JAmCeramSoc51233 ID: 295822
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TheMechanicsofIndentationInducedLateralCrackingXiChenDepartmentofCivilEngineeringandEngineeringMechanics,ColumbiaUniversity,NewYork,NewYork10027JohnW.HutchinsonDivisionofEngineeringandAppliedSciences,HarvardUniversity,Cambridge,Massachusetts02138AnthonyG.EvansMaterialsDepartment,UniversityofCalifornia,SantaBarbara,California93106Themechanicsgoverningthelateralcracksthatformwhenahardobjectplasticallypenetratesaceramicispresented.Therolesofindentationload,penetrationdepth,andworkofin-dentationareallhighlighted,asaretheinuencesoftheme-chanicalpropertiesofthematerial.Aclosedformsolutionforcrackinginducedbyexpansionofatwo-dimensionalcavityisusedtobringoutessentialfeaturesrelatedtoparametricde-pendenceandscaling.Thethree-dimensionalaxisymmetric J.Am.Ceram.Soc.,[5]1233 1238(2005)DOI:10.1111/j.1551-2916.2005.00281.x D.MarshallcontributingeditorFellow,AmericanCeramicSociety.Authortowhomcorrespondenceshouldbeaddressed.e-mail:xichen@civil.colum-bia.edu ManuscriptNo.10970.ReceivedApril12,2004;approvedNovember29,2004. theplasticzoneuponunloading.Cracksarethenintroducedandenergyreleaseratesandmodemixitiesascertained.Todeterminethestresses,thecylindricalcavitydepictedinFig.3isexpandedbyinternalpressure,,toitscurrentradiusstartingfromzero.Attentionisrestrictedtoelastic-perfectlymaterialsgovernedbyaMisesyieldsurface(withYoungsmod-ulus,,Poissonsratio,,andyieldstress,).Intheelasticallyincompressiblelimit(1/2),theplane-strain(orcavitation)problem,19,20canbesolvedinclosedform.Thesolutionrevealsthatthecavityexpandsatconstant(cavitation)pressure:.Thecoefcientisweaklydependentontheyieldstrain,with(0.0010.003),3.70.Theradiusoftheplasticzoneisrelatedto9(1)Thecavityplasticzonesizeismuchlargerthattheplasticzonesurroundinganindent.However,largeplasticstrainsarecon-nedtoradiilessthanabout3.Incylindricalcoordinatesthein-planestressesoutsidetheplasticzone()uponunloadingare (2)whereistheworkperunitlengthrequiredtocreatethecavity(3)and 1pl 8(4)Acrackisinsertedalongwithitsleftendattakensuchthatthecrackliesoutsidetheplasticzone(Fig.3).Thenormalandshearstressesactingonthecrackplanepriortoitsformationarereadilydeterminedfrom(2)as cos2sin2(5)whereisdenedinFig.3.Thenormalstressispositivefory4p/4.Sincethedeformationsaccompanyingtheforma-tionofthecrackareelastic,andignoringthesmallinteractionofthecrackwiththecavityitself,themodeIandIIstressintensityfactorsattheright-handcracktip()canbewrittenas KI;KIIÞWa3=2I;qII cos2sin2 c=2aþxc=2axs (6)Thelimitsofintegrationareconsistentwithasthevariable.Thecorrespondingenergyreleaserateis: GW2=ðEa3Þ¼ (7)andtheassociatedmeasureofmodemixtan KIIKI¼ (8) Fig.2.Plotoflateralcrackdimension()versusload()forfourbrittlematerials:comparisonbetweentheoryandexperiments.ThesymbolsrepresentexperimentaldataobtainedfromMarshalletalandsolidlinesaretheoreticalpredicationfromthepresentanalysis.(a)MgFandZnSand(b)Asandsodalimeglass. Fig.1.Schematicshowingofthelateralcracksystem.Thesphericalindentation(withload)onanelastic plasticsolidleavesanimpressionwithprojectedcontactradius.Thelateralcrackisannularwithlength(orradius),whichformsjustoutsidetheplasticzone,atdistancebelowthefreesurface.1234JournaloftheAmericanCeramicSocietyChenetal.Vol.88,No.5 Thenormalizedenergyreleaserateandthemodemixareplot-tedasafunctionofcracklengthinFig.4forcracksatvariousdepthsbelowthesymmetryplane:allhavingtheirleftendat.Bysymmetry,thecrackalong0)isneces-sarilyaModeIcrack.Atalldepths,theenergyreleaseraterstincreasesandthenfallsasthecrackgrows,similartoindenta-tion-inducedcracks.Theprecedingresultsilluminateindentation-inducedcrackscaling(althoughprecisedetailsdependonthedimensionalityofeachproblem).Forexample,toexposetheroleofthecavitysizeandtheyieldstress,byusingand),in(7)andFig.4canbere-expressedas:/[(3.70].Alternatively,toexpresstheenergyreleaserateexclusivelyintermsoftheworktocreatethecavity,relevanttoimpactindentationofasubstrate,thenbecomes.Counterpartstothesenor-malizationswillbepresentedbelowforindentation-inducedcracks.III.BasicRelationsforIndentationofaHalf-SpaceWhenthepenetrationofasphericalorconicalindenterintoanelastic/perfectly-plastichalf-spaceisplasticitydominated,well-knownresultsrelatetheindentationload,,contactradius,andindentationdepth,(9)Forshallowindentscreatedbyasphericalindenterofradius(10)whileforconicalindenterswithconeangletan(11)Inordertoobtainrepresentativeresultsandmakeconnectionsbetweensphericalandconicalindentations,theeffectofplasticpile-upisignored.Theeffectofpile-up(orelasticsink-in)andworkhardeningarediscussedinSectionV.Thehardnessofthematerialis(12)Forconicalindenters,thecoefcientdependsweaklyontheyieldstrainandtheconeangle:formanyapplicationsitissuf-cienttouse,3.Thischoicealsoappliesforsphericalind-enters(withminordeparturesdependentontheyieldstrainand).Procedurestoaccountfortheeffectsofyieldstrainandstrainhardeningwillbediscussedinthenextsection.Theworktocreateanindent,volume,is(13)Forfuturescalingpurposes,notethattheworkcanbeexpressedintermsof,and,independently.Forsphericalindentation P24pRksY¼ p4ksY (14)Forconicalindentations p3 tan tan tan(15)Neglectingtheweakdependenceofonindentergeometryandyieldstrain,Eqs.(8)and(13)implyanequivalencebetweenconicalandsphericalindentation:namely,indentswithidenticalcontactradiusandvolumerequirethesameloadandworkandhavethesamehardness.Thecorrespondingstresseldsoutsidetheplasticzonecreatedbytheindentarealsoessentiallythesame.Therelationbetweentheconeangleandforthesphericalindenter,suchthatthetwoindentationshaveidenticalaandV,isplottedinFig.5.TheconicalequivalentoftheBerkovichindentercorrespondsto0.44,whilethatfortheVickersindentercorrespondsto0.49.IV.Indentation-InducedCracksinElastic-PerfectlyPlasticMaterialsLateralcrackingfromindentationscanbeanalyzedintwoparts.(i)Determinationofthestressesinducedbyindentationofan Fig.4.(a)Thedimensionlessenergyreleaserate,),and(b)themodemixphaseangle,,asafunctionofnormalizedlateralcrackradius,,forvariousdepthsofthecrackbelowthesymmetricplane. Fig.3.Schematicdrawingofcrackingoutsideacylindricalcavity.May2005TheMechanicsofIndentationInducedLateralCracking1235 elastic-perfectlyplastichalf-spaceafterunloading(withnocrackpresent).(ii)Calculationofthestressintensityfactorsforanannularcrackinthestresseldcreatedbytheindent.Thecracksareassumedtoformoutsidetheplasticzone,wheretheresidualstressesactingonthecrackplanearetensile.Bothpartsoftheproblemareanalyzednumerically,byusingtheniteelementcodeABAQUSStandard,andcarriedoutwithinanitestrainframework,accountingfornitegeometrychanges.TheenergyreleaserateiscalculatedfromtheJ-Integralaroundthecrackfront.Thedistributionoftheresidualstresscomponentsur-roundingasphericalindentation,followingloadremoval,isil-lustratedinFig.6.Theplasticzoneextendsouttoabouttwicethecontactradius.Themaximumtensileoccursneartheboundaryofthiszoneatadepthroughlyequaltothecontactradius.Cracksdestinedtopropagateparalleltotheinterfacearemostlikelytobenucleatedinthisrelativelysmallregion.ItwillemergethatthedepthofanannularcrackparalleltothesurfacecorrespondingtomodeIconditionsisalsoatthisdepth.ThenormalizationoftheenergyreleaserateattheoutertipoftheannularcracksurroundingthesphericalindentinFig.1is: GW2=ðEa5Þ¼f ca; aR; da; (16)Thisscalingisanalogoustothatforthemodelcylindricalvoid-inducedcrackexceptthathereisthetotalworktocreatetheindentation,nottheworkperunitlength.Oneimportantim-plicationof(16),affectingtrendsincracksizewithmaterialproperties,isthatisnotexplicitlydependentontheyieldstrength(itentersimplicitlythroughitsinuenceon,aswellason,ifisprescribed).Moreover,directnumericalcalcula-tionhasveriedthatthereisnodependenceof.Thenormalizationfortheconicalindenterhasthesameformas(16)withthedependenceonreplacedbyadependenceonResultsforthenormalizedenergyreleaserateasafunctionofcracklength,computedfromtheniteelementanalysis,arepresentedinFig.7forfourvaluesof(allwith1and1.5).ThelefttipofeachcracklieswithinthezoneofhightensilestressnotedinconjunctionwithFig.6.ThenormalizedenergyreleaserateandthemodemixforcracksatvariousdepthsbelowthefreesurfaceareplottedonFig.8(ineachcasefor0.7and1.5).Notethatthecrackdepthassoci-atedwithboththelargestenergyreleaserateandwithmodeIconditionsis1;thesetwofeaturesholdforawiderangeof.Thus,cracksatthisdeptharefavorednotonlybecauseitcoincideswiththelargesttensilestressbutalsobecause,oncenucleated,cracksmustpropagateinmodeI(paralleltothesur-face).Subsequently,wefocusonsphericalindentationwith0.5sincethischoicecorrespondscloselytoeitherBerkovichorVickersindenters(cf.Fig.5).ComparisonstobemadelaterwithexperimentaldatawillbebasedonVickersindentations.Thenormalizedenergyreleaserateasafunctionofcracklengthfor0.5,1,and1.5isplottedonFig.9.Thepeakenergyreleaserateat0.6is max11(17)ThefollowingfunctionhasbeenttotheresultsinFig.9totherightofthepeak(intherange0.64): 10e(18 Fig.5.Therelationshipgoverningconical(coneangle)andsphericalindentation(withcontactradius/indenterradiusratio),suchthatthetwoindentationscreatethesamecontactradiusandvolume. Fig.6.Contourplotoftheresidualstresscomponentnormaltothelateralcracksystem()causedbysphericalindentation,with0.9. Fig.7.Normalizedenergyreleaserate)asafunctionoflat-eralcracklengthforthreevaluesofcontactradiuscausedbysphericalindentation.Thecracksarealignedwiththepreferredcrackingpath,whereattainsmaximumand0(forgiven),allwith1and1.5(seeFig.8).1236JournaloftheAmericanCeramicSocietyChenetal.Vol.88,No.5 Theaccuracyofthisttingfunctionisevidentinthegure.However,itisnotusefulforassessingtrendsincrackingwithmaterialproperties,discussedinthenextsection.Instead,apowerlawrelationshipisrequired.Thebestttoapowerlawisgivenby: (18asalsoplottedonFig.9.Thetisgoodbutimperfect.Overtherelevantrangeofcracksize(4),itisdeemedsufcientlyaccuratetocapturethematerialpropertytrendsdescribedbelow.isthemodeItoughnessofthematerial,anecessaryconditionforthepropagationofacrackismax.Conse-quently,from(17),theworkofpenetrationatthecrackingthresholdis,(19)Assumingthattheworkexceedsthethreshold,thecrackprop-agatesoutwardsandthenarrestsatthecracklengthcorre-spondingto,whichby(18)is 65ln (20orby(18)is (20Insummary,foreitheraBerkovichoraVickersindentation(orasphericalindentationwith0.5),thepotentialcrackplanewhichsimultaneouslyhasthelargestenergyreleaserateandmodeIconditionsislocatedbelowthesurfaceatdepth,Thereisamaximumenergyreleasegivenby(17).Ifthetough-nessexceedsthismaximum,acrackwillnotform.Ifitislower,acrackmaynucleate.Ifitdoes,itwouldpropagateandarrestatanouterradius,where1.5andisgivenby(20).V.ApplicationandComparisonwithMeasurementsThenon-dimensionalparametercharacterizingtheindentation,),canbeexpressedinthefollowingalternativewaysthathighlighttherolesoftheload,,theindentradius,,andtheworkofindentation,,respectively: tan ðpksYÞ3=2P1=2E¼ tan ðpksYÞ2aE¼ tan (21)Byusing(21)toconvertthelateralcracksizesolutionin(20)toatrendincracksizewithVickersindentationload,thetrendinthecracksizewithmaterialproperties,atxedload,canbeas-certainedas: 0:19 tan (22)Notethatthecracksizeincreasesasthetoughnessdecreasesandastheyieldstrengthincreases.Incorporatingthedatafortough-nessandhardnessfromMarshalletalthecracklengthspre-dictedbythepresentmodelcanbesuperposedonFig.2.Notethat,inallcases,themeasuredtrendsincracklengtharecon-sistentwithformula(22).Thatis,theformulapredictsthatthecracksinZnSshouldbeslightlylargerthanthoseinMgF(Fig.2(a)),andthecracksinAsbeappreciablylargerthan Fig.8.(a)Thedimensionlessenergyreleaserate,),and(b)themodemixphaseangle,,asafunctionofnormalizedlateralcracklength,,forvariousdepthsofthecrackbelowthefreesurface(for0.7and1.5). Fig.9.Thenormalizedenergyreleaserate)asafunctionoflateralcracklengthforVickersindenter(0.5,1,and1.5).Symbolsrepresentresultsobtainedfromtheniteelementanalysis,andsolidlineisthefunctionaltfromEq.(18May2005TheMechanicsofIndentationInducedLateralCracking1237 thoseinglass(Fig.2(b)).Bothpredictionsareinaccordancewiththemeasurements.Giventheuncertaintyintheyieldstrength(hardness)andtoughnessforthesematerials,thecor-respondenceisconsideredadequateforfurtherapplicationofthemodel.Thecrackingthatoccursuponprojectileimpactcanbepre-dictedbyrelatingtheplasticworkofpenetration,,tothekineticenergyoftheprojectile:0.8/2,(withtheprojectilemassanditsvelocity):neglectingstrainratehard-ening,,24tan (23)Sincethematerialbetweenthecrackandthesurfaceissuscep-tibletoremoval(Fig.1),thevolumeofmaterialremovedperimpact,impact,scalesas:impact tan (24)Consequently,tominimizetheerosionrate,thematerialshouldhavehightoughness.Theinuenceoftheyieldstrengthisneg-ligible:counteringthecommonlyheldbeliefthattheerosionre-sistanceisimprovedbychoosingmaterialswithhighhardness.Anisotropyofthetypefoundinthermalbarrieroxideswillaf-fectthespecificsbutnotthetrendswithmaterialproperties.Theimplicationsarediscussedelsewhere.16,17TheforegoingresultscanbeelaboratedbyusingJohnsons(p.167)solutionstoaccountforthedependenceofindenterelasticity,coneangle,orthedepthofasphericalin-dentation.Fromseveralsources,Fig.10reproducesthespreadassummarizedbyJohnson.Hereisamodulusdepend-entonbothsubstrate()andindenter()elasticproper-ties:1.Notethatreducestotheplanestrainmodulusofthesubstratewhentheindenterisrigid(asinthepresentstudy).JohnsonalsodiscussesTaborsrulesforassigningavaluetoforstrainhardeningmateri-als.Forexample,foraconewith19.7,representativeofaBerkovichindenter,Taborrecommendsthatisidentiedwiththestressatacompressivestrainof8%.Forasphericalindenter,istakenasthestressatacompressivestrain0.2VI.ConclusionAcombinedanalytical numericalanalysishasbeenpresentedofthelateralcracksthatformwhenahardobjectplasticallypen-etratesaceramic.TheradiiofcrackspredictedtoformuponVickersindentationhavebeencomparedwithexperimentalmeasurements.Anadequatecorrespondencebetweenthemeas-urementsandpredictionsprovidesarationaleforusingthemodeltopredicttrendsincrackingforvariousscenarios.Thebasicformulasrelatingthecrackradiustotheloadandworkofpenetrationhavebeenusedtopredicttrendswithmaterialprop-ertiesfortwosituations:(a)staticpenetrationuptoaspeciedloadand(b)impactbyaprojectile.Inbothcases,crackingisdiminishedbyincreasingthetoughnessoftheceramic(con-sistentwithmanypreviousndings).Theyieldstrengthhasessentiallynoeffect.ReferencesR.F.CookandG.M.Pharr,DirectObservationandAnalysisofIndentationCrackinginGlassesandCeramics,J.Am.Ceram.Soc.,787 817(1990).G.R.Anstis,P.Chantikul,B.R.Lawn,andD.B.Marshall,ACriticalEvaluationofIndentationTechniquesforMeasuringFractureToughness:I,Di-rectCrackMeasurements,J.Am.Ceram.Soc,533 45(1981).B.R.Lawn,A.G.Evans,andD.B.Marshall,Elastic/PlasticIndentationDamageinCeramics:TheMedian/RadialCrackSystem,J.Am.Ceram.Soc574 80(1980).D.B.Marshall,B.R.Lawn,andA.G.Evans,Elastic/PlasticIndentationDamageinCeramics:TheLateralCrackSystem,J.Am.Ceram.Soc,561 6(1982).B.R.LawnandE.R.Fuller,EquilibriumPenny-LikeCracksinIndentationFracture,J.Mater.Sci.,2016 20(1975).B.R.LawnandA.G.Evans,AModelforCrackInitiationinElastic/PlasticIndentationFields,J.Mater.Sci.,2195 201(1977).D.B.Marshall,ControlledSurfaceFlawsinCeramics:AComparisonofKnoopandVickersIndentation,J.Am.Ceram.Soc.,127 33(1983).A.G.EvansandE.A.Charles,FractureToughnessDeterminationbyIn-dentation,J.Am.Ceram.Soc.,371 6(1976).K.Niihara,AFractureMechanicsAnalysisofIndentation-InducedPalmq-vistCracksinCeramics,J.Mater.Sci.Lett.,221 3(1983).J.Lankford,IndentationMicrofractureinthePalmqvistCrackRegime:Im-plicationsforFractureToughnessEvaluationbytheIndentationMethod,J.Mater.Sci.Lett.,493 5(1982).A.G.EvansandT.R.Wilshaw,Quasi-StaticSolid-ParticleDamageinBrit-tleSolids,ActaMater.,939 45(1976).B.R.Lawn,IndentationofCeramicswithSpheres:ACenturyofHertz,J.Am.Ceram.Soc.,1977 85(1998).A.VasinontaandJ.L.Beuth,MeasurementofInterfacialToughnessinThermalBarrierCoatingSystemsbyIndentation,Eng.Fract.Mech.,843 60(2001).M.D.DroryandJ.W.Hutchinson,MeasurementoftheAdhesionofaBrittleFilmonaDuctileSubstratebyIndentation,Proc.R.Soc.LondonA4522319 41(1996).A.G.Evans,D.R.Mumm,J.W.Hutchinson,G.H.Meier,andF.S.Petit,MechanismsControllingtheDurabilityofThermalBarrierCoatings,Progr.Mater.Sci.,505 53(2001).X.Chen,R.Wang,N.Yao,A.G.Evans,J.W.Hutchinson,andR.W.Bruce,ForeignObjectDamageinaThermalBarrierSystem:MechanismsandSimula-tions,Mater.Sci.Eng.A352,221 31(2003).X.Chen,M.Y.He,I.Spitsberg,N.A.Fleck,J.W.Hutchinson,andA.G.Evans,MechanismsGoverningtheHighTemperatureErosionofThermalBar-rierCoatingsUsedinGasTurbines,Wear256,735 46(2004).X.Chen,J.W.Hutchinson,andA.G.Evans,SimulationoftheHighTem-peratureImpressionofThermalBarrierCoatingswithColumnarMicrostructure,ActaMater.,565 71(2004).Y.Huang,J.W.Hutchinson,andV.Tvergaard,CavitationInstabilitiesinElastic PlasticSolids,J.Mech.Phys.Solids,223 41(1991).R.Hill,TheMathematicalTheoryofPlasticity.ClarendonPress,Oxford,1950.H.Tada,P.C.Paris,andG.R.Irwin,TheStressAnalysisofCracksHand-book.ASMEPress,NewYork,2001.K.L.Johnson,ContactMechanics.CambridgeUniversityPress,Cambridge,1985.ABAQUS,ABAQUS5.8UsersManual.ABAQUSInc.,Pawtucket,RI,1998.X.ChenandJ.W.Hutchinson,ParticleImpactonMetalSubstrateswithApplicationtoForeignObjectDamagetoAircraftEngines,J.Mech.Phys.Sol-,2669 90(2002).D.Tabor,HardnessofMetals.OxfordUniversityPress,Oxford,1951. Fig.10.Thespreadofexperimentaldatain(shownintheshadedregion)measuredfromconicalandsphericalindentations,whicharesummarizedbytheclassictextbookofJohnson.Thedimensionlessindentationstrainistanforconicalindenterandforsphericalindenter,respectively. Duringimpact,althoughamajorityofthekineticenergyisconvertedtotheplasticworkofpenetration,acertainamountislostinformsofstresswaveandreboundenergy.1238JournaloftheAmericanCeramicSocietyChenetal.Vol.88,No.5