Maximum Cross Section Reduction Ratio of Billet in a Single Wire Forming Pass Based on Unified Strength Theory. Xiaowei Li1,2, a

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1 Inenainal Fum n Enegy, Envinmen and Susainable evelpmen (IFEES 06 Maximum Css Sein Reduin Rai f Bille in a Single Wie Fming Pass Based n Unified Sengh They Xiawei Li,, a Shl f Civil Engineeing, Panzhihua Univesiy, Panzhihua 67000, China epamen f Tunnel and Undegund Engineeing, Suhwes Jiang Univesiy, Chengdu, 6003 China a lilixxxwww@6.m Keywds: Unified Sengh They, Reduin Rai, Wie awing, Wie Exusin Absa. F baining he maximum ss sein eduin ai (MCSRR f any bille, he equain f esimaing he MCSRR f bille in a single wie fming pass was bained based n unified sengh hey. I is pved ha he equain f he MCSRR f bille in wie dawing pass is simila ha in a single wie exusin pass. Inluded he nibuin f bh inemediae pinipal shea sess and vaying ensin-mpessin-ai (TCR maeial mehanial ppey, he equain develped an easnably apply a vaiey f maeials. If he TCR f he maeial is equal, he MCSRR f bille in a single wie fming pass is In addiin, he anges f he MCSRR f bille f bh duile and bile maeials in a single wie fming pass ae bained. Induin Wie fming is lassified as wie dawing and exusin, and eaed as hee-dimensinal axisymmei pblem in analysis. Wie dawing is a mealwking pess used edue he ss-sein f a wie by pulling he wie hugh a single, seies f, die(s. Alhugh simila duing pess, dawing is diffeen fm exusin, beause duing dawing pess he wie is pulled, ahe han pushed, hugh he die. awing is usually pefmed a m empeaue, hus lassified as a ld wking pess. F simpliiy, suppse ha []: he inne wall f nial die is smh; he adial diein f plasi flw is wad he viual apex f he inne wall f nial die; and he maeial f bille is igus-plasi. Meve, basially, idenial sengh in ensin and mpessin f maeial is adaped by invesigas, whih will esul in nsideably es f paial pupse, f, pulled exuded hugh nial die, maeial f bille will develp Baushinge effe, and he TCR f duile meal vaies fm 0.77, bu is n always equal. Eve sine he unified sengh hey was pesened, eseahes have pdued fuiful wks in many fields [,3]. Assuming he yield sufaes ae nvex, he unified sluin f he MCSRR f bille in a single wie fming pass is pesened based n unified sengh hey, whih inludes he nibuins f bh inemediae piniple shea sess and he vaying sengh in ensin and mpessin, and is suiable f bh wie dawing and exusin. I is pved ha he pevius sluins f bille wih idenial sengh in ensin and mpessin in a single wie dawing exusin pass ae all speial ases f he unified sluin. Unified Sengh They Based n win shea sengh hey, M.H. Yu [4] develped unified sengh hey. The hey inludes he nibuin f maeial sengh, and uses a unifm mehanial mdel desip he plasi feaues f maeial. The mahemaial expessin f unified sengh hey is simple and given as fllws: F = (b + 3 =, + b 06. The auhs - Published by Alanis Pess ( (

2 F b = + 3 ( + b =, (. ( + b + = 3 ( + τ ( + τ = B B, / =, B = / τ. (3 whee, is he ensile ulimae sengh f a maeial, he mpessive ulimae sengh, τ he shea ulimae sengh, he TCR, vaying fm f duile meal maeial, f bile meal maeial, and geneally less han 0.5 f geehnial maeials, B ensin shea ai, and b paiipain fa f inemediae pinipal shea sess. Analysis n wie dawing f inwad adial flw When wie is pulled hugh smh nial die, adial flw is indued, pining he viual apex f inne wall f nial die. The shemai diagam f seady-sae axisymmei pblem, used ay u he develpmen f inwad - adial -flw wie dawing, is pled as Fig.. x x i x i θ Fig. Shemai iagam f wie dawing whee is he diamee f bille a he die exi, i a he die eny, and a any pin wihin die; x is he disane beween he viual apex f he inne wall f nial die and he die exi, x i he disane beween ha and he die eny, and x he disane beween ha and any pin wihin die; is he psiive diein f adial pinipal sess a any pin wihin die, θ he psiive diein f he iumfeenial. Ading elas-plasi hey, he elains f psiive sesses n he sufaes f mi-bdy laed in any pin f bille wihin he egin f die an be bained as: = > = 3 = θ. Ading he equain, we an ge: - θ >0. By he iein ndiin f unified sengh hey, ne f Eqn. and Eqn. an be apppiaely seleed be used ay u analysis. Subsiuing bh - θ >0 and > 0 in he iein ndiin f Eqn., given as: θ θ = θ = 0. ( > Ading Eqn.4, he Eqn. is used analysis, subsiuing = > = 3 = θ in he Eqn., hen, he fllwing equain an be given as: θ =. (5 560

3 Fm Eqn.5, i is bvius ha he yield sess duing wie dawing pess is independen f he paamee fa f inemediae piniple shea f b. Ading elas-plasi hey [5], he diffeenial equain f hee-dimensinal axisymmei pblem an be bained as: d dx ( + x θ = 0. (6 Subsiuing Eqn.5 in Eqn.6, and give: x d dx ( + = 0 +. (7 when, Inegaing Eqn.7, and he fllwing equain an be g as: ( ln x = ln +. (8 Subsiuing bh x 0 and - / (- < 0 in Eqn.8, and we an ge he fllwing equain as: ( = e x. (9 MCSRR f bille in a single wie dawing pass The inegal nsan in Eqn.9 an be deemined by he sess ndiins a die eny. Cmbining equain x = x i, = 0 wih Eqn.9, e - an be bained as: ( e = x i. (0 The adial piniple sess expessin an be bained by subsiuing Eqn.0 in Eqn.9 as fllw: x = x i (. ( A die exi, x = x, = ; a die eny, x = x i, = i, wing x / x i = / i, he adial pinipal sess expessin a die exi an be bained by subsiuing fmula x / x i = / i in Eqn. as fllw: = i (. ( The adial pinipal sess,, shuld be less han he ensile yield sengh f maeial,. Ading Eqn., he fllwing equain an be hen g as: e i (. (3 Ading Eqn.3, he fllwing equain an be bained as: 56

4 i. (4 The MCSSR f bille an be defined as: R=- / i, Then, he fllwing equain an be g by subsiuing Eqn.4 in R=- / i : R =. (5 i Using =, and finding he limi f Eqn.5, he Eqn. 5 f he effiien f MCSSR f bille based n Yu s unified sengh hey an be hanged in an equain based n Yu s unified yield iein. I is fund ha he effiien f MCSSR f bille in a single wie dawing pass based n Yu s unified yield iein is a nsan, whih is I had been eified ha he Mises and Tesa yield ieins ae jus speial ases f Yu s unified yield iein. Theefe, if he ensin sengh a maeial is equal is mpessin sengh, he MCSSR f bille in a single wie dawing pass is MCSRR f bille in a single wie exusin pass The pinipal sesses in bille duing wie exusin pess mee = > = 3 = θ. Theefe, he Eqn.9 is appliable f esimaing he pinipal sess in bille duing wie exusin pess. The inegal nsan in Eqn.9 an be deemined by he sess ndiins f x = x and = 0 a die exi. Subsiuing x = x and = 0 in Eqn.9, and leading : ( e = x. (6 The adial piniple sess expessin an be bained by subsiuing Eqn.6 in Eqn.9 as fllw: x = x (. (7 The adial pinipal sess expessin a die eny an be bained by subsiuing x / x i = / i, in Eqn.7 as fllw: i i = (. (8 The adial pinipal sess f i shuld be n less han he ensile yield sengh f maeial, -. Ading Eqn.8, he fllwing equain an hen be g as: i (. (9 Rewiing Eqn.9, and, hen Eqn.4 an be g again. Simulaneusly, Eqn.5 an als be bained. Thus i an be pved ha he MCSRR f bille in a single wie dawing pass is equal ha in a single wie exusin pass, suppsing he die is simila. 56

5 Cnibuins f a he MCSRR f bille in a single wie fming pass The uve f he maximum eduin, R vesus he TCR,, is shwn in Fig.. Suh fllwing sluins an be shwed fm Fig. as: R ineases wih ; when ineases, he value f R is 0.63; if he bille in a single wie fming pass is bile maeial, i.e., vaying fm , he elaive ange f he value f R is ; if he bille in a single wie fming pass is duile maeial, i.e., vaying fm 0.77, he elaive ange f he value f R is R Fig. Relainship beween R and Cnlusins The MCSRR f bille in a single wie dawing pass is equal ha in a single wie exusin pass. If he ensin sengh f a bille s maeial is equal is mpessin sengh, he MCSRR f bille in a single wie fming pess is If he bille is a bile maeial, he elaive ange f he MCSRR f bille in a single wie fming pess is If he bille is a duile maeial, he elaive ange f he MCSRR f bille in a single wie fming pess is Aknwledgemens This wk was finanially supped by he Fund Pgam f Sihuan Pvine Key Lab f Cmpehensive evelpmen and Uilizain f Indusial Slid Wase (SC_FQWLY0305, SC_FQWLY0403, Sihuan Eduain epamen Fund Pgam (6ZB0475, and Panzhihua Univesiy Fund Pgam (bkqj Refeenes [] R.A.C.Slae: Engineeing Plasiiy They and Appliain Meal Fming Pesses (Mehanial Pess, China 983. (in Chinese [] M.H. Yu, S.Y. Yang, C.Y. Liu: China Civil Engineeing Junal, Vl. 30 (997 N., p.4. (in Chinese [3] J.H. Zha: Sengh They and Is Appliains (Siene Pess, China 003. (in Chinese [4] M.H. Yu: Unified Sengh They and is Appliains (Spinge, Gemany 004. [5] R. Wang, Z.H. Xing, W.B. Huang: Fundain f Plasi Mehanis (Siene Pess, China 98. (in Chinese 563