复合型裂纹的断裂准则.doc

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FractureandDamageMechanicsChapterFiveFracturecriterionformixedmodecrack

ChapterFiveFracturecriterionformixedmodecrack

Inthematerialmechanics,forthemultiaxialstressstate,fourstrengththeorieshavebeendeveloped.Inthefracturemechanicsforthemixedmodecrackproblem,weneedtodevelopthefracturetheoryaccordingly.Manyfracturetheorieshavebeendeveloped.Twokeyquestionsmustbeanswered.

（1）Whatdirectiondoesacrackpropagatealong?

（2）Whatisthecriticalcase?

Inwhatfollows,fivetheorieswillbeintroduced.

§5-1Maximumnormalstresscriterion

MaximumstresscriterioncanbeappliedtothemixedmodecrackofmodeIandmodeII.

Theasymptoticstresssolutionis

Byapplicationofthecoordinatetransformationformulas,wecanobtaintheexpressionsofthreestresscomponentsinthepolarcoordinates（r,q）.Thecircumferentialnormalstressis

Thecircumferentialnormalstressintensityfactorisdefinedas

Hence,canbewrittenas

Assumptions:

（1）Crackinitiationdirectionisthedirectionofthemaximum;

（2）Whenreachesitscriticalvalue,breakoccurs.isamaterialconstant.

Thecrackinitiationanglecanbedeterminedfrom

Theresultis

Thecriticalconditionis

Determinationof:

FormodeIcrack,,,,thecriticalconditionreducesto

Notethatisamaterialconstant.When,and,therestillprevails.Themaximumstresscriterionisexpressedas

ApplicationtomodeIIcrack:

ForamodeIIcrack,and.Thecrackinitiationanglecanbesolved,.From

onecanobtainthat

,

ThefracturecriterionforModeIIcrackcanbederivedfromthemaximumstresscriterionthat

Itisconvenientfortheengineeringapplication.However,thereisnodifferencebetweentheplanestressandplanestrain.

§5-2Maximumnormalstraincriterion

Nearthecracktip,thecircumferentialnormalstrainis

,forplanestress;,,forplanestrain.

Thecircumferentialnormalstrainintensityfactorisdefinedas

Then,

Assumptions:

（1）Crackinitiationdirectionisthedirectionofthemaximum;

（2）Whenreachesitscriticalvalue,breakoccurs.isamaterialconstant.

Thecrackinganglesatisfies

Thecriticalvaluecanbedeterminedby.ForModeI,,.Itcanbeobtainedthat

Themaximumnormalstraincriterionis

Nowtheplanestressandplanestraincanbedistinguished.

§5-3Strainenergydensityfactortheory

StrainenergydensityfactortheorywasproposedbyProf.G.C.Sihthatcanbeappliedtothethreedimensionalproblem.

When,,,,theasymptoticstresssolutionis

Thestrainenergydensitywis

Thestrainenergydensitywcanbeexpressedintheformof

where

strainenergydensityfactor

Assumptions:

itisphysics,notmathematics.

（1）CrackinitiationdirectionisthedirectionoftheminimumS;

（2）Whenreachesitscriticalvalue,breakoccurs.isamaterialconstant.

Thecrackinganglecanbesolvedfrom

Thecriticalconditionis

DeterminationofSc:

FormodeI,itcanbederivedthat

Theminimumstrainenergydensityfactorcriterioncanbeexpressedas

S£Sc,i.e.,.

ModeIIcrack:

,

Take.Thereis

Recallthatforthemaximumnormalstresscriterion,thereis

Tworesultshavelittledifference.

§5-4Modifiedmaximumnormalstresscriterion

Sometimethemaximumnormalstresscriterionisnotsogood.Amodifiedmaximumnormalstresscriterionhasbeenproposed.

Ithasbeenknownthatinviewof

astrainenergydensityfactorSisdefined.ForthemixedmodeofmodeIandII,Scanbewrittenas

Let.

FordifferentvaluesofC,wecanobtainagroupofcurvescalledasisolinesofstrainenergydensity.

Thecircumferentialnormalstressis

Let,.

Let.Thisgives

Ontheisolinesofthestrainenergydensity,,thecircumferentialnormalstressis

Thecircumferentialnormalstressintensityfactorisidenticalwith§5-1.

Assumptions:

（1）Crackinitiationdirectionisthedirectionofthemaximumontheisolineofthestrainenergydensity.Thecrackinitiationanglecanbedeterminedfrom

（2）Whenreachesitscriticalvalue,breakoccurs.

ItcanbederivedfromModeIproblemthat

Thefracturecriterionis

§5-5Energyreleaseratetheory

Nearthecracktip,thestressesinthepolarcoordinatesare

Let

Thereresults

Energyreleaseratealongtheangleq:

Gdenotestheenergyreleaseratealongthedirectionq=0.Nowweneedtoknowtheenergyreleaseratealongthedirectionq.

Itisknownthat

,

Recallthedefinitionsofand.Itisknownthat

Comparingtwocases,weknowthatandarethestressintensityfactorsofthevirtualcrack.Thestressfieldsfortwocracksarecompletelysame.TheconclusionisthattheenergyreleaseratealonganglefortherealcrackisequaltotheenergyreleaserateGalongitsowndirectionforthevirtualcrack.Hence,wehave

Assumptions:

（1）Crackinitiationdirectionisthedirectionofthemaximum.Thecrackinitiationanglecanbedeterminedfrom

（2）Whenreachesitscriticalvalue,thebreakoccurs.

Inasameway,itisobtainedthat

Thecrackinganglesatisfiestheequation

Thefracturecriterionis

§5-6Fatiguecrackpropagationproblem

Fatigueprocess:

（1）Fatiguecrackinitiationperiod:

empiricalformula（Miner’slinerdamageaccumulationtheory）ordamagemechanics;

（2）Fatiguecrackpropagationperiod:

fracturemechanics.

maximumstress;,minimumstress;,meanstress;

stressamplitude;,cyclicstressratio.

Inafatigueprocess,thestressintensityfactoralsovarieswithtimet.

ThefatiguecrackpropagationratedependsontheamplitudeofSIF.

Experimentalresult:

RegionI:

smallcrack,microscopiceffectisimportant.

RegionII:

crackstablepropagation.

RegionIII:

crackinstablepropagationtofailure.

Parisequation:

1960s,LehighUniversity,USA

FortheregionII,therelationcanbegivenby

straightline

ParametersCandncanbedeterminedbytheexperimentaldata,whichdependonthestressratioR,materialproperty,temperatureandsoon.

ThefatiguecrackgrowthlifecanbecalculatedbyusingtheParisequation.

TherearemanyimprovementsforParisequation.

第五章完

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