Thermo-Mechanical Processing of Dual-Phase Steelsand Its

Thermo-Mechanical Processing of Dual-Phase

Steels and Its Effects on the Work Hardening

Behaviour

By

Hossein Seyedrezai

A thesis submitted to the

Department of Mechanical and Materials Engineering

in conformity with the requirements for

the degree of Doctor of Philosophy

Queen’s University

Kingston, Ontario, Canada

December 2014

Copyright© Hossein Seyedrezai, 2014

To my Mother and Father

i

Abstract

This thesis focuses on understanding the relationship between the microstructure

and the different work hardening mechanisms of DP steels. Through the applica-

tion of various thermo-mechanical processing schedules prior to inter-critical (IC)

annealing, five distinctly different microstructural variants were produced. The work

hardening behaviour of the five microstructural variants was examined in terms of

the true work hardening rate, θ, the instantaneous work hardening exponent, n, and

the dislocation annihilation factor, h. Additionally, back stresses were measured in

selected microstructural variants having similar martensite volume fraction of ∼15%,

using a custom-made in-plane forward-reverse shear testing fixture.

At small strains (<2%), the work hardening behaviour was found to be domi-

nated by the introduction of back stresses and the generation of GNDs in the ferrite

matrix. The work hardening response at this stage was characterized by θǫp=0.5%

and a minimum value in the instantaneous work hardening exponent, nmin. Both of

these parameters were determined to be functions of√

f/d (f is the volume fraction

and d is the size of martensite particles), the mean ferrite grain size as well as the

morphology and spatial distribution of martensite particles.

At higher strains (2-3%), a maximum value in the instantaneous work hardening

exponent is reached (nmax). This parameter, which can be considered as the work

hardening capacity of the material, was found to be a function of mean ferrite grain

size but is independent of√

f/d. The relative contribution of back stresses was

also found to reach a constant value at a similar von Mises equivalent strain. This

observation suggests that at strains above those associated with nmax, other work

ii

hardening mechanisms become more important.

At strains over 4%, dislocation annihilation by dynamic recovery becomes the

controlling factor for the rate of work hardening. This phenomenon is described by

the dislocation annihilation factor, h, and is a function of√

f/d, the mean ferrite

grain size as well as the morphology and spatial distribution of martensite particles.

Finally, it was concluded that the ideal DP microstructure will contain a uniform

distribution of fine, equiaxed martensite particles in a fine, equiaxed ferrite matrix.

iii

Co-Authorship

This dissertation is based on the following manuscripts which are published, sub-

mitted or will be submitted for publication:

Chapter 3

H. Seyedrezai, A.K. Pilkey, J.D. Boyd, “Effect of Pre-IC Annealing Treat-

ments on the Final Microstructure and Work Hardening Behaviour of a

Dual-Phase Steel”, Published in: Materials Science and Engineering A,

vol. 594 (2014) pp. 178–188

Chapter 4

H. Seyedrezai, A.K. Pilkey, J.D. Boyd, “Effect of Ferrite Grain Size and

Spatial Distribution of Martensite Particles on the Work Hardening Be-

haviour of a Dual-Phase Steel”, Submitted to: Materials Science and

Engineering A (currently under review)

Chapter 5

H. Seyedrezai, A.K. Pilkey, J.D. Boyd, “Measurement of Back Stress

Contribution to Work Hardening”, Will be condensed and submitted to:

Materials Science and Engineering A or Acta Materialia

The work presented in this dissertation is original and my own with co-authors

acting in an advisory capacity.

iv

Acknowledgements

I would like to start by thanking my supervisors Professor Doug Boyd and

Professor Keith Pilkey for their guidance and support during my work at Queen’s

University. They were with me every step of the way and they gave me the knowl-

edge and understanding required to be a better researcher. The completion of this

thesis would have not been possible without their continued advice, patience, encour-

agement and friendship.

I am grateful to Professor Bradley Diak for his advice on mechanical testing proce-

dures and equipment. A big thank you also goes to Mr. Charlie Cooney. His endless

help and guidance in the lab made all of my experimental challenges easier. Further-

more, the McLaughlin Hall Machine Shop members, especially Mr. Andy Bryson, are

appreciated for their understanding and support. I would also like to acknowledge

the financial support of the AUTO21 NCE and the Natural Sciences and Engineering

Research Council of Canada (NSERC).

Of course, many thanks go to my friends and colleagues who made my journey

at Queen’s more rewarding. There are simply too many people to name but specifi-

cally, I am grateful to (alphabetical order): Grant Bell, Jasmine Chiang, Christopher

Cochrane, Andrew Sloan and Christopher Walasek.

Finally, I am forever grateful to my family; to my mom and dad, who have been

providing me with endless love, support and encouragement all of my life, without

which, I would have not been the man that I am today. And to my sister, Shadi, who

has always been there for me with her kindness and care.

v

Table of Contents

Abstract ii

Co-Authorship iv

Acknowledgements v

List of Tables xi

List of Figures xiii

List of Abbreviations xviii

List of Symbols xix

Chapter 1: Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Dual-Phase Steels in the Automotive Industry . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2: Literature Survey 8

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Processing of DP Steels . . . . . . . . . . . . . . . . . . . . . . . . . . 8

vi

2.2.1 Inter-Critical Annealing . . . . . . . . . . . . . . . . . . . . . 10

2.3 Mechanical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1.1 Residual Stresses . . . . . . . . . . . . . . . . . . . . 15

2.3.1.2 Transformation Dislocations . . . . . . . . . . . . . . 16

2.3.2 Work Hardening . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2.1 Back-Stress Hardening . . . . . . . . . . . . . . . . . 24

2.3.2.2 Geometrically Necessary Dislocations . . . . . . . . . 28

2.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter 3: Effect of Pre-IC Annealing Treatments on the Final Mi-

crostructure and Work Hardening Behaviour of a Dual-

Phase Steel 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Heat Treatments . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.3 Microstructure Characterization . . . . . . . . . . . . . . . . . 43

3.2.4 Mechanical Testing . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.2 Uniaxial Tensile Behaviour . . . . . . . . . . . . . . . . . . . . 53

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.1 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 56


3.4.2.1 True work hardening rate . . . . . . . . . . . . . . . 59

vii

3.4.2.2 Instantaneous work hardening rate . . . . . . . . . . 63

3.4.2.3 Dislocation annihilation factor . . . . . . . . . . . . . 65

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Chapter 4: Effect of Ferrite Grain Size and Spatial Distribution of

Martensite Particles on the Work Hardening Behaviour

of a Dual-Phase Steel 74

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


4.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.2 Thermo-Mechanical Processing . . . . . . . . . . . . . . . . . 77

4.2.3 Microstructure Characterization . . . . . . . . . . . . . . . . . 79

4.2.4 Mechanical Testing . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81



4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91



4.4.2.1 True Work Hardening Rate . . . . . . . . . . . . . . 96

4.4.2.2 Instantaneous Work Hardening Rate . . . . . . . . . 100

4.4.2.3 Dislocation Annihilation Factor . . . . . . . . . . . . 102

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

viii

4.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Chapter 5: Measurement of Back Stress Contribution toWork Hard-

ening 112

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112


5.2.1 Shear Specimen Geometry . . . . . . . . . . . . . . . . . . . . 118

5.2.2 Shear Fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.3 Stress and Strain Measurements . . . . . . . . . . . . . . . . . 126

5.2.4 Shear Experiments . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2.5 Shear Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 132

5.2.5.1 Calibration Experiments . . . . . . . . . . . . . . . . 132

5.2.5.2 Back Stress Experiments . . . . . . . . . . . . . . . . 135

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3.1.1 Strain Maps . . . . . . . . . . . . . . . . . . . . . . . 135

5.3.1.2 Shear Stress - Shear Strain Plots . . . . . . . . . . . 140

5.3.1.3 Back Stress Calculations . . . . . . . . . . . . . . . . 143

5.3.2 Back Stress Experiments . . . . . . . . . . . . . . . . . . . . . 145

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Chapter 6: Complementary Discussion 169

6.1 Microstructure Evolution During IC Annealing . . . . . . . . . . . . . 169

6.2 Effect of Microstructural Parameters on Work Hardening of DP steels 171

ix

6.3 Practical Implications of the Present Research . . . . . . . . . . . . . 178

6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Chapter 7: Conclusions and Future Work 182

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.2 Original Contributions to the Field . . . . . . . . . . . . . . . . . . . 184

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Appendix A: Specimen Designation Conversions 189

Appendix B: Kocks-Mecking Fitting Procedure 190

Appendix C: Technical Drawings of the Shear Fixture 191

x

List of Tables

3.1 Chemical compositions of DP780-CR and IF steel sheets (in wt. %). . 41

3.2 Summary of specimens and their respective heat treatment schedules 43

3.3 Martensite particle measurements for microstructures shown in Figure

3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Carbide particle measurements for microstructures shown in Figure 3.2 52

3.5 Martensite particle measurements for three different starting condi-

tions after IC annealing at 735◦C . . . . . . . . . . . . . . . . . . . . 58

3.6 Work hardening rates, θ, at plastic strain of 0.5% and dislocation an-

nihilation factor, h, for the uniaxial tensile data shown in Figure 3.4. 68

4.1 Chemical composition (in wt. %) of DP780-CR and DP780-HB steel

sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Summary of TMP schedules for the three microstructure variants . . 78

4.3 Ferrite and martensite measurements for microstructures shown in Fig-

ure 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Uniaxial Tensile parameters calculated from Figure 4.4a . . . . . . . . 91

4.5 The dislocation annihilation factor, h, and the uniform elongation val-

ues for the three microstructural variants of Figure 4.4. . . . . . . . . 105

5.1 Chemical composition of the steels used in this research . . . . . . . . 128

5.2 Details of the shear testing calibration experiments . . . . . . . . . . 130

5.3 Microstructural parameters of the five DP780 microstructural variants

with similar volume fraction of martensite particles . . . . . . . . . . 130

xi

5.4 Details of the back stress experiments for the DP780 microstructural

variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.5 Comparison of the experimental back stress values (0.002 offset method)

with the permanent softening back stress defined as 0.5 × ∆τPS for

DP780 microstructural variants after average real forward pre-shear

strain of 0.263 ± 0.015 . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.6 Comparison of the experimental back stress values (0.002 offset) with

the predictions of the B & S model for DP780 microstructural variants

after average real forward pre-shear strain of 0.263 ± 0.015 . . . . . . 156

5.7 The Bauschinger stress parameter (β0.002σ ) obtained from forward-reverse

shear tests of DP780 microstructural variants and IF steel after average

real forward pre-shear strain of 0.263 ± 0.015 . . . . . . . . . . . . . 162

A.1 Specimen designation conversions . . . . . . . . . . . . . . . . . . . . 189

xii

List of Figures

1.1 Plots of the uniform elongation values for different steel microstructures

with respect to their tensile strength . . . . . . . . . . . . . . . . . . 2

2.1 Schematic of a typical industrial galvannealing schedule for the pro-

duction of dual-phase steel . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Tensile behaviour of DP steels in comparison with other HSLA steels 15

2.3 Mobile dislocations near ferrite/martensite interface . . . . . . . . . . 17

2.4 Application of EBSD in calculating GND distribution . . . . . . . . . 18

2.5 Micro-pillar test results showing variations of ferrite strength at ferrite

grain interior and at the locations close to the ferrite/martensite interface 19

2.6 An example of a Hollomon analysis of DP steel’s tensile behaviour . . 21

2.7 An example of Jaoul-Crussard analysis of DP steel’s tensile behaviour 22

2.8 Plastic inhomogeneity in a DP microstructure captured by a Digital

Image Correlation (DIC) technique . . . . . . . . . . . . . . . . . . . 22

2.9 Variation in the magnitude of back stresses as a function of pre-strain

in a DP steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Schematic representation of the evolution of SSD and GND densities

with strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 SEM micrographs of the CR, AT, Q2T, Q7T and Q12T conditions . . 48

3.2 SEM micrographs of the CR+IC25, AT+IC20 and Q7T+IC35 mi-

crostructural variants at both low and high magnifications . . . . . . 51

xiii

3.3 Bright field TEM image of a thin foil and a carbon extraction replica

specimens obtained from the Q7T+IC35 microstructural variant . . . 52

3.4 Engineering stress-strain tensile curves and the instantaneous work

hardening versus engineering strain plots for the CR+IC25, AT+IC20

and Q7T+IC35 microstructural variants . . . . . . . . . . . . . . . . 54

3.5 Engineering stress-strain curves for CR+IC microstructural variants

tested along RD and TD directions . . . . . . . . . . . . . . . . . . . 55

3.6 SEM micrograph of the Q2T+IC20 (30) specimen. The heat treat-

ment history involves the QT (920◦C) pre-treatment followed by IC

annealing at 720◦C for 30 minutes. . . . . . . . . . . . . . . . . . . . 57

3.7 The true work hardening rate, θ, plotted as a function of√

f/d for

the CR+IC, AT+IC and QT+IC microstructural variants at plastic

strains of 0.5%, 2% and 5% . . . . . . . . . . . . . . . . . . . . . . . 60

3.8 Minimum and maximum instantaneous hardening exponents, nmin and

nmax, plotted as a function of√

f/d for CR+IC, AT+IC and QT+IC

microstructural variants . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.9 Dislocation annihilation factor, h, plotted as a function of√

f/d for

the CR+IC, AT+IC and QT+IC microstructural variants . . . . . . 67

4.1 SEM micrographs of the HB, AT and QT conditions . . . . . . . . . 82

4.2 SEM micrographs of the CR, AT+CR8 and QT+CR8 conditions . . . 84

4.3 SEMmicrographs of CR+IC725, AT+CR8+IC720 and QT+CR8+IC720

microstructural variants at both low and high magnifications . . . . . 87

4.4 Engineering stress-strain tensile curves and the instantaneous work

hardening vs. engineering strain plots for the CR+IC725, AT+CR8+IC720

and QT+CR8+IC720 microstructural variants . . . . . . . . . . . . . 90

xiv

4.5 Microstructure of the QT+CR8 specimen heated to the IC annealing

temperature and immediately quenched without any holding time . . 94

4.6 The true work hardening rate, θ, plotted as a function of√

f/d for the

CR+IC, AT+CR8+IC and QT+CR8+IC microstructural variants at

plastic strains of 0.5% and 2% . . . . . . . . . . . . . . . . . . . . . . 97

4.7 Minimum and maximum instantaneous hardening exponents, nmin and

nmax, plotted as a function of√

f/d for the CR+IC, AT+CR8+IC and

QT+CR8+IC microstructural variants . . . . . . . . . . . . . . . . . 101

4.8 Dislocation annihilation factor, h, plotted as a function of√

f/d for

the CR+IC, AT+CR8+IC and QT+CR8+IC microstructural variants 104

5.1 Schematic of forward-reverse deformation test results . . . . . . . . . 113

5.2 Geometry of the shear specimen . . . . . . . . . . . . . . . . . . . . . 121

5.3 The shear fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.4 The shear test assembly procedure . . . . . . . . . . . . . . . . . . . 125

5.5 Example of imposed grid pattern: “Full Grid” specimen used for cal-

ibration tests and “Camera Grid” specimen used in both calibration

and back-stress experiments . . . . . . . . . . . . . . . . . . . . . . . 127

5.6 Details of the offset and the softening (τF−τR) methods used in analysis

of the shear results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.7 IF steel specimen after forward shear (5mm displacement) . . . . . . 136

5.8 Distribution of normal and shear strain in IF steel specimen prior to

reverse loading (5mm forward displacement) . . . . . . . . . . . . . . 137

5.9 Distribution of normal and shear strain in IF steel specimen at the

end of the reverse loading after the initial forward deformation (5mm

forward-5mm reverse displacements) . . . . . . . . . . . . . . . . . . 139

xv

5.10 Shear strains measured in the left shear zone plotted against shear

strains in the right shear zone . . . . . . . . . . . . . . . . . . . . . . 140

5.11 Total shear and accumulative shear plots for IF steel . . . . . . . . . 142

5.12 Back stress (τ offsetB ) measurements plotted against forward pre-shear

strain for the calibration IF steel specimens . . . . . . . . . . . . . . 144

5.13 τF −τR plotted against reverse shear strains for the calibration IF steel

specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.14 Accumulative shear stress-shear strain (raw) plots for DP780 microstruc-

tural variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.15 Evolution of Back stress (τ 0.002B ) as a function of forward pre-shear for

the DP780 microstructural variants . . . . . . . . . . . . . . . . . . . 150

5.16 τF − τR plotted against reverse shear strains for all forward pre-shear

experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.17 Bauschinger stress parameter, β0.002σ , plotted as a function of forward

pre-shear strain for DP780 microstructural variants . . . . . . . . . . 162

6.1 The true work hardening rate at ǫP = 0.5% (θǫP=0.5%), and the min-

imum instantaneous work hardening exponents, nmin, for the five DP

steel microstructural variants investigated in this research . . . . . . . 173

6.2 The maximum instantaneous work hardening exponents, nmax, for the

five DP steel microstructural variants investigated in this research . . 175

6.3 Uniform elongation values of the five DP steel microstructural variants

plotted against their respective tensile strengths . . . . . . . . . . . . 178

B.1 Example of the Kocks-Mecking fitting procedure . . . . . . . . . . . . 190

C.1 Drawing of the Shear Specimen . . . . . . . . . . . . . . . . . . . . . 192

xvi

C.2 Drawing of the Shear Fixture’s Base . . . . . . . . . . . . . . . . . . 193

C.3 Drawing of the Shear Fixture’s Cover . . . . . . . . . . . . . . . . . . 194

C.4 Drawing of the Shear Fixture’s Plunger . . . . . . . . . . . . . . . . . 195

C.5 Drawing of the Shear Fixture’s Spacer . . . . . . . . . . . . . . . . . 196

C.6 Drawing of the Shear Fixture’s Connector to the Instron . . . . . . . 197

xvii

List of Abbreviations

A1 Eutectoid transformation temperature

A3 Austenite/Austenite+Ferrite transformation temperature

AHSS Advanced High Strength Steel

B & S Brown and Stobbs (model)

DP Dual-Phase (steel)

GND Geometrically Necessary Dislocations

HSLA High-Strength Low-Alloy (steel)

IC Inter-Critical (annealing)

IF Interstitial-Free (steel)

KM Kocks-Mecking (model)

SSD Statistically Stored Dislocations

Stage 1, 2, 3 Stages 1, 2 and 3 of work hardening for DP steels based on the

Jaoul-Crussard analysis

Stage I, II, III Stages I, II and III of work hardening for single crystals

Stage A, B, C Stages A, B and C of work hardening for DP steels based on the

instantaneous work hardening exponent analysis

UTS Ultimate Tensile Strength

xviii

List of Symbols

b Burgers vector

C carbon content, wt.%

D ferrite grain diameter (size)

d martensite particle diameter (size)

E modulus of elasticity

f volume fraction of martensite

h dislocation annihilation factor

n instantaneous work hardening exponent

m Ludwik work hardening exponent

r martensite particle radius

βOffsetσ Bauschinger stress parameter (offset method)

γ shear strain

γp plastic shear strain

γ∗

p unrelaxed plastic shear strain

ǫ normal strain

ǫt martensitic transformation strain in ferrite

ǫp normal plastic strain

θ true work hardening rate

θ0 athermal work hardening rate

∆τPS permanent softening

λg geometric slip distance

xix

ν Poisson’s ratio

µ shear modulus

ρ dislocation density (general term)

ρGND geometrically necessary dislocations (GNDs) density

ρSSD statistically stored dislocations (SSDs) density

ρtotal total dislocation density

σ normal stress

σres equivalent residual stress

σ0 baseline strength

σB normal back stress

σS non-directional strengthening component

σv scaling stress (Kocks-Mecking model)

σy yield stress

τ shear stress

τB shear back stress (general term)

τOffsetB shear back stress (offset method)

τF forward flow stress

τUnloadF forward unloading stress

τR reverse flow stress

τOffsetR reverse yield stress (offset method)

φ accommodation factor (Brown and Stobbs model)

xx

CHAPTER 1

Introduction

1.1 Background

1.1.1 Dual-Phase Steels in the Automotive Industry

Through the years, the automotive industry has been continuously working to

improve the performance of vehicles while satisfying stricter safety requirements. Ad-

ditionally, rising oil prices, increasingly stringent emissions regulations as well as

the environmental consciousness of consumers demand significant weight reductions

to increase fuel efficiency while reducing green house gas emissions. The challenge,

therefore, is to achieve this weight reduction goal while concurrently enhancing the

safety and crashworthiness of the vehicle.

With improvements in processing technologies, many new lightweight materials,

mainly aluminum alloys, are emerging as alternatives to conventional high strength

steels. To ensure competitiveness of ferrous alloys, new grades of steel are being

produced which have combinations of superior strength, good formability and good

failure properties. These new steels, referred to as Advanced High Strength Steels

(AHSS), allow for the implementation of thinner gauge and higher strength steel

sheets for structural components. As seen in Figure 1.1, the AHSS grades have higher

uniform elongation values for a given tensile strength compared to conventional steels,

such as HSLA [1].

The present research is focused on an AHSS grade called Dual Phase (DP) steel

which consists of ferrite and martensite phases. This steel was originally developed

1

Chapter 1 Page 2

Figure 1.1: The so called “banana diagram” which plots the uniform elongation of different steelmicrostructures with respect to their tensile strength. Adopted from [1]

in the 1960s but it gained significant attention almost a decade later following the

energy crisis of the 1970’s and the dramatic increase in fuel prices. Between 1977 and

1981, three major conferences were held on the processing and mechanical behaviour

of DP steels. The results presented in these conferences form the basis of the ma-

jority of research in this field [2–4]. The widespread application of DP steels in the

automotive industry, however, didn’t occur until the early 2000’s. In the past decade,

a significant percentage of the body-in-white (BIW) is progressively being made from

AHSS grades, including DP steels.

Compared to conventional HSLA, DP steels demonstrate continuous yielding,

very high initial work hardening rates, high ultimate tensile strength (UTS)-to-yield

strength ratios and moderate uniform elongation values. The combination of these

properties make DP steels advantageous for many forming processes at moderate

strains. The formability of DP steels at large strains, however, can become limited

mainly by the susceptibility of this material to development of significant void damage

Chapter 1 Page 3

emanating from martensite particles in the ferrite matrix.

1.2 Motivation

It is known that the improved mechanical behaviour of DP steels is due to the

plastic incompatibility between the soft, deformable ferrite and hard, non-deformable

martensite. This incompatibility leads to the introduction of long-range internal back

stresses as well as the generation of geometrically necessary dislocations (GNDs) into

the ferrite matrix. These two phenomena further work-harden the ferrite matrix and

result in the significantly high initial work hardening rate of DP steels.

The work hardening of DP steels is known to be a function of the volume frac-

tion, f , and size, d, of the martensite particles. However, the effects of morphology

and spatial distribution of martensite particles on work hardening has received very

little attention in the research literature. It is apparent that in order to improve the

mechanical behaviour of DP steels even further, it is essential to have a clearer under-

standing of the effects of all microstructural features on individual strain hardening

mechanisms.

Specifically, the majority of experimental research on the work hardening of DP

steels has focused on the work hardening effects due to the generation of GNDs while

little data exists on the back stress hardening in these steels. This is generally due

to the difficulties associated with the application of forward-reverse deformation to

sheet samples, which is required for the measurement of back stresses. Information

on the effects of microstructural parameters on the development of back stresses is

also lacking and any further insight into this area would be of significant value.

Chapter 1 Page 4

1.3 Research Objectives

The three fundamental objectives of the present research are as follows:

I Investigate the relationship between thermo-mechanical processing (TMP) pa-

rameters and the microstructure of DP steels in terms of ferrite grain size as well

as size, volume fraction, morphology and spatial distribution of martensite par-

ticles. This task was done by applying different thermo-mechanical processing

(TMP) schedules prior to the IC annealing step in order to produce five distinct

DP microstructural variants.

II Examine the relationship between the microstructural parameters, particularly

ferrite grain size as well as volume fraction, size, morphology and spatial distri-

bution of martensite particles, and the work hardening rate at both small and

large plastic strains. Application of the Ashby’s work hardening model to the

results will be used to determine the influence of microstructural parameters on

the work hardening due to the GNDs.

III Study the effects of microstructural parameters on the evolution of back stresses

in the five distinct DP microstructural variants by means of a custom in-plane

shear testing fixture capable of applying forward-reverse shear deformation. This

study will be the first comprehensive study of its kind reported in the literature.

Chapter 1 Page 5

1.4 Organization of the Thesis

The contents of this thesis is organized as followed:� Chapter 2 provides a review of the relevant literature. As this thesis is written in

a manuscript format, the literature review chapter was intentionally kept rela-

tively short to avoid repetition. Additionally, the review of literature concerning

the measurements of back stresses is presented in Chapter 5.� Chapter 3 presents a paper published in Materials Science and Engineering A [5],

which reports the results of a study on a cold-rolled DP780 starting material.

Through application of different heat treatments prior to inter-critical (IC) an-

nealing, three distinctly different microstructural variants were produced with a

range of martensite volume fractions and with different ferrite grain size as well

as size, morphology and spatial distribution of martensite particles. The evolu-

tion of microstructures at different stages of thermo-mechanical processing was

investigated. The work hardening behaviour of the three microstructural vari-

ants was then examined in terms of the true work hardening rate, θ = dσ/dǫ, the

instantaneous work hardening exponent, n = dlogσ/dlogǫ and the dislocation

annihilation parameter, h = −(θ − θ0)/(σ − σ0).� Chapter 4 presents a second paper submitted to, and currently under review

at, Materials Science and Engineering A. This chapter describes the effects of

introducing an additional 80% cold-rolling step between the pre-heat treatment

step and IC annealing. Two distinctly different microstructural variants were

produced having a range of martensite volume fractions and with different fer-

rite grain size as well as size and spatial distribution of martensite particles.

Chapter 1 Page 6

The yielding and work hardening behaviour of these microstructures were com-

pared with that of the cold-rolled (untreated) and IC annealed microstructural

variant from Chapter 3. The structure of this paper is similar to Chapter 3,

however, the emphasis is on the significant refinement of the ferrite grain size

obtained through the interaction between austenite transformation and ferrite

recrystallization, as well as its effect on the work hardening behaviour.� Chapter 5 details the design, manufacture, implementation and application of

an in-plane shear testing fixture capable of performing forward-reverse shear

deformation on the five DP microstructural variants discussed in Chapters 3

and 4. Measurements of back stresses obtained from the shear tests are pre-

sented and explained with respect to the differences in selected microstructural

parameters among the five DP microstructural variants.� Chapter 6 provides a complementary discussion to the thesis considering the

collective results of Chapters 3–5. Additional comments on the practical impli-

cations of the critical findings in this research are also given.� Chapter 7 presents the primary conclusions of the present research as well as

the original contributions to this field of study. Some suggestions for future

work are also provided.

1.5 References

[1] http://worldautosteel.org/, Online Source, 2014.

[2] A. T. Davenport (Ed.), Formable HSLA and Dual-Phase Steels, The Metallurgical

Society of AIME, Warrendale, PA, 1977.

[3] R. A. Kot, J. W. Moris (Eds.), Structure and Properties of Dual-Phase Steels,

The Metallurgical Society of AIME, Warrendale, PA, 1979.

Chapter 1 Page 7

[4] R. A. Kot, B. L. Bramfitt (Eds.), Fundamentals of Dual-Phase Steels, The Met-

allurgical Society of AIME, Warrendale, PA, 1981.

[5] H. Seyedrezai, A. K. Pilkey, J. D. Boyd, Mater. Sci. Eng. A 594 (2014) 178–188.

CHAPTER 2

Literature Survey

2.1 Introduction

Dual-Phase (DP) steels are one grade of the first-generation Advanced High

Strength Steels (AHSS). Their microstructure comprise non-ferritic phase (NFP) par-

ticles distributed in a ferrite matrix. Traditionally, martensite is the only NFP; how-

ever, in steels with a more complicated processing history, other NFPs such as bainite

and retained austenite may also be present. A typical DP steel contains a low carbon

content (∼0.1 wt.%) and ∼1-2 wt.% manganese. Other alloying elements, such as

chromium, molybdenum and silicon, are also present in DP steels. In addition to

increasing the strength, these alloying elements also ensure a high hardenability of

the steel which is required for the martensitic transformation at typical industrial

cooling rates.

2.2 Processing of DP Steels

To produce DP steel sheet on a commercial scale, cold-rolled sheet is passed

through a continuous galvanizing/galvannealing line [1–3]. A schematic of this thermo-

mechanical processing (TMP) route is presented in Figure 2.1. The starting cold-

rolled sheet generally contains a mixture of ferrite, pearlite and sometimes bai-

nite. During soaking at the inter-critical (IC) annealing temperature, nucleation and

growth of austenite particles takes place and the starting microstructure transforms

to a mixture of ferrite and austenite (Section 2.2.1). After a short soaking time of

8

Chapter 2 Page 9

2-5 minutes, the sample is cooled and passed in succession through a zinc pot (at

∼460◦C) and a galvannealing furnace, and then finally cooled to room temperature.

The final microstructure consists of ferrite, martensite and bainite. The formation

of bainite during galvannealing is not desirable as it reduces both the strength and

the high initial work hardening rate often observed in classic ferrite-martensite DP

microstructures. The steel chemistry, therefore, is of particular importance as the

steel should have such a combination of carbon and manganese content that ensures

high hardenability during the galvannealing process. Note that, as shown in Figure

2.1, at the end the soaking (IC annealing) step, the specimen is initially slow cooled

to the quenching temperature and then rapidly cooled into the zinc bath. The slow

cooling process promotes the formation of “epitaxial ferrite” through partial rever-

sion of austenite, which results in an additional carbon enrichment of the remaining

austenite and therefore, higher hardenability [3].

In a laboratory and research setting, as was the case for the present research, it

is more practical to quench the specimen from the IC annealing temperature into

water at room temperature. This quenching method is also beneficial for studying

the effects of the DP microstructure on mechanical behaviour since the final ferrite-

martensite microstructures are less complex than the industrial ferrite-martensite-

bainite microstructures. Hence, the majority of publications in this field use the

direct water quenching method from the IC annealing temperature.

From Figure 2.1, it follows that a large number of processing parameters are

present in the TMP schedule of DP steels. These parameters include time and tem-

perature of the IC annealing treatment, as well as the heating rate to and cooling

rate from the IC annealing temperature. Additionally, the microstructure prior to the

IC annealing can have a significant effect on the final DP microstructure. In Section

Chapter 2 Page 10

Figure 2.1: Schematic of a typical industrial galvannealing schedule for the production of dual-phase steel. A: Austenite, B: Bainite, F: Ferrite, M: Martensite and P: Pearlite [Modified from1].

2.2.1, the IC annealing process as well as the influence of each of these processing

parameters will be discussed.

2.2.1 Inter-Critical Annealing

Inter-critical (IC) annealing is the key step in the TMP of DP steels. The process

involves heating the steel into the two-phase austenite-ferrite region, i.e. between the

A1 and A3 temperatures, and holding for a relatively short period of time to allow

for the austenite transformation process to occur. The austenite transformation is

diffusion-controlled, and therefore, occurs by a nucleation and growth process. The

transformation depends on the availability of:

Chapter 2 Page 11

(i) Suitable nucleation sites: These sites are locations within the matrix that are

associated with high surface/interfacial energies, which assists the heterogeneous

nucleation of austenite by lowering the critical nucleation activation energy, ∆∗

G.

(ii) Sufficient carbon content: Austenite is a carbon-rich phase that requires prox-

imity to high carbon content sources for its growth.

In the starting DP steel microstructure, the carbon-rich constituents are pearlite,

individual cementite particles and martensite. Therefore, when the ferrite matrix is

fully recrystallized prior to reaching the austenite transformation temperature, the

ferrite/cementite interface in pearlite colonies and cementite particles located on fer-

rite grain boundaries, are the two preferred nucleation sites for austenite [4, 5]. This

preference has been attributed to the additional surface energy associated with grain

boundaries which reduces the required activation energy for austenite nucleation,

∆G∗. Note that cementite particles which are not located on ferrite grain boundaries

remain intact during IC annealing. This observation can be attributed to the lack of

additional surface energy associated with grain boundaries, which results in a larger

critical austenite nucleation energy (∆G∗ ), thereby rendering these particles ineffec-

tive for the nucleation of austenite [4–6]. It is evident that the final DP microstructure

(after IC annealing) is directly related to the initial starting microstructure through

the variations in composition, size, morphology and spatial distribution of the start-

ing constituents. Indeed, a large number of studies have focused on optimizing the

final DP microstructure through the addition of various “pre-heat treatments” prior

to IC annealing [7–9].

The above discussion does not consider the effect of deformed microstructure

prior to IC annealing. If the ferrite matrix is not fully recrystallized before reaching

the IC annealing temperature, it has been demonstrated that the stored energy of

Chapter 2 Page 12

the unrecrystallized ferrite grains leads to an additional reduction of the activation

energy for austenite nucleation (∆G∗), thereby inducing a significant nucleation of

austenite on cementite particles located on sub-grain boundaries of unrecrystallized

ferrite grains [4, 5, 10]. To maintain an unrecrystallized, or partially recystallized

ferrite matrix prior to reaching the IC annealing temperature, the heating rate to the

IC annealing temperature represents the critical parameter, such that relatively high

heating rates (>10◦C/sec) are usually required [11].

This change in the austenite nucleation process caused by the presence of un-

recrystallized ferrite grains can result in a significant refinement of the final DP mi-

crostructure. Hence, in recent years, a number of researchers have combined a form of

high deformation process with IC annealing [12–16]. Note that the austenite trans-

formation and ferrite recrystallization processes have been shown to be competing

processes. That is, austenite nucleation results in a partial relaxation of the high

strain energies associated with deformed ferrite grains, which in turn lowers the driv-

ing force for ferrite recrystallization [17]. Moreover, austenite grains that are formed

during IC annealing have a pinning effect on the ferrite grain boundaries which con-

sequently renders these boundaries immobile. For these reasons, in microstructures

where the ferrite matrix is not recrystallized prior to reaching the IC annealing tem-

perature, it is possible to observe a number of unrecrystallized ferrite grains in the

final DP microstructure [17].

Following austenite nucleation, the austenite growth process takes place in three

different stages [18–22]:

1. Complete transformation of pearlite (or bainite) into austenite: This process, in

which a high carbon content austenite is produced, is often controlled by carbon

diffusion. Since the diffusion distances are short (on the order of inter-lamellar

Chapter 2 Page 13

spacing), the time required for completion of this stage is also very short, e.g.

on the order of ∼15 seconds during annealing at 780◦C [18].

2. Growth of austenite into ferrite: Depending on the IC annealing temperature,

this process is controlled by either carbon or manganese diffusion [18, 20, 21]

and can take hours to fully complete.

3. Manganese redistribution in austenite to eliminate the manganese concentration

gradient: This process is extremely slow due to the low diffusion rate of Mn in

austenite and is generally not completed even after 24 hours [18].

Note that in microstructures where ferrite recrystallization is not completed prior

to reaching to the IC annealing temperature, the sub-grain boundaries of the deformed

ferrite matrix also provide paths of accelerated diffusion for carbon and other alloying

elements, thereby enhancing the kinetics of the austenite growth process [10, 23].

During the IC annealing treatment, the critical parameters to consider are the

temperature and time of the process as they can influence the volume fraction, size

and strength of the martensite particles. Additionally, the cooling rate from the IC

annealing temperature to room temperature is also very important. To ensure the

production of martensite, the cooling rate must be fast enough to suppress diffusional

transformations. This is defined by the steel’s hardenability which is a function of

alloy composition. Slower cooling rates will result in the formation of other non-

ferritic phases (NFP’s), namely bainite and epitaxial ferrite, which can significantly

change the mechanical properties, e.g. existence of epitaxial ferrite has been shown

to improve the work hardening and uniform elongation of steel while decreasing the

tensile strength [24–27]. Slower cooling rates are often encountered during continuous

galvannealing/galvanizing lines (Figure 2.1).

Chapter 2 Page 14

2.3 Mechanical Behaviour

DP steels exhibit a characteristic tensile behaviour which is of great interest from

a metal forming point of view. This includes continuous yielding, a very high initial

work hardening rate which transitions to moderate hardening levels at larger strains,

and a low yield-to-tensile strength ratio. Figure 2.2 compares the tensile behaviour

of a typical DP steel with two different HSLA steels (SAE 950X and 980X). The

continuous yielding of DP steel is evident. Moreover, at similar strength levels, the

DP steel exhibits more uniform elongation. In general, these improvements in the

plastic behaviour of DP steels are associated with two phenomena:

1. The martensitic transformation after IC annealing is accompanied by a volume

expansion of 3-4% which is resisted by the ferrite. Consequently, residual elastic

stresses are developed in both the ferrite matrix (tensile) and martensite par-

ticles (compressive). Additionally, mobile dislocations are introduced into the

ferrite near ferrite/martensite interfaces in order to relax these elastic stresses.

These two phenomena result in continuous yielding of DP steels at relatively

low stresses, discussed further in Section 2.3.1.

2. During the plastic deformation of a DP steel strain inhomogeneity is introduced

in the structure due to the plastic incompatibility between the constitutive

phases, i.e. hard non-deforming martensite embedded in soft-deformable ferrite.

As explained in Section 2.3.2, this incompatibility leads to the development

of back-stresses and the generation of the geometrically necessary dislocations

(GNDs) in the ferrite matrix, both of which can act as significant work hardening

mechanisms.

Chapter 2 Page 15

Figure 2.2: Tensile behaviour of DP steels in comparison with other HSLA steels [modified from28]

2.3.1 Yielding

The continuous yielding of DP steels at low stresses has been found to be due to

the volume expansion associated with the martensite transformation [18, 24, 29–33].

This volume expansion results in the introduction of residual internal stresses (Section

2.3.1.1) as well as mobile dislocations near ferrite/martensite interfaces in the ferrite

matrix (Section 2.3.1.2).

2.3.1.1 Residual Stresses

The transformation strain, ǫt, associated with the martensitic transformation is a

function of the carbon content of the alloy and can be estimated as [31]:

ǫt = 0.0058 + 0.0045C (2.1)

Chapter 2 Page 16

where C is the carbon content in weight percent. This lattice strain will be resisted by

the ferrite matrix and therefore tensile and compressive residual stresses are developed

in the ferrite and martensite, respectively. The residual stresses can be approximated

by combining Equation 2.1 with the Eshelby method. Such a calculation was done by

Sakaki et al. [31]. Assuming that this stress is approximately isotropic, it was found

that for ferrite, the equivalent residual stress, σres, is given by:

σres =E

1− νǫ0(

rmr)3 (2.2)

where rm is the martensite particle radius and r is the distance from the

ferrite/martensite interface. Residual stresses can assist the deformation in ferrite

since they are of a tensile nature. Their effectiveness, however, is only limited to the

very early stages of straining (∼1-2%) as they will start to be gradually consumed

throughout the structure as the strain is increased [24, 30].

2.3.1.2 Transformation Dislocations

From Equation 2.2, the magnitude of the residual stresses outside the martensite

particles can be very large which leads to the plastic relaxation of the surrounding fer-

rite. However, since the residual stress has a cubic relationship with r, the yielding of

the ferrite only takes place at the areas close to the interface [31]. This localized plastic

yielding results in the introduction of mobile dislocations near the ferrite/martensite

interfaces in ferrite grains [32, 33], as shown in Figure 2.3, which thereby leads to the

suppression of Luders-band formation. Additionally, these transformation-induced

dislocations can be easily moved upon the application of small stresses and therefore

enhance plastic deformation by reducing the yield strength [28, 29, 32, 33]. These

Chapter 2 Page 17

Figure 2.3: Mobile dislocations near ferrite/martensite interface [32]

dislocations become immobilized when they interact with other dislocations. How-

ever, the immobilization path length is not constant for all dislocations due to their

heterogeneous distribution, which results in a gradual reduction in the number of

mobile dislocations and consequently continuous yielding [24] for the DP structure.

The mobile, unpinned transformation dislocations are associated with lattice rota-

tions and therefore can be detected using high resolution EBSD techniques.

Calcagnotto et al. [34], for example, studied two DP steels with different volume

fractions of martensite particles and were able to calculate the density of the trans-

formation dislocations within the microstructure. In their work, Calcagnotto et al.

referred to the transformation dislocations as GNDs (similar terminology is used in

Figure 2.4). They used Kernal Average Mis-orientation (KAM) data around the

measured point while disregarding mis-orientations larger than 2 degrees as they

Chapter 2 Page 18

(a) (b) (c)

Figure 2.4: Application of EBSD in GND analysis of DP steel. (a) Image quality (IQ) map, (b)Kernal Average Mis-orientation (KAM) map, (c) Calculated GND distribution map from KAM data.Notice high density of GND’s near ferrite/martensite interface in comparison with ferrite/ferritegrain boundaries as well as interior of ferrite grains [34].

were assumed to belong to grain boundaries. An example of their results is shown

in Figure 2.4. Notice the higher GND densities near ferrite/martensite interfaces in

comparison with the ferrite grain interiors. Ramazani et al. [35, 36] also used an

EBSD technique on a number of DP steels with various ferrite and martensite grain

sizes and confirmed the findings of Calcagnotto et al. [34]. Moreover, they found that

within a ferrite grain, the area surrounding a martensite particle that is affected by

transformation dislocations is a function of martensite particle size but is independent

of ferrite grain size. It follows that for a reduced ferrite grain size, and at a constant

volume fraction of martensite, f , a larger area fraction of each ferrite grain is affected

by these transformation dislocations. This concept is particularly important in ultra-

fine ferrite grains with an average size ∼1 micron where it has been demonstrated that

entire ferrite grains can be affected by transformation dislocations, thereby resulting

in a more homogenous distribution of dislocation density [12, 34].

Chapter 2 Page 19

Figure 2.5: Compression engineering stress vs. engineering strain curves for three different micro-pillar specimens within the same ferrite grain, with one located in the interior and the other twowere made from areas close to the ferrite/martensite interface [39].

It has been shown that a minimum volume fraction of martensite is often required

to produce continuous yielding behaviour [37, 38]. This minimum is associated with

an increase in the density of the transformation dislocations as a function of the

martensite volume fraction.

The transformation dislocations have been found to cause a local increase in fer-

rite strength [39]. Specifically, in a recent study by Ghassemi-Armaki et al. [39],

micro-pillar specimens were extracted from the areas near the martensite particles as

well as regions within the ferrite interior. As shown in Figure 2.5, the micro-pillars

that were made from areas close to a ferrite/martensite interface exhibited distinctly

higher strengths compared to the ferrite interior specimen; an observation which was

attributed to the presence of mobile dislocations in these regions.

Chapter 2 Page 20

2.3.2 Work Hardening

For many materials, the work hardening response has been traditionally expressed

using a simple power law relationship (Hollomon rule):

σ = Kǫn. (2.3)

When the material follows the Hollomon rule, the plot of log dσ/dǫ versus log ǫ

should be linear with slope of n. However, this is not the case for DP steels, as shown

in Figure 2.6. To better describe the work hardening of DP steels, an analysis based

on Ludwik’s work hardening rule is often used, given by:

σ = σ0 +Kǫmp (2.4)

where ǫp is the plastic strain. Differentiating Equation 2.4 with respect to ǫp results

in a linear relationship between log dσ/dǫ and log ǫp at a constant m, such that:

log dσ/dǫ = log (K.m) + (m− 1) log ǫp . (2.5)

Analysis of the stress-strain data of a DP steel using Equation 2.5 is referred to

as the Jaoul-Crussard analysis and it reveals three different stages of work hardening

with distinct linear slopes [9, 24, 32]. An example of such an analysis is shown in Fig-

ure 2.7. These different stages indicate that in addition to the conventional strength-

ening effects, i.e. solid solution hardening, grain boundary hardening (Hall-Petch

effect) and precipitation hardening, other hardening mechanisms are also present in

DP steels. The additional work hardening contributions are due to the mechanical

differences between the constituent phases; Namely, ferrite is a soft, deformable phase,

Chapter 2 Page 21

Figure 2.6: An example of a Hollomon analysis of DP steel’s tensile behaviour. It is evident thatthe relationship between log dσ/dǫ and log ǫ is not linear [40].

whereas martensite is very strong and non-deformable. Therefore, deformation of the

ferrite is restricted by the presence of martensite particles and this leads to plastic

inhomogeneity in the structure. This phase incompatibility has been experimentally

observed by Digital Image Correlation (DIC) techniques, where the local strains are

mapped by introducing a grid onto the surface of the specimen during an in-situ

tensile test [41, 42], as shown in Figure 2.8. Notice the high local strain values near

ferrite/martensite interfaces compared to the interiors of ferrite grains.

There are two important consequences of the presence of plastic inhomogeneity in

the microstructure:

1. Back-stresses are generated in the ferrite due to dislocation pileups at the fer-

rite/martensite interfaces. The back stresses will result in additional hardening

of the steel [43–45], discussed further in Section 2.3.2.1.

2. The strain gradient is accommodated by the introduction of “geometrically

necessary dislocations” (GNDs) which further strengthens the material [30, 33]

Chapter 2 Page 22

Figure 2.7: An example of Jaoul-Crussard analysis of DP steel’s tensile behaviour. The threestages of work hardening (unique to DP steels) are indicated in this plot [Modified from 32].

(a) (b)

Figure 2.8: (a) SEM image of a DP steel microstructure with martensite volume fraction of 20%.(b) Strain measurement obtained from Digital Image Correlation (DIC) technique. Local averagestrain: 0.215 [42].

Chapter 2 Page 23

(see Section 2.3.2.2).

From Figure 2.7, the three stages of work hardening in DP steels are:� Stage 1 is associated with homogenous deformation of ferrite and martensite

which is followed by a elastic-to-plastic transition of the ferrite. During this

stage, the residual stresses present in the matrix due to martensitic transforma-

tion (see Section 2.2) are gradually consumed [32, 46].� Stage 2 corresponds to inhomogeneous deformation of the ferrite matrix. During

this stage, which is unique to DP steels, the plastic incompatibility between fer-

rite and martensite [9, 32] leads to the additional contributions of back stresses

and GNDs to the overall work hardening rate. Consequently, the decrease in

work hardening rate with strain is slower in Stage 2 than Stages 1 and 3. The

second stage of work hardening is more pronounced in DP steels with more

inhomogeneous deformation of ferrite [32, 38, 47]. This stage continues until

about 2-4% strain [47], at which point the back stress contribution [48, 49] and

the GND hardening effects saturate [50] .� Stage 3 is associated with a more homogenous deformation of ferrite and marten-

site [9], the formation of dislocation cell substructure in ferrite, and the onset

of dynamic recovery effects [9, 46, 51]. The work hardening of DP steel in this

stage is similar to that of other structural steels [24, 32, 47]. The dislocation

cell structure is fully developed in this stage, noting that the dislocation cell

structure is finer near martensite/ferrite interfaces [32].

The microstructure of DP steel has been shown to significantly affect its work

hardening behaviour [9, 12–14, 30, 33, 35, 47, 52–57]. The work hardening rate has

been found to a be direct function of the martensite volume fraction, f [30, 33,

Chapter 2 Page 24

47, 52–54], and an inverse function of the martensite particle size, d [52, 58, 59].

Furthermore, the spatial distribution and morphology of martensite particles have

also been shown to influence the work hardening rate [9, 35, 52, 56, 57, 60, 61]

although due to various definitions of these parameters in different publications, a

clear relationship cannot be established. The effect of ferrite grain size, D, on the work

hardening rate of DP steels has also been studied [12–14]. However, in these studies,

other microstructural parameters, such as size, morphology and spatial distribution

of martensite particles, are usually also changed along with the ferrite grain size;

therefore, a definitive conclusion on the effect of ferrite grain size alone cannot be

made.

To provide a better understanding of the effects of microstructure on the work

hardening behaviour of DP steels, the two additional work hardening mechanisms of

back stress hardening and GND hardening will be reviewed in more detail in Sections

2.3.2.1 and 2.3.2.2.

2.3.2.1 Back-Stress Hardening

Through a series of publications in the 1970s [43, 44, 62], Brown and Stobbs

provided a comprehensive understanding of the deformation mechanism of a soft

matrix embedded with hard, non-deformable second phase particles. Their work

provides the framework required for understanding the evolution of back-stresses in

DP steels and many other materials with second-phase hard particles.

In terms of dislocation theory, bypass of a dislocation from a non-deforming par-

ticle leaves an Orowan loop around the particle. Two different internal stresses are

then developed as a result of Orowan loops [43]: (a) short range internal stress, some-

times referred to as source shortening stress and (b) long range internal stress which

Chapter 2 Page 25

is known as back stress. The former stress, is essentially a locally varying stress field

with zero mean value. This stress reduces the effective inter-particle spacing which

forces successive dislocations to stand off from an Orowan loop around a particle. The

source shortening stress is frictional in nature and is non-directional [62–64]. The long

range internal stress, on the other hand, is the global image stress that is imposed on

the ferrite matrix. For a first approximation, using Eshelby’s approach, Brown and

Stobbs calculated the back-stress as:

τB = 2µφfγp (2.6)

where, µ is the shear modulus, φ is the morphology-dependent accomodation factor

(typically assumed ∼ 0.5 for spherical particles), f is the volume fraction of the

second-phase and γp is the plastic shear strain. Note that Equation 2.6 is based on

the assumption that both the particle and the matrix have similar elastic moduli. This

is a correct assumption in terms of DP steels; however, in other materials, a correction

to Equation 2.6 has to be made to account for differences in the elastic moduli of the

constituent phases. The back stress increases linearly with the plastic strain, until

at relatively small strains, local plastic relaxation via introduction of secondary slip

dislocations occurs in the soft matrix, near the hard particles’ interface [44]. Through

this plastic relaxation, a large amount of the elastic stresses are relieved. Due to the

interaction between secondary and primary slip systems, the material work hardens

very rapidly and complete relaxation of back-stresses is not possible. Consequently,

Equation 2.6 should be adjusted to include only the contribution from the unrelaxed

portion (γ∗

p) of total plastic shear strain (γp). The secondary slip dislocation array was

found to be identical to the geometrically necessary dislocations (GNDs) proposed

by Ashby [65, 66](Section 2.3.2.2). Therefore, using Ashby’s work hardening model

Chapter 2 Page 26

[65, 66], unrelaxed plastic shear strain, γ∗

p , can be calculate as:

γ∗

p ≅ α

[

8γpb

πr

]1/2

(2.7)

where α is a constant describing forest hardening strength (typically ∼ 0.3-0.4), b is

the Burgers vector and r is the particle radius. Combining Equations 2.6 and 2.7, it

follows that in the presence of plastic relaxation, the back stress can be written as:

τB = 2αµφf

[

8γpb

πr

]1/2

. (2.8)

A forward-reverse loading test is often employed to measure the back stresses

from the difference observed in the stress levels before and after strain reversal. A

comprehensive review of these tests is given in Chapter 5. In this chapter, only

the important results of these tests are provided. Gerbase et al. [30] measured the

back-stress evolution of a DP steel and found that in agreement with the Brown and

Stobbs model [43, 44], the magnitude of the back stress increases with the amount of

pre-strain. These results are shown in Figure 2.9. An interesting feature in this figure

is the saturation of back-stress values at relatively small pre-strain values. Similar

observations were also reported by Li and Gu [67], Tomota [68] and Han et al. [69].

Gerbase et al. attributed this saturation to the occurrence of other competing process

such as the plastic deformation and/or fracture of the martensite particles [30].

Studies concerning the effects of microstructural parameters of DP steels on back

stress hardening are limited. The back stress has been found to increase with the

volume fraction of martensite [67, 70, 71]. With respect to other microstructural

parameter, Erdogan and Priestner [71] observed a higher back stress contribution

in a DP microstructure with a finer martensite particles size. They attributed this

Chapter 2 Page 27

Figure 2.9: Variation in the back stress parameter, σB, as a function of pre-strain [30].

effect to a more efficient stress transfer from ferrite to martensite due the higher

overall interfacial area of smaller martensite particles at similar volume fractions. In

another study, Goel et al. [70] demonstrated that when the martensite particles are

too closely spaced, partial cancelation of the internal stresses takes place due to the

mutual interactions between the stress fields associated with the dislocation pileups

at the neighbouring particles. Although, these three studies [67, 70, 71] are very

useful for providing a general understanding of contribution of back stress to the

work hardening behaviour of DP steels, a more systematic and comprehensive study

of the effect of microstructural parameters on the back stresses is lacking, which forms

one of the main objectives of the present research.

Chapter 2 Page 28

2.3.2.2 Geometrically Necessary Dislocations

As discussed in Section 2.3.2.1, the magnitude of elastic back stresses increases

very rapidly with increasing plastic strain. Using a copper/silica system, Brown

and Stobbs [44] showed that eventually at relatively small strains, the relaxation of

elastic back stresses occurs by the formation of secondary slip dislocations. Ashby

demonstrated that the accumulation of secondary dislocations can be related to the

gradient of strain by simple geometrical considerations [65, 66]. Therefore, these

dislocations are referred to as “geometrically necessary dislocations” (GNDs) and

their density can be calculated as [66]:

ρGND = (1

λg)4γ

b(2.9)

where γ is the shear strain, b is Burgers vector and λg is the geometric slip distance.

For spherical particles with raduis r and volume fraction f , this distance is equal to:

λg =r

f. (2.10)

In areas away from the hard particles, the dislocation accumulation is statistical

in nature, and these dislocations are referred to as “statistically stored dislocations”

(SSDs). In addition to GNDs, the SSDs are also accumulated near the marten-

site/ferrite interface. To calculate the overall dislocation strengthening contribution

to work hardening, both SSDs and GNDs should be included in the Taylor relation-

ship, given by:

τdislocation = αµb√

ρTotal (2.11)

Chapter 2 Page 29

where ρTotal is the overall dislocation density, i.e. ρTotal = ρSSD + ρGND. The effec-

tiveness of GNDs depends on their slip distance; the smaller it is, the larger their

contribution to overall hardening. As shown in Figure 2.10, when the density of SSDs

becomes larger than that of the GNDs, the hardening mechanism is controlled by

SSDs and consequently the work hardening is reduced. At the early stages of de-

formation, the density of the GNDs dominates the structure, which means they can

provide a significant amount of hardening. As strain is increased, so too does the

density of SSDs. Above a critical value of strain, the work hardening is controlled by

the SSDs.

The Ashby work hardening model has been successfully used to describe the work

hardening of DP steels. Balliger and Gladman [33] showed that in agreement with

Ashby’s work hardening model (Equations 2.9 and 2.10), an increase in volume frac-

tion of martensite or a decrease in martensite particle size results in a higher work

hardening rate of DP steels at small strains. Therefore, Balliger and Gladman [33]

described the work hardening as:

dσ

dǫ= 0.78k

µb1/2

ǫ1/2

√

f

d(2.12)

where k is a constant of order 1, f is the volume fraction and d is the average

diameter of the martensite particles. Similar observations were made by Lanzillotto

and Pickering [54]. Both the works of Balliger and Gladman [33] and Lanzillotto and

Pickering [54] are highly regarded in the field of DP steels; therefore, the parameter√

f/d is often used to describe the work hardening of DP steels at low strain, e.g.

[52]. At higher strains (<∼3%), in agreement with Ashby’s work hardening theory

(see Figure 2.10), it has been found that the additional work hardening effects due

to GNDs (as well as back stresses) become less significant and the work hardening

Chapter 2 Page 30

Figure 2.10: Schematic representation of the evolution of SSD and GND densities with strain [66].

behaviour of DP steels becomes less affected by the presence of martensite [19, 24,

30, 32, 35, 52].

Unlike the effects of volume fraction and size of martensite particles, the influence

of other DP microstructural parameters on the GND hardening, such as ferrite grain

size as well as the morphology and spatial distribution of martensite particles, has

not been reported in detail in the literature. A possible explanation for this lack of

Chapter 2 Page 31

information is that generally the DP microstructural parameters are coupled with

each other which makes the study of individual effects particularly challenging. Due

to the importance of these parameters, however, a systematic study is much needed

to provide a better understanding of the influences of ferrite grain size as well as

morphology and spatial distribution of martensite particles on the GND hardening

of DP steels. As a result, this constitutes one of the main objectives of the present

research.

Chapter 2 Page 32

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CHAPTER 3

Effect of Pre-IC Annealing Treatments on

the Final Microstructure and Work

Hardening Behaviour of a Dual-Phase

Steel

Abstract

This paper investigates the relationship between the microstructure and the work

hardening behaviour of a dual-phase (DP) steel. Various DP microstructures were

systematically produced by applications of different pre-IC annealing heat treatments

as well as changing the IC annealing temperature. It was found that various austenite

nucleation sites such as grain boundaries, prior pearlite colonies, martensite particles

and cementite particles, have different nucleation and growth effectiveness which sig-

nificantly influences the microstructure after IC annealing. Following a quantitative

analysis of all microstructures, the effect of microstructural parameters including

martensite particle size, volume fraction as well as their spatial distribution and mor-

phology, on the mechanical behaviour of DP steels is examined by considering true

work hardening rate, instantaneous work hardening rate and the dislocation annihila-

tion factor from the Kocks-Mecking analysis. These analyses reveal that at constant

ratios of volume fraction to size of martensite particles, there are significant differ-

ences in all three work hardening parameters. It is proposed that these observations

are due to the effects of morphology and spatial distribution of martensite particles.

Furthermore, it was shown that the contribution of martensite particles to work hard-

ening behaviour, via geometrically necessary dislocations, is only significant at the

early stages of deformation.

37

Chapter 3 Page 38

3.1 Introduction

There is a continuing trend in the automotive industry to move towards lighter,

more fuel-efficient vehicles. To ensure competitiveness of ferrous alloys, new grades of

Advanced High Strength Steels (AHSS) are being developed with a superior combina-

tion of strength and formability. The present research investigates dual-phase (DP)

steels which contain non-ferritic phase (NFP) particles distributed in a ferrite matrix.

In traditional DP steels, martensite is the only NFP. However, for more complicated

processing routes, other NFPs such as bainite are also present. The standard pro-

cessing of DP steels involves inter-critical (IC) annealing of cold-rolled ferrite-pearlite

steel in the ferrite-austenite phase field. During this treatment, austenite nucleates

at the interfaces between ferrite and cementite particles, either as individual particles

or within pearlite colonies [1–3]. The carbides provide the elevated carbon content

required for the growth of austenite. However, their effectiveness as austenite nu-

cleation sites varies with their location within the microstructure. Earlier studies

[4, 5] have shown that pearlite is the first phase that is dissolved and replaced by

austenite. This process is very rapid and occurs within a few seconds of heating

to the IC temperature [5]. The effect of individual carbide particles on austenite

nucleation, however, is more complex. Following pearlite colonies, grain boundary

carbides are primary austenite nucleation sites, whereas grain interior carbides are

less effective for austenite nucleation [5–7]. This difference can be attributed to the

extra surface energy associated with the grain boundaries. It is therefore expected

that the starting microstructure, i.e. prior to IC annealing, directly influences the

final DP microstructure, a correlation that has been demonstrated in several studies

[1, 3, 8–13].

Chapter 3 Page 39

DP steels exhibit continuous yielding, low yield-to-tensile strength ratios, large

uniform elongations and very high initial work hardening rates. The continuous yield-

ing of DP steels at low stresses is associated with the presence of mobile dislocations

at ferrite/martensite interfaces as well as residual stresses produced by quenching

from the IC annealing temperature [11, 14–18]. On the other hand, the high initial

work hardening rate is a direct consequence of the strain incompatibility between fer-

rite and martensite which results in load transfer between soft, deformable ferrite and

hard, non-deformable martensite [10, 11, 18–20]. Furthermore, this strain incompat-

ibility results in the introduction of additional hardening mechanisms in the ferrite

matrix due to the development of back stresses and the production of geometrically

necessary dislocations (GNDs) near particle interfaces [11, 17, 18, 20–22]. However,

as total plastic strain is increased, the effectiveness of these additional hardening

mechanisms is reduced and subsequently the work hardening behaviour of DP steels

is determined by the balance between the accumulation of statistically stored dislo-

cations [20, 21, 23–25] and dislocation annihilation due to dynamic recovery [26, 27].

Microstructural parameters such as ferrite grain size as well as the volume fraction,

size, morphology and spatial distribution of martensite particles are known to have

a significant effect on the tensile behaviour of DP steels [10, 11, 18, 24, 28–36]. The

volume fraction of martensite, f , is found to directly affect the amount of load transfer

between ferrite and martensite [10], the yield strength [11, 12, 21, 22, 29, 32, 33] as

well as the work hardening rate due to back stresses [20, 24, 30] and GND hardening

[9, 11, 20, 31]. The mean size of martensite particles, d, is known to inversely affect the

tensile properties, in particular work hardening due to GNDs [11, 31]. Consequently,

on the basis of Ashby’s work hardening model for the deformation of plastically

inhomogeneous alloys [37, 38], work hardening models for DP steels typically consider

Chapter 3 Page 40

the parameter√

f/d as the main microstructural parameter that controls the uniaxial

tensile behaviour [11, 24, 31]. In addition, the spatial distribution and morphology of

martensite have also been shown to influence the yield strength [1, 36], work hardening

rate [8, 10, 23, 24, 35] and load transfer between ferrite and martensite [10]. The effects

of DP steel microstructural parameters on dislocation annihilation effects by dynamic

recovery have not been studied extensively. There are only a limited number of studies

[26, 27] that model the dislocation annihilation effect using a Kock-Mecking approach

for DP steels [39–41] but these models are very general and do not incorporate the

effects of microstructural parameters.

The majority of reported studies on the work hardening behaviour of DP steels

do not distinctly separate the effects of martensite morphology and spatial distribu-

tion from that of martensite size and volume fraction. Therefore, the objectives of

the present study are to utilize pre-IC annealing heat treatments to produce distinct

variations in the morphology and spatial distribution of martensite particles at a con-

stant√

f/d ratio, and to investigate the effects of these microstructural parameters

on the work hardening behaviour.

3.2 Experimental Procedure

3.2.1 Materials

The material used for this study was provided by US Steel Canada as cold-rolled,

commercial DP780 grade, 0.95mm thick sheet (DP780-CR). The composition is given

in Table 3.1. Blanks of size 100mm by 20mm were cut from the CR sheet with the

long axis either along the rolling direction (RD) or the transverse (TD) direction.

Only a limited number of blanks were prepared along TD. Therefore, all specimens

Chapter 3 Page 41

Table 3.1: Chemical compositions of DP780-CR and IF steel sheets (in wt. %).

Steel C Mn P S Si Cu Ni Cr Mo N V B Ti Nb

DP780-CR 0.09 2.1 0.012 0.006 0.02 0.03 0.01 0.26 0.29 0.004 0.001 0 0.001 0.002IF 0.004 0.12 0.003 0.008 0.008 0.019 0.014 0.011 0.005 0.02 0.02 0.003 0.063 0.005

are assumed to be RD, unless specified otherwise. In addition to the DP780-CR

material, an interstitial free (IF) steel sheet was examined for comparison purposes.

Its composition is also listed in Table 3.1.

3.2.2 Heat Treatments

Two different “pre-treatments” were applied to the CR material in order to pro-

duce additional microstructural variants prior to the IC annealing process:

(a) Austemper (AT). The aim of this treatment was to produce a bainitic structure.

The processing involved full austenitization followed by an isothermal hold in the

bainitic transformation region. Austenitization treatments were performed at

920◦C for 30 minutes in a Lindberg 54232 tube furnace under continuous argon

flow to avoid oxidization of the specimen’s surface. The end-to-end variation in

the temperature along the length of the heat treating blanks was 10◦C. The bai-

nite hold step was carried out at 500◦C for 20 minutes in a salt bath of potassium

nitrate and sodium nitrate mixture. The temperature gradient over the sample

dimensions in the salt bath was determined to be 2◦C. The cooling rate during

the transfer of specimens from the austenitization furnace to the bainite hold

salt bath was 13 K/s. The bainite hold temperature was selected to be midway

between the bainite start (575◦C) and martensite start (430◦C) temperatures cal-

culated using Steven and Haynes [42] and Andrews [43] formulae, respectively.

Chapter 3 Page 42

All specimens were water quenched to room temperature after the bainite hold

treatment. The resulting microstructures will be referred to as AT.

(b) Quench and Temper (QT). The aim of this treatment was to produce a tem-

pered martensite structure which contains carbide particles with morphologies

and distributions that are distinctly different from the AT microstructure. This

involved a full austenitization treatment, a subsequent water quench to complete

the martensitic transformation and finally a temper at 600◦C for 1 hour in order

to produce a fully tempered martensite microstructure. Similar to the proce-

dure for the austenitization of AT specimens, both austenitization and tempering

treatments of the QT samples were performed in the Lindberg 54232 tube furnace

for 30 minutes under continuous argon flow. Three different austenitization tem-

peratures of 920, 970 and 1020◦C were used in order to change the prior-austenite

grain size. The resulting microstructures will be referred to as QT.

The final processing stage involved IC annealing of pre-treated specimens (CR, AT

or QT) such that the final microstructures will be referred to as CR+IC, QT+IC

and AT+IC, respectively. The QT microstructures are further categorized into Q2T,

Q7T and Q12T based on the austenitizing temperature, i.e. 920, 970 and 1020◦C,

respectively. IC annealing was performed at different temperatures of 720, 725, 730

and 735◦C, in order to produce microstructures with various volume fractions of

martensite. IC annealing treatments were done in a salt bath with an average heating

rate of 19 K/s. The end-to-end variation in the temperature along the length of the

heat treating blanks was 2◦C. The IC annealing time began 40 seconds after immersing

the specimens in the salt bath. This timing was chosen based on the average time

required for specimens to reach the Ac1 temperature, of 700◦C, determined using

Andrews’ formula [43]. The majority of specimens were annealed for 2 minutes except

Chapter 3 Page 43

Table 3.2: Summary of specimens and their respective heat treatment schedules. AC: Air Cooled,WQ: Water Quenched.

Sample DesignationPre-treatment IC Annealing

Temp. 1 Time 1 Notes 1 Temp. 2 Time 2 Notes 2 Temp. Time Notes(◦C) (min) (◦C) (min) (◦C) (min)

CR+IC20 – – – – – – 720 2 WQCR+IC25 – – – – – – 725 2 WQCR+IC30 – – – – – – 730 2 WQCR+IC35 – – – – – – 735 2 WQAT+IC20 920 30 AC 500 20 WQ 720 2 WQAT+IC35 920 30 AC 500 20 WQ 735 2 WQQ2T+IC20 920 30 WQ 600 60 WQ 720 2 WQQ2T+IC20 (5) 920 30 WQ 600 60 WQ 720 5 WQQ2T+IC20 (10) 920 30 WQ 600 60 WQ 720 10 WQQ2T+IC20 (30) 920 30 WQ 600 60 WQ 720 30 WQQ2T+IC25 920 30 WQ 600 60 WQ 725 2 WQQ2T+IC35 920 30 WQ 600 60 WQ 735 2 WQQ7T+IC20 970 30 WQ 600 60 WQ 720 2 WQQ7T+IC25 970 30 WQ 600 60 WQ 725 2 WQQ7T+IC35 970 30 WQ 600 60 WQ 735 2 WQQ12T+IC20 1020 30 WQ 600 60 WQ 720 2 WQQ12T+IC25 1020 30 WQ 600 60 WQ 725 2 WQQ12T+IC35 1020 30 WQ 600 60 WQ 735 2 WQ

some Q2T+IC specimens which were also treated for 5, 10 and 30 minutes in order

to investigate the effect of IC annealing time. Table 3.2 summarizes the thermal

processing history of each specimen1. Note that in all stages of heat treatments, the

processing temperatures were measured at the centre of the heat treating blanks using

a K-type thermo-couple.

3.2.3 Microstructure Characterization

Metallography specimens were cut from the middle of the heat treated blanks

using a Struers Accutom precision cut-off machine. For RD specimens, the RD/ND

1Note to the reader (not included in the published paper): The specimen naming convention usedhere is different from the rest of the thesis; however, to avoid altering the contents of the publishedwork, this naming convention remains unchanged here. Please refer to Appendix A for completedetails.

Chapter 3 Page 44

sections were investigated while for TD specimens, the plane of interest was TD/ND

where ND is the normal, through-thickness, direction. After mounting, specimens

were polished down to a 0.06 micron surface finish using standard metallography

procedures. They were then etched using either 2% Nital for 13 seconds or 4% Picral

plus zaphiran chloride addition for 40 seconds (with 3 seconds of 2% Nital pre-etch).

The former etchant revealed ferrite grain boundaries as well as martensite and carbide

particles, while the latter only etched carbides. The metallographic specimens were

then examined using a JEOL 840 scanning electron microscope (SEM) at a working

distance of 15mm and with an accelerating voltage of 20kV. Selected specimens were

also etched with the LePera’s etchant [44] for 40 seconds (with 3 seconds of 2% Nital

pre-etch) and examined using a Zeiss Axiophot optical microscope.

The average size and volume fraction of martensite particles and carbide particles,

were measured for all microstructures. Volume fraction measurements were obtained

using a manual point counting technique with a minimum of ∼4000 points in ac-

cordance with ASTM-E562. Measurements of the average particle size, i.e. mean

particle diameter, were made following ASTM-E112 using the intercept method with

three concentric circles. Carbide particle measurements were performed on selected

Picral-etched microstructures, i.e. CR+IC25, AT+IC20 and Q2T+IC25, Q7T+IC35,

Q12T+IC35. For martensite particle analysis, Nital-etched specimens were used.

Since Nital etches both martensite and carbide phases, it was important to establish

a criterion for distinguishing between these two phases. During carbide measurements

(using Picral-etched specimens), it was observed that carbides are generally circular

in shape and very small in size (∼0.07 ± 0.01 microns). Therefore, for martensite

particle measurements using Nital-etched specimens, any particle that fit this carbide

Chapter 3 Page 45

description (i.e. very small and circular) was not included2.

One specimen, Q7T+IC35, was selected for a detailed transmission electron mi-

croscope (TEM) study. Samples were ground down to ∼40 microns, punched into

3mm discs and finally twin-jet electropolished using a solution of 5% perchloric acid

in 95% glacial acetic acid at a temperature of 16.5C and voltage of 27 V. Thinned

samples were then examined using a Philips CM20 TEM at an accelerating voltage

of 200 kV. In addition to thin foil specimens, carbon extraction replicas were also

prepared to investigate carbides in more detail. For the carbon extraction replicas,

a bulk steel sample was first polished down to a 0.06 microns surface finish using

standard metallography procedures. Next, the specimen was heavily etched with 2%

Nital for 30 seconds. A carbon layer was then deposited on the surface using a JEOL

JEE-400 vacuum evaporator. This layer was scored into 2mm by 2mm squares. The

sample was then etched in 5% Nital and subsequently dipped into distilled water

(with a few drop of ethanol) to separate the replica films. The replicas were removed

from the water on 3mm diameter copper grids, dried, and examined in the TEM.

3.2.4 Mechanical Testing

Sub-size uniaxial tensile specimens with a gauge length of 25mm were water-jet cut

from the heat treated blanks according to the ASTM-E8 standard. Uniaxial tensile

tests were performed at room temperature using an Instron 8521 hydraulic testing

machine at a constant extension rate of 0.75mm/min. Strain measurements were

obtained using a 25mm gauge Instron clip-on extensometer. All tests were carried out

2The results of such analysis were validated by performing separate measurement of carbides,martensite and total NFP (both carbides and martensite) volume fractions on the same SEM mi-crographs of selected Nital-etched specimens and comparing the results with the carbide volumefraction measurements of the same microstructures using Picral-etched specimens. Minimal differ-ences were observed in the measurements obtained from these different techniques. Therefore, itwas concluded that the procedure used for carbide and martensite particle measurements is valid.

Chapter 3 Page 46

to failure. True stress (σ) versus true strain (ǫ) tensile curves were calculated along

with the work hardening rate, θ = dσ/dǫ , and the instantaneous work hardening

exponent, n = dlogσ/dlogǫ [45].

3.3 Results

3.3.1 Microstructures

The starting microstructures, i.e. prior to the IC annealing treatment, are shown

in Figure 3.1. The following observations can be made from these micrographs:� The CR microstructure (Figure 3.1a) consists of a heavily deformed mixture

of ferrite, pearlite and small amounts of martensite. Consequently, this mi-

crostructure has the finest ferrite grains in comparison with the other pre-treated

materials.� The AT microstructure (Figure 3.1b) is composed of upper bainite (UB), i.e.

carbide free ferrite laths with martensite/retained austenite (M/A) between

the laths, granular bainite (GB), i.e. carbide-free irregular ferrite grains with

M/A, and also individual martensite (M) particles. This microstructure is in

agreement with the published CCT and TTT plots of steels with similar com-

positions [46, 47]. In terms of the non-ferritic phases, this material has the

coarsest microstructure among the pre-treated specimens.� All three QT microstructures (having different austenitization temperatures)

show a tempered martensite structure with a uniform spatial distribution of

fine carbides located at the prior austenite grain boundaries as well as prior

martensite lath boundaries. Additionally, from Figures 3.1c to 3.1e, it can be

Chapter 3 Page 47

seen that the austenitization temperature does not have a significant effect on

the microstructure other than an increase in the prior austenite grain size.

IC annealing was carried out on the pre-treated materials (shown in Figure 3.1) at

various temperatures, as summarized in Table 3.2. Three conditions were chosen for

the base comparison: CR+IC25, Q7T+IC35 and AT+IC20. These microstructures,

shown in Figure 3.2, were selected due to the similarity in their martensite volume

fractions (average of 14.7 ± 3.2 %). Measurements of volume fraction and average size

are given in Tables 3.3 and 3.4 for the martensite and carbide particles, respectively.

Table 3.3 also includes descriptions of the morphology and spatial distribution of

martensite particles. The following observations can be made:� In microstructure CR+IC25 (Figures 3.2a and 3.2d), the martensite particles

are primarily located at the ferrite grain boundaries and they are mostly aligned

along the rolling direction. This corresponds directly with the distribution of

carbides/pearlite colonies in the starting cold-rolled (CR) microstructure (Fig-

ure 3.1a). The martensite particles in this microstructure have a rather uniform

shape and size, i.e. they are mostly equiaxed and there is a small variation in

their size. Finally, the ferrite matrix is fully recrystallized as demonstrated by

equiaxed ferrite grains and a lack of visible prior deformation. Note that fer-

rite grains have a bimodal size distribution, i.e. ferrite grains along the prior

pearlite bands are generally smaller in size compared to those located in prior

pearlite-free regions.� Microstructure AT+IC20 (Figures 3.2b and 3.2e) consists of elongated marten-

site particles positioned along prior bainite lath boundary locations of the AT

microstructure. In addition, smaller martensite particles are also present at the

Chapter 3 Page 48

(a) (b)

(c) (d)

(e)

Figure 3.1: Microstructures after pre-heat treatments. (a) CR, (b) AT, (c) Q2T (920C), (d) Q7T(970C) and (e) Q12T (1020C). SEM-SE, 2% Nital etch. In the CR and AT microstructures, thepearlite (P), martensite (M), upper bainite (UB) and granular bainite (GB) constituents are marked.White phases are carbides and martensite, the black phase is ferrite.

Chapter 3 Page 49

prior austenite grain boundaries. These smaller particles resemble the marten-

site particles in the CR+IC25 microstructure (Figures 3.2a and 3.2d). Namely,

they have a more equiaxed shape in comparison to those along prior bainite

laths. The spatial distribution of martensite particles is uniform within the

ferrite matrix. Finally, ferrite grains are generally similar in size to those of the

CR+IC25 microstructure.� In microstructure Q7T+IC35 (Figures 3.2c and 3.2f) the majority of marten-

site particles are located at the prior austenite grain boundaries and corners,

while few particles are present at prior martensite lath boundaries. In this mi-

crostructure, very large martensite-free regions exist within the ferrite matrix in

comparison to the other microstructures (CR+IC25 and AT+IC20 in Figures

3.2a and 3.2b, respectively). These particle-free regions are further divided into

ferrite sub-grains corresponding to the prior martensite lath boundaries. Note

that among the QT+IC microstructures, as the austenitization temperature is

elevated, there is an increasing trend in the size of both martensite particles and

martensite-free regions. However, these variances are minimal. Hence, only one

representative QT+IC microstructure is shown in Figure 3.2.� In all microstructures, there is a uniform distribution of very fine particles

throughout the ferrite matrix. Figures 3.2d-3.2f highlight these fine particles

(arrows) in higher magnification SEM micrographs. The size and location of

these particles (in comparison with the microstructures prior to IC annealing

in Figure 3.1), suggests that they are carbides. To confirm this inference, the

Q7T+IC35 microstructure was selected for further analysis using TEM. Fig-

ure 3.3a shows a bright field image of a thin foil specimen, while Figure 3.3b

was obtained from a carbon extraction replica and only shows the particles of

Chapter 3 Page 50

interest that were extracted from the matrix. In Figure 3.3a, carbide parti-

cles are clearly visible (shown by arrows) at ferrite sub-grain boundaries (prior

martensite lath boundaries). Using selected area diffraction analysis of the car-

bon replica specimen, the aforementioned particles were indexed as cementite.

From Table 3.4 it can be observed that the carbide particles generally have a

similar average size and volume fraction in all three microstructures.

Chapter 3 Page 51

(a) (d)

(b) (e)

(c) (f)

Figure 3.2: Final dual-phase microstructures after the inter-critical (IC) annealing. (a,d)CR+IC25, (b,e) AT+IC20 and (c,f) Q7T+IC35. Carbides (indicated by arrows) and martensitephases appear white, while the ferrite matrix is black. SEM-SE, 2% Nital etch.

Chapter 3 Page 52

Table 3.3: Martensite particle measurements for microstructures shown in Figure 3.2

Microstructure Vol. Frac., f Size, d√

f/d Morphology Spatial Distribution(%) (microns) (%.microns−1)

CR+IC25 15.1 ± 1.8 0.50 ± 0.06 5.5 ± 0.5 Equiaxed Along rolling directionAT+IC20 16.2 ± 1.0 0.44 ± 0.05 6.1 ± 0.4 Elongated/

EquiaxedUniform

Q7T+IC35 12.9 ± 3.0 0.54 ± 0.10 4.9 ± 0.9 Irregular Along prior austenitegrain boundaries, largeparticle-free regions

Table 3.4: Carbide particle measurements for microstructures shown in Figure 3.2

Microstructure Carbide Vol. Frac., f Carbide Size, d(%) (microns)

CR+IC25 1.7 ± 0.3 0.06 ± 0.01AT+IC20 2.4 ± 0.3 0.08 ± 0.01Q7T+IC35 2.5 ± 0.7 0.08 ± 0.02

(a) (b)

Figure 3.3: (a) Bright field TEM image of Q7T+IC35 thin foil specimen. Carbide particles areindicated by arrows. (b) Bright field TEM image of Q7T+IC35 carbon extraction replica specimen.

Chapter 3 Page 53

3.3.2 Uniaxial Tensile Behaviour

To determine the effects of microstructure on the uniaxial tensile behaviour, the

same three microstructures with similar volume fractions of martensite (Figure 3.2)

were used. Figure 3.4 shows the engineering stress-strain curves (3.4a) as well as

the instantaneous hardening exponent versus engineering strain plots (3.4b) for mi-

crostructures CR+IC25, AT+IC20 and Q7T+IC353 which have an average martensite

volume fraction of 14.7 ± 3.2 %. These microstructures also have similar√

f/d values

(see Table 3.3).

It can be seen in Figure 3.4 that the three microstructures have similar ultimate

tensile strengths. However, despite having a similar volume fraction of martensite

(and√

f/d values), they exhibit different tensile properties in terms of yield strength,

uniform elongation and work hardening rate. The CR+IC25 microstructure exhibits

the lowest yield strength, largest uniform elongation and highest sustained work hard-

ening rate throughout the deformation, corresponding to the best combination of

tensile properties. In contrast the Q7T+IC35 microstructure, with the highest yield

strength, smallest uniform elongation and lowest work hardening rate, possesses the

least desirable tensile behaviour. Furthermore, in Figure 3.4b, it is evident that the

AT+IC20 has the smallest value of n at low strains while the Q7T+IC35 becomes

the microstructure with the lowest n at higher strains.

Based on the micrographs in Figure 3.2, it can be speculated that the superior

tensile behaviour of the CR+IC25 microstructure is due to the directional effect

caused by the inhomogeneous distribution of martensite particles (banding) along

3Note that all three QT+IC microstructures, i.e. Q2T+IC35, Q7T+IC35, Q12T+IC35, exhibitvery similar uniaxial tensile behaviour, suggesting that the austenitization temperature does nothave a significant effect on the mechanical behaviour of the final DP microstructures with QT pre-treatments.

Chapter 3 Page 54

0.00 0.05 0.10 0.15 0.20 0.25

200

400

600

800

CR+IC25 AT+IC20 Q7T+IC35

Engi

neer

ing

Stre

ss (M

Pa)

Engineering Strain(a)

0.00 0.05 0.10 0.15 0.20 0.25

0.1

0.2

0.3

0.4

CR+IC25 AT+IC20 Q7T+IC35

Inst

anta

neou

s W

ork

Har

deni

ng, n

(MPa

)

Engineering Strain(b)

Figure 3.4: (a) Engineering stress-strain tensile curves for selected DP microstructures shown inFigure 3.2 with an average martensite volume fraction of 15%. Arrows indicate uniform elongation.(b) Instantaneous hardening vs. engineering strain plots obtained from stress strain curves shownin (a).

Chapter 3 Page 55

0.00 0.05 0.10 0.15 0.20 0.250

200

400

600

800

CR+IC25 RD (U.EL=0.153) CR+IC30 RD (U.EL=0.132) CR+IC35 RD (U.EL=0.100) CR+IC25 TD (U.EL=0.153) CR+IC30 TD (U.EL=0.132) CR+IC35 TD (U.EL=0.099)

Engi

neer

ing

Stre

ss (M

Pa)

Engineering Strain

Figure 3.5: Engineering stress-strain curves for CR+IC microstructures tested along RD and TDdirections.

the tensile direction, i.e. rolling direction of the sheet. To investigate this possibility,

two groups of specimens were prepared from the initial CR material: one with the

loading direction along the RD and a second set along the TD. These specimens

were then IC annealed at three different temperatures of 725, 730 and 735◦C. The

results of these RD versus TD uniaxial tensile tests are plotted in Figure 3.5. With

the exception of a small difference in ultimate tensile strength, it is evident that the

tensile behaviour is similar in both RD and TD directions. In particular, the uniform

elongation, i.e. strain at the UTS, is virtually identical in each sample pair. This

similarity in RD versus TD tensile response suggests that the superior behaviour

of the CR+IC25 microstructure, in comparison with QT+IC and AT+IC20, is not

simply a result of the banding of martensite particles in the loading direction of RD

specimens.

Chapter 3 Page 56

3.4 Discussion

3.4.1 Microstructure

It is evident from Figures 3.1 and 3.2 that the microstructures prior to and after

IC annealing correlate directly with one another. In the final DP microstructures,

martensite particles are primarily located at ferrite grain boundaries as well as other

locations of high carbon concentration such as previous pearlite colonies, carbides

and martensite particles. It is known that the rate of austenite transformation from

pearlite colonies and martensite particles is very rapid [5]. In the case of individual

carbide particles, however, the situation is more complicated. Referring to the QT mi-

crostructures (Figures 3.1c-3.1e), the microstructure prior to IC annealing is tempered

martensite with a uniform spatial distribution of very fine carbide particles through-

out the structure. It is known that carbide particles are a primary nucleation site

for austenite since they provide a high concentration of carbon required for austenite

nucleation [5–7]. However, from Figures 3.2c and 3.2f, it can be observed that not all

of the carbide particles seem to have the same effectiveness for austenite nucleation.

That is, the majority of martensite particles in the final DP microstructure are lo-

cated along prior austenite grain boundaries, whereas there are a very limited number

of martensite particles at ferrite sub-grain boundaries and most of the sub-grain car-

bides appear to be unaffected by the IC annealing process. This discrepancy may be

explained by considering the grain boundary effect in lowering the critical nucleation

energy. That is, the formation of austenite on grain boundaries reduces the surface

energy associated with these grain boundaries, and therefore, austenite nucleation on

the ferrite grain boundaries (high energy) can decrease the total surface energy of

the system more effectively than nucleation on the ferrite sub-grain boundaries (low

Chapter 3 Page 57

Figure 3.6: SEM micrograph of specimen Q2T+IC20 (30). The heat treatment history involvesthe QT (920◦C) pre-treatment followed by IC annealing at 720◦C for 30 minutes.

energy). Furthermore, for the case of the CR starting microstructure, there is an

additional effect of ferrite recrystallization due to the heavily deformed nature of the

material. As proposed by Huang and coworkers [2], recrystallization can affect the

austenite transformation rate by changing the effectiveness of ferrite grain boundaries

as nucleation sites; namely, a moving grain boundary is not a suitable nucleation site

for austenite.

It is worth mentioning that after IC annealing for longer times, e.g. 30 minutes,

the sub-grain carbide particles are fully dissolved and replaced by austenite. This

is illustrated in Figure 3.6 for microstructure Q2T+IC20 (30), which has a similar

processing history to Q2T+IC20 but with 30 minutes of IC annealing instead of 2

minutes. This suggests that the sub-grain carbides eventually become nucleation sites

for austenite, albeit at rather long IC annealing times.

In addition to the effect of microstructure on the nucleation of austenite particles,

the effect on austenite growth should also be considered. The QT microstructure

consists of carbide particles that are uniformly distributed throughout the ferrite

matrix, whereas in the CR and AT microstructure, carbides are primarily located in

pearlite colonies or bainite packets. Additionally, the CR and AT microstructures

Chapter 3 Page 58

Table 3.5: Martensite particle measurements for three different starting materials after IC annealingat 735◦C

Microstructure Vol. Frac., f Size, d√

f/d(%) (microns) (%.microns−1)

CR+IC35 21.2 ± 1.9 0.54 ± 0.03 6.2 ± 0.3AT+IC35 28.7 ± 2.4 0.56 ± 0.11 7.1 ± 0.7Q7T+IC35 12.9 ± 4.2 0.54 ± 0.10 4.9 ± 0.9

also contain small amounts of martensite and M/A particles, respectively. Since the

rate of austenite growth from pearlite colonies and martensite particles is known to

be very rapid [5], it is expected the austenite transformation is faster for both the

CR and AT starting conditions in comparison with QT.

In summary, it can be seen that the starting microstructure prior to the IC an-

nealing will affect the kinetics of both austenite nucleation and growth, leading to the

observed differences in martensite volume fractions for the various starting microstruc-

tures subjected to common IC annealing treatments. Accordingly, the martensite vol-

ume fraction was found to be the highest for the AT starting material while the QT

starting microstructure produced the smallest martensite content. This relationship

is summarized in Table 3.5 for the case of IC annealing at 735◦C for 2 minutes.


In Figure 3.4, it can be seen that the CR+IC25, AT+IC20 and Q7T+IC35 mi-

crostructures exhibit different tensile behaviours in terms of uniform elongation and

work hardening rates, despite having similar volume fractions (and√

f/d values) of

martensite particles (see Table 3.3). These differences in uniaxial tensile behaviour

are therefore likely due to the other microstructural parameters such as spatial distri-

bution and morphology of the martensite particles. The Q7T+IC35 microstructure

with the least uniform spatial distribution of elongated martensite particles produces

Chapter 3 Page 59

the most inferior uniaxial behaviour. The large particle free regions in the ferrite

matrix are likely a contributing factor for the inferior performance of this microstruc-

ture in comparison with the other two. Between the CR+IC25 and AT+IC20, the

former has more equiaxed and fewer large martensite particles, which could explain

the better uniaxial tensile behaviour of this microstructure. Moreover, martensite

particles are more similarly sized in the CR+IC25 microstructure when compared to

AT+IC20, which could also produce differences in tensile behaviour.

By including the complete data sets of tensile work hardening properties for each

of the heat treatment conditions (Table 3.2), the effect of microstructure on the work

hardening behaviour can be investigated in more detail by considering the variations

with√

f/d for three parameters: (i) the true work hardening rate, θ = dσ/dǫ, (ii)

the instantaneous work hardening rate, n = dlogσ/dlogǫ and (iii) the dislocation

annihilation factor from the Kocks-Mecking model, h = −(θ − θ0)/(σ − σ0) .

3.4.2.1 True work hardening rate

From the uniaxial tensile test data of all microstructures (Table 3.2), a series

of master curves can be produced, such that the true work hardening rate of each

microstructure, θ = dσ/dǫ , is plotted as a function of√

f/d at plastic strains of

0.5%, 2% and 5% (Figure 3.7). It can be seen that for the QT+IC and AT+IC

microstructures, at early stages of deformation (ǫp = 0.5%), θ follows a near linear

relationship with√

f/d . This is in agreement with Ashby’s work hardening model

[37, 38], where the work hardening rate of a DP steel is directly related to the density

of geometrically necessary dislocations (GND’s), which is in turn proportional to the

martensite volume fraction, f , and inversely related to the martensite particle size, d

[9, 11, 20, 31].

Chapter 3 Page 60

2 4 6 80

4000

8000

12000

16000 p

at p

last

ic s

train

of 0

.5%

(MPa

)

( f/d )1/2 (%.micron-1)1/2

CR+IC AT+IC QT+IC IF

(a)

2 4 6 81500

3000

4500

6000

7500p

at p

last

ic s

train

of 2

% (M

Pa)

( f/d )1/2 (%.micron-1)1/2


(b)

Figure 3.7: Variation in the true work hardening rate, θ, with the microstructural parameter,√

f/d for plastic strains of (a) 0.5% (b) 2% and (c) 5%. Full symbols represent θ for microstruc-tures with martensite volume fractions larger than 10%, while the half-full symbols are θ values formicrostructures with martensite contents of less than 10%.

Chapter 3 Page 61

2 4 6 81000

1500

2000

2500

3000

3500

p

at p

last

ic s

train

of 5

% (M

Pa)

( f/d )1/2 (%.micron-1)1/2


(c)

Figure 3.7 (Continued)

As deformation continues (ǫP=2%), the dependence of θ on√

f/d becomes less

significant, particularly in the case of microstructures with martensite volume frac-

tions larger than 10% (shown as solid symbols in Figure 3.7). For microstructures

with less than 10% martensite (half-filled symbols), θ still has a direct relationship

with√

f/d . Finally, at larger strains (ǫp = 5%), θ becomes independent of√

f/d for

all microstructures regardless of the martensite volume fraction. This is in agreement

with previously published models for work hardening of DP steels [20, 21, 23–25].

Note that the work hardening rate of IF steel is also shown in Figure 3.7, which

illustrates that DP steel microstructures generally have higher work hardening rates

than IF steels, a difference that increases as a function of accumulated plastic strain.

In Figure 3.7, it is evident that at the early stages of deformation (ǫp = 0.5 or

2%), there is a different relationship between θ and√

f/d for each microstructure

Chapter 3 Page 62

group. Additionally, at larger strains (ǫp =5%), θ reaches a different saturation value

for each of the microstructure groups. This saturation value of θ is the highest for

the CR+IC microstructure while AT+IC has the lowest value. It can therefore be

concluded that in addition to size and volume fraction of martensite particles, θ is

also dependent upon martensite morphology and spatial distribution.

Note that, for the CR+IC microstructures, θ is not a function of√

f/d at any

of the values of plastic strain plotted in Figure 3.7. This suggests that the work

hardening of these microstructures is primarily controlled by other factors such as

ferrite grain size. Studies are currently underway to confirm this effect.

Finally, when comparing the tensile behaviour of multiple microstructures, it is

important to consider the effect of changes in martensite strength caused by varia-

tions in the martensite volume fraction. That is, due to a lower carbon content, the

strength of martensite is decreased by increasing the martensite volume fraction as

well as the IC annealing temperature. Changes to the martensite strength will re-

sult in different levels of strain partitioning between ferrite and martensite. However,

based on the results of Mazinani and Poole [10], for the volume fractions of martensite

investigated in this paper (maximum of 28.7 ± 2.4), it can be assumed that marten-

site remains elastic throughout the applied deformation. Therefore, the variations

of strain partitioning among different specimens should be minimal4. Additionally,

it follows that Ashby’s work hardening model is valid for the volume fractions of

martensite investigated here, which means that the generation of GND’s is a function

of geometry rather than the strain partitioning level between the constituents [11, 38].

4Note to the reader (not included in the published paper): Following the work reported inChapter 4, this statement was found to be inaccurate. The stress partitioning between ferrite andmartensite is indeed a function of their respective strengths, both of which are changing in differentmicrostructures. This will be discussed in more detail in Chapter 4.

Chapter 3 Page 63

Note that at higher volume fractions of martensite (not investigated here), marten-

site is expected to deform plastically to some extent, such that the work hardening

behaviour could be affected by variations in martensite strength.

3.4.2.2 Instantaneous work hardening rate

The uniaxial tensile data can also be analyzed using the instantaneous hardening

rate, n = dlogσ/dlogǫ. This parameter provides an instantaneous mathematical

representation of the work hardening rate during uniaxial tensile deformation (see

Figure 3.4b). During the transition from elastic to plastic deformation, it is generally

observed that n decreases rapidly from 1 (elastic behaviour) to a minimum value,

referred to as nmin. Upon further straining, n increases to a maximum value, nmax, and

then gradually decreases again until the onset of plastic instability. The increase from

nmin to nmax takes place within the early stages of deformation (on average, ǫ = 2.2%

± 0.7%). It has been previously reported in the literature that the contribution of

martensite particles to the work hardening of DP steels is most prominent during the

early stages of deformation [20, 21, 23–25]. Therefore, the instantaneous hardening

rate analysis based on nmin and nmax can provide a suitable method for identifying

effects of microstructure on tensile behaviour. The results of such an analysis are

shown in Figure 3.8.

In Figure 3.8, the nmin (solid symbols) and nmax (empty symbols), are plotted

for each microstructure as a function of√

f/d. A direct relationship between nmin

and√

f/d is evident, a trend that agrees with the previously established DP work

hardening models (i.e. particle hardening mechanism) suggested by Ashby [37, 38].

Additionally, it is evident from Figure 3.8, that the AT+IC microstructures generally

have smaller nmin values in comparison with CR+IC and QT+IC (regardless of the

Chapter 3 Page 64

2 4 6 80.0

0.1

0.2

0.3

CR+IC AT+IC QT+IC

Inst

anta

neou

s W

ork

Har

deni

ng, n

(MPa

)

( f/d )1/2 (%.micron-1)1/2

Figure 3.8: Minimum and maximum instantaneous hardening exponents, nmin and nmax, as afunction of

√

f/d. Full symbols represent nmin while empty symbol show nmax. The half-fullsymbols are nmin values for microstructures with martensite content less than 10%.

austenitization temperature). This suggests that in addition to the size and volume

fraction of martensite particles, the work hardening of DP steels (at early stages) also

depends on the morphology and/or spatial distribution of martensite particles. Al-

though this effect has been proposed in the literature before [1, 8, 24], to the authors’

knowledge, this is the first systematic study that demonstrates such behaviour.

From Figure 3.8 it is evident that nmax (empty symbols) does not depend on the

volume fraction and size of martensite particles because all of the specimens with the

same starting microstructures produce similar values of nmax, regardless of the value of√

f/d. Moreover, the nmax value for each microstructure group differs from the others;

that is, the CR+IC specimen group has the highest nmax while this parameter is the

lowest for the QT+IC microstructures. In addition to the size and volume fraction of

Chapter 3 Page 65

martensite particles, these microstructure groups also differ from one another in terms

of martensite particle morphology and spatial distribution. Hence, the variation in

nmax can be attributed to these latter two parameters.

The results presented in Figure 3.7 correspond with the findings of this instan-

taneous work hardening analysis. That is, at early stages of deformation, θ follows

virtually the same relationship with√

f/d as does nmin. At ǫp = 2%, the dependence

of θ on√

f/d becomes less significant, particularly for microstructures with marten-

site volume fractions larger than 10% (full symbols). Finally, at larger strains (ǫp =

5%), θ becomes independent of√

f/d for all microstructures. This change in depen-

dency agrees with the literature [20, 21, 23–25] and can be explained using Ashby’s

work hardening model [37, 38], where the work hardening contribution of GNDs is

only significant in the early stages of plastic deformation when their respective dislo-

cation density, ρGND, is larger than that of the statistically stored dislocations (SSDs),

ρSSD.

3.4.2.3 Dislocation annihilation factor

The third approach for studying the effect of microstructure on the work hardening

behaviour of DP steels is based on the Kocks-Mecking (KM) model [39–41]. In brief,

the evolution of the dislocation density with strain is assumed to be a function of a

dislocation storage term (athermal) and a dislocation annihilation term (strain rate

and temperature dependent). According to the KM model, the work hardening rate,

θ, follows a linear relationship with stress during stage III of work hardening [41],

defined by:

θ = θ0 −θ0σV

(σ − σ0) (3.1)

Chapter 3 Page 66

where θ0 is the athermal hardening rate and σ0 is the flow stress due to strength-

ening mechanisms that do not include dislocation-dislocation interactions, such as

friction stress, back stresses, solid solution hardening and composite effects. σV is a

microstructure-dependant scaling factor that accounts for dynamic recovery effects

due to dislocation annihilation and is function of the strain rate and temperature.

For the sake of brevity, the θ0/σV will be referred to as the dislocation annihilation

factor, h, throughout this discussion. The KM model was originally proposed for fcc

materials; however, it has also been successfully applied to bcc structures such as

multiphase steels [48, 49].

In order to apply the KM model to the results of the current study, σ0 has to

be calculated. For DP steels with additional complications due to the presence of

residual stresses as well as initial mobile dislocations in the structure prior to loading

[11, 14–18], the calculation of σ0 is not trivial. A simpler approach is to assume that

the standard 0.2% proof stress represents the onset of plastic deformation. While

this may be considered an oversimplification of the problem, due to its consistency,

the 0.2% proof stress can provide a good starting point for comparison of different

microstructures. Note that the determination of σ0 can affect the value of θ0 obtained

from fitting Equation 3.1 to the data. Therefore, the KM model here is only used to

compare the dislocation annihilation terms of the various DP microstructures.

By fitting Equation 3.1 to the stress-strain data (see Appendix B for an example

of the fitting procedure), the relationships between h and√

f/d can be found for

each microstructure group as shown in Figure 3.9. It can be seen that the dislocation

annihilation rate, h, increases with increasing values of√

f/d. Additionally, the

AT+IC microstructure generally has smaller h values in comparison with QT+IC and

CR+IC. All of the DP steel microstructures, regardless of their martensite volume

Chapter 3 Page 67

2 4 6 810

20

30

40

50

Anni

hila

tion

Fact

or, h

( f/d )1/2 (%.micron-1)1/2


Figure 3.9: Dislocation annihilation factor, h, as a function of√

f/d. Full symbols represent h formicrostructures with martensite volume fractions larger than 10%. Half full symbols show h valuesfor microstructures with martensite content of less than 10%.

fraction, have larger dislocation annihilation rates when compared to the IF steel.

The KM analysis can be used to provide a better explanation of the differences

between the uniaxial tensile responses of the CR+IC25, Q7T+IC35 and AT+IC20

microstructures. Table 3.6 summarizes the annihilation factors (h) obtained from this

analysis. As mentioned earlier, due to the uncertainties associated with the definition

of σ0, true values of θ0 (athermal hardening rate) cannot be obtained from the KM

analysis presented here. However, the work hardening rate at a very small plastic

strain of 0.5% can be considered an alternative parameter for comparing the initial

dislocation storage rate among various microstructures. The values of this parameter

are also listed in Table 3.6.

It is evident from Table 3.6 that the CR+IC25 and Q7T+IC35 microstructures

Chapter 3 Page 68

Table 3.6: Work hardening rates, θ, at plastic strain of 0.5% and dislocation annihilation factor,h, for the uniaxial tensile data shown in Figure 3.4.

Microstructure θ at ǫp = 0.5% (MPa) Dislocation Annihilation Factor, h

CR+IC25 10460 25.68AT+IC20 7530 21.53Q7T+IC35 6620 26.28

have similar h values while the AT+IC20 microstructure has the lowest dislocation

annihilation rate among the three microstructural variants. In contrast, the disloca-

tion storage rate, represented here by θ at ǫp = 0.5%, is similar for both AT+IC20

and Q7T+IC35, while the CR+IC25 microstructure has the highest value. It follows

that the overall work hardening behaviour of a material is a function of both the dis-

location storage and dislocation annihilation terms. For microstructures with similar

h, CR+IC25 and Q7T+IC35, the material with a higher θ is expected to deform to

larger strains due to its higher initial work hardening capacity. This trend is clearly

visible in Figure 3.4. Considering the case of microstructures with similar values of θ,

AT+IC20 and Q7T+IC35, it can be seen that the AT+IC20 microstructure exhibits

a larger uniform elongation value since the true work hardening rate can be sustained

at higher plastic strains due to a lower dislocation annihilation factor, h.

3.5 Conclusions

The following conclusions can be drawn from the results and analysis presented

in this paper:

1. Various austenite nucleation sites exist in the DP steel microstructures during

IC annealing. The most effective ones are pearlite colonies, previous martensite

particles and carbides located at ferrite grain boundaries. The grain-interior

Chapter 3 Page 69

carbides are not very effective for austenite formation.

2. The starting microstructure can affect the austenite transformation during IC

annealing by changing the kinetics of both austenite nucleation and growth.

Consequently, the three main DP steel microstructural variants, CR+IC, QT+IC

(regardless of austenitization temperature) and AT+IC, differ significantly from

one another in terms of martensite particle volume fraction, size, morphology

and spatial distribution.

3. The three main DP steel microstructural variants exhibit very different uniax-

ial tensile behaviours. The QT+IC microstructure has the most non-uniform

spatial distribution of martensite particles and produces the least desirable uni-

axial tensile properties in terms of uniform elongation and work hardening rate.

The CR+IC microstructure on the other hand, despite having a non-uniform

distribution of martensite particles, produces the best combination of uniform

elongation and work hardening rate among the three microstructural variants.

Additionally, the work hardening rate of CR+IC microstructure is not a func-

tion of√

f/d and this discrepancy was attributed to the potential effect of

ferrite grain size.

4. It was found that, in addition to the particle size and volume fraction, the

work hardening behaviour of DP steels is also influenced by the morphology

and spatial distribution of martensite particles. This influence can be observed

by changes in both initial work hardening rate (which is a function of the dis-

location accumulation rate) as well as the dislocation annihilation rate.

5. The contribution of martensite particles to work hardening behaviour, via geo-

metrically necessary dislocations (GNDs), is only significant at the early stages

Chapter 3 Page 70

of deformation. As deformation proceeds, the work hardening rate decays to a

similar value for each microstructural variant, i.e. independent of the ratio of

martensite volume fraction to average particle size (√

f/d).

3.6 Acknowledgments

The authors would like to thank the AUTO21 NCE and the Natural Sciences and

Engineering Research Council of Canada (NSERC) for their financial support of this

research.

Chapter 3 Page 71

3.7 References

[1] N. J. Kim, G. Thomas, Metall. Trans. A 12 (1981) 483–489.



(2008) 83–90.

[4] G. R. Speich, V. A. Demarest, R. L. Miller, Metall. Trans. A 12 (1981) 1419–

1428.


[6] S. Estay, L. Chengji, G. R. Purdy, Can. Metall. Q. 23 (1984) 121–130.

[7] J. Y. Joon, S. K. In, S. C. Hyung, Metall. Trans. A 16 (1985) 1237–1245.






[12] G. R. Speich, in: R. A. Kot, B. L. Bramfitt (Eds.), Fundamentals of Dual-Phase

Steels, Metall. Soc. of AIME, 1981, pp. 3–45.

[13] N. C. Law, D. V. Edmonds, Metall. Trans. A 11 (1980) 33–46.



[16] J. M. Rigsbee, P. J. VanderArend, in: A. T. Davenport (Ed.), Formable HSLA

and Dual-Phase Steels, Metall. Soc. of AIME, 1977, pp. 56–86.


Chapter 3 Page 72


1228.

[19] P. J. Jacques, Q. Furnemont, F. Lani, T. Pardoen, F. Delannay, Acta. Mater. 55

(2007) 3681–3693.



118–144.




Metall. 104 (2007) 315–323.


Int. J. Plast. 43 (2013) 128–152.


(2010) 91–98.


[26] O. Bouaziz, T. Lung, M. Kandel, C. Lecomte, Le Journal de Physique IV 11

(2001) Pr4–223–Pr4–231.

[27] M. Delince, Y. Brechet, J. D. Embury, M. G. D. Geers, P. J. Jacques, T. Pardoen,

Acta. Mater. 55 (2007) 2337–2350.


2738–2746.

[29] G. R. Speich, R. L. Miller, in: R. A. Kot, B. L. Bramfitt (Eds.), Fundamentals





Chapter 3 Page 73

[32] P. Movahed, S. Kolahgar, S. P. H. Marashi, M. Pouranvari, N. Parvin, Mater.

Sci. Eng. A 518 (2009) 1–6.

[33] A. Kumar, S. B. Singh, K. K. Ray, Mater. Sci. Eng. A 474 (2008) 270–282.

[34] E. Ahmad, T. Manzoor, M. M. A. Ziai, N. Hussain, J. Mat. Eng. Perform. 21

(2012) 382–387.


[36] A. Bag, K. K. Ray, E. S. Dwarakadasa, Metall. Mater. Trans. A 30 (1999) 1193–

1202.



[39] Y. Estrin, H. Mecking, Acta Metall. 32 (1984) 57–70.

[40] U. F. Kocks, H. Mecking, Prog. Mater. Sci. 48 (2003).

[41] H. Mecking, U. F. Kocks, Acta Metall. 29 (1981) 1865–1875.

[42] W. Steven, A. G. Haynes, J. Iron Steel Inst. 183 (1956) 349–359.

[43] K. W. Andrews, J. Iron Steel Inst. 203 (1965) 721–727.

[44] F. S. LePera, J. Metal. 32 (1980) 38–39.

[45] H. Paruz, D. V. Edmonds, Mater. Sci. Eng. A A117 (1989) 67–74.

[46] Y. Tobiyama, K. Osawa, M. Hirata, Kawasaki Steel Giho 31 (1999) 181–184.

[47] G. F. Vander Voort, Atlas of Time-temperature Diagrams for Irons and Steels,

ASM International, Materials Park, OH, 1991.

[48] J. Bouquerel, K. Verbeken, B. C. D. Cooman, Acta. Mater. 54 (2006) 1443–1456.

[49] I. Gutierrez, M. A. Altuna, Acta. Mater. 56 (2008) 4682–4690.

CHAPTER 4

Effect of Ferrite Grain Size and Spatial

Distribution of Martensite Particles on

the Work Hardening Behaviour of a

Dual-Phase Steel

Abstract

This paper reports on a study of the relationship between microstructure and

work hardening of dual-phase (DP) steel sheets. Through the addition of a cold-

rolling step between pre-heat treatment and IC annealing, three distinct DP steel mi-

crostructural variants with a significantly refined ferrite matrix were produced. These

variants differ with respect to their mean ferrite grain size, D, as well as the volume

fraction, f , mean size, d, morphology and spatial distribution of martensite particles.

The work hardening behaviour of uniaxial tensile specimens was analyzed using the

true work hardening rate, θ = dσ/dǫ, the instantaneous work hardening exponent,

n = dlogσ/dlogǫ and the dislocation annihilation factor, h = −(θ− θ0)/(σ− σ0) . At

small strains (ǫp = 0.5%), the work hardening rate was found to be dominated by the

generation of geometrically necessary dislocations (GNDs) in the ferrite grains. The

work hardening response at this stage was characterized by θ at ǫp = 0.5% and a min-

imum in the instantaneous work hardening exponent, nmin. Both of these parameters

were determined to be functions of√

f/d, the mean ferrite grain size and the spa-

tial distribution of martensite particles. At larger plastic strains (>4%), dislocation

annihilation by dynamic recovery becomes the controlling factor for work hardening.

This phenomena is described by the dislocation annihilation factor, h, and is also a

function of√

f/d, the mean ferrite grain size and the spatial distribution of marten-

site particles. The uniform elongation was found to be inversely proportional to the

dislocation annihilation factor, h. Finally, the three work hardening parameters of θ

at ǫp = 0.5%, nmin and h all exhibit similar relationship with√

f/d and a consistent

74

Chapter 4 Page 75

trend between the three microstructural variants under study. This observation sug-

gests a common role of internal stresses on the two work hardening mechanisms of

GND hardening and dynamic recovery.

4.1 Introduction

Dual-phase (DP) steels are known for their attractive combination of strength and

formability as they exhibit continuous yielding, low yield-to-tensile strength ratios,

large uniform elongations and very high initial work hardening rates. The traditional

microstructure of these steels comprises martensite particles distributed in a ferrite

matrix. The continuous yielding at low stresses is associated with the presence of

mobile dislocations at ferrite/martensite interfaces as well as residual stresses in the

ferrite matrix which are produced by quenching from the inter-critical annealing tem-

perature [1–6]. The high initial work hardening rate, on the other hand, is due to

the strain incompatibility between the soft, ductile ferrite and hard, non-deformable

martensite. This incompatibility results in additional work hardening of the ferrite

matrix due to the development of long-range elastic back-stresses and the genera-

tion of geometrically necessary dislocations (GNDs) at ferrite/martensite interfaces

[1, 2, 4, 7, 8]. It has been found that the work hardening rate of DP steels is primarily

controlled by the volume fraction, f [4, 7, 9–12], and average size, d [4, 12], of marten-

site particles through the parameter√

f/d. Other microstructural parameters, such

as the ferrite grain size [13–16] as well as the morphology and spatial distribution

of martensite particles [9, 17–21], have also been shown to be significant. However,

since these parameters often cannot be changed independently of one another, it has

proven difficult to understand their individual effects on work hardening behaviour.

In the previous work [18], three distinct microstructures were produced, while

Chapter 4 Page 76

controlling the parameter√

f/d, to investigate the effects of morphology and spatial

distribution of martensite particles on the work hardening rate of DP steels. It was

found that the work hardening rate was clearly affected by these parameters; how-

ever, this finding was not conclusive because the three microstructures had a range

of ferrite grain sizes. Therefore, the current research programme was designed to

determine the specific effects of ferrite grain size and spatial distribution of marten-

site particles on work hardening behaviour. This goal was accomplished by including

an additional cold-rolling step prior to the inter-critical (IC) annealing of two differ-

ent starting microstructures: (i) austempered (AT) and (ii) quenched and tempered

(QT). The cold-rolling step was added to refine the ferrite grain size through ferrite

recrystallization during the IC annealing process [22–26]. These two microstructures

and their tensile properties are compared to a baseline microstructure obtained by

direct IC annealing of an industrial cold-rolled ferrite/pearlite structure without any

additional pre-heat treatment. Specifically, the work hardening behaviour of each

microstructural variant was characterized using: (i) the true work hardening rate,

θ = dσ/dǫ; (ii) the instantaneous work hardening exponent, n = dlogσ/dlogǫ [27, 28];

and (iii) the dislocation annihilation parameter, h = −(θ − θ0)/(σ − σ0) [18].


4.2.1 Materials

The material used for this study was a commercial DP780 grade steel provided

by US Steel Canada in two conditions: (i) 0.95mm thick cold-rolled sheet (CR) and

(ii) 3.12mm thick hot-rolled sheet (HB). The chemical analyses of the two sample

materials are given in Table 4.1. Prior to thermo-mechanical processing, 100mm by

Chapter 4 Page 77

20mm and 110mm by 45mm blanks were cut from the CR and HB sheets, respectively,

both with the long axis oriented in the rolling direction (RD).

Table 4.1: Chemical composition (in wt. %) of DP780-CR and DP780-HB steel sheets

Steel C Mn P S Si Cu Ni Cr Mo N V B Ti Nb

DP780-CR 0.09 2.1 0.012 0.006 0.02 0.03 0.01 0.26 0.29 0.004 0.001 0 0.001 0.002DP780-HB 0.09 2.11 0.015 0.007 0.024 0.03 0.01 0.26 0.302 0.003 0.001 0.0001 0.001 0.002

4.2.2 Thermo-Mechanical Processing

The thermo-mechanical processing (TMP) schedules for both the CR and HB

starting materials are given in Table 4.2. The DP780-CR material was directly IC

annealed without any additional pre-heat treatment to produce a baseline microstruc-

ture (CR+IC). Two other microstructural variants were made from the 780-HB ma-

terial by applying a pre-heat treatment step followed by cold-rolling and then inter-

critical annealing.

The pre-heat treatment was either an austemper (AT) or quench and temper (QT)

treatment, as follows:

(a) Austemper pre-treatment (AT): HB samples were austenitized for 30 minutes at

920◦C, air cooled to 500◦C, held for 20 minutes and finally water quenched to room

temperature. Austenitization treatments were carried out in a Lindberg 54232

tube furnace under continuous argon flow to avoid oxidization of the specimen’s

surface. The end-to-end variation in temperature along the length of the heat

treating blanks was 10◦C. The 500◦C bainite-hold treatment was carried out in

a salt bath of a potassium nitrate and sodium nitrate mixture. The temperature

gradient in the salt bath over the sample dimensions was 2◦C. The cooling rate

Chapter 4 Page 78

during the transfer of specimens from the austenitization furnace to the bainite-

hold salt bath was 13 K/s. The 500◦C transformation temperature was selected

to be midway between the bainite-start (575◦C) and martensite-start (430◦C)

temperatures, calculated using the Steven and Haynes [29] and Andrews [30]

formulae, respectively.

(b) Quench and Temper pre-treatment (QT): HB samples were austenitized for 30

minutes at 970◦C, water quenched and tempered for 1 hour at 600◦C. Both austen-

itization and tempering treatments were carried out in a Lindberg 54232 tube

furnace under continuous argon flow.

Following pre-heat treatments, the AT and QT samples were cold rolled on a

lab-scale rolling mill to 80% reduction (15-20 passes), cut into 100 mm by 20 mm

blanks and labelled as AT+CR8 and QT+CR8, respectively (Table 4.2). All three mi-

crostructural variants were then IC annealed for 2 minutes and subsequently quenched

into a water and ice mixture. IC annealing was carried out at various temperatures

in the range of 715-735◦C to produce DP microstructures with a martensite volume

fraction range of 12-29%. IC annealing treatments were carried out in a salt bath with

an average heating rate of 19 K/s. The end-to-end variation in the temperature along

the length of the heat treating blanks was 2◦C. Based on the average time required

for specimens to reach the Ac1 temperature of 700◦C, determined from the Andrews

Table 4.2: Summary of TMP schedules for the three microstructure variants. WQ: WaterQuenched.

MicrostructuralVariant

Pre-treatment Cold-Roll IC AnnealingTemp. 1 Time 1 Cool 1 Temp. 2 Time 2 Cool 2 Amount Temp. Time Notes(◦C) (min) (◦C) (min) (%) (◦C) (min)

CR+IC – – – – – – 720-725 2 WQAT+CR8+IC 920 30 14 K/s 500 20 WQ 80 720 2 WQQT+CR8+IC 970 30 WQ 600 60 WQ 80 735 2 WQ

Chapter 4 Page 79

formula [30], the IC annealing time began 40 seconds after immersing the specimens

in the salt bath. The details of the TMP schedules for the three microstructural

variants investigated are presented in Table 4.2. At all stages of pre-heat treatment

and IC annealing, the temperature was measured at the centre of the heat treatment

blanks using a K-type thermocouple.

4.2.3 Microstructure Characterization

Metallographic specimens were cut from the middle of the heat treated blanks

using a Struers Accutom precision cut-off machine. The RD/ND sections were ex-

amined for all specimens, where RD is the rolling direction and ND is the normal,

through-thickness direction. After mounting, specimens were polished to a 0.06 mi-

cron surface finish using standard metallography procedures and etched in 2% Nital

for 13 seconds to reveal the ferrite grain structure as well as the martensite and

carbide particles. The metallographic specimens were examined using a JEOL 840

scanning electron microscope (SEM) under secondary electron (SE) imaging mode at

a working distance of 15mm and with an accelerating voltage of 20kV.

Identification of martensite and carbide particles followed the procedure reported

elsewhere [18]. Martensite volume fraction measurements were performed on the SEM

micrographs using a manual point counting technique with a minimum of 4000 points

in accordance with ASTM-E562. Measurements of mean ferrite grain size (diameter)

and mean martensite particle size (diameter) were made using the three concentric

circles intercept method of ASTM-E112.

Chapter 4 Page 80

4.2.4 Mechanical Testing

Sub-size uniaxial tensile specimens with a gauge length of 25mm were water-

jet cut from the heat treated blanks according to ASTM-E8. All specimens had

their long axis oriented in the RD. Uniaxial tensile tests were performed at room

temperature using an Instron 8521 hydraulic testing machine at a constant extension

rate of 0.75mm/min. Strain measurements were obtained using a 25mm gauge Instron

clip-on extensometer. For each microstructural variant, a minimum of three specimens

was tested with all tests completed to failure. Work hardening analyses were carried

out using the uniaxial tensile test data by calculating the true work hardening rate,

θ = dσ/dǫ, the instantaneous work hardening exponent, n = dlogσ/dlogǫ [27, 28] and

the dislocation annihilation parameter, h = −(θ − θ0)/(σ − σ0) [18].

Chapter 4 Page 81

4.3 Results


The microstructure of the as-received HB material is presented in Figure 4.1a.

This microstructure comprises ferrite, pearlite and small amounts of martensite. Fig-

ure 4.1b, depicts the microstructures following the austempering (AT) pre-heat treat-

ment (Table 4.2) which contains a mixture of granular bainite (GB), i.e. irregular

ferrite grains with M/A, and upper bainite (UB), i.e. ferrite laths with marten-

site/retained austenite (M/A) between the laths. These observations are consistent

with both experimental results [31–33] as well as the predications of published CCT

and TTT diagrams [34–36] for steels of similar compositions. Note that individual

martensite (M) particles are also present in the AT microstructure. These martensite

particles were formed upon quenching of the remaining untransformed austenite at the

end of the bainite-hold step. In specimens subjected to the quench and temper pre-

heat treatment (QT), Figure 4.1c, a tempered martensite microstructure is obtained

where fine carbide particles are distributed along prior austenite grain boundaries and

prior martensite lath boundaries.

Chapter 4 Page 82

(a)

(b)

(c)

Figure 4.1: SEM micrographs of (a) as-received HB, (b) AT and (c) QT conditions. Nital etch,where ferrite is dark and non-ferritic phases (pearlite, martensite and carbides) appear white.

Chapter 4 Page 83

Figure 4.2 shows the three microstructural variants following the cold-rolling step

and prior to inter-critical (IC) annealing, i.e. as-received CR microstructure as well

as the AT+CR8 and QT+CR8 microstructures. The following observations can be

made:� The CRmicrostructure (Figure 4.2a) contains ferrite, pearlite and small amounts

of martensite. All three constituents are deformed in the rolling direction (RD),

producing a visibly banded microstructure.� The AT+CR8 microstructure (Figure 4.2b) comprises both ferrite and non-

ferritic phase (NFP) particles, i.e. individual martensite particles as well as

M/A constituents of upper bainite and granular bainite phases. In comparison

with the other two microstructures, the size of the NFP particles is the largest,

the degree of deformation in individual NFP particles is low and the degree of

banding is also low.� In the QT+CR8 microstructure (Figure 4.2c), the NFP (carbide) particles are

the smallest of the three microstructural variants and show no evidence of de-

formation. The NFP particles are generally aligned along the RD, producing a

distinctly banded microstructure.

Chapter 4 Page 84

(a)

(b)

(c)

Figure 4.2: SEM micrographs after thermo-mechanical processing and prior to the IC annealing:(a) CR (b) AT+CR8 and (c) QT+CR8.

Chapter 4 Page 85

The three microstructures depicted in Figure 4.2 underwent an IC annealing treat-

ment for 2 minutes at various temperatures to obtain a range of volume fractions of

martensite. To compare the microstructures after IC annealing, representative spec-

imens with a similar volume fraction, f , of 15% were selected from each of the three

microstructural variants, i.e. CR+IC725, AT+CR8+IC720 and QT+CR8+IC720

(Table 4.3). Figure 4.3 shows the microstructures for these conditions at low (4.3a-

4.3c) and high (4.3d-e) magnifications. The mean ferrite grain size, D, mean marten-

site particle size, d, mean martensite volume fraction, f , and the mean√

f/d values

are presented in Table 4.3. From Figure 4.3 and Table 4.3, it is evident that the

martensite particles have an equiaxed morphology and a relatively uniform size dis-

tribution in all three microstructural variants. The martensite particles are also pri-

marily located at ferrite grain boundaries and corners in all cases. Hence, the spatial

distribution of the martensite particles in each microstructure is directly related to

the ferrite grain structure. Specifically, the following observations can be made:� In the CR+IC725 microstructure (Figures 4.3a and 4.3d), the ferrite matrix is

fully recrystallized with a bimodal grain size distribution, i.e. small, equi-axed

grains at prior pearlite band locations (e.g. grain #1 in Figure 4.3d) and large

grains at prior ferrite sites (e.g. grain #2 in Figure 4.3d). Consequently, the

martensite particles are spatially distributed in bands along the RD. The mean

ferrite grain size and the mean martensite particle size are the largest for this

condition.� In the AT+CR8+IC720 microstructure (Figures 4.3b and 4.3e), the ferrite ma-

trix is fully recrystallized and ferrite grains are equiaxed, uniform in size and

have the smallest average size in comparison with the other variants. As a re-

sult, no visible banding of martensite particles is present. The mean martensite

Chapter 4 Page 86

Table 4.3: Ferrite and martensite measurements for microstructures shown in Figure 4.3

Ferrite Martensite

Microstructural Size, D Vol. Frac., f Size, d√

f/d Morphology SpatialVariant (microns) (%) (microns) (%.microns−1) Distribution

CR+IC725 1.84 ± 0.25 15.1 ± 1.8 0.50 ± 0.06 5.5 ± 0.5 Equiaxed Bands along RDAT+CR8+IC720 1.09 ± 0.07 16.1 ± 1.2 0.36 ± 0.02 6.7 ± 0.3 Equiaxed UniformQT+CR8+IC720 1.51 ± 0.28 14.7 ± 2 0.34 ± 0.4 6.5 ± 0.6 Equiaxed Bands along RD

particle size is smaller than CR+IC725 but similar to QT+CR8+IC720.� In the QT+CR8+IC720 microstructure (Figures 4.3c and 4.3f), the spatial dis-

tribution of martensite particles is similar to the CR+IC725 variant in that

these particles are primarily located at ferrite grain boundaries and they are

aligned in bands along the RD. This microstructure, however, differs signifi-

cantly from CR+IC725 in terms of the ferrite matrix as three different types of

ferrite grains can be distinguished in Figure 4.3f: (i) small, recrystallized and

equiaxed, e.g. grain #1, (ii) large, recrystallized and elongated along RD, e.g.

grain #2, and (iii) bands of small, unrecrystallized ferrite grains oriented along

RD, e.g. grain #3. The mean ferrite grain size (based only on ferrite grain

boundaries and not the sub-grain boundaries)1 is smaller than CR+IC725 but

larger than AT+CR8+IC720.� In all microstructures, fine carbide particles, in the range of 50-100 nm, are

present throughout the ferrite matrix. The procedure for identifying these par-

ticles is given in [18]. The carbide particles are marked by arrows in the high

magnification SEM micrographs of Figures 4.3d-4.3f.

1To distinguish between ferrite grain boundaries and sub-grain boundaries in unrecrystallizedferrite grains, EBSD images (not presented here) were acquired for the specimen of Figures 4.3cand 4.3f. It was found that the sub-grains were generally very small in size and slightly elongatedalong the rolling direction. The recrystallized ferrite grains, however, are larger in size and are oftenequixed. This knowledge of morphology, size and spatial distribution of grains and sub-grains wasthen used to distinguish them in the SEM images.

Chapter 4 Page 87

(a) (d)

(b) (e)

(c) (f)

Figure 4.3: Final dual-phase microstructures after IC annealing. (a,d) CR+IC725, (b,e)AT+CR8+IC720 and (c,f) QT+CR8+IC720. Arrows point to the individual carbide particles insidethe ferrite matrix.

Chapter 4 Page 88


The uniaxial tensile behaviour of the microstructures shown in Figure 4.3 (with an

average martensite volume fraction of 15%) are compared in Figure 4.4. Engineering

stress-strain curves are presented in Figure 4.4a and plots of the instantaneous work

hardening exponent, n = dlogσ/dlogǫ, versus engineering strain are given in Figure

4.4b. The latter parameter provides an instantaneous mathematical representation of

the work hardening rate during uniaxial tensile deformation. The yield strength, σy,

ultimate tensile strength, UTS, and uniform elongation, i.e. strain at UTS, are listed

in Table 4.4. The following observations can be made from Figure 4.4 and Table 4.4:� The CR+IC725 microstructure demonstrates continuous yielding behaviour,

whereas both AT+CR8+IC720 and QT+CR8+IC720 microstructures exhibit

discontinuous yielding (Figure 4.4a).� Despite the differences in their yielding behaviour, all three microstructures

show similar ultimate tensile strength (UTS) values. However, the uniform elon-

gation is very different among the three microstructures, i.e. the AT+CR8+IC720

has the largest uniform elongation and QT+CR8+IC720 the lowest (Table 4.4).� In terms of the instantaneous work hardening exponent (Figure 4.4b), each

microstructure demonstrates typical three-stage behaviour, labelled by A, B and

C, respectively. That is, at the beginning of deformation, n decreases rapidly

from elastic behaviour to a minimum value, nmin (Stage A), then increases to a

maximum value, nmax (Stage B), and finally gradually decreases again until the

onset of plastic instability (Stage C). From Figure 4.4b, it is evident that at early

stages of deformation (Stage A with 0 - 2% strain), the CR+IC725 condition

has the highest value of n while AT+CR8+IC720 has the lowest n. However,

Chapter 4 Page 89

at higher strains (Stage C with strains larger than 4%), the AT+CR8+IC720

microstructure exhibits the highest value of n and QT+CR8+IC720 produces

the lowest.

Chapter 4 Page 90

0.00 0.05 0.10 0.15 0.20 0.250

200

400

600

800

CR+IC725 AT+CR8+IC720 Q7T+CR8+IC720

Eng

inee

ring

Stre

ss (M

Pa)

Engineering Strain(a)

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

BC

Inst

anta

neou

s W

ork

Har

deni

ng E

xpon

ent,

n (M

Pa)

Engineering Strain

CR+IC725 AT+CR8+IC720 QT+CR8+IC720

A

(b)

Figure 4.4: (a) Engineering stress-strain tensile curves for the three microstructural variants shownin Figure 4.3 (average martensite volume fraction of 15%). Arrows indicate uniform elongation(strain at maximum load). (b) Instantaneous work hardening exponent, n, vs. engineering strainplots obtained from stress strain curves shown in (a).

Chapter 4 Page 91

Table 4.4: Uniaxial Tensile parameters calculated from Figure 4.4a

Microstructural Yield Strength, σy UTS Uniform ElongationVariant (MPa) (MPa) (%)

CR+IC725 342 750 15.3AT+CR8+IC720 423 740 16.6QT+CR8+IC720 427 761 12.8

4.4 Discussion


The austenite transformation is a diffusion-controlled transformation which there-

fore depends on the availability of (i) suitable nucleation sites and (ii) sufficient carbon

content for growth of carbon-rich austenite. When the ferrite matrix is fully recrystal-

lized prior to reaching the austenite transformation temperature, the ferrite/cementite

interface in pearlite colonies as well as the individual cementite particles on ferrite

grain boundaries are the two preferred nucleation sites for austenite [22, 24]. This

preference has been attributed to the additional surface energy associated with grain

boundaries which reduces the required activation energy for austenite nucleation,

∆G∗. If, however, the ferrite recrystallization is not completed before reaching the IC

annealing temperature, the stored energy of the unrecrystallized ferrite grains leads

to an additional reduction of the activation energy for austenite nucleation (∆G∗),

thereby inducing a significant nucleation of austenite on the cementite particles that

are located on the sub-grain boundaries of unrecrystallized ferrite grains [22–24].

Moreover, the sub-grain boundaries of the deformed ferrite matrix also provide paths

of accelerated diffusion for carbon and other alloying elements, thereby enhancing the

kinetics of the austenite growth process [23, 37].

With this background, it is possible to understand the evolution of the three final

Chapter 4 Page 92

DP microstructures shown in Figure 4.3. In the CR+IC725 specimen, the starting

microstructure before IC annealing (CR) consists of a banded ferrite and pearlite

mixture oriented in the RD as well as a small number of individual martensite particles

(Figure 4.2a). The pearlite colonies and martensite particles are known to be a very

efficient nucleation sites for austenite; therefore, during IC annealing, they rapidly

transform to austenite [22]. As a result, the final CR+IC725 microstructure comprises

bands of martensite particles at recrystallized ferrite grain boundaries with a spacing

and spatial distribution similar to that of the pearlite bands in the starting CR

microstructure (Figures 4.3a and 4.3d).

For the AT+CR8+IC720 condition (Figure 4.3b and 4.3e), the starting microstruc-

ture prior to IC annealing (AT+CR8) comprises ferrite grains and NFP particles

(M/A) with a low degree of banding (Figure 4.2b). During heating of the AT+CR8

microstructure to the IC annealing temperature, two processes occur simultaneously:

(i) complete recrystallization of ferrite leading to a fine grained, equiaxed ferrite

matrix, and (ii) tempering of M/A constituents to produce ferrite and spheroidized

cementite. Both of these effects were confirmed by heating specimens to the IC

annealing temperature and quenching immediately to avoid the start of austenite

transformation. Thus, the uniform distribution of suitable austenite nucleation sites

(equiaxed, recrystallized ferrite grain boundaries) as well as carbon sources (carbides

due to tempering of M/A particles) results in a uniform distribution of austenite

grains throughout the microstructure after the IC annealing process (Figures 4.3b

and 4.3e).

In the case of the QT+CR8+IC720 specimen (Figure 4.3c and 4.3f), the start-

ing microstructure prior to the IC annealing, QT+CR8, contains fine cementite

particles that are distributed in a deformed ferrite matrix and located along prior

Chapter 4 Page 93

austenite grain boundaries and prior martensite lath boundaries (Figure 4.2c). When

heated to the IC annealing temperature (Figure 4.5), the majority of ferrite grains

in the QT+CR8 remain unrecrystallized, while the few recrystallized ferrite grains

exhibit significant growth. Comparing Figure 4.5 with Figure 4.3c suggests that

these large recrystallized grains correspond to the large ferrite grains in the final

QT+CR8+IC720 microstructure. Furthermore, during the IC annealing treatment,

the austenite nucleation primarily takes place on the carbide particles that are located

on the unrecrystallized ferrite grain (and subgrain) boundaries, due to the high strain

energy at these locations. Since the majority of these grain (and subgrain) bound-

aries are elongated along the RD, the spatial distribution of martensite particles in

the final DP microstructure is also banded along the RD. Finally, a comparison of

Figures 4.3c (and 4.3f) with Figure 4.5 reveals that some of the unrecrystallized fer-

rite grains do not recrystallize during the 2-minute IC annealing treatment, possibly

due to the pinning effect of the austenite grains (formed during IC annealing) on the

ferrite grain boundaries. Additionally, the competition between austenite transfor-

mation and ferrite recrystallization can also lead to an incomplete recrystallization of

ferrite grains [25]. That is, austenite nucleation results in a partial relaxation of the

high strain energies associated with deformed ferrite grains which in turn lowers the

driving force for ferrite recrystallization [26].

As a final observation, the cementite particles that are not located on ferrite grain

boundaries (recrystallized or unrecrystallized) in the three starting microstructures

(Figure 4.2) remain intact during IC annealing, as shown in Figures 4.3d-4.3f (marked

with arrows). This outcome is expected [22, 24] since the lack of additional surface

energy associated with the grain (and subgrain) boundaries results in a larger critical

austenite nucleation energy (∆G∗), thereby rendering these particles ineffective for

Chapter 4 Page 94

Figure 4.5: Microstructure of the QT+CR8 specimen heated to the IC annealing temperature andimmediately quenched without any holding time.

the nucleation of austenite.


As shown in Figure 4.4 and Table 4.4, at similar volume fractions of martensite

( 15%), the three microstructural variants exhibit similar UTS values while their yield-

ing behaviour, work hardening rate and uniform elongation values are significantly

different. In terms of yielding behaviour, it is well known that the continuous yielding

of DP steels results from the presence of mobile dislocations at ferrite/martensite in-

terfaces which are produced by the volume expansion associated with the martensitic

transformation during quenching from the IC annealing temperature [1–6]. The den-

sity of these transformation dislocations is a function of the martensite volume fraction

(f) and a minimum volume fraction of martensite is needed to produce continuous

yielding behaviour [38]. Since f is similar in all three microstructures investigated

here, the discontinuous yielding in two of the microstructures, i.e. AT+CR8+IC720

and QT+CR8+IC720, requires further explanation. Using EBSD measurements, Ra-

mazani et al. [39] found that, within a ferrite grain, the area surrounding a martensite

Chapter 4 Page 95

particle that is affected by transformation dislocations is independent of ferrite grain

size. It follows that for a reduced ferrite grain size, and at a constant f , a larger

area fraction of each ferrite grain is affected by these transformation dislocations.

This concept is particularly important in ultra-fine ferrite grains with an average size

∼1 micron where it has been demonstrated that entire ferrite grains can be affected

by transformation dislocations, thereby resulting in a more homogenous distribu-

tion of dislocation density [15, 40]. As the entire ferrite grain is influenced by the

transformation dislocations, a greater degree of interaction is possible between these

dislocations which could lead to their partial immobilization, and consequently, dis-

continuous yielding behaviour. Therefore, the discontinuous yielding behaviour of

the QT+CR8+IC720 and AT+CR8+IC720 variants could be caused by their smaller

mean ferrite grain size in comparison to CR+IC725.

To understand the work hardening behaviour of the three microstructural variants,

Ashby’s hardening model [41, 42] can be used where the work hardening rate is directly

related to the density of geometrically necessary dislocations (GNDs). The dislocation

density is proportional to the volume fraction, f , and inversely related to the size, d

of martensite particles [4, 9, 12]. Since flow stress is a function of the square root of

the dislocation density, the√

f/d parameter is often selected to describe the Ashby

work hardening mechanism [4]. From Table 4.3, the√

f/d values are similar in the

AT+CR8+IC720 and QT+CR8+IC720 microstructures, but smaller in CR+IC725.

This is in contrast with the trends observed in Figure 4.4b for the instantaneous work

hardening exponent (n). That is, at early stages of deformation (0 - 2% strain), the

CR+IC725 condition has the highest value of n while AT+CR8+IC720 has the lowest

n. At higher strains (4% and larger), the AT+CR8+IC720 microstructure exhibits

the highest value of n and QT+CR8+IC720 shows the lowest. It follows that the

Chapter 4 Page 96

differences observed in the uniaxial tensile behaviour of these three microstructures

cannot simply be explained by the effect of√

f/d. Therefore, other microstructural

parameters, namely the spatial distribution of martensite particles as well as the

average ferrite grain size, should also be considered. To investigate this hypothesis,

three different parameters were used to analyze the work hardening behaviour of

the three microstructural variants in more detail: (i) the true work hardening rate,

θ = dσ/dǫ, (ii) the instantaneous work hardening exponent, n = dlogσ/dlogǫ, and

(iii) the dislocation annihilation factor, h = −(θ − θ0)/(σ − σ0).

4.4.2.1 True Work Hardening Rate

Figure 4.6 shows a series of master curves where the true work hardening rate,

θ = dσ/dǫ, of all microstructures produced in this study (Table 4.2) are plotted

against√

f/d at two plastic strains of 0.5% (Figure 4.6a) and 2% (Figure 4.6b).

Note that for microstructures exhibiting discontinuous yielding, θ is plotted as half-

full symbols. It can be seen that at small strains (ǫp = 0.5 %), θ follows a linear

relationship with√

f/d for all of the three microstructural variants, regardless of the

yielding type (continuous or discontinuous). This trend indicates that, in accordance

with Ashby’s work hardening model [41, 42], the generation of GNDs is a significant

work hardening mechanism at small strains in all of the microstructures. It is also

evident from Figure 4.6a that the dependence of θ on√

f/d differs for each of the three

microstructural variants. Since, the three microstructural variants have similarly

sized martensite particles with an equiaxed morphology, the differences in the work

hardening behaviour at small strains can be related to variations in (i) the spatial

distribution of martensite particles and (ii) the ferrite grain size.

Specifically, comparing AT+CR8+IC and QT+CR8+IC variants at a similar

Chapter 4 Page 97

2 4 6 80

5000

10000

15000

20000

25000

p a

t pla

stic

stra

in o

f 0.5

% (M

Pa)

(f/d )1/2 (%.micron-1)1/2

CR+IC AT+CR8+IC QT+CR8+IC

(a)

2 4 6 80

5000

10000

15000

20000

25000

at p

last

ic s

train

of 2

% (M

Pa)

(f/d )1/2 (%.micron-1)1/2

p


(b)

Figure 4.6: Relationship between true work hardening rate, θ, and the microstructural parameter√

f/d for plastic strain of (a) 0.5% and (b) 2%. Half-full symbols show θ for the microstructureswhich exhibit discontinuous yielding.

Chapter 4 Page 98√

f/d, the latter exhibits a larger θ at low strains (Figure 4.6a). This observation

can be related to the inhomogeneous spatial distribution of martensite particles in

the QT+CR8+IC variant which produces regions of higher strength within the mi-

crostructure, where bands of unrecrystallized ferrite grains contain a large number of

martensite particles, as well as softer areas, where bands of large, recrystallized ferrite

grains have a limited number of martensite particles. These distinct regions cause

additional plastic incompatibility in the microstructure, and therefore, an increase

in the amount of internal stresses [15]. Since the generation of GNDs are associated

with the plastic relaxation of internal stresses [41–43], a higher density of GNDs is

expected in the ferrite matrix of the QT+CR8+IC variant. Similar effects have been

reported by Azizi-Alizamini et al. [44] in a ferrite/cementite microstructure with a

bimodal size distribution of ferrite grains.

In terms of the CR+IC and QT+CR8+IC variants, both have a very similar

spatial distribution of martensite particles. From Table 4.3 it can be seen that for the

selected microstructural variants with an average martensite volume fraction of 15%,

the average ferrite grain size is smaller in AT+CR8+IC720 than in CR+IC725. In a

given microstructural variant, the changes in the volume fraction of martensite (and

therefore√

f/d) is not expected to result in a significant difference in the average

ferrite grain size [45]. Therefore, it follows that regardless of the value of√

f/d,

the average ferrite grain size is also generally smaller in AT+CR8+IC compared to

CR+IC. Consequently, the smaller θ in the AT+CR8+IC variant can be associated

with its finer ferrite grain size. The influence of ferrite grain size on the work hardening

response of DP steels has been reported before [13–16]; however, the present study

is unique since this effect is investigated independent of the value of√

f/d. The

reduction of θ with decrease in ferrite grain size can be explained by two effects.

Chapter 4 Page 99

First, upon refinement of the ferrite matrix, a larger area fraction of ferrite grains is

expected to be affected by the transformation dislocations (formed upon quenching

from IC annealing). The transformation dislocations cause a local increase in ferrite

strength [46]; therefore, as the ferrite grain size is decreased, the strength of the

ferrite is increased by the effect of the transformation dislocations [15]. In addition

to the Hall-Petch effect, this also leads to a smaller stress/strain incompatibility

between ferrite and martensite [13], thereby reducing the magnitude of the internal

stress and consequently, GND hardening. Second, since the martensite particles are

primarily located at ferrite grain boundaries in both microstructural variants, the

inter-particle spacing of martensite is expected to decrease with a reduced ferrite grain

size. These more closely spaced martensite particles are reported to result in mutual

interaction and cancelation of the stress fields associated with dislocation pileups at

the neighbouring particles, thereby reducing the magnitude of overall internal stress

[47], and hence, GND hardening.

The ferrite grain size effect could also explain the lower θ data for AT+CR8+IC

since it has the smallest average ferrite grain size of all three microstructural variants.

However, in this case, it is impossible to clearly separate out the effects of ferrite

grain size and spatial distribution of martensite particles in AT+CR8+IC as these

two microstructural parameters are coupled with one another.

From Figure 4.6b it is evident that as the plastic strain is increased to 2% (ǫp =

2%), the dependence of θ on√

f/d is absent and θ becomes essentially similar for

all three microstructural variants. This common upper limit, suggests that at higher

plastic strains, the work hardening due to GNDs is not the primary work hardening

mechanism; hence, other work hardening effects (not exclusive to DP steels) become

more prominent. This transition in the work hardening mechanism will be discussed

Chapter 4 Page 100

in more detail in Section 4.4.2.2.

4.4.2.2 Instantaneous Work Hardening Rate

The instantaneous hardening (n) curve of a DP steel, as seen in Figure 4.4b, can

be divided into three stages. In Stage A, n decreases rapidly from elastic behavior

to a minimum value of nmin. Upon further straining (Stage B), n increases to a

maximum value of nmax, and finally during Stage C, n gradually decreases until

the onset of plastic instability. The three stages of n corresponds directly with the

three work hardening stages typically observed in the Jaoul-Crussards (JC) analysis

[2, 19, 48, 49]. Stage A generally corresponds to the elastic to plastic transition of

ferrite [2, 19, 48]. Stage B is attributed to the plastic incompatibility between ferrite

and martensite [2, 19] which leads to the additional contributions of back-stresses and

GNDs to the overall work hardening rate, thereby increasing n from nmin to nmax.

Finally, Stage C is associated with a more homogenous deformation of ferrite and

martensite [19], the formation of dislocation cell substructure in ferrite, and the onset

of dynamic recovery effects [2, 48].

In Figure 4.7, the nmin (solid symbols) and nmax (empty symbols) values associated

with all of the DP microstructures investigated in this study (Table 4.2) have been

plotted as a function of√

f/d. Note that for microstructures exhibiting discontinuous

yielding, nmin is plotted as half-full symbols. It can be seen that nmin follows a

linear relationship with√

f/d, regardless of the yielding behaviour, i.e. continuous

or discontinuous. Additionally, for a given√

f/d, the dependence of nmin on√

f/d

differs for each of the three microstructural variants. These observations are similar

to those of the true work hardening rate (θ) at ǫp = 0.5% (Figure 4.6a), signifying

the dominance of the GND hardening mechanism at low strains. In comparison

Chapter 4 Page 101

4 6 80.0

0.1

0.2

0.3

0.4

Inst

anta

neou

s W

ork

Har

deni

ng, n

(MP

a)

(f/d )1/2 (%.micron-1)


Figure 4.7: Relationship between minimum, nmin (closed) and maximum, nmax (open) instanta-neous work hardening exponents and

√

f/d. Half-full symbols show nmin for microstructures whichexhibit discontinuous yielding.

with the AT+CR8+IC, the inhomogeneous spatial distribution of martensite particles

in QT+CR8+IC variant results in a larger GND hardening effect, and therefore, a

higher value of nmin. On the other hand, comparing the CR+IC and QT+CR8+IC

microstructural variants, the smaller ferrite grain size in the latter leads to a higher

strength in the ferrite grains [15], thereby lowering the plastic incompatibility between

ferrite and martensite [13]. As a result, the magnitude of the GND hardening, and

consequently the value of nmin, is reduced in the QT+CR8+IC variant compared to

CR+IC.

An interesting feature in Figure 4.7 is the lack of dependence between nmax and√

f/d. Moreover, nmax is comparable for all three microstructural variants. In Figure

4.6b, a similar relationship was observed for the true work hardening rate (θ) at 2%

Chapter 4 Page 102

plastic strain, i.e. θ is independent of√

f/d and the microstructural variant. This

outcome implies that at strain levels associated with nmax (∼2-3%) the additional

work hardening effects due to GNDs are less significant and other work hardening

mechanisms, not specific to DP steels, are more important [2, 7–9, 20, 50]. Among the

three microstructural variants, the primary non-martensite related parameter is the

mean ferrite grain size. For this reason, it is hypothesized that nmax is a function of the

ferrite grain size. In the three microstructural variants, the average ferrite grain size is

very small and ranges from 1 to 1.8 microns. It is possible that despite the differences

in the work hardening behaviour at the beginning of the deformation (as demonstrated

by nmin and θ at ǫp = 0.5%), the entire ferrite grains are filled with GNDs as the

deformation proceeds due to the small mean size of ferrite grains. Therefore, the

work hardening of the three microstructural variants become relatively similar at

larger strains associated with nmax (∼2-3%), which leads to the comparable nmax

values observed in Figure 4.7. This argument is further strengthened by considering

the results of previous work [18], where nmax was found to be an inverse function of

ferrite grain size in materials with a larger ferrite grain size.

4.4.2.3 Dislocation Annihilation Factor

Stage C of work hardening in DP steels (above 4% strain) is associated with the

formation of dislocation cell substructures and the influence of dislocation annihi-

lation through dynamic recovery effects at high strains [2, 48]. To investigate this

phenomenon in more detail, the Kocks-Mecking (KM) approach can be applied where

the evolution of dislocation density with strain is assumed to be a function of a dis-

location storage term (athermal) and a dislocation annihilation term (strain rate and

temperature dependent) [51–53]. According to the KM model, the work hardening

Chapter 4 Page 103

rate, θ, follows a linear relationship with stress during Stage III of work hardening in

polycrystalline materials [18, 52], given by:

θ = θ0 − h(σ − σ0) (4.1)

where θ0 is the athermal hardening rate and σ0 is the flow stress due to strength-

ening mechanisms that do not include dislocation-dislocation interactions, such as

friction stress, back stresses, solid solution hardening and composite effects. The

parameter h is a microstructure-dependant “annihilation factor” that accounts for

dislocation annihilation and is a function of the strain rate and temperature.

To calculate h for the current data set, using Equation 4.1, a straight line was fit

to the θ vs (σ−σ0) plots at high strains (>4%). An example of the fitting procedure

is provided in Appendix B. For consistency, a 0.2% proof stress was used for σ0.

Since the objective was to calculate h, i.e. the slope of the fitted line, and not θ0,

this oversimplification is deemed appropriate (more details on this fitting procedure

can be found elsewhere [18]). The calculated h values for all of the microstructures

in Table 4.2 are plotted in Figure 4.8. It is evident that within each microstructural

variant, the dislocation annihilation factor, h, increases with√

f/d. Moreover, at a

given√

f/d, h increases in the order of AT+CR8+IC, QT+CR8+IC and CR+IC. The

values of the dislocation annihilation factor (h) and uniform elongations are compared

in Table 4.5 for the three selected microstructural variants with average martensite

volume fraction of ∼15%. It is evident that uniform elongation is inversely related to

h, which indicates the significance of the dislocation annihilation process to the overall

work hardening response of DP steels. This inverse relationship is understandable,

since as shown in Figure 4.8, all three microstructural variants have similar nmax,

Chapter 4 Page 104

4 5 6 7 8

20

40

60

80

100


Ann

ihila

tion

Fact

or, h

(f/d )1/2 (%.micron-1)

Figure 4.8: Relationship between dislocation annihilation factor, h, and√

f/d.

despite their differences in nmin. The parameter nmax can be considered as the work-

hardening capacity of the DP steel and the common value of this parameter implies a

similar dislocation generation rate in all three microstructural variants at the strains

associated with nmax. Consequently, at strains larger than nmax, what differentiates

the work hardening rate among the three microstructural variants is the dislocation

annihilation rate (h). It follows that the microstructural variant having a smaller h

can sustain a higher work hardening rate up to larger strains, which translates to

higher values of uniform elongation.

Dislocation annihilation due to dynamic recovery is controlled by the ability of dis-

locations to cross slip [54]. Cross slip is a stress-assisted process; therefore, when the

lattice is strongly stressed due to the presence of non-deformable hard particles, dis-

location annihilation occurs with greater ease [55, 56]. Based on the model developed

Chapter 4 Page 105

Table 4.5: The dislocation annihilation factor, h, and the uniform elongation values for the threemicrostructural variants of Figure 4.4.

Microstructural Variant h Uniform Elongation(%)

CR+IC725 25.7 15.3AT+CR8+IC720 22.9 16.6QT+CR8+IC720 27.8 12.8

by Brown and Stobbs [43, 57], the internal stresses are direct functions of f , while

they are inversely proportional to d. This results in the observed relationship between

h and√

f/d in Figure 4.8. Furthermore, the internal stresses, which are a direct con-

sequence of plastic incompatibility between ferrite and martensite, are also affected

by the average ferrite grain size (through variations in ferrite strength) and the spa-

tial distribution of martensite particles. Specifically, comparing the AT+CR8+IC

and QT+CR8+IC microstructural variants, the inhomogeneous spatial distribution

of martensite particles in QT+CR8+IC, i.e. bands of unrecrystallized ferrite grains

with large number of martensite particles as well as regions with large, recrystallized

ferrite grains with limited martensite particles, produces an additional plastic incom-

patibility in the microstructure which leads to increased internal stresses. It follows

that these higher internal stresses enhance cross slip in QT+CR8+IC, resulting in a

larger h. On the other hand, between the CR+IC and QT+CR8+IC microstructural

variants, the smaller ferrite grain size in the latter leads to a higher strength of ferrite

grains [15], thereby lowering the plastic incompatibility between ferrite and marten-

site [13], and hence, reduced internal stresses. Accordingly, cross slip is expected to

be more difficult in QT+CR8+IC which leads to a smaller h value.

A notable finding from the work hardening analysis of the three microstructural

variants is that the three work hardening parameters of θ at ǫp = 0.5%, nmin and

h all show similar relationships with√

f/d (Figures 4.6a, 4.7 and 4.8, respectively).

Chapter 4 Page 106

Additionally, the trend observed between the individual microstructural variants is

identical for these three work hardening parameters, i.e. at a given√

f/d, θ at ǫp =

0.5% , nmin and h increase in the order of AT+CR8+IC, QT+CR8+IC and CR+IC.

These observations are intriguing as the work hardening mechanisms associated with

these parameters are not similar, i.e. GND hardening for ǫp = 0.5% and nmin, and

dynamic recovery for h. These two work hardening mechanisms, however, are linked

with each other through the internal stresses that are produced due to the incompat-

ibility between ferrite and martensite. That is, the GND hardening corresponds to

the plastic relaxation of internal stresses [41–43] while dynamic recovery is enhanced

by unrelaxed portion of the internal stresses. Additional research is currently under-

way to study the effect of microstructural parameters on these internal stresses using

forward-reverse in-plane shear experiments.

4.5 Conclusions

1. Through the addition of a cold-rolling step between pre-heat treatments and

IC annealing, three distinct DP microstructural variants, having a significantly

refined ferrite grain size were produced. At a similar volume fraction of marten-

site, the variants differ with respect to their mean ferrite grain size and spatial

distribution of martensite particles.

2. At small strains (ǫp = 0.5%), the work hardening behaviour was found to be

dominated by the generation of geometrically necessary dislocations (GNDs)

in the ferrite grains and near the martensite particles. The work hardening

response at this stage was characterized by θ at ǫp = 0.5% and a minimum in

the instantaneous work hardening exponent, nmin. Both of these parameters

Chapter 4 Page 107

were determined to be functions of√

f/d, the mean ferrite grain size and the

spatial distribution of martensite particles. At higher strains (2-3%), however,

all three microstructural variants reach a similar work hardening limit, denoted

by a maximum value in the instantaneous work hardening exponent, nmax. This

parameter, which can be considered as the work hardening capacity of the DP

steel, was found to be independent of√

f/d.

3. At larger plastic strains (>4%), dislocation annihilation by dynamic recovery

becomes the controlling factor for the rate of work hardening. This phenomenon

is described by the dislocation annihilation factor, h, and is a function of√

f/d,

the mean ferrite grain size and the spatial distribution of martensite particles.

It was also shown that, for the three microstructural variants under study,

the uniform elongation is inversely proportional to the dislocation annihilation

factor, h.

4. The three work hardening parameters of θ at ǫp = 0.5%, nmin and h all ex-

hibit similar relationship with√

f/d and consistent trends between the three

microstructural variants under study. This observation suggest a common role

of internal stresses on the two work hardening mechanisms of GND hardening

and dynamic recovery. That is, the GND hardening corresponds to the plastic

relaxation of internal stresses while dynamic recovery is enhanced by unrelaxed

internal stresses.

4.6 Acknowledgments

Authors would like to acknowledge the financial support of the AUTO21 NCE

and the Natural Sciences and Engineering Research Council of Canada (NSERC).

Chapter 4 Page 108

4.7 References



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(2014) 3608–3621.

[27] H. Paruz, D. V. Edmonds, Mater. Sci. Eng. A A117 (1989) 67–74.

[28] A. K. Sachdev, Acta Metall. 31 (1983) 2037–2042.

[29] W. Steven, A. G. Haynes, J. Iron Steel Inst. 183 (1956) 349–359.

[30] K. W. Andrews, J. Iron Steel Inst. 203 (1965) 721–727.

[31] J. S. Kang, J. B. Seol, C. G. Park, Mater. Charact. 79 (2013) 110–121.

[32] P. C. M. Rodrigues, E. V. Pereloma, D. B. Santos, Mater. Sci. Eng. A 283 (2000)

136–143.

[33] F. G. Caballero, H. Roelofs, S. Hasler, C. Capdevila, J. s. Chao, J. Cornide,

C. Garcia-Mateo, Mater. Sci. Technol. 28 (2012) 95–102.

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[34] G. F. Vander Voort, Atlas of Time-temperature Diagrams for Irons and Steels,

ASM International, Materials Park, OH, 1991.

[35] P. A. Manohar, T. Chandra, C. R. Killmore, ISIJ Int. 36 (1996) 1486–1493.

[36] R. R. Mohanty, O. A. Girina, N. M. Fonstein, Metall. Mater. Trans. A 42 (2011)

3680–3690.

[37] M. Goune, P. Maugis, J. Drillet, J. Mater. Sci. Technol. 28 (2012) 728–736.


Metall. 104 (2007) 315–323.


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[44] H. Azizi-Alizamini, M. Militzer, W. J. Poole, Scripta Mater. 57 (2007) 1065–

1068.

[45] M. Calcagnotto, D. Ponge, D. Raabe, ISIJ Int. 52 (2012) 874–883.


Mater. 62 (2014) 197–211.


[48] W. R. Cribb, J. M. Rigsbee, in: R. A. Kot, J. W. Morris (Eds.), Structure and


[49] Y. Tomita, K. Okabayashi, Metall. Trans. A 16 (1985) 865–872.


Chapter 4 Page 111

[51] Y. Estrin, H. Mecking, Acta Metall. 32 (1984) 57–70.

[52] U. F. Kocks, H. Mecking, Prog. Mater. Sci. 48 (2003).

[53] H. Mecking, U. F. Kocks, Acta Metall. 29 (1981) 1865–1875.

[54] M. A. Meyers, K. K. Chawla, Mechanical Metallurgy: Principles and Applica-

tions, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984.

[55] O. Bouaziz, D. Barbier, J. D. Embury, G. Badinier, Philos. Mag. 93 (2013)

247–255.

[56] G. Fribourg, Y. Brechet, A. Deschamps, A. Simar, Acta. Mater. 59 (2011) 3621–

3635.


CHAPTER 5

Measurement of Back Stress Contribution

to Work Hardening

5.1 Introduction

The high initial work hardening rates of dual-phase (DP) steels have been at-

tributed to the plastic incompatibility between the soft ferrite and hard martensite.

This plastic incompatibility leads to an introduction of long-range internal elastic

stress into the ferrite matrix [1, 2] which is often referred to as “back stress”. The

back stress is directional in nature as it inhibits the further motion of dislocations in

the forward direction while it assists dislocation slip in the reverse direction. Due to

this directionally of back stress, its effect can be observed in forward-reverse deforma-

tion tests. In these tests, the specimen is first deformed to a pre-defined strain value

and then the strain path is reversed to continue deformation in the reverse direction.

An example of the stress-strain behaviour in a forward-reverse test is presented in

Figure 5.1. Two important effects are observed during reverse loading: (a) the tran-

sient softening effect, which is described by the reduced yield stress, i.e. σyR < σU

F ,

as well as the continuous yielding (regardless of the yielding response during forward

loading), and (b) the permanent softening effect, ∆σPS, which is represented by the

difference between the forward and reverse flow curves, typically measured after the

point where parallelism is established. Both of these effects are associated with the

directional internal stresses, i.e. back stresses, within the matrix. Additionally, during

forward loading, pileup of dislocations at barriers results in an increased resistance

112

Chapter 5 Page 113

Figure 5.1: An example of forward-reverse deformation test results. Transient and permanentsoftening effects are clearly visible. Note that the compressive portion of the flow curve has beenmirrored to allow for easier comparison of forward and reverse flow curves.

to dislocation motion, while upon load reversal, dislocations encounter a weaker set

of obstacles of random strengths. Therefore, the transient softening during reverse

loading is also associated with this reduced resistance to the dislocation motion and

the statistical sampling of the obstacles by the dislocations [3].

To calculate the back stress from the forward-reverse tests, the dislocation-based

model of Ibrahim and Embury can be used [4]. In this model, which was originally

developed for ferrite-cementite systems, the flow stress is assumed to be a linear sum-

mation of the initial flow stress, σ0, the non-directional work hardening contributions1,

σS, and the directional back stress component, σB:

σF = σ0 + σS + σB . (5.1)

1This term includes contributions from both statistically stored dislocations (SSD) as well asgeometrically necessary dislocations (GNDs).

Chapter 5 Page 114

During reverse loading, the back stress assists the motion of dislocations and

therefore the reverse flow stress can be written as:

σR = σ0 + σS − σB (5.2)

Assuming the structure that gave rise to σS remains unchanged upon strain re-

versal, the back stress can be calculated as:

σB =σF − σR

2. (5.3)

It follows that in order to measure back stress, σF and σR should first be deter-

mined. In a pioneering study, Atkinson et al. [3] showed that taking the permanent

softening parameter (∆σPS) as σF − σR, correct measurements of back stress can be

made. This approach has been adopted by a number of researchers [4–10]. However,

more recently, using careful X-ray measurements of internal stress, Wilson and Bate

[11, 12] have demonstrated that this method underestimates the value of the back

stress. They offered an explanation that during reverse straining up to the point of

parallelism between forward and reverse flow curves, the dislocation substructure that

gives rise to the non-directional hardening component (σS) is actually modified and

therefore permanent softening is not solely a function of the back stress. Specifically,

in the TEM studies of Gardey et al. [13, 14], changes in the dislocation substructure

upon reverse loading were clearly observed for the case of DP steels. Additionally,

such effects were also manifested in the reverse flow curve by the appearance of a

work hardening stagnation region, i.e. a transitionary negative work hardening rate,

as well as a lack of parallelism between the forward and reverse flow curves [13, 14].

A practical difficulty in using the permanent softening parameter to calculate the

Chapter 5 Page 115

back stress is the large strains required to achieve the parallelism between forward

and reverse flow curves. For this reason, as well as the aforementioned inaccuracies

associated with the permanent softening method, in recent years, and particularly in

the case of DP steels, the back stress has instead been calculated from the difference

between the forward unloading stress (σUF ) and the reverse yield stress (σy

R) [2, 6, 15–

19]. An inherent complication in such analyses is the lack of a clear yield point during

reverse loading due to the presence of the transient softening effects. To overcome

this uncertainty, the offset method is often utilized. Different true-strain offset values

have been reported, e.g. 0.0005 [15, 16], 0.001 [6, 17, 18] and 0.002 [2, 19]. Although

it has been shown that the choice of the offset strain changes the absolute magnitude

of the calculated back stress, the trends observed between back stress and pre-strain

remain unaffected [18].

Another complication in determining σR is the plastic relaxation that occurs dur-

ing unloading [10, 11, 20, 21]. This effect has been associated with the relaxation and

runback of the repulsive pileup dislocations during unloading [20, 21]. Moreover, this

plastic relaxation has also been attributed to the local variations of internal stress

and obstacle strength [11]. In the absence of forward-reverse test equipment, several

researchers used unload-reload type tests to measure the back stress from the inelastic

unloading behaviour after different amounts of pre-straining [22–24]. Although useful,

this approach clearly produces an overestimation of the back stress since the inelastic

relaxation is only associated with the movement of a limited number of dislocations

which have favorable local effects.

Traditionally, the forward-reverse deformation test comprises a tension-compression

strain path. However, in the case of sheet materials, this type of testing is limited

by the buckling of the specimen during compressive loading. Therefore, alternative

Chapter 5 Page 116

testing methods such as torsion testing [19], constrained uniaxial tension-compression

testing [25] and more recently, in-plane shear testing [13, 14, 18, 26] have been uti-

lized. Shear testing is particulary attractive due to the possibility of achieving large

strains normally unattainable by uniaxial tensile testing. Moreover, in contrast to

torsion testing, shear testing has the advantage of a more versatile specimen form

(sheet vs. tube) as well as uniform strain across the gauge section.

To describe the back stress contribution to flow stress, the model developed by

Brown and coworkers is commonly used [3, 27, 28]. In this model, which is based

on the original work of Orowan, it is assumed that a dislocation pile up at obstacles

results in both short- and long-range internal stresses in the matrix. Both of these

internal stresses increase rapidly with plastic strain and at relatively small value of

pre-strain, plastic relaxation effects such as introduction of geometrically necessary

dislocations (GNDs) takes place near the hard particles [29, 30]. Due to the interaction

between GNDs and the dislocations on the primary slip system, the material work

hardens rapidly and a complete relaxation of internal stress is not possible. The

unrelaxed portion of plastic strain, γ∗

P , therefore continues to contribute to the back

stress [3, 27, 31]. Under these conditions, hardening effects due to both back stresses

and GNDs should be taken into account. The GND hardening is calculated using

Ashby’s work hardening model, as described in Chapters 4 and 5. For the back

stress, Brown and Stobbs [28] assumed that the unrelaxed plastic shear strain, γ∗

P , is

a function of the local matrix strength near particles which is affected by the density

of the GNDs . From this model, the back stress is given by:

τB = 2µφfγ∗

P = 2αµφf

[

8γpb

πr

]1/2

(5.4)

where α is a constant describing forest hardening strength (typically ∼ 0.3-0.4), µ is

Chapter 5 Page 117

the shear modulus2, φ is the morphology-dependant accommodation factor (typically

assumed ∼ 0.5 for spherical particles), f is the volume fraction of hard particles, b is

the Burgers vector, r is the particle radius and γp is the plastic shear strain.

Experimental research on back stress effects in DP steels has been limited. The

majority of these publications have focused on providing a general understanding

of back stresses [6, 15, 18], while studies concerning the effect of microstructural

parameters on back stress hardening are limited. As expected, the back stress has

been found to increase with the volume fraction of martensite [2, 10, 16, 17]. However,

in terms of other microstructural parameters, such as the size and spatial distribution

of martensite particles, the reported studies are not systematic such that a clear

conclusion regarding their influence on back stress hardening cannot be made [16, 17].

In the present study, the primary objective was to provide a more comprehensive

understanding of the relationship between back stress hardening and microstructural

parameters in DP steels, specifically ferrite grain size as well as the size, morphology

and spatial distribution of martensite particles. To accomplish this goal, an in-plane

shear fixture was designed and built in order to perform forward-reverse tests on five

different DP780 microstructural variants with similar volume fractions of martensite

(described in Chapters 3 and 4). In the remainder of this Chapter, the design, con-

struction and calibration of the shear fixture is described, followed by a presentation

and discussion of forward-reverse shear test results obtained using the five DP780

microstructural variants.

2 Equation 5.4 is based on the assumption of similar elastic moduli for both the matrix (µ) and

hard particle (µ∗). If this is not the case, then a correction factor of µ∗

µ∗−φ(µ∗

−µ) , must be included

in Equation 5.4.

Chapter 5 Page 118


The design of the shear fixture is presented in Sections 5.2.1 and 5.2.2. A summary

of the stress and strain measurement procedure is provided in Section 5.2.3. Finally,

the testing routine as well as the analysis protocols are discussed in Sections 5.2.4

and 5.2.5, respectively.

5.2.1 Shear Specimen Geometry

The design criteria for the shear specimen were as follows:

1. Simplicity: The specimen design shall be such that calculations of shear strain

and stress are straightforward. A sample geometry that produces a “simple

shear” deformation was therefore selected.

2. Strain homogeneity: The developed strain field within the shear zone shall be

as uniform as possible and with the minimum amount of inhomogeneities such

as end effects. The FEM calculations of Bouvier et al. [32] showed that reducing

the width-to-height ratio of the shear gauge area (w/H) results in a more uni-

form and homogeneous distribution of the shear stress field. Consequently,

Bouvier et al. [26] used the ratio of 1:5 in their design and obtained a rela-

tively uniform distribution of shear at the areas away from the edges. Similar

considerations were used in the present design.

3. Balanced moments: The shear zone in the specimen shall be symmetrical such

that the net moment within the setup remains zero. Therefore, in the shear

test developed here, a specimen with two symmetrical and identical shear zones

was used.

Chapter 5 Page 119

4. Heat treating restrictions: The maximum specimen width, H, is limited to

52mm due to dimensional constraints of the heat treating facility. Coupled

with criterion #2, this constraint limits the shear zone width to a maximum of

10 mm.

5. Load limit: The maximum design load during shear testing shall preferably not

exceed the capacity of the load cell, i.e. 100 KN. A specimen with a larger shear

zone has the benefit of a larger area of uniform shear distribution. However,

when using high-strength steel sheets at typical thickness of 1-2mm, the load

can rise dramatically with an increase in the specimen size. Consequently, in

the present specimen design, a balance between size of the shear zone and load

capacity was considered.

The geometry of the designed shear specimen is shown in Figure 5.2 for before

(5.2a) and after (5.2b) ideal deformation. The shape and proportions of the shear

specimen were based on the design of Bouvier et al. [26]. This design has the benefits

of symmetry (criterion #3), simplicity (criterion #1) and a previously-established

strain homogeneity (criterion # 2). However, the shear specimen used in the present

study differs from that of Bouvier et al. in several ways. The Bouvier et al. design

is based on a shear zone width-to-height ratio (w/H) of 0.2. Based on Criterion

#4, H was chosen as 50mm. The Bouvier et al. design therefore results in a shear

zone width of 10mm. However since decreasing the w/H parameter has the beneficial

effect of reduced end effects [32], in the present study, w was reduced slightly to 8mm

in order to lower w/H to 0.16 for a more uniform strain distribution. Assuming a

maximum shear strength of 600MPa (equivalent to ∼1000MPa in uniaxial tension)

and a thickness of around 1.5mm, from this design, the maximum shear load of

600MPa×(50mm×1.5mm)×2=90 kN will be below the limits of the load cell (100

Chapter 5 Page 120

kN), therefore satisfying Criterion #5. Finally, all remaining dimensions other than

H and w were selected to allow for a proper placement of eight 10mm holes (in the

gripping regions of the specimen), according to design standards, i.e. a hole spacing

of 2.5 × hole diameter and an edge separation of 1.25 × hole diameter. A detailed

drawing of the shear specimen is given in Appendix C.

Chapter 5 Page 121

(a)

(b)

Figure 5.2: Schematic of the shear specimen (a) before and (b) after ideal deformation. Thetheoretical shear zones are shown in (b) as the hatched areas. See Section 5.2.3 for details.

5.2.2 Shear Fixture

A custom-made shear fixture having the following specifications was designed and

manufactured to perform the shear tests:

Chapter 5 Page 122

1. Easy assembly: The fixture assembly prior to the testing shall be easy and

straightforward.

2. Interruption-free strain reversal: The strain reversal shall be able to be per-

formed without disassembling and reassembling of the setup.

3. Compatibility: With only minimal adjustments, the fixture shall be compatible

with the available Instron hydraulic machine.

4. Optical access to the shear zone: The shear zone shall be visible during the test

so that strains can be instantaneously measured using an imposed grid pattern.

5. Safety: Due to very high loads encountered during the test, the setup shall meet

all applicable safe practice requirements.

The designed shear fixture, made from untreated 4140 alloy steel (nominal hard-

ness of Rockwell B90), is shown in Figure 5.3. The fixture consists of a fixed grip

(Figure 5.3a) and a moving grip (Figure 5.3b). The fixed grip includes the base

and the cover while the moving grip comprises the plunger and the spacer. The

spacer is designed to stabilize the plunger by restricting the out-of-plane deforma-

tion. The detailed drawings of the individual components of the shear fixture can

be found in Appendix A. Note that in Figure 5.3, the two shear zones as well as the

gripping regions are marked. Additionally, as will be explained in Section 5.2.4, all

contact/moving surfaces in the fixture are lubricated with dry graphite to reduce the

frictional effects. These surfaces are labeled “L” in Figures 5.3a and 5.3b.

The specimen mounting procedure is shown in Figure 5.4 and is as follows:

1. With the aid of four high strength bolts (Class 10.9), the spacer and specimen

are aligned and placed together on the plunger (Figure 5.4a and b).

2. The bolts are loosely secured using compatible high strength nuts (SAE Grade 8).

Chapter 5 Page 123

(a)

(b)

Figure 5.3: The shear fixture. (a) Fixed grip with the base and the cover. (b) Moving grip with theplunger and the cover. Surfaces that are indicated with letter “L” are lubricated with dry graphiteprior to the start of the test.

Chapter 5 Page 124

3. The plunger is moved to the zero position and the specimen holes are aligned

with the fixed grip base (Figure 5.4c).

4. With the specimen aligned properly, plunger bolts are tightened using a torque

wrench with 53ft.lb of torque, i.e. recommended torque for dry, unlubricated

M10 Class 10.9 bolts.

5. The fixed grip cover is placed on the rest of the fixture and is secured using four

additional high strength bolts and nuts, again with 53 ft.lb of torque (Figure

5.4d).

Chapter 5 Page 125

(a)

(b)

(c) (d)

Figure 5.4: Assembly procedure for the shear test. Refer to text for details.

Chapter 5 Page 126

5.2.3 Stress and Strain Measurements

For a first approximation, the shear strain can be measured from the crosshead

displacement data. Based on the geometry of the specimen, and under ideal shear

conditions where: (a) shear occurs purely within the theoretical shear zone, shown in

Figure 5.2b, and (b) all other strains, except the primary shear strain (γxy), are zero,

the raw shear strain can be expressed by:

γxy =∆y

w. (5.5)

To provide a more accurate measurement of the strain, a square grid pattern of

dots with 2 mm by 2 mm spacing was applied to the surface of every shear specimen.

This gridding was done using a custom made marker press attached to a computer-

controlled microscope stage. The strain field, including normal and shear components,

can be calculated by comparing the grid before and after deformation. Two types of

grid patterns were used:

1. Full Grid: For the calibration experiments, the complete strain field was mea-

sured using a 12×25 point grid pattern that was imposed on both sides, i.e. left

and right, of the shear specimens (Figure 5.5a). To calculate the strains, the

specimen was scanned before and after deformation using a HP Scanjet G4050

scanner at a resolution of 600dpi.

2. Camera Grid: For all specimens used in the non-calibration experiments, the

exact strain at the point of the strain path reversal during forward-reverse tests

was measured using a 5× 25 point grid pattern applied on both gauge sections

(left and right). The pattern, which is shown in Figure 5.5b, is placed such that

it will be visible during the test through the viewfinder of the shear fixture.

Chapter 5 Page 127

(a) (b)

Figure 5.5: Example of imposed grid pattern: (a) “Full Grid” specimen used for calibration testsand (b) “Camera Grid” specimen used in both calibration and back-stress experiments

The patterns are captured prior to the start of the test and at the point of

strain path reversal using a Nikon D70 digital camera at an effective resolution

of ∼ 32microns/pixel.

Following each test, before and after grid pattern images of the shear specimen

were analyzed using Image Pro Plus software to separate the grid pattern from the

rest of the image. The grid patterns were then indexed for the coordinates of each dot

using the ImageJ software package. Next, the indexed grid patterns were divided into

2 point by 2 point quadrilateral elements to calculate normal and shear strains, i.e. ǫx,

ǫy and γxy, using an in-house, custom made MATLAB code based on reference [33].

The resulting shear strains are referred to as real shear strains. The measurement

of real shear strains is a relatively tedious and time-consuming task. Therefore, as

discussed in Section 5.2.5, the real shear strains were only measured at critical points

along the deformation path, such as the point of forward unloading (strain reversal

after forward deformation).

It was assumed that shear takes place on the X-Y plane and along the total width

of the specimen, H (Figure 5.2b), hence, the shear stress was calculated as:

τxy =Load

H.t(5.6)

Chapter 5 Page 128

where t is the specimen thickness.

5.2.4 Shear Experiments

The shear experiments are divided into two groups:

1. Calibration Experiments: A series of experiments were performed on a

1.70mm thick, IF steel sheet obtained from CanmetMATERIALS. The chemical

composition of this steel is listed in Table 5.1 and the calibration test parame-

ters are given in Table 5.2. The primary objectives of these experiments were

to: (a) measure the strain field distribution during/after shear testing; and (b)

establish a protocol for detailed analysis of the stress-strain data. In the latter

experiments, the specimen is first deformed with a prescribed displacement in

the forward direction. The resulting strain associated with this forward dis-

placement is referred to as “forward pre-shear strain”. Next, the deformation

direction is reversed and is continued in the reverse direction up to a set reverse

displacement. Four different forward displacements of 1mm, 3mm, 5mm and

7mm were applied (see Table 5.2). Separate specimens were used for each test,

i.e. no cyclic test was performed. Repeat tests were performed on separate

specimens in all testing conditions to assess test variability.

2. Back Stress Experiments: Five different DP780 microstructural variants

Table 5.1: Chemical compositions of the steels used in this research (in wt.%). CR: Cold-Rolled,HB: Hot Band, F: Ferrite, P: Pearlite

Material C Mn Si Cr Mo V Ti Nb Constituents Thickness

DP780CR 0.09 2.1 0.02 0.26 0.29 0.001 0.001 0.002 F/P 0.95 mmDP780HB 0.09 2.11 0.024 0.26 0.302 0.001 0.001 0.002 F/P 3.12 mmIF 0.004 0.12 0.008 0.011 0.005 0.02 0.063 0.005 F 1.70 mm

Chapter 5 Page 129

underwent forward-reverse shear tests in order to calculate and compare the

development of back stress as a function of ferrite grain size as well as the size,

morphology and spatial distribution of martensite particles. These microstruc-

tures, described in Chapters 3 and 4, have a similar volume fraction of marten-

site (∼15%) and they include the microstructures with the DP780CR starting

condition, i.e. CR+IC725, AT+IC720 and QT+IC735 (Chapter 3), as well as

microstructures with the DP780HB starting condition, i.e. AT+CR8+IC720

and QT+CR8+IC720 (Chapter 4). The chemical compositions of the starting

DP780CR and DP780HB materials are listed in Table 5.1, while selected mi-

crostructural parameters of these five DP780 variants are summarized in Table

5.3.

For each DP780 microstructural variant, the evolution of back stress as function

of pre-shear strain was measured using forward-reverse shear tests with four

different forward displacements. Individual specimens were used for each test,

i.e. no cyclic tests were performed. The test parameters are listed in Table 5.4.

To assess reproducibility, repeat tests were performed for 2 of the test conditions,

i.e. 5mm and 7mm forward displacement, for each of the five microstructural

variants.

The shear tests were performed at a constant displacement rate of 0.5mm/min.

Under the assumption of ideal shear conditions (see Section 5.2.3), the width of the

shear gauge area, w, can be taken as 8mm (Figure 5.2b) and the ideal shear strain

rate can be calculated as:

Chapter 5 Page 130

Table 5.2: Details of the shear testing calibration experiments. Individual samples were used foreach test. The “Reverse Displacement” is the total displacement following the reversal of the straindirection, including the unloading step from forward deformation.

Objective Material Grid Pattern Forward Displacement Reverse DisplacementAfter Strain Reversal

(mm) (mm)

Strain Map IF Full Grid 5 -Full Grid 5 5

Back Stress IF Camera Grid 1 3Camera Grid 3 4Camera Grid 5 4Camera Grid 7 4

Table 5.3: Microstructural parameters of the five DP780 microstructural variants with similarvolume fraction of martensite particles (∼ 15%).

Ferrite MartensiteMicrostructure Grain Size, Vol. Fraction, Size, Morphology Spatial Distribution

D (µm) f (%) d (µm)

CR+IC725 1.84 ± 0.25 15.1 ± 1.8 0.50 ± 0.06 Equiaxed Bands along RDAT+IC720 2.00 ± 0.28 16.2 ± 1.0 0.44 ± 0.05 Elongated,

EquiaxedUniform(No banding)

QT+IC735 2.77 ± 0.65 12.9 ± 3.0 0.54 ± 0.10 Irregular Along prior austen-ite grain boundaries(No banding)

AT+CR8+IC725 1.09 ± 0.07 16.1 ± 1.2 0.36 ± 0.02 Equiaxed Uniform(No banding)

QT+CR8+IC725 1.51 ± 0.28 14.7 ± 2.0 0.34 ± 0.04 Equiaxed Bands along RD

dγxydt

=1

w.∆y

∆t

=0.5mm.min-1

8mm

= 0.0625min-1≈ 0.001sec-1 .

(5.7)

To compare this rate with a standard uniaxial tensile test, the equivalent shear

rate can be calculated using the von Mises equivalent strain:

Chapter 5 Page 131

Table 5.4: Details of the back stress experiments for the DP780 variants. Individual samples wereused for each test. For specimens with underlined displacement amounts, a complete repeat testwas performed. The “Reverse Displacement” refer to the total displacement immediately after thereversal of the strain direction, hence including the unloading step from forward deformation.

Material Grid Pattern Forward Displacement Reverse DisplacementAfter Strain Reversal

(mm) (mm)

CR+IC725 Camera Grid 1.2 2.5Camera Grid 3 3.5Camera Grid 5 4Camera Grid 7 4

AT+IC720 Camera Grid 1.2 2.5Camera Grid 3 3.5Camera Grid 5 4Camera Grid 7 4

QT+IC735 Camera Grid 1.2 2.5Camera Grid 3 3.5Camera Grid 5 4Camera Grid 7 4

AT+CR8+IC720 Camera Grid 1.2 2.5Camera Grid 3 3.5Camera Grid 5 4Camera Grid 7 4

QT+CR8+IC720 Camera Grid 1.2 2.5Camera Grid 3 3.5Camera Grid 5 4Camera Grid 7 4

e =

√

2

3(ǫ2x + ǫ2y + ǫ2z) +

1

3(γ2

xy + γ2yz + γ2

xz) . (5.8)

This results in an equivalent strain rate of 5.7 × 10−4s-1 which is similar to that

used in the uniaxial tensile tests reported in Chapters 3 and 4.

The second important shear testing parameter is the lubrication of the shear fix-

ture. To reduce the friction between moving surfaces, dry graphite was used as a

lubricant and it was applied only to the moving surfaces of the shear fixture (ex-

cluding the gripping areas). These surfaces are marked with letter “L” in Figure 5.3.

Following the completion of each test, the fixture was fully cleaned using ethyl-alcohol

Chapter 5 Page 132

and relubricated for the next test.

5.2.5 Shear Data Analysis

5.2.5.1 Calibration Experiments

In the first set of calibration experiments, IF steel samples with the Full Grid

patterns were analyzed to calculate and construct strain distribution maps within

the specimen at (i) the end of the forward deformation test and (ii) the end of the

reverse shear test after the forward deformation.

In the second set of calibration experiments, the stress-strain data from shear

testing of IF steel at various forward-reverse shear displacements (Table 5.2) were

used to measure the evolution of back stress as a function of forward pre-shear strain.

In accordance with the most recent publications involving DP steels [2, 6, 15–19],

especially Aouafi et al. [18] which employed a shear fixture similar to the one herein,

the offset method was used to calculate the reverse yield strength, τOffsetR . This

approach is preferable to the permanent softening method since it is known that the

large reverse strains required for the calculation of permanent softening changes the

dislocation substructure that gives rise to non-directional strengthening components

[13, 14], thereby rendering the measurements of back stress inaccurate [11, 12].

In the offset method, as shown in Figure 5.6a, the back stress is defined as half

of the difference between the forward unloading stress (τUnloadF ), i.e. flow stress at

strain reversal, and the reverse offset stress (τ offsetR ), i.e. onset of non-linearity during

reverse loading. That is:

τOffsetB =

τUnloadF − τOffset

R

2. (5.9)

Chapter 5 Page 133

Three different strain offset values of 0.0005, 0.001 and 0.002 were used to in-

vestigate the effect of offset amount on the measurements of back stress. Note that

in plotting the stress-strain data, the choice of the shear strain type, i.e. raw (from

crosshead displacement data) versus real (from grid point data), can be important as

the slope of the elastic region can be affected. However, in the offset method, used to

calculate the onset of the non-linearity, the absolute value of the elastic region’s slope

is not crucial; it is the deviation from the linearity that is of interest. Therefore, for

consistency, the raw shear stress-strain data were used to calculate the reverse offset

stresses in all specimens. Additionally, for each sample, the real forward pre-shear

strains were also measured from the the grid patterns at the end of the forward defor-

mation, i.e. forward unloading point, as described in Section 5.2.3. The back stress

data were then plotted with respect to the real forward pre-shear strains.

A second method to analyze the forward and reverse shear data is to measure the

differences between the forward, σF , and the reverse, σR, flow curves, i.e. τF − τR.

This analysis technique is referred to henceforth as the “Softening Method” and its

details are shown in Figure 5.6b. The softening curves (τF − τR) were calculated

for all forward pre-shear experiments and the results were plotted with respect to

the reverse shear strains (see Figure 5.6b for the definition of reverse shear strain).

Note that the softening curves are useful since they show both the effects of transient

softening (at small reverse shear strains) and permanent softening (at large reverse

shear strains where the forward and the reverse flow curves become parallel).

Chapter 5 Page 134

0.30 0.35 0.40 0.450

20

40

60

80

100

120

140

160

180

OffsetB

= ( UnloadF

- OffsetR

)/2

She

ar S

tress

(MP

a)

Accumulative Shear Strain (Raw, from Displacement Data)

Forward unloading curve

Reverse flow curve, R

Reverse yield stress, OffsetR

Linear reverse loading + offset

Forward unloading stress, UnloadF

Forward flow curve, F

(a)

0.0 0.2 0.4 0.6 0.80

50

100

150

200

Reverse shear strain

She

ar S

tress

(MP

a)


Softening curve F -

R

Reverse flow curve, R

Forward flow curve, F

(Uninterrupted Sample)

(b)

Figure 5.6: Details of the shear results analysis: (a) The offset method. The stress-strain curvesare for the IF specimen with 3mm forward displacement and 4mm reverse displacement. (b) Thesoftening (τF − τR) method. The stress-strain curves are for two IF specimens, one with uninter-rupted 7mm forward only shear, and another with 3mm forward displacement and 3mm reversedisplacement.

Chapter 5 Page 135

5.2.5.2 Back Stress Experiments

In the back stress experiments, the back stresses were measured for the five DP780

microstructural variants using the offset method. The procedure was identical to

that of the calibration experiments (Section 5.2.5.1), with the exception that only the

0.002 offset was employed. This choice was based on the results of the calibration

experiments (Section 5.3.1.3) as well as the literature [6, 16–18].

In addition to the back stress, the softening curves (τF − τR) were also calculated

for all forward pre-shear experiments and the results are plotted against the reverse

shear strain for all five DP780 microstructural variants.

5.3 Results

5.3.1 Calibration

5.3.1.1 Strain Maps

A specimen at the end of the forward shear and prior to the reverse shear (5mm

displacement) is shown in Figure 5.7. The shear zones can be recognized as the un-

shadowed regions inside the grid pattern. Additionally, significant rigid-body transla-

tion and rotation effects can also be observed at the top and bottom of the specimen

(marked with arrows). Following calculation of the strain in each 2 × 2 set of dots,

the strain maps of Figure 5.8 can be produced. These maps display both normal

strains (ǫx and ǫy) as well as the in-plane shear strain (γxy). Figure 5.8, shows the

following:

1. The shear strain is primarily confined to the ideal shear zones (marked by dotted

lines, see Figure 5.2b for details). However, closer examination of Figure 5.8c

Chapter 5 Page 136

Figure 5.7: IF steel specimen after forward shear (5mm displacement). Arrows show the areaswith significant amount of rigid body translation and rotation. For strain maps of this specimen seeFigure 5.8.

also reveals that:

(a) The shear strains extend by distances of ∼ 5mm on both sides of the ideal

shear zones, although the magnitude of these shear strains is relatively

small.

(b) At the edges of the ideal shear zones, the shear strains appear to be slightly

larger than the rest of the shear zone. The grid points in these regions show

signs of additional smearing, possibly due to heavy gripping. The smeared

grid points likely cause the observed larger than expected strain values.

2. At the top and bottom of the ideal shear zones, the shear strains are very

small and, as shown in Figure 5.7, these areas are associated with rigid-body

translations and rotations (marked with arrows).

3. Large normal strains are mainly localized to the outside corners of the ideal

shear zones (Figure 5.8a and b).

The strain maps produced by forward shearing with 5mm displacement followed

by reverse shearing of equal amount (5mm) are presented in Figure 5.9. Similar to

Chapter 5 Page 137

(a)

(b)

(c)

Figure 5.8: Distribution of normal and shear strain in IF steel specimen prior to reverse loading(5mm forward displacement): (a) ǫx, (b) ǫy and (c) γxy. Black dotted lines show the ideal shearzone.

Chapter 5 Page 138

the results of Figure 5.8, it can be seen that with the exception of the areas close to

the free edges, the in-plane shear strain is relatively uniform across the shear zone.

In terms of normal strains, both ǫx and ǫy are relatively small within the gauge area,

although ǫy shows slight inhomogeneity near the edges of the shear zone.

From the above observations, it can be concluded that at the center of the ideal

shear zone, and away from the edges and corners, the deformation is relatively uniform

and of in-plane shear type, therefore satisfying design Criterion #2 (Section 5.2.4).

The other important design parameter was the symmetry of the shear zones (Criterion

#3). To investigate this in more detail, IF steel specimens with varying amounts of

forward and reverse shear strains were analyzed. Figure 5.10 shows the results of

this analysis, where values of shear strains in the left shear zone are plotted against

the ones in the right shear zone. It is evident that a linear, 1:1 relationship exists

between the shear strains in the two shear zones; therefore, design Criterion #3 is

also satisfied.

Chapter 5 Page 139

(a)

(b)

(c)

Figure 5.9: Distribution of normal and shear strain in IF steel specimen at the end of the reverseloading (5mm forward-5mm reverse displacements): (a) ǫx, (b) ǫy and (c) γxy. Black dotted linesshow the ideal shear zone.

Chapter 5 Page 140

0.0 0.1 0.2

-0.2

-0.1

0.0

xy (Right Shear Zone)

xy (L

eft S

hear

Zon

e)

IF Steel1

Figure 5.10: Shear strains measured in the left shear zone plotted against shear strains in the rightshear zone. Each data point belongs to an individual specimen.

5.3.1.2 Shear Stress - Shear Strain Plots

Figure 5.11 shows the shear stress - shear strain plots of the IF steel specimens.

In this figure, the individual specimen results (with different amounts of forward pre-

shear strains) are presented, i.e. no cyclic testing was performed. Additionally, raw

strains (calculated from Equation 5.5) are used. Two types of stress-strain plots are

presented here: the total strain plots (Figure 5.11a) and accumulative strain plots

(Figure 5.11b). The difference in these two plots is the way that the reverse loading

data are presented. In the former type, the stress and strain values are plotted without

any modification, i.e. considering signs (positive or negative) and without any offset

in the strains. In the latter type, the absolute values of stress and strain are used and

the reverse curves are offset to the point of zero stress after unloading. Consequently,

Chapter 5 Page 141

in the accumulative strain plots, the reverse stress-strain curves are mirrored in the

forward quadrant and the strain values are the accumulative strains, i.e. forward

strain plus the absolute value of the reverse strain.

From Figure 5.11, the following observations can be made:

1. The unloading curve is linear at the start but becomes non-linear at lower

stresses and some inelastic relaxation takes place upon unloading.

2. Upon reverse loading, both transient softening and permanent softening are

observed in all specimens.

3. The specimens with larger amounts of forward pre-shear strains exhibit the

work hardening stagnation effect, i.e. a transitionary negative work hardening

rate (marked with arrows in Figure 5.11b).

4. The measured stress-strain curves have a high degree of reproducibility. The

repeat tests in each condition, i.e. 1mm, 3mm, 5mm and 7mm forward dis-

placements, show less than ∼5% scatter in the forward loading direction as well

as the subsequent reverse straining path.

Chapter 5 Page 142

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

-200

-100

0

100

200 IF Steel

She

ar S

tress

(MP

a)

Total Shear Strain (Raw, from Displacement Data)(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.20

50

100

150

200

IF Steel

Forward Reverse

She

ar S

tress

(MP

a)


WHS

WHS

WHS

(b)

Figure 5.11: Raw Shear Stress - Shear Strain plots for IF steel: (a) Total shear plots and (b)accumulative shear plots. The work hardening stagnation is marked by “WHS”.

Chapter 5 Page 143

5.3.1.3 Back Stress Calculations

For the calibration experiments, three different offset values of 0.0005, 0.001 and

0.002 were used to calculate the back stress in the calibration experiments. Figure

5.12 presents the measured back stress values with respect to the real forward pre-

shear values (obtained from grid point data) for all IF steel calibration specimens.

The following observations can be made:

1. The back stress is an increasing function of the forward pre-shear strain. The

rate of increase is relatively high at small pre-strains and it decreases at higher

pre-strains.

2. The measured back stress depends on the offset value. Larger offset values

result in smaller back stresses. A total difference of ∼10 MPa can be observed

between the back stresses measured using the 0.0005 and 0.002 offsets.

3.

4. The measured back stresses have a high degree of reproducibility, i.e. less than

∼10% scatter in each of the pre-shear strain conditions.

The back stress results presented here are comparable with previously published

data in the literature [6, 16–18] in terms of both the magnitude and the trend with

respect to the amount of pre-shear strain.

In the literature, the 0.0005 and 0.001 offset values are commonly used to deter-

mine the back stress [6, 16–18]. Comparing raw and real shear strains in the present

study, these offset values translate into ∼0.002 raw shear. For this reason, and since

the offset value does not change the overall trends between back stress and pre-shear

strain (Figure 5.12), only the offset value of 0.002 is used to measure back stress in

the “Back Stress Experiments” (Section 5.3.2).

Chapter 5 Page 144

0.0 0.1 0.2 0.3 0.410

15

20

25

30

35

40

IF SteelB

ack-

Stre

ss,

Offs

etB

(MP

a)

Forward Pre-Shear Strain (Real, from Grid Point Data)

0.0005 Offset 0.001 Offset 0.002 Offset

Figure 5.12: Back stress (τoffsetB ) measurements results of calibration IF steel specimens plottedagainst forward pre-shear strain.

In addition to back stress, softening curves (τF − τR) were also calculated and the

results are plotted in Figure 5.13 with respect to the reverse shear strain (see Figure

5.6b for details). The following observations can be made from Figure 5.13:

1. At small reverse shear strains, τF − τR is a decreasing function of the reverse

shear strain.

2. The magnitude of τF −τR increases for larger values of forward pre-shear strain,

i.e. it increases in the order of 0.023, 0.149 and 0.262 forward pre-shear strain.

3. At large reverse shear strains, the rate of decrease in τF − τR (with respect to

reverse shear strain) is significantly reduced. Specifically, in specimens with

0.149 and 0.262 forward pre-shear strain, the τF − τR value is slightly raised

Chapter 5 Page 145

0.0 0.1 0.2 0.30

50

100

150

200

Forward Pre-Shear Strain(Real, from Grid Point Data)

0.023 0.149 0.262

IF Steel

F - R (M

Pa)

Reverse Shear Strain (Raw, from Displacement Data)

Figure 5.13: τF − τR plotted against reverse shear strains for all forward pre-shear experiments

by an increase in the reverse shear strain. This is a manifistation of the work-

hardening stagnation effect which was also observed in Figure 5.11b.

4. The measured τF −τR values have a high degree of reproducibility, i.e. less than

∼10% scatter in each of the pre-shear strain conditions.

5.3.2 Back Stress Experiments

The shear stress versus raw shear strain (obtained from crosshead displacement

data) plots of the DP780 microstructural variants, including the repeat tests, are

presented in Figure 5.14. The following observations can be made:

1. In terms of forward straining, all microstructures show relatively similar strength

levels (∼450-500 MPa) at the maximum forward strain (raw forward shear strain

Chapter 5 Page 146

0.0 0.2 0.4 0.6 0.8 1.0 1.20

100

200

300

400

500

She

ar S

tress

(MP

a)


CR+IC725

Forward Reverse

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.20

100

200

300

400

500

She

ar S

tress

(MP

a)


AT+IC720

Forward Reverse

(b)

Figure 5.14: Accumulative shear stress-shear strain (raw) plots for DP780 variants: (a) CR+IC725,(b) AT+IC725, (c) QT+IC735, (d) AT+CR8+IC720 and (e) QT+CR8+IC720.

Chapter 5 Page 147

0.0 0.2 0.4 0.6 0.8 1.0 1.20

100

200

300

400

500

She

ar S

tress

(MP

a)


QT+IC735

Forward Reverse

(c)

0.0 0.2 0.4 0.6 0.8 1.0 1.20

100

200

300

400

500

She

ar S

tress

(MP

a)


AT+CR8+IC720

Forward Reverse

(d)


Chapter 5 Page 148

0.0 0.2 0.4 0.6 0.8 1.0 1.20

100

200

300

400

500

She

ar S

tress

(MP

a)


QT+CR8+IC720

Forward Reverse

(e)


of ∼0.8).

2. The CR+IC725, AT+IC720 and QT+IC735 specimens exhibit continuous

yielding during forward straining, whereas yielding is discontinuous for

AT+CR8+IC720 and QT+CR8+IC720, as discussed in Chapter 4.

3. For all five microstructural variants, the unloading curve is linear at the be-

ginning but becomes non-linear at lower stresses, i.e. some inelastic relaxation

takes place during unloading.

4. Upon reverse loading, regardless of the initial forward yielding behaviour (con-

tinuous versus discontinuous), all microstructure variants exhibit a strong tran-

sient softening effect, i.e. continuous yielding at a reduced stress.

Chapter 5 Page 149

5. Permanent softening occurs in all five microstructural variants and at all pre-

shear strains.

6. No visible hardening stagnation effect is observed for any condition.

7. In the specimens where repeat testing was performed, both forward and reverse

curves are highly reproducible with less than ∼ 5% scatter.

In Figure 5.14, some of the forward and reverse flow curves exhibit sharp stress

drops of various magnitudes. In contrast, no stress drops were observed in the case of

IF steels. It is assumed that these stress drops are associated with slippage of the spec-

imen within the grips [32]. The stress drops are particularly frequent in the CR+IC725

microstructure where the specimens were very flat. In all other microstructures, the

specimens where warped due to more extensive heat treating schedules. It is postu-

lated that the warped specimens had improved gripping due to a spring effect, thereby

producing significantly fewer and smaller drops during loading.

For all five DP780 microstructural variants, the evolution of back stress with re-

spect to the pre-shear strain was calculated using a 0.002 offset method (see section

5.2.5.1 for details) and the results are presented in Figure 5.15. The following obser-

vations can be made:

1. For all five DP780 microstructural variants, the back stress increases continu-

ously with increasing forward pre-shear strain.

2. For all five DP780 microstructural variants, the rate of increase in back stress

decreases with increasing forward pre-shear strain.

3. The back stress is higher for all five DP780 microstructural variants compared

with the IF steel.

4. For very small pre-shear strains, all five DP780 microstructural variants have

Chapter 5 Page 150

0.0 0.1 0.2 0.3 0.40

20

40

60

80

100

120

140

160

Bac

k-S

tress

, 0.

002

B (M

Pa)


CR+IC AT+IC QT+IC AT+CR8+IC QT+CR8+IC IF

Figure 5.15: Evolution of Back stress (τ0.002B ) as a function of forward pre-shear for DP780 mi-crostructural variants. IF steel results are also plotted for reference.

relatively similar back stress values. However, with increasing forward pre-

shear strain, the back stress measurements separate into three groups: the

CR+IC725 variant has the highest value of back stress, the AT+IC720 variant

has the lowest and the other three variants (QT+IC735, AT+CR8+IC720 and

QT+CR8+IC720) fall between these two extremes.

The softening curves (τF − τR) were also calculated for all five DP780 microstruc-

tural variants and the results are plotted in Figure 5.16 against reverse shear strain.

The following observations can be made:

1. For all conditions, the τF −τR parameter decreases with increasing reverse shear

strain and permanent softening is observed, i.e. the τF − τR reaches a plateau

at high reverse shear strains.

Chapter 5 Page 151

2. No work hardening stagnation effect, i.e. increase in τF − τR, is observed for

any condition.

An alternative method to measure the back stress is through the permanent soft-

ening parameter, ∆τPS. From the softening (τF −τR) curves of Figures 5.13 and 5.16,

the permanent softening parameter, observed at the plateau of the τF −τR curves, was

measured for IF steel and DP780 microstructural variants at a forward displacement

of 5mm (average real forward pre-shear strain of 0.263 ± 0.015). Using Equation 5.3,

the “permanent softening back stress” was then calculated as 0.5 × ∆τPS [3]. The

results are given in Table 5.5, along with the experimental back stresses measured

from the same samples using the 0.002 offset method. From Table 5.5 is can be seen

that:� For all conditions, the permanent softening back stress (0.5 × ∆τPS) is sig-

nificantly smaller than the back stress value measured from the 0.002 offset

method.� Both the permanent softening back stress (0.5 × ∆τPS) and the back stress

values measured from the 0.002 offset method show similar trends among the

five DP780 microstructural variants.

Chapter 5 Page 152

0.0 0.1 0.2 0.30

100

200

300

400

500 Forward Pre-Shear Strain(Real, from Grid Point Data)

0.026 0.126 0.248

CR+IC725

F - R (M

Pa)

Reverse Shear Strain (Raw, from Displacement Data)(a)

0.0 0.1 0.2 0.30

100

200

300

400


0.24 0.141 0.269

F - R (M

Pa)

Reverse Shear Strain (Raw, from Displacement Data)

AT+IC720

(b)

Figure 5.16: τF − τR plotted against reverse shear strains for all forward pre-shear experiments.(a) CR+IC725, (b) AT+IC725, (c) QT+IC735, (d) AT+CR8+IC720 and (e) QT+CR8+IC720

Chapter 5 Page 153

0.0 0.1 0.2 0.30

100

200

300

400


0.018 0.137 0.259

QT+IC735

F - R (M

Pa)

Reverse Shear Strain (Raw, from Displacement Data)(c)

0.0 0.1 0.2 0.30

100

200

300

400


0.042 0.159 0.290

AT+CR8+IC720

F - R (M

Pa)

Reverse Shear Strain (Raw, from Displacement Data)(d)


Chapter 5 Page 154

0.0 0.1 0.2 0.30

100

200

300

400


0.033 0.155 0.281

QT+CR8+IC720

F - R (M

Pa)

Reverse Shear Strain (Raw, from Displacement Data)(e)


Table 5.5: Comparison of the back stress values (0.002 offset method) with the permanent softeningback stress defined as 0.5 × ∆τPS . Measurement were done for DP780 microstructural variants andIF steel after 5mm forward displacement (average real forward pre-shear strain of 0.263 ± 0.015).

Microstructural Back Stress (Offset Method), 0.5 × ∆τPS

Variant τ0.002B (MPa) (MPa)

CR+IC725 138 ± 3 33 ± 2AT+IC720 103 ± 3 19 ± 0QT+IC735 117 ± 1 23 ± 1AT+CR8+IC720 125 ± 1 31 ± 0QT+CR8+IC720 125 ± 1 35 ± 0IF Steel 25 ± 0 7 ± 0

Chapter 5 Page 155

5.4 Discussion

As shown in Figure 5.15, the back stress is an increasing function of the pre-shear

strain in all specimens, i.e. IF steel as well as five DP780 microstructural variants. The

rate of increase in back stress, however, decreases at larger forward pre-shear strain.

Using the Brown and Stobbs model [28], this decreased rate of change is associated

with the establishment of plastic relaxation effects such as the introduction of GNDs

near hard particles. Plastic relaxation is often not complete, such that the unrelaxed

portion of plastic strain results in the development of additional back stress [3, 27, 31]

through Equation 5.4. In addition to back stress effects, plastic relaxation of elastic

stresses through the generation of GNDs also produces significant work hardening in

the matrix; therefore, this contribution should be accounted for using Ashby’s GND

hardening theory, as previously done in Chapters 3 and 4.

To compare the measured back stresses with the Brown and Stobbs (B & S) model

[3, 27, 31], Equation 5.4 was applied to all five DP780 microstructural variants at the

forward pre-shear strains associated with a forward displacement of 5mm (average real

shear strain of 0.263 ± 0.015). For the volume fraction (f) and size (d) of martensite

particles, the values given in Table 5.3 were used, along with the following values:

α = 0.35 (halfway within the range of 0.3-0.4 [34–38]), µ = 80 GPa [35], b = 2.48×1010

m [38], and φ ≈ 0.5 [27]. The calculated back stress values of the B & S model are

listed in Table 5.6, alongside the experimentally measured back stress values (using

a 0.002 offset method). It is evident that in all five DP780 microstructural variants,

the experimentally measured back stresses fall within the range of the B & S model

predictions. This observation also supports the choice of a 0.002 offset strain in

calculating the experimental back stresses.

Chapter 5 Page 156

Table 5.6: Comparison of the experimental back stress values (0.002 offset) with the predictions ofthe B & S model for DP780 microstructural variants after 5mm forward displacement (average realforward pre-shear strain of 0.263 ± 0.015).

Microstructural B & S Model Back Stress (Offset Method),Variant (MPa) τ0.002B (MPa)

CR+IC725 105 138 ± 3AT+IC720 125 103 ± 3QT+IC735 88 117 ± 1AT+CR8+IC720 145 125 ± 1QT+CR8+IC720 132 125 ± 1

For the relative magnitudes of back stresses in five DP780 microstructural vari-

ants, the trends among the predictions of the B & S model are different from those

observed experimentally. Presumably this difference is due to the limitations of the

B & S model which only includes the effects of volume fraction and size of marten-

site particles. In reality, at relatively similar volume fractions of martensite, the five

DP780 microstructural variants differ significantly in their ferrite grain size as well

as the size, morphology and spatial distribution of martensite particle (Table 5.3). It

follows that these microstructural differences are the primary reason for the differing

magnitudes of experimental back stresses amongst the five DP780 microstructural

variants (Figure 5.15), recognizing that the magnitude of the back stress is generally

a function of three parameters: (i) the strain (and stress) incompatibility between

the ferrite and martensite, which can be affected by a number of parameters such

as the strength difference between these constituents, (ii) the efficiency of the load

transfer from the soft ferrite to the hard martensite, as taken into account by the

morphology-dependant parameter, φ in Equation 5.4, and (iii) the degree of relax-

ation of elastic stresses associated with the pileup of dislocation at particles. In the

B & S model, this relaxation phenomenon is accomplished through plastic relaxation

effects such as the generation of GNDs (as described earlier). Additionally, Goel

Chapter 5 Page 157

et al. [2] demonstrated that when the martensite particles are closer to each other,

i.e. smaller inter-particle spacing, further relaxation occurs by the mutual interac-

tions of the stress fields associated with the pileups at neighbouring particles, thereby

resulting in a partial reduction of the internal stresses in the ferrite matrix [2, 39].

From Figure 5.15 it is evident that the AT+CR8+IC720 and

QT+CR8+IC720 variants (both starting from the as-received HB material) have

smaller back stresses in comparison with CR+IC725. Specifically, comparing

QT+CR8+IC720 and CR+IC725 (Figure 4.3 in Chapter 4), it can be seen that while

both variants have a banded spatial distribution of martensite particles with a similar

morphology (equiaxed), they particularly differ with respect to their ferrite grain size.

On average, the ferrite grain size is smaller in QT+CR8+IC720, particulary due to

the presence of very small and sometimes unrecrystallized (but fully recovered) fer-

rite grains within martensite bands (Table 5.3). As discussed in Chapter 4, a smaller

ferrite grain size reduces the magnitude of internal stresses for two reasons. First, in

a ferrite matrix with finer grains, a larger area fraction of ferrite grains are affected by

the transformation dislocations that are formed upon quenching after IC annealing

[40, 41]. The transformation dislocations cause a local increase in ferrite strength

[42]. Consequently, the strength of ferrite is increased as the ferrite grain size is de-

creased3 [41]. This leads to a smaller stress/strain incompatibility between ferrite

and martensite [44], thereby reducing the internal stresses. Second, since martensite

particles are primarily located at ferrite grain boundaries in both of these microstruc-

tures, the inter-particle spacing of martensite is expected to decrease with a reduced

ferrite grain size, resulting in a partial cancelation of the internal stresses due to the

3For the range of ferrite grain sizes investigated here, the variation in the Hall-Petch effect isnot significant and therefore is not considered, e.g. using a Hall-Petch coefficient of 4 MP.mm1/2

[41, 43], a difference of 10 MPa can be calculated between the ferrite yield strength of CR+IC720and QT+CR8+IC720.

Chapter 5 Page 158

mutual interactions between the stress fields associated with pileups at neighbouring

particles [2, 39].

In Figure 5.15, the back stresses of the AT+CR8+IC720 and QT+CR8+IC720

variants are observed to be relatively similar. From Chapter 4 (Figure 5.3), both

variants comprise equiaxed martensite particles located at ferrite grain (and sub-

grain) boundaries. Therefore, in these microstructures, the spatial distribution of

martensite particles is directly related to ferrite grain size distribution, which is

uniform and equiaxed in AT+CR8+IC720 while it is banded and heavily bimodal in

QT+CR8+IC720. The inhomogeneous spatial distribution of martensite particles in

QT+CR8+IC leads to the presence of locally hard regions within the microstructure,

where bands of unrecrystallized ferrite grains with large number of martensite parti-

cles are located, as well as softer areas, containing bands of large, recrystallized ferrite

grains with few martensite particles. These distinct regions cause an additional de-

gree of plastic incompatibility in the microstructure, and therefore, an increase in

the magnitude of internal stresses [41]. Furthermore, the average ferrite grain size is

larger in the QT+CR8+IC720 variant (Table 5.3), which could lead to higher back

stress in this microstructure. The similar back stresses observed in Figure 5.15 are

therefore unexpected and could possibly be due to the heavily bimodal size distri-

bution of ferrite grains in QT+CR8+IC720. That is, although the average ferrite

grain size is larger in the QT+CR8+IC720 microstructure, there are bands of very

fine ferrite grains with large numbers of martensite particles at the grain boundaries

which lead to a significantly smaller inter-particle spacing and thereby, reduction of

back stresses in QT+CR8+IC720.

Regarding the CR starting microstructures in Figure 5.15, i.e. CR+IC725,

QT+IC735 and AT+IC720, the back stress is observed to increase in the order of

Chapter 5 Page 159

AT+IC720, QT+IC735 and CR+IC725. The larger magnitude of back stress in

CR+IC725 relative to QT+IC735 can be explained by the spatial distribution of

martensite particles. In both microstructures, the majority of martensite particles

are located at ferrite grain boundaries while virtually no martensite particle exists

inside ferrite grains. Moreover, specifically in the QT+IC735, due to the larger fer-

rite grain size, the ferrite grain boundaries are completely covered by inter-connected

martensite particles (Figure 4.2). It follows that the inter-particle spacing between

martensite particles is very small, which results in a smaller back stress in QT+IC735

compared to CR+IC725.

With respect to AT+IC720, a close examination of Figures 4.2 (Chapter 3) and

5.3 (Chapter 4) reveals that this microstructure has two distinct features that differ-

entiate it from all of the other DP780 variants: (i) the elongated morphology of the

majority of martensite particles, and (ii) the presence of martensite particles inside

ferrite grains. The different morphology of martensite particles clearly affects the load

transfer between ferrite and martensite, and therefore, the magnitude of back stress

through parameter φ in Equation 5.4. On the other hand, the existence of martensite

particles inside ferrite grains reduces the inter-particle spacing of martensite particles

compared to the other two CR starting microstructures, thereby resulting in a smaller

back stress in the AT+IC720 variant.

From Figure 5.15, it is evident that the back stress values for the IF steel are

significantly smaller than for the DP780 variants, which can be explained by the lack

of hard particles in the IF steel microstructure. In IF steels, the back stress originates

from the polarized heterogeneous dislocation cell substructures that are formed during

forward loading [13, 45] and its magnitude is generally smaller in comparison with

those introduced by hard particles, such as the martensite islands in DP steels.

Chapter 5 Page 160

An alternative method to measure the back stress is through the permanent soft-

ening parameter, ∆τPS. From Table 5.5, it can be seen that for all conditions, the

permanent softening back stress (0.5 × ∆τPS) is significantly smaller than both

the experimental (0.002 offset) and the B & S model back stresses. This difference

can be attributed to the changes in the non-directional hardening component (σS of

Equations 5.1 and 5.2) during reverse straining to the large reverse strains required

for the measurements of the permanent softening parameter, as suggested by Wilson

and Bate [11, 12]. This change has been experimentally reported in the TEM stud-

ies of Gardey et al. where they observed the partial dissolution of dislocation cell

structures during reverse straining at large reverse strains [13, 14]. This phenomenon

can also result in the appearance of the work hardening stagnation effect [13, 14], as

observed in the case of IF specimens, where the softening curves (Figure 5.13) show a

slight increase with increasing reverse shear strain. Therefore, in conjunction with the

above argument, the experimental results presented here confirm that the permanent

softening back stress is not an accurate representation of the actual back stress due

to the changes associated with the the non-directional hardening component (σS) at

larger reverse strains.

Finally, the importance of back stress to overall strength can be realized from the

parameter known as the Bauschinger stress parameter, βOffsetσ , which is defined as

[15]:

βOffsetσ =

τUnloadF − τOffset

R

τUnloadF

. (5.10)

The Bauschinger stress parameter provides a convenient method for presenting

the contribution of the back stress hardening relative to other hardening mechanisms.

This parameter was calculated for all microstructures (using a 0.002 offset) and the

Chapter 5 Page 161

results are presented in Figure 5.17. Additionally, for the specimens with a forward

displacement of 5mm (average real forward pre-shear strain of 0.263 ± 0.015), the

β0.002σ values are listed in Table 5.7. From Figure 5.17, it can be seen that for all

conditions, the β0.002σ follows a similar trend with forward pre-shear strain as the

back stress (5.15). The magnitudes of β0.002σ for the DP780 microstructural variants

(∼ 0.3−0.5) are in agreement with the available literature [15, 18] and suggest a strong

contribution of the back stresses to the overall strength of DP steels. Moreover, as

expected, β0.002σ is significantly smaller for IF steels in comparison with the DP780

microstructural variants. The more interesting finding from Figure 5.17, however, is

that β0.002σ reaches a saturation limit for all specimens at high strains (above ∼0.1).

This saturation effect has been reported before and is believed to be associated with

the initiation of alternate relaxation processes such as plastic deformation and/or

cracking of hard particles [4, 7, 15]. A saturation in β0.002σ is very intriguing because

it suggests that after a certain amount of shear deformation, the contribution of back

stress to overall strength is constant. The saturation of β0.002σ occurs at shear strains

in the order of ∼ 0.1, which translates to a von Mises equivalent strain of ∼0.05-

0.06. Interestingly, these strain levels are close to those associated with the transition

from GND dominated Stage B work hardening, to Stage C where non DP-specific

hardening mechanisms are more prominent (see Chapter 4). This correspondence

suggests that both the saturation of β0.002σ and the transition from Stage B work

hardening to Stage C are due to a similar mechanism, such as the plastic deformation

and/or fracture of martensite particles. These effects have both been experimentally

observed in the literature before [44, 46, 47].

Finally, from Equation 5.10, the β0.002σ parameter was defined as the normalized

back stress value with respect to the forward unloading stress. As the forward flow

Chapter 5 Page 162

0.0 0.1 0.2 0.3 0.4

0.2

0.3

0.4

0.5

0.6

0.7

Bau

schi

nger

Stre

ss P

aram

eter

, (M

Pa)


CR+IC AT+IC QT+IC AT+CR8+IC QT+CR8+IC IF

Figure 5.17: Evolution of Bauschinger stress parameter (β0.002σ ) as a function of forward pre-shear

strain for DP780 microstructural variants. IF steel results were also plotted for reference.

Table 5.7: The Bauschinger stress parameter (β0.002σ ) obtained from from forward-reverse shear

tests of DP780 microstructural variants and IF steel after 5mm forward displacement (average realforward pre-shear strain of 0.263 ± 0.015).

Microstructural Bauschinger stress parameter,Variant β0.002

σ

CR+IC725 0.57AT+IC720 0.48QT+IC735 0.51AT+CR8+IC720 0.51QT+CR8+IC720 0.51IF Steel 0.29

Chapter 5 Page 163

stress is an ever increasing function of strain, the saturation of β0.002σ suggests that

for the case of back stresses, a saturation limit is not reached and only the rate

of back stress increase is significantly reduced at higher strains. This behaviour

is understandable since as deformation continues, the martensite particles still act

as barriers to dislocation motion and therefore result in dislocation pileups despite

their plastic deformation and/or fracture. The plastic deformation and/or fracture of

martensite particles have two consequences. First, it reduces the magnitude of plastic

incompatibility between ferrite and martensite, which leads to a smaller increase in

the internal stresses as deformation proceeds. Second, it causes an additional plastic

relaxation of the internal stresses. Both of these effects are consistent with the results

of Chapters 3 and 4 where the work hardening contribution due to the generation of

GNDs was found to be insignificant at larger strains of magnitudes similar to those

associated with the saturation of back stress. It follows that, at higher strains, where

plastic deformation and/or fracture of martensite particles is taking place, the back

stresses are still expected to increase with applied strain, albeit at significantly slower

rates (Figure 5.15).

5.5 Conclusions

1. An in-plane shear test fixture was designed and commissioned to enable the

measurement of back stress in steel sheet through forward-reverse shear tests.

2. Forward-reverse shear tests were performed on an IF steel as well as the five

DP780 microstructural variants described in Chapters 3 and 4. The five DP780

microstructural variants had similar volume fractions of martensite but var-

ied in their average ferrite grain size as well as the average size, morphology

Chapter 5 Page 164

and spatial distribution of martensite particles. The reverse flow curves were

used to calculate and compare the back stress as well as the Bauschinger stress

parameter for each microstructure.

3. It was found that within the five DP780 microstructural variants, the CR+IC725

variant exhibited the highest value of back stress, while the AT+IC720 vari-

ant offered the lowest magnitude of back stress and the other three variants

(QT+IC735, AT+CR8+IC720 and QT+CR8+IC720) fell between these two

extremes. These differences are attributed to the variation in the average fer-

rite grain size as well as the spatial distribution and morphology of martensite

particles. The former two parameters affect the stress-strain incompatibility

between ferrite (soft) and martensite (hard), while the latter influences the ef-

ficiency of load transfer from the ferrite matrix to the martensite particles.

4. In all five DP780 microstructures, the Bauschinger stress parameter (β0.002σ )

reaches a saturation limit at a von Mises equivalent strain similar to that of

the transition from Stage B to Stage C of work hardening, suggesting that this

transition is likely due to the initiation of other plastic relaxation effects such

as the plastic deformation and/or cracking of martensite particles.

Chapter 5 Page 165

5.6 References




[3] J. D. Atkinson, L. M. Brown, W. M. Stobbs, Philos. Mag. 30 (1974) 1247–1280.

[4] N. Ibrahim, J. D. Embury, Mater. Sci. Eng. 19 (1975) 147–149.



118–144.

[6] M. T. Ma, B. Z. Sun, Y. Tomota, ISIJ Int. 29 (1989) 74–77.

[7] D. Uko, R. Sowerby, J. D. Embury, Metals Technol. London 7 (1980) 359–367.

[8] R. Sowerby, D. K. Uko, Y. Tomita, Mater. Sci. Eng. 41 (1979) 43–58.

[9] Y. W. Chang, R. J. Asaro, Metal Sci. 12 (1978) 277–284.


[11] P. S. Bate, D. V. Wilson, Acta Metall. 34 (1986) 1097–1105.

[12] D. V. Wilson, P. S. Bate, Acta Metall. 34 (1986) 1107–1120.

[13] B. Gardey, S. Bouvier, V. Richard, B. Bacroix, Mater. Sci. Eng. A 400-401 (2005)

136–141.

[14] B. Gardey, S. Bouvier, B. Bacroix, Metall. Mater. Trans. A 36 (2005) 2937–2945.

[15] K. Han, C. J. van Tyne, B. S. Levy, Metall. Mater. Trans. A 36 (2005) 2379–2384.


[17] Y. Tomota, Mater. Sci. Technol. 3 (1987) 415–421.

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[18] A. Aouafi, S. Bouvier, M. Gasperini, X. Lemoine, O. Bouaziz, in: E. Cueto,

F. Chinesta (Eds.), 10th ESAFORM Conference on Material Forming, 18-20

April 2007, volume 907 of AIP Conf. Proc., LPMTM-CNRS, Univ. Paris, Vil-

letaneuse, France, pp. 100–105.

[19] C. Zhongchun, S. Maekawa, T. Takeda, Metall. Mater. Trans. A 30 (1999) 3069–

3078.

[20] H. Kim, C. Kim, F. Barlat, E. Pavlina, M. G. Lee, Mater. Sci. Eng. A 562 (2013)

161–171.

[21] R. M. Cleveland, A. K. Ghosh, Int. J. Plast. 18 (2002) 769–785.


(2010) 91–98.

[23] Y. Bian, Microstructure and Mechanical Properties of Al and Al/Si Alloyed

TRIP-assisted Steels Produced through Galvanizing Heat Treatments, The-

sis/dissertation, McMaster University, Hamilton, ON, Canada, 2009.

[24] K. Spencer, The work hardening of austenitic stainless steel, applied to the fab-

rication of high-strength conductors, Thesis/dissertation, McMaster University,

Hamilton, ON, Canada, 2004.

[25] F. Yoshida, T. Uemori, K. Fujiwara, Int. J. Plast. 18 (2002) 633–659.

[26] S. Bouvier, B. Gardey, H. Haddadi, C. Teodosiu, J. Mater. Process. Technol. 174

(2006) 115–126.





[31] L. M. Brown, Acta Metall. 21 (1973) 879–885.

[32] S. Bouvier, H. Haddadi, P. Levqe, C. Teodosiu, J. Mater. Process. Technol. 172

(2006) 96–103.

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[33] G. A. S. Bell, The relationship between microstructure and damage evolution

in hot-rolled complex-phase steel sheet, Thesis/dissertation, Queen’s University,

Kingston, ON, Canada, 2013.


Int. J. Plast. 43 (2013) 128–152.

[35] O. Bouaziz, T. Lung, M. Kandel, C. Lecomte, Le Journal de Physique IV 11

(2001) Pr4–223–Pr4–231.

[36] M. Delince, Y. Brechet, J. D. Embury, M. G. D. Geers, P. J. Jacques, T. Pardoen,

Acta. Mater. 55 (2007) 2337–2350.

[37] O. Bouaziz, D. Barbier, J. D. Embury, G. Badinier, Philos. Mag. 93 (2013)

247–255.

[38] J. Bouquerel, K. Verbeken, B. C. D. Cooman, Acta. Mater. 54 (2006) 1443–1456.

[39] Y. Tomota, I. Tamura, Transactions of the Iron and Steel Institute of Japan 22

(1982) 665–677.


2738–2746.



Mater. 62 (2014) 197–211.


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670.

[45] B. Peeters, S. R. Kalidindi, P. V. Houtte, E. Aernoudt, Acta. Mater. 48 (2000)

2123–2133.

[46] P. J. Jacques, Q. Furnemont, F. Lani, T. Pardoen, F. Delannay, Acta. Mater. 55

(2007) 3681–3693.

Chapter 5 Page 168


CHAPTER 6

Complementary Discussion

6.1 Microstructure Evolution During IC Annealing

In Chapters 3 and 4, five distinct DP microstructural variants, namely CR+IC,

AT+IC, QT+IC, AT+CR8+IC and QT+CR8+IC, were presented. For each vari-

ant, a microstructure with a martensite volume fraction of ∼15% was selected, i.e.

CR+IC725, AT+IC720, QT+IC7351, AT+CR8+IC720 and QT+CR8+IC720, and

the evolution of microstructure during IC annealing was extensively discussed us-

ing the concepts of austenite transformation and ferrite recrystallization. It was

shown that the most suitable austenite nucleation sites are those associated with

both (i) high carbon content necessary for austenite growth and (ii) high interfa-

cial and/or stored energies required to reduce the activation energy for austenite

nucleation (∆G∗). When the ferrite matrix is fully recrystallized prior to reaching

the austenite transformation temperature, the ferrite/cementite interface in pearlite

colonies as well as the individual cementite particles on ferrite grain boundaries are

the two preferred nucleation sites for austenite [1, 2]. On the other hand, when ferrite

recrystallization is not completed before reaching the IC annealing temperature, sig-

nificant nucleation of austenite takes place on the cementite particles that are located

on the sub-grain boundaries of unrecrystallized ferrite grains [1–3].

A comparison of Figures 3.2 and 4.3 reveals that the addition of a cold-rolling step

between pre-heat treatments and IC annealing can result in significant modifications

1This specimen was referred to as Q7T+IC35 in Chapter 3

169

Chapter 6 Page 170

to the final DP microstructures. Specifically, in the microstructures with the AT

starting pre-heat treatments, i.e. AT+IC720 and AT+CR8+IC720, the latter has a

considerably refined microstructure with fine, fully recrystallized and equiaxed ferrite

grains. Moreover, in AT+CR8+IC720, the martensite particles are relatively smaller

in size and they have an equiaxed morphology (compared to the elongated morphol-

ogy in AT+IC720). The spatial distribution of martensite particles, however, are

similarly uniform in both variants. These differences can be related to the deformed

microstructure of the pre-IC annealing AT+CR8 condition (Figure 4.2) compared to

AT (Figure 3.1), for two reasons: (i) in the AT+CR8 condition, the high carbon con-

tent M/A particles are deformed and redistributed in the deformed ferrite matrix, and

(ii) upon heating to the IC annealing temperature complete ferrite recrystallization

of AT+CR leads to a fine-grained, equiaxed ferrite matrix, as compared to AT.

In the microstructural variants with the QT pre-heat treatments, i.e. QT+IC735

and QT+CR8+IC720, the addition of the cold rolling step also results in distinctly

different microstructures. During IC annealing of the QT+IC735, only a small por-

tion of cementite particles, located on the ferrite grain boundaries, can act as effective

austenite nucleation sites. This leads to a final DP microstructure where the majority

of martensite particles are located at the prior austenite grain boundaries and cor-

ners, leaving some very large martensite-free regions within the ferrite matrix. On

the other hand, in the QT+CR8+IC720 variant, the addition of a cold rolling step

after the QT pre-treatment results in a partial recrystallization of the deformed ferrite

matrix during heating to the IC annealing temperature (Figure 4.5) and therefore,

a more refined ferrite matrix compared to QT+IC735. Moreover, as carbide parti-

cles are primarily located at either the recrystallized ferrite grain boundaries or the

Chapter 6 Page 171

unrecrystallized subgrain boundaries, the majority of them can act as austenite nu-

cleation sites during IC annealing, which leads to a more uniform spatial distribution

of martensite particles in QT+CR8+IC720 compared to QT+IC720.

6.2 Effect of Microstructural Parameters on Work

Hardening of DP steels

In Chapters 3 and 4, the work hardening behaviour of the five DP780 microstruc-

tural variants were analyzed and discussed individually using three work hardening

parameters: the true work hardening rate (θ), the instantaneous work hardening rate

(n), and the annihilation factor (h). To complete this discussion, the work hardening

rate will be analyzed further here by comparing the results of Chapters 3, 4 and 5.

Figure 6.1a presents the the true work hardening rate at ǫP = 0.5% (θǫP=0.5%) for

the five microstructural variants investigated in this research, i.e. CR+IC, AT+IC,

QT+IC, AT+CR8+IC and QT+CR8+IC. In Figure 6.1b, a similar comparison is

made using the minimum instantaneous work hardening exponent, nmin. Both of

these parameters represent the work hardening behaviour of DP steels at small strains

which was assumed to be dominated by the generation of geometrically necessary

dislocations (GNDs). The different relationships of θǫP=0.5% and nmin with√

f/d for

the five microstructural variants was extensively discussed in Chapters 3 and 4 and

they were related to the effects of the mean ferrite grain size as well as the morphology

and spatial distribution of martensite particles. It is evident from Figures 6.1 that

in the variants with an additional cold-rolling pre-treatment step, i.e. AT+CR8+IC

and QT+CR8+IC, both the θǫP=0.5% and nmin plots (i) are shifted to higher values of√

f/d and (ii) are generally more sensitive to the value of√

f/d, i.e. higher slopes

Chapter 6 Page 172

with√

f/d. Both of these observations can be related to the finer microstructure of

the AT+CR8+IC and QT+CR8+IC variants compared to the AT+IC and QT+IC

(see Table 5.3). In the former two microstructural variants, a smaller martensite

particle size leads to a higher value of√

f/d. On other hand, as discussed in Chapter

4, due to the finer mean ferrite grain size in AT+CR8+IC and QT+CR8+IC, a

larger area fraction of ferrite grains is expected to be affected by the transformation

dislocations [4, 5], leading to a higher strength of ferrite grains [5] and hence, a smaller

stress/strain incompatibility between ferrite and martensite [6]. Consequently, GND

hardening is expected to be lower in AT+CR8+IC and QT+CR8+IC compared to

AT+IC and QT+IC. In addition to the ferrite grain size effect, between AT+IC and

AT+CR8+IC, the morphology of the martensite particles is different, while between

QT+IC and QT+CR8+IC, the spatial distribution of martensite particles is changed.

In Chapters 3 and 4, differing morphology and spatial distribution of martensite

particles were shown to result in additional variations in GND hardening, although

since these parameters are coupled with the effect of ferrite grain size, it is difficult

to independently discuss their influence on AT+IC versus AT+CR8+IC, as well as

on QT+IC versus QT+CR8+IC.

At larger strains (∼2-3%), the instantaneous work hardening rate reaches a max-

imum limit which is referred to as nmax. It was found that in each microstructural

variant, nmax is independent of the√

f/d value, which implies that at these strain

levels, the work hardening effects due to GNDs are less significant. This observation

correlates well with the back stress results of Chapter 5. That is, the relative con-

tribution to back stresses, i.e. Bauschinger stress parameter (β0.002σ ), was also shown

to increase very rapidly at the beginning of applied deformation, while reaching a

saturation value at strains similar to those associated with the nmax (see Section

Chapter 6 Page 173

2 4 6 80

5000

10000

15000

20000

25000

CR+IC AT+IC QT+IC AT+CR8+IC QT+CR8+IC

at p

last

ic s

train

of 0

.5%

(MP

a)

(f/d )1/2 (%.micron-1)(a)

2 4 6 80.0

0.1

0.2

0.3

CR+IC AT+IC QT+IC AT+CR8+IC QT+CR8+IC

Min

imum

Inst

anta

neou

s W

ork

Har

deni

ng, n

min (M

Pa)

(f/d )1/2 (%.micron-1)(b)

Figure 6.1: (a)The true work hardening rate at ǫP = 0.5% (θǫP=0.5%), and (b) the minimuminstantaneous work hardening exponents, nmin, for the five microstructural variants investigated inthis research.

Chapter 6 Page 174

5.4). This finding suggest that at strains associated with nmax and beyond, other

work hardening mechanisms, not specific to DP steels, become more important. In

Chapter 5, this effect was attributed to the plastic deformation and/or fracture of

martensite particles.

It was hypothesized in Chapter 4 that nmax is a function of the ferrite grain size.

The nmax values of all five microstructural variants are plotted with respect to√

f/d

in Figure 6.2. Indeed, it is evident that within the CR+IC, AT+IC and QT+IC

variants, nmax is the highest in CR+IC while it is the lowest in QT+IC, suggesting

that nmax is inversely proportional to the ferrite grain size (Table 5.3). In the case

of AT+CR8+IC and QT+CR8+IC, it can be seen that nmax is relatively similar

to the CR+IC which is attributed to the smaller ferrite grain size of these three

microstructural variants (Chapter 4).

At strains over 4%, it was assumed that dislocation annihilation by dynamic

recovery becomes the controlling factor for the rate of work hardening. Using the

Kocks-Mecking (KM) model, this phenomenon was discussed in Chapters 3 and 4

for the five DP microstructural variants. In the KM model, it is assumed that the

work hardening rate, θ, follows a linear relationship with stress during Stage III of

work hardening in polycrystalline materials, i.e. θ = θ0 − h(σ − σ0), where θ0 is the

athermal hardening rate (function of dislocation accumulation), h is the dislocation

annihilation factor and σ0 is the flow stress due to strengthening mechanisms that

do not include dislocation-dislocation interactions (Chapters 3 and 4). By fitting a

straight line to the θ vs (σ − σ0) plots of all five DP microstructural variants at high

strains (>4%), the dislocation annihilation factor (h) was calculated and was shown

to be a function of√

f/d, the mean ferrite grain size, the morphology and the spatial

distribution of martensite particles.

Chapter 6 Page 175

2 4 6 80.1

0.2

0.3

0.4

CR+IC (D=1.84 m) AT+IC (D=2.00 m) QT+IC (D=2.77 m) AT+CR8+IC (D=1.09 m) QT+CR8+IC (D=1.51 m)

Max

imum

Inst

anta

neou

s W

ork

Har

deni

ng, n

max

(MP

a)

(f/d )1/2 (%.micron-1)

Figure 6.2: The maximum instantaneous work hardening exponents, nmax, for the five microstruc-tural variants investigated in this research. The average ferrite grain size of selected microstructureswith martensite volume fraction of ∼15% was also listed for each microstructural variant.

A notable observation in Chapter 4 was that within the CR+IC, AT+CR8+IC

and QT+CR8+IC microstructural variants, the three work hardening parameters of

θ at ǫP = 0.5%, nmin and h all showed similar relationships with√

f/d (Figures 4.6,

4.7 and 4.8). Additionally, the trend observed between the individual microstructural

variants is identical for these three work hardening parameters. That is, θǫP=0.5%, nmin

and h decrease in the order of CR+IC, QT+CR8+IC and AT+CR8+IC. Similar

observations can also been made from Figures 3.7, 3.8 and 3.9, i.e. θǫP=0.5%, nmin

and h decrease in the order of CR+IC, QT+IC and AT+IC. These observations

are also consistent with the forward-reverse shear tests results, where it was shown

that the CR+IC725 variant has the highest value of back stress, the AT+IC720

variant has the lowest and the other three variants (QT+IC735, AT+CR8+IC720

Chapter 6 Page 176

and QT+CR8+IC720) fall between these two extremes. This consistent trend further

strengthens the argument made in Chapter 4 that GND hardening and dynamic

recovery effects are linked with each other through the internal stresses produced by

the incompatibility between ferrite and martensite.

From the Considere criterion, it follows that an ideal DP microstructure with the

highest uniform elongation will have a combination of a large initial work hardening

rate, and a small h value. Hence, the KM model was applied to the work hardening

data at strains larger than ∼4%, where the work hardening rate showed a linear

relationship with the stress (σ − σd). The two important parameters of the KM

model are θ0, the athermal hardening rate (function of dislocation storage), and h,

the dislocation annihilation factor. It was discussed in Chapter 3 and 4 that only h can

be reliability measured and a true value of θ0 cannot be obtained due to the complex

yielding behaviour of DP steels. However, since in KMmodel, θ0 is defined as the work

hardening rate at σ − σd=0 (i.e. the intercept of the KM fit with the ordinate), the

work hardening rate at ǫP = 0.5% was instead considered as an alternative parameter

to compare the initial dislocation storage rate among various microstructural variants

(Chapter 3).

At small strains (ǫP = 0.5%), the work hardening was found to be controlled by

the generation of geometrically necessary dislocations (GNDs) and the development

of back stresses in the ferrite grains. Additionally, it was demonstrated that at a given√

f/d, the θǫp=0.5% and nmin were functions of the mean ferrite grain size as well as the

morphology and spatial distribution of martensite particles. As the deformation pro-

ceeds, at strains of ∼2-3%, the work hardening rate reaches a maximum limit, nmax,

which is also microstructure-dependent but not a function of√

f/d. The nmax can be

considered as the work hardening capacity of the DP steel for a given microstructural

Chapter 6 Page 177

variant. From Figure 6.2, it can be seen that its magnitude is the highest (and simi-

lar) for the CR+IC, AT+CR8+IC and QT+CR8+IC microstructural variants. This

similar value of nmax suggests that as the ferrite grain size is decreased, the effects

of the average ferrite grain size and spatial distribution of martensite particles on

the work hardening are minimized. In Chapter 4, this observation was related to the

influence of transformation dislocations on the stress/strain incompatibility between

ferrite and martensite.

From the above argument, it follows that at strains larger than ∼4%, where dis-

location annihilation by dynamic recovery becomes the controlling factor for the rate

of work hardening, between CR+IC, AT+CR8+IC and QT+CR8+IC (with similar

nmax), the microstructural variant with the smallest h can sustain the work harden-

ing to the highest strains. Comparing these three microstructural variants, Figure

4.8 shows that the AT+CR8+IC has the lowest value of h parameter. This finding

was attributed to the a combined influences of small average ferrite grain size and

a uniform spatial distribution of dislocations in Chapter 4. Both of these effects re-

sult in a smaller stress/strain incompatibility between ferrite and martensite, thereby

lowering the magnitude of the internal stresses which are required for dynamic re-

covery. It follows that the AT+CR8+IC microstructural variant, with a high value

of nmax and a small h factor, is expected to have the highest uniform elongations at

similar strength levels. Figure 6.3 presents the uniform elongation values of all mi-

crostructures investigated in this thesis plotted against their respective UTS values.

It is evident that the AT+CR8+IC microstructure indeed has the largest uniform

elongation value of all five microstructural variants, particularly at higher strength

levels (> ∼750 MPa). This finding is important as it demonstrates that through

additional thermo-mechanical processing prior to IC annealing, i.e. AT+CR8, visible

Chapter 6 Page 178

600 700 800 900

0.08

0.10

0.12

0.14

0.16

0.18

0.20 CR+IC AT+IC QT+IC AT+CR8+IC QT+CR8+IC

Uni

form

Elo

ngat

ion

UTS (MPa)

Figure 6.3: Plots of the uniform elongation values of the five DP steel microstructural variantsversus their respective tensile strengths.

improvements in the uniform elongation of DP steels can be achieved in comparison

to the baseline (commercial) CR+IC microstructural variant.

6.3 Practical Implications of the Present Research

One of the primary research objectives in the field of DP steels is to produce

a microstructure that exhibits the most desirable mechanical behaviour in terms of

high strength and large uniform elongation values. Throughout this thesis research,

the underlying goal was to identify the effect of different DP steel microstructural

parameters on individual work hardening mechanisms of GND hardening, back stress

hardening and dynamic recovery. A critical finding was that certain microstructural

parameters, i.e. the average ferrite grain size, as well as the volume fraction, size,

Chapter 6 Page 179

morphology and spatial distribution of martensite particles, not only affect the initial

work hardening behaviour (GND hardening and back stress hardening), but they

also significantly influence the work hardening properties at large strains through

variations in the dynamic recovery (dislocation annihilation) process.

Despite the variation of θǫp=0.5%, nmin and h with√

f/d, each microstructural

variant exhibited similar nmax values at strains of ∼2-3% which are associated with

the transition from Stage B to Stage C work hardening. The nmax parameter is not a

function of√

f/d and for this reason, it can be considered as the work hardening ca-

pacity of each microstructural variant. An important finding in the present research

was the effect of ferrite grain size on nmax. That is, in DP steels with a refined ferrite

matrix, the influence of the ferrite grain size and the spatial distribution of martensite

particles on nmax becomes minimal, irrespective of the differences in nmin. This out-

come suggests that nmax is limited by a certain ferrite grain size; therefore, significant

improvement in the uniform elongation of a DP steel requires a microstructure with

a low h parameter, as the Considere criterion is then satisfied at larger strains. In

this research, it was shown that both the reduction of the ferrite grain size and the

uniform spatial distribution of martensite particles leads to a smaller h value, which

is the case for the AT+CR8+IC microstructural variant. It can be concluded that

in order to produce a DP steel with a large uniform elongation and a high strength,

the processing should be designed to result in a microstructure with the uniform spa-

tial distribution of martensite particles in a significantly refined ferrite matrix. To

produce such a microstructure, the AT+CR8+IC processing route could be modified

by: decreasing the initial austenitization temperature to reduce the prior austenite

grain size, thereby refining the AT microstructure, or by increasing the amount of

Chapter 6 Page 180

cold rolling after the AT pre-heat treatments and prior to the IC annealing to fur-

ther reduce the ferrite grain size. The latter modification is particularly interesting

since, as discussed in Chapter 4, one practical difficulty in the present research was

that the changes in the spatial distribution of martensite particles were also generally

accompanied by a variation in ferrite grain size, e.g. compare AT+CR8+IC with

QT+CR8+IC and CR+IC. As a result, it was impossible to investigate these two

effects independently. Presumably, the amount of the cold rolling can be changed to

50, 60 ,70 and 90%, instead of the 80% value used here, to obtain microstructural

variants with a wide range of ferrite grain sizes. This leads to a larger data set of

microstructural variants with different ferrite grain sizes as well as sizes and spatial

distributions of martensite particles, which could possibly allow for an independent

study of the separate effects of ferrite grain size and spatial distribution of martensite

particles on the work hardening properties of DP steels.

Chapter 6 Page 181

6.4 References



1385–1392.


3576.


2738–2746.


[6] M. Calcagnotto, Y. Adachi, D. Ponge, D. Raabe, Acta. Mater. 59 (2011) 658–670.

CHAPTER 7

Conclusions and Future Work

7.1 Conclusions

1. The addition of a cold-rolling step between the pre-heat treatments and IC

annealing leads to a significant refinement of the final DP microstructure com-

pared to similar processing conditions without a cold-rolling step. This refine-

ment can be explained by the interactions between the ferrite recrystallization

and austenite transformation processes.

2. At small strains (ǫp = 0.5%), the work hardening behaviour of the five DP

microstructural variants was found to be dominated by the generation of ge-

ometrically necessary dislocations (GNDs) in the ferrite grains and near the

martensite particles. The work hardening response at this stage was charac-

terized by θ at ǫp = 0.5% and a minimum value in the instantaneous work

hardening exponent, nmin. Both of these parameters were determined to be

functions of√

f/d, the mean ferrite grain size as well as the morphology and

spatial distribution of martensite particles.

3. At higher strains (2-3%), a maximum value in the instantaneous work harden-

ing exponent (nmax) is reached. This parameter, which can be considered as

the work hardening capacity of the material, was found to be independent of√

f/d. Additionally, the relative contribution of back stresses was also found

to reach a constant value at a similar von Mises equivalent strain. This ob-

servation suggests that at strains associated with nmax and above, other work

182

Chapter 7 Page 183

hardening mechanisms, not specific to DP steels, become more important. For

microstructural variants with larger ferrite grains, nmax was found to be a func-

tion of ferrite grain size while variants with a refined ferrite matrix showed no

dependence of nmax on ferrite grain size.

4. At strains over 4%, dislocation annihilation by dynamic recovery becomes the

controlling factor for the rate of work hardening. This phenomenon is described

by the dislocation annihilation factor, h, and is a function of√

f/d, the mean

ferrite grain size as well as the morphology and spatial distribution of martensite

particles.

5. The three work hardening parameters of θ at ǫp = 0.5%, nmin and h exhibit

similar relationships with√

f/d and consistent trends between the five mi-

crostructural variants under study. This observation which is in agreement

with the trends observed in measured back stresses, suggests a common role of

internal stresses on the two work hardening mechanisms of GND hardening and

dynamic recovery.

6. At a given UTS value, the AT+CR8+IC microstructural variant, comprising a

uniform distribution of fine, equiaxed martensite particles in a fine, equiaxed

ferrite matrix, exhibited the largest uniform elongation values of all five mi-

crostructural variants, particularly at higher strength levels. Having a nmax pa-

rameter similar to the other two microstructural variants with a refined ferrite

matrix (CR+IC and QT+CR8+IC), the improved uniform elongation values

for AT+CR8+IC were attributed to the smallest h parameter associated with

this microstructural variant.

Chapter 7 Page 184

7.2 Original Contributions to the Field

1. In the present research, five distinctly different DP microstructural variants

with a range of ferrite grain sizes (D), as well as volume fractions (f), sizes (d),

morphologies and spatial distributions of martensite particles were produced.

Characterizing these microstructural variants based on the√

f/d parameter

allowed for an investigation of the effects of ferrite grain size as well as mor-

phology and spatial distribution of martensite particles on the work hardening

behaviour, independent of the volume fraction (f) and the size (d) of marten-

site particles. This new characterization method is particularly useful since

modification of the former three microstructural parameters through thermo-

mechanical processing often results in a different size of martensite particles,

hence, limiting the applicability of characterizations based solely on f , as re-

ported in the literature.

2. This research utilized three different parameters to analyze the work hardening

behaviour of DP steels at various strain levels: the true work hardening rate, θ,

the instantaneous work hardening exponent, n, and the dislocation annihilation

parameter, h. A critical finding was that the microstructural parameters, i.e.

the average ferrite grain size, as well as the volume fraction, size, morphology

and spatial distribution of martensite particles, not only affect the initial work

hardening behaviour (GND hardening and back stress hardening), but they also

significantly influence the work hardening properties at large strains through the

dynamic recovery (dislocation annihilation) process.

3. Despite the variation of the work hardening parameters at both small and large

strains with√

f/d, each microstructural variant exhibited similar instantaneous

Chapter 7 Page 185

work hardening exponent values, nmax, at strains of ∼2-3%. This is a unique

observation for DP steels and suggests the presence of a work hardening ca-

pacity for each microstructural variant, independent of the√

f/d parameter.

An important finding was the effect of ferrite grain size on nmax. That is, for

microstructural variants with large ferrite grains, nmax was found to be a func-

tion of ferrite grain size while variants with a refined ferrite matrix showed no

dependence of nmax on ferrite grain size.

4. Until now, experimental research on back stress effects in DP steels has been

limited. The majority of these publications have been focused on the effect of the

volume fraction of martensite on back stress hardening and studies regarding

the influence of other microstructural parameters on back stresses have been

sparse and often inconclusive. This deficiency was addressed in the present

research by providing a better understanding of the effect of ferrite grain size as

well as the size, morphology and spatial distribution of martensite particles on

back stress hardening, through application of the in-plane forward-reverse shear

deformation tests. Furthermore, it was found that the relative contribution of

back stresses to overall work hardening (β0.002σ ) is generally significant in all

microstructures. Additionally, this contribution reached a constant value at a

von Mises equivalent strain similar to that associated with the transition from

Stage B to Stage C of work hardening in DP steels.

Chapter 7 Page 186

7.3 Future Work

The following extensions to the present study are suggested:

1. Additional thermo-mechanical processing should be done to refine both the fer-

rite grain size and the martensite particle size in the AT+CR8+IC microstruc-

tural variant in order to further improve the uniform elongation values at sim-

ilar strength levels. For example, this goal can be accomplished by decreasing

the initial austenitization temperature to reduce the prior austenite grain size,

thereby refining the AT microstructure, or by increasing the amount of cold-

rolling after the AT pre-treatments and prior to the IC annealing to further

refine the final ferrite grain size.

2. For both the AT+CR8+IC and QT+CR8+IC microstructural variants, the

amount of cold-rolling should be changed to 50, 60, 70 and 90%, instead of

the 80% value used here, thereby producing a series of microstructural variants

with a wide range of ferrite grain sizes. A larger data set of microstructural

variants with different ferrite grain sizes as well as sizes and spatial distribu-

tions of martensite particles could possibly allow for an independent study of

the separate effects of ferrite grain size and spatial distribution of martensite

particles on the work hardening behaviour of DP steels.

3. High resolution EBSD should be used to directly measure the density of the

GNDs in the five microstructural variants investigated herein. These EBSD

experiments should be carried out on an un-deformed specimen as well as spec-

imens with plastic strains of 0.5%, 3% (transition from stage B to C work

hardening) and 5%.

Chapter 7 Page 187

4. To observe and characterize the dislocation substructures formed during defor-

mation, TEM studies should be performed on the five microstructural variants

investigated herein using an undeformed specimen as well as specimens with

plastic strains of 0.5%, 3% and 5%.

5. Neutron diffraction experiments should be performed to measure the internal

stresses associated with ferrite and martensite during in-situ tensile testing of

the five selected DP780 microstructural variants (with martensite volume frac-

tion of ∼15%). The measured internal stresses could then be compared with the

experimental back stresses obtained from forward-reverse in-plane shear testing.

6. Metallography specimens should be made from interrupted shear tests of the five

selected DP780 microstructural variants at a strain level close to that associated

with the saturation of the Bauschinger stress parameter, β0.002σ . Each specimen

should then be examined in the SEM to detect the plastic deformation and/or

fracture of martensite particles.

7. The forward-reverse shear tests on the five microstructural variants should be

extended to other volume fractions of martensite in order to produce master

curves of back stresses with respect to√

f/d.

Appendices

APPENDIX A

Specimen Designation Conversions

The specimen naming convention used in Chapter 3 is different from the rest of

the thesis; however, to avoid altering the contents of the published work, this naming

convention remained unchanged in Chapter 3. Table A.1 summarizes the conversion

of these naming conventions to that of the rest of the thesis.

Table A.1: Conversion of the naming conventions used in Chapter 3 and in the rest of the thesis

Naming Convention in Chapter 3 Naming Convention in the Rest of the Thesis

CR+IC20 CR+IC720CR+IC25 CR+IC725CR+IC30 CR+IC730CR+IC35 CR+IC735AT+IC20 AT+IC720AT+IC35 AT+IC735Q2T+IC20 Q2T+IC720Q2T+IC20 (5) Q2T+IC720 (5)Q2T+IC20 (10) Q2T+IC720 (10)Q2T+IC20 (30) Q2T+IC720 (30)Q2T+IC25 Q2T+IC725Q2T+IC35 Q2T+IC735Q7T+IC20 QT+IC720Q7T+IC25 QT+IC725Q7T+IC35 QT+IC735Q12T+IC20 Q12T+IC720Q12T+IC25 Q12T+IC725Q12T+IC35 Q12T+IC735

189

APPENDIX B

Kocks-Mecking Fitting Procedure

To calculate the dislocation annihilation factor, h, using Equations 3.1 and 4.1, a

straight line was fit to the θ vs (σ−σ0) plots at high strains (>4%). For consistency,

a 0.2% proof stress was used for σ0. Since the objective was to calculate h, i.e. the

slope of the fitted line, and not θ0, this oversimplification is deemed appropriate.

0 200 4000

5000

10000

15000

d

d (M

Pa)

CR+IC725 Linear KM Fit

(MPa)

Figure B.1: Example of the Kocks-Mecking fitting procedure. The slope of the fitted line is thedislocation annihilation factor, h.

190

APPENDIX C

Technical Drawings of the Shear Fixture

191

Appendix C Page 192

Figure C.1: Drawing of the Shear Specimen

Appendix C Page 193

Figure C.2: Drawing of the Shear Fixture’s Base

Appendix C Page 194

Figure C.3: Drawing of the Shear Fixture’s Cover

Appendix C Page 195

Figure C.4: Drawing of the Shear Fixture’s Plunger

Appendix C Page 196

Figure C.5: Drawing of the Shear Fixture’s Spacer

Appendix C Page 197

Figure C.6: Drawing of the Shear Fixture’s Connector to the Instron

A word cloud of the thesis

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