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Progress in Modeling of Phase Transformation Kinetics
Available online at www.sciencedirect.com OCIENCE
d)
DIRECT.
JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2006, 1 3 ( 3 ) : 68-73
Progress in Modeling of Phase Transformation Kinetics ZHAO Hong-zhuang ,
LIU Xiang-hua,
WANG Guo-dong
(State Key Lab of Rolling and Automation, Northeastern University, Shenyang 110004, Liaoning. China) Abstract: Several methods representing the evolution of microstructure were introduced, which include the John-
son-Mehl-Avrami-Kolmogorov (JMAK) equation, Internal State Variable (ISV) framework, Koistinen-Marburger (K-M) equation, modified Magee’s rule and phase field model, etc. By combining calculation of martensite transformation kinetics, considering the selection of parameters with the effect of austenite grain size ( AGS) , some suitable ways of obtaining better results have been proposed. Key words: phase tr*ansformation: kinetics: modeling; numerical simulation
In recent decades, numerical simulation methods have been rapidly developed, and widely used to analyze the phase transformation kinetics, to predict and control the microstructure and mechanical properties and to design new materials more systematically. In fact, phase transformation is very complex; so it is difficult to calculate different phase fractions, material properties, etc. IQ order t o get reasonable results, models must be developed firstly. And the selection of parameters in the kinetics equations and other factors considered exert a great effect on the calculated results. For example, in the K-M equation which is widely used in martensite transformation, the parameter a is usually considered as a constant, from which the volume fraction calculation result differs obviously frorh the experimental one. Especially at the beginning, the volume fraction deviation approaches 0. 2. If the parameter a is fitted as a function of temperature, one can obtain more reasonable results. Therefore, a few methods were briefly reviewed and some discussions about the models were undertaken hereinafter.
1 Transformation Kinetics Models 1 . 1 JMAK equation The JMAK equation which has been presented by Johnson, Mehe, Avrami, and Kolmogorov separately is one of the most popular equations of kinet-
ics. Avrami equation and Johnson-Mehl ( J M ) equation or Johnson-Mehl-Avrami ( J M A ) equation may be classified into the same kind of JMAK equa- 171 It could be used to evaluate the volume fractions of austenite, pearlite, and bainite, etc. The JMAK equation“’] has been used widely for describing kinetics of phase transformation. When the growth rate is assumed to be a constant in all directions and the nuclei p grow as regular spheres in the matrix a, the transformed volume fraction ED
.
where I is nucleation rate, i. e. number of nuclei per unit volume per second; k is radial growth rate; and r is time since the start of the transformation. Kinetics of two-dimensional growth is
The transformed areal fraction described in Eqn. ( 2 ) is time dependent; k is the constant radial growth rate, I is the nucleation rate (number of nuclei per unit area per second). Kinetics of one-dimensional growth is (3) c a ( r )= 1 - exp( - Zkr2) I n Eqn. ( 3 ) , the nucleation rate Z has one dimensional “number of nuclei per unit length per second”.
Foundation 1tem:Item Sponsored by State High Technology Research and Development Program (863 Plan) of China (2001AA332020) Revised Date: June 16, 2005 Biography:ZHAO Hong-~huang(l966-), Male, Doctor Candidate$ E-mail: zhzl966@sohu. corn;
69
Progress in Modeling of Phase Transformation Kinetics
No. 3
T h e isothermal phase transformation behavior described by the JMA equation is [=l-exp(--K T") (4) where [ and T represent transformed volume fraction and time, respectively; K is rate parameter, a function of chemical composition and original grain size; n is reaction parameter depending on the growth geometry and the type of nucleation site and logical change. T h e values of K and n are obtained by fitting experimental dataclgl. UmemetoCZn1 developed the JMA equation by considering the original grain size a s follows :
(5) I
I
where d is original grain size; m is empirical constant. When the original grain size effect is neglected, Eqn. ( 5 ) is simplified t o Eqn. (4). T h e rule of additivity must be applied for continuous transformation. This method is based on isothermal transformation incubation period calculation to decide the onset of transformation. T h e incubation rate is the ratio of the time t , at the transformation temperature and the incubation period. T h e transformation starts when the accumulation of incubation rate becomes 1.
At,--1
c"
(6)
'fT,
r=l
T h e rule of additivity is suitable for the kinetics calculation of the same
1 . 2 ISV framework Lusk et alcZ1Ireviewed internal state variable framework. They have previously developed on the 1st international conference on thermal process modeling and computer simulation held in Shanghai, China. ISV method used the following equations to obtain the volume fraction of each phase for arbitrary cooling profiles. @A = 1-@F -@p -@B (7) &F = v F
( T>PF ( 1
@F],
&B=vB
T)
@A [ c P F . ~ (~ ~ ~~
>'-I
(8)
@F(O)=@FO
&p=Vp
(l-@p)'-'@A,
@P(o)=@W
>"' ( 1-@B )'-'@A ( TI -@I31 @B (0) = @m
( T >(@B
bon content in mass percent; and & is derivative with respect to time. T h e diffusive phase mobilities ( v F t 'up, v B > are functions of temperature while the martensite mobility is a function of carbon. T h e martensite start temperature is given a s M , , equilibrium volume fraction of ferrite a s QB+ sIas,s and parameters OeBand QeMrefer t o the enhanced nucleation of bainite and martensite due t o the presence of previously formed product phases. Parameters in the mobility and exponent functions a r e fitted with combinations of isothermal and continuous cooling dilatometry data. Lusk et allzll also intended to add a tempering equation t o this kinetics set.
(9)
+@eB
COB. sfasis 0 , T>M, d @ -~ VM (c> (OM +Q~M> e M ( c ) (I-DM (l-@M-@F-@p-@B)
9
z,,
-1 s
0.4
-Experimental result
)pM(')-l
0
(11)
50
TGM,
where @ is phase volume fraction; subscripts A , F, P, B, and M refer to austenite, ferrite, pearlite, bainite, and martensite; T is temperature; C is car-
Fig. 1
0
o a=a, x T+a2
(10)
9
m-1
1 . 3 Equations for nondiffusible transformation 1. 3. 1 K-M equation For the nondiffusional transformation, such a s martensitic transformation, the Koistinen-Marburger kinetic equationCz0+22.231 may be used a s follows: 2, = ( 1- 2, -Zr ) { 1- exp[a(M, - 2-31} ( 12) where z b, are proportion of phase martensite, bainite, and ferrite, respectively; M, is beginning temperature of the martensitic transformation; T the temperature, and a is parameter obtained by fitting experimental data. For single martensite transformation process, Zb=ZI=O, and 2, = 1 -exp[-u(M, - T I ] . Fig. 1 shows the experimental and calculation results of martensite transformation in a 1080 carbon steel, where M,=200 "C. When the parameter a is considered a s a constant (a=O. 02) , there is a big difference between the calculated and the experimental re-
a=0.020
100 150 Temperature/ e
20
Comparison between measured and calculated martensite transformation kinetics
Journal of Iron and Steel Research, International
70
sults. Especially at the beginning, the largest difference between the results approaches 0 . 2 . When a is fitted a s a function of temperature, i. e. a= a l X T + a z , the calculated result fits in with the experimental one better. T h e parameters a ] , az are fitted in with experimental results. In this condi1 9 , az=-O. 014 6. It tion concerned, a l =-0.000 is obvious that when a is considered a s a function of temperature, the transformation kinetics curve is “S” shape which frequently appeared in most phase transformation processes, and the calculated result fits the experimental data better. 1. 3. 2 M o d i f i e d Zener-Hillert equation T h e K-M equation has been widely used, but it has not considered the AGS effects. By introducing the AGS item, the following Zener-Hillert equation could be obtainedcz4].
where V, is martensite volume fraction; T is the cooling temperature; G is ASTM grain size number; a1 , a2 , a 3 , a 4 , a5 , a6 are optimum parameters fitted by experiments. 1. 3. 3 M o d i f i e d Magee’s rule One of the other methods for the diffusionless transformation is the modified Magee’s rulecz5’. (14) where is volume fraction of martensite which is the function of carbon content, temperature, and stress; J z is second invariant of deviatory stress; +h2, g 3 , (0, are all coefficients obtained from experiments.
cM
ture. T h e sharp-interface theory couples a postulate of configurational force balance t o an equation for substructure evolution. T h e phase field regularization of this model is based on the balance of two microforce systems. A link has been made between this general framework and a single Ginzburg-Landau equation. T h e latter is associated with a doublewell energy that has uneven energy wells. T h e association between the phase field and sharp-interface theories has been made. Jou and Lusk have offered numerical implementation and steady shape solutions to the phase field method.
1.5
Phase field model
Other methods
There are some other microstructure formation models that could be used for kinetics problems. These models include Monte-Carlo simulation method, molecular kinetic method, master equation ( a kinetic expression of the Monte-Carlo method), and phenomenological equation of time-evolution, etc. Monte Carlo Simulation Method is another important method, which has been performed in cases of three-, two- and one-dimensional nucleation and growthCL8].I t is a probabilistic analysis, and could not determine the causality of the phenomenon, so that there is no restriction o n the size scale of calculation space. Therefore, the Monte-Carlo model could be used t o all phenomena from the atomic scale t o the macroscopic optical scale, such a s the atomic scale structural change in phase transformation, macroscopic grain growth, and so on“”.
2 2.1
1.4
Vol. 13
Applications NCMS program
T h e software DANTEC13281 used the ISV model. D A N T E is a collaborative project which had been Phase field method is one of the four methods of conducted by the National Center for Manufacturing the phenomenological equation of time-evolution. Sciences (NCMS) in the US. T h e NCMS program is T h e other three methods are: Cahn-Hilliard nonlina collaboration among a group of companies, nationear diffusion equation, Khachaturyan‘ s diffusion eal research laboratories, and universities. T h e aim quation, and Time-dependent Ginzburge-Landau eof this project is t o develop a methodology that will quation. Phase field method offers a strong simulation tool in the area of phase t r a n ~ f o r m a t i o n ~ ~ ~ *predict ~ ~ ’ . component response ( distortion, residual stress, microstructure, etc. ) t o heat treatment. Jou and LuskCZ1introduced this method and compared JMAK kinetics with a phase field model for 2 . 2 HEARTS program microstructural evolution driven by substructure enAnother program, HEARTS ( HEAt tReaTergy. ment Simulation system) has been developed, which The micromechanical model is based on sharpmay be applied to heat treatment problems. HEARinterface and phase field theories. This method is TS was developed by Prof. Inoue and his colused to model phase transformations driven by the leagues. This program applied the modified Magee’s reduction of energy stored in the material substruc-
No. 3
Progress in Modeling of Phase Transformation Kinetics
ruleCz5'to treat non-diffusion type transformation, and the JMAK equation to treat diffusion-type transformation, which is controlled by temperature history and temperature itself. HEARTS has been used to simulate thermal variation, steel transformation products, and associated deformation of parts during heating, quenching, induction heat treatment, and other processes. Reasonable agreement was obtained"'.
2.3 Others There are many other kinds of method to describe phase transformations or the analogues of the above equations. Here only a few examples were offered. TszengCZg1 provides another version of JMAK equation to solve bainite transformation problems. Avrami's equation has been used to reproduce kinetics of pearlite-to-austenite transformation in a eutectoid steel during continuous heatingC3']. Capdevila C et alC3l1utilized the framework of JMA heterogeneous transformation kinetics theory to solve austenite-to-allotriomorphic ferrite transformation problems. Since the austenite decomposition products at temperatures above and below Ael are different, two different mathematical models have been proposed for the isothermal austenite decomposition in allotriomorphic ferrite. The extension of the JMAK theory for overall transformation kinetics has been adapted to deal with the simultaneous precipitation of many phases[3.7-321
A software called Micress ( MICRostructure Evolution Simulation Software)C251 has been developed to simulate y-a transformation. This software applies the multicomponent multiphasefield model, which is based on the reduction of total free enthalpy. Geiger J et alCss1present a computer simulation model, named Cellular Automaton ( CA ) , which aims at investigating the behavior of normal grain coarsening in 2D that corresponds well to the described physical model.
3
Discussion
The JMAK equation, one of the most popular kinetic equations, has been applied to track volume fraction kinetics of diffusive transformations and recrystallizations phenomena, such as austenite-ferrite and pearlite structure changes and vice versa. This equation was derived from idealized micromechanical considerations under isothermal conditions. But most of the processes developed under nonisother-
71
ma1 conditions. This model has also served as a basis for considering phase transformations that occur under nonisothermal conditions through an appeal t o the rule of additivityC3']. Todinov gave an exact kinetics equation in his work"]. Some limitations of the JMAK equation for describing kinetics of phase transformation were demonstrated using probabilistic analysis and Monte Carlo simulations. Todinov pointed out that JMAK equation can be regarded as a special case of the exact equation when the volume ratios of the growing nuclei are small, and that the discrepancies in the predictions from the JMAK equation depend on the number of growing nuclei in the controlled volume and do not depend directly on nucleation and growth rate. Morales et alC1llmade some comments on the errors in the Avrami plots and showed a different alternative about the treatment of the uncertainties that cause deviations at the ends of the Avrami plots. T h e deviations at the end of the Avrami plots have been mainly associated with these factors: the non-random distribution of the heterogeneous nucleation sites, the overlapping of two or more simultaneous reactions, and the effects of the experimental uncertainties in the fraction transformed calculation. The ISV method can also be used to solve problem of the precipitation of ferrite, pearlite, bainite, and martensite simultaneously. The ISV tempering model developed by Lusk et a1 is able to provide quantitatively accurate predictions of strain relaxation in tempering stage I for a class of low alloy steels. It was found that the model parameters should depend on carbon concentration but not on other alloying In studying deformation-induced martensitic transformation and transformation-induced plasticity ( TRIP in steels, the kinetics of deformation-induced martensitic transformation has usually been treated as a function of strain. However, since the martensitic transformation is promoted essentially by the shearing stress, the theoretical treatment of transformation kinetics should be based on stress rather than strainC351. During the research and development of novel materials in the past decade years, phase transformation kinetics models have been used in the research of deformation induced transformation, TRIP, and heat treatment simulation process. TRIP steels have been getting more and more concerned
72
Journal
of Iron a n d Steel R e s e a r c h , International
for their excellent mechanical and deformation behavior and favourable energy absorption function. The main factor that affects TRIP effect is controlling retained austenite ( R A ) . The RA transformation kinetics, i. e. , transformation temperature, transformation process and transformation quantity, affect the mechanical properties of TRIP steel straightly. The factors that affect RA transformation kinetics include the austenite grain size (AGS) , the alloy element content, heat treatment process, stress/strain state, etc. It is necessary to constitute further RA transformation model. Gross works concerned with simulation on heat treatment process have been contributed by scholars and all the kinetics models mentioned above have been utilized. No universal model exits that is suitable for all transformation processes. T h e aforementioned equations must be selected or modified properly in treating different processes. A suitable model is the one that can obtain the most reasonable results for the undergoing transformations. The existing models must be modified and developed; meanwhile, new models that are suitable for the research of novel steels and new materials must be developed. References: lnoue Tatsuo, Funatani Kiyoshi, Totten George E. Process Modeling for Heat Treatment: Current Status and Future D e velopments [J]. Journal of Shanghai Jiao Tong University, 2000v E ( 5 ) : 14-25. Jou Herng-Jeng, Lusk Mark T. Comparison of Johnson-Mehlc21 Avrami-Kologoromov Kinetics With a PhaseField Model for Microstructural Evolution Driven by Substructure Energy [J]. Physical Review B, 1997, 55(13): 8114-8121. ~ 3 1 Robson J D, Bhadeshia H K D H. Modelling Precipitation S e quences in Power Plant Steels: Part I , Kinetic Theory [J]. Materials Science and Technology, 1997, 13: 631-639. Lee Chan-Woo, Uhm Sang-Ho, Kim Kyoung-Min, et al. Modc41 eling of Phase Transformation Kinetics in the Coarsened Grain HAZ of C-Mn Steel Weld Considering Prior Austenite Grain Size [J]. ISIJ International, 2001, 41(11): 1383-1388. LIU Zhuang. ZENG Pan. Development of Thermal Process and c51 Numerical Modeling [J]. Journal of Shanghai Jiao Tong University , 2000, E ( 5 ) : 42-48. C61 Gottstein Guenter, Marx V , Sebald R. Integral Recrystallization Modelling [J]. Journal of Shanghai Jiao Tong University, 2000, E ( 5 ) : 49-57. Jones S J , Bhadeshia H K D H. Kinetics of the Simultaneous [71 Decomposition of Austenite Into Several Transformation Products [J]. Acta Materialia. 1997, 45(7): 2911-2920. XU Zu-yao. Phase Transformation Theory [MI. Beijing: SciC81 ence Press, 2000 ( in Chinese). GAO Ning. Study on Coupled Modeling of Quenching Process Cgl and 3-Dimensional Numerical Simulation [D]. Beijing: Tsinghua University, 2000 (in Chinese).
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Mittemeijer E J I Van Gent A , Van Der Schaaf P J. Analysis of Transformation Kinetics by Ninisothermal Dilatometry [J]. Metallurgical Transactions, 1986, 17A: 1441-1445. Valencia Morales E , Vega 1, J , Villar C E. et al. Some Comments About the Errors in the Avrami Plots [J]. Scripta Materialia, 2005, 52: 217-219. Gomez M , Medina S F , Caruana G. Modelling of Phase Transformation Kinetics by Correction of Dilatometry Results for a Ferritic Nb-Microalloyed Steel [J]. ISIJ International, 2003, 43(8): 1228-1237. Elwazri A M , Essadiqi E , Yue S. T h e Kinetics of Static R e crystallization in Microalloyed Hypereutectoid Steels [J]. ISIJ International, 2004, 44 (1): 162-170. Schroder R. Influences on Development of Thermal and Residual Stresses in Quenched Steel Cylinders of Different Dimensions [J]. Materials Science and Technology, 1985, l ( 1 0 ) : 754-764. L U O Haiwen, Sietsma Jilt, Van Der Zwaag Sybrand. Effect of Inhomogeneous Deformation on the Recrystallization Kinetics of Deformed Metals [J]. ISIJ International, 2004, 44 (11): 1931-1936. Hawbolt E B, Chao B, Brimacombe J K. Kinetics of Austenite-Ferrite and Austenite-Pearlite Transformations in a 1025 Carbon Steel [J]. Metallurgical Transactions. 1985, 16A: 565-578. Roosz A , Gdcsi Z, Fuchs E G. Isothermal Formation of Austenite in Eutectoid Plain Carbon Steel [J]. Acta Metallurgica. 1983, 31(4): 509-517. Todinov M T. O n Some Limitations of the Johnson-MehlAvrami-Kolmogorov Equation [J]. Acta Metallurgica, 2000, (48): 4217-4224, I J U Zhuang, W U Zhao-ji, W U Jing-zhi, et al. Numerical Simulation of Heat Treatment Process [MI. Beijing: Science Press, 1996 (in Chinese). Lee Jyelong, Bhadeshia H K D H. A Methodology for the Prediction of TimeTemperature-Transformation Diagrams [J]. Materials Science and Engineering, 1993, A171: 223230. Lusk Mark T, Lee Young-Kook, Jou Herng-Jeng. et al. An Internal State Variable Model for the Low Temperature Tempering of Low Alloy Steels [J]. Journal of Shanghai Jiao Tong University, 2000, E ( 5 ) : 178-184. Vincent Y ,Jullien Jean-Francois. Fouquet F, et al. Weld Models Incorporating the H A Z Phase Transformation Effects, Comparison Between Experimental and Numerical Results [J]. Journal of Shanghai Jiao Tong University. 2000, E ( 5 ) : 107-113. Kundu Saurabh. Mukhopashyay Ananya. Chatterjee Sudin, et al. Modelling of Microstructure and Heat Transfer During Controlled Cooling of Low Carbon Wire Rod CJ]. ISIJ lnternational, 2004, 44(7): 1217-1223. Lee Seok-jae, Lee Young-kook. Effect of Austenite Grain Size on Martensitic Transformation of Low Alloy Steel [A]. Proceedings of the 5th Pacific Rim International Conference on Advanced Materials and ProceSsing [C]. Beijing: T h e Chinese Society for Metals, 2004. 3169-3172. J U Dong-Ying, Narazaki Michiharu, Kamisugi Hirofumi, et aL Computer Predictions and Experimental'Verification of R e sidual Stresses and Distortion in Carburizing-Quenching of Steel [J]. Journal of Shanghai Jiao Tong University, 2000, E (5): 165-172.
Progress in Modeling of P h a s e T r a n s f o r m a t i o n Kinetics
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Pariser Gerhard. Schaffnit Philippe, Steinbach Ingo, et al. Simulation of the )CO Transformation Using the PhaseField Method [J]. Steel Research, 2001, 72(9) : 354-360. 1,IU Chun-cheng, J U Dong-ying, lnoue Tatsuo. A Numerical Modeling of Metallo-Thermo-Mechanical Behavior in Both Carburized and Carbonitrided Quenching Processes [J]. ISIJ International, 2002. 42(10): 1125-1134. Lusk Mark T. Jou Herng-Jeng. On the Rule of Additivity in Phase Transformation Kinetics [J]. Metallurgical and Materials Transactions, 1997, 28A: 287-291. Tszeng T C. Autocatalysis in Bainite Transformation [J]. Materials Science and Engineering, 2000, A293: 185-190. Caballero F G , Capdevila C , Garcia de Andres C. Influence of Pearlite Morphology and Heating Rate on the Kinetics of Continuously Heated Austenite Formation in a Eutectoid Steel [J]. Metallurgical and Materials Transactions, 2001, 32A: 1283-1291.
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[32]
[33]
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Capdevila C, Caballero F G , Garcia de Andres C. Modeling of Kinetics of Austeniteto- Allotriomorphic Ferrite Transformation in 0. 37C1-1. 45Mn-0. 11V Microalloyed Steel [J]. Metallurgical and Materials Transactions, 2001, 32A: 661-669. Kasuya T, Ichikawa K , Fuji M , et al. Real and Extended Volumes in Simultaneous Transformations [J]. Materials Science and Technology, 1999, 15: 471-473. Geiger J , Roosz A , Barkoczy P. Simulation of Grain Coarsening in Two Dimensions by Cellular-Automation [J]. Acta Metallurgica, 2001, 49: 623-629. WU W T, Tang J P, 1-i G. Recent Development of Process Simulation and Its Application to Manufacturing Processes [J]. Journal of Shanghai Jiao Tong University, 2000, E(5) : 235-24 1. Tamura 1. Deformation-Induced Martensitic Transformation and Transformation-Induced Plasticity in Steels [J]. Metal Science, 1982, 16: 245-253.
(Continued from Page 60)
3 Conclusions (1) Silicon can raise the y / ( ( r + y ) phase boundary and accelerate the pro-eutectoid ferrite precipitation. ( 2 ) Silicon can inhibit the formation of cementite during bainite isothermal transformation, decrease the soluble carbon content in ferrite, and result in the increase of soluble carbon content in retained austenite. ( 3 ) In order to bring about TRIP effect, both thermomechanical processes and chemical compositions must be controlled strictly, and it is now possible to stabilize noticeable amounts of austenite at room temperature.
C31
c41 c51
C61
c71
References C8l [l]
C2]
Tersner G , Werner E A. Modelling of Retained Austenite in Low-Alloyed TRIP Steels [J]. JOM, 1997, 49(9): 62-65. Zarei H A , Yue S. Ferrite Formation Characteristics in Si-Mn
TRIP Steels [J]. ISIJ International, 1997, 37(6): 583-589. Tanaka T, Aaronson H I , Enomoto M. Calculation of a / y Phase Boundaries in F ~ C - X I - X Systems Z from the Central Atoms Model [J]. Metall Mater Trans. 1995, 26A(3): 535-545. HSU T Y, Mou Y W. Thermodynamics of the Bainitic Transformation in F e C Alloys [J]. Acta Metall, 1984. 32(9): 1469-1481. Miettinen J. Simple Semiempirical Model for Prediction of Austenite Decomposition and Related Heat Release During Cooling of Low Alloyed Steels [J]. lronmaking and Steelmaking, 1996, 23(4): 346-356. Ghosh G, Olson G B. Simulation of Para-equilibrium Growth in Multicomponent Systems [J]. Metall Mater Trans, 2001, 32A(3) : 455-467. Enomoto M , Aaronson H I. Calculation of o+y Phase Boundaries in FeC-X Systems From the Central Atoms Model [J]. Calphad, 1985, 9 ( 1 ) : 43-58. Tsukatani I, Hashimoto S, Inoue T. Effects of Silicon and Manganese Addition on Mechanical Properties of High-Strength Hot-Rolled Sheet Steel Containing Retained Austenite [J]. ISIJ International, 1991. 3 1 ( 9 ) : 992-1000.
d)
DIRECT.
JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2006, 1 3 ( 3 ) : 68-73
Progress in Modeling of Phase Transformation Kinetics ZHAO Hong-zhuang ,
LIU Xiang-hua,
WANG Guo-dong
(State Key Lab of Rolling and Automation, Northeastern University, Shenyang 110004, Liaoning. China) Abstract: Several methods representing the evolution of microstructure were introduced, which include the John-
son-Mehl-Avrami-Kolmogorov (JMAK) equation, Internal State Variable (ISV) framework, Koistinen-Marburger (K-M) equation, modified Magee’s rule and phase field model, etc. By combining calculation of martensite transformation kinetics, considering the selection of parameters with the effect of austenite grain size ( AGS) , some suitable ways of obtaining better results have been proposed. Key words: phase tr*ansformation: kinetics: modeling; numerical simulation
In recent decades, numerical simulation methods have been rapidly developed, and widely used to analyze the phase transformation kinetics, to predict and control the microstructure and mechanical properties and to design new materials more systematically. In fact, phase transformation is very complex; so it is difficult to calculate different phase fractions, material properties, etc. IQ order t o get reasonable results, models must be developed firstly. And the selection of parameters in the kinetics equations and other factors considered exert a great effect on the calculated results. For example, in the K-M equation which is widely used in martensite transformation, the parameter a is usually considered as a constant, from which the volume fraction calculation result differs obviously frorh the experimental one. Especially at the beginning, the volume fraction deviation approaches 0. 2. If the parameter a is fitted as a function of temperature, one can obtain more reasonable results. Therefore, a few methods were briefly reviewed and some discussions about the models were undertaken hereinafter.
1 Transformation Kinetics Models 1 . 1 JMAK equation The JMAK equation which has been presented by Johnson, Mehe, Avrami, and Kolmogorov separately is one of the most popular equations of kinet-
ics. Avrami equation and Johnson-Mehl ( J M ) equation or Johnson-Mehl-Avrami ( J M A ) equation may be classified into the same kind of JMAK equa- 171 It could be used to evaluate the volume fractions of austenite, pearlite, and bainite, etc. The JMAK equation“’] has been used widely for describing kinetics of phase transformation. When the growth rate is assumed to be a constant in all directions and the nuclei p grow as regular spheres in the matrix a, the transformed volume fraction ED
.
where I is nucleation rate, i. e. number of nuclei per unit volume per second; k is radial growth rate; and r is time since the start of the transformation. Kinetics of two-dimensional growth is
The transformed areal fraction described in Eqn. ( 2 ) is time dependent; k is the constant radial growth rate, I is the nucleation rate (number of nuclei per unit area per second). Kinetics of one-dimensional growth is (3) c a ( r )= 1 - exp( - Zkr2) I n Eqn. ( 3 ) , the nucleation rate Z has one dimensional “number of nuclei per unit length per second”.
Foundation 1tem:Item Sponsored by State High Technology Research and Development Program (863 Plan) of China (2001AA332020) Revised Date: June 16, 2005 Biography:ZHAO Hong-~huang(l966-), Male, Doctor Candidate$ E-mail: zhzl966@sohu. corn;
69
Progress in Modeling of Phase Transformation Kinetics
No. 3
T h e isothermal phase transformation behavior described by the JMA equation is [=l-exp(--K T") (4) where [ and T represent transformed volume fraction and time, respectively; K is rate parameter, a function of chemical composition and original grain size; n is reaction parameter depending on the growth geometry and the type of nucleation site and logical change. T h e values of K and n are obtained by fitting experimental dataclgl. UmemetoCZn1 developed the JMA equation by considering the original grain size a s follows :
(5) I
I
where d is original grain size; m is empirical constant. When the original grain size effect is neglected, Eqn. ( 5 ) is simplified t o Eqn. (4). T h e rule of additivity must be applied for continuous transformation. This method is based on isothermal transformation incubation period calculation to decide the onset of transformation. T h e incubation rate is the ratio of the time t , at the transformation temperature and the incubation period. T h e transformation starts when the accumulation of incubation rate becomes 1.
At,--1
c"
(6)
'fT,
r=l
T h e rule of additivity is suitable for the kinetics calculation of the same
1 . 2 ISV framework Lusk et alcZ1Ireviewed internal state variable framework. They have previously developed on the 1st international conference on thermal process modeling and computer simulation held in Shanghai, China. ISV method used the following equations to obtain the volume fraction of each phase for arbitrary cooling profiles. @A = 1-@F -@p -@B (7) &F = v F
( T>PF ( 1
@F],
&B=vB
T)
@A [ c P F . ~ (~ ~ ~~
>'-I
(8)
@F(O)=@FO
&p=Vp
(l-@p)'-'@A,
@P(o)=@W
>"' ( 1-@B )'-'@A ( TI -@I31 @B (0) = @m
( T >(@B
bon content in mass percent; and & is derivative with respect to time. T h e diffusive phase mobilities ( v F t 'up, v B > are functions of temperature while the martensite mobility is a function of carbon. T h e martensite start temperature is given a s M , , equilibrium volume fraction of ferrite a s QB+ sIas,s and parameters OeBand QeMrefer t o the enhanced nucleation of bainite and martensite due t o the presence of previously formed product phases. Parameters in the mobility and exponent functions a r e fitted with combinations of isothermal and continuous cooling dilatometry data. Lusk et allzll also intended to add a tempering equation t o this kinetics set.
(9)
+@eB
COB. sfasis 0 , T>M, d @ -~ VM (c> (OM +Q~M> e M ( c ) (I-DM (l-@M-@F-@p-@B)
9
z,,
-1 s
0.4
-Experimental result
)pM(')-l
0
(11)
50
TGM,
where @ is phase volume fraction; subscripts A , F, P, B, and M refer to austenite, ferrite, pearlite, bainite, and martensite; T is temperature; C is car-
Fig. 1
0
o a=a, x T+a2
(10)
9
m-1
1 . 3 Equations for nondiffusible transformation 1. 3. 1 K-M equation For the nondiffusional transformation, such a s martensitic transformation, the Koistinen-Marburger kinetic equationCz0+22.231 may be used a s follows: 2, = ( 1- 2, -Zr ) { 1- exp[a(M, - 2-31} ( 12) where z b, are proportion of phase martensite, bainite, and ferrite, respectively; M, is beginning temperature of the martensitic transformation; T the temperature, and a is parameter obtained by fitting experimental data. For single martensite transformation process, Zb=ZI=O, and 2, = 1 -exp[-u(M, - T I ] . Fig. 1 shows the experimental and calculation results of martensite transformation in a 1080 carbon steel, where M,=200 "C. When the parameter a is considered a s a constant (a=O. 02) , there is a big difference between the calculated and the experimental re-
a=0.020
100 150 Temperature/ e
20
Comparison between measured and calculated martensite transformation kinetics
Journal of Iron and Steel Research, International
70
sults. Especially at the beginning, the largest difference between the results approaches 0 . 2 . When a is fitted a s a function of temperature, i. e. a= a l X T + a z , the calculated result fits in with the experimental one better. T h e parameters a ] , az are fitted in with experimental results. In this condi1 9 , az=-O. 014 6. It tion concerned, a l =-0.000 is obvious that when a is considered a s a function of temperature, the transformation kinetics curve is “S” shape which frequently appeared in most phase transformation processes, and the calculated result fits the experimental data better. 1. 3. 2 M o d i f i e d Zener-Hillert equation T h e K-M equation has been widely used, but it has not considered the AGS effects. By introducing the AGS item, the following Zener-Hillert equation could be obtainedcz4].
where V, is martensite volume fraction; T is the cooling temperature; G is ASTM grain size number; a1 , a2 , a 3 , a 4 , a5 , a6 are optimum parameters fitted by experiments. 1. 3. 3 M o d i f i e d Magee’s rule One of the other methods for the diffusionless transformation is the modified Magee’s rulecz5’. (14) where is volume fraction of martensite which is the function of carbon content, temperature, and stress; J z is second invariant of deviatory stress; +h2, g 3 , (0, are all coefficients obtained from experiments.
cM
ture. T h e sharp-interface theory couples a postulate of configurational force balance t o an equation for substructure evolution. T h e phase field regularization of this model is based on the balance of two microforce systems. A link has been made between this general framework and a single Ginzburg-Landau equation. T h e latter is associated with a doublewell energy that has uneven energy wells. T h e association between the phase field and sharp-interface theories has been made. Jou and Lusk have offered numerical implementation and steady shape solutions to the phase field method.
1.5
Phase field model
Other methods
There are some other microstructure formation models that could be used for kinetics problems. These models include Monte-Carlo simulation method, molecular kinetic method, master equation ( a kinetic expression of the Monte-Carlo method), and phenomenological equation of time-evolution, etc. Monte Carlo Simulation Method is another important method, which has been performed in cases of three-, two- and one-dimensional nucleation and growthCL8].I t is a probabilistic analysis, and could not determine the causality of the phenomenon, so that there is no restriction o n the size scale of calculation space. Therefore, the Monte-Carlo model could be used t o all phenomena from the atomic scale t o the macroscopic optical scale, such a s the atomic scale structural change in phase transformation, macroscopic grain growth, and so on“”.
2 2.1
1.4
Vol. 13
Applications NCMS program
T h e software DANTEC13281 used the ISV model. D A N T E is a collaborative project which had been Phase field method is one of the four methods of conducted by the National Center for Manufacturing the phenomenological equation of time-evolution. Sciences (NCMS) in the US. T h e NCMS program is T h e other three methods are: Cahn-Hilliard nonlina collaboration among a group of companies, nationear diffusion equation, Khachaturyan‘ s diffusion eal research laboratories, and universities. T h e aim quation, and Time-dependent Ginzburge-Landau eof this project is t o develop a methodology that will quation. Phase field method offers a strong simulation tool in the area of phase t r a n ~ f o r m a t i o n ~ ~ ~ *predict ~ ~ ’ . component response ( distortion, residual stress, microstructure, etc. ) t o heat treatment. Jou and LuskCZ1introduced this method and compared JMAK kinetics with a phase field model for 2 . 2 HEARTS program microstructural evolution driven by substructure enAnother program, HEARTS ( HEAt tReaTergy. ment Simulation system) has been developed, which The micromechanical model is based on sharpmay be applied to heat treatment problems. HEARinterface and phase field theories. This method is TS was developed by Prof. Inoue and his colused to model phase transformations driven by the leagues. This program applied the modified Magee’s reduction of energy stored in the material substruc-
No. 3
Progress in Modeling of Phase Transformation Kinetics
ruleCz5'to treat non-diffusion type transformation, and the JMAK equation to treat diffusion-type transformation, which is controlled by temperature history and temperature itself. HEARTS has been used to simulate thermal variation, steel transformation products, and associated deformation of parts during heating, quenching, induction heat treatment, and other processes. Reasonable agreement was obtained"'.
2.3 Others There are many other kinds of method to describe phase transformations or the analogues of the above equations. Here only a few examples were offered. TszengCZg1 provides another version of JMAK equation to solve bainite transformation problems. Avrami's equation has been used to reproduce kinetics of pearlite-to-austenite transformation in a eutectoid steel during continuous heatingC3']. Capdevila C et alC3l1utilized the framework of JMA heterogeneous transformation kinetics theory to solve austenite-to-allotriomorphic ferrite transformation problems. Since the austenite decomposition products at temperatures above and below Ael are different, two different mathematical models have been proposed for the isothermal austenite decomposition in allotriomorphic ferrite. The extension of the JMAK theory for overall transformation kinetics has been adapted to deal with the simultaneous precipitation of many phases[3.7-321
A software called Micress ( MICRostructure Evolution Simulation Software)C251 has been developed to simulate y-a transformation. This software applies the multicomponent multiphasefield model, which is based on the reduction of total free enthalpy. Geiger J et alCss1present a computer simulation model, named Cellular Automaton ( CA ) , which aims at investigating the behavior of normal grain coarsening in 2D that corresponds well to the described physical model.
3
Discussion
The JMAK equation, one of the most popular kinetic equations, has been applied to track volume fraction kinetics of diffusive transformations and recrystallizations phenomena, such as austenite-ferrite and pearlite structure changes and vice versa. This equation was derived from idealized micromechanical considerations under isothermal conditions. But most of the processes developed under nonisother-
71
ma1 conditions. This model has also served as a basis for considering phase transformations that occur under nonisothermal conditions through an appeal t o the rule of additivityC3']. Todinov gave an exact kinetics equation in his work"]. Some limitations of the JMAK equation for describing kinetics of phase transformation were demonstrated using probabilistic analysis and Monte Carlo simulations. Todinov pointed out that JMAK equation can be regarded as a special case of the exact equation when the volume ratios of the growing nuclei are small, and that the discrepancies in the predictions from the JMAK equation depend on the number of growing nuclei in the controlled volume and do not depend directly on nucleation and growth rate. Morales et alC1llmade some comments on the errors in the Avrami plots and showed a different alternative about the treatment of the uncertainties that cause deviations at the ends of the Avrami plots. T h e deviations at the end of the Avrami plots have been mainly associated with these factors: the non-random distribution of the heterogeneous nucleation sites, the overlapping of two or more simultaneous reactions, and the effects of the experimental uncertainties in the fraction transformed calculation. The ISV method can also be used to solve problem of the precipitation of ferrite, pearlite, bainite, and martensite simultaneously. The ISV tempering model developed by Lusk et a1 is able to provide quantitatively accurate predictions of strain relaxation in tempering stage I for a class of low alloy steels. It was found that the model parameters should depend on carbon concentration but not on other alloying In studying deformation-induced martensitic transformation and transformation-induced plasticity ( TRIP in steels, the kinetics of deformation-induced martensitic transformation has usually been treated as a function of strain. However, since the martensitic transformation is promoted essentially by the shearing stress, the theoretical treatment of transformation kinetics should be based on stress rather than strainC351. During the research and development of novel materials in the past decade years, phase transformation kinetics models have been used in the research of deformation induced transformation, TRIP, and heat treatment simulation process. TRIP steels have been getting more and more concerned
72
Journal
of Iron a n d Steel R e s e a r c h , International
for their excellent mechanical and deformation behavior and favourable energy absorption function. The main factor that affects TRIP effect is controlling retained austenite ( R A ) . The RA transformation kinetics, i. e. , transformation temperature, transformation process and transformation quantity, affect the mechanical properties of TRIP steel straightly. The factors that affect RA transformation kinetics include the austenite grain size (AGS) , the alloy element content, heat treatment process, stress/strain state, etc. It is necessary to constitute further RA transformation model. Gross works concerned with simulation on heat treatment process have been contributed by scholars and all the kinetics models mentioned above have been utilized. No universal model exits that is suitable for all transformation processes. T h e aforementioned equations must be selected or modified properly in treating different processes. A suitable model is the one that can obtain the most reasonable results for the undergoing transformations. The existing models must be modified and developed; meanwhile, new models that are suitable for the research of novel steels and new materials must be developed. References: lnoue Tatsuo, Funatani Kiyoshi, Totten George E. Process Modeling for Heat Treatment: Current Status and Future D e velopments [J]. Journal of Shanghai Jiao Tong University, 2000v E ( 5 ) : 14-25. Jou Herng-Jeng, Lusk Mark T. Comparison of Johnson-Mehlc21 Avrami-Kologoromov Kinetics With a PhaseField Model for Microstructural Evolution Driven by Substructure Energy [J]. Physical Review B, 1997, 55(13): 8114-8121. ~ 3 1 Robson J D, Bhadeshia H K D H. Modelling Precipitation S e quences in Power Plant Steels: Part I , Kinetic Theory [J]. Materials Science and Technology, 1997, 13: 631-639. Lee Chan-Woo, Uhm Sang-Ho, Kim Kyoung-Min, et al. Modc41 eling of Phase Transformation Kinetics in the Coarsened Grain HAZ of C-Mn Steel Weld Considering Prior Austenite Grain Size [J]. ISIJ International, 2001, 41(11): 1383-1388. LIU Zhuang. ZENG Pan. Development of Thermal Process and c51 Numerical Modeling [J]. Journal of Shanghai Jiao Tong University , 2000, E ( 5 ) : 42-48. C61 Gottstein Guenter, Marx V , Sebald R. Integral Recrystallization Modelling [J]. Journal of Shanghai Jiao Tong University, 2000, E ( 5 ) : 49-57. Jones S J , Bhadeshia H K D H. Kinetics of the Simultaneous [71 Decomposition of Austenite Into Several Transformation Products [J]. Acta Materialia. 1997, 45(7): 2911-2920. XU Zu-yao. Phase Transformation Theory [MI. Beijing: SciC81 ence Press, 2000 ( in Chinese). GAO Ning. Study on Coupled Modeling of Quenching Process Cgl and 3-Dimensional Numerical Simulation [D]. Beijing: Tsinghua University, 2000 (in Chinese).
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Mittemeijer E J I Van Gent A , Van Der Schaaf P J. Analysis of Transformation Kinetics by Ninisothermal Dilatometry [J]. Metallurgical Transactions, 1986, 17A: 1441-1445. Valencia Morales E , Vega 1, J , Villar C E. et al. Some Comments About the Errors in the Avrami Plots [J]. Scripta Materialia, 2005, 52: 217-219. Gomez M , Medina S F , Caruana G. Modelling of Phase Transformation Kinetics by Correction of Dilatometry Results for a Ferritic Nb-Microalloyed Steel [J]. ISIJ International, 2003, 43(8): 1228-1237. Elwazri A M , Essadiqi E , Yue S. T h e Kinetics of Static R e crystallization in Microalloyed Hypereutectoid Steels [J]. ISIJ International, 2004, 44 (1): 162-170. Schroder R. Influences on Development of Thermal and Residual Stresses in Quenched Steel Cylinders of Different Dimensions [J]. Materials Science and Technology, 1985, l ( 1 0 ) : 754-764. L U O Haiwen, Sietsma Jilt, Van Der Zwaag Sybrand. Effect of Inhomogeneous Deformation on the Recrystallization Kinetics of Deformed Metals [J]. ISIJ International, 2004, 44 (11): 1931-1936. Hawbolt E B, Chao B, Brimacombe J K. Kinetics of Austenite-Ferrite and Austenite-Pearlite Transformations in a 1025 Carbon Steel [J]. Metallurgical Transactions. 1985, 16A: 565-578. Roosz A , Gdcsi Z, Fuchs E G. Isothermal Formation of Austenite in Eutectoid Plain Carbon Steel [J]. Acta Metallurgica. 1983, 31(4): 509-517. Todinov M T. O n Some Limitations of the Johnson-MehlAvrami-Kolmogorov Equation [J]. Acta Metallurgica, 2000, (48): 4217-4224, I J U Zhuang, W U Zhao-ji, W U Jing-zhi, et al. Numerical Simulation of Heat Treatment Process [MI. Beijing: Science Press, 1996 (in Chinese). Lee Jyelong, Bhadeshia H K D H. A Methodology for the Prediction of TimeTemperature-Transformation Diagrams [J]. Materials Science and Engineering, 1993, A171: 223230. Lusk Mark T, Lee Young-Kook, Jou Herng-Jeng. et al. An Internal State Variable Model for the Low Temperature Tempering of Low Alloy Steels [J]. Journal of Shanghai Jiao Tong University, 2000, E ( 5 ) : 178-184. Vincent Y ,Jullien Jean-Francois. Fouquet F, et al. Weld Models Incorporating the H A Z Phase Transformation Effects, Comparison Between Experimental and Numerical Results [J]. Journal of Shanghai Jiao Tong University. 2000, E ( 5 ) : 107-113. Kundu Saurabh. Mukhopashyay Ananya. Chatterjee Sudin, et al. Modelling of Microstructure and Heat Transfer During Controlled Cooling of Low Carbon Wire Rod CJ]. ISIJ lnternational, 2004, 44(7): 1217-1223. Lee Seok-jae, Lee Young-kook. Effect of Austenite Grain Size on Martensitic Transformation of Low Alloy Steel [A]. Proceedings of the 5th Pacific Rim International Conference on Advanced Materials and ProceSsing [C]. Beijing: T h e Chinese Society for Metals, 2004. 3169-3172. J U Dong-Ying, Narazaki Michiharu, Kamisugi Hirofumi, et aL Computer Predictions and Experimental'Verification of R e sidual Stresses and Distortion in Carburizing-Quenching of Steel [J]. Journal of Shanghai Jiao Tong University, 2000, E (5): 165-172.
Progress in Modeling of P h a s e T r a n s f o r m a t i o n Kinetics
No. 3
Pariser Gerhard. Schaffnit Philippe, Steinbach Ingo, et al. Simulation of the )CO Transformation Using the PhaseField Method [J]. Steel Research, 2001, 72(9) : 354-360. 1,IU Chun-cheng, J U Dong-ying, lnoue Tatsuo. A Numerical Modeling of Metallo-Thermo-Mechanical Behavior in Both Carburized and Carbonitrided Quenching Processes [J]. ISIJ International, 2002. 42(10): 1125-1134. Lusk Mark T. Jou Herng-Jeng. On the Rule of Additivity in Phase Transformation Kinetics [J]. Metallurgical and Materials Transactions, 1997, 28A: 287-291. Tszeng T C. Autocatalysis in Bainite Transformation [J]. Materials Science and Engineering, 2000, A293: 185-190. Caballero F G , Capdevila C , Garcia de Andres C. Influence of Pearlite Morphology and Heating Rate on the Kinetics of Continuously Heated Austenite Formation in a Eutectoid Steel [J]. Metallurgical and Materials Transactions, 2001, 32A: 1283-1291.
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[32]
[33]
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Capdevila C, Caballero F G , Garcia de Andres C. Modeling of Kinetics of Austeniteto- Allotriomorphic Ferrite Transformation in 0. 37C1-1. 45Mn-0. 11V Microalloyed Steel [J]. Metallurgical and Materials Transactions, 2001, 32A: 661-669. Kasuya T, Ichikawa K , Fuji M , et al. Real and Extended Volumes in Simultaneous Transformations [J]. Materials Science and Technology, 1999, 15: 471-473. Geiger J , Roosz A , Barkoczy P. Simulation of Grain Coarsening in Two Dimensions by Cellular-Automation [J]. Acta Metallurgica, 2001, 49: 623-629. WU W T, Tang J P, 1-i G. Recent Development of Process Simulation and Its Application to Manufacturing Processes [J]. Journal of Shanghai Jiao Tong University, 2000, E(5) : 235-24 1. Tamura 1. Deformation-Induced Martensitic Transformation and Transformation-Induced Plasticity in Steels [J]. Metal Science, 1982, 16: 245-253.
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3 Conclusions (1) Silicon can raise the y / ( ( r + y ) phase boundary and accelerate the pro-eutectoid ferrite precipitation. ( 2 ) Silicon can inhibit the formation of cementite during bainite isothermal transformation, decrease the soluble carbon content in ferrite, and result in the increase of soluble carbon content in retained austenite. ( 3 ) In order to bring about TRIP effect, both thermomechanical processes and chemical compositions must be controlled strictly, and it is now possible to stabilize noticeable amounts of austenite at room temperature.
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c41 c51
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C2]
Tersner G , Werner E A. Modelling of Retained Austenite in Low-Alloyed TRIP Steels [J]. JOM, 1997, 49(9): 62-65. Zarei H A , Yue S. Ferrite Formation Characteristics in Si-Mn
TRIP Steels [J]. ISIJ International, 1997, 37(6): 583-589. Tanaka T, Aaronson H I , Enomoto M. Calculation of a / y Phase Boundaries in F ~ C - X I - X Systems Z from the Central Atoms Model [J]. Metall Mater Trans. 1995, 26A(3): 535-545. HSU T Y, Mou Y W. Thermodynamics of the Bainitic Transformation in F e C Alloys [J]. Acta Metall, 1984. 32(9): 1469-1481. Miettinen J. Simple Semiempirical Model for Prediction of Austenite Decomposition and Related Heat Release During Cooling of Low Alloyed Steels [J]. lronmaking and Steelmaking, 1996, 23(4): 346-356. Ghosh G, Olson G B. Simulation of Para-equilibrium Growth in Multicomponent Systems [J]. Metall Mater Trans, 2001, 32A(3) : 455-467. Enomoto M , Aaronson H I. Calculation of o+y Phase Boundaries in FeC-X Systems From the Central Atoms Model [J]. Calphad, 1985, 9 ( 1 ) : 43-58. Tsukatani I, Hashimoto S, Inoue T. Effects of Silicon and Manganese Addition on Mechanical Properties of High-Strength Hot-Rolled Sheet Steel Containing Retained Austenite [J]. ISIJ International, 1991. 3 1 ( 9 ) : 992-1000.