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首页 A linear theory for bending stress strain analys…

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

A linear theory for bending str…

jjzheng4335
2012-07-05 0人阅读 0 0 0 暂无简介 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

JIntegralBending�ElsevierLtdAllrightsreservedeofractutipiagatilyticairystformulaehasbeencollectedbyTadaetalFurtherdevelopmentsinelasticandelastoplasticfracturemechanicscanbefoundintheliterature–Almosteverystressanddisplacementanalysisperformedinfracturemechanics,dealswiththeneighborhoodofthecrackwhilethestressanddisplacementfieldsinthewholecrackedcontinuumhaverarelybeenconsideredButinseveralapplicationssuchasvibrationanalysisofcrackedstructures,itisessentialtohaveaload–deflectionmodelfortheentirebody$seefrontmatter�ElsevierLtdAllrightsreserved*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–BernoullibeamcontainingoneormorepairsofsymmetriccracksbyassumingthatthecrackeffectcanbetakenintoaccountbyapplyingsomemodificationsonthestressanddisplacementfieldsofanordinLlengthofthebeamMbendingmomentUdisplacementcomponentinxdirectionudisplacementofthedividerlineofthebeaminxdirectionUTadditionalstrainenergyduetothecrackVdisplacementcomponentinydirectionVdimensionlessdeformationofthecrackedbeamWdisplacementcomponentinzdirectionxccrackposition�zverticalcoordinateofthecentroidofthecrosssection�zcverticalcoordinateofthecentroidofthecrackedpartofthecrosssection�zcverticalcoordinateofthecentroidofthehealthypartofthecrosssectionaexponentialdecayrateDadditionaldisplacementofthecrosssectionabovethedividerlineexnormalstraincxzshearstrainuadditionaldisplacementofthecrosssectionabovethedividerlineatthecrackpositionjageometricalfactorforthecrosssectionofacrackedbeammPoissonratiohrotationofanormalbeamunderbendingh*additionalremotepointrotationofacrackedbeamhcrackedrotationofacrackedbeamunderbendingrxnormalstresssxzshearstresswslopeofthecrosssectionbelowthedividerlinearyEuler–BernoullibeamTheysuggestedtwoindependentfunctionsforstressanddisplacementfieldsbyguessworkInfacttheirsuggestedfunctionsareincompatibleandthemodelcanonlybeusedforaspecificapplicationsuchasestimationofthefirstnaturalfrequencyofthecrackedbeamsShenandPierrepresentedasimilarmodelforbendinganalysisofacrackedbeamwithsymmetriccracksTheyusedatwodimensionalfiniteelementmethodtoobtainparametersrelatedtothestressconcentrationprofilenearthecracktipTheyhavealsodevelopedacontinuousmodelforbendingofacrackedEuler–BernoullibeamwithasingleedgecrackSimilartoChristidesandBarrresearch,thedisplacementandstrainfieldshavebeenchosenindependentlyandthereforetheyarenotcompatibleFurthermore,thismodelisverysimilartotheirpreviousmodelexceptthemodificationsdoneinthestressdistributionfunctionduetothecrackThisfunctiondependsonsomeconstantswhichhavebeencalculatedfromthefiniteelementresultsCarneiroandInmansuggestedsomeslightmodificationsfortheShenandPierremodelinordertoimprovetheresultsSomeotherresearchersforcomputingthenaturalfrequenciesofcrackedstructuresusedalmostthesamemodel–ThintheMBehzadetalEngineeringFractureMechanics()–inthistheoryisthattheplanesectionsofbeamwhichareperpendiculartotheneutralaxisremainplaneandperpendiculartotheneutralaxisafterdeformationInthepresenceofanedgecrack,theplaneswillnotremainplaneafterdeformationparticularlyatthevicinityofthecrackduetoashearstressnearthecracktipwhichleadstowarpinginplanesectionsThus,atthevicinityofthecrackthedisplacementfieldiscompletelynonlinearFortheplanesfarfromthecracktip,thewarpingwillbesmallerandthedisplacementfiledcanbeassumedlinearInordertohaveabettersenseofthebendinginacrackedbeam,arealmodelhasbeenproducedinthisresearchandthemidspancrackbehaviorunderapurebendingmomentcanbeseeninFigThebeamismadefromalinearelasticmaterialwithlowmodulusofelasticityandaUshapenotchatthemidspanasacrackAsonecanseeinFig,nearthecrackareatheplanesectionswillnolongerremainplaneWithagoodapproximationitcanbesupposedthateachplanesectionturnsintotwostraightplanesafterdeformationThehorizontallinepassingthroughthecracktipiscalled‘‘dividerline”inthisresearchwhichisshowninFigEachstraightplanesectionturnsintotwoplaneswithdifferentslopesonebeneathandtheotherabovethedividerlineTheslopedifferencebetweenthesetwoplanesdecreaseswhilethedistancefromthecrackincreasesDifferentslopesofplanesaboveandunderthedividerlineisduetotheshearstressregimenearthecracktipTheshearstressnearthecracktipinthecrackopeningmode,modeI,canbewrittenassxy¼KIffiffiffiffiffiffiffiffiffiprpcoshsinhcoshðÞInwhichKIisthestressintensityfactor(SIF)ofthecrackandtheparametershandraredefinedinFigTheshearstressismaximumath=pwhichcorrespondstotheselecteddividerlineThedisplacementfiledofabeamwithanedgecrackcanbeapproximatedasabilinearfieldFigshowsthisapproximationgraphicallyAnotherpointofinterestistheconditionoftheneutralaxisofabeamwithanedgecrackInanEuler–Bernoullibeam,theneutralaxisremainsatthesamepositionandparalleltotheupperandlowercordsofthebeaminbendingButinabeamwithandedgecrack,theneutralaxishasadeparturefromitsoriginalpositionandremainsnolongerparalleltotheupperFigAlinearelasticcrackedbeamsubjectedtopurebendingeEuler–BernoullibendingtheoryforbeamsproposedinhasbeenusedonalargescaleaftersomedevelopmentslatethcenturyThistheorycanbeusedforlongandslenderbeamswithsmalldeformationsThebasicassumptionInthispaper,anewapproachforfindingabendingmodelofacrackedbeamhasbeenpresentedFromexperimentalobservations,abilineardisplacementfieldhasbeenintroducedforabeamwithanedgecrackThestrainandstressfieldshavebeencalculatedfromthedisplacementandtheequilibriumequationsleadingtothebendingdifferentialequationofthecrackedbeamTherequiredconstantneededinthismodelcanbeobtainedusingfracturemechanicsTheresultsofthisstudyarecomparedwiththefiniteelementandempiricalresultsforverificationFactsandfiguresneutrasystemMBehzadetalEngineeringFractureMechanics()–DisplacementfieldInEuler–BernoullibendingtheorythexaxisisassumedtobetheneutralaxisThenthelongitudinaldisplacementcanbewrittenasuðxÞ¼�zdwðxÞdxðÞInwhichu(x)andw(x)arethedisplacementsalongxandzaxis,respectivelyOnthexaxiszandwarebothzeroThisisonlytruewhentheneutralaxiscoincideswiththexaxisIfthexaxisischosentobeanywhereelse,egthebottomcordofthebeam,Eq()mustbecorrectedasfollows:whereNobeamthexThOnsubjeclaxiscannotbepredictedexactlyThisissuewillbediscussedlaterinthispaperwhilechoosingthecoordinateandlowercordsofthebeamFigshowsthelocationoftheneutralaxisofacrackedbeamafterbendingFighasbeengeneratedbyfiniteelementstressanalysisusingANSYSIntheclassicEuler–BernoullibeamtheorythexaxisistheneutralaxisHowever,inacrackedbeamthelocationoftheFigThelocationoftheneutralaxisofacrackedbeamsubjectedtoabendingloadFigAcrackedbeamparameterdefinitionuðxÞ¼uðxÞ�zdwdxðÞu(x)isthelongitudinaldisplacementalongthexaxiswconsideraslenderprismaticbeamwithanopenedgecrackasshowninFigThedisplacementfiledofacrackedinpurebendingcanbeapproximatedinabilinearformSincethepositionoftheneutralaxisisunknowninthiscaseaxisconsideredtobethesameasthedividerlineintroducedbeforeeessentialassumptionsusedinthisresearchcanbelistedasfollows:ThebeamisslenderandprismaticThecrackisconsideredtobeanopenedgenotchThedeformationsaresupposedtobesmallTheplanestrainassumptionhasbeenusedinthisresearchConsequently,thedisplacementsalongyaxishavebeenneglectedThestressesaresmallenoughandthecrackdoesnotgrowthebasisoftheaboveassumptionsanddiscussions,thefollowingdisplacementfieldisintroducedforacrackedbeamttopurebending:w¼wðxÞðaÞv¼ðbÞuðxzÞ¼uðxÞ�zwðxÞþDðxzÞhðzÞðcÞInwisthethediThcreaserespecEquilibriumequationsMBehzadetalEngineeringFractureMechanics()–NowthestrainfieldcanbeextractedfromthedisplacementfieldTheonlynonzerocomponentsofthestrainfieldareexandcxz:ex¼ouox¼dudx�zdwdx�ad�dðx�xcÞ��uðzÞ�e�ajx�xcjdhðzÞcxz¼owoxþouoz��¼owox�wþdudzhðzÞþuðzÞdðzÞ��e�ajx�xcjd�sgnðx�xcÞ��>><>>:ðÞwhered(x�xc)istheDiracDeltafunctionwhichshowsthesingularityatthecracktipinbothnormalandshearingstressfieldsNearthecrackfacewherex¼xþcorx�candz>,thenormalstressmustbezeroTherefore,thenormalstrainiszerotooThen:dudx����xc�zdwdx����xc�aduðzÞ¼!uðzÞ¼dadudx����xc�zdwdx����xc!ðÞIneacwhereSubInwhcoordA�z¼Ainordertopursuethebilinearplanesectionassumption,D(x,z)shouldbelinearalongzaxisHencethefunctionD(x,z)wouldbedefinedasfollows:DðxzÞ¼uðzÞ�e�ajx�xcjdsgnðx�xcÞðÞInEq()thefunctionu(z)isalinearfunctionofz,andaisadimensionlessexponentialdecayratewhichwillbeobtainedlaterinthispaper,andsgn(x�xc)isthesignfunctionwhichis�forx<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:ZArx�dA¼!ZAEex�dA¼!ZAex�dA¼ðÞAisthecrosssectionareaofthebeamandEisthemodulusofelasticitystitutingfromEqs()and()intoEq()onehas:Adudx��zdwdx���Acdudx����xc��zcdwdx����xc!�e�ajx�xcjd¼x¼xcðÞichAcistheareaofthecrackface,�zistheverticalcoordinateofthecentroidofthecrosssectionand�zcistheverticalinateofthecentroidofthecrackedpartofthecrosssectionFigdemonstratestheseparametersNotethatc�zcþAh�zhandA¼AcþAhFromEq(),thestaticequilibriumatx=xcgivesthefollowingresult:�InordcanbeUsingFinally�������dInwhcanbeForthdxyieldsðÞMBehzadetalEngineeringFractureMechanics()–rx¼MIgð�z�zÞþAc�b�zhA�j��ð�z�zÞþAcAð�zh��zcÞ�IgIhy�Ah�zh�dadðx�xcÞ��ð�zh�zÞhðzÞ��e�ajx�xcjd��ðÞSubstitutingEq()intoEq()thenormalstressfunctioncanbeexpressedinanexplicitformasfollows:j¼ðAc�b�zhÞIg�AAc�zð�zh��zcÞþAAc�zc�zh�AIcyAðIhy�Ah�zhÞME¼IgdwdxþjME�e�ajx�xcjdx¼xctoInwhichIhyisthemomentofinertiaofthehealthypartofthecrackedsectionaboutyaxisSubstitutingEq()intoEq()M¼�EZAhzexðxcz<Þ�dA!�ME¼ðAh�zh�IhyÞdwdx����xcðÞxcSincethecrackedsectionhasthesamemomentM,thefollowingrelationwillberesulted:ehealthyareaofthecracksection(Ah)thestraincanbeexpressedasfollows:exðxcz<Þ¼ð�zh�zÞdw����ðÞEdxAdx�xcsimplifiedintothefollowingform:M¼IgdwþAc�b�zhIg�Ac�zð�zh��zcÞþAc�zc�zh�Icy��dw����e�ajx�xcjdx¼xcðÞycybethehorizontalaxispassingthroughthecentroidofthecrosssectionasshowninFig,onehasIy¼IgþA�z,andEq()�E¼ðAz�IyÞdxþAAz�IyþAczðzh�zcÞ�AczczhþIcydx�xc�ex¼xcðÞichIandIarethemomentofinertiaofthecrosssectionandthecrackfaceaboutyaxis,respectivelyIfgistakentoMdwAc�b�zh��dw����ajx�xcjM¼�Azrx�dA¼�EAzex�dAðÞ,usingEqs()and()thebendingdifferentialequationofthecrackedbeamcanbeobtained:dxAAdx�xcWhenthebeamissubjectedonlytopurebendingthemomentequilibriumineachcrosssectionleadsto:ZZEqs()and()thenormalstrainstatementwhichwasdefinedinEq()canberewrittenasfollows:ex¼ð�z�zÞdwþAc�b�zhð�z�zÞþAcð�zh��zcÞ�ð�zh�zÞhðzÞ��dw����e

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