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1、J.Mech.Phys.So/id.i,Vol.46,No.3,pp.41I-425.1998Cc‘1998ElsevierScienceLtd.AllrightsreservedPergamonPrintedinGreatBritaur0022-5096,YS$19.00+0.00PII:SOO22-5096(97)00086-OINDENTATIONSIZEEFFECTSINCRYSTALLINEMATERIALS:ALAWFORSTRAINGRADIENTPLASTICITYWILLIAMD.NIX*$andHUAJIANGAO?*DepartmentofM
2、aterialsScienceandEngineering;fMechanicsandComputationDivision,DepartmentofMechanicalEngineering,StanfordUniversity,Stanford,CA94305-2205,U.S.A.(Received21August1997;inref?ised,fortn23Ocroh~r1997)ABSTRACTWeshowthattheindentationsizeeffectforcrystallinematerialscanbeaccuratelymodeledus
3、ingtheconceptofgeometricallynecessarydislocations.Themodelleadstothefollowingcharacteristicformforthedepthdependenceofthehardness:whereHisthehardnessforagivendepthofindentation,11,H,]isthehardnessinthelimitofinfinitedepthandh*isacharacteristiclengththatdependsontheshapeoftheindenter,t
4、heshearmodulusandH,).Indentationexperimentsonannealed(111)coppersinglecrystalsandcoldworkedpolycrystallinecoppershowthatthisrelationiswell-obeyed.Wealsoshowthatthisrelationdescribestheindentationsizeeffectobservedforsinglecrystalsofsilver.Weusethismodeltoderivethefollowinglawforstrain
5、gradientplasticity:-u20=I+&,cowhere0istheeffectiveflowstressinthepresenceofagradient,0istheflowstressintheabsenceofagradient,xistheeffectivestraingradientand1isacharacteristicmateriallengthscale,whichis,inturn.relatedtotheflowstressofthematerialintheabsenceofastraingradient.Formateria
6、lscharacterizedbythepowerlaw0,)=l&i:I,theabovelawcanberecastinaformwithastrain-independentmateriallengthscale1.ThislawresemblesthephenomenologicallawdevelopedbyFleckandHutchinson.withtheirphenom-enologicallengthscaleinterpretedintermsofmeasurablematerialparameters.(1998ElsevierScience
7、Ltd.Alirightsreserved$Authortowhomcorrespondenceshouldbeaddressed.Fax:0016507254034.E-mail:nixfccaoc.stanford.edu.411412W.D.NIXandH.GAO1.INTRODUCTIONAsdiscussedbyFleckandHutchinson(1993,1997),conventionaltheoriesofplasticitydonotincludemateriallengthscales.Forsuchtheoriestheflowstress
8、atany
