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Austenite grain growth kinetics and the grain size distribution Giumelli, Alan 1995

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AUSTENITE GRAIN GROWTH KINETICS AND THE GRAIN SIZE DISTRIBUTION B Y A L A N GIUMELLI B.Eng.,Hons., The University of Wollongong, Australia, 1991. A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTERS OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (DEPARTMENT OF M E T A L S A N D MATERIALS ENGINEERING) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A APRIL, 1995. ©Alan Giumelli, 1995. 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. Department of ^(£/fPr(_s 4. r^fKSXi^ A c s c r y - f C r ^ C i T n-<r~*(r The University of British Columbia Vancouver, Canada Date h/^ / . DE-6 (2/88) ABSTRACT The aims of this work are to assess the kinetics of austenite grain growth, and to estimate the three dimensional grain size distribution. Measurements are to be used for the validation of a statistical model for the estimation of the kinetics of austenite grain growth during hot strip rolling. A series of tests are performed with two industrial plain carbon steels in the temperature range from 950°C to 1150°C. Grain size is measured by standard manual procedures and by the use of image analysis equipment. Systematic errors in the measurements are identified. A correction is proposed to account for the systematic error in the measurement of the two dimensional grain size distribution by the use of image analysis. The kinetics of austenite grain growth are discussed and the influence of second phase particles is observed. Results are described well by a statistical model for grain growth. A number of methods for the estimation of the three dimensional grain size distribution are compared. It is concluded that the method proposed by Matsuura and Itoh is the most suitable. ii TABLE OF CONTENTS Page A B S T R A C T ii LIST OF T A B L E S vi LIST OF ILLUSTRATIONS viii N O M E N C L A T U R E x A C K N O W L E D G M E N T S xiii CHAPTER 1. INTRODUCTION 1 CHAPTER 2. L ITERATURE REVIEW 4 2.1 Modelling of Hot Strip Rolling of Steel 4 2.2 Modelling of Grain Growth 5 2.2.1 The Power Law for Grain Growth 5 2.2.2 Statistical Modelling of Grain Growth 8 2.3 Normal and Abnormal Grain Growth 12 2.4 Measurement of Grain Size 14 2.5 The Grain Size Distribution 17 2.6 Estimation of the Three Dimensional Grain Size Distribution 20 2.6.1 Methods Based on Spherical Grain Shape 20 2.6.2 The Method of Takayama et al 24 2.6.3 The Method of Matsuura and Itoh 26 CHAPTER 3. SCOPE A N D OBJECTIVES 31 C H A P T E R 4. E X P E R I M E N T A L 32 4.1 Sample Composition and Specimen Design 32 4.2 Equipment for the Thermal Treatment : 35 4.2.1 The Gleeble® 1500 Thermomechanical Simulator 35 4.2.2 The Vertical Tube Furnace 38 iii 4.3 The Thermal Cycles 42 4.4 Revealing the Prior Austenite Microstructure 46 4.5 Measurement of the Austenite Grain Size 46 CHAPTER 5. RESULTS A N D DISCUSSION I. THE KINETICS OF AUSTENITE G R A I N GROWTH 50 5.1 The Structure of the Quenched and Etched Specimens 50 5.1.1 Helium Quenched A36 Steel 50 5.1.2 Water Quenched A36 Steel 52 5.1.3 Water Quenched DQSK Steel 53 5.1.4 Water Quenched 1080 Steel 55 5.2 Measurement of the Grain Size and Estimation of the Error 56 5.2.1 Interpretation of the Structure 56 5.2.2 Measurement of the Grain Size 57 5.2.3 The 95% Confidence Intervals 59 5.2.4 Systematic Errors in Measurement of Equivalent Area Diameter.... 60 5.2.5 Systematic Errors in Measurement of Linear Intercept Length 64 5.3 The Kinetics of Austenite Grain Growth 64 5.3.1 A36 Steel Heated to Temperature at 5°C/second 64 5.3.2 DQSK Steel Heated to Temperature in the Tube Furnace 67 5.4 Heating Rate Effects 69 5.5 Models Describing the Kinetics of Grain Growth 70 5.5.1 Application of the Power Law 70 5.5.2 Application of the Statistical Model 72 5.6 The Influence of Second Phase Particles 77 5.7 The Evolution of the Grain Size Distribution 80 5.9 Solution Treatment 86 iv 5.10 Grain Growth During Hot Strip Rolling 89 C H A P T E R 6. RESULTS A N D DISCUSSION II. ESTIMATION OF THE THREE DIMENSIONAL G R A I N SIZE DISTRIBUTION.. . . 91 6.0 Introduction 91 6.1 Methods Based on the Assumption of Spherical Grain Shape 91 6.2 The Method of Takayama et al 94 6.3 The Method of Matsuura and Itoh 100 6.3.1 Application of the Method 100 6.3.2 The Shape of the Estimated 3-D Distribution 105 6.3.3 Convergence of the Distribution With Number of Size Classes 107 6.4 Comparison of the Methods of Takayama et al. and Matsuura and Itoh 110 6.5 Validation of the Method of Matsuura and Itoh With Measured 3-D Results I l l CHAPTER 7. S U M M A R Y A N D CONCLUSIONS 114 BIBLIOGRAPHY 118 APPENDIX 124 v LIST OF TABLES Table Page Table 2.1 - Grain growth exponents for the power law, for zone refined metals 7 Table 2.2 - Power law parameters reported for austenite grain growth 8 Table 2.3 - Description of geometric and arithmetic size class scales 19 Table 4.1 - Composition of steels to be used in this investigation 32 Table 4.2 - Standard thermal cycles performed on the Gleeble 42 Table 4.3 - Standard furnace thermal cycles 44 Table 4.4 - Stepped Gleeble thermal cycles 44 Table 4.5 - Continuous cooling thermal cycles 46 Table 4.6 - Summary of etching procedures 48 Table 5.1 - Correction of systematic errors for the mean equivalent area diameter. 63 Table 5.2 - Power law fitting parameters for the A36 and DQSK steels 72 Table 5.3 - Evolution of the grain size distribution 81 Table 6.1 - The 3-D grain size distribution from Takayama et al 95 Table 6.2 - The 3-D grain size distribution using the image analyser results from Matsuura and Itoh 102 Table 6.3 - The 3-D grain size distribution from the corrected 2-D distribution from Matsuura and Itoh 103 Table 6.4 - The 3-D grain size distribution predicted using the method of Takayama et al. and Matsuura and Itoh 110 Table 6.5 - Comparison of methods for Al-Sn alloy and stainless steel 112 Table A . l - A36 results, standard Gleeble thermal treatment 124 Table A.2 - A36 results, Gleeble thermal treatment 127 Table A.3 - A36 results, continuous cooling on the Gleeble 127 vi Table A.4 - A36 results, stepped Gleeble thermal treatment 128 Table A. 5 - DQSK results, tube furnace thermal treatment 129 Table A.6 - DQSK results, standard Gleeble thermal treatment 130 Table A.7 - 1080 steel results, furnace thermal treatment 130 vii LIST OF ILLUSTRATIONS Figure Page Figure 1.1- Schematic diagram of a hot strip mill 2 Figure 2.1 - The geometry of two dimensional array of grains 10 Figure 2.2 - The intersection of a sphere and a random plane 21 Figure 2.3 - Schematic illustration of the method of Matsuura and Itoh 28 Figure 4.1 - Design of the Gleeble specimens 34 Figure 4.2 - Specimens used in the tube furnace 35 Figure 4.3 - Schematic diagram of the Gleeble 36 Figure 4.4 - Schematic diagram of vertical tube furnace and specimen holder 39 Figure 4.5 - Typical furnace thermal cycle 41 Figure 4.6 - Standard Gleeble thermal cycles 43 Figure 4.7 - Stepped Gleeble thermal cycles 45 Figure 4.8 - Continuous cooling thermal cycles 47 Figure 5.1 - A36 Steel, Gleeble heated at 5°C/sec to 1150 C, 60 seconds 51 Figure 5.2 - A36 Steel, Gleeble heated at 5°C/sec to 1000°C, lOseconds 52 Figure 5.3 - DQSK Steel, furnace heated to 1050°C, 420seconds 54 Figure 5.4 - DQSK Steel, furnace heated to 1150°C, 227seconds 55 Figure 5.5 - 1080 Steel, furnace heated to 1100°C, immediately quenched 56 Figure 5.6 - A36 Steel, Gleeble heated to 1000°C, 120seconds 58 Figure 5.7 - Kinetics of austenite grain growth in A36 steel, heating rate=5°C/sec. 66 Figure 5.8 - Kinetics of austenite grain growth in DQSK steel 68 Figure 5.9 - Effect of heating rate on grain growth in A36 steel 71 Figure 5.10 - The statistical model, A36 steel, heating rate=5 C/sec 74 Figure 5.11 - The statistical model, DQSK steel 75 viii Figure 5.12 - Calculated and experimental Rnm 76 Figure 5.13 - Application of the statistical model to continuous cooling conditions.79 Figure 5.14 - Evolution of the A36 grain size distribution, normal grain growth.... 82 Figure 5.15 - Evolution of the grain structure, in the A36 steel, abnormal growth. 84 Figure 5.16 - Evolution of the A36 grain size distribution, abnormal grain growth. 85 Figure 5.17 - Austenite grain size in the as-received and solution treated A36 88 Figure 6.1 - Saltikov's and Huang and Form's 3-D grain size distribution 92 Figure 6.2 - The relationship between .syand the ratio of lm to V a o t 97 Figure 6.3 - Flow chart for applying the method of Matsuura and Itoh 101 Figure 6.4 - Relationship between and dymld^m 104 Figure 6.5 - Evolution of the 3-D grain size distribution 108 Figure 6.6 - Effect of the number of size classes on the method of Matsuura and Itoh 109 Figure 6.7 - Stainless steel, the estimated 3-D distribution compared to the measured distribution, from Matsuura et al 113 ix NOMENCLATURE A grain section area Am mean grain area b Burger's vector c lower limit of the smallest size class for the geometric scale Cn coefficient for the nth term in Saltikov's equation d grain diameter dg initial grain diameter dA equivalent area diameter dAb average equivalent area diameter dAg peak equivalent area diameter of the log normal distribution dAm mean equivalent area diameter db average diameter dg peak diameter of the log normal distribution Dgb grain boundary diffusivity dm mean diameter dR relative diameter dv equivalent volume diameter dVb average equivalent volume diameter dVg peak equivalent volume diameter the of log normal distribution dVm mean equivalent volume diameter f(d) distribution of grain diameters fAi fraction of grains measured with equivalent area diameter in size class i fACi corrected fraction of grains measured with area diameter in size class i x fa calculated fraction of grains with equivalent area diameter in size class i fpi fraction of grains in size class i, in contact with the edge of the measurement field fi fraction of grains of size i fvi fraction of grains with equivalent volume diameter in size class i F driving force for grain growth Fij driving force for growth of grains of size / relative to size j FP pinning force / number of grain faces Jl number of faces for grains from size class i k total number of grain size classes K grain growth constant (theoretical) Ka grain growth constant (power law) lm mean linear intercept m grain growth exponent Mgfy grain boundary mobility N^i measured number of grain sections of size class i Nj^a corrected number of grain sections of size class i Npi number of grains of size class i which touch the edge of the field of measurement Ni number of grains in size class i Ntot total number of grains per unit volume NVti number of grains per unit volume after truncation correction P pinning parameter xi Pp(d) probability that a grain is in contact with the edge of the field of measurement PM^R) probability that the relative diameter is equal to dR Psid) probability that the section diameter is less than d for a sphere Q apparent activation energy for grain growth (power law) r average particle radius R gas constant R2 correlation coefficient Rg average grain radius R( radius of a grain of size class i Rlim limiting grain radius Rmax maximum grain radius s standard deviation of the log normal grain size distribution sA standard deviation of the log normal equivalent area grain size distribution sv standard deviation of log normal equivalent volume grain size distribution S2 sum of the differences squared t time T temperature V grain volume W(j probability that grains i and j are neighbors A width of size class for arithmetically scaled distribution a multiplying factor for geometrically scaled size distribution 7g£ grain boundary energy v volume fraction of precipitates xii ACKNOWLEDGMENTS This work is a part of a group project to model the industrial process of hot strip rolling. In completing this thesis, I have worked closely with other members of the research group, in order to make progress toward our common goal. The cooperative atmosphere has been the catalyst for many group discussions regarding experimental methods and the interpretation of results. I would like to thank all members of the hot strip mill modelling group for their contributions to this work. In particular, Dr. Matthias Militzer deserves a special mention for his guidance and support throughout the course of this research. I am also grateful for the technical assistance provided by Mr. Rudy Cardeno and Mr. Binh Chau and for the support and guidance of Dr. Bruce Hawbolt. I would like to acknowledge the financial support provided by the American Iron and Steel Institute and the United States Department of Energy. I owe thanks also to my wife, Katrina, for the patience and support she has so generously shown me, especially during the later stages of this work. xiii CHAPTER 1 INTRODUCTION Steel is a versatile material which has found widespread use because of its exceptional mechanical and magnetic properties, its relatively low cost, and the ease with which it can be used in manufacturing processes such as forming, welding and machining. The steel industry has experienced great successes as a result of these characteristics. However, in recent times, economic developments have generated a highly competitive atmosphere in many industries, and steel producers have been forced to do battle with each other, and with other materials producers for a share of a market which is demanding higher quality products at lower cost. This has driven steel producers to improve their operations by the application of improved and advanced technology. An example of this is process modelling, which links measurable processing parameters to the final properties of the product. Process models are an excellent knowledge base for off line process simulation, and ultimately, it is likely that models will be used for on line process control. The main steps in the production of steel are refining of the material in the molten state, followed by casting to form a solid product, then deformation of the solid to give the specified mechanical properties, and a more useful product shape. If the desire is to produce steel in form of sheet, the first stages of deformation will be performed by hot rolling. The process of hot strip rolling is shown schematically in Figure 1.1. A steel slab is heated to temperature in a reheat furnace. Deformation begins in the roughing mill where the slab is reduced in thickness by a series of rolling passes. The slab leaves the roughing null with Utile change in width, but with significantly increased length and reduced 1 2 thickness; at this stage the steel is referred to as transfer bar. The next stage of deformation is the finishing mill, at the end of which the material is referred to as strip. Frequently, the aim is to complete all deformation while the steel is austenitic. The subsequent decomposition of the austenite can be controlled by cooling at an appropriate rate, using water sprays on the runout table. At the end of the runout table the material is coiled on the down coiler. Roughing mill Reheat furnace Figure 1.1- Schematic diagram of a hot strip mill. Most rolling passes are carried out at an elevated temperature, where recrystallization occurs, so that large amounts of deformation are possible. Other metallurgical phenomena which accompany hot rolling are grain growth, phase transformation, and in some steels, precipitation. The microstructure of the finished product dictates its properties; the control of the metallurgical phenomena can ensure that the product meets the property requirements specified by the customer. Process modelling of hot strip rolling is a sensible approach to achieving the goals of improved product quality and reduced cost by the application of improved technology. The modelhng task requires the luiking of the process parameters with the kinetics of the metallurgical phenomena. The kinetics of grain growth assume significant importance in the evolution of the structure during hot strip rolling. This thesis examines the kinetics of austenite grain growth, measured experimentally, under conditions of time and temperature which are relevant to the 3 process of hot strip rolling. Measurements are to be used to develop and validate a model to describe the kinetics of austenite grain growth during hot strip rolling. The model to be used will have a fundamental basis, and the phenomenon of grain growth will be treated as the evolution of a distribution of three dimensional grains. Grain size measurements are generally performed on planar sections through the structure, and direct measurement of the three dimensional grain size is a difficult task. However, procedures have been reported to estimate the three dimensional grain size from two dimensional measurements. These methods will also be examined in this work. The grain growth model is one component of the process model being developed by an engineering research team at the University of British Columbia (UBC); the author of this work is a member of that team. The U B C team is working in conjunction with United States Steel and the National Institute of Standards and Technology. The aim of the U B C group is to develop a versatile and fundamentally based model for the prediction and control of the microstructure and mechanical properties of steel during hot rolling. The model will be developed for a range of mill processing conditions and steel compositions, including plain carbon, interstitial free and microalloyed grades. In this stage of development of the process model, investigations have been directed toward the plain carbon steels. The deliverable from this research will be a PC (personal computer) based model, which will link the processing variables to the final microstructure and properties of the product. Application of this model will allow the steel producer to deliver a higher quality product to the customer, at lower cost. CHAPTER 2 LITERATURE REVIEW 2.1 Modelling of Hot Strip Rolling of Steel Several workers have developed process models for the prediction and control of microstructure and mechanical properties during hot strip rolling of steel [1-5]. Process models which have been developed are generally based on empirical relationships which relate processing parameters to material structure and properties. A more desirable model would have a fundamental basis. Such fundamental models will be capable of accurately describing the thermal history of the strip, the rolling forces at each rolling pass, and the kinetics of the microstructural phenomena of recrystallisation, grain growth, precipitation and phase transformation, which accompany hot strip rolling. The kinetics of the microstructural events are modelled by investigating each event separately, using laboratory equipment which is able to approximate the thermal and mechanical conditions in the mill. The individual events are combined using a model which describes the thermal and mechanical history of the strip. Since the microstructural evolution can be quantified and related to the stresses of deformation, the mechanical properties at each stage can be anticipated; consequently, the mill forces can be predicted based on the properties. The resulting model must then be validated by performing measurements on the mill under industrial conditions to verify temperature and force predictions. The structure must also be assessed in the hot rolled product, to verify microstructural predictions. This approach is being taken by the U B C team. Here, one of the metallurgical events to be modelled is examined; that is, the kinetics of austenite grain growth. 4 2.2 Modelling of Grain Growth 2.2.1 The Power Law for Grain Growth Burke and Turnbull were among the first to model the kinetics of grain growth [6]. They assumed that the driving force for growth is the reduction in grain boundary area and the subsequent release of boundary surface energy, 7^. The model was proposed for the case of normal grain growth in a pure homogeneous material and the assumption was made that the only forces acting on boundaries are those due to surface curvature. It was also assumed that the average radius of curvature of a boundary is proportional to the grain diameter, d, and that the rate of change of the diameter is proportional to the driving force for growth, F, as described by ^ = k.F dt 2.1 where k is a constant of proportionality. Since the driving force for growth is assumed to be directly related to boundary area per unit volume, it can be shown that F is proportional to the inverse of grain diameter, d. By integration of equation 2.1, Burke and Turnbull were able to derive the parabolic equation for the kinetics of isothermal grain growth, d2-dQ=Ktexp(-Q/RT) 2 .2 where dg is the grain size at time equal to zero, t is time, AT is the growth constant, Q is the activation energy for grain growth, R is the gas constant and T is absolute temperature. The grain boundary energy, ygb, is included in the constant, K, and the exponential term accounts for the temperature dependence of the rate of boundary migration. Feltham in 1957 [7] and Hillert in 1965 [8] derived equations of the form of equation 2.2 to describe the kinetics of grain growth. Feltham focused on the relationship 6 between driving force and boundary mobility, whereas Hillert concentrated on the effect of a distribution of grain sizes. The parabolic growth law has been found to apply to the increase in mean cell size of a froth of soap bubbles [9]. This implies that the analysis is geometrically correct, since in a network of soap bubbles, as in a network of grains, the driving force for growth is the decrease in boundary surface area [6]. Monte-Carlo simulations of growth have also yielded growth kinetics which approach those predicted by the parabolic law [10]. Unfortunately, experimental results often do not obey the parabolic equation. For this reason, the original equation [6] is often modified and used in the form, dm-d(? = K0t.exp(-Q/RT) 2.3 where m is the grain growth exponent and K0 is the fitting constant. The equation is referred to as the power law for grain growth. A short summary of experimental results from very pure zone refined materials, which have been described using the power law, was compiled by Anderson [10] and is shown in Table 2.1. The table demonstrates that m is rarely equal to two, even for extremely pure metals. It is likely that this is due to the influence of imperfections such as solute atoms and voids which limit boundary mobility, or have a boundary pinning effect [6,7,10,16]. 7 Metal, zone refined Exponent, m Reference A l 4.0 [11] Fe 2.5 [12] Pb 2.5 [13] Pb 2.4 [14] Sn 2.3 [14] Sn 2.0 [15] Table 2.1 - Grain growth exponents for the power law, for zone refined metals [10]. To explain the deviation from parabolic growth kinetics, it is necessary to examine the assumption that the only forces which influence boundary motion are due to surface curvature. In Burke and Turnbull's derivation [6], this assumption leads to the proportional relationship between driving force and the inverse of grain size, the integration of which leads to the exponent of two. Atkinson [16] suggests that the mechanisms which influence the value of m include the effect of void size and distribution, the presence of solute atoms, solute segregation, second phase particle size and distribution, and texture effects. A study of these effects would be difficult to perform, due to the small size of the voids and particles, and the low concentrations of solute which are thought to influence grain growth kinetics. The power law has often been used to describe the kinetics of austenite grain growth in process models for the hot rolling of steel [1,3-5]. The kinetics of austenite grain growth are usually investigated by reheating and holding specimens of steel at a constant temperature for varying times. A more elaborate cycle involving deformation as 8 well as heat treatment may be used to replicate grain growth during hot rolling. The specimens are quenched and etched after the discrete holding time to reveal the grain structure and allow the austenite grain size to be measured. A table of data relating grain size to time at temperature is produced and models are developed by fitting the results to the power law. The parameters which can be used for fitting are m, K0 and Q. Table 2.2 is an overview of literature results reported for plain carbon steels. C,wt% Mn,wt% m K0 ,|imm/sec Q ,kJ7mol Reference 0.22 0.9 2 1.44xl0 1 2 266 [17] 0.77 7.5 4.2X1027 400 [3] 0.69 0.76 6 8.2xl0 2 4 400 [18] 0.08 to 0.30 to 10 3.87X1032 7M100°C 400 [5] 0.89 1.68 1.31x10s2 7/<1100°C 914 Table 2.2 - Power law parameters reported for austenite grain growth. The use of the power law for m¥l is empirical. Therefore, the resulting equations should only be used over the range of experimental conditions for which the model has been validated. Since it is impossible to carry out experiments for all steels and all conditions encountered in a hot strip mill, it is desirable to develop a more versatile model which is based on fundamental principles, which can be validated with limited data. 2.2.2 Statistical Modelling of Grain Growth Several workers have reported the development of fundamentally based statistical grain growth models [19-23]. The factors which are known to influence grain growth kinetics are the driving force for grain growth, the mobility of the grain boundary, and the 9 distribution of grain sizes in the structure. A statistical model is able to account for each of these. To understand the influence of the width and shape of the grain size distribution, it is important to understand the geometry of a network of grains. A number of investigations have been directed towards this task [24-27]. A grain boundary has a surface energy and the arrangement of the boundaries is such that the boundary area tends to be minimised [6]. It is helpful to visualize this effect as being equivalent to surface tension. Atkinson [16] has summarized the geometry of the structure and the trends which are a consequence of the driving force for growth, as illustrated for a two dimensional structure in Figure 2.1. Generally, in such an array, only three grain boundaries meet at a corner, with the angle between them being 120°. In a three dimensional array, three grain boundaries meet at a grain edge at angles of 120°. At a corner, four edges meet at 109°28'. A polycrystalline metal is made up of grains which vary in size and shape. To satisfy the geometrical considerations, grain boundaries must be curved surfaces. The smallest grains will have boundaries with the lowest radius of curvature, with their centre of curvature inside the grain. Larger grains will have boundaries with a higher radius of curvature, with their centre of curvature outside the grain. The motion of a grain boundary will reduce grain boundary area if the boundary is curved; the boundary moves toward the centre of curvature. Since the driving force for growth is the liberation of energy by the reduction of boundary area, grains which are large relative to the grains around them tend to grow and grains which are small tend to shrink. Therefore, an important factor to consider when modelling the kinetics of grain growth is the width and shape of the grain size distribution. 10 Figure 2.1 - The geometry of two dimensional array of grains. Abbruzzese and Liicke [20,21] have demonstrated the use of a statistical model which has been developed from the work of Hillert [8]. Abbruzzese and Liicke were able to use the statistical model to describe grain growth under the influence of texture and in the presence of second phase particles. For austenite grain growth in low carbon steels on the hot strip mill, the pinning effect due to second phase particles is of primary importance. The statistical model for grain growth which accounts for the pinning effect of second phase particles is based on the following assumptions: • The driving force for growth, F, is the liberation of grain boundary energy, ygb. • Grains of size / can be described using which is the radius of a sphere of the same volume as the grain. The relationship between the grain boundary surface area and the radius can also be approximated by assuming the grain shape to be spherical. 11 • The grain size distribution can be approximated by subdividing the distribution into discrete grain size classes. • If a grain from size class i is larger than its neighbor from size class j, a driving force for growth will exist, which can be calculated using Fij = Vgb yRj Rt ) Fjj = 0 if Fy < 0 for Rt > Rj Fij Fji 2.4 where P is related to the boundary pinning force. • The growth rate for grains in size class i can be determined by assuming grains from class i are surrounded statistically by grains from all other classes. Growth rate is calculated using dRi — ^Mg^WijFij 2.5 where is the probability that a grain from class i is in contact with a grain from class j and Mgb is the grain boundary mobility. Grain boundary mobility is a function of the grain boundary diffusivity, the magnitude of the burgers vector, and the temperature. 12 • The probability that a grain from class i is in contact with a grain from class j can be calculated assuming that spherical grains are randomly distributed, using wa = • fvjRj 2.6 where/yy is the fraction of grains of size class j. The fitting parameter for the model is P. The model is used by performing a calculation for each time step to determine the number of grains entering or leaving each size class. Smaller grains tend to srmnk and disappear while larger grains tend to grow, so that the mean grain size increases with time. The model treats grains as three dimensional objects. The unique growth kinetics for each grain size class are determined by considering the influence of grains from all surrounding size classes. Therefore, the width and shape of the three dimensional grain size distribution is important. To validate the statistical model, it is necessary to estimate the three dimensional grain size distribution. This will be described in a later section. 2.3 Normal and Abnormal Grain Growth Grain growth models are generally used to describe normal grain growth, which is the continuous increase in mean grain size with time. Atkinson [16] describes normal grain growth as scaling, since the grain structure can be made to appear identical at any point in time by changing the magnification of the image. During normal grain growth, the width and shape of the grain size distribution do not change significantly with time. Abnormal grain growth can also occur, when discontinuous growth kinetics are observed. During abnormal growth, the largest grains in the structure tend to grow much faster than the grains around them, and the grain size distribution broadens. The growth 13 behaviour is unstable until the larger grains consume almost all of the smaller grains and a structure consisting of uniformly large grains has been established. It is generally accepted that abnormal grain growth is due to the pinning effects of second phase particles, as discussed in recent review articles by Gladman [28], and Worner and Hazzledine [29]. It is generally accepted that much of the progress in this field has resulted from the work of Zener, as quoted in a classic paper on microstructure by Smith [30]. As a result, the effect of particle pinning has become known as Zener pinning. A moving boundary can be pinned by a second phase particle or pore in its path. When a distribution of particles are present, a net pmning force will exist, the magnitude of which, according to Zener, can be calculated from j-, 3 v v Ar 2.7 where r is the particle radius, v is the second phase particle volume fraction and ygb is the grain boundary energy [30]. Grain growth in the presence of particles will be affected if the magnitude of the pinning force is significant compared to that of the driving force for growth. The growth rate can approach zero as the grain radius tends toward a linnting value, Rnm. Zener proposed that Rnm can be calculated from the pinning force. Since the pinning force is related to the pinning parameter, P, used in the statistical model for grain growth [20], it is possible to write _ J _ Rlim ~ p 2.8 A grain size distribution which has a mean grain size approaching the lairing value can become stagnant. However over time, second phase particles tend to either coarsen or 14 dissolve, depending on the equilibrium solubility of the second phase. As this happens, the size and distribution of the second phase particles will change. Since the pinning force is dependent on the size of the particles, the j^iting grain radius, Rnm, will also change with time. The grains which have the highest driving force for growth are those which are the largest relative to the grains around them. They are the first to break free from the piruiing particles and they consume the matrix of smaller grains which remain pinned. Gladman [31,32] proposed the expression 6RQV (3 - 2 r RQ 1 K l 2 -Rmax -) for the relationship between the size of the pinning particles, r, the volume fraction of the second phase, v, the average grain radius, R0 and the radius of the largest grains present, By predicting the coarsening behavior of second phase particles, and choosing an appropriate ratio of RQ to RMAX, Gladman was able to predict the time and temperature for the onset of abnormal grain growth for a number of steels. In plain carbon steels, Gladman showed that the likely identity of the second phase particles which cause abnormal grain growth is aluminum nitride. This result has been reported also by other workers [33,34]. 2.4 Measurement of Grain Size A grain is a three dimensional object with a size which can be characterized by its volume. However, for the sake of simplicity, grain size tends to be described using units of length. Most measurements of grain size are obtained from a planar section through the three dimensional structure. The measured value, which is a two dimensional result, is not equal to the true three dimensional grain size. In this work, grain size measurements are to 15 be used to validate a statistical model for grain growth. It is the three dimensional grain size distribution that is of interest since the statistical model is fundamentally based. The three dimensional grain size can be described by the equivalent volume diameter, dv, which is the diameter of a sphere with the same volume as the grain [35], as defined by 2.10 where Vis the grain volume. The three dimensional grain size distribution, f(dv), can be obtained by measuring the volume of a significant number of grains. However, grain volume in an opaque material is not easy to measure. Individual grains would have to be observed by complete disintegration of the specimen, or by using an x-ray technique, or by serial sectioning. For most investigations of grain growth kinetics, these procedures are impractical. As a result, the three dimensional grain size distribution, f(dv), is often estimated from the two dimensional measurement [36-39]. Measurement of the two dimensional grain size can be made by determining the area of each grain visible in the planar section through the structure. The two dimensional grain size can be expressed as the equivalent area diameter, dA, which is defined by 2.11 where A is grain area [35]. By determining dA for a large number of grains, it is possible to construct a two dimensional grain size distribution,/^), which can be used directly to estimate the three 16 dimensional distribution [36,37,39]. The two dimensional arithmetic mean equivalent area diameter, dAb, is defined as the sum of all grain diameters divided by the number of grains. Similarly, a three dimensional mean diameter, dVb, can be defined. The mean grain area, Am, is defined as the sum of the area of all grains, divided by the total number of grains. By the substitution of Am for A in equation 2.10, the geometric mean equivalent area diameter, dAm, can be calculated. The mean area can be determined by using quantitative image analysis equipment to measure the area of each grain, as described by A S T M E1382 [40]. Mean grain area can also be determined using Jeffries' method, as described by A S T M E l 12 [41]. A simple count of the number of grains is all that is required, and the mean area is calculated by dividing the image area by the number of grains. The major difference between the measurement of mean area by image analysis and by Jeffries' method is the treatment of grains which touch the edge of the field of view. Using the image analyser, only whole grains are measured. Therefore grains which touch the edge of the frame are ignored. Using Jeffries' method, each grain which touches the edge of the frame is counted as a half grain. An alternative two dimensional grain size measurement is Heyn's linear intercept method. Again, measurements can be made using image analysis equipment, as described in A S T M E1382 [40] or by manual techniques, as described by A S T M E l 12 [41]. Both methods require that lines of known length are superimposed on the planar section of grain boundaries, and each intersection of a line and a boundary is counted as one intercept. The mean linear intercept length, lm, is the total line length divided by the number of intercepts. A number of methods have been proposed which use the linear intercept diameter for the estimation of the three dimensional grain size distribution [35,38]. 17 Two dimensional area or intercept measurements of grain size are performed on a random planar section through a three dimensional structure. The number of grains which are measured must be significant so that errors due to the randomness of the section are avoided. A S T M E1382 [40] and E l 12 [41] suggest that a minimum of five separate fields are measured, and that at least fifty grains are visible in each field [40,41]. The methods previously described are satisfactory for cases where the grain structure is relatively uniform. For structures with a grain size distribution which is bimodal (composed of fine and coarse grains), A S T M E l 181 contains some additional suggestions [42]. For structures where the fine and coarse grains are distinctly separate, a mean grain size can be determined in each region by measuring the grain size in each region using standard techniques already described. The area fraction of each region can be determined by image analysis measurement, or estimation using comparison techniques. The overall mean grain size can be calculated by combining the mean and area fraction of each region. 2.5 The G r a i n Size Distribution It is generally thought that during normal grain growth, the three dimensional grain size distribution is approximately log normal [7,22,23,38,43]. A log normal distribution can be described by the peak grain size, d„, and the standard deviation, s, [23,38] using 2.12 18 The average grain size, db, is given by dfr = 2Zfidt = dgexp{{lns)2/2J i=l 2.1.3 For a three dimensional distribution, Equation 2.12 and 2.13 are modified by the addition of the subscript, V. For example, d becomes dv. The mean equivalent volume diameter, dVm, is given by The two dimensional grain size distribution which exists during normal grain growth can also be approximated by a log normal distribution, as demonstrated in the hterature [7,22,23,38]. For a two dimensional distribution, the subscript, A, would be used in equations 2.12 and 2.13. Other distributions have been proposed [16,35] which describe experimental data with a limited improvement in accuracy, but the log normal distribution is characteristically simple and convenient. 2.14 19 Parameter Arithmetic Geometric Number of grains Nt Nj, Fraction of grains fi fi Lower limit of smallest size 0 c class, i=l Upper class limit LA c.a1 Lower class limit (i-l).A cat'1 Class midpoint (i-l/2).A c.a1-1/2 Class width A q.(l-cW) Upper limit of largest size k.A c.ak class, i=k Table 2.3 - Description of geometric and arithmetic size class scales A grain size distribution can be approximated by subdivision of the distribution into discrete size classes. The class limits are usually arranged using an arithmetic or a geometric scale, as described by Table 2.3 [35]. The arithmetic scale is described by the class width, A, and the number of classes, k. The geometric scale is described by the lower limit of the smallest size class, c, the geometric multiplying factor, a, and the number of classes, k. 2.6 Estimation of the Three Dimensional Grain Size Distribution A number of mathematical conversions have been proposed for estimating the three dimensional grain size distribution from two dimensional measurements. Some methods are based solely on the measurement of the two dimensional linear intercept length 20 [44,45], while others are based on the measurements of area [36,37,39], or on the combined measurement of intercept and area [38]. 2.6.1 Methods Based on Spherical Grain Shape Saltikov [36] and Huang and Form [37,46] assumed the grain shape to be spherical. Saltikov proposed a method which requires the measurement of the two dimensional grain size distribution, f{dA). The method is used to apply a correction to the number of grains observed in each size class. Huang and Form proposed a method which is similar to the method of Saltikov, but with an additional correction applied. Use of Saltikov's method requires that the following assumptions are made. Firstly, grains are assumed to be spherical in shape. Secondly, the measured distribution can be represented by dividing the data into a number of discrete size classes. And finally, the three dimensional grain size for each size class is not a range of grain sizes, but it is the upper limit of that size class. That is, the number of grains in the size class in the two dimensional distribution is equal to the number of grains which have a diameter which is between the upper and lower hmit of the size class. But, in the calculated three dimensional distribution, all grains in a size class are assumed to have a diameter equal to the upper limit of that class. The method is based on the probable diameters which result from intersection of a plane and a sphere, as illustrated by Figure 2.2. The shape of the intersection between a plane and a sphere of diameter, dv, is a circle. The diameter of the circle, dA, will be less than or equal to dy. By definition, dv is the equivalent volume diameter, while dA is the equivalent area diameter. Figure 2 . 2 - The intersection of a sphere and a random plane 22 It can be easily shown that the probability, Ps(dA), that the measured diameter is between zero and dA, is given by [35] 2.15 Hence the probability that the measured diameter is between dAi and dA^ will be the difference between P^{dA\) and P${dA^). For a distribution of spherical grains of different sizes, the problem becomes more complex. A grain section with measured diameter, dA, must be from a grain which has a true diameter, dv, that is greater than or equal to the measured one. Considering the largest size class, the true number of grains in this class will be greater than the observed number, since some of the grain section diameters measured in lower size classes are due to truncation of grains from the largest size class. The number of grains per unit volume which are estimated to be in the largest class, NVtk, is determined by applying a probability correction to the number of grains observed, N^, as described by ( 1-Hdvk-l) 2.16 where dy^ is the upper limit of the size class for the measured data [36]. Only one correction is applied to the largest size class, since it is assumed that there is no chance that grain sections observed in that size class actually belong to spherical grains from a larger size class. For all size classes smaller that the largest one, other corrections must be applied. The number of grains observed in classes smaller than the class of interest, which actually belong in the class of interest, is accounted for. This increases the number of grains in the size class. Then, the number of grains observed in the 23 class, which belong in size classes larger than the class of interest, is determined using equation 2.15. This number is subtracted from the result. A general equation can be written, where Cn is the coefficient for the nth term and k is the number of size classes. The work of Saltikov is not unique in its approach to estimating the three dimensional distribution, as others have used a similar theory [47,48]. Methods were developed in the earlier part of this century at a time when advanced means of calculation were not available, and tables of coefficients were generally reported. Each method can only be applied to a distribution described by a specific arrangement of size classes. For example, Saltikov provided coefficients for twelve size classes which are geometrically scaled with a multiplication factor, a, equal to 1 0 0 1 [36]. Scheil provided coefficients for fifteen arithmetically scaled classes [47]. Saltikov's method applies a correction only for the probability that the planar section reveals a diameter which is less than the diameter of the grain; this is called truncation [35]. A second correction should also be applied to account for the fact that it is more likely that a large grain will be intersected by a planar section than a small grain; this a correction for sampling [35]. Huang and Form [37,46] proposed a method which is identical to Saltikov's, except that it accounts for sampling as well as truncation. If a cube of unit volume is considered, the correction for the sampling is estimated by dividing the number of grains after truncation correction by the size class diameter. 2.17 2.18 24 The main problem reported for methods based on the assumption of spherical grain shape is the prediction of a negative number of grains for the smaller size classes. Huang and Form reported such a result [46]. Aaron et al. suggested that negative counts can occur if some of the smallest grains are not detected when the measurements are being performed [49]. The smallest classes are most affected, since errors are compounded by the nature of the corrections applied. The biggest source of error is likely to be the assumption that the grain shape is spherical, since an array of spherical grains cannot fill space [49]. A number of methods have been proposed which make use of more realistic grain shapes [38,39,45,50]. 2.6.2 The Method of Takayama et aL A method has been proposed by Takayama et al. to estimate the three dimensional grain size distribution from the measured mean area , Am, and the mean linear intercept length, lm, [38,51]. This method assumes that all grains are tetrakaidecahedral in shape, and therefore the grains are space filling. In addition, the grain size distribution is assumed to be log normal and grains of different size are distributed randomly in space. The first step taken by Takayama et al. [38,51] in the development of the method was the determination of the probability distribution of the linear intercept length. The intersection of a random line with a solid object of volume, V, will produce an intercept of length, /. From V, the equivalent volume diameter, dv, can be easily calculated. If a number of random lines are constructed, a probability distribution which relates / to dy can be determined. Takayama et al. performed a computer simulation using this approach to determine the probability distribution for the tetrakaidecahedral grain shape and for a log normal distribution of grain sizes [52]. The mean linear intercept length was determined to be a function of the width of the grain size distribution. An equation was fit to the 25 simulation results relating the standard deviation of the log normal distribution, sv, and the peak volumetric grain size, dVg, to the mean linear intercept, lm. The mean area, Am, and mean linear intercept, lm, were related to the arithmetic mean volume diameter, dvt), using the relationships described by DeHoff [53]. By applying the characteristic equation of a log normal distribution (Equation 2.12), it was possible to relate the mean area, Am, and the mean linear intercept, lm, directly to the peak grain size, dVg, and the standard deviation, sv, as described by "m = 0.6066Wy^exp(5(lnsv)2 / 2 ) 2.19 Am = 0.48610Liy^2exp(4(ln^) 2.20 These equations can be rearranged and combined with equation 2.14 to give ( sv =exp 21rJ 1.14935 2.21 0.82575A 2.5 li -exp 3(hu v) 2 ^ 2 . 2 2 which can be applied to the measured results, to estimate the three dimensional grain size distribution. Jeffries' method for measuring mean area, Am, and Heyn's method for measuring mean linear intercept, lm, are used [41]. These parameters are easily determined with or without image analysis equipment, making it possible to obtain a quick estimate of the three dimensional grain size distribution. However, it should be emphasised that this 26 method is limited by the assumption that the grain size distribution is log normal, and that all grains are tetrakaidecahedral in shape. In fact, a range of grain shapes have been observed [24-27,43,54]. 2.6.3 The Method of Matsuura and Itoh An alternative method for estimating the three dimensional grain size distribution, based on a range of grain shapes, has been proposed by Matsuura and Itoh [25,39]. They used twelve different types of regular, equiaxed polyhedra to represent the range of grain shapes observed in a polycrystaUine solid [39]. By examining the work of Rhines and Patterson [43], they were able to determine a simple relationship between the number of grain faces, / , the equivalent volume diameter for the grain, dy, and the mean equivalent volume diameter for the entire distribution, dyb, as described by Matsuura and Itoh [39] used a similar approach to that taken by Takayama et al. [52] to determine a probability distribution for the equivalent area diameter. The intersection of a random plane with a solid object having volume, V, will result in a polygon with area, A, and appropriate values of dy and dA can be easily calculated. If a large number of random planes are generated, a probability distribution for the equivalent area diameter can be constructed. Matsuura and Itoh chose to describe the probability distribution in terms of a relative diameter, dA/dv, which can also be denoted as dR. They performed computer simulations using the twelve grain shapes to determine the probability distribution, PM{dR), as a function of the number of grain faces. The relative diameter distribution curves were characterised by the position and magnitude of the peak 2.23 27 probability and the maxirnum value of the relative diameter, dRM. These parameters were used to develop general equations to describe the shape of the curves. The method is applied by measuring the two dimensional grain size distribution, f{dA), using quantitative image analysis. The measured distribution is then divided into a number of discrete size classes. Each size class, /, is made up of grains having a measured diameter between dAi and dA^.iy A l l grains in size class i for the three dimensional distribution are represented by the upper limit of the size class, dVi, which is equal to dAi. It is assumed that a three dimensional size class contains grains of only one shape, and the number of faces, Jt , can be calculated using equation 2.23, where dAb is used as a first approximation for dvb. There is a finite probability that the relative diameter is equal to any value between zero and dRMi. The measured fraction of grains in each size class, fAi, is assumed to be equal to the sum of the probability distributions for each size class, / Q . The method is illustrated for three size classes in Figure 2.3 [39]. In the figure, the hatched region under each curve is representative of the integration of the probability function from each size class, over the range of diameters in the second size class. The results from each integration can be multiplied by the true three dimensional fraction of each size class (fvbfv2 and/yj), then summed, to determine the fraction of grains observed in the two dimensional distribution, in the second size class, fA2- In reality, it is the two dimensional distribution that is most easily measured, and the true three dimensional fractions are initially not known. 28 Figure 2.3 - Schematic illustration of the method of Matsuura and Itoh [39] for the case of three grain size classes. 29 The summation of the integrations is described again by the equation, 2.24 where k is the number of size classes, and fyj is the true fraction of grains in each size class. The fraction of grains which should be observed in the two dimensional distribution in size class i will be equal to fCi. Note that the last term in the equation accounts for the probability that a grain belonging to size class (/-l) has produced a section area with an equivalent area diameter that belongs in size class i. This is illustrated in Figure 2.3, where the tail of the probability function from size class 1 is integrated from 1.0 to dRMj, to account for the section diameters which will be observed in the two dimensional distribution, in size class 2. The three dimensional grain size distribution is known when values of fVi are determined for size classes from i=l to i=k. Equation 2.24 is used to create a matrix of k equations with k unknowns. A solution is determined for/y; to fy^ by nrinmising the sum of the squares, S2, where Values of fVi which are determined by solving the matrix of equations are used to calculate a revised mean volumetric grain size, dyb. New values of Jj are determined using equation 2.23 and probability functions are also redefined. The new functions are used in 2.25 30 equation 2.24 to solve again for fVi. The calculations are repeated until the values of fVi are seen to converge. The method of Matsuura and Itoh does not rely on unrealistic assumptions regarding the type of grain size distribution or the grain shape. To apply the method, the user needs to create a routine using computer code, making the method less convenient than the method of Takayama et al. [38,51]. CHAPTER 3 SCOPE AND OBJECTIVES The goal of this research is the measurement of austenite grain growth kinetics in plain, low carbon steels. Relevant time and temperature conditions are selected to reflect the industrial process of hot strip rolling. Area and linear intercept grain size measurements are made and used for the validation of a statistical model for grain growth. This study has two objectives. Firstly, austenite grain size will be measured in two commercial plain carbon steels. Experiments will be performed over a range of temperature and time conditions which are relevant to hot strip rolling. The measured results will be used to quantify the kinetics of austenite grain growth. Secondly, an appropriate method for the estimation of the three dimensional grain size distribution will be determined. The methods proposed by Saltikov [36], Huang and Form [37], Takayama et al. [38] and Matsuura and Itoh [39] will be compared. The most suitable methods will be used to estimate the three dimensional grain size distribution, to be applied to the statistical model for grain growth described by Abbruzzese and Liicke [20,21]. 31 CHAPTER 4 EXPERIMENTAL 4.1 Sample Composition and Specimen Design Samples of two plain carbon steels, denoted as A36 and DQSK, were obtained from United States Steel (US Steel) Gary Works. Specimens were cut from the samples, to be thermally cycled by heating in a tube furnace or by resistance heating in a thermomechanical simulator. Specimens from a third steel, denoted as 1080, were thermally cycled in a previous study [55]. Measurements from the 1080 steel were used for comparing the methods for estimating the three dimensional grain size distribution. The chemical compositions of the three steels used in this study, in weight percent, are reported in Table 4.1. The A36 and DQSK steels differ primarily in carbon content and manganese content, with the A36 being more highly alloyed. A low alloy content makes the austenitic microstructure difficult to reveal, affecting the specimen shape and the design of experiments. The 1080 steel was chosen for its higher alloy content since the austenitic microstructure in this material can be more easily revealed. A l l steels are aluminum killed. The chemical analysis of the 1080 steel is included in Table 4.1. Steel C Mn P S Si Cu Ni Cr A l N A36 0.17 0.74 .009 .008 .012 .016 .010 .019 .040 .0047 DQSK 0.038 0.30 .010 .008 .009 .015 .025 .033 .040 .0052 1080 0.78 0.68 .024 .014 0.25 .035 Table 4.1 - Composition of steels to be used in this investigation. 32 33 Inspection of the Fe-Fe3C phase diagram shows that for these carbon levels, the equilibrium microstructure obtained below 727°C is a combination of ferrite and iron carbide [56], these products resulting from the decomposition of high temperature austenite. In order to reveal the location of the prior austenite boundaries, it is necessary to quench at a high rate so that the decomposition of austenite does not produce a structure which is dominated by polygonal ferrite. During the transformation, ferrite nucleates at the austenite grain boundaries. If the amount of ferrite can be limited to between 5% and 15%, it will outline the prior austenite boundaries. Alternatively, if a higher quench rate is achieved, the diffusional transformation of austenite can be averted and a martensitic structure will result. Certain etchants have been used to reveal the prior austenite boundaries in the martensitic transformation product. Samples of the A36 and DQSK steels were received in the form of transfer bar, taken at the end of rough rolling on a hot strip mill. While many specimens were thermally cycled in the as-received condition, some of the A36 specimens were solution treated in an air furnace for 3 hours at 1200°C, prior to being tested. These specimens were sealed in quartz tubes under a partial vacuum to avoid excessive oxidation and decarburization. The specimen temperature was not monitored directly, but furnace temperature was maintained within 10°C of the set point. After holding for 3 hours, specimens were quenched by breaking the quartz tubes in a bath of room temperature water. The size and shape of specimens machined from the transfer bar were determined by the equipment to be used to apply the thermal cycle. The A36 tests and a few DQSK tests were performed on a Gleeble® 1500 thermomechanical simulator using tubular specimens so that a quenching fluid could be passed down the axis after the thermal cycle. The Gleeble designation is a registered trade name for the thermomechanical simulator, marketed by Dynamic Systems Inc., Troy, N Y . The minimum tubular wall thickness was limited to the diameter of ten austenite grains, in order to rninimize the effects of the free 34 surface on the grain growth kinetics [6]. The maximum wall thickness was timited by the quench rate required to reveal the structure. Two tubular geometries were chosen, each 20mm in length with an inner diameter of 6mm. The wall thickness was either 1mm or 2mm. The design of a Gleeble specimen is shown in Figure 4.1. Figure 4.1 - Design of the Gleeble specimens. A vertical tube furnace was also used, to perform many of the tests for the DQSK steel. The specimen thickness was again limited to a rmnimum of ten grain diameters and a maximum determined by the required quench rate. Specimens were rectangular in shape, 15mm wide, 25mm high and either 1.5mm or 3mm thick. A 2mm diameter hole was drilled through the thickness at the top of each specimen so that it could be hung in the furnace. Figure 4.2 illustrates the specimen design for the vertical tube furnace. 35 15 mm E £ LO Figure 4.2 - Specimens used in the tube furnace 4.2 Equipment for the Thermal Treatment 4.2.1 The Gleeble® 1500 Thermomechanical Simulator Tubular specimens were thermally cycled on the Gleeble by resistance heating to temperatures in the range of 900°C to 1200°C, using soak times extending to 1200 seconds. It was possible to impose carefully controlled thermal cycles using computer controlled resistance heating capabilities of the Gleeble. During heating, the specimens were held in a vacuum of less than 10~4 Torr to minimize decarburization and oxidation. Figure 4.3 is a schematic diagram of the specimen support in the Gleeble test chamber. Each thermal cycle, including the quench, was programmed into the Gleeble control system. Specimens were prepared by cleaning and spot welding the thermocouple wires. Wires were welded at the specimen mid-length with each wire aligned with the tube axis and in contact electrically, only on the specimen surface. 36 Figure 4.3 - Schematic diagram of the specimen support in the Gleeble test chamber. 37 During the thermal cycle, the temperature was controlled by a feedback system operating at a sampling rate of 100Hz. Temperature was monitored using the output of the Pt/Pt-10%Rh, Type S, thermocouple intrinsically spot welded to the specimen surface. The thermocouple wires were 0.25mm in diameter (0.010"). Heating rates of 5°C/second, 50°C/second, 100°C/second and 300°C/second were employed to heat specimens to a holding temperature. Control of temperature during heating was excellent for the lower heating rates of 5°C/second and 50°C/second. However, reproducible temperature overshoot was observed for the higher heating rates, since a time delay exists between the application of power and the resultant temperature change. The magnitude of the temperature overshoot was dependent on the heating rate, holding temperature and the specimen design. Temperature control during the isothermal holding period was excellent, always being within 2°C of the desired temperature. Most specimens were quenched by passing hehum down the tube axis. Higher quench rates of up to 250°C/second were achieved by also directing hehum onto the outer specimen surface. Water was used as an alternative to hehum when the maximum quench rate was required; rates as high as 600°C/second were achieved. However water quenching was not a favorable option since water vapor contaminated the Gleeble vacuum system. 4.2.2 The Vertical Tube Furnace The DQSK steel required an extremely high quench rate to retain the outline of the prior austenite microstructure. The vertical tube furnace was used for most of the DQSK tests to avoid the problems associated with water quenching in the Gleeble, and to achieve the highest possible quench rates (approaching 1500°C/second). The heating rate and thermal history of specimens heated in the tube furnace was dictated by the laws of heat 38 transfer. During the thermal cycle, oxidation was minimized by directing argon past the specimen at a constant flow rate of 3 litres/minute. Figure 4.4 is a schematic diagram of the furnace arrangement. The furnace consists of a vertical alumina tube, 900mm in length, with an inner diameter of 38 mm and an outer diameter of 44 mm. The tube is electrically heated using a Super Kanthal element. Temperature is controlled by a proportional digital system with thermocouple feed back, the position of the control thermocouple being outside of the alumina tube. The furnace was set to 22°C above the desired specimen temperature, this offset being experimentally determined. The furnace temperature was allowed to stabilize for a period of at least 8 hours at each temperature before the tests could be performed. The furnace rheostat was set so that the controller was on a heating cycle and on a cooling cycle for equal periods of time, the control limits being within ±5°C. The specimen was hung in the furnace from a chromel wire 1.2mm thick. The wire was also used to support a Type S thermocouple, electrically insulated by a twin bore alumina sheath. The chromel wire and thermocouple arrangement were housed in an open ended quartz tube with an inner diameter of 12mm and an outer diameter of 14mm. The quartz tube allowed for the motion of the apparatus within the alumina furnace tube during quenching. Above the furnace, a support arrangement was used to locate the specimen at the same height in the furnace for every test. 39 Thermocouple, type S Figure 4.4 - Schematic diagram of the vertical tube furnace and specimen holder. 40 The specimen temperature was measured using a Pt/Pt-10%Rh, Type S, thermocouple mtrinsically spot welded on the centre of the broad face, 3mm up from the specimen's base. The thermocouple wire diameter was 0.25mm. Temperature was monitored using a digital display with cold junction compensation. The thermocouple output was recorded by plotting the millivolt signal directly on a Kipp and Zonan, having a pen speed of 2mm/second and operated on a 20mV full scale. The specimen thermal history was monitored during heating, soaking and quenching, the chart recorder plots being used to quantify the thermal cycle. Heating rates were a function of the furnace temperature and specimen thickness. The vertical position of the specimen was kept constant for all tests, at a location 25mm from the furnace temperature control thermocouple. Soak temperatures were 1000°C, 1050°C, 1100°C and 1150°C. Due to the expected grain size, specimens 3mm thick were used at the highest temperature. At lower temperatures, 1.5mm thick specimens were used. The surface of the specimens generally reached the desired temperature within 3 minutes of entering the furnace. After reaching temperature, specimens were held for periods of up to 420 seconds. The reproducibility of the thermal history for each specimen was verified by comparing the chart recorder plots in the temperature range above 900°C. Each specimen was assumed to have reached temperature when the temperature interpreted from the plot was within 5°C of the desired soak temperature. Figure 4.5 illustrates a typical furnace thermal cycle. Following the isothermal soak, specimens were quenched by emersion in an agitated solution of iced salt water, maintained at a temperature of approximately -10°C. The salt water bath was placed at the bottom of the vertical tube, as illustrated in Figure 4.4, so that the transport time from the furnace to the quenchant was minimized. 41 Figure 4.5 - Typical furnace thermal cycle 42 4.3 The Thermal Cycles Most thermal cycles involved simply heating specimens to the desired temperature, holding, then quenching. However, some Gleeble tests used more elaborate treatments, such as, multiple heating rates and hold times, and periods of controlled cooling prior to quenching. The heat treatments can be divided into four groups: 1. Standard Gleeble thermal cycles, 2. Furnace tests, 3. Stepped Gleeble thermal cycles, and 4. Continuous cooling tests. Figure 4.6 shows a schematic time versus temperature plot for a standard Gleeble thermal cycle. Table 4.2 gives an overview of all standard Gleeble tests which were performed. Specimen Wall Heat. Hold T, °C Hold t, seconds. thick., Rate mm °C/sec. A36 as-received 2 5 900 120 A36 as-received 2 5 950 10, 120, 600 A3 6 as-received 2 5 1000 10, 60,120, 600, 900 A36 as-received 2 5 1050 1,10, 60,120, 300, 750 A36 as-received 2 5 1100 1,10, 30, 60, 120, 300, 600 A3 6 as-received 2 5 1150 1,10, 30, 60,120, 300, 450 A36 as-received 2 5 1200 300, 600 A3 6 as-received 2 50 950,1050,1150 120 A36 solution treated 2 50 950,1050,1150 120 DQSK as-received 1 5 1050 120 DQSK as-received 1 5 1100 60, 120 DQSK as-received 1 5 1150 60 Table 4.2 - Standard thermal cycles performed on the Gleeble. time Figure 4.6 - Standard Gleeble thermal cycle 44 A typical furnace thermal cycle is illustrated by the schematic time versus temperature plot shown in Figure 4.5. Table 4.3 summarizes the standard furnace cycles which were used. Specimen Aim hold T, °C Aim hold t, seconds. DQSK, 3mm thick 1150 0, 60, 120, 240, 420 DQSK, 3mm thick 1100 0, 60, 120, 240, 420 DQSK, 1.5mm thick 1100 0, 420 DQSK, 1.5mm thick 1050 0, 60, 120, 240, 420 DQSK, 1.5mm thick 1000 0, 60, 120, 240, 420 Table 4.3 - Standard furnace thermal cycles Stepped thermal cycles applied to A36 specimens on the Gleeble are described in Figure 4.7 and Table 4.4. Specimen HeatRatei, °C/sec. Hold T\ °C Holdfi. sec. Heat.Rate2 °C/sec. PeakT °C. Hold T2°C Hold t2 sec. A3 6 as-received, 5 900 120 100 1100 1100 1, 10, 30, 2mm tube 120,600 A36 as-received, 5 900 120 100 1100 1050 1, 10, 30, 2mm tube 120, 750 Table 4.4 - Stepped Gleeble thermal cycles 45 time Figure 4.7 - Stepped Gleeble thermal cycle 46 Tests performed to examine grain growth kinetics under conditions of continuous cooling are described in Figure 4.8 and Table 4.5. Specimen Hold time, sec. Quench Temp.,°C A36, as-received, 2mm wall 1, 5, 10, 20, 45, 60 1120, 1110, 1100, 1080, 1030, 1000. Table 4.5 - Continuous cooling thermal cycles 4.4 Revealing the Prior Austenite Microstructure Specimens to be used for grain size measurement were cut at the thermocouple location, to ensure a known thermal history. Specimens were mounted in a cold setting aery he resin and subsequently ground and polished to a l[im diamond finish. A number of different etchants were used, depending on the condition of the specimen. These etching procedures are described in Table 4.6. 4.5 Measurement of the Austenite Grain Size Polished and etched specimens were photographed on a Leitz Vario-Orthomat® metallograph system using black and white Polaroid® Type 55 film at magnifications from x80 to x900. The revealed austenite grain boundaries were traced on transparent plastic film using felt tipped pen. From the traced austenite structure, the mean grain area, Am, was determined by Jeffries' method [41]. Each whole grain was counted once and each partial grain, cut by the edge of the field of measurement, was counted as a half grain, as described by A S T M standards [41]. From the mean area, the mean equivalent area diameter, dAm, was determined using equation 2.11. 47 cooling, 2°C/sec. time Figure 4.8 - Continuous cooling thermal cycle 48 Specimen Condition Etchant Reference He quenched A36 4% picric acid in ethanol. Immerse at room temperature for 1 to 5 minutes. [56] H 2 0 quenched A3 6 Saturated aqueous picric acid or lOOmL of saturated aqueous picric acid with 4g of sodium dodecylbenzene in lOOmL H,0 and 10 drops of Triton X-100 surface active agent. Immerse and swab at 60°C to 80°C for up to 10 minutes. [56] [57] DQSK (coarse grains) Immerse in 2% Nital, room temperature, 20 to 30 seconds, followed by lOg sodium metabisulfite in lOOmL H,0, room temperature, 10 to 30 seconds. Do not clean after latter step. [58] DQSK (fine grains) 4% picric acid in alcohol. Immerse at room temperature for 1 to 5 minutes. [56] 1080 4g picric acid and 50g NaOH in 250mL H,0. Immerse, boil and swab sample, 10 minutes then light lum polish, followed by light 1% Nital etch. [58] Table 4.6 - Summary of etching procedures. A number of the traced microstructures were examined using image analysis equipment with C I M A G I N G systems SIMPLE© software. The system uses a screen of 480 by 640 colour pixel points. The area of each whole grain.was measured individually from the traced microstructure using standard A S T M methods [40]. An image processing routine was developed so that the width of grain boundaries detected by the system was 49 on average, 2 pixels. The area of a grain was defined as its net internal area plus half the area of its boundary, so that the pixels taken up by the boundary could be accounted for. It was verified that the sum of the area of each grain was equal the total area of the image. The image analyzer did not measure the area of partial grains. That is any grains which touched or were cut by the edge of the field of measurement were ignored. From the mean area, Am, the mean equivalent area diameter, dAm, was determined. From each measured grain area, A, an equivalent area diameter, dA, was also determined, and the two dimensional grain size distribution,/^), was assessed. The mean linear intercept length, lm, was measured on some of the traced microstructures using Heyn's method, as described by A S T M standards [41]. A rectangular grid of 12 lines was placed randomly over the tracing. Each intersection of a grain boundary with a line from the grid was counted as 1. A line which touched a boundary but did not cross it was counted as a half, and a line which exactly intercepted a grain corner, where three boundaries were seen to meet, was counted as 1.5. The mean linear intercept length was determine by dividing the total grid length by the number of counted intercepts. The mean linear intercept was also determined for some specimens from the traced structure using the image analyzer and standard A S T M methods [40]. The analyzer used a grid made up of all of the pixels on the screen. An image processing routine was used to reduce the width of each boundary to 1 pixel. The mean intercept length was calculated as the total number of pixels in the screen divided by the number of pixels which made up the boundaries. CHAPTER 5 RESULTS AND DISCUSSION I. THE KINETICS OF AUSTENITE GRAIN GROWTH 5.1 The Structure of the Quenched and Etched Specimens 5.1.1 Helium Quenched A3 6 Steel Specimens of the A36 steel were heat treated on the Gleeble then quenched using helium, and etched in 4% picral. Appropriate quench rates were determined to be between 100°C/second and 200°C/second. Figure 5.1 is a photomicrograph of an A36 specimen heated to 1150°C at a rate of 5°C/second and held for 60 seconds before being quenched at a rate of 120°C/second. The mean equivalent area diameter, dAm, is 144.0fJ.rn. The structure is approximately 10% ferrite, with the remainder being a combination of pearlite, bainite and martensite. During the transformation, ferrite, which is white in the print, has nucleated on the prior austenite grain boundaries in the form of grain boundary allotriomorphs; almost all boundaries appear to be decorated in this way. Due to the high cooling rate, secondary Widmanstatten side plates have grown out from the grain boundary ferrite, creating a feathery appearance. At the boundary between the growing ferrite and the austenite, pearlite has nucleated, and it is visible as the dark structure inside the austenite grains. Within some grains, acicular regions are also visible and are thought to be upper bainite. However, detail regarding these non-equihbrium phases is unimportant for the purposes of grain size measurement, as long as the prior austenite boundaries are decorated with ferrite. Near the top left corner of the print, a lightly etched region within a larger grain is visible, and laths of martensite are easily distinguished. 50 5 1 500pjn Figure 5.1 - A36 Steel, Gleeble heated at a rate of 5°C/sec to 1150°C and held for 60 seconds. He quenched. Magnification is approximately x85. Figure 5.1 shows features typical of all of the hehum quenched A36 specimens. Larger grained specimens generally contained more regions of martensite within the austenite grains, while smaller grained specimens had higher proportions of ferrite and pearlite. The rate of formation of ferrite increased with decreasing austenite grain size, requiring higher quench rates to obtain the necessary austenite grain boundary identification. Specimens with an austenite grain size (mean equivalent area diameter), dAm, of less than 20(im transformed at such a high rate that the formation of excessive ferrite could not be avoided without the use of a water quench. 52 5.1.2 Water Quenched A36 Steel Fine grained specimens of A36 were water quenched in the Gleeble attaining cooling rates between 300°C/second and 600°C/second. Figure 5.2 is a photomicrograph of an A36 specimen heated to 1000°C at a rate of 5°C/second and held for 10 seconds. The specimen was water quenched and etched in aqueous picric acid. The mean equivalent area diameter of the austenite grains, dAm, is 14.5p:m. 50um Figure 5.2 - A36 Steel, Gleeble heated at a rate of 5°C/sec to 1000°C and held for lOseconds. H 2 0 quenched. Magnification is approximately x820. The water quenched structure is almost entirely martensitic. The prior austenite boundaries appear to be dark since they have been preferentially attacked by the etchant. Some boundaries are decorated by nucleating allotriomorphic ferrite, but significant 53 growth has been prevented by the high quench rate. The action of the etchant is likely to be due to the segregation of solute to the austenite boundaries, and the lack of solute redistribution during the martensitic transformation [57]. Residual strains from the transformation may also promote the etchant attack. 5.1.3 Water Quenched DQSK Steel DQSK specimens were water quenched after heat treatment in the Gleeble or in the vertical tube furnace. Specimens with mean austenite grain size, dAm, of less than 40(i.m were etched using a 4% picral solution, while those with larger grains were etched in nital, then sodium metabisulfite. Figure 5.3 is a photomicrograph of a fine grained DQSK specimen heated to 1050°C in the tube furnace, held for 420 seconds before water quenching. The specimen is etched in 4% picral to give mean equivalent area diameter, dAm, of 36.2|im. The prior austenite boundaries are decorated by ferrite, but the ferrite distribution is non-uniform along the boundaries. Within the austenite grains, the darker structures of pearlite and bainite are visible. The rate of the transformation from austenite to ferrite is limited by the rate of redistribution of alloying elements (carbon and manganese). Since the alloy content of DQSK is relatively low, the transformation occurs very quickly. The low alloy content also affects the quality of the etch. The quench rate was achieved by immersing the specimen in an agitated, iced solution of saturated salt water at -10°C. Although the DQSK steel did not etch as well as the A36 steel, some signs of the location of the prior austenite boundaries are present. However, a more ideal structure would be preferred. 54 l (X) | im Figure 5.3 - DQSK Steel, furnace heated to 1050°C and held for 420seconds. H 2 0 quenched. Magnification is approximately x420. The rate of the transformation for more coarsely grained austenite was lower and a different microstructure was observed. Figure 5.4 is a photomicrograph of a DQSK specimen heated to 1150°C in the vertical tube furnace and held for 227 seconds before being quenched in the iced salt water solution. The specimen was etched using nital then sodium metabisulfite, showing a mean equivalent area diameter, dAm, of 168.0|im. The structure is similar to that of the helium quenched A36, in that ferrite has nucleated first as grain boundary allotriomorphs from which secondary Widmanstatten side plates have grown. Adjacent to the decorated austenite boundaries, regions of darker etching pearlite can be seen. Also, grains of polygonal ferrite have formed within some of the smaller grains, nucleating at the interface between the grain boundary ferrite and the 55 receding austenite. As observed in the A36 steel, some of the intermediate regions may be upper bainite. 50f)Lim Figure 5.4 - DQSK Steel, furnace heated to 1150°C and held for 227seconds. H 2 0 quenched. Magnification is approximately x85. 5.1.4 Water Quenched 1080 Steel The 1080 steel was used in a previous grain growth study [55] because it could be quenched to martensite and etched to reveal prior austenite boundaries. The structure is shown in Figure 5.5, for a specimen heated to 1100°C and immediately water quenched. This heat treatment produced a mean equivalent area diameter, dAm, of 67.2|im. 56 500|jm Figure 5.5 - 1080 Steel, furnace heated to 1100°C and immediately H 2 0 quenched. Magnification is approximately X85. A number of dark, parallel streaks can be seen. These are thought to be MnS inclusions, as they were visible prior to etching. 5.2 Measurement of the Grain Size and Estimation of the Error 5.2.1 Interpretation of the Structure A transparent tracing of the prior austenite grain structure was produced from each photomicrograph, to be used for the measurement of grain size. Depending on the microstructure, it was sometimes difficult to determine the location of the boundaries. As a guide, the following assumptions were made to assist in the interpretation. Firstly, grains do not exist within the boundaries of another grain. Secondly, grain corners are at the 57 intersection of three boundaries; the angle between the boundaries is assumed to be approximately 120°. Finally, the grain shape is equiaxed. The water quenched A36 and 1080 specimens were relatively easy to interpret since the structure was predominantly martensitic and the appropriate etchant produced an image with good contrast. The DQSK and the hehum quenched A36 steels were examined on the assumption that the grain boundaries were highhghted by the nucleation and growth of ferrite allotriomorphs. In some cases, extensive growth of Widmanstatten side plates complicated the structure. In regions where little ferrite was visible, the orientation of the pearlite provided some clues as to the prior austenite structure. In all cases, when the visible structure was discontinuous, the typical geometry of a network of grains was considered. 5.2.2 Measurement of the Grain Size The results of the grain size measurements are summarized in the Appendix. Most specimens were measured using A S T M standard methods, which recommend that at least fifty grains be visible in each field [40,41]. For some specimens, high magnification was required in order to accurately interpret the structure and as a result, fewer grains were visible in the field of view. The magnification was chosen so that a compromise was reached between the number of grains visible and the practical interpretation of the structure; both factors contribute to errors in the measurement. In situations where the total number of grains from all fields was less than 100, the results were considered to be only an estimate of the grain size rather than a statistically valid measurement. Some specimens had a structure of distinctly separated fine and coarse grains, as illustrated in Figure 5.6. The area fraction of each region was measured using the image analyser, by A S T M standard methods [42]. 58 Mean equivalent area diameter, dAm, and mean linear intercept length, lm, were determined by the use of manual methods and by the use of image analysis [40,41]. A more complete description of the measurement methods used is found in Section 4.5. 250|jm Figure 5.6 - A36 Steel, Gleeble heated to 1000°C and held for 120seconds. He quenched. Magnification is approximately x410. 59 5.2.3 The 95% Confidence Intervals For each of the measured parameters, a 95% confidence interval was calculated. The measured value in each field was assumed to be a variable distributed normally about a true mean value. The student's ^-distribution was used for the calculation of the confidence interval, as described by any general statistics text [65]. In most cases, five measurement fields were used. The 95% confidence intervals for the manually measured values of linear intercept length, lm had a maximum of ±14.0%, a minimum of ±2.2% and a mean value of ±8.2%. Similarly, the 95% confidence intervals for the manual measurement of the mean grain area, Am, had a maximum of ±23.8% and a nidnimum of ±1.4%, with the mean value being ±10.6%. From the mean area, an equivalent area diameter, dAm, was calculated; it is this value that is actually of interest. Since the equivalent area diameter is a function of the square root of the area, the confidence intervals become smaller and are a maximum of ±12.3% and a minimum of ±0.7%, with the mean value being ±5.3%. The mean equivalent area diameter measured on the image analyser had a 95% confidence interval with a maximum value of ±12.3%, a minimum of ±1.0% and a mean value of ±5.5%. A complete list of the 95% confidence intervals are reported with the results in Tables A.1 to A.7 in the Appendix. The confidence intervals are a reflection of the variability of the structure since they are calculated from measurements made on a number of different fields from the same specimen. A large interval indicates that the grain size in a specimen has some degree of non-uniformity while a small value indicates that the structure is relatively uniform. Generally, the larger confidence intervals correspond to specimens where the austenite grain structure was more difficult to reveal. 60 5.2.4 Systematic Errors in Measurement of Equivalent Area Diameter Comparison of the equivalent area diameter measured manually and that obtained on the image analyser reveals that the manual measurement was consistently higher, by an average of 7%. Calibration checks showed that the area measured for a single object by each method was the same. The differences between the mean equivalent area diameters were found to be a result of the treatment of grains which touch the edge of a field. Using Jeffries' method [41], grains which touched the edge of the frame were counted as half grains. Using the image analyser, grains which touched the edge were ignored. This implies that in order to get the higher result, the structure intersected by the edge of the field consistently had larger grains than the structure in the centre of the image. In order to explain the larger grains at the edge of the image, the following argument must be considered. A single large grain on a plane is more likely to be intersected by a random line than a single small grain. In fact, the probability of intersection is directly proportional to the grain diameter. If it is assumed that the spatial distribution of grains in a structure is random, it follows that the probability that a grain will be intersected by the edge of a field is proportional to the diameter of the grain, since the location of the edge of the field is essentially a random line. Obviously, the probability of intersection is also proportional to the number of grains of that diameter which are present in the distribution. Therefore, large grains are intersected by the edge of the field more often than small grains. The grain size distribution measured by image analysis ignores the edge grains and therefore skews the result towards a distribution with a lower mean. This argument applies to all equiaxed structures and it can account for almost all of the discrepancy between the two mean equivalent area diameter results. Less grains are measured by the analyser than by the manual method. The size distribution of the grains which are ignored by the analyser can be estimated by assuming that the probability that 61 they are in contact with the edge of the field, Pp{d), is proportional to the equivalent area grain diameter, dAi, and to the fraction of grains in the two dimensional distribution, fAi, as described by k ^fAjdAj j=1 5.1 As an example, the specimen of 1080 steel heated to 1050°C and quenched had a mean equivalent area diameter, dAm, of 50.3|im, measured on 1142 grains using the image analyser. The mean equivalent area diameter measured by Jeffries' method [41] was 53.6(im from 1319 grains. Note that the grains cut by the edge of the field are counted as half grains by Jeffries' method. Therefore, 354 grains were ignored in the image analyser measurement. The grain size distribution of the missing edge grains can be calculated by determining the number of edge grains in each size class, NFi, using 354 ^ F i ~ k fAAAI ^fAjdAj j=1 5.2 To remain consistent with Jeffries' method, half of this number can be added to the number of grains measured in each size class, NAi, by the image analyser, to obtain the corrected number of grains, NFi NAci=-r+NAi 2 5.3 The correction which is proposed here is consistent with Jeffries' method, since only half of the edge grains are added. Rejection of all edge grains, as is done during the image 62 analysis, skews the distribution to a lower mean since more large grains are rejected than small ones. On the other hand, consideration of all of the edge grains would skew the distribution toward a higher mean. The intermediate approach of considering half of the edge grains is the most appropriate because it will give the best approximation of the true two dimensional grain size distribution. The corrected fraction of grains in each size class can then be determined by r _ NAa JACi ~ k 1NACJ i = 1 5.4 The true fraction of grains in each size class, fAi, is initially not known, but the fraction measured by the image analyser can be used as a first approximation. The corrected fraction, fAa, w u ^ b e c l ° s e r to the true fraction, and so equations 5.2, 5.3 and 5.4 can be applied iteratively, by substituting fACi for fAi in equation 5.2. The corrected grain size distribution converges to a final solution in approximately ten iterations. The final distribution will be closer to the true two dimensional grain size distribution than the measured result obtained from the image analyser, since the skewing effect of the edge of the field of measurement is accounted for. Returning to the example, a corrected mean equivalent area diameter of 52.8|im is calculated which compares well with the Jeffries value of 53.6|im. Without the applied correction, the discrepancy between the two results is 6.0%. With the correction applied, the discrepancy is reduced to 1.5%, which in view of the required accuracy is negligible. Results from other examples are reported in Table 5.1. The difference between the results are reported as a percentage of the Jeffries result. 63 Steel Temp time Jeffries Jeff. Image L A . diff. corrected diff. °C sec. no. analyzer no. % result, % dAm, urn grains dAm,\im grains dAm, | lm 1080 1050 0 53.6 1313 50.3 1142 -6.1 52.8 -1.5 1080 1050 120 59.8 1055 55.3 885 -7.6 58.8 -1.6 1080 1050 420 70.7 756 64.7 629 -8.5 67.8 -4.2 A36 1000 600 92.3 229 81.5 185 -11.7 87.4 -5.3 A36 1150 1 99.8 237 89.7 193 -10.2 103.4 +3.6 A36 1150 60 144.0 281 134.6 206 -6.5 149.8 +4.0 DQSK 1000 0 29.0 352 27.3 299 -6.1 27.7 -4.5 DQSK 1000 120 32.3 356 31.3 271 -2.9 32.4 +0.3 DQSK 1150 420 178.1 298 162.1 231 -9.0 174.6 -2.0 Table 5.1 - Correction of systematic errors for the mean equivalent area diameter The corrected distribution can be assumed to be closest to the true two dimensional grain size distribution. The mean equivalent area diameter, dAm, from Jeffries' method [41] is always closer to the true mean equivalent area diameter than the result obtained directly from the image analyser. The corrected mean result is generally within the 95% confidence hmits of the Jeffries result. A l l results are affected by the statistics of the measurement and Table 5.1 demonstrates that the agreement between Jeffries' results and the corrected mean equivalent area diameter improves as the measured number of grains increases. The scatter in the relative magnitude of the difference between Jeffries' results and the true mean result can be attributed to the influence of measurement statistics. 64 5.2.5 Systematic Errors in Measurement of Linear Intercept Length Comparison of linear intercept results determined on the image analyser and by manual methods also show that the manual result is larger, by an average amount of approximately 15%. The difference can be attributed to the way that each method determines the number of intercepts. Manually, the intersection between the grid and a grain boundary is counted as one intersection. Using the image analyser, the grid length is considered to be the total number of pixels on the screen, and that is 640 by 480. The number of intercepts is counted as the number of pixels which make up the area of the grain boundaries. The matrix of pixels in the screen can be assumed to be a grid of 480 horizontal lines that are 640 pixels in length. Using Heyn's method, a grain boundary or a portion thereof, that is horizontal on the screen will only constitute a single intercept. Using the image analyser, a horizontal boundary will be counted by the analyser as several intercepts, with the number of intercepts being equal to the number of pixels in the horizontal portion of the boundary. Therefore the image analyser counts more intercepts than Heyn's method. The mean linear intercept length is equal to the total line length divided by the number of intercepts. Therefore, the analyser result is lower than the Heyn's measurement. The true mean linear intercept length, lm, is closest to the value obtained manually, using Heyn's method. Linear intercept measurements performed with the image analyser should thus be viewed with caution. The argument above could also be applied by assuming that the image analyser screen is a grid of 640 vertical lines, 480 pixels in length. 5.3 The Kinetics of Austenite Grain Growth 5.3.1 A36 Steel Heated to Temperature at 5°C/second Mean equivalent area diameter measured by Jeffries' method [41] is plotted against time in Figure 5.6, for as-received A36 steel specimens heated at a rate of 5°C/sec. The symbols are the data points and the lines show the fit obtained using the power law, to be 65 described in Section 5.5. The scatter of results is reasonable when the magnitude of the errors is considered. At all temperatures, the grain size is seen to increase with time and the growth rate increases with rising temperature. At 1000°C for 60 seconds and 120 seconds and at 1050°C for 10 seconds and 60 seconds, two grain sizes are reported to exist simultaneously, for each specimen. Under these conditions, abnormal grain growth was observed and separate, measurable groups of grains were present. The grain size distribution appeared to be bimodal. As an example, Figure 5.6 is a photomicrograph of the specimen held at 1000°C for 120 seconds. The data appears to be divided into two sets, with one group of results being less than 50|im and the other being greater than 80(im. The finer structures were observed after heat treatment at lower temperatures and shorter times. To explain the discontinuous growth between 50|im and 80|nm, the presence of a second phase precipitate, having a boundary pinning effect is assumed. At 950°C the pinned structure is stable for times up to 600 seconds. At 1000°C and 1050°C the fine structure is unstable and with time, abnormal grain growth begins as the pinning particles dissolve or coarsen leading to a reduction of the pinning force [31,32]. In the coarse structure, the data shows, that the rate of grain growth slows down substantially with increasing time; at times greater than 450 seconds at all temperatures, a hmiting grain size is being approached. At 1000°C and 1050°C there is little growth of the coarse structure, once abnormal grain growth has ceased. In the specimens held at 1100°C and 1150°C, a substantial amount of grain growth has occurred during heating, since the initial grain sizes are greater than 80(J.m. It is likely that abnormal grain growth has occurred during the heating period. 66 Figure 5.7 - Kinetics of austenite grain gowth in A36 steel, heating rate=5°C/second. Lines are the power law, for ^ > 8 0 L i m , /•22-d08'22=1.51E47xexp(-840/i?r) for dA m<80Lim, d3-37-d0337=5.46E54.*.exp(-1291/i?r) 67 5.3.2 DQSK Steel Heated to Temperature in the Tube Furnace Results from the DQSK specimens heated in the vertical tube furnace are plotted in Figure 5.8. The mean equivalent area diameters were measured manually using Jeffries' method and curves were fit using the power law, to be discussed in section 5.5. At 1000°C and 1050°C significant grain growth was not observed. At 1100°C abnormal growth was observed and a single curve does not fit well to the stepped nature of the data. Unlike the A36 steel, measurements for the DQSK could not be made on the fine and coarse structure separately. At 1150°C, significant growth has occurred during heating since the initial grain size is greater than 120(im. At all temperatures, the grain size is approaching a limiting value, although at 1100°C and 1150°C, after 450 seconds, grain growth is continuing, but at a decreasing rate. Investigations of grain growth kinetics in plain carbon steels reported by other workers show similar results to those reported here [31,55,60]. Austenite grain size is generally reported in the range of 10|im to 250|Lim and abnormal growth is often observed when samples are heated in the temperature range of 900°C to 1150°C. Results differ primarily in the time dependence of growth. For both steels in this investigation, the growth rate is seen to decrease significantly with time and a limiting grain size is quickly approached. Some other investigations have reported similar growth behaviour [31,55], while others have not observed a limiting grain size [60]. Abnormal growth and a limiting grain size are phenomena which are typically observed when second phase particles are present. In a steel similar in composition to the A36, Gladman investigated abnormal growth [31] and was able to quantitatively describe the phenomena by considering the pinning effects of aluminium nitride. His calculations cannot be applied to this study because the particle size and volume fraction of the A1N in the as-received condition in the A36 and DQSK steels is not known. 68 200 20 H o - h 1 1 1 1 1 1 0 100 200 300 400 500 600 Time, seconds Figure 5.8 - Kinetics of austenite grain growth in DQSK steel, heated in the tube furnace. Lines are the power law, d5M-d 5-66=5.02E56.texp(-1271/fl7) 69 Variability in the growth kinetics of austenite in the DQSK and the A36 steel can be partially accounted for by differences in the puuiing effects. The influence of pinning particles is determined by the amount of aluminium and nitrogen in the steel, the precipitation kinetics of A1N and the size distribution of the austenite grains, as described by equations 2.4, 2.7, and 2.8. Each of these factors is affected by the thermal and mechanical history of the material. Further, the content of alloying elements such as carbon and manganese has an influence on the kinetics of growth since the boundary energy and mobility are affected. 5.4 Heating Rate Effects Differences in the heating cycle can have a significant influence on the growth kinetics. Figure 5.9 is a comparison of austenite grain growth in the as-received A36 steel, heated to temperature by different heating cycles. The slowly heated specimens were heated at a constant rate of 5°C/second to 1100°C and held. Rapidly heated specimens underwent a stepped heating cycle, and were heated initially at 5°C/second to 900°C and held for 120 seconds, then heated again at 100°C/second to 1100°C. The curves in the plot are the fit of the power law to the data. The initial grain size in specimens heated at the 100°C/second was 47|i.m while in those heated at 5°C/second the initial grain size was 88p:m. The apparent j^ i t ing grain size was lOOiim in the quickly heated specimen and 140|Lim in the slowly heated one. These different sizes can be explained by grain growth during heating, and the influence of A1N particles. Nucleation effects during the transformation to austenite are likely to be ^significant due to the identical thermal cycles up to 900°C. The difference in initial grain size can be explained by grain growth occurring during heating since the rapidly heated specimen spent less time at high temperature, and therefore less growth occurred. The effect of holding time at 900°C can be ignored with respect to grain growth since at this temperature, limited grain coarsening has been observed. The different limiting grain size 70 is evidence that the dispersion of particles is affected by the heating cycle, since Zener has described the limiting grain size as function of particle volume fraction and radius [30]. If equilibrium conditions exist, particle volume fraction is unchanging, and the smaller j^ i t ing grain size must be due to a finer dispersion of closely spaced particles. The influence of second phase particles will be discussed in greater detail in Section 5.6. 5.5 Models Describing the Kinetics of Grain Growth The power law and the statistical model for grain growth were fit to data from the A36 and the DQSK steels. Measured values of mean equivalent area diameter, dAm were used to determine the constants for the fit of the power law. Estimated values of the average equivalent volume diameter, dvb, were used to fit the statistical model. The statistical model of Abbruzzese and Liicke [20,21] was applied by Militzer et al. [61,62]. 5.5.1 Application of the Power Law The power law for grain growth was fit to the results of mean equivalent area diameter measured for the DQSK steel heated in the tube furnace and the A36 steel heated in the Gleeble at a rate of 5°C/second and 100°C/second. In each case the parameters of m, K0 and Q were used to fit the power law equation by a least squares procedure. The sum of the squares was minimised using a solving routine on a Microsoft Excel® spreadsheet. Separate curves were needed to fit to fine and coarse austenite grain growth A3 6 steel heated at 5°C/second. Table 5.2 is a summary of the parameters determined to describe the results. 71 180 160 H I 140 120 100 80 60 40 20 0 • 5°C/sec(1100°C) cf 2 2-d0 8 2 2=1.51 E47.f.exp(-84CW?7) 100°C/sec (1100°C) du 9 -L1 1 4 9=1.94E68.f.exp(-1089/flT) 0 ~ I 1 1 1 1 1 1 100 200 300 400 500 600 700 800 time, seconds Figure 5.9 - Effect of heating rate on austenite grain growth in A36 at 1100°C. 72 Condition m Kn, |imm/sec. Q, kJ/mol. Fine A36, 5°C/sec 3.37 5.46E+54 1291 Coarse A36, 5°C/sec 8.22 1.51E+47 840 A36, 100°C/sec 14.85 1.94E+68 1089 DQSK 5.66 5.02E+56 1271 Table 5.2 - Power law fitting parameters for the A36 and DQSK steels, for the equation, dm-d0m=K(pxTp(-Q/RT). Use of the power law with m not equal to 2 is empirical. When m is used as a fitting parameter, the equation becomes a curve fitting empirical expression for describing the austenite growth kinetics, as shown in Figure 5.7, 5.8 and 5.9. However, the reported values obtained for m, K0 and Q are widely scattered. The values obtained for the A36 steel heated to temperature using two different heating rates are entirely unrelated, despite the fact that results were obtained using the same steel. The equations cannot be used outside of the experimental conditions for which they were determined. For these reasons, the development of a more fundamentally based relationship is of considerable importance. 5.5.2 Application of the Statistical Model The statistical model was applied by Militzer et al. [61] to the results for the A36 steel heated at 5°C/second and 100°C/second, and the DQSK steel. The calculations are based on the average equivalent volume diameter, dy^, estimated from the measured mean equivalent area diameter, dAm, by the method of Takayama et al. [38]. The grain boundary energy, y ^ , is calculated as a function of the carbon content, as described by Gjostein [63]. Consequently, a value of 7^=0.7Jhr 2 is used for the A36 and 73 7g£=0.8Jrrr2 is used for the DQSK steel. The grain boundary mobihty, Mgb, is approximated by that of pure austenitic iron and is calculated using the grain boundary diffusivity, Dgb, and the Burgers vector, b, from Frost and Ashby [64], Dahb2 kT 5.5 Mobihty decreases quickly as the temperature decreases. The pimiing parameter, P, was estimated, in order to fit the model to the data of each isothermal test series, for each steel. The fit of the model is shown in Figure 5.10 for the A36 steel heated at 5°C/second in the Gleeble, and in Figure 5.11 for the DQSK steel heated in the tube furnace. The model fits well to the growth kinetics of the coarse A36. At this stage, no attempt to fit the model to the abnormal growth condition has been reported for the A36 steel. For the DQSK, all data, including the abnormal growth results were desrcibed using the model. Since during abnormal growth the pinning force changes with time, at 1100°C, two values of the pmning parameter were used, resulting in a good fit for the abnormal grain growth data. The discontinuity in the curve is a result of the sudden change in P, related to the changing pinning behaviour of the A1N particles in the material. For the A36 and the DQSK steel results, the statistical model is more effective than the power law in describing the experimental data. The j^ i t ing grain radius, Rnm, was estimated from the experimental results. According to equation 2.7, Rnm is the reciprocal of the pinning parameter, P , so that the limiting grain radius can be predicted from the pinning force estimations made with the statistical model. Figure 5.12 is a comparison of the experimental and predicted values of Rlim, showing good agreement between the two. 74 240 0 100 200 300 400 500 600 700 800 time, seconds Figure 5.10 - The statistical model and the results for the A36 steel for a heating rate of 5°C/second. Lines are generated by the model [61,62]. 75 Figure 5.11 - The statistical model and the results for DQSK steel. Lines are generated by the model [61]. 0 20 40 60 80 100 120 140 Rlim (experimental), (im Figure 5.12 - Predicted and experimentalRu [61,62]. 77 The results indicate that the statistical model can be applied to describe the grain growth kinetics using a single fitting parameter, P. Furthermore, the value of P can be used to predict the jjiniting grain radius with reasonable accuracy, demonstrating a consistency between the experimental evidence and the model's theoretical basis. Since the statistical model is developed from fundamental principles, it can be extrapolated to describe grain growth kinetics under a wide range of conditions, as long as reasonable assumptions are made regarding the pinning forces which exist. The aim in applying the statistical model is to describe the kinetics of grain growth during hot strip rolling, under conditions which are not isothermal. A test series was performed with the A36 steel to measure the kinetics of grain growth during continuous cooling conditions. During hot rolling, between the roughing mill and the finishing mill, a typical temperature change from 1120°C to 1000°C can occur over a period of approximately 60 seconds. A36 steel specimens were heated to temperature using a stepped thermal cycle to 1120°C and a continuous cooling cycle of 2°C/second was imposed. The statistical model was applied by Militzer et al. [61], employing the temperature dependence of the pinning parameter, P, determined during stepped isothermal tests. Figure 5.13 illustrates the results. Good agreement is obtained, demonstrating that the statistical model can be used to model the kinetics of grain growth under non-isothermal conditions. 5.6 The Influence of Second Phase Particles Most of the grain growth kinetics observed in this investigation were influenced by the presence of second phase particles, since at all temperatures, a tendency toward a limiting grain size was observed. The A36 and DQSK steels contained similar proportions of aluminium and nitrogen. 78 75 980 1000 1020 1040 1060 1080 1100 1120 1140 Temperature, °C Figure 5.13 - Application of the statistical model to describe continuous cooling conditions [61]. 79 The temperature at which both constituents were completely soluble in both steels is approximately 1170°C, according to the solubility product, log 1 0[Al][N] = - - — + 1.48 -» 5.6 reported by Gladman [32], where [Al] is weight percent aluminum, [N] is weight percent nitrogen, and T is temperature in Kelvin. In the as-received condition, the dispersion of A1N in each of the samples, water quenched off the hot strip mill, is not known. Gladman has reported that A1N precipitates slowly during cooling, but more quickly during heating [32]. Even if no precipitation occurred in the steels during the initial quench on the mill, it is certain that a dispersion of particles exists during the isothermal holding period in the furnace heated DQSK and in the slowly heated A36, due to the low heating rates employed. Even specimens of A36 heated at 100°C/second were first taken to 900°C at a rate of 5°C/second and held for 120 seconds. It is likely that these also had a dispersion of A1N particles present during isothermal grain growth. The observed abnormal growth kinetics and the tendency to approach a jj^ting grain size are evidence of the presence of A1N particles. Since the initial particle size distribution is not known, accurate prediction of the kinetics of precipitation is not possible. However Gladman's expression [32] can be used to estimate the size of the particles which are present when abnormal growth is observed. In the A36 steel heated at 5°C/second and held at 1000°C for 60 seconds, if it is assumed that the initial ratio of the largest to the average grain radius is 2.0, and A1N is in equilibrium, then the critical particle radius is approximately 30nm. Transmission electron microscopy would be required to observe a particle of this size; such an investigation is difficult to perform and is outside the scope of this work. 80 It seems likely that both of the as-received steels are in a supersaturated condition with respect to soluble aluminium and nitrogen, consistent with water quenching of the transfer bar on the hot strip mill; but the degree of supersaturation is not known. The rate at which solutes can precipitate and coarsen, and the rate at which precipitates can dissolve, is diffusion limited. In either case, the distribution of particles is changing with time, as is the pinning force. The observed phenomena of abnormal grain growth is due to the dynamic nature of the pinning force. The limiting grain sizes are also likely to change with time, as the distribution of pmning particles tends to dissolve or coarsen. Further, the effect of heating rate on the Ijmiting grain size in the A36 steel can be attributed to a change in the A1N particle distribution during heating. Additional studies are required to disclose detail of the nature of this process. 5.7 The Evolution of the Grain Size Distribution The two dimensional grain size distribution was measured using the image analyser. Changes in the distribution during grain growth will be discussed for the case of normal grain growth, as occurs in the A36 steel heated to 1150°C at 5°C/sec, and for abnormal grain growth as observed in the A36 steel heated to 1000°C at 5°C/second. As a measure of the log normality of each distribution, a correlation coefficient, R2, will be reported [65]. Calculation of R2 is described by R2 = i ( z , - X f c ) ( i - - F j •i=i where X ; is the measured fraction of size class i and 7; is the calculated fraction of size class /, determined from the log normal distribution described by equation 2.11, the peak, dAg, and the standard deviation, sA. Xb is the average of all Xt and Yb is the average of all 81 Yt. The log normal distribution is determined by solving for the best fit of the log normal curve to the measured results. A least squares solution, similar to that used to fit the power law, is applied, where the sum of the difference between calculated and measured fractions is minimised. Values of the standard deviation, sA, can be calculated directly from the measured distribution using the general equation, s = exp 2 / i ( l ^ ) - l n G / ) ) ' i=l 5.8 A l l diameters are mean equivalent area diameters, dAm. A l l standard deviations and diameters in this section are measured using the image analyser. Figure 5.14 shows the evolution of the measured two dimensional grain size distribution during normal grain growth at 1150°C, at times of 1 second, 60 second and 450seconds. The curves shown are the best fit log normal distributions. Table 5.3 summarizes the relevant parameters. Steel Temp,°C time, sec. dAm, (im i?2 A36 1150 1 89.7 1.71 0.96 A36 1150 60 134.6 1.58 0.99 A36 1150 450 187.0 1.62 0.97 A36 1000 10 14.4 1.59 0.97 A36 1000 120 47.8 2.40 0.85 A36 1000 600 81.5 1.63 0.97 Table 5.3 - Evolution of the grain size distribution 82 0.20 o 0.15 H •+—» o CO Lt 0.10 0.05 H 0.00 450 sec. i r - ' i ' ' ' ' i ' ' ' ' i ' ' 1 ' i ' r 20.0 31.3 48.8 76.3 119.2 186.3 291.0 454.8 Grain size, EQAD, |nm Figure 5.14 - Evolution of the A36 2-D grain size distribution at 1150°C. 83 During normal grain growth at 1150°C, the relative width of the distribution does not change greatly with time, but the peak steadily increases. The shape of the distribution is closely approximated by a log normal distribution at all times. The size distribution evolves by the shrinkage and disappearance of the smaller grains and the coarsening of large grains. The distribution observed at 1 second is the widest, possibly due to the after effects of abnormal grain growth during heating. However, the distribution width is observed to be stable and constant at 60 seconds and 450 seconds. Therefore, after a short time at 1150°C, normal grain growth is observed to occur by scaling, as described by Atkinson [16]. Photomicrographs showing the structure before, during and after abnormal grain growth in the A36 steel held at 1000°C, are shown in Figure 5.15. At 10 seconds, the specimen was water quenched so that the fine grains could be revealed. At 120 seconds and 600 seconds, a helium quench was employed. The development of the structure over time is visibly dramatic, being uniform and fine at 10 seconds, bimodal at 120 seconds and finally uniform and coarse at 600 seconds. The evolution of the two dimensional grain size distribution during abnormal growth at 1000°C is shown in Figure 5.16. Initially, at 10 seconds, the structure is uniform and fine. At the onset of abnormal growth, the largest grains in the structure break away from the changing dispersion of pinning particles, since they have the greatest driving force for growth. This is evident in the widening of the distribution and the appearance of the bimodal structure at 120 seconds. The plot at 120 seconds suggests the existence of two peaks, consistent with the observed bimodal structure. (a) Uniform fine structure at 10 seconds (b) Bimodal structure at 120 seconds (c) Uniform coarse structure at 600 seconds 250um Figure 5.15 - Evolution of the grain structure, in the A36 steel held at 1000°C, showing abnormal grain growth. Magnification is approximately x410. 0.25 85 0.20 H 10 sec. c 0.15 o cd Lt 0.10 H 0.05 H 0.00 i 1 1 i i i i i r 4.0 6.3 9.8 15.3 23.8 37.3 58.2 91.0 142.1 222.0 Grain size, EQAD, (xm Figure 5.16 - Evolution of the A36 2-D grain size distribution at 1000°C. 86 The reduced correlation coefficient in Table 5.3 demonstrates that the distribution deviates from log normality. When the pinning particles are sufficiently dissolved or coarsened, all grains are able to grow, and the fine grains quickly disappear. The structure regains its uniform appearance as the distribution narrows and the distribution shape becomes log normal again at 600 seconds. 5.9 Solution Treatment Thus, having found a significant effect of A1N precipitates on the grain growth behaviour, attempts were made to eliminate or minimise the effect by performing a solution treatment. A number of A36 specimens were soaked in a furnace at 1200°C for 3 hours to dissolve the A1N precipitates, then water quenched to prevent their precipitation during cooling. A test series was performed to compare the kinetics of austenite grain growth in the heat treated and the as-received specimens. The results of these tests are shown in Figure 5.17. The specimens were heated at 50°C/second to temperatures of 950°C, J 1050°C and 1150°C, and held at each temperature for 120 seconds. An intermediate heating rate was chosen to reduce the precipitation of A1N during heating and to minimise temperature over shoot. The solution treated specimens displayed a grain size which was larger than that obtained in the as-received specimens for all thermal cycles employed. Since the solution treatment was designed with the intent of minimising the pinning effect of the second phase particles, it is not surprising that the solution treated specimens have a larger grain size; a higher rate of grain growth would be expected with the reduced influence of second phase particles. Differences in the structure of the specimens prior to the thermal cycle may also have some influence on the observed differences in grain size. Prior to thermal cycling, the structure of the as-received A36 steel was mostly acicular ferrite and pearlite, while the 87 solution treated specimens were almost all martensite. The prior austenite grain size of the as-received structure also appeared to be much finer than that of the solution treated material. During the transformation back to austenite on heating, the initial structure of the as-received specimen would be expected to have a refining effect on the nucleation of the new phase, since the as-received structure would be expected to offer more nucleation sites for transforming to austenite. A larger number of nuclei could explain the finer grain sizes. Further tests would be required in order to conclusively state whether nucleation effects, or the effect of second phase particle dominate the grain growth kinetics. It is likely that both have a some influence. The structure of the as-received specimen after heating to 950°C was bimodal, showing signs of abnormal grain growth, whereas, the structure in the solution treated specimen heated to 950°C was uniform. This result can only be explained by an initial difference in the dispersion of second phase particles. Therefore, from these results, it can be concluded that the solution treatment reduced the effect of second phase particles. Both sets of results show an inflection in the plot at 1050°C. The inflection indicates that, despite the anticipated difference in the initial particle distribution and structure, a similar trend with respect to the kinetics of grain growth is occurring in each type of specimen. Few specific statements can be made regarding the kinetics of isothermal growth at each temperature, since only one holding time has been used. 88 Figure 5.17 - Comparison of austenite grain size in as received and solution treated A36 steel, heated to 950°C, 1050°C and 1150°C and held for 120 seconds. 89 At 1150°C, the grain size is much larger than in the specimens held at lower temperatures. This result can be explained by a the combined effect of increased A1N solubility and increased atomic mobility. The solution temperature of A1N in the A36 steel is approximately 1170°C, according to the solubility product reported by Gladman [32]. Therefore at 1150°C, much of the A1N will be already in solution, or in the process of dissolving. With fewer second phase particles present, the rate of grain growth is higher since the pinning force is reduced. Also with increasing temperature, grain boundary mobility is higher, thereby increasing the rate of grain growth. 5.10 G r a i n Growth During Hot Strip Rolling In conventional process models the power law is used to describe grain growth kinetics. It is usually fit first to results similar to those reported here, and then extrapolated to describe grain growth occurring under the conditions of hot strip rolling. In this work however, it has been observed that the kinetics of grain growth differ when the heating cycle is changed or when the as-received specimens are solution treated prior to being used for grain growth tests. The power law has been fit with different parameters, to the results obtained even for the same material, heated to temperature by different thermal cycles. This scatter in the fitting parameters cannot be described from any fundamental perspective. During conventional grain growth tests, rapid growth occurs during heating and the initial grain size is large. A smaller initial grain size can be achieved by deformation and recrystalhzation of the specimen. However, quenching the structure after the test, to reveal the prior austenite grain size, then becomes difficult. Therefore, the early stages of grain growth, which are relevant to the industrial process of hot strip rolling, are rarely measured. In laboratory tests, the specimen must be held at temperature for extended periods so that significant growth is observed and a reasonable time dependence for the change in grain size can be formulated. 90 Under hot strip rolling conditions, deformation occurs in each of the rolling passes, and grain growth occurs in the time between stages of deformation, when recrystallization is complete. In a typical rolling mill, interpass times range from 1 second to 20 seconds and the delay between the roughing mill and finishing mill is typically 60 seconds. Hold times for laboratory studies of grain growth are usually in the range of 500 seconds to 1000 seconds [3,4,18], or longer [5]. Thus an extrapolation of the empirical power law to mill conditions appears to have little validity. Differences in the grain growth kinetics have been described with respect to the influence of second phase particles. Abnormal grain growth and the attainment of a limiting grain size are rarely observed during hot rolling of plain carbon steels, providing evidence that second phase particles have little influence. Before rolling, the steel is soaked at between 1200°C and 1300°C in a reheat furnace. At this temperature, ALN is usually completely soluble. During rolling, the temperature of the material, and the solubility of A1N drop, but Gladman states that precipitation of A1N during cooling is slow [31]. Therefore it would be expected that A1N particles have limited effect until the later stages of rolling, at lower temperatures where austenite grain growth is not significant. Militzer et al. have described the application of a statistical model to the prediction of austenite grain growth kinetics in A36 and DQSK steels during hot strip rolling [61,62]. It is reported that a pjjming parameter of P=0 is required to model grain growth occurring between the roughing and the finishing mills, which corresponds to unpinned grain growth. Such a situation is realistic, since in the hot strip rnill, it is likely that little or no ALN particles will precipitate between the roughing and fMshing mills, in the corresponding temperature range from 1120°C to 10Q0°C. The kinetics of austenite grain growth can then be described by Burke and Turnbull's parabolic growth kinetics [6], as discussed by Militzer et al. [61]. CHAPTER 6 RESULTS AND DISCUSSION II. ESTIMATION OF THE THREE DIMENSIONAL GRAIN SIZE DISTRIBUTION 6.0 Introduction Methods for the estimation of the three dimensional grain size distribution were applied to two dimensional results obtained from the 1080, the A36 and the DQSK steel. The 1080 steel was used for much of the work since the microstructure was easily interpreted and a large number of grains could be measured. The three dimensional grain size distribution was described using the mean equivalent volume diameter, dVm and the log normal standard deviation, sv. 6.1 Methods Based on the Assumption of Spherical Grain Shape The methods of Saltikov [36] and Huang and Form [37,46] were applied to the two dimensional grain size distributions measured on the image analyzer for the 1080 steel, heated to 1050°C and held for 0 seconds, 120 seconds and 420 seconds. Figure 6.1 shows the measured distribution and the distribution as predicted by each method, for the specimen held for a time of 0 seconds. The correlation coefficient, R2, for the fit of the measured distribution to a log normal distribution is 0.99. The results from each method are quite different, with the mean equivalent volume diameter, dym, from Saltikov's method [36] being equal to 70.2|im and that obtained from Huang and Form's method [37,46], equal to 59.0|im. 91 92 0.24 0.22 -0.20 -0.18 -0.16 0.14 0.12 0.10 0.08 0.06 -0.04 -0.02 -0.00 -0.02 -0.04 -0.06 H Measured 2-D data Saltikov's method Huang and Form's method ~ l 1 1 1 1 1 1 1 1 6.3 9.8 15.3 23.8 37.3 58.2 91.0 142.1 222.0 Grain size, EQVD, Lim Figure 6.1 - Saltikov's [36] and Huang and Form's [37] 3-D grain size distribution compared to the measured 2-D distribution for 1080 steel heated to 1050°C and quenched. 93 In Saltikov's original publication [36], his method required that the grain size distribution be represented by size classes geometrically spaced with a multiplying factor, a, of 1 0 0 1 (1.2589) since application of the method required the use of reported tables of coefficients. In this case, a equal to 1.25 has been used. The mathematics of the procedure were set up using a Microsoft Excel® spreadsheet so that any multiplying factor could be chosen, since the principles of the calculation are independent of a. The major difference between the two methods is that Saltikov [36] corrects for sampling only, while Huang and Form [37,46] correct for both sampling and truncation. Up to the point where the truncation correction is applied, the two methods are identical. The truncation correction increases the number of smaller grains and decreases the number of large grains, since it accounts for the probability that a grain is sectioned, based on the grain diameter. It is therefore expected that the mean predicted by Saltikov is greater than that predicted by Huang and Form; this is seen to be the case. The width of the estimated distribution cannot be assessed by calculation of the log normal standard deviation because the negative number of grains in the smallest size classes makes such a calculation impossible. The prediction of a negative number of grains, as shown in Figure 6.1, illustrates one of the major problems with both of these methods. Even when an artificially generated, perfect log normal distribution was used as input for the methods, a negative number of grains was predicted in the smaller size classes. This is obviously an unrealistic result. Huang and Form [46] and Aaron et al. [49] have reported similar negative counts when using these techniques. Both methods operate by correcting the number of grains in each size class, by application of an equation of summation, as described by equation 2.17. The number of terms in the equation is equivalent to the number of corrections applied, and is a function of the relative magnitude of the size class, with the largest class having a single correction and the nth largest class having n corrections. Therefore, any errors which result from 94 assumptions inherent in each method are compounded for the smallest size classes. The assumption that grains are spherical in shape is incorrect, as stated by Aaron et al. [49], because an array of spherical grains cannot fill space. The methods are based on the probability that an object is sectioned to give a plane of lesser diameter. It has been shown that such probabilities are a function of object shape [36,39]. It is the probability calculation resulting from the assumption of spherical grain shape that directly introduces the errors observed. 6.2 The Method of Takayama et al. The three dimensional grain size distribution was estimated from the measured results of mean linear intercept length, lm, and mean area, Am, using the method of Takayama et al. [38]. This method was applied to all of the A36 specimens heated at 5°C/second and 100°C/second, and to the 1080 steel specimens held at 1050°C. Some of the results for the A36 steel heated at 5°C/second are shown in Table 6.1. Mean linear intercept, lm, was determined by manual measurement using Heyn's linear intercept method [41]. The measurement involved the placement of a grid over a photomicrograph of the structure so that number of intercepts per unit length could be determined. Mean area was measured by a manual procedure using Jeffries' method [41]. Photomicrographs of the structure were used to determine the number of grains per unit area. The correlation coefficient, R2, for the fit of the measured distribution to a log normal one is reported for specimens where the two dimensional distribution was also determined. Values of the standard deviation of the log normal distribution, sv, were calculated directly from the measured results by using equations 2.21 and 2.22. Calculated values of sv are reported in Table 6.1, as are overall mean values. The calculated and mean values of Sy were used to determine dym. 95 Steel 7/,°C t R2 From c ale. sv From i lean sv sec. / | im (im 2 dVm |_im sv dvm (im sv A36 1000 10 0.97 11.6 165 17.7 1.31 17.7 1.33 A36 1000 600 0.97 78.8 6687 105.6 1.57 112.7 1.33 A36 1100 1 73.4 6036 102.6 1.50 107.0 1.33 A36 1100 10 67.2 5836 108.5 1.15 105.3 1.33 A36 1100 30 88.7 10387 146.2 1.02 140.4 1.33 A36 1100 60 103.6 13108 157.9 1.32 157.7 1.33 A36 1100 120 106.4 13593 159.4 1.36 160.6 1.33 A36 1100 300 112.6 14363 159.1 1.48 165.1 1.33 A36 1100 600 113.5 15315 168.4 1.38 170.5 1.33 A36 1150 1 0.96 77.7 7819 125.6 1.15 121.8 1.33 A36 1150 60 0.99 116 16287 175.2 1.34 175.8 1.33 A36 1150 450 0.97 169 33506 247.1 1.42 252.2 1.33 Table 6.1 - The three dimensional grain size distribution estimated using the method of Takayama et al. [38]. There is significant scatter in the sv results, as reported in Table 6.1. The scatter predicted by the method is not consistent with experimental observations of the two dimensional grain size distribution. It is reasonable to assume that the width of the two dimensional distribution is a strong function of the width of the true three dimensional distribution. During normal grain growth in the A36 at 1150°C, the two dimensional distribution width was seen to be fairly constant, sA being 1.71 at 1 second, 1.58 at 60 seconds and 1.62 at 450 seconds (refer to the discussion in Section 5.7). From Table 6.1, the variation in sy ranges from 1.15 to 1.42. 96 The log normal standard deviation, sv, is dependent on the ratio of lm to the square root of Am, as shown in Figure 6.2. Scatter in the measured results of lm and Am introduced unacceptable variability in the estimated value of sv. The 95% confidence intervals for lm were calculated to be approximately 10%. From Figure 6.2, it is apparent that a 10% variation in the measurements would introduce a 30% difference in sv. Accuracy in the determination of lm and Am would improve with an increase in the number of grains measured. The mean equivalent volume diameter, dVm, is a function of sv. In order to remove the unrealistic variability of sv and its subsequent influence on dVm, an average value of the standard deviation was used. Table 6.1 reports dVm results obtained for the A36 specimens, determined by using an average standard deviation. The average standard deviation of the log normal distribution is estimated to be equal to 1.33. Equation 2.22 leads to ^Vm = 1-22^Am 6.1 The complete list of results from the A36 steel heated at 5°C/second shows that the average of the standard deviation is approximately 1.33 for each of the five different temperatures: At 950°C the mean is 1.31; at 1000°C it is 1.38; at 1050°C it is 1.34; at 1100°C it is 1.32 and at 1150°C it is 1.26. This indicates that the scatter in sv can be explained by the statistical variability of the measurement of lm and Am. When enough measurements are used together, a statistically stable result for sv can be determined; this value approaches 1.33. It is unfortunate that the required number of grains seems to be of the order of 1000, since it is not practical to propose that every measurement of grain size be made on such a large number of grains. 97 2.4 I 2.2 2.0 1.8 H 1.6 H 1.4 H 1.2 1.0 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 / divided by the square root of A 1.25 m 1/2 Figure 6.2 - The relationship between sy and the ratio, / m / (A m ) , using the method of Takayama et al. [38]. 98 The variability of the lm and Am measurement was smaller for the 1080 steel specimens, since the structure was more clearly revealed, and a much larger number of grains were measured for each of these specimens. Linear intercept length and mean area were determined manually [41] and by using image analysis equipment [40]. The method of Takayama et al. [38] was applied to both sets of results. The systematic error in the measurement of mean linear intercept, lm, and mean area, Am, by the image analyzer was discussed in Section 5.2. The linear intercept length measured on the image analyzer is much lower than that measured manually, leading to smaller values of sv, as explained by Figure 6.2. Mean area is also lower but the difference is not as great. If the image analyzer results are used then a value of sv equal to 1.15 is estimated. Using manual methods, sv is estimated to be equal to 1.43. Comparison of the calculated values of dVm and reveals that there is only a minor difference in the mean volume diameters. Since the estimated three dimensional distribution is very sensitive to the ratio of lm to the square root of Am, the systematic error in lm manifests itself in an under estimation of sv. Therefore, measured values of lm and Am from the image analyzer cannot be used with the method of Takayama et al. [38] to estimate the three dimensional grain size distribution. The method of Takayama et al. [38] applied to the 1080 steel using the manual measurements produced reasonable results: In a specimen heated to 1050°C and immediately quenched, the equivalent volume diameter, dVm, was estimated to be 64.3|im with a standard deviation of 1.41; in a specimen held at 1050°C for 120 seconds, dVm was 71.2(j,m and sv was 1.44 and in a specimen held at 1050°C for 420 seconds, dVm was 83.6|im and sv was 1.46. As would be expected during normal grain growth, the distribution width is estimated to change little, with the mean grain size steadily increasing. 99 If the mean value of sv, equal to 1.43, is used with the 1080 steel manually measured results, then equation 2.22 becomes dVm = U9dAm 6 2 From equation 6.1 and 6.2, it can be said that the mean equivalent volume diameter, dVm, predicted by the method of Takayama et al. [38] is approximately 20% greater than the mean equivalent area diameter, dAm, determined by Jeffries' method [41], for the cases examined. In addition, comparison of dVm to lm reveals that dVm is consistently 50% greater than lm. The approximate general equation, dVm - l.2dAm = \.5lm 6 3 can be written. In summary, there are two main problems with the method of Takayama et al. [38]. Firstly, the method requires that the grain size distribution is log normal. Examples discussed in this section were observed to have log normal grain size distributions in two dimensions; for these cases, the method is therefore apphcable. However, several specimens were observed to have grain size distributions which were not log normal (abnormal grain growth was observed) and in these cases, use of the method of Takayama et al. [38] is not valid. Secondly, the method is very sensitive to the ratio of lm to the square root of Am. During measurement of grain size in the A36 and DQSK steels, errors in the order of 10% were encountered because only a hmited number of grains could be measured. Errors of this size introduce unreasonable scatter in the calculated values of sv and dVm. The method works reasonably well with microstructures which are easily measured, and with 100 other structures when a large number of grains are analyzed. Thus, successful application of the method is dependent on good measurement statistics. 6.3 The Method of Matsuura and Itoh 6.3.1 Application of the Method The three dimensional grain size distribution was also estimated using the method of Matsuura and Itoh [39]. The method was applied to results obtained from the 1080 steel, the A36 steel heated at 5°C/second, and the DQSK steel heated in the furnace, as summarized in Table 6.2. The mean equivalent volume diameter, dVm, and the standard deviation of the log normal distribution, sv, were calculated from the estimated distribution. A flow chart shown in Figure 6.3 demonstrates how the method is applied, using a programming routine written in FORTRAN. As a first assessment of the method of Matsuura and Itoh [39], the two dimensional grain size distribution measured directly by the image analyzer was used as input. The method consistently predicted that the three dimensional grain size distribution has a higher mean than the two dimensional grain size distribution. The width is predicted to narrow slightly. An average conversion factor of 1.23 can be calculated by assessment of the ratio of the mean equivalent area diameter (obtained from the image analyzer), dAm, to the mean equivalent volume diameter, dVm. From Table 6.2, it is obvious that there is some variability in the results, and for the wider distributions, a higher ratio has been calculated. A simple conversion factor can be used for cases where the width of the grain size distribution does not vary substantially; this is the case, where grain growth is normal and the grain size distribution is increasing by scaling. (^""start Determine 2-D distribution and represent in k discrete size classes as single columned matrix, A . Calculate dAb, and assume equal to dvp Calculate the number of grain faces for each size class, Ji, using equation 2.19 and dvb. t Determine the relative diameter distribution curve for each grain shape from values of Ji. Integrate over each distribution curve for each grain size class and place results in a /cx/c matrix, B. Solve matrix problem C . B = A by determining least squares solution for matrix C. i Use single columned solution matrix, C, to calculate new result for dVb. Compare new dvb to previous dvb. <^STOP^) Figure 6.3 - Flow chart of the procedure for applying the method of Matsuura and Itoh [39]. 102 Measured 2-D Estimated 3-D Steel T aim t No. sv °C sec. grains Lim | im 1080 1050 0 1142 50.33 1.73 62.85 1.59 1.25 1080 1050 120 885 55.26 1.70 68.37 1.57 1.24 1080 1050 420 629 64.69 1.63 78.21 1.49 1.21 A36 1000 10 250 14.39 1.59 17.74 1.51 1.23 A36 1000 120 300 47.79 2.40 71.06 2.25 1.49 A36 1000 600 185 81.50 1.63 104.0 1.54 1.28 A36 1150 1 193 89.65 1.71 109.8 1.61 1.22 A36 1150 60 206 134.6 1.58 166.4 1.48 1.24 A36 1150 450 115 187.0 1.62 227.0 1.52 1.21 DQSK 1000 0 299 27.62 1.36 31.84 1.28 1.17 DQSK 1000 120 271 31.33 1.38 36.39 1.29 1.16 DQSK 1000 420 199 33.51 1.35 38.62 1.27 1.15 DQSK 1150 0 184 113.8 1.51 136.9 1.43 1.20 DQSK 1150 120 161 145.7 1.50 177.3 1.42 1.22 DQSK 1150 420 231 162.1 1.58 197.9 1.49 1.22 Table 6.2 - 3-D distribution from the image analyzer results, for 1080 steel, A36 steel and DQSK steel, after Matsuura and Itoh [39]. Figure 6.4 shows the relationship between the distribution width, sA, and the ratio of dAm to dVm. From the plot, it appears that a linear relationship exits between sA and dVm/dAm. In order to confirm this relationship, it would be desirable to estimate the three dimensional grain size distribution for specimens having a two dimensional log normal 103 standard deviation, sA, in the range from 1.8 to 2.4. However, none of the distributions assessed in this work had those values of sA. An alternative route to assessment of the conversion ratio in this range would be the artificial generation of distributions by computer simulation. The method of Matsuura and Itoh [39] should actually be applied to the true two dimensional grain size distribution, rather than to the distribution measured on the image analyzer. As discussed in Section 5.2, a systematic error is made when the two dimensional grain size distribution is measured. A correction can be applied to account for the error, thus obtaining the true two dimensional grain size distribution. The method of Matsuura and Itoh [39] was applied to a number of cases where this correction was applied, as reported in Table 6.3. Corrected 2-D Estimated 3-D Steel T . aim t No. SA sv °C sec. grains |im fim 1080 1050 0 1142 52.83 1.14 66.14 1.60 1.25 1080 1050 120 885 58.84 1.71 74.25 1.58 1.26 1080 1050 420 629 67.67 1.63 81.85 1.50 1.21 A36 1000 600 185 87.37 1.65 111.8 1.57 1.28 A36 1150 1 193 103.4 1.75 128.1 1.63 1.24 A36 1150 60 206 149.8 1.62 189.7 1.51 1.27 Table 6.3 - 3-D distribution from corrected 2-D distribution, after Matsuura and Itoh. 104 1.55 Standard deviation of the 2-D log normal distribution, s. Figure 6.4 - The relationship between sA and dVm/dAm obtained from the application of the method of Matsuura and Itoh [39] to measurements made on the 1080 steel, the A36 steel and the DQSK steel. 105 Comparison of Table 6.2 with Table 6.3 reveals that the ratio of dVm to dAm for corresponding specimens is almost the same, with the average ratio from Table 6.3 being slightly higher at 1.25. For the same specimens from Table 6.3, the ratio is 1.24. This difference can be explained by the relationship between the distribution width and the ratio of dVm to dAm, as described by Figure 6.4. The corrected two dimensional distributions are slightly wider than the measured distributions; this leads to the wider three dimensional distribution estimated by the method of Matsuura and Itoh [39]. The difference between the dVm/dAm ratios from each Table is not substantial and the conversion factor of 1.23 from all of the results in Table 6.2 is still approximately true. This result compares well with the result of approximately 1.2, found using the method of Takayama et al. [38]. A more detailed comparison of the method of Takayama et al. [38] and the method of Matsuura and Itoh [39] will be made in Section 6.4. 6.3.2 The Shape of the Estimated 3-D Distribution The distribution estimated by the method of Matsuura and Itoh [39] was generally similar in shape to the measured distribution. Distributions which were measured as log normal in two dimensions, remained so after the method was applied. Figure 6.5 shows the three dimensional grain size distribution in the A36 steel held at 1000°C for 10 seconds, 120 seconds and 600 seconds, as representative examples. The log normal distribution (shown) was fit by the method of least squares, by rninimizing the value of S2 calculated using equation 2.21. The measured two dimensional grain size distribution and its corresponding log normal curve are also shown; the measured results are plotted as symbols and the log normal curve plotted as dotted lines. The two dimensional and three dimensional distributions at 10 seconds and 600 seconds are reasonably well approximated by the log normal curve. At 10 seconds, R 2 for 106 the two dimensional distribution is 0.97 and 0.94 for the three dimensional distribution. At 600 seconds, R 2 for the two dimensional distribution is 0.97 and 0.88 for the three dimensional distribution. Both examples show that the estimated distribution tends to be more scattered than the measured one. The method seems to exaggerate deviations from log normality. For the case of abnormal grain growth at 120 seconds, the initial distribution is not well approximated by a log normal distribution, as R 2 is 0.85. The estimated three dimensional distribution deviates even further from log normality, with R 2 equal to 0.62. This result is not unreasonable since the grain size distribution during abnormal grain growth is not expected to be log normal; a bimodal grain structure was actually observed. The scatter in the estimated distribution is likely to be due to approximations inherent in the calculation method, and inaccuracies due to the measurement statistics. One error in the calculation method is the determination of the number of faces of a grain. In reality, the number of faces of a grain, / , must be an integer whereas the calculation permits / to be equal to any decimal value greater than 4. A lower limit of 4 is chosen since it is impossible for a polyhedron to have less than 4 sides. Furthermore, in order to smplify the calculation of the probability distribution function for each grain shape, Matsuura and Itoh assumed that the grain shape could be approximated by planar faced, regular polyhedra. In reality, grains have curved surfaces and are unlikely to be entirely regular. The assumption that each three dimensional size class can be represented by a single grain shape is also an over simplification, which could be the source of some error. The scatter in the estimated three dimensional distribution is a magnification of the initial measured scatter. The two dimensional grain size distributions were determined from a limited number of grains and the initial scatter in the measured distributions is a result of this. It follows that if the two dimensional distribution could be more accurately 107 determined, the scatter in the three dimensional distribution could be minimized. In order to obtain a good approximation of the two dimensional distribution, it is likely that 30 or more size classes should be measured, with, on average, 50 to 100 grains for each size class. Therefore, an impractical number of grains, between 1500 and 3000, would need to be measured. Other investigators, in studies directed specifically towards investigation of the grain size distribution, have performed measurements of the distribution with the number of grains in this range [25,43,46,66]. In this study, approximations of the two dimensional grain size distribution were usually obtained from 200 to 300 grains. 6.3.3 Convergence of the Distribution With Number of Size Classes The stability of the method of Matsuura and Itoh [39] was assessed by varying the width and number of size classes used to predict the mean equivalent volume diameter in the 1080 steel specimen, held at 1050°C for 0 seconds. Both arithmetically scaled and geometrically scaled size classes were assessed, with a total of 1142 grains being measured in the two dimensional distribution. Figure 6.6 shows the convergence of the mean diameter. The arithmetically scaled and geometrically scaled size classes produced similar results. For the number of size classes greater than 20, the method predicted a mean equivalent volume diameter of approximately 62|im. The results predicted by the geometrically scaled distribution for less than 20 size classes were less stable than those obtained from the arithmetically scaled distribution; but the differences were not significant. Both types of distributions provided mean values which were within 3% of each other, when the number of size classes were greater than 10. The log normal standard deviation, sv, of the predicted distribution showed similar trends. 0.25 4.0 6.3 9.8 15.3 23.8 37.3 58.2 91.0 142.1 222.0 Grain size, EQAD, Lim Figure 6.5 - Evolution of the 3-D grain size distribution for A36 steel held at 1000°C from the method of Matsuura and Itoh [39]. 109 Figure 6.6 - Effect of the number of size classes on the method of Matsuura and Itoh [39]. 110 6.4 Comparison of the Methods of Takayama et al. and Matsuura and Itoh The mean equivalent volume diameter, dVm, and the standard deviation, sv, predicted by each of the methods for the estimation of the three dimensional grain size distribution are compared in Table 6.4 using the results from the 1080 steel held at 1050°C. The results measured manually by Heyn's method and by Jeffries' method [41] were used with the method of Takayama et al. [38]. The true, corrected two dimensional grain size distribution was used with the method of Matsuura and Itoh [39]. The mean values estimated by the method of Takayama et al. [38] and the method of Matsuura and Itoh [39] are in very close agreement, with the difference being on average, 3%. The estimation of the log normal standard deviation is a different matter. The standard deviations predicted by each method differ by approximately 15%. By considering the sensitivity of the method of Takayama et al. [38], and by comparison of the estimated value of sv to the measured value of sA for each method, the standard deviation estimated using the method of Matsuura and Itoh [39] seems to be the most reasonable. Hold No. True 2-D Takayama Matsuura and time grains et al. [38] Itoh 39] sec. |im dvm (im sv dvm | im sv 0 1142 52.8 1.74 64.3 1.41 66.1 1.60 120 885 58.8 1.71 71.2 1.44 74.3 1.58 420 629 67.6 1.63 83.6 1.46 81.9 1.50 Table 6.4 - Three dimensional grain size distribution predicted using the method of Takayama et al. [38] and Matsuura and Itoh [39] for 1080 steel held at 1050°C. I l l 6.5 Validation of the Method of Matsuura and Itoh With Measured 3-D Results As seen from the above discussion, the method of Matsuura and Itoh [39] is the most reasonable so far reported in the literature. Matsuura et al. [25] confirmed this by validation of the method with three dimensional measurements. The three dimensional grain size distribution was measured in a specimen of stainless steel by complete disintegration, and individual measurement of each grain volume. The two dimensional grain size distribution was also measured on a section through the specimen, by the use of image analysis equipment. Approximately 1600 grains were measured. The results were reported in the form of eleven arithmetically scaled size classes. The measured distribution compares well to that predicted by the method of Matsuura et al. [39], as shown in Figure 6.7. Very little literature is available which reports the measurement of the three dimensional grain size distribution, probably because such a task is difficult to perform. However, three dimensional measurements have also been made by Williams and Smith [26] using stereo radiography on an aluminum-tin alloy. Two dimensional data was obtained on the same specimen by Aaron et al. [49], who compared a number of estimation methods based on the assumption of spherical grain shape. The actual measured two dimensional data is not reported, but it can be estimated by back calculation using the results from Saltikov's method [36]. Therefore, it is possible to obtain two dimensional and three dimensional measurements for the alummum-tin specimen; that data can be used to further validate the method of Matsuura and Itoh [39]. The data from the aluminum tin alloy is far from ideal, since the two dimensional results are in the form of back calculated values, and only nine size classes are given. With only nine size classes, it is likely from Figure 6.6 that the method of Matsuura and Itoh [39] has not converged to a stable solution. The three dimensional results are also somewhat dubious, since the distribution was measured in three dimensions by x-ray 112 stereo radiography, with the size of each grain being determined by comparison with a sphere of known diameter. The accuracy of such a method is not likely to be high and only seven measured size classes were reported. Even so, an estimation of the three dimensional grain size distribution obtained using the method of Matsuura and Itoh [39] compares favorably to the measured results, as shown in Table 6.5. Measured 3-D Matsuura and [25,49] Itoh [39] Material dvh,\im *v dvh,\im sv Al-Sn 414 1.43 424 1.34 St. Steel 460 1.53 464 1.50 Table 6.5 - Comparison of methods for Al-Sn alloy and stainless steel. 113 0.30 0.25 H 0 200 400 600 800 1000 1200 1400 Grain size, EQVD, urn Figure 6.7 - Comparison of the measured 3-D grain size distribution with the estimated distribution, for stainless steel, using the method of Matsuura and Itoh [39], from Matsuura et al. [25]. CHAPTER 7 SUMMARY AND CONCLUSIONS The objectives of this work were firstly, to quantify the kinetics of austenite grain growth in two plain carbon steels, and secondly, to determine an appropriate method for the estimation of the three dimensional grain size distribution. For the purpose of investigating the kinetics of grain growth, the grain size was measured in specimens of A36 and DQSK steel, heated to temperatures between 950°C and 1150°C, and held for periods up to 900 seconds. Linear intercept, lm, and mean area, Am, were measured using the Heyn and Jeffries manual methods and by the use of image analysis equipment, as describe by A S T M standards [40-42]. Mean equivalent area diameter, dAm, was calculated from mean area, Am. Standard statistical methods were used to determine 95% confidence intervals of approximately ±10% for the mean linear intercept, lm, and the mean equivalent area diameter, dAm, determined using both types of measurement methods. Systematic errors in the determination of the linear intercept length and the mean equivalent area diameter were identified and discussed. The true linear intercept is closest to that determined manually by Heyn's method [41]. The true mean area can be determined by the application of an iterative correction to measurements determined by image analysis [40]. The number of grains which touch the edge of the image being measured must be known. Mean area determined by Jeffries' method [41] is a good first approximation of the true mean area. The best type of measurement to perform is the measurement of the two dimensional grain size distribution by the use of the image analyzer, since the true two dimensional grain size distribution can then be determined from this result. Such a 114 115 measurement is not difficult to perform and the necessary image analysis equipment is becoming commonly available, as a standard metallographic laboratory tool. The kinetics of austenite grain growth were observed in the A36 steel and in the DQSK steel. In both materials abnormal growth and a limiting grain size were observed. Growth kinetics in the A36 steel were influenced by heating rate and by solution treatment of the specimens. The kinetics of grain growth were adequately interpreted with respect to the influence of second phase particles, which were concluded to be aluminum nitride, from the work of Gladman [28]. The phenomenon of abnormal grain growth and a hndting grain size were also explained by the presence of aluminum nitride particles. The kinetics of grain growth observed under laboratory conditions are expected to be quite different from those observed in a hot strip mill, because the thermal and mechanical history obtained during hot rolling is difficult to reproduce. Both the dispersion of second phase particles and the width of the grain size distribution influence the kinetics of grain growth and both are influenced by the thermal and mechanical processing history. The power law was fit to results of mean equivalent area diameter obtained for the A36 and the DQSK steels. The fitting constants used were found to be dependent on the thermal history. Because the exponent, m was greater than two, no fundamental basis for the power law equation was found. It must be concluded that the power law is an empirical relationship, which should not be extrapolated to model grain growth for conditions outside of those for which the results have been obtained. The statistical model described by Abbruzzese and Lticke [20,21] was applied to the results from the A36 and the DQSK steels by Militzer et al. [61,62]. The pmiing parameter, P, was used to fit the model and the parameter could be interpreted from a theoretical perspective [61,62]. The pirating parameter, P, was also estimated for hot strip 116 rolling conditions [61,62]. The results obtained in this work were therefore used to successfully validate the statistical model. Thus, as anticipated in the first objective, the kinetics of austenite grain growth have been quantified. The results have been used to validate a statistical model, which can be used to model the kinetics of austenite grain growth during hot strip rolling. The evolution of the measured two dimensional grain size distribution was described for a case of normal growth and abnormal growth in the A36 steel specimens heated at 5°C/second. Under conditions of normal grain growth at 1150°C, the distribution width and shape were seen to change little over time, with the distribution being closely approximated by a log normal curve. The mean of the distribution was observed to increase by scaling. At 1000°C and after 10 seconds, the distribution shape was also apparently log normal. Under conditions of abnormal growth at 120 seconds, the distribution was seen to widen dramatically, and then at 600 seconds, to return to a width and shape similar to that initially observed. Methods for estimation of the three dimensional grain size distribution were compared by applying each to results obtained from the 1080, the A36 and the DQSK steels. It was determined that the methods of Saltikov [36] and Huang and Form [37], based on the assumption that the grain shape is spherical, were unacceptable; both methods predicted a negative number of grains for the smallest size classes observed. The method of Takayama et al. [38] was found to be sensitive to the ratio of linear intercept to the square root of the mean grain area. When the method's sensitivity was dampened by the use of a mean value for the log normal standard deviation, sv, reasonable estimations were obtained for the equivalent volume diameter, dVm. The method was found to be unsuitable for determining the width of the distribution and could not be applied to distributions which were not log normal. 117 The method of Matsuura and Itoh [39] was found to be appropriate for calculating the mean equivalent volume diameter and the width of the distribution. The true, corrected two dimensional grain size distribution must be used as input. Some scatter was observed in the histograms predicted by the method, but the distribution shape generally appeared to be reasonable. The scatter was a results of the statistics of the measurements, which were performed using a limited number of grains, due practical considerations. Some scatter could also attributed to inaccuracies in the simplifying assumptions, inherent in the procedure. The method was applied to measured distributions where the shape of the distribution was log normal, and also to distributions where the shape was not log normal. It was found that a stable solution was predicted when greater than ten size classes were employed. The method was originally validated using measured three dimensional results by Matsuura et al. [25]. A second example of a measured three dimensional distribution was reported in the literature [49] and this was also used to validate the method. The methods of Takayama et al. 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Miller, Influence of Austenitizing Time and Temperature on Austenite Grain Size of Steel, Trans.ASM, Vol.43, 1951, pp260-289. [61] M . Militzer, A . Giumelh, E.B. Hawbolt and T.R. Meadowcroft, University of British Columbia, Internal Report, Unpublished, 1994. [62] M . Militzer, A . Giumelli, E.B. Hawbolt and T.R. Meadowcroft, Austenite and Ferrite Grain Size Evolution in Plain Carbon Steel, 36th Metal Working and Steel Processing Conference, Baltimore, 1994. [63] N .A. Gjostein, H.A. Domain, H.I. Aaronson and E. Eichen, Acta metall, Vol.14, 1966, ppl673. [64] H.J. Frost and M.F. Ashby, Deformation-Mechanism Maps, Permagon Press, Oxford, U K , 1982. [65] R .V . Hogg and J. Ledolter, Engineering Statistics, Macmillan, 1987. [66] D.A. Aboav, The Arrangement of Grains in a Polycrystal, Metallography, Vol.3, 1970, pp383-390. APPENDIX Temp Spec. time HR. No. 95% d-Am A nm 95% cond. °a grns conf conf °c sec sec |im (±%) |im (im2 (±%) 900 As rec. 120 5 506 9.17 17.5 11.87 110.59 17.5 950 As rec. 10 5 733 9.06 4.7 11.38 101.80 4.1 950 As rec. 120 5 492 11.05 8.2 13.90 151.66 9.4 950 As rec. 600 5 491 11.27 3.3 13.91 151.97 4.5 1000 As rec. 10 5 324 11.58 7.6 14.48 164.65 12.3 1000 As rec. 60 5 309 12.81 5.4 14.83 172.65 12.5 1000 As rec. 60 5 61 63.74 87.95 6075.9 1000 As rec. 120 5 286 14.06 12.6 17.23 233.16 23.8 1000 As rec. 120 5 281 71.99 8.88 93.13 6812.4 10.2 1000 As rec. 600 5 229 78.84 13.9 92.28 6687.4 16.3 1000 As rec. 900 5 264 81.93 8.5 96.09 7251.1 13.7 Table A.1 - A36 austenite grain size results obtained by standard Gleeble thermal treatment and manual measurements, using Heyn's and Jeffries' procedures [41]. 124 125 Temp Spec. time H.R. No. 95% d-Am A "-m 95% cond. °C/ gms conf conf °c sec sec |lm (±%) |im (im 2 (±%) 1050 As rec. 1 5 143 15.08 4.6 18.87 279.79 1.65 1050 As rec. 10 5 48 21.14 27.96 614.15 1050 As rec. 10 5 350 64.79 7.2 83.45 5469.4 9.76 1050 As rec. 60 5 14 28.70 44.99 1590 1050 As rec. 60 5 284 76.55 9.3 92.64 6740.4 9.93 1050 As rec. 120 5 265 84.98 9.3 105.1 8668.4 8.30 1050 As rec. 300 5 316 86.58 12.1 110.0 9499.3 11.6 1050 As rec. 750 5 291 97.79 7.9 114.6 10315 13.1 1100 As rec. 1 5 307 73.39 12.7 87.67 6036.3 11.0 1100 As rec. 10 5 328 67.15 8.7 86.20 5836.2 22.3 1100 As rec. 30 5 289 88.68 3.0 115.0 10387 4.0 1100 As rec. 60 5 229 103.6 9.4 129.2 13108 12.9 1100 As rec. 120 5 265 106.4 9.3 131.6 13593 8.3 1100 As rec. 300 5 209 112.6 3.0 135.2 14363 6.7 1100 As rec. 600 5 196 113.5 2.3 139.6 15315 10.1 Table A.l(cont) - A36 austenite grain size results obtained by standard Gleeble thermal treatment and manual measurements, using Heyn's and Jeffries' procedures [41]. 126 Temp Spec. time HR. No. 95% dAm A nm 95% cond. °c/ grns conf conf °c sec sec |im (±%) |im [im2 (±%) 1150 As rec. 1 5 237 77.7 11.3 99.8 7819.1 18.5 1150 As rec. 10 5 318 105.2 7.2 135.3 14366 15.9 1150 As rec. 30 5 293 108.7 8.7 140.9 15592 13.9 1150 As rec. 60 5 281 116.0 4.4 144.0 16287 7.8 1150 As rec. 120 5 258 150.1 9.0 191.6 28821 6.6 1150 As rec. 300 5 295 167.3 13.7 196.1 30189 7.0 1150 As rec. 450 5 222 169.2 5.1 206.6 33506 5.9 1200 As rec. 300 5 172 183.9 10.3 235.8 43685 4.9 1200 As rec. 600 5 253 206.4 6.7 244.4 46916 11.9 950 As rec. 120 50 523 30.5 732.2 950 Sol.trt. 120 50 219 47.2 1748.5 1050 As rec. 120 50 309 54.8 2358 1050 Sol.trt. 120 50 177 72.4 4116.8 1150 As rec. 120 50 132 189.7 28263 1150 SoLtrt. 120 50 72 257.0 51815 Table A.l(cont.) - A36 austenite grain size results obtained by standard Gleeble thermal treatment and manual measurements, using Heyn's and Jeffries' procedures [41]. 127 Temp time H.R. No. Sy 95% °c/ conf °c sec sec grns. \\m | i m 2 (±%) 1000 10 5 250 14.4 1.59 162.63 21.4 1000 120 5 47.8 2.40 1793.8 1000 600 5 185 81.5 1.63 5217.1 17.5 1150 1 5 193 89.7 1.71 6312.2 23.0 1150 60 5 206 134.6 1.58 14234 8.6 1150 450 5 115 187.0 1.62 27471 9.3 Table A.2 - A36 austenite grain size results obtained by standard Gleeble thermal treatment and measurement using the image analyzer [40]. Quench Spec. cool No. 95% d-Am A nm 95% Temp, cond. time, grns conf conf °c sec |im (±%) |im i im 2 (±%) 1120 A R l 470 38.7 6.0 49.7 1938 in 1110 A R 5 285 49.7 6.0 63.8 3196 1A 1100 A R 10 269 50.7 8.3 65.7 3386 5.5 1080 A R 20 263 51.2 5.6 66.4 3463 7.4 1030 A R 45 243 55.0 4.9 69.1 3748 9.95 1000 A R 60 228 55.3 7.1 71.3 3995 4.6 Table A.3 - A36 austenite grain size results obtained by continuous cooling on the Gleeble, and measurement by Heyn's and Jeffries' procedures [41]. 128 Temp, Spec. time, No. 95% d-Am 95% cond. gms conf conf °c sec (im (±%) (lm (im2 (±%) 1050 AR 1 371 46.4 9.2 55.9 2455.1 15.8 1050 AR 10 251 47.0 2.8 60.8 2903.1 9.9 1050 AR 30 232 55.0 3.8 63.2 3140.8 5.9 1050 AR 120 406 62.5 8.0 76.2 4564.4 12.7 1050 AR 750 330 71.9 14.0 84.6 5615.6 16.4 1100 AR 1 427 38.9 10.3 46.6 1706.5 23.0 1100 AR 10 222 59.4 9.6 72.3 4102.9 12.1 1100 AR 30 135 68.0 5.3 82.9 5397.5 4.5 1100 AR 120 226 73.0 10.6 91.4 6559.8 14.2 1100 AR 600 251 78.9 10.6 97.0 7383.0 18.3 Table A.4 - A36 austenite grain size results obtained by stepped Gleeble thermal treatment and manual measurement by Heyn's and Jeffries' procedures [41]. 129 Temp time No. d-Am A 95% grns Jeff. Im.An. Im.An. conf °c sec |im |Xm | im 2 (±%) 1000 0 352 29.0 27.2 580.7 9.8 1000 50 395 30.6 29.7 691.3 6.8 1000 122 356 32.3 31.3 767.9 1.1 1000 245 266 33.4 32.5 827.9 7.7 1000 412 256 34.0 33.3 868.3 1.4 1050 0 397 30.6 28.9 654.2 12.0 1050 61 311 30.9 29.4 679.9 20.3 1050 118 276 32.8 31.1 758.8 15.8 1050 236 298 35.3 33.5 879.6 6.8 1050 420 226 36.2 33.8 897.0 9.1 1100 0 261 37.4 34.8 952.8 10.2 1100 68 299 45.2 42.8 1436.8 10.2 1100 124 249 66.8 64.0 3219.8 1100 244 295 79.0 72.8 4160.4 1100 424 368 101.9 87.3 5101.7 19.8 1150 0 246 124.6 112.4 9917.5 9.9 1150 55 271 148.0 135.8 14486 10.6 1150 113 218 165.0 145.5 16618 11.6 1150 227 335 168.0 152.8 18342 11.6 1150 410 298 178.1 163.1 20894 9.6 A.5 - DQSK austenite grain size results obtained by tube furnace thermal treatment and measurement using Jeffries' procedure [41] and the image analyzer [40]. 130 Temp time No. gms Jeff. Jeff. °c sec (im (im 2 1050 60 173 37.5 1107 1100 60 462 78.3 4813 1100 120 183 113.5 10126 1150 60 197 155.3 18937 Table A.6 - DQSK austenite grain size results from standard Gleeble thermal treatment and measurement by Jeffries' procedure [41]. Temp time No. 95% dAm 95% dAm 95% grns conf Jeff. Jeff. conf Im.An Im.An. conf °c sec |im (±%) fim | im 2 (±%) |im | i m2 (±%) 1050 0 1313 43.8 2.2 53.6 2258.2 6.6 50.3 1953.7 7.5 1050 120 1055 49.3 4.0 59.8 2810.4 6.2 55.3 2405.6 7.4 1050 420 756 58.5 5.8 70.7 3922.0 9.2 64.7 3273.9 8.2 Table A.7 - 1080 steel austenite grain size results from furnace thermal treatment applied in previous work [55], and measurement by Heyn's and Jeffries' procedure [41], and by the image analyzer [40]. 

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