Microstructure-Property Relationships and Constitutive Response of Plastically Graded Case Hardened Steels

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Microstructure-Property Relationships and Constitutive Response of Plastically Graded Case Hardened Steels
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1 online resource (157 p.)
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english
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Klecka,Michael A
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University of Florida
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Subhash, Ghatu
Committee Members:
Ifju, Peter
Arakere, Nagaraj K
Myers, Michele Viola

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Subjects / Keywords:
carbide -- element -- finite -- functionally -- graded -- gradient -- hardened -- hardness -- indentation -- modeling -- plastically -- steel
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, Ph.D.
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Abstract:
Case hardened materials, popularly used in many demanding engineering applications such as bearings, gears, and wear/impact surfaces, have high surface hardness and a gradient in material properties (hardness, yield strength, etc.) as a function of depth; therefore, they behave as plastically graded materials. In the current study, two different commercially available case carburized steels along with two through hardened steels are characterized to obtain relationships among the volume fraction of subsurface carbides, indentation hardness, elastic modulus, and yield strength as a function of depth. A variety of methods including micro-indentation, nano-indentation, ultrasonic measurements, compression testing, rule of mixtures, and upper and lower bound models are used to determine the relationships for elastic modulus and compare the experimental results with model predictions. In addition, the morphology, composition, and properties of the carbide particles are also determined. The gradient in hardness with depth in graded materials is commonly determined using micro-indentation on the cross-section of the material which contains the gradation in microstructure or composition. In the current study, a novel method is proposed to predict the hardness gradient profile using solely surface indentations at a range of loads. The method does not require the graded material to be sectioned, and has practical utility in the surface treatment industry. For a material with a decreasing gradient in hardness, higher indent loads result in a lower measured hardness due to the influence of the softer subsurface layers. A power-law model is presented which relates the measured surface indentation hardness under increasing load to the subsurface gradient in hardness. A coordinated experimental and numerical study is presented to extract the constitutive response of graded materials, utilizing relationships between hardness, plastic deformation, and strain hardening response. The average plastic strain induced by an indent is shown to be an effective measure of the representative plastic strain, which is used in order to relate hardness to yield strength in both virgin and plastically deformed materials. It is shown that the two carburized steels contain gradients in yield strength, but constant strain hardening exponent with depth. The resulting model of material behavior is used to characterize the influence of specific gradients in material properties on the surface indentation behavior under increasing indentation loads. It is also shown that the response of the material is not greatly influenced by strain hardening exponent, while a gradient in strain hardening ability only has minimal impact. Gradients in elastic properties are also shown to have negligible influence for a fixed gradient in hardness. The depth of subsurface plastic deformation is shown to increase with sharper gradients in hardness, but is not altered by gradients in elastic properties. The proposed approach is not specific to case hardened materials and can be used to determine the subsurface hardness gradient for any graded material.
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In the series University of Florida Digital Collections.
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Includes vita.
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Statement of Responsibility:
by Michael A Klecka.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Subhash, Ghatu.

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MICROSTRUCTURE-PROPERTYRELATIONSHIPSANDCONSTITUTIVERESPONSEOFPLASTICALLYGRADEDCASEHARDENEDSTEELSByMICHAELA.KLECKAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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2011MichaelA.Klecka 2

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ACKNOWLEDGMENTS Thankstomyparents,aswellasmywife,fortheirsupport,understanding,andencouragementduringmytimeasagraduatestudent.Theyhavealwaysbeenreassuring,andneverdoubtedmyabilitiesasaresearcher.Withoutthem,mygraduatecareerwouldhavebeenamuchlessenjoyableprocess.Specialthankstomyadvisor,ProfessorGhatuSubhash,forbeingagreatmentorandfriendthroughoutmyacademiccareer.Hisunwaiveringenthusiasmforresearchhashelpedtokeepmemotivatedduringthevarietyofresearchprojectswehavepursuedtogether.Histimelycommentsandsuggestionsonmanuscriptrevisionsaswellascountlessfruitfuldiscussionshavebeenagreatbenettomyongoingeducation.AnotherspecialthankstoProfessorNagarajArakereforhissupportandguidancethroughoutthelastfewyears.Hisextensivebackgroundinbearingmaterialsandtestingprocedureshasallowedmetolearnsomethingneweverytimeweconverse.Thankstomycommitteemembers,Dr.PeterIfjuandDr.MichelleManuel,fortheirtimespentreviewingmyresearch.Theirinsightfulcommentsandsuggestionsregardingmydissertationaregreatlyappreciated.Iwouldalsoliketothankmyofceandlabmates,whomhaveassistedoncountlessoccasions,whetheritbehelpingtodesignanexperiment,repairingequipment,proofreadingmanuscripts,orsimplylendingsupport.Thecooperativeenvironmentofourworkplaceallowedmetolearnagreatdealaboutmorethanjustmyownresearch,whichalwayskeptthingsinteresting.ThisresearchwaspartiallysupportedbyNationalScienceFoundationAwardCMMI-0927849underprogramofcerDr.ClarkV.CooperinadditiontoasubcontractfromAirForceSBIRgrantthroughUESInc.Dayton,OH.TheauthorsincerelyacknowledgesMr.RobertWolfeofTimkenCo.,Canton,OHandDr.NelsonFosterofAFRL,WPAFB,Dayton,OHandMr.HiteshTrivediofUESInc.,Dayton,OHforsupplyofmaterialsforthisresearch. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 14 1.1GradedMaterials ................................ 14 1.2Applications ................................... 15 1.3Characterization ................................ 16 1.4CurrentStudy .................................. 17 2MICROSTRUCTURE-PROPERTYRELATIONSHIPS .............. 20 2.1Background ................................... 20 2.2Materials .................................... 23 2.3Experimental .................................. 27 2.3.1CarbideDistribution .......................... 27 2.3.2Microindentation ............................ 28 2.3.3Nanoindentation ............................ 29 2.3.4CompressionTesting .......................... 32 2.3.5UltrasonicDeterminationofElasticProperties ............ 33 2.4Discussion ................................... 34 2.4.1CompositeModelforModulus ..................... 36 2.4.2HardnessYieldStrengthRelationship ................ 38 2.5Summary .................................... 40 3DETERMINATIONOFSUBSURFACEHARDNESSGRADIENTSINPLASTICALLYGRADEDMATERIALSVIASURFACEINDENTATION .............. 55 3.1Background ................................... 55 3.2Materials .................................... 57 3.3SurfaceIndentationScheme ......................... 57 3.4Results ..................................... 60 3.4.1SurfaceHardness ........................... 60 3.4.2DetectingChangesinGradientTrends ................ 64 3.5Summary .................................... 65 4

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4DETERMININGTHECONSTITUTIVERESPONSEOFGRADEDMATERIALS 74 4.1PastMethods .................................. 74 4.2HardnessandYieldStrengthRelationship .................. 76 4.3Procedure .................................... 81 4.3.1ExperimentalMethod .......................... 81 4.3.2FiniteElementModel .......................... 84 4.4HomogeneousMaterials ............................ 85 4.4.1Results ................................. 85 4.4.2CompressionTestCross-Section ................... 88 4.5DeterminingtheConstitutiveResponseofGradedMaterials ........ 89 4.5.1IndentMappingGradedMaterials ................... 90 4.5.2SelectingMaterialProperties ..................... 92 4.5.3Results ................................. 93 4.6Summary .................................... 95 4.6.1RepresentativePlasticStrain ..................... 95 4.6.2ConstitutiveResponseofGradedMaterials ............. 96 5MODELINGGRADEDMATERIALSURFACEHARDNESSASAFUNCTIONOFINDENTATIONLOAD .............................. 109 5.1Background ................................... 109 5.2Procedure .................................... 112 5.2.1FiniteElementModel .......................... 112 5.2.2ParametricStudyofHardnessGradients ............... 113 5.3ResultsandDiscussion ............................ 114 5.3.1LinearGradientinHardness ...................... 114 5.3.2PerfectlyPlasticandHighStrainHardening ............. 116 5.3.3GradientinStrainHardening ..................... 117 5.3.4GradientinElasticModulus ...................... 118 5.3.5Power-LawModel ............................ 119 5.3.6SubsurfaceStressandStrainFields ................. 120 5.3.6.1Hardnessgradients ..................... 121 5.3.6.2Perfectlyplasticbehavior .................. 122 5.3.6.3Gradientinstrainhardeningexponent ........... 122 5.3.6.4Gradientinelasticmodulus ................. 123 5.4Summary .................................... 124 6CONCLUSION .................................... 142 6.1GradedMaterials ................................ 142 6.2MicrostructuralCharacterization ....................... 142 6.3HardnessGradientDetermination ...................... 143 6.4ConstitutiveResponse ............................. 143 6.5FutureWork ................................... 144 6.5.1NitridedandDuplexHardenedMaterials ............... 144 5

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6.5.2MicrostructureModeling ........................ 145 REFERENCES ....................................... 149 BIOGRAPHICALSKETCH ................................ 157 6

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LISTOFTABLES Table page 2-1Nominaltoolsteelandstainlesssteelcompositions. ............... 53 2-2Distributionofcarbidevolumefractionwithdepth. ................. 53 2-3Materialpropertiesforvariouscarbidestested. .................. 53 2-4Elasticpropertiesdeterminedviaultrasonicmeasurements. ........... 54 2-5Calculatedeffectiveelasticmodulus. ........................ 54 2-6Comparisonofyieldstrength,hardness,andconstraintfactor. .......... 54 3-1Summaryofmaterialsectionsandrelevantproperties. .............. 73 4-1Averageplasticstrainasafunctionofhardeningexponent. ........... 108 4-2Summaryofrepresentativeplasticstrainvaluesfoundinliterature. ....... 108 5-1Summaryofmaterialparametersconsideredduringparametricstudy. ..... 141 7

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LISTOFFIGURES Figure page 2-1CrosssectionofthroughhardenedM-50showinguniformcarbidedistribution. 42 2-2CasehardenedM-50NiLcrosssectionshowingcarbidedistributionwithinthe2.5mmgradedlayer. .............................. 42 2-3CasehardenedP675crosssectionshowingcarbidedistributionwithinthe1.5mmgradedlayer. ................................. 43 2-4P675`monolithic'casematerialcarbides. ..................... 44 2-5SEMimageoftemperedmartensitestructure. ................... 45 2-6MeasuredsubsurfacedistributionofcarbidesinM-50,M-50NiL,andP675. .. 46 2-7Hardnessprolesasafunctionofdepthfromthesamplesurface. ....... 47 2-8Nanoindentationsonsteelmatrixandindividualcarbides. ............ 47 2-9Indentationload-displacementcurvesfordifferentcarbidespeciesinM-50NiLandP675. ....................................... 48 2-10Compressivestressstrainresponseforthevarietyofmaterialstested. ..... 49 2-11M-50NiLhardnessproleandcarbidevolumefractionasafunctionofdepth. 50 2-12P675hardnessproleandcarbidevolumefractionasafunctionofdepth. ... 50 2-13Subsurfacehardnessvaluesasafunctionofcarbidevolumefraction. ..... 51 2-14Calculatedeffectivecompositemodulusasafunctionofdepth. ......... 52 3-1HardnessproleswithdepthforP675,M-50NiL,andM-50materials. ..... 67 3-2Surfacehardnessvs.loadforM-50throughhardenedandthehardestsectionsofbothgradedmaterials. .............................. 68 3-3Surfacehardnessvs.loadforallsectionstested. ................. 69 3-4Normalizedsurfacehardnessasafunctionofindentationloadforallsectionsandmaterialstested. ................................. 70 3-5Normalizedsurfacehardnessasafunctionofnormalizedindentationloadforallsectionsandmaterialstested. .......................... 71 3-6Trendsinhardnessgradientandpowerlawexponent. .............. 72 4-1Relationshipbetweenhardnessandowstressusingrepresentativeplasticstrain. ......................................... 97 8

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4-2Averageplasticstrainasafunctionofhardeningexponent. ........... 97 4-3Schematicofsectionedindentgeometryandsubsurfaceindenteld. ...... 98 4-4SubsurfacehardnessincreasebeneathlargemacroindentationonP675corematerial. ........................................ 98 4-5SubsurfacehardnessincreasebeneathlargemacroindentationonM-50throughhardenedmaterial. .................................. 99 4-6Examplegeometryandmeshusedforindentationmodeling. .......... 100 4-7FlowcurvesforP675coreandM-50throughhardenedmaterials. ....... 100 4-8SubsurfacestraincontourplotforM-50throughhardenedmaterial,alongwithstrainvaluesalongthecenterline. ......................... 101 4-9ComparisonbetweenexperimentalandcalculatedsubsurfacehardnessincreaseforM-50. ....................................... 101 4-10Alternativestress-straincurvesforthethroughhardenedM-50withvarietyofstrainhardeningvalues. ............................... 102 4-11ComparisonbetweenexperimentalandcalculatedsubsurfacehardnessincreaseforM-50forvariouslevelsofstrainhardening. ................... 102 4-12PredictedandmeasuredhardnessvalueswithintheplasticzoneofthemacroindentonP675core. ................................. 103 4-13Stress-straincompressioncurvefortheP675corematerialathighstrainlevels. 103 4-14Subsurfacehardnessbeneaththemacro-indentationongradedP675. ..... 104 4-15Subsurfacehardnessbeneaththemacro-indentationongradedM-50NiL. ... 104 4-16RelationshipbetweenH=yandE=ycalculatedbytheexpandingcavitymodelforvariousvaluesofn. ............................... 105 4-17FlowcurvesasafunctionofhardnessforthegradedP675. ........... 105 4-18Subsurfaceplasticstrainsalongcenterlinebeneaththemacro-indentationongradedP675. ..................................... 106 4-19Experimentalandnumericalsubsurfacehardnessbeneaththemacro-indentationongradedP675. ................................... 106 4-20FlowcurvesasafunctionofhardnessforthegradedM-50NiL. ......... 107 4-21Experimentalandnumericalsubsurfacehardnessbeneaththemacro-indentationongradedM-50NiL. ................................. 107 9

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5-1RelationshipsbetweenH=yandE=ydeterminedviasimulationsofindentationsonstrainhardeningmaterials. ............................ 126 5-2Subsurfacehardnessprolesasafunctionofdepthforthevariousexperimentalspecimens. ...................................... 127 5-3Simulatedsubsurfacehardnessgradientswithstartingsurfacehardnessof900kg/mm2. ..................................... 128 5-4Surfacehardnessasafunctionofindentationloadforgradedmaterialswithstartingsurfacehardnessof900kg/mm2. ..................... 129 5-5Flowcurvesforperfectlyplasticandstrainhardeningmaterialswithxedelasticmodulusof200GPa. ................................. 129 5-6Surfacehardnessasafunctionofindentationloadforperfectlyplasticandstrainhardeningmaterials. ............................. 130 5-7Subsurfacetrendsinstrainhardeningexponent. ................. 131 5-8Flowcurvesformaterialswithgradientinstrainhardening. ........... 131 5-9Surfacehardnessunderincreasingloadformaterialswithgradientsinstrainhardeningbehavior. ................................. 132 5-10Subsurfacetrendsinelasticmodulus. ....................... 133 5-11Surfacehardnessunderincreasingloadloadforelasticallygradedmaterials. 134 5-12Normalizedsurfacehardnessasafunctionofnormalizedindentationloadforthemodeledgradedmaterials. ........................... 135 5-13Trendinhardnessgradientandpowerlawexponent. ............... 135 5-14Subsurfaceplasticstrainforuniformhardnessandgradedmaterials. ...... 136 5-15Subsurfaceplasticstrainalongcenterlinebeneathindentat50mdepth. ... 136 5-16SubsurfacevonMisesstressalongcenterlinebeneathindentat50mdepth. 137 5-17SubsurfaceplasticstrainandvonMisesstressalongcenterlinebeneathindentat50mdepthonperfectlyplasticgradedmaterials. ............... 138 5-18SubsurfaceplasticstrainandvonMisesstressalongcenterlinebeneath50mindentformaterialswithgradientsinstrainhardeningexponent. ........ 139 5-19SubsurfaceplasticstrainandvonMisesstressalongcenterlinebeneathindentat50mdepthformaterialswithgradientsinelasticmodulus. ......... 140 6-1HardnessproleofanitridedthroughhardenedM-50material. ......... 147 10

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6-2Convertingamicrographofcarbidesintoniteelementmodel. ......... 147 6-3Plasticstrainssurroundingcarbideparticles. ................... 148 11

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMICROSTRUCTURE-PROPERTYRELATIONSHIPSANDCONSTITUTIVERESPONSEOFPLASTICALLYGRADEDCASEHARDENEDSTEELSByMichaelA.KleckaAugust2011Chair:GhatuSubhashMajor:MechanicalEngineeringCasehardenedmaterials,popularlyusedinmanydemandingengineeringapplicationssuchasbearings,gears,andwear/impactsurfaces,havehighsurfacehardnessandagradientinmaterialproperties(hardness,yieldstrength,etc.)asafunctionofdepth;therefore,theybehaveasplasticallygradedmaterials.Inthecurrentstudy,twodifferentcommerciallyavailablecasecarburizedsteelsalongwithtwothroughhardenedsteelsarecharacterizedtoobtainrelationshipsamongthevolumefractionofsubsurfacecarbides,indentationhardness,elasticmodulus,andyieldstrengthasafunctionofdepth.Avarietyofmethodsincludingmicroindentation,nanoindentation,ultrasonicmeasurements,compressiontesting,ruleofmixtures,andupperandlowerboundmodelsareusedtodeterminetherelationshipsforelasticmodulusandcomparetheexperimentalresultswithmodelpredictions.Inaddition,themorphology,composition,andpropertiesofthecarbideparticlesarealsodetermined.Thegradientinhardnesswithdepthingradedmaterialsiscommonlydeterminedusingmicroindentationonthecross-sectionofthematerialwhichcontainsthegradationinmicrostructureorcomposition.Inthecurrentstudy,anovelmethodisproposedtopredictthehardnessgradientproleusingsolelysurfaceindentationsatarangeofloads.Themethoddoesnotrequirethegradedmaterialtobesectioned,andhaspracticalutilityinthesurfaceheat-treatmentindustry.Foramaterialwithadecreasinggradientinhardness,higherindentloadsresultinalowermeasuredhardnessduetothe 12

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inuenceofthesoftersubsurfacelayers.Apower-lawmodelispresentedwhichrelatesthemeasuredsurfaceindentationhardnessunderincreasingloadtothesubsurfacegradientinhardness.Acoordinatedexperimentalandnumericalstudyispresentedtoextracttheconstitutiveresponseofgradedmaterials,utilizingrelationshipsbetweenhardness,plasticdeformation,andstrainhardeningresponse.Theaverageplasticstraininducedbyanindentisshowntobeaneffectivemeasureoftherepresentativeplasticstrain,whichisusedinordertorelatehardnesstoyieldstrengthinbothvirginandplasticallydeformedmaterials.Itisshownthatthetwocarburizedsteelscontaingradientsinyieldstrength,butconstantstrainhardeningexponentwithdepth.Theresultingmodelofmaterialbehaviorisusedtocharacterizetheinuenceofspecicgradientsinmaterialpropertiesonthesurfaceindentationbehaviorunderincreasingindentationloads.Itisalsoshownthattheresponseofthematerialisnotgreatlyinuencedbystrainhardeningexponent,whileagradientinstrainhardeningabilityonlyhasminimalimpact.Gradientsinelasticpropertiesarealsoshowntohavenegligibleinuenceforaxedgradientinhardness.Thedepthofsubsurfaceplasticdeformationisshowntoincreasewithsharpergradientsinhardness,butisnotalteredbygradientsinelasticproperties.Theproposedapproachisnotspecictocasehardenedmaterialsandcanbeusedtodeterminethesubsurfacehardnessgradientforanygradedmaterial. 13

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CHAPTER1INTRODUCTION 1.1GradedMaterialsGradedmaterialshaverecentlydrawnincreasingattentionduetotheirimprovedperformanceinabroadassortmentofapplications[ 1 5 ].Benetsincludesuperiorresistancetofatigue,wear,impact,andabrasivedamage.Theyarefrequentlyclassiedaselasticallygraded,plasticallygraded,orfunctionallygraded,duetospatialvariationinassortedmaterialproperties.Thesepropertiestypicallyvaryinthematerialasafunctionofdepth,consistingofgradientsinelasticproperties(e.g.,Young'smodulus)and/orplasticproperties(e.g.,yieldstrength,strainhardeningbehavior).Manyclassesofmaterialscanbecreatedwithgradedcongurations,includingmetals,ceramics,polymers,andbiologicalmaterials,andmixturesofthesecomponentsarefrequentlyutilized.Often,twoormoredifferentmaterialsarecombinedinordertoexploitthepositivebehavior/propertiesofindividualconstituentswhileattemptingtoovercomeinherentweaknesses.Someimportantattributesforenhancedperformanceincludehardness,toughness,strength,fatigueresistance,wearresistance,energyabsorption,thermalproperties,andcorrosionresistance.Gradedmaterialsfrequentlycontainreinforcingphasesand/orgradientsincompositionandmicrostructurewhichcanbebothlinearornon-linear,smoothorabrupt,andgradualorstep-wise.Eventhinlmsareatypeofgradedmaterial,consistingofonlytwolayersthelmandthesubstrate.Finally,thegradientsmaybecreatedthroughmanydifferentmethods,includingcasting,deposition,powderprocessing,andheattreatment.Someexamplesofgradedmaterialsincludehardceramicparticlesdistributedwithinaductilemetallicorpolymericphase,wherethevolumefractionoftheinclusionsisvariedwithdepth.Additionally,thesizeoftheparticlescanbealteredtoachievevariousresults.Gradientsinalloyingelementscanbeintroduced,resultinginchangesinstrength,corrosionresistance,andevenelectricalproperties.However,gradedmaterials 14

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canalsoexistwithoutchangesincomposition.Forexample,varyinggrainsizewithdepthcanbeusedtocontrolpropertiessuchashardnessandstrength.Shot-peenedmaterialsarealsoconsideredplasticallygraded,withaworkhardenedexterioroveraductilecorematerial.Insomedesigns,acombinationofthesevariablesmaybeusedinordertocreatecomplexvariationsinanynumberofproperties.Withrespecttoenhancedperformance,ithasbeendemonstratedthatgradientsinelasticpropertiescanbedesignedinordertosuppresscrackinitiationduringHertzianindentationofbrittlematerials[ 4 5 ].Gradientsinplasticpropertieshavebeenusedinordertominimizedamageduetoindentationwithbothbluntandsharpindenters[ 6 8 ].Thelocationofmaximumstresscanbealtered,oftentimesshiftedawayfromthesurface,thuslimitingcrackformationanddamagepropagation.Thisindicatesthatdifferentgradientsinmaterialpropertiescanbetailoredtotspecicapplications. 1.2ApplicationsGradedmicrostructurescreatedthroughvariousheattreatmentprocedureshavebeenusedtotailorthepropertiesofengineeringmaterialssincetheiron-age,withearlierexamplesofsuchfunctionallygradedmaterialsdatedbacktothedawnofcivilization[ 9 10 ].Interestingly,theseprinciplesofheattreatmentarepracticedthroughoutthesteelindustryeveninmorerecenthigh-technologyapplications.Commonprocessingmethodsincludeinductionhardening,carburizing,nitriding,aswellasothersurfacetreatments.Thesecasehardeningprocessescanbeappliedtoawidevarietyofalloysteels,resultinginanassortmentofgradientsandpropertiesfordemandingapplications.Dependingonthenatureoftheheat-treatmentandslightvariationsincomposition,complexgradientsinplasticproperties(yieldstress,workhardeningbehavior,ultimatestrength)canbeintroducedintheouterlayersofamaterial.Morespecically,carburizationisacasehardeningprocedurewhichresultsinamaterialwithgradientsinmicrostructure,composition,andmaterialpropertiesas 15

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afunctionofdepth.Theresultingmicrostructuretypicallyincludeswelldistributedcarbides,whichprecipitateduringheat-treatment,surroundedbyasteelmatrix.Thecarbidesareamuchharderphasethanthesurroundingmatrix,andtheirdistributionvariesasafunctionofdepth,resultinginagradientinmechanicalproperties.Carburizedsteelshavebecomeincreasinglyimportantindemandingengineeringapplicationsincludinghighperformancebearings,gears,andwear/impactsurfaces.Thesematerialshaveahardexteriorwhichgraduallytapersintoasofter/ductilecorematerialbeneaththesurface.Ithasbeenshownthatagradedmaterialofthiscongurationfrequentlyoutperformsmaterialswithuniformcompositioninapplicationsinvolvinghighcontactstresses[ 3 ].Thegradientinpropertieswithdepthredistributesstressesinducedbycontact,thusallowingforhigheroperatingloadswhilesimultaneouslylimitingplasticdeformation.Thegradedcongurationisanimprovementuponasimplesurfacecoating,whichmayleaveanabrupttransitioninmaterialpropertiesbetweencoatingandsubstrate. 1.3CharacterizationDeterminingthebehaviorofsuchgradedmaterialsisessentialfordesignofengineeringmaterialswithtailoredpropertiesapplicabletomanyhighperformanceapplications.Althoughgradientsaretypicallyintroducedbydesign,characterizationofthevariousmechanicalandmaterialparametersisparamountinunderstandingandimprovingtheperformanceofgradedmaterials.Simplyobservingthematerialusingopticalorelectronmicroscopycanidentifythesize,shape,anddistributionofthevariousconstituents,providedtheycanberesolvedusingthesemethods.Thesemicroscopicapproachesrequiresamplestobepreparedappropriately,typicallybycross-sectioningandpolishing,inorderforasuitableimagetobeviewed.Chemicalcompositionscanalsobedeterminedonthecross-section,usingmethodssuchaselectronmicroprobeanalysis,aswellaswavelengthandenergydispersiveX-rayspectroscopy. 16

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Indeterminingmechanicalproperties,gradedmaterialscannotbereadilycharacterizedusingstandardtestingprocedures,duetotheirnon-homogeneousconstruction.Ifagradedsectionweretestedusingconventionalcompression,tension,torsion,andbendingtests,theresultswouldonlyprovideaveragedpropertiesfromthecombinedbehaviorofthevariouscomponents.Instead,testingofindividualconstituents/phasesisoftenaviableoptionfordeterminingtheupperandlowerbounds.Forexample,puresamplesofeachcomponentcanbespeciallyproducedfortesting.Withtheresultingproperties,aruleofmixturescanbeusedinordertoestimatethebehaviorofthecombined,gradedmaterial.Whilethisprovidesabaselinefordesignofgradients,itcanonlybeusedwhenthetwophasescanbesynthesizedorseparatedreadily.Indentationmeasurementsareexceptionallyversatileandfrequentlyusedasacharacterizationtoolforgradedmaterials,astheyallowforsamplingofmaterialpropertiesacrossmanysizesandscales[ 11 12 ].Nano-andmicro-indentationmethodsprovidetheabilitytoextractpropertiesfromindividualcomponentsorphases,whilelargermacro-indentationcanprovidepropertiesrelatingtothecombinedbehavior.Withrecentdevelopmentsininstrumentedindentation,theload-displacementbehaviorcanberelatedtoelasticproperties,aswellashardness. 1.4CurrentStudyThisdissertationdetailsacoordinatedexperimentalandnumericalstudytodeterminethecompleteconstitutiveresponseofgradedmaterials.Twocasehardenedcarburizedsteels(onestainlesssteelandonetoolsteel)commonlyutilizedinhighperformancegearandbearingracewayapplicationsareusedasmodelmaterials.However,themethodsandanalysisareintendedtobeapplicabletoavarietyengineeringmaterialsirrespectiveofthenatureofthegradientsinthematerialproperties.Chapter 2 containsacomprehensivecharacterizationofthevariationofcarbidevolumefractionanditsrelationshiptomechanicalpropertyvariationasafunctionof 17

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depthinthetwocarburizedsteels.Forcomparisonpurposes,twoadditionalsteelswithnominallynovariationincarbidevolumefractionwithdepth(i.e.,throughhardened)arealsoanalyzed.Multiplecharacterizationandtestingtechniquesareemployed,andtheresultsareanalyzedtodeterminethevariousgradientsinmechanicalproperties.Attentionispaidtoboththeindividualconstituents(i.e.,carbidesandmatrix),aswellastheglobalbehaviorunderindentation,compressiontesting,andultrasonicmeasurements.InChapter 3 ,amethodisintroducedfordeterminingsubsurfacehardnessgradientsusingonlysurfacehardnessmeasurements,withouttheneedforcross-sectioning.Thisnovelmethodhaspracticalutilityinthesurfaceheat-treatmentindustry,wheretheeffectivenessofanintendedtreatmentprocesscanbeassessedrapidlybysimplesurfacehardnessmeasurementsinordertodeterminetheresultinggradientinmechanicalpropertyvariationwithdepth.Theresultalsoprovidesinsightintothelevelofinuencethedeeperlayersofagradedmaterialmayhaveonsurfacedeformation,andhowtheseverityofthesubsurfacehardnessgradientinuencesthesurfacehardnessmeasurement.Chapter 4 presentsacombinedexperimentalandnumericalmethodusefulinextractingmechanicalpropertiesfromgradedmaterials,usingthecarburizedsteelsamplesasmodelmaterials.Themethodfocusesonmacro-indentationasameansforinducingplasticdeformationintobothgradedandhomogeneousmaterials,whileattentionispaidtotherelationshipsbetweenhardnessandyieldstrengthinordertoextractthemechanicalbehavior(strainhardeningcharacteristics).Theresultsincludethestress-strainbehavior(owcurves)asafunctionofdepthforthegradedmaterials.Thenalchapter(Chapter 5 )utilizestheknowledgegainedinthepreviousthreechaptersinordertoconstructamodeloftheindentationbehaviorofgradedmaterials.Avarietyofgradientpropertiesareconsidered,includingthosewithgradientsinhardness,gradientsinelasticproperties,andgradientsinplasticproperties(specically,strain 18

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hardeningbehavior).Attentionispaidtoboththesurfacehardnessunderincreasingindentationloads,aswellassubsurfacedeformation.TheresultsarerelatedbacktothesurfaceindentationbehaviorrstdescribedinChapter 3 19

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CHAPTER2MICROSTRUCTURE-PROPERTYRELATIONSHIPS 2.1BackgroundWhiletherehavebeenavarietyofstudiesongradedmaterialsinthepast[ 3 6 8 13 15 ],fewhavediscussedtherelationshipsamongmicrostructure,compositionalgradation,andmechanicalpropertiesinsuchmaterials.Thosewhichhaveconsideredmicrostructuralfeaturesoftenutilizedspeciallycraftedmaterialspreparedexclusivelyforthestudyofparticulargradientsinproperties(e.g.,gradientingrainsize).Thisapproachprovidesatechniqueforisolatingtheinuenceofindividualcharacteristics(i.e.,inuenceofgrainsizeandcompositiononmodulus,yieldstrength,hardeningcoefcientvariation,etc.).Forinstance,Choietal.[ 6 7 ]conductedacombinedniteelementandexperimentalanalysisonananocrystallinenickel-tungstenalloywithgradientsingrainsize.Thematerialdisplayedalinearvariationinyieldstrengthasafunctionofdepthduetovariationingrainsizedistribution.Thestudyinvestigatedtheindentationresponseofthematerialintermsofsub-surfacestressdistributionandpile-upbehaviorasafunctionofyieldstrengthgradientwithdepth.However,studieswhichconcentrateongradientsinasinglematerialpropertymayhavedifcultypredictingtheresponseofmaterialscontainingcomplexsimultaneousgradientsinmicrostructure,composition,grainsize,volumefractionofinclusions,etc.Thepresentanalysisintendstoshedlightonthesecomplexrelationshipsbetweenthevariousattributescontributingtooverallmechanicalproperties.Carburizationisacasehardeningprocedurecommonlyusedinthemanufactureofhighperformancesteelcomponents.Thissurfaceheat-treatmentmethodresultsinamaterialwithgradientsinmicrostructure,composition,andmaterialpropertiesasafunctionofdepth,thusbehavingasafunctionallygradedmaterial.Theresultingmicrostructuretypicallyincludeswelldistributedcarbides,whichprecipitateduringheat-treatment,surroundedbyasteelmatrix.Thecarbidesareamuchharderphase 20

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thanthesurroundingmatrix,andtheirdistributionvariesasafunctionofdepth,resultinginagradientinmechanicalproperties.Carburizationitselfisalongstandingprocess,datingbacktotheiron-ageandbeginningwithmethodsofpackcarburizationusedtoimprovethehardnessandwearresistanceofavailablesteelsusedintoolsandothercomponents[ 9 10 16 ].Whencomparedtohighcarbonsteels,carburizedsteelstypicallyhaveimprovedfracturetoughness,duetotheductilecorematerial,aswellasimprovedimpacttoughness[ 17 ].Caremustbetaken,however,inordertoavoidpotentialpitfallsassociatedwithimproperprocessingconditions.Difcultiesincludeoxidationproblemsduringheattreating,decarburization,andquenchcracksinimproperlydesignedcomponents.Currentcleancastingmethods,tightalloycontrol,andvacuumcarburizingproceduresresultinconsiderablyimprovedcharacteristicsascomparedtopastmethods.Furthermore,inthehighperformancesteelsconsideredinthecurrentstudy,thealloyingprocesshasbeenextensivelystudied,resultingintheabilitytopredictperformancebasedonprocessingconditions[ 18 20 ].Despitetheseimprovementsinheattreatmentandprocessing,thegradientinmechanicalpropertiesisrarelyemphasized.Whilestudiesinthepasthavebeeninterestedinextractingthemechanicalpropertiesofcarburizedmaterials,rarelyarethemicrostructureandmechanicalpropertyvariationwithdepthincludedinthecharacterization.Withamicrostructureconsistingofahardceramiccarbidephasesurroundedbyasteelmatrix,carburizedsteelsarerelatedtocompositematerials.Examplesofothermaterialswithsimilarmicrostructureincludemetalmatrixcomposites(MMC)andcementedcarbides(cermets)usedincuttingandgrindingtools.Shenetal.[ 21 22 ]analyzedaluminumalloyreinforcedbysiliconcarbideparticles,focusingonrelationshipsbetweentensileandindentationbehaviorasafunctionofparticlesizeandvolumefraction.Theydemonstratedthatprocessinginducedbrittledamageinthesiliconcarbideparticlesresultedinreducedtensilestrength,butdidnotsignicantlyaffectthe 21

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indentationbehavior.Inafollow-upanalysis,ShenandChawla[ 23 24 ]attemptedtopredicttensilestrengthbasedonindentationhardnesstesting,however,theresultsfrequentlyoverestimatedtheoverallcompositestrength.AsimilaranalysisbyRavichandran[ 25 ]attemptedtocreateclosed-formequationsinordertopredicttheowstressandstress-strainbehaviorofcompositesconsistingofcoarserigidparticlesinaductilematrix.Theresultswerecompared,withgoodagreement,toniteelementsimulationsofcompositematerials.Onlyrecentlyhavealimitednumberofstudiesconsideredanalyzinghardenedsteelsinamannersimilartometalmatrixcomposites.Youngetal.[ 26 ]usedX-raydiffractiontoobservetheloadvariationbetweenaferritematrixandcementite(ironcarbide)particlesinhigh-carbonsteels.Theirsamplescontainedacarbidevolumefractionontheorderof30%,resultinginconsiderablyalteredbehaviorwhencomparedtothesteelmatrixalone.Similarly,Delinceetal.[ 27 ]constructedamicrostructurebasedmodelfortheoptimizationofdual-phasesteels(ferriteandmartensite).Theydemonstratedtheinuenceofinclusionsizeandvolumefractionontheoverallperformanceofhigh-carbonsteels.Whilethesestudieshavetreatedcarburizedmaterialsasacomposite,literatureonthecharacterizationofmicrostructuralgradationasafunctionofdepthissparse.Thischaptercontainsacomprehensivedescriptionofthevariationofcarbidevolumefractionanditsrelationshiptomechanicalpropertyvariationasafunctionofdepthinthecaselayeroftwodifferentcarburizedsteelswhicharecommonlyusedinthehighperformanceballbearingandgearindustries.Forcomparisonpurposes,twoadditionalsteelswithnominallynovariationincarbidevolumefractionwithdepth(i.e.,throughhardened)arealsoanalyzed.Multiplecharacterizationmethodsareemployed,andtheresultsareanalyzedtodeterminethegradientinmechanicalproperties.Attentionispaidtoboththeindividualconstituents(i.e.,carbideandmatrix), 22

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aswellastheglobalbehaviorunderindentation,compressiontesting,andultrasonicmeasurements. 2.2MaterialsFourdifferentsteelswereexaminedinthecurrentstudy:twothroughhardenedsteelsandtwocasehardenedcarburizedsteels.TherstthroughhardenedsteelwasM-50toolsteel,whichhasauniformcarboncontentof0.85wt.%throughoutitsdepth.Amongthecasecarburizedsteelswere(i)M-50NiL,whichisavariantontraditionalM-50toolsteelhavinganincreasednickelcontent(3.40wt.%)andlowerinitialcarboncontent(0.13wt.%)toaidincarburizationandsubsequentheattreatment,and(ii)Pyrowear675(P675forshort),astainlesssteelwith13.0wt.%chromiumcontent,designedforenhancedcorrosionresistance.ThenominalcompositionsofthesethreematerialsaresummarizedinTable 2-1 .Whiletheabovethreematerialsarecommerciallyavailable,thefourthmaterialconsideredinthisstudywasaspeciallyproducedcarburizedandheat-treatedP675withnearlyuniformhighhardness(nearthatofthecaselayerinP675)andnearlyuniformcarboncontent.Inorderforthecarbideconcentrationtoremainrelativelyconstantthroughoutthethicknessofthismaterialaftercarburization,thespecimenthicknesswasrestrictedtoonly2.54mm.ThesurfacehardnessofthismaterialnearlymatchedthatofthestandardP675,withonlya10%decreaseinhardnessintheinnerregion.Thissampleof`monolithic'P675casematerialwasusedasarepresentationofthehardeneduppermostcaselayerofP675andwillbeincludedinlateranalysis.AsimilarmaterialwasnotavailablefortheM-50NiL.AllmaterialswereinitiallyproducedusingtheVIMVAR(vacuuminductionmelting,vacuumarcremelting)method,resultinginexceptionallyhomogenizedmicrostructureswithconsistentchemicalcompositionandcleanliness.WhilethecarburizationandheattreatmentproceduresforM-50NiLandP675werenotrevealedbythesupplier(consideredproprietaryinformation)ageneraldiscussionofheattreatmentandtemperingisbenecialindescribingtheevolutionofthenal 23

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microstructure.Thetypicalcasehardeningprocessinvolvesvacuumcarburizationfollowedbymultipletemperingcyclesinordertoproducethenalmicrostructureconsistingofnelydispersedcarbideswithinatemperedmartensitematrix.UnlikethethroughhardenedM-50steel,whichfrequentlycontainslarge(greaterthan5m)primarycarbidesprecipitatedduringinitialcasting,thesecarburizedmaterialscontainmainlysmallercarbideswhichonlyformduringsubsequenttemperingcycles(duetothelowerinitialcarboncontentduringcasting).Thesecarbidesareontheorderof1to3minsize,andwelldistributed.Additionally,duetothecarburizingprocedurebeingdiffusioncontrolled,thereexistsagradientinthedistributionofthecarbidesfromcaselayertocore,withthecaselayerhavingalargevolumefractionofcarbides,whichdecreasesgraduallywithdepth.Inadditiontothegradientincarbidevolumefraction,aresidualcompressivestressdevelopsintheoutercaselayersofthecarburizedmaterials.Thisisduetothevolumetricexpansionfromthemartensitictransformationwhichbeginswhenthesamplesarequenchedfollowingcarburization[ 10 16 ].Additionalmartensitedevelopsduringthesub-zerotreatment,whichaidsinconvertinganyretainedaustenite.Duetothegradientincarboncontent,theexpansionoftheoutercaselayersisgreaterthantheinnermaterial.However,giventhattheoutermaterialisconstrainedbythecore,aresidualcompressivestressdevelops.Aportionofthisresidualcompressivestressisrelievedduringthesubsequenttemperingcycles(duringwhichthecarbideprecipitatesform),thoughagoodportionoftheresidualstressremains.Thedirectionofthisresidualcompressivestressisparalleltothesurface,whilerelativelylittleresidualstressexistsnormaltothesurface.Themagnitudeofthisresidualstresscanbeontheorderof400MPanearthesurface,andrapidlydecreaseswithdepth[ 28 29 ].Thisstressstatehasbeenshowntoincreasetheperformanceofcarburizedmaterialsbydecreasingtheeffectivemeantensilestressinducedduringrollingcontactloading.Conversely,thethroughhardenedM-50and`monolithic'P675casematerialsdonotdevelopnotable 24

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residualcompressivestressesduringheattreatment,duetothelackofgradationincarbondistribution.Inbearingringsandraceways,acompressivestressinthehoopdirectionalsodevelops,duetothegeometry(expansionisconstrainedduetotheringshape).Thisisespeciallybenecialinracewayswhicharefrequentlypress-tontoshaftsinordertoavoidanymotion.Thepress-t,whichistypicallyextremelystrong,createsatensilestressinthehoopdirection,whichiscounteredbytheresidualcompressivestressesfromheattreatment.Inordertocharacterizethesize,shape,anddistributionofthecarbideparticleswithinthegradedregionsofP675andM-50NiL,aswellasthethroughhardenedM-50,specimenswereextractedfromtheseregionsandpolishedonthecross-sectionusingstandardmetallographictechniques.Toenhancethecontrastofthecarbideparticleswhenviewedunderanopticalmicroscope,nalpolishingusingchemical-mechanicalpolishing(CMP)mediawasemployed.TheCMPcontainedamixtureof0.05mcolloidalsilicaandaluminainasolutionwithapHof8.5.Thisprocedurelightlyremovesthesteelmatrixmaterial,whileleavingthehardcarbidephasevisiblewithwell-denedboundaries.TheresultingmicrostructurefromallfourmaterialsisshowninFigures 2-1 through 2-4 underdarkeldillumination(carbidesappearbright,matrixappearsdark).InsetwithintheseguresarehighmagnicationimagestakenunderdarkeldandNomarskiillumination(differentialinterferencecontrast,DICcarbidesappearraised)tobetterillustratethesizeandshapeofthecarbides.ThethroughhardenedM-50material,showninFigure 2-1 ,containsamixtureoflargecarbides(5to10minsize),andsmallercarbides(lessthan5m),withnearlyuniformvolumefractionthroughoutthethickness.Thelargecarbidesformduringcastingofthismaterial,duetothehigherinitialcarboncontent,whilethesmallersecondarycarbidesprecipitateduringtemperingcycles[ 17 30 ].Furthermore,thelargecarbidesareoftenorientedinbands(Figure 2-1 ),whichisaresultofanyrollingand 25

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forgingoperationsperformedonthematerialaftercastingbutpriortoheattreatment.IthasbeenshownthatthesecarbidestendtobeoftheformM23C6andM7C3,whereMrepresentsametalandCrepresentscarbon[ 31 ].Thesecarbidesareamixtureofbothironandchromiumcarbides,whileotheralloycarbides(MoandV)arealsolikelypresentinsmallquantities.ThemicrographsinFigures 2-2 and 2-3 containtheentiregradedregionsofM-50NiLandP675.Theseguresrevealthehighconcentrationofcarbidesnearthesurfaceofthegradedmaterials,withadecreasingconcentrationofcarbideswithdepth.ThecarbidesshownintheM-50NiLsamplesaresmaller,typicallyontheorderof1minsize(Figure 2-2 ).IthasbeenshownthroughbothX-raydiffractionandelectronbackscatterdiffraction(EBSD)thatthesesmallroundcarbidesarevanadiumcarbides,mainlyintheformofVCandV7C8[ 17 32 ].Theseareamuchhardercarbidespecies,aswillbediscussedingreaterdetailinalatersection.Additionally,thegradedlayerappearstoextendtoadepthofmorethan2mm.Alternatively,thecarbidesintheP675includebothelongated/rodlikeparticlesaswellassmallerrounded/globularparticles(Figure 2-3 ).Thelargestcarbidesareontheorderof3to4minsize,whiletheroundedcarbidesareslightlysmaller(1to2minsize).IthasbeenreportedfromEBSDthattheelongatedcarbidestendtobeoftheM7C3composition,whilethemoreroundedcarbidestendtobeM23C6[ 17 32 ].Duetothehighchromiumcontentinthissteel,itisbelievedthatthesecarbidesareCr7C3andCr23C6,althoughotheralloycarbidesarelikelypresentinsmallerquantities.Furthermore,aswillbeshownlater,thereappearstobeahigherconcentrationofthesecarbidesinthecaselayerascomparedtotheM-50NiLmaterial,althoughthegradedlayerdoesnotextendasdeep(approximately1.5mmdepth).The`monolithic'P675casesampleshowninFigure 2-4 containsasimilarconcentrationofcarbidesasthesurfaceofthegradedP675.Theconcentrationappears 26

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toremainrelativelyconstantthroughoutthethicknessofthespecimen.Additionally,thesizeandshapeofthecarbidesmatchthosefoundinthegradedsection.Finally,thegrainsizeofthesteelmatrixiscomparableinallfourmaterialsandontheorderof4to5m.Itshouldbenoted,however,thateachgrainwillcontainevidenceofavarietyofmartensiticplatestructures,showninFigure 2-5 forthroughhardenedM-50,whicharetheresultofheattreatment[ 10 ].Subsequenttemperingservestoprecipitatethecarbidephase,whilethenalmicrostructurestillcontainsevidenceofthistransformation.Thecarbidestendtoformatgrainboundaries,allowingforestimationoftheoriginalgrainsize;however,distinctgrainboundariesaredifculttorecognize. 2.3Experimental 2.3.1CarbideDistributionOpticalmicroscopywasutilizedtodeterminethevolumefractionofcarbidesinthegradedregionsofbothcarburizedandthroughhardenedmaterialsutilizingtheguidelinesinASTMstandardE-1245.Thisisageneralmethodfordeterminingthevolumefractionofinclusionsinanymaterial,andspeciesthatatleastsiximagesmustbeanalyzedforproperstatisticalcoverage.Theareaofthecarbidesismeasured,plusthetotalareaofthesampleimage.ThevolumefractionisthenequaltotheareafractionsimplyasVcarbides=Acarbides=Acarbides AtotalAminimumoftenimageeldsofsize200mby100mwereanalyzedatagivendepth.Thedepthatthecenterofeachimagewasnotedasthelocationofthemeasurement.Thesmallverticalsizeoftheimageeldwasusedinordertominimizeerrorduetothecontinuouslyvaryingvolumefractionasafunctionofdepth.Nodistinctionwasmadebetweenthepossiblecarbidespecies;rather,thetotalvolumefractionofcarbideswasanalyzedatagivenlocation.TheresultingcarbidecontentvariationasafunctionofdepthisshowninFigure 2-6 andTable 2-2 27

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Figure 2-6 revealsthatthevolumefractionofcarbidesatthesurfaceofP675ismorethantwiceasdenseascomparedtotheM-50NiLmaterial;however,thedepthofthegradedlayerintheM-50NiLmaterialisnearlydoublethatoftheP675.ThethroughhardenedM-50maintainsanearlyconstantvolumefractionaround20%,withonlyslightdecrease(approximately2vol.%)whenapproachingdepthsgreaterthan1mm.Finally,thespeciallyproducedP675`monolithic'casecontainsasimilarmaximumcarbidevolumefractionastheoutercaseofthegradedP675,withonlyaminordecreasetowardthecenterofthe2.54mmthicksample.Whiletheseresultsillustratetheextentofthegradedlayerinthecarburizedmaterials,theresultingvariationinmaterialpropertiesasafunctionofdepthcannotbedirectlyinferred.Tofullycharacterizethesegradientsinmechanicalproperties,additionaltestingwasconductedonthesamecrosssections. 2.3.2MicroindentationToevaluatetheinuenceoftheabovedistributionofcarbidesonthemechanicalbehaviorofthesematerials,microindentationhardnessmeasurementswereconductedonthecrosssection.Vickershardnesswasmeasuredusingastandardtable-topindentationtester(WilsonTukon2100B)atanindentationloadof200gwith15stotalloadingtime.Indentswereplacedat100mspacingthroughtheentiredepthofthegradedmaterialasshownintheinsetofFigure 2-7 .Theloadof200gwaschosenbecauseitwasfoundtobethesmallestindentwhichcanbereliablymeasuredusinganopticalmicroscopewhileallowingareasonablyclosespacing(asperASTME384).Itshouldbenotedthatindentationsproducedatthisloadhadindentdiagonalsizesontheorderof20mandlarger,encompassingboththematrixmaterialaswellasanumberofcarbides,thusgivingameasureofthehardnessduetothecombinationofthetwo.Aminimumofveindentswereperformedateachdepth,andtheresultingsubsurfacehardnessprolesareshowninFigure 2-7 28

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Concerningtheresidualstressesinthecarburizedmaterials,ithasbeenshownthatindentationhardnessmeasurementsproducedusingasharpindentertiparenotinuencedgreatlybyresidualcompressivestresses,particularlywhentheresultingindentationimpressionismeasureddirectly(i.e.,notestimatedfromaninstrumentedindentationload-displacementcurveusingindentergeometry)[ 33 34 ].Moreover,thesectioningandpolishingproceduremayalsorelievetheresidualstress.Assuch,thehardnessdatafromthecross-sectionspecimenspresentedinFigure 2-7 canbeconsideredanappropriaterepresentationofthesubsurfacehardnessgradients.Also,notethatthethrough-carburized`monolithic'P675casesample(whichshouldnotdevelopanynotableresidualstresses,similartothethroughhardenedM-50)nearlymatchestheappropriatehardnessofthetoplayerofthegradedP675materialbasedonthevolumefractionofcarbides,furtherdemonstratingthatthemeasuredsubsurfaceindentationhardnessisaccurateandnotinuencedbyresidualstress.MuchlikethecarbidedistributionsshowninFigure 2-6 ,thehardnessgradientintheM-50NiLmaterialismoregradualthanintheP675,whilethethroughhardenedM-50exhibitsconstanthardnesswithdepth.ThesurfaceregionoftheM-50NiL,wherethecarbidedistributionremainedwithintherangeof12vol.%overadepthof300m(Figure 2-6 ),alsomaintainsarelativelyconstanthardnesslevelnear825kg/mm2.Interestingly,thetrendsinsubsurfacehardnessforallofthematerialsappearverysimilartothetrendsincarbidedistributionshowninFigure 2-6 2.3.3NanoindentationTogainfurtherunderstandingofthepropertiesoftheconstituentsineachofthematerials,instrumentedindentationwasconductedonindividualcarbideparticlesandthematrixusingaHysitronTriboIndenterataloadof1mNusingaBerkovichindentertip.Becauseitisdifculttopreciselyplaceanindentonagivencarbideparticle,anarrayof150indents(15x10gridpattern)wasconductedinthecaselayerofeachmaterialandonlythoseindentswhichwerefullycontainedwithincarbideparticles 29

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wereanalyzedtodeterminethehardnessandelasticmodulusofcarbides.WiththehighvolumefractionofcarbidesinthecaseregionofP675,atotalofeightindividualindentsoccurredentirelywithincarbideparticles.FortheM-50NiLwithlowercarbidevolumefraction,veindividualindentsfellwithincarbideparticles.Figure 2-8 showsSEMimagesofnanoindentationsondifferentcarbides.Notethatthetypicalindentationsizeisestimatedtobelessthan300nm,whereastheparticlesizeswerewellabove1m(Figures 2-1 through 2-4 ).Therefore,theselectedfewindentationswerefullycontainedwithineachparticle.Theresultingload-displacementdatawereutilizedinordertodeterminethehardnessandthemodulususingthefamiliarOliver-Pharrmethod[ 35 36 ].Exampleload-displacementcurvesareshowninFigure 2-9 forcarbidesfoundinbothgradedmaterials.HardnessvaluesforthecarbidesinthecaselayerofP675werefoundtobeintherangeof19.02.0GPa,whilethemodulusvalueswereintherangeof29015.0GPa.Thesevaluesrelatewelltothosefoundintheliteratureforinstrumentedindentationonchromiumcarbides[ 37 38 ].ForthehardervanadiumcarbidesfoundinM-50NiL,thehardnessapproached30GPawithamodulusnear500GPa.Veryfewstudieshavereportedhardnessandmodulusvaluesforvanadiumcarbides;nevertheless,thesevaluescomparewelltomaterialpropertiesbasedonrst-principlescalculations[ 39 40 ].AcomparisonoftherelevantmaterialpropertiesavailableinliteratureandthecurrentstudyisshowninTable 2-3 .AlsoincludedarethehardnessandmodulusvaluesmeasuredforthecarbidesfoundinthethroughhardenedM-50material.ItshouldbenotedthatthePoisson'sratiosusedforconversionbetweenreducedmodulus(commonlymeasuredviainstrumentedindentation)andelasticmoduluswere0.21forvanadiumcarbidesand0.33forchromiumcarbides,asfoundinliterature[ 37 40 ].Theequationrelatingreducedmodulustoelasticmoduluswas[ 35 ]: 1 Er=1)]TJ /F3 11.955 Tf 11.96 0 Td[(2i Ei+1)]TJ /F3 11.955 Tf 11.96 0 Td[(2s Es(2) 30

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whereEristhereducedmodulus,EiandiaretheelasticmodulusandPoisson'sratioofthediamondindentertip(1140GPaand0.07,respectively)andEsandsaretheelasticmodulusandPoisson'sratioofthematerialbeingtested.Inasimilarmanner,themodulusandhardnessvaluesforthesteelmatrixweredeterminedviainstrumentedindentation(againat1mNload).Themoduluswasfoundtobeintherangeof190to240GPaforallthreematerials,whilehardnesswasintherangeof4to9GPa(400to915kg/mm2).However,duetothelargevolumefractionofcarbides,anumberoftheseindentsmayhavebeeninuencedbyundetectedsubsurfaceandnearbycarbides.Inordertominimizethiseffect,theupperquartiledatawaseliminated,asitwaslikelyinuencedassuch.Testswereadditionallyconductedinboththecaseregionandthecore,todetermineifanyvariationinthesteelmatrixpropertiesexistsbetweenlocations.WithinbothouterandinnerregionsofthethroughhardenedM-50material,thesteelmatrixmoduluswasfoundtobe21010GPa,whilehardnesswas62GPa(610kg/mm2).InthecarburizedM-50NiL,thecaseregionshowedanelasticmodulusof22020GPaandhardnessof7.01.0GPa.Thesevaluesmaybeslightlyelevatedduetothecarbidesinthisregion.Conversely,thecoreregionwhichcontainsfewercarbidesshowedamodulusof20012GPaandahardnessof6.50.5GPa.Conductingindentsinthecoreregionataslightlyhigherload(10mN)providedsimilarelasticproperties,butwithanaveragehardnessof5.0GPa,whichmorecloselyresemblesthehardnessdeterminedviaVickersindentation(Figure 2-7 ).Thisminordifferenceislikelyattributedtotheindentationsizeeffect(ISE)[ 41 42 ].Finally,instrumentedindentationinthecaseregionofP675showedamodulusof21710GPaandhardnessof9.01.0GPa.Thesevalues,again,arepossiblyelevatedduetothehighnumberofcarbidesinthecaseregion.Inthecore,themoduluswasamoreconservative2038.0GPa,whilehardnesswas7.00.5GPa.Indentsat10mN 31

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loadgaveanaveragehardnessof6.0GPa,whileVickershardnessinthecoreregionwasontheorderof4.5GPa.Thisdifference,again,islikelyattributedtotheISE. 2.3.4CompressionTestingDuetothecontinuousgradationincarbidevolumefraction,itisdifculttoextractspecimensofconstantcompositionatanygivendepthinordertoobtainthestress-strainresponse.However,portionsoftheuniformhardnesscorematerialwerelargeenoughtobeextractedfrombothP675andM-50NiLandwerepreparedforcompressiontesting.Cylindricalspecimensofsize5mmdiameterx10mmheightwerecarefullyextractedviaelectricaldischargemachining(EDM)andthenlightlygroundandpolishedtoremoveanyheataffectedzone.Straingageswerebondedonthesespecimensinordertomonitorthedeformationduringloading.SpecimensfromthethroughhardenedM-50and`monolithic'P675casematerialwerealsopreparedforcompressiontestinginasimilarmanner.The`monolithic'P675casematerialprovidedameanstomeasurethecompressivebehaviorofthecaseregionofthegradedP675.CompressiontestingwasconductedinanMTSloadframewitha100kNcapacityloadcell.Thestraingagesweremonitoredprimarilytoinsurethemostaccuratemeasurementofelasticpropertiesforthematerials.Beyondastrainofapproximately5%,thegagesfrequentlypeeledoffofthesamples,astheadhesivewasratedonlytothisstrainlevel.Beyondthisstrain,themachineload-displacementdatawasutilizedtomonitortheplasticbehaviorofthematerials.AtotalofthreecorespecimensfrombothM-50NiLandP657,three`monolithic'P675casesamples,andthreethroughhardenedM-50specimensweretested,withtheresultingtrue-stressvs.true-straindataaveragedandshowninFigure 2-10 .Fromthecompressiondata,theelasticmodulusoftheP675corematerialwasdeterminedtobe192GPa,whilethemodulusofthe`monolithic'P675casematerialwas230GPa.Thisincreaseinmodulusisattributedtothelargevolumefractionofcarbides(withhighmodulusnear300GPa,Table 2-2 )inthecaselayer.Theyield 32

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strengthofthecasematerial(3.0GPaat0.2%offset)wasfoundtobeslightlymorethandoublethatofthecore(1.2GPaat0.2%offset).Thistwo-foldincreaseinyieldstrengthbetweencoreandcaseiscomparabletotheincreaseinhardnessbetweenthetwosections,whichisalsodouble(900kg/mm2forcaseand433kg/mm2forcore).Thisstrengthincreaseisaccompaniedbyadecreaseinductility,wherebythe`monolithic'casesamplesfailedcompletely(i.e.,fragmented)beyondacompressivetruestrainof3%.Thecorematerial,however,displayedconsiderableductility,andcouldbedeformedwithoutfailurebeyond20%strain.ThisdataclearlyillustratestheextremerangeofplasticbehavioroftheP675material.SimilartotheP675corematerial,theM-50NiLcorehadanelasticmodulusof195GPaandyieldstrengthof1.3GPaat0.2%offset.Additionally,thehardeningbehaviorofthismaterialwascomparabletotheP675core,andwasductilebeyond20%truestrain.ThethroughhardenedM-50materialhadanelasticmodulusof216GPaandyieldstrengthof2.8GPaat0.2%offset,butfailedbeyond5%strain.ItshouldbenotedinFigure 2-10 ,thestrainisonlyplottedto7%simplyforeasiercomparisonoftherelativeelasticandplasticpropertiesofallmaterials. 2.3.5UltrasonicDeterminationofElasticPropertiesAsasupplementalmethodforextractingelasticpropertiesofthe`monolithic'caseandcorematerialsofP675,aswellasthroughhardenedM-50,ultrasonicmeasurementswereconductedtodeterminethelongitudinalandshearwavevelocities.TheintentwastoeventuallydetermineYoung'smodulus,shearmodulus,andPoisson'sratio,inordertocomparethesewiththeotherexperimentalmethodsusedinthisinvestigation.Notethatonlynon-gradedmaterialswereusedforthispartoftheanalysis,becauseoftheirhomogenousstructure.Ultrasonicmeasurementswereconductedusing10MHzlongitudinaland5MHzshearwavepiezoelectrictransducersalongwithastandardultrasonicpulser/receiver(Olympus5072PR)andappropriatecouplinguid.Theresultingwavereectionswerecapturedusingahigh-speeddigitaloscilloscope 33

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forlateranalysis.Thetimerequiredforeachpulsetotravelthroughthespecimen,reectfromthebackside,andreturntothepulser/receiveristhenusedtocalculatethelongitudinalandshearwavevelocities.Singlereections,aswellasmultiplereectionswereanalyzedinordertomoreaccuratelymeasurethetraveltime.Thismethodwasrepeatedatmultiplelocationswithineachsample,andthetraveltimeforthewavemotionaveraged.Testingwasperformedonthesamesamplesthatwereusedforcompressiontesting.TheresultinglongitudinalandshearwavevelocitiesforthematerialsareshowninTable 2-4 .Fromtheultrasonicmeasurementsofshearandlongitudinalwavevelocity,anestimateoftheelasticpropertiesofthesamplescanbedetermined[ 43 44 ].First,thePoisson'sratiowascalculatedusingtheequation, =1)]TJ /F2 11.955 Tf 11.95 0 Td[(2(VS=VL)2 2)]TJ /F2 11.955 Tf 11.95 0 Td[(2(VS=VL)2(2)whereVSandVLaretheshearwaveandlongitudinalwavevelocities,respectively.Next,theelasticmodulus,E,andshearmodulus,G,aredeterminedusingthefollowingformulae[ 43 ]: E=V2L(1+)(1)]TJ /F2 11.955 Tf 11.96 0 Td[(2) (1)]TJ /F3 11.955 Tf 11.96 0 Td[()(2)or E=3V2SV2L)]TJ /F4 7.97 Tf 13.15 4.71 Td[(4 3V2S V2L)]TJ /F2 11.955 Tf 11.95 0 Td[(V2S(2)and G=V2S(2)ThecalculatedvaluesforelasticpropertiesareshowninTable 2-4 .ThedensityvaluesrequiredforcalculationweredeterminedusingtheArchimedes'method. 2.4DiscussionThedepthsofthegradedsectionsofbothmaterialswereshowninthemicrographspresentedinFigures 2-2 and 2-3 .Theseobservationswereconrmedthrough 34

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theanalysisofthecarbidevolumefractionasafunctionofdepthinFigure 2-6 .Furthermore,thereisanoteworthyconnectionbetweenthecarbidevolumefractionandsubsurfacemicro-hardness(Figure 2-7 ).Bycomparingtheseresults,arelationshipbetweencarbidevolumefractionandhardnesscanbedetermined.TheserelationshipsareshowninFigures 2-11 and 2-12 ,wherebothhardnessandcarbidevolumefractionareplottedsimultaneouslyforthegradedmaterialsasafunctionofdepth.Figures 2-11 and 2-12 illustratethatthereisadirectrelationshipbetweenthelocalsubsurfacecarbideconcentrationandthemicro-hardnessvalueatthatdepth.Inordertoarriveatarelationshipbetweenhardnessandcarbidedistribution,thesubsurfacehardnesswasplottedasafunctionofcarbidevolumefractioninFigure 2-13 ,whichindicatesalinearrelationshipbetweenthetwovariablesforbothmaterials,withM-50NiLhavingasteepertrendthanP675.Despiteconsiderabledifferencesinhardnessvaluesforthecarbidesinthetwocarburizedmaterialsaswellaslargedifferencesinthevolumefractionofcarbides,bothmaterialsreachcomparablyhighsurfacehardnessvalues.Interestingly,ifthetrendlinesinFigure 2-13 areextrapolatedoutto100%volumefraction(i.e.,purecarbide)theresultinghardnessisontheorderof17.3GPa(1760kg/mm2)forP675and28.3GPa(2900kg/mm2)forM-50NiL,bothofwhichcorrespondtothevaluesdeterminedviananoindentationandthosefoundinliterature[ 37 40 ]forthecarbidesinthesetwomaterialsasreportedinTable 2-3 .ThisresultiscomparabletotheanalysisofXuandAgren[ 45 ]whichconsideredtungstencarbidedistributedinacobaltmatrix.TheiranalysiswasbasedonthehardnessmodeloriginallyproposedbyLeeandGurland[ 46 ],whichconsideredamodiedruleofmixturesforthepredictionofhardnessofcementedcarbides.Theseanalysesutilizednumerousspecimenswithvaryingvolumefractionofthecarbidephaseovertheentirerangeofpossibleconcentrations(frompurematrixtopurecarbide).Onthecontrary,thecurrentanalysisisabletoextractasimilartrendfromasinglespecimenwithagradientincarbideconcentration.Suchinformation 35

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isnoteasilyavailableformanymaterials,andisofimmensevalueforthemodelingcommunity.Thesendingsadditionallysuggestthatthecarbidehardnesscanbeestimatedfromplotsofhardnessvs.carbidevolumefraction,whichisbenecialwheninstrumentedindentationisnotavailable. 2.4.1CompositeModelforModulusSimilartotheconnectionbetweenhardnessandcarbidevolumefraction(Figure 2-13 ),arelationshipisexpectedbetweenthecarbidevolumefractionandthemodulusvaluesdeterminedfortheindividualconstituentsviananoindentation.Morespecically,thegradientinmoduluswithdepthofthegradedmaterialisinuencedbythehighstiffnessofthecarbidesdependingonthevolumefractionofcarbidespresentatagivendepth.Anestimateofthecompositeelasticmodulusasafunctionofdepthcanthusbedetermined.Theanalysiscanbeattemptedusingmodelswithvaryingdegreesofcomplexity,startingwithasimpleruleofmixtures[ 47 48 ], Ee=EmVm+EiVi(upperbound,parallelmodel)(2)and 1 Ee=Vm Em+Vi Ei(lowerbound,seriesmodel)(2)wherethesubscriptm,i,andedesignatematrix,inclusion(carbides),andeffectivemodulusofthecomposite,andVisthevolumefraction.AmorecompletemodelbyHashinandShtrikman[ 49 ]wasoriginallydevelopedforthemodulusofcementedcarbides,andhasbeenshowntobemoreappropriatethanasimplerule-of-mixtureswhendeterminingtheelasticbehaviorofcompositesconsistingofahardinclusionphaseinamatrix/bindermaterial.Themodelpredictstheupperandlowerboundsfortheeffectivebulkmodulus(Ke)andeffectiveshearmodulus(Ge)fromthevolume 36

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fractionoftheconstituentmaterialsas Ke,upper=Ki+Vm 1 Km)]TJ /F2 11.955 Tf 11.95 0 Td[(Ki+3Vi 3Ki+4Gi (2a)Ke,lower=Km+Vi 1 Ki)]TJ /F2 11.955 Tf 11.96 0 Td[(Km+3Vm 3Km+4Gm (2b) Ge,upper=Gi+Vm 1 Gm)]TJ /F2 11.955 Tf 11.95 0 Td[(Gi+6(Ki+2Gi)Vi 5Gi(3Ki+4Gi) (2a)Ge,lower=Gm+Vi 1 Gi)]TJ /F2 11.955 Tf 11.95 0 Td[(Gm+6(Km+2Gm)Vm 5Gm(3Km+4Gm) (2b)whereGistheshearmodulusandKisthebulkmodulus.Finally,theeffectiveelasticmodulusisdeterminedforeitherupperorlowerboundvaluesfrom Ee=9KeGe 3Ke+Ge(2)Withamodulusof195GPaforthesteelmatrixofthethreematerials,andvaluesof290GPa,500GPa,and290GPaforthecarbidesinM-50,M-50NiL,andP675(determinedfromnanoindentation,Table 2-3 ),respectively,thecompositemodulusvalueswereestimatedusingtheaboverelationships.FortheuppermostcasematerialofP675withmaximumcarbideconcentrationof26.6vol.%,theeffectivemodulusfromtheruleofmixturesis270GPafromtheupperboundparallelmodeland232GPafromthelowerboundseriesmodel.Similarly,fromtheHashinandShtrikman[ 49 ]model,theupperandlowerboundsfortheP675caselayerare228GPaand226GPa.Interestingly,fortheM-50NiLmaterialwithamaximumcarbidecontentofonly12.7vol.%inthecaselayer,theeffectivemodulusiscalculatedtobe280GPaupperboundand238GPalowerboundfromtheruleofmixtures,and228GPaupperboundand223GPalowerboundfromtheHashinandShtrikmanmodel.AlthoughthemaximumcarbidecontentisonlyhalfofthatintheP675,themodulusvaluesare 37

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showntobesimilarduetothehighstiffnessofthevanadiumcarbides(Table 2-3 ).TheeffectivemodulusvaluesaresummarizedinTable 2-5 alongwiththeresultsfromthecompressiontesting,forcomparison.Comparingthecompositemoduluscalculationstothemodulusvaluesforthe`monolithic'P675casematerialfromthecompressiontestdata(230GPa)andtheultrasonictestdata(230GPa),itappearsthattheequationspredictvalueswhichareclosetotheexperimentaldata,withtheexceptionoftheupperboundruleofmixtures.ThisisadditionallyreectedintheM-50throughhardenedmaterial,wherethecompressiontestdata(216GPa)andtheultrasonictestdata(219GPa)correspondwelltothecalculatedmodulus.Whiletheultrasonictestdataforthecorematerialsappearsslightlyelevated,thiswaslikelyduetothesmallsamplesizetestedwherediscrepanciesinthicknessmayinuencecalculationsofelasticproperties.NotingthattheupperboundfromtheHashinandShtrikman[ 49 ]modelbestmatchesboththecompressiontestdataandultrasonictestdata,thisupperboundvaluewasselectedtorepresenttheeffectivemodulusofbothmaterials.Fromthisanalysis,themodulusasafunctionofdepth(andthus,carbidevolumefraction)wasdetermined.ThisresultisplottedinFigure 2-14 2.4.2HardnessYieldStrengthRelationshipAsmentionedpreviously,theincreaseinyieldstrengthbetweenP675coreandcaseiscomparabletotheincreaseinhardnessbetweenthetwosections(3.0GPacaseyield,1.2GPacoreyield,at0.2%offset;900kg/mm2casehardnessand433kg/mm2corehardness).Thesevaluesmayfollowthewell-knownTaborrelationshipbetweenhardnessandyieldstrength,H=Crat8%strainforastrainhardeningmaterial.Therelationshipprovidesanindicationthatthehardnessisdirectlyrelatedtotheyieldstrength.Sucharesulthasbeendemonstratedforavarietyofmaterials[ 50 ],andisbestdescribedbytherelationshipofTabor[ 51 ].Tabor'srelationship 38

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expressedthatthehardnessofamaterialwhichexhibitsperfectly-plasticbehavior(i.e.,nostrainhardening)isontheorderofthreetimestheyieldstrength.Furthermore,Tabordemonstratedthatforastrainhardeningmaterialtheyieldstrengthtakenatarepresentativestrainof8%relatestothehardnessofthematerialbyafactorofapproximately3(sometimes2.75or2.9,dependingonindentergeometry).ForM-50NiL,the8%offsetowstressfromthecompressiontestsofthecorematerial(Figure 2-10 )was1605MPa,whilethehardnessofthismaterialwas500kg/mm2(4.9GPa),whichgivesaconstraintfactorC=H=rof3.05.Similarly,theowstressfortheP675corematerialat8%offsetwas1500MPa,whilethehardnesswas433kg/mm2(4.25GPa).ThisyieldsaconstraintfactorCof2.84,suggestingthatTaborsruleholdsforbothductilecorematerials.NotethatTaborsuggestedavalueofapproximately3.0forstrainhardeningmaterials.ForthehardcaselayerofP675andthethroughhardenedM-50,asimilaranalysisgeneratesconstraintfactorsof2.5and2.4,respectively.Theowstressandhardnessusedtodeterminethesevaluesareasfollows:FortheP675`monolithic'casematerial,theowstressat8%was3500MPa(estimated,duetothecompressionsamplefailingbeforethisstrainlevel)whilethehardnesswas900kg/mm2(8.83GPa).ForthethroughhardenedM-50,theowstressat8%was3300MPa,whilethehardnesswas810kg/mm2(7.92GPa).Thus,itappearsthatthehighhardnessmaterials(caselayerandthroughhardened)arebestrepresentedwithaconstraintfactornear2.5.WhilethisislowerthanthatsuggestedbyTabor,theoriginalworkconsideredonlyductile/softmetals.Fromthisanalysis,itispossibletoestimatethestrengthoftheM-50NiLcasematerial,whichwasnotavailableforcompressiontesting.Giventheconstraintfactorof2.5forthehardcasematerials,andthehardnessoftheM-50NiLcaselayer(825kg/mm2or8.09GPa),thestrengthat8%offsetisgivenas3230MPa.ConsideringthatthisbehavioriscomparabletotheM-50throughhardenedsteel,theyieldstrength 39

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ofthecaselayerofM-50NiLisestimatedat2.8GPa.AcomparisonoftheseresultsisincludedinTable 2-6 .Overall,itappearsthathardnesscanprovideareasonableestimateofthestrengthofthesematerials,butonlywhentheconstraintfactorismodiedforthehighhardnesscasematerials. 2.5SummaryThecurrentanalysisutilizedmultipleexperimentaltechniquesinordertocharacterizethegradientsincompositionandmaterialpropertiesincarburizedsteels.Theelasticpropertiesofthetwoconstituentphases(carbidesandsteelmatrix)weredeterminedviainstrumentednanoindentation,andtheresultingdatawasutilizedwithseveralmodelsforcompositemodulusinordertoestimatethevariationinelasticpropertiesasafunctionofcarbidevolumefraction.Itwasshownthatthehighelasticmodulusofthecarbidesresultsinanincreasedelasticmodulusinthecaseregion,wherethevolumefractionofcarbidesishigh.Theresultscomparedwelltotheelasticmodulusdeterminedfrombothcompressiontestingandultrasonicmeasurementsonsamplescomposedofuniformcarbidevolumefraction.Additionally,thegradientinsubsurfacehardnesswascorrelatedwiththegradientinsubsurfacecarbidevolumefraction.Despitesizabledifferencesinhardnessvaluesdeterminedforthecarbidesinthetwocarburizedmaterials,inadditiontolargedifferencesinthevolumefractionofcarbides,theybothreachcomparablyhighsurfacehardnessvalues.Forbothmaterials,alinearrelationshipbetweenthetwoparameterswasfound.Whenextrapolatedtotheoreticalpurecarbideproperties,theresultinghardnessvaluesmatchthatdeterminedviananoindentationontheindividualcarbides.Thissuggeststhatthecarbidehardnesscanbeestimatedfromplotsofhardnessvs.carbidevolumefraction,whichisbenecialwheninstrumentedindentationisnotavailable.ItwasdeterminedthattheductilecorematerialsfollowedTabor'sruleverywell,withconstraintfactorscloseto3.0.Thehigh-hardnesscasematerials,however,indicated 40

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thattheconstraintfactormustbereducedtoavalueof2.5inorderfortherelationshipH=Crtohold.Usingthisvalue,anestimateofthestrengthcanbeobtainedfromhardnessvalues.Overall,theresultsprovidemultiplemethodsfordeterminingrelevantpropertiesofcarburizedmaterialsasafunctionofdepth,whileconsideringgradientsinseveralmaterialparametersoccurringsimultaneously.Theseresultsareinvaluableformodelingandoptimizingtheperformanceofcasehardenedmaterialsforuseinbearings,gears,andothercontactsurfaces. 41

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Figure2-1. CrosssectionofthroughhardenedM-50showinguniformcarbidedistribution(darkeldimagecarbidesappearbright).InsetimagestakenunderdarkeldandDIC(Nomarskiilluminationcarbidesappearraised)toshowsizeandshapeofindividualcarbidesinadditiontobanding/orientationofprimarycarbides. Figure2-2. CasehardenedM-50NiLcrosssectionshowingcarbidedistributionwithinthe2.5mmgradedlayer.InsetimagestakennearsurfaceunderdarkeldandDICtoshowsizeandshapeofindividualcarbides. 42

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Figure2-3. CasehardenedP675crosssectionshowingcarbidedistributionwithinthe1.5mmgradedlayer.InsetimagestakennearsurfaceunderdarkeldandDICtoshowsizeandshapeofindividualcarbides. 43

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Figure2-4. P675`monolithic'casematerialcarbides.Shape,size,anddistributionmatchthatoftheuppercaseofgradedP675. 44

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Figure2-5. SEMimageoftemperedmartensitestructureinM-50.Sampleetchedin5%nitalsolution. 45

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Figure2-6. MeasuredsubsurfacedistributionofcarbidesinM-50,M-50NiL,andP675. 46

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Figure2-7. Hardnessprolesasafunctionofdepthfromthesamplesurface. Figure2-8. Nanoindentationsonsteelmatrixandindividualcarbides. 47

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Figure2-9. Indentationload-displacementcurvesfordifferentcarbidespeciesinM-50NiLandP675. 48

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Figure2-10. Compressivestressstrainresponseforthevarietyofmaterialstested,including`monolithic'P675caseandhomogeneouscoreP675material,M-50NiLcore,andM-50throughhardenedmaterials. 49

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Figure2-11. M-50NiLhardnessprole(lledshapes)andcarbidevolumefraction(X's)asafunctionofdepth. Figure2-12. P675hardnessprole(lledshapes)andcarbidevolumefraction(X's)asafunctionofdepth. 50

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Figure2-13. Subsurfacehardnessvaluesasafunctionofcarbidevolumefraction. 51

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Figure2-14. Calculatedeffectivecompositemodulusasafunctionofdepth.Theupper-boundHashinandShtrikman[ 49 ]modelforeffectivecompositemodulusisused. 52

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Table2-1. Nominaltoolsteelandstainlesssteelcompositions.Valuesshowninweightpercent.Notethatcarboncontentisinitialcontent,andincreasesaftercarburization. AlloyCarbonMnCrMoVCoNi M-500.800.354.04.251.00M-50NiL0.130.204.14.251.253.40P6750.070.6513.01.800.605.42.60 Table2-2. Distributionofcarbidevolumefractionwithdepth. DepthfromSurface(m)M-50ThruCarbideVol.%M-50NiLCarbideVol.%P675CarbideVol.%P675MonolithicVol.% 2520.40.212.70.626.60.926.20.810020.40.412.40.624.20.925.90.830020.20.612.00.819.80.624.80.450020.20.611.10.615.20.724.00.475020.00.410.00.29.30.623.00.4100019.10.58.60.23.70.322.10.8130018.50.96.40.21.10.121.80.5180017.80.41.80.20.60.122.80.5 Table2-3. Materialpropertiesforvariouscarbidestested. MaterialCarbideSpeciesHardness(GPa)Modulus(GPa)Reference ChromiumCarbide,NickelMatrixCermetCr7C318.41.5340.34.9Hussainovaetal.[ 38 ]ChromiumCarbide,NickelMatrixCermetCr7C321.53.534535Hussainovaetal.[ 37 ]HighChromiumD2ToolSteelM7C318.22.429417Casellasetal.[ 52 ]Pyrowear675Cr7C3/Cr23C6192.029015CurrentStudyFirstPrinciplesStudyVC28.5518.5Liuetal.[ 40 ]FirstPrinciplesStudyVC29.0SimunekandVackar[ 39 ]M-50NiLVC273.549030CurrentStudyM-50ThroughHardenedM7C3/M23C6163.029015CurrentStudy 53

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Table2-4. Elasticpropertiesdeterminedviaultrasonicmeasurements. MaterialDensity(kg/mm3)Long.Vel.(m/s)ShearVel.(m/s)Poisson'sRatioElasticModulus(GPa)ShearModulus(GPa) P675Core7750591032200.2920580.5MonolithicP675Case7720620033600.2923088.5M-50NiLCore7970580031500.2920279.5M-50ThroughHard8000605032500.3021984.5 Table2-5. CalculatedeffectiveelasticmodulusforP675,M-50NiL,andthroughhardenedM-50. CarbideVol.%RoMUpperBound(GPa)RoMLowerBound(GPa)H-SUpperBound(GPa)H-SLowerBound(GPa)CompressionTest(GPa) M-50NiLCore0.9195195195195195M-50NiLCase12.7280238228223n/aP675Core0.6195195195195190P675Case26.6270232228226230M-50Thru20.4265218216215216 Table2-6. Comparisonofyieldstrength,hardness,andconstraintfactor. MaterialYieldStrengthCompression(GPa)VickersHardness(kg/mm2)ConstraintFactor(C=H=r) M-50NiLCore1.35003.05M-50NiLCase2.8(est.)8252.5(est.)P675Core1.24332.84P675Case3.09002.5M-50Thru2.88102.4 54

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CHAPTER3DETERMINATIONOFSUBSURFACEHARDNESSGRADIENTSINPLASTICALLYGRADEDMATERIALSVIASURFACEINDENTATION 3.1BackgroundIntheprevioussection,itwasshownthatcarburizedsteelsexhibitagradientinmechanicalpropertiesasafunctionofdepthduetocarburizationandsubsequentheattreatment.Dependingontheapplication(bearings,gears,andloadingconditions),thedepthofthecarburizationcanbealteredtoproducethedesiredgradientinthematerial.Thisisaccomplishedbyincreasingthetimeallowedfordiffusionofcarbonduringcarburization.Furthermore,changesintemperingconditionsinuencethegradientinhardness.Typically,theseconditionsareselectedinanappropriatemannerastoproduceapredictablehardnessgradient.However,conditionsareoccasionallyappliedincorrectly.Furthermore,thelargenumberofvariablesassociatedwithprocessingandcasehardeningresultinslightvariationsinthenishedproduct,especiallywithmaterialssourcedfromdifferentsuppliers[ 16 ].Thus,rapidlyassessingthesegradientsisimportantfordeterminingwhetherthetreatmentwassuccessful.Forcasehardenedmaterials,determininghardnesswithdepthofthegradedlayerisreadilyavailableviasectioningandindentation[ 15 ].Thisdataistypicallyemployedasasimplemeanstodeterminethepenetrationdepthofaheattreatmentprocedure.Sectioningisadestructivetestingmethod,however,meaningthatthecomponentmustberemovedfromservice.Anon-destructivemethodwouldbepreferable.Nayebietal.[ 53 55 ]haveanalyzedsurfaceheat-treatedtoolsteelswithagradientinhardness/yieldstrength.Theydevelopedaprocedurewhichutilizedinstrumentedindentationinordertopredictthehardnessvs.depthproleofaheattreatedsteel.Aniteelementmodelwasemployedinordertocalibratetheindentationresults.Thismethodwasbasedonanunderstandingoftheinuenceofsofter(deeper)layersonsurfaceindentationsofincreasingload/depth.Theyshowedthatthedecreasinghardnessgradientofthegradedmaterialresultsinaperceivedreductioninsurface 55

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hardnesswithincreasingindentationdepth.Byutilizingamodiedruleofmixtures,theobservedsurfacehardnesswasrelatedbacktothecumulativehardnessgradientinuencingtheindent.Usingthismethod,theywereabletopredictthehardnessvs.depthprolesforfourdifferentcasehardenedmaterials.Theseincludedbothsharpgradients(suchasthosefoundinlm/substratesystems)andavarietyofnitridedsteelswithgradedlayersof100mthickness.Similarly,Guetal.[ 15 ]usedinverseanalysistocharacterizematerialswithgradedlayerscomposedofyttriapartiallystabilizedzirconia(PSZ)andmetallicbondcoat(NiCrAlY)depositedbyplasmaspraying.Theexperimentalprocedureutilizedmultiplesphericalindenterswithradiusonthesameorderasthedepositedlayerthickness.Afterconductingindentsofvariousdepths/loads,aniteelementmodelwasdevelopedtodeterminethepropertiesofthegradedmaterial.Thesestudiesutilizedinstrumentedindentation,niteelementmodels,multipleindentergeometries,oracombinationofthesemethods.Theyalsorelyontailormadematerials(e.g.,plasmacoating,depositionmethods,etc.)designedexclusivelyforcalibrationofthemethods.Amorecommonmethodfordeterminingthevariationinhardnesswithdepthistosection,polish,andindentacross-sectionofthegradedmaterial.Whiletestinginthismannerisperfectlysuitableatthematerialdevelopmentalstage,thisrequirementbecomeslaborintensiveandprohibitivelyexpensivewithincreasingnumberofsamplesoratproductionlevels.Alternatively,thecurrentstudyaimstodevelopanapproachfordeterminingthenatureofgradientsbeneaththesurface,solelyfromsurfacehardnessmeasurements.Suchamethodhaspracticalutilityinthesurfaceheat-treatmentindustry,wheretheeffectivenessofanintendedheat-treatmentprocesscanbeassessedbysimplesurfacehardnessmeasurementsinordertodeterminetheresultinggradientinmechanicalpropertyvariationwithdepth.Theresultwillalsoprovideinsightintothelevelofinuencethedeeperlayersofagradedmaterialmay 56

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haveonsurfacedeformation,andhowthegradientseverityinuencesthesurfacehardnessmeasurement. 3.2MaterialsThethreematerialscharacterizedintheprevioussection(casehardenedM-50NiLandP675,alongwiththroughhardenedM-50)wereemployedasmodelmaterialsinthecurrentstudy.Asthesearecommerciallyavailablebearingsteels[ 17 32 ],theimportanceofthemethoddevelopedhereappliesdirectlytotheheattreatingandprocessingindustry.Thehardnessgradientsdescribedearliercontaininterestingaspectsforuseinthecurrentstudy.ThesubsurfacehardnessprolesofthethreematerialsareshowninFigure 3-1 ,withsomeadditionswhichwillbedescribedinthefollowingsection.NotethestartingsurfacehardnessoftheP675ismorethan100kg/mm2greaterthanthatoftheM-50NiLmaterial(930kg/mm2forP675vs.825kg/mm2forM-50NiL).WhiletheP675hasasteep,decreasinggradientovertherst1.5mmdepth,theM-50NiLhasagraduallydecreasinghardnessproleover2.5mmdepth.Additionally,theM-50NiLhardnessprolecontainsaregionofnearlyconstanthigh-hardnessnearthetopsurfaceoveradepthof0.3mm.Thisappearsasaplateauinthehardness-depthprole.Alsoincludedisthethrough-hardenedM-50steel,whichdisplaysaconstanthardnessnear810kg/mm2throughoutthethicknessandprovidesabaselinetowhichthegradedmaterialswilllaterbecompared. 3.3SurfaceIndentationSchemeWhilethemethodofsectioning,polishing,andindentingrevealsthegradientinhardnessonthecross-sectionofaspecimen,thegoalofthisstudywastoanalyzewhetherthesubsurfacegradientscanbecapturedusingonlysurfaceindentations.Suchamethodcanfacilitatethepredictionofgradientsproducedduringheattreatmentwithouttheneedforcross-sectioning.Assuch,thetestingschemeconsistsofaseriesofstaticindentationtestsatincreasingloadsonthetopsurfaceofthegradedmaterial. 57

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Inordertoencompassawiderangeofloads,indentationswereperformedusingbothastandardcommercialtable-topVickershardnesstester(WilsonTukon2100B,forloadsrangingfrom200gto50kg)andatable-topuniversaltestingmachineloadframe(MTSAllianceRT/30,forloadsabove50kg)withVickersindentertipmountedanddriveninloadcontrol.Theresultingindentationsweremeasuredusinganopticalmicroscopeinordertocalculatehardness.Thisprovidedasimpleexperimentalsetup,withoutrequiringload-displacementdatacommontoinstrumentedindentation.Itshouldbenotedthatevenunderhighindentationloads,theresultingindentdepthreachedamaximumof125m,whilethedepthoftheentiregradedregionisgreaterthan2mm.Therefore,onlyaportionofthesubsurfacegradedmaterialisaffectedbythesurfaceindentation.Toexpandtheavailablegradedmaterialsforanalysis,additionalsampleswereextractedfromvarioussub-sectionsofthegradedmaterialsbyremoving(grindingandpolishing)aportionofthesurfacegradedlayer.Thisresultedinadditionalgradedspecimenswithdifferentstartingsurfacehardnessvaluesanddifferentstartinghardnessgradientswithdepth.Theobjectivewastounderstandhowthesetwofactors(i.e.,surfacehardnessandsubsurfacehardnessgradient)inuencetheobservedchangeinsurfacehardnessvalueswithincreasingindentationload.Preparationofthesesectionsofnewgradedmaterialsfromtheas-suppliedcarburizedspecimensrequirespreciseremovalofprescribedamountsofmaterialinordertoarriveatregionswithdesiredstartinghardnessandhardnessgradientofinterest.Thisprocessissimilartoanynalmachiningorgrindingoperationwhichmayoccurafterapartisheattreated,thusremovingaportionoftheuppercaselayer.Thisoccursquitefrequentlyduringthemanufacturingofbearingsandgearsinordertomeetspecieddimensionaltolerancesfollowingheattreatment.Additionally,hardenedpartsareoccasionallyremanufacturedfromusedparts.Thisprocesstypicallyinvolveslightmachiningofthedamagedouterportiontorevealanewsurface.Whilethisremovesaportionofthehighhardness 58

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case,thecostsavingstypicallyoverridesthisconsequence.Furthermore,differentsizepartsarecommonlyheattreatedinthesamemanner.Subscalebearingsandtestsamplesmaycontainslightlyalteredhardnessprolesduetochangesingeometry.Thisemphasizestheneedforarapidmethodforcheckingthehardeningresultspriortosubsequenttesting.TheopensymbolsinFigure 3-1 indicatethelocationsofthesupplementarysectionswithdifferentstartingsurfacehardnessandinitialhardnessgradients(i.e.,localchangeinhardnessatthesurfaceasafunctionofdepth,Hv/x)extractedforthisstudy.Table 3-1 providestherelevantdetailsalongwiththeslopesofthehardnessgradientsforeachlocation.Furthermore,becausetheP675andM-50NiLcontaindifferentinitialgradientsandsurfacehardnessvalues,separatingtheeffectsofthesetwoparametersrequiredtestingofsectionswithdifferingstartinghardnessyetcomparablegradients(e.g.,P675-AandM50NiL-C,Table 3-1 ).Theuseofthesesectionsaidedindeterminingthesensitivityofthemethodtoslightchangesininitialhardnessgradientandstartingsurfacehardness.Finally,sectionsP675-CandM50NiL-Dwereselectedduetotheirproximitytothetransitionbetweenthegradedlayerandcorematerialofconstanthardness.Theimplicationsofthistransitionwillbediscussedinalatersection.Itwasnotedinthepreviouschapterthattheresidualstressesfromcarburizationwouldhaveinuencedanalysisviainstrumentedindentation,however,becausetheseindentsaremeasuredoptically(notfromload-displacement),theyshouldprovideanaccuratemeasurement[ 33 34 ].Furthermore,itwasshowninChapter 2 thatthethrough-carburized`monolithic'P675casesample,whichdoesnotdevelopanynotableresidualstresses,matchedtheappropriatehardnessofthetoplayerofthegradedP675material,indicatingthatthemeasuredindentationhardnessisaccurateandnotinuencedbyresidualstress. 59

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3.4Results 3.4.1SurfaceHardnessThevariationinsurfacehardnessvalueswithincreasingloadontheas-receivedspecimenswithhighestsurfacehardnessareshowninFigure 3-2 forbothP675andM-50NiLmaterials(P675-AandM50NiL-A)inadditiontothethroughhardenedM-50.Bothmaterialsshowhigherhardnessatlowloads(200g),revealingaslightindentationsizeeffect(ISE).AstheloadPincreases,thetrendinhardnessbecomesmorestable,reectingthenatureofthegradientbeneaththesurface.Byconductingdeeper(i.e.,higherload)surfaceindents,theinuenceofthegradientinsubsurfacehardnesscanbeinferred.Asexpected,thereisanapparentreductioninhardnessastheindentationloadisincreased.Thisbehaviorisanticipated,resultingfromtheinuenceofsoftersubsurfacelayers,whichismoreevidentintheP675specimenduetothesharperinitialhardnessgradient(Figure 3-1 ).Withtheadditionalsectionsproducedbycarefullyremovingportionsoftheoutercasematerial,afullrangeofbehavior(varyinggradientsandvaryingstartingsurfacehardness)iscreated.TheseresultsareshowninFigure 3-3 .WhilethedatainFigures 3-2 and 3-3 areanindicatorofhowthematerialgradientinuencesthemeasuredsurfacehardnessunderincreasingindentationload,thisdatadoesnotallowfordirectcomparisonbetweensectionswithdifferentstartingsurfacehardnessvalues.Inordertoallowthedatafrommanysections/materialswithdifferentsurfacehardnesslevelsandgradientstobeplottedandcomparedsimultaneously,anormalizationschemewasadoptedinvolvingthesurfacehardnessofeachspecimentested.Thisnormalizationutilizesthesurfacehardnessfromeachsection(orsub-section),measuredatanindentationloadof200g.Thisloadwaschosentominimizetheinuenceofthesubsurfacegradientsinmeasuredhardness.PlotsofnormalizedhardnessasafunctionofindentationloadareshowninFigure 3-4 forthevarioussub-sectionsofthetwomaterials,aswellasforthethrough 60

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hardenedM-50steelwhichcontainsnohardnessgradientwithdepth.Allmaterialsshowhigherhardnessatlowloads,duetotheindentationsizeeffect(ISE).AstheloadPincreases,thetrendinnormalizedhardnessreectsthenatureofthegradientbeneaththesurface.Theresultingdatahavebeentwithpower-lawtrendlinesdenedbyH=)]TJ /F2 11.955 Tf 9.3 0 Td[(0.06Pb+1.04whereHisthenormalizedhardness(i.e.,hardnessatthatloadoverthehardnessat200g).Thesetrendsareusedtoillustratetheextenttowhichsubsurfacehardnessgradientsandstartingsurfaceharnesses(at200g)inuencethenalobservedsurfacehardnessmeasurementunderincreasingindentationload.Thepower-lawwaschosenduetoitssimplicityandabilitytotthevarietyofgradedsections.However,thereisquestionastowhethertheindentationsizeeffectobservedinthelowloadregionforallsectionsinuencestheoverallanalysis.Furthermore,thecurvetdenedaboveisnotdimensionless,asithasunitsofload.Inordertocorrecttheseoversights,theanalysiswasrepeated,butwithastartingsurfaceloadof1kg.Thisloadwaschosentominimizetheinuenceoftheindentationsizeeffect.Itadditionallyallowsforeasiernormalization.Theindentsproducedat1kghavediagonalsizesof30morlarger,whilethegrainsizeisanorderofmagnitudesmaller.Inordertocreateadimensionlessrelationshipforanalysis,bothindentationhardnessandloadwerenormalizedbasedonthestartingsurfacehardnessandstartingload(1kg)foreachsection.TheresultingplotsofnormalizedhardnessasafunctionofnormalizedindentationloadareshowninFigure 3-5 .ComparedtoFigure 3-4 wheretheindentationsizeeffectwasprevalentatlowloads,byplottingthecurveswithastaringloadof1kg,theISEhasbeensignicantlyreduced.ThisisclearlyevidentintheplotofM-50throughhardenedmaterialwhichshowsrelativelyaconstantvalueofhardnesswithload.AlthoughbothP675andM-50NiLappeartoshowaslightindentationsizeeffect,thedecreaseinhardnesswithincreaseinloadisprimarilyduetothegradientin 61

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hardnesswithdepth.Theresultingdatahavebeentwithpower-lawtrendlinesdenedbyH=)]TJ /F2 11.955 Tf 9.3 0 Td[(0.06Pb+1.06whereHisthenormalizedhardness(i.e.,hardnessatthatloadoverthehardnessat1kg)andPisthenormalizedindentationload.Consideringthatthepowerlawcurvetstillbestrepresentsthesetrends,itappearsthattheindentationsizeeffecthasminimalimpactontheoveralltrends.ThisislikelyduetotheISEbeingsimilarinallthreeofthematerialsandacrossallsections.RecallingtheindicatedstartingslopeslistedforthedifferentsubsectionsinTable 3-1 (i.e.,startinggradientinhardness),atrendbeginstoemergefromtheplotsshowninFigure 3-5 .Materialswithnegligiblegradientsinhardnessimmediatelybeneaththesurface(e.g.,surfaceofM-50NiLlabeledM50NiL-A)showminimalchangeinhardnesswithincreasingindentationload,nearlymatchingthebehaviorofthethrough-hardenedM-50materialwhichcontainsnogradient.However,astheseverityofthesubsurfacehardnessgradientincreases(i.e.,Hv/x),thereductioninsurfacehardnesswithincreasingloadbecomesmoreprevalent.Forexample,theM50NiL-Bsectionwithamildgradient(-112Hv/mm)showsadecreaseinthesurfaceindentationhardnessofnearly10%whenloadisincreasedfrom1kgto300kg,whiletheP675-Bsectionwithaseveregradient(-290Hv/mm)decreasesinhardnessby15%acrossthesameloadrange.Whilethistrendisexpected,interestingly,itisalsonotedthatthestartingsurfacehardnessvaluehaslittleinuenceonthetrendsinnormalizedhardnessvs.load.Thisbehaviorismostevidentwhencomparingdatabetweenspecimensofsimilargradientfromtwodifferentmaterialswithnoticeablydifferentstartingsurfacehardnessvalues.Forexample,theP675-AsectionhasnearlythesamehardnessgradientasM50NiL-C(i.e.,-180Hv/mmvs.-196Hv/mm,respectively),butvastlydifferentstartingsurfacehardnessvalues(920kg/mm2forP675-Avs.648kg/mm2forM50NiL-C).Nevertheless,theygeneratecomparabletrendsinnormalizedsurface 62

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hardnessunderincreasingindentationload.Thisresultindicatesthattheresponseoftheplasticallygradedmaterialisnotinuencedgreatlybytheabsolutesurfacehardnessvalue,butisinsteadsensitivetothesharpnessofthegradientinsubsurfacehardnessimmediatelybeneaththeindentedregion.Thesetrendsinhardnessasafunctionofincreasingindentationloadformthebasisforananalyticalmethodusefulfordeterminingunknownhardnessgradientsinamaterialbymeansofonlysurfaceindentation.Inthismethod,thegradientinhardnessasafunctionofdepthisextractedfromthecurvesshowninFigure 3-1 andplotted,asshowninFigure 3-6 .Thelocations(i.e.,depths)ofthesubsectionsareindicatedbyopensymbolsforeachofthematerials.Alsoincludedinthisgureisaplotofthepower-lawexponentsb(giveninTable 3-1 )fromthecorrespondingtestsections(fromFigure 3-5 )asafunctionofhardnessgradient.Interestingly,theplotofhardnessgradientvs.exponentvaluesalsofollowsapower-lawcurvet,Hv x=23500b3.0providingarelationshipbetweenthehardnessgradientandthebehaviorofthesurfacehardnessunderincreasingindentationload.ItcanbeseenfromFigure 3-6 andTable 3-1 thatthehighertheseverityofthelocalsubsurfacehardnessgradient,thehigherthebvalue.Irrespectiveofthematerialandstartingsurfacehardness,allofthebvaluesshareacommoncurve.Althoughfurtherstudyisrequiredinordertodeterminewhetherthisrelationshipholdsforagreatervarietyofmaterials,theproceduredescribedabovecanbeappliedtoanygradedmaterial.Furthermore,anychangeinindentergeometrywouldaffectmeasuredhardnessvalues[ 56 ]andtrendsandthereforemaywarrantanewanalysis.Forexample,thedeformationbeneathasharpindentermayextenddeeperthanthatofabluntindenter,thusthemeasuredhardnesswillbeinuencedbythelower(softer) 63

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subsurfacelayers.Thiswouldaltertheeffectsofsubsurfacegradientsonmeasuredhardnessunderincreasingload. 3.4.2DetectingChangesinGradientTrendsInadditiontopredictingthehardnessprolesingradedmaterials,thistechniquehasdemonstratedtheabilitytodetectchangesinthetrendsinsubsurfacehardnessgradients.Suchbehavioroccursinregionswherethegradedlayertransitionsintotheconstanthardnesscorematerial.Forexample,considertheplotsofP675-CandM50NiL-D(Figure 3-5 )whicharenearthecoreregion.Thetrendsinthedatafromthesetwosectionsinitiallyappearsimilartothedatafromothersections;however,thereisadiscontinuityatwhichthenormalizedhardnesstrendsbecomeconstant.Thepointatwhichthisbehaviorbeginsdirectlyreectstheapproximatelocationwherethegradientinhardnessceasestoexistandaconstantcorehardnessbegins.IthasbeennotedthatthedepthatwhichtheplasticzoneextendsbeneathaVickersindentationisontheorderofsixtoseventimestheindentdepth[ 57 58 ].Whilethisisapplicableforhomogenousmaterials,slightchangesintheplasticzonehavebeenshownforindentationongradedmaterials[ 8 ].Assuch,thisrelationshipwillbeutilizedhereonlyasaruleofthumbwhennotingthedepthofindentationinduceddeformation.FromthegeometryoftheVickersindenter,theapproximateindentationdepthisrelatedtothemeasuredindentdiagonalbyafactorofseven(i.e.,theratioofdepthtodiagonallengthfortheVickersindentergeometryis1/7).Thus,anapproximatedepthoftheplasticzonecanbepredictedfromthemeasuredindentdiagonal.Whilethisdoesnottakeintoconsiderationelasticrecovery,itprovidesareasonableestimateofthedepthofanindentationandtheassociatedsubsurfaceplasticdeformationzone.Forthetwosectionsconsidered(P675-CandM50NiL-D),thetransitiontothecorehardnessleveloccursatadepthofapproximately500to600mbeneaththetestedsurfaceregion(Figure 3-1 ).Whenthesurfaceindentsreachanapproximatedepthofone-seventhofthisvalue(correspondingtotheplasticzonedepthof500to600m), 64

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thetrendsinsurfacehardnesswithincreasingloadreachaconstantvalue(i.e.,atregionsofP675-CandM50NiL-DtrendsinFigure 3-6 ).Thissensitivityofthemethodtoachanginghardnessgradientprovidesanexcellentmeansfordetectingdiscontinuitiesinhardnessgradientthroughtheuseofonlysurfaceindentation. 3.5SummaryTheresultsprovidedhereindicateapracticalrelationshipbetweenthehardnessmeasuredviasurfaceindentationunderincreasingloadsandtheseverityofthesubsurfacegradientinhardness.Theapproachprovidesarapidmethodforpredictingthegradientsproducedduringsurfaceheattreatment,consistingofonlyaseriesofindentationsatincreasingloadsonthesamplesurface,requiringonlyahardnesstesterandminimalsamplepreparation.Theresultsarethencomparedtothepredetermined(calibrated)power-lawtrendsinordertoestimatethelocalhardnessgradient.Here,largerexponentbvaluesindicatemoreseveregradients,whilebvaluesapproachingzerosignifyeitherathrough-hardenedmaterialorasaturationofthecasehardeningprocedure(e.g.,the`plateau'regionoftheM-50NiLsurface).Thus,themethodcanbeusedtoexamineiftheheattreatmentprocesshasbeenproperlyimplementedonapart,andisadditionallyusefulindetermininghowmuchofthegradedlayerhasbeenremovedduringnalmachiningofacomponent.Finally,anydiscontinuitiesnotedinthetrendsofnormalizedhardnessvs.loadcanbeattributedtodramaticchangesinthesubsurfacehardnessgradientorproximitytothecoreregionwithconstanthardness.Theseresultswillbeevaluatedusingniteelementanalysisinthefollowingsectionsinordertodeterminetheinuenceofparticulargradientsinmaterialpropertiesonthesurfacehardnessbehavior.Forexample,Chapter 2 determinedthatagradientinelasticpropertiesexistsalongsidethegradientinhardnessandyieldstrengthinbothP675andM-50NiL.Therefore,theinuencefromtheindividualcomponentswillbecharacterizedintermsofbothsurfaceindentationbehavior,aswellassubsurface 65

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deformation.Thisadditionallymayindicatewhethertheanalysispresentedhereisapplicabletoawiderrangeofmaterialsandgradientsnotavailableinthecurrentstudy. 66

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Figure3-1. HardnessproleswithdepthforP675,M-50NiL,andM-50materials(trend-linesincludedforclarity).Unlledsymbolsindicatelocationsoftestsections.Insetmicrographrevealsindentsonthecross-sectionofaspecimen. 67

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Figure3-2. Surfacehardnessvs.loadforM-50throughhardenedandthehardestsectionsofbothgradedmaterials. 68

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Figure3-3. Surfacehardnessvs.loadforallsectionstested.Hardnessgradientvalue(Hv/mm)isindicatednexttoeachsamplelabel. 69

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Figure3-4. Normalizedsurfacehardnessasafunctionofindentationloadforallsectionsandmaterialstested.Thevaluesshownwiththecurvesindicatetheinitialhardnessgradientforeachsection,i.e.,changeinhardness(kg/mm2)overchangeindepth(mm). 70

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Figure3-5. Normalizedsurfacehardnessasafunctionofnormalizedindentationloadforallsectionsandmaterialstested.Thevaluesshownwiththecurvesindicatetheinitialhardnessgradientforeachsection,i.e.,changeinhardness(kg/mm2)overchangeindepth(mm). 71

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Figure3-6. Trendsinhardnessgradientandpowerlawexponentforbothmaterials.Arrowsindicatetheaxistowhichthedatarefersto. 72

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Table3-1. Summaryofmaterialsectionsandrelevantproperties. Material/SectionDepthfromOriginalSurface(m)SurfaceHardnessHv@1kg(kg/mm2)HardnessGradient(Hv/mm)PowerLawbValue P675-A0920-1800.205P675-B350822-2900.224P675-C1625488-1550.175M50NiL-A0820-260.077M50NiL-B600778-1120.166M50NiL-C1450648-1960.207M50NiL-D2200515-1300.181 73

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CHAPTER4DETERMININGTHECONSTITUTIVERESPONSEOFGRADEDMATERIALS 4.1PastMethodsCompressiontestingwasperformedinSection 2.3.4 onmaterialsextractedfromthecoreregionsofcarburizedM-50NiLandP675.Additionally,compressiontestswereconductedonthroughhardenedM-50andaspeciallyheat-treatedP675withpropertiesmatchingthatofthecaseregion.Thesematerialswereespeciallyhelpful,becauseitisoftendifculttoextractmaterialsforcompressionortensiontestingfromwithinthegradedregionofacarburizedmaterial.Whilethesetestsprovidedamethodfordeterminingthecompressiveow-curvesfortworegionsofthesamples,itdoesnotrepresenttheentireconstitutiveresponseofthegradedmaterial.Speciallyheat-treated`through-carburized'materialscouldbeproducedwithvaryingcompositionsinordertocharacterizemoreofthegradedlayers.Elghazaletal.[ 59 ]attemptedthisprocedureusingthin-wallcylindricalcompressionspecimensofAISI9310steelwhichwerecarburizedtovariouslevels(whileremainingrelativelyhomogeneousduetothinsectionsize).Theiranalysischaracterizedthesmall-straincompressivebehaviorforawiderangeofcarbonconcentrations,rangingfrom0.35to0.86wt.%.Thestudy,however,didnotextendthesetrendstohighstrainvaluesencounteredinthecurrentanalysisbecausethethin-wallsampleshadatendencytobuckleprematurely.Furthermore,producingenoughofthesesamplestorepresenttheentirerangeofcarbidevolumefractionsencounteredinthegradedmaterialwouldbecostlyandtimeconsuming.Investigationsintoalternativemethodsforextractingtheconstitutiveresponseofgradedmaterialsfrequentlyfocusoninstrumentedindentation[ 15 60 63 ].Nakamuraetal.[ 60 ]conductedaninverseanalysisoffunctionallygradedmaterialsusinginstrumentedballindentation.Theprocedureutilizedload-displacementinformationprovidedatvariousloads,combinedwitharuleofmixturesinordertodeterminethe 74

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effectivepropertiesofagradedmaterialconsistingofmetallicandceramiccomponents.Sphericalindentersofmultiplesizeswererequiredinordertoaccuratelypredictmechanicalpropertiesofcoatingsandlayeredmaterials.Similarly,Fischer-Cripps[ 61 ]utilizedinstrumentedballindentationcombinedwithniteelementanalysisinordertodeterminetheelasticmodulusvariationinelasticallygradedmaterials.Guetal.[ 15 ]extendedtheinverseanalysisofNakamuraetal.[ 60 ]tocharacterizematerialswithgradedlayers,inadditiontothinlmsdepositedonsteelsubstrates.Themodelmaterialwasacontinuouslygradedlayercomposedofyttriapartiallystabilizedzirconia(PSZ)andmetallicbondcoat(NiCrAlY).Thematerialwaspreparedusingaplasmasprayingprocess,producingalineargradientbeginningwithpurePSZonthesurface,transitioningtopureNiCrAlYonthebottom.Inaddition,homogeneousthinlmsofthetwoconstituentsweredepositedonsteelsubstratesasbaselinematerials.Theexperimentalprocedureutilizedmultiplesizesofsphericalindenters.Afterconductingindentationsatvariousdepths/loads,acombinedinverseanalysisandniteelementmodelwasusedtodeterminethepropertiesofthegradedmaterial.Inordertorepresentthegradient,theniteelementmodelcontainedcontinuouslychangingmaterialpropertiesfortheelementsasafunctionofdepth.Becauseonlythepropertiesoftheupperandlowerboundsofthegradedmaterial(i.e.,thinlmsofpurePSZandNiCrAlY)wereknown,intermediatelayerswereprescribedunknownmaterialproperties.Thesepropertieswereback-calculatedthroughinverseanalysis,comparingtheexperimentalindentationload-displacementcurvestomodeledindentsintheniteelementanalysis.Theendresultwasthecompletecharacterizationofthecompressivestress-strainrelationshipsforthelayersofthegradedmaterialasafunctionofdepth/composition.Similarly,Nayebietal.[ 53 ]createdanumericalapproachforindentationofnitridedsteels,consistingofarelationshipbetweenload-displacementandstrainhardeningexponent.Themethodutilizedinstrumentedindentation,combinedwithniteelement 75

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analysisofthethin-lmandsubstrategeometry.Amodiedruleofmixturesrelatedtheindentationbehaviortothepropertiesofthelmandsubstrate.Theresultwaslaterexperimentallyveriedusinganitridedsteelwithnitridelayerthicknessofapproximately100m.IncontrasttoNayebietal.[ 53 ],CaoandLu[ 62 ]aswellasSwaddiwudhipongetal.[ 63 ]utilizedniteelementanalysisofindentation(conicalgeometry)combinedwithreverseanalysisalgorithmstopredicttheplasticpropertiesofgradedsurfaces.Theydemonstratedthattheresultingload-displacementcurvescanvarywithindentationdepth.Thisresultindicatesthattheconstitutiveresponsedeterminedviainstrumentedindentationisnotnecessarilyauniquesolution,butmayvarywithmaterialproperties.Themajorityofthesestudiesreliedheavilyoninstrumentedindentationload-displacementcurves,typicallyrequiringmultipleindentergeometries,andrelatingthebehaviortovariationsineitherelasticorplasticpropertychangesinthinlmsandgradedmaterials.Fewoftheproceduresanalyzedtheactualplasticdeformationinthematerials,insteadfocusingondimensionlessanalysisandcurvettingtopredictindentationbehavior.Themethodpresentedhereintendstoestablishaprocedureforextractingmechanicalpropertiesfromagradedmaterial,usingthecarburizedsamplesasmodelmaterials.Thismethodwillutilizemacro-indentationasameansforinducingplasticstrainintobothgradedandhomogeneousmaterials.Attentionispaidtotherelationshipsbetweenhardnessandyieldstrengthinordertoextractthemechanicalbehavior(strainhardeningcharacteristics).Thisapproachisintendedtobeapplicabletorealisticengineeringmaterialsirrespectiveofthenatureofthegradientsinthematerialplasticproperties. 4.2HardnessandYieldStrengthRelationshipAsbackgroundfortheexperimentalandnumericalwork,adiscussionregardingtherelationshipbetweenindentationhardnessandmechanicalpropertiesisappropriate.ThisrelationshipwasbrieydiscussedinSection 2.4.2 inordertoestimatethestrength 76

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oftheM-50NiLcasematerialwherecompressionsampleswerenotavailable.Thisbackgroundwillformthebasisofthemethodfordeterminingtheconstitutiveresponseofgradedmaterials.Asarststep,theapproachwillbeconrmedusingthetwohomogeneous,throughhardenedmaterialswhichwerecharacterizedinSection 2.3.4 .AswasmentionedinSection 2.4.2 ,thefamiliarTabor'srule[ 51 ]relatinghardnesstoyieldstrengthsuggestsaconstraintfactorgivenbyC=H=y.Ithasbeendemonstratedthatthevalueoftheconstraintfactorisapproximately3forperfectlyplasticmaterials[ 50 56 64 65 ],whilematerialswhichexhibitstrainhardeningarebetterdescribedatacharacteristicplasticstrainvalue.Thisvalue,oftentermedrepresentativeplasticstrain,modiestheaboverelationshiptoC=H=rwhereristherepresentativeowstresscorrespondingtoarepresentativeplasticstrain,r.Taborsuggestedauniversalrepresentativestrainvalueof0.08forVickersindentationsonstrainhardeningmaterials.Tabor'ssuggestedrepresentativestrainvalueisnotrelatedtoanymeasurablequantityduringindentation,butinsteadprovidesastatistical`bestt'representation.Itwasalsodemonstratedthatthisvalueisindependentoftheinitialplasticstraininamaterial.Thus,theindentationhardnessofanundeformed/virginmaterialcanbepredictedby H=Crj=r(4)andtheindentationhardnessofapreviouslyplasticallydeformedmaterialcanbepredictedby H=Crj=r+p(4)wherepistheamountofinitialplasticstrain.Whentheplasticstrainiszero(i.e.,undeformedmaterial),therelationshipsimpliestobecomeEquation 4 .ThisrelationshipbetweenhardnessandowstressisshownschematicallyinFigure 4-1 .Itshouldbenotedthattherepresentativeplasticstrainisconstantregardlessofindentationsize/depth.Thisobservationappliestohomogeneousmaterials,specicallywhentheindentergeometryisself-similar.TheVickersindentergeometryisconsidered 77

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self-similar,asistheconicalindentergeometry,becausetheratioofprojectedareatoindentationdepthisaxedvalue.Additionally,theratioofcontactareatodepthisalsoxed.Anexampleofanindentergeometrywhichisnotself-similarisaspherical(ball)indenter,becauseratioofprojectedareatoindentdepthvaries.AlthoughitwasshowninSection 2.4.2 (Table 2-6 )thataxedconstraintfactorof3appropriatelyrepresentstheductilecorematerials,thisdoesnotholdforthehighhardnesscaseandthrough-hardenedmaterials.Inlightofthis,thereissomequestionastowhetherTabor'sruleisapplicableforthewiderangeofmaterialbehaviorencounteredincarburizedmaterials.Furthermore,therehasbeensignicantdiscussionregardingtheselectionofanappropriaterepresentativestrainvalue,r.Themajorityofstudiesagreethatthevalueisusedtorepresentthedegreetowhichanindentationover-predictstheyieldstrengthofastrainhardeningmaterial,however,nonecanagreeonasinglevalue[ 66 68 ].Forexample,Daoetal.[ 67 ]suggestedarepresentativeplasticstrainof0.033basedonacomprehensivecomputationalstudyofstrainhardeningmaterials.Thisvalueallowedforthecreationofadimensionlessrelationshipfortheindentationloadingresponse,whileremainingindependentofthestrainhardeningabilityofthematerial.Bucailleetal.[ 69 ]extendedthisresulttoincludeconicalindenterswithvariedincludedangle,inordertodeterminethedependenceoftherepresentativestrainonindentergeometry.Ogasawaraetal.[ 68 ]attemptedtoextendtherangeofmaterialscharacterizedbyDaoetal.[ 67 ],andproposedarepresentativeplasticstrainvalueof0.0115.Theirapproachwasanattempttoextractplasticpropertiesfrommaterialsusingonlysingleinstrumentedindentationtests[ 70 71 ].However,ithasbeenshownthatitispossibleformaterialswithdifferentmechanicalpropertiestoproducethesameindentationload-displacementbehavior[ 72 73 ],limitingthecredibilityofsingleindentationmethods. 78

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Chaudhri[ 66 ]suggestedthattherepresentativestrainshouldinsteadbethemaximumstrainvaluefoundbeneathanindent,indicatingarangeofvaluesbetween0.25and0.36forannealedcopper.Thisvaluewasdeterminedbyexperimentallyobservingtheincreaseinhardnessofthesubsurfaceregionbeneathanindentationaftersectioningandpolishing.Thesubsurfacehardnessvalueswerethenrelatedtoindentationhardnessmeasurementsconductedoncompressionspecimensdeformedtoprescribedamountsofstrain.However,ithasbeenshownthatstrainvaluesbeneathsharpindentationsareoftengreaterthanthisrangeofvaluesreportedbyChaudhri[ 67 69 74 ].Srikantetal.[ 75 ]utilizedasimilarapproachofsectioningandpolishinginordertoobservetheregionbeneathalargeVickersindentation.Thestudyutilizedaluminumalloysagedfordifferenttimesinordertoobtainmaterialswithdifferentstrainhardeningbehaviorwhilekeepingtheyieldstrengthxed.Again,hardnesstestingconductedoncompressionspecimenswasusedtorelatetheindentationsubsurfacehardnessvaluestoplasticstrainvalues.Theinuenceofthestrainhardeningexponentonsubsurfacehardeningwasstudied,andindicatedthatgreaternvaluesresultinashallowerdeformationeld.Becausetheextentofindentationinducedplasticstrainwasdependentonthestrainhardeningexponent,thisndingcastsdoubtontheabilityofasingle,universalrepresentativestrainvaluetobeapplicabletoarangeofmaterialswithdifferenthardeningbehavior.Tekkaya[ 76 ]demonstratedthattherelationshipbetweenhardnessandyieldstrengthremainsapplicableformaterialscontainingpreviouslyinducedlevelsofplasticstrain.Thestudyutilizedtheextrusionprocessinordertoprovidesampleswithwellcharacterizedplasticstrainlevels,andfoundarelationshipsimilartoTabor'satanoffsetequivalentplasticstrainof0.112withavalueof2.475fortheconstraintfactor.Thisvaluewasshowntoapplyforcoldformedmaterialswithinitialstrainvaluesrangingfrom0.02uptoamaximumof1.6.Taboralsosuggestedasimilarapproach,inwhich 79

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therepresentativeplasticstrainisaddedtothealreadypresentplasticstraininordertodeterminetheincreaseinhardness[ 51 ].ThiswaspreviouslyshowninEquation 4 .Cahoonetal.[ 77 78 ]attemptedtomodifyTabor'sruleinordertodeterminetheyieldstrength(at0.2%offset)ofstrainhardeningmaterials,insteadofthestrengthatanoffsetrepresentativestrain.Theirndingswereimportantindemonstratingthatthestrainhardeningexponentinuencedthepredictedyieldstrengthforarangeofmaterialsincludingcoldworkedsamples.Thus,insteadofasinglexedvaluefortherepresentativeplasticstrain,amaterial-dependentvaluemaybemoreappropriate.Forexample,Jayaramanetal.[ 79 ]analyzedtheaverageequivalentplasticstrain,denedas p=1 VpXiVi(4)whereiistheequivalentplasticstrainatthecenterofanelementi,Viisthevolumeofthatelement,andVpisthetotalvolumeoftheplasticallydeformedareabeneathanindent.Theydemonstratedthat,foraxedangleconicalindenter,theaverageequivalentplasticstrainisdependentonthestrainhardeningexponentn,whileremainingindependentoftheyieldstrengthandelasticmodulus.TherangeofvaluesareshowninFigure 4-2 andTable 4-1 foraconicalindenterwithhalf-includedangleof70.3degrees.Thisgureshowsaverageplasticstrainvaluesrangefrom0.067forperfectlyplasticbehavior(nostrainhardening,n=0),to0.036formaterialswithastrainhardeningexponentn=0.2.Interestingly,thesevaluesallfallbelowtherepresentativestrainusedbyTabor.Antunes[ 80 ]similarlyillustratedamaterial-dependentrepresentativeplasticstrainwithdependenceonstrainhardeningexponent.Theanalysisutilizedniteelementsimulationsofmaterialswithawiderangeofmechanicalbehavior,includingYoung'smodulusvaluesfrom50to600GPa,yieldstressvaluesfrom0.3to10GPa 80

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andstrain-hardeningexponentsfrom0to0.6.Theresultingrepresentativestrainvalueswereintherangeof0.034to0.042.AsummaryoftheseproposedvaluesofrepresentativeplasticstrainisgiveninTable 4-2 .Duetothelargerangeofrepresentativestrainvaluesfoundintheliterature,itwouldappearthatnosingle/universalvalueprovidesanadequatemethodforrelatingindentationhardnessandowstress.Thisisprimarilyduetothefactthatthesevaluesaretypicallydeterminedusingttingparametersoraveragevaluesbasedonmeasurementsforanarrowrangeofmaterialproperties,insteadofactualrepresentationsofindentationinduceddeformation.Assuch,theyarenotbasedonanymeasurablequantityproducedduringindentation.Thisapproachismisleading,becausetheabilityofamaterialtoresistdeformationcertainlyinuencesthemeasuredhardness.Instead,avaluewhichismaterial-dependentmaybemoreappropriate.Becauseofthelimitationsofusinguniversalvaluesofrepresentativestrain,theaverageplasticstrainapproachsuggestedbyJayaramanetal.[ 79 ]willbeutilizedinthecurrentstudy.Theaverageplasticstrainisameasurablequantity(Equation 4 )whichwasshowntodependonthestrainhardeningbehaviorofamaterial(Figure 4-2 ).Inthefollowingsections,thisapproachwillbeexploredingreaterdetail,rstusingthematerialpropertiesofthehomogeneousmaterialsdeterminedearlier.Theanalysiswillthenbeextendedtothedeterminationoftheconstitutiveresponseofgradedmaterials. 4.3Procedure 4.3.1ExperimentalMethodAcoordinatedexperimentalandnumericalapproachwillbeutilizedinthecurrentstudyinordertoexploretherelationshipbetweenamaterial'splasticproperties(yieldstrengthandstrainhardeningbehavior)andtheindentationhardnessmeasure.ThehomogenousP675coreandthroughhardenedM-50materialswillbeemployed,becausetheirstrainhardeningbehaviorwaspreviouslydeterminedfromcompressiontesting(Section 2.3.4 ). 81

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Duringtheexperimentalwork,alargemacro-indentwillrstbeproducedoneachmaterialinordertoinduceadistributionofplasticstrainsinthesubsurfaceplasticzonebeneaththeindentedregion.Thespecimensarethencarefullysectionedandprogressivelystep-polishedinordertorevealthesubsurfaceregiondirectlybeneaththesurfaceindent.ThisapproachisshownschematicallyinFigure 4-3 .Onthenalpolishedcross-section,Vickersmicroindentationsareperformedintheplasticzonebeneaththemacroindent.Themicrohardnessvaluesthusobtainedprovideanindicationoftheextentofplasticdeformation,andthusanindicatoroftheapproximatelocationreachedalongthestress-straincurveduetoworkhardening.ThismethodofobservingthesubsurfacedeformationbeneathindentationshasbeendemonstratedbyChaudhri[ 66 ]aswellasKoeppellandSubhash[ 81 ],asameanstodetermineplasticzonedepth.Intheory,ifaperfectly-plasticmaterialweretestedinthismanner,thesubsurfacehardnesswouldnotbeexpectedtoincrease(i.e.,noworkhardening).Conversely,amaterialwithatendencyforstrainhardening(i.e.,largestrainhardeningexponentn),alargeincreaseinsubsurfacehardnessshouldbeobserved.Furthermore,duetothedecreasingsubsurfacestrainlevelsasafunctionofdepthbeneaththesurfaceindent,theamountofhardeningisexpectedtodecreasewithdepth,thusprovidingmultiplelocationsalongthestress-strainresponse.AmongthematerialstestedinthisanalysiswerethethroughhardenedM-50materialwithuniformhighhardnessof807kg/mm2,inadditiontothecoreregionofP675withuniformhardnessof433kg/mm2.Surfacemacro-indentationswerecreatedinthesamemannerasthosefoundinChapter 3 ,usingastandardVickersdiamondindentermountedinauniversaltestingmachineanddrivenunderloadcontrol.Next,specimenswerecarefullysectionedclosetotheindentusingalow-speedsawalongonediagonaloftheindentation.Thesectionedspecimenswerethenmountedandcarefullystep-polishedusingprogressivelysmallerpolishingmedia,asperstandard 82

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metallographicpractices.Finalpolishingwascompletedwith0.25mdiamondpaste,withtheresultingsubsurfacecross-sectionaligneddirectlyalongthemacro-indentdiagonal.Thepolishedsubsurfaceregionbeneathalarge(210kgload)indentontheP675corewasshownpreviouslyinFigure 4-3 .Theindentsbeneaththemacro-indentwereproducedataloadof200g(15sdwelltime),andspaced100maparttoavoidedgeeffectsandinteractionsbetweenneighboringindents.Thelineofindentsdirectlybelowthetipofthesurfaceindentationwasanalyzed,asthisrepresentstheregiongreatestinducedsubsurfacestrain.ThemeasuredincreaseinsubsurfacehardnessfortheseindentsisshowninFigure 4-4 .Here,themaximumincreaseinhardnessisshowntobe40kg/mm2.Theundeformed,orvirginmaterialhardness,isincludedforreference.Asplasticstrainsdecreasewithdepth,thesubsurfacehardnessvaluesapproachthevirginhardness,untilreachingtheedgeoftheplasticzoneatadepthofapproximately1000m.ThesameprocedurewasrepeatedonthethroughhardenedM-50,withasurfacemacro-indentationproducedat75kg.TheresultingsubsurfaceincreaseinhardnessisshowninFigure 4-5 .Here,themaximumincreaseinhardnesswas60kg/mm2,andtheplasticzonedepthwasapproximately400m.ThisissmallerthantheplasticzoneintheP675,duetothesmallersurfacemacro-indent.WhileFigures 4-4 and 4-5 provideageneralindicationoftheextentofstrainhardeningandplasticzonedepthinducedbythemacro-indentation,itisdifculttoextractthestrainlevelthatcorrespondstothemeasuredsubsurfaceowstresses.Althoughtheremaybeanalyticalmodelswhichcanestimatethesevaluesforhomogeneousmaterials,suchanapproachwouldbeunusableforthegradedmaterialsduetothechangingmaterialpropertieswithdepth.Thus,toobtainthesubsurfacestraineld,acombinedexperimentalandcomputationalapproachwasutilized.Thisapproachallowsforanalysisoftherepresentativestrainandconstraintfactorforbothhardandsoftmaterials.Asbrieydiscussedearlier,itwasshownthatthesoftmaterialstTabor'srule 83

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fairlywell,whilethehardmaterialsrequiredanadjusted(lower)constraintfactor.Forthepresentanalysis,however,insteadofthexed0.08offset,anewmaterial-dependentrepresentativestrainvaluewillbedetermined. 4.3.2FiniteElementModelInordertosimulatetheaboveexperimentalapproach,anaxisymmetricniteelementmodelwasusedtorepresenttheindentationprocessinABAQUSStandardv6.7.Themodelmadeuseofarigidconicalindenterwithhalf-includedangleequalto70.3degrees.TheresultingindentationfromthisgeometrymatchesthatcreatedbythestandardVickersdiamondindenter,producingacomparableprojectedareaatanygivenindentationdepthandthusappropriatemeasureofsimulatedhardness.TheuseofaxisymmetricindentationmodelingasarepresentationofboththeVickersandBerkovichindentergeometrieshasbeenshowntobeappropriatebyChoietal.[ 6 7 ],SureshandGiannakopoulos[ 82 ],aswellasothers[ 12 83 ].Theindenterwasallowedtotranslatenormaltothesamplesurface,includingfrictionlesscontact,whichhasbeenshowntobeappropriateforindentersoflargeincludedangle[ 14 84 85 ].Bilinearaxisymmetricelements,withanemeshinthevicinityoftheindentertip,wereusedinordertodiscretizethespecimengeometry.Theoverallsamplesizewasmodeledtobeaminimumof12timeslargerthantheindentedregion,toavoidtheinuenceofboundaryeffectsonthesimulatedindentation,whileaminimumof25elementswereincontactwiththeindenteratmaximumindentationdepth.AnexampleofthisgeometryandmeshisshowninFigure 4-6 .ThevonMises(J2)yieldcriterionwasusedwithanisotropicpower-lawhardening(i.e.,=Kn)model.Largedeformationanalysiswasalsoincorporatedusingthe*nlgeomcommandinABAQUS,asthishasbeenshowntoimprovethequalityofresults[ 84 ],particularlywhenmodelingindentationsoflargesize/depth.Thestress-strainowcurvesdeterminedearlierviacompressiontestingwillbeusedasinputsintothemodel.Thesearemodeledaselastic-plasticpower-law 84

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hardeningmaterials.TheowcurvesforthroughhardenedM-50andthecoreP675materialareshownhereinFigure 4-7 .IndicatedwiththeguresarethestrainhardeningexponentnandelasticmodulusE.Thelargemacro-indentsarethensimulatedtothesamedepthsastheexperimentalindents,andthenretracted.Thisallowsforthesubsurfaceplasticstrainsdeterminedfromthemodeltobecompareddirectlytothesubsurfacehardnessincreasemeasuredfromtheexperiments.AnexampleofthesubsurfacestraincontoursisshowninFigure 4-8 fortheM-50throughhardenedmaterial,alongwiththesubsurfacestrainvaluesalongthecenterline. 4.4HomogeneousMaterials 4.4.1ResultsUsingthesubsurfaceplasticstrainvaluesdeterminedbythesimulationasafunctionofdepthbeneaththemacro-indent,theincreaseinsubsurfacehardnesscanbecalculatedfromEquation 4 .ThisprocedurewasshownschematicallyinFigure 4-1 .Here,therepresentativeplasticstrainrisaddedtothesubsurfaceplasticstrainpatagivenlocation(inducedbythemacro-indent,Figure 4-8 ).Therepresentativeowstressratthatpointiscalculatedbasedontheprescribedpower-lawhardeningmodel=Kn,andisnextmultipliedbytheconstraintfactortoarriveatthenewhardnessvalueduetoworkhardening.Theprocedureisrepeatedthroughouttheplasticzone,untiltheplasticstrainsreachzeroandthehardnessmatchesthevirginmaterialhardness(i.e.,endoftheplasticzone).TheconstraintfactorisdeterminedusingEquation 4 ,whichisthesameasEquation 4 whenthereisnoinitialplasticstrain(i.e.,undeformedmaterial).Forahomogeneousmaterial,boththeconstraintfactorandrepresentativestrainarexed,andnotdependentoninitialplasticstrain.Furthermore,theyareindependentoftheindentationsize/depth,duetotheself-similarityoftheconicalindentergeometry(aswellasVickersindentergeometry). 85

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Foranundeformedmaterial,theconstraintfactorCisthusdeterminedfromtheknownvirginhardnessofthematerial,theowcurvedenedbythepower-lawhardeningmodel,andrfromFigure 4-2 andTable 4-1 .FortheM-50throughhardenedmaterial,therepresentativeplasticstrainris0.055fromthestrainhardeningexponentnof0.050,whiletheconstraintfactorC=H=riscalculatedtobe2.44basedonthevirginhardness.Thus,forhomogeneousmaterialsthechangeinsubsurfacehardnessatagivenlocation(depth)canbecalculatedandcomparedtotheexperimentalresults.ThiscomparisonisshowninFigure 4-9 forM-50usingtheaverageplasticstrain(r=0.055)fromFigure 4-2 .Therefore,Figure 4-9 indicatesthattheaverageplasticstrainvalueprovidesanappropriaterepresentativestrainvalueforthecalculationofhardness.Inordertoillustratetheimportanceoftheselectedstrainhardeningexponentonthesubsurfacehardnessvalues,additionalowcurvesforM-50weregeneratedandmacro-indentssimulatedusingthesamemodel.TheseowcurvesallpredictthesamevirginhardnessfromtherelationshipH=Cattheroffsetandaconstraintfactorof2.44,however,theresultsshowdifferenttrendsinsubsurfacehardeningbehavior.ExampleowcurvesareshowninFigure 4-10 forstrainhardeningexponentsnof0.010,0.025,0.050,and0.070.Forthesevaluesofn,therepresentativeplasticstrainrvalues(averageplasticstrain,Figure 4-2 )are0.064,0.061,0.055,and0.051,respectively.Aftersimulatingthemacro-indentsforthesealternateowcurves,theincreaseinsubsurfacehardnesswasagaindeterminedandisshowninFigure 4-11 .Thegureindicatesthatonlytheappropriatestrainhardeningexponent,0.050,predictstheexperimentallymeasuredincreaseinsubsurfacehardness.Outsideoftheplasticzone,however,allowcurvespredictthesameundeformed/virginhardness.ThecombinedexperimentalandnumericalprocedurewasnextappliedtothesofterP675corematerial.Usingthestrainhardeningexponent0.064determinedviacompressiontesting,therepresentativeplasticstrainis0.052andconstraintfactor 86

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is2.90.Themacro-indentwassimulatedtomatchthedepthoftheexperimentalindentation,whichwasproducedataloadof210kgresultinginanindentationdepthof135m.Fromtheplasticstraindistributiondeterminedbythemodel,thesubsurfacehardnessincreasewasagaincalculatedusingEquation 4 ,andplottedinFigure 4-12 alongwiththeexperimentalresults.Theresultingsubsurfacehardnessincreasematchestheexperimentallyobservedvalues,againdemonstratingthattheaverageplasticstrainisavalidrepresentativeplasticstrain.Furthermore,sincethisprocedureworkedequallywellforboththehighhardnessM-50andlowhardnessP675core,itcanbeappliedintheanalysisofbothcaseandcoreregionsofthecarburizedmaterials.Oneitemtonote,however,isthesubsurfacehardnessvalueclosesttothemacro-indent(Figure 4-12 ).Here,thecalculatedhardnessvaluescontinuetofollowtheincreasingtrendinplasticstrain,yettheexperimentalvaluedoesnotfollowthistrend.Someexplanationofthisbehaviormaybeofferedbyre-examiningtheowcurvesdeterminedbycompressiontesting.AsnotedinChapter 2 ,thecorematerialscouldtypicallybetestedtohighstrainvalues,beyond20%,whilethehighhardnessmaterials(throughhardenedM-50and`monolithic'P675case)commonlyfailedbeyondonly3%strain.Fromcompressiontesting,thestrainhardeningexponentwasdeterminedbyttingapowerlawinthelowerrangeofplasticstrains,typicallybetweenthe2%and7.5%strain.Thisapproachisappropriate,becausethevaluesareintherangeofthoseusedastherepresentativeplasticstrain(Figure 4-2 ).However,becausethemacro-indentontheP675corewasespeciallylarge(135mdepth),thesubsurfaceplasticstrainsareincreasinglylarge.Atthelocationsoftherstandsecondsubsurfacemicro-indents,theplasticstrainvaluesdeterminedbythemodelwere55%and20%,respectively.Thismakesthehighstrainrangeoftheowcurvesimportanttounderstandingthehardeningbehaviorinthishighstrainregiondirectlybeneaththemacro-indent.Thisissuewasnotobservedbeneaththemacro-indenton 87

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theM-50material,duetothesmallerindentused.FortheM-50macro-indent,therstsubsurfacemicro-indentwasintherangeof17%strain.Toexplorethis,thecompressiontestdatafortheP675coreisplottedinFigure 4-13 uptothemaximumstraintestedduringcompression.Interestingly,whenapproachingaplasticstrainof20%,therateofstrainhardeningbeginstodecrease,droppingbelowthatpredictedbythestrainhardeningexponentn=0.064,providinganexplanationtothemaximumhardnessvaluereached.Inthiscase,thecompressivebehaviorathighstrainsmatchesthatoftheVocestrainhardeningmodel[ 86 ],whichincludesa`saturationstress'afterwhichamaterialnolongerhardens.However,becausetherepresentativeplasticstrainwasformulatedforpowerlawhardeningmodels,thisapproachwillcontinuetobeused. 4.4.2CompressionTestCross-SectionItwasmentionedpreviouslythatChaudhri[ 66 ]andSrikantetal.[ 75 ]utilizedindentationondeformedcompressionspecimensinordertoestimatehardeningbehavior.Followingthisprocedure,oneoftheP675corecompressionspecimenswassectionedandpolishedalongtheaxisparalleltotheloadingdirection,andmicro-indentationtesting(againataloadof200g)wasconductedonthissection.Becauseoffriction(evenwhenlubricated)betweenthecompressionplatensandsample,thedistributionofstrainsistypicallynotexpectedtobeuniformwithinacompressionspecimen.Assuch,theglobalstrainvalue(approximately30%forthisspecimen)cannotbeassumedtorepresentthestrainatalllocationswithinthespecimen.Instead,theaverageindentationhardnesswasdeterminedfromatotalof35indentsdistributedthroughoutthecross-section.Theresultingaveragehardnessofthedeformedspecimenwas46012kg/mm2.Interestingly,thisvaluenearlymatchesthemaximumhardnessobservedbeneaththemacro-indentonthismaterial(470kg/mm2).Thesendingsagaindemonstratethattheamountofstrainhardeningdecreasesathigherstrains. 88

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ThisexperimentalprocedurecouldnotbeattemptedonthehighhardnessM-50and`monolithic'P675casecompressionsamples,becausethemaximumstrainsencounteredundercompressionweretypicallybelow5%.Furthermore,thesesamplesfailedbeyondthisrange,resultinginfragmentationofthesamples. 4.5DeterminingtheConstitutiveResponseofGradedMaterialsThecombinedexperimentalandnumericalmethoddescribedintheprevioussectionwillnowbeextendedinanefforttodeterminetheconstitutiveresponseofgradedmaterialsasafunctionofdepth.Asintheprevioussection,itisshownthatastrainhardeningmaterialwilldisplayincreasedhardnesswhenplasticallydeformed,however,itisimportanttonotethatthematerialnowcontainschangingpropertiesasafunctionofdepth.Becauseagradedmaterialcannotbeeasilytestedviacompression,theindentationprocedureoutlinedinSection 4.4 isutilizedinordertointroduceplasticstrainswithinthegradedlayers,thusallowingasinglegradedsampletobeusedindeterminingthemechanicalresponse.Usingtheprocedureoutlinedforhomogeneousmaterials,theindentationmappingmethodwillbeappliedtomaterialswithgradientsinmechanicalproperties.Thesamemethodofusingalargemacro-indentationwillbeemployedinordertoinducelargesubsurfaceplasticstrains.Here,amacro-indentationwillbeplacedonthesurfaceofthegradedmaterial,inthedirectionofthegradientinhardness.Astheplasticpropertiesvarywithdepthinthegradedmaterial,theplasticallydeformedregionbeneaththemacro-indentwillundergodifferentamountsofplasticstraindependingontheplasticpropertiesateachdepth,aswellasthedecreasingthestraingradientbeneaththemacro-indentation.Followingthemacro-indentation,thesampleissectionedandpolishedtotheregionbeneaththecenteroftheindentation.Micro-indentsarethenperformedintheplasticzonebeneaththemacro-indent.Atthemeasuredpointswithinthisregion,itisexpectedthatthematerialwillstrainhardenaccordingtotheplasticresponseateachpoint,and 89

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willdisplayanincreaseinhardnessabovetheundeformedsubsurfacehardnessprole.Inthissense,theincreaseinhardnessatagivenlocation/depthmustfallontheowcurvedenedatthatdepth.Thus,themicro-hardnessvaluesobtainedprovideamapoftheowstressreachedalongthestress-straincurvesforeachlayer.Bygeneratingamapofmicro-hardnessvaluesfortheentireplasticallydeformedregionbeneathamacro-indent,theentireowstressresponseasafunctionofdepthcanbepredictedforthegradedmaterial.However,inordertofullygeneratethestress-straincurves,thestrainlevelcorrespondingtotheabovemeasuredhardnessvaluesmustbeknown.Inordertodeterminethisinformation,thecomputationalapproachearlierutilizedforhomogeneousmaterialswillbeextendedinordertomodelthegradedmaterialanditshardeningbehavior. 4.5.1IndentMappingGradedMaterialsVirgin,undeformedsamplesofthehardestsectionsofbothP675andM-50NiL(P675-AandM-50NiL-A,fromChapter 3 )wereselectedfortesting.Macro-indentswereconductedonthesurface,inthedirectionofthedecreasinggradientinhardness.Theresultingindentimpressions,producedataloadof330kgfortheP675and300kgfortheM-50NiL,were130mand140mdeep,respectively.Thesehighindentloadswerechoseninanefforttodeformalargeportionofthegradedlayerineachmaterial.Followingthesamepracticesoutlinedearlier,thesesamplesweresectionedandcarefullypolisheduntiltheregiondirectlybeneaththemacro-indentwasexposed(Figure 4-3 ).Onthiscross-section,micro-indentsataloadof200gwereusedtoprobetheincreaseinhardnessintheplasticallydeformedregion.TheresultingincreaseinsubsurfacehardnessisshowninFigure 4-14 forP675andFigure 4-15 forM-50NiL.Includedwiththeseguresaretheundeformedsubsurfacehardnessproles,inordertoillustratetheextentofhardening.NotethatthemaximumamountofhardeninginboththehomogenousP675core(Figure 4-4 )andthegradedP675section(Figure 4-14 )isontheorderof45kg/mm2. 90

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BecausetheincreaseinhardnessisindicativeofthestrainhardeningbehaviorofthematerialandwasapproximatelyequalbetweenbothsectionsofP675,itisreasonabletoassumethatthevalueofthestrainhardeningexponentnremainsconstantthroughoutthismaterial.Thisassumptionwillbeinvestigatedinthefollowingsections.Anumberofotherstudieshavealsoconsideredthestrainhardeningexponentntobenearlyconstantforcasehardenedsteels[ 59 87 ].Forexample,usingthin-wallcylindricalcompressionspecimensofAISI9310steelwhichwerecarburizedtovariouslevels(whileremaininghomogeneousduetothinsectionsize),Elghazaletal.[ 59 ]determinedthatasingle/xedvalueofthestrainhardeningexponentnwasvalidforalargerangeofcarbonconcentrationsrangingfrom0.35to0.86wt.%.Incontrast,theonlyparameterwhichwasfoundtovaryasafunctionofcarbonconcentrationwasthestrengthcoefcientK.Withtheexperimentallydeterminedsubsurfaceincreaseinhardness,theniteelementmodeldevelopedearlierwillbeappliedinordertodeterminethesubsurfacestraindistributionbeneaththemacro-indentations.AswasshowninFigure 4-11 ,themodelwillonlypredictthecorrectsubsurfacehardnessincreasewhentheappropriateowcurvesareselected.However,forthegradedmaterialtheselectedowcurvesmustbeaccurateateverylocationthroughoutthegradedlayer.Atanygivenpoint,theincreaseinyieldstrengthduetoplasticdeformationcorrespondstothestrainhardeningbehavioratthatpoint,aswellaslevelofplasticstrain,whichisalsodependentonthestrainhardeningbehaviorinthegradedlayer.Thus,ifincorrectstress-strainresponseisusedasinput,theresultingsubsurfacehardnesswillnotmatchtheexperimentallymeasuredvalues.Therefore,correctowcurvesmustbeselected,otherwisetheanalysismustberepeateduntilthenumericalandexperimentalvaluesmatch,atwhichpointthecorrectcurveshavebeendetermined.Foranentirelyunknownmaterial,thisbecomesaniterativeprocess.Toavoidextensiveiteration,compressiontestingofthe 91

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corematerials,aswellasM-50and`monolithic'P675provideabasisfortheselectionofmaterialproperties. 4.5.2SelectingMaterialPropertiesFromthecompressiontestingandassociatedanalysisinSections 2.3.4 through 2.4.2 ,thestress-strainowcurvesforbothcaseandcorelayersofP675weredetermined,inadditiontothecorelayerofM-50NiL(caselayerstrengthwasonlyestimatedfromhardness).However,inordertomodeltheresponseofthesegradedmaterials,theowcurvesasafunctionofdepthmustbepredictedandassignedtotheniteelementmodel.Gaoetal.[ 56 ]developedanewexpandingcavitymodel(ECM)whichrelatesindentationhardnesstostress-strainbehaviorforelastic-plasticpower-lawhardeningandelasticlinear-hardeningmaterials.ThemodelbuildsuponJohnson'sECM[ 88 ]forperfectly-plasticmaterials,whichwasderivedbydescribinganexpandinghemisphericalhydrostaticcoreofmaterialbeneathapenetratingindenter.TheGaoetal.ECMincorporatesintothismodelthebehavioroflinearlyelastic,power-lawhardening(orlinear-hardening)plasticity[ 89 ].Thismethodisbenecial,becausepreviousECM'shavebeenshowntounderrepresentmaterialswithapropensityforstrain-hardening[ 90 ].Theratioofhardnesstoyieldstrengthwasdenedas H y=2 31)]TJ /F2 11.955 Tf 13.47 8.09 Td[(1 n+3 4+1 n1 3E ycotn(4)wherenisthestrainhardeningexponent,Etheelasticmodulus,Hthehardness,ytheyieldstrengthofthematerial,andthehalfincludedangleoftheconicalindenter(70.3degrees).Usingtheseparameters,theelasticportionoftheowcurveissimply=Eandtheplasticportionisdenedbythepowerlawmodel=Kn.ThestrengthcoefcientKisdeterminedfromtheintersectionoftheelasticandplasticcurvesfrom K=En1)]TJ /F4 7.97 Tf 6.59 0 Td[(ny(4) 92

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thusprovidingthecompleteowcurve(bothelasticandplasticportions)fromtheselectedproperties.Alternatively,theplasticportionoftheowcurvecanbeexpressedwithoutrequiringthestrengthcoefcientas =E yn)]TJ /F4 7.97 Tf 6.59 0 Td[(1(4)TherelationshipsbetweenH=yandE=ycalculatedbytheGaoetal.ECM[ 56 ]areshowninFigure 4-16 foravarietyofstrainhardeningexponentsn.ItshouldbenotedthathardnessinEquation 4 isdenedasthemeancontactpressure,P=(r2),whichistheindentationloaddividedbytheprojectedindentationarea.ThisdiffersfromVickershardness,whichusesthecontactareaforcalculation.Thetwovaluesdifferbyapproximately7.3%.Vickershardnesscanbeconvertedtotheprojectedareahardnessby Hprojected=9.81HVickers cos22 180(4)ForbothP675andM-50NiL,thesubsurfacehardnessisknownforalllocations(Figure 2-7 ).Theelasticmodulusisalsoknownasafunctionofdepth,determinedinChapter 2 (Figure 2-14 ).Therefore,withaprescribedstrainhardeningexponentn,theowcurvescanbedenedatanylocationusingtheaboverelationships.Thisprocedureisrepeatedforeverydepthwithinthegradedregion,untiltheowcurves(modulus,yieldstrength,strainhardeningresponse)arecreatedforeverylayer. 4.5.3ResultsTheowcurvevariationasafunctionofdepth(hardness)isusedasinputfortheniteelementmodelinacontinuouslygradedfashionwithdepthinordertomatchtheexperimentallymeasuredsubsurfacehardnessgradients.Forexample,theowcurvesforgradedP675areshowninFigure 4-17 forstrainhardeningexponentnof0.064.Beyondastrainlevelof20%,thestrainhardeningexponentisreducedtozero,followingthebehaviordiscussedinFigure 4-13 .Indicatedwitheachcurveisthehardnessvalueitrepresentsinunitsofkg/mm2. 93

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Recallingthattherepresentativeplasticstrainrisafunctionofthestrainhardeningexponentnonly,theselectedowcurveswithxednasafunctionofdepthalsohavexedr(Figure 4-2 ,forP675withn=0.064,r=0.052).Fortheundeformedmaterial,theconstraintfactorCiscalculatedfromtheknownsubsurfacehardnessvaluesandtheowcurvesfromFigure 4-17 .ThevaluesofCvaryfrom2.40inthecaselayer,to2.90inthecore.Thesurfacemacro-indentationisthensimulatedtoadepthmatchingtheexperimentallyproducedindent.TheresultingsubsurfaceplasticstrainsdeterminedfromthemodelforthegradedmaterialareshowninFigure 4-18 .ThisinformationisusedtopredicttheincreaseinsubsurfacehardnessasafunctionofdepthfromEquation 4 .Theincreaseinhardnessisdeterminedatagivenlocationbycalculatingtherepresentativeowstressratthatlocation(usingthesumofrandp).Thissum,multipliedbytheconstraintfactordeterminedforthatlocation,indicatestheextentofhardnessincreaseduetoworkhardening.ThecalculatedincreaseinsubsurfacehardnessforthegradedP675isshowninFigure 4-19 .Goodagreementisshownbetweentheexperimentalandnumericalresults.TheprocedureisrepeatedforthegradedM-50NiLmaterial,beginningwiththeowcurvesbasedonthesubsurfacehardnessprole(Figure 2-7 ),thevariationinelasticmodulus(Figure 2-14 ),thestrainhardeningexponentn=0.056(Figure 2-10 ),andEquation 4 .TheseresultsareshowninFigure 4-20 ,withhardnessvaluesindicatedoneachcurveinunitsofkg/mm2.Usingtheprescribedowcurves,thesurfacemacro-indentissimulatedtoadepthmatchingtheexperimentalindentationinordertodeterminethesubsurfaceplasticstraindistribution.Basedonthestrainhardeningexponent,therepresentativeplasticstrainrfromFigure 4-2 andTable 4-1 is0.054.Thisvalueisaddedtotheplasticstrainateachpointinthesubsurfaceandthenconvertedtothenewhardnessvalue.ThemodelpredictionsareshowninFigure 4-21 alongwiththeexperimentallymeasuredvalues. 94

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Forthehigh-strainregion,itwasnecessaryforthestrainhardeningexponenttobereducedtozerobeyondastrainlevelof15%.ThisbehavioragreeswiththeM-50NiLowcurvefoundinFigure 2-10 athighstrains. 4.6SummaryThisanalysispresentedamethodforextractingmechanicalpropertiesfrombothgradedandhomogeneousmaterialsusingmacro-andmicro-indentationasameanstointroduceandmeasuresubsurfaceplasticstrains.Theapproachavoidstherequirementforinstrumentedindentation,whichtypicallyrequiresmultipleindentergeometriesandextensivedimensionlessanalysisinordertopredicttheplasticresponse.Theapproachisapplicabletoavarietyofrealisticengineeringmaterialsirrespectiveofthenatureofthegradientsinmaterialproperties. 4.6.1RepresentativePlasticStrainUnlikepreviousstudieswhichhaveattemptedtodetermineauniversalvaluefortherepresentativeplasticstrainofindentation,thecurrentstudyexaminedtheuseofamaterial-dependentrepresentativeplasticstrainvalue.Thisvalue,ortheaverageplasticstrainbeneathanindentation,hasbeenshowntodependonthestrainhardeningabilityofamaterial,andwasshownheretobevalidinthepredictionofindentationhardnessfromstress-strainowcurvesonbothundeformed/virginandplasticallydeformedmaterials.Theprocedureincludedtheuseofalargemacro-indentationinordertoinduceagradientinplasticdeformation,followedbysectioningandmicro-indentationinthesubsurfaceregionbeneaththisindent.Usingtheconceptofconstraintfactorrelatingindentationhardnesstoowstress,themethodwasabletoaccuratelypredicttheincreaseinsubsurfacehardnessofplasticallydeformedstrain-hardeningmaterials,thusdemonstratingthattheaverageplasticstraincanbeusedastherepresentativeplasticstraininordertoestimatetheindentationhardnessofapreviouslyplasticallydeformedmaterial. 95

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4.6.2ConstitutiveResponseofGradedMaterialsByapplyingthesetechniquestotheanalysisofgradedmaterials,theconstitutiveresponseofbothP675andM-50NiLweredeterminedasafunctionofdepth.Therelationshipbetweenhardness,yieldstrength,andstrainhardeningexponentwasusedinordertopredictthehardeningbehaviorofasafunctionofdepthwithinthegradedlayer.Itwasfoundthatthetwocarburizedsteelscontaingradientsinyieldstrength,butconstantstrainhardeningexponentwithdepth. 96

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Figure4-1. Relationshipbetweenhardnessandowstressusingrepresentativeplasticstrain. Figure4-2. Averageplasticstrainasafunctionofhardeningexponentnfora70.3degreeconicalindenter. 97

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Figure4-3. Schematicofsectionedindentgeometryandsubsurfaceindenteld. Figure4-4. SubsurfacehardnessincreasebeneathlargemacroindentationonP675corematerial. 98

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Figure4-5. SubsurfacehardnessincreasebeneathlargemacroindentationonM-50throughhardenedmaterial. 99

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Figure4-6. Examplegeometryandmeshusedforindentationmodeling. AP675Core BM-50ThroughHardenedFigure4-7. FlowcurvesforP675coreandM-50throughhardenedmaterials. 100

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Figure4-8. SubsurfacestraincontourplotforM-50throughhardenedmaterial,alongwithstrainvaluesalongthecenterline. Figure4-9. ComparisonbetweenexperimentalandcalculatedsubsurfacehardnessincreaseforM-50. 101

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Figure4-10. Alternativestress-straincurvesforthethroughhardenedM-50withvarietyofstrainhardeningvalues.Allcurvesrepresentthesamehardness,yetdifferentsubsurfacehardeningcharacteristics. Figure4-11. ComparisonbetweenexperimentalandcalculatedsubsurfacehardnessincreaseforM-50forvariouslevelsofstrainhardening. 102

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Figure4-12. PredictedandmeasuredhardnessvalueswithintheplasticzoneofthemacroindentonP675core. Figure4-13. Stress-straincompressioncurvefortheP675corematerialathighstrainlevels. 103

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Figure4-14. Subsurfacehardnessbeneaththemacro-indentationongradedP675. Figure4-15. Subsurfacehardnessbeneaththemacro-indentationongradedM-50NiL. 104

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Figure4-16. RelationshipbetweenH=yandE=ycalculatedbytheexpandingcavitymodelforvariousvaluesofn. Figure4-17. FlowcurvesasafunctionofhardnessforthegradedP675. 105

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Figure4-18. Subsurfaceplasticstrainsalongcenterlinebeneaththemacro-indentation(130mdepth)ongradedP675. Figure4-19. Experimentalandnumericalsubsurfacehardnessbeneaththemacro-indentationongradedP675. 106

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Figure4-20. FlowcurvesasafunctionofhardnessforthegradedM-50NiL. Figure4-21. Experimentalandnumericalsubsurfacehardnessbeneaththemacro-indentationongradedM-50NiL. 107

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Table4-1. Averageplasticstrainasafunctionofhardeningexponentnfora70.3degreeconicalindenter. StrainHardeningExponentnAveragePlasticStrain 0.0000.0670.0250.0610.0500.0550.0750.0500.1000.0460.1250.0420.1500.0390.1750.0370.2000.036 Table4-2. Summaryofrepresentativeplasticstrainvaluesfoundinliterature. ReferenceRepresentativePlasticStrainr Tabor[ 51 ]0.08Johnson[ 64 ]0.07Daoetal.[ 67 ]0.033Ogasawaraetal.[ 68 ]0.0115Chaudhri[ 66 ]0.25.36TekkayaandLange[ 76 ]0.112Antunesetal.[ 80 ]0.034.042 108

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CHAPTER5MODELINGGRADEDMATERIALSURFACEHARDNESSASAFUNCTIONOFINDENTATIONLOAD 5.1BackgroundFiniteelementanalysishasbecomeausefultoolfordesignandanalysisofgradedmaterialsforuseinawidevarietyofapplications.Paststudies,discussedinthepreviouschapter,haveemphasizedtheextractionofmaterialpropertieswithinthegradedregion[ 15 60 62 91 93 ],whileothershaveexaminedtheinuenceofdifferentmaterialparametersondeformationbehavior[ 4 8 14 ].Forexample,Stephensetal.[ 8 ]conductedsimulationsofbluntindenterspenetratingmaterialswithgradientsinyieldstrengthandelasticmodulus.Theyconcludedthatanappropriatelygradedmaterialcanwithstandhighercontactloadsbyshiftingthelocationofinitialsubsurfaceyieldingawayfromthesurface.Similarly,Penderetal.[ 4 5 ]haveshownthatagradientinelasticpropertiescanbeusedinordertosuppresscracksduringHertzianindentationofbrittlematerials.Choietal.[ 6 7 ]conductedanin-depthanalysisofplasticallygradedmaterialswithlineargradientsinyieldstrengthandnovariationinelasticmodulusorstrainhardening.Thisfundamentalstudyfocusedontheevolutionofstraineldsbeneathanindentationforprescribedchanges(assumedtobelinear)intheyieldstressasafunctionofdepth.TheanalysiswasveriedinacompanionpaperdetailingaspeciallysynthesizednanostructuredNi-Walloywithalineargradientinhardness(andyieldstrength)from5GPaontheoutersurfacetoslightlymorethan9GPaintheinteriorofthematerial.Thegradientwasachievedthroughalinearvariationinthegrainsizefrom90nmontheoutersurfaceto20nmintheinterioratadepthof20m.Usingthismodelmaterialandanelasticpower-lawhardeningconstitutivemodel,theyinvestigatedthepile-upbehavioranddepthvs.hardnessrelationships.However,manygradedmaterialsinengineeringpracticearesignicantlymorecomplexthanthemodelmaterialproducedbyChoietal.[ 6 7 ].Thegradientsinthecompositionandmicrostructureofamaterialcanproduce 109

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complexvariationsinyieldstrength,workhardeningbehavior,andultimatestrengthasafunctionofdepth.Giannakopoulos[ 14 ]combinedanalyticalandnumericalmethodstoanalyzethedeformationinducedbysharpindentationonplasticallygradedmaterials.Theresultsincludedthevariationininstrumentedindentationload-displacement,aswellassubsurfacestressdistributions.However,theanalysisfocusedonlyonperfectly-plasticmaterials(i.e.,nostrainhardeningbehavior)inadditiontogradientsinelasticproperties.Similarly,Weissenbeketal.[ 94 ]conductedanumericalanalysisofindentationonalayerednickelaluminumoxidesystem.Attentionwaspaidtothestressvariationsinthetwocomponents,basedondifferentlayeringpatterns.Themajorityofthesepaststudieshavefocusedprimarilyonthesubsurfacedistributioninstresses,whilenonehaveemphasizedthemeasuredsurfacehardnessasafunctionofincreasingindentationload.Chapter 3 introducedanempiricalmethodfordeterminingsubsurfacegradientsinhardnessbyusingonlysurfaceindentationmeasurementsunderincreasingindentationloads[ 95 ].Itwasnotedthattheprocedurehaspracticalutilityintheheat-treatmentindustry,wheretheeffectivenessofanintendedheat-treatmentprocesscanbeassessedrapidlyusingsimplesurfacehardnessmeasurementsasameanstodeterminetheresultinggradientinmechanicalpropertyvariationwithdepth.Inordertobetterunderstandthetrendsinsurfacehardnessunderincreasingloads,amodelofthethisbehaviorwouldbebenecial.Thismayprovideinsightintotheinuenceofsubsurfacehardnessgradientsonthebehaviorofthesurfaceunderindentationandcontactloads,inadditiontothebenets(ordrawbacks)whencomparedtoathroughhardenedmaterialofthesameoralteredsurfacehardness.ThismayadditionallyallowtheapproachoutlinedinChapter 3 tobegeneralizedforothergradientswhichwerenotavailableinthecurrentstudy. 110

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Onlyalimitednumberofstudieshavecombinedbothsimulationsandexperimentalcharacterizationofgradedmaterials[ 6 7 15 53 ].Frequently,experimentalcharacterizationreliesonmaterialsspeciallycreatedinthelaboratoryinordertoachievespecicgradientsinmaterialproperties.Thistypeofmaterialprovidesameansforisolatingtheinuenceofindividualcharacteristics(e.g.,modulus,yieldstrength,andhardeningcoefcient).Whilespeciallytailoredmaterialswithgradientsinselectedmaterialpropertiesarevaluableforthestudyofindividualcharacteristics,avarietyofengineeringapplicationsalreadybenetfrommaterialscontaininggradedcongurations(e.g.,carburizedsteel).Chapter 2 indicatedthatthecarburizedsteelsinthecurrentstudycontaingradientsinmultiplematerialpropertiessimultaneouslyasafunctionofdepth.Theseincludedgradientsinelasticmodulus,duetothedistributionofcarbideswithhighstiffness,inadditiongradientsinyieldstrengthandhardness.Thus,itmaybebenecialtodeterminetheindividualinuenceofeachmaterialpropertygradientonthesurfacehardnessunderincreasingindentationload.ThismayallowthemethoddescribedinChapter 3 tobeextendedtoawiderrangeofmaterials.Additionally,itmaybepossiblethatthemethodcouldbeusefulinindicatingmoreaboutthenatureofhardnessgradientsthansimplythesubsurfacehardnesstrend.Inthissection,theindentationofgradedmaterialswillbeassessedusingtheniteelementmodelproposedinthepreviouschapter.Thisparametricstudywillconsiderthevarietyofgradientsfoundfrequentlyingradedmaterials,includingthosewithgradientsinhardness,gradientsinelasticproperties,andgradientsinplasticproperties(specically,strainhardeningcapacity).Thesewillbeconsideredseparately,inordertoquantifytheinuenceofindividualparameters.Attentionwillbepaidtoboththesurfacehardness,aswellassubsurfacedeformation. 111

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5.2Procedure 5.2.1FiniteElementModelTheniteelementmodelconstructedandevaluatedinthepreviouschapterwasagainutilizedtorepresentthegradedmaterialsinthecurrentchapter.Themodelconsistedofanaxisymmetricindenterwithincludedangleof70.3degrees,andmaterialbehaviorfollowingpower-lawhardening.Thecarburizedmaterialscharacterizedinthepreviouschapterswereusedasstartingpointsforselectionofgradientsofinterest.However,insteadoffocusingonsubsurfacedeformationandstrainhardeningbehavior,attentionisnowpaidtothesurfacehardnessofeachgradedmaterialwithincreasingindentationload.Simulatedsurfacehardnesswasdeterminedbymonitoringthecontactareabetweentherigidindenterandsampleduringanindentation.Theloadrequiredtoproducetheindentwasalsomonitoredsimultaneouslyinordertoestablishameasureofindentationhardness.Theresultingvaluesarecalculatedinthesamemannerasexperimentalhardness,usingthecontactareadividedbyload.Itwasnotedthatminimalpile-upand/orsink-inoccurredduringindentation,thus,thehardnessvalueswerenotgreatlyaffected.Suchbehaviorisprimarilyproblematicinmaterialswithhighlevelsofstrainhardening,particularlythosewithincreasingsubsurfacehardnessgradients(i.e.,softcasematerialoverhardenedcore)[ 6 7 ].Priortoconductingtheparametricstudyofgradedmaterials,avarietyofhomogeneousmaterialsweresimulatedinordertodetermineappropriatematerialpropertiesrequiredfortherangeofhardnessvaluesexploredinthecurrentstudy.Thepropertiesoftheseuniformmaterialsconsistedofarangeofyieldstrengths,elasticmoduli,andstrainhardeningexponents,whichfollowtheelastic-plasticpower-lawhardeningmodel(elasticportion=Eandplasticportion=Kn).ThehardeningcoefcientKofthestress-strainresponsecanbedeterminedfromtheintersectionoftheelasticandplastic 112

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owcurvesasK=En1)]TJ /F4 7.97 Tf 6.59 0 Td[(nythusprovidingthecompletestress-straindescription(bothelasticandplasticportions).Indentationsonthehomogeneousmaterialsweresimulated,andtheloadrequiredtoproduceanindentwasdeterminedalongwiththecontactareabetweentheindenterandsample(fromindentergeometry).Theresultingrelationshipsbetweenhardness,yieldstrength,elasticmodulus,andstrainhardeningexponentareshowninFigure 5-1 .Here,higherstrainhardeningresultsinanincreaseinhardness.Similarly,hardnessincreaseswithincreasingelasticmodulus.ThedatainFigure 5-1 willbeutilizedasthebasisforselectionofelasticandplasticstress-strainresponsesforsimulationsthroughouttheremainderofthechapter.ItshouldbenotedthatthevaluesshowninFigure 5-1 deviateslightlyfromthosedeterminedbytheGaoetal.expandingcavitymodel[ 56 ]discussedinthepreviouschapter(Figure 4-16 ).Thisdiscrepancyisduetotheinabilityoftheniteelementmodeltocapturetheindentationsizeeffect(ISE),primarilyforhighhardnessmaterials(greaterthan700kg/mm2).Assuch,thesimulatedyieldstrengthmustbeslightlyelevatedinordertomatchtheexperimentalhardnessvalues,butdoesnotaffectthehardeningbehavior. 5.2.2ParametricStudyofHardnessGradientsTofullyunderstandtheeffectofsubsurfacehardnessgradientsonmaterialbehavior,abroadrangeofgradientsinmechanicalpropertieswereutilized.Theintentistoencompassthegradientsinpropertiesfoundinthecarburizedsteelsaswellasothergradientsofinterest.Forexample,therangehardnessgradientsfoundinbothM-50NiLandP675(Chapter 3 )areshownhereinFigure 5-2 .Overall,onlyonemechanicalpropertywasallowedtovarywithdepthduringeachgivenanalysis,whileatotalofthreedifferentgradienttypeswerestudiedduringsubsequentinquiries.Therstinvestigationfocusedonlineargradientsinhardnessrangingfromathroughhardenedmaterialtosharplydecreasinghardnessgradients. 113

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Positivegradientsinhardnesswerenotconsideredinthecurrentstudy.Thesubsurfacehardnessgradientswillbeexpressedasthechangeinhardnessasafunctionofdepth(i.e.,Hv/mm),andrangedfromzero(throughhardened)to-500kg/mm2permm.Thestartingsurfacehardnesswasxedat900kg/mm2(8.83GPa),asthisvalueisbetweenthesurfacehardnessoftheM-50NiL-AandP675-Asamples.ThestrainhardeningexponentnandelasticmodulusEwerealsoxed.ThesimulatedhardnessgradientsareshowninFigure 5-3 .Atagivenlocation/depthwithinthegradedmaterial,thehardnessvalueisprescribed(fromFigure 5-3 ).Additionally,fortherstanalysistheelasticmoduluswasxedat200GPaandthestrainhardeningexponentwasxedat0.064.Thus,appropriatematerialpropertiesareselectedinordertomatchthedesiredhardnessateachspecicdepth.Thisprocessisrepeatedthroughouttheentiregradedmaterialusinganumericalsubroutinewhichassignsthehardnessvariationtoindividualnodesasafunctionofdepthinordertocreateacontinuouslygradedmaterial.Indentationsarethensimulatedonthesurfaceofthegradedmaterials,inthedirectionofthedecreasinghardnessgradient.Asintheprevioussection,theloadrequiredtoproduceeachindentaswellastheindentdepthismonitoredinordertodeterminethedecreaseinsurfacehardnessasafunctionofincreasingindentationload.Insubsequentanalyses,additionalparameterswillbevaried,includinggradientsinbothelasticmodulusandstrainhardeningbehavior.ThevariousmaterialparametersconsideredareshowninTable 5-1 .Allmaterialshavethesamestartingsurfacehardnessvalueof900kg/mm2. 5.3ResultsandDiscussion 5.3.1LinearGradientinHardnessTheanalysisinthissectionexhibitslinearhardnessgradientsastheonlypropertyvaryingwithdepth.Theelasticmoduluswasheldconstantat200GPa,andplasticstrainhardeningexponentnxedat0.064(samehardeningexponentthatwas 114

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determinedfortheP675materialinChapter 4 ).Fromthemodelsofthesematerialswithgradientsrangingfrom0to-500,theresultingsurfacehardnessdataunderincreasingindentationloadisshowninFigure 5-4 formaterialswithstartingsurfacehardnessof900kg/mm2.Hardnessvaluesareintermsofloaddividedbycontactarea(matchingthemethodinwhichVickersindentationhardnessismeasured).Additionally,thehardnessgradientisindicatednexttoeachsection.ThetrendsshowninFigure 5-4 appearsimilartothoseproducedexperimentallyinbothP675andM-50NiLinChapter 3 ,withtheexceptionoftheinitialISEevidentintheexperimentalsections(Figure 3-3 ).ThislimitationarisesbecausetheniteelementsimulationcannotcapturetheISE.However,itwasshowninChapter 3 thatthisdoesnotgreatlyinuencethesurfacehardnessanalysis.ItshouldbenotedthatscatterintheFEAhardnessdatacanbeattributedtoloaddiscrepanciesaselementsgaincontactwiththeindenter.Whenanewnodeiscontactedduringtheindentationprocess,theloadtendstoelevateslightly.Meshrenementwasshowntohaveminimaleffectonscatter,whileonlyincreasingcomputationtime.Ingeneral,morethan20elementswereincontactwiththeindenterduringatypicalsimulation.AcomparisonbetweentheexperimentaldatainFigure 3-3 andthesimulatedhardnessdatainFigure 5-4 showscomparabletrendsinsurfacehardnessathighloads.Forexample,theP675-Bsamplewithasubsurfacehardnessgradient(Hv/mm)of-290exhibitsadecreaseinsurfacehardnessontheorderof80kg/mm2betweenlowandhighloadindents,whilethesimulatedhardnessgradientof-300exhibitsadecreaseinsurfacehardnessof86kg/mm2overacomparableindentationloadrange.Additionally,M-50NiL-C,withsubsurfacehardnessgradientof-196,displaysadecreaseinsurfacehardnessof54kg/mm2,whilethesimulatedhardnessgradientof-200exhibitsadecreaseinhardnessof50kg/mm2overthesameloadrange.Consideringthatthesimulationscontainonlygradientsinhardness,whereasbothP675andM-50NiLalsocontaingradientsinelasticproperties,thesimulatedgradients 115

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agreewellwiththeexperimentaltrendsinsurfacehardnessforgradedmaterials.Thus,theanalysiswillbeextendedtoincludetheinuenceofadditionalmechanicalproperties,includingchangesinstrainhardeningbehaviorandelasticproperties. 5.3.2PerfectlyPlasticandHighStrainHardeningWhileintheprevioussectionthestrainhardeningexponentnwasxedat0.064,itisimportanttodeterminewhetherthisvaluehasanyinuenceonthesurfacehardnessunderincreasingloads.Inthissectionthesameanalysiswasrepeated(includingaconstantelasticmodulusof200GPa),however,twodifferentstrainhardeningexponentswereconsidered.First,amaterialwithastrainhardeningexponentofzerowasmodeled,i.e.,perfectlyplasticbehavior.Inthisregard,thestrengthofthematerialdoesnotincreasebeyondtheyieldstrengthwithadditionalstrain.Next,amaterialwithastrainhardeningexponentof0.1wasconsidered,resultinginincreasedhardeningbehaviorcomparedtotheP675.Thisisanimportantvariable,sincedifferentgradedmaterialsmayexhibitalteredstrainhardeningbehavior.Forbothofthesesimulatedmaterials,thehardnessgradientsmatchedthoseoftheprevioussection(i.e.,Figure 5-3 ).Examplestress-strainowcurvesforthesetwomaterialsareshowninFigure 5-5 .Eachcurveislabeledwiththehardnessvalueitrepresentsinkg/mm2.Figure 5-6 containsthesurfacehardnessasafunctionofincreasingindentationloadforbothmaterials,withtheperfectlyplasticmaterialdenotedbyopensymbols,andthestrainhardeningmaterialbylledsymbols.Interestingly,Figure 5-6 indicatesthatthesurfacehardnessbehaviorisapproximatelythesameregardlessofthestrainhardeningexponentofthematerial.ThesetrendsalsomatchthedatashowninFigure 5-4 ,whichhadastrainhardeningexponentof0.064.Thisresultisconvenient,becausedespitethedifferentstrainhardeningexponentsinP675andM-50NiL(Chapter 4 ),theymaybeanalyzedtogether.Theonlynoteddifferencebetweenthetwotrendsoccursinthehigh 116

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loadrangeofthematerialwithgreatestsubsurfacehardnessgradient(Hv/mm=-500).Theresultingdiscrepancyisonlyontheorderof10kg/mm2ataloadof300kg. 5.3.3GradientinStrainHardeningWhileitwasdeterminedinChapter 4 thatthestrainhardeningexponentndoesnotvarywithdepthineitherP675orM-50NiL,itisworthconsideringhowthisparameterinuencestheindentationbehaviorofgradedmaterials.Thismaybeimportantinothermaterials,especiallythosecontainingmixturesofcomponentswithdifferinghardeningcharacteristics.Insteadofvaryingthesubsurfacehardnessasbefore,inthisanalysisthehardnessgradient,Hv/mm,wasxedat-300withastartingsurfacehardnesslevelof900kg/mm2.Twobaselinematerialswereconsidered.Thesecontainedconstantstrainhardeningexponentsof0.064and0.1,matchingthevaluesconsideredintheprevioustwosections.Extendingthesetwomaterials,gradientsinstrainhardeningexponentwerenextintroduced.Twoofthesehadincreasingtrendsinstrainhardeningasafunctionofdepth,whiletwohaddecreasingtrendsinstrainhardening.ThesetrendsareshowninFigure 5-7 .Again,itshouldbeemphasizedthatdespitethegradientsinn,thehardnessgradientremainsxedat-300.Thesetrendsresultininterestingowcurves,showninFigure 5-8 .Indicatedwitheachcurveisthehardnessvalueitrepresents.TheresultingsurfacehardnessunderincreasingindentationloadsisshowninFigure 5-9 .Becausethetrendsappearverysimilar,onlythetwomostextremegradients(oneincreasingandonedecreasing)havebeenplottedalongwiththeuniformbaseline.Despitehavingsharpgradientsinstrainhardeningexponent,neitherofthematerialsdeviatesignicantlyfromthematerialwithconstantstrainhardeningexponent.Onlybeyondanindentationloadof150kgaredifferencesnotedbetweenthetrends.Interestingly,thesurfacehardnessdecreasesmorerapidlywithincreasingloadinthematerialwithincreasingstrainhardeningexponentasafunctionofdepth.Conversely,thematerialwithdecreasingstrainhardeningexponentshowslesshardnesslossat 117

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highloads.Inotherwords,atagivenindentationload,indentpenetrationwillbeslightlygreater(i.e.,softermaterial)inthematerialwithanincreasingsubsurfacetrendinstrainhardeningexponent.Alookintothesubsurfacedeformationmaylendinsightintothistrend.Thisisincludedinalatersection.Toconrmthistrendisnotsimplyanartifactoftherangeofstrainhardeningexponentsconsidered(withstartingsurfacen=0.1),anadditionalgradientinsubsurfacestrainhardeningexponentwasmodeled.Thisadditionalmodelconsistedofaperfectlyplasticsurface(n=0)withincreasingsubsurfacestrainhardeningbehaviorwithdepth.Thegradientincludedanincreaseinstrainhardeningexponentof0.2permm,matchingthemaximumgradientshowninFigure 5-7 .TheresultingsurfacehardnessunderincreasingindentationloadhasbeenincludedinFigure 5-9 (shownasX's),andisfoundtoagreewiththeoriginalincreasingexponenttrend,indicatingthatonlythegradientinninuencesthisbehavior,notthemagnitude. 5.3.4GradientinElasticModulusItwasshowninChapter 2 thatagradientinelasticpropertiesisfoundinthecasehardenedsteelsduetothechangingvolumefractionofthecarbidephaseasafunctionofdepth.Thesecarbideshaveelasticpropertieswhicharegreaterthanthesurroundingsteelmatrix(i.e.,highelasticmodulus),andcanbeontheorderof300GPaforthecarbidesfoundinP675and500GPaforthecarbidesfoundinM-50NiL[ 96 ].Thehighvolumefractionofthesecarbidesinthecaselayerhasbeenshowntoaltertheelasticpropertiesofthematerials,increasingthemodulusbeyondwhatistypicalinalowcarbonsteel.Inlightofthisinformation,thissectioncontainsananalysisofmaterialswithgradientsinelasticmodulus.ThegradientssimulatedareshowninFigure 5-10 .Thesematerialsallpossessthesamestrainhardeningbehavior(n=0.064),andaxedhardnessgradientHv/mmof-300,withastartingsurfacehardnesslevelof900kg/mm2asintheprevioussection. 118

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Figure 5-11 indicatesthesurfaceindentationhardnessofthesimulatedelasticallygradedmaterials,allwithahardnessgradientof-300.Asintheprevioussection,thetrendsappearverysimilar,soonlythetwomostextremegradients(oneincreasingandonedecreasing)havebeenplottedalongwiththenon-elasticallygradedbaseline.Fromthisanalysis,itappearsthatneitheranincreasingnordecreasinggradientinelasticpropertieshasanyinuenceonthesurfacehardnessunderincreasingindentloads.TheresultsconrmthatthemethodpresentedinChapter 3 isnotinuencedbytheelasticallygradedpropertiesofthecarburizedmaterials,butinsteadonlyrepresentativeofthesubsurfacehardnessgradients. 5.3.5Power-LawModelChapter 3 introducedamethodforttingthesurfacehardnessdatausingapower-lawcurvet.Thetechniquewasappliedtonormalizedplotsofsurfacehardnessvs.indentationload,inordertoproduceadimensionlessrelationship.Thismethodwillbeagainappliedheretothesimulatedindentationsperformedonthevariousgradedsections.Sinceitwasshownthatthesurfacehardnessisnotinuencedgreatlybysubsurfacegradientsinstrainhardeningexponentorelasticmodulus,thetrendswillbeappliedtothegradedmaterialsoutlinedinSection 5.2.2 withconstantelasticmodulusandstrainhardeningexponent(E=200GPaandn=0.064).TheresultinginformationisshowninFigure 5-12 ,wherethesurfacehardnesshasbeennormalizedbasedonthexedsurfacehardnessforallsections(900kg/mm2).Again,thepower-lawcurvethasbeenappliedtotheresults.ThettingparametersrequiredweredenedbyH=)]TJ /F2 11.955 Tf 9.3 0 Td[(0.006Pb+1.0Thetrendtstheresultswell,althoughthe-500gradientdoesdeviateslightly.However,thisgradientismoreseverethanthosefoundintheexperimentalsectionsofP675andM-50NiL. 119

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Theoverallresultisinteresting,asitdemonstratesthatthepower-lawusedinChapter 3 isasuitablettingmethod.Furthermore,sincethepresentanalysishasshownthereisnoinuencefromgradientsinelasticproperties,andminimalinuencefromgradientsinstrainhardeningbehavior,themethodcanbeextendedtoawidevarietyofmaterialsregardlessofthenatureofsubsurfacegradients.Onlythesubsurfacehardnessgradientinuencestheoveralltrendsinsurfaceindentationhardnessasafunctionofincreasingindentationload.Itispossibletocreatearelationshipbetweenthecoefcientbandthehardnessgradients.ThisresultisshowninFigure 5-13 asaplotofhardnessgradientvs.power-lawexponent,whichisshowntofollowapower-lawcurvet,Hv x=1640b2.6+50providingarelationshipbetweenthesubsurfacehardnessgradientandthebehaviorofthesurfacehardnessunderincreasingindentationload.AsinChapter 3 ,itshouldbenotedthatanychangesinindentergeometrywouldaffectmeasuredhardnessvaluesandtrends,andthereforewouldwarrantanewanalysis[ 56 95 ]. 5.3.6SubsurfaceStressandStrainFieldsInordertobetterunderstandthetrendsinsurfacehardnessbehaviordescribedintheprevioussections,itmaybebenecialtoobservethesubsurfacedeformationoccurringbeneathindentationsinthevariousgradedmaterials.Besidesevaluatingtheplasticzonesizeforthedifferentgradedmaterials,thismayalsodemonstratewhetherparticulargradientsinmechanicalpropertiesinuencetheextentofsubsurfacedeformationduringindentation.Tocomparethevarietyofgradients,axedindentationdepthof50misusedwhenplottingsubsurfacestrainsandstresses.Theequivalentplasticstrainisplottedtoprovideanindicatorofthedepthofplasticity,andplasticstrainsabove0.002are 120

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consideredyieldedmaterial(accordingtothestandard0.2%offsettypicallyusedduringanalysisofductilematerials).Additionally,thevonMisesstressisplottedasthesubsurfacestresseldtoindicatetheamountofhardening,whichisaffectedbychangesinhardeningexponent. 5.3.6.1HardnessgradientsAsanexampleofthesubsurfacedeformation,Figure 5-14 containsacomparisonbetweenthesubsurfaceequivalentplasticstrainlevelsintheuniform(throughhardened)and-500hardnessgradientmaterials(bothwithconstantE=200GPa,n=0.064,andindentdepth50m).Here,theplasticdeformation(denedasplasticstrainsbeyond0.002)extends50mdeeperinthegradedmaterialthantheuniformone.Tobetterillustratethesetrendsforthewideselectionofgradientsinmechanicalproperties,itisbenecialtoinsteadplotthesubsurfacedataalongthecenterline(i.e.,directlybeneathindentertip).Forexample,Figure 5-15 showsthesubsurfaceequivalentplasticstraindistributionsalongthecenterlineforthegradedmaterialsdescribedinSection 5.2.2 ,whileFigure 5-16 showsthesubsurfacevonMisesstressdistributionalongthesamelocation.AllofthesematerialscontainedconstantE=200GPaandn=0.064,withindentdepthof50m.Depth,plottedalongthex-axis,ismeasuredasdistancefromtheindentertip.Thedistancefromthesurfacewouldbethisvalueplusanadditional50m.Figure 5-15 indicatesthatplasticdeformationpenetratesdeeperinthegradedmaterialsatagivenindentationdepth.Theplasticzonedepthincreaseswithseverityofsubsurfacehardnessgradient.Chapter 3 discussedthedepthoftypicalplasticzones,estimatingthemtobeontheorderofsixtimestheindentationdepth.Figure 5-15 showsthatthedepthofplasticdeformationis250mforthehomogeneousmaterial,increasinguptoamaximumof300mforthesharpestgradient.Theseresultinplasticzoneswhicharebetweenveandsixtimestheindentationdepth. 121

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Figure 5-16 showsthesubsurfacevonMisesstressbeneaththeindenter,alongthecenterline.Duetothehighstressconcentrationatthetipofthesharpindenter,thematerialdirectlybelowtheindenteryieldsimmediatelyandhardensconsiderably.Becauseallofthematerialshavethesamesurfaceproperties,themaximumstressvaluematchesbetweenallofthecases.Beyondthisregion,thestressdependsonboththelevelofplasticstrain,aswellasthehardeningexponent.Becausenisxed,theextentoftheincreaseinstressaboveyieldisonlyafunctionofstrain.Thus,thesubsurfacestressfollowstheinitialgradientinyieldstrengthplustheincreasefromhardening.Thetrendcontinuesuntiltheendoftheplasticzone(appearsasakneeinthedata),afterwhichtheelasticstressestapertozero. 5.3.6.2PerfectlyplasticbehaviorTheperfectlyplasticmaterialsresultininterestingbehavior,becausetheyieldstrengthdoesnotincreasewithaddedplasticstrain.Thus,nohardeningoccursduringindentation.Thesubsurfaceplasticstrain,showninFigure 5-17 forallofthegradients,hasasimilarappearancewhencomparedtotheprevioussection(Figure 5-15 ).However,thedepthofplasticdeformationisslightlylessthanthepreviousmaterial.Forthehomogeneousperfectlyplastictestcase,theplasticzoneextendstoadepthof246m,whilethesharpestgradient(Hv/mm=-500)hasaplasticzonewhichextendsto285mbeneaththesurface.Figure 5-17 alsoshowsthesubsurfacevonMisesstressbeneaththeindenterfortheperfectlyplasticmaterial.Becausethereisnohardening,thestressdistributionwithintheplasticzonefollowsthelinearyieldstressgradient(i.e.,linearhardnessgradient).Beyondtheplasticzone,thestressesareelasticanddecreasetozerointhefareldregion. 5.3.6.3GradientinstrainhardeningexponentForthegradientinstrainhardeningexponent,allofthesectionscontainedthesamesubsurfacehardnessgradientof-300kg/mm2permm.Thegradientinhardening 122

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behaviorwithdepthwasshowntohaveaslightinuenceonthesurfaceindentationhardnessunderincreasingload.Despitethedifferentsurfacebehavior,thesubsurfacedistributionofplasticstrains(Figure 5-18 )areremarkablycomparableforallofthesections.Here,theplasticzonedepthisnearlythesame,varyingbyonlyafewmicrons.Thesubsurfacestresses,alsoshowninFigure 5-18 ,demonstrateinterestingbehaviorfortheincreasinganddecreasinggradients.Sincethesubsurfaceplasticstrainsareapproximatelyequalinallofthematerials,thealteredsubsurfacestrainhardeningbehaviorresultsinnotablechangesinsubsurfacestress.Interestingly,thelinescrossatadepthof200mfromthesurface(150mbeneaththeindentertip).Thislocationcorrespondstoanequivalentplasticstrainvalueof0.040(i.e.,4%plasticstrain).Beforethispoint,thematerialwithincreasingngradienthasahigherstresslevel,whilepastthispointthematerialwithdecreasingngradienthasahigherstresslevel. 5.3.6.4GradientinelasticmodulusFinally,thegradientinelasticmoduluswasconsidered.Recallthatneithertheincreasingnordecreasinggradientsinelasticmodulushadanyinuenceonthetrendsinsurfaceindentationhardnesswithincreasingloads.Furthermore,thesectionsanalyzedallcontainedthesamehardnessgradient.Thisndingmayleadtotheassumptionthatthesubsurfacestrainandstressdistributionswouldshowminimaldifferences;however,thisisnotthecase.Thestrainalongthecenterlineoftheelasticallygradedmaterials,showninFigure 5-19 ,indicatesthattheplasticstrainsextenddeeperinthematerialwithanincreasingelasticmodulusbeneaththesurface.Conversely,thematerialwithdecreasingsubsurfacemodulushasaslightlysmallerplasticzone,comparedtotheuniformmaterial.ThegradientinelasticpropertiesfoundinthecarburizedsteelsanalyzedChapter 2 consistedofahighmoduluscase(ontheorderof240GPa),taperingintoalowermoduluscorematerial(approximately195GPa),indicatingthatthegradientfoundin 123

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thecarburizedmaterialsmaybebenecialinlimitingthedepthofplasticdeformationduringindentation.However,thesubsurfacestressesintheelasticallygradedmaterials,foundinFigure 5-19 ,showtheoppositebehavior.Despitethereduceddepthofplasticdeformation,thematerialwithdecreasingelasticmodulushasahigherstresslevelthroughouttheplasticzonecomparedtotheuniformincreasinggradient. 5.4SummaryIndentationhardnessisawidelyusedmethodforcharacterizingmaterials,andisespeciallyusefulforgradedmaterialswherestandardtestsamplesareunavailable.Forexample,Chapter 3 demonstratedthatthesubsurfacegradientsinhardnesscanbepredictedbyusingsurfaceindentationmeasurementsunderincreasingindentationloads.Thecurrentinvestigationutilizedniteelementanalysisinordertobetterunderstandthesurfaceindentationbehaviorofmaterialswithgradientsinhardness,gradientsinelasticproperties,andgradientsinplasticproperties(specically,strainhardeningcapacity).Thendingsareespeciallyrelevant,becauseChapter 2 indicatedthatcarburizedsteelsexhibitgradientsinmultiplematerialpropertiessimultaneouslyasafunctionofdepth,includingchangesinelasticmodulusduetothedistributionofcarbideswithhighstiffness,inadditiontoyieldstrengthandhardnessgradients.Itwasdeterminedthatthesurfacehardnessbehaviorunderincreasingloadsisnotinuencedbytheelasticallygradedpropertiesofthematerial,whileuniformchangesinstrainhardeningexponentalsoshowednoinuence.Onlyslightchangesinthesurfacehardnessbehaviorwerenotedinmaterialswithsharpgradientsinstrainhardeningexponent.ThisndingconrmedthatthemethodpresentedinChapter 3 wasnotinuencedbytheelasticgradientsthatoccurinthecarburizedmaterials,butinsteadonlyrepresentativeofthesubsurfacehardnessgradients.Furthermore,thisindicatedthattheanalysismethodmaybeextendedtoawiderrangeofgradedmaterials.Lastly,thesubsurfacedeformationrelatedtothegradientsinmaterialpropertieswasevaluated.Itwasshownthatplasticdeformationextendsslightlydeeperingraded 124

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materialsthaninhomogeneousmaterials,includingthosewhichbehaveasperfectlyplasticbeyondyielding.Itwasalsoshownthatforaxedgradientinhardness,gradientsinstrainhardeningbehaviorhavenoinuenceonthedepthofsubsurfaceplasticdeformation.Finally,adecreasinggradientinsubsurfaceelasticmodulus,comparabletothosefoundinthecarburizedmaterials,wasshowntodecreasethedepthofplasticdeformation. 125

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Figure5-1. RelationshipsbetweenH=yandE=ydeterminedviasimulationsofindentationsonstrainhardeningmaterials. 126

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Figure5-2. Subsurfacehardnessprolesasafunctionofdepthforthevariousexperimentalspecimens.Hardnessgradientvalue(Hv/mm)isindicatednexttoeachspecimenlabel. 127

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Figure5-3. Simulatedsubsurfacehardnessgradientswithstartingsurfacehardnessof900kg/mm2. 128

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Figure5-4. Surfacehardnessasafunctionofindentationloadforgradedmaterialswithstartingsurfacehardnessof900kg/mm2. APerfectlyplasticbehavior BStrainhardeningexponentn=0.1Figure5-5. Flowcurvesforperfectlyplasticandstrainhardeningmaterialswithxedelasticmodulusof200GPa. 129

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Figure5-6. Surfacehardnessasafunctionofindentationloadforperfectlyplastic(opensymbols)andstrainhardening(lledsymbols)materials. 130

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Figure5-7. Subsurfacetrendsinstrainhardeningexponent. AIncreasingnwithdepth BDecreasingnwithdepthFigure5-8. Flowcurvesformaterialswithgradientinstrainhardening. 131

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Figure5-9. Surfacehardnessunderincreasingloadformaterialswithgradientsinstrainhardeningbehavior. 132

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Figure5-10. Subsurfacetrendsinelasticmodulus. 133

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Figure5-11. Surfacehardnessunderincreasingloadloadforelasticallygradedmaterials. 134

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Figure5-12. Normalizedsurfacehardnessasafunctionofnormalizedindentationloadforthemodeledgradedmaterials. Figure5-13. Trendinhardnessgradientandpowerlawexponent. 135

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Figure5-14. Subsurfaceplasticstrainforuniformhardnessandgradedmaterials.Indentdepth50m. Figure5-15. Subsurfaceplasticstrainalongcenterlinebeneathindentat50mdepth. 136

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Figure5-16. SubsurfacevonMisesstressalongcenterlinebeneathindentat50mdepth. 137

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Figure5-17. SubsurfaceplasticstrainandvonMisesstressalongcenterlinebeneathindentat50mdepthonperfectlyplasticgradedmaterials.Dashedarrowsindicatetheaxistowhichthedatarefersto. 138

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Figure5-18. SubsurfaceplasticstrainandvonMisesstressalongcenterlinebeneath50mindentformaterialswithgradientsinstrainhardeningexponentn.Dashedarrowsindicatetheaxistowhichthedatarefersto. 139

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Figure5-19. SubsurfaceplasticstrainandvonMisesstressalongcenterlinebeneathindentat50mdepthformaterialswithgradientsinelasticmodulus.Dashedarrowsindicatetheaxistowhichthedatarefersto. 140

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Table5-1. Summaryofmaterialparametersconsideredduringparametricstudy. GradientParameterHardnessGradient(Hv/mm)StrainHardeningExponent(n)ElasticModulus(E,GPa) Hardness0to-500Fixed,0.064Fixed,200Hardness0to-500Fixed,0.0Fixed,200Hardness0to-500Fixed,0.1Fixed,200StrainHardeningFixed,-3000.10.2mm-1Fixed,200ElasticModulusFixed,-300Fixed,0.06420040mm-1 141

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CHAPTER6CONCLUSION 6.1GradedMaterialsGradedmaterialshavebecomeincreasingimportantforhighperformanceengineeringapplicationsduetotheirimprovedresistancetofatigue,wear,impact,andabrasivedamage.Theyfrequentlycontaingradientsinbothelasticandplasticmechanicalproperties,whichtypicallyvaryasafunctionofdepth.Casehardened,carburizedsteelsthatcontainhighsurfacehardnessandagradientinmaterialproperties(hardness,yieldstrength,etc.)havebecomepopularforuseindemandingapplicationsincludingbearingsandgearsduetotheirincreasedperformanceunderhighcontactloads.Althoughgradientsaretypicallyintroducedbydesign,characterizationofthevariousmechanicalandmaterialparametersisparamountinunderstandingandimprovingthedesignandperformanceofgradedmaterials.Tothisend,acombinedexperimentalandnumericalstudywaspresentedwhichdeterminedthecompleteconstitutiveresponseofgradedmaterialsasafunctionofdepth.Twocasehardenedcarburizedsteelscommonlyutilizedinhigh-performancegearandbearingracewayapplicationswereusedasmodelmaterials.Themethodsandanalysisareapplicabletoavarietyengineeringmaterialsirrespectiveofthenatureofthegradientsinthematerialproperties. 6.2MicrostructuralCharacterizationMultipleexperimentaltechniqueswereutilizedtocharacterizethegradientsincompositionandmaterialpropertiesincarburizedmaterials.Theelasticpropertiesofthecarbidesandsteelmatrixweredeterminedviainstrumentednanoindentation,andtheresultingdatawereutilizedwithseveralmodelsforcompositemodulusinordertoestimatethevariationinelasticpropertiesasafunctionofcarbidevolumefraction.Itwasshownthatthehighelasticmodulusofthecarbidesresultsinanincreasedelasticmodulusinthecaseregion,wherethevolumefractionofcarbidesishigh.Theresults 142

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comparedwelltotheelasticmodulusdeterminedfrombothcompressiontestingandultrasonicmeasurementsonsamplescomposedofuniformcarbidevolumefraction.Thegradientinsubsurfacehardnesswascorrelatedwiththegradientinsubsurfacecarbidevolumefraction.Forbothmaterials,linearrelationshipsbetweenthetwoparametersweredetermined.Whenextrapolatedtotheoreticalpurecarbideproperties,theresultinghardnessvaluesmatchthosedeterminedviananoindentationontheindividualcarbides.Overall,theresultsprovidemultiplemethodsfordeterminingrelevantpropertiesofcarburizedmaterialsasafunctionofdepth,whileconsideringgradientsinseveralmaterialparametersoccurringsimultaneously.Theseresultsareinvaluableformodelingandoptimizingthedesignandperformanceofcasehardenedmaterialsforuseinbearings,gears,andothercontactsurfaces. 6.3HardnessGradientDeterminationAnovelmethodwasproposedtorapidlypredictthesubsurfacehardnessproleingradedmaterialsusingsolelysurfaceindentationsatarangeofloads.Themethodisnon-destructiveanddoesnotrequirethegradedmaterialtobesectioned.Foramaterialwithadecreasinggradientinhardness,higherindentloadsresultinalowermeasuredhardnessduetotheinuenceofthesoftersubsurfacelayers.Apower-lawmodelwaspresentedwhichrelatesthemeasuredsurfaceindentationhardnessunderincreasingloadtothesubsurfacegradientinhardness.Theapproachprovidesarapidmethodforpredictingthegradientsproducedduringsurfaceheattreatment,consistingofonlyaseriesofindentationsatincreasingloadsonthesamplesurface,requiringonlyahardnesstesterandminimalsamplepreparation. 6.4ConstitutiveResponseAcoordinatedexperimentalandnumericalstudywaspresentedtodeterminethecompleteconstitutiveresponseofgradedmaterials,usingthecarburizedsteelsasmodelmaterials.Themethodexaminedtheconceptofrepresentativeplasticstrain 143

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inducedbyindentation,whichisusedinordertorelatehardnesstoyieldstrengthinbothvirginandplasticallydeformedmaterials.Itwasshownthatthetwocarburizedsteelscontaingradientsinyieldstrength,butconstantstrainhardeningexponentwithdepth.Theresultingmodelofmaterialbehaviorwasusedtocharacterizetheinuenceofspecicgradientsinmaterialpropertiesonthesurfaceindentationbehaviorunderincreasingindentationloads.Itwasshownthattheresponseofthematerialisnotgreatlyinuencedbystrainhardeningexponent,whileagradientinstrainhardeningabilityonlyhasminimalimpact.Gradientsinelasticpropertieswerealsoshowntohavenegligibleinuenceforaxedgradientinhardness.Thedepthofsubsurfaceplasticdeformationwasshowntoincreasewithsharpergradientsinhardness,butisnotalteredbygradientsinstrainhardening.Theresultsmaybeextendedtotheoptimization,design,andselectionofdifferentgradientsinordertomeettherequirementsoftheintendedapplication. 6.5FutureWork 6.5.1NitridedandDuplexHardenedMaterialsInadditiontocarburizing,nitridinghasemergedasanimportantprocessingtechniqueforimprovingtheperformanceofgearsandbearingraceways.Theresultingmaterialshaveextremelyhighsurfacehardnessvalues,andverysharpsubsurfacegradientsinhardness.Furthermore,therecenttrendofapplyinganitridedlayertoanalreadyhardened(eithercarburizedorthroughhardened)materialhasshownpromiseinimprovingthelifeofmaterialssubjecttohighcontactloads.Thisprocedureisreferredtoasduplex-hardening,andextractingtheconstitutiveresponseofthenitridedlayermaybeanimportantparameterforunderstandingperformance.Asanexample,asectionofanitridedplusthroughhardenedM-50materialwasobtained.Thissamplewassectionedandpolished,andtheresultingsubsurfacehardnessisshowninFigure 6-1 .Thegureillustratestheexceptionallyhighsurfacehardness(near1300kg/mm2,or12.8GPa)whichcanbereachedvianitriding.The 144

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hardnessrapidlydecreasesbeneaththesurface,matchingthatofthestandardthroughhardenedM-50(800kg/mm2,or7.8GPa)beyond200mdepth.ThetechniquesdevelopedinChapter 4 canbeappliedtothismaterialinordertoextracttheconstitutiveresponseofthenitridedlayer.However,becausethislayerhasamuchsmallerdepthcomparedtoacarburizedsections,smallerindentationsmaybenecessaryinordertoinduceplasticityintothecorrectregion.Oncethebehaviorofthenitridedmaterialischaracterized,modelingthesurfaceindentationbehavior,asinChapter 5 ,willindicatehowbothsurfacehardnessandsubsurfacedeformationareaffectedbythepresenceofthethin,high-hardnesslayer. 6.5.2MicrostructureModelingAsidefromthegradientsinmaterialproperties,thecarburizedandthroughhardenedsteelsinthisstudyhavebeenmodeledashomogeneous.Morespecically,thebehaviorofcarbidesandmatrixwerenotincludedindividually.Chapter 2 discussedtheinuenceofthecarbidesonoverallmaterialproperties,includingelasticmodulusandhardness,howeverformodelingpurposestheywerenotconsideredseparately.Furthermore,itwouldbedifculttoproducecompletemodelsofgradedmaterialscontainingvariedvolumefractionofcarbideswithdepth,nottomentionextremelycostlyintermsofsimulationtime.However,asmall-scalelookintothecontributionsandbehaviorofbothcarbidesandsteelmatrixmaybeofaddedbenet.ThiscanincludethevariouspropertiesdeterminedinChapter 2 .Forexample,thecarbidesfoundinP675andM-50NiLhaveelasticmoduliontheorderof300GPaand500GPa,respectively.Theresultsmayindicatewhethertheproperties,orientation,anddistribution(volumefraction)ofcarbidesinuencelocaldeformationbehavior.Asapreliminaryinvestigation,a2Dniteelementmodelwasconstructedconsistingofindividualcarbidesandmatrix.Representativeimages,likethoseinFigures 2-1 through 2-3 ,wereusedinordertocreateageometryapproximatelymatchingreal 145

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microstructures.Forexample,Figure 6-2 showsanimageofagroupingofcarbidesinthroughhardenedM-50andthesketchedoutlinestobemodeledasseparatecomponents.Usingthissketch,aniteelementmeshwasgeneratedforbothportionsandmaterialpropertiesappliedtothecarbides(E=300GPa)andmatrix(E=200GPa,y=2.5GPa,n=0.05).Insteadofsimulatingtheindentationeventasintheprevioussections,auniformfar-eldstresswasappliedtothemodeledsection.Interestingly,localyieldingoccursinthevicinityofthecarbidesevenbeforetheglobalfar-eldstressreachestheyieldstressofthematrix.Thisindicatesthatitispossibleforlocalplasticitytooccurevenunderelasticloadingconditions.Forexample,Figure 6-3 showsthelocalplasticstrainssurroundingoneofthecarbidesfromthemodeledsection.Mostbearingsandgearsaredesignedtooperateintheelasticrange,sothislocalplasticityisanimportantdesigncharacteristic.Thisbehaviorcanbeinvestigatedfurther,includingtheinuenceofcarbidesize,shape,distribution,andproperties.Forexample,uniformlydistributedsmallcarbides,likethosefoundinM-50NiL,maybepreferredovercloselyspacedcarbideslikethosefoundinP675.However,thedifferencesinmaterialpropertiesbetweenthecarbidesinthesetwosteelsalsomustbetakenintoaccount.Overall,theseresultsmayprovideadditionalguidelinesforoptimizingdesignandperformanceofcarburizedsteelsforenhancedresistancetocontactloading. 146

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Figure6-1. HardnessproleofanitridedthroughhardenedM-50material. Figure6-2. Convertingamicrographofcarbidesintoniteelementmodel. 147

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Figure6-3. Plasticstrainssurroundingcarbideparticles. 148

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BIOGRAPHICALSKETCH MichaelKleckacompletedhisB.S.andM.S.degreesatMichiganTechnologicalUniversityinHoughton,Michigan.Hismaster'sthesisfocusedonthegrainsizedependenceofscratchinduceddamageinbrittleceramics.Shortlythereafter,hemovedtoGainesville,FloridaandbeganworkonhisdoctoraldegreeinthemechanicalengineeringdepartmentattheUniversityofFlorida.Hisworkinitiallyfocusedonhighstrain-ratebehaviorofstructuralceramics.Afterayearofresearchandcompletingqualifyingexaminations,heswitcheddirectionsandbeganworkonaprojectsponsoredbyTimken,Pratt&Whitney,andAirForceResearchLabspertainingtocharacterizationofhighperformancebearingracewaymaterials.DuringhistimeatUF,hehasauthoredthreepapersinpeer-reviewedjournals(withafourthinprogress)andco-authoredanadditionalthreepapers. 157