CONSTITUTIVE MODELING OF STRESS-STRAIN CURVES OF HOT DEFORMED IF AUSTENITE

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1 Prodings of COBEM 25 Coyright 25 by ABCM Novmbr 6-11, 25, Ouro Prto, MG CONSTITUTIVE MODELING OF STRESS-STRAIN CURVES OF HOT DEFORMED IF AUSTENITE R. E. Lino Univrsidad Fdral d Minas Grais. Dartamnto d Engnharia Mtalúrgia d Matriais. R. Esírito Santo 35, sala 26- Cntro Blo Horizont, Minas Grais, Brazil ronyduardo@yahoo.om.br R. Barbosa Univrsidad Fdral d Minas Grais. Dartamnto d Engnharia Mtalúrgia d Matriais. R. Esírito Santo 35, sala 26- Cntro Blo Horizont, Minas Grais, Brazil rbarbosa@dmt.ufmg.br Abstrat. Industry forming rosss suh as hot rolling ar oml in natur. During a rolling shdul, svral ross variabls intrat simultanously rsulting in a dsird rodut with rtain sha and mhanial rortis. Stting u th rolling mill rquirs ris knowldg of th loads ndd to sha th mtal in th svral asss. This in turn dmands th ability to rdit th strngth of th matrial whn dformd to a valu of strain, strain rat and tmratur. Finit lmnt analysis is oftn usd in r-st modls to adjust rolling mill osition. Thy howvr nd to b fd with onstitutiv quations rlating th strsss rquird to dform a rtain mtal undr th usual ross variabls. This ar shows how a st of strss-strain urvs an b modld so that both hardning and softning mhanisms ommonly rsnt during hot dformation ar takn into aount. Th modl rditions ar omard to a st of litratur data in ordr to b validatd. Rasonabl agrmnt btwn ublishd rsults and rditd valus wr obtaind indiating how ffiint th modl is to assss valus of strsss undr hot working. Kywords: Modling of strss-strain urvs, IF stls, Hot torsion, Hot rolling 1. Introdution Constitutiv quations rlating strsss to strain rats and tmraturs hav bn routinly mloyd to stimat hot forming loads (Rao and Hawbolt, 1992; Davnort t al., 2; Jonas, 2; MQun and Ryan, 22a). From all rssions ublishd in th litratur (MQun, 22b), robably th most usd is th hyrboli sin funtion rlating Z, th Znr-Hollomon aramtr, to th strss, Z ( Q RT ) A( sinhασ ) n & (1) = / = df Hr, Q df is th ativation nrgy for hot dformation, A, α and n ar onstants, th othr symbols hav thir usual maning. This rssion givs a straight lin in a log Z vrsus log[sinh(ασ) lot irrstiv of th lvl of th strsss, whras th owr and onntial laws loos linarity at high and low strsss rstivly. Th ak strss rsulting from th ourrn of dynami rrystallization, DRX, and also th stady stat strss assoiatd with th omltion of first wav DRX, hav bn sussfully modld by th abov rssion. Q df an also b stimatd for a sris of C-Mn and miroalloyd stls as a funtion of hmial omosition, as ublishd in th litratur (Cho t al., 21). Modling strss strain urv rquirs, in addition, th us of an volution quation suh as forwardd by Sllars (1985). Th uros of this ar is thrfor twofold; first, to simulat isothrmal strss strain urvs during hot dformation of doubl stabilizd Ti-Nb IF stl, following th mthod roosd by Sllars and, sond to rdit valus of strsss during ontinuous ooling as it ours in hot rolling. 2. Erimntal Produr Th hmial omosition of th IF stl usd in this work is 24mC, 9mN and 1.7Mn,.17Si,.6P,.2S,.21Al,.14Ti and.11nb, all numbrs in wight%. Samls for torsion rimnts, 15mm of lngth and 1mm in diamtr, wr ut at ¼ of th thiknss of a hot rolld billt of 1mm 2 ross stion. Th simns wr mahind with thir ais aralll to th billt rolling dirtion. Th rimnts wr arrid out in a omutr ontrolld srvohydrauli mahin quid with an infrard furna. All tsts wr rformd in argon atmoshr and th tmratur was ontrolld by a roml-aluml thrmooul wldd at th saml surfa. Th simns wr initially r-hatd at 1473K (12 o C) for 9s (15 minuts). Torqu and angular dislamnt wr onvrtd to quivalnt strss-strain urvs in th usual way (Filds and Bakofn, 1957).

2 Prodings of COBEM 25 Coyright 25 by ABCM Novmbr 6-11, 25, Ouro Prto, MG Non-isothrmal tsting was ondutd to dtrmin th valus of Tnr and Ar3, as rortd lswhr (Borato t al., 1988). Th samls wr r-hatd as just dsribd and lt to ool at a rat of 1K/s (1 o C/s). Dformations of.3 quivalnt strains wr alid to th saml at tim intrvals of 3s. Th first dformation was givn at aroimatly 1453K (118 o C) and th last at about 873K (6 o C), ovring tmraturs tyial of hot rolling in th austnit and a rasonabl rang of tmraturs in th frrit rgion. Th valus of Tnr and Ar3 masurd wr 1218K (945 o C) and 123K (75 o C), rstivly. Isothrmal tsting was rformd to obtain strss-strain urvs at onstant strain rat and at svral tmraturs. Samls wr r-hatd as dsribd abov and thn lt to ool to tsting tmratur at a rat of 1K/s (1 o C/s). All samls wr kt at 9s (3 minuts) rior to twisting romoting tmratur qualization throughout th simn bfor dformation, t for th as of tsts ondutd at 1473K (12 C). Tsts wr rformd at tmraturs of 1123, 1173, 1223, 1273, 1323 and 1373K (85, 9, 95, 1, 15 and 11 o C) and strain rats of.1, 1, 3 and 1s -1. All samls wr dformd to an quivalnt tru strain of 7, allowing th ourrn of full DRX and th ahivmnt of th stady stat in all tsts. 3. Rsults and disussion 3.1. Modling isothrmal strss-strain urvs Figur 1 shows, as full lins, th masurd strss-strain urvs for samls tstd at a strain rat of 1s -1 and tmraturs ranging from 1173 to 137K (9 to 11 o C). All urvs dislay a haratristi ak strss,, indiating ourrn of DRX. Th simns wr urosly dformd to total strains around 7. in ordr to guarant full DRX vn at th lowst tmratur. Th irls in Fig. 1 rrsnt alulatd valus giving a qualitativ viw of how wll th modl rsntd hr an simulat isothrmal strss-strain urvs. STRESS, MPa = 1s -1 Tmratur, o C Masurd Prditd STRAIN Figur 1. Comarison btwn masurd and rditd strss-strain urvs for tsts ondutd at 1s -1 and svral tmraturs. Tabl 1 rsnts a summary of th quations mloyd in this work to modl th strss-strain urvs. Th dndn on Z of th strss at onstant strain is givn by th sinh law, as shown by Eq. (1) and th ativation nrgy for hot dformation was stimatd from log [sinh (ασ) vrsus log (1/T), at onstant strain rat, taking α as.12 MPa -1, as suggstd in th litratur (MQun, 22b) for svral stls and n was alulatd for a givn tmratur from th slos of log (sinh (ασ)) vrsus log () lots. Th valus of Q df and n masurd in this work wr 35 ± 12kJ/mol and

3 Prodings of COBEM 25 Coyright 25 by ABCM Novmbr 6-11, 25, Ouro Prto, MG 4.8 ±.2, rstivly. Ths valus omar favorably to 32kJ/mol and 4.5, th formr alulatd from rssion rortd in th litratur (Mdina and Hrnandz, 1996). Th ritial strain for th initiation of dynami rrystallization,, givn by Eq. (3) and (4), is of imortan sin it dtrmins th orrt st of quations to b usd as a funtion of th strain. This ritial valu was masurd from th strss-strain urvs following a mthod rsntd by Poliak and Jonas (Poliak and Jonas, 23). For strains lowr than th strsss wr alulatd using Eq. (5). All th symbols in this rssion hav thir usual maning, t for C, a funtion of th initial urvatur of th strss-strain urv, as rrsntd by th strss at a onstant strain for a givn Z, in this as, half th valu of th ak strain. Th onstant m in Eq. (5) was masurd as.5 ±.1, in agrmnt with rortd valu drivd from mor fundamntally basd modls. At th initiation of th stady stat strss, Eq (5) will rodu a saturation strss, σ, givn by th rssion in Eq. (7). This strss is a funtion of Z and its masurmnt for ah Z valu was arrid out following mthod rntly ublishd in th litratur (Zahiri t al. 25) For strains highr than, softning by DRX was takn into aount in th modling of th urvs as shown by Eq. (8). Th kintis of DRX annot b masurd dirtly from mtallograhy obsrvation on qunhd samls of IF stls sin austnit transforms on ooling to frrit, irrstiv of th ooling rat mloyd. Thrfor, DRX was masurd from th sha of th strss-strain urvs, as rortd in th litratur (Sllars, 1985). Th kintis of DRX follows, as td, an Avrami ty rssion and th tim onnt masurd hr was m quals 1.4 ±.1. Tabl 1. List of quations mloyd in this work to simulat isothrmal strss-strain urvs of Ti-Nb IF stl. Evnt Equation Znr-Hollomon aramtr Z = & [ Q /( RT) Q = (35 ± 12) kj /( molk) = a Critial strain for th initiation of dynami rrystallization a =.6 ±.1 ( masurd) Strss at a onstant Z in th dynami rovry rgion ( < ) Strss at onstant Z in th dynami rovry and dynami rrystallization rgion ( ). σ = σ (2) (3) 2.16 = 1.31 Z (4) ( σ σ )[ 1 ( C ) σ = σ + m =.5 ±.1 ( masurd) 1 σ σ C = ln 1 σ σ = / 2 σ σ = 83sinh σ = 83sinh = 83sinh 1/ m 12.2 [ Z / [ Z / [ Z / m { σ + ( σ σ )[ 1 ( C ) } = ( σ σ ) 1 k( ( )/ & ) m = 1.4 ±.1 ( masurd) k = 246 & 1.5 ss m { [ } 3 [ 1 /( RT ) 11 83sinh [ Z /3, ss m σ (5 (6) (7) (8) σ = (9) 3.2. Modling strss-strain urvs during ontinuous ooling at tmraturs highr than Tnr Equations rsntd hr an simulat isothrmal strss-strain urvs for th rsnt hmial omosition of IF stl, as just dsribd. In addition, if ould with a suitabl modl, thy an also b mloyd to stimat rolling loads. Litratur data on rolling loads, usually rrsntd as a funtion of a man flow strss (MFS), although abundant

4 Prodings of COBEM 25 Coyright 25 by ABCM Novmbr 6-11, 25, Ouro Prto, MG for C-Mn and Nb miroalloyd stls, ar rlativly sar for IF stls. For this rason, in ordr to omar rditions of th rsnt modl with strss-strain urvs masurd during ontinuous ooling, this ar rfrs to data ublishd by Najafi-Zadh t al. (1992) to validat th work arrid out hr. Ths authors usd thr tys of IF stls: Ti, Ti- Nb and Nb. Thy rformd two tys of torsion tsts: on to masur Tnr and a sond ty to simulat a hot stri rolling ross. In thir work, th Ti and th Ti-Nb stls dislayd quivalnt lvls of MFS whn subjtd to similar shduls. Th Nb IF stl, howvr, dislayd signifiantly highr valus of MFSs suggsting that Nb inrasd ronoundly th strngth lvls of th stl. It was also shown that Tnr ourrd whnvr Nb was rsnt, this valus bing around 1213 to 1233K (94 to 96 o C). At tmraturs abov Tnr, thr was, as td, full rrystallization btwn dformation, whras blow Tnr thr was an aumulation of strain from ass to ass triggring DRX in th sond dformation and dislaying a dro in th MFS valu of th following ass. Figur 2 shows a omarison btwn MFS alulatd from th rsnt modl and th valus rortd in th ar just itd for asss givn at tmraturs highr than Tnr, that is, in th full rrystallization rgion. Clarly, th rditd valus ar onsistntly highr than thos rortd in th litratur by about 3%, although th sha of th alulatd MFS vrsus 1/T urv was idntial to th urv rsntd in th litratur. This disrany, howvr, an b aountd for if diffrns in hmial omosition btwn alloys, th on usd to obtain th data rsntd in th litratur and that mloyd hr, ar onsidrd. In fat, th alloy usd in th omarison shown in Fig. 2 and mloyd by Najafi-Zadh, Yu and Jonas had.15% Mn,.35% Ti and.28% Nb whras th alloy usd hr has 1.71% Mn,.14 Ti and.11% Nb. Clarly, th rsn of 1 tims mor Mn and 5 tims mor Nb in th alloy tstd hr afftd th ovrall strngth of th samls dformd in this work. In ordr to assss th fft of ths diffrns on th valus of MFS an rssion rortd by Siiliano t al. (1996), was usd. 15 Ar3 MFS, MPa 1 5 Tnr Ar1 Najafi-Zadh, 1992 Prditd Corrtd /T, K -1 Figur 2. Comarison btwn rditd man strss valus alulatd using th modl rsntd in this work and rimntal data availabl in th litratur (Najafi-Zadh t al., 1992) during ontinuous ooling and dformations givn at tmraturs highr than Tnr. MFS = σ M ( [ Nb +.137[ Mn [ Ti) (1) Hr, σ M stands for a flow strss alulatd aording to Misaka (Siiliano, Jr., 1996). Th valu of σ M dnds on C ontnt, strain and strain rat and, as C ontnt for th stl tstd by Najafi-Zadh, Yu and Jonas is of th sam ordr of th on for th stl tstd hr, th valus of strsss alulatd from Misaka s quation rodud similar valus in both ass. Thrfor, a orrtion fator ould b alulatd from th art of Eq. (1) within th arnthsis. This art of th quation rdits a fator of.797 for th Ti-Nb stl usd by Najafi-Zadh, Yu and Jonas and of 1.48 for th rsnt work, if it is takn into th rssion th % wights of Ti and Nb in ss to stabiliz C and N in th IF matri and not th gross nominal omosition. Th amount of Ti or Nb in ss was obtaind by

5 Prodings of COBEM 25 Coyright 25 by ABCM Novmbr 6-11, 25, Ouro Prto, MG subtrating from th bas omosition th amount Ti or Nb ombind with N, S or C, as dtaild lswhr (Najafi- Zadh, 1992). Hn, th strngthning fator du to diffrnt amounts of Ti and Nb in ss to stabiliz C and N in both alloys is 1.48/.797 or 1,31, los to th 3% diffrn found hr. This orrtion fator was thn alid to omar th MFSs of th two alloys, showing an agrmnt btwn alulatd and litratur rsults as ditd by th blak irls in Fig. 2. Th agrmnt btwn masurd and rditd valus just shown validats th us of rsnt modl to rdit strsss at tmratur rangs highr than Tnr. At tmraturs lowr than Tnr, howvr, furthr study must b arrid out onrning th kintis of dynami and mtadynami softning rior to th usag of th Eqs. rsntd in Tabl (1) and, larly, it is out of th so of suh short not, bing th subjt, nonthlss, addrssd in a sarat ovr. 4. Conlusions Th following onlusions an b drawn from this work: A siml modl was rsntd hr to rdit isothrmal strss-strain urvs of a Ti-Nb IF stl. Th rsults hav shown that th modl an b usd to omut isothrmal strss-strain urvs bing th agrmnt btwn masurd and rditd strsss rasonabl, rgardlss of th lvl of th strain, wathr smallr or largr than, and th valus of strain rat and tmratur at whih th tst was ondutd; Th modl was validatd by omaring valus of MFS alulatd hr with thos rortd by Najafi-Zadh t al. (1992) at tmraturs highr than Tnr during ontinuous ooling rimnts simulating th roughing stag of hot rolling of lats and stris. Th rditd MFS valus wr systmatially highr by a fator of about 3% than th data rortd in th litratur. Ths disranis, howvr, wr rlatd to diffrns in hmial omosition rgarding Mn, Ti and Nb ontnts in both alloys. On diffrns in hmistry wr rorly takn into onsidration, masurd and rditd valus of MFS ould b aroriatly omard. 5. Aknowldgmnts Th authors aknowldg CNPq and PRPq-UFMG for th finanial suort givn to this rsarh. 6. Rfrns Borato, F., Barbosa, R., Yu, S. and Jonas, J. J., 1988, Efft of Chmial Comosition on th Critial Tmratur of Miroalloyd Stls, Prodings on th Int. Conf. Physial Mtallurgy of Thrmomhanial Prossing of Stls and Othr Mtals (THERMEC 88), Vol. 1, Tokyo, Jaan, Cho, S-H., Kang, K. B. and Jonas, J. J., 21, Mathmatial Modling of th Rrystallization Kintis of Nb Miroalloyd Stls ISIJ Intrnational, Vol. 41, Davnort, S.B., Silk, N.J., Sarks, C.N. and Sllars, C.M., 2, Dvlomnt of onstitutiv quations for modlling of hot rolling, Matrials Sin and Thnology, Vol. 16, Filds, D.S. and Bakofn, W. A., 1957, Dtrmination of strain-hardning haratristis by torsion tsting, Amrian Soity for Tsting and Matrials, Vol. 57, Jonas, J.J., 2, Th hot stri mill as an rimntal tool, ISIJ Intrnational, Vol.4, MQun, H. and. Ryan, N.D., 22, Constitutiv analysis in hot working, Matrials Sin and Enginring A, Vol. A322, MQun, H., 22b, Elvatd tmratur dformation at forming rats of 1-2 to 1 2 s -1, Mtallurgial and Matrials Transations A, Vol. 33, Mdina, S.F. and Hrnandz, C.A., 1996, Th influn of hmial omosition on ak strain of dformd austnit in low alloy and miroalloyd stls, Ata Matrialia, Vol. 44, Najafi-Zadh, A.; Yu, S. and Jonas, J.J., 1992, Influn of hot stri rolling aramtrs on austnit rrystallization in intrstiial fr stls, ISIJ Intrnational, Vol. 32. n. 2, Poliak, E. I. And Jonas, J.J., 23, Initiation of Dynami Rrystallization in Constant Strain Rat Hot Dformation, ISIJ Intrnational, Vol Zahari, S.H., Davis, C.H.J. and Hodgson, P.D., 25, A mhanial aroah to quantify dynami rrystallization in olyrystallin mtals, Srita Matrialia, Vol. 52, Rsonsibility noti Th authors ar th only rsonsibl for th rintd matrial inludd in this ar.

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