《斷裂力學(xué)》課件-第4講 線彈性斷裂力學(xué)和應(yīng)力強(qiáng)度因子

斷裂力學(xué)

第三講裂紋尖端應(yīng)力場，應(yīng)力強(qiáng)度因子有哪幾種典型的裂紋擴(kuò)展形式？I、II、III型裂紋的裂紋尖端應(yīng)力應(yīng)變計(jì)位移的分布形式？為什么用裂紋強(qiáng)度因子可以表征線彈性材料的斷裂過程？裂紋強(qiáng)度因子與能量釋放率之間的關(guān)系問題應(yīng)力；變形；應(yīng)變；應(yīng)力應(yīng)變關(guān)系；平衡方程；平面應(yīng)力和平面應(yīng)變問題；二維問題彈性解；彈性理論簡述Airy應(yīng)力函數(shù)；Airy函數(shù)的復(fù)變函數(shù)表示形式；Westergaard應(yīng)力函數(shù)；三種裂紋尖端的線性彈性應(yīng)力場解；LINEARELASTICFRACTUREMECHANICS(LEFM)

ForLEFMthestructureobeysHooke’slawandglobalbehaviorislinearandifanylocalsmallscalecracktipplasticityisignoredThefundamentalprincipleoffracturemechanicsisthatthestressfieldaroundacracktipbeingcharacterizedbystressintensityfactorKwhichisrelatedtoboththestressandthesizeoftheflaw.Theanalyticdevelopmentofthestressintensityfactorisdescribedforanumberofcommonspecimenandcrackgeometriesbelow.ThethreemodesoffractureModeI-Openingmode:wherethecracksurfacesseparatesymmetricallywithrespecttotheplaneoccupiedbythecrackpriortothedeformation(resultsfromnormalstressesperpendiculartothecrackplane);ModeII-Slidingmode:wherethecracksurfacesglideoveroneanotherinoppositedirectionsbutinthesameplane(resultsfromin-planeshear);andModeIII-Tearingmode:wherethecracksurfacesaredisplacedinthecrackplaneandparalleltothecrackfront(resultsfromout-of-planeshear).Inthe1950sIrwin[7]andcoworkersintroducedtheconceptofstressintensityfactor,whichdefinesthestressfieldaroundthecracktip,takingintoaccountcracklength,appliedstresssandshapefactorY(whichaccountsforfinitesizeofthecomponentandlocalgeometricfeatures).

TheAirystressfunction.Instressanalysiseachpoint,x,y,z,ofastressedsolidundergoesthestresses;sx

sy,sz,txy,txz,tyz.Withreferencetofigure2.3,whenabodyisloadedandtheseloadsarewithinthesameplane,saythex-yplane,twodifferentloadingconditionsarepossible:LINEARELASTICFRACTUREMECHANICS(Contd.)1.planestress(PSS),whenthethicknessofthebodyiscomparabletothesizeoftheplasticzoneandafreecontractionoflateralsurfacesoccurs,and,2.planestrain(PSN),whenthespecimenisthickenoughtoavoidcontractioninthethicknessz-direction.Intheformercase,theoverallstressstateisreducedtothethreecomponents;sx,sy,txy,since;sz,txz,tyz=0,while,inthelattercase,anormalstress,sz,isinducedwhichpreventsthezdisplacement,ez=w=0.Hence,fromHooke'slaw:sz=ν(sx+sy)whereν

isPoisson'sratio.Forplaneproblems,theequilibriumconditionsare:IfistheAiry’sstressfunctionsatisfyingthebiharmoniccompatibilityConditionsThenForproblemswithcracktipWestergaardintroducedAiry’sstressfunctionasWhereZisananalyticcomplexfunctionAndare2ndand1stintegralsofZ(z)ThenthestressesaregivenbyOpeningmodeanalysisorModeIConsideraninfiniteplateacrackoflength2asubjectedtoabiaxialStateofstress.Defining:BoundaryConditions:AtinfinityOncrackfacesByreplacingzbyz+a,originshiftedtocracktip.Andwhen|z|0atthevicinityofthecracktipKImustberealandaconstantatthecracktip.ThisisduetoaSingularitygivenbyTheparameterKIiscalledthestressintensityfactorforopeningmodeI.Sinceoriginisshiftedtocracktip,itiseasiertousepolarCoordinates,UsingFurtherSimplificationgives:FromHooke’slaw,displacementfieldcanbeobtainedasTheverticaldisplacementsatanypositionalongx-axis(q=0)isgivenbyThestrainenergyrequiredforcreationofcrackisgivenbytheworkdonebyforceactingonthecrackfacewhilerelaxingthestressstozeroSlidingmodeanalysisorMode2Forproblemswithcracktipundershearloading,Airy’sstressfunctionistakenasUsingAir’sdefinitionforstressesUsingaWestergaardstressfunctionoftheformBoundaryConditions:AtinfinityOncrackfacesWithusualsimplificationwouldgivethestressesasDisplacementcomponentsaregivenbyTearingmodeanalysisorMode3Inthiscasethecrackisdisplacedalongz-axis.HerethedisplacementsuandvaresettozeroandhenceUsingWestergaardstressfunctionasTypesofFracturebyLoadingDirectionmodeImodeIImodeIIIm=I,II,IIIi,j=1,2,3AsymptoticStressFieldinModeIStressIntensityFactorm