THERMEC 2006The kinetics of phase transformations in medium-carbon forging steels (MCFS) have been

modeled based on CALPHAD multicomponent thermodynamics and the classical nucleation-growth

theory. New treatments include the time dependency of parabolic growth rate of proeutectoid ferrite

(α) , which account for the soft impingement effect by carbon enrichment in austenite (γ). And a

potential transition of γ/α interface equilibrium has also been considered depending on temperatures

and velocity of the moving interface. To make a realistic prediction of the onset of pearlite (P)

transformation, a normal distribution of γ grain size has been assumed and successive α→P

transformation kinetics in each grain size have been summated.

The developed program coupled with thermodynamic solver, 'ThermoCalc', calculated the isothermal

kinetics of MCFS and has been found to predict well the effect of minor difference of chemical

composition / holding temperaturesPeer reviewe

Bardow, André

Madlener, Reinhard

Majewski, Dinah Elena

Monti, Antonello

Müller, Dirk

Streblow, Rita

THERMEC 2006The kinetics of phase transformations in medium-carbon forging steels (MCFS) have been
modeled based on CALPHAD multicomponent thermodynamics and the classical nucleation-growth
theory. New treatments include the time dependency of parabolic growth rate of proeutectoid ferrite
(α) , which account for the soft impingement effect by carbon enrichment in austenite (γ). And a
potential transition of γ/α interface equilibrium has also been considered depending on temperatures
and velocity of the moving interface. To make a realistic prediction of the onset of pearlite (P)
transformation, a normal distribution of γ grain size has been assumed and successive α→P
transformation kinetics in each grain size have been summated.
The developed program coupled with thermodynamic solver, 'ThermoCalc', calculated the isothermal
kinetics of MCFS and has been found to predict well the effect of minor difference of chemical
composition / holding temperaturesPeer reviewe

Tanaka, Kouji

Hara, Masashi

Yogo, Yasu

Nakanishi, Kou

Capdevila, Carlos

Digital.CSIC

  PHASE TRANFORMATION MODELING OF MEDIUM-CARBON FORGING STEEL K. Tanaka1,2, a, M. Hara1,2,b, Y. Yogo1,c , K. Nakanishi1,2,d, and C. Capdevila3,e 1Toyota Central R&D Labs., Inc., 41-1, Yokomichi, Nagakute, Aichi, Japan 2CREST, Japan Science and Technology Agency, 1-1-2, Ohtemachi, Chiyoda-ku,Tokyo, Japan 3Centro Nacional de Investigaciones Metalúrgicas(CENIM), CSIC, Avda. Gregorio del Amo, 8. 28040, Madrid, Spain akoutnk@mosk.tytlabs.co.jp, bmhara@mosk.tytlabs.co.jp, cyogoyasu@mosk.tytlabs.co.jp, dkou-nakanishi@mosk.tytlabs.co.jp, eccm@cenim.csic.es  Keywords: phase transformation, kinetics, modeling, medium-carbon steel, multicomponent thermodynamics, proeutectoid ferrite, pearlite  Abstract. The kinetics of phase transformations in medium-carbon forging steels (MCFS) have been modeled based on CALPHAD multicomponent thermodynamics and the classical nucleation-growth theory. New treatments include the time dependency of parabolic growth rate of proeutectoid ferrite (α) , which account for the soft impingement effect by carbon enrichment in austenite (γ). And a potential transition of γ/α interface equilibrium has also been considered depending on temperatures and velocity of the moving interface. To make a realistic prediction of the onset of pearlite (P) transformation, a normal distribution of γ grain size has been assumed and successive α→P transformation kinetics in each grain size have been summated. The developed program coupled with thermodynamic solver, 'ThermoCalc', calculated the isothermal kinetics of MCFS and has been found to predict well the effect of minor difference of chemical composition / holding temperatures.  INTRODUCTION Medium carbon steels containing 0.2~0.5 mass percent carbon are used in a large quantity as materials for hot forging parts. A prediction of the microstructure consisting of proeutectoid ferrite (α), pearlite (P), and bainite (B) has long been a desired technology, which would be the basis for balancing mechanical properties and good machinability of final products. The α/P/B fraction after forging is mainly affected by both carbon and manganese content, austenite (γ) grain size (AGS), and cooling schedules. Thermodynamics in Fe-Mn-C ternary system have been well discussed in terms of partitioning of Mn, and the parameters describing diffusion-controlled transformation can now be calculated by CALPHAD method for the ternary and even higher-order system. However, there is still discussion on the type of equilibrium operating at the real interface between α and γ. On the other hand, AGS and cooling rate are strongly related to hot working conditions and have been evaluated in the course of thermal-mechanically coupled FEM analysis. Senuma et al.[1] reported experimental equations for AGS for low carbon steel based on the relationship between rolling conditions and fractions of dynamic/static recrystallized γ grain, and Yogo[2] et al. recently modified the equation to be applicable to forging process. A lot of work has been devoted to understand the kinetics of α, P and B transformations in the framework of the classical nucleation-growth theory, and there are some integrated models[3-6] reported  to be able to predict the evolution of complex α+P+B structure. These models so far have been developed for low-carbon steel strips, and therefore, focused on the prediction of α grain size as Materials Science Forum Vols. 539-543 (2007) pp. 2443-2448online at http://www.scientific.net© (2007) Trans Tech Publications, SwitzerlandAll rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without thewritten permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 161.111.90.248-27/02/07,11:19:16)   well as the fraction of widmanstätten α or B. In view of the circumstances, we have firstly constructed an integrated physical model of the successive α→P transformation for medium-carbon forging steel (MCFS). And a trial program empowered by CALPHAD method has been developed to evaluate the usefulness of such computerized predictions. This article presents the newly introduced features of treating the above diffusion-controlled phenomena in the model, together with the current status in predicting α→P transformation behaviors of MCFS by the program. The possible B transformation from the remaining γ are also to be considered and will be included in a separate paper. MODEL DESCRIPTION Thermodynamic Calculations.  Besides C content it is important to calculate the thermodynamic effect of Mn and/or Cr that exerts, through the strong interaction with C in γ, a decisive influence on the microstructure of MCFS. The CALPHAD method is a standard tool worldwide for calculating multicomponent thermodynamics. One of the commercial packages, ThermoCalc, was introduced in the present modeling work with the help of programming interface, TQ-i. Linking dynamically with ThermoCalc and the steel database TCFe3, driving force of α and cementite (θ), equilibrium compositions with γ, and other thermodynamic parameters were always calculated for Fe-C-Si-Mn-Cr system. It should be mentioned that the newest version of TQ-i has been upgraded to manage para-equilibrium (PE) constraint. As discussed later, the interface compositions were also calculated under negligible-partitioned local equilibrium (NPLE). Incubation Time.  The incubation time (τ) of isothermal α transformation was calculated in Eq. 1 according to the pillbox model[7]. The diffusivity of carbon in γ (DCγ), and the maximum driving force available for α nucleation (∆GV) were input from the above thermodynamic solver. The γ/α interfacial energy (σγ/α) of 0.705J/m2 and the volume occupied by iron atom (να) of 8.785x10-30m3 were taken as constants, while the average lattice parameter a was estimated from the experimental equations[8] including effect of alloying elements. XC0 is the initial C content and other symbols have the usual meanings.  Growth of Proeutectoid Ferrite.  The nucleation rate of α (If) on γ boundary surface was calculated following the classical nucleation theory. The fitting parameters were taken from Liu's work[9]. Assuming carbon-diffusion-controlled growth of α, the parabolic growth rate constant (α1) was expressed in Eq. 2, and numerically solved via the subroutine by Capdevila[10]. The choice of interface carbon composition set (XCγ/α, XCα/γ) has been controversy, and those in PE constraint would be the most reasonable for an interface moving at higher velocities under high supersaturations. Hillert[11] suggested, however, at a later stage of isothermal transformation, there may be a gradual shift of the interface equilibrium from PE to the one with a spiky partitioning of Mn. The treatment here proposed is that the transition from PE to NPLE occurs when the velocity of (1) (2)   =−−γγγααγγαγγ αααπCC/C/CC/C/CDerfDexpXXXXD242 121121220412VCC/GXDkTa∆⋅⋅⋅=αγαγνστTHERMEC 20062444   interface (vα) becomes lower compared to Mn diffusivity. Although the boundary diffusivity of Mn is not known, it was estimated as 5 times as the bulk one in γ (DMnγ). Thus the critical ratio p for the transition was defined in Eq. 3 where t is holding time. The soft impingement effect (SIE) is pronounced and have to be properly modeled for the α growth of MCFS. Andrés[12] proposed a 1-D analytical model after Gilmour’s work, but in the present model a regressive SIE treatment has been implemented to consider C enrichment caused by a 3-D "α-shell" growing from γ grain boundary. In Eq. 2, the C content away from interface (XCγ) should increase if considered in a finite γ grain. This could be conveniently monitored by the change in the average C content in remaining γ (Xγbar) from the volume fraction of transformed α (Vf ) as shown in Eq. 4. The increase of Vf  increases Xbarγ and makes α1 time-dependent. In this sense, If and DCγ were recalculated every time of updating Xbarγ. Onset and Growth of Pearlite.  The transformation of P from the interface between preceding α and γ has been modeled based on Capdevila's comprehensive work[13]. The critical velocity (vθ) of moving γ/α interface that allows θ nucleation was given by Eq. 5 in which the lattice parameter of γ (aγ),  the driving force (∆GθV), and α/θ interfacial energy (σα/θ) of 0.68J/m2 were included. In general, C content in γ necessary for the nucleation of θ is represented by the equilibrium content (XCγ/θ) and the previous models had determined the end of α and the start of P when the two conditions were fulfilled; * thermodynamic condition XCγ/θ < Xbarγ * kinetic condition vα < vθ  However, Xbarγ is just an average and does not necessarily corresponds to the one at θ nucleation site. And it would be more likely a situation that the smaller γ grain size (AGS) is, the faster Xbarγ increases even if the same thickness of α covers γ grain surface. Thus in our model, the critical enrichment factor C has been introduced in the new thermodynamic condition. Simply assuming a normal distribution, Eq. 6 has been checked for discrete AGS  around the average with 2.5 µm increment. The nucleation rate of pearlite (Ip) was formulated in the similar way as proposed by Reed and Bhadehia[14]. For the growth of P, diffusion-controlled mechanism in Fe-Mn-C system have been discussed extensively[15,16], and the two growth rate in LE or NPLE mode (GLE, GNPLE) were 1105 αtDνDpγMnαγMn == (3) (4) fCfCbarVXVXX−−=1 0 γαγ/( )θααγθαθθγγγθσσψσπψ∆/////cos)(cos)()(2116 533=−⋅−=KkTGXDavVCC(5) ( ) 00 CbarCC XXXXC −≤− γθγ / (6) Materials Science Forum Vols. 539-543 2445   proposed to operate depending on temperature. These are given in Eq. 7 and Eq. 8, respectively. The expressions for lamellar spacing S, Scrit and parameters in GLE including Mn partitioning coefficient (KP) were taken from ref.[17].  One of the two growth rates was selected depending above or below the "NPLE eutectoid temperature" at which XCγ/α equals to XCγ/θ  under NPLE constraint. Calculation of Integrated Kinetics Isothermal transformation kinetics has been calculated basically according to the procedure by Umemoto[6]. An ellipsoidal α having an aspect ratio of 3:1 was assumed to grow at the parabolic rate of 3α1 along γ grain boundary. As discussed, α1 changes with time and so that increases of α radius in every time increments were summated to get a present radius. Details of calculating volume fraction are in the literature and would not be repeated here. In contrast the growth rate of P is constant and, using Eq. 9[13], the volume fraction (VP) was analyzed for a semi-spherical geometry. γ grain boundary area (Sγ) per unit volume was estimated for a truncated octahedron grain and related to α/γ interface area (Sα/γ) with a simple expression; Sγ(1-Vf)2/3. VPeq is the equilibrium volume fraction of P and φ is an integral constant given by y/G where y is a distance from α/γ interface. Inside the program, first a normal distribution of AGS and corresponding volume contributions (fd) are generated from an input of average AGS. Linking Thermocalc’s functions, thermodynamic parameters, incubation time, nucleation and growth rates are deduced and then the calculation of Vf starts on the smallest AGS. It runs recursively inputting new DCγ, If, α1, Xbar and checking the PE to NPLE transition with p. Once the conditions for the onset of P are fulfilled, the Vp calculation is invoked and it finishes if the total volume fraction exceeds fd. After completion of the same procedures on all discrete AGS, volume fractions are summated along the calculated time series to get the whole transformation kinetics. RESULTS AND DISCUSSION An evaluation of the foregoing models was carried out first for 0.37C-0.56Si-1.45Mn-0.04Cr (mass%) of which isothermal kinetics was investigated in detail[10]. The calculated and observed volume fraction are compared in Fig. 1. Experimentally, the α growth in (a) at slightly above its eutectoid temperature (968K) shows a much slower kinetics than the constant parabolic law. At a later stage, the growth is especially retarded and the calculation reproduces the kinetics due to both the SIE model and PE to NPLE transition. ][. ////SSXXXXSSSDG critCCCCcNPLE −−−= 1720 γαγθθγαγθαγ][//01112SSXXXSSDKG critMnMnMnBMnpLE −−=θγαγθαδ)exp(.RTDKBMnp 14940102232 7 −⋅= −δ(7) (8) { }[ ]∫ −−−=−=103232 231311φφφπγαdtGItIGftIGGtfSVVPPPVeqPP)}](exp{[),,(),,(exp /(9) THERMEC 20062446   At a lower temperature in (b), a period of simultaneous α retarding and gradual P development is expressed in the calculation and the experiment reveals such a behavior. The final volume fraction at the end of isothermal holding agrees reasonably well with the observed one. Contrary to (a), the program predicts no transition of γ/α interface equilibrium in (b). At present, p and C are fitted and constant throughout the prediction. These values need to be related to physical quantities deduced from atomistic diffusion theory. In Fig. 1(c) and (d), a comparison of kinetics at 928K has been made on similar steels within the same compositional standard. It should be pointed out that a minor deviation of Mn content yields a noticeable difference of kinetics. The current program only calculates the thermodynamic effect of Mn, which slows diffusional growth by changing equilibrium C concentration of γ in particular under NPLE constraint. Even so the agreements are passable in both steels and prove the usefulness for Figure 1  The comparison of calculated and measured change in volume fraction of proeutectoid ferrite and pearlite. (a) 0.39C-0.50Si -1.51Mn-0.14Cr at 973K, (b) the same steel at 914K,    (c) 0.39C-0.51Si -1.29Mn-0.15Cr at 928K, and (d) 0.39C-0.50Si -1.51Mn-0.14Cr at 928K. (c) (a) (b) (d) Materials Science Forum Vols. 539-543 2447   predicting difference among commercial distributing MCFS. Still the discrepancy is large for P in the case of higher Mn content. This suggests more modeling work is necessary in order to expand the applicable composition range and to make the program a lot more practical in predicting phase transformation. SUMMARY (1) The integrated phase transformation model has been developed for medium-carbon forging steel. New treatments include the time-dependent growth rate of α in the SIE model and checking PE to NPLE constraint of γ/α interface. The both were made possible through the recursive evaluation of C enrichment in γ and the CALPHAD thermodynamics. (2) The calculation reproduced well the slow α growth above eutectoid temperatures, as well as the successive α→P transformation kinetics by introducing a distribution of AGS. The program showed an ability of predicting the difference in transformation kinetics caused from a minor compositional deviation. References [1]  T. Senuma, H. Yada, Y. Matsumura and T. Nimura: Tetsu-to-Hagane, Vol. 70(1984), p.2112. [2]  Y. Yogo, K. Ishikawa and H. Yoshida: CAMP-ISIJ: Vol. 14(2001), p.424. [3]  S.J. Jones and H.K.D.H. Bhadeshia: Acta Mater., 45(1997), p2911. [4]  M. Suehiro, K. Sato, Y. Tsukano, H. Yada, T. Senuma and Y. Matsumura, Trans. ISIJ, Vol. 27(1984), p.439. [5]  S. V. Parker: PhD thesis, University of Cambridge, 1997. [6]  M. Umemoto: "Proc. Int. Symp. on Mathematical Modeling of Hot Rolling of Steel", Montréal, (1990), p.404. [7] W.F. Lange, M. Enomoto and H.I. Aaronson, Metall. Trans. Vol. 19A(1988), p.427. [8] H.K.D.H. Bhadeshia, S.A. David, J.M. Vitek and W.Reed: Mater. Sci. Technol., Vol. 7(1991), p.686 [9] X. Liu, P. Kayjalainen and J.S. Perttula: "Proc. 2nd Int. Conf. on Modelling of Metal Rolling Processes", London, p.489. [10] C. Capdevila, private communication. [11] M. Hillert: Scripta Mater: Vol. 46(2002), p.447. [12] C.G. Andrés, C. Capdevila, F.G. Caballero and H.K.D.H. Bhadeshia: Scripta Mater: Vol. 39(1998), p.853. [13] C. Capdevila, F.G. Caballero and C.G. Andrés: Acta Materialia, Vol. 50(2002), p.4629. [14] R.C. Reed and H.K.D.H. Bhadeshia: Mater. Sci. Technol.: Vol. 8(1992), p.421. [15]  M. Hillert and L. Hölund: Scripta Mater: Vol. 50(2004), p.171. [16]  C. Capdevila, F.G. Caballero and C.G. Andrés: Scripta Mater: Vol. 50(2004), p.175. [17]  M. Takahashi: Research Paper, University of Cambridge, 1988. THERMEC 20062448

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