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Modeling of austenite decomposition in an AISI 4140 steel Chipalkatti, Jayprakash 1999

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MODELING OF AUSTENITE DECOMPOSITION IN AN AISI 4140 STEEL By Jayprakash Chipalkatti B. Tech., The Indian Institute of Technology, Mumbai, India, 1994 A THESIS SUBMITTED IN THE PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF METALS AND MATERIALS ENGINEERING) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Mar. 1999 © Jayprakash Chipalkatti, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The final mechanical properties and also the distribution of residual stresses in a heat treated product, depend upon the various microstructural constituents. In heat treatment operations, in order to predict the final properties, it has become increasingly important to be able to predict the final microstructure distribution for a given thermal history. Mathematical models of microstructure prediction are a useful tool for this purpose as they eliminate the need for experimentation. The characterization of the kinetics of austenite decomposition to equilibrium and/or non equilibrium phases in a low alloy steel, such as AISI 4140, by performing isothermal transformation experiments is time consuming and hence not cost effective. Further, to generate meaningful and accurate data with respect to the various stages of the progress of the austenite decomposition, many experiments are required. Kirkaldy's model, involving the transformation kinetics of the various possible reaction products, provides a very useful tool in this regard as this model is based on only the chemical composition and the austenite grain size of the steel. By observing certain limitations of this model, especially for low alloy steels, Li and co-workers have proposed modifications to this model. The Kirkaldy model along with the model proposed by Li and co-workers is assessed in the present work. The purpose of this study was to characterize the austenite decomposition kinetics under continuous cooling conditions in a AISI 4140 steel by applying Kirkaldy's model and Li's model and to test the results of these models by perforating GLEEBLE controlled continuous cooling experiments on this steel. The validity of the Kirkaldy model and the Li model was tested in two different ways. First, the calculated response ii was compared with the published TTT diagram for a given chemistry and y grain size for an AISI 4140 steel. The calculated response was subsequently tested using the experimentally measured CCT data for a different AISI 4140 steel. As the characterization of continuous cooling kinetics requires knowledge of the phase diagram, a mathematical model based on thermodynamic equations is used to derive the phase diagram for this steel. The model is based on the equality of chemical potentials for each of the alloying elements in the phases that are in equilibrium. The results of this model are then used in modeling the isothermal and continuous cooling transformation kinetics. It was observed that Kirkaldy's and Li's models both yield a reasonably good prediction of the TTT curve for this steel, when compared to a published TTT diagram for this steel. Under continuous cooling conditions, the microstructures predicted by the Li model are closer to the experimentally observed microstructures than are the Kirkaldy model predictions. iii Table of Contents Abstract ii Table of Contents iv List of Tables vii List of Figures ix List of Symbols xii Acknowledgements xvi 1. Introduction and Objectives 1 2. Mathematical Modeling of Phase Equilibria 6 2.1 Background 6 2.2 Thermodynamic Model 10 2.2.1 Mathematical Expressions for Chemical Potentials 10 2.2.2 Assumptions in the Thermodynamic Model 11 2.2.3 Calculation of the Orthoequilibrium (y/y+a Phase Boundary) 12 2.2.4 Calculation of the Orthoequilibrium y/y+cementite Phase Boundary 13 2.2.5 Calculation of the Orthoequilibrium Aei temperature 16 iv 2.3 Calculation of the Equilibrium Phase Diagram for 4140 Steel 16 2.4 Summary 19 3. Modeling Isothermal Transformation Kinetics 28 3.1 Kirkaldy's Isothermal Transformation Model 29 3.2 The Kirkaldy Model Assumptions 29 3.3 Kinetics of Isothermal Austenite to Ferrite, Pearlite and Bainite Reactions 31 3.4 Critical Transformation Temperatures 33 3.5 The Modified Kirkaldy Model by Li et al: 35 3.6 Results of the Isothermal Transformation Kinetics Models 37 4. Modeling CCT Kinetics 49 4.1 Description of the CCT Model: 49 4.1.1 Assumptions in the CCT Model: 49 4.1.2 Calculation of Kinetics of the Transformation under Continuous Cooling Conditions 50 5. Experimental Testing and Validation of CCT Model Results 54 5.1 Sample Preparation 55 5.2 CCT Test Conditions 56 5.3 Microstructural Characterization 57 5.4 Determination of the prior austenite grain size .; 57 5.5 Measurement of hardness 58 5.6 CCT Experimental Results 58 5.6.1 CCT Diagram 58 5.6.2 Determination of the Continuous Cooling Transformation Kinetics from the CCT Data 60 5.6.3 Prior Austenite Grain Size 62 5.6.4 Microstructural Characterization 63 5.7 Comparison of the CCT Model with the Experimental CCT Data 63 5.8 Sensitivity of the Kirkaldy Model 67 5.8.1 Sensitivity of the Kirkaldy Model to Austenite Grain Size, G 67 5.8.2 Sensitivity of the Kirkaldy Model to the Chemical Composition 68 6. Summary and Conclusions 84 7. Future Work 87 8. Bibliography 88 A - l . Appendix 1: Program used to calculate the TTT data for 4140 steel using Kirkaldy's equations 91 A-2. Appendix 2: Program used to calculate the CCT data for 4140 steel using Kirkaldy's equations 96 vi List of Tables Table 2.1 Composition of AISI 4140 steel used in this work 20 Table 2.2 Standard free energy change between a and y phases, for various elements114'191 20 Table 2.3 Standard magnetic entropy of pure iron , (SaFe)mag [ I 4 ' 1 9 1 21 Table 2.4 Self interaction coefficients of alloying elements in a and y phases114'191 21 Table 2.5 Interaction coefficients between alloying elements and carbon1'4'191 22 Table 2.6 Standard free energy change between cementite and y phased14'191 22 Table 2.7 Interaction coefficients between alloying elements and carbon.114,191 23 Table 3.1 Composition of the AISI 4140 steel for the published T T T [ 2 6 ] 41 Table 3.2 Comparison of Nose Temperatures calculated from the Kirkaldy modelt?1, the Li model1131 and the published (experimental) data 41 Table 3.3 Comparison of times for 1 % normalized transformed fraction at various temperatures obtained from the Kirkaldy model171, the Li model1131 and the published (experimental) data1261 42 Table 3.4 Comparison of times for 99 % transformed fraction (of austenite) at various temperatures obtained from the Kirkaldy model[71, the Li model[131 and the published (experimental) data*261 43 Table 5.1 Continuous cooling transformation test conditions employed for the AISI 4140 Steel 69 vii Table 5.2 Comparison of ferrite+pearlite, bainite and martensite volume fractions between experimental data, Kirkaldy model calculations171 and Li model calculations1131 Table 5.3 Comparison of hardness of the final microstructure calculated from the Kirkaldy model[71, the Li model1131 and the experimental data. List of Figures Fig. 1.1 Various interactions linking the thermal field, mechanical behavior and the phase transformation field in a typical heat treatment operation 5 Fig. 2.1 Schematic diagram of a portion of the Fe-C phase diagram illustrating the phase boundaries at various temperatures, under isothermal conditions 24 Fig. 2.2 Algorithm for the calculation of the Aej and the Acm (or Aecm) temperatures 25 Fig. 2.3 Phase diagram showing the Cj, Cya and the C / c m phase boundaries, calculated from the thermodynamics model for the composition of AISI 4140 steel employed in this work (given in Table 2.1) 26 Fig. 2.4 Equilibrium volume fractions of ferrite, pearlite and bainite as a function of temperature for the AISI 4140 steel, calculated from the phase diagram illustrated in Fig. 2.3 27 Fig. 3.1 Schematic diagram of the TTT curve illustrating the assumptions used in the transformation kinetics model 44 Fig. 3.2 Algorithm for the calculation of TTT diagram using the Kirkaldy model[7] and the Li modelt131 employed in this work 45 Fig. 3.3 Value of the term I(X) and S(X) representing the sigmoidal behaviour of the transformation in Kirkaldy's[?1 and Li's equations[13], as a function of the normalized fraction, X 46 Fig. 3.4 The published TTT diagram for AISI 4140 steel[26] for the composition: Fe -0.37% C-0.77% Mn- 0.98% Cr- 0.21% Mo Grain Size: 7-8 Austenitized at 843°C (1550°F) 47 Fig. 3.5 The TTT diagram calculated using the Kirkaldy model[7] and the Li model[13] for the AISI 4140 steel and ASTM Grain Size number of 7.5 reported for the published TTT diagram^261 shown in Fig. 3.4. Subscript Notation: F L S : Start line for ferrite by the Li Model [13 ], F K S : Start line for ferrite by the Kirkaldy Model[7] PLS: Start line for pearlite by the Li Model [13 ], PK S: Start line for pearlite by the Kirkaldy Model [ 1 3 ] B L S: Start line for bainite by the Li Model [13 ], BK S; Start line for bainite by the Kirkaldy Model [ 1 3 ] FLf: Finish line for ferrite by the Li Model [13 ], FKf: Finish line for ferrite by the Kirkaldy Model171 PLf\ Finish line for pearlite by the Li Model [ I 3 ], PKf: Finish line for pearlite by the Kirkaldy Model [ , 3 ] i ri31 K B f: Finish line for bainite by the Li Model1 , B f: Finish line for bainite by the Kirkaldy Model [ 1 3 ] 48 Fig. 5.1 Geometry of the GLEEBLE specimen used in the CCT tests 72 Fig. 5.2 Schematic representation of the continuous cooling tests 72 Fig. 5.3 Experimental Dilation versus temperature data for various continuous cooling conditions: a) . for cooling rates 0.1°C/s, 0.25°C/s, 0.5°C/s, 0.75°C/s. b) for cooling rates l°C/s, 2°C/s, 6°C/s, 12°C/s 73 Fig. 5.4 Experimental Continuous Cooling Transformation (CCT) diagram for the 4140 steel employed in this study, showing the start of transformation temperatures for ferrite, bainite and martensite transformations at various cooling rates, for the measured ASTM grain size number 9.5 74 Fig. 5.5 Experimentally determined dilation data for the 4140 steel used in this study, for cooling rate of 0.5°C/s a) showing transformation start and stop temperatures b) showing calculated kinetics of transformation of austenite to ferrite and pearlite , then bainite 75 Fig. 5.6 : a) Photomicrograph of an air cooled sample of AISI 4140 steel (austenitizing condition: 850°C for 3 min.) etched with a solution of saturated aqueous picric acid along with a solution of sodium benzene sulphonate as a wetting agent, for 1 hour, b) Outline of prior austenite grain boundaries used for grain size measurement 76 Fig. 5.7 Photomicrographs showing microstructures of continuously cooled samples of the AISI 4140 steel austenitized at 850°C for 3 min and etched with 2% Nital. a) At a cooling rate of 0. l°C/s : The Microstructure shows ferrite(white phase) and equiaxed pearlite (black phase). X400, VHN: 239. b) At a cooling rate of 0.25 C/s : The Microstructure shows ferrite(white phase), equiaxed pearlite (equiaxed dark phase) and upper bainite (acicular gray etching phase). X400, VHN: 296 77 Fig. 5.7 Photomicrographs showing microstructures of continuously cooled samples of the AISI 4140 steel austenitized at 850°C for 3 minutes and etched with 2% Nital. c) At a cooling rate of l°C/s : The Microstructure shows bainite (acicular gray etching phase) and few nodules of pearlite (darker etching phase). Very few areas with ferrite are also present. X400, VHN: 365. d) At a cooling rate of 12°C/s : The Microstructure shows bainite (acicular dark phase) and martensite (light gray matrix phase). X400, VHN: 504 78 Fig. 5.8 CCT diagram calculated from the Kirkaldy Model [ ? 1, at various cooling rates, showing the start temperatures for various transformations and the lines corresponding to 25%, 50%, 75% and 99% of austenite transformed 79 Fig. 5.9 Comparison of Vicker's hardness numbers calculated using the Kirkaldy model[7], the Li model[13] and the experimentally measured data 80 Fig. 5.10 Value of the term, 2 ( G' I ) / 2 which represents the effect of austenite grain size in Kirkaldy's[7] equations, as a function of ASTM grain size No. G 81 Fig. 5.11 Distribution of the final microstructure as a function of cooling rate (°C/s) for austenite grain size numbers of 6, 8 and 10, based on Kirkaldy's equations171 for the 4140 steel used in this work: a) For the formation of ferrite+pearlite; b) For the formation of bainite; c) For the formation of martensite 82 Fig. 5.12 Calculated kinetics of transformation of austenite, from the Kirkaldy model[?1, at l°C/s, for least hardenable (composition A) and most hardenable (composition B) compositions for 4140 steel. Composition A: Fe- 0.37 %C- 0.75 %Mn- 0.15 %Si- 0.8 %Cr- 0.15% Mo Composition B: Fe- 0.43 %C- 1.0 %Mn- 0.35 %Si- 1.0 %Cr- 0.28 %Mo 83 List of Symbols* A tf: Interaction parameter of element / with element j in cementite Aes,: The equilibrium temperature at the y/y+a phase boundary, °C Aei: The equilibrium temperature at the y+a/y+cementite phase boundary, °C Acm (or Aecm) '• The equilibrium temperature at the yly+cementite phase boundary, °C Bs: The bainite start temperature, °C Co : The Mean composition of the alloy, wt. pet CJ: Concentration of carbon in the a phase in equilibrium with y, wt. pet C°: Concentration of carbon in the y phase in equilibrium with a, wt. pet Cyem: Concentration of carbon in the cementite phase in equilibrium with y, wt. pet ADexp: Experimental dilation, mm ADlhermal: Dilation due to thermal contraction, mm ADlrans: Dilation due to transformation, mm G: ASTM austenite grain size number 0 G / : Free energy in standard state at infinite dilution for element, / in phase $ J/Mole A°Gi or ° G / / _ ^ 2 : Free energy change in the transformation between 01 and <f>2, J/Mole A0Hi: Enthalpy change associated with A°Gt, J/Mole HV (Bainite) : Vickers hardness number of bainite HV(ferrite+pearlite) : Vickers hardness number of the mixture of ferrite and pearlite HV (Martensite) : Vickers hardness number of martensite * These symbols are given in alphabetical order. Ms: The martensite start temperature, "C Mf: The martensite stop temperature, °C n : Exponent of undercooling, AT Q : Activation energy, cal/mol R : Universal Gas Constant, cal Mole"1 K"1 T: Temperature, °C or K TFS : Ferrite start temperature under continuous cooling conditions, °C TBS '• Bainite start temperature under continuous cooling conditions, °C TN : Nose temperature for a given reaction on the TTT diagram, °C AT: Undercooling or difference between two successive temperatures, °C ts,717^: Time for transformation start under isothermal conditions, s tstCCT: Time for transformation start under continuous cooling conditions, s TF: Time for a certain volume fraction of ferrite transformed, s tp: Time for a certain volume fraction of pearlite transformed, s XB- Time for a certain volume fraction of bainite transformed, s Vr: Cooling rate at 700°C, °C/hr X: Normalized volume fraction of a given phase Xi: Mole fraction of element / X*j: Mole fraction of element / in phase ^ 7;: Concentration of metal element i in cementite XFE- Equilibrium volume fraction of ferrite XFE(TI) : Equilibrium volume fraction of ferrite at temperature 77 XFE(TZ) : Equilibrium volume fraction of ferrite at temperature T2 XPE'. Equilibrium volume fraction of pearlite X"FE- Equilibrium weight fraction of ferrite XF : True volume fraction of ferrite XpT: True volume fraction of pearlite XFIT : True volume fraction of ferrite in the first time segment in continuous cooling XFN : Normalized volume fraction of ferrite XFIN : Normalized volume fraction of ferrite in the first time segment in continuous cooling X'FIN '• Equivalent or virtual volume fraction of ferrite at temperature T2, assuming the entire reaction occurred at temperature T2 XF2N '• Normalized volume fraction of ferrite in the second time segment in continuous cooling XpN : Normalized volume fraction of pearlite IM: Chemical Potential of element /, J/ Mole MM3C '• Chemical Potential of the metal (M) carbide, J/ Mole Yt: Activity Coefficient for element i £ij: Interaction Parameter of element i with element j £*ij: Interaction Parameter of element / with element j in phase 0 amix: Thermal expansion coefficient of the mixture of various phases, mm mm'1 °C GCF+P: Thermal expansion coefficient of ferrite+pearlite, mm mm"1 °C"1 xiv as: Thermal expansion coefficient of bainite, mm mm" C - 1 c OLA. Thermal expansion coefficient of austenite, mm mm Acknowledgements I would like to express my sincerest gratitude to my supervisor, Professor E. B. Hawbolt for giving me the opportunity to work with him. His excellent guidance and continuous assistance has helped me throughout the course of this work. I also wish to thank him for the financial support in the form of a graduate research assistantship, which helped me in pursuing a graduate degree at UBC. I also extend my sincere thanks to Dr. M. Militzer for helpful discussions. I will always cherish the enjoyable company of my friends at UBC and their support in the time of need. I would like to especially thank Joy deep Sengupta and S. Govindarajan for their invaluable support. Last but not the least, the continuous support and tolerance shown by my wife, Girija and my family throughout the ups and downs during the course of this work can not be put into words: I thank them from the bottom of my heart. xvi 1. INTRODUCTION AND OBJECTIVES Heat treatment often constitutes an important step in the processing of engineering components. In a typical heat treatment operation, a component is subjected to a thermal cycle (involving heating and cooling) in order to obtain a certain desired distribution of microstructure which results in the desired mechanical properties. In the past, appropriate heat treatment operations were selected on the basis of trial-and-error procedures. The process parameters that were required to obtain the desired mechanical properties were based on qualitative information and experience. However, over the years, due to the increasing demands for high productivity, improved property control, cost reduction and low energy consumption, there has been an increasing need for quantification of heat treatment processes and formulation of mathematical models which enable prediction of the final properties, given the process parameters. In a typical heat treatment operation, the thermal history that the component is subjected to, the phase transformations occurring within and the resulting mechanical properties are mutually related. The various interactions between the thermal, microstructural and the mechanical fields in a typical heat treatment operation are shown in Fig. 1.1. The interaction between the thermal field and the microstructural field is important from the point of view of this work. The various phase transformations that generate the required microstructure link the process parameters to the final properties. Thus, in order to be able to predict the microstructure distribution accurately, it is important to have an understanding of the mechanisms of the phase transformations. Due to the interaction of various alloying elements and their effect on the thermodynamic stability of the phases, steels with different alloying elements yield 1 different transformation characteristics. Hence, understanding the role of each of these alloying elements constitutes an important aspect of understanding the phase transformations. Modeling transformation kinetics in steels involves predicting the final microstructure for a given thermal history. As the temperature distribution in a component subjected to the heat treatment is changing with time, the model must take this into account. There are several transformation kinetics models, combining both a theoretical formulation based on nucleation and growth theory and empiricism based on fitting equations to experimentally obtained datafl"6]. However, given the complexity of the transformation processes, most of these models contain parameters which need to be found experimentally to explain the effect to the alloy of interest. This presents an important drawback to these models from an industrial point of view, as it is expensive and time consuming to carry out this experimentation. Hence, it is desirable to use a model that can yield the transformation characteristics of steel, given only it's composition and thermal history. Kirkaldy's transformation kinetics model ^ is one such model; it incorporates the incubation and transformation period and enables prediction of the TTT diagram of a steel, given its composition and prior austenite grain size. These equations can then be coupled with the additivity principle^1 to predict the microstructure under continuous cooling conditions. The author found that specific examples of the application of Kirkaldy's equations'71 in predicting the microstructures under continuous cooling conditions are limited. Kirkaldy's equations'71 have also been used for predicting the weld HAZ 2 microstructurestl0'11]. Pan and Watt [ 1 2 ] have tested this algorithm for plain carbon steels showing it to produce satisfactory results. Kirkaldy et a/. [9] have used their model primarily to predict the microstructure developed in a Jominy bar, for example, for a low alloy steel such as AISI 4140. However, a detailed comparison of the final microstructural distribution with the one predicted by the model, was not performed. The calculated CCT diagrams used by Kirkaldy et a/. [9] to predict the microstructure were based on the cooling conditions encountered in the Jominy hardenability test and not those obtained using controlled cooling rates. Thus, the overall objective of this work was to characterize the isothermal and continuous cooling transformation kinetics for a low alloy steel, AISI 4140, using available transformation kinetics models^7'131 and to validate the results of the models using the TTT data available in the literature and the controlled cooled experimental CCT data. The primary objectives of this work are: 1. To determine the phase diagram and the transformation temperatures for this steel using a mathematical model based on thermodynamics'14'151. 2. To use the information from the phase diagram to obtain isothermal Time-Temperature-Transformation (TTT) data for an AISI 4140 heat treatable steel with the help of existing mathematical models'^ 7'131. 3. To develop a continuous cooling transformation (CCT) diagram for the AISI 4140 steel using the TTT data and assuming additivity holds. 4. To validate the results obtained from these models using the data available in the literature and the experimentally measured continuous cooling transformation data for this steel. 3 To achieve these objectives for the AISI 4140 steel, the following methodology was applied: 1. A mathematical thermodynamics model for the ortho-equilibrium1 condition'14'151 was used to develop the temperature/composition boundaries for the y/a+y and y/a+cementite phases. Using these phase boundaries and the Lever law (carbon mass balance), the equilibrium volume fractions of the phases were determined. 2. These results were used in the austenite decomposition kinetics to predict the TTT diagram for the AISI 4140 steel using equations originally proposed by Kirkaldy et alP^ and modified by Li et a/.'131. Both models, the one by Kirkaldy et a/.'71 and that by Li et a/.'131, were coupled with Scheil's additivity principle'81 to predict the transformation start times and the subsequent microstructure evolution under given continuous cooling conditions. 3. Thermally programmed dilatometer tests were performed using a GLEEBLE 1500 thermomechanical simulator to determine the final microstructure distribution and hardness for a range of continuous cooling tests. The experimental results were used to test the transformation kinetics predictions made by the Kirkaldy'71 and the L i [ 1 3 1 models. 1 Ortho-equilibrium is an equilibrium condition at the transformation interface in which the carbon and the substitutional alloying elements completely partition across the interface. 4 Mechanical Behavior Thermal Field Fig. 1.1 Various interactions linking the Thermal Field, Mechanical Behavior and the Phase Transformation Field in a typical heat treatment operation. 5 2. MATHEMATICAL MODELING OF PHASE EQUILIBRIA To model the kinetics of decomposition of austenite to equilibrium (ferrite and pearlite) or non equilibrium (bainite and martensite) phases, it is important to know the equilibrium phase boundaries, particularly the temperatures (e.g. Ae^, Aei and Acm '), the composition of these phases and the volume fractions of the individual phases. Empirical equations have been developed to describe the equilibrium temperatures, but they are limited by the range of compositions for which they were tested'16'17]. Thus, it is desirable to use mathematical models based on fundamental principles of thermodynamics. In this work, the phase diagram for an AISI 4140 steel is developed using a thermodynamic model [14,151. This chapter is dedicated to the calculation and derivation of the phase diagram for the AISI 4140 steel. First, brief information about the various temperature regions and stable phases is provided. Then, the calculation procedure for the orthoequilibrium y/y+a, yly+cementite and y+a/y+cementite phase diagram boundaries are described. Finally, the phase diagram for the 4140 steel derived from this calculation is presented. 2.1 Background: To quantify the austenite decomposition kinetics in steels, it is important to understand the thermodynamics of the various phases and to identify the temperature range in which individual phases are stable. Heat treating (hardening) of steel involves quenching from the austenite phase field and the associated decomposition of the parent phase. In steels, the transformation from austenite to ferrite, pearlite and upper bainite are 1 The ' y 4 c m ' temperature is alternatively called the 'Ae^ temperature. 6 diffusion controlled and occur at relatively high temperatures (generally above 450°C)2. The transformation to lower bainite has both diffusional and non-diffusional characteristics and that to martensite is non-diffusional, the latter occurring at lower temperatures (below Ms) [ 1 8 ]. The sequence of transformation reactions that takes place in the Fe-Fe3C system during the isothermal decomposition of austenite, when austenite is quenched to and allowed to transform at a specific temperature, is illustrated in Fig. 2.1. Referring to Fig. 2.1, if the steel has a mean composition of Co, the phases that are stable at various temperatures are: 1. At Ti > Ae3 (or Aej at the eutectoid composition) austenite (j) is the stable phase. 2. If the steel is quenched to T2 (Aej< T2 <Ae3), austenite transforms to ferrite (a) until the composition of the remaining austenite and ferrite becomes Cya and Cj, respectively. 3. If the steel is quenched to T3 (Bs < T3 < Aei), austenite transforms to ferrite of composition Cra until the remaining austenite has composition Cfem and further transforms to pearlite. 4. If the steel is quenched to T4(MS<T4< Bs), austenite transforms to bainite. 5. If the steel is quenched to T5 (M/< T5 < Ms), austenite transforms to martensite; the amount of martensite formed is dependent on the degree of supercooling below Ms. If held for a long period of time at this temperature, the remaining austenite will transform to bainite. 6. If the steel is quenched to Tg< Mj, austenite completely transforms to martensite. Thus, the various transformation reactions that occur under isothermal conditions and the range of temperatures in which these reactions occur, can be summarized as follows: 2 This temperature limit is dependent upon composition, austenite grain size and the cooling rate. 7 1. y-Hz+y'3(Ae3<T<Aei) 2. /-^a+P(Aei <T<BS) 3. y->B(Bs<T<Ms) 4. y->M+ B(Mf<T< Ms) 5. y->M(T<Mf) Under continuous cooling conditions, however, a mixture of the above phases can be obtained. For a multi-component system, Fe-C-M, where M refers to a substitutional alloying element such as Mn, Ni, Si, Cr etc., the transformation temperatures are different from those of the Fe-Fe3C system. This is because the alloying elements influence the thermodynamics stability range of the various phases. The alloying elements also affect the diffusion processes by suppressing or enhancing the transformations, thus having an influence on the kinetics of the transformation. By influencing the nucleation and growth kinetics of the austenite decomposition products, the alloying elements also have a direct influence on the hardenability of steels. Each alloying element influences it in a different manner and to a different extent. In order to describe the equilibrium phase diagram using thermodynamic models, it is necessary to have accurate thermodynamic data. Uhrenius[19] has compiled a large amount of thermodynamic data relating to the binary Fe-M and ternary Fe-C-M systems. Using this thermodynamic data, Hillert[20] has given a comprehensive review of the methods of prediction of phase equilibria. According to Hillert[20], the thermodynamic modeling involves expressing in an appropriate mathematical form, the Gibbs free energy or chemical potential of the phases which are in equilibrium at a given temperature. For 3 Where y' is the austenite with the phase boundary composition. 8 the Gibbs free energy function, minimization of the free energy function under some constraints is the requirement for the equilibrium condition. This method requires a significantly large computer capacity. The method using chemical potential functions is obtained by solving a system of nonlinear equations, each corresponding to a specific alloying element. Kirkaldy and Beganis [ l 5 \ by using the method of chemical potentials, devised both an analytical method and a numerical procedure for the calculation of the Ae3 temperature in the Fe-C-M multi-component system. The analytical formula was found to yield accurate results for additions of up to 1 wt. pet. Si, 2 wt. pet. Mo and 6 wt. pet. Cr, Ni, Mn and Cu, the total alloying additions not exceeding 10 wt. pet.. The numerical algorithm was applicable for a wider range of chemistries. In the numerical algorithm, the chemical potential of a given phase is expressed in terms of the mole fraction and activity coefficients of the various alloying elements in that particular phase by means of Wagner's regular solution model'211. The equilibrium temperature is found by solving a system of nonlinear equations generated by equating the chemical potentials for each element (Fe, C and M) in the a and the y phase. Although the actual algorithm was proposed only for the calculation of the Ae3 temperature, it proved to be a pioneering one and similar methodologies were adopted by Hashiguchi et a/. [ M 1 and L i ' 2 2 1 for the calculation of other equilibrium temperatures (such as Aei and Acm) and by Bhadeshia'231 for the calculation of the solidus and the liquidus temperatures in alloy steels. In this work, the algorithm proposed by Hashiguchi et « / . [ I 4 ] is used for the calculation of the Ae3, Ae\ and Acm temperatures for the AISI 4140 steel. The thermodynamic data compiled by Uhrenius^191 and modified by Hashiguchi et a/.'141 is used for the calculations. 9 2.2 Thermodynamic Model: 2.2.1 Mathematical Expressions for Chemical Potentials: In the thermodynamic model the elements are designated with numbered subscripts. The calculation of the phase diagram has been made for the composition of the AISI 4140 steel used in this work. The subscript notation, the element and it's composition in the AISI 4140 steel used in this work are given in Table 2.1. The chemical potentials of the iron, carbon and the alloying elements are expressed using the Regular Solution Model, as used by Kirkaldy et al}l5\ and have the following form: Hi =° Gt +RTlnXj + RTlnyt (2.1) where, for element /, °G, is the standard free energy, Xt is the mole fraction and # is the activity coefficient at temperature T. Using Wagner[2IJ expansions for the activity coefficients for iron (/=0 ) gives lny0 =(-l/2)YjsikXiXk =(-l/2JZe„X?-X^sHX (2.2) i,k=\ i=l i=2 and for carbon and other substitutional elements ( i=\ to n), 7 / 1 ^ = 2 ^ = ^ 1 + ^ (2.3) k=l where is the interaction parameter of an element, i, with an element, k. Using these expansions and substituting for the lnyt terms, the expressions for the chemical potentials take the form: 10 H0 =° G0+RTlnX0 — ^ X f - R T X ^ X , 2 i=7 i=2 (2.4) H, =" G, +RTlnX!+RTYjeliX, (2.5) / = / juf = G, + RTlnXl + £nXj + suX, (for i = 1 to n) (2.6) The chemical potentials of iron carbide, HFeic and alloy carbides, /JM3C, are expressed using the regular solution modelpl], as given by the following equations: MFe3c=°GFEIC +^RTlny0 +1(1 - y0)^w0ly, (2.7) MM3C=°GM3C +^RTlnyi + ^ 8 K--Xv^ (fori = 2ton) (2.8) where °GFe3C and °GM3C are the standard free energy terms associated with the iron and alloy carbides, yo and yt are the concentrations in wt. pet. of iron and the alloying elements in the carbides and wy is the interaction parameter of an element / with an element j. 2.2.2 Assumptions of the Thermodynamic Model: 1. It is assumed that the matrix of interaction parameters is symmetric. Hence, s jj = £jt (for /, j = 1 to n) (2.9) 2. The interaction of one substitutional element with another substitutional element is assumed to be zero. 11 2.2.3 Calculation of the Orthoequilibrium (y/y+a) phase Boundary: The Ae3 phase boundary is a temperature/composition boundary describing the y phase that is in equilibrium with the a phase. The calculation of this boundary involves employing equilibrium conditions for the ^and the a phase. The algorithm for calculating the Ae3 temperature is shown in Fig. 2.2 and is briefly explained below: 1. Taking a trial value of temperature, T, all the temperature dependent thermodynamic parameters are calculated including the standard free energy of iron and the alloying elements in the a and y phases and the interaction parameters of the alloying elements, ey. 2. In the orthoequilibrium condition, the nominal alloy composition of the alloy lies on the y surface of the a-y phase diagram. Hence the mole fractions of carbon and all of the alloying elements in the gamma phase, X/, are equated to the nominal composition of the alloy, Xt: X\=Xt (for i = I ton) (2.10) 3. The mole fractions of carbon and all of the alloying elements in the alpha phase are found using Kirkaldy's[15] approximate partition coefficient formulae between the alpha and the gamma phase given by: A°Ga~*r XJexp(—i— + e^XJ) *> = R T A o G ^ r (f°r 1 = V (U1> l + 6riX'exp(-^-) 12 A°Ga^r X*exp( '- + er,Xr) *? = R T ,c^y (fori = 2ton) (2.12) l + c-Xfexpf-^r-) These expressions are based on the underlying assumption that carbon, as well as the alloying elements, fully partition and are in complete local equilibrium across the y/a interface. 4. The values of X", Xj and all of the thermodynamic parameters are substituted in the equation for the equilibrium of iron across the a/y interface expressed by the following equation: i - V x r 1 + ^"(Xt)2-LfsUX])2=0 (2.13) * i=l * 11=11 This is the key equation in solving for the orthoequilibrium Ae3 temperature and is termed Goa~*y=0, wherever it is referenced. This is a single nonlinear equation in only one variable, T, and is solved by the Newton-Raphson Method[24]. The solution of this equation is the orthoequilibrium Ae3 temperature. 2.2.4 Calculation of the Orthoequilibrium y/y+cementite Phase Boundary: The Acm phase boundary is a temperature/composition boundary describing the y phase that is in equilibrium with the iron and alloy carbides, [Fe,M]3 C phase. Thus, 13 calculation of this temperature involves employing equilibrium conditions for the y and the [Fe,M]3 C phases. The algorithm for the calculation of the Acm temperature is shown in Fig. 2.2 and is briefly explained below: 1. Taking a trial value of temperature, T, all of the temperature dependent thermodynamic parameters are calculated, including the standard free energy of iron carbide and the alloyed carbides, °GFe3c and °GM3C, respectively, the standard free energies of iron, carbon and the alloying elements, °Gj the interaction parameters of the alloying elements, Sy, and the interaction parameters of the iron with the alloying elements in the carbides, wy. 2. In the orthoequilibrium condition, the nominal alloy composition of the alloy lies on the y surface of the y-cementite phase diagram. Hence the mole fractions of carbon and all of the alloying elements in the gamma phase, X?, are equated to the nominal composition of the alloy, Xh as is done in the calculation of the Ae3 temperature. 3. For the calculation of the mole fractions of the alloying elements in the cementite phase, a simplified procedure is used. The chemical potentials of the carbides are expressed as follows[14]: 4MFC3C =3JuFe+M1 (2-14) 4JUM3C=3VM+MI (2-15) where Mis the alloying element. Subtraction of Eqn. (2.15) from Eqn. (2.14) and substitution for juo, Hi and ju; yields, 14 R T l n [ ( p ^ ] = ( i G _Gq _ L G l M ± G -G,-lGl)-w0ly0 (x,/x0) 3 3 3 3 3 + TJwoJyJ+RT[£nX, +euxi+x£e(lxJ +^fJ£jjX2J] (2.16) j=2 j=2 Z H For low alloy steels, using the approximations Xj«\, Yt « 1, XQ = 1 and Yo = 1, and rearranging yields, Y=BX" (2.17) X'= C,(4X,-1) ' 3B,(X,-C,) + 4C,-1 Thus, knowing the value of X/i the 17 s can be calculated using the above expression. 4. The values of 7„ Xf and all of the thermodynamic parameters are substituted in the equation of equilibrium of iron across the y/cementite interface expressed by the following equation: 4<uFe3c-3/uy0-tf =0 (2.19) This is the key equation in solving for the orthoequilibrium Acm temperature and is termed F(f~*:em==0, wherever it is referenced. This is a single nonlinear equation in only one variable, T, and can be solved by the Newton-Raphson Method[24]. 15 2.2.5 Calculation of the Orthoequilibrium Ae1 temperature: At the Aej temperature, austenite is in equilibrium with both ferrite and the alloyed cementite. Thus, after the calculation of the Ae3 and Acm temperatures, the calculation of the Ae; temperature is straightforward. Keeping the temperature, T, and the carbon content in austenite, X/, as variables, the orthoequilibrium y/y+a and the orthoequilibrium y/y+cementite phase lines are calculated. The temperature and the carbon content at the intersection of these lines are assumed to be the eutectoid temperature, Aej, and the eutectoid carbon content, respectively. 2.3 Calculation of the Equilibrium Phase Diagram for 4140 Steel: In order to calculate the area of interest in the equilibrium phase diagram for the AISI 4140 steel, the Ae3 and Acm temperatures were calculated by varying only the carbon content and keeping the composition of the substitutional alloying elements constant. The resulting phase diagram showing the Cjt Cra and Crcem lines is shown in Fig. 2.3. The calculated Ae3 temperature for this steel is 766°C. The Aei temperature, which is the intersection of the Ae3 and Acm phase lines, is 716°C. However, it should be noted that the eutectoid temperature does not correspond to a single value but to a temperature range in an Fe-C-M multi-component system. In this model, however, the calculations corresponding to the Fe-C-M (n-dimensional) system are shown on a 2-dimensional plot and the eutectoid temperature shown corresponds to the maximum temperature of this temperature range. The phase diagram for this steel is used with the Lever law and the densities of the phases to determine the equilibrium volume fractions of austenite, ferrite and pearlite in the a+yand y+Fe3C two phase fields. 16 For the AISI 4140 hypoeutectoid steel, in the temperature range from Ae3 to Ae}, austenite partially transforms to ferrite; the equilibrium weight fraction of ferrite is determined by using the Lever law of the carbon mass balance, (2.20) where X"FE is the equilibrium weight fraction of ferrite, Cj is the wt. pet. of carbon in the alpha phase in equilibrium with the / phase, Cra is the wt. pet. of carbon in the /phase in equilibrium with a and Co is the initial wt. pet. carbon in the steel. In the temperature range from Aei to Bs, austenite completely transforms to ferrite and pearlite and the equilibrium weight fraction of ferrite is determined by using the following equation: where X*FE is the equilibrium weight fraction of ferrite, Cfem is the wt. pet. concentration of carbon in the y phase in equilibrium with the cementite phase; it is determined by extrapolating the Acm boundary below the Aei temperature, as shown in Fig. 2.4. As a first approximation, the Crcem phase boundary, linearly extrapolated to lower temperatures, is expressed by the following equation: cem 0 (2.21) cem cem T = 552.86%C +548.64 (2.22) 17 where T is the extrapolated Acm temperature and %C is expressed in wt. pet. The minimum temperature, 490°C, below which austenite completely transforms to bainite was then determined using this diagram. The calculated values of the equilibrium volume fractions of ferrite, pearlite and bainite for the 4140 steel are shown in Fig. 2.4. The densities of ferrite and pearlite used for this calculation were 7.86 and 7.81 gm/cm3, respectively'251. The density of bainite was assumed to be equal to that of pearlite. It is recognized that bainite is not considered to be an equilibrium transformation product, however, its inclusion in Fig. 2.4 is necessary in order to be consistent with the assumptions employed in modeling the isothermal transformation kinetics, as described in the next chapter. The ferrite volume fraction obtained above and below the Aei temperature, is expressed as a polynomial function of temperature using the following equations: X F E = -(6xlO~5)T2 + (0.075)T-23.8 + 1.566 (for Ae3 <T < Ae,) (2.23) X F E = -(8x 10~5 )T2 + (0.0117 )T-3.8222 + 0.03 (for Ae, <T < BNose) (2.24) Where XFE is the equilibrium volume fraction of ferrite, T is the temperature of transformation in °C. The maximum ferrite fraction is obtained at the Ae; temperature, below which, the pearlite transformation starts. As this is the equilibrium amount of ferrite given by thermodynamics, the calculated maximum fraction of ferrite formed under CCT conditions should not exceed this value. 18 2.4 S u m m a r y : In this chapter, a method for calculating the equilibrium Cjt Cra and Crcem phase boundaries is described. The algorithm used in this work was proposed by Hashiguchi et alJu^ using the thermo-chemical data compiled by Uhrenius[19]. Using this method, the portion of interest of the equilibrium phase diagram for AISI 4140 steel has been derived. The calculated Ae3 temperature for this steel is 766°C and the Aej temperature, which is the intersection of the Ae3 and Acm phase lines, is 716°C. It must be mentioned, however, that in a Fe-C-M multi-component system, such as the 4140 alloy steel, there is a temperature range corresponding to the equilibrium condition. However, in this work, the equilibrium temperatures are represented on a two dimensional plot and are the maximum temperatures for the transformation range. By employing Lever Law calculations, the equilibrium weight fractions of ferrite, pearlite and bainite are derived and are then corrected to equilibrium volume fractions by using the appropriate density values. The equilibrium volume fractions of ferrite and pearlite are then expressed as a function of temperature. Kirkaldy's'71 and Li's' 1 3 1 equations for calculation of TTT diagrams are expressed as a function of the volume fraction transformed of various phases. The derived equilibrium volume fraction data is then incorporated in the model for calculation of the TTT diagram using Kirkaldy's'71 and Li's' 1 3 ] equations, which are described in the next chapter. 19 Table 2.1 Composition of AISI 4140 steel used in this work. Element Fe c Mn Ni 4 Cr Si Mo Cu S P Wt% 0.39 0.88 0.23 0.86 0.28 0.162 0.19 0.009 0.013 Subscript 0 1 2 3 4 5 6 - - -Table 2.2 Standard free energy change between a and y phases, for various elements[14'19] Element AG?-" (J/mol) Fe 8933-14.406+(12.083 X 10E-3 X T 2 )-(11.51 X 10E-6 X T 3 ) +(5.23 X 10E-9 X T 4 ) (T< 1000K) 71659-216.84+(24.773 X 10E-2 X T 2 )-(12.661 X 10E-5 X T 3 ) +(24.397 X 10E-9XT4) (T>= 1000K) C -65562+23.815 X T Mn -20520+4.088 X T +1500 X (S a F e )ma g , Si 7087-7.125 X T Ni 12950 + 5.02 X T+383 X (S a F e )mag Cr -1534-19.472 X T +2.749 X TlnT Mo 310-0.285 X T +400 X (S a F e)ma g 4 Since the 4140 steel used in this study is a Cr-Mo steel, the 0.23 wt pet Ni is there as an impurity. 20 Table 2.3 Standard magnetic entropy of pure iron , (S aF e)i (S Fe)mag -11.906 X 10E-4 X T +8.272 X 10E-6 X T 2 -15.079 X 10-9 X T 3 +12.857 X10E-12XT 4 (T< 1075K) 208.24-36710/T-23.973 InT (1075 < T < 1500) 7.87 ^.18 X 10E-4XT (1500K<T) Table 2.4 Self Interaction coefficients of alloying elements in a and y phases Element oY.. t ii s a i i C 4.786+5066/T 1.3 Mn 2.406-175.6/T 3/082-4679/T+l 509.8(SaFe)Magrr Si 26048/T -13.31+44088/T Ni -721.7/T 2.041-2478/T+385.5(SaFe)MagAr Cr 7.655-3154/T-0.661 (InT) 2.819-6039/T Mo -2330/T -0.219-4772/T+402(SaFe)Mag/T 21 Table 2.5 Interaction coefficients between alloying elements and carbon'14' Element C -Mn -4811/T Si 14795/T Ni 5533/T Cr 14.19-30210/T Mo -10715/T Table 2.6 Standard free energy change between cementite and y phase. Element (4/3)°GCEM M C . / 3 - ° G 1 ' M - (1/3)°G y c (J/mol) Fe 1.3220-64.718T+7.481TlnT Mn -14263+10T Si 28535 Cr -24418+16.61T-2.749TlnT Ni 20338-2.368T Mo 19644-0.628T 22 Table 2.7 Interaction coefficients between alloying elements and carbon. Element W0i Mn 8351-15.188T Ni 0 Cr 1791 Mo 0 23 CL, T 5 M f T 6 Co Weight Percentage Carbon Fig. 2.1 Schematic diagram of a portion of the Fe-C phase diagram illustrating the phase boundaries at various temperatures, under isothermal conditions. 24 Take Trial value of Temperature, T Ae3 From the given Composition, Calculate Xt. xr=x, Calculate Thermodynamic Parameters: °Gry, sn Calculate approximate value of Substitute in AG0a~*r Is AGoa~*r=0? Calculate Thermodynamic Parameters: °GM3r^, Wy Calculate Approximate Value of Y, Substitute in AGJ^"1 Is AG0y^em = 0 ? No Yes Stop End Yes No Fig. 2.2: Algorithm for the calculation of the Ae3 and the Ac temperatures, employed in this work. 25 Fig. 2.3 Phase diagram showing the CJ, CYa and the Crxm phase boundaries, calculated from the thermodynamics model for the composite of AISI 4140 steel employed in this work (given in Table 2.1). 1 0.9 0.8 0.6 E 0.5 | 0.4 UJ" 0.3 0.2 0.1 0 • Ferrite • - • - Pearlite Bainite Ae-, ^ N o s e Ae 3 Bs A / / • — i 1 1 — 1 490 540 590 640 690 740 790 840 890 Temperature (°C) Fig. 2.4 Equilibrium volume fractions of ferrite, pearlite and bainite as a function of temperature for the AISI 4140 steel, calculated from the phase diagram illustrated in Fig. 2.3. 3. Modeling Isothermal Transformation Kinetics In this chapter, the transformation kinetics model originally proposed by Kirkaldy et alP^ and modified by Li et alP^ is used to derive the TTT diagram for the chemistry and austenite grain size of an AISI 4140 steel[26] which has a published TTT diagram[26]. Kirkaldy et al.[7] developed this model (for a wide variety of plain carbon and low alloy steels) by calibrating the isothermal transformation equations for various reaction products using the TTT diagrams available in the literature. The advantage of this model is that the entire TTT diagram can be calculated using only the chemical composition and the austenite grain size. Kirkaldy et al.l7] found that when applied to CCT conditions, the model predicted results for plain carbon steels were in very good agreement with the published TTT diagrams for these steels. In order to improve the applicability of Kirkaldy's equations for the low alloy steels, Li et a/.'13^ in a recent publication, have proposed a new set of equations. Li et a/. [ 1 3 ] used a different approach by calibrating the CCT kinetics equations on the basis of CCT diagrams available in the literature and then back calculating the TTT diagram. Li et a/.1-131 found that these equations, when used to determine the continuous cooling transformations (CCT), yielded better final microstructure predictions for the more hardenable steels, such as 4140. This chapter is dedicated to a brief description of the above mentioned two models and a comparison of the model predicted TTT diagrams with a literature TTT diagram for an AISI 4140 steel. 28 3.1 Kirkaldy's Isothermal Transformation Model [ 7 J: Kirkaldy et al.,[1] adopted Kennon's[27] hypothesis that separate ' C curves exist for the ferrite, pearlite and bainite reactions and proposed kinetic expressions for mixed and competing transformation products. These- expressions are based on the Zener formulation and combine the incubation as well as the growth period. The general form of their growth rate equation was based on a rate equation of the form, size m — = D(G,T,X)Xm(l-X)p (3.1) dt where D(G,T,X) is the coefficient involving the effect of ASTM austenite grain G, temperature of the transformation, T, and the transformed volume fraction, X, and and p are the exponents which are empirically determined. Based on this rate equation, they proposed three separate equations, one for the ferrite, one for the pearlite and one for the bainite transformations and calibrated these equations on the basis of a large number of TTT diagrams available in the literature^261. 3.2 Kirkaldy Model 1 7 1 Assumptions : In order to simplify the mathematical description of the TTT diagram, the following assumptions were employed by Kirkaldy et alP^ in developing the isothermal transformation kinetics model. 1. The TTT diagram is assumed to consist of three separate ' C curves above the martensite start temperature, Ms. The curves are associated with the high temperature austenite-to-ferrite (A to F), the intermediate temperature austenite-to-pearlite (A to P) and the lower temperature austenite-to-bainite (A to B) transformations. The initiation 29 (start) times for these individual diffusional transformations (taken as 1% transformed) for the temperature versus time plots are denoted as Fs, Ps and Bs and the transformation completion times (taken as 99% transformed) as F/, P/and B/, respectively. 2. As a consequence of the above stated assumption, it is assumed that any particular microstructure formed under isothermal conditions can not consist of three diffusional reaction products. Hence, no bainite forms at transformation temperatures at which the Ps curve precedes the Bs curve. Similarly no pearlite forms at transformation temperatures at which the Bs curve precedes the Ps curve. Thus, ferrite can be found in combination with either pearlite or bainite, but not with both. However, it is recognized that the final microstructure obtained under continuously cooled conditions can include ferrite and pearlite (formed at higher temperatures) and bainite (formed at lower temperatures). 3. The austenite-to-ferrite and austenite-to-pearlite reactions are treated as separate reactions, meaning that their transformed fractions are expressed in terms of their normalized fractions. Hence, the actual ferrite or pearlite fraction is obtained by multiplying the normalized fraction by the equilibrium amount obtained at a given temperature, this being determined from the thermodynamic multi-component phase boundaries (those developed in chapter 2). 4. The Kirkaldy equations^ 71 assume that the ferrite and pearlite reactions are controlled by phase boundary diffusion. Hence, the driving force for these transformations is proportional to the third power of the undercooling. The bainite reaction is assumed to be controlled by volume diffusion. Hence the driving force for this transformation is proportional to the second power of the undercooling. 30 A schematic diagram exhibiting the shape of the resulting TTT diagram, illustrating the assumptions involved in this model, is shown in Fig. 3.1. In this diagram, although the bainite start curve appears to be asymptote to Ms, the amount of martensite transformed depends only upon the supercooling below the Ms and the retained austenite can still transform to bainite. The same overall assumptions were employed in Li's' 1 3 1 suggested modification to the Kirkaldy model[7], which will be treated in a later section. 3.3 Kinetics of the Isothermal Austenite to Ferrite, Pearlite and Bainite Reactions: The following equations were proposed in Kirkaldy's model'71 for each of the austenite to ferrite, pearlite and bainite reactions. These equations relate to the transformation time and transformed volume fraction in terms of the composition (in wt. pet.), prior austenite grain size (ASTM grain size number, G), undercooling and the effective diffusion resistance due to the presence of alloying elements. The equations have the following form: For the austenite to ferrite reaction, 59.6(%Mn ) + 45(%Ni) + 67.7(%Cr ) + 244(%Mo) (0.3 )2(G'l)/2(Ae3 -Tf exp(-23500 /RT') I(X ) (3.2) For the austenite-to-pearlite reaction, 1.79 + 5.42(%Cr + %Mo + 4%Mo. %Ni) 2(G-')/2(Aei-T)3D I(X) (3.3) 31 Where D, the effective diffusion coefficient, is defined as, 0.01%Cr + 0.52%Mo + , „, (3.4) D exp(-27500/RT) exp(-37000/RT) and I(X) is related to the transformed volume fraction, X, and is defined as, I(X)= I y0.66(l-X) / j _ y \0.66X (3-5) JX0.66(l-X,(j_X) For the austenite-to-bainite reaction, (2.34 + 1 O.J %C + 3.8%Cr + 19%Mo).10 ^ ( 3 6 ) TB- 2(G-')/2(Bs-T)2exp(-27500/RT) To account for the sluggish termination of the bainite reaction, the integral in this reaction has been modified to given by the following equation: _ \exp[X2(1.9%C + 2.5%Mn + 0.9%Ni +1.7%Cr + 4%Mo-2.6)] ' ' J -y0.66(l-X)^j _ ^j).66X ' ' In equations 3.2, 3.3 and 3.6, TF, tp, TB are the times required for the transformation to volume fraction, X, for ferrite, pearlite and bainite, respectively, T is the transformation temperature in °K, Ae3tAei and Bs are the equilibrium ferrite start, pearlite start and bainite start asymptotes respectively, G is the ASTM austenite grain size number, R is the universal gas constant and I(X) is the integration term which reproduces the sigmoidal behavior of the transformation. 32 It must be remembered that the ferrite and pearlite reactions are assumed to be independent reactions, both going to completion. If the true transformed fraction of ferrite and pearlite is to be expressed, the following substitutions must be made, (3.8) FE (3.9) (i-xFE) where XFT and XpT are the true volume fractions of ferrite and pearlite respectively, Xf1 and XpN are the normalized volume fractions respectively, and XFE is the equilibrium fraction of ferrite determined using the Lever law. A similar approach is used for the combined ferrite and bainite reactions. 3.4 Critical Transformation Temperatures: In order to fully describe the TTT diagram, it is first necessary to identify the temperature regions in which the various reactions occur. The Ae3 and Aei temperatures are determined by using the composition dependent thermodynamic model, as described in Chapter 2 of this thesis. As the austenite-»bainite transformation is a diffusional transformation and bainite often nucleates on the original austenite grain boundaries, the austenite grain size is an important parameter in the determination of the onset of this transformation. However, Kirkaldy et a/. [7] have stated that the bainite start asymptote is not very well established by theory. Hence, they used the following grain size independent equation 33 based on their own regression formula derived from the U.S.S steel Atlas and the Steven and Hynes[29] equation, Bs= 637-58(wt % C)-35 (wt % Mn)-15 (wt %Ni)-34 (wt %Cr)-41 (wt %Mo) (3.10) For the martensite transformation, the extent of transformation depends only on the undercooling below the Ms. Steven and Hynes[291, Andrew's [ 1 7 ] and Carpella[30J have developed empirical relationships relating Msio the composition. Kung and Rayment[31] have reviewed the empirical formulae and assessed their validity against experimentally determined Ms temperatures. They proposed that the most accurate prediction of the Ms temperature was obtained using their modification of an existing formula proposed earlier by Andrew's[17] and is given by Eqn.(3.11). Ms=539-423(wt %C)-30.4(wt %Mn)-12.1(wt %Cr)-17.7(wt %Ni)-7.5(wt %Mo) +10 (wt %Co)-7.5 (wt %Si) (3.11) As the austenite—>martensite transformation is a shear transformation, the Ms temperature is independent of the prior austenite grain size. A computer code was written in FORTRAN 77 to calculate the TTT curve for any given composition of steel. The input to this code is the chemical composition, austenite grain size, critical transformation temperatures and the equilibrium volume fractions of ferrite and pearlite at various temperatures. The output of this program is the time corresponding to the various transformed fractions of austenite. An algorithm of this program is presented in Fig. 3.2 and the code is presented in Appendix 1. 34 3.5. The Modified Kirkaldy Model 1 7 1 by Li etal. 1131: As stated earlier, the Kirkaldy model'71 provides very useful equations for the calculation of isothermal transformation kinetics. The parameters in these equations, such as the activation energy term, the individual effects of alloying elements, etc., were determined by fitting these equations to a large number of experimental TTT diagrams in the literature'261. It should be noted that there are several uncertainities in the determination of phase transformation start times (1% volume transformed). As in the published TTT diagrams, the start times are determined by metallographic observations, its determination for 1% volume transformed is not precise. Li et a/. [13 ], in a recent publication, have stated that CCT diagrams are a more reliable tool for calibration. As the CCT diagrams are obtained by dilatometry and metallographic examinations, the volume fractions of the various phases are more precisely determined and thus exhibit more accurate information about the transformation kinetics. Thus, using the CCT diagrams as a tool to calibrate the model, Li et a/. [131 have proposed a set of modified equations to represent the kinetics of the ferrite, pearlite and bainite reactions. These equations were then used to back calculate the TTT diagram. It must be mentioned that their model is based on Kirkaldy's[?1 formulation of the transformation kinetics equations and hence is an attempt to modify the original Kirkaldy'71 model. A brief explanation of the modified model is given below. The equations proposed by Li et a/.'131 have the following form; a complete description of the rationales behind the formulation of this model can be found elsewhere'221. 35 For the austenite-to-ferrite reaction, expf. 1.00 + 6.31(%C ) + l. 78(%Mn ) + 0.31(%Si) +1.12(%Ni) + 2.70(%Cr ) + 4.06(%Mo )J (3.12) S(X) 204,G(Ae3 - T) exp(-27500/RT) S(X) For the austenite-to-pearlite reaction, exp[-4.25 + 4.12(%C ) + 4.36(%Mn ) + 0.44(%Si) +1.71(%Ni) + 3.33(%Cr ) + 5.19y[Mo )J 20J2G(Ae, -T)3 exp(-27500 / RT) (3.13) For the austenite-to- bainite reaction, exp[-10.23 +10.18(%C) + 0.85(%Mn) + 0.55(%Ni) + 0.90(%Cr) + 0.36(%Mo)J v , T B = TT^P ; S(X) 2029G(BS-T)2 exp(-27500/RT) (3.14) where, S(X)= [ . ... ^ T - T T (3.15) lx0A(i-X)(i-xf- 4X For the pearlite and bainite reactions, X is the normalized fraction and all other symbols have the same meaning as in the original Kirkaldy model[7]. The salient features of the modified model are, 1. The modified model[13^ was formulated assuming that the effect of the alloying elements is multiplicative and not additive, as was assumed in the original Kirkaldy model[7]. 36 2. The term which represents the sigmoidal effect of the transformation, I(X), was replaced by S(X) (as given in Eqn.3.15 ) to improve the reaction rate kinetics. 3. From Kirkaldy's equations'71, the temperature at which the reaction time is a minimum, can be found by differentiating the time with respect to temperature and setting it to zero. This temperature, termed the 'nose' temperature, can be expressed by using the following equation, Q = ^S- (3.16) * AT Where Q is the activation energy, AT is the undercooling and TN is the nose temperature. Thus, by calibrating this equation with the nose temperatures available for the various published TTT diagrams, an optimum value of the activation energy of 27500 cal/mole was determined and is used in the model. 3.6 Results of the Isothermal Transformation Kinetics Models 1 7' 1 3 1: The sigmoidal shape of the transformation C curve in Kirkaldy's[7] and Li's [ 1 3 ] equations is represented by the expression I(X) and S(X), given in Eqns. 3.5 and 3.15, respectively. Thus, in order to calculate the times corresponding to a given normalized fraction, X, the value of this expression corresponding to various values of the normalized fraction must be found. To calculate the value of this function, numerical integration of these functions was carried out using the trapezoidal rule with a step size of 10"4. The results of this calculation are shown in Fig. 3.3. It can be seen that the ratio of the time to finish (X=0.99) to the incubation time (X=0.01) is 3.33/0.33 = 10 in Kirkaldy's model. Thus, the Kirkaldy model[7] predicts that at all temperatures, and for all reactions, the 37 time required for completion of the transformation is 10 times that required for the incubation time. For the model proposed by Li et al[u\ however, the ratio of the transformation finish to the start times is about 19. Thus the transformation rates predicted by this model are slower, as compared to the Kirkaldy Model[7]. In order to test the validity of the Kirkaldy model[7] and the Li model[13] against published data, each model was run to calculate the TTT diagram for the chemistry given in Table 3.1 and the austenite grain size of ASTM No. 7.5, of the 4140 steel published in the literature^261. The experimentally obtained published TTT diagram for the 4140 steel is shown in Fig. 3.4. The calculated TTT diagrams for both the Kirkaldy[7] and the L i [ 1 3 ] models, the same steel chemistry and ASTM austenite grain size number of 7.5 are shown in Fig. 3.5. From the published diagram (Fig. 3.4) it can be seen that the experimental ferrite and pearlite reactions are extremely slow as compared to the bainite reaction. Similar, though not identical boundaries can be seen from the model- calculated values shown in Fig. 3.5. Both the Kirkaldy model[7] and the Li model[13] give a very good prediction of the overall shape of the ' C curves corresponding to the individual reactions. Comparing the model predicted values with the values obtained from the published diagram, the model predicted incubation times (taken as 1% volume transformed) for the ferrite, pearlite and bainite reactions, by both models, are reasonably good. The Li model[13] predicted incubation times for the ferrite reaction are slightly longer. However, both models predict different transformation finish times, nose temperatures and the bay region and neither of the models can be said to accurately match that of the published TTT diagram[26]. A comparison of the model predicted nose temperatures with those shown in the published TTT diagram[26] is shown in Table 3.2. A 38 comparison of the model predicted and experimental start (1% transformed) and finish (99% transformed) times for the various reactions at specific temperatures is shown in Table 3.3 and 3.4, respectively. It can be seen that the experimental finish times for all the reactions are at least 50-100 times greater than the start times. Thus neither of the two models tested here, can be said to correctly predict the kinetics of transformation and both overpredict the transformation rates. The differences apparent between the model predicted and the experimental TTT data may be attributed to the metallographic method for the determination of the published 4140 TTT diagram[26l In the determination of start times in these diagrams, specimens were cooled by quenching small samples from an austenitizing salt bath into an isothermal transformation bath for a given holding time. The transformation start time is then obtained by water quenching these small samples after increasing isothermal holding time until the desired 1% volume fraction of the transformed phase is observed. Since the cooling rate experienced during the transfer from the austenitizing bath to the isothermal holding bath is not infinite and decreases as the isothermal holding temperature is approached, there is considerable uncertainty as to the actual thermal history and the associated transformation start times. The times corresponding to the various transformed fractions are also determined in a similar way. In order to determine the constituents of the microstructure, the characterization of the microstructure is done with the help of hardness measurements for specimens with various thermal histories. Therefore, the resulting TTT diagrams must incorporate a reasonable error bar with respect to the times and therefore are less reliable for any direct comparison with respect to the times corresponding to a model-predicted given fraction transformed. 39 However, considering the fact that the TTT curve is generated based on theoretical equations and that these equations make use of only the chemistry and austenite grain size, these models gives a useful basis for calculating the complete isothermal transformation kinetics for the low alloy steels such as AISI 4140, examined in this study. 40 Table 3.1 Composition of the AISI 4140 steel for the published TTT1 Element Fe C Mn Cr Mo Wt% Rest 0.37 0.77 0.98 0.21 Table 3.2 Comparison of nose temperatures calculated from the Kikaldy model[7], the Li model[13] and the published (experimental) data. T nose (Ferrite) Kirkaldy Li Published data 581 590 490 T n o s e (Pearlite) 595 574 650 T n 0 S e (Bainite) 466 466 490 41 Table 3.3 Comparison of times for 1 % normalized transformed fraction at various temperatures obtained from the Kirkaldy model[7], the Li model[13] and the published (experimental) data[26]. Time (s) for Ferrite- 1% transformed Temperature (UC) Kirkaldy Li Published 750 510 1120 75 700 13 32 15 650 5 13 7 600 3 10 4 550 3 12 3 500 5 20 3 Time (S) for pearlite- 1% transformed Temperature (UC) Kirkaldy Li Published 700 6620 661 177 650 263 76 49 600 34 41 115 550 35 41 49 Time (S) for Bainite -1% transformed Temperature (UC) Kirkaldy Li Published 500 48 5 3 450 4 4 4 400 7 7 6 350 20 22 6 42 Table 3.4 Comparison of times for 99 % normalized transformed fraction at various temperatures obtained from the Kirkaldy model[7], the Li model[13] and the published (experimental) data[26]. Time (S) for pearlite- 99% transformed Temperature (°C) Kirkaldy Li Published 700 66400 13072 1274 650 2640 1494 400 600 1530 824 2976 550 1900 816 14384 Time (S) for Bainite - 99% transformed Temperature (°C) Kirkaldy Li Published 500 204 103 10000 450 162 82 356 400 292 147 233 350 850 429 233 43 M s Time Fig. 3.1: Schematic diagram of the TTT curve illustrating the assumptions used in the Kirkaldy's[7] and Li's [ 1 3 ] transformation kinetics models. 44 I Input Parameters • Steel Chemistry • ASTM Grain Size No. • Ae3 and Aej temperatures • equilibrium fractions of ferrite and pearlite as a function of temperature Take Trial Value of Temperature, T starting atAe3 Determine • BS,MS • FC, PC and BC in Eqn. 3.1, 3.2 and 3.4 • Undercooling AT for each reaction • Grain Size term, Activation Energy term ± • For every normalized fraction, X, of ferrite, pearlite and bainite, calculate fraction of austenite transformed • Determine the time for transformation for ferrite, pearlite and bainite in terms of the normalized fractions as well as the true fractions. IsT-Ms? No Yes Fig. 3.2 Algorithm for the calculation of TTT diagram using the Kirkaldy model[7] and the Li model[13] employed in this work. 45 0-C , , , , , , , : • 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Fraction, X Fig. 3.3 Value of I(X) and S(X), the term representing the sigmoidal behaviour of the transformation C curve in Kirkaldy's[7] and Li's [ 1 3 ] equations as a function of normalized fraction, X. i i I Fig. 3.4 The published TTT diagram for AISI 4140 steel[26] for the composition: Fe -0.37% C-0.77% Mn- 0.98% Cr- 0.21% Mo Grain Size: 7-8 Austenitized at 843°C (1550°F). 47 200 4-1 10 100 1000 10000 100000 Time (Seconds) Fig. 3.5 The TTT diagram calculated using the Kirkaldy model[7] and the L i model[13] for the AISI 4140 steel and A S T M Grain Size number of 7.5 reported for the published TTT diagram[26] shown in Fig. 3.4. Subscript Notation: FLs: Start line for ferrite by the L i Model[13], FKs: Start line for ferrite by the Kirkaldy Model[7] PLs: Start line for pearlite by the L i Model[13], PKs: Start line for pearlite by the Kirkaldy Model[7] BLs: Start line for bainite by the L i Model[13], BKs: Start line for bainite by the Kirkaldy Model[7] FLf: Finish line for ferrite by the L i Model[13], FKf: Finish line for ferrite by the Kirkaldy Model[7] PLf: Finish line for pearlite by the L i Model[13], PKf: Finish line for pearlite by the Kirkaldy Model[7] BLf: Finish line for bainite by the L i Model[13], BKf: Finish line for bainite by the Kirkaldy Model[7] 48 4. MODELING CCT KINETICS In the previous chapter, Kirkaldy's171 and Li's [ I 3 ] transformation kinetics equations were used to develop a TTT diagram for the 4140 steel which was then compared to a published TTT curve. In this chapter, these same equations are used in conjunction with the additivity rule to predict the kinetics of transformation under continuous cooling conditions. This chapter describes the method of modeling the CCT kinetics from Kirkaldy's'71 and Li's' 1 3 1 equations. 4.1 Description of the CCT Model: To quantify the transformation to ferrite, pearlite, bainite and martensite under CCT conditions, the continuous cooling curve for a given cooling rate was approximated as a series of isothermal steps of infinitesimally short duration and the volume fraction of austenite transformed and the true fraction of a given phase formed (depending on the temperature) is calculated in each of these time steps. The volume fraction is then summed to get the final fraction of austenite and the particular phase formed. 4.1.1 Assumptions in the CCT model: 1. It is important to emphasize that the isothermal transformation kinetics can be used to predict continuous cooling transformations using the additivity principle'81 only if the instantaneous transformation rate is solely a function of the fraction transformed, X, and the temperature of transformation, T. As Kirkaldy's [ 7 ]and Li's' 1 3 1 transformation kinetics equations, which describe the incubation period as well as the transformation period, obey this condition, 49 these equations can be used to predict the CCT kinetics using the additivity principle. 2. It is assumed that all of the austenite is transformed under continuous cooling conditions and at room temperature no retained austenite is present in the final microstructure. Once the martensite start temperature is reached, any previous reaction is terminated and all the remaining austenite is transformed to martensite as the sample cools to room temperature1. 4.1.2 Calculation of Kinetics of the Transformations under Continuous Cooling Conditions: The CCT kinetics is calculated using the Kirkaldy's[7] and Li's [ 1 3 ] isothermal transformation equations adopting the additivity principle to describe the combined incubation and transformation period. A brief description of the model calculation is given below. 1. Under CCT conditions, it is first necessary to estimate the transformation start temperatures for the various reactions. The start temperatures in this model are found by using Scheil's[8] additivity principle applied to the incubation period. This principle can be mathematically expressed by Eqn. 4.4, 1 While making this assumption, it is recognized that M/may be below room temperature. However, the alternative diffusional transformation to bainite is expected to be very slow. 50 where tst (T) and tst are the times required to start the transformation isothermally and under continuous cooling conditions, respectively. The summation is evaluated for each of the ferrite, pearlite and bainite reactions. It is important to note that the value of Tj17 is taken as Ae3 for the ferrite transformation, Aet for the pearlite transformation and Bs for the bainite transformation. In addition, the value of tsJ77 is taken as the incubation time corresponding to a true fraction, X=0.01 for the ferrite and bainite reactions taken from the TTT model developed in Chapter 3. For the pearlite reaction, this value is the difference between pearlite and ferrite start times, which represents the contribution made only by the pearlite reaction. When this sum attains a value of 1, the corresponding temperature is taken as the start temperature for that particular reaction. 2. The kinetics of the transformation is calculated by integrating Kirkaldy's'71 and Li's [ 1 3 ] reaction rate equations along the thermal path. The procedure involves dividing the continuous cooling path into small isothermal segments. The time step, At, is taken as 0.001 sec. and for a given cooling rate the temperature step, AT, is calculated. At the transformation start temperature it is assumed that the true fraction transformed has a value of 0.01. An example of this procedure for the austenite-to-ferrite transformation is presented below. Kirkaldy's reaction rate equations'71 are in terms of the normalized fraction transformed. Hence, the fractions of various reaction products are calculated in terms of the normalized fractions and then converted into true fractions by using the equilibrium fractions. In the first time segment after incubation, At, at temperature, Ti, the normalized fraction of ferrite formed is calculated by using the following equation: 51 XNF, =0.01 + — At (4.2) dt where, Xs'Fi is the normalized fraction of ferrite and dX_ _ (0.3)2«G-'>/2)(Ae3 -T)3 exp(-23500/RT) x < i 6 6 ( I . x ) _X)0.66X ( 4 dt ~ 59.6(%Mn) + 1.45(%Ni) + 67.7(%Cr) + 244(%Mo) is calculated at X=0.0\. Thus, the normalized fraction can then be converted into the true fraction by using the relation, XTFI=X^.XFE(T,) (4.4) where, XTFi is the true fraction of ferrite, XNpi is the normalized fraction of ferrite and XFE(TI) is the equilibrium fraction of ferrite at temperature, T{. In the second cooling segment, At2, since each isothermal temperature will produce a different fraction of ferrite, it is necessary to change XSFI to its equivalent fraction at Tj given by, where X'NFj is the equivalent fraction at T2, XFE(TI) and XFE(T2) are the equilibrium fractions of ferrite at T] and T2 respectively. The new fraction of ferrite at T2 is then calculated by, 52 (4.6) in which , dX/dt is calculated at X=X'NFI-This procedure is continued until the pearlite start temperature, 7> , is reached, or the bainite start temperature, TBCCT, is reached, or the calculated normalized fraction, X'NF, attains a value of 1. The procedure for the calculation of pearlite and bainite transformation kinetics is similar with the following equations being integrated along the thermal path, For pearlite transformation kinetics, dX (0.3)2(( > ^(Ae, —T)D ^o.66(i-x) ,j _ ^ XQMX (47) dt ~ 1.79 + 5.42(%Cr + %Mo + 4%Mo. %Ni) For bainite transformation kinetics, dX _ (0.3)2<(G-')/2)(BS-T)2exp(-27500/RT) x0M"-x)(l -X)0MX dt ~ (2.34 + 10.1%C + 3.8%Cr + 19%Mo)10-4 expfAX2] (4.8) where, A = 1.9%C + 2.5%Mn + 0.9%Ni +1.7%Cr + 4%Mo -2.6 (4.9) Thus, by integrating Kirkaldy's'7^ and Li's [ 1 3 ] transformation kinetics equations along the thermal path for a given cooling rate, the true fractions of ferrite, pearlite and bainite in the final microstructure is obtained. The integration along the thermal path is performed till Ms is reached after which it is assumed that the rest of the austenite is transformed to martensite. The values of the fraction of various constituents in the final microstructure are compared with the experimental data as described in the next chapter. 53 5. EXPERIMENTAL TESTING AND VALIDATION OF CCT MODEL RESULTS It was initially intended to experimentally generate the complete TTT diagram for the 4140 steel being examined. However, the published TTT diagram'26^ for an AISI 4140 steel confirmed that at certain temperatures the transformation finish times are of the order of 104 to 105 seconds. After a few preliminary TTT tests, it was confirmed that the times to complete the isothermal transformation into ferrite, pearlite and bainite were very long and hence TTT tests would be very time consuming. Further, to get sufficient data for the determination of complete isothermal transformation kinetics, it was necessary to quench the samples at various intervals at various temperatures and thus would have required an even larger number of experiments. As the overall intention of this work was to assess the original Kirkaldy model'7] and the modified Li model[13] for predicting the CCT kinetics, it was decided to only obtain experimental CCT data to compare with model predictions. Thus, to experimentally generate the required CCT data, continuous constant cooling rate transformation tests were performed using the GLEEBLE 1500 thermo-mechanical simulator. These tests were performed to determine the transformation start times under various cooling rates and to determine the final microstructure and the hardness resulting from these cooling rates. In this chapter, first the details of the sample preparation, the test conditions and the metallographic procedures used to characterize the microstructures for the AISI 4140 steel used for this study, are presented. Then the experimental data is analyzed to obtain experimental CCT results. The two CCT models developed using Kirkaldy's[7] and 54 Li's [ 1 3 ] transformation kinetics equations (described in chapter 4) are used to predict the final microstructural distribution for the 4140 steel using the experimental thermal history. These predictions are then compared with the experimental data to test the model predictions. 5.1 Sample Preparation: To choose a design for the steel sample, it was necessary to ensure that the sample geometry provided sufficient wall thickness for microstructural analysis, while ensuring that the temperature gradients introduced during continuous cooling tests were minimized. A cylindrical specimen was selected for the CCT test specimen with a 6 mm I.D., a 8 mm O.D. and 20 mm in length. The schematic diagram of the cylindrical test specimen is shown in Fig. 5.1. To obtain the test sample from the available 1 in. diameter steel rod, the rod was cut into four equal quarters; each of these pieces was machined to a test specimen. This procedure was adopted in order to avoid possible centerline segregation in the center of the original rod. To ensure dissolution of existing alloy carbides in the original rod, the machined samples were enclosed in a quartz tube, which was then evacuated and sealed and homogenized in a electric element heating furnace at 1200° C for 2 hours. After homogenizing, the quartz tube was removed from the furnace, immediately broken as the specimens were quenched in water for subsequent GLEEBLE testing. 55 5.2 CCT Test Conditions: During the austenite decomposition the volume of the specimen changes since the product phases have different specific volumes than the parent phase. The relative volume change due to a phase transformation can be obtained by measuring the change in the diameter of the specimen, i.e., measuring the diametral dilation. The resulting diametral dilation in the cylindrical test specimen was measured at midlength of the specimen in tests performed on the GLEEBLE 1500 thermo-mechanical simulator. This data monitored the progress of the transformation and made it possible to identify the phases transformed in the final microstructure. Each test specimen was resistance heated in the GLEEBLE 1500 to the selected austenitizing temperature, held for a desired amount of time and then subsequently control cooled with the measured cooling rate. The temperature was monitored and controlled using a chromel-alumel thermocouple spot welded to the outside surface of the specimen at the same mid-axis position where the diametral dilatometer is attached. The AISI 4140 steel samples were austenitized at 850° C 1 for 3 minutes, being heated to this temperature at a rate of 5°C/s. The samples were then cooled at 7 different cooling rates, ranging from 0.1°C/s to 12°C/s, in order to obtain the desired transformation kinetics data. The cooling rates examined for the CCT tests for the 4140 steel were selected to duplicate those used in the published CCT diagram[26] for this steel. The CCT test conditions are summarized in Table 5.1 and are schematically shown in Fig. 5.2. 1 This temperature is more than 50 degrees above the Ae3 temperature for this steel. 56 5.3 Microstructural Characterization: After testing, the samples were cut near the thermocouple location along the transverse axis, cold mounted and polished to a 1 p.m diamond finish. The samples were then etched using an immersion or swab etch solution of 2% Nital for about 5-10 seconds. The sample surface was then washed with water and denatured alcohol and air dried for examination. To improve the clarity of the microstructure, the polishing and etching sequence was repeated at least three times for each sample. The resulting microstructure was then characterized and photographed as close as possible to the thermocouple location. For a few test samples the microstructures were examined using the scanning electron microscope to determine i f any alloy carbides had remained undissolved, during the initial austenitizing treatment. 5.4 Determination of the Prior Austenite Grain Size: As the rate of transformation is a function of the prior austenite grain size, it was necessary to metallographically determine the prior austenite grain size. A CCT sample corresponding to a cooling rate of 12°C/s, having a mixture of bainite and martensite in the microstructure, was chosen for this purpose. The sample was polished to a 1 urn diamond finish and then etched for 45 to 60 minutes by immersing in a solution of saturated aqueous picric acid containing sodium dodecyclebezene sulfonic acid as a wetting agent [ 3 2l The etching process was carried out under ultrasonic cleaning conditions. The wetting agent helps expedite the attack of the saturated picric acid solution on the bainite outlining the original austenite grain boundaries. The austenite grain size was quantified by measuring the total number of grains (number of internal 5 7 plus lA the number of edge grains) in a field of view of a given magnification containing at least 300-400 grains and employing the ASTM standard procedure for the austenite grain size determination. Three such fields of view were used for the grain size determination. 5.5 Measurement of Hardness: The hardness of the microstructure was also measured to help identify the constituents of the final microstructure. The sample preparation for hardness testing was the same as that used for microstructural characterization. For each of the CCT samples, Vickers diamond hardness (VHN) values were determined by taking at least 6 readings across the thickness of the sample at a load of lKg. The mean, maximum and minimum hardness values were recorded. 5.6 CCT Experimental Results: 5.6.1 CCT Diagram: In order to generate the CCT diagram for the AISI 4140 steel employed in this study (the composition of which is given in Table 2.1), a number of dilatometric continuous cooling tests were performed, as described in Table 5.1. The experimental dilation associated with the ensuing continuous cooling transformations in each test is shown in Fig. 5.3 (a) and (b). For the 4140 steel used in this study, it can be seen that the high temperature dilation curves show an austenite-to ferrite and pearlite transformation only at cooling rates from 0.1°C/s to 0.75°C/s. The dilation due to both the high temperature ferrite plus pearlite reactions and the low temperature bainite reactions exists 58 at cooling rates between 0.25°C/s to 0.75°C/s. At all cooling rates above l°C/s, the microstructure consisted predominantly of only low temperature reactions involving bainite and martensite. Thus, in this steel, for the given austenitizing conditions and thermal history, at low cooling rates ferrite is the predominant phase and it appears with pearlite and bainite. At higher cooling rates bainite is the predominant phase and it appears with ferrite, pearlite and martensite. To determine the transformation start temperatures and the relative fractions of the various phases formed, the thermal expansion coefficients of the parent phase (austenite) and the product phases (ferrite+pearlite and bainite) were determined. As the thermal expansion coefficient of a particular phase depends upon the dilation being measured over a specific temperature range, the dilation data, which is expressed in arbitrary size units, must be converted to mm units . The thermal expansion coefficient of the parent (austenite) and the product (ferrite+pearlite and bainite) phases were determined from the slope of D-D (/Do vs temperature plots. The measured average value for the thermal expansion coefficient for the parent phase (austenite) was found to be 2.11 xlO'5 mm mm"1 °C"1. The thermal expansion coefficients of ferrite+pearlite and bainite were found to be 1.66 xlO'5 mm mm"1 °C"' and 1.48 xlO"5 mm mm"1 °C"', respectively and were determined from the continuous cooling tests at cooling rates of 0.1°C/s and l°C/s, respectively. These tests were chosen since in these tests, the austenite is almost completely transformed to a particular product phase (either ferrite+pearlite or bainite). The onset of the y—>a and y—>B transformations were determined from the 2 The dilation data is converted using the following expression: D(mm)=2.88315 (D M e a S ) + Do Where D M e a s is the measured dilation in arbitrary units and D 0 is the initial diameter of the sample in mm. 59 deviation of the diametral dilation data from that based only on the thermal contraction of austenite alone. An experimental CCT diagram showing the transformation start temperatures (A"=0.01) and the time-temperature response at various cooling rates is shown in Fig. 5.4. No significant variation in the ferrite and bainite transformation start temperatures, as a function of cooling rate, was observed for the experimental cooling rates employed; this is consistent with that shown in the published CCT diagram[26] for this steel in which the ferrite and bainite start temperatures are essentially constant for these cooling rates. Hence, for the cooling rates examined, the transformation start temperatures were expressed as a linear function of the cooling rates using the following equations: For the ferrite start temperature, T„.= (-0.6754)(CR) + 686.95 (for allC.R < 1°C/s) (5.1) For the bainite start temperature, TBs = (-0.89040(CR) + 497.59 (for all CR. > O.fC/s) (5.2) It should be emphasized that these relations are true only at temperatures far from the nose temperatures. 5.6.2 Determination of the Continuous Cooling Transformation Kinetics from the CCT Data: The CCT dilation data and the thermal history of each CCT curve were used to calculate the kinetics of the decomposition of austenite and the fraction of each phase 60 formed in the final microstructure. The procedure involved in the calculation of the transformed fraction is briefly outlined below: 1. At each temperature, the difference between the two successive temperatures, AT, and the difference in the experimental dilation, ADexp, is obtained from the experimental data. 2. The experimental dilation, ADexp, is corrected by subtracting the dilation due to the thermal contraction, ADthermal, to obtain the dilation due only to the transformation, ADtra". During the transformation, the thermal expansion coefficient, amix, of the transforming structure was determined in terms of the expansion coefficients of austenite, OCA, ferrite+pearlite, CCF+P, and bainite, ae, using the following expression: amix= Xf+PCCT aF+P + (XA-XF+PCCT) CCb+(1-XA) aA (5.3) Where Xp+p and XA are the true volume fractions of ferrite+pearlite and austenite, respectively and XA-XF+P is the true fraction transformed to bainite. This expression was applicable only till the Ms temperature is reached. After the Ms temperature is reached, the rest of the austenite is assumed to be transformed to martensite. 3. The corrected dilation due to transformation is then used to estimate the fraction of austenite transformed. First, the total change in the dilation due to transformation is approximated and then incremental changes were made until the final fraction of austenite transformed attained a value of 1. As an example, the dilation data and the calculated kinetics of the austenite transformation, as a function of temperature for a CCT test at 0.5°C/s, are shown in 61 Fig. 5.5. For this test, both high temperature and low temperature transformations exist. First a true fraction of 18% of the austenite transforms to a mixture of ferrite and pearlite and the remaining austenite then transforms to bainite. 5.6.3 Prior Austenite Grain Size: As the kinetics of the diffusion controlled transformations are dependent on the prior austenite grain size, its measurement is necessary for the prediction of continuous cooling (CCT) kinetics. The austenitizing temperatures used for the 4140 steel in the CCT tests was 850°C with a holding time for each sample of 3 minutes. The microstructure of the CCT sample corresponding to a cooling rate of 12°C/s was used to measure the prior austenite grain size, as shown in Fig. 5.6(a); the outlines of the original austenite grain boundaries are shown in Fig. 5.6(b). The microstructure in three identical tests was examined and the three mean grain sizes were found to be 13.44, 13.51 and 14.75 um. The associated equivalent A S T M grain size number was calculated to be between 9 and 10; a mean value of 9.5 was used for the calculations. 62 5.6.4 Microstructural Characterization: The microstructures corresponding to cooling rates of 0.1, 0.25, 1 and 12°C/s are shown in Fig. 5.7 (a) through (d) along with the corresponding Vickers hardness values (Load:l Kg.). The microstructure at 0.1°C/s shows the formation of grain boundary ferrite and pearlite. Extensive formation of pearlite was observed only at this cooling rate; at cooling rates of 0.25°C/s, 0.5°C/s and 0.75°C/s, pearlite nodules were observed along with bainite. The microstructure at a cooling rate of 0.25°C/s [Fig.5.7 (b)], shows grain boundary ferrite, acicular bainite and few dark etched patches of pearlite. The light Nital etched microstructure obtained at a cooling rate of l°C/s shown in Fig. 5.7 (c), shows mainly gray etched acicular bainite and very little darker etching pearlite. The microhardness (50 gm.) of the acicular gray etched areas was found to be 350 to 380 VFIN, consistent with the presence of bainite. In this microstructure, some white patches of ferrite are also probably present. However, the ferrite fraction is very small. The microstructure at 12°C/s shown in Fig. 5.7 (d), shows dark etched bainite in a gray lightly etched martensite matrix. The microhardness (50 gm.) of the gray matrix phase in this microstructure was found to be 570-580 V H N , which confirmed the presence of the martensite phase. In this microstructure, a few very small white areas of ferrite, are present which outline the original austenite grain boundaries. 5.7 Comparison of the CCT Model with the Experimental CCT Data: The CCT model results, calculated using the Kirkaldy model'7 1, can be expressed in the form of a CCT diagram, as shown in Fig. 5.8. The CCT diagram shows lines corresponding to transformed fractions of 1%, 25%, 50%, 75% and 99% of austenite. The 63 model calculated ferrite, pearlite and bainite transformation start temperatures are also superimposed on this diagram. The distribution of final microstructure at a given cooling rate can be expressed by following the particular cooling curve and then determining at what fraction of austenite does it leave a particular reaction field. A comparison of the microstructure distribution obtained from the experimental data and that calculated for the same 4140 alloy using both Kirkaldy's'71 model and Li's model[13] for an ASTM grain size number of 9.5, is shown in Table 5.2. The experimentally obtained fractions are only approximate numbers indicating the relative amounts of various phases. The experimental ferrite+pearlite fractions at cooling rates above l°C/s, are approximately determined from the microstructures, as it was difficult to determine its content either metallographically or from the dilation data. It can be seen that Kirkaldy's model'71 predictions consist mostly of only ferrite+pearlite and bainite in the final microstructure. Martensite is predicted to be present only at 12°C/s. The experimental data, however, shows some martensite along with bainite at cooling rates above l°C/s. This data comparison indicates that the Kirkaldy model'71 overpredicts the kinetics of the ferrite and the bainite reactions. The overprediction of the reaction rates is due to the term I(X), which represents the sigmoidal nature of the transformation. As discussed in chapter 3, the value of this term for transformation finish times is only 10 times that of the transformation start times for all reactions. This overestimates the reaction rates. The Li model'131 predicted ferrite+pearlite fractions are in reasonable agreement with the experimental ferrite+pearlite fractions. Also, the overall trend in predicting a mixed microstructure consisting of bainite and martensite, above l°C/s, is shown in the Li model'131. This is attributed to the formulation of the S(X) in Li's equations'131, which 64 when compared to I(X), gives much slower reaction rates. It must also be stated that the differences observed in the actual numbers corresponding to the experimental and model calculated values may be attributed to the method of calibration of the Li model. Although the Li model'131 is calibrated on CCT diagrams generated by dilatometry and metallography, the cooling conditions encountered in these diagrams may not be controlled, which would give a combination of linear and Newtonian cooling for a given cooling program. The model in this work is used to predict the microstructure under controlled cooling conditions with a constant cooling rate. The model performance is also evaluated by comparing the model calculated and measured Vickers hardness values. The Creusot-Loire laboratories1331 have developed empirical equations relating the hardness of a microstructure to the composition (the material property) and the cooling rate (the particular thermal history). These equations were based on a statistical analysis of a large number of CCT diagrams. Kirkaldy et al.[71 in evaluating the performance of their model for predicting the mechanical properties, have used these equations for the prediction of Vicker's hardness values for a given distribution of microstructure. These equations are given as follows: HV( Martensite ) = 127 + 949%C + 27%Si + ll%Mn + 8%Ni + 16%Cr + 21Log10Vr ( 5 A ) LW (Bainite) = -323 + 185%C + 330%Si + 153%Mn + 65%Ni + 144%Cr + 191%Mo + Logw[Vr(89 + 53%C - 55%Si - 22%Mn - 10%Ni - 20%Cr)] ( 5 5 ) HV (Ferrite + Pearlite ) = 42 + 223%C + 53%Si + 30%Mn +12.6%Ni + 7%Cr + 19%Mo + Logl0fVr(10-19%Si + 4%Ni + 8%Cr + 130%Mo)J 65 Where, HV (Martensite), HV(Bainite) and HV(Ferrite+Pearlite) are the Vickers hardness numbers of martensite, bainite and ferrite and/or pearlite structures, and Vr is the cooling rate in °C/hr at 700°C. The resultant hardness for a mixed microstructure for a given cooling rate is calculated by taking a volumetric weighted average of the various phases. The following equation is used to calculate the resultant hardness, [Vol. fraction of ( ferrite + pearlite )J.HVferrile+pearlile + Vol. fraction of bainite J.HVBalnile [Vol. fraction of (martensite)J.HVmarlenxile (5_ j) A comparison of calculated hardness values of the microstructure predicted by the Kirkaldy model171, the Li model[13] and the measured mean hardness values is shown in Table 5.3 and is plotted in Fig. 5.9. The range of experimentally measured hardness values obtained for a particular sample is shown in the form of error bars. For low cooling rates (less than l°C/s), there is good agreement between the Li model113] and the Kirkaldy modelt7] calculated values and the experimental data. This is because of the better agreement between the model calculated and experimental ferrite+pearlite fractions. Further, the hardness of the ferrite+pearlite microstructure (calculated by Eqn. 4.14) has the smallest value amongst the three microstructural constituents (ferrite+pearlite, bainite and martensite) and hence affects the final hardness value the least. On the other hand, martensite makes maximum contribution to the hardness of the final microstructure. As the Kirkaldy model[7] predicted microstructure distribution does not consist of any martensite, the hardness values are much smaller. Above cooling rates of l°C/s, Li's model[13] predicts more martensite in the final microstrcuture; the hardness values calculated from this model are therefore higher as 3 Vr must be the cooling rate consistent with producing the particular microstructure of interest. 66 compared to those for the Kirkaldy model^ values. For the same reason, Li model'131 values have a better match with the experimental hardness values. At higher cooling rates, i.e. at 6°C/s and 12°C/s, the Li model[13] calculated values are higher than the experimental data. This is because of the overpredicted martensite fractions in the Li model. 5.8 Sensitivity of the Kirkaldy Model: The Kirkaldy model ^ relates the time required to form a given volume fraction of a phase to the chemical composition, undercooling and the prior austenite grain size. Thus, it would be useful to check the sensitivity of this model to some of these parameters in order to determine the relative importance of each parameter. 5.8.1 Sensitivity of the Kirkaldy Model 1 7 1 to austenite grain size, G: The value of the term, 2(G"1)/2, which represents the effect of austenite grain size in Kirkaldy's^ equations is plotted as a function of the ASTM grain size number G and shown in Fig. 5.10. It can be seen that the absolute value of this term is higher for higher value of G (finer austenite grain size). The effect of changing the austenite grain size on the final microstructure is examined and shown in Fig. 5.11 (a), (b) and (c). This figure compares the volume fractions of ferrite+pearlite, bainite and martensite in the final microstructure for ASTM grain size numbers of 6, 8 and 10. As expected, as the ASTM grain size number increases, i.e., the grain diameter decreases, the final microstructure contains a greater amount of the equilibrium phases ferrite and pearlite and less of the non-equilibrium phases bainite and martensite. The critical cooling rate at which 67 martensite begins to form is also seen to be reducing from 10°C/s for G=10 (12.9 um), to 5°C/s for G=8 (20.9 um), to 3°C/s for G=6 (28.9 um) as the grain size increases. 5.8.3 Sensitivity of the Kirkaldy Model[7] to the chemical composition: In order to check the sensitivity of the Kirkaldy model'7 1 to variations in the chemistry of a 4140 steel, the model calculated austenite transformation kinetics for the extremes of the chemistry range acceptable for this grade, Composition A: Fe- 0.37 % C -0.75 %Mn- 0.15 %Si - 0.8 %Cr- 0.15% Mo and Composition B: Fe- 0.43 % C - 1.0 %Mn-0.35 %Si-1.0 %Cr- 0.28 %Mo, at a cooling rate of l°C/s, is shown in Fig. 5.12. The effect of this difference in the composition on the kinetics of transformation is noticeable, but not very significant. 68 Table 5.1 Continuous cooling transformation test conditions employed for the AISI 4140 Steel. Test Heating rate °C/s Austenitizing Condition Cooling Condition °C/s Temperature °C Time s C41CC01 5 850 180 0.01 C41CC025 5 850 180 0.25 C41CC050 5 850 180 0.5 C41CC075 5 850 180 0.75 C41CC1 5 850 180 1 C41CC2 5 ' 850 180 2 C41CC6 5 850 180 6 C41CC12 5 850 180 12 69 Table 5.2 Comparison of ferrite+pearlite, bainite and martensite volume fractions between experimental data, Kirkaldy model calculations'71 and Li model calculations[13] Cooling Rate °C/s Ferrite+Pearlite (Volume Fractions) Experimental Data Kirkaldy Model m Li Model l l 3 J 0.1 100 100 86 0.25 30 96 29 0.5 14 53 11 0.75 18 48 2 1.0 -5-10 45 1 2.0 -5 43 0 6.0 ~5 32 0 12.0 ~5 18 0 Cooling Rate °C/s Bainite (Volume Fractions) Experimental Data Kirkaldy Model l / J Li Model l l 3 J 0.1 0 0 14 0.25 70 4 71 0.5 86 47 89 0.75 67 52 97 1.0 -80-85 55 99 2.0 -58-63 57 71 6.0 -30-35 68 11 12.0 -35-40 29 3 Cooling Rate °C/s Mar tensite (Volume Fractions) Experimental Data Kirkaldy Model l / J Li ModelL 1 3 J 0.1 0 0 0 0.25 0 0 0 0.5 0 0 0 0.75 15 0 0 1.0 -5 0 0 2.0 -32-37 0 29 6.0 -60-65 0 89 12.0 -55-60 53 97 70 Table 5.3 Comparison of hardness of the final microstructure calculated from the Kirkaldy model[7], the Li model[13] and the experimental data. Cooling Rate °C/s Hardness (VHN) Experimental Data Kirkaldy Model l / J Li ModelL l j J 0.1 239 214 222 0.25 219 222 272 0.5 239 263 299 0.75 280 273 312 1.0 385 280 322 2.0 478 292 385 6.0 504 322 592 12.0 530 505 619 71 —H r— 3 -H o CN All dimensions in mm Fig. 5.1 Geometry of the GLEEBLE specimen used in the CCT tests. T(°C) 850°Cfor3min 5°C/s Time(s) Note: Not to Scale Fig. 5.2 Schematic representation of the continuous cooling tests. 72 0.03 0.025 Temperature(°C) (a) 0.03 -0.005 -0.01 J Temperature (°C) (b) Fig. 5.3 Experimental dilation versus temperature data for various continuous cooling conditions: a) for cooling rates 0. l°C/s, 0.25°C/s, 0.5°C/s, 0.75°C/s. b) for cooling rates l°C/s, 2°C/s, 6°C/s, 12°C/s. 73 Fig. 5.4 Experimental Continuous Cooling Transformation (CCT) diagram for the 4140 steel employed in this study, showing the start of transformation temperatures for ferrite, bainite and martensite transformations at various cooling rates, for the measured ASTM grain size number 9.5. 74 0.03 -i 0.025 M 0.005 -0.005 T3 O E i_ to c to 1— I— CD ' c (D to < c: o o CD 100 200 300 400 500 600 700 800 9(D0 Temperature ( C) (a) 0 100 200 300 400 500 600 700 800 900 Temperature (°C) (b) Fig. 5.5 Experimentally determined dilation data for the 4140 steel used in this study, for a cooling rate of 0.5°C/s; a) Showing transformation start and stop temperatures b) Showing calculated kinetics of transformation of austenite to ferrite and pearlite, then bainite. 75 (a) (b) Fig. 5.6 : a) Photomicrograph of an air cooled sample of AISI 4140 steel (austenitizing condition: 850°C for 3 min.) etched with a solution of saturated aqueous picric acid along with a solution of sodium benzene sulphonate as a wetting agent, for 1 hour. b) Outline of prior austenite grain boundaries used for grain size measurement. 76 -(a) (b) Fig. 5.7 Photomicrographs showing microstructures of continuously cooled samples of the AISI 4140 steel austenitized at 850°C for 3 min and etched with 2% Nital. a) At a cooling rate of 0.1°C/s : The Microstructure shows ferrite(white phase) and equiaxed pearlite (black phase). X400, VHN: 239. b) At a cooling rate of 0.25°C/s : The Microstructure shows ferrite(white phase), equiaxed pearlite (equiaxed dark phase) and upper bainite (acicular gray etching ohaset. X400. VHN: 296. 7 7 (c) Fig. 5.7 Photomicrographs showing microstructures of continuously cooled samples of the AISI 4140 steel austenitized at 850°C for 3 minutes and etched with 2% Nital. c) At a cooling rate of l°C/s : The Microstructure shows bainite (acicular gray etching phase) and pearlite (darker etching phase). Very few areas with ferrite are also probably present. X400, VHN: 365. d) At a cooling rate of 12°C/s : The Microstructure shows mainly bainite (acicular dark vihase) and martensite Client erav matrix chase). X400. VHN: 504. 78 Fig. 5.8 CCT diagram calculated from the Kirkaldy Modelm for a range of continuous cooling rates, showing the start temperatures for various transformations and the lines corresponding to 25%, 50%, 75% and 99% of austenite transformed. 79 700.00 n 100.00 0 2 4 6 8 10 12 14 Cooling Rate (°C/s) Fig. 5.9 Comparison of Vicker's hardness numbers calculated using the Kirkaldy model[7] and the Li model[13'14] and the experimentally measured data. 80 3 4 5 6 7 8 9 10 11 12 13 Austenite Grain size (ASTM No.) Fig.5.10 Value of the term, 2(U"1)/2 which represents the effect of the austenite grain size in Kirkaldy's'71 equations, as a function of ASTM grain size No. G. 0 5 10 15 20 25 Cooling Rate (°C/s) Fig. 5.11 Distribution of the final microstructure as a function of cooling rate rC/s) for austenite grain size numbers of 6, 8 and 10, based on Kirkaldy's equations^ 71 for the 4140 steel used in this work: a) For the formation of ferrite+pearlite; b) For the formation of bainite; c) For the formation of martensite 82 Fig. 5.12 Calculated kinetics of transformation of austenite, using the Kirkaldy model[7], at a cooling rate l°C/s, for the least hardenable (composition A) and most hardenable (composition B) compositions for 4140 steel. Composition A: Fe- 0.37 %C- 0.75 %Mn- 0.15 %Si- 0.8 %Cr- 0.15% Mo Composition B: Fe- 0.43 %C- 1.0 %Mn- 0.35 %Si- 1.0 %Cr- 0.28 %Mo 83 6. SUMMARY AND CONCLUSIONS The objective of this work was to characterize the kinetics of austenite decomposition under continuous cooling conditions in a low alloy AISI 4140 steel using Kirkaldy's'71 and Li's [ 1 3 ] models and to experimentally test the results of these models by performing GLEEBLE controlled continuous cooling experiments. Kirkaldy's model[7] consists of isothermal transformation equations that combine the incubation and transformation periods and relate the times corresponding to the various transformed fractions for each of the ferrite, pearlite and bainite reactions. Li's model[13] consists of similar equations describing the isothermal transformation kinetics and is an attempt to modify Kirkaldy's model[7] to improve its performance to predict CCT kinetics. As these models use information from the equilibrium phase diagram, the phase diagram for a 4140 steel was derived using a thermodynamics based model. The results of the thermodynamic model is based on the equilibrium between y, a and the cementite ([Fe,X]3C) phases. The equilibrium temperatures are calculated by equating the chemical potentials corresponding to these phases, for each of carbon and the substitutional elements and then solving the system of governing equations. The results showed that the Ae3 and Aei temperatures for this steel are 766°C and 716°C, respectively. It must also be emphasized that a low alloy steel, being a multi-component system, exhibits a temperature range for the three phase equilibrium at Aej. However, to simplify the calculation, the maximum temperature in this temperature range are calculated. The model calculated phase diagram was then used to determine the equilibrium ferrite and pearlite fractions as a function of temperature, by using Lever law calculations. The 84 calculated equilibrium volume fractions showed that the maximum volume fraction of ferrite is about 0.48, which occurs at the Aei temperature. This information was used in Kirkaldy's[7] and Li's [ 1 3 ] models to derive the TTT diagram for this steel and the results were compared to a TTT diagram available in the literature for an AISI 4140 steel. The isothermal transformation kinetics data calculated from Kirkaldy's model[7] and Li's model[13] showed that these models can be used to adequately represent the overall shape of the TTT diagram for this steelt26l The model calculated transformation start times (0.01 volume fraction transformed) for various reactions also are satisfactorily represented. The model calculated bainite transformation kinetics was observed to be faster than the combined ferrite+pearlite reaction, which is consistent with the information from the published diagram[26l However, the discrepancy between the model calculated values and those from the published diagram increases at later stages of transformation. Comparing the performance of Kirkaldy's[7] and Li's [ 1 3 ] models under isothermal conditions, neither of the models could be said to produce better results than the other. Since there are discrepancies in the model calculated nose temperatures and large errors in the transformation finish times, care has to be exercised in calculating the TTT curves for the low alloy steels using these models. Kirkaldy's[7] and Li's [ 1 3 ] models were then used to calculate the continuous cooling transformation kinetics and the results were compared with experimental observations made on a different AISI 4140 steel. The GLEEBLE controlled continuous cooling tests were performed to obtain the final microstructure and hardness of the 4140 steel for a given thermal history. The measured diametral dilation data showed that high temperature ferrite and pearlite reaction products dominate at cooling rates from 0.1°C/s 85 to 1 C/s, whereas low temperature ferrite plus bainite and martensite reaction products develop at cooling rates from 0.25°C/s to 12°C/s. The microstructures could be broadly categorized according to the cooling rates into three categories. The microstructure at 0.1°C/s showed the presence of ferrite and equiaxed pearlite. The microstructures from 0.25°C/s to 0.75°C/s exhibited presence of ferrite, pearlite and the acicular bainite phase. The microstructure obtained at l°C/s, predominantly consists of only bainite, however, this microstructure also probably consists of ferrite and pearlite. The microstructures obtained at cooling rates above 2°C/s combined a reducing amount of ferrite and acicular bainite in a martensite matrix. In order to test the validity of Kirkaldy's[7] and Li's [ 1 3 ] models under continuous cooling conditions, these equations were used to characterize the CCT kinetics by integrating the reaction rate equations along the thermal path. The resulting microstructures calculated from the Kirkaldy model[7] showed only ferrite, pearlite and bainite at all cooling rates from 0.1°C/s to 6°C/s and martensite is predicted only at 12°C/s. Thus Kirkaldy's model overestimates the reaction rates for ferrite and bainite reactions. The final microstructures calculated from the Li model[13] showed the presence of ferrite, bainite as well as martensite in the final microstructure. Especially above a cooling rate of l°C/s, the Li model[13] calculated values showed an increasing amount of martensite and a decreasing amount of ferrite and bainite which is consistent with the experimental observations. 86 7. Future Work In this work, an attempt is made to establish the applicability of the original and modified Kirkaldy models for an AISI 4140 low alloy steel for a particular austenite grain size (G=9.5). For future work, it is suggested: 1. The original Kirkaldy model is known to work well for C-Mn steels[7]. The synergistic effect of other alloying elements such as Ni, Cr, Mo etc. should be studied to improve the performance of these models. 2. A detailed and thorough analysis of the sensitivity of these models to equilibrium transformation temperatures, austenite grain size and chemical composition should be performed. 3. The applicability of these models for various other grain sizes should be established. 87 8.BIBLIOGRAPHY 1. W. A. Johnson and R. F. 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Pavaskar, "Prediction of the Equilibrium, Paraequilibrium and No-Partition Local Equilibrium Phase Diagrams For Multicomponent Fe-C Base Alloys, CALPHAD, Vol. 8, No. 2, pp. 173-186. 88 15. Kirkaldy, J. S. and Baganis, E. A., "Thermodynamic Prediction of the Ae3 Temperatures with Additions of Mn, Si, Ni, Cr, Mo, Cu", Metallurgical Transactions, Vol. 9A(4), 1978, pp. 495-501. 16. Grange, R. A., "Estimating Critical Ranges in Heat Treatment of Steels", Metal Progress, Vol. 70(4), 1961, pp. 73-75. 17. Andrews, K. W., "Empirical formulae for the Calculation of Some Transformation Temperatures", JISI, Vol. 203, 1965, pp. 721-727. 18. Anil Kumar Sinha, 'Ferrous Physical Metallurgy', Butterworths Publishers, USA, 1989. 19. Uhrenius, "A Compendium of Ternary Iron-Base Phase Diagrams", 'Hardenability Concepts with Applications to Steel', Ed: D. V. Doane and J. S. Kirkaldy, 1978, pp. 28-47. 20. Mats Hillert: "Prediction of Iron-Base Phase Diagrams", 'Hardenability Concepts with Applications to Steel', Ed: D. V. Doane and J. S. Kirkaldy, 1978, pp. 5-27. 21. Wagner, C : Thermodynamics of Alloys, Addison-Wesley Press Inc., 1952 22. M. V. Li., Ph. D Dissertation, Oregon Graduate Institute of Science and Technology, 1996. 23. A. A. B. Sugden and H. K. D. H. Bhadeshia, "Thermodynamic Estimation of Liquidus, Solidus, Ae3 Temperatures, and Phase Compositions for Low Alloy Multicomponent Steels, Materials Science and Technology, Vol. 5, Oct. 1989, pp. 977-984. 24. "Numerical recipes in Fortran 7T\ Cambridge University Press, New York, 1996. 25. E. B. Hawbolt, B, Chau and J. K. Brimacombe, "Kinetics of Austenite-Ferrite and Austenite-Pearlite Transformations in a 1025 Carbon Steel", Metallurgical Transactions, Vol. 16A, April. 1985, pp.565-578. 26. Atlas of Isothermal Transformation and Cooling Transformation Diagrams, ASM, Metals Park, OH, 1977. 27. Noel. F. Kennon, "Schematic Transformation Diagrams for Steel", Metallurgical Transactions, Vol. 9A, January 1978, pp. 57-66. 28. C. Zener, "AIME Transactions, 1946, Vol. 167, p. 513. 29. Steven, W. and Haynes, A. G., "The Temperature of Formation of Martensite and Bainite in Low-Alloy Steels", JISI, Vol. 183(8), 1956, pp. 349-359. 89 30. Carpella, L. A., "Computing A or Ms (Transformation Temperature on Quenching) from Analysis ", Metals Progress, Vol. 46, 1944, pp. 108. 31. C. Y. Kung and J. J. Rayment, "An Examination of the Validity of Existing Empirical Formulae for the Calculation of Ms Temperature'", Metallurgical Transactions, Vol. 13A, February 1982, 328-331. 32. George F. Vander voort: "Metallography, principles and practice ", New York: McGraw Hill, 1986. 33. Ph. Maynier, B. Jungmann, J. Dollet, "Creusot-Loire System for the Prediction of the Mechanical Properties of Low Alloy Steels Products", 'Hardenability Concepts with Applications to Steel', Ed: D. V. Doane and J. S. Kirkaldy, 1978, pp. 518-545. 90 APPENDIX 1 Program used to calculate the TTT data for 4140 steel using Kirkaldy's equations C THIS PROGRAM CALCULATES THE TTT DIAGRAM FOA A STEEL OF GIVEN COMPOSITION C AND AUSTENITE GRAIN SIZE BASED ON THE KIRKALDY-VENUGOPALAN MODEL REAL*4 DELTEMP,CF,CP,CB REAL*4 AE3,AEI, BS,BNOSE,MS,DELTIME,EQF,EQP, EQB REAL*4 TEMP(2000),FTTTTIME(800,100),PTTTTIME(800,100) REAL*4 BTTTTIME(800,100) REAL*4 CARB,MN,NI,SI,CHROM,MO,G INTEGER NITER COMMON/COMl/CARB,MN,NI,SI,CHROM,MO,G,DELTIME C ************************************************ C DESCRIPTION OF IMPORTANT VARIABLES: C AE3:FERRITE START TEMP C AEI:PEARLITE START TEMP C BS:BAINITE START TEMP C MS:MARTENSITE START TEMPERATURE C EQF, EQP AND EQB: EQUILIBRIUM FRACTIONS OF FERRITE,PEARLITE C AND BAINITE C FTTTTIME(I,J): TIME REQUIRED FOR FERRITE TRANSFORMATION C AT 'I'th TEMPERATURE STEP, FOR 'J' PERCENT NORMALIZED FRACTION C SIMILAR DESCRIPTION APPLIES FOR PEARLITE AND BAINITE REACTIONS. **************************************************** C COMPOSITION OF THE STEEL Q **************************************************** CARB=0.39 MN=0.88 NI=0.23 SI=0.28 CHROM=0.86 MO=0.162 C Q ************************************************** C ASTM NUMBER OF THE AUSTENITE GRAIN SIZE G=9.5 Q ************************************************** C EQUILIBRIUM TEMPERATURES AE3=766.0 AE1=716.0 BS=554.0 BNOSE=505.0 MS=300.0 C ************************************************** C OPEN OUTPUT FILE NAMED 'TTTKIRK.OUT' OPEN(UNIT=10, FILE='TTTKIRK.OUT') Q * ************************************************ C THE TTT CALCULATION IS PERFORMED FOR A MAXIMUM OF C 'NITER' NUMBER OF TEMPERATURE STEPS.'DELTEMP' DENOTES C EACH TEMPERATURE STEP NITER=1000 DELTEMP=50.0 C ******************************************************************** C DO LOOP FOR THE CALCULATION OF THE TIME CORRESPONDING TO THE VARIOUS C TRANSFORMED FRACTIONS IN A TTT CURVE. THE TTT CURVE IS DEVIDED INTO C FOUR TEMPERATURE REGIONS,viz. AE3 TO AEI, AEI TO BS, BS TO BNOSE AND C BNOSE TO MS. ******************************************************************** C DO J=l,NITER 91 TEMP(J)=(AE3-J*DELTEMP)+(DELTEMP/2) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * CALCULATION OF TTT DATA FOR THE TEMPERATURE REGION AE3 TO A E I . I N THIS TEMPERATURE REGION,AUSTENITE TRANSFORMS TO FERRITE ************************************************************** I F ( T E M P ( J ) . G T . A E I ) T H E N CALL FERCOEF(TEMP(J) ,CF,AE3) F T T T T I M E ( J , 1 ) = C F * 0 . 3 3 2 CALCULATION OF THE EQUILIBRIUM FRACTION OF FERRITE AS A FUNCTION OF TEMPERATURE. THIS FORMULA I S BASED ON FITTING THE DATA OBTAINED FROM THE THERMODYNAMICS BASED MODEL E Q F = - ( 0 . 0 0 0 0 6 * ( T E M P ( J ) * * 2 ) ) + ( 0 . 0 7 5 * T E M P ( J ) ) - 2 3 . 8 + 1 . 5 6 6 DO N = l , 9 8 I F ( ( ( ( N + 1 ) * 0 . 0 1 ) / E Q F ) . L E . 1 . 0 ) T H E N CALL I N T E G R A L ( ( ( N + l ) * 0 . 0 1 / E Q F ) , S ) F T T T T I M E ( J , N + l ) = C F * S ELSE F T T T T I M E ( J , N + l ) = l E + 6 ENDIF ENDDO ENDIF c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C CALCULATION OF TTT DATA FOR THE TEMPERATURE REGION A E I TO BS C I N THIS TEMPERATURE REGION,AUSTENITE TRANSFORMS TO FERRITE C AND PEARLITE. r ************************************************************ I F ( T E M P ( J ) . L E . A E 1 .AND. TEMP(J ) .GT.BS)THEN C EQUILIBRIUM FRACTIONS FOR FERRITE AND PEARLITE OBTAINED FROM C THE THERMODYNAMIC MODEL. E Q F = - ( 0 . 0 0 0 0 0 8 * ( T E M P ( J ) * * 2 ) ) + ( 0 . 0 1 1 7 * ( T E M P ( J ) ) ) - 3 . 8 2 2 2 + 0 . 0 3 EQP=1-EQF CALL FERCOEF(TEMP(J) ,CF,AE3) CALL PERCOEF(TEMP(J) ,CP,AEI) F T T T T I M E ( J , 1 ) = C F * 0 . 3 3 2 P T T T T I M E ( J , 1 ) = C P * 0 . 3 3 2 DO N = l , 9 8 I F ( ( ( ( N + l ) * 0 . 0 1 ) / E Q F ) . L E . 1 . 0 ) T H E N CALL I N T E G R A L ( ( ( N + l ) * 0 . 0 1 / E Q F ) , S ) F T T T T I M E ( J , N + l ) = C F * S ELSE CALL I N T E G R A L ( ( ( ( ( N + l ) * 0 . 0 1 ) - E Q F ) / E Q P ) , S ) P T T T T I M E ( J , N + l ) = C P * S ENDIF 92 ENDDO ENDIF Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C CALCULATION OF TTT DATA FOR THE TEMPERATURE REGION BS TO BNOSE. C I N THIS TEMPERATURE REGION,AUSTENITE TRANSFORMS TO FERRITE C AND B A I N I T E . f * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I F ( T E M P ( J ) . L E . B S .AND.TEMP(J) .GT.BNOSE)THEN C EQUILIBRIUM FRACTIONS FOR FERRITE AND B A I N I T E OBTAINED FROM C THE THERMODYNAMIC MODEL. E Q F = - ( 0 . 0 0 0 0 0 8 * ( T E M P ( J ) * * 2 ) ) + ( 0 . 0 1 1 7 * ( T E M P ( J ) ) ) - 3 . 8 2 2 2 + 0 . 0 3 EQB=1.0-EQF CALL FERCOEF(TEMP(J) ,CF,AE3) CALL BAINCOEF(TEMP(J) ,CB,BS) F T T T T I M E ( J , 1 ) = C F * 0 . 3 3 2 B T T T T I M E ( J , 1 ) = C B * 0 . 3 3 2 DO N = l , 9 8 I F ( ( ( ( N + l ) * 0 . 0 1 ) / E Q F ) . L E . 1 . 0 ) T H E N CALL I N T E G R A L ( ( ( N + l ) * 0 . 0 1 / E Q F ) , S ) F T T T T I M E ( J , N + 1 ) = C F * S ELSE CALL B A I N I N T ( ( ( ( ( N + l ) * 0 . 0 1 ) - E Q F ) / E Q B ) , S ) B T T T T I M E ( J , N + l ) = C B * S ENDIF ENDDO ENDIF Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C CALCULATION OF TTT DATA FOR THE TEMPERATURE REGION BNOSE TO MS. C I N THIS TEMPERATURE REGION,AUSTENITE TRANSFORMS TO B A I N I T E . Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I F ( T E M P ( J ) . L T . B N O S E .AND. TEMP(J) .GT.MS)THEN CALL BAINCOEF(TEMP(J) ,CB,BS) B T T T T I M E ( J , 1 ) = C B * 0 . 3 3 2 DO N = l , 9 8 CALL B A I N I N T ( ( ( N + l ) * 0 . 0 1 ) , S ) B T T T T I M E ( J , N + l ) = C B * S ENDDO ENDIF Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C END OF TTT DATA CALCULATION. THE TTT DATA I S WRITTEN C I N THE FOLLOWING FORMAT: C TEMPERTURE, FERRITE START T I M E , FERRITE F I N I S H T IME, C PEARLITE START T IME,PEARLITE F I N I S H T I M E , B A I N I T E START TIME C BAIN ITE F I N I S H TIME r * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * WRITE(10,*)TEMP(J) , ' * , F T T T T I M E ( J , 1 ) , * ' , F T T T T I M E ( J , 9 9 ) , * PTTTTIME(J ,1 ) , ' * , P T T T T I M E ( J , 9 9 ) , ' * ,BTTTTIME(J,1) , ' ' * ,BTTTTIME(J,99) END DO STOP END ********************************************************? SUBROUTINE FERCOEF(TEMP,C,AE3) REAL*4 DELT,ACTEN,AE3 REAL*4 TEMP,FC,C REAL*4 CARB,MN,NI,SI,CHROM,MO,G,DELTIME COMMON/COM1/CARB,MN,NI,SI,CHROM,MO,G,DELTIME DELT=((AE3-TEMP)**3) ACTEN=EXP(-23500. /(I .98*(TEMP+273.))) FC=(l/0.3)*(60*MN+2*NI+68*CHROM+244*MO) GRAIN=(2**((G-l ) /2 .0) ) C=(FC/(DELT*ACTEN*GRAIN)) RETURN END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE PERCOEF(TEMP,C,AEI) REAL*4 C , D E L T , A E I REAL*4 TEMP,PC,DIFF REAL*4 CARB,MN,NI,SI,CHROM,MO,G,DELTIME COMMON/COM1/CARB,MN,NI,SI,CHROM,MO,G,DELTIME DELT=((AE1-TEMP)**3) DIFF=EXP(27500/(1.98*(TEMP+273.0) ))+ * ( (0.01*CHROM+0.52*MO)*EXP(37000/(1.98*(TEMP+273.0) ) PC=(l/0.3)*(1.79+5.42*(CHROM+4*MO*NI)) GRAIN=(2**((G-l) /2 .0 ) ) C=(PC*DIFF/(DELT*GRAIN)) RETURN END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE BAINCOEF(TEMP,C,BS) REAL*4 C,DELT,ACTEN,BS REAL*4 TEMP,BC REAL*4 CARB,MN,NI,SI,CHROM,MO,G,DELTIME COMMON/COM1/CARB,MN,NI,SI,CHROM,MO,G,DELTIME DELT=((BS-TEMP)**2) ACTEN=EXP(-27500./(1.98*(TEMP+273.))) BC=(1/0.3)*(2.34+10.1*CARB+3.8*CHROM+19.0*MO)*(0.0001) GRAIN=(2**( (G- l ) /2 .0 ) ) C=( BC/(DELT*ACTEN*GRAIN) ) RETURN END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE INTEGRAL(EX,S) REAL*4 E X , A , S , F 1 , F 2 , S I , H REAL*4 NUM2,DENOM2,NUM1,DENOM1 INTEGER N A=0.001 NUM1=1.0 DEN0M1=( A**(0 .66*(1-A)) )*( (1-A)**(0.66*A) ) F1=NUM1/DEN0M1 N=10+ABS( ( (EX-A) /0 .01)*100 ) H=lE-8+ABS( (EX-AJ/N ) S=0.0 DO 10 1=1,N NUM2=EXP(2.658*((A+I*H)**2)) NUM2=1.0 DENOM2=( (A+I*H)**(0.66*( 1-(A+I*H) )) ) ** ( (1-(A+I*H) )**(0.66*(A+I*H) ) ) F2=NUM2/DENOM2 SI=(F1+F2)*H/2.0 S=S+SI F1=F2 CONTINUE RETURN END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE BAININT(EX,S) REAL*4 E X , A , S , F 1 , F 2 , S I , H , D E L T I M E REAL* 4 NUM2,DENOM2,NUM1,DENOM1 INTEGER N COMMON/COM1/CARB,MN,NI,SI,CHROM,MO,G,DELTIME A=0.001 NUM1=EXP(2.658*(A**2)) DENOMl=( A * * ( 0 . 6 6 * ( l - A ) ) )*( (1-A)**(0.66*A) ) F1=NUM1/DEN0M1 N=10+ABS( ( (EX-A) /0 .01)*100 ) H=lE-8+ABS( ( E X - A ) / N ) S=0.0 DO 10 1=1,N NUM2=EXP( 2.658*((A+I*H)**2) ) * ((1.9*CARB+2.5*MN+0.9*NI+1.7*CHROM+4*MO)-2.6 DENOM2=( (A+I*H)**(0.66*( 1-(A+I*H) )) ) ** ( (1-(A+I*H) )**(0.66*(A+I*H) ) ) F2=NUM2/DEN0M2 SI=(Fl+F2)*H/2.0 S=S+SI F1=F2 CONTINUE RETURN END APPENDIX 2 Program used to calculate the CCT data for 4140 steel using Kirkaldy's equations THIS PROGRAM CALCULATES THE CCT DATA FOR AN 4140 STEEL OF GIVEN COMPOSITION FOR CONSTANT COOLING RATE USING KIRKALDY'S TRANSFORMATION KINEICS FORMULAE REAL*4 CR,DELTAX,DELTIME REAL*4 AE3, A E I , BS, E Q F , X F , X P , X B , T I M E S T E P , E X , E X F , E X P , E Q F T 1 REAL*4 TEMP(200000),TIME INTEGER NITER REAL*4 CARB, MN,NI, SI,CHROM,MO, G COMMON/COM1/CARB,MN,NI,SI,CHROM,MO,G,DELTIME DESCRIPTION OF IMPORTANT VARIABLES CR=COOLING RATE G: ASTM GRAIN SIZE NUMBER DELTEMP=TEMPERATURE STEP (FIXED) DELTIME=TIME STEP (BASED ON DELTEMP AND CR) DELTAX= VOLUME FRACTION FORMED IN THE TIME STEP FSTART=FERRITE START TEMPERATURE PSTART=PEARLITE START TEMPERATURE BASTART=BAINITE START TEMPERATURE MSTART=MARTENSITE START TEMPERATURE AE3=EQUILIBRIUM TEMPERATURE FOR AUSTENITE TO FERRITE TRANS. AE1=EQUILIBRIUM TEMPERATURE FOR AUSTENITE TO PEARLITE TRANS. BS=EQUILIBRIUM TEMPERATURE FOR AUSTENITE TO BAINITE TRANS. X F , XP AND XB= TRUE FRACTIONS OF FERRITE, PEARLITE AND BANITE EQF= EQUILIBRIUM FRACTION OF FERRITE GIVEN BY PHASE DIAGRAM EX= NORMALIZED FRACTION * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * COMPOSITION OF THE STEEL * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * CARB=0.39 MN=0.88 NI=0.23 SI=0.28 CHROM=0.86 MO=0.162 PRINT*, 'PLEASE ENTER THE PRIOR AUSTENITE ASTM GRAIN SIZE NUMBER* READ*, G AE3=766.0 AE1=716.0 BS=544.0 MS=329.0 OPEN(UNIT=10,FILE=*CCT') PRINT*,'COOLING R A T E ? . . . ' READ*,CR PRINT*,'NO OF ITERATIONS ? . . . ' READ*,NITER WRITE(10,*)'GRAIN S I Z E : ' , ' ' , G WRITE(10,*)'COOLING R A T E : ' , ' ' , C R DELTIME=0.1 DELTEMP=DELTIME*CR EX=lE-8 XP=lE-8 XF=lE-8 XB=lE-8 TIMESTEP=0.0 C CALCULATION OF THE CCT DATA STARTS. THE COOLING CURVE IS DEVIDED INTO C ISOTHERMAL INCREMENTS. THE CCT DATA IS CALCULATED AT EACH TEMPERATURE STEP. DO J=2,NITER Q ********************************************* C CALCULATION OF TEMPERATURE AT THE NEW TIME STEP Q ************************************************** TEMP(J)=(AE3-J*DELTEMP)+(DELTEMP/2) TIME=(AE3-TEMP(J))/(CR) C THE VALUES OF THE EQUILIBRIUM FRACTION OF FERRITE AND PEARLITE C ARE CALCULATED FROM THE THERMODYNAMIC MODEL. I F ( T E M P ( J ) . L E . AE3 .AND. TEMP(J) .GT.AEI)THEN EQF=-(0.00006*(TEMP(J)**2))+(0.075*TEMP(J))-23.8+1.566 E L S E I F ( T E M P ( J ) . L E . A E I .AND. TEMP(J) .GT.495.0)THEN EQF=-(0.000008*(TEMP(J)**2))+(0.0117*(TEMP(J)))-3.8222+0.03 EQP=1-EQF ELSE EQF=0.01 EQP=1-EQF ENDIF Q ***************************************************** IF (TEMP(J) .LE.AE3 .AND. TEMP(J) .GT.BS)THEN TIMESTEP=TIMESTEP+1 EXF=EXF*(EQFT1/EQF) I F ( E X F . G E . 1 . 0 ) T H E N EXF=1.0 ENDIF I F ( E X F . L T . 1 . 0 ) T H E N WRITE(10,*)CR, • ' , T E M P ( J ) , ' ' , X F , 1 ' , X P , ' ' , X B , ' ',XF+XP+XB ENDIF CALL FERCOEF(TEMP(J) ,C,EXF,AE3,DELTAX) XF=XF+DELTAX*EQF EXF=XF/EQF EQFT1=EQF ENDIF * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * IF ( T E M P ( J ) . L E . A E I .AND. TEMP(J) .GT.BS)THEN I F ( E X F . G E . 1 . 0 ) T H E N IF( (XP+XF) .LE.1 .0 )THEN CALL PERCOEF(TEMP(J) ,XP/EQP,AEI,DELTAX) XP=XP+DELTAX*EQP EXP=XP/EQP WRITE(10,*)CR, ' ' , T E M P ( J ) , ' ' , X F , ' ' , X P , ' ' , X B , ' ',XF+XP+XB ELSE GOTO 100 97 ENDIF ENDIF ENDIF C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * IF (TEMP(J) .LE.BS .AND. TEMP(J) .GT.MS) THEN IF((XF+XP+XB).LE.1)THEN CALL BAINCOEF(TEMP(J),(XP+XB),BS,DELTAX) XB=XB+DELTAX PRINT*,CR, ' ' , T E M P ( J ) , ' ' , X F , ' ' , X P , ' ' , X B WRITE(10,*)CR, ' ' , T E M P ( J ) , ' ' , X B , ' ',XF+XP+XB ELSE GOTO 100 ENDIF ENDIF Q ************************************************** IF(TEMP(J) . L T . MS .AND. TEMP(J) . G E . 0)THEN XM=(1-(XF+XB+XP)) ENDIF 100 CONTINUE 200 ENDDO 300 STOP END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE FERCOEF(TEMPR,C,EX,AE3,DELTAX) REAL*4 DELT,ACTEN,AE3 REAL*4 TEMPR,FC,C REAL*4 CARB,MN,NI,SI,CHROM,MO,G COMMON/COM1/CARB,MN,NI,SI,CHROM,MO,G,DELTIME DELT=((AE3-TEMPR)**3) ACTEN=EXP(-23500./(1.98*(TEMPR+273.))) FC=(l/0.3)*(60*MN+2*NI+68*CHROM+244*MO) GRAIN=(2**( (G- l ) /2 .0 ) ) C=((DELT*ACTEN*GRAIN)/FC) DELTAX=C*DELTIME*( E X * * ( 0 . 6 6 * ( 1 - E X ) ) ) * ( (1-EX)**(0.66*EX) ) RETURN END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE PERCOEF(TEMPR,EX,AEI,DELTAX) REAL*4 C , D E L T , A E I REAL*4 TEMPR, P C , D I F F REAL*4 CARB,MN,NI,SI,CHROM,MO,G COMMON/COMl/CARB,MN,NI,SI,CHROM,MO,G,DELTIME DELT=((AE1-TEMPR)**3) DIFF=EXP(27500/(1.98*(TEMPR+273.0)))+( (0.01*CHROM+0.52*MO)* #EXP(37000/(1.98*(TEMPR+273.0))) ) PC=(1.79+5.42*(CHROM+4*MO*NI))/0.3 \ I GRAIN=(2**((G- l ) /2 .0) ) C=((DELT*(1/DIFF)*GRAIN)) /PC DELTAX=C*DELTIME*( E X * * ( 0 . 66* ( 1 - E X ) ) ) * ( ( 1 - E X ) * * ( 0 .66* E X ) ) RETURN END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE BAINCOEF(TEMPR,EX,BS,DELTAX) REAL*4 C,DELT,ACTEN,BS,DELTIME REAL*4 TEMPR,NUMX,DENOMX REAL*4 CARB,MN,NI,SI,CHROM,MO,G COMMON/COM1/CARB, MN,NI,SI,CHROM,MO,G,DELTIME DELT=((BS-TEMPR)**2) ACTEN=EXP (-27500 : / (1. 98* (TEMPR+.273 . ) ) ) BC=(l/0.3)*(2.34+10.l*CARB+3.8*CHROM+19.0*MO)*(0.0001) GRAIN=(2**( ( G - D / 2 . 0 ) ) C=((DELT*ACTEN*GRAIN)) /BC NUMX=C*DELTIME*( E X * * ( 0 . 66* ( 1 - E X ) ) ) * ( (1-EX)* * ( 0 .66* E X ) ) DENOMX=EXP((EX**2)*((1.9*CARB+2.5*MN+0.9*NI+1.7*CHROM+4*MO)-2.6) ) DELTAX=NUMX/DENOMX RETURN END 99 

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