Cumulative fatigue damage and life

eVol.20,No.I,pp.9-34,1998©htsreservedPrintedinGreatBritain0142-1123/98/$19.00+.00PU:80142-1123(97)00081-9Cumulativefatiguedamageandlifepredictiontheories:**DepartmentofMechanical,IndustrialandManufacturingEngineering,TheUniversityofToledo,Toledo,OH43606,USAtAdvancedDesign,SpicerDriveshaftDivision,DANACorporation,Holland,OH43528,USA(Received21October1996;revised22March1997;accepted15June1997)tivefatiguedamageanalysisplaysakeyroleinlifepheintroductionofdamageaccumulationconceptbyPalmgrenabout70yearsagoand'lineardamagerule'byMinerabout50yearsago,thetreatmeult,oughearlytheoriesoncumulativefatiguedamagehavebeenreviewedbyseveralresearchers,nocomprehensivereporthasappeticleprovidesacomprehensivereviewofcumulativefatiguedamagetheoriesformetalsandtheiralloys,emphaheoriesaregroupedintosixcategories:lineardamagerules;nonlineardamagecurveandtwo-stagelinearizationapproaches;lifecurvemodificationmethods;approachesbasedoncrackgrowthconcepts;continuumdamagemechanicsmodels;andenergy-basedtheories.©1998ElsevierScienceLtd.(Keywords:cumulativefatiguedamage;fatiguedamageaccumulation;cumulativedamagerules;loadinterac-tioneffects;fatiguelifepredictions)INTRODUCTIONFatiguedamageincr-lativefatiguedamageisanold,butnotyetresolvedproblem,Morethanseventyyearsago,Palmgren1sug-gestedtheconceptwhichisnowknownasthe'linearrule'.In1945,Miner2firstexpressedthisconceptinamathematicalformas:D="i.(n/Nr),whereDdenotesthedamage,andniandNfiaretheappliedcyclesandthetotalcyclestofailureunderithconstant-amplitudeloadinglevel,hen,thetreatmeult,manyrelatedresearchpapersarepublishtheprogressonthesubjectk3inacomprehensiveearlyreviewdiscussedseveralissuesrelatingtocumulativedamageinfatiguesuchasdamagecumulationprocess,damagevscycleratiocurve,ndMorrow4presenteda*ofcontemporaryapproachesforfatiguedamageanalysisemployingsmoothspecimenmaterialdataforpredictinlytheoriesoncumulativefatiguedamagehavealsobeenreviewedbyKaechele5,Manson6,Leve7,O'Neill8,Schive9,LaflenandCooklOandGolosandEllyin".However,aspointedoutbyMansonandHalford12in1986,nocomprehensivereporthasappearedrecentlytoreviewtheconsiderableeffortmadesinceSchive'tion,viewpaperprovidesacomprehensiveovmodelsdevelopedbefore1970sweremainlyphenomenological,whiloredetailedoughsomeofthecontinuumdamagemechanics(COM)modelsarealsomentioned,rtantapplicationofthesemodelshasbeenin9

ldalsobenotedthatthisreviewpaperdealswitrreviewpaperprovidesacomprehensiveoverFORE1970sThephenomenologically-baseddamagetheoriesdevelopedbefore1970swereoriginatedfromthreeearlyconcepts(discussedbelow)andattemptedtoimprovethelineardamagerule(LDR).Thesetheoriescanbecategorizedintofivegroups:thedamagecurveapproach(DCA);endurancelimit-basedapproach;S-Ncurvemodificationapproach;two-stagedamageapproach;arlyconceptalmgren1who14firstreportedthesign1938,Komm1937,Langer16firstproposedtoseparatethefatiguhreeearlyconcepts(linearsummation,changeinendurancelimitandtwo-stagedamageprocess)laidamagerulesMiner2firstrepresentedthePalmgrenlineardamageconceptinmathematicalformastheLDRpresentedby:D='ir;='in/Nf;(l)IntheLDR,themeasureofdamageissimplythecycleratiowithbasicassumptionsofconstantworkabsorptionpercycle,rgyaccumulation,therefore,eisdeemedtooccurwhen'iri=1,whereriisthecycleratiocorrespondingtotheithloadlevel,orri=(nINf);.Damagevscycleratioplot(thedamagecurveorissimplyadiagonalstraightline,independentofload-D-rcurveasitisusuallycalled)-Ndiagram,theresiduallifecurvescorrespondingtodifferentlindeficiencieswithLDRareitsload-levelindependence,lo1949,Machlin17proposedametallurgicallybasedcumulativedamagetheory,1950s,Coffinandco-workers,19expressedtheLDRintermsofplasticstrainrange,erstudy,TopperandBwontheapplicationsoftheLDRt-ever,duetotheinherentdeficienciesoftheLDR,nomatterwhichversionisused,mentalevidenceundercompletelyreversedloadingconditionofteniridi-catesthat'ir;>1foralow-to-high(L-H)loadingsequence,and'ir;<1forahigh-to-low(H-L)-StarkeytheoryToremedythedeficienciesassociatedwiththeLDR,RichartandNewmark22introducedtheconceptofdam-agecurve(orD-rdiagram)in1948andspecuisconceptandtheresultsofloadsequenceexperiments,MarcoandStarkey23proposedthefirstnonlinearload-dependentdamagetheoryin1954,representedbyapowerrelationship,Dwhere=,figure,adiagonalstraightlinerepresentstheMinerrule,whichisaspecialcaseoftheaboveequationwithXi=stratedbyFigure1,lifecalculationsbasedonMarco-Starkeytheorywouldresultin'ir;>1forL-Hloadsequence,andin'iri

Cumulativefatiguedtaneousenofthesemodels,however,takeamaterial,heoriesaccountingforloadi-DolonBothmodepoand31,32theoriesarebasedonthemodificationoftheS-Ndiagram,aclockwiserotationoftheoriginalS-NlineDolonlevelintheloadhistorymodel,apointcorrespondingIntotheCorten-point,whileintheFreudenthal-Hellerapproach,thisisselectedasthereferencethehighestreferenceafatigueislifechosenatthestresslevelcorrespondingof103_,SpitzggestedtoobtaintheslopeofthemodifiedS-Nfewlitatingbend-ofSAE4130steel,Mansonetal.34,35,alggestedthatlineapointcorrespondingontheoriginaltoafatiguelifebetween102and10pproachalsoprovidesforpredictingthereductioninendurancetoaccountnotonlyfortheloadinteractioneffect,buttoprecyclingdamage,efigures,theMinerruleisreseenthattheLDRandtheS-NlinerotatuntTwo-StagelineardamagetheoriestheLDRshortcomings,whilestillretaiingLanger'sconceptl6,Grover36con-sideredcycleratiosfortwoseparatestagesinthefatiguedamageprocessofconstantamplitudestressing:duedamageduewheretotocrackpropagation,Ncrackinitiation,N1==(l-a)Naf,efractionfactorfortheinitiationIneitherstage,3?revertedlineartoGrover'sworkandmodelanditsapplicationswerefurtherexaminedanddamagerule(DLDR)riginalversionofDLDR,theNtwostageswereseparatedbyequationsacoef-of:ficient1=Nf-PNf,6andNofthesecondstagefatiguell=PNf·6,icalrepresentatly,Bilir39two-levelcyclingonnotchedcarriedoutanexperimentalinves"00.-.-....::::lc..CT,8~0CJ:ICJ:I.-"0.-0c..

,,ANo2"c:wZNo~,,,,,,"1-+--"2-TOFAILURE~a:u>-u~,,~,,STRESS0I-I-tH+lI+H+f+++f~""'-',-TIMEC)zzW::!E~a:16N2,,,",,,,,,~",,-DOUBLELINEARDAMAGERULE",,~(N,:I'NoI6"2)N',2,,-LINEARDAMAGE'(RULE,,".Nr,2,,''C+-APPLIEDCYCLERATIO,"1/N',l·11.0Figure3IllustrationofthedoublelineardamageruleforH-Ltwo-levelloadi41,ntitativedevelopmentofthetheoryisbasedonconceptsderivedfromdislocationtationfor-mulatedisinaformsimilartothatexpressedbylinearelasticfracturemechanics(LEFM):da/dN=Cf(a)a,whereaisthecracklength,Cisaconstantandj(a)rdamagetheoryusihematicalexpressionisrepresentedby:da/dN=am+'j(ai),CURVEAPPROACH,REFINEDDOUBLELINEARDAMAGERULEANDDOUBLEDAMAGECURVEAPPROACHTheDCA,refinedDLDR,anddoubledamagecurveapproach(DDCA)weredevelopedbyManson,Half-ord,andtheirassociates1Z,curveapproachafunctionofNintheformq=BNf3(Bandf3aretwomaterialconstants).Damageisthendefinedastheratioofinstantaneoustofinalcracklength,D=cases,ao=0,andthedamagefunctionoftheDCAsimplybecomes:D=r4(3)Obviously,haseriesoftwo-leveltests,thecon-stantf3canbedeterminedfromtheslopeoftheregressionlineoftheexperimentaldata:thatis,log[log(l-rz)llogrdvs10g(N/Nz).Avalueoff3=rmore,ifareferencelevel,Nnisselected,theotherconstant,B,canthenbeexpressedasN;ore,theexponentqinEquation(3)canbewrittenasq=(N/Nr)f3,ddoublelineardamageruleTheoriginaefinedDLDR,thekneepointsinadamagevscycle-ratio(D-r)plot,whichdividethedamageprocessintotwophases,aredeterminedby:DkneeThisapproachwcog-nizedthatthemajormanifestationofdamageiscrackgrowthwhichinvolvesmanycomplicatedprocessessuchasdislocationagglomeration,subcellformation,multiplemicro-crackformationandtheinthisphenomenologicalrec-ognition,MansonandHalford44empiricallyformulatedthe'effectivecrackgrowth'modelthataccountsfortheeffectsoftheseprocesses,delisrepresentedby:(2)=A(N/NY"andrknee=1-(l-A)(N/N)OI(4)retwiricalvaluesofthesetwoconstantswerefoundtobeA=0.35anda=0.25forhighstrengthsteels1Z,l.45,-Damagecurveapproachwhereao,aandarareinitial(r=0),instantaneous,andfinal(r=1)cracklengths,respectively;andqisThisapproachisdevelopedbyaddingalineartermtotheDCAequationwithsomemathematicalmanipu-lationandcanbepresentedas:D=[(pr)k+(l-pk)~q]llk(5)

Cumulativefatiguedamageandlifepredictiontheories131.0,--------------------.B--DOUBL.£DAMAGECURVEAPPROACH---DOUBL.£W£ARDAMAGERUL.£---DAMAGEISNGL.£TERMCURVEEOUATlON)APPROACHonlN,CYCL.£RATIO1.0ComparisonoftheDDCAwithDLDRandDCAl2kisamathematicalexponenttogiveaclosefitpisaconstantmeasuredfromtheslopeofthefirstdamageaccumu-lationlineinDLDR:p---DkneeA(N/NY'-rknee-1-(1-A)(N/N)C<(6)AscanbeseenfromFigure4,theDDCArepresentsacontinuousdamagecurvewhichconformstotheDLDRlineintheearlyportionofthePhaseIregime,uatetheeffectivenessofthedevelopedDDCA,MansonandHalfordandco-workers46-48conductedcumurisonoftheexperimentalresultswiththeDDCAFigure5showsLO::.-u...0z.8",0316STAINLESSSillLNl''"~•200COO600>=IX«u...........6u-'>u..........OOCA"0~.4'"'.........u......pVI"""--,DCA.2--.4--""0.8'",.61.0"1/YCL.£FRACnONILCFIFigure5ImprovedrepresentationofH-Lloadinteractiontestsof316stainlesssteelusingDDCAascomparedwithLDRandDCAl2animprovedrepresentationofdatabyDDCAhasalsobeenappliedtotwoothermateeallload-leveldependent,butdonotmemodificationinprocedure,themeansTHEORYBui-Quocandcolleaguespresentedtheirworkdealingwithcumulativefatiguedamageoryforstress-controlledfatiguewasfirstdevelopedfromthehybridizationoffourpriordamage41modelsbyHenry26,Gatts27,Shanley4°eorieswerethencombinedintoa'unifiedtheory'theinteractioneffectundercyclicloadinginvolvingseveralstresslevelss4,,amagemodelshadalreadybeenextendedtoincludehightemperaturefatigueS9,refurthermodifiedtotakeintoaccountnotonlytheeffectsofmeanstress/strain72,butalsothe-ControlledversionThemainhypothesesinthedevelopmentofthisdamagetheoryisthatcracksgrowinginamaterialsubjectedtocyclicloadvenience,alltheparametersinthismodelwereexpressedbydimensionlessratioswithrespecttotheoriginalendurancelimit,aeo'Theseincludetheinstan-taneousendurancelimitratio,'Ye=ajaeo'theappliedstressratio,=l'=a/aeo,andthecriticalendurancelimitratio,'YecaeJaequationforeo'rentialstrengthevaluationratewasobtainedbycombiningthreefundamentaldamagetheories:y'spowerruleofcrackgrowthrateintermsofthemaximumcyclicstress;i,srelationbetweencrackgrowthandcyclicstressrange;'damagefunctiondescribedbythesecondpowergrationofthisdifferentialequationwithsomemathematicalmanipulationsresultsinthedamagefunc-tionforthestress-controlledconditionas:D=1-'VIer1-'Yec(7a)where'Yu=aJaeo,racteristicofthisequationisshowninFigure6asFigure4wheretothedoublelineardamageline,andpredictionsindicategoodagreements.

1.0,.....--------------_•1./Henrytheory0."'0.2MinertheoryYy•2.0oFigure6o0.20."'0.•0.8toCYCLERATIOCharacteristicofthehybriddamagefunction209andcomparisonwithtgey,-ControlledversionTheconversionofthestress-controlledtheorytostrain-controlledversionwasmadesimplybyreplacingthestressparametersYxinEquation(7)a,withthecorrespondingstrainparameters,Ax,whicharedefinedas:Ax=1+In(E)Eeo).Thesymbol'x'ore,thestrain-controlledver-sionofhybridtheorycanbemathematicallypresentedas:D_In(EjEeo)_theloadparameterforthefirstlevel,AI'Forthesecondloadlevel,however,theloadparameter,Az,isreplacedbyanimaginarystrain,A2',whichis,therefore,called'fictitiousload'.Todeterminethefictitiousvalue,A2',aparameterYusedinregressionanalysisisproposed:ILlAI)B,-If3Y=1+BI(-~LlA*2(8)whereBI,B2andB3areconstantstobedeterminedexperimentally;LlAisthedifferencebetweenstrainlevels:LlA=A2-AI;YandLlA*aresequence-relatedparametersdefinedasfollowsfortheL-Hincreasingstep:Y=rA-(A/Ar)mr+(1-r)A-I-In(EeJEeo)-(7b)Ar*-A2andLlA*=A*-A'Ar*-ArI2(9a)whereA=1+In(E/Eeo)andAr=1+In(ErfEeo),inwhichEeoandEe,Eecareinstantaneous,initialandcriticalstrainendurancelimit,Eistheappliedmaximumcyclicstrain,-rplotofEquation(7)bissimilartoFigure6describedbyEquation(7)uations(7)aand(7)bgiveanonlinear,soaccountfortheeffectoodelsimprovelifepredictionscomparedtotheLDR,butdeviationsfromexperimentalresultsarestillfound74,mainlydedversiontoaccountforloadinteractioneffectsToaccountforloadinteractioneffects,hefictitiousloadapproach51.55,57andanotheristhecycleratiomodificationapproach55,approach,thereisnomodificationofwhereAr*Y=A2A2'-=Af7;-andfortheH-Ldecreasingstep:(9b)IandLlA*=A-IIInthecycleratiomodificationapproach,thedamagefunctioninEquation(7)a,(7)bismodifiedbyintroduc-inganexponent,v,tothecycleratio,rTherefore,-stepcycling,visrelatedtoanotherparameter,a,bytheempiricalequation:V•(10)whereLlA=A2-AI'eexperimen-tallydeterminedfromtwo-stepfatiguetests,orempiri-callyestimatedbytakinga=proachcanbeextendedtomulti-steploadingbydefiningtheinteractionpara-meterVk'(k=2,3'00.,i)betweenanytwosuccessivestrainlevelsk-IandkinthesameformasEquation(10),butwithLlA=Ak-Ak_I'Undertheassumptionthatamulti-stepfatigueprocessaccumulatesinteraction

Cumulativefatiguedamageandlifepredictiontheories15effectaswellasdamage,theinteractionparameterappropriatefortheithloadlevelbecomes:Vi=1XV2'XV3'X.,.XVi_I'XV;'(k=2,3,...,i)(11)Iterativecalculationsfromiasimilarprocedurepresentedfor=2toi=i(i2::2)followingtwo-stepcyclingwouldprESUSINGTHECRACKGROWTHCONCEPTThecrackgrowthconceptsdevelopedin1950sand1960shaveenjoyedwideacceptancesincecracksaredirectlyrelatedtodamage,andsincemodemtech-nologyhasprovidedsophisticatedtoolsandtelmacrofatiguecrackgrowthmodelsbasedonLEFMconceptsweredevelopedintheearly1970stoaccountforloadinteractioneffectsinthecrackpropagationphase(stageII)odelsattempttoexplainmacrocrackgrowthretardatiheearly1970s,severalnewfatiguedsomearestillphenomenologi-cal,mostofthesenewermodelsatiguecrackgrowthmodelsApdelassumesthecrackgrowthratetoberelatedtotheinteractionofcrack-tidelmodifiestheconstantamplitudegrowthrateequation,da/dN=A(~Kt,byanempiricalretardationfactor,C:da/dN=CJA(~Kt]where:Ci=(rp/rmaxY'(12)Hererpiistheplasticzonesizeassociatedwiththeithloadingcycle,rmaxisthedistancefromthecurrentcracktiptothelargestpriorelastic-plasticzonecreatedbytheoverload,andpisanempiricaadelusesaneffectivestressintensityfactoratthecracktip,(~Keff);'factor,toreducetheappliedcracktipstressintensity~K;,duetotheuctionintheapplied~KisafunctionoftheinstantaneousplasticzonesizeatttheWheelermodelhowever,nhisexperimentalobservations,Elber77.78suggestedthatafatiguecrackcancloseataremotelyappliedtensilestrcktipstressintensityfactordrivingthecrackisthenaneffectivestressintensityfactorbasedontheeffectivestressrange,~Seff=Smax-Sop,rackclosuremodelshavealsobeendevelopedwhichincludethosebyNewman79,8o,Dilletaf.8I,82,ficultyinusingcrackclosuremodelsisindeterminingtheopeningstress,-man'smodeF9predictsthecrackopeningstressbyaniterativesolutionprocedureforacyctiontotheplasticityinducedcrackclosure,otherformsoffatiguecrackclosurecanarisefromcorrosion(oxide-inducedclosure),fracturesurfaceroughness(roughness-inducedclosure),andothermicrostructticalmacrocrackgrowthmodelshavealsobeenproposed89,90inwhichcrackgrowthrateisrelatedtoaneffectivestressintensityfactorrangeectivestressintensityfactorrangedescribedintermsoftheroot-mean-squarevalueofstressintensityfactorrange,~KrmS'proposedbyBar-som90isgivenby:~Krms=J(*1~K7)/n(13)where~KiisthestressodelsareempiricalanddonexponentiallawFortheaccumulationoffatiguedamageincrackinitiationandstageIgrowth,MillerandZachariah91introducedanexpmodeldamageisnormalizedas:D=a/af,whereaandafareinstantaneousandfinalcracklengths,,nthegrowthmechanismofverysmallcracks,crackpropagationbehaviorinstageIwasthenmathematicallydescribedinamannersimilartothatexpressedbyLEFMforstageIIgrowthas:dNda=

1.0..---_Jnitiation"-..J:.,........0N"Stage'..,~,PropagationIc:Cl.-...>"II"-c:00..'-..lit0CJ:CIJ:0~..Uoo1.0fractionoflifespentatthelowstrainlevel,rIFigure7SchematicrepresentationofthecumulativedamagecurvebasedonthemodifiedIbrahim-MillermodelinaL-Htwo-levelsteptest93D=!!.-=(~)(I-r)/(I-rt)afaf(15)AssummarizedinFigure8fordamagelinesatvariousstrainrangelevels,theaboveequationrepresentsabundleoflinesegmentsradiatedfrompoint(1.0,1.0)andterminatedatthephaseboundarydefinedby(NlNf,alaf)'However,-------.--........--....------.--..,10-IEE010-2..J:a.0"~0u10·3u'-°010-400.20.40.60.81.0AsummaryoftheaccumulationoffatiguedamageatIbrahimandMiller93accumulationintheinitiationphaseisnotyetestab-lished,ultiesofmodelingthisdamagephasecanhardlybeovercome,racktheoryMillerandco-workers94-101investigatedthebehaviorofveryshortcracksandproposedthatcrackinitiationoccursimmediatelyinmetalfatigue,andthatthefatiguelifetimeiscomposedentirelyofcrackpropa-gationfromaninitialdefectsize,aQ•Theearlytwophaseswererenamedasmicrostructurallyshortcrack(MSC)growthandphysicallysmallcrack(PSC)growth,CsandPSCsareelasto-plasticfracturemechanics(EPFM)wthbehaviorofMSCcracksis,however,significanseboundarybetweenMSCsandPSCs,andtamodificationofKitagawa-Takahashi!nexperimentalobservationanddataanalysis,crackgrowthmodelsforMSCsandPSCswereestab-lishedandmathematicallydescribedas96-98,IOO:dNda=A(!ly)"(d-a)forMSCs:aQ::5a::5at(16a)dNda=B(!ly)f3a-CforPSCs:at::5a::5af(16b)whereA,B,(Xandf3areconstantsobtainedbyfittingoftheexperimentaldata;!lyistheshearstrainrange,atisthecracklengthcorrespondingtophasetransitionfromMSCgrowthtoPSCpropagation,drepresentsthebarriersize,on(16)hematicalformsofEquation(16)a,(16)bseemcr,thephysicsandval-LairdmodelMaandLaird103foundthatintheshortcrackregime,similartotheMSCregiondefinedbyMiller,crackpopulation,P,islinearlyrelatedtotheappliedstrainamplitudeandusedlife,nthisconcept,MaandLairdproposedanewapproachtosummingcumulativedamageandpredictingfatiguelife,whichisformu-latedas:(17)where(!ly/2)limitisthefatiguelimitstrain,K(Cisaconstantinthestrain-lifeequation),P=C/Peritcriticalcrackpopulationatwhichfailureiseritisthedeemed,and(ntheexperimentalfindingsinRef.104,MaandLairddefined(XiastheratioofFigure8variousloadlevelsbasedonthedoubleexponentialruleproposedby

V!VlVltitINON1-PR~allySmallCracksCRACKLENGTHFigure9AmodifiedKitagawa-Takahashil1u-QdiagramshowingboundariesbetweenMSCsandPSCs,andbetweenEPFMcracksandLEFMcracks""thecurrentlyappliedstrainamplitudetothemaximumstrainamplitudeinthepre-loadinghistoryincludingthecurrentcycle:aiD=2Dcrforinitiation:D:::;r:::;1/2andDc[em(2rD=Dc+-m~)-I(20a)=(~'Yp)/(~'Yp)2i2(-loadinghistory:::;1)I]forpropagation:1/2:sr:S1(18)ThemodelrepresentedbyEquation(17)r,itshouldbepointedoutthatthismodelpredictsalongerlifeforH-Lstrainsequencewherea<1,incontradictionwiththecommonexperimentalobservationsincompletelyreversedloading.(20b)whereDc=ajar,andm=CENTTHEORIESBASEDONLIFECURVEMODIFICAnONSThelifecurvemodificationapproachesintroducedbeforethe1970spossessattractivefeaturesofrelativesimplicityinform,1970sseveraasek-PolakapproachBasedontheirexperimentalobservationsandinterpretation,rackinitiationregime,aconstantcrackgrowthratewasproposed,describedby:dN=Viforao:::;a:::;acda(19a)aekpropageti0n3tageandinthecrackpropagationregime,thedependenceofda/dNonthecracklengthwasapproximatedbyalinearrelation:dNt---------f----------i£ccIIQda=Vi+k(a-ac)forac:::;a:::;ar(19b)....whereViisthecrackgrowthrateindependentofappliedcycles,kisacoefficient,andao,acandararetheinitial,critical,andfinalcracklengths,ticalcracklength,ac,,themagnitudesofVi'rexperiments,VasekandPolakfoundthevaluesofthesethreequantitiestoincreasewithincreasingtheloadinglevel,andacwasreachedapproximatelyathalf-life(r=nlNr=1/2)uently,integratingEquation(19)a,(19)b,damageevolutionfunctionscanbeexplicitlyexpressedas:aa0.51.0Cycleretio,rFigure10SchematicrepresentationofdamagefunctionsproposedbyVasekandPolak

danyan'skneepointapproachAkneepoint-basedapproachwasinttudy,asetofisodamagelineswereintroducedwhageisthendefinedapliesanassumptionthattheenduraaticalexpressionforanyisodamagelinecaneasilybeobtainedfromthispostulation,adingsequenceincludingi(i2::1)steps,amathematicalformfortheresidualcycleratioattheithlevelcanbefound:r;=1-k-I+[r;_z+...+(rz+rj'1)"2...]"i-2}"i-I(21)where(lk=10g(N/Ne)/log(N,jNHowever,itshouldbenotedthatthisapproachk+e)fork=1,2,...,sonisthesingulaondreasonisthenonlinearityt-RotemmodelHashinandRotem107presentedadiscussionoftheS-Nlineconvergenceandrotationappirstmodel,alldamagelinespassthroughtheintersectionoftheoriginalS-NlinewiththeS-axis(calledstaticultimate).Thisapproachavoidsdetermi-nationofthecondmodel,-tially,thisisSubramanyan'sconcept,ntheproposedapproaches,HashinandRotemlO7performedanalyticalcalculationsontwo-stage,three-stage,mentswerecarriedoutbyHashinandLaird108withtwo-stagecyclingandthedatawereusedtotesttheeffec-tivenessotedresultswerefoundtobeingoodagreementwithtestdata,-Amoz'sboundtheoryFatiguedamageisnthisargument,Ben-AmozlO9introducedaconceptofbandsoeorystatesthataresiduallifelineobtainedfromthfirstapproximation,Ben-Amozrep-resentedthetwoboundsbyMinerLDR(aparalleltranslationofS-Nline)andSubramanyan'stheory(arotationaroundtheendurancelimit).arenarrowedbytheinclusionofadditionalinformtiatirimprovement110,ofthismodelwasmadebyconsideringallparameterstobefunctionsoftherandomvariablesN1,NzandNextremevaluesassociatede•Fortheseparametersinthebounds,ifwithanydesirednumberofstandarddeviationsareused,staenttowhichtheboundscanbracketnthemathematicalanalogybetweenthefatigueandcreepcumulativedamageproblems,theboundtheorywasmodifiedtopredictcreepresidualtimeilz'sapproachInagreementwithFreudenthal'sandHeller'sopinionthattheerrorsinlifepredictionsbasedonLDRareduenottoitslinearsummationbuttotheassumptionofdamage-rateindependenceofloadinglevels,Lei-pholzltZ-114resumedtheconceptofreplacingtheorig-inalS-Ncurvewithamodifiedcurve,S-N',lz'smodelisrepresentedas:N'2,=lri,(f3/N/)(22)whereN'2,isthetotalaccumulatedlife,andf3iandN/arethefrequencyofcycles(n/N'2,)andthemodifiedlifewithloadinglevelai'11describesthetypicalmannerinwhichthemodifiedS-Ncurveconvergestotheoriginalcurveatahighloadinglevel,-N'curveisdeterminedfrommulti-levelrepeatedblocktestsalongwithEquation(22).DetailsofthemethodforobtainingthemodifiedS-NcurvearereferredtoRefs112,113,erimentsshowthatthismodelcanprovideaccuratepredictionsoffatiguelivesundersvirgin$-Ncurvemodified$-Ncurves·IN·I,ONFigure11SchematicrepresentationofthemodifiedS-NcurveaccordingtotheLeipholzapproach113

Cumulatedictivetheoryisalsoexpandedtostochasticloadinghistories113,BASEDDAMAGETHEORIESSincethereportofconnectionbetweenhysteresisenergyandfatiguebehaviorbyInglisl16,lfailurecriteriabasedonstrainenergywereestablishedbyMorrowll?r,cumulativedamagethergy-baseddamageparametershavebeenproposedsuchasthosebyZuchowski119andBudianskyandO'eenrealisedthatanenergy-baseddamageparametercanunifythedamagecausedbydifferenttypesofloadingsuchasthermalcycling,creep,unctionwithGlinka'srulel2l,itispossibletoanalyzethedamag-baseddamagemodelscanalsoincludemeanstressandmultiaxialloadssincemultiax-ialfatigueparametersbasedonstrainenergyhavebeendevelopedI22,proposedbyEllyinandco-workersKujawskiandEllyin124developedapreltically,plasticstrainenergyabsorbedinacomplete,therefore,alsoreferredtoasthehysteresisenergyanddenotedbyawp•eenfoundl25thattherearetwotypesofmaterials,MasingtypeandnonMasingtype,singmaterial,themasnMasingmaterial,however,andco-workersll.126-129eemastercurveisconstructed,aterfoundthatsom~xample,Incorporatedtheeffectinth,forthelowstrainhigh-cyclefatigue,cases,thoughthemacroscopic(bulk)responseofthematerialiselasticorquasi-elastic,microscopic(local)plasticdeformationmaystillexistinthematerialduetothenonuniformityoflocalstraindistributionand/cometheseshortcomings,GolosandEllyin11,126,127modifiedtheplasticstrainenergy-basedmodelbyusingtotalstrainenergydensity,alstrainenergydensitycombinesbothplastic(awP)andelastic(awe)sticportionisthoculationofaweisobtainedfrom:aw=-2E1(aiT--+iT2)2m(23)us.~•I(i)(a)Masing-typedeformation1000100lUT•(MPa)800600l..•('lb)2.5~o1.52.02.5tot3.0(%l(b)non-MasingtypedeformationFigure12Materialsexhibitinghysteresisloopswith(a)Masing-typedeformation,and(b)nonMasingtypedeformation'26Regardlessofthetypeofenergymodel,ergymodelsareessentiallysimilartoSubramanyan'functionanalogousto~nlIfeS-NrelatIOn,~elationWhICh:-stratedinFigure13foratwo-levelloadtest,isodamagelinesintersecttheextensionoftheoriginalenergy-lifelineatthepoint(Ne*,awe*),ratherthanat(Ne,aWe)~reseveralmethodstodetermine(~e*,.awe*)is,therefore,calledthe'apparent'thecoordinatesNe*andaWe*.OnemethodisbasedonthepredictiveequationofchangeinendurancelimitsuchasBui-Quoc'rmethodl31isbasedontheuseoftherelationbetweenthethresholdstressintensityfactor,stresslimitinconjunctionwiththecyclicstress-strainaKlh,rmodifications,EllyinetaZy,126,127

-------------,--II:-,-It-~I-------'----J..---II1I'II~_..2-'~------i"...J---III1','"n2nIN•NelogNfFigure13Damagelinethrough'apparent'fatiguelimitdefinedbypoint(Ne*,~We*)124fixedthepoint(Ne*,dWe*)attheintersectionoftheoriginalenergy-lifecurveextensionwithacriticaldam-agecurvewhichdeiticalcurvecanbeexperimentallydetermined,econvergencepointisdetermined,damagelinescorre-rgy-baseddamagemodelsproposedbyEllyinandco-workerspossessfeaturessimilartoSubramanyan'r,theenergyapproacheshavenosingularityproblematendurancelimit,sincetheconvergencepointisselectedatthe'apparent'eoryLeis'33proposedanenergy-basednonlinearhistory-dependentdamagemodelwhichlinksthedamagepara-metertofatiguelifeinamannersimilartotheSmith-Watson-TopperparameterI34:iningconstantstrainamplitudetestdata,Niuetal.135,136foundthecyclicstrainhardeningcoef-ficienttochangeduringthecyclingprocess,ore,anewcyclicstress-strainrelationwasproposedas:dO"(dEp)n*-=K*-r/322(26)whereK*andn*arecyclicstrainhardeningcoefficientandexponentdeterminednearfailure(r=n/Nf=1),ressionfor13wasgivenas:(27)rementalrateofplasticstrainenergywasthenderivedas:n*r/3K*(dE)'dWI----=4-pdN1+n*2+n*D=40"'fE(2Nf?b1+40"/E/(2Nf)b1+c,(24)(28)where0"/andE/arethefatiguestrengthandductilitycoefficients,r,theexponentsb,andc,areanalogoustobutdifferentfromthefatiguestrengthexponent,b,andfatigueductilityexponent,model,b,andc,aretwovariablesrelatedtotheinstantaneousstrain-hardeningexponent,nl'through:-1c,=15andb,andtheenergyaccumulationisdefinedbyintroducingaparametercalledthefractionofplasticstrainenergy,=W/Wf=r'+/3,Finally,thefatiguedamagefunctionwasconstructedas:D=l/l(n'++13)]=r'/(n'+(29)Ol)('Ol)+n,-n,=15+n,(25)whichareassertedbyananalogytotheMorrow'perimentalobservations,Leisspeculatedthattheparameternlcanbecharacterizedasafunctionoftheaccumulationofplasticstrain,~elrepresentedbyEquation(24)iy,properlydefiningthefunctionnl=nl(IdEp)=(dmiE;4?b-y~,andn'elrepresentedbyEqu-ation(29)isanonlinear,untsfortheloadinteracmagetiontotheenergyapproachesreviewedabove,arly1970s,Bui-Quoc'37conductedanexperimental

Cumulativefatiguedamageandlifepredictionep-up(increasingloads)andstep-down(decreasingloads)berofstrainlevelsinmultiple-steptestsdidnothaveaninfluenceonthecyclicstrain-hardeningcoefficient,andthestrainratiohadlittleeffectonthecyclicstress-straincurvesandanegligibleeffectonthetotalplasticenergyatfracture,Wf'Basedontheseexperimentalobservations,amodelforcalcu-latingthevalueofWfaccumulatedduringafatiguedamageprocesswasproposed,givenby:W:2K'Mn'+ff=!.n~w.=n'+1!'rNl.-C(n'+I)(30)IIIfIwhere~Wiisthehysteresisenergyfortheithloadinglevel,andMandcarematerialconstantsintherelation:~E;Vf=r,sinceitisfoundl17,118thatthetotalplasticenergyatfailureisnotconstantformostmaterials,applicatieviewpointofcrackgrowth,Radhakrish-nan138,139postulatedthatthecrackgrowthrateispro-portionaltot-levelloadvariation,anexpressionforpredictingtheremaininglifefractionatthelastloadstepwas,therefore,formu-latedas:(31)whereWfiandWfmaretotalplasticstrainenergyatfailurefortheithandthelast(mth)levelsunderconstantamplitudecycling,rmu-lationimpliesthatfailureoccurswhplicaptsimilartoRadhakrishnan'swasalsopro-posedbyKteresisenergyofeachloadingblockwascalculatedasthesumofthemultiplicationoffrequency,nbi'bythecorrespondingplasticstrainenergydensity,~w;,asWApparently,thishypothesisdoesnotconsidertheloadb=!.~winbi'ity,however,ithasbeenexper-imentallyshown1ardingloadingsequenceeffects,Klimandefinedthedamagefractionperblockas:Db=-Wb1W=-!.~RII(32)eisdeemedtooccurwhenfRisthetotalenergyatfractureforagiven=nthisdamageaccumulationmodel=DJJfforblockloading,onecancalculatetheaccumulatedenergyfollowingasuccessiveprocedure,UUMOAMAGEMECHANICSAPPROACHESContinuumdamagemechanicsisarelativelynewsub-jectinengineeringmechanicsanddealswithtproachisdevelopedbasedontheoriginalceralco146,-cessofCOMapplicationinmodelingthecreepdamageprocesshasencouragedmanyresearcherstoextendthisapproachtoductileplasticdamage,creep-fatigueinteraction,tiontometallicmaterials,-novicandChabochel56,1one-dimensionalcase,Chaboche158postu-latedthatfatiguedamageevolutionperconductedundercompletelyreuringthechangesintensileload-carryingcapacityandusingtheeffectivestressconcept,heformulatedanonlineardamageevolutionequationasI58,159:D=1-[1-rllo-a)p/O+f3)(33)magemodelishighlynon,therefore,calledanonlinear-continuous-damage(NLCO),itallowsforthegrowthofdamagebelowtheinitialfatiguelimit,,themodelisabletotakeintoaccounttheinfluenceofinitialhardeningeffectbyintroducinganewinternalvariablewhichkeepsm,r,sinceascalardamagevariableisemployedandthemodeliswritteninitsuniaxialforminvolvingthemaximumandmeanstresses,difficultieswillinevitanfeatures,ntheCOMconcept,manyotherformsoffatiguedamageequationshavebeendevelopedafterChaboche'sworkl58posedbyLemaitreandChabochel45,.Suchmodelsincludethosepro-l63162,LemaitreandPlumtree,Wangl64,WangandLOUl65andLietal.166Basically,alltheseCOM-basendifferenceslieinthenumberandthecharacteristicsoftheparametersusedinthemodel,intherequirementsforadditionalexperiments,ndco-workersI67,168appliedtheLemaitre-Plumtreemodeltothefatiguedamageaportedimprovedlifepredictionsascom-paredtotheMinerrule'eeandO'Connorl69attemptedtoanalyzedamageaccumulationandfatiguecrackprSociel70alsoevaluatedtheChabocheandtheLemaitre-Plumtree

undtheChabochemodeltobmfcomethesedifficulties,ChowandWei171haverecentlyattemptedageneralizedthrr,duetothecomplexityofnonproportionalmultiaxialfatigueproblems,theframeworkwasalreadyproposedbyChabocheinRef.148,greateffortsarestillneededtoobtainanAMAGETHEORIESKramer'ssuifacelayerstressmodelRecognizingthatinformationfromthesurfaceofafatiguedmaterialusuallyplaysanimportantroleindamageanalysis,Kramer172introstulatedthatduringfatiguecycling,thespeci-mensuently,toattainagivenplasticstrain,morestressmustbeimpodefinedthisadditionalstressasthesurfacelayerstress,onstantamplitudecycling,thisstresswasfoundtolinearlyincreasewithappliedcycles,n,as:as=Sn,wheretheproportionalefficientcanbedescribedas:S=Kaf;,atigueprocesscontinues,thesurfacelayerstresswouldreachacriticalvalue,as*,iticalstressisfoundtobeindependentof*the=stressamplitudeandcanbeexpressedas:asSNf•Astress-lifeequationwas,therefore,derived:U2a*a=~(2Nf)-lip(34)Inthisequation,2a.//Kisequaltothefatiguestrengthcoefficienta;and-liptothefatiguestrengthexponent,eringaccumulationofthesurfacelayerstressdirectlytoquantifythedamageprocess,Kramerdefinedthedamagerulesimplyas:D='}.(as/as*)(35)FailureisdeemedtooccurwhenD=siislinearlydevelopedwithappliedcyclesasindicatedbyasi=Sn;,Equation(35)nexperimentalobservationofthesurfacelayerstressevolutionundertwo-levelcycling,Kramermodifiedthepreviousmodelto:(36)loadlevel,butsumsupnonlinearlyfromleveltolevelore,Equation(36)representsaloaddependentdamagemodelwithlinearevolution,nonlinearaccumulation,believedthemodelcouldalsobeextendedtocorrosion-fatiguedamageanalysis,ationsofthemodelrepresentedbyEquation(36)totwoandthreesteploadlevelfatigueteitanium6AL-4Valloy'basedoninternalandeffectivestressesTheconceptofinternalandeffectivestresseswasgeneratedfromthediscoverythattheaveragedislo-cationvelocityandthustheplasticstrainrateispro-portionallyrelatedtotheeffective(resolved)fectivestress,aconceptdifferentfromtheeffectivestressdefinedinCDMapproach,isequivalenttothedifferencebetweentheappliedandinternal(back)dco-workersl8°-stratedinFigure14,thehighestlevelofanelasticrange(HG)isdefinedastheinternalstress,ai'andthedifference(HC),aa-ai'givestheeffectivestress,aeff(whereaaistheappliedstressamplitude).Basedonthe'stress-dip'techniqueI84,185,aiandaeflevel,fforagivenappliedstressaa'ntheirexperimentalobservationsunderbothconstantandvariableamplitudestresscycling,dedthatinternalstress(asaresultofelasticinteractionofdislocations)isrepresentativeofthefatigueresistanceofamaterialandthattheeffectivestressaboveacriticalvalueisresponsibleforthefatiguedeformation,peculatedthatamaterialundercycling,/a//I,p0BAa-£pcurve(J:-a,e14Illustrationofinternalstressandeffectivestressonastress-plasticstrainhysteresislooplHOAccordingtothisequation,damageevolveslinearlyata

Cumulativefatiguedamageandlifepredictiontheories23reactsintwocontradictorymanners:sufferingdamageasaconsequenceoftheeffectivestress;andbeingstreernalstresswasfounchanismcanprobablybeusedtoexplainmanyphenomenaincumulativefatiguedam-ageprocessessuchasthecoaxingeffect2z,z5,fatiguelimitanditsloadhistorydependency,tion,theinternalstressconceptmayprovidephysicalinterpretationsforcycliccreepandcyclicstressrelaxation179,186,187,bar-riersforgrowthofMSCs97-99,188,189,loadinteractioneffects,r,thismethodcannotbeappliedtodamageassessmentforfatiguecrackpropagationprocess,asisofthebehaviorofinternalstressandeffectivestress,Ikaiandco-workersI8o,ocalledES-Minerrule,aaefYlogNfplotisusedasthebaselineindamagecalcu-lation,insteadoftheconventionalS-Ndiagram,andtheMinerlinearsummationisthenapplied,AnoverloaddamagemodelBothtensileandcotal.190condurformedcompletelyreversedstrain-controlledfatiguetestsonsmallsmoothspecimenswitheither10completelyreversedinitialoverstraincycles,ly,Topperandco-workers191,192carriedoutintensivestudiesonthissubjectbysubjectingsmallsmoothspecimenstouniaxialstresshistoriesconsistingofrepeatedblocks,whereeachblockcontainedemodelwhichaccountsforthemodel,damagesummationwasexpressedby:D="iDol+"iDss+"iDintwithD=1atfailure(37)whereDolisthedamageduetotheoverloads,Dssisthedamageduetothesmalleramplitudecyclesintheabsenceofoverloads(atasteady-statecondition),andDintistheadditional'interactive'riodicoverloadhistory,theinteractivedamageperblockisdeterminedthrough:(forcompressiveoverload)(fortensileoverload)(38)whereD,istheinteractivedamageduetothefirstsmallercycle(N=1)aftertheoverload,ccuratepredictionoffatiguelife,quantitiesD,magemodelaccountsfortheinteractivedamageresultingfromeithertensileand/takesintoaccountthedamagefrompost-overloadsmall-amplitudecyclesbelowtheconstantamplitudefatiguelimit,ultsinRefs191,icstrainevolutionmodelInordertorepresenttherelationshipbetweendam-ageevolutionandchangesinmechanicalproperties,Azarietal.194postulatedageneralformofdamagefunction,thatis,D=j(Y,X),whereYdenotesthedamageandXisthematerial'owsthattheproperselectionofpropertyandevolutionofitschangeenabrstudy,totalstrainrange,dE,wascontrolledandplasticstrainrange,dEp,ore,damageaccumulationwasexpressedas:D="i.(dEpIICdE-dEpodE)~(39)pf-po;whereCisaconstant,anddEpo,dEfinalvaluesofthepanddEpfareinitial,presentandplasticstrainrange,respectively,dEpisusuallyafunctionoftheappliedstrain,dE,andelapsedcycles,vendEofconstantamplitude,damagecanbeplottedwithrespectton,aunderthecurve,A,ments194showedthatthequantityA0,55).dividedbythevalueofNfisaconstant(aboutTherefore,Azarietal.'94proposedacriterionforfatiguelifepredictionundercomplexloadingexpressedas:"i.(A;fNfi)=Constant(40)Theformofthismodelissimpleander,intheconstantamplitudetests,r,thionalapproachesInadditiontotheaforementionedmodels,trtocoverasmanymodelsaspossiblewhileavoidingbeinglengthy,thissubsectionreviewstheremainingapproaches,'95postulatedthatfatiguedamagerate,asafirstapproximationcanbelinearlyrelatedwithdamageitself,Withtheinitialandfinalconditions=(D=0atr0andD=1atr=1atfailure),theintegrationofthedamagerateequationgivesadamagefunctionintheformof:=ekr;-ID"i.-k---le-forki=O(41)Thevalidityofthismodelwasexperimentallyevalu-atedandcomparedwiththreeothermodelsinRef.170

yzecomplexstrainhistories,Landgrafl96derivedadamageequatioedwithMorrow'smeanstressmodifi-cation197,thedamagerateequationwasderivedas:Dlreversal=_1_=[a/-am2NfE/E(~_~/lEe+ama/)]II(h-c)(42)wherea/,E/,ioofplastictoelasticstrainrangeprovidestheexperimentalinput./lEdeterminedfromablockofsteady-statestress-strainpand//lEri/lEeasadamageparameter,thismensideringvariableamplitudehistoriesrep-resentativeofserviceloadsituations,Kurathetal.198examinedboththeeffectofselectedsub-cyclesequencesonfatiguedamage,aswellastheoducinganinteractivefactorintoeachsummationterminlineardamagehypothesis,thedamageforablocksequencewithklevelswasdefinedas:D-~_l_~k2n.(/la)lId(43)b-;=1(2Nf);/lahwheredisaninteractiveexponentderivedfrombase-linedata(d=bl[b+c+l]inwhichbandcarefatiguestrengthandductilityexponents),and/plexloadinghistories,aneventisconsielsuggeststhatthefactor(/laJ/lah)lIdmodifiestheslopei-integraldamagemodel,itcharacterizescrackgrowthly,Pasic201attemptedt'unified'approach,damageinstress-controlledfatigueisdefinedastheproductofthelongestor'equivalent'surfacecracklengthandtheaccumulatedstrainrange,ingfracturemechanics,arelationbetwmalizedcracklengthwastheore,del,however,thanusingmacrodamageparameters,someinvestigatorsfocusedtheireffortstosearchforappro-priatemicro-variaboetal.202formulatedcumulativefatiguedamageforpersist-entslipband(PSB)typematerialsthroughPSBden-sity,herstudy,ageaccumulationwaspresumedtobeafunctionoftheappliedcyclesandtheintensityofslipb~nds,'l'(t:;>*,occurringinallgrainswithcommonN(unitvectornormaltotheslipplane).Normalizedbytheintensityattheterminationofcrackinitiationlifewherethemaximumvalueof'l'(t:;)*isreached,ascalardamagevariablewasthendefinedby:D(N,~n)=--'l'(t:;>*(n)-ck/m'l'~---(44)(N)max*NIwheremisthehardeningexponent,andcandkaretheexponentsinrelations:h(/lE)k,/lE=g(N;)

Cumulativefatiguedamageandlifepredictiontheories25havebasicloadingparametersidenticalorsimilartothoseinthereferencespectrumsuchasmaximum/minimumamplitudes,YMorethan50fatiguedamagemodelshavebeenpro-posedsincethsi-calbasis,damageexpresreviationsusedfordescribingthephysicalbasral,damagetheoriesdevelopedbefore1970saremainlyphenomenological,whilethoseafter1970saresemi-analyticalbecause,tosomeextent,theyinvolvedamagemechanism(s).Asawhole,sixmajorcategoriesincumulativefatiguedamagemodelingexist:damageevolutionandlinearsummation;eardamagecurveandtwo-stagelineariz-ationapproaches;rvemodificationstoaccountforloadinterac-tions;chesbasedoncrackgrowthconcept;basedonCDM;nnotacstnonlinearload-dependentdamagetheoryrep-resentedbythepowerrelationship,DandStarkeyin1954.=Ir/i,waspro-posedbyMarcoIntwo-stagelinearizationapproaches,thedamageprocessisdividedintotwostagesofcrackrvemodificationapproachesarebasedonmodifyingthematerialS-Ncurve,areload-leveldependent,chesbasedonthecrackgrowthconceptincludingmacrocrackgrowthretardationmodelshaveenjoyedawidedegreeofacceptancesinroachesarerelativelynewapproaches,pproacheswereoriginallydevelopedtomtivedamagetheoriesbasedonenergyhavemainlybeendevelopedsincethelate1970sandhavethepotentialtounifythedamagecausedbydifferenttypesofloadssuchasthermalcycling,manydamagemodelshavebeendeveloped,unfortunately,magemodelcanonlyaccountforoneorseveralphenomenologicalfactors,suchasloaddependence,multipledamagestages,nonlineardamageevolution,loadsequenceandinteractioneffects,over-loadeffects,hecomplexityoftheproblem,uently,thePalmgren-MinerLDRisstilldominantlyusedindesign,,themostcommonmethodforcumulativedamageassessmentusingLEFMhasbeenbasedonintegrationofaParis-typecrackgrowthrateequation,fortsinthestudyofcumulativefatiguedamageareneededinordertoprovidedesignengineerswitLEDGEMENTSFinancialsupportforthisprojectwNCESPalmgren,A.,renste-chinik,Berlin,1924,68,339-341.2Miner,M.A.,lofAppliedMechanics,1945,67,AI59-AI64.3Newmark,N.M.,gueandFractureofMetals,hnologyPressoftheMIT-Wiley,NewYork,NY,1952,pp.197-228.4Socie,row,J.D.,eofEngineering,UniversityofIllinois,Urbana,IL,December1976;[alsoinRiskandFailureAnalysisforImprovedPeifonnanceandReliability,,NewYork,1976,pp.141-194.]5Kaechele,L.,dCorporation,SantaMonica,1963.6Manson,S.S.,gJournalResearch,1964,43(Supplement),344s-352s.7Leve,H.L.,lFatigue:TheoryandDesign,,NewYork,NY,1969,pp.170--203.8O'Neill,M.J.,uresandMaterialsReportNo.326,AeronauticalResearchLaboratories,Melbourne,Australia,1970.9Schive,J.,ryGroupforAerospaceResearchandDevelopment,Paris,1972.10Laflen,k,T.S.,alAeronauticsandSpaceAdministrationCon-tractReport,,1982.11Golos,yin,F.,GenticalandAppliedFractureMechanics,1987,7,169-176.12Manson,ford,G.R.,Re-examiner-ingFractureMechanics,1986,25(5/6),539-57I.13Yang,emi,A.,Cumulativefatiguedamagemech-anismsandquantifyingparameters:lofTestingandEvaluation,inpress.14French,H.J.,ctions,AmericanSocietyofSteelTreating,1933,21,899-946.15Kommers,J.B.,dings,AmericanSocietyforTestingandMaterials,1938,38(PartII),249-268.16Langer,B.F.,urnalofAppliedMechanics,1937,59,AI60-AI62.17Lim,L.C.,Tay,g,H.S.,FatiguedtallurgicaetMaterialia,1990,38(4),595-{i01.18Coffin,L.F.,DesignaspectsofctionsoftheASME,1956,78,527-532.19Baldwin,E.E.,Sokol,fin,L.E,dings,Amer-icanSocietyforTestingandMaterials,1957,57,567-586.

mmaryofcumulativefatiguedamagetheories:workbeforethe1970sModeldeveloperYearPhysicalbasis"DRMiner1945Constantenergyabsorptionpercycle(CON)Constantdislocationgenerationpercycle(CON)DirectlyconvertedfromstressversionConceptual(CON)Endurancelimitchange(PHE)Endurancelimitchange(PHE)D="in/Nf;="ir;LDE,nLLD,nLlA,nSC,manyAppl(popular),SLDE,nLLD,nLlA,nSC,nAppI,C2Machlintheory(metallurgicLDR)Machlin1949PI2D="in;JR0208g;dtStrainversionofLDRMarco-StarkeytheoryHenrytheoryCoffin1956D="in(~E)'IaIpIe'laLDE,nLLD,nLlA,nSC,someAppl,GnLDE,LLD,nLlA,nAppi,SnLDE,LLD,nLlA,SC,fewAppl,GnLDE,LLD,nLlA,SC,fewAppl,C18MarcoandStarkeyHenryGatts1954D="ir;,;,x;>I232219551961D=~eo-(Te=(Teo"ir;I+(I-r;)fA;GattstheoryYe=-u,=Y"[1-aco_Y.'+'y,(I-Yu-IY"-IJ27Bluhm'shypothesisCorten-DolonmodelFrudenthal-HellertheoryGrover'stwo-stagedamagetheoryBluhmCortenandDolonFrudenthalandHellerGrover1961Endurancelimitchange(CON)d"=IY("JI-IYe("-IIID="im;p;nb;nLDE,LLD,nLlA,SC,nAppi,CnLDE,LLD,LlA,SC,fewAppl,GnLDE,LLD,LlA,nSC,someAppl,GTwo-stageLDE,LLD,nLlA,nSC,fewAppl,S2930311956Numberofdamagenuclei(CON)1959Fictitiouslifecurve,probabilisticD=analysis(sANA)Crackinitiationandcrackpropagation,two-stagelinearevolution(CON)Crackinitiationandcrackpropagation,two-stagelinearevolution(EMP)Crackgrowth,cracklengthasdamagemeasure(PHE)Crackgrowthanddislocation,fracturemechanics(sANA)Crackgrowthfracturemechanics(sANA)"i(nw)~;,wherew;isaninteractionfactorn1960"ia;~;=Iforinitiationstage"im·(I_~;)N;=Iforpropagationstage36Doublelineardamagerule(DLDR)Mansonetal.19661967"i2="in;00=IforphaseIN,.;Nf-PNf'm·m·"i="iPN:L6=IforphaseIIi-II,iTwo-stageLDE,LLD,nLlA,SC,someAppl,S3738fShanleytheoryShanley1952D="i((j

CumulativefatiguedamageandlifepredictiontheoriesTable2ModelSummaryofcumulativefatiguedamagetheories:DCA,refinedDLDRandDDCAcurveapproach(DCMRefinedDLDRcMansonandHalford1981Effectivemicrocrackgrowth(PHE)D='q=(N/N,)i3andf3=0.4nLDE,LLD,nLlA,44,12nSC,someAppl,GMansonandHalford1981BasedonDCAandlinearization(EMP)BasedonbothDCAandrefinedDLDR(EMP)='2.(nr/N,);N]=N-NIlB=0.65Dn='2.(nnINn);Nn=BN(N/N)u0'=0.25D,DPTwo-stageLDE,44,12LLD,nSC,nLlA,manyAppl,SMansonDoubledamagecurveandapproachHalford(DDCA)C1986='2.[(pr;)k+(I-=A(N/N;)U/[I=0.35,k=5,B(N/N;)U],B=0.65,0'=0.25,f3=0.4-Pk)rkq]Ilk,AnLDE,LLD,nLlA,nSC,someAppl,C12aCON,conceptual;PHE,phenomenological;EXP,experimental;EMP,empirical;ANA,analytical;sANA,semianalyticalbLDE,lineardamageevolution;LLD,loadleveldependent;LlA,loadinteractionaccountable;SC,smallamplitudecycledamageaccountable;Appl,application(s);S,simple;G,general;C,complicated;thesuffix'n'standsfor'not'or'non'cTheconstantswereobtainedbasedonexperimentswithMaraging300CVMsteel,SAE4130steel,andTi-6AI-4ValloyTable3Summaryofcumulativefatiguedamagetheories:hybridtheoriesModversion"Bui-Quocetal.1971Hybridization,endurancelimitchange(sANA)D=--=.:,1-Ye~r;(.1-Yecr,+(1_r,.)Y;-(Yhu)m)1Y;-nLDE,LLD,nLlA,SC,someAppl,C209m=8StrainversioncBui-Quocetal.1971TransplantedfromthestressversionD(sANA)=~='2.1-Aer;ec.+(1_.)''fr,r,A.,-1mA._(A/A)mnLDE,LLD,nLlA,SC,someAppl,C52=8nLDE,LLD,LlA,SC,fewAppl,C51,55FictitiousloadmodificationcBui-Quoc1981,.JEndurancelimitchange,toaccountD=oroamteractIoneffects(sANA)Id"Ir;-------=-------ri+(l-r;)A/-(A//Af)m.'_1/,m=8Cycle-ratiomodification"Bui-Quoc1982Endurancelimitchange,toaccountD=forloadinteractioneffects(sANA)Ir"r•.+(l-rJ'A.-(A/A)nLDE,LLD,LlA,SC,someAppl,C55,56A;-Im=8,tmaCON,conceptual;PHE,phenomenological;EXP,experimental;EMP,empirical;ANA,analytical;sANA,semianalyticalbLDE,lineardamageevolution;LLD,loadleveldependent;LlA,loadinteractionaccountable;SC,smallamplitudecycledamageaccountable;Appl,application(s);S,simple;G,general;C,complicated;thesuffix'n'standsfor'not'or'non'cTheconstantm=8wasdeterminedfromexperimentswithA-201andA-517steels282930Gatts,R.R.,urnalofBasicEngineering,1962,84,,J.1.,alsResearchandStandards,,on,T.J.,utionofMechanicalEngineeringandAmericanSocietyofMechanicalEngineers,1956,nthal,ler,R.A.,loftheAerospaceSciences,1959,26(7),431-442.32333431Freudenthal,A.M.,oquiumonFatigue,Stockholm,er-Verlag,Berlin,1956,r,ten,H.T.,Effectod-ings,AmericanSocietyforTestingandMaterials,1961,61,,S.S.,Nachigall,,d-ings,AmericanSocietyforTestingandMaterials,1961,61,679-703.

ble4Summaryofcumulativefatiguedamagetheories:recenttheoriesbasedoncrackgrowthModelModeldeveloperYearPhysicalbasis'exponentiallaw(1stversion)Doubleexponentiallaw(2ndversion)ShortcracktheoryMillerandZachariah1977Two-stagecrackgrowth(PHE)Nt.(rl+r21=Nt,1r-I)2nLDE,LLD,nLlA,nSC,afewAppl,C91IbrahimandMiller19801981Two-stagecrackgrowth(PHE&r2EMP)=(1-rt)~In-'lin---:t,t(I)(atI)atDMiller1982MSC,PSC,E-Pfracturemechanics(sANA)=~=(~afafr-oSoS(aD)atnLDE,LLD,nLlA,nSC,someAppl,C9293r)/(I-r,)dadN=A(dy)"(d-a)forMSCs:aDdadN=B(dy)f3a-CforPSCs:ataoSataoSatClearphysicalbasis,itisdifficulttodeterminethemicro-parametersinvolvedLlA,fewAppl,notuniversal,G95-99Ma-LairdtheoryMaandLaird1989Crackpopulation(PHE)D="i.(P/Pcd,)=K"i.n,[(dy/2);u;-=2Dcrforinitiation:DDcm=Dc+-[e(2r-oS(dy/2)tlmit]104Vasek-PolakmodelVasekandPolak1991DMicrocrackkineticsequivalentcrackDlength(PHE&sANA)roS1/2oSm.I)-I]forpropagatIOn:1/2roSITwo-stagenLDE,LLD,nLlA,nSC,fewAppl,C105"CON,conceptual;PHE,phenomenological;EXP,experimental;EMP,empirical;ANA,analytical;sANA,semianalyticalbLDE,lineardamageevolution;LLD,loadleveldependent;LlA,loadinteractionaccountable;SC,smallamplitudecycledamageaccountable;Appl,application(s);S,simple;G,general;C,complicated;thesuffixon'standsfor'not'or'non'Table5Summaryofcumulativefatiguedamagetheories:modelsbasedonmodifyinglife-curveModelModeldeveloperYearPhysicalbasis'anyanSubramanyanmodelHashin-HashinandRotemRotemtheoryBoundtheoryBen-Amoz1976Convergencetother;knee-point(CON)Twotypesofconvergence(CON)Upperandlowerboundsofconvergencelines(sANA)Experimentaldetermination(EXP)=I-{r;-t+[r;-2+...+(r2+ri)",...]";_2J"ha;_ILDE,LlA,nSC,someAppl,GLDE,LlA,nSC,someAppl,G1061978Formulationbasedonstaticstrengthpoint,andformulationbasedonendurancelimitpointBoundsformedbyMinerruleandSubramanyanmodel;boundsformedbyDLDRandSubramanyanmodel;andstatisticalboundsModifiedlifecurveisobtainedfromrepeatedmulti-levelblocktests1071990LDE,LlA,nSC,109-11IfewAppl,CLeipholz'sapproachLeipholz1985LDE,LlA,SC,afewAppl,G,E112,I13'CON,conceptual;PHE,phenomenological;EXP,experimental;EMP,empirical;ANA,analytical;sANA,semianalyticalbLDE,lineardamageevolution;LLD,loadleveldependent;LlA,loadinteractionaccountable;SC,smallamplitudecycledamageaccountable;Appl,application(s);S,simple;G,general;C,complicated;thesuffixon'standsfor'not'or'non'353637Manson,S.S.,Nachigall,A.J.,Ensign,che,J.C.,FururnalofEngineeringforIndustry,1965,87,,H.J"osiumonFatigueofAircraftStruc-tures,anSocietyforTestingandMaterials,Philadelphia,PA,1960,,S.S.,Interfacesbetweenfatigue,creep,ationalJournalofFractureMechanics,1966,2,328-363.383940Manson,S.S.,Freche,ign,S.R.,Applicationofadoublelineardamageruletocumulativefatigue,InFatigueCrackPropagation,anSocietyforTestingandMaterials,Philadelphia,PA,1967,,O.G"ExationalJournalofFatigue,1991,13(1),y,F.R.,Atheoryoffatiguebasedonunbondingduringreversedslip,dCorporation,SantaMon-ica,1952.

CumulativefatiguedamageandlifepredictiontheoriesTable6ModelSummaryofcumulativefatiguedamagetheories:energy-baseddamagetheoriesModeldeveloperYearPhysicalbasisaExpressionCharacteristicsbRef.29Plasticstrainenergy(hysteresisenergy)TotalstrainenergyBui-QuocmodelKujuwskiandEllyin1984Convergence,Intheplasticstrainenergyvslifediagram,isodamageLDE,LIA,SC,curvesconvergetotheapparentfatiguelimit,ratherfewAppl,G,Eplasticstrainenergy,(CON)thantotheoriginallimitConvergence,totalstrainenergy,(sANA)Constanttotalplasticenergyatfailure(CON)Crackgrowthrateisrelatedtoplasticenergy(CON)Blockspectrum,similarbasistotheabove(CON)Strainhardeningandplasticstrainenergy(sANA)Relatedtotwoexponentsinstrain-lifeequation(CON)Inthetotalstrainenergyvslifediagram,isodamagecurvesconvergetotheapparentfatiguelimit,ratherthantotheoriginallimitW,124GolosandEllyinBui-Quoc1987LDE,LIA,SC,11,126,127someAppl,G,EEnergyversionofLDR,fewAppl,GAnotherenergyversionofLDR,fewAppl,G1371973=k2K'M",+InjLlWj=~-I-kriNf~-C(n'+I)RadhakrishnanapproachRadhakrishnan1978r",=I"'ilWi=IWfjrjfm138KlimanTheoryKliman1984Dbb=-=-kLlWjnbjWfRWfRWIAlsoenergyversionofLDR,afewAppl,G+a)140NiutheoryNiuetai,1987D=

mmaryofcumulativefatiguedamagetheories:continuumdamagemechanicsapproachhemodelChaboche0Lemaitre-LemaitreandPlumtreePlumtreemodelLemaitre-LemaitreandChabochemodelChabocheWangmodelWang-LoumodelWangWangandLoferencesarethenumberofvariablesassumedinthedamagerateequationandboundaryconditionsBesidesabove,adislocationvariableisinvolved(ANA)CDMapproachinthree-dimensionalspace(ANA)D=I-D[I-r'/(l-alj'/!]+(3)=I-D(I-r)'II]+p)a)="kr;'l(l-DD=I-(I-r)]/I]+n(Rl]=Dc-(Dc-Do)(I-r)'-(3158Allareinsimilarcharacteristics163suchasnLDE,LLD,potential145,162toaccountforSCandLIA,164increasingAppl,Convenience165dependsontheparametersusedLi-Qian-LimodelLietal.1989D=I-A(fa)pSimilarcharacteristicstoabove,nAppI,CSimilarcharacteristics,nAppI,veryC166Three-dimensionalCDMmodelChowandWei1991Adamageeffectivetensorwasintroducedandageneralizedthree-dimensionalisotropicCDMmodelwasproposedbasedoneffectivestressconcept171aCON,conceptual;PHE,phenomenological;EXP,experimental;EMP,empirical;ANA,analytical;sANA,semianalyticalbLDE,lineardamageevolution;LLD,loadleveldependent;LIA,loadinteractionaccountable;SC,smallamplitudecycledamageaccountable;Appl,application(s);S,simple;G,general;C,complicated;thesuffix'n'standsfor'not'or'non'mentalMechanics,1975,15(6),-Quoc,T.,AnurnalofEngineeringMaterialsandTechnology,1979,101,-Quoc,on,A.,rnationalConfer-enceonEngineeringAspectsofCreep,icalEngineering,London,UK,1980,-Quoc,on,A.,IntlofMechan-icalEngineeringScience,1981,23(6),-Quoc,T.,sactionsofthe6thInternationalConferenceonStructuralMechanicalReactionTechnology,Paris,France,1981,L7/-Quoc,T.,Rec3rdInternationalSeminaronInelasticAnalysisandLifePredictioninHighTemperatureEnvironment,Paris,1981,-Quoc,T.,Recentdevelopmentsofcontinuousdamageappr,1982,-Quoc,on,A.,ArEngineeringDesigns,1982,71(1),-Quoc,uc,encesinLifePredictionMethods,ProceedingsoftheASMEInternationalConference,ricanSocietyofMechanicalEngineers,NewYork,NY,1983,,-Quoc,T.,Ananalysisofthefatigue/urnalofPressureVesselTechnology,1986,108,,R.,Bui-Quoc,on,A.,hanica,1987,21(2),-Quoc,T.,Gomuc,R.,Biron,A.,Nguyen,ou-nave,1..Elevatedtemperaturefatigue---{;ycleFatigue,ed.H.D.77778798081Solomon,d,,anSocietyforTestingandMaterials,Philadel-phia,PA,1988,,R.,Bui-Quoc,T.,Biron,nard,M.,Analysisoftype316stainlesssteelbehaviorunderfatigue,creepandcombinedfatigue---{;urnalofPressureVesselTechnology,1990,112(3),-Quoc,T.,Choquet,on,A.,CumulativefatiguedamageonlargeurnalofEngineeringMaterialsandTechnology,1976,98,,A.,Bui-Quoc,uc,R.,AprocedureforlournalofEngineeringMaterialsandTechnology,1990,112,,ade,C,TheeffectongthofMetalsandAlloys(ICSMA8),Proceedingsofthe8thInter-nationalConferenceontheStrengthofMetalsandAlloys,PergamonPress,Oxford,1989,r,O.E.,urnalofBasicEngineering,1972,D94(1),borg,J.,Engle,d,H.A.,Acrackgrowthretardationmodelusinganeffectivestressconcept,AFFDLTM-71-I-FBR,,W.,er-ingFractureMechanics,1970,2,,W.,geToleranceinAircraftStructures,anSocietyforTestingandMaterials,Philadelphia,PA,1971,,,AcrackclosuremodsandModelsforPredictingFatigueCrackGrowthunderRandomLoading,anSocietyforTestingandMaterials,Philadelphia,PA,1981,,,PredictionoffatiguecrackggnofFatigueandFractureResistantStructures,anSocietyforTestingandMaterials,Philadel-phia,PA,1982,,f,C.R.,Spectrumcrackgrowthprediction

CumulativefatiguedamageandlifepredictiontheoriesTable8SUinmaryofcumulativefatiguedamagetheories:otherapproachesModelModeldeveloperYearPhysicalbasisaExpressionCharacteristicsbRef.31SurfacelayerKramerstressapproachES-MinerruleIkaietal.1974Surfacelayerstresschange(PHE)D=L(erJer,*)LDE,LIA,SC,someAppl,G172,1731989InternalstressAppliedstresscanberesolvedintointernalandLDE,LIA,SC,ernalstressisfewAppl,C,Estressevolutionsrepresentativeofthefatigueresistanceofamaterial(sANA)whiletheeffectivestressisresponsibleforthefatiguedamageCrackopeningandclosure(PHE)Plasticstrainevolutionandaccumulation(PHE)Assumingalineardamagerate(CON)D=LDo'+LD"+LD;nt180-183OverloaddamagemodelTopperetal.1990LLD,LIA,SC,someAppl,G,EnLDE,LLD,nLIA,nSC,fewAppl,GnLDE,LLD,fewAppl,GLDE,nLIA,nSC,errninvolved,fewAppl,GLDE,LIA,nSC,fewAppl,G191,ionmodel1984ppoD=L(.iE-.iE)-.iEpn;-l/eFongtheoryFong1982D=L(ekr;-I)/(ek-1)forki=O195LandgrafmodelLandgraf1973Strainversionpermm-+~Dlreversal=-1=[er,.'-er(.iEofLDR2NfE,.'involvingmeanstress(CON)Plasticwork,LDR,loadinteraction(CON)CDMandfracturemechanics(ANA)Persistentslipbanddensity(PHE)InvolvingPSBparameter(ANA)Changeinresistance-to-flow(ANA)D-b-)]l/(b-196c)PlasticworkbaseddamagemodelUnifiedapproachKurathetal.1984,~,(2Nr),.ierhlid±-~,2n-'(.ier)198Pasic1992ThisisanapproachcombiningfracturemechanicswithdamagemechanicsOnlyconceptualnAppI,ro-damagemechanicsmodelInoueetal.1988L(n/Nr;)=DS=I+DiDILDE,nLLD,fewAppl,CnLDE,LLD,fewAppl,CnLDE,LLD,LIA,SC,fewAppl,CRelyingonexperiment,someAppl2021987D(N,n)=['¥*(N)/,¥*(N)rnax](nIN,)-d1m203AmodelbasedAbuelfoutouhonresistance-to-andHalfordflowAcorrectionapproachBuchetal.1989dX/dN=±J]+bexP(CX+d)2041982UsingcorrectionN"=Ncai'(Ncxp'INcal')=NeatefactorC205,206aCON,conceptual;PHE,phenomenological;EXP,experimental;EMP,empirical;ANA,analytical;sANA,semianalytical"LDE,lineardamageevolution;LLD,loadleveldependent;LIA,loadinteractionaccountable;SC,smallamplitudecycledamageaccountable;Appl,application(s);S,simple;G,general;C,complicated;thesuffix'n'standsfor'not'or'non'gueCrackGrowthunderSpectrumLoads,anSocietyforTestingandMaterials,Philadel-phia,PA,1976,,H.D.,Saff,ter,J.M.,EffectofFighterAttackSpectrumonCrackGrowth,anSocietyforTestingandMaterials,Philadelphia,PA,1980,g,ger,T.,DugdaleceringFractureMechanics,1979,11,ng,A.U.,AsimplecrackclosuremodelforptureMechanics:ThirteenthConference,ASTMSTP743.85868788AmericanSocietyforTestingandMaterials,Philadelphia,PA,1981,e,R.0.,Suresh,es,C.M.,Nearthresholdfatiguecrackgrowthin21/urnalofEngineeringMaterialsandTechnology,1980,102,e,esh,S.,SomeconsiderationsonfatiguecrackclosureurgicalTransactions,1981,13A,,chie,R.0.,Ageometricur-gicalTransactions,1982,13A,,chie,R.0.,Near-thresholdfatiguecrack

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