A numerical fracture analysis of indentation into thin hard films on soft substrates

压痕断裂的数值分析

EngineeringFractureMechanics70(2003)

1323–http://wendang.chazidian.com/locate/engfracmech

Anumericalfractureanalysisofindentationinto

thinhard?lmsonsoftsubstrates

K.Sriram,R.Narasimhan*,S.K.Biswas

DepartmentofMechanicalEngineering,IndianInstituteofScience,Bangalore560012,India

Received4December2001;receivedinrevisedform10April2002;accepted30April2002

Abstract

Inthispaper,?niteelementsimulationsofsphericalindentationofathinhard?lmdepositedonasoftsubstratearecarriedout.Theprimaryobjectiveofthisworkistounderstandthemechanicsoffractureofthe?lmduetoformationofcylindricalorcircumferentialcracksextendinginwardsfromthe?lmsurface.Also,theroleofplasticyieldinginthesubstrateontheabovemechanicsisstudied.Tothisend,theplasticzonedevelopmentinthesubstrateanditsin?uenceontheloadversusindentationdepthcharacteristicsandthestressdistributioninthe?lmare?rstexamined.Next,theenergyreleaserateJassociatedwithcylindricalcracksiscomputed.ThevariationofJwithindentationdepthandcracklengthisinvestigated.Theresultsshowthatforcrackslocatedneartheindenteraxisandatsmallindentationdepth,Jdecreasesoverarangeofcracklengths,whichimpliesstabilityofcrackgrowth.Thisregimevanishesasthelocationofthecrackfromtheaxisincreases,particularlyforasubstratewithlowyieldstrength.Finally,amethodforcombiningexperimentalloadversusindentationdepthdatawithsimulationresultsinordertoobtainthefractureenergyofthe?lmisproposed.

Ó2002ElsevierScienceLtd.Allrightsreserved.

Keywords:Thinhard?lms;Indentation;Finiteelements;Fracture;Energyreleaserate

1.Introduction

Itisrealisedtodaythatoneofthemoste?ectivewaystoprotectanengineeringcomponentoperatinginerosive,corrosiveorhightemperatureenvironmentunderheavycontacttractionistocoatitthinlywithahardmateriallikediamondoraceramic.Hardcoatingsonsoftsubstratesmaybesingleormulti-layeredepitaxialwiththicknessrangingfromafewnm(asinthecaseofdiamond-likecarboncoatingsusedinmagneticrecording)to100lm.Insuchsituations,themechanicalintegrityofthe?lmshouldbepreserved.

Thestressesresultingfromcontacttractionsonthin?lmsaregenerallyveryhigh.Thehardnessofthe?lmenablesittoresistplasticyieldingthatiseasilyinducedintheunderlyingsubstrateevenwhenthetractionsaresmall.However,undertheseconditions,di?erentfailuremodesmayoperateinthe?lm.

The*Correspondingauthor.Tel.:+91-80-394-2959;fax:+91-80-360-0648.

E-mailaddress:narasi@mecheng.iisc.ernet.in(R.Narasimhan).

0013-7944/03/$-seefrontmatterÓ2002ElsevierScienceLtd.Allrightsreserved.

doi:10.1016/S0013-7944(02)00112-1

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?rsttypeoffailureisassociatedwithspallingofthe?lm.Withincreasingtraction,crackswhichmayberadial[1,2]orcircumferential[3]extenddeeperintothe?lmandreachthe?lm–substrateinterface.Ifsubstantialyieldingoccursinthesubstrate,thecracksmayde?ectandpropagatealongtheinterface[1].Thereathirdpossibilitywhereinthecrackmayextendintothesubstrate.

Thepropertieswhichneedtobedeterminedinordertodesignacoatedsystemarethemodulusandstrengthofthe?lmandfracturetoughnessofboththe?lmaswellastheinterface.Thee?ectivenessofconventionaltechniques(suchasmicro-hardnesstest)todeterminethepropertiesofthe?lmareseverelylimitedduetothesheersmallnessofthe?lmthickness.Also,introductionofasubmicrometricpre-crackfordeterminationofthefracturetoughnessofthe?lmisrestrictedduetolimitationsinfabrication.Thepeeltest,scratchtestandbendingtestmaybeconductedonacomparativebasis,toyieldameasureofadhesionofthe?lmtothesubstrate[4].However,itsrelationshiptomorefundamentalpropertiessuchasstrengthandfracturetoughnessoftheinterfaceisasyetnotclear.Inthiscontext,itmustbementionedthatWeiandHutchinson[5]haverecentlyexaminedthemechanicsofthepeeltest,whileVolinskyetal.[6]havereviewedthin?lmadhesionandproposedrelationshipsbetween?lmconstitutivepropertiesandinterfacialtoughness.

Inviewoftheaboveproblemsassociatedwiththedimensionalsmallnessofthin?lms,depthsensingnanoindentationhasbecomeapopulartechniqueinrecenttimestoprovidequanti?cationoftheme-chanicalpropertiesofthe?lmandtheinterface.Thus,determinationoftheelasticmodulusandhardnessofacoatedsamplefromnanoindentationload–displacementcharacteristicsisnowwellestablished[7],giventheareafunctionassociatedwiththeindentertip.However,methodsforobtainingthefracturetoughnessofthe?lmorinterfacefromthesetestresultsarenotwelldeveloped.Sinceinmoststructuralapplications,the?lmisbrittle,thispersistsasanimportantobstacletothin?lmdesign.

OneoftheearliestapproachestoresolvingthisproblemwasduetoMarshallandLawn[8]whogaveasemi-empiricalformulationforestimatingthefracturetoughnessfromdataobtainedduringindentationbyaVickersindenter.However,thisanalysisisapproximateandalsodoesnottakecognizanceofthefactthatcrackgrowthmayproducediscontinuitiesorstepsintheload–displacementcharacteristics.Suchstepsinnanoindentationtestrecordshavebeenobservedexperimentally[3,9].AschematicofanexperimentalloadversusindentationdepthcurvecontainingastepwillbepresentedlaterinSection4.5.

Theoccurrenceofthesestepsinload–displacementcurveshasbeenshowntobeassociatedwiththeformationofcircumferentialcracksaroundtheindentedzone[3,10,11].Further,thesecracksgrowalongcylindricalsurfacesthroughthethicknessofthe?lm[3,10].Suchtypeofcrackingmayindeedbeanaturaloccurrencein?lmshavingacolumnargrainstructurewiththeaxesoftheelongatedgrainsbeingper-pendiculartothe?lmsurface[3,12].However,theinterpretationofexperimentaldatatoobtainthefracturetoughnessofthe?lmhasbeenbasedonapproximaterelationships[2]andaposteriorimeasurementofcracklengthsfromthecoatedsamplewhichisadi?culttask.

Itmustbenotedthatemissionofdislocationsunderneaththeindentermayberesponsibleforthediscontinuitiesintheload–displacementcharacteristicsatindentationdepthsoftheorderofafewang-stromsasdiscussed,forexample,byTadmoretal.[13].Thisgivesrisetostrainlocalizationandhardnessgradientsnearthe?lmsurface.Thecontinuum(fracturemechanicsbased)viewpointadoptedinthispaperwouldbeapplicableonlytosituationswherethe?lmthicknessandindentationdepthsareatleastanorderofmagnitudemorethantheabovesizescales.

The?rststeptowardsprovidingasoundfracturemechanicsmethodologyforanalysingindentationtestresultswastakenbyWeppelmannandSwain[3].Theyconductedadetailedstudyofthestress?eldsthatdevelopduringsphericalindentationofthinhard?lmsdepositedonsoftsubstrates.Thefocusoftheirworkwastoinvestigatethee?ectsofindenterradiusto?lmthicknessratioandplasticyieldinginthesubstrateontheabove?elds.Theweightfunctionmethodwasemployedtocomputethestressintensityfactorsasso-ciatedwithcylindricalcracksinthe?lm.However,asystematicinvestigationofthemechanicsoffractureinthe?lmduringindentationwasnotperformedintheirwork.Further,theuseoftheweightfunction

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methodentailstheassumptionsofplanestrain(whereas,inreality,thedomainisaxisymmetric),andsmallgeometrychangesintheindentedzone.

Hence,theprimaryobjectiveofthispaperistoinvestigatethemechanicsoffractureinthinhard?lmsonmoreductilesubstratesduetooccurrenceofcylindricalcracksduringsphericalindentation.Tothisend,thestress?eldsinthe?lm,aswellastheenergyreleaserateassociatedwithcylindricalcracksatdi?erentradiallocationsandindentationdepthsarestudiedusing?niteelementanalyses.Animportantaimofthisworkistoassesstheroleofplasticyieldinginthesubstrateontheaboveresults.Hence,theJ2?owtheoryofplasticityisemployedtodescribetheconstitutiveresponseofthesubstrateanddi?erentyieldstrengthsareconsidered.The?lmisassumedtobeisotropicandlinearelasticandtobefreeofresidualstresses.Theenergyreleaserateiscomputedfromthe?niteelementresultscontinuouslythroughouttheindentationanalysisof?lmswithpre-existingcylindricalcracksusingthenumericallyaccuratedomainintegralfor-mulation[14].Finally,amethodisproposedbywhich?niteelementsimulationscanbeemployedtoin-terpretindentationtestresultsandyieldinformationrelatedtothefracturebehaviourofthe?lm.

2.Numericalprocedure

Inthiswork,axisymmetric?niteelementanalysesofsphericalindentationofthe?lm–substratesystemiscarriedout.IntheanalysesreportedinSection3,the?lmisassumedasuncracked,whereas,cylindricalcracksextendinginwardsfromthe?lmsurfacearoundthezoneofindentationaremodelledinSection4.AnupdatedLagrangian?niteelementformulation[15]isemployedinalltheseanalyses.

AschematicoftheaxisymmetricgeometryconsideredinthesimulationsundertakeninSection3isshowninFig.1.Theratioofthe?lmthicknesstoindenterradius,t=Ri,ischosenas1.5.Thefocusofthepresentworkisonbrittlecrackingofthe?lmandnotondebondingofthe?lm–substrateinterface.Hence,the?lmisassumedtobeperfectlybondedtotheinterface.Theratiosoftheradialdimensionof

the

Fig.1.Schematicshowingthe?lm–substratesystemalongwiththesphericalindenterinther–zplane.

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?lm–substratesystemandthesubstratethicknesstotheindenterradius,Rs=Riandts=Ri,aretakenas300and150,respectively,sothatboundarye?ectsdonotin?uencethestressdistributionnearthezoneofindentation.Thecontactbetweenthe(rigid)indenterandthe?lmisassumedasfrictionlessandismodelledusingaslidelineapproachalongwithapenaltyformulation[16].

The?lmisassumedtobelinearelastic,whereasa?nitedeformation,J2?owtheoryofplasticitymodel

[15]isemployedforthesubstrate.ThematerialpropertiesarechosentoberepresentativeofaTiN?lmonasteelsubstrate(asintheexperimentsreportedin[3]).Thus,theratiooftheYoung’smodulusofthesubstratetothatofthe?lm,Es=Ef,istakenas0.4.ThePoisson’sratiosofthe?lmandsubstrate,mfandms,areassumedas0.2and0.3,respectively.Further,theratioofthetangentmodulustoYoung’smodulus,Ets=Es,ofthesubstrateischosenas0.04.Inordertostudythee?ectofplasticyieldinginthesubstrateontheindentationmechanics,threevaluesofinitialyieldstrainofthesubstrate,rys=Es¼0:001,0.003and0.009areconsidered.The?niteelementmeshesusedintheanalysesarecomprisedoffour-nodediso-parametricquadrilateralelementsandarewellre?nednearthezoneofindentation.

Itmustbenotedthatsinceacontinuumwithhomogeneousmaterialpropertiesisperformedhere,theresultsobtainedwoulddependonlyontheratiosofdi?erentlengthdimensionsandnotontheirabsolutevalues.Thus,forexample,fromanalyticalsolutionsforcontactbetweenasphereandanelastichalf-space[17],aswellasfromdimensionalconsiderations,itmaybeexpectedthattheloadPduringindentationofa?lmbondedtoasoftsubstrate(asinFig.1)wouldhavethefollowingfunctionalform,

????htEsrys2P¼EfRif;;;;othermaterialconstants:ð1ÞtRiEfEs

Here,h=tistheratiooftheindentationdepthto?lmthicknessandisameasureofthecompressiveaxialstraininthe?lmattheindenteraxis.Itshouldbementionedthatadditionallengthscaleconsiderationsmaybeimportantatsubmicronthicknesseswherelocalizedstraingradientsinthesubstratemaybecomedominant.

3.Stressanalysisofuncracked?lm–substratesystem

Itisimportantto?rstunderstandthedevelopmentofplasticyieldinginthesubstrateduringindentationanditsin?uenceontheloadversusindentationdepthcharacteristicsandstressvariationsinthe?lm.Thiswouldenableinterpretationofthetrendsinvariationofenergyreleaseratewithindentationdepthandcracklengthinthefractureanalysistobeperformedlater.Hence,adetailedanalysisofindentationofanuncracked?lm–substratesystemisundertakeninthissection.

3.1.Loadversusindentationdepth

ThevariationofnormalizedloadP=ðEfR2iÞwithindentationdepthh=tuptoamaximumindentationdepthofhm=t¼0:05and0.2arepresentedinFig.2(a)and(b),respectively.Inthese?gures,thesolidlinescorrespondtoaninitialyieldstrainofthesubstrateofrys=Es¼0:009andthedashedlinestorys=Es¼0:001.Boththeloadingandunloadingsegments(startingfrommaximumindentationdepth)areshowninthese?gures.ItcanbeobservedfromFig.2(a)thatforrys=Es¼0:009,theloadingandunloadingcurvescoincidewitheachother,indicatingareversible(elastic)responseofthe?lm–substratesystem.However,forrys=Es¼0:001,theunloadingcurvedi?ersfromtheloadingcurveevenathm=t¼0:05(seeFig.2(a)),whichisattributedtothedevelopmentofplasticyieldinginthesubstrate.Asthenormalizedindentationdepthisincreased,itmaybenoticedfromFig.2(b)thattheunloadingsegmentsforbothcasesdeviatesubstantiallyfromtheloadingcurves.

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Fig.2.Normalizedloadversusindentationdepthforrys=Es¼0:001and0.009correspondingtomaximumindentationdepthof(a)hm=t¼0:05and(b)hm=t¼0:2.Theunloadingcurvesfrommaximumindentationdeptharealsoshown.

ItcanbeobservedfromFig.2(a)and(b)thatthepeakloadatagivenhm=tincreaseswiththeyieldstrengthofthesubstrate.Also,thePversushcurvebecomesalmostlinearatlargeh=t(seeFig.2(b)).Insordertocharacterizethise?ect,variationsofthetypeP=ðEfR2iÞ¼Aðh=tÞ,whereAisaconstant,were?ttedtotheloadingcurves.Thevaluesoftheindexscorrespondingtodi?erentrangesofindentationdeptharesummarizedinTable1forthethreevaluesofrys=Es.Itmust?rstbenotedthattheindexsforsphericalTable1sValuesofexponentsobtainedby?ttingcurvesofthetypeP=ðEfR2iÞ¼Aðh=tÞtothenumericalresultscorrespondingtodi?erentrys=Esandrangeofh=t

Rangeofh=trys=Es

0.0010.0030.009

0–0.051.41.461.48

0.05–0.11.21.291.4

0.1–0.21.01.131.2

6