Fatigue crack nucleation assisted by thermal activation

疲劳裂纹

MaterialsScienceandEngineeringA272(1999)5–9

http://wendang.chazidian.com/locate/msea

Fatiguecracknucleationassistedbythermalactivation??

M.E.Fine*,V.Stolkarts,L.M.Keer

NorthwesternUni6ersity,2145SheridanRoad,E6anston,IL60208-3109,USA

Abstract

ExperimentalevidenceispresentedthatveryearlyinthefatiguelifetimeofSn–Pbeutecticsolderthefatiguedamageisgovernedbymicrocracknucleation.Amodelforthermallyassistedfatiguecrackinitiationispresentedusingstandardequationsfornucleationinphasetransformations.Predictionsofthemodelareingoodagreementwithnucleationdataforthissolder.©1999ElsevierScienceS.A.Allrightsreserved.

Keywords:Fatigue;Crack;Nucleation;Activation;Thermal;Solder

1.Introduction

PhasetransformationshavebeenamajorelementinHerbHerman’steachingandresearchfromthebeginninggoingbacktohisPhDThesisandhissub-sequentpapersonGuinier–PrestonzoneformationinAlbasealloys[1–5].Muraandcoworkers[6,7]showedthatinitialfatiguecrackformationisanucle-ationprocessinthatthereisanenergybarriertotheformationofafatiguecrack.Thepredictedinitialsizeisinthenanometerrangeandthusverydif?culttoinvestigate.Whenfatiguecracksare?rstseenevenwiththehighestresolutiontechniqueavailable,muchcrackgrowthbeyondtheinitiationstagehasalreadyoccurred.Kwonetal.[8,9]observedfatiguecracksassmallas100nmindepthalongslipbandsinCubytransmissionelectronmicroscopeexaminationofshadowedreplicasoffatiguedspecimensurfaces.Thekineticsofnucleation,however,canbestudiedwith-outknowingthejustnucleatedcracksizeifthefa-tiguecracksinitiatedarenoninteracting.Somemeasuresofthecrackdensitiesforgivenfatiguecon-ditions(cyclenumber,frequency,amplitude,meanstress,waveform,temperature)areneeded.Thesitua-tionissimilartonucleationofcrystalsinglassceram-

ThispaperisdedicatedtoProfessorHerbertHermanontheoccasionofhis65thbirthday.

*Correspondingauthor.Tel.:+1-847-491-4322;fax:+1-847-491-7820.

E-mailaddress:m-?ne@nwu.edu(M.E.Fine)

??

icsbecausethemaximumnucleationratetemperatureismuchbelowthemaximumcrystalgrowthratetem-perature[10].Specimensmaybenucleatedatalowtemperatureandthengrownatahighertemperatureuntilthecrystalsareobservable.Excellentnucleationkineticdatahavebeenobtainedinthisway[10].

IneutecticSn–Pbsolderthefatiguecracksinitiateatinterfacesandgrowtotheextremitiesoftheindi-vidualfacetsinitiallywithminimalinteractionbe-tweenneighboringcracks[11].ThefatiguedamageinSn–Pbeutecticsolderischaracterizedbyadecreaseinmaximumloadinstraincontrolledtests.Afteraninitialtransient,thedecreaseinloadoccursatacon-stantrate,i.e.d|/dN(where|isthenominalmaxi-mumtensilestressinthecycleandNisthecyclenumber)isconstant,untilarapiddropinloadoccurswhenunstablecrackpropagationtofailureoccurs.AnexampleisshowninFig.1.Shortfatiguecracksatinterfacesarereadilyseenveryearlyinthefatiguecyclingclosetotheonsetofthesteadystatesloperegion.Thisregioncannotbefatiguecrackpropaga-tioncontrolledbecausetherateofdropwouldin-creaseasthecracksgetlongerduetotheincreaseinDK,thestressintensityrange.Alsotherateofnucle-ationwouldincreasewithcyclingifdislocationaccu-mulationoccursatinterfacesandaconstantslopewouldnotbeobserved.Thehomologoustemperatureforthissolderisgreaterthan0.6atroomtempera-ture,sodynamicrecoveryisexpectedtooccurgivingasteadystatedislocationstructureatthemaximumloadpointinthecycle.

0921-5093/99/$-seefrontmatter©1999ElsevierScienceS.A.Allrightsreserved.PII:S0921-5093(99)00456-6

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6M.E.Fineetal./MaterialsScienceandEngineeringA272(1999)5–9

InthispaperweapplythemathematicsofnucleationtoasetoffatiguecracknucleationdatadeducedforSn–Pbeutecticsolder.

2.Themathematicsoffatiguecracknucleationapplica-bletoSn–Pbeutecticsolder

IntheMura–Tanaka–Nakasonetheoryoffatiguecracknucleationthefreeenergyinitiallyincreaseswithcracksizebecauseoftheneedtocreatesurfaces.Intheirmodelduringfatiguecyclingthenumberofdislo-cationsalongaslipbandincreasesasthecyclenumberincreases.Whentheenergyofthedislocationsinaregionofthebandreachesacriticalvalue,thenafatiguecrackofcriticalsizeformsspontaneously.Thepredictedcracksizeisapproximatelyanorderofmag-nitudesmallerthanthesmallestcracksobservedbyKwoninthislaboratory.Athighhomologoustempera-tures,suchasroomtemperatureinSn–Pbsolders,recoverypreventsanincreaseindislocationdensityafterasteadystatedislocationstructureisreachedandifthemaximumappliedstressisnothighenoughtofracturethematerial,thenthermalactivationaidsthecracknucleation.Thisisthebasisofthefollowingdevelopment.

Thefreeenergychange,DG,toformacrackofradius,a,followingthetheoryofMuraetal.,is:DG=?W1?W2+(2ks?ki)ya2

(1)

TheMuramodelistwodimensionalsothesurfacetermhasbeenmodi?edforthreedimensions,thatisthecrackhasbeenassumedtobepennyshapedofarea,ya2,insteadofaline.Sincetwosurfacesareformed,thefreesurfaceenergy,ks,ismultipliedby2.Thefatiguecrackisassumedtoformonaninterface,sothe

Fig.1.Evolutionofpeakstresswithcyclingat0.6%strainrangeand1sramptime.

freeenergyoftheareaofinterfacelost,ki,mustbesubtracted.W1isthedislocationenergylostwhenthecrackformsandW2isthestoredelasticenergyreleasedoncrackformation(excludingdislocationenergy).Theformerisaconstantwithanincreaseincyclenumbersinceasteadystatehasbeenassumed.Duringthecycledislocationmotionoccurstoaccomplishtheplasticdeformationbutthedislocationstructureatmaximumstressinthecycleisassumedconstantinthemodel.Ifaplanepennyshapedcrackisperpendiculartotheappliedstress,thenthestoredelasticenergy,W2,isgivenby:

W8(1?w2)2=

3E

|2a3

(2)

where|,inthepresentcase,isthemaximumappliedstressinthecycle,EisYoung’smodulusandwisPoisson’sratio.

ThenDGmaybedifferentiatedwithrespecttoaandsetequaltozerotoobtainthenucleationconditions:dDG/da=?8(1?w2)|2a2/E+2aky=0

(3)

wherek=2ks?ki.SolvingEq.(3)forthecriticalcracksize,a*gives:a*=

Eky4(1?w2)|2

(4)

Substitutinga*intoEq.(1)togiveDG*,theactiva-tionenergyforfatiguecracknucleation,gives:E2k3y3

DG*=12(1?w2)24

?W1

(5)

Therateofnucleation,i.e.thenumberofnucleatedcracksperunitvolumepercycle,dn/dN,isgivenby:dnkT?DdN=pt??G*hexpkT

??

(6)

wherepisthenumberofpotentialnucleationsitesperunitvolume,tistimeinsequaltothatportionofthecyclewherethestressisnearmaximum,kisBoltz-mann’sconstant,TistheabsolutetemperatureandhisPlanck’sconstant.InEq.(6)kT/histhefrequencyofatomicvibrations,exp(?DG*/kT)istheprobabilitythatagivennucleationsitewillnucleateacrackwhenthestressisatornearthemaximuminthecycle.Thenumberofpotentialnucleationsitesisconsideredlargecomparedtothenumberofcracksnucleated,i.e.pisassumedconstant.

Thedensityofanassemblyofpennyshapedmicroc-rackswithanaverageradiusaisgivenby??=na3,wherenisthenumberofmicrocracksperunitvolume.Theincreaseinmicrocrackdensitypercycleis:

D??cycle=

d??dndN=dN

3

(7)

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M.E.Fineetal./MaterialsScienceandEngineeringA272(1999)5–97

wheretherateofnucleation,dn/dN,isgiveninEq.(6).Therateofnucleationofmicrocracksisafunctionofshearstrainrangepercyclealongaslipplane.Themicrocrackdensityincreasepercycle,D??cycle,isafunctionofDkcycle:D??cycle=g(Dkcycle)

(8)

BarenblattandBotvina[12]arguedthatamicrocracknucleationgoverneddamageprocesscouldbeconsid-eredself-similar.Self-similaritymeansthatifwecom-paretwomicrocrackdensityincrementsD??cycle1andD??cycle2,correspondingtotwodifferentrangesofshearstrainDkcycle1andDkcycle2,thentheratioofD??cycle1overD??cycle2isafunctionoftheratioofDkcycle1overDkcycle2,i.e.:D??cycle1

??Dkcycle1cycle2=g

cycle2

(9)

Itcanbeshown,similartoBarenblattandBotvina[12],thatfromEq.(9)anincreaseinmicrocrackdensitypercycleisequalto:D??cycle=

d??

=(Dkcycle)??dN

(10)

where??isamaterialconstantandNisthecyclenumber.ThisisaCof?n–Mansontyperelationshipformicrocracknucleation.ThenmicrocrackdensityafterNcyclesisequalto:??=D??cycleN=(Dkcycle)??N

(11)

BudianskyandO’Connell[13]usedtheself-consistentmodeltoderiveexpressionsfortheeffectiveelasticandshearmoduliasfunctionsofmicrocrackdensity,??,forbodiescontainingarandomdistributionof?atcircularcracks??asfollows:E=E1?16(1?(w¯)2)(10?3¯w)45(2?w¯)

??

v¯=v??

1?

32(1?w¯)(5?w¯)

45(2?w¯)

??

n

n

(12)

wheretheeffectivePoisson’sratio,w¯,canbefound

from:??=

45(w?w¯)(2?w¯)

16(1?(w¯)2)(10w?¯w

(1+3w))(13)

wherevisshearmodulusofdamage-freematerial.

Underuniaxialloadingtheeffectivestressisgivenby:|¯=|

E

E

(14)

whereE

/Eisafunctionofmicrocrackdensity??,whichisgiveninEq.(11).EstimationsofDkcycleand??inEq.(11)canbemadebyanalyzingtheexperimentallydeter-minedvaluesofeffectivestress,|¯,asafunctionof

numberofcyclesN.ThentheGibbsfreeenergyofmicrocracknucleation,DG*,canbedeterminedbycombiningEqs.(6)and(7)andEq.(10).Itisgivenby:DG*=?kTln

??

h(Dkcycle)??

ptkTa3

(15)

wherethemicrocrackradius,a,isgiveninEq.(4).

Thedislocationenergy,W1,canbefoundbyequat-ingexpressionsforDG*inEqs.(5)and(15):E2k3Wy3

h(Dkcycle)??1=12(1?w2)24+kTln

??

ptkTa3

(16)

3.AnalysisoffatiguedatatoobtainnucleationdataAnoftenusedmeasureoffatiguedamageinstraincontrolledexperimentsofeutecticSn–Pbsolderisthedeclineinpeakstress.Atypicalcurveshowingtheevolutionofpeakstresswithcyclingfor63Sn–37Pbsolderhasthreedistinctregionsrepresentingdifferentstagesoffatigue(Fig.1).The?rststageistheshortest,usuallynomorethan100cycles,andischaracterizedbyaninitialsharpdropinpeakstressfollowedbycontinuousreductionatadecliningrate.Thesecondstagehasthelongestdurationandistypi?edinthissolderbyanalmostconstantrateofdecline,i.e.thisregionisrepresentedbyanearlystraightline.Finally,thethirdstageshowstheaccelerateddeclineinpeakstressleadingtoultimatefailure.Similartocreep,thesethreestagescanbecalledtransient,steadystateandtertiaryfatigue.Theinitialrapiddropinpeakstressisduetomicrostructuralchanges.Themicrostructure,exceptformicrocracking,isinsteadystateaftertheinitialdrop.

Steadystatecreepisusuallyexplainedonthebasisofabalancebetweenthecompetingprocessesofstrainhardeningandrecovery.Steadystatefatigueineutectictin–leadsolder,onthecontrary,cannotbeat-tributabletoabalancebetweenhardeningandreduc-tioninoverallstressduetomicrocracks,sincenoisotropichardeningwasobservedinourexperiments,i.e.W1isconstant,andkinematichardeningshouldbeexcludedduetothesymmetrybetweenloadingandunloadingpasses.Whiletheoverallmaximumstressisreducedfromtheformationoffatiguecracks,thelocalmaximumstressingoodmaterialisassumedtoremainconstant.

Aconstantrateofpeakstressdeclineduringsteadystatefatigueexcludesmicrocrackgrowthasthemainsourceofdamageevolution,since,ifthiswerenottruetheratewouldnotbeconstant,butwouldincreasewithcracklengthfollowingaParislikerelationship.If,insteadystatefatigue,therewasnotasteadystatemi-crostructure,thenthenucleationratewouldbeex-pectedtoincreaseordecreasewithcycling,depending

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8M.E.Fineetal./MaterialsScienceandEngineeringA272(1999)5–9

onwhetherthedislocationdensitywasincreasingordecreasingwithcyclingorwhethersomeotherchangeinmicrostructurewasoccurringsuchasphasegrowth.Cutiongco[11]observedmicrocracksincriss-crossingshearbandsonthesurfaceoffatigued63Sn–37Pbsolder.Themicrocrackswerelocatedbe-tweenSn–SngrainboundariesandSn–Pbphaseboundaries.Thenetworkofmicrocrackseventuallylink-uptoformmacrocracks.Thisphysicalevidenceallowsustomakeanassumptionthatmicrocrackgrowthin63Sn–37Pbsolderislimitedtotheareaofgrainfacets,anddamageevolutionduringsteadystatefatigueisdominatedbymicrocracknucleationasstatedinthemodel.

Cutiongco[11]collectedextensiveexperimentaldataonfatigueinSn–Pbeutecticsolder.Heperformedisothermalstraincontrolledcyclictestsat25,50,65,80and100°C.Thestrainrangewasfrom0.3to1%andtheramptimefrom0.5to120s.AllshowedsimilarbehaviortothedataplottedinFig.1.Analy-sisofthesedatashowedthattheloadingamplitudehadamajoreffectontherateofmicrocracknucle-ationinSn–Pbeutecticsolder,whiletheloadingfre-quency,temperature(inisothermalconditions)andinitialmicrostructureplayedonlymarginalrolesovertherangesexamined.By?ttingtheexperimentaldataontheevolutionofpeakstresswithcyclingtothevaluesobtainedfromEq.(14),itwasdeterminedthatinEq.(11)??=3andDkcycle=5.1Dm,whereDmistheappliedtensilestrainrange.Itfollowsthatthein-creaseinmicrocrackdensitypercycleisequalto:D??cycle=133Dm3

(17)

andtheGibbsfreeenergyinEq.(15)canbewrittenas:

DG*=?kTln

??

133h(Dm)3

ptkTa3

n

(18)

SincemostofthemicrocracksineutecticSn–Pbsolderinitiateatinterfacesandgrowtotheextremi-tiesoftheindividualfacets,anaverageradiusacanbetakentobeequaltoahalfoftheaveragefacetsize.If,forexample,a=3×10?6andDm=0.006,thenanincreaseinthenumberofmicrocracksinonesawtoothedcyclefollowsfromEq.(7)andisequaltodn/dN=D??cycle/a3=133Dm3/a3=1012m?3.

Thepredictedradiusofanucleatedmicrocrackac-cordingtoEq.(4)amongotherthingsdependsonthesurfaceandinterfaceenergiesandappliedstress.Whiletheappliedstressdeclineswithcycling(Fig.1),atthemicrolevelitisconsideredconstantasalreadydiscussed.Taking|=45MPa,k=1Jm?2,E=35GPaandw=0.37,thenucleatedcrackradiusa*=15mm.Thisvalueismuchlargerthanthe100nmsizecracksobservedinCu.Themaximumlocalstressattheinterfacewherethecracknucleatesmaybelarger

than45MPa.However,doublingthestressreducesthepredictedcracksizetoonly4mm.Thektermmaybetoolarge.Dislocationsareexpectedtosegre-gatetointerfacesduetoslipdiscontinuity.ThiswouldincreasetheatomicmismatchattheinterfaceandincreasekiinEq.(1).Reducingkto0.1Jm?2reducesa*to1.5mm.Acombinationofthetwoeffectscouldreducea*toavaluemoreinlinewiththesmallestcracksobservedinCu.However,thefa-tiguecracknucleationinCuisexpectedtobedomi-natedbyadifferentprocess,adislocationaccumulationprocess,andthejustnucleatedcracksinthissoldermayactuallybelargerthaninCu.Carefulresearchisneededtomeasurethesmallestcracksob-servableandtodeterminethedislocationstructureatandnearinterfacesduringfatigue.

Ifweassumethematerialisdividedinto6-sidedcubes6mmonaside,thenumberofsharedinter-faces,p,isapproximately14×1015m?3.Ift=0.1s,a=1.5×10?6mandDm=0.006,thentheGibbsfreeenergyofactivationatroomtemperature,DG*:?0.4×10?20J.Thedislocationenergythencanbewrittenas:

W:E2k3y3

112(1?w2)24

+DG

(19)

Ifk=0.1Jm?2,thenW1:10?12J.WithoutthedislocationterminDG*,thepredictednucleationratewouldbeessentiallyzero;W1issubstantiallylargerthan?DG*.Whilethedislocationenergyhasbeentakenasconstant,inSn–Pbeutecticsolderatroomtemperatureafteraninitialtransient,becauseofthehighhomologoustemperature,asteadystatedisloca-tionstructuresuchasacellstructurewillform.Thesteadystatedislocationdensitymaybelargerthantheinitialdislocationdensity.Fatiguecracksareas-sumedtoformwherenodesofthedislocationstruc-tureareclosetointerfaces.Thedislocationenergylostwhenacrackforms,W1,mustbelargeenoughtosuf?cientlyreducetheactivationenergytogivetheobservednucleationrate.

4.Conclusions

ThesteadystatefatiguedamageregioninSn–Pbeutecticsolder,whered|/dN(|isthemaximumstressandNisthecyclenumber)isconstant,wasinterpretedasbeingduetoaconstantrateoffatiguecracknucleation.Themodelpresentedtreatsmicroc-racknucleationwhenthedislocationdensitystaysal-mostconstantduetorecovery.Themodelallowspredictionsofthesizeofthenucleatedmicrocrackandthedislocationenergylostwhenacrackforms.Itfurtherleadstoestimationsofthetotallengthofthedislocationsneededtobeconcentratedatthenu-

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M.E.Fineetal./MaterialsScienceandEngineeringA272(1999)5–99

cleationeventasfollows:takingthedislocationenergy,W1,tobe10?12J,theenergyperatomiclengthofdislocationtobe5×10?19J[14],andtheatomicdistancealongthedislocationtobe0.3nm(theaveragebetweenSnandPb)thetotallengthofdislocationsneededtobeconcentratedinthenucleationeventis600mm.Thiswouldbeintheformofapileuporadislocationcellortangle.Transmissionelectronmicro-scopeexaminationtodeterminethedislocationar-rangementsneartheinterfacesisneededtogivefurtherguidancetothetheorypresentedaswellastheinitialcracksizedeterminationalreadymentioned.Possiblyothermaterialswherethereisaconstantd|/dNregionearlyinthefatiguelifecanbetreatedsimilarly.

Acknowledgements

TheauthorsaregratefultotheSemiconductorRe-searchCorporationforsponsoringthisresearch.

.

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