dispersion of ultrasonic surface waves in a steel epoxy concrete bonding layered medium based on analytical experimental and numerical study

Furthermore,through obtaining the phase velocity of surface wave propagating in the medium usingNDE, the elastic property of the adhesive material can be estimated properly[17].. In bond

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Evaluation www.elsevier.com/locate/csndt

Yang Shen ∗ ,1, Sohichi Hirose2, Yuya Yamaguchi3

Dept of Mechanical & Environmental Informatics, Tokyo Institute of Technology, Japan

Article history:

Available online 1 August 2014

Asawidelyusedretrofittingapproachappliedinreinforcedconcrete(RC)structures,e.g.RCbridges,longspanRC build-ings, the epoxy-bonded steelplate strengthening method is effectiveand low cost [3,13] However, using epoxy as the adhesivebetweenconcreteandsteelwillmeetthedeterioration problemduetoepoxy’s aging[14,19]andthe delamina-tionontheinterfaces[10].UltrasonicNondestructiveEvaluation(NDE)isnowappliedfrequentlyinmulti-layeredbonding materials andcomposite materials[4,8,11],inparticular,to detectthedelamination orthebondingquality ofthelayered medium Furthermore,through obtaining the phase velocity of surface wave propagating in the medium usingNDE, the elastic property of the adhesive material can be estimated properly[17] Fora material with aging problemlike epoxy,

* Corresponding author Tel.: +81 03 5734 2692; fax: +81 03 5734 2692.

E-mail address:shen.y.aa@m.titech.ac.jp (Y Shen).

1 Doctoral student.

2 Professor.

3 Master student.

http://dx.doi.org/10.1016/j.csndt.2014.07.002

2214-6571/©2014 Published by Elsevier Ltd.

Fig 1 Transducer setting and wave propagation in a steel–epoxy–concrete 3 layers model.

knowingitsmaterialpropertybeforetheoccurrenceofseveredeteriorationissignificantinengineering.Therefore,a feasi-bledetectionmethodforelasticpropertyofepoxylayerinthesteel–epoxy–concretebondinglayeredmediumisindemand

tobedeveloped

In bonding layered medium, wave propagates dispersively Wave dispersion in bonding layered medium needs to be studiedwellsincethecompositemediumwithdifferentmaterialswillhavesignificantlydifferentdispersionproperties.[6]

utilized a laser generatedsurface wave toobtain dispersion property ofa copper–epoxy–aluminum compositespecimen Some researchers[9]conductedresearchandfield testinmulti-layerpavementstructures Throughanalytical calculation,

[5] mfounded that even a slightalteration of layers’ elastic constants can significantly affect wave’sdispersion property Therefore,beforetheNDEtest,acomprehensiveunderstandingofwavedispersioninsteel–epoxy–concretebondinglayered mediumisnecessary

Through generatedsurfacewave testandspectralanalysis, theexperimentaldispersion curvesare abletobeobtained Afteraninversionprocessbetweentheanalyticaldispersioncurvesandtheexperimental ones,layer’selasticpropertycan

beestimated.TheSpectralAnalysisofSurfaceWave(SASW)methodwasoriginallyappliedingeotechnicalresearch[15]in whichtheacousticwavefrequencyisunder100 Hz.Also,pavementstructureswereusuallytestedthroughthismethod[9], withthefrequencynormallyunder500 Hz.Inhighfrequencyrange(500 kHz∼4000 kHz),severalattemptshavealsobeen made forlayered metal specimens [16].In both low andhighfrequency range,thismethod can be applied successfully, however, in the frequency range about 10 kHz∼300 kHz,which is an approximately range forsteel–concrete composite material, rareworkhasbeendone.Theultrasonictransducerwithitsworkingfrequencyaround200 kHz willbesuitable forthesurfacewave testonthesteel–epoxy–concretebondinglayeredmedium,butthematerialconcrete’sinhomogeneity willchallengetheeffectofultrasonicNDEtest.Formateriallikeconcrete,whichissignificantlyimportantininfrastructure construction,nevertheless,theNDEattemptsarefewandalwaysnoteasy[12].TheultrasonicNDEstudyonlayeredmedium whichconsistsofconcretehasalmostnotyetbeenconcerned,although,inmanyfields,itisurgentlytobeconducted

In thisstudy, firstly,a model of2 layers overlying ona solid half spacewill bebuilt, basedon whichthe dispersion equation willbededucedandanalyticaldispersioncurvescanbeplotted.Thenthemodesandshapesofdispersioncurves influenced by layers’ material properties willbe discussed Meanwhile an ultrasonic NDEtest on a steel–epoxy–concrete bonding specimen willbe taken TheSASW methodwill beapplied into dataprocessingto obtainthe experimental dis-persion curvesofthespecimen Throughan inversionprocess basedona variancefunctionformulti-modes betweenthe analytical andexperimental dispersioncurves,theelasticpropertyoftheepoxy layercanbeestimated, whichwillbe in-troduced in the Explicit Finite Element Method (EFEM) model as the material property setting Through the 3-D EFEM simulation, thesurfacewave propagationinsidethe specimencan bevisualized, andthenumericaldispersioncurvescan alsobe plottedthroughthespectrumanalysisofnumericalwaveforms.Thenumericalresultcanbeusedtofurtherverify theaccuracyofthematerialpropertyestimationapproach

2 Analysis of dispersion curves

Wave propagationina semi-infinitesolid covered by multilayers ofuniformthicknesshasbeenstudied by many re-searchers before.Thedeductionofthedispersion equation formulti-layered medium canalsobe foundin[18].Here,for brevity, onlythefinal formofthedispersionequation ofthemodelof2layersoverlyingonahalfspace(Fig 1)isshown

inAppendix A

Tosolvethedispersionequation, animprovedbisectionmethodbasedalgorithmisproposedformulti-rootssearching Then, thedispersioncurves,whichindicatetherelationbetweenwavenumberorfrequencyversus phasevelocity,can be plotted

Thedispersioncurvesofmulti-layeredmediumarequitesensitivetolayer’smaterialproperties.Thestiffening(longitude andtransversephasevelocitiesoflowerlayerarelargerthanthoseofupperlayer)andsoftening(phasevelocitiesoflower

Table 1

Material parameters for analytical dispersion curves.

Layer’s material Model P-wave velocity

CL(m/s

S-wave velocity

CT(m/s

Density (kg/m 3 )

Thickness (mm)

value, wherek iswavenumber,H is1st layer’s thickness.

layeraresmallerthanthoseofupperlayer)mediacanhavesignificantlydifferentdispersionproperties.Forinstance,ina copper–epoxy–aluminumcompositemedium [17],inwhichthethirdlayeraluminumhasthehighestphasevelocity,only onemodecanbeactivatedintheconcernedfrequencyrange,andinhighfrequencyrange,wave propagateswithRayleigh wave speed of the first layer; in a softening layered medium however, due to a low phase velocity of the half-space, whenthephasevelocitynearlyequalsthetransversevelocityofthehalf-space,themodewillceasetopropagate,whichis called“cutoffeffect”[2].Here,amodelofsteel–epoxy–concretelayeredmediumisstudied,whichbelongstothesoftening mediumasconcreteinthebottomandsoftestmaterialepoxyinthemiddle

InTable 1,fourmodels ofmaterialpropertyare presented.Inallofthesemodels,the propertiesofsteellayerareset

tobeunchangeable.Tounderstandtheeffectofdifferentmaterialpropertiesofepoxylayertodispersioncurves,thephase velocitiesofepoxyareraisedinsequencefromModel1toModel3,whereallofthesesettingsarebasedonarealpossibility

ofepoxymaterial.Thematerialofconcreteisapproximatedashomogeneousinthoseanalyticalmodels,andbecauseitis difficulttobepreciselyqualifiedonitselasticconstants,weproposeanenhancedconstantssettingofconcreteinModel4

toseeitsinfluence

Fig 2showstheanalyticaldispersioncurvesofthefourmodelsinTable 1.FromFig 2,wecanseethatinfinitemodes existin theSteel–Epoxy–Concrete layeredmedium The 1stmode starts fromtheRayleigh wavespeed ofhalf-space(3rd layer) CR3,andothermodesstartfromtransversewavespeedofhalf-space(3rdlayer)CT3 withcutofffrequencies.Allthe modes asymptotically trend to the transverse wave speed of 2nd layer CT2 asfrequency tends towards infinity The 1st mode performsa hook-likecurve inlow frequencyrange,wheretheminimum pointofthe “hook”isdeterminedby the 2ndlayer’stransversewavespeedCT2 also

InModel1(Fig 2a),phasevelocitiesofallthemodesdecreasefasterthantheydoinother modelsasthelowestphase velocityofthe2ndlayerisassumed.Aninterestingphenomenonisthatthedispersioncurveofthe4thmodeseemstobe

cut whenits phasevelocityreachesCT3,andbeexcitedagainwhenfrequencyincreases.Thisphenomenonagainconfirms thatthephasevelocityofguidedwavesmustbelessthantheshearwavevelocityofthelastlayerinastratifiedhalf-space Otherwise,theenergyofthisguidedwavewillbeinfinitewhenthedepthz tendstowardsinfinity[5].InModel2(Fig 2b), dueto theincrementofelasticconstants ofepoxy layer,the“hook”of the1stmodecurve ispulled upto meetthe2nd mode closely.TheninModel3(Fig 2c),witharelativehighelasticconstantsofepoxy,the1stmodehascrossedoverthe 2nd modeandreachescloselyto CT3,whichisthelimitationofthewavespeed inthismedium.InModel4(Fig 2d),the highelasticconstantsofconcretehasclearlyliftedthecurvesofallthemodesupasthemaximumvaluesofthosecurves are determinedbythematerialpropertyofhalf-space,however,theminimumvalues,determinedbythesecondlayer,are almostsameastheonesinModel3

Different frequencyranges ofdispersion curvesare sensitiveorinsensitivetodifferentparameters.Inrelativehigh fre-quencyrange(above230 kHz),theoretically,thedispersioncurvesaresensitivetothechangeofelasticproperties:CLand

CTofepoxylayer.However,duetothedecreaseof1stmode’svelocity,andmoreenergycontributionofotherhighermodes, thosemulti-modes arehardtobedetectedanddistinguished.Whenthefrequencygoesevenhigher,forawavewithvery shortwavelength,thelayered half-spaceisequivalentto ahomogeneoushalf-spacecomposed ofthe firstlayer’smaterial only.Hence,althoughtheoretically,thereisnomodecanbeexcitedwhosephasevelocityisgreaterthanCT3,itstillshould

be consideredthatthelimitofthephasevelocityinhighfrequencyrangeisCR1,whichistheRayleighwavespeedofthe first layer.Thepracticalmeasurementalsoshowsthat,inhighfrequencyrange,thedetectablewave’sspeed isquiteclose

to theRayleighwave speedofsteel Therefore,fortheparameterofepoxy’s elasticconstants,we foundthat therangeof

20 kHz∼160 kHz,whichisaround the“hook” shapepartofthe 1stmode,isthe mostdistinguishableandsensitive part

of dispersioncurveswithevensmallchangeof elasticconstants values.Afterknowing eachparameter’s effecton disper-sioncurvesanditssensitivitytothoseparameters,wedecidetouseultrasonictransducerof200 kHzcentralfrequencyas transmitterandreceiver,andwe canroughlypredicttheshape oftheanalyticaldispersion curvesandtherangeofphase velocitiesbeforeconductingcalculation,whichcanefficientlyimprovetheinversionprocessinthelaterwork

Asteel–epoxy–concrete3layerbondingspecimenhasbeenmanufacturedapproximatingtothecompositestructureon

RC bridge strengthened by steel plates.Firstly a concreteblock (400 mm×400 mm×150 mm) was casted, then itwas supported aboveasteelplate(500 mm×500 mm×4.5 mm)withagapof5 mm,afterthattheepoxywas injectedinto thegapevenly,asshowninFig 3a.Whentheepoxyfinishedhardening,theultrasonictransducerscanbesetonthesteel plate(Fig 3b)

Through compressiontest conductedonsample concretecylinders,withdiameterof100 mm,length of200 mm,and density of2400 kg/m3,the compressive elastic modulus (E c) ofthe concrete isobtained as E c=28.92Gpa. However, in thisstudy,thedynamic modulus(E d) ofconcreteratherthanthestaticcompressivemodulus(E c)shouldbe used.Several attemptshavebeenmadetocorrelatestaticcompressive(E c)anddynamic(E d)moduliforconcrete.Thesimplestofthese empiricalrelationsisproposedbyLydonandBalendran[1]:

AccordingtoEq.(1),thedynamicmodulusoftheconcreteusedinthespecimenisE d=34.84Gpa.WithagivenPoisson’s ratioof0.2,thelongitudinalandtransversephase velocityoftheconcreteusedinthespecimenisabout: CL=4000 m/s;

CT=2450 m/s Thesevalues willbe used inthe followinginversion process, asthe knownparameters of theanalytical dispersioncurves

AsshowninFig 3b,twonormal-typeultrasonictransducersoffrequencyof200 kHzareusedinthisNDEtest.A pitch-catch methodis applied,withonetransducer astransmitterandanotherone asreceiver Tokeep goodsignal coherence, thetransmitterandreceiverarethesametype.Thediameterofthetransducer’scontactsurfaceis34.2 mm.Theyare verti-callysetontheplaneofthespecimen,toproducenormal-typeultrasonicwave.Asconductingmedium,Glycerineispasted betweentransducersandspecimen.Thecontactpressureoftransmittingandreceivingtransducerisproducedbytheir self-weightof325 g.Anintegratedhighpowerultrasonicpulser&receiverisutilizedinthistest.Thegeneratedfrequencyrange

ofpulser is30 kHz∼10 MHz, andthedetectablefrequencyrangeofreceiveris300 Hz∼30 MHz.Inthistest,onecycleof pulsewave(rectangularwave)withfrequencyof200 kHzisemployedasincidentwaveandwillbegeneratedbythepulser Thepulser&receiveriscontrolledbyaportablecomputerwithparametersettingandpulser’striggerreleasing.Thereceived signal isalsorecordedinthePCwithreal-time waveformpresentingandFFTspectrumanalyzing.Thisultrasonictesting systemislight-weight,portableforfieldtesting;meanwhile,ithasa wideworkingfrequencyrangefordifferentobjective

tobedetected,aslongascorrespondingworkingfrequencies’transducersareutilized

Toknowthetransducers’characteristicmorespecifically,apulse-echotesthasbeenconductedonahomogeneous Poly-methylmethacrylate(PMMA)block,withthedimensionof400 mm×152 mm×152 mm.Identicalwiththefollowingtests, one cycleof200 kHz pulsewave isgeneratedby thepulser,whichistransmitted bythetransducer andreflectedby the opposite face (152 mm)of the block, then received by the same transducer Fig 4a showsthe waveform ofthe 1st re-flected wave;Fig 4bshowsitsFourierspectrum,fromwhichwecanseethattheeffectivefrequencyrangeofthistypeof transducerisfrom50 kHzto350 kHz,astheincidentwaveis200 kHz

Fig 3 Steel–epoxy–concrete specimen (a) and ultrasonic transducers (b) on it.

As shownin Fig 5,thereceiver ispositioned ontwo spotssymmetricallyoffthecenterlinewitha distance D apart.

Thesetwospotsare markedasR1 andR2,andthepositionoftransmitterismarkedasT.Duringthetest,thereceiveris firstlypositioned onR1,detectingthe surfacewave signal fromR1,then itisshiftedto R2,keepingthesamepulsewave fromthetransmitteronT.Althoughthe signalsfromR1 andR2 are notrecorded simultaneously,by keepingthe pulser’s parametersanddistancebetweentransmitterandreceiveridentical,thismethodofsinglereceivershiftingcanbeequivalent

tothedetectionusingdoublereceiverssimultaneously

Fig 5 shows the plan of transducer setting in the experiment Because our interested frequency range is about

20 kHz∼160 kHz, and especially 20 kHz∼60 kHz, in which the most distinguishable (sensitive) part: the bottom of the 1stmode’s“hook”exists,soarelativelylargespacingbetweenreceiversisadopted.Inlowfrequencyrange(below50 kHz), thecorrespondinglongwavelengthrequiresarelativelarge receiverspacing, thatcanprobethewavesignaldistinctly.For example,at f=20 kHz,thewavelengthisabout75 mm,towhichacomparativereceiverspacingdistanceisneeded How-ever, based onpractical experience fromrepeatedly testing, a toolarge spacing maycause significant signal attenuation from R1 to R2, which isa negative effect inspectral analysisafterwards In the experiment, thespacing D between R1

andR2 is shifted from15 mm, withevery 15 mm’sinterval, until75 mmto find amostproper distancefrommultiple concerns.Meanwhile,tocomparethedatacollectedfromtwooppositedirections,thetransmitterisalsopositionedontwo symmetricalspots.Theoretically,ifthematerialisisotropicandhomogeneous,andtheepoxylayerhasuniformthickness, theresultsobtainedfromtheoppositedirectionshouldbethesame

4 Spectral analysis of surface waves

From theNDE test,the waveformdatain timedomain canbe obtained However, itisdifficult toextract information regardingtothedispersionpropertiesdirectlyfromthetimedomainsignals.TheFastFourierTransform(FFT)andSpectral AnalysisofSurfaceWaves(SASW)areneededinthesignal processing.Then,infrequencydomain,thephasedifferenceof

Fig 4 Pulse-echo test on a homogeneous PMMA block: a, received waveform; b, Fourier spectrum.

thetworeceiverscanbeobtained,fromwhich,therelationbetweenphasevelocityandfrequencyorwavenumber,namely, dispersioncurvescanbeplottedexperimentally

Thetransducers’coordinatesarealreadyshowninFig 1.Ifwetransformthetimedomainwaveformintofrequency do-mainusingFFT,wecanhavethecross-powerspectrum,S x1 , x2( f),betweenthetwosignalsfromR1 andR2withdistance D,

whichisdefinedas

S x1 , x2(f) =1

n

n



i=1



R1(f) 

i· R∗

2(f) 

i



(2)

where, R1(f) and R2(f) correspond to the Fouriertransforms of time recordsfrom two receivers located a distance D

apart.Thebarabove S x1 , x2( f)correspondstothefrequency-domainaverageofseveralrecords.Parametern isthenumber

ofrecordsaveraged,inthisstudy,n=3 isadopted.TheasteriskaboveR2(f)correspondstothecomplexconjugateoperator Anotherimportantfunctionhereisthecoherencefunction, γ2(f),whichcharacterizesthesignals’reliability.Itis calcu-latedfrom:

γ2(f) = |S x1 , x2(f) |2

where,A x1( f)andA x2( f)correspondtotheaveragedautopowerspectraofrecordsfromreceiver1andreceiver2, respec-tively.Theautopowerspectrumforarecord A xk(f)isdefinedas:

A xk(f) =1

n

n



i=1



Rk(f) 

i· R∗

k(f) 

i



FromRHSofEq.(4),we canseethateverything isaveragedbyn times,whichmeansthevalue ofcoherencefunction

γ2(f)isa measurementofexperimentalrepeatability Iftherecordedsignalsare reliable,thenonlytiny differenceexists amongn times repeat,andthevalueof γ2(f)willapproachtotheunity

For each frequency f , the phase shift ϕ can be picked from the cross-power spectrum The cross-power spectrum

S x1 , x2( f)isacomplex-valuedparameter.Therefore,thephaseiscalculatedfrom:

ϕ =tan−1Imag[S x1 , x2(f) ]

Knowingthephase,thetraveltimet canbecalculatedby:

t= ϕ

andthephasevelocityc canbeobtainedby:

c= D

t =x1−x2

ϕ

2π f

=2πf x1−x2

where,D=x1−x2isthedistancebetweenthereceivers

From Eq.(7),the relationbetweenphasevelocity c andfrequency f isrevealed,accordingto whichtheexperimental dispersioncurvescanbeplotted

5 Estimation of elastic properties

Fig 6a showsthe earlier arrived waveforms observed by the receiver located on R1 andR2 when the transmitteris located on the left andright handside respectively (see Fig 5asreference) Fig 6bshows the later arrived waveforms observed in the two opposite settings From both Figs 6a andb, we can clearly see that the waveforms obtained from the two opposite position settings ofthe transducer are quite close, which revealsa good “homogeneity”from different areasofthespecimen Inordertofocusonthesurfacewave onlybutnotthereflected wave,awaveformofashorttime period(about130 μshere)iscropped.Theattenuationofthesurfacewaveafter60 mm’spropagationisclearthroughthe amplitudecomparisonbetweenwaveformsinFig 6aandb

Fig 7aandbshowtheFourierspectraofthewaveformswithtransmitteronLHS.TheFourierspectrumgivesaclearsight thatmainenergyofthosewavesisdistributedbetween0 kHz and300 kHz,amongwhich,aroundverylowfrequencynear

0 kHzand150 kHz,theenergydistributionislowwithsomenotableamplitudedrops.Usually,forultrasonictransducer,the performance inverylowfrequencypartistricky,limitedbytheworkingrangeofthetransducer.Inthespectral analysis, thewaveform witha generalevenenergydistributiononconsidered frequencyrangeispreferred.The amplitudedropin frequencydomainwillbringphaseinformationloss,whichwillbediscussedlater

WiththeFourierspectrum,accordingtoEq.(2),wecanhavethecross-powerspectrum,S x1 , x2( f),fromwhichthephase differenceofeachfrequencycanbeobtained(Fig 9a).WithEq.(3),thecoherencefunction, γ2(f)(Fig 8)canbecalculated Foreach positionsetting,the same testwillbe repeatedforn times, inthisstudy,n=3 Sincethe spectrumanalysisis basedonthe averagedresultofthesen times tests,thefunction value of γ2(f)isa judgeofcoherence ofthosetests If thetest isrepeatable, namely,the measureddata isstable, thecoherence value willconvergetowards one.In Fig 8,the coherence valuesofboththeopposite settingsare closeto 1below300 kHz, andbecome unstableabove300 kHz Thus, only frequency components lower than 300 kHz are adopted.However, even in thisrange, we can alsosee that atthe

Fig 6 Received waveforms when D=60 mm a, earlier arrived waveforms; b, later arrived waveforms.

frequencies near0 kHz, and150 kHz, the coherence value drops by 10% moreor less, dueto the relatively low energy distributionaroundthere

IfweunfoldthephasedifferenceinFig 9afromperiodicphasetocontinuousone,wecanredrawthecontinuousphase differenceasshowninFig 9b.Afterunfolding,thedatapointsonA(98 kHz)andB(195 kHz)inFig 9aarethenlocatedon

acontinuousphasedifferencecurveasinFig 9b.Withthecontinuousphasedifferencecurve,theexperimentaldispersion curvesthencanbeplotted(Fig 10)accordingtoEq.(14).InFig 9bandFig 10,theresultsagreewellwhenthetransmitter

issetoppositely,whichshowsagoodhomogeneityofthespecimen.FromFig 10,wecanseethatinthefrequencyrange near0 kHzandaround150 kHz,theresultsdonotbehaveasexpected,causedbytheamplitudedropinfrequencydomain (Fig 7)andcoherencevaluedropinFig 8

Whenboththeanalytical(Fig 2)andexperimental(Fig 10)dispersioncurveshavebeenobtained,aninversionprocess based on a variance function for multi-modes can be applied to find the mostappropriate analytical curve, which has minimumdifferencewiththeexperimentalone.Thevariancefunctioncanbeexpressedas:

F var=

N

i=1[cexp(i) −c modex ant (i) ]2

N

where,i representseachdatapointinthecalculatedfrequencyrangeandN isthetotalnumberofdatapointsutilizedin the inversionprocess.Thecexp(i)istheexperimentalphase velocityobtainedfromspectrumanalysis, andthec modex ant (i)is theanalyticalphasevelocityofthemostadjacentmodewiththeexperimentalvalue.Whentherearemulti-modesexisting

inthefrequencyrange,thealgorithmwillcalculateandcomparethedistancesofthosemodes’ valuetotheexperimental value, andpick up the mostadjacent one into the calculation of F var The value of c ant modex(i) is also affected by several variables such as phasevelocity of materials, andlayer’s depth,aswe discussed in Section 2.Here, asthe epoxy layer’s depthisalreadyknown,weonlyconsidertwoparametervariables:theepoxylayer’slongitudinalvelocityCL2andtransverse velocity CT2.Namely,thevariancefunctioncanbedescribedas:

Fig 7 Fourier spectra of waveforms a, receiver on R1, transmitter on left; b, receiver on R 2 , transmitter on left.

Fig 8 Coherence function values in the cases of transmitter on left and right.

Asimplexmethod[7]basedminimizationprocessisthenappliedtofindtheminimumvalueofF varfromaninitialguess

of CL2 and CT2.Avery smallcriticalvalue will beset tocease theminimumvalue searching.Fig 11 givestheinversion resultofoneofspecimens.AninitialguesshasbeensetasCL2=1600 m/s andCT2=800 m/s.After32loopsofiteration, themostappropriateanalyticalcurveshavebeenfound.Thevariancefunction valueofthelastloop, namely,theerrorof the inversionprocess is 9.0e−4 Thefrequency rangeofthe inversion processis from15 kHz to230 kHz, excludingthe verylowfrequencypointswithlowcoherencevalue.Fig 11usesthesameexperimentaldataaspreviousfigures.Fromthis figure,wecanseethattheinterferenceofthe2ndwavemodehasinfluencedthemeasureddataaround150 kHz,inother

Fig 9 a, Periodic phase difference; b, continuous phase difference.

Fig 10 The experimental dispersion curves.

words, inthisfrequencyrange,the 2ndmode ismore easily tobe excited thanthe 1stmode.Hence, the phasevelocity measuredaround150 kHzismainlyfromthe2ndmode,whichisfasterthanthe1stmode

Before we conduct the inversion process, we need to check the experimental dispersion curve manually andget rid

of those obviouslydeviated phasevelocity points refer to thecoherence value spectrum andphase difference spectrum Thisprocedure canefficientlyincreasetheaccuracyoftheinversionandreducetheconvergencetime Hence,obtaininga well-recognizedexperimentaldispersioncurveisthebasicpremiseofthefollowinginversionprocess