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FatiguethresholdR-curvesforpredictingreliabilityofceramicsundercyclicloadingJ.
J.
Kruzica,*,R.
M.
Cannonb,J.
W.
AgerIIIb,R.
O.
Ritchieb,caDepartmentofMechanicalEngineering,OregonStateUniversity,204RogersHall,Corvallis,OR97331,USAbMaterialsSciencesDivision,LawrenceBerkeleyNationalLaboratory,Berkeley,CA94720,USAcDepartmentofMaterialsScienceandEngineering,UniversityofCalifornia,Berkeley,CA94720,USAReceived7December2004;receivedinrevisedform7December2004;accepted8February2005Availableonline2April2005AbstractFormonolithic,grain-bridgingceramics,thecrack-sizedependenceofthefatiguethresholdduringbridgingzonedevelopmentpresentsadicultyinitsapplicationindesign.
ThispaperdemonstrateshowafatiguethresholdR-curvemayexpressthiscrack-sizedependence,analogoustothetraditionalfracturetoughnessR-curve,andmaybeusedtopredicttheendurancelimitandretainedstrengthundercyclicloadingconditions.
Furthermore,thefatiguethresholdR-curvemaybededucedfromthebehav-iorofmillimeter-scale,through-thickness,fatiguecracksviaanaccuratelymeasuredcrackbridgingstressprole.
Bothanaluminaceramicwithlargesteady-statebridgingzones($2mm),wherethepredictedandexperimentallymeasuredfatiguethresholdR-curvesagreewelloverarangeofcracksizesfrom0.
06to7mm,anda(Y2O3-MgO)-dopedSi3N4ceramic,wherebridgingzonesaremuchshorter($100lm),areinvestigated.
Theapproachprovidesausefulalternativetoperformingdicultshort-crackfatigueexperiments,particularlyformaterialswithsmallbridgingzones.
2005ActaMaterialiaInc.
PublishedbyElsevierLtd.
Allrightsreserved.
Keywords:Ceramics;Mechanicalproperties(fatigue,R-curves,crackbridging)1.
IntroductionWhilegrainbridginghasprovedtobeapotentmech-anismforcreatingtoughceramics[1–9],undercyclicloadsthesematerialsaretypicallysusceptibletoprema-turefailurebysubcriticalcrackpropagationowingtodegradationofthebridges[9–18].
Moreover,asbridgingonlydevelopswithcrackextension,thereisaninherenteectofcracksizeonthematerialtoughness[1–5]and,aswell,onthecyclicfatigue-crackgrowthresistance[11]forcracksshorterthanthesteady-statebridging-zonelengths.
Suchcrack-sizeeectsonfracturetoughnessareexpressedintheformofaresistance-orR-curve,whichallowsforfailurepredictionsbasedontheloadingconditionsandinitialawsize[19].
Forbridgingceram-ics,methodshavebeendevelopedforpredictingsuchR-curvesbasedonmeasuredbridgingstressfunctionsforcrackswithsteady-statebridgingzones[2,20–22].
However,suchmethodsarenon-conservativeforpre-dictingreliabilityundercyclicloadingconditionsowingtodamageinthebridgingzone;andthereiscurrentlynoanalogousmethodologyforusewithfatigue.
Further-more,asbridgingisachievedwithweakgrainbound-ariesthatpermitintergranularfracture,theabsolutethresholdforinitialcrackextensionisreducedversusthatfornonbridgingmaterialsandisfartooconserva-tiveforanalternativedesigncriterion.
Accordingly,thegoalofthepresentpaperistodeve-lopaframeworkforthepredictionofreliabilityinbridgingceramicsundercyclicloadingconditionsbasedonafatiguethresholdR-curve[23–25],whichgivesthethresholdasafunctionofcrackextension.
The1359-6454/$30.
002005ActaMaterialiaInc.
PublishedbyElsevierLtd.
Allrightsreserved.
doi:10.
1016/j.
actamat.
2005.
02.
018*Correspondingauthor.
Tel.
:+15417377027;fax:+15417372600.
E-mailaddress:kruzicj@engr.
orst.
edu(J.
J.
Kruzic).
ActaMaterialia53(2005)2595–2605www.
actamat-journals.
comproposedmethodisanalogoustothetraditionalfrac-turetoughnessR-curveapproach,andisachievedusingmeasuredbridgingstressdistributions.
2.
BackgroundOneconsequenceofmakinghigh-toughnessceramicshasbeenthatsuchmaterialsbecomesusceptibletofati-guefailureduetoprogressivedegradationofthetough-eningundercyclicloading.
Thisisincontrasttountoughenedceramicswhichareessentiallyimmunetocyclicfatigue,withintheprovisothattheymayexperi-enceenvironmentallyassistedsub-criticalgrowthduringcyclicloading.
Fatigue-crackgrowthratesinceramicsataspecicloadratio(R=Kmin/Kmax)tendtofollowaclassicalParispower–lawrelationship:dadNADKm;1whereAandmarescalingconstants,da/dNisthegrowthrate,DKisthestress-intensityrange(KmaxKmin),andKmaxandKminare,respectively,themaxi-mumandminimumvaluesappliedduringaloadingcycle.
TheParislawexponentistypicallyverylargeforceramics,i.
e.
,m$15–100;consequently,propagat-ingfatiguecracksquicklygrowtofailureunlesstheyareinadecayingDKsituation.
Becauseofthis,thefati-guethreshold,DKTH,belowwhichcracksarepresumednottopropagateundercyclicloadsisoftenconsideredasacriticalparameterinthefatigueofceramics.
1Inlightofthis,theprecisedenitionofthefatiguethresholdisacentralissue.
Themajorityoffatigue-crackgrowthtestsperformedtodateonceramicmaterialshaveusedmillimeter-scale(socalled''large'')cracksgrowinginstandardfracturemechanicsspecimens[9,12–16,29–31].
Theseexperimentstypicallyarebelievedtorepresentthebehaviorofcrackswithsteady-statebridgingzones,wherebridgesarecreatedanddestroyedataroughlyequalrateandthezonesizeisequilibratedateachKvalue.
Crack-sizeeectsonthefatiguethresholdarethusignoredinsuchexperiments,andthemeasuredfatiguethresholdsarenon-conservativewhencomparedtothosefromsmallercrackswithdeveloping,ortran-sient,bridgingzones[11].
Sincerelevantcracksizesinceramicsforstructuralusewilllikelybeseveralordersofmagnitudesmallerthanthoseinsuchtestspecimens,eectsofcracksizemustbeaddressedifthefatiguethresholdistobeusedasadesignparameter.
Oneapproachtoassessingtheeectsofdeveloping,ortransient,bridgingzonesonthefatiguethresholdistoconductdirectexperimentsonfunctionallysmallorshortcracks.
Inthecontextofthispaper,cracksareconsideredshortorsmallifcomparableinsizeto,orlessthan,oneofthreecharacteristicdimensions,namely,themicrostructure(microstructurally-small),theextentoflocalinelasticityaheadofacracktip(mechanically-short/small),and/ortheextentofthezonecausingcrack-tipshieldingorbridgingbehindacracktip(functionally-short/small)[32,33].
Shortcracksmeetsuchconditionsinonlyonedimension,i.
e.
,thecracklength,andsmallcracksmeetoneormoresuchconditionsinbothdimensions.
Afunctionally-shortcrackhaslimitedcrack-tipshielding,yetsamplesthemicrostructurestatisticallybecauseofitsextensivecrackfront[34].
Conversely,functionally-smallcracksaresmallinalldimensions,astypiedbyasmall,semi-ellipticalsurfaceaw.
Whensmallcracksarecompara-bleinsizetomicrostructuralentities,inadditiontoreducedshielding,biasedsamplingofmicrostructurallyweakpathsmayoccur.
Duetotheserestrictionsinshieldingandmicrostructuralsampling,thecrack-growthresistancesofmicrostructurallysmallcracksgenerallytendtobemorevariablebutattheextremeareamongthelowest.
Inaddition,residualstresseseitherfromcontactdamageorfromthermalexpansionmismatchmayactmorestronglyonsmallcracks;ifunappreciatedorunaccounted,theylowertheeectiveoroperationalR-curveforsmallcracks[35–37].
Whiledirectshortorsmallcracktestingseemsattractive,therearemanyexperimentaldicultiesincreating,propagat-ing,andmonitoringverysmallfatiguecracksinceram-ics.
Indeed,formaterialswithsteeplyrisingR-curves,mostattemptsatsuchexperimentshavefoundthatthepracticalcracksizesfortestingweretoolongcom-paredtothebridging-zonelengthtonoticeanysmall-crackeects[27,38–40].
Recently,wehaveproposedanalternativeapproach[11]whichrstinvolvesdeterminingthesteady-statecontributionofbridging,KSS,br,inreducingtheappliedstressintensity;superpositionisthenusedtosubtractthisbridgingstressintensityfromthemeasuredsteady-statethreshold,KTHSS;max,toobtainanintrinsicthresholdforthetipofanunbridgedcrack,KTHo;max,using[11]:KTHo;maxKTHSS;maxKSS;br:2Inceramicsthethresholddependsmorestronglyonthevalueofthemaximumstressintensity,Kmax,asopposedtothestress-intensityrange,DK,perse[9,12–14,41];consequently,fatiguethresholdsareexpressedhereintermsofKTHmax,notingthatthismayeasilyberelatedtoDKthroughtheloadratio,R=Kmin/Kmax.
Also,sincethereisnocyclicfatigueeectintheabsenceofbridgingforthemostbrittleceramics,KTHo;maxmaybethoughtofastheintrinsictoughnessofthematerial,Ko,viz.
:KTHo;maxKo:31Alternatively,thefatiguethresholdcanbedescribedintermsofthemaximumstressintensity,KTHmax,asEq.
(1)canberewrittenasda/dN=A0(DK)p(Kmax)nwhereA0=A(1R)nand(n+p)=m[26,27].
Withbrittlematerialssuchasceramics,n)p[28].
2596J.
J.
Kruzicetal.
/ActaMaterialia53(2005)2595–2605Thisapproachgivesaworst-casefatiguethreshold,Koforabridgingmaterialbelowwhichcracksofallsizesshouldnotpropagate,providedtheyarelargeenoughtostatisticallysamplethe''continuum''microstructure.
Adisadvantageofdesigningwiththismethod,however,isthatitistooconservative.
Inrealitymostshort/smallcrackswillarrestafterashortamountofsubcriticalgrowthduetotheincreasedbridging[11];accordinglyamethodologyisneededthatpredictstheconditionsun-derwhichshort/smallfatiguecrackswillarrestversusgrowtofailure.
Inordertoaddresstheseproblems,inthepresentworkweproposetheuseofafatiguethresh-oldR-curveanddemonstratehowthismaybederivedfromtraditionallong-crackfatigueexperimentsbydeterminingthebridgingstressesfornear-thresholdsteady-statefatiguecracks.
3.
Procedures3.
1.
MaterialsAcommercial99.
5%purealumina(AD995,CoorsTechnicalCeramicsCo.
,OakRidge,TN)waschosenasamodelmaterialduetothefactthatitexhibitslargesteady-statebridgingzones($2mm)nearthefatiguethreshold[11].
Thispermitteddirectmeasurementsoffatiguethresholdsoverarangeofcracksizesthatwereundertransientconditions,i.
e.
,wherethebridgingzonewasstillbeingdeveloped.
SuchexperimentsprovideddataforthefatiguethresholdR-curvewhichwasthencomparedwithpredictionsderivedfromadeducedbridgingstressdistribution.
Thealuminahadawidegrainsizedistribution,withanominal,averagesizeof$25lm;morecompleteinformationonthematerialmaybefoundinRef.
[11].
Additionally,agaspressuresintered,self-reinforcedSi3N4(with5wt%Y2O3and2wt%MgO,added,designatedYM-Si3N4)ceramicwasusedtodemonstratemethodsforpredictingthefatiguethresholdR-curveinamaterialwithsmallbridg-ingzones($100lm)whereshort/smallcracktestingisprohibitivelydicult.
Thereinforcinggrainshadameandiameterof0.
28lmwithanaspectratioof$7;detailsontheprocessingandmicrostructureofthismaterialcanbefoundelsewhere[42–44].
3.
2.
FatiguethresholdexperimentsFatigue-crackgrowthexperimentswereconductedonstandardcompact-tension,C(T),specimens(width,W%17and19mmfortheAl2O3andSi3N4,respec-tively;thickness,B%3–3.
5mm)ingeneralaccordancewithASTMstandardE647.
Completedetailsofthefa-tigue-crackgrowthproceduresareinRef.
[11];abriefsummaryofissuespertinenttothemeasurementoffati-guethresholdsispresentedhere.
Fatiguecrackswereini-tiatedfromstraightmachinednotches(lengthao%4–5mm)undercyclicloadingconditions(25Hzsinewave,loadratio,R=0.
1),afterwhichthecracksweregrowntoaspeciedlength,asmonitoredusingback-facestraincompliancemethods[45].
Notchrootradii,q,rangedfrom$15to150lm,withthesharpestnotchesusedforthesmallestcracksizes.
Inallcases,datacollectiondidnotbeginuntiltheamountoffatigue-crackexten-sionfromthenotch,Daf,exceededq,atwhichpointtheinuenceofthenotcheldonthestressintensitycouldbeconsideredtobenegligible[46,47].
Inordertomeasurethefatiguethreshold,theappliedstress-intensityrangewasreducedataroughlyconstantDK-gradient(=[dDK/da]/DK)of0.
08mm1.
Basedonpreviousresults[11],thisDK-gradientwaslowen-oughinAD995aluminatoachievesteady-statebridgingzonesforcrackswithDaf>2mmintherangeofgrowthratesfrom$108to1010m/cycle.
Usingthreemeth-ods,thefatiguethresholdwasmeasuredasafunctionofcrackextensionforDafrangingfrom60lmto6.
5mm,withthethresholdoperationallydenedasthelow-eststressintensityatwhichthefatigue-crackgrowthratecouldbemeasuredanddoesnoteverexceed$1010m/cycle.
FortheSi3N4,whichhasmorethananorderofmagnitudesmallerbridgingzone($100lm),onlythesteady-state(i.
e.
,long-crack)fatiguethresholdwasobtained.
3.
3.
QuanticationofbridgingstressesinAD995-Al2O3Asthebridgingstressdistribution,rbr(x),foranear-thresholdsteady-statefatiguecrackinAD995aluminawasdeterminedinapreviousstudy[11],themethodsareonlyoutlinedhere.
First,afatiguecrackwasgrownfor$3mm,i.
e.
,throughtheentirebridging-zonelength,nearthefatiguethreshold(da/dNL,integratingEq.
(7)givesthesteady-statebridgingcontribution,KSS,br,whichpertainsatKmaxandisinsensitivetocracklength,whileforDaL=2mm.
Becausetypicalfatiguethresholdsaremeasuredonnominallysteady-statecrackswhicharegenerallymanymillimetersinlength,Fig.
1(a)illustratesthenecessityofconsideringthefatiguepropertiesforrealisticawsizesinceramics,i.
e.
,inthetenstohundredsofmicrometers.
OneapproachtothisproblemistheuseofafatiguethresholdR-curve,showninFig.
2intermsofKTHmaxforAD995alumina.
ThedatapointsinFig.
2representmeasuredfatiguethresholdsplottedasafunctionofthefatiguecrackextension,Daf,fromthestarternotchwhenthethresholdwasmeasured.
ForcracksizeswhereDaf>L,thesefatiguethresholdscorrespondtosteady-statefatiguecracks,wherebridgesarecreatedandex-haustedatroughlyequalratesandthenatureofthebridgingzoneuniquelycorrespondstotheappliedDKlevel.
Suchsteady-statefatiguethresholdsshouldnotbehistory-orsize-dependent,meaningthatsteady-stateconditionsarealwaysmaintainedwhileapproachingthefatiguethreshold(aconditionmoreoftenassumedthanproven[11]).
Fortransientcracks(Daf102lm.
J.
J.
Kruzicetal.
/ActaMaterialia53(2005)2595–26052601plotted(Fig.
6)forstrengthversusinitialawsize,aicomputedusing:rKRpap;9adKappdadKRda:9bHereandinFig.
6,theawsize,a,referstothefulllengthofasurfaceaw,orhalf-lengthofasubsurfaceaw;theinitialawistakentobefreeofbridging.
Fig.
6demonstratestheutilityofthefatiguethresholdR-curvesbyplottingthepredictedmaximumcyclicstress,rmax,justsucienttocausefatiguefailure,essen-tiallythefatigueendurancestrengthforR=0.
1,versustheinitialawsize,ai.
Theendurancestrengthwascal-culatedanalogouslytothewaymonotonicstrengthispredictedfromfracturetoughnessR-curves:rmaxKTHmaxpap;9cdKmaxdaappdKTHmaxda;9dasshownschematicallyinFig.
7.
Usingsuchrelation-ships,predictionsofreliabilityrelevanttocyclicloadingconditionscanbereadilydeterminedintheformoftheendurancestrengthasafunctionofawsizeandcom-paredwithstrengthsformonotonicloading;thisisshowninFig.
6.
Formaximumapplied(cyclic)stressesbelowthislevel,crackswouldeithernotpropagate,orarrestaftersomeamountofcrackextension,whileforstressesabovethispredictedendurancestrength,thestructureshouldfailaftersomeamountofcycling.
DuetothehighParisexponentsinceramics,thenumberofcyclestofailurewouldbeexpectedtobesmall,whichprovidesthemotivationforreliabilitypredictionsbasedonfatiguethresholds.
Forcomparisontwovaluesof02468CRACKEXTENSION,a,af(mm)0246MAXIMUMSTRESSINTENSITY,Kmax(MPa√m)intrinsictoughnessfatiguethresholdR-curve(Kmax)R=0.
1fracturetoughnessR-curve(KR)R=1.
0unstablefracturestableformonotonicloadingunstableforcyclicloadingstableforcyclicandmonotonicloadingAD995alumina04080FATIGUECRAC(a)(b)KEXTENSION,af(m)12002468MAXIMUMSTRESSINTENSITY,Kmax,(MPa√KoYM-Si3N4estimatedfracturetoughnessR-curve(KR)R=1.
0fatiguethresholdR-curve(Kmax)R=0.
1stableforcyclicandmonotonicloadingstableformonotonicloadingunstableforcyclicloadingunstablefracturem)Fig.
5.
PlotofthefatiguethresholdR-curveandfracturetoughnessR-curvefor(a)AD995aluminaand(b)YM-Si3N4,illustratingregimesofstablebehaviorforbothmonotonicandcyclicloadingconditions.
110100100010000INITIALFLAWSIZE,ai(m)10100100010000STRENGTH,σmax(MPa)AD995aluminafatigueR=0.
1typicalsapphirefractureR=1YM-Si3N4fatigueR=0.
1AD995aluminafractureR=1YM-Si3N4fractureR=1transgranularSi3N4fractureR=1virginretainedvirginretainedFig.
6.
PredictedendurancestrengthsasafunctionofinitialawsizeforAD995aluminaandYM-Si3N4basedonthefatiguethresholdR-curvesinFig.
4.
Forcomparison,thepredictedoverloadfailurestrengthsforAD995Al2O3,basedonmeasuredR-curvedatafromFig.
5[11],andforsapphirearealsoshownbothforvirginmaterialandfortheas-retainedstrengthforawsgrowntoarrestundercyclicloading.
SimilarstrengthsareshownfortheYM-Si3N4,basedontheestimatedR-curvefromFig.
5andforaSi3N4thatexperiencestransgranularfracture.
Finally,retainedoverloadstrengthsarepredictedforbothAD995aluminaandYM-Si3N4,representativeofsamplesthatwerefatiguecycledattheendurancestrengthandwhichexperiencedsomecrackextensionpriortoarrest,givinglowerstrengththanvirginmaterial.
2602J.
J.
Kruzicetal.
/ActaMaterialia53(2005)2595–2605strengthundermonotonicloadingareshown:oneforvirginmaterialandthesecondforthecaseinwhichaninitialawhasrstgrownundercyclicloadinguntilar-restatthefatiguethresholdcondition,termedthere-tainedstrength.
Forthelatter,thestartingcracksizeforthemonotonicreloadingisconservativelytakenastheinitialawsizeplustheincrementofcrackextensionundercyclicloadingattheendurancestrength(i.
e.
,afinFig.
7),butforbothpartsallbridgingisdiscounted.
AlsoshowninFig.
6aretheexpectedpropertiesforsapphirebasedondatafromRefs.
[58–61],andfortransgranularSi3N4[57].
ThestrengthsofbothsinglecrystalandpolycrystallineAl2O3arereducedbytestinginmoistenvironments,whicheveninuencestheR-curves,asshownrecentlyinRef.
[56].
Tomakecom-parisonsmoremeaningful,strengthsforsapphirewerecomputedusingatoughnessof0.
75Kc;thisisusedasslowcrackgrowthcurvesshowthatthesubcriticalcrack-growththresholdsoccuratabout0.
5Kc,whereKcisforfractureininertconditionsorathighervelocity[59,62–64],andthepresenttestingisatintermediatevelocities.
NotethatalthoughbothAD995aluminaandYM-Si3N4havesimilarintrinsicfatiguethresholds,KTHo;max,andlongcracktoughnesses,theSi3N4-basedceramichasmuchhigherpredictedmonotonicstrengthsandendurancelimitsatrelevantawsizesduetothemoresteeplyrisingR-curves.
Thisdemonstrateshowsimplyusingtheintrinsicfatiguethresholdasadesignparame-terwouldbeoverlyconservativesincesubcriticallygrowingcrackswilloftenarrestinmaterialswithrisingfatiguethresholdR-curves.
ItisalsoevidentfromthesecurvesthatamaterialwiththerapidlyrisingandrobustR-curvecanprovidesignicantadvantageovertheuntoughenedmaterialinbothmonotonicandfatiguesituations.
Incontrast,forthisAl2O3material,theben-etsoftougheningarevirtuallyeliminatedforanyawsizesofinterestbycomparisonwiththebehaviorofsap-phireundercyclicloading.
Toreinforcethesecompari-sonsitisnotedthattheunreinforcedmaterialsarenotexpectedtoexhibitcrackgrowthinducedbycyclicfati-gue,sothatthecurvesshownasvirginstrengthessen-tiallyalsodenotetheenduranceandretainedstrengths.
5.
2.
EectsofloadratioWhenconsideringtheexpectedeectofloadratioonthefatiguethresholdR-curvesseeninFig.
4,itisnotedthatthefracturetoughnessR-curve,whichexperiencesnocyclicloading,isessentiallydeterminedataloadra-tioofR=1,i.
e.
,theminimumandmaximumloadsareequal(Fig.
5).
ItisbeexpectedthenthattheR=0.
1fa-tiguethresholdR-curveshouldbeconservativeforloadratioshigherthan0.
1sincethefatiguebehaviorshouldapproachmonotonicfractureastheloadratioisin-creased.
Thisisbecauseastheloadratioisincreased,thevalueofDK,whichdeterminestheamountofbridg-ingdegradation,fallsforagivenvalueofKTHmax[41].
In-deed,thisnotionissupportedbyfatiguedataforbridgingceramicswhichshowanincreaseintheKTHmaxfa-tiguethresholdwithincreasingloadratiooverarangeofRfrom0.
1to0.
9[14,65,66].
Thus,forpositiveloadra-tioslessthanone(i.
e.
,tension–tensionloading),itisex-pectedthatmeasurementsattheminimumexpectedserviceloadratiowouldresultinaconservativefatiguethresholdR-curveformakingpredictionsofreliability.
Itthenfollowsthatpredictionsforpositiveloadra-tiosmaybenon-conservativeforreversedloading(i.
e.
,negativeloadratios)sinceatagivenKmaxvalue,DKwouldbehigherandtheamountofbridgedegradationwouldbeexpectedtobemoresevere.
Unfortunately,experimentaldataforR<0islimited(e.
g.
,Refs.
[67,68])anddirectcomparisonstodatafor0However,thisnotionwouldbeinagree-mentwithcompression-compressionfatigueresultswhereRvaluesexceedone;here,thefatigueresistanceislowerathigherloadratios(rangingfrom2to8)[69],suggestingthatcompressivecyclingisprogressivelymoredamagingwithincreasingloadratio.
5.
3.
FinalcommentsOnlyonesampleoftheSi3N4wasusedinthepresentworkandservedtodemonstratetheapplicabilityofthe0FATIGUECRACKEXTENSION,afMAXIMUMSTRESSINTENSITY,Kmaxσmax<σenduranceσmax=σenduranceaiaffatiguethresholdR-curveFig.
7.
SchematicillustratinghowthefatigueendurancestrengthmaybefoundusingthefatiguethresholdR-curveforaawwithinitiallength,ai.
Additionally,thenalcracklengthaftercrackarrest,af,maybedetermined,whichmaybeusedtosubsequentlyestimatetheretainedfracturestrengthusingthefracturetoughnessR-curve.
J.
J.
Kruzicetal.
/ActaMaterialia53(2005)2595–26052603proposedmethods.
ToobtainaconservativefatiguethresholdR-curvefordesignpurposes,however,moreexperimentswouldbeneededtoaccountforscatterinthefatiguethreshold,asseenforAD995aluminainFigs.
1(a)and2.
Moreover,aswithAD995alumina,caremustbetakentodeterminethepredictedfatiguethresholdR-curveusingconservativevalues.
Themethodspresentedherearestrictlyonlyapplica-bletofunctionally-short/smallcracks,andtheintegralinEq.
(7)shouldberedonetoreectthepertinentgeome-try.
Wherecracksizesapproachthescaleofcharacteris-ticmicrostructuralfeatures,additionaleectsduetolimitedsamplingofthemicrostructureandresidualstressesmayplaymoredominantroles.
Lawnandcoworkers[35–37]haverecognizedandmodeledtheef-fectsoftheadditionaldrivingforcesduetoresidualstresses,e.
g.
,fromthermalexpansionmismatch.
Tensilestressesthatactaroundagrainunderresidualcompres-sionaidinitialextensionofasmalladjacentaw,withthiseectdecayingasthecrackgrowsoutoftheresidualeld.
Ifunaccounted,thiscanreducetheeectiveR-curveforextensionoverdimensionsoftheorderofthesizeofthestressedgrain,eventotheextentofallow-ingsomespontaneousfracture.
However,asthecrackgrowsbeyondthisresidualeld,theeectiveR-curvecanriseveryrapidlyandmayquicklyapproachthatforashortcrack.
Inthatinstance,theeectistocausealimitingstrengthatsmallawsizesratherthantheusualcontinuedriseinthestrength-awsizecurve.
Sucheectscouldbeaddedanalyticallyincombinationwiththeexperimentally-derived,short-crackR-curves.
AlsochemicalormicrostructuralheterogeneitiesmayreduceearlypartsofanR-curveandsimilarlylimittheachiev-ablestrengthatsmallsizes.
Finally,theprinciplesdescribedheremaybeex-tendedtootherbridgingmaterialsandshouldnotbelimitedtoceramics.
Forbrittleandsemi-brittlematerialssuchasbridgingintermetalliccompoundsandbrittlemetalalloys(e.
g.
,belowtheductile-brittletransitiontemperature)whichhavesimilarlyhighParisexponents(e.
g.
,seeRefs.
[70,71]),thefatiguethresholdshouldalsobethemostrelevantdesignparameterforcyclicloadingconditions,andasimilarmethodologycouldeasilybeadoptedprovidedaccuratebridgingstressprolescanbedetermined.
Furthermore,somehigh-cyclefatiguesituations,e.
g.
,bladesingas-turbineengines,involveveryhighcyclingfrequenciessuchthatevenrelativelyductilemetalssuchastitanium-andnickel-basealloyswillfailquicklyifthefatiguethresholdisexceeded,pro-vidinglimitedusefulservicelife.
Formaterialsusedinthesehigh-cyclefatigueapplicationsthatexperiencecrackbridging,methodssimilartothosedescribedherecouldbeapplied.
Itisimportanttonote,however,thatasmaterialsbecomemoreductiletherewillbeanincreasingroleofcrackclosureinaectingthefatiguethresholdwhichmustbeconsideredaswell.
6.
ConclusionsBasedonanexperimentalstudyoffatigue-crackgrowththresholdsingrain-bridgingceramics(Al2O3andSi3N4),thefollowingconclusionsaremade:1.
Theeectofcracksizeonthethresholdforfatigue-crackgrowthmaybeexpressedintermsofafatiguethresholdR-curve,whichisanalogoustotheclassicalfracturetoughnessR-curveandmaybeusedforpre-dictionsoftheendurancestrengthundercyclicload-ingconditions.
2.
Usingmeasuredbridgingstressproles,thefatiguethresholdR-curveforshortcracksmaybepredictedbasedsolelyonresultsfrommanymillimeterlong,through-thickness,fatiguecracks.
Directcompari-sonsofpredictionstoexperimentallymeasuredfati-gue-crackgrowththresholdsinAD995Al2O3overarangeofcracksizesdemonstratetheaccuracyofsuchmethods.
3.
Themethodspresentedhereoerabasisforanalter-nativetodicultandcostlyshort/smallcrackfatigueexperimentsneededtodeterminethefatiguebehav-ior,i.
e.
,thethresholdforcrackgrowthandtheretainedstrengthafterfatiguecrackarrest,atrelevantcracksizesforceramicmaterials.
However,workremainstoassessmodicationsproposedforapplica-tiontomicrostructurally-smallcracks.
4.
Thesemethodsprovideatangiblemeanstoassesstheexpecteddegradationoftougheningmechanismsforcracksinceramicssubjectedtocyclicloading.
Althoughthecrackingresistanceisinvariablydegradedbycyclicloading,insomeinstancessuchdegradationosetsanybenetsoftougheningforawsizesofanyinterest;whereas,insituationswithsucientlyrapidlyrisingandresilientR-curves,thenevenwiththedegradedbridging,shortcrackswillarrestafterasmallamountofgrowth,leavingsignif-icantusefulresidualstrength.
Suchbehaviorcanbequantitativelydescribedbasedonrelativelyaccessi-blemeasurementsofeitherthresholdR-curvesdirectlyorbridgingfunctionsfromlong,near-thresholdfatiguecracks,andthesemethodsmaybeextendedtobridgingmaterialsotherthanceram-icsaswell.
AcknowledgmentsThisworkwassupportedbytheDirector,OceofScience,OceofBasicEnergySciences,DivisionofMaterialsSciencesandEngineeringoftheUSDepart-mentofEnergyunderContractNo.
DE-AC03-76SF00098.
TheauthorsgratefullyacknowledgeDrs.
M.
J.
HomannandR.
L.
Satet(UniversityofKar-2604J.
J.
Kruzicetal.
/ActaMaterialia53(2005)2595–2605lsruhe)forprovidingtheSi3N4materialandMs.
K.
L.
BreedenforaidinanalyzingtheRamanspectra.
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