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Scripta Materialia 65 (2011) 373–375 www.elsevier.com/locate/scriptamat
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Reply to comments on “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, Ont., Canada M5B 2K3 b CANMET-Materials Technology Laboratory, Natural Resources Canada, 183 Longwood Road South, Hamilton, Ont., Canada L8P 0A1 Received 18 April 2011; revised 6 May 2011; accepted 9 May 2011 Available online 14 May 2011
In a recent publication, Van Bohemen and Hanlon [1] commented on the physical aspects and the input parameters of the model of Sidhu et al. [2, p. 74]. In this reply, we present a detailed discussion and rebuttal of the points raised by Van Bohemen and Hanlon. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Displacive models; Bainite steels; Phase transformation kinetics; Nucleation sites
In a recent publication, Van Bohemen and Hanlon [1] raised some points of discussion about our model for predicting the volume fraction of bainite [2]. Specific comments were on the following: (1) The expression for the number density of potential nucleation sites (Ni), that are initially present in the austenite involves an exponential function and, further, that when T > Th, the model predicts a small but finite value of Ni. (2) The Th value of 629 °C is taken as a constant for all three steels, and this can mean that the model and its predictions are questionable. (3) There is a lot more focus on the initial kinetics, and the model does not predict this well for all the experimental data. (4) The scaling function used to limit the maximum volume fraction of bainite has no physical meaning. In the following paragraphs, we present a reply to these points. (1) In the formulation of Sidhu et al. [2], B Ni ¼ ; Lap
ð1Þ
where B is a temperature-dependent parameter that controls the initial kinetics. This parameter is given by the equation
⇑ Corresponding author. E-mail: [email protected]
T ; B ¼ C 1 exp C 2 Th
ð2Þ
where T is the isothermal transformation temperature, Th is the highest temperature at which bainite formation takes place, and C1 and C2 are material constants. Along with the formulation of this equation, the parameters in this equation are discussed in detail in the paragraphs below, to address the various points raised in Ref. [1]. The exponential relation of B (and thereby Ni) with the material constants and the isothermal transformation temperature is similar to the other works in the literature. For example, Gaude-Fugarolas and Jaques [3] proposed Ni to be an exponential function of temperature and the material’s properties, viz. the yield strength of austenite and the tensile strength of the parent austenite. In the model of Sidhu et al. [2] the impact of such material properties is collectively represented by the material constant C2. Similarly, Chester and Bhadeshia [4] have also included an additional exponential term for temperature that can be interpreted as a contribution of Ni. Additionally, we would also like to highlight the fact that the fitting values of B clearly follow the temperature-dependent exponential formulation (see Fig. 1a in Ref. [2]). Hence, while Van Bohemen and Hanlon [1] argue that the exponential formulation of B is incorrect, we are convinced that there is nothing wrong with this formulation of B. In fact, the expression of Ni (Ni = a(Th – T)/Vb), as presented by Van Bohemen
1359-6462/$ - see front matter Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2011.05.016
374
G. Sidhu et al. / Scripta Materialia 65 (2011) 373–375
and Sietsma [5], gives a negative value of Ni when T > Th, a completely non-physical situation. The choice and inclusion of Th in the exponent in Eq. (2) is the other issue that has been raised in Ref. [1]. Specifically, it has been mentioned that when T > Th, the model will give a finite non-zero value of Ni, which is physically incorrect. We would like to point out that when T = Th, using C1 = 1.004e30 and C2 = 100.49 for steel 1 (see Table 1 in Ref. [2]), B = 2.3e14. This implies a very small value of B (O(1e14)), so for all practical purposes we can say Ni = 0. The effect of a larger T on Ni for three different Th values is shown in Figure 1. It is evident that in all three cases, as T increases beyond Th, Ni approaches zero. It is a trivial task to show that the volume fraction of bainite, f, in Eq. (7) of Ref. [2] approaches zero, since the rate parameter, m, which is a linear function of Ni, is approximately zero due to the large negative value of C2 (see Table 2 in Ref. [2]). In other words, the predicted volume fraction of bainite is infinitesimally small when T P Th and cannot be experimentally measured, i.e. fmax 0 when T P Th. Nevertheless, for a stricter mathematical formulation, to ensure that f = 0 when T P Th we could write Eq. (1) more precisely as B Ni ¼ ; Lap N i ¼ 0;
T < T h; T P T h;
ð3aÞ ð3bÞ
Alternatively, in view of the fact that Ni should be an integer number, Eq. (1) can be written as bBc Ni ¼ ; Lap
ð4Þ
where b c is a flooring function. The above equation, valid for all isothermal transformation temperatures, mathematically ensures that when T P Th, Ni = 0 instead of Ni being a very small number close to zero. Nevertheless, as mentioned earlier, we would like to reiterate the fact that in the original formulation of model [2] the predicted volume fraction of bainite is
Figure 1. Ni as a function of the isothermal transformation temperature using three different values of Th.
infinitesimally small when T > Th and for all practical purposes f 0. (2) With respect to the choice of Th = 629 °C for all three steels, we agree that Th would be different for the three steels. Now, the first thing that we would like to emphasize is that Th is used only as a normalization constant in Eq. (5) of Ref. [2] (Eq. (2) in this paper). This has been clearly stated in Ref. [2] and is reiterated here for clarity. The normalization is to make the exponent dimensionless. In principle, any arbitrary normalization temperature larger than the range of T can be used. Further, such changes in Th will only require retuning the model constant C2 in this expression. Besides, this model constant is in any case going to be different for different steels. Secondly, we arrived at the temperature of 629 °C by considering only C in the three steels. Specifically, based on the fact that in Ref. [5], for steels with carbon content varying from 0.46 to 0.66%, Th varies between 629 and 603 °C, i.e. by about 4%, and noting that in this study the steels we are investigating have carbon contents in the range 0.38–0.44%, variation in Th is estimated to be smaller than 4%. Hence, we have used a constant Th for all three steels. We did not use the correlations of Steven and Haynes [6] or Van Bohemen [7] because the former correlation [6] does not account for the presence of Si and Van Bohemen’s [7] regression expression does not include the effects of V (although we concede that V is present in only one of our steels). Nevertheless, as mentioned earlier, a change/correction in Th can be absorbed by the model by tuning the material constants. The constants C1–C4 are material-dependent constants, as is evident from the following observations: (i) in the model presented in Ref. [2], these constants are different for each steel (see Table 2 in Ref. [2]); and (ii) considering steels 1 and 2, from the experimental point of view, these steels have been heat treated in the same manner by the research group of Bhadeshia [8]. Now, for the same heat treatment, the fact that there are differences in the initial kinetics as well as the final maximum volume fraction of bainite (fmax) (0.37 for steel 1 at T = 352 °C and 0.45 for steel 2 at T = 351 °C) [9,10] can only mean that these variations are because of the variations in the composition of the steels. A detailed discussion of this was not possible in Ref. [2] due to length constraints. (3) With respect to the comment on the focus in Ref. [2] being on the initial kinetics only, it is rightly so because the model underperforms largely in this area and performs extremely well over the rest of the transformation. In fact, the maximum volume fraction of bainite is very well predicted for most cases. This is clearly evident in the R2 value in Figure 1b for all three steels and also in Figures 2–4 in Ref. [2]. As mentioned in Ref. [2], the fact that the model does not take into account the effects of alloying elements may be the reason for the underperformance of the model in predicting the initial transformation kinetics. Also, as mentioned by Van Bohemen and Hanlon [1], the role of autocatalysis in displacive transformations may also be a contributing factor. We acknowledge that the model does not perform well in terms of the initial kinetics for all the presented experimental data. However, to conclude that the model
G. Sidhu et al. / Scripta Materialia 65 (2011) 373–375
has no predictive capabilities is an overstatement, especially since we adequately validated the model (see Figs. 2–4 in Ref. [2]) for three different steels. We would also like to point out that such model discrepancies are present in the works of other researchers as well [8,11]. Additionally, there is the important consideration that there are variations in the experimental data itself (see the scatter in the experimental data of steel 1 at T = 352 °C in Fig. 2 of Ref. [2], the scatter of data in Fig. 4 of Ref. [8] etc.). Thus, there must be room for these experimental fluctuations when judging the predictive capabilities of any model. (4) Finally, regarding the use of a “non-physical” scaling function to predict fmax, we introduced this scaling function to modify and extend the model of Van Bohemen and Sietsma [5] to take into account the incomplete reaction phenomenon that prevents a 100% transformation [8,12]. This was mentioned in our paper [2]. We must highlight the fact that, unlike the claims in Ref. [1], the model of Van Bohemen and Sietsma [5] cannot predict the incomplete reaction phenomena for any steel. This is briefly explained below. The expression for f in Ref. [5] is f ¼
1 expðjð1 þ kÞtÞ 1 þ k expðjð1 þ kÞtÞ
ð5Þ
From the claims in Ref. [1] that their physically based approach is more suitable for description of the incomplete transformation phenomenon, they would need to adjust the parameters j and k in various steels (small positive non-zero values, one would assume, considering the negative sign in the exponents of Eq. (5)) to obtain a value of f < 1, i.e. an incomplete reaction phenomenon. However, no matter how small j and k are made, it is obvious that in the limit as t approaches infinity, f approaches 1 in Eq. (5). In other words, according to their model [5], for a sufficiently long isothermal transformation time one would expect a 100% bainite transformation.
375
The scaling function in our model [2] addresses the issue of an incomplete reaction phenomenon precisely. This scaling function was derived from the experimental data (see Fig. 1b in Ref. [2]) of the maximum volume fraction of bainite in the investigated steels and is not an arbitrary mathematical function. In summary, we agree with Van Bohemen and Hanlon [1] that the model has some quirks in the published format. However, we have adequately justified the formulation and the application of the expression for the number density of potential nucleation sites initially present in the austenite. Further, we have given sufficient evidence of the performance of the model for three different steels at several isothermal transformation temperatures. Hence, we strongly disagree with the doubts raised about the predictive capabilities of the model in Ref. [2]. [1] S.M.C. Van Bohemen, D.N. Hanlon, Scripta Mater., SMM-11-392R1, 2011. [2] G. Sidhu, S.D. Bhole, D.L. Chen, E. Essadiqi, Scripta Mater. 64 (2011) 73. [3] D. Gaude-Fugarolas, P.J. Jaques, ISIJ Int. 46 (2006) 712. [4] N.A. Chester, H.K.D.H. Bhadeshia, J. Phys. IV France 7 (1997) C5–C41. [5] S.M.C. Van Bohemen, J. Sietsma, Int. J. Mater. Res. 99 (2008) 739. [6] W. Steven, A.G. Haynes, J. Iron Steel Inst. 183 (1956) 349. [7] S.M.C. Van Bohemen, Metall. Mater. Trans. A 41 (2010) 285. [8] H. Matsuda, H.K.D.H. Bhadeshia, Proc. Roy. Soc. Lond. A 460 (2004) 1707. [9] H.K.D.H. Bhadeshia, J. Phys, J. Phys. Paris C43, Colloq. C4 (1982) 443. [10] G.I. Rees, H.K.D.H. Bhadeshia, Mater. Sci. Technol. 8 (1992) 985. [11] M.J. Santofimia, F.G. Caballero, C. Capdevila, C. Garcia-Mateo, C.G. de Andres, Mater. Trans. 47 (2006) 2473. [12] H.K.D.H. Bhadeshia, Bainite in Steels, The Institute of Materials, London, 2001.
Scripta Materialia 65 (2011) 373–375 www.elsevier.com/locate/scriptamat
Comments and Reply
Reply to comments on “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, Ont., Canada M5B 2K3 b CANMET-Materials Technology Laboratory, Natural Resources Canada, 183 Longwood Road South, Hamilton, Ont., Canada L8P 0A1 Received 18 April 2011; revised 6 May 2011; accepted 9 May 2011 Available online 14 May 2011
In a recent publication, Van Bohemen and Hanlon [1] commented on the physical aspects and the input parameters of the model of Sidhu et al. [2, p. 74]. In this reply, we present a detailed discussion and rebuttal of the points raised by Van Bohemen and Hanlon. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Displacive models; Bainite steels; Phase transformation kinetics; Nucleation sites
In a recent publication, Van Bohemen and Hanlon [1] raised some points of discussion about our model for predicting the volume fraction of bainite [2]. Specific comments were on the following: (1) The expression for the number density of potential nucleation sites (Ni), that are initially present in the austenite involves an exponential function and, further, that when T > Th, the model predicts a small but finite value of Ni. (2) The Th value of 629 °C is taken as a constant for all three steels, and this can mean that the model and its predictions are questionable. (3) There is a lot more focus on the initial kinetics, and the model does not predict this well for all the experimental data. (4) The scaling function used to limit the maximum volume fraction of bainite has no physical meaning. In the following paragraphs, we present a reply to these points. (1) In the formulation of Sidhu et al. [2], B Ni ¼ ; Lap
ð1Þ
where B is a temperature-dependent parameter that controls the initial kinetics. This parameter is given by the equation
⇑ Corresponding author. E-mail: [email protected]
T ; B ¼ C 1 exp C 2 Th
ð2Þ
where T is the isothermal transformation temperature, Th is the highest temperature at which bainite formation takes place, and C1 and C2 are material constants. Along with the formulation of this equation, the parameters in this equation are discussed in detail in the paragraphs below, to address the various points raised in Ref. [1]. The exponential relation of B (and thereby Ni) with the material constants and the isothermal transformation temperature is similar to the other works in the literature. For example, Gaude-Fugarolas and Jaques [3] proposed Ni to be an exponential function of temperature and the material’s properties, viz. the yield strength of austenite and the tensile strength of the parent austenite. In the model of Sidhu et al. [2] the impact of such material properties is collectively represented by the material constant C2. Similarly, Chester and Bhadeshia [4] have also included an additional exponential term for temperature that can be interpreted as a contribution of Ni. Additionally, we would also like to highlight the fact that the fitting values of B clearly follow the temperature-dependent exponential formulation (see Fig. 1a in Ref. [2]). Hence, while Van Bohemen and Hanlon [1] argue that the exponential formulation of B is incorrect, we are convinced that there is nothing wrong with this formulation of B. In fact, the expression of Ni (Ni = a(Th – T)/Vb), as presented by Van Bohemen
1359-6462/$ - see front matter Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2011.05.016
374
G. Sidhu et al. / Scripta Materialia 65 (2011) 373–375
and Sietsma [5], gives a negative value of Ni when T > Th, a completely non-physical situation. The choice and inclusion of Th in the exponent in Eq. (2) is the other issue that has been raised in Ref. [1]. Specifically, it has been mentioned that when T > Th, the model will give a finite non-zero value of Ni, which is physically incorrect. We would like to point out that when T = Th, using C1 = 1.004e30 and C2 = 100.49 for steel 1 (see Table 1 in Ref. [2]), B = 2.3e14. This implies a very small value of B (O(1e14)), so for all practical purposes we can say Ni = 0. The effect of a larger T on Ni for three different Th values is shown in Figure 1. It is evident that in all three cases, as T increases beyond Th, Ni approaches zero. It is a trivial task to show that the volume fraction of bainite, f, in Eq. (7) of Ref. [2] approaches zero, since the rate parameter, m, which is a linear function of Ni, is approximately zero due to the large negative value of C2 (see Table 2 in Ref. [2]). In other words, the predicted volume fraction of bainite is infinitesimally small when T P Th and cannot be experimentally measured, i.e. fmax 0 when T P Th. Nevertheless, for a stricter mathematical formulation, to ensure that f = 0 when T P Th we could write Eq. (1) more precisely as B Ni ¼ ; Lap N i ¼ 0;
T < T h; T P T h;
ð3aÞ ð3bÞ
Alternatively, in view of the fact that Ni should be an integer number, Eq. (1) can be written as bBc Ni ¼ ; Lap
ð4Þ
where b c is a flooring function. The above equation, valid for all isothermal transformation temperatures, mathematically ensures that when T P Th, Ni = 0 instead of Ni being a very small number close to zero. Nevertheless, as mentioned earlier, we would like to reiterate the fact that in the original formulation of model [2] the predicted volume fraction of bainite is
Figure 1. Ni as a function of the isothermal transformation temperature using three different values of Th.
infinitesimally small when T > Th and for all practical purposes f 0. (2) With respect to the choice of Th = 629 °C for all three steels, we agree that Th would be different for the three steels. Now, the first thing that we would like to emphasize is that Th is used only as a normalization constant in Eq. (5) of Ref. [2] (Eq. (2) in this paper). This has been clearly stated in Ref. [2] and is reiterated here for clarity. The normalization is to make the exponent dimensionless. In principle, any arbitrary normalization temperature larger than the range of T can be used. Further, such changes in Th will only require retuning the model constant C2 in this expression. Besides, this model constant is in any case going to be different for different steels. Secondly, we arrived at the temperature of 629 °C by considering only C in the three steels. Specifically, based on the fact that in Ref. [5], for steels with carbon content varying from 0.46 to 0.66%, Th varies between 629 and 603 °C, i.e. by about 4%, and noting that in this study the steels we are investigating have carbon contents in the range 0.38–0.44%, variation in Th is estimated to be smaller than 4%. Hence, we have used a constant Th for all three steels. We did not use the correlations of Steven and Haynes [6] or Van Bohemen [7] because the former correlation [6] does not account for the presence of Si and Van Bohemen’s [7] regression expression does not include the effects of V (although we concede that V is present in only one of our steels). Nevertheless, as mentioned earlier, a change/correction in Th can be absorbed by the model by tuning the material constants. The constants C1–C4 are material-dependent constants, as is evident from the following observations: (i) in the model presented in Ref. [2], these constants are different for each steel (see Table 2 in Ref. [2]); and (ii) considering steels 1 and 2, from the experimental point of view, these steels have been heat treated in the same manner by the research group of Bhadeshia [8]. Now, for the same heat treatment, the fact that there are differences in the initial kinetics as well as the final maximum volume fraction of bainite (fmax) (0.37 for steel 1 at T = 352 °C and 0.45 for steel 2 at T = 351 °C) [9,10] can only mean that these variations are because of the variations in the composition of the steels. A detailed discussion of this was not possible in Ref. [2] due to length constraints. (3) With respect to the comment on the focus in Ref. [2] being on the initial kinetics only, it is rightly so because the model underperforms largely in this area and performs extremely well over the rest of the transformation. In fact, the maximum volume fraction of bainite is very well predicted for most cases. This is clearly evident in the R2 value in Figure 1b for all three steels and also in Figures 2–4 in Ref. [2]. As mentioned in Ref. [2], the fact that the model does not take into account the effects of alloying elements may be the reason for the underperformance of the model in predicting the initial transformation kinetics. Also, as mentioned by Van Bohemen and Hanlon [1], the role of autocatalysis in displacive transformations may also be a contributing factor. We acknowledge that the model does not perform well in terms of the initial kinetics for all the presented experimental data. However, to conclude that the model
G. Sidhu et al. / Scripta Materialia 65 (2011) 373–375
has no predictive capabilities is an overstatement, especially since we adequately validated the model (see Figs. 2–4 in Ref. [2]) for three different steels. We would also like to point out that such model discrepancies are present in the works of other researchers as well [8,11]. Additionally, there is the important consideration that there are variations in the experimental data itself (see the scatter in the experimental data of steel 1 at T = 352 °C in Fig. 2 of Ref. [2], the scatter of data in Fig. 4 of Ref. [8] etc.). Thus, there must be room for these experimental fluctuations when judging the predictive capabilities of any model. (4) Finally, regarding the use of a “non-physical” scaling function to predict fmax, we introduced this scaling function to modify and extend the model of Van Bohemen and Sietsma [5] to take into account the incomplete reaction phenomenon that prevents a 100% transformation [8,12]. This was mentioned in our paper [2]. We must highlight the fact that, unlike the claims in Ref. [1], the model of Van Bohemen and Sietsma [5] cannot predict the incomplete reaction phenomena for any steel. This is briefly explained below. The expression for f in Ref. [5] is f ¼
1 expðjð1 þ kÞtÞ 1 þ k expðjð1 þ kÞtÞ
ð5Þ
From the claims in Ref. [1] that their physically based approach is more suitable for description of the incomplete transformation phenomenon, they would need to adjust the parameters j and k in various steels (small positive non-zero values, one would assume, considering the negative sign in the exponents of Eq. (5)) to obtain a value of f < 1, i.e. an incomplete reaction phenomenon. However, no matter how small j and k are made, it is obvious that in the limit as t approaches infinity, f approaches 1 in Eq. (5). In other words, according to their model [5], for a sufficiently long isothermal transformation time one would expect a 100% bainite transformation.
375
The scaling function in our model [2] addresses the issue of an incomplete reaction phenomenon precisely. This scaling function was derived from the experimental data (see Fig. 1b in Ref. [2]) of the maximum volume fraction of bainite in the investigated steels and is not an arbitrary mathematical function. In summary, we agree with Van Bohemen and Hanlon [1] that the model has some quirks in the published format. However, we have adequately justified the formulation and the application of the expression for the number density of potential nucleation sites initially present in the austenite. Further, we have given sufficient evidence of the performance of the model for three different steels at several isothermal transformation temperatures. Hence, we strongly disagree with the doubts raised about the predictive capabilities of the model in Ref. [2]. [1] S.M.C. Van Bohemen, D.N. Hanlon, Scripta Mater., SMM-11-392R1, 2011. [2] G. Sidhu, S.D. Bhole, D.L. Chen, E. Essadiqi, Scripta Mater. 64 (2011) 73. [3] D. Gaude-Fugarolas, P.J. Jaques, ISIJ Int. 46 (2006) 712. [4] N.A. Chester, H.K.D.H. Bhadeshia, J. Phys. IV France 7 (1997) C5–C41. [5] S.M.C. Van Bohemen, J. Sietsma, Int. J. Mater. Res. 99 (2008) 739. [6] W. Steven, A.G. Haynes, J. Iron Steel Inst. 183 (1956) 349. [7] S.M.C. Van Bohemen, Metall. Mater. Trans. A 41 (2010) 285. [8] H. Matsuda, H.K.D.H. Bhadeshia, Proc. Roy. Soc. Lond. A 460 (2004) 1707. [9] H.K.D.H. Bhadeshia, J. Phys, J. Phys. Paris C43, Colloq. C4 (1982) 443. [10] G.I. Rees, H.K.D.H. Bhadeshia, Mater. Sci. Technol. 8 (1992) 985. [11] M.J. Santofimia, F.G. Caballero, C. Capdevila, C. Garcia-Mateo, C.G. de Andres, Mater. Trans. 47 (2006) 2473. [12] H.K.D.H. Bhadeshia, Bainite in Steels, The Institute of Materials, London, 2001.