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An Investigation of the Suitability of Using AISI 1117 Carbon Steel in a Quench and Self-Tempering Process to

Satisfy ASTM A 706 Standard of Rebar

by

Matthew Allen

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Materials Science and Engineering University of Toronto

Copyright by Matthew Allen, 2011

ii

An Investigation of the Suitability of Using AISI 1117 Carbon Steel

in a Quench and Self-Tempering Process to Satisfy ASTM A 706

Standard of Rebar

Matthew Allen

Master of Applied Science

Graduate Department of Materials Science and Engineering University of Toronto

2011

Abstract

Experiments were conducted to investigate the potential of using a quench and self-tempering

heat treatment process with AISI 1117 steel to satisfy the mechanical properties of ASTM A 706

rebar. A series of quenching tests were performed and the resulting microstructure and

mechanical properties studied using optical microscopy, microhardness measurement, and tensile

tests. The presence of martensite throughout the samples contributed to the enhanced strength

and strain-hardening ratio (tensile to yield strength) of the material. The experimental results

showed that AISI 1117 is capable of meeting the ASTM standard. In addition to the

experiments, a computer model using the finite difference method and incorporating heat transfer

and microstructure evolution was developed to assist in future optimization of the heat treatment

process.

iii

Acknowledgments First, I would like to thank my supervisor Professor Zhirui Wang for his invaluable guidance and

technical insight throughout this endeavor. I would also like to extend my sincere appreciation

for his continuous encouragement and reinforcement from which I found a reliable source of

motivation.

I would also like to convey my thanks to Anand and Jeff Persad from A&C Tool. It was through

their generosity that I was able to perform my experiments. As always, it was a privilege and a

pleasure to work with them.

Special thanks also go to Don Allen from BLM. His support and assistance helped to make my

research possible.

Last but certainly not least, I would like to thank my wife Amy and family Charles, Darlene,

Jack, Bill, and Quinn for their unconditional love and support. I would especially like to thank

my wife Amy for her strength throughout this challenging time and for all the sacrifices she

made to make this endeavor possible. I am truly fortunate to have such an amazing partner in

life.

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Table of Contents

1 Introduction ................................................................................................................................ 1

1.1 Mechanical Properties of Rebar Overview ......................................................................... 1

1.2 Chemical Composition of Rebar Overview ........................................................................ 1

1.3 Rebar Production Methods ................................................................................................. 2

1.3.1 Work-Hardening ..................................................................................................... 2

1.3.2 Microalloying .......................................................................................................... 2

1.3.3 Quench and Self-Tempering (QST) ........................................................................ 3

1.4 Overview of Rebar Standards ............................................................................................. 4

1.4.1 United States Rebar Standards ................................................................................ 6

1.4.2 Canadian Rebar Standards ...................................................................................... 6

1.4.3 Australian/New Zealand Rebar Standards .............................................................. 7

1.4.4 Japan Rebar Standards ............................................................................................ 7

1.4.5 German/European Rebar Standards ........................................................................ 7

2 Objectives ................................................................................................................................... 7

3 Quench and Self-Tempering Model ........................................................................................... 8

3.1 Heat Transfer Problem ........................................................................................................ 8

3.2 Finite Difference Method .................................................................................................. 10

3.2.1 Discretization ........................................................................................................ 11

3.2.2 Implicit Form of Finite Difference Equations ...................................................... 12

3.2.3 Tridiagonal Matrix ................................................................................................ 13

3.3 Microstructure Evolution .................................................................................................. 13

3.3.1 Diffusional Transformation .................................................................................. 14

3.3.2 Diffusionless Transformation: .............................................................................. 20

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4 Computer Model ...................................................................................................................... 21

4.1 Model Overview ............................................................................................................... 22

4.2 Input Parameters ............................................................................................................... 22

4.2.1 General Parameters ............................................................................................... 23

4.2.2 Model Parameters ................................................................................................. 23

4.2.3 Material Parameters .............................................................................................. 23

4.2.4 Isothermal Data ..................................................................................................... 24

4.3 Temperature Field ............................................................................................................. 24

4.4 Phase Formation ................................................................................................................ 26

4.4.1 Applying Scheils Additivity Principle ................................................................. 26

4.4.2 Applying the Avrami Equation: The Growth Function ........................................ 28

4.4.3 Incorporating Transformation of Multiple Phases ................................................ 30

4.5 Latent Heat Generation ..................................................................................................... 31

5 Materials and Experimental Procedure .................................................................................... 32

5.1 Materials ........................................................................................................................... 32

5.2 AISI 1117, ASTM A 706, and Microalloyed Comparison ............................................... 33

5.3 Microstructure Characterization: Optical Microscopy ..................................................... 34

5.4 Mechanical Property Characterization .............................................................................. 36

5.4.1 Microhardness ....................................................................................................... 36

5.4.2 Tensile Testing ...................................................................................................... 36

5.5 Heat Treating .................................................................................................................... 38

6 Results and Discussion ............................................................................................................. 43

6.1 Computer Model Results .................................................................................................. 43

6.1.1 Model Issues ......................................................................................................... 43

6.1.2 Comparison of Model with Experimental Results ................................................ 45

6.2 Experimental Results ........................................................................................................ 47

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6.2.1 Hardness Tests ...................................................................................................... 47

6.2.2 Microstructure ....................................................................................................... 49

6.2.3 Tensile Test Results .............................................................................................. 53

7 Conclusions .............................................................................................................................. 60

7.1 Computer Model Conclusions .......................................................................................... 60

7.2 Experimental Conclusions ................................................................................................ 60

8 Future Work ............................................................................................................................. 62

9 References ................................................................................................................................ 63

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List of Tables

Table 1.1 Mechanical Requirements of World Rebar Standards .................................................... 5

Table 1.2 Chemical Requirements of World Rebar Standards ....................................................... 6

Table 4.1 Computer Model Input Parameters ............................................................................... 23

Table 5.1 Chemical Composition of 3/8" Dia. AISI 1117 ............................................................ 33

Table 5.2 Chemical Composition of 1" Dia. AISI 1117 ............................................................... 33

Table 5.3 AISI 1117, ASTM A 706, Microalloyed Chemical Composition Comparison ............ 34

Table 5.4 Tensile specimen dimensions ....................................................................................... 37

Table 5.5 Experiment 1 heat treatment details .............................................................................. 39





Table 6.1 Inputs for model and experimental comparison ........................................................... 46

Table 6.2 AISI 1117 heat treatment Exp. 4 first sample batch mechanical properties ................. 54

Table 6.3 AISI 1117 heat treatment Exp. 4 second sample batch mechanical properties ............ 56

Table 6.4 AISI 1117 heat treatment Exp. 5: 3/8" bar stock mechanical properties ...................... 57

Table 6.5 1117 heat treatment Exp. 5: 1" bar stock first batch mechanical properties ................. 59

Table 6.6 1117 heat treatment Exp. 5: 1" bar stock second batch mechanical properties ............ 60

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Table 7.1 Comparison of ASTM A 706 requirements and experimental results .......................... 61

ix

List of Figures

Figure 3.1 Overview of QST Model ............................................................................................... 8

Figure 3.2One-dimensional heat transfer in infinite cylinder ......................................................... 9

Figure 3.3 Discretization of Cylinder ........................................................................................... 12

Figure 3.4: Example of Sigmoidal Form of Phase Transformation .............................................. 15

Figure 3.5 Approximating a Cooling Curve Using Isothermal Steps ........................................... 15

Figure 3.6 Combining Transformation-Time Curve Segments: Additivity Rule ......................... 17

Figure 4.1 QST Computer Model GUI ......................................................................................... 21

Figure 4.2 High-Level Flow Diagram of Computer Model .......................................................... 22

Figure 5.2 As received AISI 1117 perpendicular to rolling direction .......................................... 35

Figure 5.3 As received AISI 1117 parallel to rolling direction .................................................... 35

Figure 5.5 Microhardness measurement arrangement .................................................................. 36

Figure 5.6 Tensile specimen ......................................................................................................... 37

Figure 5.7 Tensile Test Machine and Extensometer ..................................................................... 38

Figure 6.1 Example of Fe-Fe3C Phase Diagram ........................................................................... 44

Figure 6.2 Example TTT curve ..................................................................................................... 45

Figure 6.3 Experimental results .................................................................................................... 47

Figure 6.4 Model results ............................................................................................................... 47

Figure 6.5 Vickers hardness profile from Exp. 1 .......................................................................... 48

x

Figure 6.6 Vickers hardness profile from Exp. 2 & 3 ................................................................... 49

Figure 6.7: AISI 1117 1 dia. bar austenized and air cooled ........................................................ 50

Figure 6.8 AISI 1117 1 bar quenched: centre ............................................................................. 50

Figure 6.9 AISI 1117 1" bar quenched: surface ............................................................................ 50

Figure 6.10 A2 micrograph: air cooled ......................................................................................... 52

Figure 6.11 Q5 micrograph: 3 second quench .............................................................................. 52

Figure 6.12 Q3 micrograph: full quench ....................................................................................... 52

Figure 6.13 Q4 micrograph: full quench ....................................................................................... 52

Figure 6.14 Q6 micrograph: 30 second quench ............................................................................ 52

Figure 6.15 Q7 micrograph: 15 second quench ............................................................................ 52

Figure 6.16 AISI 1117 engineering stess-strain curves from Exp. 4: 1st sample batch ............... 54

Figure 6.17 AISI 1117 engineering stress-strain curves from Exp. 4: 2nd sample batch ............. 55

Figure 6.18 AISI 1117 engineering stess-strain curves from Exp. 5: 3/8" bar stock .................... 57

Figure 6.19 AISI 1117 1" bar stock: sample A1 ........................................................................... 58

Figure 6.20AISI 1117 3/8" bar stock: sample A2 ......................................................................... 58

Figure 6.21 AISI 1117 1" bar stock mechanical properties comparison ...................................... 59

xi

List of Appendices

Appendix 1: Mill Report AISI 1171 1 ..................................................................................... 66

Appendix 2: Mill Report AISI 1171 3/8 .................................................................................. 67

Appendix 3: Example Mill Report from US Rebar Producer .................................................... 68

Appendix 4: Computer Program - Main Function ..................................................................... 69

Appendix 5: Ferrite Growth Function ....................................................................................... 94

Appendix 6: Max Ferrite Function ............................................................................................ 95

Appendix 7: Pearlite Growth Function ...................................................................................... 96

Appendix 8: Martensite Growth Function ................................................................................. 97

Appendix 9: DeltaPhase Function ............................................................................................. 98

Appendix 10: Carbon Equivalent Equations ............................................................................... 99

1

1 Introduction Around the world, concrete structures are reinforced by deformed steel bars, or rebar. The

design requirements of these structures are typically defined by building codes based on regional

structural practices and environmental demands. In turn, these codes often reference some

standard of rebar that define such characteristics as, dimensions, mechanical properties, and

chemical composition. The following sections provide a brief overview of these characteristics

including commentary on some of the variations between different world rebar standards.

1.1 Mechanical Properties of Rebar Overview Strength and ductility are critical mechanical properties defined in rebar standards. Primarily,

rebar is classified or graded by its yield strength. For instance, the rebar standard in Canada,

CSA-G30.18, defines two minimum yield strength levels, 400 and 500 MPa [1]. Many standards

also reference a minimum tensile strength level either directly or indirectly. The Japanese rebar

standard, JIS G 3112, states a minimum tensile strength level while the Australian/New Zealand

rebar standard, AS/NZS 4671, defines tensile strength indirectly by specifying a minimum strain

hardening ratio in reference to the yield strength (i.e. ratio of tensile strength to yield strength)

[2, 3]. Ductility is commonly defined quantitatively by % elongation to fracture or the ratio of

tensile to yield strength, and qualitatively by a bendibility test. For example, the US rebar

standard, ASTM A 615, requires a minimum percent elongation in an eight inch tensile test

sample which varies according to bar diameter and strength level [4]. Similarly, bend tests

which specify varying sizes of mandrels and degrees of bend, require that cracks do not form on

the bent surface of the bar.

1.2 Chemical Composition of Rebar Overview There are two general forms of carbon steel rebar that differ mainly by chemical composition:

weldable and non-weldable. This distinction is based on the maximum carbon and carbon

equivalent levels. A clear distinction is made between these forms under the US standards for

rebar. For example, ASTM A 615, which makes no provisions for weldability, does not include

a maximum carbon or carbon equivalent requirement. In comparison, ASTM A 705, defines

specific carbon and carbon equivalent levels in order to enhance the materials weldability [5]. It

2

should be noted that the method of calculating the carbon equivalent differs slightly from

standard to standard, with some including more alloy elements than others.

1.3 Rebar Production Methods The general method of producing rebar involves forming the steel through a series of

successively smaller grooved rolls. However, apart from a common method of obtaining the

basic shape, there are three methods used to achieve the required mechanical properties. These

include:

Work-hardening

Microalloying

Quench and Self-Tempering

1.3.1 Work-Hardening

Work-hardening is a process that involves twisting the rebar after it has cooled (also known as

cold-working). By plastically deforming the rebar, a higher level of yield strength is achieved.

Unfortunately, there are several drawbacks to this process. Given that the rebar has been

permanently strained, the ductility is reduced. Additionally, because this process is performed

off-line from the rolling mill, additional equipment and labor is required. The combination of

poor mechanical properties and increased manufacturing costs, make this alternative an

unpopular process.

1.3.2 Microalloying

As the name implies, the microalloying process uses small additions of certain alloy elements,

typically vanadium or niobium, to increase the strength of the rebar. The microalloying elements

have a threefold influence on the mechanical properties of the steel including:

1. Grain size refinement during thermo-mechanical hot forming.

2. Lowering the austenite to ferrite transition temperature (reduces grain growth rate).

3. Precipitation hardening (impediment to dislocation movement).

3

This process achieves the desired mechanical properties of strength and ductility; however, the

expensive alloy additions significantly raise the manufacturing costs. Microalloyed rebar is

typically sold with a $40 to $60 USD/ton premium over regular (non-weldable) rebar [6]. This is

the most commonly used process for producing ASTM A 706 rebar in North America.

1.3.3 Quench and Self-Tempering (QST)

Quench and self-tempering is another production method used to increase the strength of rebar

while limiting the level of carbon and carbon equivalent. In this process, immediately after the

rebar exits the final rolling stand in the mill it travels through a series of quenching devices. The

particular design of the quenching device is unique to the equipment manufacturer, but in

general, high-pressure water is applied against the surface of the bar in order to cool it. In a very

short time, this rigorous quenching reduces the surface temperature of the bar to a point where

martensite develops. After exiting the quenching area, the remaining heat from the core of the

bar conducts outwards and tempers the martensitic surface layer, which helps to increase its

ductility. Following this, the bar continues to air-cool until finally the core transforms into a

mixture of proeutectoid ferrite and pearlite. Due to the variation in microstructure over the

cross-sectional area of the bar, the resulting product can be described as having a composite

microstructure. Similar to how reinforced concrete, a composite material, obtains its final

properties by a combination and interaction of concrete and steel, quench and self-tempered

rebar achieves its beneficial qualities from the combination of the constituent properties of the

strong tempered martensite surface and ductile pearlite/ferrite core. Of the three production

methods discussed, quench and self-tempering is the most cost effective. It requires a moderate

capital investment in new equipment and increase in operating costs; however, it saves on

additional labor costs over the off-line work-hardening method and the expensive alloy additions

used by the microalloying method. Although this process improves many of the mechanical

properties of the rebar, it unfortunately has a negative effect on the strain-hardening ratio.

4

Of the various methods of producing weldable rebar, the quench and self-tempering process is a

predominantly unfamiliar process in North American. Some contributing factors that explain

this unfamiliarity include:

1. The greater strain-hardening ratio requirement for ASTM A 706 rebar

2. The small market size of A 706 rebar

3. The inadequate space and associated cost required to retrofit existing mills

To start, the ASTM A 706 strain-hardening ratio of 1.25 is higher than many other world rebar

standards. Section 1.4 below provides a brief description and comparison of various world

standards. Since the QST process was initially developed in Europe where there is a lower

strain-hardening ratio requirement, the typical chemical composition used does not create any

barrier for achieving this property level. Seeing as this requirement is easily met, there is little

motivation for experimenting with different compositions to try to increase it.

Secondly, it is estimated that A 706 makes up a much smaller, albeit growing portion of the rebar

market [7]. If A 706 makes up only a small portion of a manufactures total production, the

payback period required for such an investment might not justify the prioritization of such a

capital expenditure.

Finally, the ASTM A 706 standard was first published in the 1970s around the same time that the

QST process was first used in Europe. Few rebar mills being built after the introduction of QST,

means that equipment layouts were not originally designed for this technology. This creates

another hurdle for retrofitting existing mills considering the quenching line would have to be

installed between the last mill stand and the cooling bed. Without the necessary space available,

the expenditure to install this technology is greatly increased.

1.4 Overview of Rebar Standards High-strength, low-alloy rebar is normally characterized by restricted carbon and carbon

equivalent (CE) levels, greater ductility requirements, and controlled strength ranges. Many of

5

the differences between rebar standards can be related to the intended purpose and region that

they were developed for, and relate mainly to the weldability and ductility properties of the steel.

Increasing the amount of carbon or other elements like manganese is a relatively cheap and easy

way to increase the strength of steel. Unfortunately, it has a negative effect on ductility and

weldability. Ductility is an important aspect in the design of earthquake resistant structures.

Also important in seismic design is the ability to accurately predict how a structure will behave

under real world conditions. This is motivation for having a tightly controlled strength range (as

opposed to requiring only a minimum level). Weldability on the other hand, is a desirable

characteristic in rebar that allows for efficient prefabrication of reinforcing structures in

workshops and durability during transportation and handling to the construction site. However,

carbon and other elements impair the ability to effectively weld steel. To ensure weldability, a

low level of carbon (< 0.3%) and CE (a weighted average of a group of elements that have a

similar effect as carbon:

6

Table 1.2 Chemical Requirements of World Rebar Standards

Carbon Manganese Phosphorus Sulfur Silicon CE*

US (A 706) 0.30 1.50 0.035 0.045 0.50 0.55

Canada 0.30 1.60 0.035 0.045 0.50 0.55

Australia/NZD 0.22 - 0.050 0.050 - 0.49

Japan 0.32 1.80 0.040 0.040 0.55 0.60

Germany 0.22 - 0.050 0.050 - 0.50

*Note: Method of calculating CE differs between standards and therefore cannot be directly compared in the above table. Refer to Appendix 10 for the list CE equations used in each standard.

1.4.1 United States Rebar Standards

In the United States, the International Building Code (IBC) which is adapted and enacted by state

governments, references American Concrete Institutes standard ACI 318 for the construction of

structural concrete [8]. In turn, ACI 318 references ASTM standards for rebar.

There are two ASTM standards for plain carbon rebar, A 615 and A 706. Both of these

standards include similar groups of requirements such as: dimensions, chemical composition,

and mechanical properties. However, the main difference between the two is that A 706 has

controlled tensile properties (meaning both minimum and maximum values are specified) and

greater restrictions to alloy levels. The motivation behind the development of standard A 706,

including mention of the various organizations involved, is provided in the report from ACI

Committee 439 [9]. As previously mentioned the strain-hardening ratio of 1.25 required by A

706 presents a challenge in the use of the quench and tempering process.

1.4.2 Canadian Rebar Standards

The Canadian standard for carbon-steel rebar is CSA-G30.18. This standard is very similar to

the ASTM standards both in terms of organization and requirements owing to the collaboration

between the organizations. However, as it relates to the topic of quench and tempering, the

strain-hardening ratio specified in the Canadian standard is 1.15.

7

1.4.3 Australian/New Zealand Rebar Standards

The carbon-steel rebar standard for Australia and New Zealand is AS/NZS 4671. This standard

differs from the previous ones in that they define grades both by minimum yield strength and

ductility class. There are three ductility classes; however, the seismic class has the greatest

strain-hardening ratio requirement of 1.15.

1.4.4 Japan Rebar Standards

In Japan the standard for carbon rebar is JIS G 3112. This standard takes a different approach

from the ones discussed previously. Instead of citing a specific minimum strain-hardening ratio,

a range of yield strength is given along with a minimum tensile strength. For the purpose of this

discussion, the minimum strain-hardening ratio inferred by these ranges can be determined by

dividing the tensile strength by the maximum yield strength. Among all of the grades, the

highest maximum strain-hardening level is 1.13.

1.4.5 German/European Rebar Standards

To describe rebar requirements in Europe, two types of standards need to be included in the

discussion. The first type is the European Standard EN 10080. This standard was prepared by

the European Committee for Standardization (CEN) through mandate by the European

Commission and the European Free Trade Association. This is considered a harmonized

standard, meaning that member nations of the CEN are bound to implement these standards. It

provides a general set of specifications and definitions, however, no specific grades or bar sizes

are given. For this reason rebar cannot actually be ordered to this standard. Instead, the second

type of standard is needed, which have been developed through the member countrys

standardizing bodies, and comply with the requirements set forth in EN 10080. In Germany this

standard is DIN 488. The minimum strain-hardening ratio stated in this standard is 1.05 [10, 11].

2 Objectives The objective of this work is to first, determine if AISI 1117 steel can achieve the mechanical

properties of ASTM A 706 rebar, and secondly, to develop a preliminary model which will act as

a tool in future work to optimize the composition and process. In determining the mechanical

response of AISI 1117, an operating window will be determined. The steel will be subjected to a

8

quenching condition, which although is not entirely practical due to quenching time for the

continuous operating setting of the rolling mill (i.e. there is a very short time available to quench

due to mill speeds), will serve to eliminate the chemical composition as the barrier for this

process.

3 Quench and Self-Tempering Model The QST model incorporates two major components, heat transfer, and microstructure evolution.

Using a finite difference method, these components are coupled and solved to determine the

temperature and microstructure fields throughout a cylindrical bar. The output of the model is

the quenching and tempering temperatures, the volume percent of each phase formed, the

continuous cooling curves at the centre and surface of the bar, and the sigmoidal curve showing

the transformation rates of each microstructure at the centre and surface of the bar.

Figure 3.1 Overview of QST Model

3.1 Heat Transfer Problem The type of heat transfer problem being modeled is a one-dimensional, cylindrical, transient

conduction problem. In order to reduce the complexity of the model, a cylinder is used as the

rebar analogue (i.e. ignore the affects of the rebar deformations). Furthermore, because the

cylinder, or round bar, produced by rolling is very long compared to its diameter, the effects of

Heat Transfer

Infinite CylinderOne-dimensional

Microstructal Evolution

Diffusional Transformation (Ferrite, Pearlite, Bainite)Avrami EquationAdditivity Principle

Diffusionless Transformation (Martensite)Koistinen & Marburger

QST Model Finite Difference

Method

9

heat transfer at the ends (head and tail) are ignored. And since this form of heat treatment is an

inline process, being applied continuously along the length of the symmetric bar, it is assumed

that heat conducts only in the radial direction.

Figure 3.2One-dimensional heat transfer in infinite cylinder

The overall behavior of the system can be expressed using the conservation of energy:

= + Where Et is the rate of change of thermal energy being stored in the bar, Ein and Eout are the rate that thermal energy is entering and leaving the bar respectively, and is the rate of thermal

energy being generated within the bar (i.e. latent heat of transformation). Initially; however, the

generation term will be neglected (incorporated in later part of model). Below is the differential

form of Fouriers Law governing the transient conduction in a one-dimensional, infinite cylinder:

=

Where is the density of the material, Cp is the specific heat of the material, and k is the

conductivity of the material.

Boundary Conditions:

Center: =0 = 0

The center of the bar is treated as adiabatic due to symmetry.

R

(, 0) =

(, ) = ( )

10

Surface: = = [(, ) ]

Where is the temperature of the adjacent fluid and h is the convection coefficient.

Initial Conditions:

The model assumes that the initial temperature is uniform through the diameter of the bar.

3.2 Finite Difference Method The model employs the finite difference method to solve the nondimensional version of

Fouriers Law discussed in Section 3.1. The finite difference method is a numerical method of

approximating the solution to differential equations by using simple difference expressions in

place of difference quotients. For example, the first derivative of a function may be represented

by the following expression:

() = lim0 ( + ) ()

This function can be approximated by simply selecting a value for h (step size). Of course the

step size is proportional to the magnitude of error; the smaller the step size the smaller the error.

This is the general concept behind the finite difference method. There are; however, different

variations and types of finite difference schemes that can be utilized. For instance, there are

three common forms of finite differences: 1. forward difference, 2. backward difference, and 3.

central difference.

1. Forward Difference: () = ( + ) () 2. Backward Difference: () = () ( ) 3. Central Difference: () = + 12 ( 12 ) There are also expressions for higher order finite differences. For example the expression below

is a second-order central difference:

11

"() = ( ) 2() + ( + )2

The solution to these methods can also be characterized as explicit or implicit. The explicit

method uses the forward difference approach to approximate the solution at the next step value

based on the adjacent current step values. On the other hand, the implicit method uses a

backward difference approach to approximate the solution at the next step value based on the

adjacent next step values. Although the implicit method is more computationally complex,

unlike the explicit method, it is unconditionally stable. The advantage of this is that the selection

of step size, both in time and space, are less of a concern in regards to acquiring a convergent

solution. This adds to the robustness of the model by ensuring that a meaningful output is

generated.

3.2.1 Discretization

The first step in applying the finite difference method is to divide the geometry of the cylinder

into small segments. For example in Figure 3.3 a cylinder is divided into segments using

concentric rings. At the centre of each segment a node represents the point for which a value

will be calculated. Since the heat transfer problem being solved is one-dimensional, the

temperature throughout the cylinder is dependent only on the radial dimension (i.e. the distance

from the centre of the bar). This means that the discretization can be simply represented by a

single node for each radial segment as shown to the right in Figure 3.3. The thickness of each

segment, r, is the same throughout the cylinder with the exception of the surface and centre

segments. These areas require special consideration. Because heat transfer between the cylinder

and surrounding fluid occurs at the surface, to more accurately determine the thermal conditions

in this region of the cylinder, this node is assigned a thickness half of that of the interior nodes

(r/2). Also, since there is a boundary condition at the centre of the cylinder, a node is required

at that point. And because the centre is symmetric, using a thickness of r/2 will actually result

in a central nodal segment equal to r.

12

Figure 3.3 Discretization of Cylinder

3.2.2 Implicit Form of Finite Difference Equations

Using a backward difference approximation for the time derivative and a second-order central

difference approximation for the space derivative, the implicit form of the finite difference

equations for the heat problem are:

Surface Node: (1 + 2 + 2)0+1 21+1 = 2 + 0 Internal Node: 1+1 + (1 + 2)+1 +1+1 = Central Node: 21+1 + (1 + 2)+1 = Where the superscript (p) over temperature represents a time reference, and the subscript (i) after

temperature is in reference to the node. For example, p+1, indicates the temperature one time

step ahead of the current, and i+1 indicates one node towards the centre. It should also be

noted that the surface node has i=0 and the central node has i=Nr, where Nr represents the

number of nodes used in the model.

Also, Fo is known as the Fourier number: = 2

And Bi is known as the Biot number: =

rCENTRE SURFACE

r/2r/2NODES

13

The Fourier number is a dimensionless form of time. Conceptually, it represents the relationship

between the rate of conduction to the rate of thermal energy storage. A Fourier number of 0

would indicate that no thermal energy is being conducted versus a value of 1 which would

indicate the change in thermal energy stored is entirely due to conduction. The Biot number is a

dimensionless number that represents the ratio of heat transfer resistance at the surface (h) versus

within the bar (k). A Biot number much greater than 1 indicates that the temperature field will

vary significantly throughout the bar.

3.2.3 Tridiagonal Matrix

As can be seen in the implicit form of the finite difference equations, the new (in terms of time

step) temperature of a given node is dependent on the new temperature of the adjacent nodes.

However, since these new temperatures are unknown the system of equations must be solved

simultaneously. To accomplish this, matrix inversion is used. The following represents the

matrix form of the system of finite difference equations:

0+11+1

1+1+1

=

(1 + 2 + 2) 2 0 0 0

(1 + 2) 0 00 00 0 0 0 0 2 (1 + 2)1

(2) + 0

1

1

3.3 Microstructure Evolution Early development of computational models to couple heat transfer with microstructure

evolution focused primarily on eutectoid carbon steel [12-17]. The absence of ferrite formation

with this composition of steel reduced the complexity of the model so that only the

transformation of austenite to pearlite or martensite was of concern. These two phases, pearlite

and martensite, represent two significantly different forms of transformation which are often

classified as diffusional and diffusionless respectively.

14

3.3.1 Diffusional Transformation

Diffusional transformation refers to phase transformations that are characterized by the diffusion

of carbon and are thus dependent on both temperature and time. This type of transformation

includes the decomposition of austenite into; ferrite, pearlite, and bainite, depending on the

chemical composition and rate of cooling. From the models previously mentioned, it has been

well established that the relation proposed by Johnson and Mehl, and Avrami to describe

isothermal transformations can be combined with the additivity principle proposed by Scheil to

apply isothermal data to predict continuous cooling conditions [18-22]. In the sections to follow,

these components of diffusional transformation modeling are described in further detail.

3.3.1.1 Avrami Equation

An Avrami type equation, shown below, which is also commonly referred to as the Johnson-

Mehl-Avrami (JMA) or Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation, is often used to

describe the S-shaped or sigmoidal form of a phase transformation-time curve.

= 1 exp() Equation 3.1 Avrami Phase Transformation

Here Vi is the volume fraction of the new phase formed after time t. The coefficient B is a

kinetic parameter which includes nucleation and growth rates, and the coefficient k is related to

the geometry of the growing phase (i.e. polyhedral, plate-like, or lineal). Below is an illustration

of a typical phase transformation-time curve.

15

Figure 3.4: Example of Sigmoidal Form of Phase Transformation

The Avrami equation; however, cannot be directly applied for modeling microstructure evolution

of a continuous cooling processes since it is based on isothermal conditions. In order to

overcome this problem, a method used to approximate a continuous cooling curve, as described

in an early model by Agarwal and Brimacombe, is to divide the curve into many small

isothermal steps [12]. Then at successive steps after transformation has begun, the Avrami

equation can be employed to calculate the incremental formation of the new phase. The figure

below illustrates the approximation of the cooling curve using a series of isothermal time steps.

Figure 3.5 Approximating a Cooling Curve Using Isothermal Steps

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10 100

Volu

me F

ract

ion

Time (sec)

Tem

pera

ture

Time

TTT curve

Cooling curve

16

3.3.1.2 Additivity Principle

An important distinction should first be made between possible applications of the additivity

principle. In the example discussed later, the additivity principle is applied to the transformation

process (i.e. after transformation has begun). However, another possible application, as

proposed originally by Scheil, is that an additivity principle can be used to describe the

incubation period and to predict the start of transformation of a continuous cooling process using

isothermal data. This is achieved by dividing each time step of the cooling curve with the

associated start time of transformation (obtained from a TTT curve) in order to calculate the

proportion of the incubation period consumed (called the fractional nucleation time). Then,

when the sum of these proportions is equal to unity the incubation period is complete. This can

be expressed as:

= 1

Equation 3.2 Condition for Start of Transformation

Where t is the time step and s is the transformation start time at a given temperature. Although

the additivity principle has been employed in numerous models to predict the start of

transformation [7, 23], in their study of the decomposition of austenite to pearlite of a eutectoid,

plain carbon steel, Hawbolt et al showed that the additivity principle lead to an over estimation

of the incubation time [14]. On the other hand, the study showed that the transformation stage

was accurately described by the additivity principle. However, because it is time consuming and

expensive to experimentally determine the start of transformation for a continuous cooling

process, and since isothermal data is readily available in the literature for a variety of chemical

compositions [24], the error associated with using the Scheil additivity principle for estimating

the incubation period is seen as an acceptable compromise for this model.

Using the additivity principle to describe the transformation process requires that the

transformation be a function of the temperature and the amount of transformation product

already present [25].

= (,)

17

The figure below is helpful for illustrating the additivity principle. For the first time step, the

phase transformation follows the T1 transformation-time curve and so the percent of phase

transformed is easily calculated using the Avrami equation. However, for the next time step,

starting at t1, there is a decrease in temperature to T2. At this new temperature, the

transformation will continue with the new phase forming at a rate according to the T2 curve.

However, some consideration must be made regarding the time value used in the application of

the Avrami equation for T2. The starting point on the T2 curve must correspond to the volume

percent of phase already transformed during the first time step. However, if the second time step

were to start at the end of the first time step, t1, then the starting volume percent of phase

transformed would be incorrect (i.e. it would correspond to the intersection of t1 and T2). To

properly account for this a fictitious time, t2, must be calculated which corresponds to the

volume percent of phase transformed at the end of the first time step but relating to the T2 curve.

Continuing in a similar manner, the total phase transformation can therefore be described by

combining the segments of the transformation-time curves.

Figure 3.6 Combining Transformation-Time Curve Segments: Additivity Rule

In their assessment of the additivity principle for predicting the kinetics of austenite to pearlite

decomposition in eutectoid, plain carbon steel, Kuban et al were able to justify the additivity

condition shown above [25]. Furthermore, their experiments showed that the condition for

applying the additivity principle as proposed by Avrami and later by Cahn did not hold [19-21,

26]. Kuban et al instead suggested an effective site saturation criterion to explain the success

The rate of transformation changes with temperature

Segments showing amount of phase transformed in each time step

Time

Vo

lum

e F

ract

ion

18

of the additivity principle in the use of predicting continuous-cooling kinetics, which states that

the later growth of the early nuclei dominates the transformation event. This assessment of the

additivity principle was later expanded by Kamat et al to include proeutectoid, plain carbon steel

[27].

3.3.1.3 Solving for the coefficients of the Avrami equation

In the studies of transformation kinetics previously mentioned, the coefficients of the Avrami

equation were found experimentally [12, 14]. However, as discussed before, this process is very

time consuming. To allow the model to be easily expanded for various chemical compositions,

an alternative method of solving these coefficients using isothermal data has been used [17, 23,

28, 29].

The transformation start and finish curves on a TTT diagram represent some volume fraction

close to 0 and 1 respectively of the new phase being formed. For instance, a typical start curve

may represent the point when the volume fraction of the new phase formed is 0.005 and a finish

curve may represent the point when the volume fraction of the new phase is 0.995. Using these

start and finish volume fraction values along with the corresponding start and finish times, two

expressions of the Avrami equation, as shown below, can be written for any particular

temperature [29].

= 1 exp(()()) = 0.005 = 1 exp(()()) = 0.995

Where Vs is the volume fraction represented by the start of transformation curve, s is the time

corresponding to the start of transformation, and Vf and f are similarly representative of the

finish transformation curve. Thus for a given temperature, T, there are two equations and two

unknown material parameters, B(T) and k(T). Using these equations, an expression for the

material parameters can be written as:

19

() = ln ln(1 )ln(1 )ln Equation 3.3

() = ln(1 )()

Equation 3.4

The final component required to use the Avrami equation for a continuous cooling process is an

appropriate expression for time. Recall that in the Avrami equation, the time variable represents

the time required from the beginning of transformation at a particular temperature (i.e

isothermal) to form a certain volume fraction of new phase. However, since the additivity

principle is employed to allow the continuous cooling curve to be estimated by a series of small

isothermal steps, the time to be used must account for the change in transformation rate. As

illustrated in the description of the additivity principle, at a new temperature (or say the next time

step), the rate of transformation changes but starts at the point corresponding to the volume

fraction already formed. In other words, to calculate the total volume fraction of new phase

formed after the current time step the time value used in the Avrami equation is equal to the

actual time step, tj, plus a fictious time, tj, that represents the volume fraction of phase formed

at the end of the previous time step but at the current transformation rate (i.e. according to the

current temperature). The expression is shown below:

= + Equation 3.5

Where tj is derived from the Avrami equation for the volume fraction present at the previous

time step, Vj-1, but with material parameters at the current temperature:

1 = 1 exp(() ()) Equation 3.6

20

Therefore Equation 3.5 can be written as:

= + ln( 11 1)() 1

()

Equation 3.7

Therefore obtaining the unknowns, s,f, and T from the TTT and continuous cooling curves at

the current isothermal time step, and subbing into Equation 3.3, Equation 3.4, and Equation 3.5

to find the material parameters, the volume percent formed at the end of the current time step can

be found using Equation 3.8 below. = 1 exp(() ())

Equation 3.8

3.3.2 Diffusionless Transformation:

The formation of martensite is called a diffusionless transformation since its rapid cooling from

austenite prevents the diffusion of carbon from taking place. Instead, the FCC austenite

undergoes a polymorphic transformation to a body-centred tetragonal (BCT) martensite. This

transformation process is therefore dependent only on temperature as opposed to time.

3.3.2.1 Koistinen and Marburger Equation

In many continuous cooling models involving diffusionless transformation, [16, 17, 23, 28-30],

the empirical relation developed by Koistinen and Marburger, shown below, has been used to

calculate the volume fraction of martensite formed [31].

= {1 exp[( )]}(1 ) Equation 3.9

21

Here, VM, is the volume fraction of martensite formed, is a constant equal to 1.10x10-2, Ms is

the martensite start temperature, and (1-Vi) is the austenite that has not been transformed into

some other phase and hence is available to form martensite.

4 Computer Model A model was created using MATLAB to couple temperature and microstructure evolution. The

first attempt focused on eutectoid, plain carbon steel for its abundance of data available in

literature and relative simplicity due to the absence of proeutectoid ferrite formation. This model

served to verify the programming and use of the fundamental principles discussed above and

acted as the basis for development of more complex systems. Later development included the

addition of different, non-eutectoid, steel compositions. Figure 4.1 below shows the graphical

user interface for the computer program. The computer code for the main model function along

with all sub functions can be found in Appendix 4 to 9.

Figure 4.1 QST Computer Model GUI

22

4.1 Model Overview

The operation of the model is briefly described here followed by a flow chart which provides a

general overview of the program. Individual components are described in greater detail in later

sections.

1. Input parameters are entered

2. The temperature field along the radius of the bar is calculated

3. The amount of new phase formed based on the temperature field is found

4. Latent heat generated from phase transformation is determined

5. The temperature change due to the latent heat generated is added to the temperature field

and the amount of new phase formed is recalculated

6. The time step is incremented and steps 2 through 5 are repeated

7. The program ends when the user defined cooling time has been reached

Figure 4.2 High-Level Flow Diagram of Computer Model

4.2 Input Parameters

The table below lists the input parameters of the model, organized by user and program defined

parameters, as well as by parameter type. In addition to the input parameters listed here, various

characteristics of the program can also be defined based on user selection. This group of

characteristics allows the user to customize the output of the program and to define what type of

1.Input parameters

2.Calculate

temperature field

3.Calculate the

volume % of new phase formed

4.Calculate latent heat generated

6.Increment time

step

Defined time reached?

Change in temp below min criteria

5.Add increase in temperature to

temperature field

No

Yes Yes 7.End Program

No

23

analysis to perform; however, since the fundamental behavior of the model is not affected they

are not included here.

Table 4.1 Computer Model Input Parameters

User Defined Internal to Program

General Parameters

Bar diameter Initial bar temperature Water temperature Air temperature Quenching time Air cooling time Heat transfer coefficient (water) Heat transfer coefficient (air)

Model Parameters Number of nodes Time step

Material Properties (steal)

Density Specific heat Conductivity Thermal diffusivity

Isothermal Data (from TTT) Pearlite start time Pearlite finish time Martensite start time

4.2.1 General Parameters

The general parameters allow the user to define the overall process to be analyzed. Depending

on the user selected attributes of the analysis, as mentioned previously, some of the general

parameters may not be required. For example, if the user does not select the air cool option then

air temperature, air cooling time, and heat transfer coefficient (air) are not required.

4.2.2 Model Parameters

The model parameters allow the critical attributes of the finite difference method including,

number of nodes and size of time step, to be varied in order to determine their affect on the

results. As will be shown later, this allows convergence tests to be performed.

4.2.3 Material Parameters

Although accuracy is somewhat reduced over a large temperature range, the model was

simplified by assuming the material properties such as steal density and specific heat were

constant. Since the overall behavior of the model could still be demonstrated effectively this

24

way, incorporating variable material properties was seen mostly as an exercise in programming

and so was left for future work.

4.2.4 Isothermal Data To enter the material properties, a published TTT figure for 1080, eutectoid steel was converted

to a series of coordinates using a publically available MATLAB m-file. Using a scan of the TTT

figure, once the axes had been defined, coordinates were generated by manually selecting a

series of points along the desired curve. In this manner the pearlite start curve, pearlite finish

curve, and martensite start temperature were entered into the model.

4.3 Temperature Field

As mentioned in the finite difference method and heat transfer sections, the temperature at each

node along the radius of the bar is calculated simultaneously (implicit method). Therefore the

temperature field is expressed in the form of a vector. Along with the assumption that the

temperature is initially constant throughout the bar, the expression for the temperature field is

shown below:

= []1[] The tridiagonal matrix, [Tri], is assembled and consists only of the grid Fourier (Fo) and grid

Biot (Bi) numbers which are calculated directly from the input parameters.

= 2

=

= 1 + 2 + 2 2 0 0

1 + 2 0 0 1 + 20 0 2 1 + 2

The matrix, [C], is simply the previous or initial temperature field with an expression added to

the first term, or surface node, resulting from the convection between the steel and the water.

25

= + 2+1

The following section of code illustrates how the model calculates the temperature field. The

first section assembles the tridiagonal matrix. This is followed by the initialization of the [C]

vector. Finally a loop is used to calculate the temperature field after each time step. It should be

noted that there is additional code within the loop for other functions (i.e. to calculate phase

formation), however, it has been removed to simplify this explanation.

% Z matrix is the subdiagonal coefficients of the tridiagonal matrix E=sparse(2:Nr,1:Nr-1,ones(1,Nr-1),Nr,Nr); Z=E*-Fo; Z(Nr,Nr-1)=(-2*Fo);

% Y matrix is the diagonal coefficients of the tridiagonal matrix G=ones(1,Nr); X=G*(1+(2*Fo)); X(1,1)=(1+(2*Fo)+(2*Fo*Bi)); Y=sparse(1:Nr,1:Nr,X,Nr,Nr);

% U matrix is the superdiagonal coefficients of the tridiagonal matrix W=sparse(1:Nr-1,2:Nr,ones(1,Nr-1),Nr,Nr); U=W*(-Fo); U(1,2)=(-2*Fo);

% N matrix is the assembled tridiagonal coefficient matrix M=Z+Y+U; N=full(M);

% The following initializes the TempField vector with the initial bar temp for each node A=ones(1,Nr)*InitialBarTemp; TempField=A'; % The vector C is created which incorporates the effects of convection at the surface node of the bar B=[(2*Fo*Bi*WaterTemp),zeros(1,Nr-1)]; C=TempField+B';

% The loop is used to calculate the temperature field at each time step. The number of steps is determined by

The

tridiagonal

matrix is

formed

26

dividing the user inputed quench time divided by the size of time step. For example if quench time is 10 seconds and the time step is 0.1 seconds, than the number of time steps is 10/0.1 = 100. In that example the loop will iterate 100 times. % The order in which these two lines of code appear in the loop is critical. Since the initial temperature field has been calculated above, outside the loop, in the first time step the [C] matrix is calculated and then the new temperature field is found. This temperature field (TempField) is then the input for [C] on the next iteration. for k=1:(QuenchTime/DeltaT)

. . . C=TempField+B'; TempField=N\C; . . . end

4.4 Phase Formation

Once the initial temperature field has been calculated the next step is to determine the volume

fraction of new phase formed at each node. It should be noted that an assumption of the model is

that initially the entire materiel consists of austenite. As described previously in the section on

diffusional transformation, new phase does not begin to form until the incubation period is

complete. Scheils additivity principle is applied using isothermal data obtained from the TTT

diagram to predict the end of the incubation period (i.e. the start of the continuous cooling,

transformation time). With the Scheil condition satisfied, the Avrami equation is then used to

determine the amount of phase formed.

4.4.1 Applying Scheils Additivity Principle

In order to incorporate this principle in the model, the fractional nucleation time (FNT = (

))

must be calculated at each time step and then added with the values of all previous time steps to

27

determine if the condition in Equation 3.2 has been met for the start of transformation. To

accomplish this, two vectors are used. The first vector holds the values of the FNTs for each

node for the current time step. The second vector, Scheil vector, is created to keep a running

sum of the FNTs for each node. The Scheil vector is initialized as a series of zeros, but at each

time step, once the FNTs are calculated they are added to the Scheil vector.

To calculate the FNT, the isothermal start time, s, needs to be interpolated from the TTT data.

Since the isothermal start time is dependent on temperature, the temperature field is provided as

an input to a function which applies linear interpolation to return the corresponding

transformation start times. The FNT can then be calculated for each node by dividing the time

step by the transformation start times. The code used in the computer model is shown below.

for k=1:(QuenchTime/DeltaT)

% The vector D calculates the fractional nucleation time for each time step by dividing the time step by the transformation start time for the given temperature (TempField). The transformation start time is interpolated from the TTT data (Temp & LogTime) D=bsxfun(@rdivide,DeltaT,(interp1(Temp,LogTime,TempField))); % For some nodes that fall outside the TTT data (i.e. do not have a transformation start time) the result of the above is NaN. These entries are replaced with zeros RepNaN=isnan(D);

D(RepNaN)=0;

% The Scheils vector takes a running total of the FNTs (i.e. sums all previous FNTs) Scheils=Scheils+D; % To eliminate rounding error a loop is used to set any nodes (1:Nr) that have a Scheils value greater than 1 to 1 (i.e. in reality there should never be a Scheils value greater than one). This makes later calculations easier since only a single value of Schiels is considered (i.e. Scheils=1).

for j=1:Nr if Scheils(j)>=1 Scheils(j)=1; else Scheils(j)=Scheils(j); end end

end

28

In the above expression, the interp1 function takes the TempField vector and interpolates the

transformation start times using the TTT data points; Temp and LogTime. Note that since

TempField is a vector of dimension Nr (number of nodes), the result of performing interp1 is

also a vector of the same dimension. The bsxfun function is used to apply an entry-by-entry

manipulation of the interp1 vector. In this case it is dividing the time step, DeltaT, by each entry

of the interp1 vector. The result is a vector, FNT, with values calculated for each node.

4.4.2 Applying the Avrami Equation: The Growth Function

At each time step a growth function is applied to calculate the volume fraction of new phase

formed at each node. However, there are a number of conditions incorporated into this function

that must be met on a node-by-node basis before the volume fraction formed is determined.

These conditions are listed below:

1. The Scheil condition must be equal to or greater than unity

As described before, this is the critical condition that indicates that phase formation can take

place

2. The temperature of the node must be greater than the martensite start temperature

This condition ensures that the function is not applied below the martensite start temperature

martensite formation (diffusionless transformation) requires a different function

3. The temperature of the node must be less than or equal to the Ac1 temperature

This condition is used to avoid errors from developing due to irregularities outside the

temperature range of the TTT data

4. The transformation start temperature at the node must be less than or equal to the elapsed time

This is to avoid errors from occurring when the cooling curve has dropped below the nose

of the TTT curve

Assuming that all of these conditions have been met, it is straightforward to solve the variables

of the Avrami equation (B(T), k(T), and tj) and then the resulting volume fraction of new phase

formed.

29

The growth function begins by using the temperature field to interpolate the transformation start

and finish time for each node. This results in the creation of a transformation start and finish

time vector (ts and tf). Next, using a loop, the function constructs a growth vector (Fp) one node

at a time. It does this by calculating the material parameters based on the temperature and

isothermal data of each node (B(T) and k(T)), and then the transformation time followed by the

volume fraction using the Avrami equation. The iteration of the loop calculates the volume

fraction of successive nodes resulting in the final assembly of the growth vector. With each time

step, a new temperature field is passed to the growth function to determine the corresponding

phase formation. Below is the main part of code used to calculate the pearlite transformation.

function [ Fp,ts,ElapsedTime ] = PearliteGrowth2( Temp,LogTime,TempField,Temp1,LogTime1,DeltaT,k,Nr,Ms,Scheils,Fp,SteelType_PopupmenuStatus,PearliteStartTempUpper,PearliteStartTimeUpper,PearliteStartTempLower,PearliteStartTimeLower ) %This function calculates the % of Pearlite transformation at each node

if SteelType_PopupmenuStatus==1 || SteelType_PopupmenuStatus==2

%The transformation start and finish times are interpolated from the TTT data for the current temperature field

ts=interp1(Temp,LogTime,TempField); tf=interp1(Temp1,LogTime1,TempField);

%The MaxTTTtemp is interpolated from the TTT data and used as a condition in the following loop so that only values in the temperature field vector (TempField) that have a valid start and finish transformation temp will be used

MaxTTTtemp=min(max(Temp),max(Temp1));

%The elapsed time is another condition used in the following loop to ensure that only the portion of the cooling curve that has intersected the transformation start time are used

ElapsedTime=DeltaT*k;

%The loop functions to solve the Avrami equation for each node (j=1:Nr) and then to assemble the values into a vector (Fp(j)) for j=1:Nr

%The if statement incorporates the four conditions that were described above if ((TempField(j)>Ms) && (TempField(j)=1) && (ts(j)

30

B=(reallog(reallog(0.995)/reallog(0.005)))/(reallog(ts(j)/tf(j)));

A=(-reallog(0.995))*((ts(j))^(-B)); tj=DeltaT+((reallog(1/(1-Fp(j)))/A)^(1/B)); Fp(j)=1-exp(-A*tj^B); else

%If the conditions are not met for a given node the amount transformed is the same as the previous amount Fp(j)=Fp(j);

end

end end

4.4.3 Incorporating Transformation of Multiple Phases

For non-eutectoid steels, multiple growth functions are used. However, as austenite is

transformed into some phase, the austenite available for future transformation is reduced. This

reduction in available austenite must be accounted for in order to calculate the proper amount of

phase being formed. The following code illustrates how the computer model incorporates this

consumption of austenite and how it affects the amount of subsequent phase that can be

transformed.


. . . %The ferrite growth function is first applied. [Ff]=FerriteGrowth(FerriteStartTemp,FerriteStartTime,TempField,PearliteStartTemp,PearliteStartTime,DeltaT,k,Nr,MartensiteStartTemp,Scheils,Carbon,Ff,SteelType_PopupmenuStatus,PearliteStartTempUpper,PearliteStartTimeUpper);

Xf=bsxfun(@times,FerriteMax(TempField,Carbon,Nr),Ff);

%The pearlite growth function is then applied. [Fm]=PearliteGrowth2(PearliteStartTemp,PearliteStartTime,TempField,PearliteFinishTemp,PearliteFinishTime,DeltaT,k,Nr,MartensiteStartTemp,Scheils,Fm,SteelType_PopupmenuStatus,PearliteStartTempUpper,PearliteStartTimeUpper,PearliteStartTempLower,PearliteStartTimeLower);

%The amount of pearlite that can form is dependent on how much austenite is available and therefore, depends on how much ferrite has been formed. The pearlite found in the function above is

31

multiplied by the amount of austenite left after being transformed to ferrite (1-Xf) Xm=bsxfun(@times,(1-Xf),Fm);

%The martensite growth function is applied next.

[FM]=MartForm(Nr,TempField,Xm,FM,MartensiteStartTemp,Xf);

%The remaining austenite available after ferrite, pearlite and martensite transformation is then calculated so that it can be used as an output for the model

Xa=1-Xf-Xm-FM; . . . end

4.5 Latent Heat Generation When phase formation occurs, latent heat is generated. As previously mention in the description

of the conservation of energy and the heat transfer problem, this generation term was initially

neglected. However, once the temperature field and corresponding phase transformation is

found, this information can be used to determine the amount of heat generated at each time step.

The following provides details of the code used in the computer model to incorporate this latent

heat generation.


%Place holders are created to keep a running total of the phase formed before the current time step. This will be used to determine the delta (change in) phase formed for the current time step.

Xfprev=Xf; Xmprev=Xm; FMprev=FM;

%The following calculates the change in enthalpy related to formation of ferrite. DeltaHf=(1.082*10^2)-(0.162*(TempField+273))+((1.118*10^-4)*(bsxfun(@power,(TempField+273),2)))-((3*10^-8)*(bsxfun(@power,(TempField+273),3)))-(bsxfun(@rdivide,3.501*10^4,(TempField+273))); %The following calculates the change in enthalpy related to formation of pearlite. DeltaHp=(1.56*10^9)-((1.5*10^6)*TempField);

32

%The delta for each phase is determined so that it can be incorporated with the change in enthalpy in following equations. [DeltaXf,DeltaXm,DeltaFM]=DeltaPhase(Nr,Xf,Xfprev,Xm,Xmprev,FM,FMprev);

%Using the change in enthalpy and the delta phase formed, the change in temperature due to latent heat generation is calculated for each phase. %Change in temperature due to latent heat generated from ferrite formation.

DHFf=bsxfun(@times,DeltaHf,DeltaXf); Qdotf=bsxfun(@rdivide,DHFf,DeltaT); TempQf=bsxfun(@rdivide,Qdotf,(density*Cp))*DeltaT;

%Change in temperature due to latent heat generated from pearlite formation.

DHFm=bsxfun(@times,DeltaHp,DeltaXm); Qdotm=bsxfun(@rdivide,DHFm,DeltaT); TempQm=bsxfun(@rdivide,Qdotm,(density*Cp))*DeltaT;

%Change in temperature due to latent heat generated from martensite formation.

DHFmM=DeltaFM*DeltaHM; QdotM=bsxfun(@rdivide,DHFmM,DeltaT); TempQM=bsxfun(@rdivide,QdotM,(density*Cp))*DeltaT;

%A new temperature field is calculated which adds the change in temperature due to the latent heat generated from the formation of each phase. TempField=TempField+TempQf+TempQm+TempQM;

end

5 Materials and Experimental Procedure

5.1 Materials The material studied in this research is AISI 1117 steel. The SAE/AISI 11xx series of steel is

classified as resulfurized steel because it can contain between 0.08% to 0.33% sulfur. The

intention of the added sulfur is to form excess manganese sulfide (MnS) inclusions which

enhance the machinability of the steel. The higher manganese content of AISI 1117 compared to

the 10xx series is also desirable in that it increases the strength of the steel due to solid solution

strengthening. This is beneficial when carbon levels must be kept low in order to achieve the

desired weldability requirement.

33

The AISI 1117 steel used in these experiments was supplied as cold rolled bars from the

Taubensee Steel & Wire Company. Two diameters of bar stock were supplied, 3/8 inch and 1

inch, with the chemical compositions (weight percentage) provided in Table 5.1 and Table 5.2

respectively. Note that the test reports are also included in Appendix 1 and 2 for reference. One

bar of each diameter was supplied and then divided into smaller pieces to produce all required

samples for this research. In other words, chemical compositions described below are

characteristic of all samples used in this work.

Table 5.1 Chemical Composition of 3/8" Dia. AISI 1117

C Mn P S Si Ni Cr Mo Al Cu V Nb

0.160 1.100 0.007 0.100 0.180 0.040 0.070 0.010 0.003 0.120 0.001 0.001

Table 5.2 Chemical Composition of 1" Dia. AISI 1117

C Mn P S Si Ni Cr Mo Al Cu V Nb

0.160 1.100 0.011 0.113 0.220 0.040 0.050 0.010 0.002 0.090 0.001 0.001

5.2 AISI 1117, ASTM A 706, and Microalloyed Comparison AISI 1117 was selected as the topic of this research due to the fact that it satisfies the major

chemical requirements of ASTM A 706 grade rebar but also provides unique mechanical

properties as compared to typical microalloyed rebar compositions currently used in North

America. Table 5.3 provides a comparison between these three compositions. Note that the

elements provided in the table represent all requirements defined in ASTM A 706 with the

exception of Vanadium and Niobium. These elements are provided here because they are critical

additions for the microalloyed composition.

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Table 5.3 AISI 1117, ASTM A 706, Microalloyed Chemical Composition Comparison

C Mn P S Si V Nb

AISI 1117 (3/8 sample)

0.16 1.10 0.007 0.100 0.18 0.001 0.001

ASTM A 706 (maximums)

0.30 1.50 0.035 0.045 0.50 N/A N/A

Microalloyed 0.27 1.17 0.012 0.026 0.30 0.020 0.030

The microalloyed composition provided in the table above represents an average from various

mill test reports collected from various steel producers in the United States. It should be

emphasized that the weight percentage of vanadium and niobium are averages and that typically

only one will represent the majority of the microalloying addition.

5.3 Microstructure Characterization: Optical Microscopy Microstructures were examined under an optical microscope. Micrographs were captured with a

digital camera connected to the microscope. The following describes the process used for

sample preparation.

Bar samples were cut using a hacksaw and by stopping intermittently to cool the cut surface with

water to avoid significant temperature gain. The samples were than cold mounted in epoxy to

avoid inducing microstructure changes. Following mounting, the samples were mechanically

ground starting with 120 grit sandpaper and continuing with progressively finer grits (240, 400,

600, 800, and 1200). After using the 1200 grit sandpaper the samples were then polished using

rotating polishing clothes along with a1 m alumina slurry. Finally, the samples were etched

using a 4% Nital solution. The grain size of the samples was determined using an intercept

approach. Several lines of different orientations were drawn over the optical micrographs. The

line length was then divided by the number of grains intercepted by the line to find the average

grain size.

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The microstructure of the as received 3/8 inch bar stock is show in Figure 5.2 and Figure 5.3. It

consists of a Proeutectoid Ferrite (light regions) and Pearlite (dark regions) mixture.

Figure 5.1 As received AISI 1117 perpendicular to rolling direction

Figure 5.2 As received AISI 1117 parallel to rolling direction

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5.4 Mechanical Property Characterization

5.4.1 Microhardness

A Shimazu microhardness tester was used to measure the Vickers hardness of the samples. The

sample surfaces were ground using a similar method described in the micrograph preparation up

to the 1200 grit sandpaper. Measurements were taken using a load of 2.942 N applied for 10

seconds. The micrometers (both X and Y axis) attached to the sample table of the tester were

used to determine the centre of the samples and to increment the sample between readings. In

this way measurements were taken at equal increments along the radius of the sample. In

addition, as illustrated in Figure 5.5, measurements along two radii 90 degrees apart were taken

and then averaged.

Centre Pos. 1 Pos. 2 Pos.3

C 1 + 12 2 + 22 3 + 32

Figure 5.3 Microhardness measurement arrangement

5.4.2 Tensile Testing

Tensile specimens were machined to have a shape shown in Figure 5.6. Two versions were

produced: one for the 3/8 inch bar stock and the other for the 1 inch bar stock. The 3/8 inch

samples were machined to have a gauge diameter of 0.25 inches. The 1 inch samples were

reduced to a nominal diameter of 0.50 inches with a gauge diameter of 0.375 inches. Table 5.4

C a1 a2 a3

b1

b2

b3

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provides the dimensions of these two sample sizes. All samples were 6 inches in length with

smooth, cylindrical ends.

Table 5.4 Tensile specimen dimensions

A G D R

3/8 Bar Stock 1 1.000 0.005 0.250 0.005 3/16

1 Bar Stock 2 1.500 0.005 0.375 0.005 3/8

Figure 5.4 Tensile specimen

All tensile tests were performed at room temperature on a 100 KN MTS (Material Testing

System model 630) testing machine with the load axis parallel to the longitudinal axis of the

bar.

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Figure 5.5 Tensile Test Machine and Extensometer

A load cell measured the force and an extensometer, attached to the specimen, measured the

strain during testing. The measured values were captured by a data acquisition program on a

computer. This data was then exported to a spreadsheet to generate the engineering stress-strain

curves and used to determine the 0.2% yield stress, UTS, and elongation to failure of the

specimens.

5.5 Heat Treating For the quenching experiments all samples were heated in a Lindberg Tube Furnace. While in

the furnace, samples were placed on a bed of steel turnings (1045) to reduce decarburization

affects. Cold, clean tap water was used as the quenchant in all experiments. Table 5.5 to Table

5.9 that follows provides the details of the heat treatment process used in each experiment. Two

general groups of experiments were performed; the first to produce samples for hardness testing,

and the second to produce samples for tensile testing. The samples for the tensile tests were

longer than those for the hardness tests (6 inches compared to 3 inches). Although all samples

were austenized at 900C, the 1 inch diameter bars were held at this temperature for 45 minutes

and the 3/8 inch samples were held for 20 minutes. It is also important to note that during

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Experiments 2 to 4, the still water in the tank was not changed between successive samples being

quenched. This practice was changed for Experiment 5 so that a separate tank was used for each

sample (each tank having the same quantity and temperature of water). The reasoning for this

change in procedure will be discussed further in the following results section. Finally, one other

critical difference to be noted is regarding the two versions of the tensile test specimens. All 3/8

inch bar stock samples were machined after being quenched (to 1/4 inch gauge diameter). In

contrast, the 1 inch diameter bar stock samples that were machined to a nominal diameter of 1/2

inch with a gauge diameter of 3/8 inch was done so before being quenched. Regarding the

sample IDs, in experiments 1 to 4, the Q indicates that the sample has been quenched and the

A indicates the sample was air cooled. In experiments 4 and 5, the number before the decimal

refers to the quenching time and the M indicates that the samples were machined prior to heat

treatment.

Table 5.5 Experiment 1 heat treatment details

Experiment 1: 1 Dia. Samples for Hardness Tests

Sample ID Furnace

Temp (C) Time @ Temp

(min) Quenching Time (sec)

Samples Inserted Before Furnace Turned On (Y/N)

Q1 900 45 Full (5mins) Y

Q2 900 45 Full (5mins) Y

A1 900 45 0 (air cooled) Y

Notes: - All three samples were placed in furnace together

- Samples quenched in container (2L) with water from tap flowing in

- Q in sample ID refers to quench and A refers to air cooled

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Experiment 2: 3/8 Dia. Samples for Hardness Tests

Sample ID Furnace




Q3 900 20 Full (5mins) N

Q4 900 20 Full (5mins) N

Q5 900 20 3 N

A2 900 20 0 (air cooled) N

Notes: - All four samples were placed in furnace together

- Samples quenched in tank (25L) with still water

- Q3 and Q4 were placed in separate tanks but Q5 was quenched in same tank as Q4 (without changing water)



Experiment 3: 3/8 Dia. Samples for Hardness Tests

Sample ID Furnace




Q6 900 20 30 N

Q7 900 20 15 N

Notes: - Samples placed in furnace together


- Q7 was quenched immediately after Q6 in same tank without replacing water


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Experiment 4: 3/8 Dia. Samples for Tensile Tests

Sample ID Furnace




10.1 900 20 10 N

10.2 900 20 10 N

15.1 900 20 15 N

15.2 900 20 15 N

30.1 900 20 30 N

30.2 900 20 30 N

A.1 900 20 0 (air cooled) N

A.2 900 20 0 (air cooled) N

Full1 900 20 Full (5mins) N

Full2 900 20 Full (5mins) N

Notes: - First four samples (10.1, 15.1, 30.1, A.1) were placed in furnace at same time

- Second set of four samples (10.2, 15.2, 30.2, A.2) were placed in furnace after first set had been removed and furnace was allowed to come back up to temperature (900C)

-Third set of samples (Full1 and Full2) were placed in furnace after it had come back up to temperature


- For each set, samples were quenched consecutively in same tank of water starting with 10 than 15 than 30 (approximately 2L of cold tap water was added to tank between the 15 and 30 samples)

- Water in tank was replaced for fully quenched samples

- Full1 was quenched in tank first, being held in the vertical position for about 1 minute before allowing to sit on bottom of tank

-Full2 was quenched immediately after Full1 in same tank but with 2L of new cold water added

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