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首页 A linear theory for bending stress strain analy…

A linear theory for bending stress strain analysis of a beam with an edge crack.pdf

A linear theory for bending str…

上传者: jjzheng4335 2012-07-05 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《A linear theory for bending stress strain analysis of a beam with an edge crackpdf》,可适用于工程科技领域,主题内容包含JIntegralBendingElsevierLtdAllrightsreservedeofractutipiagatilyticairystfo符等。

JIntegralBendingElsevierLtdAllrightsreservedeofractutipiagatilyticairystformulaehasbeencollectedbyTadaetalFurtherdevelopmentsinelasticandelastoplasticfracturemechanicscanbefoundintheliterature–Almosteverystressanddisplacementanalysisperformedinfracturemechanics,dealswiththeneighborhoodofthecrackwhilethestressanddisplacementfieldsinthewholecrackedcontinuumhaverarelybeenconsideredButinseveralapplicationssuchasvibrationanalysisofcrackedstructures,itisessentialtohaveaload–deflectionmodelfortheentirebody$seefrontmatterElsevierLtdAllrightsreserved*CorrespondingauthorTel:Emailaddress:aebrahimimehrsharifedu(AEbrahimi)EngineeringFractureMechanics()–ContentslistsavailableatScienceDirectEngineeringFractureMechanicsdoi:jengfracmechgoverningequationforstressstatenearthecracktipThelinearelasticassumptionhasbeenusedinWilliamsworkLater,someresearcherstriedtofindasolutionforthisequationSedovpointedoutthegeneralsolutionforaninternalcrackusingplanestatestressassumptionforsymmetric(modeI)andantisymmetric(modeII)casesThissolutionhasbeenderivedonthebasisofcomplexvariabletechniquessuggestedbyMuskhelishviliandWestergaardAndhasleadtothedefinitionofthestressintensityfactorHowever,thereexistsafewexactclosedformanalyticalsolutionsapplicabletocracksininfinitylargebodiesunderpuretensionThenecessityoffindingstressanddisplacementfieldsnearthecracktipinrealcasespersuadetheresearcherstodevelopnumericalandempiricalmethodsAvastnumberofempiricalandnumericalformulationshasbeenreportedforseveralcontinuawithdifferentformsofcrackundervariousformsofloadingThemostimportantandusefulformoftheseIntroductionCracksresultedfromfatigueisonbeenstudiedbymanyresearchersFStressintensityfactoratthecracktheoriesaboutfatiguelife,crackpropVeryfirstattemptsforfindinganana,WilliamssuggestedanAthemajorproblemsinstructuresandmachineryDifferentaspectsofcrackshaveremechanicsscience,isthedirectoutcomeoftheseattemptssaveryimportantsubjectdiscussedinthefracturemechanicsAlargenumberofonandfractureisbaseduponthestressintensityfactoratthevicinityofthecracklsolutionforstressanddisplacementfieldsnearthecracktip,wereperformedinressfunctionforaninfinitethinplatewithathroughcrackandderivedageneralAlineartheoryforbendingstress–strainanalysisofabeamwithanedgecrackMBehzad,AMeghdari,AEbrahimi*SharifUniversityofTechnology,MechanicalEngineeringDepartment,AzadiAvenue,Tehran,IranarticleinfoArticlehistory:ReceivedJuneReceivedinrevisedformMayAcceptedJuneAvailableonlineJulyKeywords:ElasticityFracturemechanicsabstractInthispaper,anewlineartheoryforbendingstress–strainanalysisofacrackedbeamhasbeendevelopedAdisplacementfieldhasbeensuggestedforthebeamstrainandstresscalculationsThebendingdifferentialequationforthebeamhasbeenwrittenusingequilibriumequationsTherequiredconstantforthismodelisalsoobtainedfromfracturemechanicsThebendingequationhasbeensolvedforasimplysupportedbeamwithrectangularcrosssectionandtheresultsarecomparedwithfiniteelementandempiricalresultsThereisanexcellentagreementbetweentheoreticalresultsandthoseobtainedbynumericalandempiricalmethodsThemodeldevelopedinthisresearchisasimpleandpreciseapproximationofthebehaviorofthecrackedbeamsinbendingjournalhomepage:wwwelseviercomlocateengfracmechMBehzadetalEngineeringFractureMechanics()–NomenclatureacracklengthAcrosssectionalareaAcareaofthecrackfaceAhareaofthehealthypartofthecrosssectionBwidthofthebeamDheightofthebeamci(i=,,)integrationconstantsEmodulusofelasticityF(ad)crackshapefactorGshearmodulush(z)heavyside(unitstep)functionIymomentofinertiaofthesectionaboutyaxisIcymomentofinertiaofthecrackedpartofthesectionaboutyaxisIgmomentofinertiaofthesectionaboutgaxisJsstrainenergyreleaseratefunction(Jintegral)Ki(i=I,II,III)stressintensityfactorsPossiblythelackofsuchmodelsisduetothecomplexity,nonlinearityandsingularityofthestressanddisplacementfieldsnearthecracktipUsinglinearelasticfracturemechanics,thestressnearthecracktipmergestoinfinitythusasingularityoccursatthecracktipFurthermoreinelasticfracturemechanics,thestressdecaysproportionalto=ffiffiffirpwhereristhedistancefromthecracktipAccordingly,itishardtodevelopacomprehensiveandexactmodelwhichcandescribeallphenomenaoccurringatthevicinityandfarfromthecracksimultaneouslyForthefirsttime,DimaragonassuggestedalocalflexibilitymethodforbendinganalysisforcrackedbeamswhilecalculatingthedynamicresponseofacrackedEuler–BernoullibeamInthisassumptionthecrackedbeamconsistsoftwohealthypartsconnectedbyarotationalspringatthepositionofthecrackThestiffnessoftheconnectingspringcanbecalculatedusingtheconceptofJintegralfromfracturemechanicsThislocalflexibilityideahasbeenfollowedbyseveralresearcherstillnow–AlthoughtheirapproachisverysimpletheresultsareonlyreliableatpointsfarfromthecrackRecently,someresearchersimprovedthismethodbyusingacombinationoftorsionalandlinearspringsatthecrackpositionThenTimoshenkobeammodelisusedinsteadofEuler–Bernoullimodel,However,theseresultsarenotaccurateforthecrackneighbourhoodSomeresearcherspreferredtouseacontinuousdisplacementandstressfieldratherthanthelocalflexibilitymodelSuchanapproachcanleadtoabetterresultatthevicinityofthecrackChristidesandBarrdevelopedacontinuoustheoryforbendingvibrationofauniformEuler–BernoullibeamcontainingoneormorepairsofsymmetriccracksbyassumingthatthecrackeffectcanbetakenintoaccountbyapplyingsomemodificationsonthestressanddisplacementfieldsofanordinLlengthofthebeamMbendingmomentUdisplacementcomponentinxdirectionudisplacementofthedividerlineofthebeaminxdirectionUTadditionalstrainenergyduetothecrackVdisplacementcomponentinydirectionVdimensionlessdeformationofthecrackedbeamWdisplacementcomponentinzdirectionxccrackpositionzverticalcoordinateofthecentroidofthecrosssectionzcverticalcoordinateofthecentroidofthecrackedpartofthecrosssectionzcverticalcoordinateofthecentroidofthehealthypartofthecrosssectionaexponentialdecayrateDadditionaldisplacementofthecrosssectionabovethedividerlineexnormalstraincxzshearstrainuadditionaldisplacementofthecrosssectionabovethedividerlineatthecrackpositionjageometricalfactorforthecrosssectionofacrackedbeammPoissonratiohrotationofanormalbeamunderbendingh*additionalremotepointrotationofacrackedbeamhcrackedrotationofacrackedbeamunderbendingrxnormalstresssxzshearstresswslopeofthecrosssectionbelowthedividerlinearyEuler–BernoullibeamTheysuggestedtwoindependentfunctionsforstressanddisplacementfieldsbyguessworkInfacttheirsuggestedfunctionsareincompatibleandthemodelcanonlybeusedforaspecificapplicationsuchasestimationofthefirstnaturalfrequencyofthecrackedbeamsShenandPierrepresentedasimilarmodelforbendinganalysisofacrackedbeamwithsymmetriccracksTheyusedatwodimensionalfiniteelementmethodtoobtainparametersrelatedtothestressconcentrationprofilenearthecracktipTheyhavealsodevelopedacontinuousmodelforbendingofacrackedEuler–BernoullibeamwithasingleedgecrackSimilartoChristidesandBarrresearch,thedisplacementandstrainfieldshavebeenchosenindependentlyandthereforetheyarenotcompatibleFurthermore,thismodelisverysimilartotheirpreviousmodelexceptthemodificationsdoneinthestressdistributionfunctionduetothecrackThisfunctiondependsonsomeconstantswhichhavebeencalculatedfromthefiniteelementresultsCarneiroandInmansuggestedsomeslightmodificationsfortheShenandPierremodelinordertoimprovetheresultsSomeotherresearchersforcomputingthenaturalfrequenciesofcrackedstructuresusedalmostthesamemodel–ThintheMBehzadetalEngineeringFractureMechanics()–inthistheoryisthattheplanesectionsofbeamwhichareperpendiculartotheneutralaxisremainplaneandperpendiculartotheneutralaxisafterdeformationInthepresenceofanedgecrack,theplaneswillnotremainplaneafterdeformationparticularlyatthevicinityofthecrackduetoashearstressnearthecracktipwhichleadstowarpinginplanesectionsThus,atthevicinityofthecrackthedisplacementfieldiscompletelynonlinearFortheplanesfarfromthecracktip,thewarpingwillbesmallerandthedisplacementfiledcanbeassumedlinearInordertohaveabettersenseofthebendinginacrackedbeam,arealmodelhasbeenproducedinthisresearchandthemidspancrackbehaviorunderapurebendingmomentcanbeseeninFigThebeamismadefromalinearelasticmaterialwithlowmodulusofelasticityandaUshapenotchatthemidspanasacrackAsonecanseeinFig,nearthecrackareatheplanesectionswillnolongerremainplaneWithagoodapproximationitcanbesupposedthateachplanesectionturnsintotwostraightplanesafterdeformationThehorizontallinepassingthroughthecracktipiscalled‘‘dividerline”inthisresearchwhichisshowninFigEachstraightplanesectionturnsintotwoplaneswithdifferentslopesonebeneathandtheotherabovethedividerlineTheslopedifferencebetweenthesetwoplanesdecreaseswhilethedistancefromthecrackincreasesDifferentslopesofplanesaboveandunderthedividerlineisduetotheshearstressregimenearthecracktipTheshearstressnearthecracktipinthecrackopeningmode,modeI,canbewrittenassxyKIffiffiffiffiffiffiffiffiffiprpcoshsinhcoshðÞInwhichKIisthestressintensityfactor(SIF)ofthecrackandtheparametershandraredefinedinFigTheshearstressismaximumath=pwhichcorrespondstotheselecteddividerlineThedisplacementfiledofabeamwithanedgecrackcanbeapproximatedasabilinearfieldFigshowsthisapproximationgraphicallyAnotherpointofinterestistheconditionoftheneutralaxisofabeamwithanedgecrackInanEuler–Bernoullibeam,theneutralaxisremainsatthesamepositionandparalleltotheupperandlowercordsofthebeaminbendingButinabeamwithandedgecrack,theneutralaxishasadeparturefromitsoriginalpositionandremainsnolongerparalleltotheupperFigAlinearelasticcrackedbeamsubjectedtopurebendingeEuler–BernoullibendingtheoryforbeamsproposedinhasbeenusedonalargescaleaftersomedevelopmentslatethcenturyThistheorycanbeusedforlongandslenderbeamswithsmalldeformationsThebasicassumptionInthispaper,anewapproachforfindingabendingmodelofacrackedbeamhasbeenpresentedFromexperimentalobservations,abilineardisplacementfieldhasbeenintroducedforabeamwithanedgecrackThestrainandstressfieldshavebeencalculatedfromthedisplacementandtheequilibriumequationsleadingtothebendingdifferentialequationofthecrackedbeamTherequiredconstantneededinthismodelcanbeobtainedusingfracturemechanicsTheresultsofthisstudyarecomparedwiththefiniteelementandempiricalresultsforverificationFactsandfiguresneutrasystemMBehzadetalEngineeringFractureMechanics()–DisplacementfieldInEuler–BernoullibendingtheorythexaxisisassumedtobetheneutralaxisThenthelongitudinaldisplacementcanbewrittenasuðxÞzdwðxÞdxðÞInwhichu(x)andw(x)arethedisplacementsalongxandzaxis,respectivelyOnthexaxiszandwarebothzeroThisisonlytruewhentheneutralaxiscoincideswiththexaxisIfthexaxisischosentobeanywhereelse,egthebottomcordofthebeam,Eq()mustbecorrectedasfollows:whereNobeamthexThOnsubjeclaxiscannotbepredictedexactlyThisissuewillbediscussedlaterinthispaperwhilechoosingthecoordinateandlowercordsofthebeamFigshowsthelocationoftheneutralaxisofacrackedbeamafterbendingFighasbeengeneratedbyfiniteelementstressanalysisusingANSYSIntheclassicEuler–BernoullibeamtheorythexaxisistheneutralaxisHowever,inacrackedbeamthelocationoftheFigThelocationoftheneutralaxisofacrackedbeamsubjectedtoabendingloadFigAcrackedbeamparameterdefinitionuðxÞuðxÞzdwdxðÞu(x)isthelongitudinaldisplacementalongthexaxiswconsideraslenderprismaticbeamwithanopenedgecrackasshowninFigThedisplacementfiledofacrackedinpurebendingcanbeapproximatedinabilinearformSincethepositionoftheneutralaxisisunknowninthiscaseaxisconsideredtobethesameasthedividerlineintroducedbeforeeessentialassumptionsusedinthisresearchcanbelistedasfollows:ThebeamisslenderandprismaticThecrackisconsideredtobeanopenedgenotchThedeformationsaresupposedtobesmallTheplanestrainassumptionhasbeenusedinthisresearchConsequently,thedisplacementsalongyaxishavebeenneglectedThestressesaresmallenoughandthecrackdoesnotgrowthebasisoftheaboveassumptionsanddiscussions,thefollowingdisplacementfieldisintroducedforacrackedbeamttopurebending:wwðxÞðaÞvðbÞuðxzÞuðxÞzwðxÞþDðxzÞhðzÞðcÞInwisthethediThcreaserespecEquilibriumequationsMBehzadetalEngineeringFractureMechanics()–NowthestrainfieldcanbeextractedfromthedisplacementfieldTheonlynonzerocomponentsofthestrainfieldareexandcxz:exouoxdudxzdwdxaddðxxcÞuðzÞeajxxcjdhðzÞcxzowoxþouozowoxwþdudzhðzÞþuðzÞdðzÞeajxxcjdsgnðxxcÞ>><>>:ðÞwhered(xxc)istheDiracDeltafunctionwhichshowsthesingularityatthecracktipinbothnormalandshearingstressfieldsNearthecrackfacewherexxþcorxcandz>,thenormalstressmustbezeroTherefore,thenormalstrainiszerotooThen:dudxxczdwdxxcaduðzÞ!uðzÞdadudxxczdwdxxc!ðÞIneacwhereSubInwhcoordAzAinordertopursuethebilinearplanesectionassumption,D(x,z)shouldbelinearalongzaxisHencethefunctionD(x,z)wouldbedefinedasfollows:DðxzÞuðzÞeajxxcjdsgnðxxcÞðÞInEq()thefunctionu(z)isalinearfunctionofz,andaisadimensionlessexponentialdecayratewhichwillbeobtainedlaterinthispaper,andsgn(xxc)isthesignfunctionwhichisforx<xcandforx>xcTheapplicationofsignfunctionisduetothefactthattheadditionaldisplacementfunctionhasadiscontinuityatthepositionofthecrackandthesignofitsvaluechangeswhenpassingthroughthecracktipviderlineFigshowstheseparametersgraphicallyeadditionaldisplacementoftheplanesectionabovethedividerlinehasitsmaximumvalueatthecrackfacesanddesgraduallywithdistancefromthecracktipThisadditionaldisplacementisanonlinearandcomplexvariablewithttoxHereinthisresearchanexponentialregimehasbeenassumedforfunctionD(x,z)alongthexaxisFurthermore,dividerlineInanEuler–Bernoullibeamtheorybyneglectingtheshearstresseffectonehasw(x)=dwdxInacrackedbeamtheshearstressnearthecracktipcannotbeignoredthusw(x)isdifferentfromdwdx,howeverfarfromthecracktheshearingstressdecreasesgraduallyandw(x)tendstobeequaltodwdxh(z)istheunitstepfunctionwhichisequaltozeroforz<andforz>AccordinglythetermD(x,z)h(z)canbeconsideredastheextradisplacementoftheplanesectionsabovehichu,v,warethedisplacementcomponentsalongx,yandzaxisEq(c)isthekeyequationinthisresearchu(x)longitudinaldisplacementofthedividerlinealongthexaxisandw(x)istheslopeoftheplanesectionsbelowtheFigGraphicalrepresentationofacrackedbeamdeformationfieldandremotepointrotationhcrosssectiononehasthestaticequilibriuminthexdirectionif:ZArxdA!ZAEexdA!ZAexdAðÞAisthecrosssectionareaofthebeamandEisthemodulusofelasticitystitutingfromEqs()and()intoEq()onehas:AdudxzdwdxAcdudxxczcdwdxxc!eajxxcjdxxcðÞichAcistheareaofthecrackface,zistheverticalcoordinateofthecentroidofthecrosssectionandzcistheverticalinateofthecentroidofthecrackedpartofthecrosssectionFigdemonstratestheseparametersNotethatczcþAhzhandAAcþAhFromEq(),thestaticequilibriumatx=xcgivesthefollowingresult:InordcanbeUsingFinallydInwhcanbeForthdxyieldsðÞMBehzadetalEngineeringFractureMechanics()–rxMIgðzzÞþAcbzhAjðzzÞþAcAðzhzcÞIgIhyAhzhdadðxxcÞðzhzÞhðzÞeajxxcjdðÞSubstitutingEq()intoEq()thenormalstressfunctioncanbeexpressedinanexplicitformasfollows:jðAcbzhÞIgAAczðzhzcÞþAAczczhAIcyAðIhyAhzhÞMEIgdwdxþjMEeajxxcjdxxctoInwhichIhyisthemomentofinertiaofthehealthypartofthecrackedsectionaboutyaxisSubstitutingEq()intoEq()MEZAhzexðxcz<ÞdA!MEðAhzhIhyÞdwdxxcðÞxcSincethecrackedsectionhasthesamemomentM,thefollowingrelationwillberesulted:ehealthyareaofthecracksection(Ah)thestraincanbeexpressedasfollows:exðxcz<ÞðzhzÞdwðÞEdxAdxxcsimplifiedintothefollowingform:MIgdwþAcbzhIgAczðzhzcÞþAczczhIcydweajxxcjdxxcðÞycybethehorizontalaxispassingthroughthecentroidofthecrosssectionasshowninFig,onehasIyIgþAz,andEq()EðAzIyÞdxþAAzIyþAczðzhzcÞAczczhþIcydxxcexxcðÞichIandIarethemomentofinertiaofthecrosssectionandthecrackfaceaboutyaxis,respectivelyIfgistakentoMdwAcbzhdwajxxcjMAzrxdAEAzexdAðÞ,usingEqs()and()thebendingdifferentialequationofthecrackedbeamcanbeobtained:dxAAdxxcWhenthebeamissubjectedonlytopurebendingthemomentequilibriumineachcrosssectionleadsto:ZZEqs()and()thenormalstrainstatementwhichwasdefinedinEq()canberewrittenasfollows:exðzzÞdwþAcbzhðzzÞþAcðzhzcÞðzhzÞhðzÞdwe

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