IDENTIFICATION OF THE FLOW STRESS MODEL FOR THE STRIP MATERIAL SUBJECT TO BULK DEFORMATION *

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1 Informtyk w Technologii Mteriłów Pulishing House AKAPIT Vol. 14, 214, No. 2 IDENTIFICATION OF THE FLOW STRESS MODEL FOR THE S MATERIAL SUBJECT TO BULK DEFORMATION * MATEUSZ AMBROZIŃSKI 1 *, ROMAN KUZIAK 2 1 AGH University of Science nd Technology, l. Mickiewicz 3, 3-59 Krkow, Polnd 2 Institute for Ferrous Metllurgy, Gliwice, Polnd *Corresponding uthor: mmroz@gh.edu.pl Astrct The prolem of determintion of the flow stress of sheet mteril for lrger s ws investigted. Plstometric tests for the smple mde s pile of disks cut from the strip were performed. In order to void sliding the discs were plced in tue mde of the IF steel nd CuCr lloy. Flow stress of the IF steel nd CuCr lloy ws determined on the sis of compression tests for cylindricl smples. With the dt otined it ws possile to perform the nlysis to identify the model for steel using inverse nlysis for the compression of the ssemled smples. The developed model ws used in two cse studies. The first ws simultion of stmping of the utomotive prt. The second ws simultion of strength of the nchor-concrete joint, in which the expnsion sleeve ws mde of the steel. Key words: sheet metl, compression tests, inverse nlysis 1. INTRODUCTION The prolem of identifiction of flow stress model for metllic mterils is well studied in the scientific literture. In ulk forming processes plstometric compression tests re used nd inverse nlysis is pplied for n ccurte interprettion of the results. The uthors lgorithm for the inverse solution ws descried y Szelig et l. (26). Flow stress model for sheet forming is usully uilt on the sis of tensile tests. More dvnced experimentl techniques like i-xil tension test re recently used, see for exmple (Vegter & vn den Boogrd, 26). Aprt from the flow stress, these tests generte informtion regrding the nisotropy nd llow to formulte the flow rule (Bnic, 21). Smll s tht cn e pplied to the smples re the min limittion of these tests. Beyond this, present * This pper ws presented t the 21st Conference KomPlsTech, which ws held on Jnury 19-22, 214 in Wisł Mlink. work ws done for the steels. steel microstructure is otined in the continuous nneling process fter cold rolling, when the thickness of the strip is smll. It is not possile to cut from the strip stndrd cylindricl smples for compression on the Gleele simultor. Intensive serch for experimentl methods, which llow to determine mechnicl properties of strip mteril to lrger s, hs een recently oserved. Such need results from oth: the specific chrcter of sheet forming in which locl s cn exceed those chieved in tensile tests, extensively developing metl forming methods of ulk-sheet forming (Schneider & Merklein, 211; Merklein et l., 212; Merklein et l., 213). The ojectives of the present work were formulted with the ove remrks in mind. To otin lrger s compression tests of pile of discs mde of the investigted strip were performed. To ISSN

2 void slipping of discs the pile ws put in tue mde of mteril with known flow stress. Inverse nlysis ws pplied for interprettion of the results ccounting for the influence of the externl tue mteril. The developed flow stress model ws pplied to simultions of the stmping process of n utomotive prt nd deformtion of the expnsion of sleeve used to connect n nchor with concrete plte. 2. EXPERIMENT 2.1. Mteril nd test conditions Compression tests for pile of roundels cut from the strip were performed. Since roundels during compression were sliding ginst ech other, the pile ws put in tues mde oth of the IF steel nd CuCr lloy (Fig. 1). Two kinds of the tue mteril with vrious flow stress were used to determine the extent to which the results cn e ffected. Compression of ulky cylindricl smples mde of IF steel nd CuCr lloy ws performed first to determine the flow stress model for these mterils in the temperture rnge 2-2 o C nd the rte in the rnge.1 1 s -1. Following this, piles of disks in IF steel tue were compressed to the totl of 1 with the sme deformtion conditions. tempertures of 2 o C, 1 o C nd 2 o C nd t rtes of.1 s -1,.1 s -1 nd 1 s Results The selected results of force mesurements in the compression tests for the ssemled smples re shown in Fig. 2. The results nlysis shows tht the difference etween lods otined for IF steel nd CuCu sleeve is very smll temperture 2 o C rte.1 s -1 IF.1 s -1 IF.1 s -1 IF 1 s -1 CuCr.1 s -1 8 Fig. 1. Schemtic illustrtion of the smple collected of steel disks in the sleeve mde of the IF steel or CuCr lloy. The tested mteril ws steel strip contining.22%c, 1.5%Mn,.9%Si,.14%P,.5%S,.3%Cr,.2%Ni,.1%Mo,.4%Ti, 1.7%Al. IF steel used for the sleeve, which prevents disks sliding, contined.3%c,.15%mn,.23%si,.17%p,.12%s, >.28%Ti,.3%N. CuCr lloy contined.81%cr,.26%fe nd lnce Cu. Thickness of the steel strip ws 1.45 mm. Solid smples dimension ws φ mm for the IF steel nd φ mm for the CuCr lloy. Disks with φ mm were cut from the steel strip nd put into IF steel or CuCr tue with φ8 φ mm. These smples were compressed on the Gleele 38 simultor t IF 2 o C IF 1 o C IF 2 o C CuCr 2 o C Fig. 2. Selected results of mesurements of compression force for the pile of steel in IF steel tue nd CuCr lloy tue, temperture 2 o C () nd rte.1 (). For ll tests the temperture ws mesured with thermocouple spot welded to the side of the smple. The results of the temperture mesurements re shown in Fig. 3. Anlysis of the mesured tempertures shows tht in cse of tests with low rte the control system ws cple of compensting het genertion due to plstic work nd friction, s well 1

3 s the het trnsfer to the surrounding, nd resonly stle temperture ws mintined. In fst tests (1 s -1 ) the temperture increse ws oserved. To compenste the influence of the temperture vritions, the inverse nlysis ws pplied. In cse of the direct prolem model the mesured tempertures in the inverse nlysis were used s Dirichlet oundry condition. temperture, o C s -1.1 s -1 1 s -1 Fig. 3. Results of the temperture mesurements during compression tests. 3. IDENTIFICATION OF THE FLOW STRESS MODEL 3.1. Inverse lgorithm Interprettion of the results of compression tests ws sed on the inverse nlysis cple of eliminting, or t lest minimizing the influence of such disturnces s inhomogeneity of s, stresses nd tempertures, see (Gvrus et l., 1996; Gelin & Ghouti, 1994, Szelig et l., 26) nd llows flow stress to e determined independently of the method of testing. Cpilities of the inverse nlysis to improve interprettion of the results of plstometric tests were confirmed y Gwąd et l. (25). The two-step lgorithm descried in detils y Szelig et l. (26) ws used in the present work. The qudrtic norm of the error etween mesured nd clculted compression lods ws used s the ojective function: 2 Nt 1 Ns m c 1 Fij F ij Φ= m Nt i= 1 Ns j= 1 F ij (1) where: Nt numer of tests, Ns numer of lod mesurement smpling points in one test, F m, F c, - mesured nd clculted lod, respectively. Direct prolem model is used to clculte F c in the ojective function (1). This model is sed on the FE therml mechnicl solution for metl forming prolems. Detils of this solution re given y Pietrzyk (2) nd re not repeted here. Friction coefficient of.4, determined erlier for the cold compression tests performed on Gleele 38, ws used in the present work. In the present work the inverse nlysis hd n dditionl tsk, which ws to ccount for the influence of the IF steel nd CuCr lloy sleeve. Therefore, inverse nlysis for the compression test of the solid IF steel nd CuCr lloy cylindricl smples ws performed in the first plce nd the flow stress model for this mteril ws determined. This model ws implemented in the FE code, which ws used to simulte compression of the smple ssemled of steel roundels nd IF steel or CuCr lloy sleeve Inverse nlysis for the IF nd CuCr smples The selected results of the inverse nlysis for the IF steel nd CuCr lloy re shown in figure 4. These curves represent flow stress s function of nd they cn e considered property of the mteril for isotherml, constnt rte conditions. The functions introduced in the finite element progrm in the tulr form will give perfect greement etween mesured nd predicted lods. The flow stress in the finite element simultions hs to e descried in wide rnge of tempertures nd rtes. Therefore, the description of this reltion in the form of lgeric or differentil equtions is needed. Thus, the next step of the inverse nlysis involved determintion of the coefficients of the function descriing the flow stress. The ehviour of polycrystls during plstic deformtion depends on mny fctors, the influence of which should e ccounted for, which is difficult. To overcome this prolem polycrystls re descried y flow (stress-) curve, which represents sttisticlly ll mentioned phenomen. Lrge numer of flow stress models for metl forming were pulished in the scientific literture in the lst two decdes. 11

4 rte.1 s -1 IF 2 o C 1 o C 2 o C CuCr 2 o C rte.1 s IF 2 15 o C IF 1 o C 1 IF 2 o C 5 CuCr 2 o C rte 1 s IF 2 15 o C IF 1 o C 1 IF 2 o C 5 CuCr 2 o C c Fig. 4. Flow stress vs. in the tulr form determined in the first step of the inverse nlysis for the IF steel nd the CuCr lloy for the rte of.1 s -1 (),.1 s -1 () nd 1 s -1 (c). These models re chrcterised y vrious complexity of mthemticl formultion nd vrious predictive cpilities. There were severl ttempts to clssify these models, see for exmple (Grosmn, 1997; Schindler et l., 1994; Schindler & Hdsik, 2), ut there is still lck of convincing hints for selection of the most pproprite model for prticulr ppliction. However, the results of preliminry inverse nlysis (figure 4) show tht the investigted steel does not soften during deformtion t low tempertures. Therefore, the simplest version of the Hensel nd Spittel (1979) eqution ws selected in the present work: m ( ) exp( ) n σ p = Aε exp qε ε βt (2) where: A, n, q, m nd β - coefficients, T - temperture in o C. Approximtion of the flow stress curves using eqution (2) ws pplied nd the coefficients otined from the pproximtion were used s strting point for optimiztion of function (1). Coefficients in eqution (2) were determined y serching for the minimum of the ojective function defined s squre root error etween flow stress clculted from eqution (2) nd otined from the preliminry inverse nlysis. It llowed to decrese the computing times significntly. The coefficients otined from optimiztion for the IF steel nd the CuCr lloy re given in the first nd the second row of tle 1, respectively. The finl vlue of the ojective function (1), which cn e referred to s the mesure of the ccurcy of the inverse nlysis, is given in the lst column of this tle. Tle 1. Coefficients in eqution (2) for the IF steel nd steel otined from the inverse nlysis. mteril A n q m β Φ IF CuCr Inverse nlysis for the steel Similr nlysis ws performed for the smples ssemled from the steel disks inserted in the IF steel tue or the CuCr tue. Knowing the flow stress of the tue mterils the flow stress model for the steel ws determined. The results of this nlysis in the form of the flow stress re shown in figure 5. Full inverse nlysis gve coefficients of eqution (2) for the steel. They re presented in the lst row of tle 1. 12

5 temperture 2 o C rte.1 s -1 c.1 s -1.1 s -1 1 s rte 1 s -1 2 o C 1 o C 2 o C o C 1 o C 2 o C Fig. 5. Flow stress vs. in tulr form determined in the first step of inverse nlysis for the steel for the temperture of 2 o C () nd the constnt rte of.1 s -1 () nd 1 s -1 (c). Decrese of flow stress for lrger s is the inccurcy of eqution (2). Since s in ulksheet forming processes, s well s in stmping processes, cn even exceed 1, ppliction of eqution (1) for these processes cn led to erroneous results. Therefore, simpler model which is n extension of the Hollomon eqution ws dditionlly considered in the present pper: n m Q σ p = Aε ε exp (3) RT ( + 273) where: Q ctivtion energy. As previously, the inverse method ws pplied to determine coefficients of eqution (2) nd these coefficients for the steel re given in tle 2. It cn e seen tht in the investigted rnge of prmeters the steel shows negligily smll sensitivity to the rte (m =.3) nd smll sensitivity to the temperture (Q = 663.2). Tle 2. Coefficients in eqution (3) for the steel otined from the inverse nlysis. Steel A, MP n m Q, J/mol Φ VERIFICATION AND VALIDATION OF THE MODEL 4.1. Verifiction of the model Verifiction of the models ws mde y comprison of forces mesured in the tests nd clculted using the FE code with equtions (2) implemented in the constitutive lw. Figure 6 shows the comprison results for the IF steel. The results nlysis shows tht good ccurcy ws reched for the whole rnge of rtes nd tempertures. Similr comprison of mesured nd clculted forces for the smples mde of the steel disks in the IF steel sleeve ws then performed nd the results re shown in figure 8. Agin eqution (2) with coefficients in tle 1 ws used to descrie flow stress of the IF nd steels in the constitutive model. The nlysis shows tht ccurcy is not s good s for solid IF steel smples ut it is still cceptle Vlidtion of the model Flow stress vlues for the steel otined from the tensile nd compression tests re compred in figure 8. Apprently, for s elow.25 very good greement etween the results otined from oth tests is oserved. Compression test gve relistic results up to the of.4. Aove this vlue the lost its cpility to con reltive flow 13

6 of the disks, see cross section of the smple with the vlue of.9 (figure 1) rte.1 s -1 IF 2 2 o C 2 o C rte.1 s -1 IF rte 1 s -1 IF c 2 o C 2 o C o C 2 o C Fig. 6. Comprison of mesured vs. clculted forces (solid lines) using the FE code with eqution (2) implemented in the constitutive lw (dotted lines), for the rte of.1 s -1 (),.1 s -1 () nd 1 s -1 (c); IF steel in CuCr sleeve temperture 2 o C.1 s -1.1 s Fig. 7. Comprison of mesured forces (solid lines) with clculted using the FE code with eqution (2) implemented in the constitutive lw (dotted lines) t the temperture of 2 C for the rte of.1 s -1 nd.1 s -1 ; steel disks in the CuCr steel sleeve in IF steel sleeve 2 2 o C rte.1 s -1 2 o C in IF steel sleeve 2 2 o C rte.1 s -1 2 o C c Fig. 8. Comprison of mesured vs. clculted forces (solid lines) using the FE code with eqution (2) implemented in the constitutive lw (dotted lines), for the rte of.1 s -1 (),.1 s -1 () nd 1 s -1 (c); steel roundels in the IF steel sleeve. flow stress,mp in IF steel sleeve rte 1 s o C 2 o C temperture 2 o C rte.1 s -1 compression tension Fig. 9. Comprison of stress- curve of compression test nd tension test for the steel t temperture of 2 C nd rte.1 s -1. Flow stress in compression ws determined y the inverse nlysis for the steel roundels in the IF steel sleeve nd in the CuCr sleeve. 14

7 INFORMATYKA W TECHNOLOGII MATERIAŁÓW rotry punch (Merklein et l., 212). Reduction in force during the forming is the min dvntge of the oritl forming, which is schemticlly shown in figure 13. Initil thickness of the steel strip ws 2 mm. Fig. 1. Cross section of the smple ssemled of the steel disks in the IF steel sleeve fter compression with the of.9. Fig. 12. Comprison of mesured min (Experimentl vlues) with clculted using the FE code (Simultion vlues). Fig. 13. Schemtic illustrtion of the oritl forming of the steel (α =.5, φ =.5 ). The finl thickness of the disk vried etween 2.6 mm t the rim nd 1.55 mm in the centre. Clculted distriution of the effective is show in figure 14. In oth considered processes it is seen tht s re much lrger thn in the stmping Fig. 11. Steps in mnufcturing of the cr ody prt ) pre-stmping, ) cutting hole, process nd they exceed lso those c) finl stmping. otined in the compression of disks in sleeve. It mens tht extrpolthe oritl forming ws nother process which tion eyond the experimentl dt ws needed, ut ws simulted using the mteril dt otined in the the rnge of this extrpoltion ws much lower for present work for the steel. This process conthe compression of disks in sleeve s compred sisted in chnging the shpe of the metl disk using with the tensile tests. 15 Flow stress model developed from the compression tests (eqution (3) with coefficients given in tle 2) ws implemented in the FE code Aqus nd two processes were simulted. The first ws stmping of cr ody prt nd the second ws oritl forming of disc. The stmping process of crsh ox element (prt of the cr ody) ws crried out in three stges. Cutting hole in the prt ws not simulted. Susequent steps considered in simultions re shown in figure 11. The results otined from the numericl simultions were compred with the s mesurement in the prt mnufctured in one of the stmping compny. Assuming the plne deformtion, mjor nd minor s were mesured nd then compred with those otined from the numericl clcultions. The comprison results re shown in figure 12. The nlysis of s shows tht loclly they exceed mximum otined in the tensile tests ut they fll within the rnge of s otined in the compression of disks in sleeve.

8 Fig. 14. Clculted distriution of the effective in the disk mde y the oritl forming. 5. CONCLUSIONS Compression tests for pile of disks cut from the strip inserted in the IF steel or CuCr lloy tue sleeve llows to determine flow stress of the strip mteril to the s of out.5. Test performed for the steel disks inserted in the sleeve llows to otin lrger s s compred with the smple without sleeve. Inverse nlysis llows to eliminte the influence of the sleeve mteril nd to determine rel flow stress of the disk mteril. Very good greement etween the results from the tensile nd compression tests ws otined for lower s. It supports n opinion tht, when the inverse nlysis is pplied, comprle results re otined from different tests. In the investigted rnge of prmeters, the steel showed negligily smll sensitivity to the rte nd smll sensitivity to the temperture. Due to its lrger flow stress, lrger deformtions could e otined for the CuCr lloy sleeve s compred with the IF steel sleeve. It is expected tht ppliction of the hrder mteril for the sleeve would llow the totl in the tests to increse even further. ACKNOWLEDGEMENTS The finncil support provided y NCN, project no. N N , is cknowledged. REFERENCES Bnic, D., 21, Sheet metl forming processes: Constitutive modeling nd numericl simultion, Springer, Berlin, Heidelerg. Gvrus A., Mssoni E., Chenot J.L., 1996, An inverse nlysis using finite element model for identifiction of rheologicl prmeters, Journl of Mterils Processing Technology, 6, Gwąd, J., Kuzik, R., Mdej, Ł., Szelig, D., Pietrzyk, M., 25, Identifiction of rheologicl prmeters on the sis of vrious types of compression nd tension tests, Steel Reserch Interntionl, 76, Gelin, J.C., Ghouti, O., 1994, An inverse method for determining viscoplstic properties of luminium lloys, Journl of Mterils Processing Technology, 34, Grosmn, F., 1997, Appliction of the flow stress function in progrmmes for computer simultion of plstic working processes, Journl of Mterils Processing Technology, 64, 1997, Hensel, A., Spittel, T., 1979, Krft- und Areitsedrf Bildsmer Formgeungsverfhren, VEB Deutscher Verlg fur Grundstoffindustrie, Leipzig. Kowlski, B., Sellrs, C.M., Pietrzyk, M., 2, Development of computer code for the interprettion of results of hot plne compression tests, ISIJ Interntionl, 4, Merklein M., Plettke, R., Opel, S., 212, Oritl forming of tilored lnks from sheet metl, CIRP Annls Mnufcturing Technology, Merklein M., Hgenh H., Schneider T., 213, Sheet-ulk metl forming processes - stte of the rt nd its perspectives, Kolleck R., (ed.): Proc. Conf. TTP Tools nd Technologies for Processing Ultr High Strength Mterils, Grz, Pietrzyk, M., 2, Finite element simultion of lrge plstic deformtion, Journl of Mterils Processing Technology, 16, Schindler, I., Klier, J., Borut, J., 1994, Predikce deformcnich odporu pri vysokoredukcnim tvreni, Proc. Conf. METAL, Ostrv, Schindler, I., Hdsik, E., 2, A new model descriing the hot stress- curves of HSLA steel t high deformtion, Journl of Mterils Processing Technology, 16, Schneider, T., Merklein M., 211, Sheet-ulk metl forming of preformed sheet metl prts, Key Engineering Mterils, 473, Szelig, D., Gwąd, J., Pietrzyk, M., 26, Inverse nlysis for identifiction of rheologicl nd friction models in metl forming, Computer Methods in Applied Mechnics nd Engineering, 195, Vegter, H., vn den Boogrd, A.H., 26, A plne stress yield function for nisotropic sheet mteril y interpoltion of ixil stress sttes, Interntionl Journl of Plsticity, 22, IDENTYFIKACJA MODELU NAPRĘŻENIA UPLASTYCZNIAJĄCEGO DLA MATERIAŁU W FORMIE TAŚMY PODDAWANEGO ODKSZTAŁCENIOM OBJĘTOŚCIOWYM Streszczenie W prcy rozwżono prolem wyznczeni modelu nprężeni uplstycznijącego dl mteriłu w formie tśmy poddwnego dużym odksztłceniom plstycznym. Wykonne zostły próy ściskni próek zroionych z krążków wyciętych z tśmy. Ay uniknąć przesuwni się krążków względem sieie umieszczono je w rurkch ze stli IF lu ze stopu CuCr. Nprężenie uplstycznijące stli IF i stopu CuCr wyznczono n podstwie ściskni próek cylindrycznych zroionych z tych mteriłów. Znjąc to nprężenie, zstosowno nlizę odwrotną do identyfikcji modelu nprężeni uplstycznijącego stli n podstwie ściskni próki z krążków umieszczonych w rurce. Oprcowny model wykorzystno w symulcji dwóch procesów. Pierwszym yło 16

9 tłoczenie części ndwozi smochodu ze stli. Drugim przykłdem ył proces ksztłtowni elementu wykonnego z tśmy ze stli. Received: April 3, 214 Received in revised form: June 9, 214 Accepted: June 16,