Modeling Constitutive Behavior and Hot Rolling of ...

Modeling Constitutive Behavior and Hot Rolling of Steels A Thesis Submitted for the Degree of

Doctor of Philosophy in the Faculty of Engineering

By

M. P. Phaniraj

Centre for Advanced Study

Department of Metallurgy INDIAN INSTITUTE OF SCIENCE Bangalore- 560 012, India December 2004

Synopsis

Constitutive equations are used in estimating the forces required to carry out any deformation process. The first part of this thesis is dedicated to developing constitutive equations for steels under hot working conditions. The second part deals with the simulation of the hot strip mill. The constitutive behavior of steels is complex especially under hot working conditions and has been the subject of study for many decades. A number of models are available in literature to predict flow stress of steel during hot deformation. In recent years, neural networks have also been used. Quantitative assessment of these models shows that the prediction errors range from 2-60% of the mean flow stress, when used over a range of strain rates (2-120 s-1), temperatures (900-1100 °C) and strains until 0.8. A neural network model, which can be used to predict flow stress for carbon steels, ranging from 0.03%C to 0.34%C, is proposed. The network is able to simulate the flow stress behavior with an average error of 3.7% of the mean flow stress using strain, strain rate, temperature and chemical composition as inputs. The effect of chemical composition can be included by either taking individual alloying elements as separate inputs or using the carbon equivalent. It is found that the effect of manganese on flow stress is best accounted for, by using carbon equivalent as input. The network is able to interpolate not only over the domain of strain rates and temperatures but also over the domain of carbon equivalents in which it is trained. Neural network models reported in literature have different structures i.e. number of layers and neurons, for different grades of steel. A single neural network structure, 3:4:1 is devised, which models the flow behavior better than existing semi-empirical models.

Its

application is demonstrated for four carbon steels, a microalloyed steel, an austenitic stainless steel and a high-speed steel. The neural network structure is less complex than the models available in literature. This network can be used as an alternative to semi-empirical models for constitutive behavior of steels. As an application of the constitutive equation, the simulation of deformation during hot rolling of vanadium microalloyed steel is chosen. Constitutive equations for a number of carbon

Synopsis

ii

steels are available in literature. However, the vanadium microalloyed steels have seldom been studied for their constitutive behavior. Further, the effect of vanadium on the hot strength of austenite is not clear in literature. Hot compression tests were carried out in the temperature range 850-1150 °C at 0.1-60 s-1 strain rates. The temperatures and strain rates cover the range in which hot rolling of this steel is carried. The hyperbolic sine equation and neural networks were used to model the constitutive behavior. Evaluating constitutive parameters, n, Q and A, using the usual log-log plots does not yield constant values. The n, Q and A values were determined simultaneously using a non-linear optimization procedure and were fitted as functions of strain. These relations, in conjunction with the hyperbolic sine equation describe the constitutive behavior of the steel. The 3:4:1 neural network is also used to predict the constitutive behavior. The network predicts flow stresses and peak strains with greater accuracy. Vanadium increases the peak strain, peak stress and mean flow stress of austenite significantly. The hot strip mill (HSM) of Jindal Vijaynagar Steels Limited (JVSL) at Toranagallu, Karnataka, India, is simulated. The temperature measurements in the mill are available only at the end of roughing mill and at the end of the finishing mill. Therefore, the heat transfer in the hot strip mill starting from the end of roughing to the end of deformation in the last stand of the finishing mill was simulated. DEFORM-2D, commercial finite element software was used to simulate heat transfer and deformation. Special programs in the UNIX environment were developed in order to adapt the software, suited for forming, to tandem rolling. The mill data for vanadium microalloyed steels were not available, however the simulations were carried out for plain carbon steel strips. Heat transfer coefficients in various zones of the HSM and the friction factors between the strip and roll were optimized to predict the rolling loads and finish rolling temperatures reasonably. Published work on HSM simulation is validated either with a single grade of steel or with one or two strip thicknesses. However, the variation in mill data is such that simulation of one or two strips is inadequate to give confidence in the model parameters like friction factor etc. In the present study, the applicability of the model was investigated for 18 plain carbon steel strips of final thickness 2.0-4.0 mm with 0.025-0.139%C. The loads and finish rolling temperatures were predicted with an accuracy of ±15% and ±15 °C, respectively. Comparing model predictions with industrial data indicated that the γ→γ+α transformation occurred in the finishing mill. Besides measurement errors in loads and temperatures, there are other sources of errors that are inherent to the hot strip mill. These are due to variation in roll gap setting, roll speed, roll flattening, change in roll diameter etc. Error analysis shows that uncertainty in calculated loads is greater than 6%.

Synopsis

iii

In between the stands of the finishing mill, the strip undergoes microstructure changes. The evolution of microstructure, namely recrystallisation and grain growth, was simulated using microstructure models available in literature. DEFORM-HT, an add-on heat treatment module with the commercial finite element software, DEFORM, was used to simulate the microstructure in between the passes. Incomplete recrystallisation retains a part of the strain in the strip. This could lead to an increase in the rolling load, or decrease in load due to dynamic recrystallisation or transformation, when the temperatures are close to transformation temperatures.

The

calculation of retained strain was not a feature of DEFORM-HT. A suitable modification was made to the program to incorporate the strain retained because of incomplete recrystallisation. The type and extent of recrystallisation and austenite grain size were obtained for plain carbon steel strips of final thickness 2.0, 3.5 and 4.0 mm. A change in the initial temperature of rolling by ~70 °C affects the final austenite grain size by ~9 µm. The final austenite grain size is determined by metadynamic recrystallisation occurring in the first few stands.

The

microstructure evolution equations reported by two different groups of researchers were used in simulations. It is observed that evolution equations from both groups lead to similar values of rolling loads. The microstructure is validated against experimental data for ferrite grain size and mechanical properties. The need for precision in controlling coiling temperature and cooling profiles on the run out table for hot rolled low carbon steel strips is investigated. It is claimed in literature that a high degree of automation and control of the run out table is required to control the yield strength variation to within 20 MPa. However, calculations based on models available in literature predict that the variation in strip temperature encountered in a typical run out table does not affect the mechanical property significantly. This is shown to be true even from experimental data available in literature. Thereby, even a coarse control over the run out table is adequate to achieve the desired yield strength. It is noted, however, that precise control is needed in HSLA steels.

Acknowledgements

I will cherish for life, the moments that I spent with Prof. Lahiri as guide and philosopher. Discussions with him not only help in thinking clearly but also increase the self-confidence. This has been the case with the over forty students and project assistants, whom I witnessed in the past five years, as with me. Needless to say, the thesis has been possible only because of his tutelage. Prof. K. Chattopadhyay, present Chairman and Prof. K. Natarajan, past Chairman of the Department of Metallurgy, for extending the facilities of the Department. Prof. Y.V.R.K Prasad, for useful discussions and for ready access to the facilities in the Processing Science laboratory. I wish to thank Prof. Raman for useful inputs on my thesis. I thank Prof. Seshan for extending the facilities of his laboratory Mr. Sasidhara; experiments using DARTEC and INSTRON would not have been possible but for him. Mr. Shamanna, for carrying out the experiments in the Foundry laboratory. I am also thankful to the staff of the workshop in the Department of Aerospace Engineering. JVSL authorities, especially Dr. Manjini, JVSL-HSM, who took time out of his busy schedule and made available the mill data, strip samples and any related information as and when I asked for it! Ravi Aggarwal made my visit to JVSL useful and enjoyable. Dr. Shamasundar, Mahesh, Chandrasekhar and others at AFTC; their cooperation and assistance was important during the initial stages of this work. Dipak, Ashwin, Sunil, Srikant, Pranesh and Binod whose assistance at different stages in the ‘larger’ rolling project were immensely helpful. I wish to specially thank Binod; we could experiment much more with DEFORM. There would hardly be any student who crossed the portals of this department and did not carry with him a little bit of Prof. Ranganathan. I am one of them. Interactions with him, though few, will be cherished forever. I wish to thank Prof. Chokshi and Aarti, Prof. Surappa, Prof. G.S. Gupta, Prof Subramanian and Prof. Abinandanan for their friendly overtures. Dr. Ravi, Dr. Avadhani, Dr. B.V. Narayana, Mr. Babu, Mr. Deshpande, Mr. A.V. Narayana, and

Acknowledgements

v

several others made life in the department pleasurable. The affectionate and caring nature of Prof. Chattopadhyay added to the ‘family feeling’ that is unique to the department. The Dept. staff: Mr. Padai, Mr. Gurulinga, Mr. Krishnamurthy, Mr. Srinivasa Murty, and Mr. Muniramappa, Mr. Mollaiah, Mr. Shiva, Mr. Gangadharaiya, Mr. Raghu, Mr. Narayanappa for being resourceful. Mr. Sudhir, Mrs. Thelma, Mrs. Sarojini and Mr. Srinivas made official work easy. Thanx are due to many friends; their magnanimity made me feel more at home in the institute. Phanikumar has been an intimate friend and a peer, at once. Shankara made learning music possible, thanx to the practice we did in his room. Breaks from IISc were movies, Coffee day, Barista, Javacity, Kadambam, Strand and other bookstalls; Guru was always available. Dream rides on my Kawasaki-4S to many places near Bangalore were possible, thanx to Sudhir and his CD100. Hospitality and warmth, Ram’s forte, will always be remembered. The trek to Tekkal will especially bring him to memory. Thanx to Kottada for ‘being there’, always. The two o’clock tea meeting that we continue to have will remain fresh in memory, ever. Chokshi’s lab was my post-lunch and dinner adda with Shankara, and Kottada, Sudhir, Shantanu, and others. Thanx to Atul for making it possible. Life in the lab was enjoyable, thanx to Subrata, Yogi, Pranesh, Eswar, Kajal, Mahajan, Azad, Manas, Vikas and many others. Having tea at half past three was a lab ritual and it served as a major get-together when our lab was 14 strong. I enjoyed it until the day when it was only two of us, me and Vikas. I wish to specially thank Subrata for taking time out of his schedule for proof reading my papers and thesis. That we could be close friends at work place, sharing the same table for 5 long years, can be credited only to him. Friends@dept_over_the_years: Ramakrishnan, Nagendra, Gandhi, Tania, Sandip, Manivelraja, Pragga, Bsrao, Viji, Dheepa, NRS, Mantha, Anandh, Sathya, Basa, Lnrao, Jeykumar Sharmishtha, Joy, Tripti, Suresha, Kavita, Deb, Victoria, Deep, Kris, Saravanan, Partho, Radhika, Kaythi, Rejin, Sudarshan, Prasad, Suresh (polymer, sms), Srikant, Yogesh, Rama, Padai and many others for making the stay in the department lively. Friends@iisc: Prithu, Raghu, Santosh, Diwan, Jeevan, Pramod, Arun rao, Tambat, Sai, Murali, Gvsk, Krishnan, Senthil, Saikat and others made the stay at iisc enjoyable. I would like to specially thank Sabita, CP and Mrs. Phanikumar for their timely help and cooperation. I can never adequately express my gratitude to Gnyanambal maami for introducing me to the world of Carnatic music. Her untiring efforts not only made us learn but also perform!

Acknowledgements

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The Ramakrishna math and Swami Harshananda’s lectures made life more meaningful. I shall remain eternally grateful to them. Prabhanna, Deepthi-vahini, Bhavya and Balram provided me the ‘much-needed break’, as Bhavya tried hard to convince me, from the routine at IISc. Sisters: Harini, Lavanya and Vasuki for their support and understanding. My nephew, Bhargav, for posing questions that bordered between amusing and annoying. I owe a great debt of gratitude to my parents who let me free. There are many more than I have recounted above from whom I have taken help in some form or other. I wish to thank all of them. M. Phaniraj Bangalore, December 2004

Contents

Synopsis

i

Acknowledgements

iv

Contents

vii

1

Introduction………………………………………………………………………1

2

Neural network model for carbon steels 2.1

Introduction………………………………………………………………….……4

2.2

Constitutive behavior models…………………..…………………………………4

2.3

2.4

2.5

3

2.2.1

Phenomenological…………………………………………………...……4

2.2.2

Semi-empirical……………………………………………………………6

2.2.3

Neural networks…………………………………..………………………8

Neural network model for carbon steels……………………...…………………12 2.3.1

The database…………………………………..…………………………12

2.3.2

The model………………………………………..………………………13

2.3.3

Training the neural network…………..…………………………………13

2.3.4

Chemical composition as input……….…………………………………15

Results and discussion…………………………...………………………………15 2.4.1

Effect of chemical composition…………………………………………19

2.4.2

Initial microstructure as input…………………...………………………22

Conclusions……………………………………………………...………………22

A neural network structure for carbon and alloy steels 3.1

Introduction…………...…………………………………………………………24

3.2

The database…………………..…………………………………………………25

3.3

The neural network model………………………………………………………26

3.4

Results and discussion…………...………………………………………………26

Contents 3.5

4

viii Conclusions………………………...……………………………………………32

Vanadium microalloyed steel: Hot compression behavior and modeling 4.1

Introduction……………………………………………………………………...33

4.2

Experimental…………………….………………………………………………34

4.3

Results…………….……………………………..………………………………35 4.3.1

Adiabatic heating………………………..………………………………35

4.3.2

Stress-strain behavior……………………………………………………38 4.3.2.1 DRX……………………………..………………………………38

4.4

Constitutive modeling……………………………...……………………………38 4.4.1

Semi-empirical approach………………………..………………………38 4.4.1.1 Model predictions……….…………………….…………………43

4.4.2

Neural network model………….……………………………..…………44 4.4.2.1 Model predictions…….…………………………………………44

4.5

Discussion…………………………….…………………………………………46 4.5.1

Effect of vanadium………………………………………………………48 4.5.1.1 Effect of precipitates………….…………………………………50

4.6

5

Conclusions……………………………………………...………………………50

Thermomechanical modeling of a hot strip mill Part-I Prediction of loads and finish rolling temperatures 5.1

Introduction…………………………………………………….………………..52

5.2

Intercritical deformation…………………………………………………..……..53

5.3

5.2.1

Microscopy………………………………………………………..……..53

5.2.2

Tensile tests……………………………………………………………...53

The model………………………………………………..………………………55 5.3.1

Heat transfer and deformation…………………………...………………57

5.3.2

HTC in different zones of the mill………………………………………60 5.3.2.1 Convection to air……………………………………..………….60 5.3.2.2 Descaler box……….……………………………….………….60 5.3.2.3 Roll gap………….…………………………………………….61

5.3.3

Emissivity………………………………………………………………..61

5.3.4

Coiling………………………………..………………………………….62

5.3.5 Microstructure…………………………………...……………..………..62 5.4

Input data……………………………………………...……………….………..62

Contents

ix 5.4.1

Hot strip mill data………………………………………………………..62

5.4.2

Thermal properties………………………………………………………65

5.4.3

Constitutive equation……………………………………………………66

5.4.4

Microstructure models…………………………………………………..67

5.5

Solution procedure………………………………………………………………68

5.6

Sensitivity analysis………………………………………………………………68 5.6.1 Number of mesh elements..………………………..………………………68 5.6.2

Friction factor and roll gap HTC…………………………………………..69

5.7

Simulation strategy………………………………………………………………70

5.8

Simulation results…………………………………………………………….….72 5.8.1

Heat transfer…………………………………………………………..…72

5.8.2 Strain and strain rate……………………………………………….…….74 5.8.3 Loads………………………………………………………………….…76 5.8.4

Roll pressure…………………………………………………………….76

5.8.5 Microstructure…………………………………………………………...78 5.9

Errors inherent in load prediction……………………………………………….78 5.9.1

Change in roll diameter………………………………………………….78

5.9.2

Roll flattening……………………………………………………………78

5.9.3 Work roll temperature…………………………………………………...79 5.9.4 Thickness error…………………………………………………………..79 5.10

Discussion…………………………………………………………………….…79 5.10.1 Modifying Shida’s equation……………………………………………..79 5.10.2 Strain induced transformation………………………….………………..81 5.10.3 Effect of oxide layer on roll chill………………………………………..82 5.10.4 Roll-gap heat transfer coefficient………………………………………..83

5.11

6

Conclusion………………….……………………………………………………84

Thermomechanical modeling of a hot strip mill Part-II Microstructure evolution 6.1

Introduction………………..……………………….……………………………85

6.2

Model……………………….……………………..…………………………….86 6.2.1

Recrystallisation…….…………………..……………………………….86 6.2.1.1 SRX………….………………..…………………………………87 6.2.1.2 MDRX……….…………………………………………………..87

Contents

x

6.3

6.4

6.5

7

6.2.2

Grain growth…………………………………………………………….87

6.2.3

Partial recrystallisation……………………………………………..……87

Simulation results………………………………………………………………..90 6.3.1

Recrystallisation and grain growth………………………………………90

6.3.2

Austenite grain size……………………………………………………...93

Discussion……………………………………………………………………….94 6.4.1

Choice of microstructure model…………………………………………96

6.4.2

Microstructure after ROT………………………………………………..96

Conclusions………………………………………………………………….…100

Relevance of cooling control on a run out table in a hot strip mill 7.1

Introduction…………………………………………………………………..101

7.2

Experimental data on cooling rate dependence of ferrite grain size…………...102

7.3

Yield strength predictions……………………………………………………...103

7.4

Cooling rate deviation in the strip……………………………………………...104

7.5

Coiling temperature…………………………………………………………….106

7.6

Discussion……………………………………………………………………...107

7.7

Conclusions…………………………………………………………………….108

8

Summary………………………………………………………………….……109

9

Scope for further work………………………………………………………..111

References……………………………………………………………………………..113 List of publications……………………………………………………………………126

Chapter 1 Introduction

Hot working processes such as rolling, extrusion, or forging are used in the first step of converting a cast ingot into a wrought product. They are usually carried out at temperatures above 0.6Tm and at high strain rates in the range 0.5-500 s-1 [1]. Determination of the load required to carry out these processes is of prime importance. The load depends on the flow stress of the material besides the geometry of deformation and the friction at the tool-workpiece interface. The flow stress depends on the initial microstructure and, the strain, strain rate and temperature applied during deformation. The equation that relates the flow stress to these variables is known as the constitutive equation and is given as

σ = f (S , ε , ε&, T )

(1.1)

where, ε is strain, ε& is strain rate, T is temperature and S denotes the microstructure variables such as grain size and dislocation density.

For steels, the hot working temperatures are,

typically, in the austenite phase field. The stress-strain behavior of steels, in general, follows two types of behavior viz. dynamic recrystallisation (DRX) and dynamic recovery (DRY), as shown in Fig. 1.1. DRX is characterized by a steady rise in flow stress with strain to a peak

σ

σs (Dynamic recovery)

σp σc

σss (Dynamic recrystallisation)

σy εy εc εp Figure 1.1

ε

Schematic of the types of stress-strain behavior and the characteristic points on

stress strain curve.

Introduction

2

followed by a steady drop to a certain value at higher strains. DRY is characterized by a steady rise in stress followed by a plateau when the steady state stress is reached. Modeling constitutive behavior for hot working conditions has been a topic of research for a few decades [2-7] now, and still is pursued actively. This fact goes to indicate the complex nature of constitutive behavior of steels. Most of the models consist of two expressions for the evolution of stress: the first predicts the work hardening and dynamic recovery region and the second predicts the softening caused by dynamic recrystallisation, with the transition at the critical strain, εc, or peak stress, σp. The reduction in stress following the peak is determined by an Avrami-type of equation. The accuracy of the prediction depends on the proper choice of σy,

σs, σss, εc, shown in Fig. 1.1, and activation energy of deformation, Qdef. Often, correct estimation of these values from the experimental data is difficult. Recently, neural networks are being increasingly [8-11] used to model constitutive behavior of steels.

The neural network represents a generalized non-linear curve fitting

technique. The advantage with neural networks is that there is no need to assume prior form of relationship between the parameters.

Further, it can approximate any complex non-linear

function [12]. Neural network models for different grades of steels are found in literature [8,1011]. However, it is observed that the neural network structure, i.e. the number of neurons and layers, is different for different grades of steels. It is worthwhile to investigate if a single neural network structure can serve as a model for any grade of steel. Typically, hot compression/torsion tests are carried out to generate stress-strain data needed to develop a constitutive equation. Such data are available in literature for a number of carbon steels. However, a model is lacking that can accommodate the effect of chemical composition and predict the flow stress for these varied data. Neural networks are known to approximate fairly complex data and can be used to develop such a model. Vanadium microalloyed steels are commonly hot rolled because of the superior mechanical properties they offer [13]. However, the constitutive behavior of these steels has seldom been studied. Some researchers [14] have used constitutive equations developed for carbon steels to model the deformation behavior of vanadium microalloyed steels. Others [1517] have reported indicating that vanadium increases the strength but they differ in the extent to which it increases the strength. Thereby, investigating the deformation behavior of vanadium microalloyed steels is of interest. The neural network structure discussed above can be used to model the constitutive behavior of this steel.

Introduction

3

The need for accurate constitutive equations is more due to the increasing use of finite element methods in simulating metal forming processes. The hot rolling of strips is one of the more common processes to be simulated using finite element methods [18-24]. Typically, in a hot strip mill the steel slab is reduced in thickness to nearly 1/100th its thickness at a series of rolling stands in tandem. The objective in simulating the process is, primarily, to predict the rolling loads and the finishing rolling temperatures. This involves determining a number of parameters such as the heat transfer coefficients (HTC) in the different zones in a hot strip mill, friction factor between roll and strip, etc. Empirical equations relating the heat transfer coefficients to the mill parameters such as rolling pressure [20], strip speed [25] etc., are available in literature. However, most investigators have selected heat transfer coefficients such that the simulation results agree well with the mill data. The friction factor between the roll and strip is also chosen with the same objective. Published work dealt with HSM simulation either of a single grade of steel or confined to one or two strip thicknesses. However, the variation in mill data is such that simulation of one or two strips is inadequate to give confidence in the model parameters like HTC, friction factor etc. There are simple formulae available in statistics [26] that determine the data size required for a given confidence interval. In between rolling passes, the deformed strip undergoes changes in the microstructure due to recrystallisation and grain growth. Often, there is little time for complete recrystallisation to occur. As a result of incomplete recrystallisation, some strain is retained in the strip, which influences the load in the next stand. With strain retained in the strip, when the strip temperature falls close to the transformation temperature, as it does in the later stands in the finishing mill, strain induced transformation from austenite to the ferrite [27] can occur. This has significant effects on the load and the microstructure [28]. The present study includes the simulation of strips with their finish rolling temperature close to the transformation temperature. Mechanical properties of low carbon steels with predominantly ferrite microstructures are controlled by the ferrite grain size. The ferrite grain size depends on the austenite grain size, retained strain before cooling, and the cooling rate on the run out table. Several models [29-31] are found in literature, which relate the grain size to these parameters. A number of papers [3233], however, aim at achieving extremely precise control of cooling rate on the run out table. It is of interest to determine how precise a cooling rate is desired given the allowable variation in the mechanical properties of low carbon steels.

Chapter 2 Neural network model for carbon steels

2.1 Introduction There are has been a consistent effort over the past two decades towards developing constitutive models which would be give a complete mathematical description of the flow curves. This has been mostly due to the increasing use of FEM in solving a wide variety of industrial problems. However the need for greater accuracy and precision in prediction persists. Hodgson et al. [10] have reported that for an off line hot rolling mill simulator the need is of a constitutive model which can predict flow stress within 5%, over a wide range of deformation conditions. This implies that for high temperature, low strain rate conditions the error in flow stress prediction should be as low as 5 MPa or lesser. In literature, methods adopted to develop constitutive equations for various grades of carbon steels fall into three categories viz. Phenomenological, Semi-Empirical and ANN. In this chapter the applicability of the models based on these methods is critically reviewed. Steels have been studied for their hot working behavior for many decades now. Thereby, data in the form of experimental stress-strain curves is available in literature for a number of carbon steels. A neural network model developed using stress-strain data for carbon steels, available in literature, is described in the following sections. Such a model would be industrially useful for e.g. in optimizing rolling schedules of new grades of steel, which otherwise would have involved carrying out laborious hot deformation tests for every new grade of steel.

2.2 Constitutive behavior models 2.2.1 Phenomenological The basic idea in this approach is that the evolution of dislocation density during deformation can be represented as a sum of two independent terms: dislocation storage or multiplication, and

Neural network model for carbon steels dislocation annihilation.

5

The former represents work hardening and the latter, dynamic

recovery. The models [34-37] differ in their definition of these terms. Estrin and Mecking (EM) [36] assumed the contribution due to work hardening to be constant. The term associated with dynamic recovery term is assumed to follow first order kinetics i.e. to be linear in ρ, the dislocation density. The evolution equation then is

dρ = k − k2 ρ dε

(2.1)

where, k is a constant and k2 is a function of strain rate and temperature because dynamic recovery is thermally activated. The mechanical strength [36] of obstacles to dislocation glide is related to the dislocation density as

σ = αGb ρ

(2.2)

where, G is the shear modulus, b is the magnitude of Burger’s vector and α is a numerical constant of order unity. Combining Eq. (2.1) and Eq. (2.2), and rearranging,

θσ = A − Bσ 2

(2.3)

where, θ = dσ dε is the work hardening coefficient, A and B are constants that are a function of strain rate and temperature. Eq. (2.3) after integration becomes [38]

σ = [σ s2 + (σ y2 − σ s2 ) exp(−2 Bε )]

1/ 2

(2.4)

where, σy is the initial flow stress when ε = 0 and σ s =

A B is the saturation stress in the

absence of dynamic recrystallisation. The EM model considers only the work hardening region. Kong and Hodgson [38] used Eq. (2.4) to predict the stress-strain curves for carbon steels. They modified the EM model to incorporate dynamic recrystallisation as follows

σ = [σ s2 + (σ y2 − σ s2 ) exp(−2 Bε )]

when ε