第1講 斷裂力學導論

斷裂力學

第一講導論：什么是斷裂力學？斷裂力學研究什么？斷裂力學和材料力學有什么聯系和區別？從強度理論到斷裂理論斷裂力學和破壞力學（failure)之間的關系斷裂力學從材料力學、彈性力學發展起來固體力學中最活躍的研究領域實用性：大量的工程應用要求研究材料破壞；跨學科：力學、材料、物理、環境、分析計算方法、測試技術；科學、工程結合形形色色的斷裂現象疲勞斷裂脆性斷裂斷裂的扳手韌性試件的斷裂LibertyShipsinWWIIFactsonLibertyShipsTherewere2,751LibertyShipsmanufacturedbetween1941-1945.Crackspropagatedin400oftheseshipsincluding145catastrophicfailures;only2existtodaywhicharesea-worthy.ThehullsofLibertyShipsfracturedwithoutwarning,mainlyintheNorthAtlantic.Thesearethefirstshipsmassproducedwithwelds.ThelowtemperaturesoftheNorthAtlanticcausedthesteeltobebrittle.Fracturesoccurredmainlyinthevicinityofstressraisers.Theproblemmaybepreventedbyemployinghigherqualitysteelsandimprovementofthedesignoftheship.TheAlohaBoeing737AccidentOnApril28,1988,partofthefuselageofaBoeing737failedafter19yearsofservice.Thefailurewascausedbyfatigue(multi-sitedamage).TheAlohaBoeing737Accident一個不斷發展的學科：

強度問題、裂紋問題、斷裂問題、疲勞、損傷、分析和規范、結構完整性問題，等等損傷分布與裂紋共生的結構工程問題:結構承載能力(Structuralsafetyandintegrity).結構承載時間(Durability).結構可信度(Reliability).安全性評估(Confidencelevel).經濟問題:建造成本和獲取手段(capitaloracquisitioncost);運行代價(operationalcost)，廢棄代價(disposal/recyclingcost)成本優化和使用壽命優化(Optimizecapitalcostsandlifecyclecost)結構完整性和壽命評估系統從強度理論到裂紋力學應力狀態和主應力參數強度函數SomeCriteriapredictmaterialsfailurebyyielding,othersbyfractureAgivenmaterialmayfailbyyieldingorfracturestrengthdependingonitspropertiesandthestateofstressFailurecriterionforisotropicmaterialisexpressedastheprincipalnormalstresseshavereachedacriticalfailurestrength(c)

f(1,2,3)=c常用的強度準則最大主應力準則；最大剪切應力準則；最大主應變準則；最大總應變能準則；最大畸變應變能準則；最大VonMises應力準則；Etc.孔洞導致應力集中-Inglis理論FracturestressforrealisticmaterialInglis(1913)analyzedfortheflatplatewithanellipticalholewithmajoraxis2aandminoraxis2b,subjectedtofarendstressThestressatthetipofthemajoraxis(pointA)isgivenby

Theratioisdefinedasthestressconcentrationfactor,Whena=b,itisacircularhole,thenWhenbisveryverysmall,Inglisdefineradiusofcurvatureas

Andthetipstressas

Inglis理論推廣到裂紋玻璃纖維的斷裂強度（示意圖）Griffith關于裂紋擴展的概念Griffith’sEnergybalanceapproachFirstdocumentedpaperonfracture(1920)ConsideredasfatherofFractureMechanicsAAGriffithlaidthefoundationsofmodernfracturemechanicsbydesigningacriterionforfastfracture.Heassumedthatpre-existingflawspropagateundertheinfluenceofanappliedstressonlyifthetotalenergyofthesystemistherebyreduced.Thus,Griffith'stheoryisnotconcernedwithcracktipprocessesorthemicromechanismsbywhichacrackadvances.Griffith’sEnergybalanceapproach(Contd.)Griffithproposedthat‘Thereisasimpleenergybalanceconsistingofthedecreaseinpotentialenergywithinthestressedbodyduetocrackextensionandthisdecreaseisbalancedbyincreaseinsurfaceenergyduetoincreasedcracksurface’Griffiththeoryestablishestheoreticalstrengthofbrittlematerialandrelationshipbetweenfracturestrengthandflawsize‘a’

Griffith’sEnergybalanceapproach(Contd.)Theinitialstrainenergyfortheuncrackedplateperthicknessis(2.14)Oncreatingacrackofsize2a,thetensileforceonanelementdsonellipticholeisrelaxedfromtozero.Theelasticstrainenergyreleasedperunitwidthduetointroductionofacrackoflength2aisgivenby

(2.15)

Griffith’sEnergybalanceapproach(Contd.)Externalwork=(2.16)Thepotentialorinternalenergyofthebodyis

Duetocreationofnewsurfaceincreaseinsurfaceenergyis(2.17)Thetotalelasticenergyofthecrackedplateis

(2.18)

Griffith’sEnergybalanceapproach(Contd.)Thevariationofwithcrackextensionshouldbeminimum

Denotingasduringfracture(2.19)forplanestress(2.20)forplanestrainTheGriffiththeoryisobeyedbymaterialswhichfailinacompletelybrittleelasticmanner,e.g.glass,mica,diamondandrefractorymetals.Griffith’sEnergybalanceapproach(Contd.)GriffithextrapolatedsurfacetensionvaluesofsodalimeglassfromhightemperaturetoobtainthevalueatroomtemperatureasUsingvalueofE=62GPa,Thevalueofas0.15Fromtheexperimentalstudyonsphericalvesselshecalculatedas0.25–0.28However,itisimportanttonotethataccordingtotheGriffiththeory,itisimpossibletoinitiatebrittlefractureunlesspre-existingdefectsarepresent,sothatfractureisalwaysconsideredtobepropagation-(ratherthannucleation-)controlled;thisisaseriousshort-comingofthetheory.ModificationforDuctileMaterialsFormoreductilematerials(e.g.metalsandplastics)itisfoundthatthefunctionalformoftheGriffithrelationshipisstillobeyed,i.e..However,theproportionalityconstantcanbeusedtoevaluategs(providedEisknown)andifthisisdone,onefindsthevalueismanyordersofmagnitudehigherthanwhatisknowntobethetruevalueofthesurfaceenergy(whichcanbedeterminedbyothermeans).Forthesematerialsplasticdeformationaccompaniescrackpropagationeventhoughfractureismacroscopicallybrittle;Thereleasedstrainenergyisthenlargelydissipatedbyproducinglocalizedplasticflowatthecracktip.IrwinandOrowanmodifiedtheGriffiththeoryandcameoutwithanexpressionWheregprepresentsenergyexpendedinplasticwork.Typicallyforcleavageinmetallicmaterialsgp=104J/m2andgs=1J/m2.Sincegp>>gswehave

StrainEnergyReleaseRateThestrainenergyreleaserateusuallyreferredto

Notethatthestrainenergyreleaserateisrespecttocracklengthandmostdefinitelynottime.Fractureoccurswhenreachesacriticalvaluewhichisdeno