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Theultrasonicwavepropagationincompositematerial
anditscharacteristicevaluation
JunjieChang,ChangliangZheng,Qing-QingNi
1.Introduction
FRPcompositematerialswereappliedtovariousfields,suchasaircraftandspacestructures,becauseoftheexcellentcharacteristics,e.g.,light-weight,highratioofrelativeintensityandhighratioofrelativerigidity.DespiteFRPhavingsuchoutstandingcharacteristic,cracksinthematrixandfracturesofthefibermakedebondingsuchkindofdamageeasytooccurbetweenthefiberandthematrix,orthemulti-layers.Thesedamagesaredifficulttobedetecteddirectlybyvisualinspectionfromthesamplesurface,causingtroubletoensurethereliabilityandsafetyofthecompositematerialandstructures.Meanwhile,healthmonitoringtechnologiesofmaterialsareindispensable.Amongthem,theultrasonichealthmonitoringtechnologyattractslotsofattentionsinrecentyears.Simulationsbyfiniteelementmethodhavebeenperformedforthedevelopmentofapparatusforultrasonicdamage-detection,suchasultrasonicpictureinspectionandultrasoniclaser,andfortheverificationoftheirsafetyandvalidity.Researchesandcalculationsonthepropagationanalysisoftheultrasonicwaveinfiberstrengtheningcompositematerialshavebeenwellconductedandreported[1–8].
Onthesolidinterface,twokindsofboundariescanbeconsidered.Oneisliquidcontactinwhichthinlubricantisplaced,andonlypowerandpositionmovementperpendiculartotheinterfacearetransmitted.Theotheroneiscompletesolidcombination,whichpowerandpositionmovementbothperpendiculartoandparalleltotheinterfacearetransmitted.
Fiberstrengtheningcompositematerial,theinterfacebetweenthefiberandthematrixcanbeconsideredtobesolidcontact.Inthecaseof,debondingexistingbetweenthematrixandthefiber,fewliteratureswerefound,sincetheconversionsofthetransmittedwavemode,reflectionwavemodeandreflectionpulsephase(waveform)maketheanalysisverycomplicated.Providedthisproblemtobesolved,thequalityofthematerials,tosomeextent,canbeestimatedfromthesoundimpedanceofthereflectorandthetransmissionobject,andtheoptimaldamage-detectionmethodcanbealsoassumedinasimulation.
Inthisresearch,inthesimulationofthetechniquemonitoringthehealthbyanultrasonicwavemethod,theultrasonicwavedistributionpatternwasanalyzedwiththebasictheoryforwavepropagationbyusingthemodelforfiberstrengtheningcompositematerial.Namely,itaimsatobtainingtheamplitudeofthereflectionwaveandtheamplitudeofatransmittedwave,whenthelongitudinalwavehasunitamplitudeincidenceinmodelcompoundmaterial.Inthecaseofanultrasonicwavepropagationinsideamodelmedia,theratesofthereflectivelongitudinal,reflectivetraversewave,transmissionlongitudinalwaveandatransmissiontraversewavegeneratedatageneralincidenceangleintheinterface(afiberandexfoliation)wereanalyzedandreflectivecoefficientandatransmissioncoefficientweregotten,
respectively.Visualizedstudiesseparatingintoalongitudinalwaveandatraversewavewerecarriedout,andthemechanismsofalongitudinalwavedistributionandatraverse-wavedistributionwereelucidatedwhentheultrasonicwavepropagatedinsideacompositematerial.
2.Ultrasonicwaveequations
Considerasinglefibercomposite,i.e.,asinglefiberisembeddedinamatrix.TwodimensionsanalysisisconductedasshowninFig.2.Inthiscase,whenanultrasonicwavepropagatesinthissolidmedia,fromHooke’slaw,thestress–strainrelationshipfortwo-dimensionalplanestraininanisotropicmediaiswrittenasfollows[2]:
(1)
(2)
(3)
(4)
WherekandlareLame′constants,andtheTsuperscriptdenotesthetransposition.
Theultrasonicwaveequationsofmotionfortwodimensionalplanestraininanisotropicmediaareasfollows:
(5)
Where,thefirsttermontheleft-handsideofEq.(5)correspondstoalongitudinalwave,andthesecondtermcorrespondstoatransversewave.
isdensity.Ifthelongitudinalwavevelocity
andtransversewavevelocity
areintroducedtheultrasonicwaveequationsofmotionfortwo-dimensionalplanestraincanberewrittenby
(6)
Inthecaseofaplaneadvancingwave,thefollowingformulaisusedtocalculatefortheoscillatingenergygeneratedbytheultrasonicwaveperunittime:
(7)
3.Resultsofanalysisandsimulation
3.1.Transmissionenergyindifferentinterfaceshapes
Whenanincidentverticalwaveisobliquelyirradiated,fourwavesasshowninFig.3,i.e.,reflectedlongitudinalwave,reflectedtransversewave,transmittedlongitudinalwaveandtransmittedtransversewave,wouldappearontheinterface.Inotherwords,theshapeoftheinterfacebetweenepoxyandglassmayinfluencethepropagationoftheultrasonicwave.Forthisreason,themodelwithdifferentinterfaceshapesasshowninFig.1wasusedtoinvestigatetheinfluenceofinterfaceshapeonwavepropagationbehavior.Thevolumefractionproportionofbothmaterialsis1:1,despiteofthedifferentinterfaceshapesofthethreemodels.Thatistosay,theglass-volume-percentageofallthemodelsis50%.ThepropertiesofeachmediumusedintheanalysisareshowninTable1.Asaboundaryconditionofthemodel,absorptionisconsideredontherightandleftedge,whileitissymmetrical(theroller)ontheupanddowndirection.TheanalyticconditionandtheinputparameterswereshowninTable1.
Fig.2showsthetransmissionenergyoftheultrasonicwavepropagationforthesefourmodelsshowninFig.1.
Fig.1.Fourdifferentinterfaceshapesbetweenepoxyandglass.
Herethetransmissionenergywasdefinedbytheaverageenergyperunitarea,lJ/mm2,atthereceiveredge.Asseen,inModel1,theincidentultrasonicwaveisperpendiculartotheplaneinterface,andtransmittedwaveoccursalongwholeplane,sothatthetransmissionenergyisfarlargerthanthatintheothermodels.Thefull-reflectiontakesplaceinpartofinterfaceinbothModel2andModel3whentheincidenceangleislargerthanthecriticalanglebecausetheultrasonicwaveradiatesobliquelyonaconvexorconcaveinterface.Aboutonethirdoftheincidentwaveexperiencesfull-reflectioninModel2andModel3.However,thetransmissionenergyofModel3islargerthanthatofModel2.AsecondpeakappearsinthetransmissioncurveofModel3.Peak1isareflectedwavethatpropagatesasasecondarywavesourceneartheup-down-wardinterface(intheglassregion),whilepeak2isatransmittedwaveinthecentralpartoftheglassregion.Thereasonmightbethatneartheinterface,arefractiveindexdistributionoccurs,resultingintheappearanceofthescatteredwaves,includingrefractionandreflectionwaves.
Thefull-reflectiontakesplaceininterfaceofModel4(incidenceangleis45_).Allprimaryincidentwaveswerereflected,andtheverysmalltransmissionenergythatshowsasfigureisbecausethedispersionwaveandthereflectedwavepenetratedthepartassecondarywavesourcefromtheverticalneighborhood.
3.2.Influenceofdifferentfiberconditions
Refractiveindexdistributionoccursnearthesecondphaseboundaryduetothesecondphasecompounding,resultingintheappearanceofthescatteredwaves,includingrefractionandreflectioninthecompositematerialsstrengthenedbyfibers.Inthenext,thescatteringoftheultrasonicwaveshowninFig.1willbetakenintoconsideration.Thescattersoccurduetofibersembeddedincompositematerials.Theincidentwave
,propagatinginmatrixregion,isasinusoidalwave.Whentheincidentwavereachesthefiber,someistransmittedintothefiber,andtheotherisreflectedonthefiber/matrixinterface,andbecomesasecondarywavesource.Accordingtotheoverlappingprincipleofwavefunctions,thewholewavefunction
canbeexpressedasasumoftheincidentwave
andthescatteredwave
.
(8)
Wherethescatteredwave
includesallthewavesscatteringcomponentsgeneratedduetotheinterfacefromtheknownwave
.
ThemodelfigureofthecompositematerialsfortheinvestigationofthescatterswasdesignedaswhatshowninFig.3,wherethreefiberswithdifferentshapeswereembeddedinthematrix.Thesizeofthemodelwas
.Theboard-shapedglassfiberwiththickness
wasembeddedinthecenterofthematrixofepoxyinModel1,andwasobliquelyembeddedinModel2.Acolumnshapedglassfiberwithadiameter
wasembeddedinthecenterofmatrixinModel3.Theabovethreemodelshadacommonfiberpercentageof20.TheanalyticconditionandtheinputparameterswereshowninTable1.
ForthemodelsinFig.3,whentheincidentwaveontheleft-handsideoftheglassregionarrivedatthefirstinterfacebetweentheepoxyandglass,thetransmittedwaveandthereflectedwavearose.Thenthereflectedwavepropagatedtotheincidenceside,whilethetransmittedwavepropagatedtothereceiversideandarrivedatthesecondinterfaceoftheglassandepoxythroughtheglassregion.
Thesecondtransmittedwaveandthesecondreflectedwavearoseatthesecondinterface,andamultiplexreflectionoccurredintheglassregion.Fortheboard-shapedfiber(planefiber)andthecolumn-shapedfiber(cylindricalfiber),Fig.4showsthecomparisonsoftheanalyticresultsinthecasesofModel1(fiberthickness
),Model2(fiberthickness
,
_)andModel3(fiberdiameter
)inFig.3,withanequivalentfibervolumefractionbutwithadifferentshape.Asseenfromthefigure,thetransmissionenergyoftheModel1isfarlargerthanthatModel2andModel3.
FromFig.4,whichembeddedtheboard-shapedfiber,twoenergypeakswereclearlyobservedbytransmissionenergycurveinModel1andModel3.InModel1,thestrongpeakscorrespondtothefirsttransmittedwave,andfourweakpeaksareascribedtothefirstreflectedwavebytheglassfiber.InModel3,thefirstenergypeakresultedfromatransmittedwavethroughtheglassfiberregion,whilethesecondenergypeakwasduetothewavepropagatingthroughtheupperandlowerregionsoftheepoxy.Consequently,itcanbeunderstoodwhythetransmissionenergyfortheboard-shapedfiberislargerthanthatofthecolumn-shapedfiber,whenthefibervolumefractionwasthesame.
4.Behaviorofwavepropagationincompositematerial
4.1.Analysismodelandultrasonicpropagationsimulation
Mostoffiberreinforcedcompositesmaterialmaybeconsideredasaninhomogeneousbodymicroscopically,andahomogeneousonemacroscopically.Forthecompositeswithfibers,thefiberarraymodelwillbeusefultotakeintoaccountofthereflectionand/ortransmissionofmultiinterfaces.Inordertoevaluatethemacroscopiccharacteristicofsuchacompositematerial,atwo-dimensiondomainwithdifferentfiberarrayswasproposedasshowninFig.5.Inthismodel,circularglassfiberswereembeddedwithhexagonalintheinterioroftheepoxymatrix.Thesizeofthemodelwas
;thefiberdiameterisd.Anincidentwaveof100MHzwasused.Themodelforanalysiswasdividedinto
elements(1,72,80,000totalelements).Inordertoaccountforthelossofloadcarryingcapacityofthefailedelements,thestiffnessofsuchelementsarereducedbytheuseofnextmethod.
Fig.6showstheseriesofstressdispersionpatternsduringtheultrasonicwavepropagationformodeloffiberreinforcedcompositesinFig.5(fiberdiameter
,withoutattenuation).Whentheultrasonicwavewaspropagatedoutreachedthefiber,thereflectedwave,thetransmittedwave,anddispersionwavewereappearedclearly(Fig.6(a)).Ifawavemotionarrivedattheinterfacebetweenthefiberandthematrix,partofthewavewasreflectedasasecondarysourcewave,andatthesametimeadispersionwavewasgeneratedaroundthefiber.Theotherpartofthewavewastransmittedfiberandpropagatedtoreceiverside.Themultiplexreflectiontookplaceinteriorofthefiber(Fig.6(b)).Moreover,thewavewhichspreadsthecircumferenceofthefiberinterfereseachotheramongfibers,thepropagationsituationoftheultrasonicwavebecomefurthercomplicatesthanthatofbefore(Fig.6(c)–(e)).Fromtheseresults,theinfluenceoffiberonpropagationanddispersionofanultrasonicwaveinacompositematerialcouldbevisualizedandunderstood.
4.2.Influenceoffiber-volume-percentageandwithattenuationinmatrix
Whendiameteroffiberischangedby
andattenuationwith/withoutattenuationinmatrix,whichinvestigateshowthepropagationactionoftheultrasonicwaveinadistributedcompositematerialmodel.Figs.7and8haveshownthetimehistorycurveofreflectionenergywith/withoutattenuationinepoxymatrix,thatduringtheultrasonicwavepropagationformodeloffiberreinforcedcompositesinFig.5,respectively.Fig.9hasshownthetimehistorycurveoftransmittedenergywithattenuationinepoxymatrix.Fig.10hasshownthatcomparisonoftransmissionenergyratiowithand/orwithoutattenuationduringtheultrasonicwavepropagationformodeloffiber-reinforcedcompositesinFig.5,respectively.Afigureincasewithoutattenuationinepoxymatrixisomitted.
Ifthewith/withoutattenuationinepoxymatrixiscompared,thepeakvalueofreflectedenergycurve(inthecaseoffiberdiameter
)withattenuationinepoxymatrix(attenuationcoefficient120dB/m/MHz)issmallerabout30%thanthatwithoutattenuationinepoxymatrix.Moreover,althoughthereflectedenergycurveinthefigureisdisplayedonlytotwopeaks,the2ndpeakvalueislargerthanthe1stpeakvalue.The1stpeakvalueistheenergyofthereflectedwavefromafiber3,andthe2ndpeakvalueistheenergyofthereflectedwavefromfibers1and6(Fig.5).Disorderaroseonthesubsequentreflectiveenergycurve,andregularitywaslost.Moreover,itfollowsontheincreaseinfibersdiameter(fibercontent)thattheenergyofareflectedwaveincreasesirrespectiveofwith/withoutattenuationinepoxymatrix.
Inthecasewithattenuationinepoxymatrix,atforthetransmittedenergyhistorycurve,andthepeakvalue(inthecaseoffiberdiameterd=2k)inthetransmittedenergycurveisabouthalfofthatwithoutattenuation,andthegradeofinfluencebyattenuationinepoxymatrixshowup.Itbecomesclearerfromthe
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