Upper bound analysis of equal channel angular extrusion

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1 Downloded from orbit.dtu.dk on: Jun 06, 018 Upper bound nlysis of equl chnnel ngulr extrusion Lptev, Alexnder M.; Perig, Alexnder V.; Kkvs, Pnyiotis A.; Anyfntis, Konstntinos; Erfort, Yuri A. Published in: Proceedings of the 8th Interntionl Conference on Reserch nd Development in Mechnicl Industry Publiction dte: 01 Document Version Publisher's PDF, lso known s Version of record Link bck to DTU Orbit Cittion (APA): Lptev, A. M., Perig, A. V., Kkvs, P. A., Anyfntis, K., & Erfort, Y. A. (01). Upper bound nlysis of equl chnnel ngulr extrusion. In Proceedings of the 8th Interntionl Conference on Reserch nd Development in Mechnicl Industry Generl rights Copyright nd morl rights for the publictions mde ccessible in the public portl re retined by the uthors nd/or other copyright owners nd it is condition of ccessing publictions tht users recognise nd bide by the legl requirements ssocited with these rights. Users my downlod nd print one copy of ny publiction from the public portl for the purpose of privte study or reserch. You my not further distribute the mteril or use it for ny profit-mking ctivity or commercil gin You my freely distribute the URL identifying the publiction in the public portl If you believe tht this document breches copyright plese contct us providing detils, nd we will remove ccess to the work immeditely nd investigte your clim.

2 8 th Interntionl Conference Reserch nd Development in Mechnicl Industry RDMI September 008, Užice, Serbi UPPER BOUND ANALYSIS OF EQUAL CHANNEL ANGULAR EXTRUSION Alexnder M. Lptev 1, Alexnder V. Perig 1, Pnyiotis A. Kkvs, Konstntinos N. Anyfntis 3, Yuri A. Erfort 1 1 Donbss Stte Engineering Acdemy, Krmtorsk, UKRAINE, lptev@dgm.donetsk.u Technologicl Eductionl Institute of Ptrs, Ptrs, GREECE, kkvs@uptrs.gr 3 Ntionl Technicl University of Athens, Athens, GREECE, knyf@centrl.ntu.gr Summry: The reltive pressure nd totl sher t equl-chnnel ngulr extrusion (ECAE) in nonrectngulr die were determined by upper bound nlysis nd discrete velocity field. The obtined dt were compred with the slip line solution of V. Segl. Good greement between the two results ws found. Physicl modeling by plsticine confirmed the ppernce of ded zone t ECAE. The development of ded zone ws nlyticlly investigted. Keywords: Equl Chnnel Angulr Extrusion, Upper-Bound Anlysis, Physicl Modeling INTRODUCTION Equl Chnnel Angulr Extrusion (ECAE), lso known s Equl Chnnel Angulr Pressing (ECAP), involves one or severl psses of extrusion of lubricted billet in die with two intersecting chnnels of equl crosssection (Fig. 1). This process hs been very promising for production of bulk ultr fine-grined mterils with specil properties. The lrge cumultive sher strin t ECAE leds to refinement of grins nd to increse of their boundry length. The obtined mterils show the combintion of very high strength nd ductility [1]. P 1 θ θ 3 4 l 1 h A C β β O θ V 1 V 1 [V -3 ] [V n 3 h D B [V 1- ] [V -4 ] b l n [V 1 ] c Figure 1: ECAE lyout: 1 nd 4 - intersecting chnnels, - punch nd 3 - smple; b rigid blocks prtitioning scheme; c velocity hodogrph For n nlyticl investigtion of ECAE pressure nd resulting sher Segl V. M. pplied the slip line pproch [,3]. Beyerlein I. J. nd Tomé C. N. used kinemtic nlysis of strins during ECAE [4]. Alkort J. et l. [5], Altn B. S. et l. [6], Lptev A. M. et l. [7,8] nd Eivni A. R. et l. [9] studied ECAE by upper bound

3 nlysis. For visuliztion of metl flow during ECAE nd indiction of possible problems t rel process Mnn R. et l. crried out physicl modeling of this process [10]. In the present work we combine the physicl modeling nd theoreticl nlysis of ECAE by upper bound theory nd discrete velocity field. The development of ded zone ws investigted by optimiztion of nlyticl solution.. PHYSICAL AND MATHEMATICAL SIMULATION.1. Plsticine modeling First of ll the mteril flow t ECAE ws studied using plsticine. The rectngulr plsticine billet with crosssection of 30x0 mm ws extruded in the wood die s it is shown in Fig.. The die hd two equl chnnels with intersection ngle of θ=105. The mteril flow ws recorded by cmer through the trnsprent wll of the die mde from Plexigls. The tlcum powder ws used s lubricnt. The ppernce of the ded zone ner the externl die corner ws observed. The ded zone ws symmetricl t the beginning of ECAE. The ded zone retined its length in the entrnce chnnel during process development, but its length in outlet cnnel becme lrger. As result the ded zone trnsformed to symmetricl one. The gp between the plsticine billet nd the die wll ws lso found in the outlet chnnel. Gp Ded zone Ded zone Ded zone.. Upper bound nlysis Figure : The plsticine flow during ECAE According to the upper bound pproch tril velocity field hs to be introduced. The velocity field cn be continuous, discontinuous or mixed. In the present work the discontinuous velocity field ws used. The D plne model of smple ws divided into 4 rigid tringulr sections, s shown in Fig. 1b. The ppernce of symmetricl ded metl zone ws ssumed becuse its symmetry leds to violtion of mteril incompressibility t used rigid blocks division. This ssumption is in contrst to results of plsticine modeling, but it cn be used s first pproximtion. Opposite the more complex blocks division should be tried. The length of symmetricl ded zone ws chrcterized by h x. Here is the chnnel width; x 0;1, 0 is the reltive length of the ded zone in both entrnce nd outlet chnnels. The friction only long the lines AC nd DB ws tken into ccount by Tresc friction lw. Corresponding velocity hodogrph is shown in Fig. 1c. Further the ECAE pressure on the line AO ws clculted. The extruded mteril ws discussed s rigid-plstic with no strin-hrdening. The friction forces were ssumed s independent on sliding velocity. The blnce of externl nd internl powers t plstic deformtion ws expressed by eqution l1 V 1 l3v 3 l4v 4 mk l AC ldb 1 p V1 k V, (1) where p is pplied pressure; V 1 is velocity in entrnce nd output chnnels; k is sher strength of extruded mteril; m 0; 0,5 is friction fctor in Tresc lw expressed by eqution f mk ; li j is the length of join interfce of blocks i nd j ; V i j is reltive sliding velocity of these blocks. The terms in eqution (1) were expressed s functions of velocity V 1 nd reltive length of the ded zone x h. After substitution of obtined reltionships in eqution (1) nd lgebric trnsformtion, the formul for clcultion of reltive pressure ws derived p 1 x tg ctg x mctg x, () tg ctg x where is n intersection ngle between entrnce nd outlet chnnels (Fig. 1). 37

4 Equl chnnel ngulr extrusion is technique for grin refinement. Therefore the estimtion of resulting plstic ECAE sher is lso importnt. The totl ECAE sher is the sum of the shers on the discontinuity lines CO nd DO in Fig. 1b, i.e (3) It is known tht n i j [ V i j ] V i j, (4) where V n i j is velocity component orthogonl to discontinuity line l ij [11]. Using hodogrph in Fig. 1c the following reltionship ws obtined x 1 ctg x. (5) tg ctg According to the upper bound theory the best pproximtion of the rel p k nd corresponds to the minimum of expression (). The dependence of p k on reltive length of ded zone x, friction fctor m nd ngle ws numericlly investigted. The corresponding digrms re presented in Fig p 105 p 10 p b c Figure 3: Reltive ECAE pressure p/ versus x nd m t different chnnels intersection ngle Anlysis of () shows tht minimum of this function tkes plce when x 1 m 1 tg tg 1 m 1 tg 1 m tg. (6) In sme cses the globl minimum of p k is not reched t positive vlues of x. Then the miniml vlues of p k correspond t given m to x=0 (Fig. 3). This mens tht the ded zone does not lwys pper. The development of ded zone cn be seen in Fig. 4. The corresponding dt were clculted by (6) with restriction of 0 x 1. Thus ded zone lwys ppers t nonzero friction when 90, but its ppernce tkes plce only t m 0.13 if 105 nd t m 0.5 if 10. This result coincides well with our previous finite element simultions of ECAE [8] x=h/

5 Figure 4: Influence of friction nd die ngle on ppernce nd reltive height of ded zone 3. DISCUSSION The obtined results were compred with Segl s slip lines solution [, 3]. Corresponding to slip line nlysis the reltive ECAE pressure cn be clculted by formule [] p cot m sin sin cos, (7) where 1 rccosm.the summry sher is [3] cot. (8) The results of comprison re presented in Fig. 5. The good greement especilly for plstic sher ws found. p/ 1,55 1,45 1,35 1,5 1,15 1,05 0,95 Upper Bound, θ=90 Slip Lines, θ=90 γ,05,00 1,95 1,90 1,85 1,80 1,75 1,70 1,65 1,60 Upper Bound, θ=90 Slip Lines, θ=90 d p/ p/ 1,30 1,0 Upper Bound, θ=105 Slip Lines, θ=105 1,10 1,00 0,90 0,80 0,70 1,00 Upper Bound, θ=10 0,90 Slip Lines, θ=10 0,80 0,70 0,60 0,50 0,40 b c γ γ 1,80 1,75 1,70 1,65 1,60 1,55 1,50 1,45 1,40 1,35 1,60 1,55 1,50 1,45 1,40 1,35 1,30 1,5 1,0 1,15 1,10 Upper Bound, θ=105 Slip Lines, θ=105 e Upper Bound, θ=10 Slip Lines, θ=10 f Figure 5: Comprison of upper bound nd slip lines solutions t different ngels θ nd friction fctors m As expected, the vlues of reltive ECAE pressure obtined by upper bound nlysis re slightly higher thn corresponding ones derived by slip line theory. The reltive divergence of results ws evluted by formul R R R 100, (9) % SL UB SL where R SL nd R UB re vlues, obtined by slip line nd upper bound theory respectively. The evlution shows tht t two-wy p/ clcultions in the rnge of m 0; 0,3 is not more thn 7,5% t θ=90, 9,5% t θ=105, nd 13,3% t θ=10. Thus the mximum disrrngement in reltive ECAE pressure does not exceed 39

6 13,3%. Similr comprison shows tht the lrgest t two-wy clcultions in the rnge of m 0; 0,3 is =5,5% t θ=90, =3,75% t θ=105 nd =5,5% t θ=10. Thus the mximum disrrngement in summry sher does not exceeds 5,5%. Both theories predict sufficient decrese in p/ nd when θ ngle grows from 90 to 10. The increse of friction fctor leds to rise of the extrusion pressure nd to remrkble decrese in totl sher. 4. CONCLUSION The ppliction of the upper bound theory to nlysis of ECAE in rectngulr nd nonrectngulr dies llows correctly describe the essentil fetures for this process like ppernce of ded zone nd its increse depending on externl friction. Physicl modeling by plsticine confirms the formtion of ded zone. Also the increse of ECAE pressure nd decrese of totl plstic sher t rise of friction is well predicted. Finlly the decrese of both ECAE pressure nd summry sher t growing chnnel intersection ngle is correctly modeled. The upper bound results bsed on discontinuous velocity field re in good greement with clssicl Segl s slip line solution. The proposed upper bound pproch cn be pplied t further nlysis of ECAE in the dies with more complex geometry. REFERENCES [1] VALIEV, R. Z., LANGDON, T. G.: Principles of equl-chnnel ngulr pressing s processing tool for grin refinement. Progress in Mterils Science, Vol. 51 (006), No 7, pp ISSN ISSN: [] SEGAL, V. M.: Engineering nd commerciliztion of equl chnnel ngulr extrusion (ECAE). Mterils Science nd Engineering A, Vol. 386 (004), No. 1-, pp ISSN [3] SEGAL, V. M.: Slip line solutions, deformtion mode nd loding history during equl chnnel ngulr extrusion. Mterils Science nd Engineering A, Vol. 345 (003), No. 1-, pp ISSN [4] BEYERLEIN, I. J., TOMÉ, C.N.: Anlyticl modeling of mteril flow in equl chnnel ngulr extrusion (ECAE). Mterils Science nd Engineering A, Vol. 380 (004), No. 1-, pp ISSN: [5] ALKORTA, J., SEVILLANO, J. G.: A comprison of FEM nd upper-bound type nlysis of equl-chnnel ngulr pressing (ECAP). Journl of Mterils Processing Technology, Vol. 141 (003), No. 3, pp ISSN [6] ALTAN, B. S., PURCEK, G., MISKIOGLU, I.: An upper-bound nlysis for equl-chnnel ngulr extrusion. Journl of Mterils Processing Technology, Vol. 168 (005), No. 1, pp ISSN [7] LAPTEV, A. M., VYAL, E. Yu., PERIG, A. V.: An nlysis of equl-chnnel ngulr extrusion using rigid blocks method (in Russin lnguge). In: Improvement of processes nd mchines for metls plstic working in metllurgy nd mchine building, Krmtorsk, Donbss Stte Engineering Acdemy, (006), pp ISBN [8] PERIG, A. V., KAKAVAS, P. A., ANYFANTIS, K. N., LAPTEV, A. M.: Mechnics of the equl chnnel ngulr extrusion. In: Proceedings of XXXVI Interntionl Summer School-Conference "Advnced Problems in Mechnics- 008", St. Petersburg, Russi, July 008, St. Petersburg, IPME RAS, (008), pp ISBN [9] EIVANI, A. R., TAHERI, A. K.: An upper bound solution of ECAE process with outer curved corner. Journl of Mterils Processing Technology, Vol. 18 (007), No. 1 3, pp ISSN [10] MANNA, R.; AGRAWAL, P.; SUSHANT, J.; MUDDA, B. K.; MUKHOPADHYAY, N.; SASTRY, G.V.: Physicl modeling of equl chnnel ngulr pressing using plsticine. Script Mterili, Vol. 53 (005), No. 1, pp ISSN [11] STEPANSKY, L. G.: Clcultions of plstic metl working processes (in Russin lnguge). Moscow: Mshinistroenie, pp