复合型裂纹的断裂准则

1、000009 ChapterFiveFracturecriterionformixedmodecrackInthematerialmechanics,forthemultiaxialstressstate,fourstrengththeorieshavebeendeveloped.Inthefracturemechanicsforthemixedmodecrackproblem,weneedtodevelopthefracturetheoryaccordingly.Manyfracturetheorieshavebeendeveloped.Twokeyquestionsmustbeanswer

2、ed.Whatdirectiondoesacrackpropagatealong?Whatisthecriticalcase?Inwhatfollows,fivetheorieswillbeintroduced.5-1MaximumnormalstresscriterionMaximumstresscriterioncanbeappliedtothemixedmodecrackofmodeIandmodeII.TheasymptoticstresssolutionisTOC o 1-5 h zKA.B.3AK.A038、g=i=cos-(1-sinsin)-iisin(2+coscos)xx伽

3、r222兀r222KAA3AKAA3Ag=.icos(1+sinsin)+.iisincoscos-yy兀r2222兀r222KAA3AKAA3Ag=i=cossmcos+u=cos(1-sinsin)xy兀r2222兀r222Byapplicationofthecoordinatetransformationformulas,wecanobtaintheexpressionsofthreestresscomponentsinthepolarcoordinates(r,A).ThecircumferentialnormalstressisgAA11AcosK(1+cos0)-3Ksin02兀r

4、221”isdefinedasThecircumferentialnormalstressintensityfactorKKrtO10=2cos2Ki(1+cos0)-3化sin0FractureandDamageMechanicsChapterFiveFracturecriterionformixedmodecrack Hence,ocanbewrittenaseeKo=ee=ee2兀rAssumptions:CrackinitiationdirectionisthedirectionofthemaximumK;eeWhenKreachesitscriticalvalueK,breakocc

5、urs.Kisamaterialeeeeceecconstant.Thecrackinitiationangleecanbedeterminedfrom0dKd2Kee二0,ee0deae2TheresultisKsine+K(3cose1)二0I0II0eeKcos0(13cose)+Ksin*(9cos0+5)0I20II20Thecriticalconditionissine)=K0eecee3K=cosKcos20Kemax2I22IIDeterminationofK:eecFormodeIcrack,K丰0,K=0,e二0,thecriticalconditionreducestoI

6、II0K=K=KemaxIceecNotethatKisamaterialconstant.When,K丰0andK丰0,therestillprevailseecIIIK=K.ThemaximumstresscriterionisexpressedaseecIcecos02(KIecos2022IIsine)K0IcApplicationtomodeIIcrack:ForamodeIIcrack,K=0andK丰0.Thecrackinitiationanglecanbesolved,III0=70.5。.From0003K=cos亠(Kcos2亠Ksin0)=K=K0max2I22II00

7、0cIconecanobtainthat1.149K二K,K二0.87K,IIcIcIIcIc-00a.a.Oo=_7O.5Fig.5-1ThefracturecriterionforModeIIcrackcanbederivedfromthemaximumstresscriterionthatKKIIIIcItisconvenientfortheengineeringapplication.However,thereisnodifferencebetweentheplanestressandplanestrain.5-2MaximumnormalstraincriterionNearthec

8、racktip,thecircumferentialnormalstrainis&=丄(G-VG)00E001rr1TOC o 1-5 h z11000K(1+cos0)v(3一cos0)cos一K3sin0cos+v(3cos01)sini12ii2121,forplanestrain.1vK*isdefinedas00EE=E,v=v,forplanestress;E=,v1111V21ThecircumferentialnormalstrainintensityfactorK*=limf2兀rs_12E1nnnK(1+cos0)v(3-cosn)cos-K3sin0cos+v(3cos0

9、-1)sinI12II212Then,K*S_00002兀rAssumptions:CrackinitiationdirectionisthedirectionofthemaximumK*;00WhenK*reachesitscriticalvalueK*,breakoccurs.K*isamaterialTOC o 1-5 h z0000c00cconstant.Thecrackingangle0satisfies0dK*门d2K*门00_0,00060602K_0,0_0.ItII0ThecriticalvalueK*canbedeterminedbyK.ForModeI,00cIccan

10、beobtainedthat1-vK*_1K00cEIc112(1-v)1ThemaximumnormalstraincriterionisK(1+cos0)-v(3-cos0)cos002-K3sin0cosII000+v(3cos0-1)sin2100200ThecriticalconditionisS=S(0)=Smin0cDeterminationofS:cFormodeI,itcanbederivedthatTheminimumstrainenergydensityfactorcriterioncanbeexpressedasSS,i.e.,上竺aK2+2aKK+aK2+aK2K2.

11、III22II33IIIIcc1-2v11I12ModeIIcrack:K=K=0,IIII0=arccos(匕兰),K03iic=：3(1-2v)Ky2(1-v)-v2IcTakev二1.Thereis30=-83o37,K=0.9K0IIcIcRecallthatforthemaximumnormalstresscriterion,thereis0=-70.5o,K=0.87K0IIcIcTworesultshavelittledifference.5-4ModifiedmaximumnormalstresscriterionSometimethemaximumnormalstresscr

12、iterionisnotsogood.Amodifiedmaximumnormalstresscriterionhasbeenproposed.IthasbeenknownthatinviewofSw=rastrainenergydensityfactorSisdefined.ForthemixedmodeofmodeIandII,ScanFractureandDamageMechanicsChapterFiveFracturecriterionformixedmodecrack7-bewrittenasS二aK2+2aKK+aK211I12III22IILetw=C=constant.SS1

13、r=aK2+2aKK+aK2二F(6)wCC11I12III22IIFordifferentvaluesofC,wecanobtainagroupofcurvescalledasisolinesofstrainenergydensity.ThecircumferentialnormalstressisG66116离严2Ki(1+cos6)-3Kiisin6Let16f(6)二2cos2(1+cos6),36fi(6)一-cos2sin6睬土Kifi(6)+Kifi(6)Let士Kifi+%(6).Thisgives66Ontheisolinesofthestrainenergydensity,

14、thecircumferentialnormalstressis%Cf(6)ThecircumferentialnormalstressintensityfactorKisidenticalwith5-1.66K二limf2rg二Kf(6)+Kf(6)66a66IIIIIIrtOAssumptions:(1)CrackinitiationdirectionisthedirectionofthemaximumKontheisolineofthe66strainenergydensity.Thecrackinitiationangle6canbedeterminedfromO亠0062Fractu

15、reandDamageMechanicsChapterFiveFracturecriterionformixedmodecrack (2)WhenKreachesitscriticalvalueK,breakoccurs.eeeecK=Kf(e)+Kf(e)=KeemaxII0IIII0eecItcanbederivedfromModeIproblemthatK二KTOC o 1-5 h zeecIcThefracturecriterionisKf(e)+Kf(e)KII0IIII0Ic5-5EnergyreleaseratetheoryNearthecracktip,thestressesi

16、nthepolarcoordinatesare11e=cosK(1+cose)-3KsineeeGre、：2兀厂22iii11ecosKsine+K(3cose-1)兀r2211Let1K=cosK(1+cose)-3Ksine22iii1K=cosKsine+K(3cos-1)22iiiThereresultsKGeeGrKEnergyreleaserateGalongtheangle:eGdenotestheenergyreleaseratealongthedirection=0.NowweneedtoknowtheenergyreleaserateGalongthedirectione.

17、eItisknownthatK=limf2兀rg11rT0yXe=0RecallthedefinitionsofKandK.ItisknownthatieiieK=lim2兀rq,K=lim2兀rqIe-oeeIie-oreya*-xvirtualcrack/Fig.5-2Comparingtwocases,weknowthatKandKarethestressintensityfactorsofIeIIethevirtualcrack.Thestressfieldsfortwocracksarecompletelysame.Theconclusionisthattheenergyreleas

18、erateGalongangleefortherealcrackisequaltotheeenergyreleaserateGalongitsowndirectionforthevirtualcrack.Hence,wehaveK+1G=(K2+K2)e8pieiieAssumptions:(1)CrackinitiationdirectionisthedirectionofthemaximumG.ThecrackinitiationeangleeocanbedeterminedfromdGe=0,ae叫0ae2(2)WhenGreachesitscriticalvalueG,thebreak

19、occurs.eecInasameway,itisobtainedthatG=K+1K2ec8pIcThecrackingangleesatisfiestheequationoK2sin0(1+cos0)2KK(sin26cos26cos0)+K2sin0(1-3cos0)二0I00III000II00Thefracturecriterionis1(1+cos0)K2(1+cos0)4KKsin0+K2(53cos0)K240I0III0II0Ic5-6FatiguecrackpropagationproblemFatigueprocess:Fatiguecrackinitiationperiod:empiricalformula(Minerslinerdamageaccumulationtheory)ordamagemechanics;Fatiguecrackpropagationperiod:fracturemechanics.o,maximumstress;maxo,minimumstress;mino=(o+o),meanstress;m2maxminmtn-,cyclicstressratio.Ao=oo,stressamplitude;maxminmaxInafatig