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Scattering of SH-wave from interface cylindrical elastic inclusion with a semicircular disconnected curve

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Scattering of SH-wave from interface cylindrical elastic inclusion with a semicircular disconnected curveof,SH,a,from,wave,with,WAVE

    Scattering of SH-wave from interface

    cylindrical elastic inclusion with a

    semicircular disconnected curve App1.Math.Mech.Eng1.Ed.,

    DOI10.1007/s1048300806091

    ?EditorialCommitteeofApp1.

    Springer-Verlag2008

    2008,29(6):779786

    Math.Mech.and

    AppliedMathematics

    andMechanics

    (EnglishEdition)

    ScatteringofSHwavefrominterfacecylindricalelasticinclusionwith

    asemicirculardisc0nnectedcurve

    ZHAOJia-xi(赵嘉喜),QIHui(齐辉),SUShengwei(苏胜伟)

    (CivilEngineeringCollege,HarbinEngineeringUniversity,Harbin150001,P.R.China) (CommunicatedbyGUOXing-ming)

    AbstractScatteringofSHwavefromaninterfacecylindricalelasticinclusionwitha semicirculardisconnectedcurveisinvestigated.Thesolutionofdynamicstressconcen- trationfactorisgivenusingtheGreen'sfunctionandthemethodofcomplexvariable functions.First,thespaceisdividedintoupperandlowerpartsalongtheinterface.In thelowerhalfspaceasuitableGreen'sfunctionfortheproblemisconstructed.Itisan essentialsolutionofthedisplacementfieldforanelastichalfspacewithasemicylindrical

    hillofcylindricalelasticinclusionwhilebearingoutplaneharmoniclinesourceloadat

    thehorizontalsurface.Thus,thesemicirculardisconnectedcurvecanbeconstructed whenthetwopartsarebondedandcontinuousontheinteffaceloadingtheundetermined

    anti-planeforcesonthehorizontalsurfaces.Also,theexpressionsofdisplacementand stressfieldsareobtainedinthissituation.Finally,examplesandresultsofdynamicstress concentrationfactoraregiven.Influencesofthecylindricalinclusionandthedifierence parametersofthetwomediatorsarediscussed.

    Keywordsdynamicstressconcentrationfactor(DSCF),asemicirculardisconnected curve,Green'sfunction,scatteringofSHwave

    ChineseLibraryClassification0347.3

    2000MathematicsSubjectClassification73J20

    Introduction

    Withthecomprehensiveapplicationofnew

    tractedconsiderableattention.SinceWilliams[1]

    materials,interface~acturemechanicshasat

    firstinvestigatedthestaticproblemofinterface

    crackintheisotropicmateria1.whichpointedoutthatthestressfieldofthecracktipremains singularityofoscillationforplaneproblems.WiththeincreasingdynamicproblemsofIFM, interfacedynamicfracturewassetup.Forthedynamicproblem,thesingularityisthesameas thatofthestaticproblem.Forthesolvedproblems.itisoftensupposedthatthemaincrack iseitherstraightorcurvilineartl.However.manykindsofflawstructures.suchasholesor

    elasticinclusion,appearinmaterials.Moreover,materialswithdifferentstructuresnearthe interfacemaybedamagedwhenexperiencingaloading,andsplitfromtheedgewhichleadsto cracks.Underthissituation,thestressfieldwillhaveSomechangesandthenewstressconcen- trationisformed.However.theflawstructuresanddisconnectedcurveneartheinterfacehas scarcelybeenstudied.

    ReceivedMay10,2007/RevisedApr.28,2008

    ProjectsupportedbytheNaturalScienceFoundationofHeil0ngjiangProvince(NO.A0206) CorrespondingauthorZHAOJiaxi,Doctor,F~mail:zhaojiaxi1980@126.com

780ZHAOJia-xi,QIHuiandSUSheng-wei

    ThispaperinvestigatestheproblemofSHwavescatteringbyaninterfacecylindricalelas

    ticinclusionwithasemicirculardisconnectedcurve,andgivesthesolutionofdynamicstress concentrationfactorbyusingtheGreen'Sfunctionmethod.ThereisaspecialGreen'Sfunc

    tion,whichistheessentialsolutionofthedisplacementfieldforanelastichalfspacewitha semi:cylindricalhillofcylindricalelasticinclusionwhilebearingout

    planeharmoniclinesource

    loadatanypointofitshorizontalsurfaceconstructedforthepresentproblem.Intermsofthe solutionofSH-wavescatteringbyasemi

    cylindricalhillofcylindricalelasticinclusioninhalf

    spaceandasemi-circularholeinhalfspace[7j.thematerialisdividedintotwopartsalongthe interfaceandthesemicirculardisconnectedcurvecanbeconstructedloadingtheundetermined

    antiplaneforcesonthehorizontalsurfaces.Finally,numericalresultsofdynamicstresscon

    centrationfactorareobtained.andtheinfluencesbythecylindricalinclusionandthedifierence

    parametersofthetwomediatorsarediscussed.

    1Expressionofproblem

    FigureX(a)presentSaninterfacecylindricalelasticinclusionwithasemicirculardisconnected

    curve.Wedividethisintotwopartsalongtheinterface.Figurel(b)standsfortheupperpart, whichhasasemicircularholeinhalfspace.Figure1(c)standsfortheunderpartwitha semi-cylindricalhillofcylindricalelasticinclusion.

    (a)Themode1(b)Theupperpart(c)Theunderpart

    Fig.1Themodelofaninterfacecylindricalelasticinclusionwithasemicirculardisconnectedcurve

    2Greenfunction

    2.1Fundamentalequation

    ThedependenceofdisplacementfunctionWontimeise一乱andWsatisfiesthefollowing

    governingequation:

Here,cs

    02W

    +

    02

    

    W

    +k2W:0

    a2'Oy2(1)

    whereisthecircularfrequencyofdisplacementW(x,Y,t)andCs standsfortheshearwavevelocity;andP,presentthemassdensityandtheshearmodulusof

    elasticityrespectively.'

    Equation(1)canbeexpressedinthecomplexplain(,)as 02W

    +_

    1

    4aa'.

    Thecorrespondingstressexpressionscanbewritteninpolarcoordinatesystemas

    

    (+OWe).(e).

    ScatteringofSHwavesfrominterfacecylindricalelasticinclusion781 2.2Green'SfunctionI

    Figure2presentsasemicylindricalhillofcylindricalelasticinclusioninunderspace,

    while

    bearingoutplaneharmoniclinesourceloadathorizontalsurface.Wledivideitintotwodis

    

    tricts(IandIII)whichsatisfytheboundaryconditions: G1=(.?,r=R;

    l=

    )

,r=R;(4)

    =6(z?z0),0=0,7r,lZ0l>R.

    

    C

    Fig?2Themodelofasemi-cylindricalhillofcylindricalelasticinclusioninhalfspace

    Inacompletehalfspace,the

    describedasincidentwave(i)

    disturbanceimpactedbyline

    andexpressedintheform:

    ?=(--Z01)

    sourceloading(Iz?zo)canbe

    (5)

    ,

    I

    ,

    nordertosatisfythestr

    ,.,

    essfreeconditiononthehorizontalinterface,thescatteringwave

    s)whichisexcitedbyW(i)canbeassumedtobe ?m砩卜一m=0'.Il.I

    whereAmareunknowncoefficientstobedeterminedbytheboundary

    maximumvalueofthestandingwavewiththenumericvalueof1. 0.

    (6)

    condition,w0isthe

    IndistrictIII,astandingwavefunctionisconstructedwhichsatisfytheconditionofstress

    freeonCboundary,

    Cboundary.Accordingtothe stressanddisplacementcontinuouson

    methodofRef.f81,Wetakeitas

    stw0cmJ?m-l(k3R)-Jm+l(k3nR)amnJ~(n~--O01n+l,l,,=..nV0,n3Ljjj where,

    f1,m=n;

    t1e-i(m-n)~",m?n.

    (+(.)=(s?

    ,

    +=).

    Equation(9)istheinfinitealgebraicseries,andAm,canbedetermined

    TheGreen'SfunctioninmediumIis

    G1?+?'(1Z--Z01)+?+).2.3Green'SfunctionII

    Figure3standsforasemicircularholeinupperspace.WithRef.usingthecomplex

782ZHAOJia-xi,QIHuiandSUSheng-wei

    Fig.3Themodelofasemi

    circularholeinhalfspace

    functionmethod,theGreen'SfunctioninmediumIIis G2=+:2#2

    (2Z--Z01)

    B+

    ).(11)

    whereBmcanbedeterminedbytheboundarycondition inRef.[7.

    3ScatteringofSH-wavebyaninterfacecylindricalelasticinclusionwith

    asemicirculardisconnectedcurve

    3.1IncidenceofSHwave

    Figure4presentstheincidenceofSHwaveinthewholespace.ThewavefieldsW(?,

    (,

    (f)

    canbegivenbyRef.[9.Thescatteringwave,indistrictIandII,Canalsobedeter

    minedbythesamemethodintheGreen'SfunctionIandGreen'SfunctionIIseparately.

每一.

    Fig.4ThemodeloftheincidenceofSHwave

    3.2Integralequationsfordeterminingunknownforces Theincidentwave.reflectedwaveandscatteringwaveinmediumIandtherefractedwave

    andscatteringwaveinmediumIIcanbeobtainedfromtheabovediscussions.Accordingto

    Ref.[7,applyingtheGreen'Sfunctionandthemethodofcombination,thebi

    mediamaterials

    canbedividedalongtheinterfaceintotwoparts:uppermediumIIandlowermediumI(see

    Fig.5).

    Thetotaldisplacementandstressonthelower

    r0sectionplaneare

    Fig.5Formingbi-mediamaterials

    ()=W(i)+(r)+

    ,

    )(i1.

    I

    (r)

    7-|ez|ez'

    (12)

    Thetotaldisplacementandstressontheupper

    seetionplaneare

    (.):()+w

    ,'=f)(13)

    Tocombinethetwohalfspaces,itisnecessary

    tosatisfythecontinuityconditionalongthein

    terrace.Meanwhile,apairofunknownforcesis

    ??一

    ScatteringofSH-wavesfrominterfacecylindricalelasticinclusion783 loadedontheinterfacetomeetthecontinuityconditionsofdisplacementandstressonthese

    regions.Theintegralequationstodetermineunknownforcesf1to,0o)canbeexpressedas /fl(r0,7r)[G1(r,7r;ro,7r)+G2(r,7r;r0,7r)0

    R

    ,o.

    +/fl(r0,0)[Gl(r,7r;ro,0)+G2(r,7r;r0,0)]dr0

    =

    ()

    /fl(r0,7r)[Gl(r,0;r0,7r)+G2(r,0;r0,7r)]dr0

    R

    ,o.

    +/fl(r0,0)[Gl(r,0;r0,0)+G2(r,0;r0,0)]dr0

    R

    =

    ().

    TheintegralequationsareN'edholmequationsofthefirstkindthatpossessinferiorsingu

    larityonthehalfinfiniteregion.Directdiscretemethodisusedheretodefinetheaddedforces

    ,1atdiscretepoints.

    4Dynamicstressconcentrationfactor(DSCF)

    InthepresenceofthesteadyincidentSHwaveandwiththestressfreeconditionoilthe surfacesofthecircularcavity,thedynamicstressconcentrationfactorcanbewrittenas :

    }~02/7o

    Here,?isthestressaroundtheedge,=11W0standsforthelargestamplitudeofincident stress.

    InmediumIwhenr=R.thestressaroundthecylindricalelasticinclusionis 7_5=+,1(ro.)等讥+R..,l(ro-.)0G1(r,0,r0,

    InmediumII,whenr=R,thestressaroundtheedgeofthesemicircularholeis 0

    11)

=+)Oa2(r,0,ro,7r)O0

    dro.(17)

    )0)0s)

    Itshouldbepointedoutthatthestressattheexternalpointofthedisconnectivecurve (pointA)isthenominalvalue.Itistheprogressivevalueinsteadoftheanalyticone. 5Numericalexamplesandanalyses

    Inthissection.numericaIexamplesareprovidedtoshowtheeffectofthedifferentparame- ters.Here,;presentstheDSCFaroundtheedgeofthesemicircularholeinmediumIIat theangleof0._180.,;presentstheDSCFaroundthecylindricalelasticinclusioninmedium Iattheangleof180.-360..Figures6-9pointoutthegraphicalresultsofDSCF.Figures 10

    13givethetendencyofDSCFattheinterfacepointfponitA1.Figure14showsoneextreme situationtochecktheaccuracyandrationality.

    Finally,wecangetthefollowingresults:

784ZHAOJia-xi,QIHuiandSUSheng-wei

    f11ThescatteringofSH

    wavebyaninterfacecylindricalelasticinclusionwithasemicircular

    disconnectedcurveisaffectedby,,,,klR,ando,where,=k2/kl,theratioof thewavenumberinmediumIItomediumI:=2/,theratiooftheshearmodulusin

    mediumIItomediumI;=/k,theratioofthewavenumberinmediumIIItomediumI:

    =3/l,theratiooftheshearmodulusinmediumIIItomediumI;klR,theincidentwave number;andO,theincidentangle.

    f21Figures68givethegraphicalresultsofDSCFaffectedbyincidentSHwavevertically.

    ItshowsthatthesemicirculardisconnectedcurvelargelyaffectedthevalueofDSCFatthe interfacepoint.whenlR=0.1,====1.0,thevalueattheinterfacepointis 3.63whichis80%largerthantheresultoftheinterfacecircularcavityinRef.I71.WhenSH waveincidencesinlow~equency,hasthesamenumericvalueasRef.[7]atangleof45 135..However,attheotherpoint,;"islargerthanthatinRef.[7].Asthepointapproaches theinterfacegradually.theDSCFisgettinglargerduetothesemicirculardisconnectedcurve.

    hasthesimilartendencyas".Inlowincident~equency,hasthesamenumericvalue asthatinRef.I9latangleof225.315..However,inthehighincidentfrequencysituationthe DSCFappearswavvandthetendencyget.scomplex.butthemaximumvaluealsoappearsat. theinterfacepoint.

    (3)Figure9givesthegraphicalresultsaffectedbyincidentSH-wavewhenn-m45..Com- paredwiththeverticalincidentwave.theDSCFhassophisticatedchanges.Themaximum valuealsooccursat.theinterfacepoint,butthevalueis.10%20%whichislowerthanwhen affectedbyincidentSH-wavevertically.

    f4)Figures1011givethetendencyofDSCFat.theinterfacepointwhenaffectedbyincident SH_wavevertically.:ItindicatesthattheDSCFappearst.o:havesympatheticvibrationwithlow

    ~equency..Fromthefigures+when1R=0.60.8,theDSCFget.sitsmaximumvalue.When =

    5.0,theDSCFisseveraltimesthanthevalueswhen=1.0sotheaugmentationfor ;canacceleratethefailureofmaterials.

    (5)Figures.1213present.thetendencyofDSCFat.theinterfacepointwithimpacted byincidentSHwavevertically.When1R=0.1.theDSCFget.sitsmaximumvaluewith =7.0,which,thevaluecanbe78largerthanthevalueinthecaseof=1.0.Also,with

    theaugmentationfor,themaximumvalueisgettinglarger,thevalueforatthesame timeislarger.Whenincidentfrequencyishigh.theconcussionandwavemotionappears.and themaximumvalueislessthanlowfrequencycases.

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