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plane analysis of semi-infinite crack in piezoelectric strip

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plane analysis of semi-infinite crack in piezoelectric strip

    plane analysis of semi-infinite crack in

    piezoelectric strip

    App1.Math.Mech.Eng1.Ed.,32(1),7582(2011)

    DOI10.1007/s1048301113959

    ?ShanghaiUniversityandSpringer-Verlag

    BerlinHeidelberg2011

    AppliedMathematics

    andMechanics

    (EnglishEdition)

    Anti-planeanalysisofsemi-infinitecrackinpiezoelectricstrip

    Jun-hongGUO(郭俊宏),PingLIU(刘萍),

    ZixingLU(卢子兴),TaiyanQIN(秦太验).

    (1.InstituteofSolidMechanics,BeijingUniversityofAeronauticsandAstronautics, Beijing100191,P.R.China;

    2.Collegeofsciences,ChinaAgriculturalUniversity,Beuing100083,PR.China)

    AbstractUsingthecomplexvariablefunctionmethodandthetechniqueofthecon. formalmapping,thefractureproblemofasemi-infinitecrackinapiezoelectricstripis studiedundertheantiplaneshearstressandtheinplaneelectric1oad.Theanalytic

    solutionsofthefieldintensityfactorsandthemechanicalstrainenerKyreleaserateare presentedundertheassumptionthatthesurfaceofthecrackiselectricallyimpermeable. Whentheheightofthestriptendstoinfinity.theanalyticsolutionsofaninfinitelylarge piezoelectricsolidwithasemi.infinitecrackareobtained.Moreover.thepresentresults canbereducedtothewell.knownsolutionsforapurelyelasticmateria1intheabsenceof theelectricloading.Inaddition.numericalexamplesaregiventoshowtheinfluencesof theloadedcracklength,theheightofthestrip,andtheappliedmechanical/electricloads onthemechanicalstrainenergyreleaserate.

    Keywordspiezoelectricstrip,semiinfinitecrack,complexvariablefunctionmethod

fieldintensityfactor,mechanicalstrainenergyreleaserate

    ChineseLibraryClassification0346.1

    2010MathematicsSubjectClassification74R10,74E15

    1Introduction

    Withtheextensiveapplicationofpiezoelectricmaterials,thefractureproblemsofthepiezo

    electricmediainvolvingdefects,suchascracks,holes,andinclusions.haveattractedtheat

    tentionofmanyresearchers[I-41.

    Recently,basedontheelectricallyimpermeableboundary

    condition.?ngandGao[5Jinvestigatedtheantiplaneproblemofonecrackandtwocracks

    originatingfromacircularholeinpiezoelectricsolidsbythecomplexvariablefunctionmethod

    andobtainedtheanalyticsolutionsofthefieldintensityfactorsandtheenergyreleaserate undertheremotelyuniforminplaneelectricloadingandtheanti

    planemechanicalloading.

    Then,usingthetechniqueofthemappingfunctionandthecomplexvariablefunctionmethod, Gnoeta1.6Jstudiedtheanti

    planeproblemoftwonon-symmetrica1cracksemanatingfroman

    }ReceivedSept.6.2010/RevisedNov.16.2010

    ProjectsupportedbytheNationalNaturalScienceFoundationofChinafNos.10932001and 110720151,theScientificResearchKeyProgramofBeijingMunicipalCommissionofEducation

    fNo.KZ20l0l0005003),andthePh.D.InnovationFoundationofBeijingUniversityofAeronautics

    andAstronauticsfNo.3003511

    COrrespondingauthorZixingLU,Professor,E.mail:luzixing@buaa.edu.an

    76JunhongGUO,PingLIU,Zi-xingLU,andTai-yanQIN

    ellipticalholeinpiezoelectricmaterialsandderivedtheanalyticsolutionsofthefieldintensity factorsandtheenergyreleaserateatthecracktip.Furthermore.basedontheStrohtype

    formulism.Gnoeta1.'7一驯proposedananalyticmethodtosolvetheanti

    planeproblemsoftwo

non

    symmetricalcracksemanatingfromanelliptica1holeandmultiplecracksemanatingfrom acircularholeinapiezoelectricmaterialundertheelectricallypermeableandsemi

    permeable

    boundaryconditionsrespectively.Theydeducedtheanalyticsolutionsofthefieldinten. sityfactorsandtheenergyreleaserateatthecracktip.Forthesemiinfinitecrackproblem.

    Razzaqeta1.usingtheStrohformulaanalyzedthegeneralizedtwodimensionalproblemof

    asemi

    infinitecrackinthepiezoelectricmediaundertheexactelectricalboundarycondition andpresentedtheexplicitexpressionsoftheGreenfunctionandfieldintensityfactors.Liand FlanlUJandLitllJconsideredthestaticanddynamicproblemsofanimpermeablesemi

    infinite

    crackinapiezoelectricmaterial,respectively.Theyaddressedthefieldintensityfactorsandthe

    mechanicalstrainenergyreleaserateunderapairofeoneentratedforcesandfreechargeson thecracksurface.LiandMatagaanalyzedthedynamicresponseofasemi

    infinitecrackunder

    theelectricallypermeableboundarycondition[1andtheelectricallyimpermeableboundary condition[13J.Liueta1.l4Jinvestigatedthefractureproblemofasemi

    infinitecrackinpiezoelec

    tricmaterialsundertheremotelyuniformanti

    planemechanicalandin-planeelectricloadings.

    Theypresentedthesolutionsinseriesformofthedisplacementfunctionandtheelectricpo

    tentialfunction.However.theresearchmentionedaboveismainlyforcedonaninfinitelylarge

    piezoelectricsolidwithdefects.Forasemi

    infinitecrackinapiezoelectricstrip.totheauthors'

    knowledge,noworkhasbeendoneinthepreviousliteratures.

    Inthispaper,usingthetechniqueoftheconformalmappingandthecomplexvariable

functionmethod,weconsiderthefractureproblemofasemiinfinitecrackinapiezoelectric

    stripunderuniformanti.planemechanicalloadandthein-planeelectric1oadonthecrack surface.Theanalyticsolutionsofthefieldintensityfactorsandtheenergyreleaserateatthe cracktiparegivenundertheelectricallyimpermeableboundarycondition.Inaddition,the numericalexamplesareconductedtorevealtheinfluenceoftheloadedcrack1ength.theheight

    ofthestrip.andthemechanica1/electricloadingsonthemechanicalstrainenergyreleaserate. 2Descriptionofproblem

    Consideratransverselyisotropicpiezoelectricstripwiththepolingdirectionalongthepos

    itivezaxisandtheisotropicplaneinthexYplane.Theantiplanedisplacementw(x,Y)and

    thein.planeelectricpotentialv(x.y1arecoupled.

    Thereisasemi.infinitecrackembeddedinthecenterofthepiezoelectricstrip.asshownin Fig.1.Hstandsfortheheightfromthecracksurfacetotheuppersurface.Assumethatthe cracksurface(a?x?0)issubjectedtotheanti

    planeshearstressT0andthein-planeelectric

    loadingDo.Theboundaryconditionscanbewrittenas

    (1)

    Inthiscase,allthephysicalquantitiesareindependentofz.Thus,theconstitutiveequations aresimplifiedas

    Owa

    '

    Owa'

    一一n'

    (2)

    =,DIlD一一=

    ll,00=

    :

    ?.:协日

    z<

    <<<

?<

    J,-_-______.--,--.-______

    .,cc)

    5JM

    一如c善一c;;

    _)

    lll_

    Anti--planeanalysisofsemi--infinitecrackinpiezoelectricstrip Fig.1Asemiinfinitecrackinapiezoelectricstrip

    andtheequilibriumequationsare

    C44W+el5V.:0

    ,

    el5V.WgllV=0

    where(Tij,D,W,anddenotethestress,theelectricdisplacement,theanti

    planedisplacement,

    andtheinplaneelectricpotential,respectively,cij,eij,andaretheelasticconstant, the

    piezoelectricconstant,andthedielectricpermittivity,respectively. IfC44gll+e}5?0,Eq.(3)canbefurtherwrittenas

    V.W=0,V.=0(4)

    Thus,thecouplingproblemcanbefinallytransformedintotheboundaryvalueproblem,whic

    h

    satisfiesEq.(4)andtheboundarycondition(1).

    Accordingtothecomplexvariablefunctiontheory,thesolutiontoEq. (4)canbethereal

    partortheimaginarypartofananalyticfunction.Assumethat()and1(Z)areanalytic functions,wherez=X+iy.Thus,wehave

    w=Re[Ul(z)],=Re[Ol(z)],(5)

    inwhichRedenotestherealpartoftheanalyticsolution.SubstitutingEq.(5)intoEq.(2)

    leadsto

    Fromtheboundarycondition(1),wefind =

    [(+)+e(i+:)]

    D=[el5(U~-,)()]

    c44()+e15()=2iT0,el5(uI)(el:):2iD0

    3Solutiontoproblem

    3.1Introductionofconformalmappingandsolution

    Weintroducethefollowingconformalmapping[15]:

    Hl

    n

    [+().1J,L\l(/(8)

    whichmapstheregionoftheplaneintotheinnerofaunitcircle7inthe(plane.The

    conversemappingofEq.(8)is

    ()=(9)

    ??

    一一

    +

    

    q)

    +

    一十

    .4)

    212

    ==

    

    ,??-??J(1??IL

    JunhongGUO,PingLIU,ZixingLU,andTai.yanQIN FromEq.(9),=0correspondsto=-1,Y=0+andX=acorrespondto.,andY=0 andX=-acorrespondton,where

    rl

    n=

    {I

    n=

    e-~'a/H+2i

    Weintroducethefollowingnotations

    2en|H

    

    e-mn}H2i\

    2eTra/H

    f()=[(()=U((),1()=(()], 1UI(z)=?=.

    SubstitutingEq.(11)intoEq.(7),multiplyingbothsidesoftheequationby

    (a)da

    2~i(ae)'

    (10)

    where(isanarbitrarypointinl(1<1,andintegratingaroundtheunitcircle,thenbythe

    Cauchyintegralformula,wehave

    

    wehave

    F(<)=

    )=H[

    FromEq.(12),weobtain ln(_()+ln(_1)

    C441

    2)

    1e5

    C

    2i

    44~11+ei5'

NotingEq.(11)andinsertingEq.(14)intoEq.(6)resultin

    (12)

    (13)

    (14)

    (15)

    llI

    ,,Je

    ,L,I

    ,,

    西

    ,??????,,????L

    

    2

    =:

    

    

    

    (

    =ll

    .

    _'

    Anti?,planeanalysisofsemi--infinitecrackinpiezoelectricstrip

    3.2Fieldintensityfactors ByRefs.[5and6,thecomputationalformulaofthefieldintensityfactorsintheeplane

    canbeexpressedas k3=

    ()=(Do)2(16)

    FromEqs.(8),(13),and(16),theanalyticexpressionofthefieldintensityfactorscanbe

    obtainedas

()=(Do)v/2Hn

    WhentheheightHofthestriptendstoinfinity,Eq.(17)reducestotheanalyticexpression ofthefieldintensityfactorsatthecracktipforasemiinfinitecrackininfinitelypiezoelectric

    solids,i.e.,

    ()=V/(.)(18)

    Ifwetakenoaccountoftheelectricfield,Eq.(18)agreeswellwiththeresultofpureelastic materials[15J,

    whichvalidatesthecorrectnessofthepresentresultstosomeextent.

    3.3Mechanicalstrainenergyreleaserate

    Dilerentfractureparametersareregardedasthefracturecriteriaofpiezoelectricmaterials bymanyresearchers,suchasthestressintensityfactor,thetotalenergyreleaserate,theenergy densityfactor,theloca1energyreleaserate,themechanicalstrainenergyreleaserate,andthe crackopeningdisplacement(C0D).However,thesecriteriaarenotconsistentwiththeprevious

    experimentalresults.Comparingthedifferentfracturecriteriawiththeexperimentalresults. FlangandLiu[pointedoutthatthefracturecriteria,suchasthemechanicalstrainenergy releaserate,thelocalenergyreleaserate,andtheCOD,agreewellwiththeexperimental results.Therefore.themechanica1strainenergyreleaserateischosenasthefracturecriterion toanalyzethefracturebehavioratthecracktipinthiswork.

    Letk8denotethestrainintensityfactoratthecracktip.ByEq.f2),kscanbewrittenas e15kD+11

    ks=———_————一

    el5+/'~iIC44

    Themechanicalstrainenergyreleaserate[6canbeexpressedas

    门一su

    T'

    (19)

    (20)

    SubstitutingEqs.(17)and(19)intoEq.(20),wecanobtainthemechanicalstrainenergyrelease

rateatthecracktipasfollows:

    Gm:4

    7r(e;5+/~11C44)(e15ToDo+~11n.

    3.4Numericalexamples

    Toanalyzetheeffectsoftheheightofthestrip.theloadedcracklengtha.andthe

    mechanica1/electricloadsonthemechanicalstrainenergyreleaserate,wetakePZT

    5Hasa

    modelmaterialwiththefollowingmaterialconstants[:

    fC44:3.53×i0N/m,el5=17.0C/m,

    11:1.51×10c/(v-m),G:5.0N/m,(22)

    JunhongGUO,PingLIU,Zi-xingLU,andTaiyanQIN

    whereGcristhecriticalenergyreleaserate.

    Figure2showstheinfluenceoftheheightofthestriponthemechanicalstrainenergyrelease rateunderdifferentmechanical-electricloadingsforagivena=0.01m.ItCallbeseenthatthe mechanicalstrainenergyreleaseratedecreasesastheheightHofthestripincreases,which indicatesthatthenarrowerthepiezoelectricstripis.theeasieritisforthematerialtofail. Whentheheightofthestripissmallerthantheloadedcracklength.anincreaseoftheheight ofthestripgreatlyaffectsthemechanicalstrainenergyreleaserate.Ontheotherhand.when theheightofthestripislargerthantheloadedcracklength,theincreaseoftheheightofthe striphasalittleeffectonthemechanicalstrainenergyreleaserate.Especially,themechanical strainenergyreleaserateapproachesaconstantwhenHtendstoinfinity,whichcorrespondsto thecaseofasemi

    infinitecrackinaninfinitepiezoelectricsolid.Inaddition.ifHiSconsidered asaninvariablevalue.themechanicalstrainenergyreleaserateincreaseswithg/abecoming smal1.i.e..abecominglong.Theresultindicatesthatanincreaseoftheloadedcracklength caneasily1eadtocrackpropagation.

    H}

    Fig.2VariationofG/awitharatioH/a

    Figures3and4showtheeffectsofthemechanicalloadingandtheelectricloadingonthe mechanicalstrainenergyreleaserateforgivena=0.01mandH=0.05m.Itcanbefound

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