Mathematical Modelling of Isothermal Decomposition of Austenite in Steel

Abstract

The main goal of this paper is mathematical modelling and computer simulation of isothermal decomposition of austenite in steel. Mathematical modelling and computer simulation of isothermal decomposition of austenite nowadays is becoming an indispensable tool for the prediction of isothermal heat treatment results of steel. Besides that, the prediction of isothermal decomposition of austenite can be applied for understanding, optimization and control of microstructure composition and mechanical properties of steel. Isothermal decomposition of austenite is physically one of the most complex engineering processes. In this paper, methods for setting the kinetic expressions for prediction of isothermal decomposition of austenite into ferrite, pearlite or bainite were proposed. After that, based on the chemical composition of hypoeutectoid steels, the quantification of the parameters involved in kinetic expressions was performed. The established kinetic equations were applied in the prediction of microstructure composition of hypoeutectoid steels.

1. Introduction

The research of the mathematical simulation of microstructure distribution in steel is one of the highest-priority research areas in the simulation of phenomena of the heat treatment of steel. By using the additivity rule and kinetic equations of isothermal decomposition of austenite, it is possible to calculate kinetics of austenite decomposition at continuous cooling of steel. The prediction of isothermal decomposition of austenite can be applied for understanding, optimization and control of microstructure composition and mechanical properties of steel [1,2,3,4].

The most common method of computer prediction of isothermal decomposition of austenite results is based on the chemical composition of steel by using time-temperature-transformation (TTT) diagrams [5].

Studies of the kinetics of isothermal decomposition of austenite have been intensified in the course of some pioneering studies on the isothermal decomposition of austenite [6,7,8].

The prediction of microstructure composition is usually based on semi-empirical methods derived from kinetic equations of microstructure transformation [9]. To describe the transformation kinetics by mathematical methods, a semi-empirical approach is employed using the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation together with additivity rule [10,11].

The phase transformations can be categorized into two categories: reconstructive phase transformations and displacive phase transformations. Decompositions of austenite into ferrite and pearlite in steels are typical examples of reconstructive phase transformations, while martensite, bainite, and Widmanstatten ferrite phase transformations can be recognized as displacive phase transformations [12].

The formation of ferrite occurs by nucleation at the austenite grain boundaries. After that the ferrite grows inside the austenite grains. The rate of volume fraction of the ferrite is a function of the nucleation rate and the velocity of the ferrite/austenite interface. The nucleation rate is primarily a function of the undercooling below the A_e3 temperature and the grain size of austenite [13,14].

The nucleation mechanism of pearlite involves the formation of two phases, ferrite and cementite. The nucleation of cementite is a rate-limiting step in hypoeutectoid steels. The proeutectoid ferrite nucleates first and continues to grow with the same crystallographic orientation during the pearlite formation. For hypereutectoid steels, the role of the nucleation of ferrite is a limited process in comparison with the roles of the cementite nucleation. In eutectoid steel, the pearlite nucleation is assumed to occur at the austenite grain corners, edges, and boundaries.

Two different theories are proposed for the growth of pearlite. The Zener–Hillert theory assumes that the volume diffusion of carbon in the austenite is the rate-controlling mechanism [15,16]. In addition, Hillert theory assumes that grain boundary diffusion of the carbon atoms is the rate-controlling mechanism. The nucleation rate of pearlite follows the general nucleation theory [16].

The Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory predicts the overall transformation rate on the basis of nucleation and growth rates. It is the most widely used model to describe the austenite–pearlite transformation kinetics [15].

Bainite was discovered nearly eight decades ago [17]. The research work carried out in the field of bainite is immense [17,18,19,20]. A qualitative theory to explain bainite formation still remains a subject of controversy [18,21]. One theory suggests a diffusion-controlled transformation where bainitic growth occurs by a diffusional ledge mechanism, while the other suggests that the bainite reaction is a displacive transformation [18]. Both theories have assumed models to predict the transformation kinetics [22,23]. The growth of bainite and Widmanstatten ferrite requires the partitioning of interstitial carbon. Because of this reason, their growth is controlled by diffusion of interstitial atoms of carbon [12].

Computer simulation of isothermal decomposition of austenite in steel is still a complex problem. The dependence of the physical quantities involved in the kinetic expressions of austenite decomposition has not yet been sufficiently defined in the literature. Efforts are being made to predict the dependence of physical quantities in the kinetic expressions of austenite decomposition on chemical composition.

This work proposes inversion methods for quantification of kinetic parameters and setting the kinetic expressions in the prediction of isothermal decomposition of austenite into ferrite, pearlite, or bainite.

The proposed method of setting kinetic relations can be used in the calculation of characteristic kinetic parameters for other groups of steel. The established model can be used for computer simulation of austenite decomposition in other steels with similar chemical compositions.

2. Materials and Methods

2.1. Methods for Estimation of Kinetic Parameters of Austenite Isothermal Decomposition

2.1.1. Kinetics Expressions of Austenite Decomposition in an Incremental Form

Kinetics of isothermal decomposition of austenite can be defined by Avrami′s isothermal equation:

$$X=1-\exp \left(-k{t}^n\right)$$

(1)

where X is transformed part of the microstructure, t is time, and k and n are kinetic parameters. By extracting the time component, Equation (1) can be written as:

$$t=\frac{1}{k^{\frac{1}{n}}}{\left(\ln \left(\frac{1}{1-X}\right)\right)}^{\frac{1}{n}}$$

(2)

In computer-based mathematical analysis, it is convenient to define the kinetics of austenite decomposition in an incremental form. By differentiating Avrami′s equation, it follows that:

$$\frac{dX}{dt}=\exp \left(-k{t}^n\right)nk{t}^{n-1}$$

(3)

After introducing Equation (2) in Equation (3) and a short rearrangement, it follows that:

$$\frac{dX}{dt}=n{k}^{\frac{1}{n}}{\left(\ln \frac{1}{1-X}\right)}^{1-\frac{1}{n}}\left(1-X\right)$$

(4)

Equation (4) can be written in an incremental form, and the volume fraction ΔX^(N) of austenite transformed in the time interval Δt^(N) can be calculated as [3]:

$$\Delta X^{\left(\mathrm{N}\right)}=n{k}^{\frac{1}{n}}{\left(\ln \frac{1}{1-X^{\left(\mathrm{N}-1\right)}}\right)}^{1-\frac{1}{n}}\left(1-X^{\left(\mathrm{N}-1\right)}\right)\Delta t^{\left(\mathrm{N}\right)}$$

(5)

where X^(N−1) is the volume fraction of austenite transformed in previous N − 1 time intervals. Kinetic parameters k and n can be evaluated inversely by using data of time of isothermal transformation. The total volume fraction of austenite transformed during isothermal decomposition can be calculated as:

$$X={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}\mathsf{\Delta }{X}^{\left(\mathrm{N}\right)}$$

(6)

2.1.2. Ferrite Transformation

The ferrite transformation takes place by the mechanism of nucleation and growth, with the following assumed kinetic parameters [13]:

$$n_{\mathrm{F}}=4$$

(7)

$$k_{\mathrm{F}}=\frac{\pi }{3}{I}_{\mathrm{F}}S{G}_{\mathrm{F}}{}^3$$

(8)

where S is the surface of austenite grain suitable for nucleation, while I_F is the nucleation rate and G_F is the growth rate defined as:

$$I_{\mathrm{F}}=T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{Q_{\mathrm{dif}}}{RT}\right)\exp \left(-\frac{k_1}{RT{\left(\Delta T\right)}^2}\right)$$

(9)

$$G_{\mathrm{F}}=\frac{c_{\mathsf{\gamma }}-c_0}{c_{\mathsf{\gamma }}-c_{\mathsf{\alpha }}} D_0\exp \left(-\frac{Q_{\mathrm{dif}}}{RT}\right)\frac{1}{y^D}$$

(10)

In Equations (9) and (10), T is temperature, ΔT is the undercooling below the critical temperature A_e3, D₀ is the material constant, R is the universal gas constant, Q_dif is the diffusion activation energy, while c₀, c_α, and c_γ are the concentrations of steel, ferrite and austenite at the boundary with ferrite, respectively. The effective diffusion length is defined as [7,13]:

$$y^{\mathrm{D}}=\frac{k_2}{{\left(\Delta T\right)}^{n_1}}$$

(11)

where k₁, k₂ and n₁ are the kinetic parameters dependent on chemical composition of steel. After introducing Equations (9)–(11) in Equation (8), and after some modification, Equation (8) can be rewritten as [14]:

$$k_{\mathrm{F}}=S{D}_0^4\exp \left(\frac{-4Q_{\mathrm{dif}}}{RT}\right)\exp \left(\frac{-k_1}{RT{\left(A_{\mathrm{e}3}-T\right)}^2}\right){\left(\frac{c_{\mathsf{\gamma }}-c_0}{c_{\mathsf{\gamma }}-c_{\mathsf{\alpha }}}\right)}^3{\left(\frac{{\left(A_{\mathrm{e}3}-T\right)}^{n_1}}{k_2}\right)}^3$$

(12)

To determine the values of the constants k₁, k₂ and n₁, it is first necessary to determine the value of the coefficient k_F for three temperatures, and then to solve a system of three equations with three unknowns (k₁, k₂, n₁).

It was assumed that the ferrite transformation does not take place to the end, but to the maximum volume V_max = V_rF × V, when the normalized volume fraction of ferrite can be defined as [18]:

$${\xi }_{\mathrm{F}}=\frac{X_{\mathrm{F}}}{V_{\mathrm{rF}}}$$

(13)

In Equation (13), X_F is the volume fraction of ferrite and V_rF is the relative volume of ferrite. The linear temperature dependence of the volume V_rF can be evaluated using an Fe-Fe₃C diagram with the following assumptions: at temperatures A_e3 and B_s, the volume V_rF is equal to 0, while at temperature A_e1, it takes the maximum value V_rF = c₀/0.8 (Figure 1).

$$V_{\mathrm{rF}}=a_1+a_{2}T, \mathrm{for} A{\mathrm{e}}_3 > T \ge A{\mathrm{e}}_1$$

(14)

where:

$$a_1=\frac{A_{\mathrm{e}3}}{A_{\mathrm{e}3}-A_{\mathrm{e}1}}\frac{c_0}{0.8}$$

(15)

$$a_2=\frac{-1}{A_{\mathrm{e}3}-A_{\mathrm{e}1}}\frac{c_0}{0.8}$$

(16)

$$V_{\mathrm{rF}}=a_3+a_{4}T, \mathrm{for} A{\mathrm{e}}_1 > T \ge T_F$$

(17)

where:

$$a_3=\frac{-T_{\mathrm{F}}}{A_{\mathrm{e}1}-T_{\mathrm{F}}}\frac{c_0}{0.8}$$

(18)

$$a_4=\frac{1}{A_{\mathrm{e}1}-T_{\mathrm{F}}}\frac{c_0}{0.8}$$

(19)

The real volume of ferrite can be written as:

$$\mathrm{d}{V}_{\mathrm{F}}=\left(\frac{V_{\max }-V_{\mathrm{F}}}{V_{\max }}\right)\mathrm{d}{V}_{\mathrm{Fe}}=\left(\frac{V_{\mathrm{rF}}V-V_{\mathrm{F}}}{V_{\mathrm{rF}}V}\right)\mathrm{d}{V}_{\mathrm{Fe}}$$

(20)

$$\mathrm{d}{V}_{\mathrm{F}}=\left(1-{\xi }_{\mathrm{F}}\right)\mathrm{d}{V}_{\mathrm{Fe}}$$

(21)

The extended volume of ferrite is defined as:

$$\mathrm{d}{V}_{\mathrm{Fe}}=\frac{4}{3}\pi I_{\mathrm{F}}S{G}_{\mathrm{F}}{}^{3}V{\left(t-t_{\mathrm{i}}\right)}^3\mathrm{d}t$$

(22)

where t_i is the incubation time. After introducing Equation (22) in Equation (21), it follows that:

$$V_{\mathrm{rF}} \mathrm{d}{\xi }_{\mathrm{F}}=\left(1-{\xi }_{\mathrm{F}}\right)\frac{4}{3}\pi I_{\mathrm{F}}S{G}_{\mathrm{F}}{}^3{\left(t-t_{\mathrm{i}}\right)}^3\mathrm{d}t$$

(23)

$$V_{\mathrm{rF}} \mathrm{d}{\xi }_{\mathrm{F}}=\left(1-{\xi }_{\mathrm{F}}\right)4k_{\mathrm{F}}{\left(t-t_{\mathrm{i}}\right)}^3\mathrm{d}t$$

(24)

$$V_{\mathrm{rF}} \frac{\mathrm{d}{\xi }_{\mathrm{F}}}{\left(1-{\xi }_{\mathrm{F}}\right)}=4k_{\mathrm{F}}{\left(t-t_{\mathrm{i}}\right)}^3\mathrm{d}t$$

(25)

After integrating Equation (25), it follows that:

$$-\ln \left(1-{\xi }_{\mathrm{F}}\right)V_{\mathrm{rF}}=k_{\mathrm{F}}{t}^4$$

(26)

For small values of the normalized volume fraction of ferrite, Equation (26) can be written as:

$$-\ln \left(1-X_{\mathrm{F}}\right)=k_{\mathrm{F}}{t}_{\mathrm{i}}^4$$

(27)

At any temperature, knowing an incubation time of ferrite transformation, found out from the IT diagram, the value of the kinetic parameter k_F can be written as:

$$k_{\mathrm{F}}=\frac{-\ln \left(1-X_{{\mathrm{Ft}}_{\mathrm{i}}}\right)}{{\left(t_{\mathrm{i}}\right)}^4}=\frac{-\ln \left(1-0.01\right)}{{\left(t_{\mathrm{i}}\right)}^4}$$

(28)

From Equation (26), follows the normalized volume fraction of ferrite which is:

$${\xi }_{\mathrm{F}}=1-\exp \left(-\frac{k_{\mathrm{F}}}{V_{\mathrm{rF}}}{t}^4\right)$$

(29)

Equation (29) can be written in an incremental form, when the normalized volume fraction of ferrite formed by the mechanism of nucleation and growth in the time interval Δt^(N) can be calculated as:

$$\Delta {\xi }_{\mathrm{F}}^{\left(\mathrm{N}\right)}=4{\left(\frac{k_{\mathrm{F}}}{V_{\mathrm{rF}}}\right)}^{\frac{1}{4}}{\left(\ln \frac{1}{1-{\xi }_{\mathrm{F}}^{\left(\mathrm{N}-1\right)}}\right)}^{\frac{3}{4}}\left(1-{\xi }_{\mathrm{F}}^{\left(\mathrm{N}-1\right)}\right)\Delta t^{\left(\mathrm{N}\right)}$$

(30)

At any time of transformation, the real volume fraction of ferrite can be calculated as $X_{\mathrm{F}}={\xi }_{\mathrm{F}}{V}_{\mathrm{rF}}$ (Equation (13)).

2.1.3. Bainite Transformation

The bainite transformation begins at a temperature B_s. Like ferrite transformation, it does not take place to the end, but to the maximum volume V_max = V_rB × V, when the normalized volume fraction of bainite can be defined as [18]:

$${\xi }_{\mathrm{B}}=\frac{X_{\mathrm{B}}}{V_{\mathrm{rB}}}$$

(31)

where V_rB is the relative volume of bainite, while X_B is the volume fraction of bainite defined as:

$$\mathrm{d}{\xi }_{\mathrm{B}}{V}_{\mathrm{rB}}=\left(1-{\xi }_{\mathrm{B}}\right)I_{\mathrm{B}}u\mathrm{d}t$$

(32)

where u is the volume of the structural unit of bainite. The nucleation rate is defined as [14]:

$$I_{\mathrm{B}}=T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{ Q_{\mathrm{dif}}+k_6}{RT}\right)\exp \left(-\frac{k_7}{RT{\left(\Delta T\right)}^2}\right)\exp \left(-\frac{t_{\mathrm{i}}}{t}\right)$$

(33)

where k₆ and k₇ are the kinetic parameters dependent on chemical composition of steel. After introducing Equation (33) to Equation (32), it follows that:

$$\frac{\mathrm{d}{\xi }_{\mathrm{B}}\Delta V_{\mathrm{rB}}}{\left(1-{\xi }_{\mathrm{B}}\right)}=T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{ Q_{\mathrm{dif}}+k_6}{RT}\right)\exp \left(-\frac{k_7}{RT{\left(\Delta T\right)}^2}\right)\exp \left(-\frac{t_{\mathrm{i}}}{t}\right)u\mathrm{d}t$$

(34)

$$V_{\mathrm{rB}}{{\displaystyle \int }}^{}\frac{\mathrm{d}{\xi }_{\mathrm{B}}}{\left(1-{\xi }_{\mathrm{B}}\right)}=T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{ Q_{\mathrm{dif}}+k_6}{RT}\right)\exp \left(-\frac{k_7}{RT{\left(\Delta T\right)}^2}\right)u{{\displaystyle \int }}^{}\exp \left(-\frac{t_{\mathrm{i}}}{t}\right)\mathrm{d}t$$

(35)

If the following is accepted [24]:

$$\underset{0}{\overset{t}{{\displaystyle \int }}}\exp \left(-\frac{t_{\mathrm{i}}}{t}\right)dt\approx k{t}^n$$

(36)

Equation (36) can be rewritten as:

$$-\ln \left(1-{\xi }_{\mathrm{B}}\right)V_{\mathrm{rB}}=T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{ Q_{\mathrm{dif}}+k_6}{RT}\right)\exp \left(-\frac{k_7}{RT{\left(\Delta T\right)}^2}\right)k_8{t}^{n_{\mathrm{B}}}$$

(37)

where k₈ is the kinetic parameters. The time of bainite transformation can be expressed by:

$$t={\left[\frac{-\ln \left(1-{\xi }_{\mathrm{B}}\right)V_{\mathrm{rB}}}{T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{ Q_{\mathrm{dif}}+k_6}{RT}\right)\exp \left(-\frac{k_7}{RT{\left(\Delta T\right)}^2}\right)k_8}\right]}^{\frac{1}{n_{\mathrm{B}}}}$$

(38)

For small values of the normalized volume fraction of bainite it can be taken that $\ln \left(1-{\xi }_{\mathrm{B}}\right)V_{\mathrm{rB}}\approx \ln \left(1-X_{\mathrm{B}}\right)$; therefore, Equation (38) can be rewritten as:

$$t={\left[\frac{-\ln \left(1-X_{\mathrm{B}}\right)}{T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{ Q_{\mathrm{dif}}+k_6}{RT}\right)\exp \left(-\frac{k_7}{RT{\left(\Delta T\right)}^2}\right)k_8}\right]}^{\frac{1}{n_{\mathrm{B}}}}$$

(39)

As a rule, at low temperatures austenite is completely transformed into bainite, when it can be assumed that V_rB ≈ 1 and $\xi$_B ≈ X_B. With this assumption, the kinetic parameter n_B can be defined as:

$$n_{\mathrm{B}}\approx \frac{2.661}{\log \left(t_{0.99}\right)-\log \left(t_{\mathrm{i}}\right)}$$

(40)

where t_i is the incubation time and t_0.99 is the finish time of the isothermal bainite transformation found out from the IT diagram. The denominator of Equation (39) is a function of temperature; therefore, for the incubation time and constant temperature, Equation (39) can be rewritten as:

$$t_{\mathrm{i}}=t_{0.01}={\left[\frac{-\ln \left(1-X_{\mathrm{B}}\right)}{k_{\mathrm{B}}}\right]}^{\frac{1}{n_{\mathrm{B}}}}$$

(41)

At any temperature, knowing an incubation time of bainite transformation, found out from the IT diagram, the value of the kinetic parameter k_B can be expressed by:

$$k_{\mathrm{B}}=\frac{-\ln \left(1-X_{{\mathrm{Bt}}_{\mathrm{i}}}\right)}{{\left(t_{\mathrm{i}}\right)}^{n_{\mathrm{B}}}}=\frac{-\ln \left(1-0.01\right)}{{\left(t_{\mathrm{i}}\right)}^{n_{\mathrm{B}}}}$$

(42)

For 99% of austenite transformed into bainite, Equation (38) can be written as:

$$t_{0.99}={\left[\frac{-\ln \left(1-{\xi }_{\mathrm{B}}\right)V_{\mathrm{rB}}}{k_{\mathrm{B}}}\right]}^{\frac{1}{n_{\mathrm{B}}}}={\left[\frac{-\ln \left(1-\frac{X_{\mathrm{B}}}{V_{\mathrm{rB}}}\right)V_{\mathrm{rB}}}{k_{\mathrm{B}}}\right]}^{\frac{1}{n_{\mathrm{B}}}}$$

(43)

where the linear temperature dependence of the volume V_rB is assumed:

$$V_{\mathrm{rB}}=a_5+a_{6}T.$$

(44)

Coefficients a₅ and a₆ can be determined by corresponding values of the volume V_rB on two different temperatures in IT diagram. Based on Equations (39) and (41), the kinetic parameter k_B can be written as:

$$k_{\mathrm{B}}=T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{ Q_{\mathrm{dif}}+k_6}{RT}\right)\exp \left(-\frac{k_7}{RT{\left(\Delta T\right)}^2}\right)k_8$$

(45)

With the previously determined kinetic parameter n_B, the defined temperature dependence of the volume V_rB and with the known values of the constants k₆, k₇ and k₈, the kinetics of the bainite transformation is completely defined. To determine the values of the constants k₆, k₇ and k₈, it is first necessary to determine the value of the coefficient k_B for three temperatures, and then to solve a system of three equations with three unknowns.

Based on Equations (46) and (47), the volume fraction of bainite and the normalized volume fraction of bainite can be determined by:

$$X_{\mathrm{B}}=V_{\mathrm{rB}}\left(1-\exp \left(-\frac{k_{\mathrm{B}}}{V_{\mathrm{rB}}}{t}^{n_{\mathrm{B}}}\right)\right)$$

(46)

$${\xi }_{\mathrm{B}}=\frac{X_{\mathrm{B}}}{V_{\mathrm{rB}}}=1-\exp \left(-\frac{k_{\mathrm{B}}}{V_{\mathrm{rB}}}{t}^{n_{\mathrm{B}}}\right)$$

(47)

Equation (47) can be written in an incremental form, when the normalized volume fraction of bainite formed in the time interval Δt^(N) can be calculated as:

$$\Delta {\xi }_{\mathrm{B}}^{\left(\mathrm{N}\right)}=n_{\mathrm{B}}{\left(\frac{k_{\mathrm{B}}}{V_{\mathrm{rB}}}\right)}^{\frac{1}{n_{\mathrm{B}}}}{\left(\ln \frac{1}{1-{\xi }_{\mathrm{B}}^{\left(\mathrm{N}-1\right)}}\right)}^{1-\frac{1}{n_{\mathrm{B}}}}\left(1-{\xi }_{\mathrm{B}}^{\left(\mathrm{N}-1\right)}\right)\Delta t^{\left(\mathrm{N}\right)}$$

(48)

At any time of transformation, the real volume fraction of bainite can be calculated as $X_{\mathrm{B}}={\xi }_{\mathrm{B}}{V}_{\mathrm{rB}}$ (Equation (31)).

2.1.4. Pearlite Transformation

In the remaining undercooled austenite that has not transformed into ferrite or bainite, at temperatures lower than A_e1, the pearlite transformation takes place. The kinetics of pearlite transformation is independent of the kinetics of the previous ferrite or bainite transformation. At temperatures A_e1 > T ≥ B_s the remaining volume available for pearlitic transformation is V_rP1·V, while at temperatures T < B_s the remaining volume is V_rP2 × V, where V_rP1 = 1 − V_rF and V_rP2 = 1 − V_rB.

For pearlite transformation by the mechanism of nucleation and growth, the following kinetic parameters are assumed:

$$n_{\mathrm{P}}=4$$

(49)

$$k_{\mathrm{P}}=\frac{\pi }{3}{I}_{\mathrm{P}}S{G}_{\mathrm{P}}{}^3$$

(50)

where S is surface of austenite grain suitable for nucleation, while I_P is the nucleation rate and G_P is the growth rate defined as [14]:

$$I_{\mathrm{P}}=T^{-\frac{1}{2}}{D}_0\exp \left(-\frac{Q_{\mathrm{dif}}}{RT}\right)\exp \left(-\frac{k_3}{RT{\left(\Delta T\right)}^2}\right)$$

(51)

$$G_{\mathrm{P}}=\Delta T{D}_0\exp \left(-\frac{Q_{\mathrm{dif}}}{RT}\right)\left(c_{\mathsf{\gamma }\mathsf{\alpha }}-c_{\mathsf{\gamma }\mathrm{Fe}3\mathrm{C}}\right)$$

(52)

In Equations (51) and (52), ΔT is the undercooling below the critical temperature A_e3, while c_γα and c_γFe3C are the concentrations of austenite at the boundary with ferrite and cementite, respectively. k₃ is the kinetic parameters dependent on chemical composition. After introducing Equations (51) and (52) into Equation (50) and after some modifications, it can be rewritten:

$$k_{\mathrm{P}}=S{D}_0^4\exp \left(\frac{-4\left(Q_{\mathrm{dif}}+k_5\right)}{RT}\right)\exp \left(\frac{-k_3}{RT{\left(A_{\mathrm{e}1}-T\right)}^2}\right){\left(c_{\mathsf{\gamma }\mathsf{\alpha }}-c_{\mathsf{\gamma }\mathrm{Fe}3\mathrm{C}}\right)}^3{\left(A_{\mathrm{e}1}-T\right)}^3{k}_4^{-4}$$

(53)

To determine the values of the constants k₃, k₄ and k₅, it is first necessary to determine the value of the coefficient k_P for three temperatures, and then to solve a system of three equations with three unknowns.

The real volume of pearlite can be written as:

$$\mathrm{d}{V}_{\mathrm{P}}=\left(1-{\xi }_{\mathrm{P}}\right)\mathrm{d}{V}_{\mathrm{Pe}}$$

(54)

where the normalized volume fraction can be expressed by:

$${\xi }_{\mathrm{P}}=\frac{X_{\mathrm{P}}}{V_{\mathrm{rP}1}}$$

(55)

The extended volume of pearlite is defined as:

$$\mathrm{d}{V}_{\mathrm{Pe}}=\frac{4}{3}\pi I_{\mathrm{P}}S{G}_{\mathrm{P}}{}^{3}V{\left(t-t_{\mathrm{i}}\right)}^3\mathrm{d}t$$

(56)

After introducing Equation (56) into Equation (55), it follows that:

$$V_{\mathrm{rP}1} \mathrm{d}{\xi }_{\mathrm{P}}=\left(1-{\xi }_{\mathrm{P}}\right)\frac{4}{3}\pi I_{\mathrm{P}}S{G}_{\mathrm{P}}{}^3{\left(t-t_{\mathrm{i}}\right)}^3\mathrm{d}t$$

(57)

$$V_{\mathrm{rP}1}{ \mathrm{d}\xi }_{\mathrm{P}}=\left(1-{\xi }_{\mathrm{P}}\right)4k_{\mathrm{P}}{\left(t-t_{\mathrm{i}}\right)}^3\Delta \mathrm{d}t$$

(58)

$$V_{\mathrm{rP}1} \frac{\mathrm{d}{\xi }_{\mathrm{P}}}{\left(1-{\xi }_{\mathrm{P}}\right)}=4k_{\mathrm{P}}{\left(t-t_{\mathrm{i}}\right)}^3\mathrm{d}t$$

(59)

After integrating Equation (59), it follows that:

$$-\ln \left(1-{\xi }_{\mathrm{P}}\right)V_{\mathrm{rP}1}=k_{\mathrm{P}}{t}^4$$

(60)

For small values of the normalized volume fraction of perlite, Equation (60) can be written as:

$$-\ln \left(1-X_{\mathrm{P}}\right)=k_{\mathrm{P}}{t}_{\mathrm{i}}^4$$

(61)

At any temperature, knowing the incubation time of pearlite transformation, found out from the IT diagram, the value of the kinetic parameter k_P can be written as:

$$k_{\mathrm{P}}=\frac{-\ln \left(1-X_{{\mathrm{Pt}}_{\mathrm{i}}}\right)}{{\left(t_{\mathrm{i}}\right)}^4}=\frac{-\ln \left(1-0.01\right)}{{\left(t_{\mathrm{i}}\right)}^4}$$

(62)

From Equation (60) follows the normalized volume fraction of pearlite, which is:

$${\xi }_{\mathrm{P}}=1-\exp \left(-\frac{k_{\mathrm{P}}}{V_{\mathrm{rP}1}}{t}^4\right)$$

(63)

As for ferrite and bainite transformation, Equation (63) can be written in an incremental form. The normalized volume fraction of pearlite formed by the mechanism of nucleation and growth in the time interval Δt^(N) can be calculated by:

$$\Delta {\xi }_{\mathrm{P}}^{\left(\mathrm{N}\right)}=4{\left(\frac{k_{\mathrm{P}}}{V_{\mathrm{rp}1}}\right)}^{\frac{1}{4}}{\left(\ln \frac{1}{1-{\xi }_{\mathrm{P}}^{\left(\mathrm{N}-1\right)}}\right)}^{\frac{3}{4}}\left(1-{\xi }_{\mathrm{P}}^{\left(\mathrm{N}-1\right)}\right)\Delta t^{\left(\mathrm{N}\right)}$$

(64)

The presented method for estimation of k_P and ξ_P is also valid at temperatures lower than B_s. In that case, in the above equations, the relative volume V_rP1 should be replaced by the relative volume V_rP2.

2.2. Materials

With the aim of qualitatively and quantitatively defining the influence of chemical composition on the isothermal decomposition of austenite, the values of kinetic parameters were investigated on a number of hypoeutectoid, low-alloy steels [25]. Their composition is shown in

Table 1. Chemical composition of studied steels (balance Fe).

Designation (DIN)	Chemical Composition, wt. %
	C	Si	Mn	P	S	Cr	Cu	Mo	Ni	V
42CrMo4	0.38	0.23	0.64	0.019	0.013	0.99	0.17	0.16	0.08	<0.01
Ck45	0.44	0.22	0.66	0.022	0.029	0.15	-	-	-	0.02
28NiCrMo74	0.30	0.24	0.46	0.030	0.025	1.44	0.20	0.37	2.06	<0.01
34Cr4	0.35	0.23	0.65	0.026	0.013	1.11	0.18	0.05	0.23	<0.01
25CrMo4	0.22	0.25	0.64	0.010	0.011	0.97	0.16	0.23	0.33	<0.01
36Cr6	0.36	0.25	0.49	0.021	0.020	1.54	0.16	0.03	0.21	<0.01
41Cr4	0.44	0.22	0.80	0.030	0.023	1.04	0.17	0.04	0.26	<0.01

.

3. Results

Section 2.1 presents methods for estimating kinetic parameters, which completely define the kinetics of austenite isothermal decomposition into ferrite, pearlite and bainite. The calculated values of the kinetic parameters depend on the chemical composition, i.e., they are valid only for one steel.

The critical temperatures of austenite decomposition were calculated based on Equations (65) and (66) [26], and Equation (67) [27].

$$\begin{array}{cc}\hfill A_{\mathrm{e}3}=883.49& -275.89\%\mathrm{C}+90.91{(\%\mathrm{C})}^2-12.26\%\mathrm{Cr}+16.45\%\mathrm{C}\%\mathrm{Cr}-29.96\%\mathrm{Mn}+23.50\%\mathrm{C}\%\mathrm{Mn}\hfill \\ & +8.49\%\mathrm{Mo}-10.80\%\mathrm{C}\%\mathrm{Mo}-25.56\%\mathrm{Ni}+14.71\%\mathrm{C}\%\mathrm{Ni}+1.45\%\mathrm{Mn}\%\mathrm{Ni}+0.76{(\%\mathrm{Ni})}^2\\ & +13.53\%\mathrm{Si}-3.47\%\mathrm{Mn}\%\mathrm{Si}\end{array}$$

(65)

$$\begin{array}{cc}\hfill A_{\mathrm{e}1}=727.37& +13.40\%\mathrm{Cr}-1.03\%\mathrm{C}\%\mathrm{Cr}-16.72\%\mathrm{Mn}+0.91\%\mathrm{C}\%\mathrm{Mn}+6.18\%\mathrm{Cr}\%\mathrm{Mn}-0.64{(\%\mathrm{Mn})}^2\hfill \\ & +3.14\%\mathrm{Mo}+1.86\%\mathrm{Cr}\%\mathrm{Mo}-0.73\%\mathrm{Mn}\%\mathrm{Mo}-13.66\%\mathrm{Ni}+0.53\%\mathrm{C}\%\mathrm{Ni}+1.11\%\mathrm{Cr}\%\mathrm{Ni}\\ & -2.28\%\mathrm{Mn}\%\mathrm{Ni}-0.24{(\%\mathrm{Ni})}^2+6.34\%\mathrm{Si}-8.88\%\mathrm{Cr}\%\mathrm{Si}-2.34\%\mathrm{Mn}\%\mathrm{Si}+11.98{(\%\mathrm{Si})}^2\end{array}$$

(66)

$$B_{\mathrm{s}}=830-270\%\mathrm{C}-90\%\mathrm{Mn}-37\%\mathrm{Ni}-70\%\mathrm{Cr}-83\%\mathrm{Mo}$$

(67)

The dependence of kinetic parameters of ferrite, pearlite and bainite transformation on the content of carbon, chromium, molybdenum and nickel was estimated by regression analysis (Equations (68)–(78)). Because of the similar content of manganese and silicon in studied steels, these elements were not included in the regression analysis. Based on the proposed equations, the kinetic parameters involved in mathematical model of ferrite, pearlite and bainite transformation can be calculated for any other chemical composition of hypoeutectoid, low-alloy steels (

Table 2. Kinetic parameters of austenite isothermal decomposition.

Transformation	Constant	Steel Designation (DIN)
		42CrMo4	Ck45	28NiCrMo74	34Cr4	25CrMo4	36Cr6	41Cr4
Ferrite	k1	2.5 × 105	2.5 × 105	-	2.5 × 105	2.5 × 105	2.5 × 105	2.5 × 105
	k2	3.8321 × 10−5	1.7441 × 10−4	-	9.0625 × 10−10	6.3183 × 10−8	8.8213 × 10−9	1.1941 × 10−9
	n1	4.6923	5.2095	-	2.3955	3.3200	2.5230	2.3547
	a1	8.8839	8.7592	-	7.8504	3.1064	10.0511	14.7870
	a2	−0.0115	−0.0114	-	−0.0101	−0.0039	−0.0130	−0.0195
	a3	−1.8688	−4.4695	-	−1.7667	−1.4606	−1.5163	−1.7387
	a4	0.0032	0.0070	-	0.0030	0.0024	0.0027	0.0031
Pearlite	k3	27,988,177	76,767,479	16,492,913	31,355,038	43,316,272	56,142,707	93,514,410
	k4	1.1450 × 10−11	9.8057 × 10−5	2.1918 × 10−19	6.1325 × 10−12	8.6385 × 10−17	2.5088 × 10−17	5.4094 × 10−12
	k5	51,728	−92,146	214,273	47,324	149,610	151,595	46,211
Bainite	k6	−99,033	51128	−101,946	−15,089	−77,442	−34,123	−16,704
	k7	3.2308 × 108	2.4803 × 109	2.0943 × 109	2.5027 × 109	2.5966 × 106	2.2482 × 109	1.6819 × 109
	k8	5.4488 × 107	1.1746 × 1020	8.1265 × 107	8.0362 × 1014	6.6340 × 1010	2.7482 × 1012	4.2311 × 1013
	nB	1.4923	2.30043	1.09856	1.58602	1.12141	1.51777	1.74637
	a5	2.989028	1.000000	-	2.349733	2.794667	3.587317	2.881892
	a6	−0.003041	0.00000	-	−0.002139	−0.002667	−0.003902	−0.002973

).

$$k_2=\exp \left(-4.02-11.11\%\mathrm{C}-1.99\%\mathrm{Cr}+20.76\%\mathrm{Mo}-40.99\%\mathrm{Ni}\right)$$

(68)

$$n_1=6.23-2.25\%\mathrm{C}-0.81\%\mathrm{Cr}+5.17\%\mathrm{Mo}-8.83\%\mathrm{Ni}$$

(69)

$$k_3=106003024.20-51200507.88\%\mathrm{C}-19751485.43\%\mathrm{Cr}-256581910.00\%\mathrm{Mo}+25605277.23\%\mathrm{Ni}$$

(70)

$$k_4=\exp \left(-33.63+61.93\%\mathrm{C}-15.96\%\mathrm{Cr}+6.20\%\mathrm{Mo}-3.60\%\mathrm{Ni}\right)$$

(71)

$$k_5=97606.14-488262.55\%\mathrm{C}+134064.43\%\mathrm{Cr}+28612.33\%\mathrm{Mo}+29497.32\%\mathrm{Ni}$$

(72)

$$k_6=190361.38-288009.61\%\mathrm{C}-76052.87\%\mathrm{Cr}-693123.59\%\mathrm{Mo}+78021.58\%\mathrm{Ni}$$

(73)

$$k_7=\exp \left(10.37+24.69\%\mathrm{C}+1.24\%\mathrm{Cr}-9.98\%\mathrm{Mo}+2.63\%\mathrm{Ni}\right)$$

(74)

$$k_8=\exp \left(81.35-73.62\%\mathrm{C}-16.43\%\mathrm{Cr}-127.88\%\mathrm{Mo}+14.57\%\mathrm{Ni}\right)$$

(75)

$$n_{\mathrm{B}}=1.57+1.74\%\mathrm{C}-0.47\%\mathrm{Cr}-1.79\%\mathrm{Mo}+0.16\%\mathrm{Ni}$$

(76)

$$a_5=-1.0130+3.7800\%\mathrm{C}+1.8340\%\mathrm{Cr}+4.2012\%\mathrm{Mo}+0.4655\%\mathrm{Ni}$$

(77)

$$a_6=0.003525-0.006847\%\mathrm{C}-0.002777\%\mathrm{Cr}-0.006653\%\mathrm{Mo}-0.001100\%\mathrm{Ni}$$

(78)

4. Discussion

The values of kinetic parameters given in Table 2 were verified by comparing the modeled curves of the isothermal transformation (IT) diagram of steel 42CrMo4, 36Cr6, Ck45, and 28NiCrMo74 with those obtained experimentally. In Figure 2, Figure 3, Figure 4 and Figure 5, the dashed lines show the experimental IT diagram, while the mathematically determined times of start (incubation time) and times of finish of the isothermal austenite decomposition, t_0.01 and t_0.99, are shown by solid lines. Additionally, Figure 2 shows curves corresponding to bainite volume fraction of 25%, 50%, 75% and 90%.

The times of start and finish of isothermal decomposition of austenite were calculated based on Equations (30), (48) and (64), and the known values of the kinetic parameters.

Kinetic parameters of ferrite, bainite and pearlite transformation, k_F, k_B and k_P, were calculated by Equations (12), (45) and (53), respectively. Other physical quantities used in the developed mathematical model are shown in

Table 3. Physical quantities used in modelling of austenite isothermal decomposition.

Quantity Value	Units	Description
D0 = 2.3 × 10−5	m2 s−1	Material constant
Qdif = 1.48 × 105	J mol−1	Diffusion activation energy
R = 8.314	J mol−1 K−1	Universal gas constant
S = 170153	m−1	Surface of austenite grain suitable for nucleation
cγ = 1186.661 exp(−7.2834 × 10−3 T)	wt.% C	Concentration of austenite
cα = 0.1592−1.3423 × 10−4 T	wt.% C	Concentration of ferrite
cγα = 9.6782−8.82 × 10−3 T	wt.% C	Concentration of austenite at the boundary with ferrite
cγFe3C = −0.5248 + 1.28 × 10−3 T	wt.% C	Concentration of austenite at the boundary with cementite

.

Figure 2, Figure 3, Figure 4 and Figure 5 show that differences between times of transformations in experimentally and mathematically determined IT diagrams are not relevant. Therefore, it is seen that the kinetic parameters involved in an established mathematical model of ferrite, pearlite and bainite transformation can be successfully determined on the basis of Equations (68)–(78) with high accuracy. Developed model avoids the use of simple empirical expressions in predictions of isothermal decomposition of austenite.

Since the developed model is written in incremental form, it is suitable for predicting austenite decomposition during the continuous cooling of steel using Scheil′s additivity rule. Additionally, it is very easy to extend this approach in the prediction of the kinetics of austenite decomposition for other types of steel.

5. Conclusions

In this paper, the equations for the estimation of microstructure constituents’ volume fractions after the isothermal decomposition of austenite have been proposed. Isothermal decomposition of austenite implies quenching of steel from the austenite range to the temperature of isothermal transformation where all austenite decomposes at a constant temperature.

The inversion methods for the calculation of characteristic variables in the mathematical model of kinetics of austenite decomposition were developed.

The mathematical model was verified by the comparison of experimentally and mathematically determined IT diagrams of steel. It can be concluded that characteristic parameters included in the mathematical model of ferrite, pearlite and bainite transformation can be successfully evaluated by the proposed method.