Modeling Bainite Formation in Steels - DiVA portal1205497/FULLTEXT01.pdf · Bainite is a phase...

Modeling Bainite Formation in Steels

Lindsay Leach

Doctoral Thesis in Materials Science and Engineering

Stockholm 2018

ii

Lindsay Leach, Modeling bainite formation in steels

Unit of Structures Department of Materials Science and Engineering School of Industrial Engineering and Management KTH Royal Institute of Technology SE-100 44 Stockholm, Sweden

ISBN 978-91-7729-779-6

© Lindsay Leach, 2018

Akademisk avhandling som med tillstånd av Kunglinga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av doctorexamen fredagen den 15 juni 2018 kl 10.00 i sal B3, Brinellvägen 23, Kunglinga Tekniska Högskolan, Stockholm

This thesis is available in electronic format at kth.diva-portal.org

Printed by Universitetsservice US-AB, Stockholm, Sweden

iii

To my parents

iv

Abstract

This study examines the fundamental aspects of bainite formation in a guided effort to lay a foundation for development of a model capable of predicting bainite formation. In the first part of this study, the tenets of an existing model for growth, developed by Zener and later modified by Hillert are examined. A number of interacting and adjustable parameters are identified namely, diffusivity, driving force, radius of the ferrite plate tip, interfacial carbon content, and thermodynamic barriers. Amongst which, there are a number of assumptions which are no longer justifiable because of the availability of software and databases from which more accurate calculations can be obtained. The approximation of the driving force is one such example. Another is the carbon content in ferrite, which was assumed negligible. The capillarity effect of the curved ferrite interface had initially been assigned by Zener as having a fixed, optimal value at the maximum growth rate. Although this principle was kept it led to quite different results when the earlier approximations were removed. It is shown that the shape of the C-curves is largely dependent on the diffusivity.

The second focus of this thesis is aimed towards developing a model for the start temperatures of bainite, WBs and is achieved twofold. The first procedure was to develop an empirical model. Experimental information on which it is based refers to the start conditions for both bainite and Widmanstätten ferrite. A systematic approach is adapted by which Fe-C is the basis and the effect of alloying elements are evaluated separately from ternary alloys. Regression analysis of data on five ternary systems with Mn, Ni, Si, Cr and Mo gives separate coefficients. A linear empirical equation for the WBs is defined from their sum which was possible because their effects were independent. Carbon had by far the largest effect and Ni the smallest. The equation has good agreement with data but further improvement can be achieved with more reliable experimental data. The second procedure is directed towards a thermodynamic description of the start condition. The critical driving forces corresponding to the critical conditions depicted in the empirical equation are calculated. The results are presented with a dependence on temperature and the same could be translated to a carbon dependence. The critical driving force increases substantially with decreasing temperature and increasing carbon content and the effect of alloying elements is varied.

In the final section, the growth model is further developed. A more generalized expression for the barrier is formulated and together with the capillarity effect, consists of the energy requirement to move the growth interface. This is balanced with the available driving force and is solved with an optimization procedure. The predicted C-curves are compared with experimental results and reasonably good agreement is found.

v

Sammanfattning

I denna studie undersöktes de grundläggande aspekterna av bildning av bainit för att lägga grunden till en modell som kan förutsäga bainitbildningen. I den första delen granskades en befintlig modell för tillväxt utvecklad av Zener och senare modifierad av Hillert. Ett antal interagerande och justerbara parametrar identifierades såsom diffusivitet, drivande kraft, radien på ferritskivans spets, fasgränsens kolhalt och en termodynamisk barriär. Det finns ett antal antaganden som inte längre behövs tack vare tillgången till programvara och databaser med vilka mer noggranna beräkningar kan utföras. Approximationen av drivande kraften är ett exempel. Ett annat är kolhalten i ferrit, vilken antogs försumbar. Kapilläreffekten av det krökta ferritfasgränsen hade initialt bestämts av Zener till ett betsämt, optimalt värde vid den maximala tillväxthastigheten. Även om denna princip behållits så förändrades resultatet när de tidigare approximationerna avlägsnades. Det visade sig också att formen på C-kurvorna är starkt beroende av diffusiviteten. I den andra delen av avhandlingen utvecklades en modell för prediktering av starttemperaturen för bildning av bainit och Widmanstättenferrit, WBs. Först utvecklades en empirisk modell baserad på experimentell information av startförhållanden för både bainit och Widmanstättenferrit. Den baserades på experimentell data för Fe-C och effekten av legeringselementen utvärderades separat för ternära legeringar där en regressionsanalys av data från fem ternära system med Mn, Ni, Si, Cr och Mo gav separata koefficienter. En linjär empirisk ekvation för WBs presenterades, oberoende av interaktion mellan ingående legeringselement, där C har störst effekt och Ni minst. Ekvationen har god överenstämmelse med tillgänglig experimentell data men för ytterligare förbättring krävs mer tillförlitliga experimentell data. Efter det utvecklades en termodynamisk baserad modell där den kritiska drivande kraften som motsvarar de kritiska förhållandena som beskrivs av den empiriska ekvationen bestämdes. Den kritiska drivande kraften presenterades som funktion av temperatur och kolhalt och visade sig öka väsentligt med minskande temperatur och ökande kolhalt men effekten av legeringselement varierade. I den sista delen utvecklades tillväxtmodellen vidare. Ett mer generaliserat uttryck för barriären formulerades som består av kapilläreffekten och energin som åtgår för förflyttning av fasgränsen. Det balanseras av den tillgängliga drivande kraften och en lösning optimerades fram. De predikterade C-kurvorna jämfördes och visade en bra överensstämmelse med experimentell data.

vi

List of appended papers:

I. Modeling C-curves for the growth rate Widmanstätten and

Bainitic ferrite in Fe-C alloys.

L. Leach, M. Hillert, A. Borgenstam Metall. Mater. Trans. A, 2016, vol. 47, pp19 -25.

II. C-curves for Lengthening of Widmanstätten and Bainitic ferrite. J. Yin, L. Leach, M. Hillert, A. Borgenstam Metall. Mater. Trans. A, 2016, vol. 48, pp3997 -4005.

III. Empirical methods to predict bainite start conditions. L. Leach, P. Kolmskog, L. Höglund, M. Hillert, A. Borgenstam Submitted manuscript

IV. Critical driving forces for formation of bainite. L. Leach, P Kolmskog, L. Höglund, M. Hillert, A. Borgenstam Submitted manuscript

V. Representing lengthening rate of bainitic ferrite – a part of the

steel genome. L. Leach, J. Ågren, L. Höglund, M. Hillert, A. Borgenstam Submitted manuscript

The author’s contribution to appended papers:

I. Literature survey, analysis of data and writing of manuscript. II. Calculations in the modeling section, took part in writing.

III. Calculations and data analysis, took part in writing. IV. Calculations and thermodynamic modeling, took part in writing. V. Initial calculations, modeling, took part in writing.

vii

Contents Chapter 1 Introduction 1

1.1. Overview on bainite 2 1.1.1. The diffusional hypothesis 3 1.1.2. The diffusionless hypothesis 4

1.2. Scope and Objectives 4

Chapter 2 Growth rate model 6

2.1. Zener-Hillert model 7 2.2. Trivedi model 8 2.3. Modification of Zener-Hillert model 10 2.4. Model parameters 13

2.4.1. Effect of optimal radius 13 2.4.2. Effect of diffusivity 14

2.5. Experimental information 15

Chapter 3 Bainite start temperatures 18

3.1. Experimental methods 18 3.1.1. T methods 18 3.1.2. C methods 19

3.2. Empirical modelling of start temperature 19

Chapter 4 Thermodynamic barriers 26

4.1. Driving force for a diffusional process 29 4.2. Alloying element barriers 31 4.3. Driving force for a diffusionless process 35 4.4. Analytical model 38 4.5. Effect of barrier on growth modeling 39

Chapter 5 Bainite formation model 41

5.1. Energy dissipation during growth 41 5.2. Optimized parameters 42 5.3. Prediction of C-curves 43 5.4. Software package 44

Chapter 6 Concluding remarks and future outlook 45

6.1. Concluding remarks 45 6.2. Outlook 45 Acknowledgements

Bibliography

Chapter 1 Introduction

1

Chapter 1

Introduction teel production is an industry that is at the core of the modern day circular economy. Steel products reach many aspects of our everyday lives with functionalities that range from automotive and construction

applications to components in domestic appliances, electrical equipment, tools and mechanical machinery. Furthermore, high demanding applications as would be for aerospace materials utilize steel. Given its central position as product supplier to related industries, its far-reaching influence subsequently impacts economic growth. With steel production having increased markedly and nearly doubling in tonnage between 2001 and 2011 to about 1.5 million tonnes [1], there is a recognized need to direct the efforts of both industry and stakeholders towards sustainable performance. The major sustainable advantage of steel is its reuseability. For instance, within the construction industry, steel offers a flexible and adaptable solution to the architectural demands of buildings. At the end of their lifespan, steel products can be reclaimed and recycled or reused without loss of properties. Therefore the design of steel and material properties in general, by computational tools, is an important part of research efforts supporting development of steels. Implementation of empirical procedures to determine material properties can lead to inefficiency and unsustainable product usage as well as wastage of energy expenditure during production. There is therefore a manifest need to develop a more conceptual framework founded in theoretical principles. This project was conducted as a part of the Materials Genome to support the development of computational tools that can be used in the design of bainitic steels. To this end available computational resources such as Thermo-Calc [2] are used to perform calculations, thus reducing the cost and time associated with experimental trials. The process can be envisioned in the framework of Cohens design reciprocity in Fig. 1 showing the connection between the four elements of processing, structure, properties and performance. The philosophy guides the design process which could be directed either towards the materials performance through a cause-effect relation or by investigating the routes of achieving the required performance through a goal-means relation. The linkage tools connecting the four elements are creators, translators and performers. The modeling work presented in this thesis is within the creator category, which links processing conditions to structure.

S


2

Figure 1. Materials design philosophy showing the reciprocal relation between fundamental design aspects [3].

1.1. Overview on bainite

Bainite is a phase transformation product of austenite that occurs at intermediate temperature ranges between pearlite and Martensite, see Fig. 2. Hultgren observed it as a mixture of ferrite and cementite and called it troostite [4]. It was later named in honor of Bain who first identified its Time-Temperature-Transformation (TTT) behavior [5]. With the discovery came a new field of research that has seen development of steels with impressive properties. Bainitic steels have several desirable and beneficial properties such as good strength, toughness, hardness and wear resistance that render it preferable to other steels. Its beneficial properties are retained even when in combination with other phases as is the case for dual-phase steels or Transformation Induced Plasticity (TRIP) Steels. The reasons why this could be beneficial to industry are manifold with the foremost being an increase in profitability. Novel uses for bainite include its use as a rail steel, owing to superior wear resistance, toughness and strength. They also find use in as forged sheet steel automotive parts which demand high strength, high formability and high energy absorption at fast deformation rates.

With such wide and far-reaching interest, the need to describe and control the design of bainitic steels by researchers and manufacturers alike has grown. To achieve this, one has to understand the mechanism and conditions

Processing

Structure

Properties

Performance


3

of bainite formation. Although several explanations to this end have been put forth, there remains a divide in opinion and divergent views have emerged. Despite having been the subject of research that spans several decades, all attempts so far to describe the constitution and mechanism of bainite have fallen into either the diffusional hypothesis or the diffusionless hypothesis and there remains no single-standing formalism for bainite.

Figure 2. Upper bainite formed during isothermal annealing of a Fe-0.3C-0.5Si steel

at 723K (450�) [6].

1.1.1. The Diffusional hypothesis

Hultgren proposed that bainite forms as plates of Widmanstätten ferrite [7] and that the edgewise growth is controlled by carbon diffusion in austenite. The lengthening rate is substantially larger than the lateral growth which is attributed to low mobility of the broad coherent interfaces. The austenite ferrite interface grows with carbon content that deviates from equilibrium and is subject to capillarity effects, alloying element interactions and kinetic effects. There is no partitioning of substitutional alloying element and this condition is referred to as paraequilibrium. According to proponents of the diffusional paradigm, the transformation products of Widmanstätten ferrite and acicular ferrite in the two variants of upper and lower bainite form a continuous series of products [8]. Evidence for this is given by the fact that bainite ferrite bears morphological resemblance to Widmanstätten ferrite. Hillert observed that the lengthening rates of Widmanstätten ferrite and bainite varied smoothly over a range of temperatures [9] and took this as evidence that there is no change in transformation mechanism. The bainitic ferrite transformation is therefore the same diffusional process as the


4

Widmanstätten ferrite transformation. Carbide formation in upper and lower bainite is a secondary reaction following primary growth of the ferrite plate.

1.1.2. The Diffusionless hypothesis

The similarity of bainite to martensite is one of the main factors prompting the diffusionless hypothesis according to which bainite is formed through a displacive process and the transformation interface behaves similarly to the martensitic interface. In the early work by Oblak and Heheman [10], they performed a detailed microstructural study examining Widmanstätten ferrite, upper and lower bainite from which they suggested that the structural differences between the three phases indicates that they are not a continuous series of transformation products. They instead proposed that the growth process occurs by repeated nucleation of substructural units growing to a limited size determined by strain accumulation and with continued growth limited by strain relaxation. The ferritic component forms with a carbon supersaturation which is then relieved by the formation of carbides [10] [11]. Several studies have pointed to the presence of a surface relief on Widmanstätten ferrite [12] [13] [14] yet along with this, Ko and Cottrell’s study showed that it was accompanied by slow growth implying some diffusion. Proponents of the diffusionless theory have largely built their theories based on these early works.

1.2. Scope and objectives

This thesis presents a model for bainite founded in theoretical principles and developed through thermodynamic analysis of experimental data. The endeavor as a whole is a part of the larger scope of the materials genome and the objective of the work is to develop a predictive tool for bainite formation in steels.

To sufficiently describe the formation of bainite, three aspects are dealt with. The first is the growth rate of bainitic ferrite. A diffusional approach is adapted for this purpose and is dealt with in Chapter 2. A modification to the original Zener-Hillert formulation of growth rate is presented and the implications are described. The second aspect is the bainite or Widmanstätten ferrite start temperature which is generalized to their transformation conditions. Chapter 3 demonstrates that the criterion for formation is not solely based on temperature but as a conjugate equivalent in the composition variable. With a select set of experimental information, an


5

empirical model describing the start temperature is constructed. Particular attention is given to the effect of alloying elements. The third aspect dealt with is a thermodynamic treatment describing thermodynamic barriers and is an integral part of the model. The phenomenon is shown to be necessary to sufficiently describe both the start and the growth rates, especially when dealing with alloyed steels. It arises from the fact that bainite forms at non-equilibrium conditions and thus some ‘barrier’ must be surmounted to correctly describe the formation conditions. It is chosen to represent the barrier as a function of composition, as is illustrated in Chapter 4.

The work culminates with a consolidation of the above mentioned into a model that depicts bainite formation in alloyed steels. Calculations are executed in a Thermo-Calc based software package developed specifically for the model. The underlying objective, to aid in the development of advanced technologies for advanced steels, is thus achieved by the resulting computational tool.

Chapter 2 Growth rate modeling

6

Chapter 2

Growth rate model

The overall kinetics of a phase transformation is established in accordance with the kinetics governing the reaction and is typically represented by a Time-Temperature-Transformation diagram. Kinetic modeling of bainite is intimately connected to the assumed mechanism of the reaction. Therefore, within the diffusionless school of thought, the growth of bainite is a discontinuous process. It is described as an autocatalytic process which proceeds with repeated nucleation of subunits. The subunits accumulate to form a sheaf and the overall growth is thus dictated by the rate of nucleation of these subunits [15] [16]. On the other hand, the diffusional school prescribes a growth process whereby the growth interface advances with no discontinuities and the morphology is often described as having a plate or lath morphology. Several works have identified continuous lengthening. Eichen studied the growth interface with thermionic emission and the results showed continuous lengthening [17]. Nemoto’s thermionic emission study on Fe-C-Ni alloys also showed continuous growth [18], seen in Fig. 3. Using laser confocal microscopy, Wan and coworkers monitored the lengthening rate of ferritic plates emanating from the grain boundary during continuous cooling [19]. Using the same technique, Hu et al. measured and found a constant lengthening rate for plates emanating from the grain boundary and from preformed plates [20]. Kolmskog et al. observed constant lengthening of acicular units [21].

Figure 3. Plate lengthening measured by thermionic emission [18].


7

Given the experimental evidence for diffusional growth of bainite, it has been described rather well with the diffusional model. Quidort and Brechet for instance, studied isothermal growth of bainite with a kinetic model based on the Trivedi approach with paraequilibrium conditions and the assumption that carbon diffusion at the interface is the controlling mechanism [22]. The diffusional model has been adopted in the present study on the basis of experimental evidence.

2.1. Zener-Hillert model

Zener presented an equation for the edgewise growth of a plate precipitate and it was later modified by Hillert to what is now referred to as the Zener-Hillert model. The main tenet defining bainitic formation in the Zener-Hillert model is that the process is controlled by diffusion of carbon through the austenite away from the advancing interface [23] [13]. It is given as the maximum lengthening rate �� ,

�� = ��(�� /��)

�� (1)

where R is the molar gas constant, D is the carbon diffusion coefficient in austenite, T is the temperature, σ is the interfacial energy, Vm is the molar

volume of ferrite. The supersaturation term, (��/� − ��)/��contains the

equilibrium mole fraction of carbon in austenite ��/�and the nominal carbon

content �� . There is local equilibrium at the interface hence an equilibrium

composition ��/�corresponding to a planar interface. However, for a plate

precipitate with a finite radius of curvature, the curvature of the advancing interface creates a pressure and causes a non-equilibrium composition at the interface. Equilibrium carbon contents are attained at an infinite plate tip radius i.e. for a flat interface. The concentration gradient created ahead of the interface is illustrated schematically in Fig. 4 and provides a driving force for the diffusion of carbon.


8

Figure 4. Concentration profile for growth. The arrow shows the direction of advance

of the interface.

The capillarity effect is distinctly accounted for in the Zener-Hillert model. Plate lengthening occurs at a constant rate whereby the plate tip is adjusted to some optimal value for that temperature. Zener proposed that this value should be the solution to a maximized velocity and calculated the optimal plate tip radius as 2�� , with �� being the critical radius. The precipitation of carbide is a secondary process and does not interfere with the advancing interface as long as they are not nucleated too close to the ferrite tip. It is thus sufficient to use as a general growth model for the overall transformation though it should be mentioned that the presence of carbides will most likely alter the diffusion field.

2.2. Trivedi model

Trivedi [24] [25] considered the growth of plate precipitates and proposed a more rigorous treatment for the growth of bainite than was dealt with in the Zener-Hillert model, which was weak in his view due to assumptions. The objections were that: the approximation that the effective diffusion distance is proportional to the curvature of the plate tip leads to errors as the proportionality constant is not simply a numerical constant, that the effect of a varying interfacial concentration on the concentration gradient at the plate tip is not properly accounted for and that the concentration profile assumed was approximate. Detailed accounts of these and other objections are discussed in Ref. [25]. The fundamental principle of Travedis’ model is that carbon diffusion at the interface is the rate limiting factor. There are four main assumptions on which the model is based, namely, that the plate is parabolic in shape, that the diffusion coefficient of carbon has no

��/�

��

��

BCC $ Carb

on c

onte

nt

Distance

FCC γ


9

concentration dependency, that the carbon concentrations are a function of the capillarity effect and that the elastic strain and surface anisotropy effects are negligible. Furthermore, the analysis is based on the premise that local equilibrium applies at the interface. The result of the analysis was an implicit relation between the supersaturation and the growth rate and gave solutions for an interface with an unvarying concentration and capillarity effect. The maximum velocity is attained at some optimal tip curvature. In subsequent work [24] they considered kinetics at the interface and included a term for this in their solution,

Ω� = ()* +,+-./(*01 + (�/�3) Ω�45(*) + (�3/�)Ω�46(*)7 (2)

Where Ω� is the supersaturation * is the peclet number a dimensionless parameter given by ��/29, �3 is the critical radius, 45 and 46 are parameters dependent on *, �3 is the critical velocity of a flat interface controlled entirely by interface kinetics. The three terms thus represent the proportion of the supersaturation consumed by diffusion, interface kinetics and capillarity, see Fig. 5. The Zener-Hillert model on the other hand equally divides the supersaturation between diffusion and capillarity. Simonen et al. analyzed experimental growth rates using Trivedi’s model [26] and showed that Gibbs free energy was dissipated mainly due to diffusion of carbon in austenite.

Figure 5. Dissipation of supersaturation in Trivedi model [26].

A simplified version of the model was later given [27] and has been used in several investigations [22] [28] [29]. However, it was later shown [30] that there is a limitation in the solution at low supersaturations and it is more reliable at high supersaturations. Figure 6 demonstrates this effect.


10

Figure 6. Model predictions for velocity at low and high supersaturation [30].

2.3. Modification of Zener-Hillert model

There are a number of simplifications in the Zener-Hillert formulation and an improved solution can be achieved through a review of the approximations and assumptions. A reviewing of the underlying principles are discussed in detail in Paper I. In his model for the lengthening rate of plate-like precipitates, Zener proposed that a spherical nucleus grows into a parabolic plate and the plate tip adjusts itself according to a particular size. The criterion is that the maximum lengthening rate occurs when the radius of the plate tip becomes twice the value of the nucleus radius before growth begins. The radius where growth begins or where velocity is zero, is referred to as the critical radius. At this radius all the Gibbs free energy consumed during the process is used to increase the surface area and the nucleus is in an unstable equilibrium, particles with a radius smaller than the critical value shrink and dissolve as their growth would result in an increase in the Gibbs free energy of the system [13]. Particles with a larger radius will grow. According to Zener, the current density of atoms can be expressed both as a function of the concentration gradient and as a function of the diffusion coefficient,

Current density = (:6 − :5); = −9 <3<� (3)

Where ; is the interface velocity, :5 and :6 are the concentration of solute atoms in the alloy and parent phase respectively and 9 is the diffusion coefficient. It follows from equation 3 that the velocity is given by,

; = �3@�3A

<3<� (4)


11

The concentration gradient can in turn be related to the effective diffusion

distance L and is equivalent to ∆3C and equation 4 becomes

; = ∆33@�3A

C (5)

However, growth of precipitates usually occurs with a curved interface. Due to the fact that the growth interface is not planar, the concentration of solute i.e. carbon in the parent phase is changed. A planar surface is equivalent to an interface with infinitely large radius and where the carbon concentration in austenite is in equilibrium. For a curved interface, the Gibbs-Thomson effect applies and consequently, the concentration difference is altered by the interface curvature and the actual value is obtained from the Gibbs-Thomson equation as:

∆: = (1 − �� ⁄ )∆:E (6)

Where ∆:E is the concentration difference of carbon in the austenite at the interface according to equilibrium and further away in the austenite if the effect of the curved interface is neglected. �� is the critical radius and its value will be 2�� when the Gibbs-Thomson equation, based on dilute solution thermodynamics is used. However, by removing the dilute solution assumption, the maximum velocity will not appear at 2�� . This is possible because the overpressure of the critical nucleus, ∆F is related to the driving force, ∆G� through

∆F = �HIJ

= ∆K��

(7)

Where L is the interfacial energy and M� is the molar volume of ferrite. By adapting Zener’s approach and given that the effective diffusion distance is proportional to the radius of curvature of the plate tip N = O�, where O = 2, Hillert introduced a growth rate equation

; = �H

�� P

�P�� Q1 − HIJH R (8)

There is a further approximation to be applied before one arrives at the classical Zener-Hillert equation. It is that for equilibrium or close to

equilibrium concentrations, the magnitude of �� − �E can be estimated

with the driving force ∆G�

�� − �E = ∆K�

�� (9)

Applying equation 9, equation 8 becomes


12

; = ∆K��

�� P

�P�� HIJ

H Q1 − HIJH R = ��

��(��

�� P)@

�P�� HIJ

H Q1 − HIJH R (10)

Another approximation used in the Zener-Hillert equation is that the carbon content in ferrite is low and certainly much lower than the nominal carbon content. Thus the supersaturation term was altered to exclude the negligible amount in ferrite.

�� ≪ �E (11)

And finally, in compliance with the maximum growth rate criterion, the optimal radius � will be 2�� and the effective diffusion distance N will be 2�, equation 10 then reduces to the classical Zener-Hillert expression for the maximum growth rate ��

�� = ��

(�� P)@

�P (12)

A refinement of the model parameters was achieved by computational thermodynamic calculations rendering possible more precise calculations of equations 7, 9 and 11. A major consequence is that the removal of the restriction that � �� ⁄ = 2 at �� reverts to an expression denoting the velocity at any ratio � �� ⁄ . In Fig. 4 of Paper I it is shown that the optimal value of � �� ⁄ at which velocity is maximum varies with temperature and carbon content. A final step was to reintroduce the capillarity effect of the growth interface. Since the curvature affects the carbon concentrations, the supersaturation term will consist of the actual interfacial carbon content ��

rather than the equilibrium counterpart �� . The final version of the equation

is thus

� = ∆K�6��

� �� P�P��

HIJH (13)

There are thus a number of parameters affecting the velocity equation which will now be examined.


13

2.4. Model parameters

2.4.1 Effect of optimal radius

The growth rates in Paper I were calculated with equation 13 in which the velocity varies with the ratio � �� ⁄ . The value at which the highest velocity is obtained is referred to as the optimal radius. Fig. 7 shows how the growth rate varies with curvature for three binary Fe-C steels. A positive growth velocity is observed beyond the critical radius, where the growth velocity is zero. The growth rate increases starting from the critical radius, reaching a maximum value before decreasing again. There are two marked points on each curve signaling an assumed optimal ratio of 2 and the actual optimal ratio, at the turning points of the curves.

Figure 7. Variation of growth rates with radius of curvature calculated for three Fe-C

steels at 400 �.

It can be seen that the highest velocity is attained at a higher optimal radius than that ascribed by Zener and the result is consistent for all three alloys. A further examination of the highest velocity in alloys with 0.3 – 1.8 mass%C is shown in Fig. 8 at temperatures between 300 and 700�.


14

Figure 8. Optimal radii as a function of temperature and carbon content.

There is rather good agreement with Zener’s postulate at the highest temperature of 700 � where the optimal radii is about 2�� . As the temperature is lowered, the optimal radius increases by a factor of 3.5. The slope of the lines indicate that the optimal radius reduces with increasing carbon in the steel and the effect is less pronounced at higher temperatures.

2.4.2 Effect of diffusivity

As the formation of bainite is a diffusion-controlled process, the diffusivity of carbon plays a central role. During the formation of bainitic ferrite, carbon diffuses away from the austenite-ferrite interface and the variable carbon contents to be considered have been illustrated in Fig. 4; the nominal carbon content of the steel, the carbon content at the interface and the carbon content in the ferrite. Since the growth process is controlled by carbon diffusion in the austenite, one can consider the diffusivity as an averaged

value 9�TT between the highest and lowest carbon contents or the maximum diffusivity as calculated with the carbon at the interface yielding the maximum diffusivity 9�� or with the nominal carbon content which gives the minimum diffusivity 9� . The expression for diffusivity, 93 used in this work is given by Ågren [31] and is expressed as a function of carbon content U3 and temperature T,

93 = 4.53 ∙ 10�\ Q1 + U3(1 − U3) �]]^.^� R exp b− Q5

� − 2.221 ∙ 10�cR (17767 −U326436)f g6h�5 (14)

According to Trivedi and Pound [32] to obtain the growth rate the diffusivity

should correspond to the weighted average value 9�TT


15

9�TT = i (�)j��@��A

�@�A (15)

The composition dependence of diffusivity in this work was examined by calculating growth rates with diffusivities according to the 3 criteria of 9� ,

9�TTand 9�� , see Fig. 9. Detailed information can be found in Paper I.

Figure 9. Growth rates of bainite calculated with the modified Zener-Hillert model for

the minimum, average and maximum diffusivity.

As seen in Figure 9, there is a direct correlation between the growth rate and the diffusivity, however, the maximum diffusivity results in growth rates that are only somewhat higher than those obtained with the average diffusivity. The difference between the curves become broader with increasing diffusivity and at lower temperatures, it would seem that there is an increasing amount of carbon trapped in the ferrite, as will be shown in the next section.

2.5 Experimental information

There are several techniques used to monitor the growth rate of bainite. With hot-stage microscopes, the lengthening of bainitic plates can be monitored in-situ during isothermal annealing [33]. This is possible due to the visible surface relief accompanying the transformation. Speich and Cohen used photomicrographs taken at various time intervals and based their lengthening rates on the location of the plates. The length of plates growing within the grain was halved to maintain consistency with grain boundary plates growing in one direction, whereas interior plates grow in two directions from both tips observed on the surface. Hawkins studied an Fe-0.9C and an Fe-0.9-1.5Mn alloy using the same technique [34] and several

0.01 0.1 1 10 100 1000-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

Deff

Do

Dmax

100

0/T

(K

)

Growth rate (µm/s)

0.3C0.6C

0.9C


16

other studies [35] [36] [37] [38] [39] [40] have been conducted with this technique. Simonen obtained the growth rates in low carbon alloys by measuring the length of the longest plate as a function of the reaction time [26]. Measurements were made on metallographically prepared samples following an austenitising, isothermal annealing and quenching treatment. The rationale behind selecting the longest plate is due to a sectioning effect caused by a stereographic projection onto the surface. Several other authors have used this technique [41] [42] [22].

Figure 10. Experimental lengthening rates compared with predictions of Zener-Hillert

model.

0.1 1 10 100450

500

550

600

650

700

750

800

850

900

950

1000

T (

K)

Lengthening rate (µm/s)

0.9C-Calculated-Do

0.9C-Calculated-Deff

0.9C-Calculated-Dmax

0.9C Hawkins

0.81C-0.23Mn-0.26Si Hillert


17

Experimental growth rates were compared with the calculations from Eq. 13 and Fig. 10 shows the result for high and low carbon alloys [41]. The C- curves were calculated with an average composition for both sets of alloys and shown as a function of three types of diffusivities outlined in Section 2.4.1.2. Growth rates obtained using an averaged diffusivity describe the high carbon alloy rather well toward low temperatures. The discrepancy for the low carbon alloy at low temperatures is possibly due to an increasing carbon supersaturation in ferrite and is discussed in Paper II. The discrepancy between the asymptotic curve and the data at higher temperatures is because no temperature limit is set hence it approaches the A3 temperature. This illustrates the need to make the model more realistic by defining a starting temperature.

Chapter 3 Bainite start temperatures

18

Chapter 3

Bainite start temperatures

The start temperature is an important design parameter for bainitic alloys and a fundamental aspect for the development of a model. The bainite start temperature is defined as the highest temperature at which bainite is observed during cooling of austenite or the upper limiting temperature for decomposition of austenite into acicular ferrite [43]. The transformation begins when supersaturated austenite reaches the start temperature, growth proceeds below this temperature. In the previous section which uses the Zener-Hillert equation to predict growth rates, the upper limiting temperature reaches the equilibrium value yet should only reach the bainite start temperature. However the referred to temperature is one of two parameters concerning the beginning of the transformation with the other being carbon content. These parameters are denoted as critical conditions and are discussed in Paper III. There are several experimental methods to determine the critical temperature or carbon content for bainite formation. These are now introduced as temperature dependent methods (T methods) or carbon content dependent methods (C methods).

This is precisely the information used to develop a model for the start temperatures, which in this work, has been done in two ways. The first procedure, as addressed in this chapter, is to derive an expression using experimental information. The second procedure is a derivation based on thermodynamics and the use of driving forces and is treated in Chapter 4.

3.1 Experimental methods

3.1.1 T methods

The critical temperature for bainite or Widmanstätten ferrite formation is determined primarily from the highest temperature where it is observed and is here referred to as the Widmanstätten/bainite start, klm. It can be determined for instance by metallographic studies where temperature intervals are measured as the transformation proceeds. Established TTT diagrams are a prominent source of information. Here the upper temperature of the C-curve gives an indication of the start temperature. When quantities


19

relating to the transformation rate are measured as a function of temperature, that quantity can be extrapolated to the highest temperature. The volume fraction evolution is one such quantity. When volume fraction is measured during isothermal heat treatment, a plot of the different amounts at their corresponding temperatures which are typically below the Bs temperature can be extrapolated to zero volume fraction. Another such quantity is the growth rate. The same extrapolation procedure can be done on growth rates measured below the Bs temperature. Widmanstätten ferrite growth rates measured with temperature can be extrapolated to the highest temperature but the accuracy of this result will depend on the length of the extrapolation. A total of 258 data points were obtained according to the above description.

3.1.2 C methods

The amount of carbon in the austenite where bainite has ceased to form is regarded as a critical condition provided that cementite does not form as this would consume the carbon. The accuracy of this particular approach is thus dependent on whether there is carbide free bainite or not which in turn is dependent on what alloying elements are present. The presence of Si and Al will delay the formation of cementite. This condition is commonly referred to as the transformation stasis or the incomplete transformation. The carbon content can be obtained from a series of measurements of the volume fraction and is identified at the plateau of a sigmoidal curve. The amount of carbon at the plateau can then be determined using a mass balance. A more direct measurement involves ATP or XRD analysis of the carbon content in retained austenite. Extrapolation of growth rates, measured for different alloys at the same temperature, gives a carbon content at zero growth rate which is identified as a critical point. From the literature survey, 344 data points were found.

3.2 Empirical modelling of start temperature

Several attempts have been made towards predicting the bainite start temperature with some arguing for a simple empirical approach and others adapting a more sophisticated approach. Bodnar et al. evaluated the Bs temperatures from 3 types of steels with relatively high Ni and Cr contents and low C content and performed a linear regression analysis with a linear composition dependency [44]. The resulting contributions was reflective of


20

the limited range of their study. Zhao et al. performed a linear regression over a much larger number of alloys, 82 which were collected from Isothermal Transformation diagrams [45] and the result can be expected to be more generally applicable. In addition to chemical composition, some authors included the effects of prior austenite grain size and cooling rate [46] [47]. Others yet argued in favor of non-linear dependencies [48] citing exponential or polynomial [45] composition dependencies. Meanwhile, Mateo and coworkers recognized the shortcoming caused by limited input data and opted for a neural network model [49]. Artificial Neural Networks (ANN) have more computational flexibility than linear regression analysis and are capable of vastly large calculation domains. The network is trained to learn a relationship between input data and the target variable using a weighting technique and back-propagation, a feedback procedure. Consequently, the functions mapping input to output are seldom linear but are combinations of linear and non-linear functions. The technique is rather suitable for handling complex problems. Consequently, it is rather difficult to extract the decisional criterion and to interpret the overall structure of the network. Additionally, no information about the significance of the parameters or weighting is provided. The present approach to describing the start temperatures is thus based on the simpler statistical approach with manifest interpretation of parameters.

The goal of this part was to develop a model for the start temperature using the available experimental information previously discussed. The procedure utilized linear regression analysis to obtain a relationship between the Bs temperature and the alloy chemical composition to determine the contribution of each alloying element. Since Bs temperature studies conducted in this manner seem to be strongly influenced by the specific set of alloys used and have the limitation in their applicability to other alloy data, the present study adapted an ordered systematic approach. The analysis was performed in consecutive order on select sets of data and it was imperative that the effect of each element is isolated as much as possible. This was achieved by first considering the effect of carbon in binary Fe-C alloys in Fig. 11 and later adding the effect of Mn, Si, Ni, Cr and Mo in ternary alloys. Similar analyses of the start temperature have not considered Fe-C alloys independently, see Fig 12. The concept was that an optimization of Fe-C-M1 data would better represent the effect of M1 than it would in an Fe-C-M1-M2-M3-M4-M5 data assuming there is no interaction effect between Mi. As carbon is the basis for this method, the procedure began with analysis of Fe-C alloy data. An important aspect of the modeling is the inclusion of data from Widmanstätten ferrite microstructures. As mentioned previously, diffusional theory regards it as bainitic ferrite due to morphological and mechanistic


21

similarities. The transformation of austenite to bainite begins with the formation of ferrite and the subsequent carbide formation is a secondary reaction. In this work information from both Widmanstätten ferrite and bainite were thus accepted.

Figure 11. Start temperatures for Fe-C alloys. The lower dashed line, ln� evaluated by

Aaronson. klm� is based on the filled symbols.

The klm� line was taken as a reference point for the ternary alloys and in Fig. 11 it is shown together with the lm� line evaluated by Steven and Haynes [50] and with the A3 line. Information on the experimental data is discussed in Paper III. Although meagre, the data was well represented by the linear relation

kln�(�) = 850 − 206: (16)

The data from Aaronson [51] agreed well with Steven and Haynes equation but was not included in the final analysis of kln� in Paper III on grounds of insufficient background information.

The carbon content dependence of WBs temperatures for several linear models [50] [52] [53] [44][ 54] [45] is shown in Fig. 12 and compared with the present result.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0300

400

500

600

700

800

900 Aaronson 1962

Aaronson 1986

Townsend & Kirkaldy 1968

Johanson 1960

T

em

pe

ratu

re (

°C)

Carbon (wt%)

A3

WBo

s

Bo

s


22

Figure 12. Linear plots for empirically derived Bs temperatures for Fe-C alloys.

An evaluation of the effect of alloying elements was performed by linear regression analysis ternary alloy data containing Mo, Cr, Mn, Ni or Si with the two parameters for carbon fixed in each analysis. A detailed description is given in Paper III. A unified equation could then be obtained from the analysis on the five Fe-C-M systems due to the fixed carbon parameter and the fixed intercept and the final equation is a combination of all the coefficients obtained,

klm (�) = 850 − 206: − 78pq − 33rs − 70:- − 75pt − 614s (17)

From the equation, it can be seen that C has by far the largest effect on the Bs temperature followed by Mn and Mo. By using a linear expression, it is assumed that there is no interactive effect between the alloying elements. Moreover, within the examined ternary systems, an indication of such an effect would be the difference between the slope of the data and that of the predicted line for the same composition. Although there was general agreement with the data, some data sets tended toward a steeper slope, alluding to a stronger carbon dependency in the Fe-C data and possibly an interactive effect though that type of assessment would require data higher in quality.

The case of Fe-C-Si is one such example of a shortcoming due to the quality of experimental data. Figure 13 shows four sets of Fe-C-Si data [55] [56] [57] [58] with the reference klm� line. The main complication arose because of conflicting information from the two sets of data with 3.85 mass% Si and 2.69 mass% Si. The two alloys are in direct contradiction with each other and the coefficients evaluated with reference to the klm� line are -56 and -32

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0300

400

500

600

700

800

900

Bain

ite

sta

rt te

mp

era

ture

(oC

)

Carbon content (wt%)

present work, WBo

S

[45][50]

[44]

[54]

[53]

[52]


23

respectively. It was thus that an exception was made to evaluate the Si coefficient by considering Fe-C-Mn-Si alloys.

Figure 13. Ternary Fe-C-Si data shown with reference to the evaluated klm� line.

There were initially three possible routes available: First is a linear regression analysis where the C coefficient is fixed and Mn and Si are simultaneously optimized. Second is an analysis where only Si is optimized and the remaining elements are fixed and thirdly, analysis whereby all 3 elements are simultaneously optimized. The following 3 equations were obtained as a result.

lh = 850,44 − 206,15: − 81,79pq − 57,614s (18)

lh = 850,44 − 206,15: − 78,04pq − 61,044s (19)

lh = 833 − 241: − 74pq − 364s (20)

With C fixed in Eqs. 18 and 19, there is a small difference in the coefficients for Mn and it is even smaller for Si. However, when all 3 elements are optimized the coefficient for Si in Eq. 20, is almost half of the previous values. Nevertheless, the selected criterion, Eq. 19, was chosen in such a way that consistency with the systematic approach was sufficiently maintained.

The overall predictions with Eq. 17 are now shown and classified according to T and C method data.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0300

400

500

600

700

800

900 Fe-0.80C-1Si [55]

Fe-0.096C-0.51Si

& Fe-0.12C-1.47Si [56]

Fe-0.90C-3.85Si [57]

Fe-0.37C-2.69Si [58]

Tem

pera

ture

(°C

)

Carbon (wt%)

A3 for Fe-C

WBo

s

Bo

s


24

Figure 14. Results of start temperature calculations with Eq. 17 compared with

experimental data according to T methods.

Figure 15. Results of start temperature calculations with Eq. 17 compared with

experimental data according to C methods.

0 100 200 300 400 500 600 700 800 900 1000

0

100

200

300

400

500

600

700

800

900

1000

Calc

ula

ted W

Bs(°

C)

Experimental WBs(°C)

T method data

New equation

RMS: 70K

0 100 200 300 400 500 600 700 800 900 1000

0

100

200

300

400

500

600

700

800

900

1000

RMS: 76K

Calc

ula

ted W

Bs(°

C)

Experimental WBs(°C)

C method data

New equation


25

The data presented in Figs. 14 and 15 is alloy steel with several alloying element additions and were thus precluded from the foregoing analysis. There is reasonable agreement with empirical predictions and experimental data. Discrepancies may be attributed to the inherent scatter in the experimental data and unaccounted for interaction effects. However, the comparisons show a correlation between the two data-sets, illustrating that the critical conditions can be defined with both parameters. In the next chapter, a third parameter, the critical driving force, is introduced to describe the start temperatures.

Chapter 4 Thermodynamic barriers

26

Chapter 4

Thermodynamic barriers

The concept of a thermodynamic barrier, as applies to the bainite transformation originates in the work of Hillert’s in which he proposed a barrier to explain the discrepancy with the A3 line. Fig 16a shows the data points used to determine the WBs line. Fe-C data at 300 and 700� was obtained from growth rates extrapolated to zero and Fe-C-Si data at 450� in which the carbon content in the remaining austenite was used. The proposed barrier was reevaluated later by Hillert and coworkers [59] and fitted with a spline function, see Fig 16b.

(a)

Figure 16. (a) Critical carbon contents obtained from extrapolation of growth rates

plotted with temperature compared with A3 and To lines [59].


27

(b)

Figure 16. (b) Barrier fitted using a spline function [59].

Other thermodynamic accounts of the limiting carbon content in austenite after bainite has formed have been proposed in the context of the Incomplete transformation (ICT) or stasis. One such is the To temperature, defined as the temperature where austenite and ferrite of the same composition have equal Gibbs free energy. It is proposed [60] that the displacive formation of bainite, occurring with a full supersaturation of carbon, can occur only below the To temperature. It was later extended to include some adjustment for the strain energy accompanying the bainitic transformation [61], under which the To is adjusted by 400J/mol to w�x. A number of recent investigations have been presented on thermodynamic theories responsible for the limiting condition of bainite formation [62] [63] [64], centered around the To , WBs and solute drag theories. Fig. 17 shows the comparisons. The authors found good agreement with the WBs barrier proposed by Hillert [59] in an Fe-C-Si alloy [65] but the predictions were not good for their Mo alloys [62] [64], demonstrated in Figs. 17 (a), (b) and (c) respectively.


28

(a)

(b) (c)

Figure 17. The remaining carbon content in austenite after bainite formation in (a)

Fe-C-Si alloys [65] and (b) (c) in Fe-C-Mo alloys compared with the kln line [62]

[64].

Other thermodynamic explanations are based on the Gibbs Energy Balance (GEB) approach [66]. It should be mentioned that the purpose seemed rather to describe the ICT phenomenon than the bainitic formation per se and its bearing on the present work is its correspondence to one of several defining criteria characteristic of the transformation.

This chapter demonstrates how the critical conditions introduced in Chapter 3 can be expressed as driving forces capable of predicting WBs temperatures. Since the driving forces are a function of the process governing the transformation, there are two options for bainite which are explored and based on diffusional or diffusionless formalisms. The driving forces are analyzed according to both processes and the results are discussed in view of the feasibility of each process.


29

4.1. Driving force for a diffusional process

In accordance with the diffusional hypothesis, described in section 1.1.1, the transformation of austenite to bainite occurs with a driving force given by

9y9 = ∑ U{�(|{

� − |{�)m (21)

where U{ is a compositional variable related to the mole fraction �{ by

U{ = �}5��~

and DFD refers to the driving force for a diffusional process.

Equation 21 gives the driving force at any temperature and at the reaction onset, the critical start conditions discussed in Chapter 3 apply and the ordinary driving force becomes a critical driving force, DFDcrit. It has in this work been computed as a function of carbon or temperature using Thermo-Calc software and the input data was the empirical representation of experimental critical temperatures and carbon contents evaluated in Paper III. The advantage to this concept is that experimental scatter is largely eliminated. This section examines the binary Fe-C system and the effect of alloying elements is examined in section 4.2.

When plotted with temperature, the klm� line as calculated is a curve which increases with DFDcrit at decreasing temperatures. It is compared in Fig. 18 with DFDcrit from T and C method data.

Figure 18. DFDcrit for Fe-C shown as the klm� line plotted with DFDcrit for multicomponent alloys as a function of temperature.

DFDcrit for Fe-C is nearly a straight line at lower temperatures but slowly begins to curve at 500� and as the temperature is increased, the curvature becomes more prominent. The shape of the curve is confirmed by the

100 200 300 400 500 600 700 800 900 10000

500

1000

1500

2000

2500

3000

DF

Dcri

t (J/m

ol)

Temperature (°C)

WBo

S


30

experimental points which show strong curvature from 700� and is attributed to the energy difference between the ferritic and austenitic states of iron. The mass of experimental information plotted is a combination of data from the T and C methods of the previous chapter. The mass of data distinctly lies above the klm� indicating that the various combinations of alloying elements yield higher driving forces. No distinction is made for the alloying elements in Fig. 18 but the shape of the calculated klm� line is supported by the data which is linear below 500� and has a non-linear deflection at higher temperatures. Since the experimental information is sourced from T and C methods, the agreement between the two methods was tested according to DFDcrit and the comparison is shown in Fig. 19.

Figure 19. Comparison of DFDcrit for alloyed steels from T and C method data with the

klm�line.

The data in the two figures in Fig. 19 show similarities and there is a rather smooth dependence on temperature with no sharp distinction between the higher temperature ranges where Widmanstätten ferrite forms and at the lower ranges where bainitic ferrite forms.

Thermodynamic properties are strongly correlated with temperature and one could hope to isolate this by taking carbon as the independent variable. It was the hope that such an approach would aid in the development of a physically based model.

200 400 600 800 10000

500

1000

1500

2000

2500

3000

T method data

DF

Dcr

it(J

/mo

l)

Temperature (oC)

200 400 600 800 10000

500

1000

1500

2000

2500

3000

D

FD

crit(J/m

ol)

Temperature (oC)

C method data


31

Figure 20. DFDcrit for Fe-C shown as the kln� line plotted with DFDcrit for high

alloyed steels [67] [68] as a function of the amount of carbon.

The kln� line in Fig. 20 shows an increase in the critical driving force with increasing carbon content. The experimental data lends some support to this trend though only very generally. The presence of alloying elements tends to increase DFDcrit as shown by all the high alloy steel data in Fig. 20 [67] [68], with the exception of some points with low driving forces which were reportedly measured within the relatively higher temperature range of 475-600�. A large deviation from the kln� line was observed and it seemed the curvature of the kln� line was transferred.

4.2. Alloying element barriers

It has already been shown in Figs. 18 to 20 that the general effect of alloying elements is to increase the critical driving force though their effect was not quantified. There were two alternatives to achieve this, either through an optimization of experimental data or by using the empirical model. The scatter and inadequate quality of experimental data narrowed it down to the latter choice. Future work on carefully conducted experiments is recommended. DFDcrit is shown with temperature and carbon content dependence for the five alloys in Figs. 21 to 25.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

500

1000

1500

2000

2500

3000

DF

Dcrit(J/m

ol)

Carbon content (wt%)

WBo

s


32

(a)

(b)

Figure 21. The effect of Mn on the critical driving force. The insert in (a) shows a

magnified view of the the effect of Mn. The dashed kln� line is for Fe-C. (b) DFDcrit as

a function of carbon shown with isotherms.

300 400 500 600 700 800 9000

500

1000

1500

2000

2500

3000

340 360 380

1400

1450

1500

1550

1600

1650

1700

1750

Fe-C

1Mn

2Mn

3Mn

4Mn

5Mn

6Mn

DF

Dcrit (

J/m

ol)

Temperature (°C)

WBo

s

0.0 0.5 1.0 1.5 2.00

500

1000

1500

2000

2500

3000

700oC

600oC

500oC

400oC

WBo

s

DF

Dcrit (

J/m

ol)

Carbon (wt%)

8wt%Mn

300oC


33

Figure 22. Effect of Si on DFDcrit as a function of temperature and carbon content. The

dashed kln� line is for Fe-C.

Figure 23. Effect of Cr on DFDcrit as a function of temperature and carbon content.

The dashed kln� line is for Fe-C.

Figure 24. Effect of Ni on DFDcrit as a function of temperature and carbon content.


300 400 500 600 700 800 9000

500

1000

1500

2000

2500

3000

DF

Dcrit (

J/m

ol)

Temperature (°C)

8wt%Si

WBo

s

0.0 0.5 1.0 1.5 2.00

500

1000

1500

2000

2500

3000

WBo

s

DF

Dcrit (

J/m

ol)

Carbon (wt%)

7wt%Si

300 400 500 600 700 800 9000

500

1000

1500

2000

2500

3000

DF

Dcrit (

J/m

ol)

Temperature (°C)

7wt%Cr

WBo

s

0.0 0.5 1.0 1.5 2.00

500

1000

1500

2000

2500

3000

WBo

s

DF

Dcrit (

J/m

ol)

Carbon (wt%)

7wt%Cr

300 400 500 600 700 800 9000

500

1000

1500

2000

2500

3000

DF

Dcrit (

J/m

ol)

Temperature (°C)

10wt%Ni WBo

s

0.0 0.5 1.0 1.5 2.00

500

1000

1500

2000

2500

3000

DF

Dcrit (

J/m

ol)

Carbon (wt%)

10wt%Ni

WBo

s


34

Figure 25. Effect of Mo on DFDcrit as a function of temperature and carbon content.


As a function of temperature, the critical driving force increased with decreasing temperature and each consecutive line is parallel to the kln� line. Termination points for each line is at the kln temperature for that alloy composition. The effects of the five alloying elements are quite varied. Cr, Mo and Si increase the driving force with reference to the kln� line with Mo having the strongest effect and Cr the least. Hillert et al. in earlier work on the thermodynamic effect of elements, found a similar effect for Cr and Mo [59]. It is an interesting finding though, that Cr would have a smaller effect than Si. In addition to exerting a positive effect, the rate of increase of DFDcrit is not the same per mass pct. The effect is greatest in the beginning at 1 mass% alloying element and decreases as the amount of alloying element is increased. The austenite formers Mn and Ni on the other hand, exhibit much different effects. Ni causes a non-linear decrease in the DFDcrit starting from kln�. In contrast to Cr, Mo and Si, the effect is minimal in the beginning but increases with more Ni content though it is small in comparison. The overall effect of Mn is to raise DFDcrit. Initially it increases up to 3 mass% Mn before decreasing again. The effects of Mn and Ni are attributed to their stabilizing effects of austenite. The strong temperature dependence above 700� persists in the alloyed systems and is most likely due to the thermodynamic properties.

As a function of carbon, the Gibbs energy difference between austenite and ferrite which causes a curvature at about 700� is observed though it is less pronounced. For the highest alloyed steels, the dependence on DFDcrit is nearly linear. The effects of Cr, Si, Mn and Mo are approximately the same when expressed this way though Ni has the smallest effect.

300 400 500 600 700 800 9000

500

1000

1500

2000

2500

3000

DF

Dcrit (

J/m

ol)

Temperature (°C)

6wt%Mo

WBo

s

0.0 0.5 1.0 1.5 2.00

500

1000

1500

2000

2500

3000

WBo

s

DF

Dcrit (

J/m

ol)

Carbon (wt%)

5wt%Mo


35

4.3. Driving force for a diffusionless process

In the diffusionless paradigm, bainite forms with full supersaturation of carbon, which diffuses out of the ferrite as a secondary process. The critical driving force for a non-diffusional process (9yr� {�) is calculated as the difference in Gibbs free energy between austenite and ferrite,

9yr� {� = G�� (w, :�,p5� � G�� w, :�,p5� (29)

According to Bhadeshias’ proposal [69], the critical driving force for diffusionless formation of bainite is a constant value of 400J/mol regardless of the composition. The value was estimated from a Gibbs free energy diagram of a Fe-C-Si-Ni alloy and later tested on a Fe-C-Mn-Si alloy.

Bhadeshia utilized the experimental information from Steven and Haynes [61] and considered two cases of the driving force for the nucleation of ferrite ∆G�. In the first case, it forms with the carbon content of the austenite ∆y�→� i.e. with a full supersaturation, and in the second case it forms with the equilibrium carbon content ∆G�→�A�� and there is continuous rejection of carbon into the austenite adjacent to the interface. The results of the tests showed that the nucleation occurs with diffusion of carbon. It was proposed that although both Widmanstätten ferrite and bainite nucleate through a diffusional process, for bainite growth proceeds with a full supersaturation controlled by interface friction and for Widmanstätten ferrite, it proceeds as a diffusional process. The criterion is determined by the available driving force and are given |∆G�→��| � ∆G� and |∆G�→�| �h�t-+�+q+-��t.�Osqs�+/gt� for Widmanstättten and bainite respectively. A test was done on a Fe-C-Si-Ni steel and the result gave an estimated value of 400J/mol for the stored energy of bainite.

Figure 26. Thermodynamic diffusionless criterion for bainite formation [69].


36

In Paper IV a comparison was made between the start temperature according to the above criterion first with the Thermo-Calc TCFE8 database [70] and later using software from Peet and Bhadeshia [71]. The critical conditions are illustrated for the 5 ternary systems in Fig 27.

Figure 27. Effect of the alloying elements Mn, Ni, Si, Mo, Cr on the critical driving force

for a diffusionless process calculated with ThermoCalc TCFE8 database.

According to the diffusionless criterion, the driving forces for formation of bainite in Fe-C is negative. Addition of the alloying elements causes a positive increase and driving forces are eventually attained when the alloy content is sufficiently large. The results therefore indicate that the formation of bainite is not thermodynamically feasible. In view of this result, further calculations were performed with the software by Peet and Bhadeshia [71] for FT0, the driving force calculated according to Bhadeshias’ software. Predictions by both databases for alloys with 0.1C, 0.5C, 1.0C and 1.2C (in mass pct.) were

300 400 500 600 700 800 900-3000

-2000

-1000

0

1000

2000

DF

Ncrit (J

/mol)

Temperature (°C)

6wt%Mn

WBo

s

300 400 500 600 700 800 900-3000

-2000

-1000

0

1000

2000

DF

Ncrit (J

/mol)

Temperature (°C)

10wt%Ni

WBo

s

300 400 500 600 700 800 900-3000

-2000

-1000

0

1000

2000

DF

Ncrit (J

/mol)

Temperature (°C)

6wt%Si

WBo

s

300 400 500 600 700 800 900-3000

-2000

-1000

0

1000

2000

DF

Ncrit (J

/mol)

Temperature (°C)

5wt%Mo

WBo

s

300 400 500 600 700 800 900-3000

-2000

-1000

0

1000

2000

DF

Ncrit (

J/m

ol)

Temperature (°C)

6wt%Cr

WBo

s


37

compared in Fig. 28. Temperatures calculated according to FT0 (solid red lines) extrapolate to those calculated according to DFNcrit on the kln� line, indicating that the databases are essentially the same. According to Bhadeshias’ proposal [60] the limit for bainite formation is at 400J/mol, the dashed line, and the critical temperatures are shown with the plus symbols. This assessment precludes any alloying element effect despite that it was tested on a Mn and Si containing alloy. In light of this, the difference in calculated Bs temperatures is shown with superimposed lines for ternary Fe-C-Mn where the critical temperatures according to FT0 clearly fall in a region with appreciable Mn contents. This affirms the need to consider the effect of alloying elements separately from that of carbon.

Figure 28. The driving force for a diffusionless process FT0 (red solid lines) for binary

alloys with 0.1, 0.5, 1.0, 1.2 C compared with the kln� line (dashed). DFNcrit for Fe-C-

Mn alloys are shown (solid lines). Bs temperatures are indicated by the plus symbols

calculated at 400J/mol and solid circles.

The analysis of alloying elements indicates that they exert an appreciable effect on driving forces and Fig. 28 particularly demonstrates the possible error that is a result of neglecting the said effect. Furthermore, the negative critical driving forces do not support the formation of bainitic ferrite as a diffusional process.

200 300 400 500 600 700 800 900-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

-F

T0 (

J/m

ol)

DF

Ncrit(J/m

ol)

Temperature (oC)

WBo

s

8Mn

0.1C0.5C

1.0C1.2C 400J/mol


38

4.4. Analytical model

The same information on dependence of DFDcrit on carbon content can be obtained by fitting of an expression to the lines in Figs. 21 to 25, the procedure is discussed in Paper IV. A description of the start temperatures with a composition dependence, may prove useful in understanding the physical conditions of the transformation although that was not the intention of the present work at this stage. The following expression was obtained

9y9� {� = �3 + l3U3 + :3U36 + ∑ (��U3U�{ { + l��U�� + :��U��6 ) (30)

Where the coefficients for carbon are given by Ac, Bc and Cc and those for alloying elements by ��l�� and :�� . The effect of alloying elements, U{ is

additive. Table 1 shows the alloying element coefficients obtained. Although several types of expressions were attempted, the final choice was based on a decision to limit the number of fitting parameters.

Table 1. Coefficients in Equation 30.

Alloy element A(J/mol) B (J/mol) C (J/mol)

C 70 -4560 178900

Cr 460000 7200 300000

Mn 482000 3600 300000

Mo 744000 31500 890000

Ni 219000 -1600 46000

Si 221000 6550 68600

For an interactive effect between alloying elements, higher order systems should be analyzed. The model is based on the empirical representation of experimental data and the coefficients defined here have no more physical significance than do those of the empirical expression. The alternative would be to perform the optimization and fitting directly on experimental data. Eq. 30 offers the advantage of a direct and relatively simple description of DFDcrit

without the need to access databases. A typical example of the fit is shown by the lines in Figure 29 comparing the result calculated with Eq. 30, the dashed lines, with that calculated with Eq. 17, the solid lines. There is good agreement at low compositions but the difference becomes more prominent as the C and Mn content is increased.


39

Figure 29. DFDcrit for a diffusional process for Fe-C-Mn up to 6 wt% Mn comparing the

dashed lines calculated with Eq. 30 with the empirical representation of data, solid

lines calculated with Eq. 17.

4.5. Effect of barrier on growth modeling

It has thus far been shown how the bainite start conditions can be expressed, as a function of temperature or composition, achieved through an empirical representation and an analytical representation using a thermodynamic variable. This has implications on the modeling of bainite and particularly the use of a critical driving force. The driving force is explicitly expressed in the growth model through the capillary effect and is the total driving force for precipitation of ferrite from austenite. By applying the critical driving forces evaluated for the start of the transformation, the model for growth rates can now be expressed as

� = (∆K��)6��

� �� P�P��

HIJH (31)

Here B represents the driving force DFD and is only DFDcrit provided it is the start of the transformation which is under consideration, at which point the velocity is zero. Its effect on the supersaturation can be schematically demonstrated with Gibbs free energy curves, see Fig. 30.

0.0 0.5 1.0 1.5 2.00

500

1000

1500

2000

2500

3000Fe-C-Mn

DF

Dcri

t (J/m

ol)

Carbon (wt%)


40

Figure 30. Effect of Gibbs free energy addition, B, on the interfacial carbon content.

The Gibbs free energy of ferrite is increased by the amount B and the total driving force reduced by the same amount. Consequently the carbon content at the interface is changed, directly affecting the velocity calculation. The effect is demonstrated in Fig. 31.

Figure 31. Growth rates calculated without a barrier (dashed curve) and with a

barrier (solid curve). Experimental rates of an Fe-1.43C alloy [33].

Gib

bs F

ree E

nerg

y

Carbon

γ α

��/� ��/�

B

γ

B

Chapter 5 Model for bainite formation

41

Chapter 5

Bainite formation model

The culmination of this work combines the three fundamental aspects of the start temperatures, critical driving forces and the growth model. The main result will be prediction of bainite C-curves, see for example Fig. 31. As seen previously, the start temperature can be directly obtained from the critical driving force but it remains to be seen how to obtain a driving force below the start. One of the previous attempts to do so was by fitting to the experimental data, as can be seen in Paper I, where an average value of the barrier was obtained for each alloy. By comparing the C-curve obtained without the barrier and that with the barrier, one can get an impression of the size of the required barrier. However, this was an initial attempt and after further attempts, it became clear that a more rigorous treatment is needed.

5.1. Energy dissipation during growth

A final refinement of the model was done by making adjustments to the growth rate model. The first deals with the driving force ∆G�. It has already been mentioned in Chapter 2 that the capillarity effect requires a driving force as given by M�L �� ⁄ . Another energy requirement of the transformation was dealt with in Chapter 4, namely that of the critical driving force. It is introduced into the growth rate model in Eq. 31 as a barrier B. Together with the critical driving force at the kln temperature, the overall energy requirement is given by the sum of the capillarity effect and the critical barrier, l�� as given by Eq. 30.

∆G� = M�L �� ⁄ + l�� (32)

The available driving force is thus consumed by these two processes and is determined by the carbon contents in the austenite and ferrite.

The second adjustment is for the barrier. At the critical point, it corresponds to the value of energy to be surmounted before the transformation proceeds. It is now proposed that the carbon diffusion inside the interface causes dissipation of Gibbs energy which alters the value of the barrier, see Eq. 33. Taking into account the flux of carbon over the austenite-ferrite interface, a general expression for the barrier is then given as


42

l = l�� + ��C~

��~ /��~

�/

��

6 (33)

l = �3 + l3U3 + :3U36 + ∑ (��U3U�{ { + l��U�� + :��U��6 ) +

��C~

��~ /��~

�/

��

6 (34)

The total barrier B is a sum of the energy loss due to flux of carbon over the interface and the barrier at the critical conditions. At the kln temperature, the velocity � is 0, the second term of Eq. 33 vanishes and the expression reduces to Eq. 30. The available driving force below the kln temperature is balanced against the energy consumptions and the final equation is,

∆G� = M�L �⁄ + �3 + l3U3 + :3U36 + ∑ (��U3U�{ { + l��U�� + :��U��6 ) +

��C~

��~ /��~

�/

��

6 (35)

5.2. Optimized parameters

The equation for velocity used is the modified Zener-Hillert expression Eq. 31 and is optimized together with Eq. 35 using the experimental information from Fe-C alloys. The growth rate information used is given in Table 2. Table 2. Chemical composition of alloys used in the optimization.

Alloy Ref. C Mn Si

Yin et al. [41] 0,5 0,003 0,031

Yin et al. [41] 0,7 0,003 0,033

Hawkins [34] 0,9 0 0

Simonen [26] 0,24 0 0

Simonen [26] 0,33 0 0

Simonen [26] 0,43 0 0

Speich Cohen [33] 0,96 0 0

Speich Cohen [33] 1.16 0 0

Speich Cohen [33] 1.43 0 0


43

Optimization procedures were performed for the parameters, L , N3 and 9. It was necessary to optimize the diffusivity of carbon, as it is better established at high temperatures above 1000K according to the available experimental data [31]. A correction term was thus introduced for the lower temperature extrapolations during the optimization. In Chapter 2 the diffusivity of carbon was calculated using a weighted average composition. It is now calculated as a fitting parameter optimized according to experimental data. The surface energy term L�� has been previously evaluated as 0,23J/mol [9] and is now evaluated as 0,098J/mol.

5.3. Prediction of C-curves Some of the data used during optimization is shown in Fig. 32 together with the fitted curves. The overall predictions are in good agreement with experimental data. There is a general tendency for low carbon growth rate data to exhibit linear behavior at the lower temperature range, which is not captured with the present model.

Figure 32. C-curves fitted to experimental data from which the parameters were

optimized.

The high carbon data is reasonably well represented by the slope of the predicted curves although a lower growth rate is required. It would seem an even larger correction factor for the diffusivity is needed to fit this data. However, this is highly subjective to the data selected in the optimization procedure and a comparison with a several different alloys is needed to reach some justifiable value, taking into account experimental errors. The low


44

carbon growth rate data has a linear dependence and cannot be represented with C-curves.

Figure 33. Experimental growth rate data compared with calculated growth rate

curves for Fe-C alloys. The low carbon alloys [26] are at the higher temperatures and

the high carbon alloys [33] are at lower temperatures.

A comparison of the experimental growth rates with those calculated according to the present approach is shown in Fig. 33 for Fe-C-Mn-Si alloys [9][41]. There is good agreement with the predictions although the discrepancy is large for some alloys at lower temperatures which may be due to experimental uncertainty though it was observed in the low carbon data of Fig. 32. This feature may also be due to carbon supersaturation in the ferrite and was investigated in the work of Paper II. The overall result, however, is satisfactory.

5.4. Software package

All thermodynamic calculations in this work were performed with Thermo-Calc software and its TCFE8 database. A software package was developed specifically for this model capable of calculating DFDcrit, Ccr or Tcrit using the formalism presented in Chapter 4. Calculations can be performed according to either a diffusional or a diffusionless transformation and the input data for the barriers or alloying element coefficients is editable. Growth rate calculations are executed according to the procedures described in this thesis.

Chapter 6 Concluding remarks and future outlook

45

Chapter 6

Concluding remarks and future outlook

6.1. Concluding remarks

The objective of this thesis has been to formulate a model capable of predicting the evolution of bainite. There are three fundamental aspects of the study. Analysis of carefully selected experimental data gave empirical relations for predicting the critical temperature. The empirical results were used throughout the work as being representative of experimental data. The relationship between WBs temperatures and alloy composition was described as linear with no allowance for interaction effects. Deviation of experimental information from this behavior could be a direct consequence of the two aforementioned conditions although it is more probable that the quality of data was low. The result thus could greatly benefit from more systematic experiments and particularly more on binary Fe-C alloys.

A thermodynamic model was described by using critical values generated from the derived expression to calculate the critical driving force for that composition. The critical conditions can be sufficiently described by temperature, carbon content or driving force. The latter two were effective means of describing the critical or WBs temperature which could be predicted reasonably well with an analytical expression.

In the final stage, a consolidated model for bainite formation utilized all the aspects developed throughout the progress of this work. An additional feature for the carbon flux at the interface was included in an energy balance approach. The model was verified with experimental growth rates and the predicted growth rates were within reasonable agreement.

6.2. Future Outlook

The proposition that there exists a barrier to the formation of bainite was explored in the contents of this study. There remains a need to understand the nature of this barrier and its physical attributes.

The entire study is based on the formation of bainitic ferrite excluding carbide formation. A more comprehensive approach requires adjustments

Chapter 6 Concluding remarks and future outlook

46

allowing for carbide formation particularly in the growth rate model since it is known that carbides act as carbon sinks.

The overall result of the present work is a generation of C-curves for growth rates which lays a foundation for development of TTT diagrams. However, a complete description involves growth and nucleation, the latter is not included in this thesis. Since it is not quite possible to directly observe nucleation experimentally, a model could be developed to meet this need. Along the same lines, a study of the evolution of the volume fraction could be developed.

47

Acknowledgements

‘The journey of a thousand miles begins with a single step’ -Lao Tzu.

As I now take the final stride of this journey, I would like to extend my deepest gratitude to the many who have supported me along the way.

I would like to express my sincere gratitude to my supervisor, Prof. Annika Borgenstam, your guidance and support has been invaluable to the success

of this work. To my co-supervisor Prof. John Ågren, thank you for your advice and help. My experience as your student has been nothing less than inspirational and your contribution to this work is highly appreciated. To

my co-supervisor Dr. Lars Höglund, I am grateful for your assuring support, patience and expertise with all the computational complexities that came along with developing this model. I would like to acknowledge Prof. Mats

Hillert for the discussions and helpful suggestions, your interest and contribution to this work is much appreciated.

To my counterpart in the bainite project, Jiaqing, thank you for the fruitful discussions and collaboration. To my colleagues and friends, Thomas, Xin,

David Dilner, Bartek, David Linder, Hossein, Fei, Sedi, Arash, Simon, Raquel, Manon, Martin, to mention a few, the many fika sessions and the shared

companionship has been a great complement to the trying times, thank you all. Dr. Knippenberg, thank you for believing in me. And to my parents, your

love and support reaches from a thousand miles away.

This project has been conducted in the VINN Excellence center Hero-m and Competence center Hero-m2 Innovation at KTH Royal Institute of

Technology. I would like to acknowledge the financial support from VINNOVA, the Swedish Governmental Agency for Innovation Systems and

Swedish industry. Funding for conference visits by the Foundation for Applied Thermodynamics, STT is kindly acknowledged.

Stockholm, 2018

48

Bibliography

[1] https://www.worldsteel.org.

[2] J.O. Andersson, T. Helander, L. Höglund, P. Shi, B. Sundman, Calphad Comput. Coupling Phase Diagrams Thermochem. 26 (2002) 273–312.

[3] A.G.B. Olson, 277 (2016) 1237–1242.

[4] A. Hultgren, J. Iron Steel Inst. 114 (1926) 421–422.

[5] E.S. Davenport, E.C. Bain, Trans. AIME 90 (1930) 117–154.

[6] J. Yin, M. Hillert, A. Borgenstam, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 48 (2017) 1425–1443.

[7] A. Hultgren, Trans. Am. Soc. Mater. 39 (1947) 915–989.

[8] G.R. Purdy, M. Hillert, Acta Metall. 32 (1984) 823–828.

[9] The Growth of Ferrite Bainite and Martensite, Stockholm, 1960.

[10] J. Oblak, R.F. Hehemann, Structure and Growth of Widmanstätten Ferrite and Bainite, 1967.

[11] L.C.D. Fielding, Mater. Sci. Technol. 29 (2013) 383–399.

[12] J. Robertson, J. Iron Steel Inst. 119 (1929) 391–424.

[13] Z. C, Trans Aime 167 (1946) 1689–1699.

[14] T. Ko, S.A. Cottrell, J. Iron Steel Inst. 172 (1952) 307.

[15] G.I. Rees, H.K.D.H. Bhadeshia, Mater. Sci. Technol. 8 (1992) 985–993.

[16] D. Gaude-Fugarolas, P.J. Jacques, Solid-Solid Phase Transform. Inorg. Mater. 2005, Vol 2 (2005) 795–800.

[17] E. Eichen, H.I. Aaronson, G. Pound, R. Trivedi, Acta Metall. 12 (1964) 1298–1301.

[18] M. Nemoto, in:, P.R. et al Swann (Ed.), High Volt. Electron Microsc., Academic Press, New York, 1974, pp. 230–234.

[19] X.L. Wan, R. Wei, L. Cheng, M. Enomoto, Y. Adachi, J. Mater. Sci. 48 (2013) 4345–4355.

[20] Z.W. Hu, G. Xu, H.J. Hu, L. Wang, Z.L. Xue, Int. J. Miner. Metall. Mater. 21 (2014) 371–378.

[21] P. Kolmskog, A. Borgenstam, M. Hillert, P. Hedström, S.S. Babu, H. Terasaki, Y.I. Komizo, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 43 (2012) 4984–4988.

49

[22] D. Quidort, Y.J.M. Brechet, Acta Mater. 49 (2001) 4161–4170.

[23] M. Hillert, The Role of Interfacial Energy during Solid State Phase Transformations, 1957.

[24] R. Trivedi, Metall. Trans. 1 (1970) 921–927.

[25] R. Trivedi, G.M. Pound, J. Appl. Phys. 40 (1969) 4293–4300.

[26] E.P. Simonen, H.I. Aaronson, R. Trivedi, Metall. Trans. 4 (1973) 1239–1245.

[27] W.P. Bosze, R. Trivedi, Acta Metall. 23 (1975) 713–722.

[28] S.X. Zhao, W. Wang, D.L. Mao, Mater. Sci. Forum 539–543 (2007) 3018–3023.

[29] N.V. Luzginova, L. Zhao, J. Sietsma, Mater. Sci. Eng. A 481–482 (2008) 766–769.

[30] M. Hillert, L. Höglund, J. Ågren, Acta Mater. 51 (2003) 2089–2095.

[31] J. Ågren, Scr. Metall. 20 (1986) 1507–1510.

[32] R. Trivedi, G.M. Pound, J. Appl. Phys. 38 (1967) 3569–3576.

[33] G.R. Speich, M. Cohen, Tms-Aime 218 (1960) 1050–59.

[34] A. Hawkins, J. Barford, J. Iron Steel Inst. 210 (1972) 97–105.

[35] H. Yada, T. Ooka, Jap. Inst. Met. 31 (1967) 766–771.

[36] R.H. Goodenow, S.J. Matas, R.F. Hehemann, Trans. AIME 227 (1963) 651–658.

[37] J. Kudrman, R. Freiwillig, Prakt. Metallogr. 7 (1970) 386–393.

[38] K. Tsuya, T. Mitsuhashi, J. Mech. Lab. Japan 1 (1955) 42–48.

[39] L. Cheng, K.M. Wu, X.L. Wan, R. Wei, Mater. Charact. 87 (2014) 86–94.

[40] M.M. Rao, P.G. Winchell, 239 (1967).

[41] J. Yin, L. Leach, M. Hillert, A. Borgenstam, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 48 (2017) 3997–4005.

[42] M. Strangwood, K. Chi, in:, Proc. Int. Conf. Solid-Solid Phase Transform. Inorg. Mater., 2005, pp. 93–98.

[43] Kaufman, in:, V.F. Zackay, H.I. Aaronson (Eds.), Decompos. Austenite by Diffus. Process., New York, 1962, pp. 313–352.

[44] R.L. Bodnar, T. Ohhashi, R.I. Jaffee, Metall. Mater. Trans. A 20 (1989) 1445–1460.

[45] Z. Zhao, C. Liu, Y. Liu, D.O. Northwood, J. Mater. Sci. 36 (2001) 5045–5056.

50

[46] S. Kang, S. Yoon, S.-J. Lee, ISIJ Int. 54 (2014) 997–999.

[47] J. Trzaska, Arch. Metall. Mater. 60 (2015) 182–185.

[48] S.M.C. van Bohemen, Mater. Sci. Technol. 28 (2012) 487–495.

[49] C. Garcia-Mateo, T. Sourmail, F.G. Caballero, C. Capdevila, C. García de Andrés, Mater. Sci. Technol. 21 (2005) 934–940.

[50] W. Steven, A.G. Haynes, J. Iron Steel Inst. August (1956) 349–59.

[51] H.I. Aaronson, Decomposition of Austentie by Diffusional Processes, Interscience, New York, 1962.

[52] J.S. Kirkaldy, D. Venugopalan, Phase Transformations in Ferrous Alloys, TMS-AIME, Warrendale, PA., 1984.

[53] M. Suehiro, T. Senuma, H. Yada, Y. Matsumura, T. Ariyoshi, Tetsu-To-Hagane/Journal Iron Steel Inst. Japan 73 (1987) 1026.

[54] T. Kunatike, Y. Okada, J. Iron Steel Inst. 84 (1998) 137.

[55] S.E. Johanson, Thesis, KTH, Stockholm, 1959.

[56] H.I. Aaronson, P.G. Boswell, K.R. Kinsman, in:, E.L. Parker (Ed.), Mech. Prop. Phase Transform. Eng. Mater., Symposium on Structure Property Relationships TMS-AIME, Warrendale, PA., 1986, pp. 467–473.

[57] G. Papadimitriou, J.M.R. Genin, Mater. Res. Soc. 21 (1983) 747–774.

[58] L.M. Pevzner, T.D. Kubyushina, G.M. Rovenskii, A.I. Samoilov, Sov. Metallogr. Treat. Met. 11 (1956) 2–20.

[59] M. Hillert, L. Höglund, J. Ågren, Metall. Mater. Trans. A 35 (2004) 3693–3700.

[60] H.K.D.H. Bhadeshia, D.V. Edmonds, Acta Metall. 28 (1980) 1265–1273.

[61] H.K.D.H. Bhadeshia, Acta Metall. 29 (1981) 1117–1130.

[62] Y. Xia, G. Miyamoto, Z.G. Yang, C. Zhang, T. Furuhara, Acta Mater. 91 (2015) 10–18.

[63] Z.Q. Liu, G. Miyamoto, Z.G. Yang, C. Zhang, T. Furuhara, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 46 (2015) 1544–1549.

[64] T. Furuhara, K. Tsuzumi, G. Miyamoto, T. Amino, G. Shigesato, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 45 (2014) 5990–5996.

[65] H.D. Wu, G. Miyamoto, Z.G. Yang, C. Zhang, H. Chen, T. Furuhara, Acta Mater. 133 (2017) 1–9.

[66] H. Chen, K. Zhu, L. Zhao, S. Van Der Zwaag, Acta Mater. 61 (2013) 5458–5468.

51

[67] F.G. Caballero, M.K. Miller, S.S. Babu, C. Garcia-Mateo, Acta Mater. 55 (2007) 381–390.

[68] F.G. Caballero, M.J. Santofimia, C. García-Mateo, J. Chao, C.G. de Andrés, Mater. Des. 30 (2009) 2077–2083.

[69] H.K.D.H. Bhadeshia, Acta Metall. 29 (1981) 1117–1130.

[70] http://www.thermocalc.com/products-services/databases/.

[71] https://www.phase-trans.msm.cam.ac.uk/map/steel/programs/mucg83.html#exam, M. Peet, H.K.D.H. Bhadeshia.

Modeling Bainite Formation in Steels - DiVA portal1205497/FULLTEXT01.pdf · Bainite is a phase...

Documents

Transcript of Modeling Bainite Formation in Steels - DiVA portal1205497/FULLTEXT01.pdf · Bainite is a phase...