Plasticity-theory-for-porous-metals

Int.J.MechanicalScience,1976,Vol.18,pp.285-291.PergamonPress.PrintedinGreatBritainPLASTICITYTHEORYFORPOROUSMETALSS.SHIMAandM.OYANEDepartmentofMechanicalEngineering,FacultyofEngineering,KyotoUniversity,Sakyo-ku,Kyoto,Japan(Received1November1975)Summary--Aplasticitytheoryforporousmetalsisproposed.Fromthestress-straincurvesforsinteredcopperwithvariousapparentdensities,thestress-straincurvesforpore-freecopperiscalculatedbyutilizingthebasicequations.Theequationsareappliedtofrictionlessclosed-diecompressionandthestressinthedirectionofcompressionisevaluatedinrelationtotherelativedensityandiscomparedwithexperimentalresults.1.INTRODUCTIONINRECENTyearspowdermetallurgytechniqueshavebeencombinedwithconventionaldeforma-tionprocessesandhavebeensuccessfulinfabricatingvariousengineeringproducts.1-5Inthisnewcombinedtechnique,sinteredporousmetalsareemployedasstartingmaterialsinordinaryworkingprocesses,suchasforgingandextrusion.Uptonow,thishasbeendevelopedandemployedwithoutanytheoreticalbackground.Theoreticalapproachesare,however,ofimportanceforanalys-ingandpredicting,forexample,theworkingloadrequiredtocauseplasticdeformationandthedensityoftheproducts.Although,foranalysingproblemsinordinarymetalworkingprocesses,varioustheoriesandmethodsofanalysishavebeendeveloped,theyarenotcapableofbeingappliedtothedeformationofporousmaterials.Inconventionalplasticitytheory,onwhichthesetheoriesandmethodsarebased,volumeconstancyisassumedforthematerialundergoingdeformation,andthisassumptionap-pliesineffecttopore-freemetalsverywell.Inthedeformationofporousmetals,however,thevol-umedoesnotremainconstant.Itwasnotuntilveryrecentlythataplasticitytheoryforporousmetalswasproposedforthefirsttime.Kuhnetal.6andGreen7haveindependentlyproposedyieldcriteriaandstress-strainrelationsforthesematerials.Bothofthetheoriessuggestthattheyieldcriterionisafunctionofthefirstinvariantofstress,J1,andthesecondinvariantofdeviatoricstress,J~,thatis,F={c~J,2+~j~}m(I)andthatthestress-straintr-e,relationsareoftheform,de,=dA(tri-q~rs)(i=1,2,3)wherea,/3and~barefunctionsoftherelativedensity,dAisanon-negativeconstantand~r~thehydrostaticstress.SinceinbothcasesdAhasnotbeendeterminedtheseapproacheshavenotbeenutilizedsatisfactorilyfortheanalysisofpracticaldeformationprocesses;theyhavebeenusedonlyforsimplestressstatessuchasuni-axialcompres-sionandtensionorplane-straincompression.Thispaperisconcernedwiththedevelopmentofbasicequationsofplasticitytheoryforporousmaterialswhicharesimilartothosedescribedaboveandsomeexamplesoftheapplicationofthistheoryarediscussed.2.BASICEQUATIONS2.1YieldcriterionForayieldcriterionforsinteredmetalswemaybeginwithequation(1),whichmayberewrittenasF=[{(o.1-02)2+(02-03)2+(0-3-~r,)2}/2+(0-~//)211/z(2)wherefrepresentsthedegreeofinfluenceofthehydrostaticstresscomponent,0-,,ontheonsetofyieldingofporousbodiesandmaybeafunctionoftherelativedensity.Fintheaboveequationmayberelatedtotheyieldstressofthematrixmetal,00.Thusequation(2)mayberewrittenasf'0o=[{(o1-o'2)2+(o'2-o3)2+(o'3-o',)2}/2+(0-,/f)2],,2(3)wheref'representstheratiooftheapparentstressappliedtotheporousbodiesandtheeffectivestressappliedtothematrix,andmaybeagainafunctionoftherelativedensity.Thefunctions/andf'canbedetermined285286S.SHIMAandM.OYANEtheoreticallyfromasimplemodelflbuttheydonotnecessarilyprovidegoodagreementbetweenequation(3)andexperimentalresults.Therefore,theywillbedeter-minedexperimentallyinalatersection.Theyieldsurfaceofaporousbodywitharelativedensitypexpressedbyequation(3)isanellipsoidwhosemajoraxiscoincideswiththe~axis(Fig.1).Ifo-0isreplacedbyo-eq,thatistheequivalentstressappliedtothematrix,thenequation(3)maybeapplicabletoporousmaterialswithwork-hardeningmatrices,(3)'2.2Stress-strainrelationsFromequation(3)wemaywrite1g=~[{(o'1-o'2)2+(o'~-o-3)2+(o%-o',)2}/2+(o-~[f)2],/2Oo.Ifgisassumedtobeaplasticpotential,thentheprincipalstrainincrementsdE~,de2andde~arederivedbypartiallydifferentiatinggwithrespecttotr,,~andtry:Og_de,=dA'~-dA{o-~-(1-2/9f2)o-m}de2=dA'0g_g=dA{o-2-(1-2/9/2)o'm}3or2Og_de3=dA'~-dA{o%-(1-2/9f:)o-~}(4)andfurtherdeo=de,+de2+de3=-do/P=dA(2/3F)o-~(5)wheredAisaproportionalityconstantwhichwillbedeterminedlater.Now,ifanequivalentstrainincrementreferredtothematrixisdenotedbyde.~,thentheplasticworkdoneperunitvolumeoftheporousbody,dW,isexpressedbydW=o-,de~+or2dE2+O-3de3=pO'e,~de,,.(6)Notethatsinceaunitvolumeofaporousbodywithat5•O'm/-~o2,Mcr3FIG.1.Schematicillustrationofyieldsurfacesforporousmaterials.relativedensitypconsistsofthematrixmaterialofvolumep,thendWisnotequaltoo-e,de,,.Substitutingequation(4)intoequation(6)andrearrang-ing,wehavedA3pde,.__.(7)2(f')2o-e.Eliminatingo-mfromequations(4)and(5)andrearrang-ing,~,~r2ando-~canbeexpressedintermsofdE~,de2,de3andde,,.Substitutingtheseinequation(6),coupledwithequation(3)',wehaved~¢--~=f'[(2/9){(de,-dez)~+(d~2-de3)~+(de3-de,)2}p+(fdeo)2],,L(8)2.3Determinationoffandf'Todeterminetheformsoffunctions/and['wemaysubstituteintoequations(4)and(5)o'2=tr3=0ando%=~,/3,thesebeingtheconditionsforuni-axialcompression.ThuswehaveThusorde,-de:92dE,2[i=(de,-d,2/2\de~-~/(9