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JOURNALOFOPTOELECTRONICSANDADVANCEDMATERIALSVol.8,No.5,October2006,p.1736-1740TheresidualstressesofFeBSi-typeinaningotmouldI.ATEFNOAEI,D.RADU,H.CHIRIACaaba“Al.I.Cuza”University,CarolIBlvd.,11,700506,Iasi,RomaniabNationalInstituteofResearchandDevelopmentforTechnicalPhysics,47MangeronBlvd.,700050Iasi,RomaniaInthispaperweanalysedtheinternalresidualstressesthatappearduringtherapidcoolingofaFeBSi-typealloy,lainintheinteriorchannelofaningotmouldwhoseexteriorwallsaremaintainedattheroomtemperature.Thetheoreticalmodelpresentedinthepaperemphasizesinacompleteandsyntheticalmanner,thespatio-temporaldistributionoftheinducedstressesinaFeBSi-typealloyduringitscoolingtotheroomtemperature,takingintoaccountboththethermalbehaviourofthealloyandthesupplementarystressesinducedbytheingot-mould,asaresultofthedifferencebetweenthethermalexpansioncoefficientsofthetwomaterialsincontact.(ReceivedSeptember6,2006；acceptedSeptember13,2006)Keywords:Residualstresses,Ingot-mould,Thermalgradients1.IntroductionTheaimofthispaperistoevaluateofthethermalstressesduringthecoolingofanalloyintheinnerverticalchannelofaningotmould,whoseexteriorwallsaremaintainedattheroomtemperature,.Moreprecisely,ourpurposeistodeterminethespatio-temporaldistributionofthestressesthatappearduringtherapidcoolingofthewholematerialtothetemperature.Thismodelcontainstwoimportanttasks:(1)Theevaluationofthespatio-temporaltemperaturesdistributionduringtherapidcooling1,2,3ofthematerialtotheroomtemperature.Fordifferentvaluesoftheradiusoftheinnerchannel,takingintoaccountthethermalconditionsonthealloy-ingotmouldinterface,weanalyzedthespatialandtemporalevolutionofthetemperature；(2)Startingfromthespatio-temporaldistributionofthetemperature,onecanobtainthestressesduetobothrapidcooling(bigthermalgradients)andconstraintsproducedtothealloybythecooledingotmouldasaresultofadifferencebetweenthethermalexpansioncoefficientsofthetwomaterials.300=wTKwTThetheoreticalmodelpresentedhereisbasedonthefollowingworkinghypotheses:(i)theform(thegeometry)oftheflowingchannelofthemeltedmetal/alloy(withacylindricalsymmetry)demandsacylindricalcoordinatessystem(),rz,withtheaxispointedverticallydownward,alongwiththeingotmouldchannelandthezr-coordinatealongitsradialdirection；(ii)weconsiderthelengthLoftheingotmouldflowingchannelmuchbiggerthanitsradius,2R()2LR>>；(iii)thematerial(madefromaFeBSialloy)hasattheinitialmomentthetemperatureand(iv)weassumethattherearenotemperaturegradientsalongtheingotmouldchannel.Thecharacteristicsofthealloyare2,3:isthespecificheat,isthethermalconductibility,1200=mTKKK3530/=pcJkg130/kWm=37.210/Mkgm=isthemassdensity,112210/alloyENm=istheYoungsmodulusandisthethermalexpansioncoefficient.Thecharacteristicoftheingotmouldare2,3:68.710alloyK=1mK2383/kW=isthethermalconductibility,15Lcm=isthelength,61710Cu1K=isthethermalexpansioncoefficient,1021310/ingotENm=istheYoungsmodulus,11Rmm=istheradiusoftheinnerverticalchanneland25Rmm=isthetotalradius.2.Therapidcoolingprocessofthesolidifiedalloy2.1.Thetemperaturedistributioninthealloy-ingotmouldsystemTemperaturedistributioninthealloy.Inthefollowingitisanalysedtherapidcoolingofthesolidifiedmetalfromthetemperaturetotheroomtemperature.Asitwillbeshown,fromamathematicalpointofview,wemayconsiderthisasaproblemofconductionandthermaltransferwithasource,anditimpliesthedeterminationofthespatio-temporaldistributionofthetemperaturegT1T(r,t)ofthesolidifiedmaterial,whichhasinthecenteroftheingotmouldschannelthetemperature(gTgT=800Kisthesolidificationtemperature)；thesourceisdistributedontheinnersurfaceoftheingotmouldswall,havingthetemperature.Theheatlossesofthealloyduetoitsforcedcoolingmaybeconsideredasuniformlydistributednegativesources.Thedeterminationofthetemperaturedemandstofindthesolutionofthethermalbalanceequationforthematerialsubmittedtoarapidcoolingprocess:wT1T(r,t)TheresidualstressesofFeBSi-typeinaningotmould1737()2111121wTTTabTtrrr=+1rR,for0,(1)where:1/()=Mpakc,()/baPLV=21AR=,12PR=andListheingotmouldslength.Thegeneralsolutionofeq.(1)isoftheform:()2210(,)/=+matwTrtTCIrbame,whereisanintegrationconstant,C()20/IrbamaremodifiedBesselfunctionsofthezeroorderandtheconstantwillbedeterminedfromthethermalconditionsontheinterfacealloyingotmould.Byimposingtheparticularconditions:m1(0,0)gTrtT=and100Trr=onecanobtaintheexpressionfortheconstant:CgwCTT=,whichleadsustotheexpressionofthetemperaturedistributioninthematerial:()2210(,)()/matwgwTrtTTTIrbame=+.(2)Thetemperaturedistributionintheingotmouldswall.Duringthecoolingprocess,theingotmouldswallreceivestheheatamountfromthecoolingalloy.Wewillconsiderthetemperaturedistributionintotheingotmouldswalloftheform4:21(,)ln()TrtArA=+2,(3)where1Aand2Aarevariablesdependingonthetimet:11()AAt,22()AAt.Boundaryconditionsforthealloyingotmouldinterface.Inordertodeterminethefinalexpressionsofthetemperaturesandwemustusethefollowingboundaryconditions:1(,)Trt2(,)Trti)Theheatfluxfromthealloyisreceivedbytheingot-mould.Thisheatflux(fromthealloy-ingotinterface)mustbecontinous.So,for,wemusthave:1rR=1211=TTrrrRrRkk,(4)whereandarethecoefficientsofthermalconductivityofthealloyandingotmold,respectively；1k2kii)Onthealloy-ingotinterface(),thetemperaturesfromtheadjacentregionsmustbeequal:1rR=112()(TrRTrR=1),(5)iii)Ontheoutersurfaceoftheingotmould,weconsiderthatwehavetheroomtemperature,thatis:22()wTrRT=.(6)Usingtheboundaryconditionsgivenby(4)and(6),weobtainthefollowingexpressionsfor1Aand2A:()()()211212211()(/)()(/)(/)=gwAtkkRTTbamIRbamexpamt×,(7)()()()2212122112()(/)()(/)(/)ln()=wgwAtTkkRTTbam×IRbamRexpamt,(8)andfromconditions(5)weobtaintheconstantasthesolutionofequation:m()()()()()220111221112(/)/(/)(/)ln/=IRbamkRkbam×IRbamRR,(9)where()211/IRbamaremodifiedBesselfunctionsofthefirstorder.Inordertodeterminethestresseswhichappearduringthecoolingprocessofthealloyplacedintotheingotmould,wewillconsiderthetemperaturedistributioninthealloy(2),wheretheconstantisgivenbythesolutionofeq.(9).m2.2.TheinternalstresseswhichappearduringtheforcedcoolingprocessInthissection,ourpurposeistoanalyzethestresseswhichappearbothduetothethermalgradientsandfromtheconstraintsproducedonthealloybythecooledingotmouldswallasaresultofthedifferencebetweenthethermalexpansioncoefficientsofthetwomaterialsincontact.Theradialtemperaturegradientsleadtotheappearanceofsomedisplacements,bothinalloy(and)andinthewalloftheingot(and).Thesedisplacementssatisfythedifferentialdisplacementsequation.Incylindricalcoordinatestheseequationsreads5:alloyrualloyzuingotruingotzu1)1()(,111alloyralloydurdTrtddrrdrdrµµ+=),2).,alloyzduconstdz=(10)foralloyandI.Atefnoaei,D.Radu,H.Chiriac17381)()10ingotrdurddrrdr=,2).,ingotzduconstdz=(11)fortheingotmouldswall.IntheaboverelationsalloyisthealloysthermalexpansioncoefficientsandµisthePoissonscoefficient.ItisassumedthatthevaluesofPoissonscoefficientforalloyandingotmouldarethesame:1/3alloyingotµµµ=6.Asitwillbeshowninsection3,thethermalgradientsintotheingotmouldswallaresmall,comparedtothethermalgradientsinthealloy；thisallowsustofurtherconsider,inthecalculationofthestresses,onlytheconstriction/dilatationeffectsoftheingotmouldoverthealloy,duetothedifferentcoolingofthetwomaterialsincontact.Thesolutionsofeqs.(10)and(11)(representingboththeradialandtheaxialdisplacementsinthealloy:andandrespectively,theradialandaxialdisplacementsintheingotmould:and)leadus6tothefollowingexpressionsofthestressesforthealloyandfortheingotmouldswall:alloyrualloyzuingotruingotzu1120(,)(,)(1)ralloyalloyalloyrralloyErtECrTrtdrrµ=,()()11(,)(,)1alloyzzalloyalloyalloyrtECETrt=µ(12)()()11201(,)(,)(1)(,)1ralloyalloyalloyalloyalloyalloyErtECrTrtdrrETrtµµ=+,()222(,)(1)ingotrringotingotrtECECr=+,()223(,)(1)ingotingotingotrtECECr=+µ),(13)2(,)2ingotzzingotrtEC=whereandarethreeintegrationconstantswhichwillbecalculatedfromtheequilibriumconditionsandfromtheconditionwhichconsidersthedifferentthermalbehaviourofthetwomaterialsincontact.Theresultantstrainduetothecoolingoftwomaterialswithdifferentthermalexpansioncoefficientswhichareincontactduringtheentirecoolingprocessisgivenbytherelation:1,C2C3C(alloyingotalloyingotT=.Inourcase,forthealloywehavealloyalloyT=andfortheingotswall,ingotingotT=,wherealloyandingotarethestrainsduetothethermalcontractioninthealloyandingotrespectivellyandingotisthethermalexpansioncoefficientoftheingot.Inthiscase,isthedifferencebetweentheTgTandtheroomtemperature,.Inordertodetermine,andwemustfindthevaluesoftheconstantsand.Forthealloyingotmouldinterfaceweimposethefollowingconditions:wT(,)alloyrrrt(,)alloyrt(,)alloyzzrt1,C2C3C1)thestrainsthatappearinthisprocessaredueonlytothedifferencebetweenthethermalexpansioncoefficientsofthealloyandingot:11()()alloyingotrrurRurRR=11)；(14)2)theequilibriumconditionsatthealloyingotmouldinterface:1(,)(,alloyingotrrrrrRtrRt=and(15)3)ontheexteriorsurfaceoftheingotmould()2rR=:2(,)ingotrrrRt0=.Fromthesethreeparticularconditions,onecanobtainthenumericalvaluesfortheconstantsandwhichallowustodeterminethestresses(12)inthealloy,duringthecooling.,1C2C3C3.ResultsanddiscussionForthegivenabovecharacterinsticsoftheFeBSi-typealloyandofaningotmouldwithcopperwall2,wecalculatedtheconstants1Aand2A,andwealsodeterminedtheconstantbysolvingeq.(9).Inthefollowing,wewillconsideraningotmouldwiththeinnerchannelsradiusm11Rm=andthethicknessofthewall；thus,wewillcalculatetheradialtemperaturedistributionatdifferentmoments.Fig.1showstheshapeoftheradialdistributionofthetemperature,bothinthetransversalsectionofthealloy(theredcurves)andintheingotmouldswall(thebluecurves),atthreedifferentmoments,i.e.:20.2tsµ=andrespectively.Asthisfigureshows,thereisadifferenceinthetemperaturebetweenthecenterofthechannelanditsinnersurface；thisdifferenceisbiggerinthefirststagesofthecoolingprocessanditbecomeslesssignificantattheendoftheprocess.Fig.1.Theradialdistributionofthetemperatureinthemouldstransversalsection.Onecanobservethatinthefirstmomentsofthealloyscooling,theamountofheatdeliveredisbigger；thisTheresidualstressesofFeBSi-typeinaningotmould1739sleadstobiggertemperaturegradients.Fordifferentvaluesof,theradialdistributionoftemperatureintheingotmouldswallevolvesasfollows:afterthetimet10.1tµ=,thedifferenceoftemperaturebetweentheinnerwalloftheingotmouldandtheexterioroneis,whilefor1227RRT=Ks30.3tµ=,thisdifferenceismuchsmaller.Fig.2showstheradialdistributionoftemperatureintheingotmouldssection(bothinthealloystransversalsectionandintheingotmouldswall)atatime(124RRT=)Ks10.1tµ=,forthreevaluesoftheradiusoftheinnerchannel:11Rmm=,13Rmm=and14Rmm=respectively.Fig.2.Theradialdistributionofthetemperatureintheingotmouldsingotsectionaftert1=0.1µs,forthreedifferentvaluesoftheinnerchannelsradius.Asonemighthaveexpected,after10.1tsµ=,thetemperatureinthecenteroftheinnerchanneloftheingotmould,havingthesmallerradius(11Rmm=)andthethicknessofthebiggerwall(4gmm=)issmallerthanthetemperatureinthecenterofthechannelhavingthebiggerradius(14Rmm=)andthesmallerthicknessofthewall(1gmm=)；thisonewillcoolslower.Onecanalsonoticethat,asthethicknessoftheingotmouldswallbecomessmaller,thetemperaturedifferencebetweentheinteriorsurfaceandtheexteriorone,becomessmaller,i.e.,for12RRT11Rmm=(4gmm=),whilefor1227RRT=K14Rmm=(1gmm=),.Asforthestresseswhichappearduringthecoolingprocess,bycalculatingtheconstantsandfromtheequilibriumconditionsandreplacingthemineq.(12),onecanobtaintheirexplicitform.Fig.3representsthegraphicspatio-temporaldistributionoftheradial(a),azimuthal(b)andaxial(c)stressesintothealloy.1210RRT=K1,C2C3C(a)(b)(c)Fig.3.Thespatio-temporaldistributionofthestressesintothealloy:(a)radialstresses,(b)azimuthalstresses,(c)axialstresses.Onecanobservethatthemagnitudeorderofthesestressesisapproximately；asacommonfeature,theyareallpositive(tensile)andtendtoasaturationvalue,correspondingtotheroomtemperature,T910Paw.Theaxialstressesareapproximatelytwotimesbiggerthantheradialones.Thus,theaxialstressesreachthesaturationflatintervalmorequicklythantheothers.Thestressesdroptoarelativelyconstantvalue,inthetemperaturerange,showsusthatthetransformed(solidified)materialhasgotamuchmoreregularstructure.Furthermore,onecannoticethat,thesmallertheradiusoftheinnerchanneloftheingotmould,thebiggerthestressesinducedintothealloy.(800300)K4.ConclusionsThetheoreticalmodeldescribedinthispaperpresentsinasyntheticalmannerthespatio-temporaldistributionofthestressesinducedinanalloyduringitscoolingtotheroomtemperature,consideringboththethermalbehaviour