THEESTABLISHMENTOFCALCULATIONFORMULAFORCASTI NG-ROLLINGFORCEOFVISCOUSFLUIDANDTHEINFLUENC EOFTECHNOLOGICALFACTORS¥Sun,Binyu;Zhang,Hon g;Zhang,Xiaoping;Shi,Rong(TaiyuanHeavyMachi neryInstitute,Taiyuan030024)Abstract:Assumi ngthedeformedbodytobeviscousfluid,acalculat ionformulaofcastingrollingforceforaluminium stripwasestablishedbasedonNavier-Stokesequa tion.Thetheoreticalresultswereverifiedbyexp eriments.Andtheinfluenceofvarioustechnologi calfactorsoncasting-rollingforcewasdiscusse d.Keywords:casting-rollingforce,viscousflui d,flowfunction1INTRODUCTIONCasting-rollingf orceisaveryimportantparameterinsuccessiveca sting-rollingprocess,whereashot-rollingform ulashaveconstantlybeenusedforthecalculation ofcasting-rollingforce,herethedifferencebet weencasting-rollingandhot-rollingisneglecte d.Infact,thephysicalmodelsintheformingproce ssesofthetwometalsarenotcompletelythesame.t houghthecontactsurfaceofthemetalhassolidifi ed,itsinsideisstillinthestateofliquid-solid viscousfluid.Butmetalistotallyinsolidstate. Thehot-rollingformulaproposednormallyisused totheconditionthatthedeformedmetalisconside redasrigidplasticandthesolutiontotheformula requiresthesimultaneousequationsofequilibri umdifferentialinplasticmechanicsandthecrite rionofplasticity.So,itisnotreasonabletouset heseformulasincasting-rollingprocess.Anurge ntneedincasting-rollingistogetacalculationf ormulatoguidetheproductionandonlinecontrol. ThisparperisbasedonNavierStokesequationofhy dromechanics.Thecalcultionformulasforcastin g-rollingforceareestablishedbyusingtheflowf unctiontosetavelocityfieldandbysimplifyingt heexternalforceconditionsinthecontactbounda ry.2THEESTABLISHMENTOFFORMULASTheNavier-Sto kesequationisasfollows:force;ρ-denstity;vi- velocitycomponent.Theflowfeatureofmetalform ingincasting-rollingprocessisextremelysimil artothatofviscousfluid.Consideringthedeform edmetalincasting-rollingareasviscousfluid,t heripetheoryinviscosityhydromechanicscanbeu sedtodiscribetheflowfeatureofthedeformedmet alincasting-rollingarea.Formula(1)isthecomm onformofN-Sequation.It'snoteasytogetthesolu tionofformula(1)analytically.But,ifsomenece ssaryassumptionsaregiven,it'salsopossibleto solveformula(1).Theseassumptionsareasfollow s:Consideringthedeformedbodyaslinearviscous fluid,theimcompressioncondition①Supportedby theNaturalScienceFundCommitteeofShanxiRecei vedAug29,1994:accptedOct.4,1994divvi=0issat isfied;thecasting-rollingareaisconsideredas 2-Dflow;theaccelerationofcasting-rollingcan beneglectedasthecastingrollingspeedisverysl ow,thatis,dvi/dt=0;thevolumeforcecanalsoben eglectedsinceitisnotasimportantascasting-ro llingforceindeformation;thevelocityfieldcan beassumedtobeapotentialfieldbutnotacurlfiel d,τ=katboundary(k:shearyieldstress).Accepti ngtheassumptiontsabove,formula(1)can2.1Velo cityFieldReplacingarcwithchordatthecontactb oundaryofthecasting-rollingarea(SeeFig.1),w ehave:Theflowfunctioncanbesetupas:Sothefoll owingvelocityfieldcanbederived:Thisvelocity fieldiskinematicpermissiblesincewecanprovet hatformula(5)satisFig.1Thecasting-rollingar eagiveahintwiththeeyesfiestheequationofcont inuity.2.2TheEstablishmentofCalculationForm ulaofCasting-rollingForcePuttingtheknownvel octityfieldintoformula(2)andarranginginorde r:Accordingtohydromechanicstheory[1],weknow thattheshearstressisindirectproportiontothe velocitygradient,sowhen(;shearstressonconta ctarcsurface)andconsideringηtobeonlythefunc tionofχIfthenipangleisverysmallandτxn=τχ(τχ :frictionalforceoncontactsurface)approximat ely,formula(6)canbewrittenasSince,taking=0, wehave;whenτχ=k,formula(9)canbeintegratedwh ereC-arbitaryconstant,determinedbyboundaryc ondition.Ifthefronttensionisconsidered,theb oundaryconditioninforwardslipareawillbep=σq -k1,whenχ=0(11)wherek1-resistancetodeformat ionatexitend;σq-fronttensionstress.Thebound aryconditioninbackwardslipareaisp=0whenχ=1( 12)Usingtheboundarycondition,theconstansinf orwardandbackwardslipareascanbeobtainedresp ectively:Puttingtheconstantsintoformula(10),thefollowingdistributionformulasofcompressionstressinforwardandbackwardslipareascanalsobeobtasined:Theforwardsliparea'slengthisg
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