

σ

^rr=

σ

rr⁽²⁾(^r2) ^{accordingto Hashinand Rosen(1964). Thus, the inplane bulk modulus MK ofthe fiber compositeis} obtained(Christensen,1979)as

MK= ^M

σ

^rr

2^M

ε

^{rr =}

σ

rr⁽²⁾

(

^r2

)

2

ε

^{0 . (28)}

Foraunidirectionalfibercompositepresentinggeneralinterfacesbetweenthematrixandthefibers,itispossibletoprovide anexplicitexpressionfortheeffectivein-planebulkmodulusas

MK=

λ

⁽²⁾+

μ

⁽²⁾+ f

χ

^with

χ

= 1

 λ

⁽¹⁾+

μ

⁽¹⁾

 ψ

−



λ

⁽²⁾+

μ

⁽²⁾



+

ω

^[1+

ψ

^]+

1− f

λ

⁽²⁾⁺²

μ

^{(2), f=}

r²₁ r²₂,

(29) inwhich

ω

=

μ

2r1,

φ

=kr1,

ψ

=

φ

−

ω

2

λ

⁽¹⁾⁺²

μ

⁽¹⁾⁺

φ

⁺

ω

^{, (30)}

andthefibertomatrixvolumefractionisdenotefalthoughintwodimensions.

Theexplicitexpressionfortheeffectivebulkmodulus(29),whichisalsosummarizedinFig.6,isparticularlyinteresting since it revealsthe connectionof the currentcontributionto the existing ones. Infact, it is relativelystraightforward to showhowtheeffectivebulkmodulusrecoverstheclassicalcasesinitssimplifiedformsasfollows.

• Perfectinterfacemodelcanberecoveredbysetting

φ

→∞sincetheorthogonalresistanceoftheinterfaceagainstopening k→∞andtheinterface remaingeometricallycoherentregardlessofthedeformation.Fortheperfectinterfacemodel, theelasticresistancealongtheinterfacevanishes andthus

μ

=0whichresultsin

ω

=0.Eventually,since

φ

→∞and thematerial parameters ofthefiber arefinite, then

ψ

=1.Inserting all theseintoEq.(29)furnishes theeffective in- planebulkmodulusfortheperfectinterface modelwhichispreciselythesameresultobtainedintheseminalworkof HashinandRosen(1964).

• Elasticinterfacemodelisgeometricallycoherentanddoesnotallowforopeningandhence,thecohesiveresistanceofthe interfacetends ktoinfinityresultingin

φ

→ ∞andconsequently

ψ

=1.However, unliketheperfectinterface model, theelasticresistancealongtheinterface

μ

^{doesnot vanishandassumes a}finite value.Thefinal expressionfor theelastic interfacemodelshallbecomparedwiththefindingsinDuanetal.(2009);2006);2007)amongothers.

Fig. 7. Fiber composite RVE under shear conditions normal to the fiber axis.

• Cohesive interface model does not possess an elastic resistance along the interface. That is, for the cohesive inter- face model

μ

=0 resulting in

ω

=0. The parameter

φ

^{however remains finite and plays a crucial role since it is}

proportional to the cohesive resistance k defining the orthogonal behavior of the interface. The parameter

ψ

^reads

ψ

=

φ

/[2

λ

⁽¹⁾+2

μ

⁽¹⁾+

φ

^{].InsertingalltheseintoEq.(29)furnishestheeffectivein-planebulkmodulusforthecohesive}

interfacemodelwhichispreciselythesameresultobtainedinHashin(1990).

• Generalinterfacemodelallowsforafiniteresistancealongtheinterface

μ

^{aswellasafiniteresistanceorthogonaltothe}

interfacek.Inthiscase,bothparameters

ω

^and

φ

^{assumefinitevaluesresultinginafinite}

ψ

^notnecessarilybeingone norsimplified.

3.2.Transverseshearing— inplaneshearmodulus

Toidentifytheinplaneshearmodulusofaunidirectionalcomposite,HashinandRosen(1964)haveproposedthebound- aryvalue problemillustrated in Fig.7, which correspond to simpleshear stress in the normal plane to the fibers.They comparedthesolutionobtainedwhentheexternalstressconditionsaresubstitutedbyboundarydisplacementsofthesame type.Theobtainedinplaneshearmodulifromthesetwoproblemspresentasmalldifference,thustheoriginalformofthe problemprovidesonly boundsonthismodulus.To resolvethisissue,ChristensenandLo(1979) havedeveloped amodi- ficationofthe originalproblem, byaddingan externalthird layer,whosepropertiesare thoseoftheunknown composite medium.This way,the compositecylinders methodis transformedto thegeneralized selfconsistent compositecylinders method.

InthemodifiedversioninChristensenandLo(1979),theboundaryvalue problemofFig.7hasuzequaltozeroevery- whereandtherestofdisplacementfieldsaregivenby

u⁽_rⁱ⁾=

4 j=1

a⁽_jⁱ⁾D⁽_jⁱ⁾r^n(ji)sin

(

²

θ )

^{, u(}_θⁱ⁾⁼

4 j=1

D⁽_jⁱ⁾r^n(ji)cos

(

²

θ )

^{, i=1,2, (31)}

u⁽_r³⁾=

ε

0

r₂ 4^M

μ



2r

r₂ +D⁽₃³⁾r³₂ r³+2



1+^M

μ

MK

D⁽₄³⁾r₂ r

sin

(

²

θ )

^{, (32)}

u⁽_θ³⁾=

ε

⁰₄^rM²

μ



2r

r2 − D⁽₃³⁾r³₂ r³+2

M

μ

MKD⁽₄³⁾r2

r

cos

(

²

θ )

, (33)

where

a⁽_jⁱ⁾= 2

λ

⁽ⁱ⁾⁺⁶

μ

^{(i)− 2n(}jⁱ⁾

 λ

⁽ⁱ⁾⁺

μ

⁽ⁱ⁾

 λ

⁽ⁱ⁾⁺⁶

μ

⁽ⁱ⁾⁻



n⁽_jⁱ⁾



²

[

λ

⁽ⁱ⁾⁺²

μ

^{(i)], (34)}

andn⁽_jⁱ⁾ arethesolutions ofthepolynomialn⁴− 10n²+9=0.Thisequationleadstofourroots,namelyn⁽_j¹⁾=3,n⁽_j²⁾=1, n⁽_j³⁾=−3,n⁽_j⁴⁾=−1.Moreover,^M

μ

^{istheunknowninplane shearmodulusofthefibercomposite.Forthese}displacement fields,thestrainsandstressesforallthephases(fiber,matrix,effectivemedium)andthegeneralinterfaceareexpressedin termsoftheunknown D⁽_jⁱ⁾ constantswiththehelp oftherelations (18)–(21).The boundaryandinterfaceconditionsthat holdintheRVEinthisproblemarethefollowing:

• Finitesolutionatthecenterofthefiber

ur,u_θ finiteatr=0 or D⁽₃¹⁾=D⁽₄¹⁾=0. (35)