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An improved model for bainite formation at isothermal temperatures
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Scripta Materialia 64 (2011) 73–76 www.elsevier.com/locate/scriptamat
An improved model for bainite formation at isothermal temperatures G. Sidhu,a S.D. Bhole,a,⇑ D.L. Chena and E. Essadiqib a
Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario M5B 2K3, Canada b CANMET-Materials Technology Laboratory, Natural Resources Canada, 568 Booth Street, Ottawa, Ontario K1A 0G1, Canada Received 8 August 2010; revised 7 September 2010; accepted 8 September 2010 Available online 16 September 2010
An improved model for bainite formation via a displacive mechanism is presented. The model incorporates a temperature-dependent expression for the potential nucleation site density and a linear scaling function to predict the maximum volume fraction of bainite. The model has been validated with respect to the experimental data of three high-silicon steels from the literature and has been found to perform reasonably well for several isothermal transformation cases. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Displacive models; Bainite steels; Phase transformation kinetics; Nucleation
Bainitic transformation has received considerable attention due to its application in the railway [1] and automotive industries [2]. From the theoretical point of view, two explanations have emerged for the kinetics of bainite transformation, namely the diffusional mechanism [3,4] and the displacive mechanism. In the displacive mechanism, pioneered by the research group of Bhadeshia, the bainite subunit grows without partitioning of alloying elements analogous to the martensitic transformation [5–9]. In a recent study, Van Bohemen et al. presented a model that performed reasonably well with low-silicon steels [10]. In their displacive model Van Bohemen et al. optimized two parameters, i.e. a temperature-dependent rate constant (j) and an autocatalytic factor (k). In this investigation, we present a modified form of the model of Van Bohemen et al. [10]. The key features of this model include: (i) a temperature-dependent expression for the number density of potential nucleation sites Ni; and (ii) a temperature-dependent scaling function in the expression for the volume fraction of bainite (f) to account for the incomplete reaction of austenite into bainite [5]. This differentiates the current model from that of Van Bohemen et al., whose expression for f predicts 100% transformation, which requires the experimental data to be normalized. The model has been validated with exper-
⇑ Corresponding author. Tel.: +1 416 979 5000x7215; fax: +1 416 979 5265; e-mail: [email protected]
imental data taken from Refs. [8,11,12] for three steels containing high levels of silicon; the compositions of these steels are summarized in Table 1. According to Van Bohemen et al. [10], the rate of change of the volume fraction of bainite can be written as: df dN ¼ V b; ð1Þ dt dt where f is the volume fraction of bainite, N is the number of nucleation sites and Vb is the average volume of the bainite subunit. In this study we have used the following expression for the average volume of bainite subunits (Vb) [13,14]: 3 T 528 Vb ¼2 lm3 ; ð2Þ 150 where T is the temperature in Kelvin at which the isothermal transformation takes place. The rate change of the nucleation sites, dN/dt, is given according to Ref. [10] as: dN kT Q ¼ ð1 f ÞN i ð1 þ kf Þ exp ; ð3Þ dt h kT where k is the Boltzmann constant, T is the temperature in Kelvin, h is Planck’s constant, k is the autocatalytic nucleation parameter (cf. Table 2), Q* is the activation energy and Ni is the number density of the potential nucleation sites that are initially present in the austenite.
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.09.009
74
G. Sidhu et al. / Scripta Materialia 64 (2011) 73–76
Table 1. Chemical composition of the three steels validated in this study. The values in the brackets indicate the references for the experimental data. Steel
Chemical compositions (wt.%)
Steel 1 [11,8] Steel 2 [8] Steel 3 [12]
C
Si
Mn
Ni
Mo
Cr
V
0.44 0.39 0.38
1.74 2.05 1.29
0.67 0.00 1.73
1.85 4.08 –
0.83 0.00 –
0.39 0.00 –
0.09 0.00 –
Table 2. Values of model constants and parameters used for the three steels. Constants Ci have been chosen in such a way as to maximize the agreement between the transient profiles of the experimental data and the calculations. Steel
C1
C2
C3
C4
k
L [lm]
uw [lm]
Steel 1 Steel 2 Steel 3
1.004 1030 4.838 1023 2.831 1026
100.49 80.549 83.865
3.3384 2.0117 1.9399
2.7195 1.8334 1.9375
90 100 32
95 38 25
0.95 0.38 0.05
In the present model we use the following expression of Singh et al. [15] for Ni: B Ni ¼ ; ð4Þ Lap is the mean linear intercept representing the auswhere L tenite grain size (cf. Table 2) and ap relates the volume of a bainite subunit (u) and its thickness uw as ap = u=u3w . In order to compute uw (cf. Table 2), an as given in Table pect ratio of 0.2/10/10 and the value of L 2 were used. It should be noted that while Singh et al. [15] assumed B to be constant, in the present study B is assumed to be a temperature-dependent function in the following form that is considered to dictate the initial reaction rate, i.e.: T ; ð5Þ B ¼ C 1 exp C 2 Th where C1 and C2 are two material constants to be determined later. The temperature Th used for normalization represents the maximum temperature at which bainite formation takes place and has been chosen as a constant with a value of 629 °C [10] for all steels in this study. This expression for Ni is different from that used by Van Bohemen et al. [10]. Their expression for the temperature- and composition-dependent Ni relied on a parameter that was determined experimentally. Further, this parameter for the martensite and bainite was related via another geometrical factor, Z, that was taken as a constant for all steels. Their expression of Ni did not account for the thickness or the volume of the bainite subunits. Further, the average value of the bainite subunit volume that Van Bohemen et al. used in their expression cancels out with the rate of change of f and their model is essentially independent of the volume of the bainite subunits. In the present model, the impact of bainite subunit volume as well as thickness has been accounted for via the parameter ap, consistent with the propositions of Matsuda and Bhadeshia [8]. The activation energy Q* of the austenite to bainite transformation is calculated using the following relation [10]: Q ¼ Q0 K 1 DGm ;
ð6Þ
where Q* is the driving pressure for bainite nucleation and Q0 is a constant, expressing the activation energy
in the limit that the driving pressure becomes zero. Furthermore, K1 is a constant of proportionality that takes the values 15.049 in this investigation. This has been obtained by a linear extrapolation of the values of K1 from the values reported by Van Bohemen et al. [10]. Van Bohemen et al. [10] suggested values of Q0 in the range 180–210 kJ mol1. In this study, a value of 188 kJ mol1 [10] has been used for all three steels at all temperatures. The driving pressure for bainite nucleation, DGm, is given by DGm = Gn 7.66(Th T) J mol1 [10]. The universal nucleation function, Gn, that was originally determined by Ali and Bhadeshia [16], has been calculated as Gn = 3.637(T 273.18) 2540 J mol1 [10]. Integrating Eq. (1) using the Eqs. (3) and (4), we obtain Van Bohemen et al.’s expression as: f ¼
1 expðjð1 þ kÞtÞ 1 þ k expðjð1 þ kÞtÞ
ð7Þ
In this work, Van Bohemen et al.’s rate parameter, j, is replaced by the new rate parameter, m, where: kT B Q m¼ ð7AÞ p exp kT V b : h La In Eq. (7) f approaches 1 as t tends to infinity, indicating a 100% transformation of the austenite phase. However, as mentioned earlier, the incomplete reaction phenomenon prevents a 100% transformation of austenite to bainite [5]. To take this fact into account Eq. (7) is multiplied by a temperature-dependent scaling function C 3 TTh þ C 4 ; where C3 and C4 are material constants. Thus, the final expression for f becomes: 1 expðmð1 þ kÞtÞ T C3 þ C4 f ¼ ð8Þ 1 þ k expðmð1 þ kÞtÞ Th Four material constants have been used in this study. These have been obtained as follows. For each steel, at a given temperature, a value of B was chosen such that the model was able to predict the initial reaction rate of the experimental data reasonably well in the graph of volume fraction f vs. time t. In choosing these tuning values of B, it was noted that for a given steel the initial reaction rate must decrease with an increase in temperature [6]. Next, a plot of B vs. the normalized temperature (T/ Th) showed that B decreased exponentially with temperature and follows the relation of Eq. (5) (cf. Fig. 1a). As mentioned earlier, this temperature dependence is
G. Sidhu et al. / Scripta Materialia 64 (2011) 73–76
As part of the process of validating the model, experimental data for three steels (Table 1) at various temperatures have been considered. In all three steels the maximum volume fraction of bainite (fmax) is predicted with reasonable accuracy. The accuracy is exceptionally good for steels 2 and 3 where fmax is accurately predicted for the temperature ranges 350–455 °C (Fig. 3) and 330– 455 °C in steels 2 and 3, respectively (Figs. 3 and 4). In steel 1, the prediction is good only in the range 345– 425 °C (Fig. 2). In all three steels, the model demonstrated a mixed performance in predicting the initial reaction rate. In steels 1 and 2, the rate of increase of f decreases with T consistent with previous findings [6]. In steel 1 the model over- and underpredicts the initial reaction rate at lower tempera-
Figure 1a. Trend of ln(B) vs. the normalized temperatures.
expected, and the fact that the constants C1 and C2 are different for each steel is also evidence that B is also dependent on the composition of the steel, i.e. these are material constants. The values of C1 and C2 for all three steels are summarized in Table 2. It should be stated that in tuning the values of B for each steel, a particular value of the autocatalytic factor (k) and the mean were assigned for each steel. These vallinear intercept L ues are the ones reported in the literature and are summarized in Table 2. In a similar manner the experimentally determined ðexperimentÞ , was plotted maximum volume fraction, fmax against the normalized temperature (T/Th) for each steel at all temperatures (cf. Fig. 1b). As seen in Figure 1b, ðexperimentÞ follows a linear trend with the temperature, fmax T, i.e.: ðexperimentÞ fmax C3
T þ C4: Th
75
Figure 2. Measured and predicted isothermal transformation kinetics of bainite for steel 1. Experimental data are from Refs. [8,11].
ð9Þ
The values of the two constants, i.e. C3 and C4, were determined from this linear trend. As in the case of C1 and C2, the fact that these constants are different for each steel indicates that these are also material constants, and they are summarized in Table 2. It should be noted that a higher-order polynomial fit of the experðexperimentÞ in Figure 1b did not result in a imental data of fmax significant improvement. Hence, the linear approximaðexperimentÞ at a given isothermal temperature tion of fmax was used here. Figure 3. Measured and predicted isothermal transformation kinetics of bainite for steel 2. Experimental data are from Ref. [8].
Figure 1b. Maximum volume fraction of bainite vs. the normalized temperature. The maximum values are obtained from Refs. [8,11] for steel 1, Ref. [11] for steel 2 and Ref. [12] for steel 3.
Figure 4. Measured and predicted volume fraction of bainite for steel 3. Experimental data are from Ref. [12].
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G. Sidhu et al. / Scripta Materialia 64 (2011) 73–76
tures (T < 380 °C) and higher temperatures (T > 400 °C), respectively. For intermediate temperatures (380– 400 °C), there is a good agreement between the initial reaction rate and the experimental data. In steel 3 there is a good prediction of the initial reaction rate up to 150 s for 330, 355 and 405 °C, and an underprediction at the highest temperature of 430 °C. At 330 °C the model predicts the transformation kinetics very well. At 405 °C and lower temperatures the model overpredicts the initial reaction rate after the first 150 s. This is probably an indicator that the material constants are not reflective of the high manganese content that impacts the initial reaction rate [6]. The discrepancy of the present model is similar to the computational results of a different model proposed by Chester and Bhadeshia [12] where the initial reaction rate is faster at 330 °C than at 355 °C. In summary, an improved model for bainite kinetics has been presented. In addition to a temperature-dependent expression for the initial nucleation site density, the model also includes a temperature-dependent scaling function to predict the maximum volume fraction of bainite at a given isothermal transformation temperature. The model has been validated with respect to the experimental data of three high-silicon steels at various temperatures and predicts fairly accurately the transformation profile. The poor predictions of the initial reaction rate at certain temperatures may be attributed to the fact that the model cannot take into account the contributions due to the other alloying elements. The authors would like to thank the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), Premier’s Research Excellence Award (PREA) and CANMETMTL. The authors would also like to thank Q. Li, A.
Machin, J. Amankrah, and A. Heim, R. Churaman for their assistance in the experiments. [1] H.A. Aglan, Z.Y. Liu, M.F. Hassan, M. Fateh, J. Mater. Proc. Technol. 151 (2004) 268–274. [2] K. Sawley, J. Kristan, Fatigue Fract. Eng. Mater. Struct. 26 (2003) 1019–1029. [3] H.I. Aaronson, in: V.F. Zackary, H.I. Aaronson (Eds.), The Decomposition of Austenite by Diffusional Processes, Interscience, New York, 1962, pp. 387–546. [4] H.I. Aaronson, G. Spanos, W.T. Reynolds J.r., Scr. Mater. 47 (2002) 139–144. [5] H.K.D.H. Bhadeshia, Bainite in Steels, The Institute of Materials, London, 2001. [6] G.I. Rees, H. K. D Bhadeshia, Mater. Sci. Technol. 8 (1992) 985. [7] M.J. Santofimia, F.G. Caballero, C. Capdevila, C. Garcia-Mateo, C.G. de Andres, Mater. Trans. 47 (2006) 2465. [8] H. Matsuda, H.K.D.H. Bhadeshia, Proc. R. Soc. London, A 460 (2004) 1707. [9] M.J. Santofimia, F.G. Caballero, C. Capdevila, C. Garcia-Mateo, C.G. de Andres, Mater. Trans. 47 (2006) 1492. [10] S.M.C. Van Bohemen, J. Sietsma, Int. J. Mater. Res. 99 (2008) 739–747. [11] M.J. Santofimia, F.G. Caballero, C. Capdevila, C. Garcia-Mateo, C.G. de Andres, Mater. Trans. 47 (2006) 2473. [12] N. Chester, H.K.D.H. Bhadeshia, J. Phys. IV, Colloq. C5 7 (1997) 41–46. [13] L.C. Chang, H.K.D.H. Bhadeshi, Mater. Sci. Technol. 11 (1995) 874–881. [14] S. V. Parker: Modelling of phase transformations in hotrolled steels, Ph.D. thesis, University of Cambridge, 1997. [15] S. B. Singh: Phase transformations from deformed austenite, Ph.D. thesis, University of Cambridge, 1998. [16] A. Ali, H.K.D.H. Bhadeshi, Mater. Sci. Technol. 6 (1990) 781–784.
Scripta Materialia 64 (2011) 73–76 www.elsevier.com/locate/scriptamat
An improved model for bainite formation at isothermal temperatures G. Sidhu,a S.D. Bhole,a,⇑ D.L. Chena and E. Essadiqib a
Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario M5B 2K3, Canada b CANMET-Materials Technology Laboratory, Natural Resources Canada, 568 Booth Street, Ottawa, Ontario K1A 0G1, Canada Received 8 August 2010; revised 7 September 2010; accepted 8 September 2010 Available online 16 September 2010
An improved model for bainite formation via a displacive mechanism is presented. The model incorporates a temperature-dependent expression for the potential nucleation site density and a linear scaling function to predict the maximum volume fraction of bainite. The model has been validated with respect to the experimental data of three high-silicon steels from the literature and has been found to perform reasonably well for several isothermal transformation cases. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Displacive models; Bainite steels; Phase transformation kinetics; Nucleation
Bainitic transformation has received considerable attention due to its application in the railway [1] and automotive industries [2]. From the theoretical point of view, two explanations have emerged for the kinetics of bainite transformation, namely the diffusional mechanism [3,4] and the displacive mechanism. In the displacive mechanism, pioneered by the research group of Bhadeshia, the bainite subunit grows without partitioning of alloying elements analogous to the martensitic transformation [5–9]. In a recent study, Van Bohemen et al. presented a model that performed reasonably well with low-silicon steels [10]. In their displacive model Van Bohemen et al. optimized two parameters, i.e. a temperature-dependent rate constant (j) and an autocatalytic factor (k). In this investigation, we present a modified form of the model of Van Bohemen et al. [10]. The key features of this model include: (i) a temperature-dependent expression for the number density of potential nucleation sites Ni; and (ii) a temperature-dependent scaling function in the expression for the volume fraction of bainite (f) to account for the incomplete reaction of austenite into bainite [5]. This differentiates the current model from that of Van Bohemen et al., whose expression for f predicts 100% transformation, which requires the experimental data to be normalized. The model has been validated with exper-
⇑ Corresponding author. Tel.: +1 416 979 5000x7215; fax: +1 416 979 5265; e-mail: [email protected]
imental data taken from Refs. [8,11,12] for three steels containing high levels of silicon; the compositions of these steels are summarized in Table 1. According to Van Bohemen et al. [10], the rate of change of the volume fraction of bainite can be written as: df dN ¼ V b; ð1Þ dt dt where f is the volume fraction of bainite, N is the number of nucleation sites and Vb is the average volume of the bainite subunit. In this study we have used the following expression for the average volume of bainite subunits (Vb) [13,14]: 3 T 528 Vb ¼2 lm3 ; ð2Þ 150 where T is the temperature in Kelvin at which the isothermal transformation takes place. The rate change of the nucleation sites, dN/dt, is given according to Ref. [10] as: dN kT Q ¼ ð1 f ÞN i ð1 þ kf Þ exp ; ð3Þ dt h kT where k is the Boltzmann constant, T is the temperature in Kelvin, h is Planck’s constant, k is the autocatalytic nucleation parameter (cf. Table 2), Q* is the activation energy and Ni is the number density of the potential nucleation sites that are initially present in the austenite.
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.09.009
74
G. Sidhu et al. / Scripta Materialia 64 (2011) 73–76
Table 1. Chemical composition of the three steels validated in this study. The values in the brackets indicate the references for the experimental data. Steel
Chemical compositions (wt.%)
Steel 1 [11,8] Steel 2 [8] Steel 3 [12]
C
Si
Mn
Ni
Mo
Cr
V
0.44 0.39 0.38
1.74 2.05 1.29
0.67 0.00 1.73
1.85 4.08 –
0.83 0.00 –
0.39 0.00 –
0.09 0.00 –
Table 2. Values of model constants and parameters used for the three steels. Constants Ci have been chosen in such a way as to maximize the agreement between the transient profiles of the experimental data and the calculations. Steel
C1
C2
C3
C4
k
L [lm]
uw [lm]
Steel 1 Steel 2 Steel 3
1.004 1030 4.838 1023 2.831 1026
100.49 80.549 83.865
3.3384 2.0117 1.9399
2.7195 1.8334 1.9375
90 100 32
95 38 25
0.95 0.38 0.05
In the present model we use the following expression of Singh et al. [15] for Ni: B Ni ¼ ; ð4Þ Lap is the mean linear intercept representing the auswhere L tenite grain size (cf. Table 2) and ap relates the volume of a bainite subunit (u) and its thickness uw as ap = u=u3w . In order to compute uw (cf. Table 2), an as given in Table pect ratio of 0.2/10/10 and the value of L 2 were used. It should be noted that while Singh et al. [15] assumed B to be constant, in the present study B is assumed to be a temperature-dependent function in the following form that is considered to dictate the initial reaction rate, i.e.: T ; ð5Þ B ¼ C 1 exp C 2 Th where C1 and C2 are two material constants to be determined later. The temperature Th used for normalization represents the maximum temperature at which bainite formation takes place and has been chosen as a constant with a value of 629 °C [10] for all steels in this study. This expression for Ni is different from that used by Van Bohemen et al. [10]. Their expression for the temperature- and composition-dependent Ni relied on a parameter that was determined experimentally. Further, this parameter for the martensite and bainite was related via another geometrical factor, Z, that was taken as a constant for all steels. Their expression of Ni did not account for the thickness or the volume of the bainite subunits. Further, the average value of the bainite subunit volume that Van Bohemen et al. used in their expression cancels out with the rate of change of f and their model is essentially independent of the volume of the bainite subunits. In the present model, the impact of bainite subunit volume as well as thickness has been accounted for via the parameter ap, consistent with the propositions of Matsuda and Bhadeshia [8]. The activation energy Q* of the austenite to bainite transformation is calculated using the following relation [10]: Q ¼ Q0 K 1 DGm ;
ð6Þ
where Q* is the driving pressure for bainite nucleation and Q0 is a constant, expressing the activation energy
in the limit that the driving pressure becomes zero. Furthermore, K1 is a constant of proportionality that takes the values 15.049 in this investigation. This has been obtained by a linear extrapolation of the values of K1 from the values reported by Van Bohemen et al. [10]. Van Bohemen et al. [10] suggested values of Q0 in the range 180–210 kJ mol1. In this study, a value of 188 kJ mol1 [10] has been used for all three steels at all temperatures. The driving pressure for bainite nucleation, DGm, is given by DGm = Gn 7.66(Th T) J mol1 [10]. The universal nucleation function, Gn, that was originally determined by Ali and Bhadeshia [16], has been calculated as Gn = 3.637(T 273.18) 2540 J mol1 [10]. Integrating Eq. (1) using the Eqs. (3) and (4), we obtain Van Bohemen et al.’s expression as: f ¼
1 expðjð1 þ kÞtÞ 1 þ k expðjð1 þ kÞtÞ
ð7Þ
In this work, Van Bohemen et al.’s rate parameter, j, is replaced by the new rate parameter, m, where: kT B Q m¼ ð7AÞ p exp kT V b : h La In Eq. (7) f approaches 1 as t tends to infinity, indicating a 100% transformation of the austenite phase. However, as mentioned earlier, the incomplete reaction phenomenon prevents a 100% transformation of austenite to bainite [5]. To take this fact into account Eq. (7) is multiplied by a temperature-dependent scaling function C 3 TTh þ C 4 ; where C3 and C4 are material constants. Thus, the final expression for f becomes: 1 expðmð1 þ kÞtÞ T C3 þ C4 f ¼ ð8Þ 1 þ k expðmð1 þ kÞtÞ Th Four material constants have been used in this study. These have been obtained as follows. For each steel, at a given temperature, a value of B was chosen such that the model was able to predict the initial reaction rate of the experimental data reasonably well in the graph of volume fraction f vs. time t. In choosing these tuning values of B, it was noted that for a given steel the initial reaction rate must decrease with an increase in temperature [6]. Next, a plot of B vs. the normalized temperature (T/ Th) showed that B decreased exponentially with temperature and follows the relation of Eq. (5) (cf. Fig. 1a). As mentioned earlier, this temperature dependence is
G. Sidhu et al. / Scripta Materialia 64 (2011) 73–76
As part of the process of validating the model, experimental data for three steels (Table 1) at various temperatures have been considered. In all three steels the maximum volume fraction of bainite (fmax) is predicted with reasonable accuracy. The accuracy is exceptionally good for steels 2 and 3 where fmax is accurately predicted for the temperature ranges 350–455 °C (Fig. 3) and 330– 455 °C in steels 2 and 3, respectively (Figs. 3 and 4). In steel 1, the prediction is good only in the range 345– 425 °C (Fig. 2). In all three steels, the model demonstrated a mixed performance in predicting the initial reaction rate. In steels 1 and 2, the rate of increase of f decreases with T consistent with previous findings [6]. In steel 1 the model over- and underpredicts the initial reaction rate at lower tempera-
Figure 1a. Trend of ln(B) vs. the normalized temperatures.
expected, and the fact that the constants C1 and C2 are different for each steel is also evidence that B is also dependent on the composition of the steel, i.e. these are material constants. The values of C1 and C2 for all three steels are summarized in Table 2. It should be stated that in tuning the values of B for each steel, a particular value of the autocatalytic factor (k) and the mean were assigned for each steel. These vallinear intercept L ues are the ones reported in the literature and are summarized in Table 2. In a similar manner the experimentally determined ðexperimentÞ , was plotted maximum volume fraction, fmax against the normalized temperature (T/Th) for each steel at all temperatures (cf. Fig. 1b). As seen in Figure 1b, ðexperimentÞ follows a linear trend with the temperature, fmax T, i.e.: ðexperimentÞ fmax C3
T þ C4: Th
75
Figure 2. Measured and predicted isothermal transformation kinetics of bainite for steel 1. Experimental data are from Refs. [8,11].
ð9Þ
The values of the two constants, i.e. C3 and C4, were determined from this linear trend. As in the case of C1 and C2, the fact that these constants are different for each steel indicates that these are also material constants, and they are summarized in Table 2. It should be noted that a higher-order polynomial fit of the experðexperimentÞ in Figure 1b did not result in a imental data of fmax significant improvement. Hence, the linear approximaðexperimentÞ at a given isothermal temperature tion of fmax was used here. Figure 3. Measured and predicted isothermal transformation kinetics of bainite for steel 2. Experimental data are from Ref. [8].
Figure 1b. Maximum volume fraction of bainite vs. the normalized temperature. The maximum values are obtained from Refs. [8,11] for steel 1, Ref. [11] for steel 2 and Ref. [12] for steel 3.
Figure 4. Measured and predicted volume fraction of bainite for steel 3. Experimental data are from Ref. [12].
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G. Sidhu et al. / Scripta Materialia 64 (2011) 73–76
tures (T < 380 °C) and higher temperatures (T > 400 °C), respectively. For intermediate temperatures (380– 400 °C), there is a good agreement between the initial reaction rate and the experimental data. In steel 3 there is a good prediction of the initial reaction rate up to 150 s for 330, 355 and 405 °C, and an underprediction at the highest temperature of 430 °C. At 330 °C the model predicts the transformation kinetics very well. At 405 °C and lower temperatures the model overpredicts the initial reaction rate after the first 150 s. This is probably an indicator that the material constants are not reflective of the high manganese content that impacts the initial reaction rate [6]. The discrepancy of the present model is similar to the computational results of a different model proposed by Chester and Bhadeshia [12] where the initial reaction rate is faster at 330 °C than at 355 °C. In summary, an improved model for bainite kinetics has been presented. In addition to a temperature-dependent expression for the initial nucleation site density, the model also includes a temperature-dependent scaling function to predict the maximum volume fraction of bainite at a given isothermal transformation temperature. The model has been validated with respect to the experimental data of three high-silicon steels at various temperatures and predicts fairly accurately the transformation profile. The poor predictions of the initial reaction rate at certain temperatures may be attributed to the fact that the model cannot take into account the contributions due to the other alloying elements. The authors would like to thank the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), Premier’s Research Excellence Award (PREA) and CANMETMTL. The authors would also like to thank Q. Li, A.
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