Shiomi1999InternationalJournalofMachineToolsandManufacture.docx
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Shiomi1999InternationalJournalofMachineToolsandManufacture.docx
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Shiomi1999InternationalJournalofMachineToolsandManufacture
InternationalJournalofMachineTools&Manufacture39(1999)237–252
Finiteelementanalysisofmeltingandsolidifyingprocesses
inlaserrapidprototypingofmetallicpowders
M.Shiomi*,A.Yoshidome,F.Abe,K.Osakada
DepartmentofMechanicalEngineering,FacultyofEngineeringScience,OsakaUniversity,Machikaneyama,
Toyonaka,Osaka560-8531,Japan
Received2February1998
Abstract
Toclarifytheformingmechanisminlaserrapidprototyping,themeltingandsolidifyingprocessesof
metallicpowdersaresimulatedbythefiniteelementmethod,andthecalculatedamountofthesolidified
massiscomparedwiththeexperimentalone.Inthesimulation,themeltedpartofthepowdersisassumed
tochangeintoasphereduetothesurfacetensionandanewfiniteelementmeshisgeneratedinthe
subsequentanalysis.Latentheatandshrinkageduetosolidificationaretakenintoaccount.Thethermal
conductivityofthepowderismeasuredforvariousdensitiesandusedinthecalculation.Temperature
distributionwithinthepowdersduringheatingandcoolingiscalculatedtoobtaintheamountofthemelted
andsolidifiedpartofthepowders.ExperimentsaremadewithCupowdersusingapulsedNd:
YAGlaser.
Itisfoundthattheamountofthesolidifiedpartafterapulseoflaserirradiationisaffectedbythepeak
powerofthelaserratherthanthedurationofirradiation.Thereisanappropriatepeakpowerofthelaser
inrapidprototypingofmetallicpowders.Thecalculatedweightsofthesolidifiedpowdersbyseveralpulses
ofthelaserbeamagreewellwiththeexperimentalones.♥1998ElsevierScienceLtd.Allrightsreserved.
Keywords:
Laserrapidprototyping;DirectSLS;Metallicpowder;FEsimulation;YAGlaser
1.Introduction
Rapidprototypingisatechnologywhichcandirectlyfabricatethree-dimensionalsolidmodels
fromCADdata.Ithasthecapabilitytoreducethecycletimeofproductdevelopmentbecausea
designedmodelinaCADsystemcanimmediatelyberealizedasaprototype.Thefirstcommercial
*Correspondingauthor.
0890-6955/98/$—seefrontmatter♥1998ElsevierScienceLtd.Allrightsreserved.
PII:
S0890-6955(98)00036-4
238
M.Shiomietal./InternationalJournalofMachineTools&Manufacture39(1999)237–252
rapidprototypingmachineusedstereolithographyandthismethodhasalreadybeenwidelyused
[1].Inthestereolithographysystem,physicalprototypesareproducedbybuildingupthinlayers
ofsolidifiedphotosensitiveliquidresinwithultravioletlaserirradiation.Thisadditivemannercan
makeacomplicatedprototypeincomparisonwithasubtractivemanner.Themajorityofcurrent
rapidprototypingmachinesusepolymericmaterialssuchasresinsandwax.Althoughprototypes
madeofnonmetallicmaterialscanbeusedforassessmentoftheshapeanddesignofthemodels,
theycannotbeappliedtotestsofphysicalpropertiesbecausethematerialoftheprototypesis
quitedifferentfromthefinalproducts.
Selectivelasersintering(SLS)thatusespowderssuchaspolymers,metalsandceramicsina
layer-by-layeradditiveprocesshasbeenproposedforrapidprototyping[2].Inthemanufacturing
ofmetallicpartsbySLS,metallicpowdersareusedeitherwithapolymericbinderorwithouta
binder.IntheindirectprocessofSLS,metallicpowdersarecoatedwithbindersandonlythe
bindersaremeltedbythelaserbeamtoconnectthepowderparticlesandbuildupmetallicparts.
Theprototypesmadebytheindirectprocessareporousandstillincludethebinders,andthey
mustbedebinderedandsinteredtobemadehard,butthisprocesscauseslossoftimeanddimen-
sionalaccuracyduetoshrinkage.Inthedirectprocess,ontheotherhand,metallicpowders
withoutabindermaybeheatedbythelaserbeam,meltedorsinteredselectively,andthus
debinderingandsinteringprocessesarenotnecessary.ThedirectprocessofSLSoffersthepossi-
bilityofmakingmoreaccurateprototypesthantheindirectprocess.
Inthedirectprocess,variouskindsofmetals,e.g.pureormultiplephasemetals,areused.Abe
andOsakada[3,4]haveusedpuremetalpowderssuchasAl,Cu,Zn,Sn,PbandFe.Theyhave
reportedholesandvoidsintheproductsanda‘balling’phenomenoninwhichmoltenpowders
changeshapetoasphereduetosurfacetension.Ko¨nigetal.[5]haveusedstainlesssteeland
bronze–nickelpowders,andalsoreportedtheballingphenomenon.Toavoidballing,Bourellet
al.[6]haveproposedatwo-phasepowderapproach:
theviscosityofasolid–liquidphasecould
reducethetendencyforballing.Someothertechniquestomakemetallicandceramicpartsusing
thedirectSLSprocesshavealsobeenproposed[7].
OneoftheimportantproblemsinthedirectSLSprocessusingmetallicpowdersisthatthe
densityofmetallicprototypesisstilllow:
onlyupto70%[8],andsubsequentprocessestomake
fullydensemetallicpartsarenecessary.CarterandJones[9]haveproposedatwo-stepprocess
inwhichaporousmodelisgeneratedbythedirectSLSsystemandcompressedbyhotisostatic
pressing.Sindeletal.[10]havecombineddirectSLSandinfiltrationofmetaltomakefullydense
prototypes.Theporosityofmetallicprototypesandtheamountofconsolidatedpartsinthedirect
processaredependentonthekindsofmetalsandoperatingparameters,suchasthelaserpower
andlaserbeamspeed.However,theoperatingparametersforeachmetalarestillsetempirically.
Inthispaper,afiniteelementsimulationiscarriedoutinordertopredictthesolidifiedamount
ofmetallicpowdersinthedirectprocessofrapidprototypingusingapulsedlaser.
2.Simulationmethod
2.1.Treatmentofplurallaserpulses
Intheprototypingprocessusingpulsedlasercycles,heatingandcoolingarealternated.Inthe
simulationoftheheatingandcoolingprocessbyonepulseofalaserbeam,theshapechangeof
M.Shiomietal./InternationalJournalofMachineTools&Manufacture39(1999)237–252
239
thepowdersbymeltingandsolidifyingisnotconsideredtocalculatetheamountofsolidified
powders.Theboundaryconditionunderthelaserbeam,however,maybechangedduringlaser
irradiationbecausethemoltenpowderschangeshapeduetosurfacetensionandshrinkage.In
thesimulationofplurallaserirradiations,theshapechangeofthemoltenpowdersduetosurface
tensionandshrinkageistakenintoaccount.
Inthefiniteelementcalculation,thelaserbeamandpowdersarefixedinthespaceandthe
laserirradiatesonapointofthepowdersurfaceasshowninFig.1.Powdersmeltedbythefirst
pulseofthelaserbeamareassumedtobecomesphericalinshapeduetosurfacetension,andthe
volumeischangedbecausetherelativedensityincreasestounity.Thesphericalmoltenmetalis
cooledandsolidifiedintoaball.Thenthesecondpulseofthelaserbeamirradiatesthesolidball
andthepowders.Theballandapartofthepowdersareheatedandmelted.Thepowdersmelted
bythesecondlaserpulseareamalgamatedintothemoltenball,formingabiggerballdueto
surfacetension,andtheballissolidifiedagain.Themelting,combiningandsolidifyingprocesses
arerepeatedoverseveralpulses.
2.2.Finiteelementequation
Axisymmetrictemperaturedistributionisconsideredinthesimulation.Thegoverningequation
ofaxisymmetricheatconductionisexpressedby
1∂
r∂r
r
∂T
∂r
+
∂2T
2
+q=C
∂T
∂t
(1)
where
isthethermalconductivity,Tisthetemperature,qistheinternalheatgenerationrate
perunitvolume,
isthedensity,Cisthespecificheat,randzarethecoordinatesandtisthe
time.Constantheatflowrateisassumedatthesurfaceasexpressedbythefollowingequation:
Fig.1.Schematicviewofshapechangeofpowdersinthedirectprocessoflaserrapidprototypingofmetallicpowders.
∂z
240
M.Shiomietal./InternationalJournalofMachineTools&Manufacture39(1999)237–252
∂T
∂r
lr+
∂T
l
+qs=0
(2)
Herelrandlzarethedirectioncosinesoftheoutwardnormaltothesurfaceandqsistheheat
flowrateperunitarea.Fig.2showstheboundaryconditionsusedinthecalculation.Theqsin
Eq.
(2)isgivenateachsurfaceasfollows:
qs=qb=4Q/(d2)onS1duringheating
qs=qh=h(T−Ta)onS1duringcooling,andS2
qs=0onS3andS4
(3a)
(3b)
(3c)
HereQisthepeakpowerofthelaser,
istheabsorptivityofthelaserenergyintothepowders,
disthediameterofthelaserbeam,histheheattransfercoefficientandTaistheatmospheric
temperature.OnthesurfaceS1underthelaserbeam,externalheatfluxbythelaserbeamisgiven
duringheatingandaheattransferconditionisusedduringcooling.OnthesurfaceS5,whichis
thecontactareabetweentheballandthepowdersafterthefirstlaserpulse,theheatconduction
conditionisusedbutthecontactresistanceofheatisneglected.
Thebackwarddifferenceschemeisusedtosolvethetransientequations.Finiteelementequa-
tionsarederivedbythemethodofweightedresidualswithGalerkin’scriterionandare
expressedas
1
t
[C]+[K]{T}n+1=
1
t
[C]{T}n+{F},
(4)
Fig.2.
Boundaryconditionsusedinsimulation.
∂zz
M.Shiomietal./InternationalJournalofMachineTools&Manufacture39(1999)237–252
241
where[C]isthecapacitancematrix,[K]istheconductancematrix,{F}istheheatloadvector,
{T}n+1and{T}narethenodaltemperaturevectorsattimetn+1(=tn+t)andtn,respectively,
and
tisthetimestep.
Whenthetemperatureofthematerialcrossesthemeltingpoint,laten
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