MATHEMATICAL MODEL OF WARM DRAWING OF MgCa0.8 ALLOY ACCOUNTING FOR DUCTILITY OF THE MATERIAL

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1 Iformatyka w Techolog Materałów Publhg Houe AKAPIT Vol.,, No. MATHEMATICAL MODEL OF WARM DRAWING OF MgCa.8 ALLOY ACCOUNTING FOR DUCTILITY OF THE MATERIAL ANDRZEJ MILENIN *, PIOTR KUSTRA, MAREK PAĆKO AGH Uverty Scece Ad Techology, Krakow, Polad *Correpodg Author: mle@agh.edu.pl Abtract The reult expermetal ad umercal aaly MgCa.8 mageum alloy are preeted paper. Bag o expermetal tet flow tre fucto ad depedece betwee ductlty tra, traxalty factor, temperature ad tra rate were obtaed. Expermet wa performed o tetg mache Zwck Z5. The algorthm baed o vere method wa ued to terpret correctly expermetal reult. The FEM modellg upettg ad teo tet wa helpful to obta codto materal fracture. The developed model mechacal properte were mplemeted to Drawgd FEM code ad mulato drawg urgcal thread were performed. The troduced approach eabled modellg drawg proce at elevated temperature, accoutg for materal fracture. Key word: MgCa.8 alloy, wre drawg, FEM, urgcal thread. INTRODUCTION Mageum alloy are more te ued aeropace, automotve, electroc etc. dutre. Thee alloy are formed mlarly to or metal, ug procee rollg, forgg, extruo, tampg ad may or techologcal procee applcable at elevated temperature (Wataebe et al., 4; Ogawa et al., ; Swotek et al., 6; Cheg, et al., 7). Performed tude by Bach et al. (6), Heuble et al. (999); Haferkamp et al. (), Wa et al. (8) have how that by certa mprovemet chemcal compoto (uually addg a mall amout Ca, L), mageum alloy acheve a hgh level bocompatblty wth huma body ad dolve body wthout gfcat medcal problem. Several ew mageum alloy for bomedcal applcato (uch a MgCa, LAE44, MgCa.8) were developed at Uverty Haover (Bach et al., 6; Thoma et al., 8). Producto urgcal thread to tegrato tue ca be a example applcato e type alloy. Thee applcato requre fe wre wth dameter from. mm to.9 mm. Due to poor formablty ad lmted ductlty mageum alloy room temperature, drawg proce to dmeo. mm dffcult. Low ductlty mageum alloy correpod to r hexagoal cloe packed tructure. Thu, proce cold formg practcally mpoble (Eckemeyer et al., 4). Vo Me (98) wrte that platc deformato eed mmum 5 depedet lp ytem. I cae mageum alloy at room temperature re are oly 3 depedet lp ytem. Therefore, t eceary to creae umber lp ytem, for tace by rag temperature. I Bach et al. work (5) a ew maufacturg techology tube made Mg alloy propoed. I th techology metal heated by a hot de ad proc ISSN

2 e warm deformato performed. The oretcal decrpto wre drawg proce wth a heated de preeted by Bach et al. (7) ad Mle ad Kutra (8). The model materal ductlty a very mportat part FE program for mulato drawg. Avalablty th model eable optmzato proce wre drawg o ba FE mulato. Fracture problem for mageum alloy are precely decrbed lterature (Eckemeyer et al., 4; Mle & Kutra, 8; Yohda, 4). However, e work accout for oly few parameter drawg, uch a de agle ad reducto rato. Mageum alloy cotag alumum ad zc (uch a AZ3) are vetgated materal, whch have a bgger platcty tha Mg-Ca alloy. However, ductlty model Mg-Ca alloy are care lterature. The yeld tre model latter alloy for warm deformato are ot avalable lterature, er. The purpoe th paper developmet mamatcal model yeld tre ad ductlty for MgCa.8 alloy, mplemetato e model to FE code ad mulato wre drawg procee heated de.. FEM MODEL OF PLASTIC DEFORMATION IN DRAWING The FE code Drawgd developed by Mle (5) ued preet work. The FE model olve a boudary problem coderg uch pheomea a metal deformato, heat trafer a de ad a wre, metal heatg due to deformato ad frcto. Soluto boudary problem obtaed by ug varato prcple rgd-platc ory: J where: V V (,, t) ddv σξ dv v ds, () tra rate, S yeld tre, effectve tra, t temperature, V volume, σ mea tre, volumetrc tra rate; S cotact area betwee alloy ad de, frcto tre, v alloy lp velocty alog area de. The frcto tre determed accordg to law:.5 ftr exp, () 3 where: f tr frcto coeffcet, ormal tre o cotact betwee deformed alloy ad de. The tre teor j calculated o ba tra rate teor j accordg to followg equato: j j j. (3) 3 The tatoary formulato th tak ued paper. The teor j calculated by tegrato alog flow le: j p p ( ) d, (4) j p where: tme cremet, j tra rate teor determed accordg to equato: j d j N j, (5) where: N fte elemet hape fucto, j odal tra rate teor for curret fte elemet, d umber ode elemet. The pot flow le are determed o ba value velocty at pot p, whch are calculated accordg to followg formula: v d N v. (6) The calculato poto ext pot (p+) flow le carred out accordg to equato: ( p ) ( p x x v ). (7) 3. THE FEM SOLUTION OF THE THERMAL PROBLEM IN METAL Thermal problem olved by applyg followg method. The paage ecto through zoe deformato mulated. For th ecto at each tme tep o-tatoary temperature problem examed: t t Q r r r d dt c d (8) 7

3 where: Qd.9 deformato power, c pecfc heat; alloy dety, tme, rmal coductvty coeffcet ( followg value are ued for MgCa.8 alloy: c = 64 J /kgk, = 738 kg /m 3, = 6 J /mk), r - y cyldrcal coordate. Heat exchage betwee alloy ad de defed a: q cov t de t (9) where: t de de temperature, heat exchage coeffcet. The geerato heat from frcto calculated accordg to formula: q fr. 9 v. () 4. FEM SOLUTION OF THERMAL PROBLEM IN THE DIE The model temperature dtrbuto de baed o oluto Fourer equato cyldrcal coordate ytem: t t t Q h () r r r y where: Q h power heatg elemet. The heat Q h geerated fte elemet, whch correpod to poto heatg devce. The boudary problem olved o ba varato formulato equato (). For area, whch are cotact wth metal, temperature alloy obtaed from oluto rmal problem for metal. m m ( m) d k, t, m k, t p p, where: tme deformato, tme cremet, ( m), (3) (m) curret (m) value tra rate curret tme, m dex umber tme tep durg umercal tegrato alog flow le. k, t, ca be Crtcal deformato fucto obta o ba expermetal reult for upettg ad teo tet at dfferet value k, t,. 6. EXPERIMENTAL STUDIES MgCa.8 mageum alloy wa ued a a tetg materal. Fgure how dmeo ad hape ample for upettg ad tele tet. Upettg tet were ued to determe flow tre model. Both upettg ad tele tet were ued to calculate coeffcet crtcal deformato fucto. The materal tet were performed a Zwck Z5 tetg mache at AGH Uverty Scece ad Techology Krakow, Polad. Reult ad codto all tet are preeted table ad. The rage temperature ad tra rate chage expermet wa elected accordace wth codto for deformato metal durg wre drawg heated de. p 5. DUCTILITY MODEL The key parameter, whch preet fracture, wa amed ductlty fucto. Th parameter determed by followg formula:, () k, t, where: k traxlty factor, p k /. Crtcal deformato fucto k t, p, obtaed ad baed o expermetal tude. I Drawgd FEM code equato () wa mplemeted a a followg tegral: Fg.. Shape ad dmeo ample for upettg tet ad tele tet. 7

4 7. YIELD STRESS MODEL Yeld tre model for aalyzed alloy wa propoed a a modfed Hael-Spttel equato: ( 4) a egeerg model proce upettg vere aaly. Reult th aaly are how fgure. A relatve error objectve fucto wa.55. The value coeffcet equato (4) are preeted table 3. Plot rela- to (4) for dfferet temperature ad tra rate are how fgure 3. where: A, m m 9 emprcal coeffcet. Expermet how that depedet tra rate low temperature (below C), coe- quetly, expreo to (4). t m 6 8 Table. Codto ad reult upettg tet MgCa.8. Tet Nr. t, C v, mm/m dh, mmm Crtcal deforma- to k Stra rate, / Shape ample after tet * (wthout crack) * (wthout crack) wa added to equa- *The detructo ample ddd ot occur. The coeffcet equato (4) were deter- med ug vere approach (Szelga & Petrzyk, 7) wth leat quare method. The objectve fucto wa formulated a root-mea- quare dfferece betwee expermetal ad pre- dcted load. I mache Zwck Z5 ample are heated toger wth deformg tool. Coe- quetly, value gradet temperature model gfcat. It make poble to ue Fg.. Force veru dplacemet tool (deformato ample) upettg tet: tet umber,3,5, 7,8 from table ; tet umber,,4,6 from table, (th le modelg, thck le expermet) ). 7

5 INFORMATYKA W TECHNOLOGII MATERIAŁÓW Table. Codto ad reult tele tet MgCa.8. Tet Nr. T, C V, mmm /m dl, mmm Crtcal deformato tra Stra rate, / / k Shape ample after tet Table 3. Coeffcet yeld tre equato (4) A m.754 m.4485 m m 4 m m 6 m m m p d e xp d kexp d t 3 8. CRITICAL DEFORMATION FUNCTION Data table ad how that crtcal defor- mato depedet o temperature, tra rate ad value k coeffcet ( upettg tet crtcal deformato hgher, tha tele tet). Due to th fact followg relatohp propoed a a fucto tra lmt: where: d - d 4 emprcal coeffcet. The followg algorthm propoed to calculate parameter equato (5):. Ug Forge tware umercal mula- to all expermetal tet have bee performed. Flow tre model wa mplemeted form equato (4) ad coeffcet from table 3 weree adopted.. Performed umercal modelg all tet (upettg ad tele) to determe chage tem- perature, tra rate ad k coeffcet from be- gg tet utl materal crackg expermet. d 4, (5) 3. Bag o aaly ample, loca- t wa oberved that crack ucleato take place to crack tato wa foud. For upettg tet a corer ample (fgure 4 ), whle te- le crackg beg ax ample. Read- wa doe e pot, whch correpod to g proce dataa from FEM model tet locato formato materal crack. Ex ample reult calculato are how fgure. 4-5 for tet umber 3 from table ad fgure. 6-7 for tett umber table. I upettg tet umber ad crack wa ot t- (developed model (5) hould gve for thoe expe- rmet ). ated, o th data wa ued oly to verfy model 73

6 Flow tre, MPa 3 5 5,,5 5 Flow tre, MPa 3 5,5,5 Stra 5 Flow tre, MPa Flow tre, MPa 5,5 Stra (c),,5,,5,5 Stra,,5,, Stra (d) Fg. 3. Stre tra curve MgCa.8 mageum alloy for temperature 3 C, 5 C, (c) C, (d) 5 C ad tra rate.,.,.5 ad -. Fg. 4. Smulato reult: dtrbuto effectve tra for tet 3 from table, hape ample after tet 3 wth marked locato crack. 4. The et parameter k, tra rate ad temperature tele ad compreo tet were ued to develop coeffcet d - d 4 fucto (5). Coeffcet were calculated ug leat quare method. Chage wa decrbed by equato (3). The um quare dfferece betwee m ad. for tra, whch correpod to calc crack oberved expermet, wa ued a a goal fucto for curret tet: m tet calc. (6) m I proce mmzato fucto (6) followg parameter equato (5) were obtaed: d =.53; d =.87; d 3 =.575; d 4 = The average relatve error approxmato.4. The depedece crtcal deformato fucto o tra rate ad temperature are how fgure 8. m 74

7 (c) (d) (c) (d) Fg. 5. The reult mulato tet umber 3 from table for area marked fg 4: chage value k, temperature, (c) effectve tra ad (d) yeld tre. Fg. 6. The reult mulato tet umber from table : dtrbuto effectve tra, hape ample after tet, dtrbuto ma tre (c) ad hape ample eck after tet (d). 75

8 9. ANALYSIS OF THE WIRE DRAWING PROCESS Developed model flow tre (4) ad ductlty (5) were mplemeted to Drawgd tware. For aaly damage materal three varat drawg proce were aalyzed. Geeral data gve for all 3 varat (commo data) are: coeffcet heat exchage wth evromet 4 W/m C; de temperature t c = 35 C, temperature wre at etrace to valley tra t = C; tal wre dameter d =.5 mm, drawg agle = 4 ; legth calbrato part de L =. mm; radu trato from cocal part calbrato r =.5 mm, coeffcet frcto f =.3; drawg agle = 4. (c) (d) Fg. 7. The reult mulato tet umber from table for ceter pot ample: chage value k, temperature, (c) effectve tra ad (d) yeld tre. Fg. 8. Depedece crtcal deformato fucto o effectve tra for k = -.33 (dotted le) ad k =.33 (old le) temperature C, C, C, 5 C ad 3 C. The followg umercal aalye were performed: Varat. Fal dameter d =.38 mm; v =.5 m /; Varat. Fal dameter d =.46 mm; v =.5 m /; Varat 3. Fal dameter d =.46 mm; v =. m /. The mulato reult are how fgure 9. Varat correpod to elogato factor.73. The expermetal kowledge baed o drawg Mg alloy how that wth uch a large elogato factor, a break wre hould be oberved. The aaly tra dtrbuto (fgure 9a) make t poble to aert that deformato localzed place applcato force wre drawg. 76

9 Th dcate break wre at output from zoe deformato. Parameter take maxmum value ( max =.6) ame place, whch mea that ductlty fucto allow to predct ot oly fracture deformato zoe, but alo outde de (fgure 9b). The ecod varat mulato wa performed for a maller elogato factor (.8). I th cae alo fracture draw wre predcted ( max =.), but locato max ax draw wre deformato zoe. I thrd vero mulato peed wre drawg wa lowered. Th led to a creae temperature metal ad coequetly, amout crtcal deformato. A a reult max =.86 ad fracture metal dd ot occur. (c) (d) Fg. 9. Reult mulato:, (c), (e) dtrbuto effectve tra,, (d), (f) dtrbuto parameter for mulato varat :, ; varat : (c), (d); varat 3: (e), (f). (e) (f) 77

10 9. CONCLUSIONS Stude dedcated to fracture materal at dfferet value temperature, tra rate ad traxlty factor howed that ductlty MgCa.8 mageum alloy trogly depedet o temperature ad tra rate. Therefore, developed yeld tre model ad ductlty fucto take to accout border effect tra rate ad temperature. Reearch performed for upettg tet ad vere method were helpful to determe coeffcet yeld tre model. Stude for upettg ad tele tet were ued to calculate coeffcet ductlty fucto. The developed yeld tre model ad ductlty fucto were mplemeted to Drawgd tware. Addtoally, aaly how that:. The FE model mageum alloy MgCa.8 wre drawg proce, model yeld tre ad ductlty model temperature rage - 3 C were developed.. Expermetal - oretcal methodology to calculate parameter emprcal yeld tre ad ductlty model MgCa.8 mageum alloy wa developed. 3. Followg developed methodology expermetal tude were performed, whch are eceary to obta parameter yeld tre ad ductlty model materal. 4. The FE mulato how example dfferet mecham materal fracture durg drawg: break wre at ext from zoe deformato ad exhauto platcty materal deformato zoe deformato. ACKNOWLEDGMENTS The author would lke to thak Mtry Scece ad Hgher Educato Polad, project o. 46/N-DFG-SFB/9/. REFERENCES Bach Fr.-W., Mle A., Kuchark R., Borma D., Kutra P., 7, Modelowae za pomocą MES proceu cągea drutów ze topu magezu wykorzytywaych w chrurg, Hutk-Wadomośc Hutcze, 74, 8- ( Polh). Bach F.-W., Kuchark R., Borma D., 6, Mageum compoud tructure for treatmet boe defect, Egeerg Bomateral, 56-57, Bach Fr.-W., Hael T., Golovko A.N., 5, The fluece chemcal compoto ad extruo parameter o mechacal properte th-walled tube made mageum-calcum alloy, Sucza problemy metalurg, Naukov vt, 8, Sytem techolog, Eckemeyer J., Guth A., Falter M., Optz R., 4, Drawg mageum wre at Ambet temperature, Proc. 6th It. Cof., Mageum alloy ad r Applcato, WILEY-VCH, Haferkamp, H., Kaee, V., Nemeyer, M., Phllp, K., Pha-Ta, T., Heuble, B., Rohde, R,., Explorato Mageum Alloy a New Materal for Implatato; Mat.- w. u. Werktftech, 3: Wley-VCH Verlag GmbH, Wehem, 6-. Heuble, B., Rohde, R., Nemeyer, M., Kaee, V., Hartug, W., Röcke, C., Haudorf, G., Haverch A., 999, Degradato mageum alloy: A ew prcple cardovacular mplat techology, Paper TCT-69,. Aual Sympoum Tracater Cardovacular Therapeutc, The Amerca Joural Cardology, Expcerpta Meda Ic., New York. Mle, A., 5, Program komputerowy Drawgd arzędze do aalzy proceów techologczych cągea welotopowego, Hutk-Wadomośc Hutcze, 7, -4 ( Polh). Mle, A., Kutra, P., 8, The multcale FEM mulato wre fracture pheomea durg drawg Mg alloy, Steel Reearch Iteratoal, 79, pec. ue Cof. Metal Formg,, Ogawa, N., Shom, M., Oakada, K.,, Formg lmt mageum alloy at elevated temperature for preco formg. Iteratoal Joural Mache Tool & Maufacture, 4, Swotek, J., Goke, J., Letzg, D., Kaer, K.U., 6, Hydrotatc extruo commercal mageum alloy at C ad t fluece o gra refemet ad mechacal properte, Materal Scece ad Egeerg A., 44, 3-9. Szelga, D., Petrzyk, M., 7, Tetg vere tware for detfcato rheologcal model materal ubjected to platc deformato, Archve Cvl ad Mechacal Egeerg, 7,, Thoma, M., Kraue, C., Borma, D., Vo der Höh N., Wdhage, H., Meyer-Ldeberg, A., 8, Bomateral comparo reorbable mageum alloy LAE44 ad MgCa,8 cocerg r mechacal properte, gradet degradato ad boe-mplatcotact after moth mplatato rabbt model, NRW - Fudametal ad Clcal Applcato, 3, 7-8. Vo Me, R., 98, Mechak der Platche FormaÈderug vo Krtalle, Z. Agrew. Math. Mech., 8, 6. Wa, Y., Xog, G., Luo, H., He, F., Huag, Y., Zhou, X., 8, Preparato ad characterzato a ew bomedcal mageum-calcum alloy, Materal ad Deg, 9, Wataebe, H., Muka, T., Ihkawa, K., 4, Dfferet peed rollg AZ3 mageum alloy ad reultg mechacal properte, Joural Materal Scece, 39, Cheg, Y. Q., Che, Z. H., Xa, W. J., 7, Drawablty AZ3 mageum alloy heet produced by equal chael agular rollg at room temperature, Materal Characterzato, 58, Yohda, K., 4, Cold drawg mageum alloy wre ad fabrcato mcrocrew. Steel Grp,,

11 MATEMATYCZNY MODEL CIĄGNIENIA NA CIEPŁO STOPU MGCA.8 UWZGLĘDNIAJĄCY PLASTYCZNOŚĆ MATERIAŁU Strezczee W ejzej pracy przedtawoo wyk badań topu magezu MgCa.8. Bazując a wykach ekperymetu opracowao model aprężea uplatyczającego oraz utraty pójośc materału w zależośc od prędkośc odkztałcea temperatury. Ekperymet wykoao a mazye wytrzymałoścowej Zwck Z5. Do aalzy wyków wykorzytao metodę vere. Modele umerycze tetów a pęczae rozcągae pomogły w określeu waruków utraty pójośc aalzowaego materału. Opracowae modele materału zamplemetowao do autorkego oprogramowaa Drawgd bazującego a metodze elemetów kończoych. Take podejśce pozwol a aalzę proceu cągea w podgrzewaych cągadłach z uwzględeem kryterum utraty pójośc materału. Receved: March 3, Receved a reved form: May 8, Accepted: May 7, 79