ISSN 0967-0912, Steel in Translation, 2017, Vol. 47, No. 3, pp. 213–218. © Allerton Press, Inc., 2017. Original Russian Text © Yu.V. Yudin, M.V. Maisuradze, A.A. Kuklina, 2017, published in Stal’, 2017, No. 3, pp. 52–56.
Describing the Isothermal Bainitic Transformation in Structural Steels by a Logistical Function Yu. V. Yudin, M. V. Maisuradze*, and A. A. Kuklina Yeltsin Ural Federal University, Yekaterinburg, Russia *e-mail:
[email protected] Received March 13, 2017
Abstract⎯A new analytical description for the bainitic transformations in steel on isothermal holding is proposed, on the basis of a logistical function of the holding time (in logarithmic form). The model is characterized by the parameters a and b, which are constant in the given experimental conditions (austenitization temperature, isothermal holding temperature and time, chemical composition of the steel, etc.). The calculated temperature dependence of a and b is in good agreement with experimental data. In further research, the physical significance of coefficients a and b in the logistical function should be refined. Keywords: bainitic transformation, steel, isothermal holding, logistical functions, modeling DOI: 10.3103/S0967091217030160
Precise mathematical description of physical processes is important in any scientific research. In steel production, it is important to model the transformation kinetics in the solid state, since correct prediction of the microstructure obtained and hence the alloys properties guides the development of manufacturing and machining processes for specific components [1, 2]. The transformation of supercooled austenite in steel may be simulated by many methods: on the basis of phase fields [3], cellular automation [4], semiempirical models [5], neural networks [6], etc. Most models are based on the classical theory of initiation and growth [7, 8] and the Kolmogorov–Johnson– Mehl-Avrami equation [9–13] (or its modification [14, 15]). That permits kinetic predictions of the formation of new phases and structural components in isothermal holding and in continuous cooling (1) P = 1 − exp(−k τ n ), where P is the proportion of transformed austenite; τ is the time; k is a temperature-dependent coefficient; and n is some exponent. In many studies, k and n are assigned specific physical meanings: k determines the conversion rate, and n determines the form of the new particles formed or the conversion mechanism [7, 8]. On that basis, it makes sense that N would be a constant integer (determined by the type of transformation, the chemical composition of the steel, etc.) over the whole isothermal transformation. In some studies, a specific value of n is established for a particular process [16, 17]. In many cases, however, the experimental kinetics of isother-
mal transformation for supercooled austenite reveals very significant deviations from the conventional theory [18–20]. (1) The exponent n may take any positive real value, depending on the type of transformation, the temperature, and other factors. In some cases, it significantly exceeds the values permitted by the theory. (2) The coefficients k and n depend on the temperature of isothermal transformation and vary nonmonotonically during the process. Even if all the conditions of the theory are satisfied, the results differ greatly from the experimental data [21, 22]. It would be useful here to find a function that, on the one hand, described the observed phenomena as precisely as possible with the minimum number of optimization parameters; and, on the other, was consistent with the conventional theory. In formal terms, the experimental kinetics of isothermal transformation for supercooled austenite corresponds to a classic sigmoid curve (S curve) typical of the logistical function proposed by Verhulst [23, 24]
F ( x) = L [1 + exp(−kx )].
(2)
The saturation limit L and growth rate k are parameters of the logistical function that may be used as optimization parameters in the mathematical description of experimental data. The basic properties of the logistical growth function are as follows [25]: (1) exponential growth is seen at the beginning of the process;
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(2) the exhaustion of available resources results in slowing of the growth; (3) the total increment is determined by saturation. These properties of the logistical growth function correspond, in basic outline, to the fundamental laws of phase transformations in a solid [7, 8]: (1) the transformation accelerates initially on account of increase in the volume of previously formed nuclei and the possible formation of new nuclei; (2) the transformation slows with exhaustion of the available nucleation sites; (3) the transformation stops when all the volume has been transformed or there are no further nucleation sites. The logistical function has been used extensively in biology, medicine, economics, sociology, and the engineering sciences (including the physical metallurgy of metals) [26–29]. In the present work, we show that the logistical function may be used in kinetic description of the isothermal transformation of supercooled austenite in steel, for the example of isothermal-bainite formation. EXPERIMENTAL MATERIALS AND METHODS For steel A2, experimental data regarding the kinetics of isothermal transformation of austenite in the temperature range corresponding to bainite formation were taken from [30]. For 50ХМФА steel (D6AC steel), such data were obtained by means of dilatometric measurements. The chemical composition of the steel samples is as follows (wt %):
A2 [30] 50ХМФА (D6AC)
C 0.57 0.49
Cr 1.21 1.11
Mo Mn Si 0.24 2.02 1.52 0.98 0.79 0.23
A2 [30] 50ХМФА (D6AC)
S+P 0.056 0.025
Al Co Ni 0.65 1.59 – – – 0.46
V – 0.11
For steel A2, the austenitization temperature is 920°C, and the isothermal holding temperatures are 210, 230, 250, and 290°C [30]. For 50ХМФА steel, the austenitization temperature is 925°C, and the isothermal holding temperatures are 300, 330, 360, and 390°C. The kinetics of isothermal bainitic transformation in 50ХМФА steel is studied by means of a Linseis L78 R.I.T.A dilatometer. The steel is cooled from the austenitization temperature to the isothermal holding temperature at a rate of 30°C/s (over a period of ≥5 h). Two methods are used for mathematical description of the isothermal transformation kinetics: by
means of Eq. (1) with constant coefficients k and n; and by means of the logistical function
P Peq = 1 − 1 {1 + exp[b + a ln(τ)]}.
(3)
Here P is the proportion of bainite formed; τ is the time, s; Peq is the maximum equilibrium proportion of bainite formed at the given isothermal holding temperature; a and b are constant in holding at specific temperature. The maximum proportion of bainite formed in steel A2 at each holding temperature is taken from [30]. For 50ХМФА steel, the proportion of bainite formed is determined by quantitative metallography of the final sample microstructure after isothermal holding. To determine a and b in Eq. (3) for each isothermal holding temperature, the sum of squares of the difference between the proportion of bainite formed according to Eq. (3) and the experimental values is minimized, over 700–1000 points. RESULTS AND DISCUSSION The experimental kinetics of isothermal bainitic transformation in steel A2 is described by means of Eq. (1) with constant k and n values. However, the results do not adequately describe the experimental data for the first interval of bainitic transformation (Fig. 1a). For example, with isothermal holding at 210°C, description of the isothermal transformation by means of Eq. (1) requires the successive use of two sets of constants: up to the transformation of 40–45% of the austenite, n = 3; thereafter, n = 1.5. We may also use variable coefficients k, n = f(τ). However, that is inconsistent with the assumptions on which Eq. (1) is based [8]. In addition, k and n depend on the temperature. Therefore, considerable errors arise on assuming that k and n are constant. If we ignore the theoretical requirements and assume that k and n are functions of the temperature and the time, the model is significantly complicated, and this approach cannot easily be used in modeling austenite transformation in continuous cooling, especially at a rate that varies over time. Those difficulties may be avoided by using the logistical function in Eq. (3). As we see in Fig. 1b, the logistical function provides a practically ideal description of the experimental data throughout isothermal holding. At 210°C, a and b are constant: a = 3.0; b = –31.1. These values correspond to the coefficients of the linearized logistical function (Fig. 1c)
ln[P (Peq − P )] = a ln(τ) + b.
(4)
The agreement of the experimental and calculation results is best in the range lnτ = 9.5–11. That corresponds to 2.5–78% austenite transformation. Thus, the greatest discrepancy between the experimental and calculation results (2–4%) is observed at the beginning and end of holding. That may be due the experimental conditions (thermal stabilization of the system at transition from continuous cooling to isothermal STEEL IN TRANSLATION
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Experiment Eq. (1), n = 1.5 Eq. (1), n = 3.0 20 000
Proportion of bainite
0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
20 000
6
40 000 60 000 80 000 τ, s (c) y = 3.00x – 31.10
–30 –31.1 –32 Experiment exp (b) = 9.36 × 10–14 exp (b) = 3.11 × 10–14 exp (b) = 1.27 × 10–14 10 000
30 000
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
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(a)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
80 000 τ, s
40 000 60 000 (b)
Experiment Logistical function 0
Proportion of bainite
(a)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Proportion of bainite
Proportion of bainite
DESCRIBING THE ISOTHERMAL BAINITIC TRANSFORMATION
50 000 τ, s (b)
70 000
90 000
Experiment a = 2.75 a = 3.00 a = 3.25 10 000
30 000
50 000 τ, s
70 000
90 000
ln (P/P0 – P)
4 Fig. 2. Influence of the coefficients a and b in Eq. (3) on the isothermal transformation kinetics: (a) influence of b (given on the curves) when a = 3; (b) influence of a when b = –31.1.
2 0 –2
8
9
10
11
ln (τ)
or
–4
P (Peq − P )] = exp(b).
–6 –8
Fig. 1. Mathematical description of the isothermal bainitic transformation in steel A2 [30] at 210°C: (a) from Eq. (1); (b) on the basis of the proposed logistical function; (c) by linearization of the experimental and calculation results on the basis of the logistical function.
holding) and the physical properties of the sample (chemical heterogeneity, nonmetallic inclusions, etc.). Nevertheless, even with this error, the mathematical description of the isothermal transformation kinetics agrees satisfactorily with experimental data: the ratio of the tabular Fisher statistic to the calculated value is more than 600, and the correlation coefficient of the experimental and calculation results for the proportion of bainite formed is 0.992. It follows from Eq. (1) that when τ = 1 s
ln[P (Peq − P )] = b STEEL IN TRANSLATION
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(6)
In physical terms, exp b may be regarded as the proportion (by volume) of bainite formed in 1 s of isothermal transformation or as the probability of transition of new-phase nucleus to stable growth in specific experimental conditions. It is clear from Fig. 2a that b affects the kinetics of isothermal transformation: with constant a (a = 3), decrease in b from –30 to –32— and hence decrease in exp b from 9.36 × 10–14 to 1.27 × 10–14—reduces the rate of new-phase formation by half and markedly increases the transformation time. The transformation kinetics also depends markedly on a (Fig. 2b): decrease in a from 3.25 to 2.75 (when b = –31.1) considerably lowers the rate of transformation. We may perhaps regard a as a kinetic parameter of the model corresponding to the transformation rate. It follows from Eq. (4) that a and b are interrelated. Their ratio determines the halflife τ0.5—that is, the time required for the completion of 50% of the transformation possible at the given temperature. This follows from Eq. (3) with P/Peq = 1/2: τ0.5 = exp (–b/a).
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0.6 0.5 0.4 0.3
210°С 230°С 250°С 290°С
0.2 0.1
Proportion of bainite
10 000
30 000
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
50 000 t, s (c)
70 000
Coefficient a
0.7
0
3.0
–5
2.5
–10
2.0
–15
1.5
–20
1.0
–25
0.5
–30
0 200
90 000
220
5.0
240 260 t, °C (d)
280
–35 300 –5 –10
4.5
300°С 330°С 360°С 390°С 200
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600
800 1000 1200 1400 t, s
Coefficient a
Proportion of bainite
0.8
0
(b)
3.5
–15
4.0
–20 3.5 –25 3.0 2.5 285
Coefficient b
(a)
0.9
Coefficient b
216
–30 305
325
345 t, °C
365
385
–35 405
Fig. 3. Mathematical description of the isothermal bainitic transformation in steel A2 (a, b) and 50XMФA steel (c, d): (a, c) comparison of the experimental and calculated transformation kinetics at different holding temperatures; (b, d) dependence of a and b on the isothermal holding temperature.
To better understand the physical significance of a and b, we need further experimental and theoretical research. The calculations show that the proposed logistical function adequately describes the experimental kinetics of isothermal austenite transformation over the whole range of holding temperature in [30] (Fig. 3a). Each temperature corresponds to particular values of a and b. Thus, it is possible to determine the temperature dependence of the coefficients a and b in Eq. (3). As we see, the temperature dependences of a and b are monotonic. Accordingly, the results may be used to simulate austenite transformation in continuous cooling, on the basis of the sum rule. In addition, changes in a and b may indicate changes in the transformation. For example, a ≈ 3 in the range 210–250°C, but a = 2.2 at 290°C. At the same time, b increases monotonically over the whole temperature range. That may indicate transition from the formation of lower bainite to the gradual formation
of upper bainite in the steel. With increase in temperature, we may also note the following: ⎯decrease in the bainite growth rate on approaching the equilibrium temperature at which bainitic transformation begins (corresponding to decrease in a); ⎯increase in the proportion (by volume) of bainite formed or the probability that new nucleation centers will be formed (corresponding to increase in absolute magnitude of b) as a result of increase in diffusional mobility of the carbon atoms. Similar results are obtained for isothermal bainitic transformation in 50ХМФА steel (Fig. 3c). As we see, the proposed logistical function provides a practically ideal description of the isothermal bainite formation at the temperatures considered. The discrepancy between the experimental and calculation results for the proportion of bainite formed is no more than 3%. In the range 300–330°C, the incubation period of the process is reduced. In the range 330–390°C, it increases again. Nevertheless, the temperature depenSTEEL IN TRANSLATION
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DESCRIBING THE ISOTHERMAL BAINITIC TRANSFORMATION
dence of a and b is monotonic (Fig. 3d). We note that a tends to increase from 4 to 4.5, while b decreases smoothly from –24 to –29. The proposed logistical function offers new scope for analysis of the structural and phase transformations in steel and for numerical modeling of heat treatment. CONCLUSIONS (1) A logistical function is proposed for the kinetic description of isothermal austenite transformation in structural steel. The argument of the function includes the logarithm of the isothermal holding time and two factors a and b, which are constant at each isothermal holding temperature. (2) The logistical function permits satisfactory description of the kinetics throughout the process, as shown for the examples of steel A2 [30] and 50ХМФА steel. The ratio of the tabular Fisher statistic to the calculated value is more than 600, and the correlation coefficient of the experimental and calculation results for the proportion of bainite formed is 0.992. (3) The logistical function permits determination of the temperature dependence of a and b. Hence, it may be used to simulate the bainitic transformation of austenite in continuous cooling. (4) Further experimental and theoretical research on the proposed logistical function should focus on refining the physical significance of a (the transformation rate) and b (the proportion of bainite formed in 1 s of isothermal transformation). ACKNOWLEDGMENTS Financial support was provided by the Ministry of Education and Science (project no. 11.146.2014/K) under decree 211 of the Russian government (contract no. 02.A03.21.0006); and also by the Russian President’s fund for young scientists (grant no. MK7929.2016.8). REFERENCES 1. Ryzhkov, V.A., Maisuradze, M.V., Yudin, Yu.V., et al., Experience in improving silicon steel component heat treatment quality, Metallurgist, 2015, vol. 59, no. 5, pp. 401–405. 2. Belikov, S.V. Sergeeva, K.I., Kornienko, O.Yu., et al., Special features of formation of structure and properties of steel with heterogeneous bainite-martensite structure for gas and oil pipeline, Met. Sci. Heat Treat., 2011, vol. 52, no. 11, pp. 581–587. 3. Meccozi, M.G., Eiken, J., Santofimia, M.J., and Sietsma, J., Phase field modeling of microstructural evolution during the quenching and partitioning treatment in low-alloy steels, Comput. Mater. Sci., 2016, vol. 112, pp. 245–256. STEEL IN TRANSLATION
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Translated by Bernard Gilbert
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