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JOURNALOFOPTOELECTRONICSANDADVANCEDMATERIALSVol.8,No.5,October2006,p.17361740TheresidualstressesofFeBSitypeinaningotmouldI.AŞTEFĂNOAEI,D.RADU,H.CHIRIACaabaAl.I.CuzaUniversity,CarolIBlvd.,11,700506,Iasi,RomaniabNationalInstituteofResearchandDevelopmentforTechnicalPhysics,47MangeronBlvd.,700050Iasi,RomaniaInthispaperweanalysedtheinternalresidualstressesthatappearduringtherapidcoolingofaFeBSitypealloy,lainintheinteriorchannelofaningotmouldwhoseexteriorwallsaremaintainedattheroomtemperature.Thetheoreticalmodelpresentedinthepaperemphasizesinacompleteandsyntheticalmanner,thespatiotemporaldistributionoftheinducedstressesinaFeBSitypealloyduringitscoolingtotheroomtemperature,takingintoaccountboththethermalbehaviourofthealloyandthesupplementarystressesinducedbytheingotmould,asaresultofthedifferencebetweenthethermalexpansioncoefficientsofthetwomaterialsincontact.ReceivedSeptember6,2006acceptedSeptember13,2006KeywordsResidualstresses,Ingotmould,Thermalgradients1.IntroductionTheaimofthispaperistoevaluateofthethermalstressesduringthecoolingofanalloyintheinnerverticalchannelofaningotmould,whoseexteriorwallsaremaintainedattheroomtemperature,.Moreprecisely,ourpurposeistodeterminethespatiotemporaldistributionofthestressesthatappearduringtherapidcoolingofthewholematerialtothetemperature.Thismodelcontainstwoimportanttasks1Theevaluationofthespatiotemporaltemperaturesdistributionduringtherapidcooling1,2,3ofthematerialtotheroomtemperature.Fordifferentvaluesoftheradiusoftheinnerchannel,takingintoaccountthethermalconditionsonthealloyingotmouldinterface,weanalyzedthespatialandtemporalevolutionofthetemperature2Startingfromthespatiotemporaldistributionofthetemperature,onecanobtainthestressesduetobothrapidcoolingbigthermalgradientsandconstraintsproducedtothealloybythecooledingotmouldasaresultofadifferencebetweenthethermalexpansioncoefficientsofthetwomaterials.300wTKwTThetheoreticalmodelpresentedhereisbasedonthefollowingworkinghypothesesitheformthegeometryoftheflowingchannelofthemeltedmetal/alloywithacylindricalsymmetrydemandsacylindricalcoordinatessystem,,rθz,withtheaxispointedverticallydownward,alongwiththeingotmouldchannelandthezrcoordinatealongitsradialdirectioniiweconsiderthelengthLoftheingotmouldflowingchannelmuchbiggerthanitsradius,,2R2LRiiithematerialmadefromaFeBSialloyhasattheinitialmomentthetemperatureandivweassumethattherearenotemperaturegradientsalongtheingotmouldchannel.Thecharacteristicsofthealloyare2,3isthespecificheat,isthethermalconductibility,1200mTKKK3530/pcJkg130/kWm37.210/Mkgmρ⋅isthemassdensity,112210/alloyENm⋅istheYoungsmodulusandisthethermalexpansioncoefficient.Thecharacteristicoftheingotmouldare2,368.710alloyKα−−⋅1mK2383/kWisthethermalconductibility,15Lcmisthelength,61710Cu1Kα−−⋅isthethermalexpansioncoefficient,1021310/ingotENm⋅istheYoungsmodulus,11Rmmistheradiusoftheinnerverticalchanneland25Rmmisthetotalradius.2.Therapidcoolingprocessofthesolidifiedalloy2.1.ThetemperaturedistributioninthealloyingotmouldsystemTemperaturedistributioninthealloy.Inthefollowingitisanalysedtherapidcoolingofthesolidifiedmetalfromthetemperaturetotheroomtemperature.Asitwillbeshown,fromamathematicalpointofview,wemayconsiderthisasaproblemofconductionandthermaltransferwithasource,anditimpliesthedeterminationofthespatiotemporaldistributionofthetemperaturegT1Tr,tofthesolidifiedmaterial,whichhasinthecenteroftheingotmouldschannelthetemperaturegTgT800Kisthesolidificationtemperaturethesourceisdistributedontheinnersurfaceoftheingotmouldswall,havingthetemperature.Theheatlossesofthealloyduetoitsforcedcoolingmaybeconsideredasuniformlydistributednegativesources.ThedeterminationofthetemperaturedemandstofindthesolutionofthethermalbalanceequationforthematerialsubmittedtoarapidcoolingprocesswT1Tr,tTheresidualstressesofFeBSitypeinaningotmould17372111121wTTTabTtrrr⎛⎞∂∂∂−−⎜⎟∂⎝⎠1rR≤≤,for0,1where1/ρMpakc,,/baPLV21ARπ,12PRπandListheingotmouldslength.Thegeneralsolutionofeq.1isoftheform2210,/−−matwTrtTCIrbame,whereisanintegrationconstant,C20/−IrbamaremodifiedBesselfunctionsofthezeroorderandtheconstantwillbedeterminedfromthethermalconditionsontheinterfacealloy–ingotmould.Byimposingtheparticularconditionsm10,0gTrtTand100Trr∂∂onecanobtaintheexpressionfortheconstantCgwCTT−,whichleadsustotheexpressionofthetemperaturedistributioninthematerial2210,/matwgwTrtTTTIrbame−−−.2Thetemperaturedistributionintheingotmouldswall.Duringthecoolingprocess,theingotmouldswallreceivestheheatamountfromthecoolingalloy.Wewillconsiderthetemperaturedistributionintotheingotmouldswalloftheform421,lnTrtArA2,3where1Aand2Aarevariablesdependingonthetimet11AAt≡,22AAt≡.Boundaryconditionsforthealloy–ingotmouldinterface.Inordertodeterminethefinalexpressionsofthetemperaturesandwemustusethefollowingboundaryconditions1,Trt2,TrtiTheheatfluxfromthealloyisreceivedbytheingotmould.Thisheatfluxfromthealloyingotinterfacemustbecontinous.So,for,wemusthave1rR1211∂∂TTrrrRrRkk,4whereandarethecoefficientsofthermalconductivityofthealloyandingotmold,respectively1k2kiiOnthealloyingotinterface,thetemperaturesfromtheadjacentregionsmustbeequal1rR112TrRTrR1,5iiiOntheoutersurfaceoftheingotmould,weconsiderthatwehavetheroomtemperature,thatis22wTrRT.6Usingtheboundaryconditionsgivenby4and6,weobtainthefollowingexpressionsfor1Aand2A211212211///−−−−gwAtkkRTTbamIRbamexpamt,72212122112///ln−−−−−wgwAtTkkRTTbamIRbamRexpamt,8andfromconditions5weobtaintheconstantasthesolutionofequationm220111221112////ln/−−−IRbamkRkbamIRbamRR,9where211/−IRbamaremodifiedBesselfunctionsofthefirstorder.Inordertodeterminethestresseswhichappearduringthecoolingprocessofthealloyplacedintotheingotmould,wewillconsiderthetemperaturedistributioninthealloy2,wheretheconstantisgivenbythesolutionofeq.9.m2.2.TheinternalstresseswhichappearduringtheforcedcoolingprocessInthissection,ourpurposeistoanalyzethestresseswhichappearbothduetothethermalgradientsandfromtheconstraintsproducedonthealloybythecooledingotmouldswallasaresultofthedifferencebetweenthethermalexpansioncoefficientsofthetwomaterialsincontact.Theradialtemperaturegradientsleadtotheappearanceofsomedisplacements,bothinalloyandandinthewalloftheingotand.Thesedisplacementssatisfythedifferentialdisplacementsequation.Incylindricalcoordinatestheseequationsreads5alloyrualloyzuingotruingotzu11,111alloyralloydurdTrtddrrdrdrµαµ⎡⎤⎢⎥−⎣⎦,2.,alloyzduconstdz10foralloyandI.Aştefănoaei,D.Radu,H.Chiriac1738110ingotrdurddrrdr⎡⎤⎢⎥⎣⎦,2.,ingotzduconstdz11fortheingotmouldswall.IntheaboverelationsalloyαisthealloysthermalexpansioncoefficientsandµisthePoissonscoefficient.ItisassumedthatthevaluesofPoissonscoefficientforalloyandingotmouldarethesame1/3alloyingotµµµ6.Asitwillbeshowninsection3,thethermalgradientsintotheingotmouldswallaresmall,comparedtothethermalgradientsinthealloythisallowsustofurtherconsider,inthecalculationofthestresses,onlytheconstriction/dilatationeffectsoftheingotmouldoverthealloy,duetothedifferentcoolingofthetwomaterialsincontact.Thesolutionsofeqs.10and11representingboththeradialandtheaxialdisplacementsinthealloyandandrespectively,theradialandaxialdisplacementsintheingotmouldandleadus6tothefollowingexpressionsofthestressesforthealloyandfortheingotmouldswallalloyrualloyzuingotruingotzu1120,,1ralloyalloyalloyrralloyErtECrTrtdrrασµ−−∫,11,,1alloyzzalloyalloyalloyrtECETrtσα−−µ1211201,,1,1ralloyalloyalloyalloyalloyalloyErtECrTrtdrrETrtθθασµαµ−−∫−,222,1ingotrringotingotrtECECrσ⎡⎤−⎣⎦,223,1ingotingotingotrtECECrθθ⎡⎣µ⎤⎦,132,2ingotzzingotrtECσwhereandarethreeintegrationconstantswhichwillbecalculatedfromtheequilibriumconditionsandfromtheconditionwhichconsidersthedifferentthermalbehaviourofthetwomaterialsincontact.Theresultantstrainduetothecoolingoftwomaterialswithdifferentthermalexpansioncoefficientswhichareincontactduringtheentirecoolingprocessisgivenbytherelation1,C2C3CalloyingotalloyingotTεεεαα∆−−.Inourcase,forthealloywehavealloyalloyTεα∆andfortheingotswall,ingotingotTεα∆,wherealloyεandingotεarethestrainsduetothethermalcontractioninthealloyandingotrespectivellyandingotαisthethermalexpansioncoefficientoftheingot.Inthiscase,isthedifferencebetweentheT∆gTandtheroomtemperature,.Inordertodetermine,andwemustfindthevaluesoftheconstantsand.Forthealloy–ingotmouldinterfaceweimposethefollowingconditionswT,alloyrrrtσ,alloyrtθθσ,alloyzzrtσ1,C2C3C1thestrainsthatappearinthisprocessaredueonlytothedifferencebetweenthethermalexpansioncoefficientsofthealloyandingot11alloyingotrrurRurRR−ε11142theequilibriumconditionsatthealloy–ingotmouldinterface1,,alloyingotrrrrrRtrRtσσand153ontheexteriorsurfaceoftheingotmould2rR2,ingotrrrRtσ0.Fromthesethreeparticularconditions,onecanobtainthenumericalvaluesfortheconstantsandwhichallowustodeterminethestresses12inthealloy,duringthecooling.,1C2C3C3.ResultsanddiscussionForthegivenabovecharacterinsticsoftheFeBSitypealloyandofaningotmouldwithcopperwall2,wecalculatedtheconstants1Aand2A,andwealsodeterminedtheconstantbysolvingeq.9.Inthefollowing,wewillconsideraningotmouldwiththeinnerchannelsradiusm11Rmandthethicknessofthewallthus,wewillcalculatetheradialtemperaturedistributionatdifferentmoments.Fig.1showstheshapeoftheradialdistributionofthetemperature,bothinthetransversalsectionofthealloytheredcurvesandintheingotmouldswallthebluecurves,atthreedifferentmoments,i.e.20.2tsµandrespectively.Asthisfigureshows,thereisadifferenceinthetemperaturebetweenthecenterofthechannelanditsinnersurfacethisdifferenceisbiggerinthefirststagesofthecoolingprocessanditbecomeslesssignificantattheendoftheprocess.Fig.1.Theradialdistributionofthetemperatureinthemouldstransversalsection.Onecanobservethatinthefirstmomentsofthealloyscooling,theamountofheatdeliveredisbiggerthisTheresidualstressesofFeBSitypeinaningotmould1739sleadstobiggertemperaturegradients.Fordifferentvaluesof,theradialdistributionoftemperatureintheingotmouldswallevolvesasfollowsafterthetimet10.1tµ,thedifferenceoftemperaturebetweentheinnerwalloftheingotmouldandtheexterioroneis,whilefor1227RRT∆Ks30.3tµ,thisdifferenceismuchsmaller.Fig.2showstheradialdistributionoftemperatureintheingotmouldssectionbothinthealloystransversalsectionandintheingotmouldswallatatime124RRT∆Ks10.1tµ,forthreevaluesoftheradiusoftheinnerchannel11Rmm,13Rmmand14Rmmrespectively.Fig.2.Theradialdistributionofthetemperatureintheingotmouldsingotsectionaftert10.1µs,forthreedifferentvaluesoftheinnerchannelsradius.Asonemighthaveexpected,after10.1tsµ,thetemperatureinthecenteroftheinnerchanneloftheingotmould,havingthesmallerradius11Rmmandthethicknessofthebiggerwall4gmmissmallerthanthetemperatureinthecenterofthechannelhavingthebiggerradius14Rmmandthesmallerthicknessofthewall1gmmthisonewillcoolslower.Onecanalsonoticethat,asthethicknessoftheingotmouldswallbecomessmaller,thetemperaturedifferencebetweentheinteriorsurfaceandtheexteriorone,,becomessmaller,i.e.,for12RRT∆11Rmm4gmm,,whilefor1227RRT∆K14Rmm1gmm,.Asforthestresseswhichappearduringthecoolingprocess,bycalculatingtheconstantsandfromtheequilibriumconditionsandreplacingthemineq.12,onecanobtaintheirexplicitform.Fig.3representsthegraphicspatiotemporaldistributionoftheradiala,azimuthalbandaxialcstressesintothealloy.1210RRT∆K1,C2C3CabcFig.3.Thespatiotemporaldistributionofthestressesintothealloyaradialstresses,bazimuthalstresses,caxialstresses.Onecanobservethatthemagnitudeorderofthesestressesisapproximatelyasacommonfeature,theyareallpositivetensileandtendtoasaturationvalue,correspondingtotheroomtemperature,T910Paw.Theaxialstressesareapproximatelytwotimesbiggerthantheradialones.Thus,theaxialstressesreachthesaturationflatintervalmorequicklythantheothers.Thestressesdroptoarelativelyconstantvalue,inthetemperaturerange,showsusthatthetransformedsolidifiedmaterialhasgotamuchmoreregularstructure.Furthermore,onecannoticethat,thesmallertheradiusoftheinnerchanneloftheingotmould,thebiggerthestressesinducedintothealloy.800300K→4.ConclusionsThetheoreticalmodeldescribedinthispaperpresentsinasyntheticalmannerthespatiotemporaldistributionofthestressesinducedinanalloyduringitscoolingtotheroomtemperature,consideringboththethermalbehaviour
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