Based on the research results, coefficients were determined in constitutive equations, describing the kinetics of dynamic recrystallization in high-carbon bainitic steel during hot deformation. The developed mathematical model takes into account the dependence of changing kinetics in the size evolution of the initial austenite grains, the value of strain, strain rate, temperature and time. Physical simulations were carried out on rectangular specimens measuring 10 × 15 × 20 mm. Compression tests with a plane state of deformation were carried out using a Gleeble 3800

B. Garbarz

M. Knapinski

T. Dembiczak

English

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107METALURGIJA 56 (2017) 1-2, 107-110T. DEMBICZAK, M. KNAPINSKI, B. GARBARZ MATHEMATICAL MODELING OF PHENOMENA OF DYNAMIC RECRYSTALLIZATION DURING HOT PLASTIC DEFORMATION IN HIGH-CARBON BAINITIC STEELReceived – Primljeno: 2016-05-21Accepted – Prihvaćeno: 2016-09-25Original Scientific Paper – Izvorni znanstveni radT. Dembiczak, Jan Dlugosz University in Czestochowa, PolandM. Knapinski, Faculty of Production Engineering and Materials Tech-nology, Czestochowa University of Technology, Czestochowa, PolandB. Garbarz, Institute for Ferrous Metallurgy, Gliwice, PolandBased on the research results, coefficients were determined in constitutive equations, describing the kinetics of dynamic recrystallization in high-carbon bainitic steel during hot deformation. The developed mathematical model takes into account the dependence of changing kinetics in the size evolution of the initial austenite grains, the value of strain, strain rate, temperature and time. Physical simulations were carried out on rectangular specimens measur-ing 10 × 15 × 20 mm. Compression tests with a plane state of deformation were carried out using a Gleeble 3800.Key words: plastic deformation, high - carbon bainite steel, physical modeling, mathematical modeling, dynamic recrystallizationINTRODUCTIONCurrently, dynamically developing industry has led to the increased demand for new types of materials. At present, steel sheets are produced from a type of alloys, which are improved by heat treatment producing high strength, characterized by a hardness of maximum 600 HV, but with limited ductility that does not guarantee high impact resistance. The Institute for Ferrous Metal-lurgy in Gliwice has developed a new type of steel, which is a high-carbon bainitic steel. The steel has high development potential as it allows the achievement of very high level of strength (tensile strength up to 2,5 GPa) and at the same time good ductility, with a rela-tively small addition of alloying elements (cumulative ratio of 5 - 6 % by mass) [1]. Research on a new gen-eration of high-carbon bainitic steel has been conducted globally for several years and the results so far indicate a high probability of practical application of this steel in the defense, mining and machinery production indus-tries [2]. There is no information in the scientific litera-ture on phenomena occurring during strain in high-car-bon bainitic steel. This study attempts to mathemati-cally describe the kinetics of dynamic recrystallization, as well as the size of dynamically recrystallized grain.METHODOLOGY AND RESEARCH MATERIALThe material used for this study is high-carbon bai-nitic steel that was developed at the Institute for Ferrous Metallurgy in Gliwice, with chemical composition shown in Table 1. ISSN 0543-5846METABK 56(1-2) 107-110 (2017)UDC – UDK 539.374:669.5:621.72.53.51:620.18=111Table 1  The chemical composition of high-carbon bainitic steel used in this study / mass % [1]C Mn Si Mo Co V Al0,83 2,20 1,65 0,39 1,58 0,092 0,039A range of studies was conducted in order to deter-mine the coefficients in the constitutive equations used for modelling mathematical phenomena that occur dur-ing hot deformation of the tested steel grade. At the be-ginning, the initial austenite grain size was defined, on the basis of experimental studies. The specimens were heated to temperatures between 900 - 1 100 °C for 60 s and then cooled with air at a speed that ensured freezing of the structure. Table 2 shows the results of metallo-graphic examinations under which the initial austenite grain size was defined.Table 2 The initial austenite grain size of the alloyTemperature / °C Grain size / μm900 29,1950 31,71 000 32,11 050 89,41 100 92,8To determine the parameters, which are necessary to initiate dynamic recrystallization, it is important to know the real course of plastic flow curves for steel. With the proper analysis of plastic flow curve, it can be determined whether there is a dynamic recrystallization of the steel. If there is, then the value of the critical strain that is required for the initiation of the process can also be determined, and subsequently a mathemati-cal description of its kinetics will be made. During the analysis of plastic flow curves, a problem arises from 108T. DEMBICZAK et al.: MATHEMATICAL MODELING OF PHENOMENA OF DYNAMIC RECRYSTALLIZATION DURING... METALURGIJA 56 (2017) 1-2, 107-110the uncontrolled temperature increase of the specimen resulting from strain. To describe the influence of tem-perature and strain rate on the value of flow stress, the following equation was used (1) [3]:  (1)where: c1, c2 – experimentally determined coefficients dependent on the material.Coefficients developed (c1=-322,9; c2=6 408,9) in equation (1) allowed the introduction of correction for metal flow stress aimed at the value of a given deforma-tion, with the assumption of temperature and strain rate. An example of adjusted plastic flow curves of high-carbon bainitic steel is shown in Figure 1.The phenomenon of dynamic recrystallization in the microstructure that takes place during hot deformation of steel was analyzed in two stages in the temperature range of 900 - 1 100 °C and a strain rate of 0,1 - 10 s-1. In the first step, on the basis of the developed flow curve that considered temperature correction and strain rate, the strain parameters essential for the phenomenon of dynamic recrystallization were determined. For this purpose, the dependency of reinforcement factor (θ) in the stress function was determined, which describes the behaviour of the material during hot deformation [4-7]:  θ=dσ / dε  (2)where: σ – value of stress flow, ε – the intensity of strain.The kinetics of recrystallization dynamics in high-carbon bainitic steel is described by the following equa-tion:  (3)where: ε – strain, εkr – critical strain determined on the basis of reinforcement, εp – peak strain defined on the basis of the flow curve, ADRX, nDRX - material constant for the particular steel grade.In the second stage of the study, the constitutive equations were expanded, describing peak strain (εp), critical strain (εkr) and strain (ε0,5) corresponding to 50 % of the dynamic recrystallization phase, taking into account the initial austenite grain size and Zener-Hollo-mon parameters [8, 9]:  (4)where:  - strain rate, Q - activation energy, R - gas con-stant, T - temperature.The value of activation energy occurring in equation (4) was determined for a temperature range of 900 - 1 100 °C and a strain rate between 0,1 - 10 s-1 in accordance with equation [4, 5]:  (5)where: σp - yield stress, α – material constant that is empirically determined for the test grade.In analysis of the right side of equation (5), it can be stated that the first part of the equation is the slope of the dependency graph of the ln(sinh(ασp)) from 1/T, while the second part of the equation presents an in-verse slope that reflects the course of dependency of the ln(sinh(ασp)) from ln .In the second part of equation (5), there is a material constant α, which can be determined from the following equation [4, 5]:  (6)The values of material constant n’ and β can be de-termined by an appropriate dependency slope of  and a dependency slope of . From the dependence presented in determined by equa-tion (5), the activation energy of high-carbon bainitic steel was 385,44 kJ/mol. Determined on the basis of Ze-ner- Holloman (eq. 4) parameter and the initial grain size of the austenite, the values of the coefficients were determined in the constitutive equation that describes peak strain at εp of high-carbon bainitic steel. To deter-mine the coefficients, the basic dependency was used, which is presented in the Figures 2 and 3.Finally, the mathematical model of the constitutive equation describing peak strain εp for the tested steel is as follows:Figure 1  Plastic flow curves of high-carbon bainitic steel with strain rate of 10 s-1 at a temperature of 900 - 1 100 °CFigure 2  The dependence of peak strain (ln εp) from an initial austenite grain size (ln d0), according to which the average value of the exponent of the initial austenite grain size was determined109T. DEMBICZAK et al.: MATHEMATICAL MODELING OF PHENOMENA OF DYNAMIC RECRYSTALLIZATION DURING...METALURGIJA 56 (2017) 1-2, 107-110  (7)The material constant ‘c’ in equation (8) was deter-mined as follows:  (8)The value of the constant ‘c’ in (equation 8) is an average value quotient of εkr / εp, determined for tem-perature 900 - 1 100 °C and strain rate 0,1 - 10 s-1. The constitutive equation finishing the critical strain εkr was determined based on equations (7) and (8) and the final for εkr is as follows:  (9)Another important equation in the mathematical model describing the dynamic recrystallization phe-nomenon is the equation for defining the strain ε0.5, at which the material is at the 50 % phase of dynamic re-crystallization [10-13]. In order to determine the value of the strain, the equation that described function of the initial austenite grain size and Zener-Hollomon param-eter was used. The values of the coefficients occurring in the equation were determined from the graphical de-pendence, shown in Figures 4 and 5.Finally for the investigated steel, strain ε0.5 was de-scribed by the equation:  (10)Using the determined constitutive equations (9 and 10) which are describing the dependence of εkr and ε0,5 of the initial grain size and the Zener-Hollomon param-eter, kinetics of dynamic recrystallization can be de-scribed with dependence:  (11)Figure 6 shows an example of metallographic imag-ing conducted during experimental studies.Linear regression of dependency from lndDRX func-tion lnZ (fig.7) was performed in order to determine the coefficients. Figure 3  The dependence of peak strain (ln εp) from the Zener-Hollomon parameter (ln Z), according to which the value of the exponent Zener-Hollomon parameter was determinedFigure 4  The dependence of half recrystallization strain (ln ε0,5) from an initial austenite grain size (ln d0)Figure 5  The dependence of half recrystallization strain (ln ε0,5) from the Zener-Hollomon parameter (ln Z)Figure 6  The microstructure of the specimens after experimental testing for T = 900 °C; ε = 0,28; έ = 0,1 s-1 dγ = 23,7 μm; area 200x; Etching with picric acid (C6H3N3O7)Figure 7  The dependence of dynamically recrystallized grains (ln dDRX) from the Zener-Hollomon parameter (ln Z)110T. DEMBICZAK et al.: MATHEMATICAL MODELING OF PHENOMENA OF DYNAMIC RECRYSTALLIZATION DURING... METALURGIJA 56 (2017) 1-2, 107-110On the basis of the linearization, recrystallized grain size, which results from dynamic recrystallization of high - carbon bainitic steel is described by the following equation:  (12)SUMMARYHigh-carbon bainitic steel is characterized by excel-lent mechanical properties and plasticity thanks to the structure of bainite in the form of plates whose thick-ness is in the range of nanometers. Thanks to this it is starting to play a more and more important role among alloy steels. This steel is of particular relevance to the defense industry, where it can be used for armor plating, with very good results. Based on the results of these experimental studies, coefficients and material con-stants in the constitutive equations were developed, thereby, describing the phenomena occurring during strain caused by heat. The same mathematical descrip-tion was made for the kinetics of dynamic recrystalliza-tion. Coefficients and material constants in the equa-tions describing the grain size achieved in dynamic re-crystallization were also developed. Based on the anal-ysis, it was found that for the tested steel, dynamic re-crystallization is more intensive when deformation in-creases. The value of critical strain (εkr) at a strain rate of 0,1 s-1 is in the range 0,11 - 0,2 at temperatures in the range 900 - 1 100 °C.REFERENCES[1] Patent application of the Institute of Ferrous Metallurgy to the The Patent Office of Republic of Poland: No. P.394037 - Stal bainityczno-austenityczna i sposób wytwarzania z tej stali blach, No. Nr P.396431 – Sposób obróbki cieplnej stali bainityczno-austenityczne[2] Skuza Z., Prusak R., Kolmasiak C.: Restructuring of the metallurgical industry in the aspect of economics system change and integration with the European Union, Metal-lurgy 46 (2007) 3, 221-224[3] Knapiński M., Kawałek A., Dyja H., Kwapisz M.: Krzywe umocnienia stali mikrostopowej przeznaczonej do pro-dukcji blach grubych w kategorii wytrzymałościowej X80, Hutnik Wiadomości Hutnicze 9 (2011), 818-821[4] Shaban M., Eghbali B.: Determination of critical conditions for dynamic recrystallization of a microalloyed steel, Mate-rials Science and Engineering A 527 (2010), 4320-4325[5] Liu J., Cui Z., Ruan L.: A new kinetics model of dynamic recrystallization for magnesium alloy AZ31B, Materials Science and Engineering A 529 (2011), 300-310[6] John J.J, Xavier Q, Lan J, Etienne M: The Avrami kinetics of dynamic recrystallization, Acta Materialia 57 (2009), 2748-2756[7] Mirzadeh H, Najafizadeh A.: Prediction of the critical con-ditions for initiation of dynamic recrystallization, Materi-als and Design 31 (2010), 1174-1179[8] Knapinski M., Dyja H., Kawalek A., Fraczek T., Laber K.: Analysis of the microstructure evolution during thermo-mechanical treatment of the steel plates in grade X80-X100, Metalurgija 52 (2013) 2, 239-242[9] Chen F., Cui Z., Chen S.: Recrystallization of 30Cr2Ni-4MoV ultra-super-critical rotor steel during hot deforma-tion. 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Mathematical modeling of phenomena of dynamic recrystallization during hot plastic deformation in high-carbon bainitic steel

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