钛基复合材料好用

JournalofProcessingTechnology113(2001)153±159

Finitemodelingofthedeformationbehaviorof

semi-solidmaterials

Jong-HoonYoona,Yong-TaekImb,*,Nak-SooKimc

RocketStructures/MaterialsDepartment,SpaceTechnologyR&DDivision,KoreaAerospaceResearchInstitute,

52Oundong,Yusonggu,Taejon305-333,SouthKorea

b

ComputerAidedMaterialsProcessingLaboratory,DepartmentofMechanicalEngineering,KoreaAdvancedInstituteofScienceandTechnology,

ME3227,373-1Gusongdong,Yusonggu,Taejon305-701,SouthKorea

c

DepartmentofMechanicalEngineering,SogangUniversity,Shinsoodong,Mapogu,Seoul121-742,SouthKorea

a

Abstract

Inthecurrentstudy,thedeformationbehaviorofsemi-solidmaterialswasmodeledusingthevonMisesyieldcriterioninwhichthesemi-solidmaterialwastreatedasasinglephasematerialwiththeincompressibilityconditioninaglobalsense.The¯owstressofthematerialwasmodeledasafunctionofstraininconsiderationofthesolidfractionandthebreakageratioofdendriticstructure.AnalgorithmbasedonmixturetheoryandD'Arcy'slawwasdevelopedtoupdatethesolidfraction,thedistributionofwhichvarieswithinthematerialduetorelativevelocitybetweenthesolidandliquidphasesduringdeformation.Theparametersinvolvedwiththeproposedmodelweredeterminedthroughaparametricstudyinwhichnumerousniteelementanalysisresultswerecomparedwiththedatafromexistingisothermalupsettingexperimentsforsemi-solidSn±15%Pballoy.Comparisonwithexperimentalresultsshowedthatthecurrentapproachisimprovedcomparedtopreviouscompressibleapproaches.Thegeneralityofthecurrentapproachwasexaminedthroughrigid±thermoviscoplasticniteelementanalysesofthesemi-solidforgingofaball-jointcaseundervariouspreheatingtemperaturesinconsiderationofthereleaseoflatentheat.Thesimulationresultsagreedwellwiththetrendoftheexperimentalndingsbutshowedsomequantitativeerrors.#2001ElsevierScienceB.V.Allrightsreserved.

Keywords:Semi-solidmaterial;Flowstress;Breakageratio;Solidfractionupdatingalgorithm;Solidication

1.Introduction

Theinternalstructureofamaterialinthesemi-solidstateiscomposedofsolidgrainssurroundedbyaliquidphase.Thus,the¯uidityoftheworkpieceisenhancedandnear-netshapemanufacturingofcomplexshapedpartsispossible.However,theexactpredictionofthematerialbehaviorofsemi-solidsisextremelydifcultduetofactorssuchasthemixedcharacteristicofsoliddeformationandliquid¯owandthechangeofmicrostructureduringdeformation.nthelastfewdecadesmanyeffortshavebeenmadetocharacterizethedeformationbehaviorofsemi-solidmateri-als.Modelingforthedeformationbehaviorofsemi-solidmaterialscanbeclassiedaccordingtotherangeofsolidfractionofthematerial.Forlowsolidfractionmaterials,LaxmananandFlemings[1]andHiraietal.[2]modeledthecharacteristicsofsemi-solidslurrybyobtainingthe

Correspondingauthor.

E-mailaddress:[email protected](Y.-T.Im).

*

experimentalrelationshipbetweenshearrateandapparentviscosity.Forhighsolidfractionmaterials,Lalli[3]andCharreyronandFlemings[4]characterizedthedeformationbehaviorofsemi-solidmaterialwithD'Arcy'slawforliquidphase¯owandcompressibleplasticdeformationtheoryforsolidgrains.Inparticular,Kumaretal.[5]representedthenon-steadystatedeformationbehaviorforlowsolidfractionsemi-solidmaterialsbyintroducinganagglomera-tionvariableintotheapparentviscosityequation.Martinetal.[6]proposedanewyieldcriterionthatallowedcompressibilityofthesolidphaseandobtainedexperimen-tallytherequiredvariablesbyrelatingacreepmodelwiththeyieldcriterion.

Gebelinetal.[7],andZavaliangosandLawley[8]analyzedtheupsettingofsemi-solidmaterialsbytheniteelementmethodbasedontheabovemodelsandpredictedtheinternaldistributionofsolidfractioninthedeformedspecimen.However,theyfailedtoaccuratelypredictthechangeofstressorloadanddeformationmodeintheanalyses.Thiswasduetothefactthatthemodelsusedwereveriedforonlyaspeciedrangeofsolidfraction,

0924-0136/01/$±seefrontmatter#2001ElsevierScienceB.V.Allrightsreserved.PII:S0924-0136(01)00677-X

154J.-H.Yoonetal./JournalofMaterialsProcessingTechnology113(2001)153±159

whileinfactthesolidfractioncancontinuouslyvaryfrom0to1duringdeformation.

Thus,foranalysisofthesemi-solidforgingprocesswithreliableaccuracy,itisnecessarytodevelopamodelthatisapplicabletothefullrangeofsolidfractionandthatcon-siderstheevolutionofthemicrostructure.Withthispurposeinmind,atheoreticalmodelbasedontheincompressibilityconditionisproposedforsemi-solidmaterialsinthecurrentstudy.Thevalidityandreliabilityoftheproposedmodelwereexaminedthroughcomparisonbetweenniteelementanalysesandexperimentalresultsavailableintheliterature.2.Characterizationmodelofthedeformationbehaviorofsemi-solidmaterials

2.1.Yieldcriterionandconstitutiveequation

nthispaper,thetheoryrelatedwiththemodelingofthedeformationbehaviorofsemi-solidmaterialswillbemen-tioned.Sincetheniteelementformulationprocedureiswellestablishedintheliterature[9],itwillbeomittedhereduetolimitedspace.

Theglobalstresseldinthesemi-solidmaterialorsemi-solidmixture,sij,canberepresentedasacombinationofthe

l

solidtermssijandliquidtermsijasfollows:

lsij ssij sij;

respectively 2s

_;sHij Feijr 2_ _ije _ij:Fee

2.2.stress

Incurrentstudy,anexpressionfor¯owstressthatconsiderstheevolutionofmicrostructureasproposedbyYoonetal.[10]wasapplied.Theparametersinvolvedwiththis¯owstressweredeterminedbyaparametricstudy.ThestressthebreakageofstructureandtheisgivenTable1,whereSisthebreakageratioofstructure,0theinitialratio,crstrain, estthefcrthecriticalsolidfraction,andK0thestrengthcoefcient.Here,thebreakageratiowasassumedtobealinearfunctionofstrainasfollows:S C1 1ÀS0

eÀ est

;

stcr ecr e est;

(5)(3)(4)

whereC1isacorrectioncoefcienttobedeterminedthrough

parametricstudy.

2.3.SolidfractionupdatingalgorithmbasedonmixturetheoryandD'Arcy'slaw

fanarbitrarybodyattimet tÃisinamixturestate,wherethedensitiesandvolumefractionsforthesolidandliquidphasesarersandfsandrlandfl,respectively,thedensityofthismixturerandthemass-weightedaverageofphasevelocitiesvcanberepresentedbyEqs.(6)and(7),respectively

r rs rl rsfs rlfl;1

v rsvs rlvl :

(6)(7)

(1)

wheresuperscriptssandlrepresentthesolidandliquidphases,respectively.

Ifthesemi-solidmaterialisconsideredfromtheglobalpointofview,itispossibletotreatitasasinglephase.Therefore,theyieldcriterioncanberepresentedbythesecondinvariantofthedeviatoricstressofsij,withtheassumptionthatthesemi-solidmaterialisisotropicandhomogeneous.Amajorfactorthatcanaffectthesizeofthisyieldsurfaceisthesolidfraction.Thus,ifaparameterFrepresentingtheeffectofthesolidfractionontheyieldcriterionisintroduced,theyieldcriterionforsemi-solidmaterialscanbeexpressedasfollows:

r r 3J23HH

sss:(2)

ijijFromEq.(2)andthenormalitycondition,the¯owruleand

effectivestrainratecanbederivedasEqs.(3)and(4),

Here,thesuperscriptshaveaglobalmeaningwhilethesubscriptshavealocalmeaning.Also,therelativevelocityofliquidphasetosolidphasevrcanbedenedasfollows:vr vlÀvs:

(8)

Table1

Expressionfor¯owstressconsideringthebreakageofdendriticstructureandthesolidfraction[10]a fFlowstress,s e

a

n

e

_m f Ksexp b e

cr

eÀ estm

_ f Kexpbse

crst

Coefficient,Kfs>fcrfs fcr

K K0exp C2 fsÀfcr K K0

1À b 1Àfs 2=31À b 1Àfcr b C2 1ÀS0 ,b 1:5,C2 4:2.

J.-H.Yoonetal./JournalofMaterialsProcessingTechnology113(2001)153±159155

theUndercontinuityequationtheassumptionforofthethesemi-solidincompressibilitycondition,mixturebecomes@fs

rÁ fsvs 0 solidphase ;(9)@fl

rÁ flvl 0 liquidphase ;

(10)

or

rÁv 0 semi-solidmixture :

(11)

Thus,fromEqs.thechange(7)ofsolidfractionwithtimecanbederived@fs to(9)asfollows:

ÀrÁfsvÀfsrlflvr

:(12)Whenthepenaltyconstantnitethesemi-solidelementanalysis,materialEq.istreated(11)isasaenforcedsinglephasewithinfractioncandescribedbeobtainedduetotheandthedistributionandchangeofsolidainthefollowingbyEq.relative(12)velocitybetweenwithanditsD'Arcythetwophasesboundaryconditions:'slaw,whichis1.D'Arcy'slaw:

flvr Àk

rp;

(13)

2.Boundaryconditions:p p0

or

@p

0;(14)

wherenormalkvector,isthepermeability,andmtheviscosity,ntheoutwardsolidSubstitutingEq.(13)ptheintopressure.

Eq.(12)anddiscretizing,thebeobtainedfractionasupdatingequationfortimeincrementDtcan follows:

fts Dt ftsÀrÁfts

vt f

srlk rp tDt:(15)InexplicitlysolvingEq.(15),thepressuregradientrpcanbe thefollowingmomentumcalculatedbyr@v

equation: vÁrv Àrp rH rg;(16)wheretionalrHisthedeviatoricstresstensor,quasi-staticacceleration.Since,themetalformingandgprocessthegravita-isanalysis.

process,thesetwotermscanbeneglectedinthea2.4.Evaluationofsolidication:releaseoflatentheattheThesolidfractionofsemi-solidtionrelativevelocitybetweenthetwomaterialsphaseschangesandsolidica-duetowasthesolidicationconsideredduringdeformation.intheTheeffectoftherelativevelocityeffectsolidcanfractionbeevaluatedupdatingbysolvingalgorithm,andthe

Fig.behavior1.Flowofsemi-solidchartformaterials.

theniteelementanalysisofthedeformationenergyequation[11].withanequivalentspecicheatmethoditsboundaryconditions.Theenergyconservationequationisgivenbelowwith1.Energyconservationequation:r krT ÀrCEP@T @r

0(17) ;CEP

CpÀL@fS

;

(18)

2.Boundaryconditions: T TÃ

orqn Àkn@T

;

(19)

wheretheksionlatentisthethermalconductivity,Cpthespecicheat,Lfrictionalofplasticheat,anddeformationrtheheatenergyintogenerationdueheattoenergyandtheconver-atThementionedeachprocedureheat.

nodeequationsinthefornitecalculatingisshownelementtheinanalysischangeofsolidfractionFig.1.

usingtheabove-3.upsetting

FiniteelementanalysisofisothermalsimplewereThevaluesofparametersinvolvedinelementdeterminedlysisconditionsanalysesofthroughaparametricthestudyusingproposedmodelnitewereisothermalobtainedsimplefromupsetting.the

experimental

Theana-

156J.-H.Yoonetal./JournalofMaterialsProcessingTechnology113(2001)153±159

Table2

Materialpropertiesusedintheniteelementanalysisofsimpleupsetting[12]

PropertyValueStrainratesensitivity,mStrengthcoefficient,K0Initialsolidfraction,fS0

_Strainrate, e

SpecimensizeFrictionfactor,mf

0.230.170.63

1:33Â10À2sÀ1

f12:7ÂH6:34mm0.3

ÂryandFlemings[12].Todeter-conditionsusedbySue

minetheparametervalues,numeroussimulationswerecarriedoutcomparedwiththeexistingexperimentalresultsintermsofloadandstroke.Table2showsthesimulationconditionsandmaterialpropertiesofsemi-solidSn±15%Pballoy.ThenallydeterminedparametervaluesarelistedinTable3.

Fig.2comparesthesimulationresultsandexistingexperimentalresultsintermsofloadandstrokewiththe

nallydeterminedparametervalues.Ascanbeseen,thesimulationresultsagreewellwiththeexperimentalresultsforbothdendriticandglobularstructures.

Fig.3showsthesolidfractionateachnodeonthesymmetriclineintheradialdirection.Thiscalculatedresultisinbetteragreementwiththeexperimentallymeasuredvaluesreproducedfrom[13]comparedtotheupper-boundanalysisofCharreyronandFlemings[4].Inparticular,theupper-boundanalysisshowsasmallervaluethanthemea-suredvalueatthemaximumradialcoordinate.Thisresultsfromtheuseofcompressibleplasticitytheory.Thus,thecurrentlyproposedmodelbasedonincompressibilitycanbeseentobemorevalidthanthecompressibleapproach.Althoughtheparametersinvolvedinthecurrentlypro-posedmodelweredeterminedthroughaparametricstudyinwhichtheloadandstrokewerecomparedbetweensimula-tionandexperiment,theseresultsshowthattheproposedmodelrepresenttheoveralldeformationbehaviorwell.Thus,itisconstruedthatthecurrentlydevelopedmodelcanbeusedsuccessfullyforthecharacterizationofthedeformationbehaviorofsemi-solidmaterials.

Table3

DeterminedparametervaluesfortheproposedmodelsParametercr estC1nmafcrF

Initialmicrostructurestate(S0)Globular S0 1 0.92.20.4

0.23(constant)À0.10.5fs0:1

Dendritic S0 0

0.92.2

0.4

eÀ est

m m0 a1À 1ÀS0 ;

crst

À0.10.4fs0:1

m0 0:23

Fig.2.Comparisonofload±strokecurvesbetweenniteelementanalysesandexistingexperimentalresults[12]forsimpleupsetting.

J.-H.Yoonetal./JournalofMaterialsProcessingTechnology113(2001)153±159157

Fig.3.(a)Showing:normalizedliquidfractionvs.radialdistancefromthecenterforadendriticSn±15%Pbspecimencompressedtoe 0:52(mf 0:3,_ 1:33Â10À2);(b)aschematicdiagramofmeasuringpointlocations. e

4.Non-isothermalniteelementanalysisofsemi-solidforging

4.1.Simulationconditions

ThecurrentlyproposedmodelwasimplementedintoCAMPform2D[14],atwo-dimensionalniteelementsimu-lationtoolformetalformingprocesses.Thesemi-solidforgingofaball-jointcasewassimulatedundervariousdiepreheatingtemperaturestotesttheapplicabilityoftheproposedmodel.Theforgingconditionsandmaterialprop-ertiesofAl2024alloyusedinsimulationsareshowninTables4and5,respectively.Thevaluesofthestrengthcoefcientandstrainratesensitivityofsemi-solidAl2024alloyforT 550 Cand625 T 650 Cwerefoundfrom[10,16],respectively.Fortherangeoftemperature

Table4

Forgingconditionsforthesemi-solidforgingofaball-jointcaseBilletsize

Initialsolidfraction,fS0Initialtemperature,T0Dievelocity,vp

Dieandpunchpreheatingtemperature,Td

f35ÂH28mm0.226408C1.0mm/s

500,550and6008C

550 T 625 C betweenthesetwotemperatureranges,thesevalueswereinterpolatedasshowninEq.(21)

r TÀ428:5

m 0:167;

(20)K0 61:8exp À5:16Â10À3T ;T 550 C;m À1:282 0:00282T;K0 12:362À0:017362T;

550 T 625 C;

(21)

m À343:95 1:0719TÀ8:33Â10À4T2;K0 À287:75 0:98363TÀ8:33Â10À4T2;

625 T 650 C:

(22)

Table5

Thermalpropertiesoftheworkpiece(Al2024)andthedies(AISIH13)[15]

Workpiece

Conductivity,k(N/sK)

DensityÂspecificheat;rÂC(N/mm2K)Latentheat,L(Nmm/kg)

Heattransfercoefficient,h(N/smmK)RadiationcoefficientÂBoltzmanconstant;eÂs(N/smmK4)

100.02.943394Â1060.042

6:804Â10À12

Die30.54.882

Notrequired0.42

158J.-H.Yoonetal./JournalofMaterialsProcessingTechnology113(2001)153±159

Icientngeneral,andofasemi-solidthestrainmixtureratesensitivityandmayvarywiththestrengthcoef-dition.solidtemperatureHowever,fractiontheywerefortheglobularthetemperaturerepresentedandnon-isothermalonlyincon-functionoftemperature,becausetheassolidrepresentedfractionwasintheassumedtermsfollowingto[10]:beofafs 755:4033À2:733776T 3:301456Â10À3T2

À1:3287712Â10À6T3;

Ts fs TL;

(23)

Fig.simulations4.DistributionsT550;(c)forTdieofpreheatingsolidfractiontemperaturesandtemperatureof:(a)obtainedTfromd 600;(b)d d 500

C.

Fig.temperatures5.Experimentallyobtainedof:(a)TTball-jointcasesfordiepreheatingd 600;(b)d 500

C.

whereliquidusthetemperaturesolidustemperatureTTsis823.15KandtheLis923.15K.4.2.Simulationresultsanddiscussion

nalThedistributionsoftemperaturesdiestroke(25mm)solidaccordingfractionandtothetemperaturediepreheatingatthepreheatingthetemperaturesareshowninleadFig.to4.higherAscanbeseen,lowerdieTrapidheattransferfromtheworkpiecesolidtofractionsthedies.dueWhentoaboutdwasin0.93500which8Cthemeanspredictedthatlowestsolidfractionvaluewascrackingafullysolidwithwillbehigh.stateandthattheworkpiecethepossibilityofisapproximatelysurfaceball-jointtheexperimentalresultsThispredictionshownincanFig.be5,seenwhichtoagreediecasesobtainedfromexperimentsusingdifferentshowstionFig.preheatingin6termsshowstemperatures.

oftheloadcomparisonandstroke.ofexperimentForTandsimula-d 600 calculatedTheofcauseforloadthiswascanlowerthantheactualmeasuredCvalue.thewheremicrostructuresshownbeexplainedinFig.7.Positionthroughthe(3)photographsistheregionandbecauseascontactcanbeseenbetweenthesolidthegrainsdieandtheworkpieceinitiates,forotherofpositions.therelativelylowSuchcoarseningcoolinginthisrateregionactedcomparedwerecoarsetodisturb

tothatthe

Fig.experiment.

6.Comparisonofload±strokerelationshipsbetweensimulationand

J.-H.Yoonetal./JournalofMaterialsProcessingTechnology113(2001)153±159159

Fig.(3);for7.Microstructuresadiepreheatingat:temperature(a)positionof(1);T (b)600position C.

(2);(c)positiondrelativeexternalmotiontheloadwasbetweenrequiredsolidinthegrainsactualandforgingasaresultmoregrainsproposedsizeintermsmodelofbreakageonlyconsidersratioandthedoesshapenoteffectprocess.considerofsolidAsthiseffect,thepredictedloadwaslowerinthesimulationtheforwasOncase.

theotherhandforTd struedhigherpolatedthatthethanusethe500 C,theloadcalculationofexperimentalinexactmeasurement.Itiscon-aretemperaturethestrainmajorratecausesensitivityandmaterialofthiserror.strengthpropertiesForcoefcientandTvaluesinter-d beisintherangedistribution590±5958CinintheFig.workpiececan500 beseenC,thetocoefcientintherange,andstrainwhereratethesensitivity(Eq.interpolated4.Thisvaluestemperaturerange(21))areofstrengthapplied.

5.Conclusionsandfuturework

incompressibilityconditionInthepresentstudy,anumericalrangethatisapplicablemodelbasedtotheonfulltheofmationtheofmicrostructuresolidfractionsimplebehaviorofsemi-solidwasindevelopedconsiderationoftheevolutionmaterials.todescribeAlso,isothermalthedefor-simulationsupsettingresultsresultstoexaminewereandcarriednon-isothermalthevalidityofoutandcomparedball-jointcaseforgingthedevelopedtoexperimentalmodel.andshowedthatitcanbeusedeffectivelyintheanalyThesisapproach,Fordesignwiderofapplicationthesemi-solidandbetterforgingprocess.

methodfurtherresearchfordeterminingaccuracyofanexperimentalthecurrentpropertiestoisproperlycharacterizeneeded.semi-solidalloymaterialAcknowledgements

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