应力集中公式用于任何形状的缺口在圆试样在张力下弯曲

StressconcentrationformulaeusefulforanyshapeofnotchinaroundtestspecimenundertensionandunderbendingN.A.NODAandY.TAKASEDepartmentofMechanicalEngineering,KyushuInstituteofTechnology,Kitakyushu804-8550,JapanReceivedinfinalform2June1999ABSTRACTInthisworkstressconcentrationfactors,Kt,foraroundbarwithacircular-arcorV-shapednotchareconsideredonthebasisofexactsolutionsforspecialcasesandaccuratenumericalresults.Then,asetofKtformulaeusefulforanyshapeofnotchisproposed.Theconclusionscanbesummarizedasfollows.(i)Forthelimitingcasesofdeep(d)andshallow(s)notches,thebodyforcemethodisusedtocalculatetheKtvalues.Then,theformulaeareobtainedasKtdandKts.(ii)Ontheonehand,uponcomparisonofKtandKtditisfoundthatKtisnearlyequaltoKtdifthenotchisdeeporblunt.(iii)Ontheotherhand,ifthenotchissharporshallow,KtismainlycontrolledbyKtsandthenotchdepth.(iv)Thenotchshapeisclassifiedintoseveralgroupsaccordingtothenotchradiusandnotchdepth.Then,theleast-squaresmethodisappliedforthecalculationofKt/KtdandKt/Kts.(v)Finally,asetofconvenientformulaeisproposedthatareusefulforanyshapeofnotchinaroundtestspecimen.TheformulaegiveSCFswith1%errorforanyshapeofnotch.Keywordsstressconcentrationfactor;fatiguetestspecimen;V-shapednotch;notchformulae;tension;bending;bodyforcemethod.NOMENCLATUREa=radiusofminimumsectionD=cylindricaldiameterofmaximumsectionKt=stressconcentrationfactor(SCF)basedontheminimumsectionKtd=SCFofdeepV-shapednotchKtE=SCFofanellipticalholeinaninfiniteplateunderuniformtension(=1+2E(t/r)KtH=SCFofhyperboloidalnotchKts=SCFof60°V-shapednotchinasemi-infiniteplatet=depthofnotchx=a/rwhena/r∏1.0and=2−r/awhenr/a∏1.0DKth=thresholdstressintensityfactorrangeg=E(r/t)l=relativenotchdepth,=2t/Dn=Poisson’sratio(=0.3)j=Et/rr=rootradiusofnotchnentscannotbeexplainedonlyfromthestressconcen-INTRODUCTIONtrationfactorKtwiththefatiguelimitofaplainspecimen.DependingonthedepthandsharpnessofThestressconcentrationofanotchedroundbarisespeciallyimportantintermsofafatiguetestspecimen.notches,theeffectofthegeometricaldimensionofnotchesonthefatiguestrengthisdifferent.InanumberItisknownthatthefatiguestrengthofnotchedcompo-©1999BlackwellScienceLtd.FatigueFractEngngMaterStruct22,1071–108210711072N.A.NODAandY.TAKASEofpapers,1–4Millerandco-workershavesuggestedthatKtisnearlyequaltoKtdifthenotchisdeeporblunt.Fourth,thenotchshapeisclassifiedintoseveralgroupsnotchessubjectedtofatigueareessentiallyoftwoformsinwhichthefollowingpointsapply.accordingtothenotchradiusandnotchdepth.Finally,asetofconvenientformulaeusefulforanyshapeof1Shallownotchesunderfatiguearelargelycontrollednotchisproposedbyapplyingtheleast-squaresmethodbytherootradiusrandthefatiguelimitoftheforeachgroup.ThesetofformulaegivesSCFswithmaterial.1%errorforanyshapeofnotch.2Sharpnotchesunderfatiguearelargelycontrolledbyj=E(t/r)wheretisthenotchdepthandthefatigueSTRESSCONCENTRATIONFACTORSOFSHARPlimitasexpressedbytheLEFMparameterDKth.ORSHALLOWNOTCHESToclarifythefatiguenotchmechanismindetail,itisnecessarytouseaccurateKtvalueswhenusingahighlyConsiderthestressconcentrationfactorKtofa60°V-shapednotchasshowninFig.1(c).Inthiscasethedevelopedexperimentaltechnique.However,toestimatestressconcentrationfactorsSCFofthesharpnotchismainlycontrolledbytheSCFofa60°V-shapednotchinasemi-infiniteplateKts(SCFs)fornotchedbars,Neuber’singeniousformulaehavebeenusedformorethan40years.5Theyarehavingthesameshaperatiot/r.10TheratioKt/KtsisshowninFig.2.InFig.2thelineforr/a0isobtainedconvenientforengineeringapplicationbecause:fromresultswhenr/aissmall;e.g.r/a=0.1and0.21theyhavesimpleforms;andwhichconfirmthatthetworesultsareinagreementto2theygiveapproximatevaluesforanyshapeofnotch.thethirddigit.TheKtsapproximationisbetterintensionthaninbending.InFig.2,wecanseethefollowing.Previously,theauthorshaveproposedmoreaccurateformulae6usingtheresultsofthebodyforcemethod7andcorrectingNeuber’svalues.However,theformulae1Insharpnotches(r/a∏0.1),thevalueofKt/Ktsismainlydeterminedby2t/Dalone.ThevalueofKtcanareonlyusefulforacertainrangeofnotchshapebecauseoftherestrictionofthebodyforcemethodofappli-beobtainedfromKtsinthefollowingrangedependingontheloadingconditions,i.e.cation.8,9Generally,thereisnonumericalmethodthatiseffectiveforanalysinganyshapeofthenotchwith(i)tension:2t/D∏0.8,(ii)bending:2t/D∏0.4.1%error.Inotherwords,itisdifficulttoobtainaccuratestressconcentrationfactorswhenthenotch2Inshallownotches(2t/D∏0.02),thevalueofKt/Ktsiscontrolledby2t/Dforawiderangeofa/rvalues,i.e.rootradiusisextremelylargeorextremelysmall.Inthisstudy,first,thevaluesofKtforaroundbar(i)tension:thevalueofKt/Ktsisdeterminedby2t/Dalone,whena/r0.01,withacircular-arcorV-shapednotchareconsideredforlimitingcasesofdeepandshallownotches,KtdandKts.(ii)bending:thevalueofKt/Ktsiswithinasmallrange,whena/r0.02and2t/D∏0.02,e.g.Second,itisfoundthatifthenotchissharporshallow,KtismainlycontrolledbyKtsandthenotchdepth.0.938∏Kt/Kts∏0.968.3ThestressconcentrationfactorofabluntandaThird,uponcomparisonofKtandKtd,itisfoundthatFig.1Roundspecimenswithacircular-arcoraV-shapednotch:(a)extremelybluntnotch;(b)ordinarynotch;and(c)extreme