弹性模量与硬度关系

Investigationoftherelationshipbetweenelasticmodulusandhardnessbasedondepth-sensingindentationmeasurementsY.W.Bao*,W.Wang,Y.C.ZhouShenyangNationalLaboratoryforMaterialsScience,InstituteofMetalResearch,ChineseAcademyofSciences,72WenhuaRoad,Shenyang110016,PRChinaReceived7April2004;accepted3August2004AbstractAnanalyticalrelationshipbetweenthereducedmodulusErandhardnessHforsolidmaterialsisestablishedbasedonthecon-ventionaldepth-sensingindentationmethodofOliverandPharr.ItisfoundthatthetwopropertiesarerelatedthroughamaterialparameterthatisdefinedastherecoveryresistanceRs.Thisparameterisshowntorepresenttheenergydissipationduringinden-tation.BasedonindentationmeasurementswiththeuseofaBerkovichindenter,therelationshipisgivenasEr¼0:6647ffiffiffiffiffiffiffiHRps.Alsopresentedisasimplesetofprocedurestodeterminetheareaofindent.Theproceduresrequirethreemeasuredquantities,i.e.,thepeakloadandcorrespondingdisplacementsaswellasthedepthofresidualindentation,butdonotrequirecomplicatedcurvefittingprocessandregressionanalysiswhichthemselvesinvolvethespecimenmaterial.Nano-indentationtestswereconductedusingaBerkovichindenteronfivematerialsspanningawiderangeofhardnessandplasticity.Experimentalresultsrevealedtwoimportantfeatures:(a)thereducedmoduluspredictedbythenewEr–Hrelationshipisthesameasthatobtainedbytheconventionalmethod;(b)theelasticmodulusandhardnessvaluesdeterminedbythesimplesetofproceduresarecomparabletothoseobtainedbyusingtheconventionalmethod.
2004ActaMaterialiaInc.PublishedbyElsevierLtd.Allrightsreserved.Keywords:Nano-indentation;Er–Hrelationship;Recoveryresistance;Energydissipation1.IntroductionElasticmodulusEandhardnessHaretwoessentialparametersofstructuralmaterials,andtherelationshipbetweenthemisofkeeninteresttomaterialscientists.Fromstatisticaltrend,elasticmodulusisusuallyconsid-eredtobeanincreasingfunctionofhardness[1],butthisruleneitherhasanalyticalsupportnorisgenerallyobeyed.Forexample,layeredternaryceramicsthathavelowhardnessandhighmodulusareexceptionstothisrule.AnotherE–Hrelation,oftenusedinevaluatingtheelasticmodulusofcoatings,involvesthecompressiveyieldstressryandthehalfangleofindenterh(68forVickers
indenter)[2,3]H¼ð2=3Þ½1þlnðEcosh=3ryÞ
ry:ð1ÞSinceitstheoreticalbasisisnotclear,therangeofappli-cabilityofEq.(1)mightbelimited.Asimpleexamina-tionwasmadeinthecaseofTi3SiC2whosehardness(4GPa),compressiveyieldstrength(1GPa)andmodu-lus(310GPa)wereknown[4].ByEq.(1),amodulusva-lueashighas1188GPawasobtained,whichwasmuchhigherthanthemeasuredvalueof310GPa.Theunsat-isfactorypredictionofelasticmodulusfromhardnessleadstotwoessentialquestions:(a)Doesthereexistatheoreticalandgenerallyvalidrelationshipbetweenelasticmodulusandhardness?(b)IfthereisaninherentE–Hrelationshipforsolidmaterials,whatisit?1359-6454/$30.00
2004ActaMaterialiaInc.PublishedbyElsevierLtd.Allrightsreserved.doi:10.1016/j.actamat.2004.08.002*Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](Y.W.Bao).ActaMaterialia52(2004)5397–5404[5–7].Furthermore,thedepth-sensinginden-tationtechniquedocumentedbyOliverandPharr[5]hasprovidedaclueforestablishingthetheoreticalrela-tionshipbetweenelasticmodulusandhardness.InthistechniquetheelasticmodulusofthespecimenisrelatedtothemeasuredreducedmodulusErby1Er¼1m2Eþ1m2iEi;ð2ÞwhereEandmaretheelasticmodulusandPoisson
sratioofthespecimen,andEiandmiarethoseoftheindenter.Thus,measuringthereducedmodulusbecomesthekeyfordeterminingtheelasticmodulusofthetestspecimen.SinceEiandmiareknown,theratiooftheelasticmodulustothereducedmodulus,E/Erisafunctionofboththere-ducedmodulusandPoisson
sratio,asshowninFig.1.ThecalculationimpliesthateithertheincreaseoftheEr-valueorthedecreaseofthePoisson
sratioofthetestsamplewouldleadtoanincreaseinthevalueofE/Er.Itisinterestingthat,forthereducedmodulus,thereisathresholdvalue(Er=E)dependingonthevalueofthePois-son
sratio,belowwhichthereducedmodulusishigherthantheelasticmodulusofthetestsample,ratherthanreduced.Thisworkaimsat:(a)derivingandexaminingtheinherentrelationshipbetweenErandHbasedonthedepth-sensingindentationtechnique;(b)bringingforthanewmaterialparameter,therecoveryresistance,andexploringitsphysicalsignificance;(c)proposingasim-ple,material-independentmethodtodeterminetheareaofindent.Thetheoryissupportedbynano-indentationdatameasuredonfivedifferentmaterials.2.TheoreticalrelationshipbetweenErandHTheanalysisbeginswiththeschematicsofacycleofindentationanditsdeformation[5]asshowninFig.2.Duringtheloadingandunloadingonansmooth-surfaceelastichalfspace,theloadandindenterdisplacementobeypower-lawrelations[6,8]givenas:P¼a1hm1forloading;ð3aÞP¼aðhhfÞmforunloading;ð3bÞwherePistheindentationload,htheindenterdisplace-ment,hfthedepthofresidualimpressionafterunload-ing,anda1,a,m1,mtheempiricalfittingparameters.AsshowninFig.2(a),atthepeakloadPm,themaxi-mumdisplacementhmequalsthesumofthecontactdepth,hc,andtheelasticsurfacedisplacementattheperi