CONTRIBUTION TO THE STUDY OF AUSTENITE FORMATION IN STEELS

Transcription

1 CONRIBUION O HE SUDY OF AUSENIE FORMAION IN SEELS F. G. Caballer, C. Capdevila, D. San Martín, C. García De Andrés Department f Physical Metallurgy, Centr Nacinal de Investigacines Metalúrgicas (CENIM-CSIC), Avda. Gregri del Am, 8, 8040 Madrid, Spain Abstract Mst cmmercial prcesses rely n heat treatments which cause the steel t revert t the austenitic cnditin. he initial cnditin f the austenite determines the develpment f the final micrstructure and mechanical prperties, s the mdeling f the transfrmatin int austenite is useful. In this sense, a quantitative thery dealing with the nucleatin and grwth f austenite frm a variety f initial micrstructural cnditins is vital. he main aim f this wrk is t study the mechanisms that cntrl the austenitizatin prcess in steels with different initial micrstructures. he cmpiled knwledge in literature regarding the isthermal frmatin f austenite frm different initial micrstructures (pure and mixed micrstructures), has been used in this wrk t develp a mdel fr nn-isthermal austenite frmatin in steels with initial micrstructure cnsisting f ferrite and/r pearlite. he micrstructural parameters that affect the nucleatin and grwth kinetics f austenite, and the influence f the heating rate have been cnsidered in the mdeling. Intrductin he frmatin f austenite during heating differs in many ways frm thse transfrmatins that ccur during the cling f austenite. Fr instance, the kinetics f austenite decmpsitin can be cmpletely described in terms f the chemical cmpsitin and the austenite grain size. By cntrast, the micrstructure frm which austenite may frm is mre cmplex and additinal variables are therefre needed t describe the kinetics f austenite frmatin. Factrs such as particle size, distributin and chemistry f individual phases, hmgeneity and the presence f nn-metallic inclusins shuld all be imprtant [-4]. he develpment f dual-phase steels by partial austenitizatin revived the interest fr the heating part f the heat treatment cycle in the eighties. hese steels are prduced by annealing lw carbn steels in the intercritical temperature range with the aim f btaining ferriteaustenite mixtures, and subsequent quenching t transfrm the austenite phase int martensite [5-7]. Speich et al.[] and, Garcia and DeArd [] described in detail the mechanisms that cntrl the austenite frmatin prcess under isthermal cnditins in lw carbn steels with a ferrite-pearlite initial micrstructure. Later, Rsz et al. [8] quantitatively determined the influence f the initial micrstructure n the nucleatin rate and grain grwth f austenite during isthermal treatment f a eutectid plain carbn steel. All these authrs emphasized the imprtance f the micrstructure that exists befre intercritical annealing. Hwever, little infrmatin is available abut the austenite frmatin in steels subjected t cntinuus heating. Recently, sme researchers have adpted a new apprach t the prblem using artificial neural netwrk [9, 0]. his has helped t identify the fact that a neglect f the starting micrstructure can lead t majr errrs in the predictin f the transfrmatin temperatures, smetimes by mre than 00 C. In this sense, the aim f this wrk is t evaluate

2 the influence f heating rate and micrstructural parameters n the nn-isthermal frmatin f austenite in steels with different initial micrstructures. Frm all that theretical knwledge and the experimental study f the mechanisms that cntrl the frmatin f austenite frm different initial micrstructures (pure and mixed micrstructures), kinetics thery has been develped t allw the estimatin f austenite frmatin. able I Chemical cmpsitin (mass %) Steels C Mn Si N Al P Cr Ni Nb FERR FERR FERR FERR PEARL MIX Materials and Experimental Prcedure able I lists the chemical cmpsitin f the studied steels. FERR-4 steels in able I have a full ferrite initial micrstructure. On the ther hand, tw different isthermal heat treatments were carried ut t yield in PEARL steel (able I) fully pearlitic micrstructures with different scale parameters. In this sense, specimens were austenitized fr 5 min at 73 K, isthermally transfrmed at 948 and 798 K, and subsequently cled rapidly t rm temperature. Finally, MIX steel (able I) is a lw carbn manganese steel with a ferrite plus pearlite initial micrstructure. his steel was used t validate the kinetics mdel fr austenite frmatin in steel with a mixed initial micrstructure. he material was saked at 473 K fr 0 min., ht rlled t 30 mm in several passes, and finally air cled t rm temperature. he as-rlled micrstructure f this steel frmed by 87 % ferrite and 3 % pearlite is shwn in Fig..a. Figure : Initial micrstructure f the steel used t validate the kinetics mdel fr austenite frmatin in steel with a ferrite plus pearlite initial micrstructure: (a) Optical micrgraph; (b) Scanning electrn micrgraph.

3 Specimens were plished in the usual way and finished n 0.5 µm diamnd paste fr metallgraphic examinatin. w types f etching slutin were used: Nital-pct t reveal pure and mixed micrstructures by light ptical micrscpy and slutin f picric acid in isprpyl alchl with several drps f ilella s reagent t disclse pearlite mrphlgy n a JEOL JXA- 80 scanning electrn micrscpe. Figure.b shws a scanning micrgraph f the mrphlgy f pearlite in MIX steel. he ferrite grain size in pure micrstructures was measured n micrgraphs. he average ferrite grain diameter (D) was estimated by cunting the number f grains intercepted by ne r mre straight lines lng enugh t yield at least fifty intercepts in ttal. he effect f a mderately nn-equiaxial structure was eliminated by cunting the intersectins f lines in fur r mre rientatins cvering all the bservatin fields with an apprximately equal weight. Micrstructural characterizatin results fr FERR-4 are published elsewhere []. Likewise, tw parameters, the mean true interlamellar spacing, σ, and the area per unit vlume f the PP pearlite clnies interface, S v, characterize the mrphlgy f pearlite [8]. he values f σ in all the cases (PEARL and MIX steels) were derived frm electrn micrgraphs accrding t PP Underwd s intersectin prcedure [, 3]. he values f S v were measured n scanning micrgraphs by cunting the number f intersectins f the pearlite clny bundaries with a circular test grid as reprted by Rsz et al. [8]. Apprximating the pearlite clny by a P truncated ctahedrn, the edge length f the pearlite clnies, a, is calculated frm the area PP per unit vlume with the fllwing expressin [4]: S v S PP v P ( + 3 ) ( a ) 3( + 3 ) 6 () P 3 P 8 ( a ) 4 a PP P Data fr σ, S v and a fr MIX steel are listed in able II. he same data fr PEARL steel are published elsewhere []. able II Mrphlgical characterizatin f pearlite in MIX steel Specimen σ 0-3 PP, mm S v, mm - a P 0-3, mm MIX 0.5±0.0 89±33.06±0.49 he experimental validatin f the austenite frmatin kinetics develped in this wrk was carried ut using an Adamel Lhmargy D000 high-reslutin dilatmeter. In rder t analyze the prgress f austenite frmatin in PEARL and MIX steels, interrupted heating experiments were carried ut by quenching. Heating dilatmetric curves were analyzed t determine the start temperature (Ac ) and the end temperature (Ac 3 ) f austenite frmatin and then several quench-ut temperatures were selected in rder t evaluate the prgress f the transfrmatin. Austenite, which is frmed during heating, transfrms t martensite during quenching. hus, the prgress f austenitizatin is determined thrughut the evlutin f the vlume fractin f martensite. Specimens frm interrupted heating experiments were plished in the usual way fr metallgraphic examinatin. LePera s reagent [5] was used t reveal martensite frmed during quenching. he quantitative measurement f martensite vlume fractin was carried ut by pint-cunting methd [6].

4 Figure. Optical micrgraphs frm interrupted heating samples at different stages f the reaustenitizatin prcess in MIX steel. (a) 07 K; (b) 084 K; (c) 35 K; Heating rate f 5 K/s. Figure shws micrscpic evidences f hw austenite frmatin ccurs in MIX steel fr a heating rate f 5 Ks - thrughut micrgraphs frm interrupted heating samples at different stages f the reaustenitizatin prcess. LePera s reagent [5] reveals pearlite and ferrite as a darker phase in the micrstructure, whereas martensite frmed during quenching appears as lighter regins in the micrgraphs. Micrstructure in Fig..a is frmed mainly f ferrite, pearlite and sme grains f martensite. At this quench-ut temperature, the pearlite-t-austenite transfrmatin starts. Figures.b and.c shw intermediate stages f the reactin. Results and Discussin Mdeling f Kinetics f Nn-Isthermal Austenite Frmatin in a Steel with a Ferrite Initial Micrstructure In the frmatin f austenite frm ferrite, the austenite grwth is cntrlled by prcesses at the interface and the grwth rate G is given by [7]: δν Gact α δν S H α G exp g exp exp g { } () k k k k k where δ is the bundary thickness, ν is the number f attempts t jump the bundary activatin barrier per unit time, k is the Bltzman cnstant, is the abslute temperature, G act is the free energy fr the activated transfer atms acrss the ferrite/austenite interface, S is the entrpy f activatin per atm, H is the enthalpy f activatin per atm, and g α is the Gibbs free energy difference per atm between the α and phases. he values f H and ν are uncertain but are generally assumed t be equal t the enthalpy f activatin fr grain bundary diffusin [8] and t k/h (being h Planck cnstant), respectively. he value f S is als uncertain and may be negative r psitive. If we cnsider that the maximum ferrite/austenite interface velcity fr a 00 µm ferrite grain diameter is 0.06 m/s at 3 K [9], g α J per atm, δ 5 Å, and H S J per atm, then ν exp is equal t s -. k

5 Figure 3: Gibbs free energy change fr α transfrmatin in FERR steel. Figure 3 shws the Gibbs free energy change fr the ferrite-t-austenite transfrmatin g α fr FERR steel (able I). his energy has been btained accrding t the thermdynamic calculatins prpsed by Aarnsn et al. [0, ] and Kaufman et al. []. In rder t accunt fr the effects f allying elements int calculatin, Zener factrizatin f the free energy int magnetic and nn-magnetic cmpnents has been perfrmed [3]. Assuming that site saturatin ccurs and the reactin is cntrlled by grwth, the kinetics law btained fr the three different activated grwth sites can be expressed as fllws [4, 5]: 3 exp K st + K et + K ct (3) where represents the frmed austenite vlume fractin, t is the time and K s, K e and K c are given by, K s S L 4 3 C G K e π G K c πg (4) 3 where the grwth rate f austenite (G) is given by equatin () and S, L and C respectively are the bundary area, the edge length, and the grain crner number, all per unit vlume. Assuming ferrite grains t be tetrakaidecahedra [4], S, L and C can be expressed in terms S f the average ferrite grain diameter D by: ; D L 8.5 ; and 3 D C. D he difficulties in treating nn-isthermal reactins are mainly due t the cmplex variatins f grwth rate with temperature, described in equatin (). We can nly deal with the prblem when the rate f transfrmatin depends exclusively n the state f the assembly and nt n the thermal path by which the state is reached [7]. Reactins f this type are called iskinetic. Avrami defined an iskinetic reactin by the cnditin that the nucleatin and grwth rates are

6 prprtinal t each ther (i.e. they have the same temperature variatin). his leads t the cncept f additivity and Scheil's rule [6]. Since Avrami's cnditin fr an iskinetic reactin is nt satisfied in the present case, a general equatin t describe the nn-isthermal verall ferrite-t-austenite transfrmatin in ferritic steels was derived integrating the equatin (3) ver the whle temperature range where the transfrmatin takes place [7]. In this sense, we have taken lgarithms in equatin (3), which then was differentiated, dt t K t K K d d c e s ln (5) If we cnsider a cnstant rate,, fr the heating cnditin, time can be expressed as fllws: t d dt (6) substituting int equatin (5) and integrating in [ ], 0 and [ ] s, intervals n the left and n the right sides, respectively, it can be cncluded that: ( ) + + d K K K d c e s s 0 3 (7) where s is the start temperature f the transfrmatin r temperature at which (rt f the functin represented in Fig. 3). herefre, the vlume fractin f austenite ( ) present in the micrstructure as a functin f temperature can be calculated as fllws, 0 α g ( ) ( ) + + d G D G D G D s s s exp (8) Experimental alidatin f Kinetics f Nn-Isthermal Austenite Frmatin in Steels With a Ferrite Initial Micrstructure. In the abve sectin, a mdel is presented t describe the ferritet-austenite transfrmatin during cntinuus heating in steels with a fully ferrite initial micrstructure. his mdel allws us t calculate the vlume fractin f austenite and ferrite during transfrmatin as a functin f temperature and thus t knw the austenite frmatin kinetics under nn-isthermal cnditins in full ferritic steels. Since the ferrite-t-austenite

7 transfrmatin is s rapid in these steels, then reliable measurements f the vlume fractin f austenite frmed during the partial transfrmatin cannt be experimentally btained. his renders the validatin f the kinetics mdel thrugh the cmparisn f the experimental and calculated vlume fractin f austenite frmed during the prgress f the transfrmatin t be impssible. Hwever, dilatmetric analysis can be emplyed as an alternative technique t study phase transfrmatin kinetics in steels. In this sense, the relative changes in length which ccur during the ferrite-t-austenite transfrmatin were calculated as a functin f temperature. Bth kinetics and dilatmetric calculatins will be validated by cmparing theretical and experimental heating dilatmetric curves. Experimental values f relative changes in length and linear expansin cefficients measured by high reslutin dilatmetry were accurately reprduced by calculatins [8]. Mdeling f Kinetics f Nn-Isthermal Austenite Frmatin in a Steel with a Pearlite Initial Micrstructure Nucleatin and grwth prcesses under isthermal cnditin can be described in general using the Avrami's equatin [9]: π 3 4 exp N G t (9) 3 where represents the frmed austenite vlume fractin, N is the nucleatin rate, G is the grwth rate and t is the time. Accrding t Christian [7], with a spherical cnfiguratin, an expnent f 4 in time (t) in Avrami s equatin means that the nucleatin rate ( N ) and the grwth rate (G) are cnstant in time. Rsz et al. [8] prpsed a temperature and mrphlgy dependence f f the reciprcal value f verheating ( -Ac ) as fllws: N and G as a functin N QN f N exp (0) k G QG fg exp () k where Q N and Q G are the activatin energies f nucleatin and grwth, respectively, k is Bltzmann s cnstant, and f N and f G are the functins representing the influence f the mrphlgy f pearlite and the heating rate n the nucleatin and grwth rates, respectively. Several authrs [8,9,30] reprted that the nucleatin f austenite inside pearlite takes place preferentially at the pints f intersectin f cementite with the edges f the pearlite clny. Apprximating the pearlite clny as a truncated ctahedrn, the number f nucleatin sites per

8 unit vlume is calculated as NC where a P is the edge length f the pearlite clny P ( a ) and σ is the interlamellar spacing [4]. σ Bearing in mind that the rate f nucleatin increases as the pearlite interlamellar spacing decreases and the edge length f the pearlite clny increases [3], and cnsidering that the heating rate ( ) might influence n the nucleatin rate, the functin f N in equatin (0) is assumed t have the fllwing general frm: P n p ( a ) r f N K N ( N m C ) σ () where K N, n, m, p and r are empirical parameters. hese parameters were adjusted in rder t btain gd fit between thery and the experimental austenite vlume fractin curves. In this sense, the measured values f austenite vlume fractin as a functin f temperature can be best described with n6, m, p and r 3. Austenite nuclei in pearlite grw when carbn atms are transprted by diffusin t the ferrite/austenite bundary frm the austenite/cementite bundary thrugh the austenite and frm the ferrite/cementite bundary thrugh the ferrite, resulting in a transfrmatin f the ferrite lattice t an austenite lattice [3]. As in the case f the reverse transfrmatin (austenite-tpearlite transfrmatin), the grwth rate f austenite is believed t be cntrlled by either vlume diffusin f carbn r by bundary diffusin f substitutinal allying elements [3,33,34]. If the grwth rate f austenite is cntrlled by the bulk diffusin f atms in austenite ahead f the interface, the diffusin f carbn may play a mre imprtant rle than that f substitutinal allying elements. Diffusivity f the substitutinal allying elements in austenite is far smaller than that f carbn. As a result, the substitutinal allying elements may nt diffuse a lng distance during the reactin. Hwever, as described by Prter [35], when temperature decreases, bundary diffusin f substitutinal allying elements is the dminant mechanism in the diffusin prcess. In that case, the partitining f the substitutinal allying elements is substantial during the grwth f austenite and bundary diffusin f the allying elements may cntrl the grwth rate f pearlite. he functin f G in equatin () representing the mrphlgy dependence n the grwth rate can be expressed as fllws: f G K G (3) i σ where K G is an empirical cnstant, i if the grwth rate f austenite is cntrlled by vlume diffusin f carbn and i if the grwth rate f austenite is cntrlled by bundary diffusin f substitutinal allying elements [8]. As in ferrite-t-austenite transfrmatin, a general equatin t describe the nn-isthermal verall pearlite-t-austenite transfrmatin in pearlitic steel was derived integrating the Avrami's equatin ver the whle temperature range where the transfrmatin takes place [7].

9 In this sense, we have taken lgarithms and differentiated in equatin (9). Expressing time as t leads t: d 4π N G d (4) Integrating in [ ], 0 and [ Ac, ] intervals n the left and n the right sides f equatin (4), respectively, it can be cncluded that: 4π 3 3 exp - N G d (5) 4 Ac 3 It has been assumed that at a heating rate higher than 0.5 Ks - the grwth rate f austenite wuld be mainly cntrlled by the vlume diffusin f carbn in austenite, due t the fact that the transfrmatin wuld take place mstly at higher temperatures. Cnsequently, a i value f is cnsidered in equatin (3) fr that case. On the cntrary, at heating rates lwer than r equal t 0.5 Ks - the grwth rate f austenite has been assumed t be cntrlled by bundary diffusin f substitutinal allying elements and a i value f is cnsidered in equatin (3) fr that case. he eutectid temperature Ac f the steel was btained using Andrews frmula [36]. Experimental alidatin f Kinetics f Nn-Isthermal Austenite Frmatin in Steels With a Pearlite Initial Micrstructure. he influence f mrphlgical parameters, such as interlamellar spacing and edge length f pearlite clnies, and heating rate n the transfrmatin kinetics were experimentally studied in PEARL steel [37]. analyze the prgress f pearlite-taustenite transfrmatin, interrupted heating experiments were carried ut by quenching. Specimens f PEARL steel (able I) with tw different scales f lamellar pearlite were heated at three different cnstant rates (0.05, 0.5 and 5 Ks - ). Heating dilatmetric curves were analyzed t determine the start temperature (Ac ) and the end temperature (Ac 3 ) f pearlite-taustenite transfrmatin and then several quench-ut temperatures were selected in rder t evaluate the prgress f the transfrmatin. It was fund that austenite transfrmatin starts later and appears t be slwer the carser the initial pearlite micrstructure and the higher the heating rate. Furthermre, bth start and finish temperatures slightly increase as heating rate des, but finish temperatures f the pearlite-t-austenite transfrmatin are mre sensitive t the heating rate than start temperatures. Hwever, the influence f heating rate n bth temperatures is less significant when the finer the initial pearlite micrstructure. Experimental results fr the austenite vlume fractin and critical temperatures f pearlite-taustenite transfrmatin were in gd agreement (accuracy higher than 90% in square crrelatin factr) with the predicted values frm the mdel prpsed in this wrk [37]. he accuracy f this mdel was in all the cases higher than 90% which can be cnsidered excellent fr a kinetics mdel bearing in mind the experimental difficulties fr its validatin.

10 Mdeling f Kinetics f Nn-Isthermal Austenite Frmatin in a Steel with a Ferrite Plus Pearlite Initial Micrstructure In the austenitizatin f micrstructures cmpsed f ferrite and pearlite, tw different transfrmatins are invlved: pearlite disslutin and ferrite-t-austenite transfrmatin. Bth transfrmatins take place by nucleatin and grwth prcesses. Mdeling f Kinetics f Disslutin f Pearlite. he nucleatin and grwth prcesses that cntrl the disslutin f pearlite in a steel with a ferrite plus pearlite micrstructure are the same than thse described abve fr a steel with a full pearlite initial micrstructure. herefre, P the austenite vlume fractin btained frm pearlite disslutin,, during cntinuus heating f a ferrite plus pearlite initial micrstructure is expressed as fllws: P 4π 3 3 P exp - N G d (6) 4 Ac 3 where P is the vlume fractin f pearlite present in the initial micrstructure P in MIX steel. Mdeling f Kinetics f Ferrite-t-Austenite ransfrmatin after Disslutin f Pearlite. he purpse f this sectin is t develp a mdel describing the transfrmatin prcess f a mixture f carbn enriched austenite (btained frm the disslutin f pearlite) and ferrite, during cntinuus heating. In the present mdel, the fllwing assumptins are made: Ferrite-t-austenite transfrmatin begins nce the disslutin f pearlite is cmpleted (temperature Ac P ). N nucleatin ccurs during this transfrmatin. All the nuclei are generated during the disslutin f pearlite. Only grwth f the existing austenite grains takes place. he prcess is cntrlled by the vlume diffusin f carbn in austenite. Diffusin f carbn in ferrite has been ignred since the slubility f carbn in ferrite is negligible cmpared t its slubility in austenite. he carbn cntent in the austenite after the disslutin f pearlite is assumed t have a mean value f 0.8 wt-%. he carbn inside the austenite sn will diffuse t the austenite/ferrite (/α) interface in rder t re-establish the thermdynamic equilibrium at the crrespnding temperature. A planar gemetry is cnsidered in the mdeling and lcal equilibrium is assumed. he mean carbn cmpsitin f austenite grains is given by the equilibrium carbn cncentratins crrespnding t the (α+)/ phase bundary f the equilibrium diagram f the steel.

11 he cntinuus heating curve will be expressed as a series f small isthermal steps ccurring at a successively higher temperatures with a time interval, dt, assciated with each step. At each step f the heating curve (i.e., each temperature), the time interval dt will be shrt enugh t make pssible the assumptin f steady state diffusin. herefre, the gradient f carbn in austenite is cnsidered t be cnstant at each temperature. Finally, the diffusivity f carbn in austenite is assumed t be nly dependent n the lcal cncentratin f carbn in austenite (different fr each step f the heating cycle). Figure 4. Evlutin f the gradient f carbn during ferrite-t-austenite transfrmatin ( Ac P ). During the cntinuus heating, carbn atms diffuse frm inside the enriched austenite grains t the /α interface, crssing it. hus, the interface mves frward and sme ferrite is transfrmed int austenite. his prcess cntinues until the average carbn cntent in austenite equals the carbn cntent f the steel, just when all ferrite has transfrmed ( Ac 3 temperature) and the micrstructure is fully austenitic. Under the abve cnditins, at each step f the heating cycle, the diffusin equatin fr carbn in austenite can be simplified t: D C d C dr 0 (7) where D C is the diffusivity f carbn in austenite, C is the cncentratin f carbn in austenite and r is the spatial variable nrmal t the interface /α. slve equatin (7), we need t knw the bundary cnditins f the prblem. As we stated befre, we assumed the existence f a gradient f carbn during the transfrmatin frm the center f the austenite grains t the /α interface (Fig. 4). herefre, fr temperatures higher than Ac P, c r r C C (8) α r α r C C (9)

12 c α where, C and C are the carbn cncentratin in the center and at the interface f the austenite grains, respectively. is the radius f carbn enriched austenite grains after the r α disslutin f pearlite and r is the psitin f the interface /α. Slving (7) fr the bundary cnditins (8)-(9), gives the equatin fr the gradient f carbn inside austenite grains at each temperature, dc dr α ( C c C ) (0) L he effective diffusin distance f carbn in austenite,, can be deduced frm mass carbn cnservatin. he ttal carbn cncentratin f the system at Ac P must be equal t the carbn cncentratin at any > Ac P. Applying this principle we get, L L α eut α r ( C C ) eut c α ( C C C ) () eut α With C the eutectid carbn cmpsitin (0.8 wt-%) and C the carbn cmpsitin at the interface inside the ferrite. On the ther hand, the cnservatin f the carbn cncentratin fluxes at the /α interface at each time interval is given by, α α α dc ( C C ) dr D dt C () dr r r α Replacing equatin () in (0) and the resultant in () gives, dr dt α D C c α eut c α ( C C )( C C C ) α α α eut α r ( C C )( C C ) (3) Integrating (3), an expressin fr the radius f the austenite grain during the transfrmatin is reached, r ηt α (4) where η is the parablic grwth rate cnstant f the transfrmatin,

13 c α ( ) ( ) ( eut c α C C C C C ( ) ) α α eut α C C C C η DC (5) he diffusivity f carbn in austenite is calculated fllwing the wrk f Bhadeshia [38]. In this mdel it is assumed that the average carbn cntent f austenite grains at each temperature during the transfrmatin, C, is given by the equilibrium carbn cncentratins crrespnding t the (α+)/ phase bundary f the equilibrium diagram f the steel. Its value is calculated accrding t the prcedure reprted by Shiflet et al. [39]. Als, a linear variatin f the carbn cntent in the center f the grains with temperature is cnsidered. C is assumed t be related t the carbn cncentratin in the center f the austenite grains and at the interface as fllws, C C α + C c (6) Cnsidering spherical austenite grains whse radius vary accrding t equatin (3) and taking int accunt Avrami thery fr the relatin between real and extended vlume [9], the vlume fractin grwth rate f the transfrmatin can be written as, d α dt 3 α α α π η ( ) N 6 6 ln ( α ) (8) α 3 If we cnsider a cnstant rate fr the heating cnditin, each time interval can be expressed as fllws, ( Ac ) t P ; d dt (9) Intrducing equatin (8) int (7), separating variables and integrating, an expressin fr the vlume fractin f austenite transfrmed during heating is btained, [ 6π ( N ) ] 3 α α exp η d (30) 3 3 AcP α where and are the vlume fractin f austenite transfrmed frm ferrite and the initial α N vlume fractin f ferrite, respectively, and is the number f austenite grains per unit vlume f ferrite at the end f the disslutin f pearlite which is given by,

14 N P α Ac P N( P ) d (3) P where, P and N are the vlume fractin f austenite transfrmed frm pearlite (equatin (6)), the initial vlume fractin f pearlite and the nucleatin rate per unit vlume f pearlite (equatin (0)), respectively. Experimental alidatin f Kinetics f Nn-Isthermal Austenite Frmatin in Steels With a Ferrite Plus Pearlite Initial Micrstructure. Figure 5 shws the experimental and calculated austenite frmatin kinetics pltted as a functin f temperature in MIX steel fr a cnstant rate f 5 Ks -. Experimental results fr the austenite vlume fractin are in gd agreement with the predicted values frm the mdel and its accuracy culd be cnsidered excellent. Figure 5. Experimental and calculated kinetics results fr the frmatin f austenite under cntinuus heating cnditins in MIX steel. Cnclusins. heretical knwledge regarding the isthermal frmatin f austenite frm pure and mixed initial micrstructures has been used t develp a mdel fr the nn-isthermal austenite frmatin in a wide range f steels with an initial micrstructure cnsisting f ferrite and/r pearlite. Since cnditins t apply Scheil's rule are rarely satisfied, the Avrami's equatin has been used t reprduce the kinetics f the austenite frmatin during cntinuus heating.. In steels with a full ferrite initial micrstructure, nucleatin f austenite ccurs at the α/α grain bundaries. All pssible nucleatin sites at the grain bundaries have been taken int cnsideratin in the mdeling assuming that n nucleatin barrier exists. Since ferrite/austenite bundary migrates in the absence f diffusin, the grwth f austenite has

15 been cnsidered t be cntrlled by prcesses at the interface. Experimental validatin f the mdel was carried ut by cmparisn between experimental and theretical heating dilatmetric curves. 3. In the case f pearlite-t-austenite transfrmatin, the mdel prpses tw functins, f N and f G, which represent the dependence f nucleatin and grwth rates, respectively, n the mrphlgy f pearlite and heating rate. In this sense, the influence f mrphlgical parameters, such as interlamellar spacing and edge length f pearlite clnies, and heating rate n the transfrmatin kinetics were experimentally studied in a eutectid steel. It was fund that austenite transfrmatin starts later and appears t be slwer the carser the initial pearlite micrstructure and the higher the heating rate. Furthermre, bth start and finish temperatures slightly increase as heating rate des, but finish temperatures f the pearlite-t-austenite transfrmatin are mre sensitive t the heating rate than start temperatures. Hwever, the influence f heating rate n bth temperatures is less significant when the finer the initial pearlite micrstructure. Experimental results fr the austenite vlume fractin and critical temperatures f pearlite-t-austenite transfrmatin fr a eutectid steel were in gd agreement (accuracy higher than 90% in square crrelatin factr) with the predicted values frm the mdel prpsed in this wrk. 4. In a steel with mixed initial micrstructure cnsisting f ferrite and pearlite, the kinetics f pearlite disslutin during cntinuus heating has been reprduced using the mdel fr the kinetics f the pearlite-t-austenite transfrmatin develped initially fr steels with a full pearlite initial micrstructure. Likewise, a kinetics mdel has been prpsed fr the determinatin f the vlume fractin f austenite frmed frm ferrite after pearlite disslutin as a functin f temperature fr cntinuus heating cnditins. he cmplete mdel fr austenitizatin f mixed micrstructures has been validated finding a gd agreement between experimental and calculated values (accuracy higher than 90% in square crrelatin factr). Acknwledgements he authrs acknwledge financial supprt frm Spanish Ministeri de Ciencia y ecnlgía (MA00-67). F.G. Caballer wuld like t thank Spanish Ministeri de Ciencia y ecnlgía fr the financial supprt in the frm f a Ramón y Cajal cntract (Prgrama RyC 00). References. C.I. García and A.J. DeArd, Frmatin f Austenite in.5 Pct Mn Steels, Met. rans. A, A (98), G.R. Speich,.A. Demarest and R.L. Miller, Frmatin f Austenite during Intercritical Annealing f Dual- Phase Steels, Met. rans. A, A (98), M.M. Suza, J.R.C. Guimaraes and K.K. Chawla, Intercritical f w Fe- Mn- C Steels, Met. rans. A, 3A (98), Xue-Ling Cai, A.J. Garrat-Reed and W.S. Owen, Develpment f Sme Dual-Phase Steel Structures frm Different Starting Micrstructures, Met. rans. A, 6A (985), G. Krauss, Steels: Heat reatment and Prcessing Principles, (Ohi, ASM Internatinal, 989), 74.

16 6. D.L. Gustavssn et al., Inst. Metallfrsk. Frsk. Rapp., 88 (994), L. Gadgeel, l Ally Steels, 8 (994), A. Rsz, Z. Gacsi and E.G. Fuchs, Isthermal Frmatin f Austenite in Eutectid Plain Carbn Steel, Acta Metall., 3 (983), L. Gavard et al., Bayesian Neural Netwrk Mdel fr Austenite Frmatin in Steels, Mater. Sci. echnl., (996), C.A.L. Bailer-Jnes, H.K.D.H. Bhadeshia and D.J.C. MacKay, Gaussian Prcess Mdelling f Austenite Frmatin in Steel Mater. Sci. echnl., 5 (999), F.G. Caballer, C. Capdevila, and C. García de Andrés, Mdelling f Kinetics f Austenite Frmatin in Steels with Different Initial Micrstructures ISIJ Internatinal, 4, (00), E.E. Underwd, Quantitative Sterelgy, (Reading, Addisn-Wesley Publishing C 970), S.A. Saltykv, Steremetric Metallgraphy, (Mscw, Metallurgizdat, 958), R.. De Hff and F.H. Rhines, Quantitative Sterelgy, (New Yrk, McGraw-Hill, 968), F.S. LePera, Imprved Etching echnique t Emphasize Martensite and Bainite in High- Strength Dual-Phase Steel, Metallgraphy, (979), G.F. ander rt, Metallgraphy. Principles and Practice, (New Yrk, McGraw-Hill, 984), J.W. Christian, he hery f ransfrmatins in Metals and Allys, (Oxfrd, Pergamn Press, 965), P. Shewmn, Diffusin in Slids, (New Yrk, McGraw-Hill, 963), G.R. Speich and A. Szirmae, Frmatin f Austenite frm Ferrite and Ferrite- Carbide Aggregates, rans. MS-AIME, 45 (969), H.I. Aarnsn, H.A. Dmian and G.M. Pund, hermdynamics f the Austenite- Preutectid Ferrite ransfrmatin. I, Fe-C Allys, rans. MS-AIME, 36 (966), H.I. Aarnsn, H.A. Dmian and G.M. Pund, hermdynamics f the Austenite- Preutectid Ferrite ransfrmatin. II, Fe-C-X Allys, rans. MS-AIME, 36 (966), L. Kaufman, E.. Clugherty and R.J. Weiss, he Lattice Stability f Metals-III. Irn, Acta Metall., (963), C. Zener, Kinetics f the Decmpsitin f Austenite, rans. AIME, 67 (946), J.W. Cahn, he Kinetics f Grain Bundary Nucleated Reactins, Acta Metall., 4 (956), S.F. Dirnfeld, B.M. Krevaar and F. an t Spijker, he ransfrmatin t Austenite in a Fine Grained l Steel, Met. rans. A, 5A (974), R.G. Kamat et al., he Principle f Additivity and the Preutectid Ferrite ransfrmatin, Met. rans. A, 3A (99),

17 7. C. García de Andrés et al., Mdelling f Kinetics and Dilatmetric Behaviur f Nn- Isthermal Pearlite-t-Austenite ransfrmatin in an Eutectid Steel, Scripta Mater., 39 (998), F.G. Caballer, C. Capdevila and C. García de Andrés, "Kinetics and Dilatmetric Behaviur f Nn-Isthermal Ferrite-t-Austenite ransfrmatin", Mater. Sci. echnl., 7 (00), M. Avrami, Kinetics f Phase Change II, J. Chem. Phys., 8 (940), R.R. Judd and H.W. Paxtn, Kinetics f Austenite frm Spheridized Ferrite-Carbide Aggregate, rans. MS-AIME, 4 (968), G.A. Rberts and R.F. Mehl, he Mechanism and the rate f Frmatin f Austenite frm Ferrite-Cementite Aggregates, rans. ASM, 3 (943), M. Hillert, K. Nilssn, and L.E. rndahl, Efffect f Allying Elements n the Frmatin f Austenite and Disslutin f Cementite, JISI, 09 (97), M. Hillert, he frmatin f Pearlite (Paper presented at the Decmpsitin f Austenite by Diffusinal Prcesses Cnference, Philadelphia, Pennsylvania, 9 Octber 960), M. Hillert, he Mechanism f Phase ransfrmatin in Crystalline Slids, (Lndn, Institute f Metals, 969), D.A. Prter and K.E. Easterling, Phase ransfrmatins in Metals and Ally, nd ed., (Lndn, Chapman and Hall, 99), K.W. Andrews, Empirical Frmulae fr the Calculatin f Sme ransfrmatin emperature, JISI, 03 (965), F.G. Caballer, C. Capdevila and C. García de Andrés, Influence f Pearlite Mrphlgy and Heating Rate n the Kinetics f Cntinusly Heated Austenite Frmatin in a Eutectid Steel, Met. Mater. rans., 3A (00) H.K.D.H. Bhadeshia, Diffusin f carbn in austenite, Metal Science, 5 (98), G.J. Shiflet, J.R. Bradley and H. I. Aarnsn, A Re-examinatin f the hermdynamics f the Preutectid Ferrite ransfrmatin in Fe-C Allys, Met. rans. A, 9A (978),