Computational multiple scattering analysis of elastic waves in unidictional composites.docx

Computational multiple scattering analysis of elastic waves in unidictional composites.docx

《Computational multiple scattering analysis of elastic waves in unidictional composites.docx》由会员分享，可在线阅读，更多相关《Computational multiple scattering analysis of elastic waves in unidictional composites.docx（41页珍藏版）》请在冰豆网上搜索。

Computationalmultiplescatteringanalysisofelasticwavesinunidictionalcomposites

WaveMotion（）–

ContentslistsavailableatSciVerseScienceDirect

WaveMotion

journalhomepage:

Computationalmultiplescatteringanalysisofelasticwavesinunidirectionalcomposites

TakutoSumiyaa,ShiroBiwab,∗,GuillaumeHaïatc

aDepartmentofMicroEngineering,GraduateSchoolofEngineering,KyotoUniversity,Yoshida-honmachi,Sakyo-ku,Kyoto606-8501,Japan

bDepartmentofAeronauticsandAstronautics,GraduateSchoolofEngineering,KyotoUniversity,Yoshida-honmachi,Sakyo-ku,Kyoto606-8501,Japan

cCNRS,UniversitéParis-Est,LaboratoiredeModélisationetdeSimulationMulti-Echelle,UMRCNRS8208,61avenueduGénéraldeGaulle,94010Créteilcedex,France

articleinfo

abstract

Articlehistory:

Received5June2012

Receivedinrevisedform18August2012Accepted25August2012

Availableonlinexxxx

Keywords:

Elasticwave

Fiber-reinforcedcompositematerialMultiplescattering

StopbandformationEffectivephasevelocity

Anumericalprocedureispresentedinthispaperforthetwo-dimensional,time-harmonicelastodynamicmultiplescatteringproblemsforunidirectionalfiber-reinforcedcomposites.Theproposedprocedureisbasedontheeigenfunctionexpansionofthedisplacementpotentialsandthenumericalcollocationmethodtosolvetheexpansioncoefficients,andiscapableofmodelingarbitraryfiberarrangements.Todemonstratetheapplicabilityoftheprocedure,thePandSVwavepropagationcharacteristicsinunidirectionalfiber-reinforcedcompositesareanalyzedfordifferentfiberarrangementsandfibervolumefractions.Thesimulatedresultsareshowntocapturethedetailedfeaturesofthelocalwavefieldsinthecompositesaccompanyingthemodeconversion.Fromthecomputedwavefields,theeffectivephasevelocitiesofthecompositesareidentifiedasfunctionsofthefrequency,andfoundtobeingoodagreementwiththepredictionsofamicromechanicalmodelforrandomcomposites.TheenergytransmissionspectraofthePandSVwavesarealsodemonstrated,whichexhibitthestop-bandformationforthecompositeswithregularfiberarrangements.

©2012ElsevierB.V.Allrightsreserved.

1.Introduction

Fiber-reinforcedcompositematerialshavefoundincreasingapplicationinavarietyofareassuchasaerospaceengineeringowingtotheirenhancedmechanicalproperties.Understandingoftheelasticwavepropagationcharacteristicsinthesematerialsisimportantregardingthedesignagainstdynamicloadingsaswellasthenondestructivecharacterizationusingultrasonicwaves.Elasticwavesinfiber-reinforcedcompositesundergomultiplescatteringbytheirmicrostructuresandexhibitfrequency-dependentpropagationcharacteristics[1,2].Inthenondestructivetesting,thesefeaturesneedtobeclarifiedinordertofullyinterprettheacquiredultrasonicwaveformsandtoobtaininformationofe.g.thematrix–fiberinterfacialproperties[3].

Elasticwavepropagationinfiber-reinforcedcompositeshasbeenstudiedextensively,basedondifferentversionsofmultiplescatteringtheoriesandmicromechanicalmodels.Inmultiplescatteringtheoriesforfiber-reinforcedcompositemedia[4–8],theinfinitehierarchyofthegoverningequationsistruncatedbyintroducingcertainassumptionsforthespatialcorrelationoffiberpositionstoobtaintheaveragewavefieldandtheeffectivewavenumbers.OtherexistingmodelsassumeeitherideallyperiodicmicrostructuresasapproachedbytheBloch（orFloquet）theoryandunit-cellanalysis[9–12],orrandommicrostructuresasapproachedbytheeffectivemediumtheoriesandothermicromechanicalmodels[13–19].Actual

∗Correspondingauthor.Tel.:

+81757535794;fax:

+81757535794.

E-mailaddress:

biwa@kuaero.kyoto-u.ac.jp（S.Biwa）.

0165-2125/$–seefrontmatter©2012ElsevierB.V.Allrightsreserved.

doi:

10.1016/j.wavemoti.2012.08.012

2T.Sumiyaetal./WaveMotion（）–

Fig.1.Schematicrepresentationofaunidirectionalcomposite.

fiber-reinforcedcomposites,however,oftenshowintermediatemicrostructuralfeatures.Forexample,somemetal–matrixcompositesaremanufacturedbylayingupandhot-pressingmatrix/fibermats（so-calledmonotapes）,whichcontainasinglerowofcontinuousfibers[20,21].Suchaprocessresultsininternalfiberarrangementswhicharefairlyregularandperiodicbutnotperfectlyso,sincethenumberofthestackedmonotapesisfiniteandcertaindisorderoffiberpositioningoccurs.

Inordertoinvestigatetheelasticwavepropagationcharacteristicsincompositeswitharbitraryfiberarrangements,itisdesirabletoconsidermoredirectsimulationapproaches.Inthisregard,CaiandWilliams[22–24]proposedanumericaltechniquecalledscattererpolymerizationforlarge-scalemultiplescatteringproblemsofscalarshearwavesinfiber-reinforcedcomposites.Biwaetal.[25]presentedasemi-analyticalproceduretosolvetheequationsoftime-harmonicshear-wavemultiplescatteringdirectly,basedontheeigenfunctionexpansionofthewavefieldandnumericalcollocationtechniquetosolvetheexpansioncoefficients.Usingthismethod,theinteractionphenomenaofshearwaveswithdifferentarrangementsoffibershavebeenanalyzedeffectively,includingthestop-bandformationinregularfiberarrangementsanditsdependenceontheirfinitelength[26]aswellastheinfluenceofthedisorderedfiberarrangementonthetransmissioncharacteristics[27].Furthermore,theeffectiveshear-wavephasevelocityandattenuationcoefficientofafiber-reinforcedcompositeobtainedbythesimulationshavebeencomparedfavorablywithexperimentalresults[28].

Theabove-mentionedforegoingworks,however,dealwiththeshearwavespolarizedparalleltothealignedfibers,

whicharecommonlytermedasSH（shearhorizontal）waves.Inthissituation,nomodeconversionoccursatscattering,andtheequationstobesolvedareofascalarnature.Consequently,itremainsasanintriguingtasktoexaminemoregeneralproblemsoftwo-dimensionalelasticwavesinteractingwithunidirectionalfibers,forwhichmodeconversionphenomenaarerelevant.Inthispaper,theabove-mentionednumericalprocedure[25]isextendedtothemoregeneralsettingoftwo-dimensionalelastodynamics,inordertoanalyzemultiplescatteringofelasticwavesinunidirectionalfiber-reinforcedcomposites.Althoughtwo-dimensionalmultiplescatteringproblemsforfiber-reinforcedcompositeshavebeenapproachedbytheabovementionedmathematicaltreatmentsaswellasbydifferentnumericalmethods[29–31],thesemi-analyticaleigenfunctionexpansionapproachisworththoroughexploitationforitseffectivenessandhighaccuracy.Todemonstratetheapplicationoftheproposedprocedure,themultiplescatteringsimulationsarepresentedforelasticwavesinunidirectionalcompositeswithdifferentfiberarrangements,andtheeffectivephasevelocitiesandtheenergytransmissionbehaviorareillustratedforaspecifictypeofcomposites.

2.Formulation

2.1.Governingequations

Theproblemconsideredinthispaperconsistsofthetwo-dimensionalmultiplescatteringofelasticwavesinaninfiniteelasticmedium（Laméconstantsλ1,µ1anddensityρ1）containingNcircularcylindricalelasticfibers（radiusa,Laméconstantsλ2,µ2anddensityρ2）whichareallalignedparalleltothex3-axisandarrangedarbitrarilyinthex1–x2plane,seeFig.1.Below,thepositionvectorofagenericpointisdenotedbyr,andthepositionvectorofthecenteroftheithfiberbyri（i=1,2,...,N）.ThedomainsoccupiedbythematrixandthefibersaredenotedasD1andD2,respectively.

−

Forthetime-harmonicproblemswithtime-dependenceoftheformexp（iωt）implicitlyunderstood（ω:

angular

√

frequency,i=

−1）,thegoverningequationsofthetwo-dimensionalelastodynamicsarewrittenintermsofthe

displacementvectoruby

µα∇2u+（λα+µα）∇（∇·u）+ραω2u=0,inDα（α=1,2）,

（1）where∇isthetwo-dimensionalgradientoperator,andthesubscriptαcorrespondstothematrix（α=1）orthefiber

（α=2）.TheHelmholtztheoremallowsthetwo-dimensionaldisplacementfieldsu1（r）andu2（r）tobederivedfromtwo

potentialfunctionsΦ（r）andΨ（r）as[32]

∂Φ

1

u1（r）=∂x

∂Ψ

+∂x2

∂Φ

2

u2（r）=∂x

∂Ψ

−∂x1

（2）

whereΨisthex3-componentofthevectorpotential.Forthedisplacementsandpotentials,subscriptstoidentifythepertinentdomains（D1orD2）areomittedasitisevidentinthecontext.ThesepotentialssatisfythefollowingHelmholtzequations,

2222

∇Φ+kLαΦ=0,∇Ψ+kTαΨ=0,inDα（α=1,2）,（3）

T.Sumiyaetal./WaveMotion（）–3

wherethewavenumbersandthepertinentwavespeedsaregivenby

ω

c

kLα=

Lα

ω

c

kTα=

Tα

cLα=

λα+2µα

ρα

cTα=

µαρα

（α=1,2）.（4）

Theincidentwaveisassumedtobeaplanewavepropagatinginthepositivex1-direction,eitherwithlongitudinalortransversepolarization.Thedisplacementpotentialsoftheincidentwavearegivenby

Φinc（r）=Φ0exp（ikL1x1）,Ψinc（r）=0,（5）

forthelongitudinalwaveincidence,and

Φinc（r）=0,Ψinc（r）=Ψ0exp（ikT1x1）,（6）

forthetransversewaveincidence,whereΦ0andΨ0areconstants.Thefollowingformulationappliestothesetwokindsofincidentwavesinaunifiedmanner.Hereafter,thelongitudinalandtransversewavesarealsoreferredtoastheP（compressional）andSV（shearvertical）waves,respectively,accordingtothecommonterminology.

Thedisplacementpotentialsinthematrixareexpressedbythesumoftheincidentandthescatteredfieldsas

NN

Φ（r）=Φinc（r）+Φi,sca（r）,Ψ（r）=Ψinc（