Finite Element Analysis and Optimization for the Multi- Stage Deep Drawing of Molybdenum Sheet

Transcription

1 Fnte Element Analyss and Optmzaton for the Mult- Deep of Molybdenum Sheet Heung-Kyu Km a,*, Seok Kwan Hong a, Jong-Kl Lee b, Byung-Hee Jeon c, Jeong Jn Kang a, and Young-moo Heo a a Precson Molds and Des Technology Team, Korea Insttute of Industral Technology b Department of Mechancal Engneerng, Korea Polytechnc Unversty c Department of Mechancal Engneerng and Desgn, Induk Insttute of Technology Abstract. Molybdenum, a bcc refractory metal wth a meltng pont of about 2600 C, has a hgh heat and electrcal conductvty. In addton, t remans strong mechancally at hgh temperatures as well as at low temperatures. Therefore t s a technologcally very mportant materal for the applcatons operatng at hgh temperatures. However, a multstage process s requred due to the low drawablty for makng a deep drawn part from the molybdenum sheet. In ths study, a mult-stage deep drawng process for a molybdenum crcular cup was desgned by combnng the drawng wth the ronng, whch was effectve for the low drawablty materals. A parametrc study by FE analyss for the mult-stage deep drawng was conducted for evaluaton of the desgn varables effect. Based on the FE analyss result, the multstage deep drawng process was parameterzed by the desgn varables, and an optmum process desgn was obtaned by the process optmzaton based on the FE smulaton at each stage. 1. INTRODUCTION Molybdenum, wth a meltng pont of about 2600 C, s a mechancally strong materal even at hgh temperatures as well as at low temperatures, and thus t can be used for the applcatons operatng at hgh temperatures [1-3]. However, because of the low lmtng drawng rato and the easy oxdaton of the molybdenum at hgh temperatures, a mult-stage cold formng process s requred for makng a deep drawn part from the molybdenum sheet. The manufacturng cost for the mult-stage process s related to the number of stage nvolved, and so reducng the number of stage as many as possble s one of the process desgn objectve. For that purpose, the respectve formng stages should be carefully desgned and arranged n order to avod a falure durng the specfed numbers of stages. Generally, desgn of such optmum mult-stage process requres a tremendous amount of tral-and-error usng the expermental or the analytcal approach [4]. Thus, n many cases, we have to be satsfed wth a good desgn rather than the optmum desgn. In the present paper, the mult-stage deep drawng process for a molybdenum cup was nvestgated expermentally and theoretcally usng FE analyss, and fnally the optmum desgn of the mult-stage process was attempted by applyng an optmzaton algorthm to the overall stage. Due to the tremendous amount of calculaton tme, the optmzaton was conducted under the constrant of fxed number of stage. 2. MULTI-STAGE DEEP DRAWING PROCESS AND PARAMETRIC STUDY BY FINITE ELEMENT ANALYSIS The molybdenum sheet usually has a low drawng rato, and thus t requres many drawng stages to be transformed nto a deep drawn cup shape. To estmate the maxmum strokes per stages and obtan the materal propertes for FE analyss, a tenson test of the molybdenum sheet was conducted. The test procedure wth the result and the consttutve modelng are descrbed n the followng. 873

2 2.1 Unaxal Tenson Test and Plastc Behavor Modelng Unaxal tenson test of molybdenum sheet was conducted to measure the flow stress curve. As the asreceved molybdenum sheet has the planar ansotropy, the test was conducted for the specmens cut n the three dfferent orentatons of 0, 45, and 90 from the sheet. The test results for the respectve orentaton specmens are shown n fgure 1. True stress [Mpa] o 45 o 90 o True stran [mm/mm] FIGURE 1. Stress-stran curves for 0, 45, and 90 obtaned by unaxal tenson test Though the molybdenum sheet has the ansotropy n the sheet plane as shown n fgure 1, we assume the sheet as an sotropc materal for smplcty n the present study because the 3-D FE analyss usng the ansotropc materal model s computatonally too expensve and also the ronng process, explaned n the followng, requres the local deformaton analyss pror to the ansotropy analyss. Thus, the flow stress curves from the three orentatons were averaged, smlar to the r-value averagng, as σ σ + 2σ + σ = (1) predct the falure n the drawng process because the man deformaton n the cup drawng s plane-stran stretchng nstead of unaxal tenson. Then the crtcal effectve stran s calculated, smlar to the unaxal tenson, as ε = n n (3) * 2 From equaton (2) and (3), t s assumed that the * crtcal stran n the plane-stran drawng, ε, s about /2 tmes that n the unaxal tenson, ε * 1. Therefore we can approxmate /2 tmes the lmt stran n the unaxal tenson test as the lmt stran n the sngle-stage cup drawng. Ths lmt stran provdes an mportant crteron for sngle-stage drawng process desgn. However, as dscussed below, the mult-stage deep drawng process ncludes the ronngs as well as the pure drawngs, and the falure behavor of the ronng s dfferent from that of the drawng. For the evaluaton of the falure crteron such as the crtcal stran for the drawng-ronng combned process, the falure mechansm durng the ronng as well as the ronng effect on the materal flow stress should be nvestgated. 2.2 Intal Mult- Process Desgn and Expermental Try-Out Fnal target cup dmensons and the cup desgn varables for the ntermedate stages are shown n fgure 2 and table 1. For the cup desgn varables, 8- stage process was ntally desgned as shown n table 1 based on the followng rule of thumb; drawng rato desgn consderng the formng dffculty, desgn parameter desgn consderng the process contnuty (de corner radus, ntake angle, etc.), process arrangement consderng the volume constancy, and ronng process ncluson based on the experence. Thn sheet specmen loaded n the unaxal tenson test experences the localzed neckng, and then the localzed neckng fnally leads to a falure of the specmen. If the flow stress behavor of the sheet s n descrbed as a power law, σ = Kε, the crtcal stran for the localzed neckng becomes [5] * ε 1 = 2n (2) where we denote the stretchng drecton as 1. However, the crtcal stran for the localzed neckng under plane-stran condton s necessary to (a) (b) FIGURE 2. (a) Fnal target cup dmensons and (b) cup desgn varables for the ntermedate stages 874

3 TABLE 1. Intal desgn of the cup desgn varables at each stage for 8-stage process t = 0.5(D 0 -D t ) #1 #2 #3 #4 #5 #6 #7 #8 DO t L *, R d e (degree) - For the evaluaton of the ntal desgn, the multstage deep drawng was conducted usng the machned drawng des. Because of the low drawablty of the molybdenum sheet as well as the dffculty of the deep drawng process, the expermentally obtaned product showed the falure n the wall near the cup bottom at the stage #7. The expermental try-out results are shown n fgure 3. Illlllll^^l^ FIGURE 3. Expermental try-out result at each stage for the ntal desgn 2.3 Parametrc Study by FE Analyss The rato of wall thckness to cup dameter of ths nvestgaton s relatvely large compared wth other drawn cups assumed as sheet metal formng product. In addton, the thckness of the sheet s reduced by the ronng process. Therefore, the sold element modelng seems to be more sutable for the FE analyss of the local materal flow by the ronng than the shell element modelng. In ths study, the molybdenum sheet was modeled usng multple numbers of fnte elements across the thckness drecton. Because the ronng usually provdes more unform cup heght dstrbuton than the pure drawng, and the 3-D FE analyss consderng the ansotropy of the materal s computatonally expensve for mult-stage smulaton, 2-D FE analyss based on the sotropc materal behavor was conducted n the followng parametrc study and the process optmzaton. For the FE smulaton, the commercal code MARC was utlzed. The wall thckness of the cup must be reduced from 0.2 to O.lmm durng the mult-stage deep drawng ' because the thckness of the as-receved sheet was 0.2mm. The wall thckness reducton at each stage can be accomplshed by controllng the clearance, c, between the punch and the lower de. If the clearance s smaller than the wall thckness, the current stage s regarded as an ronng process. If not the case, the current stage s regarded as a drawng process. For convenence, we consder only the cases when the clearance s smaller than or equal to the wall thckness. In that case, we assume that c = t for drawng, and c < t for ronng. For the evaluaton of the ronng effect on the fnal cup qualty, FE analyss was conducted for 6 cases of dfferent sequental clearance desgns ncludng the ntal desgn. In the present study, for convenence, only the clearances at stages #2, #3, #4 and #5 were perturbed as shown n table 2. The maxmum effectve stran after the fnal stage, ^, or the falure occurrng stage predcted by FE analyss are also shown n table 2. TABLE 2. Dfferent sequental clearance desgns at stages #2 9 #3 9 #4, and #5 9 and FE analyss predcton ( 5 : ntal desgn defned n table 2) #2 #3 #4 # Clearance selecton at the early stages should be carefully determned by consderng the fnal product qualty. In the current molybdenum cup deep drawng, t s assumed from the FE analyss result that the stage #2 should be drawng rather than ronng because the maxmum effectve stran after the fnal stage was lower or the process was more successful n cases of drawng than ronng at the stage #2. Consderng the materal flow predcton by FE analyss, t s assumed that the flange formed n the early stages hndered the materal flow durng the drawng or the ronng n the later stages. Therefore, for mprovng the materal flow, the flange should be mnmzed as much as possble by drawng n the early stages. 875

4 The maxmum stroke at each stage wthout causng falure can be controlled by the clearance. For evaluaton of the maxmum stroke, FE analyss was conducted by usng dfferent clearances for the stage #6 of the ntal desgn. Pushng the punch downward untl the falure occurred, the maxmum possble strokes were predcted for a drawng and ronng as shown n fgure 4. However, the maxmum stroke was also affected by the de corner radus, R d. Usng the corner radus of 2.0 nstead of 0.5, the maxmum strokes decreased as shown n fgure 5. evaluaton of the effect, FE analyss was conducted by usng dfferent ntake angles for the stage #6 of the ntal desgn. FE analyss result shows that the maxmum stroke ncreases as the ntake angle ncreases, as shown n fgure 6. The reason s because the frctonal force between the flange and the de hndered the materal flow nto the punch-de gap, and the flange-de contact area decreases as the ntake angle ncreases. (a) stroke=4.13 (b) stroke=5.14 (a) stroke=4.19 (b) stroke=5.14 (c) stroke=5.77 FIGURE 6. Predcted maxmum stokes durng the stage #6 for ronng; c = 0.09, (a) θ = 15, (b) θ =, and (c) θ = 60 FIGURE 4. Predcted maxmum stokes durng the stage #6 for de corner radus=0.5, (a) drawng; c =, and (b) ronng; c = OPTIMIZATION OF MULTI-STAGE DEEP DRAWING PROCESS BASED ON FINITE ELEMENT SIMULATION 3.1 Optmzaton Procedure (a) stroke=4.02 (b) stroke=4.98 FIGURE 5. Predcted maxmum stokes durng the stage #6 for de corner radus=2.0, (a) drawng; c =, and (b) ronng; c = 0.09 De ntake angle, θ, as well as the de corner radus had an effect on the maxmum stroke. For As dscussed above, varous desgn varables have an effect on the feasblty of the desred deep drawng. Therefore, the optmum mult-stage deep drawng process desgn requres the consderaton of the effects by those desgn varables. However, the effects by the varables are cross-related due to the nonlnearty n the geometry and the materal response, and so the effects by the respectve varables provde only the ntuton for process desgn. Therefore, a systematc optmzaton should be used wth the FE smulaton because the FE analyss based tral-and-error as well as the expermental approach requres tremendous amount of tme and cost for the optmum desgn of the mult-stage process. Of course, the FE smulaton should be valdated by experment f the FE analyss based optmzaton could provde a meanngful result. 876

5 In the present nvestgaton, the commercal optmzaton code SIGHT was used to desgn the optmum mult-stage process by callng MARC nternally for FE smulaton at each stage. As the optmzaton algorthm, the Adaptve Smulated Annealng (ASA) was used because t was very well suted for solvng hghly non-lnear problems and also effectve n fndng the global optmum [6]. In the ASA algorthm, the desgn, x k prevous desgn, x, as ( ) k+ 1 = k + ξ T UB LB k +1 x x x x, ξ T 1,1 [ ] where UB x and, s updated from the (4) LB x denote the upper and lower bound of the desgn, x, respectvely, and ξ T s a random varable [7]. Based on the prevous FE analyss result, punch dameter, clearance, and stroke at each stage were selected as the ndependent desgn varables for optmzaton. Because the falure was closely related wth the excessve stran, we assumed the objectve functon, ( ) ( ) f x, to be mnmzed as 1 f x = W ε + W (5) t fnal 1 max 2 fnal cup bottom fnal where ε max was the maxmum effectve stran after the fnal stage, and t fnal cup bottom the thckness of the cup bottom, whch was ntroduced to satsfy the target cup dmenson at bottom. W 1 = 10.0 and W 2 = 1.0 are the weght factor. Intal value, lower and upper bound for each desgn varable were assumed as show n table 3, 4, and 5. TABLE 3. Desgn range for punch dameter Lower Intal Upper bound value bound #2 Ø1.5 Ø3.4 Ø3.4 #3 Ø1.5 Ø2.8 Ø3.4 #4 Ø1.5 Ø2.3 Ø3.4 #5 Ø1.5 Ø1.9 Ø3.4 #6 Ø1.5 Ø1.5 Ø3.4 #7 Ø1.5 Ø1.5 Ø3.4 #8 - Ø1.5 - TABLE 4. Desgn range for clearance Lower Intal Upper bound value bound # # # # # # # TABLE 5. Desgn range for stroke Lower Intal Upper bound value bound # # # # # # # Optmzaton Result based on FE Smulaton Through teratve calculatons, an optmum multstage process was obtaned. The cup deformatons at each stage for the optmum desgn are shown n fgure 7. For comparson, the maxmum effectve strans predcted at each stage were shown n table 6 compared wth those for the ntal desgn. From the result, t s assumed that the maxmum effectve stran of the optmzed fnal cup can be reduced to 78% of the ntal desgn. Ths result suggests that the optmum process s practcally more relable than the ntally desgned process and, more mportant, the number of stage can be reduced. TABLE 6. Predcted maxmum effectve strans, ε max, after each stage for the ntal and the optmum desgn Intal desgn Optmum desgn Stran rato (A) (B) (B/A) # # # # # # #

6 FIGURE 7. FE smulated cup deformatons for the optmum desgn 4. CONCLUSIONS In the present paper, the mult-stage deep drawng usng the molybdenum sheet was nvestgated by the experment as well as the FE analyss. The parametrc study was conducted for the mult-stage deep drawng, and the optmum desgn was obtaned by the process optmzaton usng the FE smulaton at each stage. From the parametrc study by FE smulaton, t was assumed that varous desgn varables such as clearance, de corner radus, and de ntake angle had an effect on the stroke per stage, the materal flow, and the fnal product qualty. However, as the assumed effects provded only the ntuton for optmum desgn, teratve FE smulaton was conducted under the optmzaton algorthm to obtan the optmum multstage process. The present paper showed the possblty of the FE analyss based optmzaton of the hghly nonlnear process such as the mult-stage deep drawng. In the future, the falure crteron durng the ronng and the 3-D FE analyss based optmzaton should be conducted for more precse process optmzaton. REFERENCES 1. Nemat-Nasser, S., Guo, W., and Lu, M., Expermenally-Based Mcromechancal Modelng of Dynamc Response of Molybdenum, Scrpta Materala, V.40, N.7, 1999, pp Cheng, J., Nemat-Nasser, S., and Guo, W., A Unfed Consttutve Model for Stran-Rate and Temperature Dependent Behavor of Molybdenum, Mechancs of Materals, 33, 2001, pp Hollang, L., Brunner, D., and Seeger, A., Work Hardenng and Flow Stress of Ultrapure Molybdenum Sngle Crystals, Materals Scence and Engneerng A , 2001, pp Km, S.H., Km, S.H, and Huh, H., Tool Desgn n a Mult- and Process of a Rectangular Cup wth a Large Aspect Rato Usng Fnte Element Analyss, Internatonal Journal of Machne Tools & Manufacture, V.42, 2002, pp Hosford, W.F., Caddell, R.M., METAL FORMING, Prentce Hall Internatonal Inc., 1993, pp Ingber, L., Smulated Annealng: Practce versus Theory, Mathematcal Computer Modelng, 18, 1993, pp SIGHT Reference Gude, Ver.9.0, Engneous Software Inc., pp ACKNOWLEDGMENTS The authors wsh to acknowledge the support from the Korea Mnstry of Commerce, Industry and Energy and the Inter-Research Consortum for Materals & Components Technology. 878