TheFatigueofAluminumAlloysSubjectedtoRandomLoadingExperimentalinvestigationisundertakenbytheauthorstodeterminethefatiguelifeof2024-T3and6061-T6aluminumalloysG.W.BrownandR.IkegamiABSTRACT-Thispaperdescribesanexperimentalinvesti-gationwhichwascarriedouttodeterminethefatiguelifeoftwoaluminumalloys(2024-T3and6061-T6).Theyweresubjectedtobothconstant-strain-amplitudesinus-oidalandnarrow-bandrandom-strain-amplitudefatigueloadings.Thefatigue-lifevaluesobtainedfromthenar-row-bandrandomtestingwerecomparedwiththeoreticalpredictionsbasedonMinerslinearaccumulationofdam-agehypothesis.Cantilever-beam-testspecimensfabricatedfromthealuminumalloysweresubjectedtoeitheraconstant-strain-amplitudesinusoidaloranarrow-bandrandombaseexcitationbymeansofanelectromagneticvibrationsexciter.Itwasfoundthatthee-Ncurvesforbothalloyscouldbeapproximatedbythreestraight-linesegmentsinthelow-,intermediate-andhigh-cyclefatigue-liferanges.Minershypothesiswasusedtopredictthenarrow-bandrandomfatiguelivesofmaterialswiththistypeof-Nbehavior.Thesefatigue-lifepredictionswerefoundtoconsistentlyoverestimatetheactualfatiguelivesbyafactorof2or3.However,theshapeofthepredictedfatigue-lifecurvesandthehigh-cyclefatiguebehaviorofbothmaterialswerefoundtobeingoodagreementwiththeexperimentalresults.Symbolsbl,b=constantskl,k2=constantsN=numberofcyclestofailureS=stressS=yieldstress(1,N1)=referencepointone-Ndiagrame=straine=yieldstrainep=peakstraine=endurance-limitstrain=strain-hardeningexponentG.W.BrownisProfessorofMechanicalEngineering,UniversityofCali-fornia,Berkeley,Calif.R.IkegamiisResearchEngineer,StructuralDynamicsGroup,BoeingAircraft,Seattle,Wash.Paperwaspresentedat1970SESASpringMeetingheldinHuntsville,Ala.onMay19-22.r=strainvarianceSymbolsnotshownherearedefinedinthetext.IntroductionTheproblemofpredictingthefatiguelivesofmetalstructureswhicharesubjectedtorandomloadingsisgenerallysolvedbyfirstformulatinganaccumula-tionofdamagecriteria,thenapplyingthiscriteriatothespecifiedconditionsofvaryingcyclic-loadamplitude.ThefirstandstillthemostcommonlyusedcriterionforpredictingtheaccumulationofdamageinfatiguewasproposedbyA.PalmgrenandappliedbyM.A.Miner.1Thiscriterionas-sumesthattheproblemofaccumulationofdamagemaybetreatedasoneinwhichthefractionsoffa-tiguelifeusedupatdifferentloadlevelsasdeter-minedfromtheconstantamplitudee-NcurvemaybesimplyaddedtogiveanindexofthefatiguedamageandisgenerallyknownasMinerslinear-accumulation-of-damagecriteria.Thispaperdescribesaportionoftheresultsofastudy2whichwasconductedtodetermineamethodofpredictingthefatiguelivesofalumimum-alloystructureswhichweresubjectedtonarrow-bandrandomloadings.Anexperimentalprogramwascarriedouttodeterminethelivesofcantilever-beamtestspecimenswhichweresubjectedtoeithercon-stant-strain-amplitudesinusoidalornarrow-bandrandomstrain-amplitudefatigueloadings.ThefatiguelivesofthetestspecimenssubjectedtotherandomloadingswerethencomparedtopredictionsbasedontheapplicationofMinerscriteria.ExperimentalProgramThefatiguetestswereperformedontwocom-monlyusedaluminumalloys,2024-T3and6061-T6.ThemechanicalpropertiesofthesetwoalloysareExperimentalMechanics321u10.310.2TRUESTRAINLin/in)101Fig.1-Truestressvs.truestrainfor2024-T3aluminum0.5a.BeamConfigurationb.NormalizedModeShape-0.5O0.5STRAINGAGEACCEL.:!III1000-3_iirTr10-210-ITRUESTRAIN(in/inIFig.2-Truestressvs.truestrainfor6061-T6aluminum0.5NormalizedBendingStress1/stMode-0.5-1Fig.3-VibrationcharacteristicsoffatiguespecimengiveninTable1,andthetrue-stressvs.true-straincurvesareshowninFigs.1and2.Thestrain-hardeningexponent,characterizesthestress-strainrelationshipintheplasticrange.Thesepropertiesweredeterminedfromuniaxialtensiletestsusingtensionspecimensmadefromthetwoalloys.Inbothcases,thetensionspeci-mensandthefatiguespecimensweremachinedfromthesamesheetsofaluminum,withthelongitudinalaxesofthespecimensparalleltothedirectionofroll-ing.Thiswasdonetoinsureuniformitybetweenthesetwotypesoftests.Themajorityofthefatiguetestswasdoneonanelectromagneticvibrationsexciter,withasmallportionofthelow-cycleconstant-strain-amplitudetestsbeingperformedonanInstrontester.Forthetestingwhichwasperformedonthevibra-tionsexciter,thefatigue-testmodelwasacantileverTABLE1-MATERIALPROPERTIESMaterialTrueTrueStrain-ElasticYieldfracturefractureharden-modulus,stress,stress,strain,ingex-psipsipsiin./in,ponent2024-T3i0.6X10651,00090,7000.2400.1476061-T610.6X10840,50062,0000.4400.0875beamsubjectedtoabaseexcitation,asshownschematicallyinFig.3(a).Forthenarrow-bandrandomfatiguetests,theexcitationwasanar-row-bandsignalwithaGaussianbase-accelerationamplitudeanduniformspectrumoveraspecifiedfrequencybandwidth.Theexcitationbandwascenteredatthefundamentalbeamresonance.Thesinusoidalfatiguetestswereperformedwiththeex-citationfrequencyslightlyabovethefundamentalbeamresonance.Thetypeofcyclicloadingwasthereforecompletelyreversedbending.Thecantileverspecimenswereprofiledalongthelengthofthebeamtomovethemaximumbendingstressinthefirstmodeofvibrationawayfromthefixedend.Asketchofthetest-specimenconfigura-tionisshowninFig.4.Thetestspecimenswereclampedinthemiddlebyamountingfixturewhichwasattacheddirectlytothearmatureofthevibra-tionsexciter.Ascanbeseenfromthefigure,eachfatigue-testspecimencontainedtwocantilever-beamspecimenswhichwereexcitedsimultaneously.Anendmass,intheformofanEndevcoModel2216crystalaccelerometer,wasattachedatthefreeendofthecantilever-beamspecimens.Themodeshapesandcorrespondingbending-stressdistribu-tionsforthefirsttwomodesofvibrationareshowninFigs.3(b)and3(c).Atthefundamentalbeamresonance,themaximumbendingstressoccurredat322IAugust1970i-CLAMPED-oe-j+-1-I-zt-=!r-Fig.4-Fatigue-testspecimenp,fc.=_EzrFig.5-Fatiguespecimenonvibrationexciteradistanceof7/8in.fromthefixedendofthebeam.Thefrequencyofthefirstbeamresonancewasap-proximately115cps.Thespecimenswerecare-fullyhandpolishedpriortotestingtoremoveanysharpcornersandtoeliminateallvisiblesurfacescratchesintheregioninwhichthemaximumstressoccurred.Tomeasurethestrainlevelduringthefatiguetests,straingagesweremountedoneveryspecimenatthepointwherethemaximumbendingstressoc-curred.Itwasfoundthatthefatiguelifeofthestrain-gageinstallationwasgenerallymuchsmallerthanthefatiguelifeofthespecimen.Forthisreason,thesignalfromtheaccelerometermountedatthefreeendofthebeamwasusedtodeterminethetimetofailureofthefatiguespecimens.Thesignalfromtheaccelerometerwasusedtotriggerarelaywhichdeactivatedatimerwhentheaccelera-tionleveldroppedto50percentofthenominalRMSaccelerationlevel.Itwasobservedthat,atfailure,theaccelerationleveldroppedveryrapidlysothatthetimersindicatedverycloselythetotaltimetofailureofthespecimen.ThepictureinFig.5showsafatigue-testspeci-menmountedonthevibrationsexciterjustpriortotesting.Ascanbeseenfromthisfigure,specialaccelerometercableswereconstructedbysplicingstandardMicrodotaccelerometercableswithtwosmaller,moreflexible,leadwires.Thiswasdonetominimizetheeffectofthevibrationsoftheac-celerometercableonthespecimen.Althoughthespliceincreasedthenoisepickup,thesignallevelwassolargethatthisincreaseinnoisewasnotnoticeable.6080100FREQUENCY(Hz)Fig.6-Strainspectraldensity120140160Fig.7-FatiguespecimeninInstrontesterThesignalsfromthestraingagesandtheac-celerometersweremonitoredduringthetestsandrecordedonmagnetictape.Aftereachtest,therecordedsignalswereplayedbackintoawave-ExperimentalMechanicsI323analyzersystemtodeterminetheRMSlevels.Adigitalcomputerwasusedtoperformatime-seriesanalysisoftherandomsignalsobtainedfromthenarrow-bandrandomfatiguetests.Astrainspec:tral-densityplotofthestrain-gageresponseduringatypicalnarrow-bandrandomtestisshowninFig.6.Asexpected,thisplotindicatesthatthefatiguespecimencanbeconsideredtobeaverylightlydamped,single-degree-of-freedomsystem.Themostprobablefrequencyofvibrationofthisnar-row-bandresponsecanbeshowntobetheresonancefrequencyofthesystem.Therefore,thetotalnum-berofcyclestofailure(i.e.,thetotalnumberofzerocrossingswithpositiveslope)wasassumedtobethetotaltimetofailureinsecondsmultipliedbytheresonancefrequencyincyclespersecond.Intheconstant-amplitude,sinusoidalteststhetotalnum-berofcyclestofailurewasmerelythetotaltimetofailureinsecondsm